ABSTRACT MOSSBAUER SPECTROSCOPY STUDY OF AGING AND FAST NEUTRON IRRADIATION EFFECTS IN A COPPER-RICH Cu-Fe ALLOY by Thomas Hugh Gould, Jr. Chairman: Dietrich H. Vincent Mbssbauer spectroscopy was used to study the effects of both thermal and radiation environments on precipitation processes in the alloy Cu-O.6at%Fe. The first part of this study is concerned with interpreting the room temperature spectra produced by iron in a variety of configurations or'phases' in alloy samples which were solution treated, aged at temperatures in the range 285-6000C, and cold-rolled. In the second part of the study, the knowledge gained from the thermal studies is. applied to interpret the effects of fast neutrons on a variety of heat treated alloy samples. A major portion of the thermal studies was devoted to low temperature (2850-4000C) aging, since small precipitates or clusters are the most sensitive to the effects of fast neutron irradiation. The Mbssbauer spectra of samples aged at low temperatures are consistent with the growth of spherical clusters or Guinier-Preston zones. The distribution of isomer shifts and quadrupole splittings associated with local configurational variations of iron in three basic environments -- i.e., (l)iron with only Cu nearest neighbors represented by a slightly broadened singlet designated Yo; (2)'surface' iron with both Fe and Cu nearest neighbors represented by an ever changing

broad asymmetric doublet designated Y2; and (3)'volume' iron with only Fe nearest neighbors represented by a singlet designated 1 -- were treated by using Gaussian-distributed Lorentzian functions to fit the complex spectra. For this purpose, an efficient subroutine in which the Gaussian-Lorentzian convolution and its partial derivatives are expressed in terms of the complex error function was developed for use with a general least-square fitting program. The spectra associated with both the early and very late stages of aging were found to be quite amenable to quantitative phase analysis. In addition, by assuming spherical cluster growth, approximate cluster sizes (or range of sizes) can be inferred from the Mdssbauer spectra characteristic of initial and intermediate stages of aging. This was particularly valuable in evaluating the re-solution effects of fast neutron i rradi ati on. In the irradiation studies the MBssbauer technique provided an insight into the local environment of the precipitate or cluster phase that was not possible with the more conventional techniques of resistivity and magnetization which have been used in the past to study the effects of irradiation on precipitation-hardenable alloys. In the case of solution treated alloys, the Mbssbauer spectra provided information concerning both the extent of irradiation-induced aging and the approximate size of the clusters formed. The results are consistent with the formation of very small clusters of less than 12 atoms in quenched-irradiated alloys. The re-solution effects of displacement cascades or spikes on aged Cu-Fe samples was shown by Mbssbauer spectroscopy to be extremely dependent upon the stage of aging or the cluster size. Aged samples

containing very small clusters (<12 atoms) experience essentially complete dissolution of Fe clusters, whereas those with clusters of intermediate sizes experienced incomplete cluster break-up in the displacement spikes. In contrast, well-aged samples containing large (-iron precipitates did not experience any dissolution effects, but instead experienced a slight growth in the precipitate phase. M6ssbauer measurements were also useful in observing the effects of fast neutron irradiation on the equilibrium precipitate phase, d-iron.

THE UNIVERS ITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report MOSSBAUER SPECTROSCOPY STUDY OF AGING AND FAST NEUTRON IRRADIATION EFFECTS IN A COPPER-RICH Cu-Fe ALLOY Thomas H. Gould, Jr. Dietrich H. Vincent Project Director DRDA Project 035710 supported by: NATIONAL SCIENCE FOUNDATION GRANT NO. GK-17073 WASHINGTON, D.C. administered through: DIVISION OF RESEARCH DEVELOPMENT AND ADMINISTRATION ANN ARBOR January 1974

TABLE OF CONTENTS LIST OF TABLES................................................ vii LIST OF FIGURES......................................**..*- ix LIST OF APPENDICES............................................. xi CHAPTER I. INTRODUCTION...................................... 1 A. Motivation and Purpose for This Study............. 1 B. Literature Review................................. 4 1. Mbssbauer Effect Applications in Alloys Including Phase Analysis................. 4 2. Mbssbauer Spectroscopy Applications to the Study of Radiation Effects...........7........ II. BACKGROUND................... 10 A. Origin and Properties of the Fe57 MBssbauer Effect Line Structure........................... 10 1. Resonant Line Properties............... 10 2. Resonant Line Area and Phase Analysis......... 13 3. Hyperfine Interactions: Summary of Theory..... 17 B. Interpretation of Mbssbauer Spectra for Fe57 Solute Atoms in Copper and Gold................ 24 1. Au-Fe Alloys.................................. 25 2. Cu-Fe Alloys.................................. 26 C. Metallurgy of Copper-rich Cu-Fe Alloys......... 29 D. Effects of Fast Neutrons on Precipitation Hardening Alloys.................................. 32 1. Simple Theoretical Considerations 33 2. Displacement Spikes........................... 37 3. Experimental Observation.40 iii

III. EXPERIMANTAL PROCEDURE................................ 47 A. Sample Preparation................................ 47 B. MBssbauer Spectrometer............................ 50 1. Drive System................................. 51 2. Detection System............................. 58 C. Resistivity Measurements.......................... 59 D. Data Analysis..................................... 60 1. Lorentzian Fitting Functions.................. 62 2. Gaussian-Distributed Lorentzian Functions..... 63 3. Error Analysis................................ 66 E. Sample Irradiations.............................. 67 IV. EXPERIMENTAL RESULTS.................................. 70 A. Preliminary....................................... 70 B. Results of Annealing Experiments............... 73 1. Solution Annealed and Quenched Samples........ 73 2. Aged Samples............................... 80 3. Results of Resistivity Measurements......... 100 C. Discussion of Annealing Results................... 104 1. Spectra Systematics and Other Considerations.. 104 2. Metallurgical Considerations.................. 116 Aging at 6000C.............................. 116 Low Temperature Aging: MBssbauer Versus Resistivity................................. 119 Low Temperature Aging: Growth of [-iron Precipitates................................ 124 3. Summary.................................. 134 D. Results of Irradiation Experiments................ 135 1. Solution Annealed and Quenched Samples........ 136 2. Aged Samples............................... 141 E. Discussion of Irradiation Results............... 156 1. Irradiation Aging......................... 156 2. Dissolution of Precipitates................... 160 3. Summary............................. 168 v

APPENDICES A. SPECTRAL AREA CONSIDERATIONS....................... 170 B. POINT CHARGE CALCULATIONS OF QUADRUPOLE SPLITTINGS IN Cu-Fe................................ 175 C. MOSSBAUER SPECTROMETER DETAILS..................... 179 Drive System Design Considerations................. 179 Commercial components Used in Mbssbauer Spectrometer System................................ 189 D. DATA ANALYSIS DETAILS.............................. 190 Description of Programs............................ 190 The Fitting of Gaussian-Lorentzian Convolution Functions............1........................... 192 Listing of Mbssbauer Computer Programs 197 REFERENCES.................................................. 210 vi

LIST OF TABLES III-1. NATURAL IRON SPECTRUM.................................. 57 III-2. IRRADIATION SUMMARY.................................. 69 IV-1. MOSSBAUER SPECTRAL PARAMETERS FOR SOLUTION ANNEALED AND QUENCHED Cu-Fe ALLOY SAMPLES............77 IV-2. SPECTRAL PARAMETERS FOR HEAT TREATED Cu-O.6at%Fe ALLOY SAMPLES ANALYZED AS THE SUM OF INDIVIDUAL LORENTZIAN LINES................... 82 IV-3. SPECTRAL PARAMETERS FOR AGED Cu-0.6at%Fe ALLOY SAMPLES ANALYZED AS THE SUM OF TWO INDIVIDUAL LORENTZIANS AND ONE SYMMETRIC LORENTZIAN DOUBLET 84 IV-4. SPECTRAL PARAMETERS FOR HEAT TREATED Cu-O.6at%Fe ALLOY SAMPLES ANALYZED AS THE SUM OF GAUSSIAN-DISTRIBUTED LORENTZIANS................................. 86 IV-5. SPECTRAL PARAMETERS FOR ANNEALED Cu-O.6%Fe ALLOY SAMPLES ANALYZED BY THE SUM OF ONE AREA CONSTRAINED GAUSSIAN-DISTRIBUTED LORENTZIAN DOUBLET PLUS ONE G-L SINGLET............................ 90 IV-6. SPECTRAL PARAMETERS FOR Cu-O.6at%Fe ALLOY SAMPLES ANNEALED AT 4000C.....9............................ 93 IV-7. SPECTRAL PARAMETERS FOR Cu-0.6at%Fe ALLOY SAMPLES ANNEALED AT 6000C..................................... 97 IV-8. COMPARISON OF RESISTIVITY AND MOSSBAUER RESULTS FOR HEAT TREATED Cu-Fe ALLOYS......................... 101 IV-9. COMPARISON OF DIFFERENT GAUSSIAN-DISTRIBUTED LORENTZIAN FITTING MODELS FOR CLASS I SPECTRA OF TWO AGED SAMPLES.. 115 IV-10. EFFECTS OF FAST NEUTRON IRRADIATION ON THE MOSSBAUER SPECTRA OF QUENCHED Cu-Fe ALLOYS....................... 137 IV-11. THREE G-L COMPONENT SPECTRAL PARAMETERS FOR HEAT TREATED Cu-0.6at%Fe ALLOY SAMPLES BEFORE AND AFTER FAST NEUTRON EXPOSURE OF 2.7(10)19 n/cm2............... 144 IV-12. TWO G-L COMPONENT SPECTRAL PARAMETERS FOR AGED (3350C) Cu-O.6at%Fe ALLOY SAMPLES BEFORE AND AFTER FAST NEUTRON EXPOSURE OF 2.7(10)19 n/cm2............................ 145 vii

IV-13. EFFECTS OF INCREASING FAST NEUTRON EXPOSURE ON MOSSBAUER SPECTRAL PARAMETERS FOR A Cu-O.6at%Fe ALLOY AGED AT 3350C FOR 1000 MIN............................ 148 IV-14. MOSSBAUER SPECTRAL PARAMETERS FOR Cu-O.6at%Fe ALLOY SAMPLES AGED AT 4000C (CLASS II SPECTRA) BEFORE AND AFTER A FAST NEUTRON EXPOSURE OF 2.7(10)1 n/cm2....... 149 IV-15. MOSSBAUER SPECTRAL PARAMETERS FOR Cu-O.6at%Fe SAMPLES AGED AT 6000C (CLASS III SPECTRA) BEFORE AND AFTER IRRADIATION............................................ 151 IV-16. MOSSBAUER SPECTRAL PARAMETERS BEFORE AND AFTER FAST NEUTRON EXPOSURE OF 9(10)19 n/cm2 FOR A COLD-ROLLED Cu-O.6at%Fe SAMPLE AGED AT 6000C....................... 154 IV-17. IRRADIATION-INDUCED CHANGES IN RESISTIVITY OF AGED Cu-O.6at%Fe ALLOY SAMPLES.............................. 155 vi 1

LIST OF FIGURES II-1. Magnetic and Quadrupole hf Splitting of Fe57...... 21 11-2. Copper-Rich Cu-Fe Phase Diagram......................... 29 III-1. Experimental System for Mbssbauer Effect Measurements... 52 III-2. Resonance Absorption Spectrum of Natural Iron Standard................................................ 55 II-3. Resonance Absorption Spectrum of NBS Secondary Standard Sodium Nitroprusside................................... 56 III-4. Foil Holder for Resistivity Measurements................ 61 I-5. Full Width at Half Maximum of Gaussian-Lorentzian Convolution Function Versus Gaussian Width Parameter 8.. 65 IV-1. MBssbauer Absorption Spectra of Cu-O.6at%Fe Alloy Samples Representing the Quenched and Two Aging Stages of Y-iron in Copper..................................... 74 IV-2. Mdssbauer Absorption Spectra of Two Cu-O.6at%Fe Alloy Samples: (a) Quenched and (b) Quenched and Cold-rolled.. 79 IV-3. Spectra of Samples Aged at 3350C Fit to the Sum of Single Gaussian-Distributed Lorentzian Functions........ 88 IV-4. Spectra of Two Samples Aged at 335~C Fit to the Sum of a Gaussian-Distributed Lorentzian Singlet and Area Constrained Doublet.................................... 91 IV-5. Gaussian-Distributed Lorentzian Fits to 4000C-Aged Data.. 94 IV-6. Mbssbauer Absorption Spectra Showing the Various Aging Stages of Cu-0.6at%Fe................................... 98 IV-7. M5ssbauer Absorption Spectra of a Cu-O.6at%Fe Alloy Sample After Cold-rolling and then Aging at 6000C for 1000 Minutes........................................... 99 IV-8. Fraction of Iron Transformed From the Solid Solution Phase to Iron Clusters as Determined by Mbssbauer and Resistivity Measurements............................... 103 IV-9. Variation of Y and Y2 Area Fractions for Quenched Samples as a Function of Atom Percent Iron in Copper.... 106 ix

IV-10. Splitting of 2 Doublet as a Function of Aging Temperature........................................... 109 IV-11. Position of Y2 Centroid and Y2R as a Function of Aging Treatment........................................ 109 IV-12. Variation of the Area Fraction of the YL Component With Aging Treatment................................... 117 IV-13. Plot of Resistivity Versus Atom,% Fe in Solid Solution Determined from Mbssbauer Spectra of Heat Treated Cu-Fe Alloys................................................. 122 IV-14. Effects of Fast Neutron Exposure on Spectra of Solution Treated and Cold-Rolled Cu-0.6at%Fe Alloy Samples...... 139 IV-15. Irradiation-Induced Change in the 1 Spectral Component Versus Pre-irradiated Concentration of Iron in Solid Solution for Solution Treated Alloys................... 142 IV-16. Effects of Increasing Fast Neutron Exposure on the Mbssbauer Spectrum of Cu-0.6at%Fe Aged at 3350C for 1000 Minutes............................................... 147 IV-17 Irradiation-Induced Change in the ~L Spectral Component as a Function of Pre-irradiation Condition for Annealed Cu-0.6at%Fe............................................ 152 IV-18. Irradiation-Induced Change in the {L Spectral Component Versus Fast Neutron Fluence for Solution Treated and Aged Cu-0.6at% Alloy Samples........................... 153 IV-19. Net Re-solution Effects of Displacement Spikes as a Function of Pre-irradiation Heat Treatment........... 167 C-1. Waveform Amplitudes Required to Give Drive Transducer a Triangular Velocity Response Versus Frequency........ 186 C-2. Triangular and Square Wave Generator.................. 187 C-3. Mbssbauer Drive Amplifier............................. 188 x

LIST OF APPENDICES A. SPECTRAL AREA CONSIDERATIONS.............................. 170 B. POINT CHARGE CALCULATIONS OF QUADRUPOLE SPLITTINGS IN Cu-Fe....................................... 175 C. MOSSBAUER SPECTROMETER DETAILS............................. 179 D. DATA.ANALYSIS DETAILS...................................... 190 xi

CHAPTER I INTRODUCTION A. Motivation and Purpose for This Study R. L. Mossbauer's discovery of recoilless emission and absorption of nuclear gamma radiation in 1958(1) demonstrated that the theoretical limit of resolution, the y-ray's natural line width, could be achieved under certain circumstances. This discovery opened up a new area of research, nuclear y-ray resonance (Mossbauer) spectroscopy, which has found applications in nuclear physics, solid state physics, chemistry and biology. The Mossbauer spectrum from an emitting or absorbing nuclide is very sensitive to the nuclide's local electronic environment. From the hyperfine structure of such spectra it is possible to obtain information on internal magnetic fields, electric field gradients and s-electron densities at the nucleus as well as on lattice binding forces. As a consequence, the Mossbauer effect (ME) is being used, in addition to its many chemical and biological applications, as a solid state probe to investigate lattice defects including those introduced by radiation. Early defect studies by ME spectroscopy were stimulated by an interest in understanding the local environment of Mossbauer nuclei whose low lying gamma states were produced by such techniques as alpha decay, coulomb excitation, and nuclear reactions. There was concern that such methods of producing the Mossbauer level would leave the

2 nucleus in a defect environment following recoil of the resonant atom. The logical extension of such studies was to intentionally introduce the Mossbauer nucleus into a defect environment for the purpose of gaining information about radiation induced defects. Several experiments of this type have been performed and are reviewed by G. K. Wertheim.(2) Another radiation effects application, which has received almost no attention, is the use of ME spectroscopy to study solid state phase transformations induced by radiation bombardment. In this type of application it is necessary that the Mossbauer nuclide be uniquely associated with the phase(s) undergoing change. Also, for any such experiment to be successful it is necessary that the different environments of the Mossbauer nuclides induced by radiation be few in number and produce resolvable spectra. In general, for Mossbauer nuclides associated with several defect and normal lattice configurations, the resulting absorption spectrum would consist of a sum of the absorption lines characteristic of each configuration times the fraction of sites associated with that configuration. This last statement points up the difficulties of obtaining useful information in radiation damage experiments from Mossbauer spectra. First, in order for there to be a significant effect, there must be an appreciable fraction of Mossbauer nuclides intimately associated with a given radiation induced defect configuration. Second, even for a limited number of such configurations it is quite likely that the resultant M6ssbauer spectrum would not display well-defined (resolvable) structure. However, under conditions and in materials for which these problems are minimized, ME spectroscopy becomes a useful probe with

3 which to determine the local environments of absorbing (or emitting) nuclides in radiation'damaged' materials. Such detailed information is not obtainable from some of the more conventional techniques used to study radiation effects in solids (e.g. resistivity, Hall coefficient, optical emission and absorption, saturation magnetization, etc.). This work was undertaken for the purpose of determining the effectiveness of ME spectroscopy in probing the effects of fast neutron radiation on certain solid state transformations. The most promising applications seemed to be in the area of order-disorder transformations and precipitation hardening reactions. The latter type of transformation was chosen for several reasons. First, because of the added strength arising from precipitate particles in metallic alloys, age- or precipitation-hardened alloys have structural applications in reactors and the space program. In addition, it has been discovered that small coherent precipitate particles and certain solute atoms can minimize void formation in certain nickel-based alloys, making such transformations of possible importance in fast reactor materials. Finally, these alloys have been of fundamental interest in determining the basic mechanisms of radiation damage. One such alloy —copper-iron (Cu-Fe) with a few percent or less iron —appeared to satisfy the requirement of resolvable structure in the Mossbauer spectra from the three iron phases which are known to exist in this alloy system. Additional advantages of using the Cu-Fe system include: 1) Iron (Fe57) is still the most practical element to use in Mossbauer spectroscopy. It allows excellent energy resolution and high intensity measurements to be made at room temperature.

4 2) Cu-Fe samples of the size needed and with just a few percent iron (preferably enriched in Fe57) can be irradiated in fast neutron fluences of up to 1020 n/cm2 without acquiring enough long-lived radioactivity to interfere with the Mossbauer spectroscopy. 3) Dilute Cu-Fe alloys have been the subject of several investigations including studies of precipitation kinetics and neutron irradiation effects by resistivity and magnetization (3-5) measurements() and studies of the M6ssbauer spectra of unirradiated alloy samples.(6-10) Thus, comparison of Mossbauer results with those of other measurement techniques can be made. B. Literature Review 1. Mossbauer Effect Applications in Alloys Including Phase Analysis The application of Mbssbauer spectroscopy to the study of dilute impurities in metals, metallic alloys, and phase analyses of alloys (11) (1) has been reviewed by Gonser() and Wertheim.) A majority of the papers on dilute alloys are concerned with the local magnetic and electronic effects of the impurity atoms. Only those papers which are pertinent to this work are reviewed here. Of particular interest to this study are a series of papers on Mossbauer investigations of dilute concentrations of Fe in the noble metals Cu and Au, including several of their alloys. Gonser et al.(6) were the first to apply M6ssbauer spectroscopy to identify the three known iron phases in dilute Cu-Fe alloys: Fe in solid solution with Cu, paramagnetic y-Fe precipitate, and ferromagnetic a-Fe precipitate. More recently, Bennett and Swartzendruber(7) discovered a possible fourth

5'phase' of iron represented by a doublet in the room temperature M'ossbauer spectrum of a solution annealed and quenched Cu-0.5% Fe alloy. This doublet'phase' was not recognized in the earlier work of Gonser et al., since in their somewhat broadened spectrum the righthand component of the doublet was hidden at the base of the more intense central line produced by Fe in solid solution with Cu. The left-hand component of the doublet, which is essentially coincident with the y-Fe precipitate singlet evident in well annealed samples, was thought by Gonser et al. to be due to y-Fe formed during the quench. Bennett and Swartzendruber initially interpreted the doublet in quenched samples as arising from magnetic interactions within Fe clusters. More recent work by Reno and Swartzendruber(12) and Window(8-10) have shown the doublet to be quadrupolar in origin; i.e. it results from the interaction of the 14.4 Kev M6ssbauer state's quadrupole moment with the electric field gradient (EFG) produced by a non-symmetric electronic charge distribution about the Fe57 nucleus. The charge asymmetry is believed to be caused by Fe pairs and larger Fe clusters within the Cu lattice. Because of the greater solubility of iron in gold, the dilute Au-Fe alloy system allows for the study of Fe cluster interactions over a wider range of iron concentrations. As a result there have been several papers concerned with electrostatic and magnetic interactions of Fe solute atoms within the Au lattice.(13-16) A doublet analogous to that observed in Cu-Fe has also been observed in the room temperature spectra of Au-Fe alloys. The Au-Fe doublet also appears at the base of a more intense central line produced by dispersed iron atoms in the gold

6 lattice and has been interpreted by Violet and Borg,(13) Ridout,(14) and Window(15'16) as arising from quadrupole interactions caused by iron nearest neighbors. In the area of phase analysis, Mossbauer studies have been carried out on such alloy systems as iron-carbon and steel,(724) copper-nickel-iron,(7'25) iron-molybdenum, (26), copper-cobalt,(27) copper-iron(610) and nickel-gold.(28) In addition, several orderdisorder transitions have been studied by Mossbauer techniques including those occuring in such alloys as Fe-Ni,29 Ni3Mn,(30) AuMn (31) AFe,(32'33) and Fe-V.(34) In these experiments the Mossbauer effect has been demonstrated to be an effective technique for studying the formation, structure and relationships of various phases in alloys. The Mossbauer effect seems to be particularly suited for the study of precipitation reactions in alloys because it is capable of detecting minor constituents as well as extremely small particles. The work of Gonser et al.(6) and Bennett and Swartzendruber(7) initially demonstrated the sensitivity of this method for determining the fraction of iron in the two precipitate phases in Cu-Fe alloys. Window's recent publication(1) and the present study extends their work by reporting on precipitation processes of y-Fe during the early stages of aging in the copper-rich Cu-Fe system. In the ternary Cul x(Ni.7Fe 03)x system, Nagarajan and (25) Flinn(25) demonstrated that the segregation of these alloys into two phases occurs by nucleation of precipitate particles and not by'spinoidal decomposition' as originally thought. The precipitation of Co from Cu-Co solid solutions has been studied by Nasu et al.(27) using Co57 in a 2.0% Co source experiment. They were able to not only

7 determine the fraction of cobalt precipitated but also the average size (35) of the precipitate particles by observing their superparamagnetic behavior as a function of temperature. In yet another precipitation (26) ME experiment, Marcus et al. ) showed that annealing of random solid solutions of Fe-Mo at 650~C results in the precipitation of Fe2Mo. Thus, there is ample evidence that Mossbauer spectroscopy can be an effective technique for phase identification in alloys containing Mossbauer nuclides, especially Fe57. The first part of the present study is concerned with characterizing precipitation processes in a Cu-0.6 at % Fe alloy as a function of heat treatment. 2. Mossbauer Spectroscopy Applications to the Study of Radiation Effects The use of Missbauer techniques to study lattice defects was first proposed by Gonser and Wiedersich.(36) Since then there have been several papers describing ME measurements of radiation induced defects in a variety of materials. These measurements have been reviewed by Ullrich,(37) Wertheim(2) and Schroeer et al.(38) In general, ME radiation damage experiments can be classified into two distinct categories: (1) those in which the Mossbauer nuclide is introduced into a defect environment as a result of its production and (2) those in which the Mossbauer nuclide is used to probe the effects of bombarding radiation but is not produced in the process. The former category has by far received the greatest attention since many of the techniques used to produce Moissbauer source nuclei also cause their displacement from their original lattice position. Methods used to produce a coupled source nuclide-radiation defect include radioactive decay (e.g. Ref. 39), Coulomb excitation (e.g. Ref. 40), ion

8 implantation (e.g. Ref. 41), thermal neutron activation(4244) and other nuclear reactions (e.g. Ref. 45). In principle the objective of such experiments has been to create a simple one-to-one correspondence between Mossbauer source nuclide and defect. In practice such radiation damage effects have proved to be quite elusive, especially in metals. The lack of significant effects in metallic lattices suggests that the primary displaced or implanted Mossbauer nuclide returns to its normal lattice site, probably through replacement collisions (See for example Dedericks et al., Ref. 46). Such replacement collisions have been demonstrated by Berger(43) and Czjzeck and Berger(47) who observed the order-disorder effects in FeAl and Fe3Al Mossbauer sources caused by gamma recoil following thermal neutron activation of Fe7. The most significant changes in Mossbauer spectra of'damaged' materials have been observed in source experiments on oxides, salts and semiconductors. Since these experiments are not of direct importance to this work, the reader is referred to the review papers given above. To date there have been very few experiments of the second category reported in the literature. This is primarily due to the difficulty of satisfying the criteria (See pp. 2 and 3) for the observance of resolvable Mossbauer spectra from various'damaged' configurations induced by bombarding radiation. One of the few examples of this type of application is the radiation induced disorder-order transformation studies on a 50-50 FeNi alloy performed by Gros and Pebay-Peyroula(48) using electron bombardment, and by Gros et al.(49 using fast neutrons. The Mossbauer results showed that the radiation caused the growth of a new phase with large magnetic anisotropy and a

9 partially resolved spectrum exhibiting quadrupole as well as magnetic hyperfine structure. The effects of radiation, especially fast neutrons, on precipitation processes in alloys have been studied by resistivity,(5) magnetization(550) and electron microscopy(51) techniques, but not by Mossbauer spectroscopy until this present work. Results to date suggest that fast neutrons have possibly one or more of the following effects on precipitation hardening alloys: 1) precipitation of solute atoms from supersaturated solid solution caused by radiation-enhanced nucleation and/or diffusion; 2) dissolution of existing precipitate particles by displacement spikes; and 3) conversion of coherent precipitate particles into incoherent particles. All three effects are competitive and are believed to occur simultaneously during fast neutron irradiation, although enhanced precipitation is usually masked by the dissolution of precipitates caused by displacement spikes or cascades. The dilute Cu-Fe system with its two distinct precipitate phases seems to be well suited for a ME investigation of fast neutron effects 57 on precipitation processes. In this work the Mossbauer nuclide Fe7 is used as a probe in Cu-Fe absorbers to determine the fraction of iron existing in the precipitate and solid solution phases both before and following irradiation.

CHAPTER II BACKGROUND A. Origin and Properties of the Fe57 Mossbauer Effect Line Structure In iron the M6ssbauer effect (ME) is observed for recoilless gamma-ray transitions between the ground state of the Fe57 nucleus and its first excited state at an energy Eo of 14.4 Kev. Recent values for the properties of these two states are given by Stevens.(52) The first excited state has a half-life of 97.7 ns, a magnetic dipole moment ie = -.155 nuclear magnetons, a quadrupole moment Qe = 0.20 barns and a nuclear spin I = 3/2. The Fe7 ground state has a magnetic dipole moment Vg = +.904 nuclear magnetons and a spin I 1/2. g g The nuclear properties of Fe57 allow the observation of Mossbauer line structure through the three hyperfine interactions involving the nucleus and atomic electrons. These are observed as the isomer shift, magnetic hyperfine splitting and electric quadrupole splitting in Mossbauer spectra. These three interactions acting separately or in combination, along with the inherent shape of the gamma-resonance line, determine the line structure observed in Fe57 Mossbauer spectra. 1. Resonant Line Properties a. Line Shape. —As shown by Margullies and Ehrman(53) the shape of the transmission energy distribution is a function of an 10

ll energy integral over the gamma ray's emission distribution (which is Lorentzian in shape) multiplied by an exponential function containing the resonance cross section for the absorber. For a source with no resonant self-absorption and thin absorber (effective thickness T << 1; see Equation II.1 below), the transmission line* shape is Lorentzian with a width of twice that of the excited state's width, 2 ro. In the limit of zero absorber thickness the resonant line's intensity is proportional to the absorber's effective thickness, given by T = nfaot (II.1) The symbols are defined as f, the recoil-free fraction of the absorber; n, the number density of resonant atoms in the absorber; ao, the maximum resonant absorption cross section; and t, the absorber thickness. In absorbers of finite thickness, the exponential absorption term in the transmission integral distorts the transmission energy distribution of the gamma beam causing line saturation and line broadening. Fraunfelder et al.(54) have found that for T < 5 the shape of the transmission line is Lorentzian to within 1/2% but with a broadened line width approximated by rF 2ro (1.00 + 0.135T), 0 < T < 5 (11.2) Additional broadening occurs from equipment noise and resonant atom inhomogeneities in the source and absorber. *All ME spectra in the present work were taken in the transmission or absorption geometry.

12 b. Recoil-free Fraction. —The recoil-free fraction, f, appearing above in the expression for effective thickness is singly the most important parameter in Mossbauer spectroscopy. It is a measure of the probability for zero phonon processes (i.e., recoilless emission and absorption events) in a given material, and thus determines the magnitude of the effect. The recoil-free fraction is given by( f = exp[-4 2<x2>/X2] (11.3) where <x2> is the mean square amplitude of vibration of the nucleus in the direction of the ybeam and X is the wavelength of the radiation. Using the Debye model for the case of an infinite crystal lattice, the (55) recoil-free fraction can be approximated by(5 f exp[-6(Er/ked)(T/Od)], for T>ed/2 (11.4) where E = Eo/2Mc2 is the free-nucleus recoil energy, T the ambient temperature of the source or absorber, and ed the Debye temperature. Consideration of Equation (11.4) indicates that kOd should be greater than Er and T less than Od in order to obtain a significant effect. The Debye model and thus Equation (11.4) are approximations to the actual lattice frequency spectrum and apply in the strictest sense to monatomic crystals. However, the use of an "effective" Debye temperature for more complicated crystals has proven useful in characterizing M6ssbauer recoil-free fractions. For dilute impurities in metals such as Fe in copper, the "effective" Debye temperature is given by - e] H) (II.5) (I) MH1/2 (H) ^D " r M D {^'^})

13 where MI and MH are the impurity and host masses and O H) the Debye temperature of the host lattice. For Fe57 in copper Equation (11.5) yields an'effective' Debye temperature of about 3650K using the value of 343"K given by Kittel(57) for the Debye temperature of copper. Since R = 1.95(10) ev and ke 3.15(10)"2ev, the condition keD>ER is easily satisfied. Also since T<eD the Mossbauer effect for Fe in Cu is readily observable at room temperature (i.e., from Equation 11.4, f -.74). The measured f-value for isolated Fe in Cu is about 0.73 at room temperature.(58) 2. Resonant Line Area and Phase Analysis In order to perform quantitative analysis of metallurgical phases using ME spectroscopy, it is necessary to relate one or more spectral parameters to the concentration of atoms existing in a given phase. As discussed in the previous section the spectral line intensity for a thin absorber is directly proportional to the absorber's effective thickness, and thus to the Mossbauer isotope concentration. However, the absorption area is a better parameter since it is insensitive to source line shape and instrument resolution, and tends to saturate less rapidly with increasing absorber thickness than the intensity.59-61) This study makes use of spectral area ratios to determine the relative iron phase concentrations in copper as a function of thermal treatment and fast neutron exposure. The background corrected area under a well resolved absorption line associated with an ME isotope of given phase can be expressed as(60) A= fS dE[1-e-Tiki(E)] (11.6) -00

14 In this expression f5 is the recoil-free fraction for the source, Ti is the effective absorber thickness (Equation II.1) for the ith phase or resonant atom configuration, and ki(E) is the energy dependent part of the resonance absorption cross section which is given in the BreitWigner equation as 2 ki(E)= (/2) (II.7) 1 (E-Ei)2+(r/2)2 Here, r is the total width at half maximum of the excited state's energy distribution and Ei is the transition energy centroid. The resonance absorption cross section is then given by ai(E) = aoki(E) (11.8) where ao, the cross section's maximum value, is as defined in many references; e.g. see p. 12 of Reference (55). When hyperfine interactions split the absorption spectrum of the ith phase into N lines, the cross section given by Equation (11.8) can be written as i(E) = P(j)k)(E) (11.9) J1 where P() is the probability for the jth hyperfine transition between the split excited and ground states). The P()'s are normalized to N () unity; i.e. P() = 1, so that the energy integral of the cross section J remains independent of splitting. The origins of such splittings are discussed in the next section.

15 When there are several phases present, the generalized form of Equation (11.6) becomes 00 A = fjs dE[1-exp(-!Tiki(E))] (II.10) -00CO where the sum in the exponent is over the number of phases present. If the absorption lines of one or more of the phases are split, sums over the split components must be included as in Equation (11.9). Assuming that the individual components are well resolved, the partial areas given by Equation (11.6) can be considered separately, in which case for a thin absorber (Ti<<l) the exponential in Equation (11.6) can be expanded to give Ai fsj Tiki(E)dE = 2rfsTi (II.11) -00 The total area under the absorption curve is then the sum over the individual areas; i.e. A = IAi. Unfortunately many cases of phase analysis involve lines which aren't well resolved so that the details of overlap in Equation (II.10) must be considered. This is done in Appendix A for the Cu-Fe alloy absorbers used in the present study. When the absorber can not be considered as'thin' a more comprehensive evaluation of Equation (11.6) or Equation (II.10) must be made. For isolated lines an exact expression in terms of zero and first order Bessel functions has been obtained.(61) A useful expansion of A in terms of a power series in Ti is also available.(62) The inverse of this power series which is particularly convenient in obtaining phase concentrations can be approximated to within 1% by(62)

16 A. Tj ~ IAFT~ Ai<4/3 ~(II.12) i 1-Ai/4' Ai<4/3 where A = Ai/(rif). To obtain the relationship between absorption line areas and phase concentrations, consider an absorber containing several phases. Let Ni be the atom density of the ME element in the ith phase and a, the isotopic abundance of the ME nucleus. The expression for the effective absorber thickness (Equation II.1) for the ith phase becomes Ti = fiNiaot Combining Equation (11.12) and Ti from above gives A. (auot)fNi = 4 (II.13) and similarly for the other phases. Experimentally it is much easier to measure relative rather than absolute areas. Dividing the above expression by the effective thickness for the jth phase yields the desired equation for relative phase concentrations, Ni/Nj = Rij(fi/fi) (1I.14) In the above expression Rij is the ratio of spectral areas for the ith and jth phases; i.e. Rij =Ai/Aj. The ratio in the brackets containing the absolute areas in the Al's is a correction factor which approaches unity for'thin' absorbers. Systematic errors which enter into the determination of absolute absorption line areas tend to cancel in the correction factor, so that the extreme care usually required in absolute

17 ME area measurements is not necessary in measurements of relative phase concentrations. If the recoil-free fractions for the phases are approximately equal and the thin absorber criterion met, the ratio Ai/AT, where AT is the total area of the spectrum, can be used to determine the relative concentration of the ith phase. Given the total ME element concentration in the absorber, one can then determine its concentration in the phases present. 3. Hyperfine Interactions: Summary of Theory Three nuclear-atomic interactions due to near neighbors affect the Fe57 nuclear energy levels. These are the isomer shift, which measures the s-electron density at the nucleus; the electric quadrupole interaction, which measures the distortion of the iron electrons and the lattice from spherical and cubic symmetry, respectively; and the magnetic hyperfine interaction, which measures the effective magnetic field at the Fe57 nucleus. A brief description of these interactions and their observation in ME spectra is given below. More detailed discussions of these interactions in connection with Mossbauer spectroscopy are contained in several excellent reviews (e.g., see References 63 and 64). a. Isomer Shift. —This is a nuclear volume effect resulting from the fact that the ground and excited states of the nucleus have different effective charge radii. Therefore, the electrostatic interaction energy caused by the overlap of electronic and nuclear charge distributions is different for the two levels. As a result, the gamma ray energy is shifted by an amount depending on the source and absorber's

18 s-electron density. Neglecting relativistic effects the isomer shift, IS, is given by IS = 2Ze26<R2>[La(0)i2-1s,(0)I2] (11.15) where 6<R2> is the difference between the mean square nuclear charge radii in the isomeric excited and ground states, Z is the nuclear charge, and 1la(0)12 and \is(0)12 are the s-electron densities at the nuclear sites in the absorber and source, respectively. For heavy elements relativistic effects become important. Correction factors accounting for such effects have been given by Shirley.(65) Such corrections can be as large as 30 percent for heavier nuclei. The isomer shift is measured by determining the shift of the centroid of a Mossbauer spectral component with respect to zero velocity (i.e. zero Doppler shift energy of the emitted gamma). In order to obtain information on s-electron densities, relative isomer shifts for a series of compounds of a given ME isotope are measured with respect to a standard source or absorber. In this manner, changes in s-electron densities can be determined and referenced to the calculated s-electron density for a known electronic configuration of the ME element (e.g., that of a pure ionic state). The relative isomer shifts for Fe57 as an impurity in 3d transition metal host lattices have been measured in several sources.(66) The results show that the charge density at the impurity nucleus is dependent upon the number of 3d electrons, as well as on the 4s character of the host. An exact interpretation has not yet been given owing to the difficulty of calculating the distribution of electrons

19 in the 3d band and the 4s conduction band. Using the data presented by Walker et al.,(67) the isomer shift decrease of about.31 mm/sec observed in this work and by others(7'10) in going from Fe57 in copper 57 to Fe in FCC y-iron precipitates corresponds to an increase of about 6.5% in the s-electron density at the Fe57 nucleus. According to Qaim(66) this increase is probably caused by a reduction in 3d electron screening effects when going from Cu to y-Fe. An increase in the 4s character of the surrounding lattice would also contribute to the observed effect. Since there is only about a 0.5% decrease in lattice parameter from pure Cu to y-Fe, it is unlikely that the inter-atomic spacing difference would make a significant contribution. b. Magnetic Hyperfine Splitting. —Magnetic hyperfine splitting (hfs) arises from the interaction between the nuclear magnetic moment, pIS and the magnetic field at the nucleus. The Hamiltonian for this interaction can be written Hm - eff (II.16) where Heff refers to the effective magnetic field at the nucleus. The interaction splits the nuclear state with spin I into (21+1) sub-levels with the eigenvalues Em = -gPnHeffmI (II.17) where g is the nuclear g-factor, Pn the nuclear magneton, and mI is the magnetic quantum number which takes the values I, I-1,..., -I. In the ME isotope Fe57 the spin 3/2 excited state is thus split into four sublevels as shown in Figure II-1 (left side). The multipolarity of the 14.4 Kev gamma is almost exclusively M1,

20 restricting transitions to AmI = 0, ~1. The allowed transitions result in a 6-line Zeeman pattern as shown in Figure III-2 for an a-iron foil absorber. The angular dependence of the six transitions is as follows: Transition m Angular Distribution +3/2~ 1/2 +1 (3/4)(1+cos2 ) ~1/2+~1/2 0 sin20 +1/2-+1/2 ~1 (1/4)(l+cos20) Here 0 is the angle between the magnetic field and the direction of propagation of the gamma-ray. If the internal fields are randomly oriented to the line of observation of the gamma ray, as in a polycrystalline aggregate, then an average over the above distributions leads to a ratio of 3:2:1:1:2:3 for the intensities of the six lines. An orientation of 0 = 900 leads to a ratio of 3:4:1:1:4:3, while one of 0~ results in a 3:0:1:1:0:3 intensity ratio. Finite thickness effects, instrumental broadening and internal field distributions will modify these theoretical intensity ratios. Such a modification occurs for the a-iron standard used in the velocity calibration of our spectrometer where cold rolling effects in the thin foil and saturation effects resulted in the outer four lines deviating from the theoretical intensities expected for randomly oriented fields. The effective field at the Fe57 nucleus is usually large for iron in a magnetic material below its transition temperature; e.g., Heff = 330 kG for Fe in metallic form at room temperature.(68) The origin of this field in ferromagnetic materials has been investigated theoretically by Watson and Freeman.(69) The effective fields observed for Fe in metallic materials arise largely from core and conduction electron polarization caused by unpaired 3d electrons.

21 Em = - H/I EQ = eqQ/2 m I -+3 2 ~3/2 +1/2 3.. -1/2 3 /2.3/2 2.....L -+1/2 1 ~/'L'Z2,._,.\[_!/__ +1/2 ~.1/2_ ~/ Isomer Magnetic Isomer Electric Shift hfs Shift Quadrupole Splitting Figure II-1. —Magnetic hf splitting of the ground and first excited state of Fe-57 (left) and quadrupole splitting of the first excited state of Fe-57 (right). In the present study carried out at room temperature, only a-iron precipitates that form in well aged and cold-worked Cu-Fe samples exhibit a magnetic hfs structure in their Mossbauer spectra. Room temperature is well above the magnetic transition temperatures for Fe in any of its FCC'phases' with copper.(l0'70) Therefore, any multiple line structure observed for the dilute Cu-Fe alloy samples in this study, with the exception of the well-known a-iron lines, will originate from isomer shift differences and/or electric quadrupole effects from Fe in different near neighbor configurations.

22 c. Electric Quadrupole Splitting. —Electric quadrupole splitting arises from the interaction between the nuclear quadrupole moment, Q, and the electric field gradient, W, created by the 57 electronic environment of the nucleus. In Fe57 the electric quadrupole interaction splits the first excited state (I=3/2) into two doubly degenerate sublevels as shown in Figure II-1 with the eigenvalues eQVZz 2 2 e 41(V-1) [3m-I(I+l)](1+n2/3)/2 (II.18) mI = +3/2, +1/2 In Equation (11.18) Vzz is the z component of the electric field gradient (EFG) tensor in the crystal's principal axis system and r is the asymmetry parameter given by V -V n xx 0yy < (11.19) ri V O<Tj.ll (II.19) ZZ The ground state in Fe7 has no quadrupole moment since I = 1/2. Thus, the transitions between the excited and ground states give rise to a two line Mossbauer pattern with a quadrupole splitting of eQVzz 2 1/2 AEQ = E3/2-E121 = 1+n /3)/ (II.20) For cubic or spherical symmetry each of the EFG components will be zero and the quadrupole interaction vanishes. If the field gradient has axial symmetry Vxx = Vyy, n = 0 and AEQ = eQVzz/2. In this case the angular dependence of the two transitions is as follows:

23 Transition Angular Dependence +3/2+~1/2 3(1 + cos20) ~1/2++1/2 5-3cos2e Here 0 is the angle between the principal axis of the EFG and the propagation direction of the gamma-ray. For a randomly oriented polycrystalline material the relative line intensities are 1:1 (in the thin absorber limit). The origin of electric field gradients in metals and alloys has been discussed by Watson et al.(71 The EFG arises from two sources, Voc due to electrons within the atomic sphere centered at the nucleus ZZ zz and Vatt due to charges outside the atomic sphere on near neighbors. The resultant EFG and the asymmetry parameter can be expressed in the forms: V = (1-)Vla+(l-R)Vl (II.21a) and Vzz = (1-)alattV +(l-R)n Cvoc, (II.21b) where (l-R) and (1-Y,) are the Sternheimer shielding and antishielding factors, respectively.(72) These account for the polarization of the spherically symmetric core electrons caused by the asymmetric charge distributions of p- or d-electrons in the case of V~ and by lattice charges in the case of Vlatt In general, V ~c from valence electrons zz zz in non-symmetric orbitals will be large compared to Vatt due to the 1/r3 dependence of the EFG. However, the polarization factor y, may be quite large providing significant enhancement of the lattice contribution, V.

24 The observation of quadrupole effects for Fe impurities in cubic alloys using ME spectroscopy has been limited to a few systems and until very recently(15) little attention has been given to investigating their origin. The interpretation of quadrupole effects in Au-Fe and dilute Cu-Fe alloys, which display relatively large quadrupole splittings for metallic systems, is discussed in the following section. B. Interpretation of Mossbauer Spectra for Fe57 Solute Atoms in Copper and Gold The problems encountered in analyzing ME spectra for Fe57 in alloys are complex and varied. These are brought about by inhomogeneities in the near neighbor configurations of the iron solute atoms which can cause a distribution of observed isomer shifts and quadrupole and magnetic hyperfine splittings. In this section current interpretations of room temperature Mossbauer spectra of dilute Cu-Fe alloys* and the similar alloy system, Au-Fe, are discussed. Although the majority of ME studies of these two alloy systems have been concerned with low temperature magnetic properties, only those results which are germane to an understanding of the room temperature spectra are considered here. *Before 1972 ME results existed for solution annealed and well-aged Cu-Fe alloys only. These results are considered in this section. ME spectra obtained in the present study for aged Cu-0.6%Fe samples with a variety of agei g bistories and their interpretation are given in Chapter IV. Window'sk') recent study of aged Cu-l.O%Fe alloy samples is also considered in Chapter IV in conjunction with the present work.

25 1. Au-Fe Alloys Owing to its large solubility for iron and relatively large observed quadrupole splittings, the Au-Fe system has proven to be easier to study than its noble metal counterpart, Cu-Fe. The spectra above the ordering temperature in AuFe consist of an apparent single line and doublet, both of which were interpreted by Violet and Borg(13 in terms of quadrupole interactions due to near-neighbor iron atoms. The broadened central singlet has been associated with isolated Fe atoms experiencing quadrupole interactions with further than nearest-neighbor (14) Fe atoms. Ridout(14) interpreted the asymmetry and broadening of the doublet to be a consequence of there being a distribution of numbers and configurations of iron nearest-neighbors. A mechanism to produce the large electric field gradients observed in these alloys was proposed by Window(15) on the basis of a study of Fe57 in AuX alloys, X being a 3d impurity. The correlation between quadrupole splittings and residual resistivities observed in these alloys led Window to suggest that the electric field gradients at an Fe7 impurity arise from a distortion of the iron screening charge from cubic symmetry due to the increased density of states at the Fermi level on an X impurity in the nearest neighbor shell. The screening charge can be described in terms of the Thomas-Fermi model.(73) Its distortion in the presence of another 3d impurity can be explained by Friedel's virtual-bound-state (vbs) theory.(74) In order to approximate the effects of this charge distortion upon the electric field gradient, Window(5'16) invoked an effective point charge model previously used by Preston et al.(75) to explain quadrupole effects in V-Fe alloys. In this model the distortion of the screening electrons (a seven 3d

26 electrons for Fe in Au or Cu) in the presence of an Fe (or X) neighbor is simulated by an effective point charge localized on the neighboring impurity site. The magnitude of the screening charge distortion (and thus the quadrupole splitting) will depend upon the increase in the density of states on the impurity neighbor. Thus, according to Window, it is only for d impurities with a narrow virtual bound state accommodating ten 3d electrons that the distortion is large enough to produce resolvable quadrupole splittings. A further consequence of this model is that the effective point charge should decrease with increasing number of Fe (or X) impurity neighbors. Window(16) was able to explain the changes in the doublet structure for Au-Fe alloys with increasing Fe concentrations (up to " 30%) by using a reducing point charge model to account for changes in iron screening effects occurring for an increasing number of Fe nearest neighbors (i.e., increased clustering). 2. Cu-Fe Alloys The interpretation of room temperature spectra from dilute Cu-Fe alloys is complicated by excessive iron clustering caused by the very low solubility of iron in copper. Similar to the Au-Fe system, spectra from solution annealed Cu-Fe alloys display a'cluster' doublet at the base of a central resonance line due to isolated Fe atoms (e.g., see Figure IV-1) as was first noted by Bennett and Swartzendruber.(7) Early controversy over the origin (magnetic or quadrupolar) of the doublet's splitting has been experimentally resolved by Reno and Swartzendruber(l2) in favor of a quadrupole interaction. The electric

27 field gradient responsible for the interaction has been interpreted by Window(8) as having the same origin as described above for Au-Fe alloys. Dilute Cu-Fe alloys (< 1% Fe) which are rapidly quenched from solution temperature display a reasonably well defined doublet with a splitting of approximately.58 mm/sec and area fraction of 0.3 to 0.4.(7,8) Window(8) associates this doublet with Fe pair formation occurring during the quench. By reducing the quenching rates, Window found evidence for increased iron clustering from ME spectra taken at both 300"K and 1.40K. From spectra taken at 3000K, he observed that with progressively less efficient quenching the doublet shifts slightly to lower energy, decreases slightly in splitting, and increases in area fraction —all signs of Fe clustering. The small decrease in the doublet's splitting for slower quenches can be interpreted in terms of the effective point charge model used by Window(15) to simulate the screening charge distortion occurring in Au-Fe alloys when Fe clusters of a pair or more form. In the way of illustration, this model is applied in Appendix B to calculate the relative quadrupole splittings for several Fe cluster configurations that might occur in dilute Cu-Fe alloys. From the results of Table B-l it is seen that the doublet decreases in splitting but remains quite well resolved for Fe57 with up to five iron neighbors in copper. Concurrent with the doublet's decrease in splitting, Window observed a decrease in the doublet's isomer shift with increased clustering. This is to be expected since the position of the spectral component for large y-iron precipitate particles is about -0.33 mm/sec with respect to the isolated iron (solid solution) component. The resultant effect of a decrease in splitting and isomer shift is that the

28 low energy line of the doublet remains well defined, while the high energy line broadens and becomes masked by the central solid solution component. (8) The low temperature results of Window(8) also indicate the occurrence of iron clustering during quenching from solution temperature. Window fit the complex magnetic hfs spectra obtained at 1.40K to an approximate distribution of hyperfine fields.(76) Peaks in the field probability distribution can be associated with internal magnetic fields occurring for iron with n = 0, 1, 2 or more Fe nearest neighbors. A large peak at about 80 kOe was assigned to isolated iron atoms, while broader peaks at higher fields were related to iron atoms in clusters of a pair or more. These results indicate that below ordering temperatures iron atoms in small clusters or isolated in copper couple ferromagnetically in contrast to the anti-ferromagnetic coupling observed in large y-iron precipitate particles.(70) The fraction of iron with and without Fe nearest neighbors was approximated from both the 1.40K hyperfine field distributions and the spectral area fractions obtained at 3000K; results from both methods agree reasonably well over the range of quenching rates. If dilute Cu-Fe alloys are aged at intermediate temperatures, another component appears in their room temperature ME spectra.(7) This component is a single resonance line which is essentially coincident with the low energy (left) member of the'cluster' doublet and is produced by Fe5 in FCC y-iron precipitate particles. In well-aged samples which have subsequently been cold worked, the y-iron precipitates are transformed to BCC a-iron precipitates which exhibit the well known six-line iron hfs spectrum.(7

29 C. Metallurgy of Copper-rich Cu-Fe Alloys Copper-rich Cu-Fe alloys belong to the precipitation hardening category of alloys as can be seen from the phase diagram(7) in Figure 11-2. The solubility of iron in copper reduces from about 3.0 at.% at 10600C to practically zero at room temperature. Thus, only a small amount of Fe can be retained in solid solution in pure Cu. From the phase diagram it is seen that below the eutectoid temperature (8500C), the iron-rich phase in equilibrium with the copperrich solid solution consists of BCC a-Fe. Above 850"C, for alloys containing approximately 1.4 at.% or more Fe, the equilibrium iron-rich phase is FCC y-Fe. Aging below the solvus initially produces the y-Fe precipitate phase as is discussed below. 1150 950 - 750 - in 550 - c - 350 /. -- --- 1 Cu 1.0 2.0 3.0 4.0 5.0 Atom Percentage Iron Figure II-2. —Copper-rich Cu-Fe phase diagram

30 Precipitation processes in the Cu-rich Cu-Fe alloy system have been studied by several diverse techniques including the measurement of resistivity,(3) hardness(78) and magnetization(3'79) as well as X-ray diffraction(478) electron microscopy(47885) and very recently Mossbauer spectroscopy.(10) Early aging experiments(7983) showed that the initial precipitate was paramagnetic (at room temperature) and could be made ferromagnetic by cold working or extended aging at high temperatures. This led Smith(83) to suggest that the initial paramagnetic precipitate was FCC and coherent with the Cu matrix, while the ferromagnetic precipitate could be identified with the stable BCC a-iron structure. Later attempts to verify Smith's suggestion were made by Denney(4) and Newkirk.78) Measuring the saturation magnetization of 2.4% Fe samples which were aged and then cold worked to effect a y-a transformation, Denney followed the isothermal precipitation of Fe from solid solution in the temperature range 600-8000C. From these measurements he concluded that precipitation proceeded as a diffusion limited process characterized by an activation energy of 1.65 ev and a time (4) exponent of 3/2. X-ray diffraction studies by Denney(4) revealed that the paramagnetic y-phase probably forms coherently with the parent Cu lattice on the {111} and {100} planes, transforming under plastic deformation to a ferromagnetic body-centered structure. Newkirk(78) from Debye-Scherrer X-ray patterns concluded that the FCC iron-rich 0 precipitate phase possessed a lattice constant of a = 3.590 A and was identically oriented with the copper-rich solid solution phase (a = 3.615 A).

31 Further studies of precipitation processes in dilute Cu-Fe alloys were performed by Boltax.(3) Using electric resistivity and saturation induction measurement techniques, he detected the occurrence of two distinct precipitation processes in furnace cooling experiments: (1) the formation and growth of a coarse FCC Fe precipitate (possibly incoherent) above the eutectoid temperature which transforms spontaneously to a-iron at 850~C, and (2) the formation and growth of coherent y-iron precipitate below 850~C which remains indefinitely stable. Boltax also studied the precipitation kinetics of quenched 1.7% Fe samples during isothermal aging in the temperature range 200-400"C. Results from resistivity measurements analyzed by the Johnson-Mehl(84) empirical rate equation yielded a time exponent of 0.28 and activation energy of 34 kcal/mole (% 1.5 ev), the latter value agreeing with (4) Denney's(4) result. Based on the low value of the time exponent,* Boltax concluded that low temperature precipitation processes in Cu-Fe were probably the result of precipitate particles nucleating at dislocation loops formed by the clustering of quenched-in vacancies. He suggested Denney's high temperature aging results represented the coalescence stage of precipitate particle growth (i.e., growth of large particles at the expense of smaller, less stable ones). The y-a martensitic transformation has been recently studied by Easterling and Meik-Oja(85) in a Cu-1.95%Fe alloy using electron microscopy supplemented by magnetic measurements. They demonstrated that precipitation of the y-phase for this composition occurs in the *A time exponent of n = 3/2 is expected for diffusion limited growth of spherical precipitate particles.(84)

32 temperature range 550-750~C. Nucleation was found to occur homogeneously leading to fully coherent spherical precipitate particles.* With long aging times, y-iron precipitates were found to grow, lose their coherency and take on a cubic shape. After prolonged annealing, stable y-precipitates would nucleate from large cubic y-precipitates or directly on dislocations in the matrix as well as on grain boundaries. Transformation from the y- to a-phase was found to occur martensitically 0 on dislocations which traversed large y-particles of 100 A or greater following cold working. The characterization of incipient y-iron precipitate formation during low temperature aging (285-4000C) using Mossbauer spectroscopy has been made in Window's recent paper(10) and in the present work. Discussion of these results is delayed until Chapter IV. D. Effects of Fast Neutrons on Precipitation Hardening Alloys In the past, most of the studies of radiation effects in precipitation or age-hardening alloys have been performed with the viewpoint of determining the basic mechanisms of radiation effects through the unique properties of these materials, while their possible use in reactor environments was of secondary concern. Recently it has been shown that certain nickel-based alloys(86) exhibit significantly reduced swelling compared to austenitic stainless steels in extreme fast neutron environments. It is thought that small coherent precipitate particles in such alloys reduce vacancy clustering and thereby o *It should be pointed out that particles smaller than about 100 A were not resolved in their micrographs.

33 minimize void-swelling. Thus, with the advent of fast reactors, an understanding of the effects of radiation, especially fast neutrons and heavy charged particles, on precipitation processes in such alloys is again of some importance. Fast neutrons are believed to have the following effects on precipitates in alloys: (a) increase of precipitation caused by enhanced nucleation and/or diffusion; (b) dissolution of precipitates caused by multiple displacements; and (c) conversion of coherent precipitates into incoherent precipitates. These effects have been reviewed by Damask.(87) Discussions of the general theory of radiation effects in materials can be found in several books (e.g., see References 88-90). 1. Simple Theoretical Considerations Irradiation of a metal or alloy by fast neutrons will cause the atoms to be displaced from their lattice sites. The displacements are created by the scattering of a high energy neutron from a lattice atom, called the primary knock-on atom (PKA). Before coming to rest, the PKA dissipates its energy through multiple collisions with other lattice atoms which, in turn, can be displaced and initiate further collisions. The displacements occur until the initiating atom reaches a minimum or threshold energy Ed (% 25 ev for Cu). The resulting displacement cascade is responsible for the observed effects in (87) precipitation hardening alloys.(87) The number of such displacements in copper is estimated below from simple radiation damage theory. Assuming elastic scattering, the maximum energy imparted to the primary knock-on by a neutron of energy En is

34 4A ( = ~En Tmax (l+A)2 n En (II22) where A is the atomic mass of the target atom. If the scattering is isotropic in the center of mass system, the mean energy of the PKA will be just T /2. Using Equation (11.22) the mean energy may be described max by T = f- E n or f2En (11.23) (l+A)2 where a factor f is added to account for anisotropy and inelasticity in fast neutron scattering. For Cu atoms, Dienes and Vineyard8 give a nominal value of f - 0.6 for reactor neutron bombardments. A number of models have been proposed for the analysis of radiation effects. We shall use the model of Kinchin and Pease in a modified form.(91) According to this model the number of displacements originating from a PKA of energy T can be expressed as follows: K(T-Ein) v(T) = 2E (11.24) In the above expression, the damage energy (T-Ein) is the total energy available for elastic collisions, Ein is the energy lost to inelastic processes (e.g., electronic excitations), and the displacement efficiency K is a correction factor which accounts for deviation from hard-sphere scattering assumed in the original model. The average number of displacements v(E ) created by a neutron of energy En can be approximated by substituting T from Equation (11.23) into Equation (11.24) above; i.e.,

35 a a K(T-Ein) K( —n -Ein) V(En) E 2Ed (II.25) d d The values for the parameters (T-E. ) and K in the case of Cu have been obtained in a recent computer simulation of displacement cascades in FCC lattices.(9) In the report on this study, the damage energy (T-Ein) is given as a plot against the PKA energy T. The displacement efficiency K in the computer model was found to be energy invariant and to have a value of about 0.81.* The rate of production of displacements per unit volume of material in a reactor can be estimated from dN dt - N0asff v (II.26) where No is the number of atoms per unit volume, as is the average neutron scattering cross section, Pf is the fast neutron flux density, and Vf is the mean number of displacements per fast neutron scattered. As an approximation, the flux density cf is taken to be that of a fission spectrum. The parameter vf can be estimated using the method described above by setting En in Equation (11.25) equal to the average fission spectrum energy; i.e., En 2.0 Mev.(93) *The definition of displacement efficiency used here is with reference to the damage energy (T-Ein). This definition differs from the displacement efficiency defined by Beeler(92) —i.e., K(T) = v(T)/T — which was found to decrease with increasing PKA energy. From the results of a computer study of "displacement spike" effects, Beeler concluded that this decrease was primarily caused by recombination of vacancies and interstitials from different collision events; i.e., interference effects of overlapping branches of the cascade.

36 For neutrons of this energy, the average PKA energy T given by Equation (11.23) is about 36 Kev. Using this value and the computer (91) generated plots of Torrens and Robinson(9) a damage energy (T-Ein) ) 30 Kev is obtained. From Equation (11.25), the average number of displacements per fast neutron is then about 480. For the high flux irradiations in this study (see Table III-2), Equation (11.26) yields a damage rate of approximately 2.8(10)15 displacements per cm3-sec or 3.4(10)-8 displacements per atom-sec. The value of the scattering cross section a used in the above calculation is 3 barns.(4) The simple calculation of v(T) above has not considered such effects as vacancy-interstitial recombination, annealing in "thermal spikes,"(95) channeling of displaced ions between preferential lattice planes, extreme mixing of atoms in "displacement spikes,"(96) or replacement collisions at low energies. Thus, the 480 defects produced by a 2.0 Mev neutron is at best an estimate of the initial number of vacancy-interstitial pairs formed per PKA. In comparison, the computer study of Beeler 92) yielded a v(T) value of about 300 displacements from 2.0 Mev reactor neutrons. Also, the results from many experimental studies in which the displacement rate can be estimated from physical property changes generally indicate that the simple displacement models (88) overestimate the number of defects.(88) Therefore, the number calculated above is an upper limit and probably should be reduced. However, the calculation does point out the fact that many vacancies and interstitials are created by each reactor neutron interacting with a lattice atom; these defects in turn can enhance solid state transformations which are dependent upon diffusion. Also, clusters of these defects

37 which are known to form in metals can act as nucleation sites for heterogeneous precipitation of metallic phases in alloys. The above discussion has concentrated on the number of defects produced by an energetic PKA. An equally important problem is the determination of the spatial distribution and configuration of such defects. In medium- and heavy-mass metals, a PKA created by a fast neutron initially loses energy primarily by inelastic collisions with electrons until it reaches an energy of about A Kev, where A is the atomic mass of the PKA (- 64 Kev for Cu atoms). At energies of the order of 10 Kev or greater, the mean free path between PKA collisions (97) with lattice atoms is several interatomic spacings,(97) and thus the damage consists of secondary branches dispersed along the primary's path. As the PKA slows down to about 2 Kev or so, its mean free path approaches one inter-atomic distance and it collides with almost every atom in its path. The resulting damaged region is called a "displacement spike."(96) 2. Displacement Spikes The energy lost by the PKA near the end of its path length occurs both thermally and by displacements; i.e., through "thermal"(95) and "displacement"(96) spikes. In the thermal spike mechanism, energy is dissipated by electronic collisions and their subsequent lattice (89) interactions. Simple transient heat transfer calculations(89) indicate that although localized temperatures above the melting point occur, the spike's duration is of the order of only 10-11 sec so that only limited thermal diffusion of lattice atoms is expected.

38 The displacement spike as originally envisioned by Brinkman(96) 0 consists at the instant of its creation of a region (% 75 A in diameter) which contains a vacancy-rich central core surrounded by a sheath of displaced interstitials. This configuration has been confirmed in a recent computer study(92) of displacement cascades in BCC and FCC metals. Since such a configuration of defects is highly unstable, many of the interstitials will rapidly relax back to the depleted region, although probably not to their original sites. The resulting effect is the creation of a highly mixed and disordered region of the lattice. All of the displaced atoms, however, do not return to the central depleted region, as many of the interstitials are driven away leaving single vacancies and clusters of varying sizes. The vacancies can then migrate to various sinks in the lattice, form larger vacancy clusters, or collapse to form dislocation loops. Calculations of such displacement effects are extremely sensitive to the choice of the interatomic interaction potential and the subsequent determination of the mean free path between PKA displacements. Brinkman's(96'98) original calculations based on a simple Thomas-Fermi screened coulomb potential indicated that the mean free path of Cu atoms in a copper lattice is about one interatomic distance for PKA energies of less than 20 Kev. Thus an energy of ~ 20 Kev would 0 be dissipated in the spike's volume (estimated to be about 75 A). A more recent calculation by Davies and Sims(97) indicated that the mean free path (m) of Cu atoms varied approximately as tm 0 T -2.16 A/Kev

39 where T is the primary's energy. If it is assumed that a spike occurs when Zm equals the interatomic spacing, the energy dissipated in the spike would be approximately 1.2 Kev, or a factor of ~ 17 less than Brinkman's result. The spread of these two results is indicative of the uncertainties involved in the analytical models. An improved method of determining the properties of displacement spikes is by computer experiments.(92'99) Computer simulation methods use more realistic interatomic potentials and can account for such lattice effects as channeling and vacancy-interstitial recombination in determining the energy deposition of the primary and secondary knock-on atoms. One such study (Beeler(92)) simulated displacement spike effects in a-iron and copper for a range of PKA energies. The results indicated that (1) the PKA energy is completely transferred to secondary 0 knock-on atoms within a distance of 10-80 A in the range 0.5 < T < 25 Kev, (2) the cascade dimension is determined by the ranges of quasichanneled (short range) high order knock-on atoms rather than by the range of the initiating PKA, and (3) the initial defect configuration in the spike is similar to what Brinkman suggested.(96) On the basis of such a model, a precipitation hardening alloy such as Cu-Fe in a reactor environment would experience a number of 0 highly disordered regions (spikes) 10 to 80 A in diameter having high vacancy densities. The number ANs of such spikes created per unit time would be given by Nf -' Nosaf (II.27) At~~~~''

40 since one spike is expected for each neutron-PKA interaction. In the spike regions, the high vacancy concentration would result in artificially large solute diffusion coefficients, thereby enhancing precipitation in a super-saturated alloy. Dislocation loops produced in the spike region could serve as nucleation sites for heterogeneous precipitates. On the other hand, small precipitate particles enveloped by such a spike could be dissolved, leading to an overall reduction of the precipitate phase. In addition, an ordered precipitate phase could be effectively disordered by the spike mechanism. The experimental observation of these effects is discussed below. 3. Experimental Observation Experimental observations of radiation effects on precipitation processes in metallic alloys have been made by measuring electrical (5) (5,50) resistivity,(5) magnetic(5 and mechanical(l02) properties and more recently by high voltage electron microscopy.(5) In general, results from such studies have confirmed the predicted effects discussed above, however, not without some ambiguity. Observation of the first effect discussed at the end of the last section, that of enhanced-diffusion, has been made in many systems including order-disorder alloys and precipitation hardening alloys. The pioneering quantitative studies in this area have been made primarily on Cu-Zn (a-brass) and Cu-Al and to a lesser degree on the Fe-Si precipitation system. These studies, which consist of the measurement of resistivity in originally quenched alloys as a function of annealing (87) temperature and irradiation, have been thoroughly reviewed by Damask.(87) Results from these measurements indicate that a diffusion coefficient

41 20 2 of about 10 cm2/sec can be achieved by reactor irradiation alone. Such diffusion enhancement is believed to arise primarily from the (87) motion of excess vacancies.(87) In the case of precipitation hardening alloys, the amount of precipitation caused by such a small diffusion coefficient has only been observed in samples that have been quenched to retain all the solute atoms in super-saturated solid solution.(101s102) (50) solution.(01,102) Piercy's experiments( with Cu-2 wt. % Co alloy samples which exhibit superparamagnetism indicate that the precipitate particles formed as a result of reactor neutron irradiation have radii 0 less than 10 A. The second effect mentioned above is that of nucleation of precipitates on vacancy clusters or dislocations caused by displacement spikes. This effect is difficult to determine in precipitation hardening systems since the increased precipitation caused by irradiation in a quenched alloy could be caused by either enhanced diffusion or enhanced nucleation (e.g., see Reference 101 and 102 on the Cu-Be system). Successful demonstration of the effect was performed by Hull and Mogford 10) with a Fe-.004% C alloy. Using an electron microscope these workers observed an increase in the number of carbon precipitates formed during irradiation in the temperature range 60-100C over those formed during aging in the same temperature range. There have been several other experiments of this type which have demonstrated the enhancement of nucleation by irradiation —for examples, see the review paper by Damask. 87 The dissolution effect of displacement spikes on precipitates has been experimentally investigated in two systems, Cu-Co(50) and Cu-Fe.(5) In both cases, samples aged at temperatures which would

42 favor the production of small precipitate particles showed property changes upon irradiation which indicated that precipitate particles were being dissolved. In the Cu-Co (2.0 wt. % Co) system, Piercy carefully studied the break-up of precipitate particles caused by a fast neutron exposure of 3.4(10)19 nvt as a function of particle size. Co precipitates of varying sizes were produced by aging quenched samples at 395 and 600~C for varying times. During the early stages of annealing (i.e., for 0 particles of about 150 A or less), the precipitates exhibit a phenomenon (104) called superparamagnetism.(04. This phenomenon is characterized by the particles having a single magnetic domain whose direction is independent of its neighbors and is thereby free to rotate. The size of these particles and the standard deviation of the particle size distribution can be determined from the initial and final slopes of the magnetization curve.(105) Also, the number density of the particles can be obtained from the saturation magnetization once the average particle size is known. Using this method, Piercy determined the average particle size and number density before and following irradiation. From the results, he concluded that (1) only particles with a radius of less than about 0 12.4 A were effectively dissolved by neutron induced spikes and that (2) partial overlapping of a spike and precipitate particle did not cause partial dissolution. The calculated spike radius (assuming spherical spike volumes and complete overlapping) to produce these o effects was about 34 A. His second conclusion above is questionable, since according to Table 1 in his paper(50) two samples with initial o radii of about 41-47 A showed an effective decrease in average radius

43 and accompanying increase in the number of precipitate particles. It would appear that the particles were partially dissolved and new precipitates were formed in the spike region as a result of the high solute concentration existing there. Although there is some uncertainty 0 in Piercy's calculation which predicted an average spike radius of 34 A, his experimental results clearly demonstrate that spikes do dissolve 0 small precipitate particles and that particles smaller than about 13 A are unstable in a reactor. Similar radiation effects to those reported by Piercy(50) for Cu-Co were determined in a Cu-1.7% Fe alloy by Boltax.(5) Since samples of this alloy aged below the eutectoid temperature contain paramagnetic (3) y-Fe precipitate particles,() Boltax was not able to directly determine particle sizes in the manner of Piercy.(50) Instead he relied on precipitation kinetics data from a previous aging experiment (3) (Boltax (3) and the particle size data obtained from the similar Cu-Co system (0104 to estimate the average size of the y-Fe precipitates. (Both the aging and radiation dissolution results were obtained from resistivity measurements.) His size estimate was based on the growth law for spherical particles,(06) r3 = k'e(Q/RT)t (11.28) where k' was determined from the Cu-Co size data (104) and the activation energy Q was obtained from the Cu-Fe aging results. Since the results of his earlier kinetics studies (see section C of this chapter) indicated precipitation occurred on dislocation loops, the use of Equation (11.28) was not justified. However, as discussed in the previous section, electron microscopy studies of Cu-Fe alloys aged at

44 550~C have revealed the presence of spherical coherent y-precipitate particles.(85) Therefore, Equation (11.28) would apply to particle growth in the Cu-Fe system assuming the kinetics were diffusion controlled, which is probably not the case at the lower aging temperatures used in Boltax's work. In addition, a comparison of Mtssbauer and resistivity results in aged Cu-0.6% Fe alloy samples (see Chapter IV) indicates that resistivity is not a direct measure of the concentration of Fe in solid solution as was assumed by Boltax.(3'5) The effects of small iron clusters on the resistivity can not be ignored during the early stages of aging. Because of the foregoing considerations, the quantitative aspects of Boltax's work appear to be incorrect, however, his resistivity measurements of aged and irradiated Cu-Fe samples do provide a qualitative indication of the precipitation processes occurring. In general, Boltax's(5) measurements showed an increase in resistivity upon irradiation for Cu-1.7 wt. % Fe samples previously aged at temperatures in the range 250-475~C. Since an increase in resistivity indicates an increase of Fe in solid solution (or to a lesser extent a reduction in precipitate particle size), dissolution of precipitate particles in these samples did occur. Boltax also found that the resistivity of samples aged at temperatures > 500~C decreased upon reactor irradiation. He interpreted this as an indication that the precipitate particles were too large to be dissolved by a displacement spike and that the existing particles increased in size as a result of radiation induced diffusion. Unlike the model used by Piercy(50) to calculate the effective displacement spike size, Boltax used a model which accounted for

45 precipitate dissolution caused by partial spike-particle interactions. 0 His results indicate that precipitate particles less than % 50 to 100 A in radius can be partially dissolved (or broken-up) by displacement 0 spikes whose mean radius was estimated to be 35 A. Although the size of the spike agrees with Piercy's estimate, 50) the fact that particles 0 with radii greater than 13 A can be dissolved does not agree with Piercy's conclusion. There are three reasons for this disagreement: (1) Boltax's particle size estimate is in error (see Chapter IV); (2) the spike-particle interaction models used by the two investigators were different and (3) thie property changes measured in the two experiments (magnetization and resistivity) differed in their sensitivity to particle size changes. At the onset of the present study, it was anticipated that a direct determination via MUssbauer spectroscopy of Fe concentration in the phases existing in Cu-Fe before and after irradiation would clarify the above results. A comparison of the present study's results to those of Piercy(50) and Boltax(5) is made in Chapter IV. The final effect of fast neutron irradiation on precipitation hardening alloys to be considered is the conversion of a coherent precipitate to the incoherent form. It is believed that large coherent precipitates having high strain fields associated with them can be converted to the more stable incoherent form as a result of vacancy clusters created in the collision cascade.(07) The extreme case of this effect involved the partial conversion of coherent to incoherent particles in a Cu-2.4% Fe alloy by electron irradiation.(108) In this experiment, Denney(108) observed the conversion of FCC y-particles to the incoherent BCC a-form by measuring the saturation magnetization of

46 the alloy samples before and after irradiation. This effect is remarkable since high energy electrons only produce isolated defects. Denney's results have not been successfully explained to date. A similar experiment except with fast neutron bombardment was performed by Boltax(5) on Cu-Fe alloys with Fe concentrations in the range 0.8-5.0%. Similar to Denney's results, the saturation induction measurements of this investigator also showed the presence of a-Fe following the irradiation of annealed samples originally containing only y-Fe. However, in samples containing both types of precipitates, the saturation induction measurements gave ambiguous results indicating an increase in a-Fe for some samples and a decrease for others. Mossbauer measurements should be able to provide unambiguous results concerning the y-a transformation (or its reverse), since the six-line a-iron pattern is easily resolved from the single lines due to y-iron and Fe in solid solution.

CHAPTER III EXPERIMENTAL PROCEDURE A. Sample Preparation The Cu-O.6at.%Fe absorber foils were prepared by melting a mixture of 99.999% copper shot with pure iron powder enriched to 69.7% in Fe57.* The melting was performed in a vacuum induction furnace** with the sample contained in a vitrious graphite crucible. The Cu-Fe mixture was maintained in the molten condition for approximately one hour at a temperature of about 12500C. To insure a homogeneous mixture of Fe in copper, the melting procedure was repeated three times. The ingot so obtained weighed about 20 g. It was subsequently cold-rolled** to strips 13 to 20 pm in thickness from which rectangular foils measuring 0.7x0.5 in. were cut. On two occasions during the rolling procedure, the foils were subjected to re-crystallization heat treatments at 850~C to relieve strains induced by cold-rolling. Samples from several of the Cu-Fe foils were chemically analyzed to determine the Fe concentration and its homogeneity.*** The mean composition for *The enriched Fe was obtained from Oak Ridge National Laboratory, Oak Ridge, Tennessee. **The induction melting and rolling operations were performed with the assistance of L. Terner and L. Whitt at the Ford Scientific Laboratory, Dearborn, Michigan. ***The chemical analyses of the Cu-Fe samples were performed by T. F. Strow of the Ford Scientific Laboratory. 47

48 eight samples was 0.63at.% with a standard deviation of.04at.%. A spectrochemical analysis of these samples showed only trace amounts of carbon and other impurities. The majority of the Cu-0.6at.%Fe foils underwent the same heat treatment procedure. The initial heat treatment consisted of a solution anneal for three hours at 1000~C followed by rapid quenching to room temperature. For this purpose, the foils were encapsulated in Vycor capsules filled with a mixture of argon and hydrogen and then heated in a temperature controlled tube furnace.* The rapid quench was achieved by dropping the capsule with a steel weight follower onto a piece of angle iron submerged in a water bath below the furnace. Bright surfaces were maintained on all specimens prepared in this manner. Most of the samples so treated were subsequently subjected to an aging heat treatment at lower temperatures (285-6000C) using this same technique except that the Vycor capsules were not broken during the quench. Several of the 0.6% Fe foils cut from an 80 vm-thick rolled strip were first solution annealed and then cold rolled to a thickness of about 25 vm. Five such samples were then aged at 6000C, while the remainder were used to study the effects of cold-rolling on solution annealed alloys. In addition to the 0.6at.% Fe alloy samples, Cu foils containing 0.1, 0.2, 0.3, and 0.4 at.% Fe were also used in this study.** These were prepared by plating 91% Fe57-enriched Fe onto Cu foils and diffusing at 10000C in an inert atmosphere. All of the samples with lower Fe concentrations were solution annealed at 10000C. *Lindberg Heavy-Duti tube furnace and control console, Models 54032 and 59344. **Obtained from R. Knauer of the Sandia Laboratories, Albuquerque, New Mexico.

49 One of the primary considerations in the preparation of the Cu-Fe absorbers was that of thickness optimization. In this regard, the following criteria were considered: (1) The absorber thickness is chosen so as to maximize the information obtainable in the Mossbauer experiment; i.e., maximize the ratio of the maximum resonant intensity in counts per channel to the standard deviation of the counts per channel (37) (2) The effective absorber thickness T should be less than one to minimize line broadening (maximize resolution) as discussed in Chapter II and Appendix A. (3) The actual absorber thickness t should be large enough to allow resistivity measurements to be performed (see Section C of this chapter). The latter two criteria above were considered most important for this work. Preliminary tests using the resistivity measurement apparatus described in Section C of this chapter demonstrated that reproducible resistivity measurements could be performed with foils as thin as u 12 nm. Because of significant atomic absorption of the 14.4 Kev gammas in copper, it was desirable to use thin foils to maximize the Mossbauer information. With an Fe enrichment of 69.7%Fe57, foils with thicknesses in the range 12-15 vm satisfied the second criterion (.77 < T <.94), yet provided sufficient spectral intensity to allow reasonable counting time.

50 57 The source used in this study consisted of 50 mCi of Co57 dispersed in a Cu matrix.* It was prepared by plating Co57 onto a Cu foil and diffusing at 950~C in an Argon atmosphere, followed by quenching to room temperature. With the spectrometer used in this study, the 50 mCi Cu(Co57) source produced a 0.21 mm/sec corrected line width for the inner two lines of a 1/2 mil natural iron absorber. B. Mossbauer Spectrometer In studying the Mossbauer effect, the energy scan over the line structure of the resonant gamma-rays is provided via the Doppler shift by varying the relative velocity between the source and absorber. The simultaneous registration of the transmitted gamma-rays versus relative velocity can be accomplished by coupling a multichannel analyzer with a linear velocity drive 09112) In this method, the source is mounted on the drive shaft of an electro-mechanical transducer which is constrained to move in a cyclic manner between positive and negative velocity extremes. The spectra in this experiment were recorded using a multichannel analyzer operating in the time or multiscaling mode in conjunction with a linear velocity drive of the electro-mechanical feedback type.(^) The existing spectrometer is a modified version of an earlier system used in the Department of Nuclear Engineering.** A *The Cu(Co57) source was obtained from the New England Nuclear Corporation, Boston, Massachusetts. **The original drive system is described in a thesis by J. F. Ullrich.(37) Unlike the present system it derived its basic triangular drive signal from the analog address scaler sweep of the MCA.

51 block diagram of the spectrometer equipment including the gamma-ray detection system is shown in Figure III-1. A description of the design of the drive system, including schematic diagrams of all new components, is given in Appendix C along with a listing of commercial components used in the spectrometer. 1. Drive System At the onset of this project it was apparent that'equipment' broadening of spectra taken with the original ME drive system prevented the attainment of the optimum resolution necessary to separate the complex Cu-Fe structure. The major portion of line broadening was caused by the combination of (a) higher harmonic noise due to finite steps in the analog address scaler reference signal, (b) 60 cycle pick-up from the vacuum tube components in the drive amplifier, and (c) a phase shift in the drive signal following amplification. In general, the original drive amplifier contained mostly vacuum tube components so that overall system performance was inferior to similarly designed systems using solid state components. In particular, frequent tuning and repairs were required of the vacuum tube electronics. Since commercial ME drives were relatively expensive at the time, it was decided that appropriate modification could be made to the original spectrometer by employing a redesigned drive amplifier using the latest solid state components and obtaining the drive's reference signal from an independent function generator. Some of the original spectrometer's major components were retained including the moving-coil drive transducer and the multichannel analyzer-multiscaler system. In the present ME spectrometer the triangular reference signal is derived from an ultrastable function generator and not the analog

-~ —-LEAF SPRINGS — I PERMANENT FIELD MAGNET COIL) I PROPORTIONAL S SOURCE ABSORBER COUNTER PICKUP L = =4 COIL DRIVE s I I Fs~s~s~ I COIL HIGH VO LTGE _ ___SUPPLY AMPLIFIER MAGNET FIELD SUPPLY FUNCTION n TIME SINGLE GENERATO - BASE CHANNEL I U L1 GENERATOR ANALYZER DRIVE St MULTICHANNEL AMPLIFIER ANALYZER TELETYPE Figure ~ EprmnaSytmfrEfcCRT Figure Ill-1. Experimental System for Mossbauer Effect Measurements

53 address scaler sweep of the MCA as was the case for the original drive. The function generator is a modification of Cohen's design(13) which produces extremely stable triangular and square waves that are precisely in phase. The version used has a frequency and amplitude stability and linearity of better than 0.1%. The new drive amplifier performs three basic functions. First, it integrates the triangular reference signal provided by the function generator to produce the principal parabolic displacement wave form. Then it synthesizes the square, triangular and parabolic wave forms to approximate the voltage signal required to give the moving-coil transducer a linear velocity response. Finally, by comparing the output of the transducer's velocity pick-up coil (amplified about 50 times) to the triangular reference signal a correction signal is derived which provides negative feedback to control the motion of the transducer. The combined signals are boosted in the final amplification stage to provide the required driving current for the transducer. Further details of the drive electronics can be obtained from Appendix C. The analog address scaler or multiscaler (MCS) sweep is synchronized to the source motion by a trigger signal received from the function generator at the beginning of each period. When the last channel has been addressed, the MCS is automatically reset to zero channel to await the arrival of the next trigger signal. By making the product of the MCS dwell time per channel and the number of channels equal to (actually slightly less than) the drive motion's period, the velocity spectrum is recorded twice in the MCA memory, the second half being the mirror image of the first half. The operating frequency of

54 the transducer was set below its resonance frequency of 17 Hz at approximately 10 cps for the Fe57 spectra taken in this study. Initial calibration of the spectrometer's velocity scale was performed by using the room temperature hyperfine spectrum of Fe57 from a 1/2 mil natural iron foil in conjunction with the data given in Stevens.(52) One such spectrum taken with the 50 mCi Cu(Co57) source is shown in Figure III-2. Rolling and saturation effects on the intensity of the outer lines can-be seen in the figure. A comparison of the line positions and widths with their expected values is made in Table III-1 below. Using the excited and ground state splitting factors of gl = 2.2363 mm/sec and go = 3.19156 mm/sec recommended in Stevens,(52) the linearity of the spectrometer over the velocity range of about ~ 6.0 mm/sec was determined to be about.01 mm/sec or within 0.1%. The reference zero position of all spectra taken in this work is given as the centroid of the sodium nitroprusside (SNP) doublet. For this purpose, a National Bureau of Standards' secondary SNP standard absorber ^14) was used. (Its spectrum is shown in Figure III-3.) For the majority of the Cu-Fe alloy samples studied, a reduced velocity range was used since their spectra displayed no broad magnetic hyperfine splitting. In these cases calibration of the velocity scale was obtained from the splitting of the SNP doublet. Excellent agreement occurred between the channel calibration values obtained from the SNP splitting (Av = 1.7015 mm/sec) and those from the splitting of the inner two iron lines (Av = 1.6793 mm/sec). The overall performance of the modified spectrometer satisfied our initial requirements of low equipment broadening and good velocity

U') zo LUj cc j400 -5.00 -'1.00 -3.00 -2.00 -1.00.00 1.00 2.00 3.00'1.00 5.00 6.070 MIMSEC Figure III-2. Resonance absorption spectrum of natural iron standard.

56 C) I-\ LU.." X I I. -2.00 -1.00.00 1. 00 2.00 MM/SEC Figure III-3. Resonance absorption spectrum of NBS secondary standard, Sodium Nitroprusside.

57 stability. Calibrations over a two year period showed variations of only.003 and.015 mm/sec per month in the SNP doublet's splitting and centroid position parameters, respectively. The noise reduction attained over the previous drive system was significant; the observed velocity error signal was reduced from about 0.25 to 0.1% of the maximum velocity. Line broadening* as determined from the inner two lines of the metallic iron spectrum decreased from about 0.04 to 0.015 mm/sec. TABLE III-1 NATURAL IRON SPECTRUM Measured Position Expected Measured Expected mr/sec Position Line Width Line Width Line (~.003)(a) mm/sec +.008 mm/sec mm/sec 1 -5.054 -5.052.250.225 2 -2.814 -2.816.236.215 3 -0.580 -0.580.218.204 4 +1.100 +1.100.216.204 5 +3.334 +3.336.233.215 6 +5.573 +5.572.248.225 (a) (a)Reference: NBS secondary standard, sodium nitroprusside (SNP). (ISSNP-ISFe) = -0.260 mm/sec. *Equipment broadening was determined by evaluating the Am remaining after thickness corrections to the line-width were made; i.e., the width in excess of that calculated by Equation 11.2.

58 2. Detection System The method used to detect the 14.4 Kev gamma-ray of Fe57 is well established (e.g., see Reference 115). The commercial components which make up the detection system are listed in Appendix C. A two-inch-diameter krypton-filled, C02-quenched gas proportional counter with a 1/2 in. Be "thru-window" was used because of its relatively high efficiency (% 85%) for the 14.4 Kev gamma-ray. Radiation from the Cu(Co57) source was pre-filtered with a five mil Al foil to attenuate (- xl/50) the X-ray peak at 6.3 Kev, and thereby reduce the unwanted low energy contribution to the count rate. The source-detector separation was kept at 8 in. or more so that "cosine smearing" effects(116 on the Mossbauer spectrum would be negligible. The output pulses from the Kr-CO2 proportional counter were fed through a preamplifier and pulse-shaping linear amplifier into a single-channel analyzer (SCA) with a window setting at the 14.4 Kev peak. The logic pulses from the SCA were fed into the multiscaling input of the MCA and stored in the memory channel corresponding to the velocity of the source. Adjustment of the SCA window setting was easily accomplished by using the coincidence gating circuit of the MCA in the pulse-height mode. In order to obtain approximations to the absolute spectral areas as required by Equation (11.14), it was necessary to determine the high energy gamma contribution to the pulses passed by the SCA. This was done by using the method suggested by Housley et al.(117) In this method, a filter of brass 5 mils in thickness is used to absorb the 14.4 Kev gammas (passes less than.02%) while allowing all but 4% of the high energy radiation (122 and 136 Kev gammas from the Fe57

59 decay) to pass. The count rate so obtained for a particular experimental configuration, multiplied by 1.04, is then the desired background count rate which is used to obtain the signal to background ratio. Because the experimental configuration remained constant and the samples were of the same material and approximately the same thickness, this ratio required only periodic checking. C. Resistivity Measurements Resistivity measurements of all of the solution annealed and aged samples and several of the irradiated samples* were made to supplement the ME measurements and serve as a comparison with resistivity data reported by Boltax.(3'5) A detailed description of the procedure and apparatus used in the resistivity measurements is given in a report (118) by Yu. The measurements were made by the well-known potentiometer method in which voltages across an unknown sample and a standard resistor in series are determined by balancing them against a standard voltage. The foil's resistance was obtained from the product of the standard resistance and the unknown to standard voltage ratio, while its resistivity was calculated from p= 2-R (III.1) t d In this expression R is the resistance of the foil, M its mass, d its density and t is the distance between the potential terminals. Extreme *The condition of many of the irradiated foils was too rough to obtain meaningful results from resistivity measurements.

60 care was taken in cutting the foils into rectangles in order to provide a constant cross sectional area and minimize errors associated with a non-uniform electric field. Because of the unique shape of the resistivity specimens, a special four-contact sample holder had to be designed and assembled. It is shown in Figure III-4. The foil to be measured was clamped rigidly between extremely flat brass plates to which the current leads were connected. This arrangement provided a uniform electric field across the foil and also avoided having to connect the current leads directly to the fragile sample. The potential across the foil was measured between two vertically adjustable pointed screws which could be slowly brought into contact with the foil. The resistivity of each foil was determined by averaging the results of six independent measurements taken with different potential contact positions. The precision of the measurements ranged from 0.5 to 2.0%. Systematic errors in the measurements could have resulted from non-uniform foil thickness, deviations from a constant width, and contact resistance in the potential terminals. Because of the excellent agreement achieved between measurements taken with different potential contact positions, such errors are believed to be within 2.0%. D. Data Analysis All spectra obtained in this study were analyzed by the use of computerized least squares fitting techniques based on a modified Gauss(119) Newton iteration procedure.^) The fitting methods used were adapted from the computer programs of Chrisman and Tumollilo(120) and

I ir hi^^ 00, 15 I I: J —1 i Potential terminals 5/8" CuFe Current Foil tension Current Foil terminal \adjust screw terminal ~/~1 Notes:= ~l~l~ -~ ~ p^~ 11 ~~~~(1) Drawing is to scale. I I (2) Conducting components Pi^~~ ^ ~~~~are brass. (3) Base is a Phenolic fill~~r^~~~~ n^insulator. Potential contact adjust screws Figure IV-4. —Foil holder for resistivity measurements.

62 Ullrich.(37)* A short description and Fortran listing of the M6ssbauer data analysis code used in this work appear in Appendix D. Descriptions of the iteration techniques used by the program are given in the above references. The data analyses were performed on the IBM Model 360-67 computer at the University of Michigan. Plots of the data and fitted spectra were obtained using a Calcomp automatic plotter. Basically, two types of distribution functions were used in fitting the experimental data. The first consisted of the usual sum of Lorentzian shaped resonant lines, while the second consisted of a sum of convoluted Gaussian-Lorentzian functions (i.e., Gaussian distributed Lorentzians). In both cases the non-resonant background was fit to a quadratic equation to account for solid angle effects from the moving source. 1. Lorentzian Fitting Functions As discussed in Chapter II, the transmission of resonant gamma-rays through an absorber with an effective thickness of T < 5 is Lorentzian in shape. Thus, the first attempt at fitting the complex hfs spectra observed for the Cu-Fe alloys was to use a simple sum of Lorentzian single and/or pair functions. To avoid having to write a separate "theory" subroutine for each unique resonant line configuration, the following general equation was used to fit the data: *The program originally written by J. F. Ullrich in the MAD (Michigan Algorithmic Decoder) language was subsequently translated into Fortran IV by P. Merrithew.(0 ) The existing Mossbauer analysis code is a modification of these earlier programs.

63 r. 2 F(v) = B1+Cv+Dv - 2 r. 2 i^l (v-vi) (-~) - I 1(~) + - 2 2 (111.2) 1 1 L'(v-v ) (2 (v-vvd 1)2 2 where the summations are taken over N single Lorentzians and N symmetric Lorentzian doublets. In the above expression F(v) is the number of gammas transmitted at the velocity v (given in channel units); vi, ri and Ii are the centroid, linewidth (FWHM) and normalized intensity, respectively, of the ith singlet; Aj, rj and Ij are the splitting, linewidth and intensity, respectively, of the jth doublet; vd is the mutual centroid of the Np doublets; and B is the number of counts at v = X with the coefficients C and D accounting for the linear and quadratic corrections in the non-resonant background. The summation over the mutually centered doublets was a convenient way of fitting the six-line hfs spectra produced by a-iron precipitate particles in the presence of the two singlets produced by y-iron precipitates and Fe in solid solution with copper. All of the fitting parameters were varied independently. 2. Gaussian-Distributed Lorentzian Functions In the method of fitting discussed above, line broadening brought about by inhomogeneities in the Fe near-neighbor configurations was accounted for by the unconstrained broadening of the Lorentzian linewidth parameters. In order to treat such broadening in a more realistic manner, the data were also fit to a function of the form

64 N F(v) = B+Cv+Dv 2- d(v), (III.3) ci j=l where dj(v) is a convolution of Gaussian and Lorentzian functions and is given by 00j~ 1 (v 2 2 J (v-v')2+(2)2 A9j _00 For the above convolution the resonant line fitting parameters become vj, Aj and ej; the centroid, area and Gaussian width,* respectively, of the jth resonant line. The expression of Equation (111.4) represents a Gaussian distribution of Lorentzian functions of width r and position v'. The Lorentzian width was fixed in the fitting at 0.21 mm/sec, which corresponds to the minimum experimentally observed linewidth.** The total width (FWHM) of the convolution of Equation (111.4) as a function of the Gaussian width parameter e is plotted in Figure III-5. The techniques for least-squares fitting an expression of the form given by Equations (111.3) and (III.4) —in particular, the numerical evaluation of d.(v) and the obtainment of the partial derivatives required to set up the normal equations —were not found in the Missbauer literature. For this reason, a description of the *This width parameter is related to the Gaussian "full width at half maximum," W, by e = W/2Ain2. **Corresponds to the natural iron linewidth (of inner two lines) extrapolated to zero absorber thickness.

.40.36.32 3.28.24 -^ -- T=.21 mm/sec.20 1i i I.04.08.12.16.20 8 (mm/sec) Figure III-5. —Full width at half maximum (FWHM) of Gaussian-Lorentzian convolution (G-L) function versus Gaussian width parameter 0. Lorentzian FWHM r fixed at.21 mm/sec.

66 procedure used to evaluate dj(v) and its partial derivatives is given in Appendix D (see Section B). 3. Error Analysis The deviation of the measured spectrum from the fitted spectrum —or, the goodness of fit —is determined by the normalized chisquare value given by M (F -Y)2 x /N = I 1 1 (111.5) i=l 2i (M-Np In this expression, M is the number of data points; N is the number of p fitting parameters; Fi and Yi are the values for the fitted and measured spectrum at the ith data point; and (72 is the experimental variance in Y.i Termination of the iteration procedure used in the least-squares 2 analysis program occurs when - reaches a minimum value. (Actually, N2 2 convergence was assumed to occur when X- at iteration k+l minus - at N N iteration k < 10-3). 2 After a minimum in i- is achieved, the errors in the fitting parameters are determined by the method described by Cohen et al.(l22) The statistical error in the kth parameter was calculated from 2 1/2 k (Akk XN) (III.6) ck = (AN -1 th where Akk is the k diagonal element of the error matrix. (The error matrix is obtained by inverting the coefficient matrix of the normal equations —see References (37) and (120) for additional detail.) It was also necessary to determine the errors associated with certain functions of the fitting parameters; e.g., individual resonant

67 line areas and area ratios. This was accomplished using the error propagation methods described by Ku.(123) The variance in the function f, which depends on K fitting parameters Pk' was calculated from K K f2 f (a 2 f 2k.7) j=l J j<k J where ajk is the covariance of the j and kth parameters. The parameter covariances are obtained from the off-diagonal elements of the error matrix by the expression 2'jk (Ajk N E. Sample Irradiations Following heat treatment, the Cu-Fe alloy samples were irradiated in the Ford Nuclear Reactor (FNR) at the University of Michigan. The FNR is an open pool research reactor which operates at a power level of 2 MW with an average thermal flux of about 1.5(10)13 n/cm2-sec. The core consists of MTR-type, enriched U-235 fuel elements. All irradiations were made at reactor ambient temperature at two in-core locations. Initial sample irradiations were carried out in the "Central-Stringer" (CS) irradiation facility. The absorber foils were sealed in 1/2" diameter by 1 1/2" Al capsules and then placed in the CS at a vertical position of 5" below the core's mid-plane. To provide heat transfer from the foils, spring inserts were used in the capsules to assure that the Cu-Fe foils made good contact with the capsule walls. Previous thermocouple measurements(^) of in-core Al

68 sample holders indicated a maximum temperature of about 550C could be expected. Since temperatures of 120~C or greater for a duration of one month caused no significant annealing effects (as determined by their ME spectra) in quenched and irradiated Cu-Fe samples, no attempt was made to ascertain the exact sample temperatures during irradiation. The fast flux above 0.1 Mev at the irradiation position in the CS was estimated to be u 1.7(10)13 n/cm2-sec. The CS irradiation facility had three shortcomings: (a) samples had to be irradiated for one or more cycles at a time; (b) it was extremely difficult to insert the Cu-Fe foils in the required Al capsules without damaging them; and (c) the fast flux in the CS was significantly lower (uxl/2) than in the central fuel element. Therefore, subsequent irradiations were performed between fuel plates in the central fuel element (core position L-37). For these irradiations, the Cu-Fe foils were tin plated to prevent corrosion, packaged in Al foil to provide structural stability, and sandwiched between two Al plates having the same curvature as the fuel channels. An important challenge in any radiation effects experiment involving a reactor is the characterization of the fast flux c(r,E,t) and the monitoring of the total integrated fast flux or fluence ((r,E). (128) An excellent review on this subject has been given by Zijp. Because of the inherent uncertainties involved with interpreting the Cu-Fe Mbssbauer data, a careful characterization of the fast flux was not necessary in this work. For this reason only two flux measurements were made prior to the irradiations: one at the start of a fuel cycle with an In activation monitor (threshold reaction: 115In(n,n')115mIn) and the other

69 during mid-cycle with an Fe activation monitor (threshold reaction: 54Fe(n,p)54Mn). Both measurements were made at the core's mid-plane. Using the suggested values of <a>f5 168 mb28) and <c> 78 mb29) for the effective fission neutron cross sections for the above reactions, the respective fission spectrum fluxes were found to be 4f15 f4 - 2.3(10)1 n/cm2sec and f54 2.5(10)13 n/cm2sec. These results are in (130) agreement with the earlier measurements of K. Nishina.( The determination of fast neutron fluences during the irradiations was made with iron wire activation monitors, making use of the 54Fe(n,p)54Mn threshold reaction. The results of these measurements are given in Table III-2. The fluence values reported are for an assumed fission neutron energy distribution which approximates the FNR's fast flux(130) and are based on an effective cross section of 78 mb. In calculating the fast fluence from the monitor's activity, a time independent flux was assumed. TABLE III-2 IRRADIATION SUMMARY Core Irradiation Fast Average Position Time Fluence Fast Flux Central 1 cycle 2.7x19 1.3x1013 Stringer (595.3 hr.) L-37 2 cycles 9.0Q 19 2.1xl 13 (1177 hr.) 9 L-37 48 hours 4.0x1018 2.3x1013

CHAPTER IV EXPERIMENTAL RESULTS A. Preliminary This research involves the application of M6ssbauer (ME) spectroscopy (supplemented by resistivity measurements) to study the effects of thermal aging and fast neutron irradiation on precipitation processes in copper-rich Cu-Fe alloys. A Cu-0.63at.%Fe alloy was selected for use in the aging and irradiation studies, because samples containing Fe concentrations greater than about 1.0% can not be effectively quenched from the solid solution regime of the phase diagram (see Figure 11-2). The experimental program consisted of four phases — (a) sample preparation and heat treatment, (b) ME and resistivity measurements of the heat treated samples, (c) irradiation of the heat treated samples in a reactor, and (d) ME and resistivity measurements of the irradiated samples. Sample preparation details were given in Section III-A. The heat treatment procedures can be summarized as: (1) All samples were initially solution annealed at 1000~C and quenched. (2) A majority of the quenched samples were then annealed (aged) isothermally at 285, 335, 400 or 6000C for times in the range from 6 min. to 10,000 min. 70

71 (3) A few of the quenched and one of the aged samples were cold worked (% 1/5 reduction in thickness). Four of these cold-rolled samples were subsequently given an aging treatment for various times at 6000C. (4) Four Cu-Fe samples with Fe concentrations of 0.1, 0.2, 0.3 and 0.4% were solution annealed at 10000C and quenched. For convenience, the heat treated samples will be referred to below as being either quenched, aged, cold-rolled, or cold-rolled and aged. The Cu-0.6%Fe samples so prepared were investigated by Mossbauer spectroscopy and resistivity measurements. The results are given in Section B and discussed in Section C of this chapter. The ultimate objective of the above procedure was to prepare Cu-0.6%Fe alloy samples with a variety of relative concentrations of iron in solid solution, y-Fe precipitate and a-Fe precipitate for the irradiation phase of the experimental program. The precipitates so produced should have different average particle sizes, and hence would behave differently in a fast neutron radiation environment. Our goal was to study the competitive radiation effects of enhanced precipitation, dissolution of precipitate particles and r-Fe to a-Fe precipitate transformation through the changes produced in the Mossbauer spectral components associated with the phases of iron in copper. The irradiations were performed in the FNR reactor at three different fast neutron exposures. (For this purpose, three samples per heat treatment were prepared.) Details of the irradiation phase of the program were given in Section III-E. The results of the post-irradiation measurements are given in Section D and discussed in Section E of this chapter.

72 The room temperature M6ssbauer spectra for copper-rich Cu-Fe alloys are complex. Typical spectra which are representative of quenched and various aged samples are shown in Figure IV-1. The data in the figure were computer-fit to a sum of Lorentzian functions using the techniques described in Section III-D and Appendix D. For the purpose of discussion, the following notation introduced by Bennett and Swartzendruber(7) will be used to describe the ME spectra of the Cu-Fe alloy samples of this study (refer to Figure IV-1): (1) Component yo, the central peak in the figure, is produced by iron atoms which are isolated (non-interacting) in the Cu-lattice; i.e., Fe in solid solution with copper. It has an isomer shift with respect to SNP of about 0.484 (7,8) mm/sec. ( (2) Component y2 is the partially resolved doublet shown in the top spectrum of a quenched (and cold-rolled) sample. It has a splitting of about 0.58 mm/sec.(7'8) During the early stages of aging at intermediate temperatures (500~C or less), the y2 doublet decreases in splitting and shifts towards lower velocities making its resolution extremely difficult. In spectrum (b) of Figure IV-1, the y2 component was fit to a symmetrically constrained Lorentzian doublet. Although this representation is proven to be incorrect (see Section B below), it is shown here to illustrate the doublet's behavior upon aging. As discussed in Section II-B and Appendix B, the 2 doublet is produced by small Fe clusters or possibly surface atoms of large clusters in the Cu-lattice.

73 (3) Component y1, shown in spectra (b) and (c) of the figure, is essentially coincident with the lower energy (left) line of the Y2 doublet (i.e., with Y2L). It is produced by Fe in iron-rich FCC precipitate particles (traditional y-iron). In the spectrum (c) for a sample aged at 600~C for 200 minutes, y1 is the major component and represents about 60% of the iron in copper. In the intermediate case, spectrum (b), the fraction of iron in the Y1 phase, if any, is uncertain because of the difficulty in fitting individual and doublet components (Lorentzian in Figure IV-1) to the poorly resolved spectra. The problems in interpreting the spectra during the early stages of aging are discussed in Sections B and C of this chapter. (4) The a-Fe structure, not shown in the present figure, consists of a six-line pattern similar to that in pure iron metal (see Figure IV-7). It represents Fe in the stable BCC precipitate phase. The room temperature Mossbauer spectra of this study taken of either heat treated or irradiated Cu-0.6%Fe samples display a combination of the above components. The fitting of these complex spectra is discussed in the following section (as well as in Section III-D and Appendix D), while a discussion and possible interpretation of the results is given in Section C of this chapter. B. Results of Annealing Experiments 1. Solution Annealed and Quenched Samples All of the samples (a total of 60) were initially given this heat treatment to retain as much Fe in solid solution as possible prior

74 ~,ff, 7 (R) - 0.98 -, 0.94 \ I (a)Queched! V from 1000 C 0.90 0.98',, it;J'^t/ (b) 3250C I / 100 Minutes 0,94 - 0.98 Y +(R -1.00 - 0.50 0.00 0.50 1.00 1.50 2.00 MM/SEC''I Figure IV-1. —Mbssbauer absorption spectra of Cu-O.6%Fe alloy samples representing the quenched and two aging stages of Y -iron in copper. stages of Y-iron in copper.

75 to subsequent aging and/or irradiation studies. To determine the quality of the quench, resistivity measurements were made for a majority of the solution annealed samples. Mossbauer spectra were also taken of many of the samples for the purposes of correlating with the resistivity results and providing reference spectra for subsequent aging (precipitation kinetics) and irradiation studies. The resistivity values for the quenched samples ranged from v 5.8 to 6.5 pQ-cm, depending on the quality or rapidity of the quench. Since an increase in the amount of Fe in solid solution causes an increase in resistivity, the higher p values correspond to the best quenches. As demonstrated by earlier studies,(7'8) the Mossbauer spectra for the quenched samples consisted of a slightly asymmetric Y2 doublet at the base of a large yo singlet. This is illustrated in Figures IV-l(a) and IV-2. The average values for the fitting parameters of some 15 quenched samples fit to the sum of Lorentzians are as follows: L R P, PA 2 r2 r r2 Ao A2.484(2).452(5).587(3).232(4).242(6).25(1).63(1).37(1) Here, P represents the components' isomer shift with respect to sodium nitroprusside (SNP), A2 the splitting between the y2 lines, r the linewidth (FWHM), and A the area fraction. The subscript "2" refers to the Y2 doublet, while "o" refers to Fe in solid solution or y0. The (7,8) above results agree with previously published values.(7 A few of the quenched spectra were also fit to a sum of Gaussian-distributed Lorentzians (see Section III-D). These convolution functions were introduced to simulate the broadening in the spectral

76 components caused by the different Fe near neighbor configurations associated with both isolated and clustered Fe atoms. (In the case of Fe in solid solution, neighbors further than the nearest shell of 12 atoms might broaden the y line through distant quadrupole interactions.) The fitting parameters for both types of fitting schemes are summarized in Table IV-1 for 0.2%Fe, 0.6%Fe and cold-rolled 0.6%Fe samples. The Lorentzian fits consisted of an unconstrained sum of three lines, while the Gaussian-distributed Lorentzian (G-L) fits consisted of the sum of a single line plus an area constrained doublet (i.e., the lines of the doublet are constrained to have equal areas). The x i values, which are good in both cases,* are lower for the G-L fits to the data except for the.2%Fe sample. This is reasonable since one would expect configurational broadening of the resonance lines to be Gaussian rather than Lorentzian in nature. For the 0.2% Fe sample, the lower x2/N value for the Lorentzian fit reflects the fact that very little broadening in yo has occurred and also that there were no constraints in the Lorentzian fitting model. Although it has been discussed briefly in Section III-D, a further remark concerning the Gaussian width or broadening parameter e is in order. For a pure Mossbauer absorption line, the broadening parameter e should approach zero (equipment broadening was accounted for by setting r =.21 mm/sec). This condition is almost satisfied *For the number of degrees of freedom N of about 200 used in this work, values of.95 < X2/ <.05 would fall within the + 2a limits of the X2 distribution. 131) Thus, "good fits" would be those yielding a X2/N value of about unity.

TABLE IV-1 MOSSBAUER SPECTRA PARAMETERS FOR SOLUTION ANNEALED AND QUENCHED Cu-Fe ALLOY SAMPLES Position(a)(mm/sec.) Y2 Line Width(b) (m/sec.) Relative Area sRlatine (mm/sec.) ____ Split-___ ( ) tingL R x Sample o'Y2 (mm/sec.) 2 Yo 2 Yo Y2 x (1) Lorentzian fits 0.2% Fe Q.482(4).449(4).585(1).233(3).227(1).255(4).708(8).292(8) 0.97 0.6% Fe QR.481(4).449(4).586(1).234(3).265(1).249(4).718(9).282(9) 1.03 0.6% Fe Q.484(4).447(4).575(2).242(4).246(2).273(5).600(8).400(9) 1.12 (2) Gaussian-distributed Lorentzian fits 0.2% Fe Q.483(4).451(4).574(1).066(2).038(2).085(3).683(4).317(6) 1.04 0.6% Fe QR.482(4).451(4).580(1).077(2).061(2).073(2).667(4).333(5) 0.97 0.6% Fe Q.483(4).448(4).559(2).065(2).045(2).096(3).575(4).425(5) 1.07 (a)Position or isomer shift parameter referred to NBS secondary, sodium nitroprusside. (b)Lorentzian fit: FWHM r; Gaussian-distributed Lorentzian fit: width of Gaussian broadening e. (C)Key: Q = quenched; QR = quenched and cold rolled.

78 for the Yo line of the 0.2%Fe sample (see Table IV-1), as might be expected of lower concentration alloys. The total width (FWHM) of a given resonant line can be determined from Figure III-5; e.g., for the yo line of the.2% sample for which e =.038 mm/sec, the FWHM value is 0.229 mm/sec. This compares to a Lorentzian width of r =.227 mm/sec obtained for the same 0.2% sample (see Table IV-1). Comparison of the area fractions obtained from the two fitting models indicates that the o area fractions (and thus the predicted Fe concentration in solid solution) are consistently smaller for the G-L fits. This is a consequence of the G-L convolution function having less area in its "tails" than a Lorentzian of equivalent FWHM. Figure IV-2 displays typical spectra obtained for 0.6%Fe samples in the quenched and cold-rolled conditions; the fitting parameters of typical samples are given in Table IV-1. The increase in y2's splitting and its decrease in area fraction upon cold-working indicates that some of the small Fe clusters which existed in the quenched alloy are broken-up upon cold-rolling. This is consistent with Window's(8'10) observations made from both room temperature and 4.2"K spectra of a Cu-l.O%Fe alloy. Another effect of cold-working is the broadening of the yo absorption line (about 8% of the total width for the several quenched and rolled samples measured). In a recent experiment, V. A. Lagunov et al.(32) observed similar line broadening of Fe5 in deformed copper (source experiment). They ascribed this broadening to quadrupole splitting produced by the localization of the 57 Fe atoms near defects created during the cold-rolling.

79 ai -r~~~~~~~~ % 0I^~~~~~~~~I I j / a)(a)Quenched -= I V) IJ. _ a (b)Quenched and cold-rolled ru:L.00 -.50.X00 50 L.00 L.SO 2.00 MM/SEC Figure IV-2. —Mbssbauer absorption spectra of two Cu0.6%Fe alloy samples which have been (a) solution annealed and quenched and (b) cold-rolled after quenching. The spectra were fit to the sum of a Gaussian-distributed Lorentzian singlet and area constrained doublet.

80 2. Aged Samples The results of the aging studies using Mossbauer spectroscopy are summarized in Tables IV-2 through IV-7* and Figures IV-3 through IV-7. Aging (y-iron formation) in the Cu-0.6%Fe alloy may be divided into the following three stages according to the appearance of the Mossbauer spectra: (a) incipient Fe clustering characterized by an apparent three-line M6ssbauer pattern (class I spectra); (b) an intermediate stage characterized by a broadened, apparent two-line spectrum (class II spectra); and (c) the definite formation of y-iron characterized by the dominance of the y line at about.175 mm/sec** R and the reappearance of the high energy 2R satellite line (class III spectra). The first two classes of spectra occurred for samples annealed at 285, 335, and 400~C, while the class III spectra were observed only for the samples annealed at 600~C. Examples of the spectra associated with y-iron aging are illustrated in Figures IV-3 and IV-4 (class I), Figure IV-5(b-d) (class II), and Figure IV-l(c) (class III). The following is a brief discussion of the functions used to fit the three classes of spectra. *The errors or standard deviations for the parameters are given in parentheses following the parameter values. The error in line position is a combination of the error in the determination of the SNP reference "zero" position and the statistical error obtained from the fitting program (see Equation 111.6). The other error values were determined in the manner described in Section III-D.3. **By the comparison of Mbssbauer spectra of well-aged Cu-Fe alloys with the results of other measurement techniques, the resonant line at about.17 mm/sec (with respect to SNP) has been identified as being produced by FCC y-iron.(6,7)

81 The initial attempt at fitting the M6ssbauer data from the aged samples was the application of an unconstrained sum of single Lorentzians. The parameters obtained from this fitting model are listed in Table IV-2 for the 285-400"C aging data. The spectral components have been characterized in the notation of Section IV-A. They are similar in appearance to those illustrated in Figure IV-3 for the Gaussian-distributed Lorentzian fits (see below). As a first approximation, it is tempting, as the notation suggests, to subtract the area R L of the 2R satellite line from that of the (y1+Y2 ) line and designate the remaining area fraction as representing the concentration of Fe in y-precipitates. The variation of the splitting, position and relative R area of the Y2 doublet, represented by the 2R component and a hidden component under the (Y+Y2L ) or left satellite line, would be associated with the growth of small Fe clusters. The merit of this simplified model is determined through the 2 x /N values which are consistently much too large. Since the y2 doublet can be expected to decrease in splitting and shift in position (see Section II-B and Appendix B) and since the possible growth of a component representing y-iron precipitates is masked by y2L, the critical parameters in the fitting are the position and area fraction* of the central (y) peak. Since the y line represents isolated Fe atoms, its position, if properly fit, should remain constant to within *Actually, the normalized peak height was used as the intensity parameter in the Lorentzian fits. The area fraction was obtained from A.* = 1/2rir.Ii/4A where I. is the line intensity, r. the line width and A the total area of the spectrum.

TABLE IV-2 MOSSBAUER SPECTRA PARAMETERS FOR HEAT TREATED Cu-0.6%Fe ALLOY SAMPLES ANALYZED AS THE SUM OF INDIVIDUAL LORENTZIAN LINES(a) Line Position(b) (mm/s) Line Width r (mm/s) Relative Area Goodness L R L R L R 2/f Heat Treatment + Y 2 Y 2 +2 Y+ y xR/N Average of Quenched Samples from 1000~C.158(4).484(4).746(6).232(4).242(6).250(9).18(1).65(1).17(1) 1.0-1.2 2850C, 100 min.170(4).488(4).712(5).243(3).262(3).288(5).292(4).470(9).247(9) 1.77 285~C, 1000 min.175(4).489(4).696(5).246(3).261(4).298(6).310(4).42(1).27(1) 1.89 285~C, 5000 min.187(4).500(6).668(6).255(2).285(6).303(8).362(9).44(2).20(2) 1.41 3350C, 10 min.169(4).487(4).718(4).241(3).246(3).286(6).277(8).490(9).233(9) 1.37 335~C, 100 min.186(4).496(4).684(6).256(3).279(5).294(7).327(7).50(2).17(1) 1.43 3350C, 200 min.193(4).497(4).669(7).273(3).280(5).31(1).37(1).47(2).16(2) 1.23 3350C, 1000 min.197(4).500(5).655(8).291(3).288(7).31(2).41(1).50(3).09(2) 2.15 400~C, 6 min.178(4).489(4).709(5).258(3).270(4).280(8).321(6).50(1).18(1) 1.29 400~C, 10 min.185(4).493(4).694(6).262(3).273(4).29(1).332(7).51(2).16(1) 1.23 400~C, 20 min.182(4).489(4).684(8).270(5).276(6).31(2).36(1).50(3).14(2) 1.26 400~C, 100 min().176(4).461(4).273(3).321(3).44(1).56(1) 3.24 400~C, 1000 min(c).170(4).434(4).260(3).318(2).56(1).44(1) 3.63 (a)Average of the two mirror image spectra from a single measurement. (b)Line positions or isomer shift values are referred to the NBS secondary standard, sodium nitroprusside. (SNP). (C)Only two lines were able to be fit.

83 experimental error ( ~+.005 mm/sec) at.484 mm/sec. Observation of Table IV-2 indicates that the apparent yo position has shifted to higher velocities in the initial stage of aging and to lower velocities in the intermediate aging stage. In addition, if Fe clusters or y-precipitate particles are being formed, there should initially be a monotonic decrease in the concentration of the yo phase as a function of time at a given aging temperature; i.e., a decrease in the area fraction of the yo resonant line. This expectation is also not fulfilled by the single Lorentzian fitting model. The next attempt at fitting class I spectra characteristic of early aging was the use of two single Lorentzians representing yo and Y and a symmetric Lorentzian doublet representing Y2 (e.g., see Figure IV-lb). The results of this model are summarized in Table IV-3. This approach also failed for the same reasons given above for the three-line Lorentzian model. An attempt at fitting the y2 doublet "phase" to an asymmetric, area constrained Lorentzian doublet also did 2 not significantly improve upon x /N or the critical parameter values. There are two major reasons why the simple Lorentzian fitting models failed to provide adequate fits of the initial and intermediate aging spectra. One is of course poor resolution. The other is the broadening of spectral components resulting from local variations in solvent and solute atoms about the resonance atom. While the first problem will plague any fitting model, the second can be more effectively treated. In an attempt to treat the line broadening caused by such configurational variations in a more realistic manner, the spectra were fit to the sum of Gaussian-distributed Lorentzians (G-L) given by

TABLE IV-3 MDSSBAUER SPECTRA PARAMETERS FOR AGED Cu-O.6%Fe ALLOY SAMPLES ANALYZED AS THE SUM OF TWO INDIVIDUAL LORENTZIANS AND ONE SYMMETRIC LORENTZIAN DOUBLET.- _. _ __ _Position(a (mm/sec.) (b) Line Width r (mm/sec.) Relative Area 2.__________________________________ Heat (b) Splitting 2 Treatment YO'2 (mm/sec. y2 /N 2850C 100 min..165(5).498(4).456(4).532(5).192(7).265(4).279(5).067(5).47(1).46(2) 1.71 2850C 1000 min..158(5).494(4).447(4).519(5).201(5).258(4).285(5).074(5).43(1).50(2) 2.09 2850C 5000 min..166(5).495(4).436(5).459(9).231(6).280(8).287(9).019(1).43(3).38(3) 1.40 3350C 10 min..173(7).495(4).451(4).554(8).20(1).244(4).276(5).061(4).49(1).42(2) 1.36 3350C 100 min..164(4).501(4).447(5).474(7).229(5).287(6).271(9).173(9).50(2).32(3) 1.35 400~C 10 min..163(4).496(4).459(5).470(8).231(5).258(5).285(8).185(7).51(2).30(3) 1.27 (a)Isomer shift reference: sodium nitroprusside. (b) isomer shift is reported as centroid of the doublet. s2's isomer shift is reported as centroid of the doublet.

85 Equations (111.3) and (111.4). The results of the fits for class I spectra are given in Table IV-4. The data and fitted curves for three of the samples aged at 335~0 are plotted in Ficure IV-3. Comparison of Table IV-4 with the two previous tables reveals that a significant improvement in the quality of the fits, as represented by the x /N values, has occurred with use of the G-L fitting model. In addition, the critical parameters —y's position and area fraction —behave as suggested above. The position of yo remains within ~+o of its value for quenched samples, while the relative area of yo (i.e., the relative Fe concentration in solid solution) decreases in time upon isothermal aging. Because of the poor resolution (see Figure IV-3c) in the case of the sample annealed at 335~C for 1000 minutes, the position of oy was constrained at.484 mm/sec —its average value obtained from the quenched results. For the other samples, all of the parameters were allowed to vary freely during the fit. (10) Very recently Window(1 reported on an extensive Missbauer study of aging processes in a Cu-l.Oat.%Fe alloy. In this study he suggests that aging in the 300-400~C temperature range produces very small Fe clusters (12 atoms or less). If it is assumed that the Fe atoms cluster in a spherical array —for example, say 12 atoms in the nearest neighbor shell about a central atom —then on the average the Fe atoms would see only 5 iron nearest neighbors. Thus, according to the discussion in Section II-B and Appendix B one would expect a broadened, asymmetric Y2 doublet (there would of course be a variety of cluster sizes, and thus Fe configurations) and a central singlet (y ) due to isolated Fe atoms to make up the M6ssbauer spectra of Cu-Fe alloys aged at 300-4000C. For such a model, there would be very little or no

TABLE IV-4 MOSSBAUER SPECTRA PARAMETERS FOR HEAT TREATED Cu-0.6%Fe ALLOY SAMPLES ANALYZED AS THE SUM OF GAUSSIAN-DISTRIBUTED LORENTZIANS Position(a) (mm/sec.) Gaussian Width e( (mm/sec.) Relative Area Heat Hreatment Ls R L+ R L R 2 Treatment y2+y y +y1 y21 + X /N THeatmt Y2 Yo |2 Y2 1 Yo r2 Y2 Y Yo 2 Sol. anneal & quench.158(4).484(4).739(5).065(2).042(2).089(4).189(4).633(4).178(6) 1.06 2850C 100 min..178(4).485(4).703(6).066(2).053(2).112(4).310(1).421(3).269(7)11.32 2850C 1000 min..176(4).481(5).687(6).064(2).046(3).121(5).325(1).370(5).304(9) 1.38 2850C 5000 min..187(4).487(6).63 (1).069(3).065(10).14(1).373(3).30(2).33(1) 1.09 3350C 10 min..171(4).485(4).712(6).061(2).041(3).102(5).283(2).470(3).248(6)11.06 3350C 100 min..191(5).487(5).661(9).078(2).061(4).12(1).356(2).39(1).26(2) 1.08 3350C 200 min..194(5).486(6).65(1).084(3).058(4).13(1).378(3).35(2).27(4) 1.14 3350C 1000 min..202(5).484().58(3).106(3).05(1).14(2).439(6).31(8).25(7) 1.34 4000C 6 min..182(4).485(4).691(6).070(2).056(3).115(5).342(3).441(5).220(9)11.18

TABLE IV-4 —Continued Position(a) (mm/sec.) Gaussian Width e(b) (mm/sec.) Relative Area Heat HaL R L R L R Treatment 2 + O Y2 2 +Y1 Y Y2 Y2 +Y1 O 2 /N 4000C 10 min..189(4).487(5).67(1).082(2).049(4).13(1).341(2).42(1).24(2) 0.98 4000C 20 min..203(5).484(c).61(2).098(3).053(8).14(2).421(7).36(5).22(7) 1.17 (a)Position parameters are referred to the NBS secondary standard, sodium nitroprusside. (hb) b)The actual halfwidth (FWHM) of the resonance line can be obtained from the Gaussian width 0 by use of Figure 111-5. (C)Positions were constrained at 0.484 mm/sec, the isomer shift for solid solution Fe in copper, during the fitting.

88 ^~~~ ar Ii~~~~~~~~~~( iI tal /^10 lqinutes 5c I b\inutes ^ i~~~~~~~~c 100 lqnue Age at 335^ 0C/ zr~~s C3~~~~~~~~s P.00"I sum of singleUc Gausi sa cs I~S Pe ae ae ~ ~ ~ ~ -itri'.4 -

89 Y1-iron (i.e., iron surrounded completely by Fe atoms). Therefore, Window(10) fit the class I spectra which are representative of the initial aging stage to the sum of two Gaussian-distributed Lorentzian functions —a single line to represent Fe in solid solution and an asymmetric doublet constrained to have equal areas to represent Fe in clusters. In the unconstrained three line G-L fitting model of the present study, on the other hand, there has been no a priori assumption of the size of Fe clusters. The difference in the relative areas of the (y2L +1) and 2R lines would then be interpreted as representing the fraction of iron surrounded entirely by Fe nearest neighbors. The results from the three line fits in Table IV-4 indicate that iron of the Y1 variety does form during the initial aging stages. This contradicts Window's aging model. In order to obtain a direct comparison of this more general fitting model with Window's suggested two component model, several of the aged samples were re-fit to the sum of a G-L singlet and area constrained doublet. The fitting parameters from these fits are given in Table IV-5, while two example fits are illustrated in Figure IV-4. Comparison of the results given in Tables IV-4 with those of Table IV-5 indicates that in general both methods seem to fit the data 2 equally well. As might be expected, the x /N values of the unconstrained three-line fits are less than those from fits with one parameter constrained (i.e., equal line areas in the doublet), however, their differences are small. Also, the position and area fraction of the yo line behave as expected in the ~2 area constrained fit. Thus, on the

TABLE IV-5 MOSSBAUER SPECTRA PARAMETERS FOR ANNEALED Cu-O.6%Fe ALLOY SAMPLES ANALYZED BY THE SUM OF ONE AREA CONSTRAINED GAUSSIAN-DISTRIBUTED DOUBLET PLUS ONE SINGLET Position(a) (mm/sec.) Gaussian Width e (mm/sec.) Relative Area (b) y 2 L R 2 Heat Treatment Y y2 Splitting Y2 Yo y2 X2 285~C, 100 min.479(4).430(5).504(1).067(2).042(2).134(2).379(3).621(4) 1.49 285~C, 1000 min.478(4).426(5).501(1).064(2).040(2).131(2).349(3).651(4) 1.41 285~C, 5000 min.482(5).400(5).426(2).069(3).057(4).154(3).255(3).745(4) 1.09 0 335~C, 10 min.480(4).430(4).509(1).062(2).027(3).132(3).425(3).575(5) 1.20 3350C, 100 min.483(4).405(5).428(2).078(2).045(3).155(3).290(3).710(6) 1.11 335~C, 1000 min.488(5).369(5).338(2).099(3).019(8).174(5).16(1).84(1) 1.46 4000C, 10 min.484(4).402(4).428(1).081(2).032(4).170(3).327(4).673(6) 1.00 (a)Reference: sodium nitroprusside. (b)Centroid of Y2 doublet.

91 (o (a) 10 Minutes z -- cc'I (b) 100 Minutes Aged at 3350C -1.00 -.50.00.50 1.00 1.50 2.00 MM/SEC Figure IV-4. —Spectra of two samples aged at 335~C fit to the sum of a Gaussian-distributed Lorentzian singlet and area constrained doublet.

92 basis of the fitting results alone, it is not possible to choose between the two models. The interpretation of spectra representing the initial stages of aging will be considered further in the next section. Aging at 3350C for times greater than 1000 min. and at 400~C for times of the order of 100 min. results in an apparent doublet in the M6ssbauer spectra of the Cu-0.6%Fe alloy samples; i.e., class II spectra. The 400~C aging spectra have been fit to the sum of (a) two G-L distributions and (b) one single Lorentzian plus two single G-L functions. The parameters obtained in both cases are listed in Table IV-6. The data and fitted curves (3-line) for the 400"C aged samples are shown in Figure IV-5. This figure includes the spectrum of the 400~C-10 min. aged sample which is of type class I, and represents the incipient stage of Fe clustering. (Its fitting parameters were listed in Tables IV-4 and IV-5.) Because of the poor resolution of the class II spectra, the three line fits were particularly difficult to make. Two pre-conditions on the fitting model were necessary to achieve convergence, both involving the yo solid solution component. As in the case of the 335~C1000 min. aged sample, the position of y was constrained to its quenched value. The second condition involved representing yo as a Lorentzian instead of a Gaussian-broadened Lorentzian as before. This was necessary because in these fits the Gaussian broadening parameter e became negative during the first iteration of the fitting program. Even with these conditions convergence was difficult to achieve.* *It was necessary to operate the fitting program in an interaction mode via a Teletype terminal, so that the parameter increments at the end of each iteration could be modulated according to the

TABLE IV-6 MOSSBAUER SPECTRA PARAMETERS FOR Cu-0.6%Fe ALLOY SAMPLES ANNEALED AT 4000C Position(b) (mm/sec.) Line Width (mm/sec.) Relative Area Annealing (c (d (c) Times Y Y2 Y1 () o(d) Yc Y2 x/N (a) Three-line fits:(a) (10)2min..168(5).484.41(1).067(4).21(1).22(1).35(1).17(3).48(1) 1.34 (10)3min..160(5).484.39(1).057(3).20(3).162(9).466(7).10(4).43(1) 1.28 (10)4min..169(5).484.38(1).058(2).25(1).07(3).70(2).15(3).15(4) 1.35 (b) Two-line fits: (L —left, R —right) P(L) P(R) e(L) e(R) A(L) A(R) (10)2 min..180(5).461(5).065(3).106(3).461(6).539(g) 3.10 (10)3 min..175(5).437(5).069(2).116(3).548(5).452(2) 1.57 (10)4 min..174(5).433(5).060(2).102(4).707(5).293(3) 1.45 (a)The three line fits to the spectra were analyzed as the sum of two Gaussian-distributed Lorentzians (yl and y2) plus a single Lorentzian with its position fixed at.484 mm/sec. (yo). The Y2 doublet is assumed to be a broadened singlet for the purposes of fitting the data. (b)Reference: sodium nitroprusside. (C)Width (e) of Gaussian broadening. (d)FWHM of Lorentzian line.

94 Cr) rr it.-\ \ /1 00 Minutes (b)100 Minutes zCd4,0C i ^/ (d) 10000 Minutes Aged 4(c)1000 Minutes -1.0.5.00 ()100 00 Minutes~ Fig. IV. 5 Gauss.ani t. Aged 400~ttrCbu

95 The Y2 doublet evident in the quenched and mildly aged samples is no longer resolvable in the class II spectra which represent a more advanced stage of Fe clustering. The low energy line, designated as L (y2 +y1) for class I spectra appears to be dominated by yl-iron; i.e., its position and linewidth parameters are more characteristic of those observed in well-aged samples (see below and compare Tables IV-6 and IV-7). For this reason, it has been designated as 1 for the three line fits in the table. The broad line whose centroid is between the Y1 and Yo lines is designated y2 for convenience and in a qualitative manner, represents iron with a near neighbor configuration between that of yl-iron and iron in solid solution. The parameters from the two-line G-L fits are identified in Table IV-6 as either left (L) or right (R). As aging at 4000C progresses, the spectra more closely resemble two resonant lines as indicated by the reduction in x/N for the two line fits. It should be emphasized that the assignment of the fitted lines of Table IV-6 to a particular form of iron in copper is only an approximation. All of the Cu-0.6%Fe samples aged at 600"C produced a third type of Mossbauer spectrum (class III) which is characteristic of a more advanced stage of y-iron precipitation. An example spectrum is illustrated in Figure IV-l(c) for an alloy aged for 200 minutes. Table IV-7 lists the M6ssbauer line parameters obtained in fitting four such 2 variation of X /N and the parameters from previous iterations. Even though convergence was achieved, there is of course no guarantee that the minimum in the x2 surface has been reached. Many false minima in the x2 surface undoubtedly occur because of the overlapping of the y and Y2 components. The large variances and covariances in the parameters occurring in these fits, especially for yo and y2, indicate that the results can only be interpreted as being approximate.

96 spectra (aging times from 10 to 1000 minutes) to the unconstrained sum of three single Lorentzians. Since the x /N values were satisfactory for the Lorentzian fits, the subsequent application of the Gaussiandistributed Lorentzian fitting model was unnecessary. The class III spectra are characterized by relatively narrow Lorentzian lines and a large yl component (- 60% of total area) whose relative area varies R only slightly with annealing time. The reappearance of a weak 2R component in a position comparable to that for quenched samples is also a notable feature of these spectra. All three classes of Mossbauer spectra observed for aged Cu-0.6%Fe in this study are illustrated in Figure IV-6. This figure shows the effects of increasing annealing temperatures on spectra produced by samples aged for 100 minutes. For comparison, a typical spectrum of a solution annealed and quenched sample has been included in the figure. The final spectra type encountered in the aging studies of Cu-0.6%Fe was the combination of a six-line magnetic hyperfine spectrum characteristic of a-iron and the class III spectrum characteristic of advanced y-iron precipitation. One of two such spectra obtained from cold-rolled samples that were aged for 1000 min. or more at 600~C is shown in Figure IV-7. The six-line structure in the spectrum is due to Fe in BCC a-iron precipitate particles. In this study the a-precipitates were formed by the following two processes: (1) a Cu-0.6%Fe quenched sample was annealed at 600~C for 1000 minutes and then cold-rolled and (2) a quenched sample was first cold-rolled and then aged at 6000C. The former process is the conventional method for producing a-iron precipitates below the eutectoid (y-c transformation temperature).(85) Only

TABLE IV-7 MOSSBAUER SPECTRA PARAMETERS FOR Cu-0.6% Fe ALLOY SAMPLES AGED AT 6000C. (a) Position(b)(mm/sec) Line Width r (mm/sec) Relative Area Annealing L RL R R2 Time Ya + Y2 Yo Y2 Y1 + Y2 Yo 2 Y1 Y2 Y Y2 (1) AGED AT OOC. FOLLOWING QUENCH FROM 10CO~C. 10 min..181(4).482(4).773(6).250(2).266(4).23(2).574(6).395(5).032(5) 1.17 (10)2 min..178(4).479(5).775(7).238(2).248(4).23(3).631(6).342(5).027(4) 1.05 2(10)2 min..176(4).481(5).759(6).240(2).241(5).22(3).64(1).34(1).032(8) 1.25 (10)3 min..175(4).483(5).760(8).231(2).228(4).24(3).650(6).318(5).032(4) 1.15 (2) COLD-ROLLED FOLLOWING QUENCH FROM 10000C. AND THEN AGED AT 6000C. (10)2 min..179(4).483(5).771(8).242(2).244(5).23(3).638(8).334(6).028(5) 1.18 (10)3 min. f-iron.178(4).484(5).75(1).244(2).231(5).24(1).413(8).303(8).041(6) 1.06 o(-iron Centroid a34 25 A16 A.261(8) 1.69(1) 6.142(7) 10.64(1) 0.24(1) (a)Spectra were fit to a sum of Lorentzian lines using Equation (111.2). (b)Reference: sodium nitroprusside. (C)Contained both the Y-lines and six a-iron lines.

98 Solution Anneal at 1000~C & Quenched Aged 285 C 100 Minutes C0 f Aged 3350 C CC \ X 100 Minutes C 100G Minutes C\X= +/fAged 600~ C 100 Minutes 3% Absorption -i.00 -450.00.50 1i00 t.50 2.00 MM/SEC Figure IV-6. —Mbssbauer absorption spectra showing the various aging stages of a Cu-0.6%Fe alloy.

('U o 0 CY) I —-oC,) I: cr= 0) -6.00 -S.00 -,.00 -3.00 -2.00 -t.00.00 1.00 2.00 3.00 100 00 600 700 MM/SEC Figure IV-7. —Mbssbauer absorption spectrum of a Cu-0.6%Fe alloy sample which was quenched, cold-rolled and then aged at 600~C for 1000 minutes.

100 one sample was prepared in this manner. The second method was used to prepare four samples, all aged at 600~C for various times. Only the two samples aged for 1000 and 2000 minutes exhibited a-iron lines in their Mossbauer spectra. Those aged for times less than 1000 minutes produced spectra identical to the class III spectra obtained from the straightforward aging of solution annealed (quenched) samples at 600~C. The results of the cold-rolling experiments are summarized in Table IV-7 for samples that were aged for 100 and 1000 minutes. 3. Results of Resistivity Measurements The resistivity measurement apparatus and procedure were discussed in Section III-C. These measurements were made to provide a means of comparing the Mossbauer results of this study with the resistivity results of earlier aging and radiation effects studies of Cu-Fe alloys. (3)(5) The resistivity results from quenched and aged Cu-0.6% Fe samples are summarized in Table IV-8. For each sample, the Fe concentration in solid solution has been obtained from measured resistivity values using the experimental curve of Boltax(3) (also, see Linde133). For Cu-Fe alloys with Fe concentrations of <0.5 wt.% this curve is a straight line with a slope of about 8.2 iQ-cm per atom% (See Figure IV-13). Also given in the table for the aged samples is the fraction of Fe which is transformed (e + y) as determined by f: (IV.1) P Po - Pf where p0 is the as-quenched resistivity, p the aged resistivity, and

TABLE IV-8 COMPARISON OF RESISTIVITY AND MOSSBAUER RESULTS FOR HEAT TREATED Cu-Fe ALLOYS Resistivity C Heat Treatment (PQ-cm) C f Cyo Ave. Quenched 6.29(8).56 -.41(1) Quenched and Rolled 6.60(5).59 -.461(4).1% Fe Quenched 2.51(8).09 -.080(4) 2850C, 100 min. 5.57(6).47.14.265(2).35 2850C, 1000 min. 5.33(5).44.24.233(3).48 2850C, 5000 min. 4.67(5).36.41.19(1).60 3350C, 10 min. 5.52(5).46.15.294(2).30 3350C, 100 min. 4.91(4).38.35.24(1).43 3350C, 1000 min. 4.12(4).29.54.19(5).59 4000C, 10 min. 4.81(5).37.35.263(6).39 4000C, 100 min. 3.51(4).21.66.11(2).80 4000C, 1000 min. 2.86(3).13.81.07(2).91 4000C, 10,000 min. 2.58(4).10.87.09(2).84 6000C, 200 min. 3.68(3).23.214(6) 6000C, 1000 min. 3.49(3).215.200(3) * Symbols: p= resistivity, M = Mlssbauer f = fraction transformed to Fe clusters C0= Fe Concentration (atom%) in solid solution

102 Pf the equilibrium resistivity obtained from the equilibrium solubility at the aging temperature.* For comparison, the Fe concentration in solid solution determined from the Mossbauer spectra (i.e., from the y0 component of the three-line G-L fits)** is given for each sample. In addition, the fraction of Fe transformed fM, which was calculated from the o area fractions of the spectra for each sample in the quenched and aged conditions, is also listed in Table IV-8. Equation (IV-1) was used to determine fM with area fractions replacing the resistivity values. (A nominal value of Af=.94 was used to represent the equilibrium yo area fraction for the 285~-400~C aged samples). The fractional transformation values obtained from both measurement techniques are plotted as a function of isothermal annealing time in Figure IV-8. The large errors associated with the M6ssbauer results of the 4000C data are due to the very poor resolution of the class II spectra. These errors were obtained from the fitting program and do not represent any inaccuracies in the assumed model. The reasons for the significant differences in the fractional transformation values, as well as the estimated concentration of Fe in solid solution (y ), obtained from the two experimental methods will be discussed in Section IV-C. *Estimated to be about 2 vn-cm for samples aged at temperatures of 285~ - 400~C. **For the aged samples the atom concentrations (in percent) were estimated from C = 0.63A, where A is the area fraction of the y component and th8 constant 0.63 is She total atom percent Fe in th8 Cu-Fe alloy. Because of the small effective thicknesses involved, the errors associated with this approximation (< 4%) are less than the uncertainty in the fitting model. In the case of quenched samples, for which Y is the dominant component, the procedure described in Section II-A.2 was used (i.e., Equation 11.14).

103 Resistivity data: solid symbols & solid lines M6ssbauer data: open symbols & dashed lines 1.0 0.8 E / S0 0.6-/ 0:S0.4/ / L.0 285C 0 335~C ~A 3350C a 4000C 0 I~I I I. 10 100 1000 10,000 Aging Time (minutes) Figure IV-8. —Fraction of iron transformed from the solid solution phase to iron clusters as determined by MBssbauer and resistivity measurements.

104 C. Discussion of Annealing Results The Mossbauer measurements of quenched and aged Cu-0.6% Fe alloy samples indicate that a variety of complex spectra are produced as a result of Fe cluster formation in the Cu matrix. This complexity ranges from the resolved two component spectra (oy singlet and Y2 doublet) of the quenched samples to the poorly resolved three line (class I) and two line (class II) spectra of samples isothermally aged at temperatures in the range 285" - 400~C. The samples aged at 600"C produced adequately resolved three component spectra (class III) that can be interpreted from our existing knowledge of the metallurgy of Cu-Fe alloys aged at temperatures 5500C or greater.(4) (84) The interpretation of the spectra of quenched alloys has been made by Window(8) and confirmed (12) by Reno and Swartzenruber(12) using perturbed angular correlation measurement techniques (see Section II-B). Before considering the effects of fast neutrons on these alloys, it is necessary to characterize the ME spectra associated with the early stages of aging; i.e., to relate the components of the class I and II spectra to the different phases or configurations of Fe in copper. Such a characterization is made in the present section through an argument based in part upon the systematic trends of the spectra and also upon metallurgical considerations. 1. Spectra Systematics and Other Considerations The amount of Fe clustering in solution annealed and quenched alloys is a function of the Fe solute atom concentration and the quality of the quench. The former is illustrated in Figure IV-9, which shows the variation of the area fractions for the y0 singlet

105 and Y2 doublet as a function of Fe concentration. The effect of cold-rolling on the break-up of Fe clusters is also indicated in the figure for the 0.6% Fe samples. The Y2 doublet is produced by small Fe clusters of two or more atoms. As discussed in Section II-B and Appendix B (also see Window(15)), the presence of one or more Fe neighbors causes a distortion in the 3d charge distribution of the resonant atom which produces an electric field gradient at the Fe57 nucleus. This in turn splits the excited level into two doubly degenerate states which give rise to the y2 component's splitting. For the Fe concentrations considered, the doublet's splitting remained essentially unchanged (for high quality quenches) indicating that the size of clusters remained essentially constant (probably two to four atoms) with an increase in Fe concentration up to 0.63%. (The magnitude of the splitting is slightly smaller for the G-L function fits than those using simple Lorentzian functions.) The increase in y2's area fraction is then probably associated with an increase in the number of small clusters. In the case of the less rapid quenches (e.g., see the 0.6% Fe "Q" sample in Table IV-l), a decrease in the doublet's splitting occurred in conjunction with its increase in relative area, indicating larger clusters were formed. The resolution of the spectra for the quenched samples is sufficient to allow an accurate determination of the o and y2 area fractions, and through Equation (11.14), the Fe concentrations of the two configurations. The other spectra type for which an unambiguous characterization can be made is the class III spectra (see Figure IV-lc) obtained for the

106 o Gaussian-distributed Lorentzian fit * Lorentzian fit 1 Quenched and Cold-rolled.8-.7 + 0 1.53.2.2.4.8 1.0 Atom Percent Iron Figure IV-9. —Variation of YLand (area fractions for quenched samples as a function of atom percent iron in copper.

107 "well aged" Cu-0.6% Fe samples (i.e., those aged at 600~C).* In these spectra the large component (Y1) centered at approximately.17 -.18 mm/sec is mainly produced by y-iron precipitates. Electron microscopy studies(85) have revealed these precipitates to be spherical in shape (85) and fully coherent with the copper lattice.(85) The central component of these spectra is located at essentially the same position as the peak due to solid solution Fe (y ) in quenched samples, and is so identified. R The small peak designated y2 is the high energy component of the Y2 doublet observed in the quenched samples, and can be associated with Fe pairs or small clusters in the Cu matrix. The other member of the doublet is located under the large yl peak. That the yl resonance line could be fit to a Lorentzian with relatively narrow width indicates the predominance of a single iron environment. Like the spectra of quenched samples, a fairly accurate determination of phase concentrations is possible through the relative areas of the spectral components. Aging in the temperature range 285~ - 400~C produces spectra that are intermediate to those discussed above. In the very early stages, the spectra appear to have three broad, partially resolved lines as was illustrated in Figures IV-3 and IV-4. Spectra of this type were identified as class I and were fit to (a) the sum of three Gaussian-distributed Lorentzian (G-L) functions as well as (b) the sum of a G-L singlet and an area constrained doublet. The second model assumes that Fe which is not in solid solution (yo singlet) is in the form of clusters of a few atoms having a majority of copper nearest neighbors. This model was *The expression "well aged" used here implies definite y-iron (coherent) precipitate formation and should not be confused with the metallurgical connotation -- i.e., the formation of large incoherent precipitates with the concurrent reduction of hardness.

108 recently suggested by Window.(10) The first model makes no such assumption and has been adopted in the present study to characterize the Cu-0.6% Fe spectra. The results of both fitting schemes were summarized in Tables IV-4 and IV-5 and will be further compared below. The effects of increased Fe clustering on the doublet's splitting and position are shown in Figures IV-10 and IV-11, respectively. (Both the Y2 centroid (lower) and y2R position (upper) are shown in Figure IV-11). For reference the y2 splitting and position parameters for quenched samples, designated by a cross, and the 600~-aged samples are also plotted in the figures. The open symbols represent the three component model (a), while the solid symbols represent the constrained two component model (b). The dashed lines for a given aging temperature are drawn to indicate trends in the variation of the plotted parameters. In all cases the area constrained fitting model -- i.e., (b) above -- produced somewhat smaller splittings and greater decreases in the Y2 centroid's position. Actually, the average parameter values of the spectral distribution probably fall within the limits determined by the two models. Of course, all of the foregoing is predicated on the assumption that the distribution of quadrupole splittings and isomer shifts of Fe57 in a variety of Fe and Cu nearest neighbor configurations can be effectively represented by Gaussian-broadened Lorentzians. A major change in the trend of decreasing Y2 position occurs for the samples aged at 4000C for 100 to 10,000 minutes (see lower graph of Figure IV-11). Spectra of these samples are class II and were fit to both two and three components; i.e., two G-L functions or one Lorentzian constrained at.484 mm/sec plus two G-L functions. The very broad singlet occuring between ~r and the left component yl (see Figure IV-5

109.60 — ^~U^~~~ ~- 0 o 600 C C) + 5.55- 0 285335 C /35.3010 100 1000 10000 Aging Time (min.) Figure IV-10. —Splitting of'2 doublet as a function of aging treatment. 7h5 - IQ 0 6 0 -~ 600 C 6. 65'6.0,' —-- - - 10 4000c 1 2850C.55 335 C'i 45 - O~~.- O c ~ ~ a ~nA.4- 600 C 40 A 4' 285 0C -..z~~ ~400 C -- m 3350C.35 10 100 1000 10000 Aging Time (min.) Figure IV-ll. —Position of centroid and (Ras a function of aging treatment.,

110 and Table IV-6) was taken to represent y2 iron; i.e., Fe5 with both Cu and Fe nearest neighbors. In these cases, the number of Fe nearest neighbors probably exceeds 5 or 6 (see Appendix B and discussion in Section IV-C.2). The large deviation of the centroid of this broad line from the isomer shift trend of Y2 shown in Figure IV-11 indicates the failure of this simplified model to adequately characterize the class II spectra (except possibly for the 400"C - 10,000 min. sample). Certainly one would expect a monotonic decrease in isomer shift, although not necessarily linear, as Fe atoms are added to the nearest neighbor shell of the resonance atom. In the case of the 400~C - 10,000 minute sample, the broad r2 component might be half of a doublet, the other half being masked by the dominate y1 line. Two other trends in the class I spectra (Table IV-4) are important. First, the position of the left component (Y2 + yl) increases in energy upon aging (from -.16 mm/sec for quenched samples to -.20 mm/sec for the sample aged 1000 min. at 335~C). This can be interpreted as being caused by a shift in y2L, suggesting that small Fe clusters predominate. (Recall that the established position for the Yl line is at -.17 mm/sec.) The other trend is associated with the behavior of the width (Gaussian broadening) of the (Y2L + y1) line. For the 285~C aging data, this width remains fairly constant and is approximately equal to that of the quenched samples (i.e., about.065 mm/sec). Since there is also a definite increase in the Y2 area fraction and a concurrent decrease in splitting and isomer shift for these samples, it can be concluded that both the fraction of iron in clusters and average cluster size are increasing. The constancy in y2's width is then most likely caused by the simultaneous decrease

lll in the doublet's splitting and isomer shift. (This effect, as noted by Window(8) in a variety of quenched Cu-l.1% Fe alloy samples, produces a relatively narrow low energy member and broadened high energy member of the Y2 doublet -- this is certainly evident in both Tables IV-4 and IV-5). In the case of the 335"C-aging data, a marked increase in the (2L + yl) member's Gaussian width occurs with increasing annealing time. This could very well be the result of y1-iron being formed; i.e., Fe with at least its nearest neighbor shell (12 atoms) filled with other Fe atoms. The same trends in (2L + yl)'s position and width are noted for the samples aged at 400"C for times in the range of 6 - 20 minutes. However, further annealing at this temperature results in a significant change in both the position and width parameters of the (-2L + yl) component. Comparison of Tables IV-4 and IV-6, reveals that the position of (Y2L + y1) has decreased from.20 mm/sec (20 min. anneal) to about.17 mm/sec (100 to 10,000 min. anneals), while its width e has decreased from.098 mm/sec to approximately.06 mm/sec. These values are similar to those associated with the yl-iron line of the well aged samples (compare Tables IV-6 and IV-7).* Thus, aging at 400~C for times 100 minutes or greater seems to produce significant quantities of yl-iron in Cu-0.6% Fe. The growth of these precipitates is characterized by the increase in the area *Comparison of the width parameters (one Gaussian broadened, the other Lorentzian) can be made by converting e for the 400"C data to the FWHM value for the convolution function by using Figure III-5. For example, the 100 minute case (e -.067 mm/sec) yields an FWHM of about.25 mm/sec.

112 fraction of the left component in the two line fits (Table IV-6) and the decrease in the position of the right member. The foregoing discussion emphasized parameters which qualitatively characterize the stages of aging in Cu-Fe alloys. For the purposes of this study it is desirable to be able to quantitatively determine the concentration of iron in its several forms in copper; i.e., a-iron precipitate, y-iron precipitate (yl), Fe atoms with a mixed nearest neighbor configuration (Y2), and solid solution iron (yo). As discussed in Section II-A.2, the resonant line area fraction either directly ( for T << 1) or through Equation (11.14), is related to the fractional concentrations of Fe57 in an environment that gives rise to that particular spectral component. In the cases of quenched and well-aged (class III spectra) samples, the resolution between spectral components is sufficient to permit the quantitative determination of iron in its various phases (configurations). This is also true for those samples containing both y- and a-iron precipitates (see Figure IV-7 and Table IV-7). Unfortunately, however, such an unambiguous determination is not possible from the class I and II spectra associated with the early stages of aging. In the case of the partially resolved class I spectra, the relative fractions of iron in solid soultion and Fe clusters is obtainable if it is assumed that no Yl-iron has been formed (i.e., clusters consist of 12 Fe atoms or less). Then the results of the constrained two component fitting model (singlet plus an area constrained doublet) apply. On the other hand, if some clusters of iron have grown to sizes sufficient to contain yl-iron (Fe57 with all 12 nearest neighbors being Fe atoms), it would be

113 necessary to interpret the aging data using the three component Guassian-Lorentzian fitting model. In any event, the actual fraction of Fe solute atoms remaining in solid solution should be bounded by those values predicted by the two G-L fitting models (refer to Tables IV-4 and IV-5). To estimate this fraction, one could use the mean value of the o area fractions obtained from the two fitting models. The uncertainty in the Fe solute atom concentration so obtained would range from about 5% for the 2850C and 3350C - 10 min. aging data to as much as 32% for the 335"C - 1000 min. data. The resolution problem worsens in the case of class II spectra (samples aged at 400"C for 100 min. or more), and it becomes virtually impossible to make a definite phase identification from the spectra (except possibly in the case of the 400"C - (10)4 min. sample). In these cases, both two and three line fitting models were used (Section B) to approximate the spectral contributions of y1 and yo-iron, and Fe atoms with a mixed Cu and Fe nearest neighbor shell configuration (Y2). An exact quantitative determination of the Fe concentration in either cluster or solid solution configurations is not possible from these fits. However, either the "yl" component from the three line fits or the left component (yL) from the two line fits are a measure of the fraction of iron in large clusters. An increase in their area fractions indicates an increase in the number of Fe atoms in large clusters. (Consideration of cluster size for the various aging stages is made below.) It is desirable for the irradiation effects phase of this study to be able to unambiguously measure changes occuring in the number of Fe atoms associated with the small clusters formed by aging

114 or irradiation. For this purpose, the area fraction of the (2L + y1) line in class I spectra will be used. (For convenience this component will be referred to as yL.) As shown in Table IV-9, the fitting parameters of yL, especially its area, are relatively insensitive to the fitting method used. Several fitting trials (as identified in Table IV-9) incorporating various constraints in the Gaussian-Lorentzian fitting models were tried on the 285"C - 5000 min. and 335~C - 1000 min. aging data in order to determine the sensitivity of the resonant line parameters to the fitting model. The results given in the table clearly demonstrate that the mixing of the o and R Y2 components precludes an accurate determination of their relative areas. Since the YL component is relatively well separated from the remaining portion of the spectrum, its fitting parameters have remained essentially invariant of the fitting model used. As previously discussed, YL consists of the left member of the Y2 doublet representing Fe atoms with a Cu, Fe mixture of nearest neighbors plus possibly y1 representing Fe atoms with only Fe nearest neighbors. In either case it definitely represents iron in cluster form. Also, its area fraction is sensitive to small changes in the number of Fe atoms in clusters and thus can be used as an indicator of cluster formation or dissolution in the neutron irradiation studies (see Sections D and E). It is important to note that in all of the class I spectra fit by the two methods, the doubling of the area fraction of yL from the three-line model equals the 2 area fraction of the constrained two component model. The variation of the area fraction of YL with aging time at the four annealing temperatures

TABLE IV-9 COMPARISONS OF DIFFERENT GAUSSIAN DISTRIBURTED LORENTZIAN FITTING METHODS FOR CLASS I SPECTRA OF TWO AGED SAMPLES Position (mm/sec.) Gaussian Width (mm/sec.) Relative Area Type of Fit Y2 + yL y R y2 + y Y2 Y2 + Y1 Y Y2 x/N (1) Sample aged at 285~C for 5000 minutes (a).187(4).487(6).63(1).069(2).065(10).14(1).374(3).30(2).32(1) 1.09 (b).187(4).484.62(1).069(2).06(1).15(1).373(3).27(4).36(3) 1.10 (c).187(6).482(5).613(6).069(2).057(4).154(3).373(4).255(3).373(4) 1.09 (2) Sample aged at 3350C for 1000 minutes (a).202(5).493(6).60(5).107(2).06(1).14(3).440(6).36(5).20(9) 1.28 (b).202(5).484.58(4).106(2).05(1).14(2).439(6).31(8).25(7) 1.33 (c).200(6).488(5).538(6).099(2).02(1).174(5).420(6).16(1).420(4) 1.36 (a)Fit to three single unconstrained G-L functions. (b)Same as (a) except Yo line was constrained at.484 mm/sec. (C)Fit to G-L singlet (Yo) plus G-L doublet (y2) constrained to have equal areas.

116 studied is illustrated in Figure IV-12. The saturation of the 600~C data for long aging times indicates that the coalescence stage of precipitate particle growth has been reached (for discussion, see the next section). 2. Metallurgical Considerations a. Aging at 600~C. —The metallurgy of copper-rich Cu-Fe alloys aged at temperatures of 5500C and greater has been well established. Easterling(34) using thin foil electron microscopy has shown that in this temperature range nucleation of y-precipitates occurs homogeneously leading to fully coherent spherical precipitate particles. (4) The precipitation kinetics studies of Denney( (see Section II-C) in this temperature range indicate that the precipitate growth is diffusion controlled with an activation energy of 1.65 ev. The Mossbauer results obtained in this study for a Cu-0.63 at.% Fe alloy aged at 600~C (see Table IV-7 and Figure IV-12) show that the fraction of iron in solid solution (y ) and y-precipitates (y1) remains essentially constant for annealing times of about 1000 minutes or greater. This suggests precipitate growth by the coarsening mechanism; 35) i.e., the growth of large precipitate particles at the expense of smaller, less stable particles which are eventually redissolved. The extrapolation of the Yo area fractions (Table IV-7) to longer aging times indicates that about 30% of the Fe atoms should be in solid solution at equilibrium. This corresponds to an iron solubility in copper of about 0.2% at 600~C. The reappearance of the high velocity member of the y2 doublet in the 600"C-aged samples at a location comparable to yR's position in

117.75.70.65 600".60- + 6 /C' 6550 4000C/.50 o 335C.45 /A. S_.40 <,/ ~ ^~C2850.35 -.30 - a-.25.20 I I t I 1 10100 1000 10,000 Aging Time (minutes) Figure IV-12. —Variation of the area fraction of the i component with aging treatment.

118 quenched alloy samples was noted in the previous section. This is interpreted as being caused by small Fe clusters occuring in the irondepleted Cu matrix. It is interesting to compare the atom concentration of iron in the form of small clusters in the case of the 6000C samples (obtained from twice the area fraction of YR in Table IV-7) with that 2 of Y2 for the quenched Cu-0.2%Fe alloy. (As discussed above, approximately 0.2% Fe is in solid solution in the 6000C samples.) Using the results of Figure IV-9 this value is about.06 at.% for the quenched sample, while the value obtained from the 6000C data is about.04 at.%. The closeness of these values suggests a common cause, possibly the random formation of Fe pairs and small clusters in the copper matrix. Assuming random Fe clustering, the ratio AC(2)/(A(Yo) + A(12)) should be approximately equal to the sum of the probabilities for the occurrence of a cluster of n Fe atoms within one nearest neighbor shell; i.e., nP(n) (13)cn(l - c)13-n, where n represents the number of Fe atoms in a cluster and c the Fe concentration. For random Fe pair formation, by far the most probable -4 occurrence, P(2) h^ 3x (10), which is two orders of magnitude smaller than the above ratio. Thus, the Y2 doublet in both cases is not caused by random Fe clustering. There are two other possibilities for its occurrence in the 600~C samples. It could be the result of the same homogeneous process which produced clusters in the samples aged at lower termperatures, or it could be produced by residual Fe clusters leftover from small precipitates being redissolved in the coarsening process. In either event, these small clusters should be unstable in the presence of large Y oro( precipitates.

119 The Mossbauer results of this study also show that a Y-a transformation occurs in cold worked samples which were aged at 600~C for times of 1000 and 2000 minutes; the cold rolling treatment can be 85 given either after aging (as previously demonstrated ) or before aging (see Figure IV-7 and Table IV-6). The mechanism initiating this transformation has been established by Easterling and coworkers(85)(136) to be the passage of a moving dislocation through a large y-precipitate particle of a critical size (-200 R in diameter). During cold working following aging, the,-phase was found by these authors to nucleate as disks on dislocations in i-precipitate particles. Although their microstructure is unknown, the transformation mechanism in the samples of this study which were cold-worked prior to aging is probably similar to the above mechanism. Dislocations existing after cold working very likey act as nucleation sites for i-iron precipitation during annealing at 600 C. Upon quenching from the aging temperature, the dislocations could move under thermal stresses thereby initiating the'-o martensitic transformation. The fact that there were no ^-lines in the ME spectra of samples aged for less than 1000 minutes indicates that the Y-precipitate particles had not reached critical size. It is estimated that the 1000 min. anneal at 600~C produced X-precipitates of 110-200 i in (7)(86) 0 diameter(7)(86) which indicates a critical particle size of about 200 A for the transformation. b. Low Temperature Aging: Mossbauer Versus Resistivity. — Until the recent Mossbauer study by Window(10) and the present work, the only extensive study of precipitation processes in copper-rich Cu-Fe alloys for aging treatments at 4000C and below was the work reported by Boltax.'3) Since the (-precipitates (or clusters) formed at these temperatures

120 are too small to observe by electron microscopy or x-rays, previous studies have involved the measurement of either transport, magnetic or hardness properties (see Section II-C). In the case of Boltax, resistivity measurements were used to follow the precipitation of Y-iron in a Cu-1.7wt%Fe alloy after aging at temperatures from 2000 - 7000C. Resistivity measurements were also made in the present study so that the Mbssbauer results of the 0.6at% Fe alloy could be compared directly to (3)(5) the resistivity results of Boltax. The resistivity results of this study were summarized in Table IV-8 along with the corresponding Mbssbauer results; their relationship is discussed below. A discussion of precipitation kinetics based on the resistivity measurements from both studies is delayed until later in this section. The basic assumption used by Boltax to interpret the resistivity data of aged( and irradiated(5) Cu-Fe alloys (see Sections II-c and II.D) was that the only contributors to an aged alloy's residual resistivity (i.e., Le caused by the presence of impurity atoms -- in this case iron) are the Fe atoms in solid solution. In particular, the effects of small (-precipitates or clusters were ignored. Comparison of Mtssbauer and resistivity data (see Table IV-8) indicates that the above assumption is erroneous for the early stages of aging. The results show that the concentrations of Fe in solid solution predicted from the Mbssbauer spectra (using either the three line or constrained two component G-L fitting models) are significantly less than those values derived from the resistivity results. The differences are due to the inability of the resistivity measurements to distinguish between electron scattering by small Fe clusters and that by isolated

121 Fe atoms in solid solution. The Mbssbauer data indicate that the contribution to the resistivity by small Fe clusters can be significant. This effect is illustrated in Figure IV-13 where the resistivity values of aged and quenched alloys (vertical axis) are plotted against the concentration of Fe in solid solution (atom %) as determined from the area fraction of the Y component in the Mbssbauer spectra. As a reference, the straight line representing the extrapolation of measured resistivity values for Cu with impurity concentrations of Fe (<0.3at%) has been included.(3) This curve represents the effect of Fe solute atoms on resistivity. The y-intercept at 1.75 4a-cm represents the resistivity of pure copper. For comparison, both the three component (open symbols) and constrained two component (closed symbols) Gaussiandistributed Lorentzian fitting results for class I spectra have been plotted in the figure. Error bars representing the combined uncertainties in the y area fraction and the measured total Fe concentration in the alloy are indicated in the figure, except in the case of class II spectra because of their extremely poor resolution. The uncertainties associated with the resistivity values were given in Table IV-8; most are about 1%. Comparison of the experimental data of Figure IV-13 with the extrapolated residual resistivity curve provides an estimate of the excess resistivity which can not be accounted for by the iron in solid solution with copper. In the two samples where large precipitate *The abscissa values obtained from the two fitting models represent approximate upper and lower limits to the concentration of Fe in the Yo phase for sampes exhibiting class I spectra.

122 7.0 - *^ I 6.0 — *0 - / O*'-O —- t 5.0 - O A/ "E /^ ^= 8.2x - 1.75 /-cm d 4.0 Q ^+ in>~ A ~ F 3.0.^ x/ Heat Treatments: 2/ - Quenched and Cold-rolled 2.0 -/ + Quenched v.lat% Fe Quenched o Aged 285oC; 100-5000 min. G Aged 335oC; 10-1000 min. A Aged 400~C; 10-10,000 min. ^~1.0- ~t + Aged 600~C; 100 & 1000 min. Total Fe concentration 0'~ 0 0.1 0.2 0.3 0.4 0.5 0.6 Atomic Percent Iron in Solid Solution Figure IV-13. —Plot of resistivity versus atomic percent Fe in solid solution determined from ['dssbauer spectra of heat treated copper-rich Cu-Fe alloys.

123 particles (>100A) are known to have formed (the 6000C-aging results designated by a cross), the resistivity values essentially fall on the extrapolated line. This indicates that in these cases only the Fe solute atoms remaining in solution with copper are effective in scattering electrons, and not the large precipitates. The resistivity of the 0.1% Fe quenched sample is also close to the extrapolated curve since very little clustering (as determined from the Y2 component in its ME spectrum) has occured. On the other hand, deviations from the curve were significant for both the quenched and mildly aged (low temperature) Cu-O.6at%Fe alloy samples. In such cases, there exists an excess resistivity, be, which can be associated with the Fe clusters.* The effectiveness of Fe clusters or precipitates in scattering electrons (i.e., their effective contribution to the resistivity) can be determined from Figure IV-13 as a function of heat treatment. Since the difference ac, between the data points and the extrapolated curve represents the Fe solute concentration required to account for the excess resistivity, the relative effectiveness of the clusters may be defined as 6 = ac/c2 (IV.2) where c2 is the overall concentration of Fe in clusters or small precipitates as determined from the Mdssbauer spectra (i.e., yl and Y2). Values of e are given in the following table for both quenched and aged samples. *Other factors such as vacancies and strain fields associated with dislocations can cause anomolous electron scattering and thereby contribute to Ad. However, these effects should be small compared to the effect of Fe clusters (e.g., a e of only 0.044 ~-cm occurs when pure copper is cold-worked to an 87% reduction in area137).

124 Sample e Sample Quenched & Rolled.76 3350C - 1000 min..21 Quenched.66 4000C - 10 min..38 2850C - 100 min..59 400~C - 100 min..15 2850C - 1000 min..52 4000C - 1000 min..11 28500 - 5000 min..39 4000C - 10,000 min..06 3350C - 10 min..49 6000C - 200 min..05 3350C - 100 min..38 6000C - 1000 min..03 The effectiveness parameter e should decrease with increasing cluster size or with time at a given annealing temperature as indicated in the above table. For samples containing larger precipitates (6000C - 1000 min. and 4000C - 10,000 min.) e approaches zero, in which case resistivity would be a direct measure of changes in the Fe concentration in solid solution. On the other hand, at the lower aging temperatures the large values obtained for e indicate that resistivity would not be a good indicator of the concentration of Fe atoms in solid solution. This applies to most of the samples aged in the temperature range 2850 - 400~C. c. Low Temperature Aging: Growth of Y-Precipitates. —With regard to precipitate growth during the early stages of aging, Boltax(3) concluded that precipitation in Cu-Fe alloys probably occurs at dislocations formed during the pre-anneal quench from solution temperature. In this case, the precipitates would probably form as plates or disks on the dislocations. As mentioned in Section II.C, he based this conclusion on the isothermal kinetic behavior of the transformation fraction f (Equation IV.1) obtained from resistivity measurements. As will be discussed below, the Mbssbauer results of Window(10) and this study do not support Boltax's suggested growth mechanism. In order to make a direct comparison with his work, the resistivity results of the present study (Table IV-8) have been analyzed

125 in the manner of Boltax;() i.e., by application of the generalized rate equation df = K (1 - f) tn1 (IV.3) dt In the above expression, f is the fraction transformed given by Equation (IV.1), K A exp(-nQR/RT) the rate constant, n the time exponent, and Q the activation energy. The f values versus aging time from the resistivity measurements were plotted in Figure IV-8 (solid curves). The log(t) at constant f values versus reciprocal temperature produced linear plots from which the activation energy QR was found to be about 37 kcal/mole ( 1.6 ev/atom). Plotting the function log( log(1 ) ) 1-f versus log(t) (i.e., from the solution of Eq. IV.3) yielded a time constant of nv.30. Both values agree to within limits of experimental error with the results obtained by Boltax,() suggesting that identical growth processes occured in the alloys considered by the two studies. Since the resistivity data fails to directly account for Fe in small clusters which predominate at the lower aging termperatures, the resistivity derived kinetics results are not characteristic of the initial growth kinetics of Fe precipitates in copper. The combined effects of small clusters and isolated Fe atoms on resistivity are inseparable, making these kinetic parameters meaningless. This points up the general problem of trying to interpret microscopic properties through the use of bulk measurement techniques. Because of the limited number of data points and the very poor resolution of the class II spectra, it is not possible to perform an analysis of the kinetics of Fe cluster formation and growth based on ME spectral data.* However, it does appear that the ME transformation

126 curves in Figure IV-8 (dashed lines) are not of the general sigmoidal shape characteristic of nucleation and growth reactions. As discussed below, the initial clustering of iron in the temperature range 2850C - 4000C, as characterized by Mbssbauer spectroscopy, is instead suggestive of Guinier - Preston (G.P.) zone formation.** G. P. zones are generally found in alloys where the difference in the atomic diameters of the solvent and solute atoms is small or where the supersaturation is large. Iron in copper satisfies both of these conditions. Its solubility in Cu at 600~C is about 0.2% (this study) and at 4000C and below is less than.05% (extrapolated from the high temperature portion of the solvus given in the phase diagram77). In addition, the room temperature lattice parameters of copper(139) and y-iron(78) are 3.607 A and 3.590 A, respectively, the difference being only 0.5%. Since G. P. zones in general, and Fe clusters in Cu-Fe in particular, tend to form at rates faster than would be expected from the lattice diffusion coefficient,*** their formation at low temperatures is very likely the result of excess vacancies quenched in during the solution treatment. (141) The enhanced diffusion occuring at low aging *It is possible to estimate the activation energy for the clustering process from the ME transformation curves of Fig. IV-8During the initial stage of Fe clustering, the three component G-L fitting results yield an activation energy of approximately 1.1 ev/atom (or, 26 kacal/mole). **A general review of the properties and structure of GuinierPreston zones or coherent clusters may be found in Christian, (138) In general, G.P. zones are a pre-precipitate phase in the form of soluterich clusters that are coherent with the solvent's lattice. ***The lattice diffusion, coefficient as a function of temperature is given by D = Do exp(-Q/RT). Using the values of Dol.O 0 cm2/sec and Qv51.8 kcal/mole obtained for Fe diffusion in copper at high

127 temperatures in the Cu-Fe system then would be due to the movement of the excess vacancies to grain boundaries and other vacancy sinks. The low activation energy estimated from the initial portion of the ME transformation curves is comparable to that for vacancy migration and not lattice diffusion. This is consistent with the excess vacancy mechanism of G.P. zone formation. The difficulty with the excess vacancy hypothesis is that the lifetime of a vacancy is very short compared to the observed aging periods in alloys having G.P. zones. Hart(142) suggests that vacancies are retained for longer periods because of a strong attractive force between vacancies and solute atoms. The growth model proposed by Hart has vacancy and solute atom clusters migrating through the solvent's lattice picking up excess solute atoms along the way. Unfortunately an exact theoretical growth model for G.P. zones does not exist. However, the experimental evidence available on G.P. zone formation in alloys has (138) been consistent with the proposed vacancy mechanism. The small atomic size misfit of Fe in Cu as well as the electron (134) microscopy results of Easterling for alloys with resolvable Yprecipitates indicate that incipient G.P. zones or small Fe clusters formed in copper are spherical in shape. Window's(10) recent work on aging in a Cu-l.Oat%Fe alloy treats precipitate growth in this manner. On the assumption of spherical growth, it is possible to determine the number of surface atoms (i.e., Fe atoms having fewer than 12 Fe nearest neighbors) as a function of cluster size. As pointed out by Window(10) temperatures by Mullen,(140) the value of D at 3350C would be only of the order of 3(10)-19 cm-/sec. This is much too small to account for the significant clustering occuring at these low temperatures.

128 the surface atoms on small Fe clusters in Cu are responsible for the behavior of the J2 doublet in aged alloys. The calculations of Appendix B indicate that the Y2 doublet probably remains well-defined (although considerably broadened by the variety of configurations) for Fe57 with up to five Fe nearest neighbors. Thus, during the incipient stage of clustering the MBssbauer spectra would be characteristic of Fe surface atoms in clusters (those having both Cu and Fe nearest neighbors) and Fe atoms in solid solution. To illustrate the role of surface atoms in small spherical clusters, the number of Fe nearest neighbors (n.n.) of iron atoms located in the first six shells about a central atom have been determined as a function of the number of filled shells. (The atoms were built on the FCC lattice in a spherical array through seven near neighbor shells.) The results are given in the table below. # Fe Atoms in Shell 1 12 6 24 12 24 8 Shell Number 0 1 2 3 4 5 6 1 12 5 4 2 1 0 0 2 12 7 6 3 1 2 0 ~0 ~ 3 12 11 8 5 5 4 3 A) 4 12 12 8 7 5 5 3 - 5 12 12 12 9 7 7 3 6 12 12 12 10 7 7 3 7 12 12 12 12 11 9 9

129 As can be seen from the table, surface atoms having less than 12 Fe nearest neighbors predominate throughout the build up of the first six near neighbor shells. For example, with the first three shells filled there would be 24 Fe atoms with 5 Fe n.n., 6 with 8, 12 with 11 and only one with 12. Therefore, not until about six to seven shells are filled (clusters greater than,100 atoms) would one expect the )1 line due to Y-iron to be the predominant member of the YL (y2 + 1) component. As discussed previously in this section, this is believed to occur for samples aged at 4000C for times greater than 100 minutes. If the ratio of Y to 2 Fe atoms could be accurately determined 1 2 from the Mbssbauer spectral areas, then an estimate of cluster size could be made from the preceding table. Unfortunately, the superposition of 1 and T2~ as well as the successively increasing overlap of the Yo and Y2R components renders this virtually impossible. However, by making use of the earlier discussion on spectra systematics (Section C.1) and Window's electric field gradient model (Appendix B), it is possible to make certain generalizations concerning Fe cluster size in aged Cu-.6%Fe alloys: (1) The majority of clusters formed in samples aged at 2850C and for short times (t<100 min.) at 335 C most likely consist of fewer than 13 atoms. The ME spectra of these samples (Tables IV-4 and IV-5) retain a well defined "2 doublet with minimal traces of ~, indicating that Fe surface atoms predominate. Although the three component G-L fits indicate the possible presence of some yl-iron (3 to 7 % of the total Fe atoms in clusters), Window's(10) two component G-L model fits the data almost as well. Thus, very little if any 1i-iron (i.e., Fe with completed Fe n.n. shell) has been formed in these

130 samples. (2) In the samples aged for t>100 min. at 335 C and for t<100 min. at 400 C, the spectral components associated with Fe clustering are still predominately due to Fe atoms with less than a full shell (12) of iron nearest neighbors. The three component G-L fits indicate that possibly 20 - 30% of the clustered iron atoms have either a full shell or very close to a full shell of Fe nearest neighbors. A nearest neighbor shell of 10 or 11 Fe atoms would be expected to produce a quadrupole-broadened resonance line shifted slighly to higher velocities from the true yI or Y-iron line. This might be the cause of the shift and broadening occuring in )L for these samples. If so, then the size of the clusters could extend to approximately three near neighbor shells ("43 atoms). On the other hand, the observed broadening and shift in the jL component could be the result of configurational and size variations in smaller clusters, in which case )L would be mainly comprised of the left member of the /2 doublet. This possibility can not be precluded since the two component G-L fitting model produced adequate fits to the data. (3) The spectra of samples aged at 4000C for t 100 minutes show definite signs of Y1-iron formation; i.e., the'L component in either the two or three G-L line fits (Table IV-6) exhibited position and width characteristics similar to the dominant Y-iron line produced in the 600~C - aged samples. On the basis of the n.n. configurations given in the preceding table, these clusters are very likely larger 0 than six or seven n.n. shells in radius (i.e.,> 6.2 A). In the case of the 4000C - (10)4 min. sample, it is possible to make a very crude estimation of the particle size as follows. The ratio of volume to

131 surface atoms can be approximated as A + A2 4ijR /3 = R/3LR A2 4-R2~R where A1 and A2 are spectral areas for yi- and Y2-iron, respectively, and AR is the surface shell thickness (taken as the nearest neighbor spacing — 2.47 X). Using the results of Table IV-6, the above 0 expression gives R~20 A for the average precipitate radius in this alloy. It is worthwhile to make a comparison between the very approximate Fe cluster sizes inferred from the Mdssbauer data of Cu-.6%Fe with those found in the analogous Cu-Co system. (As discussed in Section II-D, the Co precipitate particles in Cu-Co are superparamagnetic which allows their sizes to be determined from magnetization measurements.) In the Cu-rich system the Co precipitates are coherent forming homogeneously in the Cu matrix, and are believed to be roughly spherical in shape.( ) In a Cu-2.0wt%Co alloy, Piercy(50) found precipitate particles of about 13 i in radius upon aging at 395 C (31zt428 hr.). This value falls within the size range estimated above for Cu-0.6%Fe aged at 4000C (1.6et <166 hr.). Comparable precipitates are also found in the two systems after aging at 6000C. The similarity in the shape and sizes of the precipitates in the two alloy systems is not surprising considering the similarity of the Cu-rich ends of their respective phase diagrams. According to the foregoing arguments, the range of precipitate or cluster sizes inferred from room temperature ME spectra using the three component G-L fitting scheme described in Section IV-B is somewhat larger than proposed by Window(10) in a recent Mdssbauer study of

132 aging and cold-working in Cu-1.0%Fe. In that study, Window concluded that aging at temperatures of 3000 to 4000C (t 180 min.) produces Fe clusters with sizes less than 13 atoms. The room temperature spectra of the class I type (Figure IV-3) were fit to the sum of an area constrained G-L doublet representing ~2-iron and a G-L singlet representing yo, thereby precluding any Y1 contribution to the spectra. Window's estimation of cluster size was based on an interpretation of the magnetic field distributions, p(H) curves, derived from fitting the low temperature (4.2 K) hfs spectra of the aged alloys. The effective fields at the iron sites in Fe clusters in Cu are believed to be due to localized moments on atoms in the nearest neighbor shell with the contributions from more distant neighbors leading to perturbations and broadening in the hyperfine field distributions. The unfolding of the extremely complex spectra associated with these fields is very difficult and in the case of Window(76) was accomplished by using a synthesis of cosine functions to simulate the field distributions resulting from a variety of Fe cluster sizes and configurations. As a further approximation, quadrupole and isomer shift effects were neglected. The magnetic field distributions so obtained from the spectra of aged Cu-Fe alloys revealed that in small clusters the individual spins tend to align parallel producing a very broad p(H) distribution with some structure. Window (8)(1) was able to associate a definite peak at v80 kOe with Fe atoms having no Fe nearest neighbors and a very broad distribution at larger H with those in small clusters. Large clusters or precipitates of i-iron, on the other hand, are known to behave antiferromagnetically (Abrahams et al.142) The critical cluster size for which antiferromagnetic ordering sets in is

133 not known. The M6ssbauer spectrum of Y-iron precipitates below the Neel temperature* consists of a broadened resonance line containing unresolved hf structure characterized by a small effective field of ~20 kOe.(70) It was the absence of a low field peak in the p(H) distributions of moderately aged alloys which led Window to conclude that virtually no Y1-iron (Fe with a completed n.n. shell of iron atoms) exists in clusters formed at low aging temperatures. In contrast this study's room temperature spectra of Cu-0.6%Fe samples aged at 400~C for t 100 minutes show substantial evidence of Y1 formation. The analogy between the Cu-rich Cu-Fe and Cu-Co alloy systems lends further support for Y1-iron formation at 4000C. The apparent discrepancy between the 3000K and 4.20K spectra could be caused by differences in the effective contributions of neighbors outside the nearest neighbor shell to the three hyperfine interactions at an Fe57 site in a cluster. In the case of magnetic effects, this would include the influence of near neighbors on the type of order observed (e.g., in small precipitates, the relatively large number of surface atoms would influence the spins of internal or 1,-iron atoms). For Fe clusters in Cu, the influence of neighbors outside the n.n. shell on the s-electron density and the electric field gradient of the resonant atom is probably much less than their magnetic ordering effects. This would explain the appearance of the apparent Y-iron component in the room temperature spectra but not in the 4.20K spectra. *The Neel temperature in Cu-rich Cu-Fe alloys appears to be dependent upon aging temperature and time. (70) This is probably caused by its dependence on particle size of the precipitate.

134 3. Summary M6ssbauer spectroscopy (at room temperature) was able to follow in a qualitative manner the early growth of coherent iron clusters in a copper-rich Cu-Fe alloy during the incipient stages of aging. The distribution of isomer shifts and quadrupole splittings associated with configurational variations of Fe in three different environments (Fe with only Cu nearest neighbors - yo, Fe with only Fe nearest neighbors - Yi, and Fe with both Cu and Fe nearest neighbors - Y2) were treated by using Gaussian-distributed Lorentzian functions to fit the complex spectra. Two fitting models were used: one consisting of three G-L singlets and the other consisting of one G-L singlet plus an area constrained G-L doublet (Window10). Phase identification as well as the quantitative determination of iron concentrations could be unambiguously made in the cases of solution treated samples (contained predominantly y with some Y2) and samples aged at 6000C (contained predominately 1 with some Yo and trace amounts of Y2). In the case of samples aged at intermediate temperatures, poor spectral resolution resulting from excessive overlap of the spectral components prevented an exact quantitative determination of iron phase concentrations. However, upper and lower bounds to the phase concentrations were established through the application of the two G-L fitting schemes. Quantitative phase analysis was also performed on aged and cold-rolled samples containing both Y- and o-iron. The Mbssbauer results are consistent with the formation of spherical Guinier-Preston zones or pre-precipitates during the early stages of aging. The predominance of the broadened doublet component due to Y2-iron in the spectra of samples aged at 4000C (t<100 min.)

135 and lower temperatures indicates that the clusters formed consist of very few Fe atoms -- possibly up to three near neighbor shells at most. Aging for longer times at 400~C results in the formation of Fe clusters that are large enough (vv5 or 6 near neighbor shells) to produce a significant Y-iron component in the 300~K M6ssbauer spectra, while aging at 600~C is known to produce precipitates v100 2 in radius. The results of the present study are in general agreement with those of Window's (10) recent work on aging in Cu-l.Oat%Fe. The Mbssbauer results for copper-rich Cu-Fe alloys do not support the dislocation-enhanced growth mechanism suggested by Boltax(3) to explain the results of resistivity measurements in the same alloy system. Comparison of Mbssbauer and resistivity results from both quenched and aged samples in the present study indicates that small Fe clusters significantly contribute to the alloy's residual resistivity, which was assumed by Boltax to be caused by isolated iron atoms only. It is the effects of very small Fe clusters, undetected in the resistivity measurements, which are responsible for the discrepancies in the results from the two measurement techniques. MBssbauer spectroscopy has the advantage of being able to detect small Fe clusters during the early stages of aging. D. Results of Irradiation Experiments The majority of the Cu-Fe alloy samples which were thermally treated in the manner described in Section IV-B were subsequently irradiated in the FNR at three different integrated fast fluxes. The results of these irradiations are described below.*

136 1. Solution Annealed and Quenched Samples Table IV-10 summarizes the effects of fast neutron exposure on the ME spectral parameters of solution treated and cold-rolled Cu-Fe alloy samples. Parameter values are listed for both pre- and postirradiation conditions. Initially, spectra from all of the quenched - irradiated samples were fit to the sum of three unconstrained Lorentzians. For the reasons given in Section IV-B, many were re-analyzed using the Gaussian-distributed Lorentzian (G-L) fitting model described in Section III-D. Representative Mbssbauer spectra of quenched and cold-rolled Cu-0.6at%Fe samples in the pre-irradiated (fit with a G-L singlet and area constrained doublet) and post-irradiated (fit with three G-L singlets) conditions are displayed in Figure IV-14. Comparison of Tables IV-4 and IV-10 as well as Figures IV-6 and IV-14 reveals that fast neutron irradiation produces similar effects on the Mbssbauer spectra of solution treated alloys as thermal aging; e.g., the increase in the area fractions of X2' and YL (Y1 +T2L), R the significant broadening of Y2R, and the slight positional shifts in 2R (negative) and /L (positive). These spectral changes signify that iron clustering has occured (see Section IV-C.1) and are direct evidence of irradiation-induced aging. The changes in spectral parameters are even more pronounced when the constrained two component G-L fitting model is applied to the post-irradiation spectra. For comparison, the results from the two fitting methods are listed in the *Following irradiation many of the thin Cu-Fe foils contained pin holes and small tears. As a result, meaningful resistivity measurements were able to be performed on only a few of the postirradiated samples.

TABLE IV-10 EFFECTS OF FAST NEUTRON IRRADIATION ON THE MOSSBAUER SPECTRA OF QUENCHED Cu-Fe ALLOYS Position(a) (mm/sec) Line Width(b) (mm/sec) Relative Area Sample Y2 + Y1 Yo y2 2 + y 2 y1 y + Yl/ Y8 2 (1) Fast neutron fluence Pt ~4(10)18 n/cm2: Quenched b.157(4) 4.075(3).060(2).074(3).163(3) 674(4).163(3) 1.02 & Rolled a.1664).48(4) 7385).0733).0541).0884).218(2) 5831).199(4) 1.11 (G-L fits) Quenched b.158(4).484(4).744(5).231(3).244(5).251(8).184(6).632(9).184(6) 1.15 o (Lorentzian a.165(4).485(4).729(6).246(4).254(6).285(7).228(4).566(5).206(7) 1.21 fits) (2) Fast neutron fluence pt12.7(10)19 n/cm2:.2% Fe "Q" b.164(4).483(4).738(4).066(2).038(2).085(3).158(3).683(4).158(3) 1.04 (G-L fits) a.175(4).483(4).722(5).072(2).042(3).100(4).242(2).553(3).206(5) 1.12 Quenched b.170(4).484(4).738(6).065(2).048(3).092(4).199(4).602(2).199(4) 1.11 (G-L fits) a.174(4).486(4).728(6).067(2).055(3).096(4).274(2).495(2).230(6) 1.20 Quenched b.152(4).480(4).742(5).234(2).265(2).249(4).138(4).723(7).138(4) 1.03 & Rolled a.164(4).482(4).730(5).260(4).266(4).271(6).239(6).58(1).183(9) 1.38 (Lorentzian fits)

TABLE IV-10 (continued) EFFECTS OF FAST NEUTRON IRRADIATION ON THE MOSSBAUER SPECTRA OF QUENCHED Cu-Fe ALLOYS Position (a) (b) Positiona) (mm/sec) Line Width( (mm/sec) Relative Area Sample y2 + Y1 Y Y +1 2o R y L + 1 /N __ __ __ _ __ ___Yo Yo 2 (3) Fast neutron fluence (tr19(10)19 n/cm2:.1% Fe "Q" b.155(4).483(4).743(4).231(2).229(1).244(3).109(3).782(2).109(3) 0.99 (Lorentzian a.168(4).486(4).714(5).240(3).236(2).276(5).216(3).601(3).183(7) 1.14 fits) Quenched b.165(4).483(4).736(5).063(2).044(2).096(3).20(1).600(5).20(1) 1.02 (G-L fits) a.178(4).485(4).704(6).077(2).057(3).112(5).313(2).434(4).254(9) 0.96 Quenched b.165(4).482(4).737(5).077(2).061(2).073(2).166(3).667(4).166(3) 0.97 & Rolled a.173(4).482(4).705(6).078(2).049(3).116(5).292(2).471(3).237(9) 0.98 (G-L fits) (a)Position referred to SNP (b)Lorentzian fit: FWHM r; Gaussian-distributed Lorentzian fit: width of Gaussian broadening e (b = before; a = after irradiation

139 Quenched c) Irradiated oZ rI a/ -tv2.7(10)19 cc ci: " \ /r~~~~~ Quenched & -^t~~~ ^ / ~~~Cold-rolled II'\ \/ ~ l Irradiated t\ \ 9(10)19 -1.00 -.50.00.50 L.00 l.50 2.00 MM/SEC Figure IV-14. —Effects of fast neutron exposure on spectra of solution treated and cold-rolled Cu-O.6at%Fe alloy samples.

140 table immediately below for a quenched Cu-0.6%Fe alloy irradiated to a fast fluence ofv9(10)19 n/cm2 (For reference, spectral parameters of the quenched alloy are also given.) PL o PR L o 9 AL o AR X/N (a) Quench.165.483.736.063.044.096.200.600.200 1.02 (b) 3 G-L.178.485.704.077.057.112.313.434.254 0.96 (c) 2 G-L.178.481.676.078.025.147.313.374.313 1.05 Here, P represents the components' isomer shift with respect to SNP, 0 the Gaussian broadening parameter, and A the area fraction. The subsscript "o" refers to Y1-iron, while "L" and "R" refer to the left (Y + Y for'b' and 2L for'a' and'c') and right ( 2R) components, respectively. As discussed previously for thermal aging, the y area fractions from the two fitting schemes represent upper (three component model) and lower (two component model) bounds to the fraction of Fe atoms with a complete Cu nearest neighbor shell. A previous comparison of the fitting schemes also revealed that the spectral parameters for YL (left-most component) in the case of class I spectra remained relatively insensitive to the fitting model used. This is also apparent in the above table for the irradiated samples. It was shown in Section IV-C that the area fraction of the YL component represents the most reliable indicator of iron cluster formation (i.e., it is only representative of Fe atoms in a cluster configuration). Therefore, in Figure IV-15 the irradiation-induced change in the YL area fraction (converted to Fe concentration) for solution treated samples has been plotted versus the initial or preirradiation concentration of Fe in solid solution CY). Data from alloy 0 samples containing 0.1 and 0.2 at% Fe as well as from 0.6%Fe-samples which had been cold-rolled following solution treatment are also

141 plotted to indicate the strong dependence of irradiation-induced clustering on the degree of supersaturation. The ordinate values, in the case of the two component model, represent one-half the concentration of Fe atoms that have left the solid solution phase to form small clusters. Resistivity measurements were successfully performed on two of the quenched samples which were irradiated at fluences of 2.7(10)19 and 19 2 9(10) n/cm. Changes in resistivity (A ) between the pre- and postirradiated samples are given in the table below. Fluence Ae(Aa-cm) ACe ACM(3 G-L) ACM(2 G-L) 2.7(10)19 n/cm2 -.44 -.054 -.067(5) -.102(8) 9(10)19 n/cm2 -.77 -.094 -.104(8) -.14(1) Also included in this table are the associated concentration changes in solid solution Fe (ACe) derived from theA/ values, along with the corresponding concentration changes (ACM) derived from the O area fractions of the Mbssbauer spectra. (Concentration values are reported in atom% Fe.) The AC values in the above table were obtained using the slope (8.21 &l-cm/at% Fe) of the electrical resistivity versus Fe concentration curve of Figure IV-13, while the ACM values were obtained using the procedure described in Section II-A. The results of both measurement techniques show that the concentration of iron in the solid solution phase of quenched alloys decreases significantly following fast neutron irradiation. 2. Aged Samples The results of the fast neutron irradiation studies of aged alloys using M5ssbauer spectroscopy are summarized in Tables IV-11

142 Open symbols: G-L Fits Closed symbols: Lorentzian Fits 8.0 7.0 6.0 -9(10)19 n/cm z 5.0- / 9 7(10)1 9 n/cm2 2 5.0 - LI- 4.0 E QO 0 - > 3.0 q 2.07 / / 7/ ^ Q.2%Fe ~ 4(10)18 n/cm2 1.0- 1%Fe 0 I I I I 0 0.1 0.2 0.3 0.4 Initial Yo Concentration (at.%) Figure IV-15. —Irradiation-induced change in the X spectral component (atom % Fe) versus pre-irradiated concentration of iron in solid solution for solution treated Cu-Fe alloys.

143 through IV-16 and Figures IV-16 through IV-18. For comparison, spectral parameters for both pre- and post-irradiation conditions have been included in the tables. The spectra associated with the irradiated samples are categorized in the manner of Section IV-B; i.e., into three classes according to their appearance. These classes as applied to both aged and irradiated samples are representative of the various stages of iron precipitation in Cu: class I (broadened, apparent three line spectra) of incipient Fe cluster formation; class II (broadened, apparent two line spectra) of an intermediate stage of cluster formation; and class III (three relatively narrow resonance lines) of large iron precipitate formation. The exposure of aged Cu-Fe samples to fast neutrons caused subtle, but significant changes in their Mbssbauer spectra which can be interpreted in terms of either increases or decreases of Fe in cluster form. The spectral changes due to irradiation are evaluated below on the basis of the knowledge gained from the previous detailed study of aged alloys. The majority of the Cu-0.6at%Fe samples were irradiated to a fast neutron exposure of approximately 2.7(10)19 n/cm2; however, many of the samples were also irradiated to fluences of 4.0(10)18 and 9(10)19 n/cm2. The effects of an exposure of 2.7(10)19 n/cm onthe 91) n/cm on the M6ssbauer spectral parameters of moderately aged alloy samples (class I spectra) are shown in Table IV-11. Like their aged counterparts, these spectra were fit to the sum of three G-L functions with the spectral components as defined previously (see Section IV-B). In the case of the 3350C - samples, the spectra were also fit with the two component G-L model (Yo + Y2); the resulting parameters are given in Table IV-12. Comparison of the two tables indicates that the

TABLE IV-11 THREE G-L COMPONENT SPECTRA PARAMETERS FOR HEAT TREATED Cu-0.6% Fe ALLOY SAMPLES BEFORE AND AFTER FAST NEUTRON EXPOSURE OF -2.7(10)19 n/cm2 Position (mm/sec.) Gaussian Width (mm/sec.) Relative Area Heat Treatment L + yR L R L R Y2y2 l Yo Y2 y2 +y1 y Y Y2 y Y Sol. anneal b.158(4).484(4).739(5).065(2).042(2).089(4).189(4).633(4).178(6) 1.06 & quench a.173(4).486(4).724(5).067(2).050(2).098(4).276(2).496(2).229(6) 1.14 285~C b.178(4).485(4).703(6).066(2).053(2).112(4).310(1).421(3).269(7) 1.32 100 min. a.178(4).486(4).712(6).078(2).059(3).111(4).319(1).421(3).261(6) 1.30 285~C b.176(4).481(5).687(6).064(2).046(3).121(5).325(1).370(5).304(9) 1.38 1000 min. a.178(4).484(5).699(6).077(2).051(3).116(4).330(2).390(5).28(1) 1.16 285~C b.187(4).487(6).63(1).069(3).065(10).141(1).374(3).30(2).33(1) 1.09 5000 min. a.184(4).482(5).67(1).076(2).057(6).129(8).361(2).33(1).31(2) 1.08 335~C b.171(4).485(4).712(6).061(2).041(3).102(5).283(2).470(3).248(6) 1.06 10 min. a.174(4).483(4).705(5).077(3).051(3).107(6).314(2).437(3).249(9) 1.13 335"C b.191(5).487(5).661(9).078(2).061(4).12(1).356(2).39(1).26(2) 1.08 100 min. al.185(4).486(5).682(8).085(2).052(4).126(7).350(2).370(7).28(1) 0.97 335"C b.202(5).484(*).58(3).106(3).05(1).14(2).439(6).31(8).25(7) 1.34 1000 min. a.186(4).484(*).63(1).096(2).047(5).150(9).410(4).27(2).32(2) 1.01 400"C b.189(4).487(5).67(1).082(2).049(4).13(1).341(2).42(1).24(2) 0.98 10 min. a.181(4).484(5).678(9).089(3).048(5).135(9).363(3).380(9).26(2) 1.19 b = before; a = after *Positions were constrained at 0.484 mm/sec., the isomer shift for solid solution Fe in copper during the fitting.

TABLE IV-12 TWO G-L COMPONENT SPECTRA PARAMETERS FOR AGED (3350C) Cu-0.6% Fe ALLOY SAMPLES BEFORE A;D AFTER FAST NEUTRON EXPOSURE OF^2.7(10)19 n/cm2 Position (mm/sec.) Gaussian Width (mm/Sec.) Relative Area Heat Treatment Splitting Y2 Heat Treatnent Splitting L R Yo Y2 Y2 2o Y2 o 2 x Sol. Anneal & quench b.483(4).450(4).575(1).065(2).035(2).091(3).622(3).378(4) 1.11 a.484(4).434(5).515(1).067(2).038(3).129(4).448.552(4) 1.17 335"C b.480(4).430(4).509(1).062(2).027(3).132(3).425(3).575(5) 1.20 10 min. a.475(4).424(4).500(1).079(2).033(3).146(3).369(4).631(7) 1.31 335~C b.483(4).405(5).428(2).078(2).045(3).155(3).290(3).710!6) 1.11 100 min. a.481(4).418(4).465(1).081(2).037(2).155(2).299(3).701!6) 1.03 335"C b.488(5).369(5).338(2).099(3).019(8).174(5).16(1).84(1) 1.46 1000 min. a.485(4).390(4).414(2).093(2).024(7).173(3).198(4).802(6) 1.04 b = before; a = after

146 paramaeters associated with /L from the two fitting models agree within experimental error as was found previously for other class I samples. Figure IV-16 illustrates the effects of increasing fast neutron exposure upon the spectra of samples aged at 335 C for 1000 minutes. The spectral parameters are listed in Table IV-13 for three different fluences. A cursory examination of the figure and table reveals that the spectra associated with increasing neutron exposure have reverted to forms characteristic of a less advanced stage of aging; i.e., they exhibit an increase in splitting between the right (Y2R) and left (h1 + 12L) components as well as decrease in area fraction of IL. On the other hand, several samples aged for short times at temperatures of 2850-4000C showed characteristics (see Tables IV-11 and IV-12) of increased aging as also occured in solution treated alloys following irradiation. This distinctly opposite behavior is related to the stage of Fe clustering in the pre-irradiated alloy. Detailed interpretation of these phenomena is delayed until all of the results have been presented. The effects of an exposure of 2.7(10)19 n/cm2 on the 4000Caged samples which exhibited class II spectra (Figure IV-5) are shown in Table IV-14. Both two and three line fits were made of the data. Although poor resolution prevents a detailed interpretation of the pre- and post-irradiation spectra, it is apparent from the decrease in the left component's area fraction (both fits) and the large increase in the broadening of Y2 (3-line fit) or the right component (2-line fit) that irradiation-induced precipitate or cluster break-up has occured. This was not the case with those samples aged at 600 C

147 %3'" By \^/ / (a) Aged 335~C; / 1000 min. 2 /, / ~, a (b) Irradiated cic'' 2.7(10)19 n/cm2 a)?,:' (c) Irradiated MM/SEC Figure IV-16. —Effects of increasing fast neutron exposure on the Mbssbauer spectrum of Cu-0.6at%Fe aged at 3350C for 1000 minutes. The spectra were fit with three Gaussian-distributed Lorentzian functions.

TABLE IV-13 EFFECTS OF INCREASING FAST NEUTRON EXPOSURE ON MOSSBAUER ( SPECTRAL PARAMETERS FOR A Cu-0.6% Fe ALLOY AGED AT 3350C FOR 1000 MIN.(a Position (mm/sec) Gaussian Width (mm/sec) Relative Area Fast Neutron L (b) R L R L Exposure 2 + 1 o Y2 Y22+ y YO 2 x2+ (n/cm2)_ Aged Sample Before.202(5).484.58(3).106(3).05(1).14(2).439(6).31(8).25(7) 1.34 Irradiation 4.0(10)8 n/cm2.197(4).484.60(1).095(3).045(8).155(12).42(1).29(9).29(6) 1.13 2.7(10)19 n/cm2.186(4).484.63(1).096(2).047(5).150(9).410(4).27(2).32(2) 1.01 9.0(10)19 n/cm2.187(5).484.65(1).080(2).039(5).153(8).367(3).28(2).35(1) 1.09 (a) (Spectra were fit to the sum of three Gaussian-distributed Lorentzian functions. (b )Constrained at the y position for quenched samples.

TABLE IV-14 MOSSBAUER SPECTRA PARAMETERS FOR Cu-O.6% Fe ALLOY SAMPLES AGED AT 4000C (CLASS II SPECTRA) BEFORE AND AFTER A FAST NEUTRON EXPOSURE OFv2.5(10)19 n/cm2 Position (mm/sec) Line Width(c) (mm/sec) Relative Area 2 Sample Yx Y Yo2 Y] YY Y X/N (a) (1) 3-line fits:( 4000C b.165(5).484.41(1).067(4.21(1).22(l).35(1).17(3).48(1) 1.34 100 min. b.164(5).484.44(1).071(4.23(1).26(1).32(1).21(2).47(1) 1.16 400"C b.160(5).484.39(1).057(3).20(3).169(9).466(7).10(4).43(1) 1.28 1000 min a.158(5).484.40(1).052(3).20(2).25(1).35(1).09(1).56(1) 1.11 ~~~~~(b) I I o~~~~~~~~~4 (2) 2-line fits: ) P(R) e(L) e(R) A(L) A(R) 4000C b.180(5).461(5).065(3).106(3).461(6).539(2) 3.10 100 min a.165(5).482(6).076(4).156.407(7).593(5) 2.55 4000C b.175(5).437(5).069(2).116(3).548(5).452(2) 1.57 1000 min a.164(5).451(5).071(2).161(4).482(6).518(7) 2.55 b-= before; a = after irradiation (a)Data fit to the sum of two Gaussian-distributed Lorentzian functions plus a Lorentzian singlet constrained at.484 mm/sec representing y 0. (b)Data fit to the sum of two Gaussian-distributed Lorentzian functions (L = left; R = right). (c)For the 3-line fits: y line width reported as FWHM r, while y, and y2 widths are reported as the Gaussian broadening parameter e.0

150 (class III spectra) and then irradiated with fast neutrons. The results for two such samples have been listed in Table IV-15, while an example spectrum was illustrated in Figure IV-lc (Section IV-A). The spectra of the irradiated 6000C-samples show a definite increase in YL s area fraction and corresponding decrease in yo; i.e., an increase of iron in precipitates at the expense of the solid solution phase. A sizable fraction ( 60%) of YL's increase, however, is due to an increase in the concentration of iron in small clusters (from 12L), although YL is comprised primarily of Y-iron (YI). The effects of fast neutrons on the Mdssbauer spectra of aged Cu-Fe alloys are most reliably characterized by changes in the area fraction of-) for the reasons detailed in Section IV-C. Figures IV-17 and IV-18 show the changes in the area fraction, converted to Fe concentration (CL), of the left-most component ([L) as a function of pre-irradiation heat treatment (IV-17) and fast neutron exposure (IV-18). A decrease in CL signifies that Fe clusters or small precipitates have been broken-up by the irradiation. As seen from the figures, this occurs for most annealed samples, except those with very small clusters (e.g., the solution annealed and 3350,10 min.- aged samples) and those with very large precipitate particles (the 6000C- aged samples). In order to determine the effects of fast neutrons on a possible Y-c( phase transformation, two identical samples which had been coldrolled and aged at 6000C for 1000 min. containing o-iron precipitates were irradiated at fluences of 2.7(10)19 and 9(10)19 n/cm2. (A spectrum of an unirradiated sample showing the o-iron six line hf structure was shown in Fig. IV-7). The pre- and post-irradiation spectral parameters of the sample given the higher exposure are listed in Table IV-16. The

TABLE IV-15 MOSSBAUER SPECTRA PARAMETERS FOR Cu-0.6% Fe SAMPLES AGED AT 600~C (CLASS III SPECTRA) BEFORE AND AFTER IRRADIATION Position (mm/sec) Lorentzian Width (mm/sec) Relative Area Sample(a) L R L R L Sample ~ Y2L + Y1 Yo Y2 + Y2 + 1 Y Y2 Y2 x /N (1) Fast fluence qt -2.7(10)19 n/cm2: 6000C b.178(4).479(5).775(7).238(2).248(4).23(3).631(6).342(5).027(4) 1.05 100 min 100 i a.172(4).481(4).758(8).251(2).255(4).24(2).661(8).293(9).046(6) 0.96 (2) Fast fluence 1t -9(10)19 n/cm2: 6000C b.181(4).482(4).773(6).250(2).266(4).23(2).574(6).395(5).032(5) 1.17 10 min 10 n a.174(4).484(4).728(8).248(2).266(5).30(2).630(8).30(1).07(1) 1.20 b = before; a = after irradiation

152 7.0* Quenched and Cold-rolled 6.0 5.0" 0 Quenched 4. - *t2.7(10)1 n/cm2 3. ~ 2.0- 6000C Ag Tim V -CD 2850C — component ( i- to %eaafci o erdtn oio335~C -3.(>r a -4. )-5.0 10 11000 10,000 Aging Time (Minutes) Figure IV-17. —Irradiation-induced change in the \L spectral component (in atom % Fe) as a function of pre-irradiation condition for annealed Cu-O.6atFe.

153 7.06.(1 5.0- ~ Solution Treated 4.( / 6000C10 Min. 4 3.0 TI o 2.0- 600 CxI | O 100 Min. 0 " 1,0- - -3' 0 { \ \ 285040 - 3350C - 100 Mi n. -1.0- I -2. — 285c -'-3. 5000 Min. 400~-100 Min. ~-4.0+ \ \r 335 - 1000 Min. -5.0 t I ~ i. 1 I (10)18 (10) 19 (10) Integrated Fast Neutron Flux (n/cm2) Figure IV-18. —Irradiation-induced change in the YL spectral component (in atom % Fe) versus fast neutron fluence for solution treated and aged Cu-0.6at%Fe alloy samples.

TABLE IV-16 MOSSBAUER SPECTRA PARAMETERS BEFORE AND AFTER FAST NEUTRON EXPOSURE OF V9(10)19 n/cm2 FOR A Cu-0.6% Fe SAMPLE AGED AT 600~C AND COLD-ROLLED Position (mm/sec) Lorentzian Width (mm/sec) Relative Area i R I R I R Type of Iron + Yl Y +Y2 + 1 YR Y2 X2/N 600~C 1000 min y-iron b.178(4).484(5).75(1).244(2).231(5).24(3).413(8).303(8).041(6) 1.06 a.170(4).483(5).72(1).252(2).242(4).28(3).49(1).24(2).08(1) 1.03 Centroid A34 A25 A16 Aa a-iron b.26(8) 1.69(1) 6.142(7) 10.64(1).24(1) a.26(1) 1.70(2) 6.14(1) 10.632(9).19(1) b = before; a = after irradiation

155 results of the lower exposure are similar but less pronounced. Like the samples aged at 6000C which contained precipitates of only Y-iron, the spectra of the cold-rolled samples show an increase in the y-iron phase and a corresponding decrease in solid solution iron as a result of irradiation. In addition, there also occurs a significant decrease in the area fraction of the six d-iron lines. This may be interpreted as evidence of either an irradiation-induced phase transformation, or simply a partial dissolution of the o-iron precipitate phase. Finally, resistivity measurements were also performed on a few of the aged-irradiated samples. Because of the delicate nature of the thin foils (N~ mil in thickness) required for the Mbssbauer measurements, only a few came through the irradiations without significant damage (i.e., pin holes and tears). Results from these samples are given in Table IV-17 below. The calculated concentration changes of Fe solute atoms were obtained using Ae/^c = 8.2xQ2-cm/at%Fe from Fig. IV-13. Making allowance for the difference in Fe concentrations, the results of Table IV-17 are in general agreement with Botax's resistivity data of irradiated Cu-1.7wt%Fe. TABLE IV-17 IRRADIATION-INDUCED CHANGES IN RESISTIVITY OF AGED Cu-O.6at%Fe Sample Fluence A (qa -cm) AC(at%Fe) 335e-10 min. 2.7(10)19 -.126 -.016 3350-1000 min. 4(10)18 +.16 +.02 3350-1000 min. 2.7(10)19 +.32 +.04 4000-100-min. 2.7(10)19 +.196 +.024 400~-1000 min. 2.7(10)19 +.124 +.015 Quenched 2.7(10)19 -.44 -.054 Quenched 9(10)19 -.77 -.094

156 E. Discussion of the Irradiation Results 1. Irradiation-Induced Aging One of the goals of the present research was to determine the extent of Fe precipitation induced by fast neutron irradiation in the copper-rich Cu-Fe system. In the past irradiation aging has been demonstrated in both the Cu-Be-102)(144) and Cu-Fe(5' systems by bulk measurement techniques -- resistivity and hardness in the former and resistivity in the latter. In neither alloy system could any conclusions regarding the size of the precipitates formed be made, and only in Cu-Fe (Boltax5) was an attempt made to correlate resistivity with the quantity of iron precipitated. As shown in Section IV-C.2b, such a correlation is tenuous since anomalous resistivity effects are caused by the small Fe clusters that are formed by the irradiation. It is shown below that M5ssbauer spectroscopy allows an estimation of not only the number of Fe atoms clustered during the irradiation of solution treated Cu-Fe alloys, but the approximate cluster size as well. The MBssbauer results of the previous section clearly demonstrate the phenomenon of irradiation aging. As a result of the irradiations, the spectra of solution treated samples (Fig. IV-14) were converted to the Class I form characteristic of the incipient stages of clustering in Cu-rich Cu-Fe alloys. (Compare spectral parameter values of Table IV-10 with those of Table IV-4, especially the 285~-100 min. and 335~-10 min. aged samples). The very small negative shift in the centroid of 12R (or the small decrease in the Y2 doublet's splitting) as well as the relatively small changes which

157 occured in the Gaussian broadening parameters indicate that the Fe clusters formed during irradiation consist of only a very few atoms. The largest clusters are very likely those which already existed in the quenched sample prior to irradiation. It is tempting to attribute the difference in area fractions of the YL (Y1 + {2L) and 2R components to larger clusters (>13 atoms) which might have been heterogeneously nucleated in displacement spikes or cascades. However, all of the other spectral parameter values (i.e., centroid positons and Gaussian broadening parameters) are suggestive of the formation of very small clusters as was discussed in Section IV-C.2 with regard to thermal aging. Also the post-irradiation spectra were able to be fit satisfactorily with Window's(10) constrained two component G-L model* with only a slightly enlarged (2 value over that of the three component fits. Therefore, it is very unlikely that Fe clusters extending beyond one nearest neighbor shell (or 13 atoms) have been formed during the irradiations. In short, the Mdssbauer results in quenched-irradiated Cu-Fe are consistent with the formation of Fe clusters of a few atoms, and not full scale precipitate particles as suggested by Boltax.(5) In addition to providing an insight into the size of Fe clusters formed by the irradiation of quenched Cu-Fe, the Mbssbauer results also provide a very good estimation of the number of Fe atoms *A comparison of the two and three component G-L fitting results for a typical quenched-irradiated 0.6at% Fe alloy was made on p. 140. The parameter values derived from both of the fitting methods indicate that only a very small change in the average size of Fe clusters occured during the irradiation. *The average size of clusters existing in solution treated alloys is believed to be less than five atoms.

158 precipitated' or clustered from the solid solution phase. The table on p. 141 compares the irradiaton-induced concentration changes (at% Fe) determined from the resistivity results and the two Mbssbauer fitting models for a typical quenched sample. The values obtained from the constrained two component fits to the ME data (2 G-L in the table) are the most representative of the depletion of iron from solid solution. The electrical resistivity results definitely underestimate the amount of iron depleted from solid solution due to the'anomalous' scattering effects of the small clusters (Section IV-C.2b). The actual number of Fe atoms per unit volume involved in the clustering process for a given irradiation can be simply determined from NC = No aCM, where No is the atom density of copper and CCM is the irradiation-induced concentration change obtained from the ME spectral areas of the Yo phase before and after irradiation. The number of primary collisions per unit volume responsible for the clustered iron can be obtained from the time integrated form of Eq. II.27; i.e., Np No c5, where $ is the integrated fast neutron flux and es the energy averaged (over fission spectrum) scattering cross section. In the table below values representing the number of Fe atoms clustered as well as the number of primary collision events responsible for the clustering are summarized for typical solution treated 0.6at% Fe samples. Fluence (1) #/cm3 Fe atoms (2) #/cm3 of (n/cm2) clustered primary collisions (1)/(2) 4(10)18 4.4(10)19 1.0(10)18 ~44 2.7(10)19 8.6(10)19 6.7(10)18 v13 9(10)19 1.2(10)20 2.3(10)19 ~ 5

159 From the last column in the above table, it is seen that the effectiveness of primary knock-ons in causing clustering decreases with increasing fluence; e.g., at the lowest fluence approximately 44 atoms were clustered per primary knock-on event, while at the highest exposure only about 5 atoms per primary were clustered. This (5) phenomenon, also observed by Boltax (5from resistivity data, is the combined result of a gradual depletion of Fe atoms from solid solution and the competitive process of irradiation-induced dissolution. The latter effect is discussed in detail later in this section. The effect of the concentration of solute atoms (i.e., the degree of supersaturation) upon irradiation-induced clustering was illustrated in Figure IV-15. As seen from this figure a depletion of Fe solute atoms leads to a decreased rate of iron clustering. As discussed in Section II-D, irradiation aging occurs primarily by the enhancement of diffusion due to the introduction of excess vacancies into the lattice by fast neutron displacements. If a nominal value for -- the number of displacements per primary knock-on event -- of 300 is taken as the number of vacancies created by one primary event, then from about 3(10)20 to 7(10)21 vacancies were created during the course of the irradiations. Most of these were probably annealed-out by migrating to various lattice sinks. However, according to Hart's(142) theory (see p. 127) many of the irradiation created vacancies should be attracted to solute atoms and Fe clusters, and thereby retained in the lattice. If so, then short term annealing at a low aging temperature, say 335 C, should cause e clustering in excess of that occuring in a solution treated sample identically aged. To test this hypothesis one of the quenched

160 irradiated samples (Pt = 2.7x1019 n/cm2) was aged for 10 minutes at 335~C. The resulting spectrum (similar to Fig. IV-3b) showed a significant increase in the area fractions of YLand y2R as well as a greater decrease in the splitting of the Y2 doublet over that of the routinely aged sample. Since these are signs of a more advanced stage of aging, it is quite possible that vacancies in Cu-Fe are attracted to Fe solute atoms and/or Fe clusters. 2. Dissolution of Precipitates The observed effects of fast neutrons on the Mtssbauer spectra of aged alloys are caused by two competing irradiationinduced processes: enhanced diffusion leading to an increase in Fe clustering, and the dissolution of existing clusters within large displacement cascades (or spikes). The former process was discussed with regard to solution treated alloys. The latter mechanism, displacement spikes, was orginally proposed by Brinkman(96) and has since been studied by computer simulation(92)(99) and also observed experimentally.(5)(50) A summary of these studies and applicable theory was given in Section II-D. The results reported in Section D.2 of this chapter show that the irradiation-induced changes in the Mbssbauer spectra of aged alloys depend upon the stage of iron clustering (i.e., cluster size) in the pre-irradiated alloy.* Those samples containing very small *As was indicated for thermally aged alloy samples, the extent and type of information (qualitative or quantitative) extractable from the ME spectra is also dependent upon the stage of clustering (i.e., it is linked to the ability to resolve the spectral components from the three classes of spectra observed). Thus, a quantitative analysis of irradiation-induced phase changes can be

161 clusters (e.g., the 3350C-10 min. and 2850C-100 min. samples in Tables IV-11 and IV-12) showed a net increase in the number of clustered atoms after an exposure of 2.7(10)19 n/cm2, indicating that more Fe atoms were clustered than were dissolved by the irradiation. The small changes that occured in the positional and width (or Gaussian broadening broadening) parameters in the spectra of these samples indicate that the average size of the clusters following irradiation changed very little from the pre-irradiation value. The effects of an increasing fast neutron exposure on an aged alloy with very small clusters can be seen in Figure IV-18 (3350C-100 min. sample). At the lowest fast neutron exposure, the relative area of YL (i.e., the concentration of Fe cluster atoms producing /L) increases, whereas at higher exposures a net decrease in )L has occured. These results indicate that early in the irradiation, the formation and growth rate of the clusters due to vacancy migration exceeds the rate of dissolution of existing clusters by the displacement cascades. When clusters reach a critical size the rate of dissolution exceeds the rate of cluster formation and growth, and a net decrease in the cluster phase occurs. According to the discussion of Section IV-C.2c, this critical cluster size is probably less than 13 atoms. performed satisfactorily in well-aged (600 C) samples and in those samples containing Fe clusters of less than about 13 atoms (i.e., solution treated and mildly aged samples). Irradiation-induced changes in Cu-Fe samples in an intermediate stage of clustering (all class II and advanced class I spectra) can only be followed in a semi-quantitative manner through changes in the area fraction of YL, the left-most component produced by both surface and volume Fe atoms in clusters.

162 The effects of irradiation on aged alloys with slightly larger clusters (but still exhibiting class I spectra) were illustrated in Figure IV-16 for the 335~C-1000 min. sample. (The associated 3-line spectral parameters were listed in Table IV-13 for all three fast neutron exposures while Table IV-12 gives the two G-L component parameters for the intermediate exposure.) These results clearly demonstrate the retrogression effects of displacement cascades or spikes on aged alloys containing small coherent clusters. (In this case the cluster size was estimated to be anywhere from about one to at most three near neighbor shells in radius; i.e., about 13 to 43 Fe atoms -- see table on p. 128). Depending upon the fitting model used, two slightly different interpretations can be made of the data. In using the two component G-L fit (yo + Y2), it is assumed that both the pre- and post-irradiated samples contain clusters of 12 atoms or smaller. With this in mind, the spectral parameters of Table IV-12 for the 3350-1000 min. sample show a definite decrease in the number of Fe cluster atoms (from the decrease in Y2's relative area) as well as a decrease in the average cluster size (from the increase in Y2's splitting and the slight positive shift in its centroid) as a result of the irradiation. The three component G-L fits (Table IV-13) possibly provide a better insight into the dissolution process, assuming the clusters to be larger than 12 atoms in the aged sample. The results of Table IV-13 show a monotonic decrease in the relative area of L and a R concurrent increase in that of R with increasing fast neutron exposure.* This could be interpreted as evidence that the dissolution or break-up of larger clusters in this alloy is incomplete; i.e.,

163 that all of the atoms in a cluster enveloped by a displacement spike do not go into solid solution but form smaller clusters in the wake of the displacement cascade. It is very possible that'thermal' spikes(95) associated with the displacements would cause localized annealing or aging of the supersaturated Fe solute atoms dispersed in the damage cascade. The large positive shift of 2R's centroid indicating a decrease in the average cluster size is consistent with this mechanism, although not necessarily conclusive proof of it. In any event, at some stage of aging only a partial break-up of clusters by displacement spikes would be expected. This is definitely the case with the Cu-0.6at%Fe samples that were aged for 100 minutes at 400 C prior to irradiation (Table IV-14). The pre- and post-irradiation spectra of these samples are of the class II type, characteristic of the early and intermediate stages of -precipitate (large cluster) growth. (Recall from Section IV-C that the left component is comprised mainly of the Y1- or Y-iron singlet while the very broad right component consists of some yo-iron and an unresolved doublet due to Y2-surface iron. The three line Gaussian-distributed Lorentzian fits were an attempt at unfolding this complex structure.) The large decrease in the relative area *The decrease shown for the Y component in the 3350-1000 min. sample could very well be due to the Inadequacy of the fitting model and not an actual decrease in Fe solute atoms. The large increase in p (Table IV-17) measured for the sample exposed to a fluence of 2.7(10)19 n/cm2 indicates that an increase in the number of solute atoms has occured; however, a sizable fraction of this resistivity increase is very likely caused by small residual clusters from the dissolution process.

164 of YL (see Table IV-14 and Figures IV-17 and IV-18) signifies the breaking-up of )-precipitates or large iron clusters by the displacement spikes. The increase in broadening of the right component as well as its positive positional shift could be due to a combination of an increase in the yo phase and a decrease in the average cluster size. Comparison of the resistivity changes (Table IV-17) observed for the 335~-1000 min. sample (AQ= +.322) and the 400~-100 min. sample (Ze= +.196) after irradiation indicates that the'dissolved' clusters in the former are more highly dispersed (i.e., they have a greater effect on the resistivity). Thus, it can be concluded that the broadening and shift in the right component of the 4000-aged samples is primarily caused by the incomplete break-up of iron clusters in the displacement spikes and the resulting smaller average cluster size. In Section IV-C.2 it was concluded that the clusters formed in the 0.6% Fe samples aged at 4000C for more than 100 minutes were of the order of 6 i or greater in radius. The displacement spikes or damage cascades would in turn have to be considerably larger than this t6 produce the observed effects. (Boltax(5) and Piercy(50) estimated the spike size to be about 35 i in radius).* Figure IV-17 shows that aging at 400~C for times of v10,000 minutes causes a decrease in the observed dissolution effects as a result of the increased size of the o precipitates. (Recall that a cluster size of about 20 A was roughly *Boltax's size value, although reasonable, was based on a faulty calculation of precipitate particle size as well as an estimation of irradiation-induced concentration changes in the o phase from rather misleading resistivity data.

165 estimated from the Mbssbauer data for this sample.) Further aging at 400 C, or an increase in annealing temperature would eventually produce clusters too large to be significantly effected by the damage cascades. This occured for all of the samples aged at 6000C (see Table IV-15 and Figure IV-17). The Mbssbauer results -- which for the class III spectra can be interpreted quantitatively -- indicate a significant decrease in Fe solute atoms and a slight increase or growth of the large Y-iron precipitates. (Some 60-70% of the increase in the YL component is due to Y -iron.) The increase in Y2's area 2 fraction and the negative shift in the Y2' centroid indicate that the small clusters observed in the 600 -aged alloys (see p.118) have grown slightly during the irradiation. The extent of irradiation-induced dissolution in the aged Cu-0.6at%Fe samples (i.e., the approximate number of iron atoms'dissolved' during the course of the irradiation) can be approximated from the changes in the spectral area of %L. (The results should be good to within a factor of about two for most of- the aged samples.) The problem is complicated because of the competing effects of irradiation-induced aging and dissolution on the concentration of Fe in clusters or precipitates. Assuming that the irradiation aging of the supersaturated solid solution can be treated independently from the re-solution of clusters, then the desired result can be obtained with the aid of Figures IV-15 and IV-17 in the following manner. The total concentration (atom %) of Fe atoms in an aged sample'dissolved' by displacement spikes can be approximated by subtracting the aging contribution to L — i.e., the change in YL expected

166 for a solution treated alloy with the same initial solute ( o) concentration as the aged sample -- from the appropriate LL values of Figure IV-17. The results are shown in Figure IV-19 as dashed lines along with the uncorrected data (solid lines). The total number of cluster atoms per unit volume'dissolved' in displacement spikes is then obtained by multiplying the concentration values of the figure by the atom density of copper (v8.5x1022). Taking the 33501000 min. sample as an example, the approximate number of Fe cluster atoms broken-up by a fast neutron exposure of 2.7(10)19 n/cm2 is about 3.7(10)19 atoms/cm3; or approximately 5 atoms'dissolved' per primary knock-on event. At the lowest exposure this value is about 30 atoms per primary event, whereas at the highest exposure it decreases to only 3 atoms per primary knock-on. This decrease in the re-solution effectiveness with increasing fast neutron exposure is most likely the result of saturation effects caused by the overlapping of the damage cascade volumes. A secondary objective of this research was to ascertain whether fast neutron irradiation could induce a Y-c transformation in well aged Cu-Fe alloys. Although not conclusive proof to the contrary, there was no evidence of the o(-iron six line structure in the spectra of a 0.6at% sample aged at 600 -1000 min. and exposed to a fluence of 2.7(10)19 n/cm2. Similar samples which had been coldrolled either before or after aging showed definite signs of dc-iron formation in their ME spectra (Fig. IV-7). Upon irradiating two of these cold-rolled samples to exposures of 2.7(10)19 and 9(10)19 n/cm2, a decrease of significant magnitude in the area fraction of o(-iron occured (Table IV-16) —an opposite effect to what was previously

167 3.0 2.0 1.010 \C) ^^^ 285~C X -10 -1.0 L3 3350C \ 3350C o 0 - 2850C \ 400 C -6.0 - -8.0-9.0 -6.0 Corrected: Dashed lines & open symbols Uncorrected: Solid lines & symbols -8.0 -9.0 ~ 10 100 1000 10,000 Aging Time (Minutes) Figure IV-19. —Net Re-solution Effects of Displacement Spikes as a Function of Pre-irradiation Heat Treatment.

168 reported.(108) Evidently Fe atoms existing in c —iron platlets within large y-iron precipitates were converted by the displacement spikes to the - form. 3. Summary The effects of fast neutron irradiation on solution treated and aged Cu-0.6at%Fe samples has been followed with some success by Mbssbauer spectroscopy. In general, the results have confirmed the conclusions of similar studies using resistivity(5) and magnetization(50) measurement techniques; i.e., that high fast neutron exposure causes both aging and re-solution effects in age-hardening alloys. In addition, the M6ssbauer technique provided an insight into the local environment of the precipitate or cluster phase in both pre- and post-irradiated alloy samples that was not possible with the other techniques. Specifically, in the case of solution treated alloys, the Mbssbauer spectra provided information concerning both the extent of irradiation-induced aging and the approximate size of the clusters formed. The results are consistent with the formation of very small clusters of less than 12 atoms in the quenched-irradiated alloys. The re-solution effects of displacement cascades caused by fast neutron interactions in aged Cu-O.6at%Fe samples was shown to be extremely dependent upon the stage of aging or the cluster size. Aged samples containing very small clusters (<12 atoms) experienced essentially complete dissolution of Fe clusters, whereas those with clusters of intermediate sizes (up to about 20 R in radius) experienced incomplete cluster break-up in the displacement cascades.

169 In contrast, well-aged samples containing large Y-iron precipitates (>50 A) did not experience any dissolution effects, but instead exhibited a very slight growth in the Y-iron phase. Finally, the Mbssbauer measurements were also able to determine the effects of irradiation on the equilibrium precipitate phase, o(-iron.

APPENDIX A SPECTRAL AREA CONSIDERATIONS The results of Chapter IV indicate that the spectral components for FCC iron'phases' in copper are not well resolved. Therefore, before applying the ideas of Section II-A.2 it is necessary to consider saturation and overlap effects in Equation (II.10) for the Cu-Fe alloy absorbers used in this study. Since there are many similar Fe configurations which produce a given absorption line, it is assumed that k.(E) in Equation (II.10) can be represented by a Lorentzian distribution with a broadened width. The results of this work indicate that Gaussian distributed Lorentzian functions provide better fits to the data; however, a broadened Lorentzian distribution is used for k.(E) in the following argument because of its mathematical simplicity. As given by Equation (II.10) the total area under an absorption spectrum is 0o A = fsJ dE[l-e T iki(E)] (A.1) where the sum over i includes any split components; e.g., the quadrupole split doublet designated r2. Since the f value for small FCC iron clusters is not known, it is assumed that the f'is for FCC iron configurations are equal to the value for isolated Fe in copper (f = 0.73). The effective thickness Ti for the ith phase can then be approximated as 170

171 T. A. T (A.2) where T is the total effective absorber thickness and Ai is the experimental area fraction for the ith spectral component (i.e. Ai = Ai/A). For the majority of the Cu-Fe absorbers used in this work, 0.6 < Ta 10. The maximum absolute value of the argument in the exponential function of Equation (A.1) will be less than Ta since the Cu-Fe spectra contain at least three partially separated components. The effective thickness per component in most annealed samples is < 0.4. Making allowances for the overlapping of the absorption lines, the following upper limit can be established: N Max[ITik (E)] < 0.5 i To obtain an expression for the error associated with neglecting overlap effects in our Cu-Fe absorbers consider the expanded form of Equation (A.1), A = fs dE{- [I 1k T kki (E)] (A.3) where the product is over N absorption lines. Keeping only terms of second order or less in Ti, Equation (A.3) can be approximated to better than 3% by f k N NN A fs [YTiki(E)- 2 TTjkJk i(E)kj(E)]dE (A.4) -ooThe integral of the first term of Equation (A.4) is given by1 The integral of the first term of Equation (A.4) is given by

172 N I = rf Ti i (A.5) 1 which is just the thin absorber result for well separated lines. Integration of the second term can be accomplished in the complex plane using the Theory of Residues. The integral can be written as f N O II = - i TT ki (E)kj (E)dE -00 or -'f N~2 2 f, N O a.2a. — II - 1 T T ~ ~ 3 ~~J dE I - j i'j [(E-E)2+ai][(E-Ej)2+a2] where a- = r /2. The path of integration is taken in the upper half of the complex plane about the poles at (E.+ia.) and (Ej+iaj). The result is fN 22 1 II = - T T{2iai a j [2iai[Ei-Ea+i(a +ai)][EJ-EE+i(a-a)] + 1 2ia.[Ej-E.+i(a.+a.)][Ej-E.-i(a.-a.)] ] ci i i Jci J N rr. II = - Tfs TiTji 2i 2 (A.6) Ai.j +(ri r /4 2 = - fs iTi; for j = i where Aij = (Ei-Ej).

173 Combining Equation (A.5) and (A.6) gives the desired approximation for the total spectral area, N T N r.r. Ti 1 N A jfs[riTi(l- 4)- - (r+j)TT 2 2j ] (A.7) 1 I j3i A i +(ri+rj) /4 The first term above represents the second order approximation to the total spectral area for N non-overlapping lines, while the second term accounts for overlapping of the ith and jth absorption lines. The fractional error in the total area cuased by ignoring the overlap terms can thus be approximated by N N r.r..8 Iv1 TiT (r +rj 2+(r +r)2 i j^ 1 i i rr) /4 (A.8) A/(frfs/2) Similarly, the fractional error in the ith spectral component's area is approximately Ti.N r+r 8 VT (r r.) A2+ 2 8i i i (A.9) 173 i(rir'/4 A/(rfs/2) (A The errors associated with neglecting overlap effects were estimated for several Cu-Fe samples which were representative of the various heat treatments. This was accomplished by using the experimental values obtained for the linewidth ri (for the Gaussian-distributed Lorentzian fits this was taken as the FWHM of the convolution —see Chapter III, Section D) and the spacings A..j in the above equations. In general, the spectra contained three partially resolvable components

174 (e.g., see Figure IV-l), although the lower energy (left) spectral line is most likely a superposition of two individual lines. In determining the "overlap" errors, it was treated as a single component. Except for the samples annealed at 400~C, these errors amounted 4% or less. In the case of those samples annealed at 400~C for periods of 100 minutes or greater, the "overlap" errors approach 10% because of the extremely poor spectral component resolution. In all cases these errors are in the same direction, i.e., neglecting c causes an underestimation of the atom concentration. As a result, "overlap" errors tend to cancel when differences in identical spectral components are taken to determine phase changes; e.g. before and following irradiation.

APPENDIX B POINT CHARGE CALCULATIONS OF QUADRUPOLE SPLITTINGS IN Cu-Fe According to Window(8'5) the screening charge distortion that occurs when two or more Fe atoms cluster in a Cu or Au lattice produces an electric field gradient (EFG) that gives rise to a resolved doublet in their Mossbauer spectra. To estimate the EFG caused by Fe clustering in Cu, we assume that the screening charge distortion can be approximated by an effective point charge q located along the lines joining adjacent Fe atoms. Only nearest-neighbors are considered in the calculation, since the EFG varies as 1/r3 and the Fe screening charge does not extend appreciably past the nearest-neighbor shell. For a distribution of n equal point charges q about a test nucleus, the ijh element of the EFG is given by(26) n 2 -5 V - l q(3xkikj-rkij)rk5 (B.1) k=l where rk is the distance between the test point and the kth charge, and (XklXk2,Xk3) is the position of the kth charge. As discussed in Chapter II (see Equation 11.18), an electric field gradient vE gives rise to a quadrupole splitting in Fe57 spectra of eQV 2 1/2 AE = ~(1-) (B.2) 175

176 where n is given by Equation (11.19). The principal value Vzz and the asymmetry parameter r are determined by diagonalizing the EFG whose components are obtained from Equation (B.1). From Equation (B.2) the splitting observed when an Fe57 nucleus in Cu has one Fe nearest-neighbor atom is AE1 = Qlql, where Q1 = 3 eQ/<r > and ql is the effective charge. As the number of Fe nearestneighbors increases, the effective charge localized along an Fe-Fe direction would be expected to decrease. The seven screening electrons of the test atom would be expected to be distributed according to the density of available d states on the twelve nearest neighbors. Assuming the charge is distributed uniformly among the available states and assuming the effective charge qn can be approximated by the difference between the charge localized on each of the n Fe-Fe axes and that localized on each of the (12-n) Fe-Cu axes, the effective charge on each of n Fe neighbor axes is given by II+oa qn (12-n) + n ql (B) where a is the ratio of the densities of states (Fe to Cu) and ql is the effective charge for one Fe neighbor. This is the same model as used by Window to fit Au-Fe spectra.(15'l6) For his Au-Fe data, a = 17 was found to give the most consistent fits to spectra for alloys with Fe concentrations in the range 2-31%. This value is about four times that expected from calculations, which is not too surprising since the radial distribution of the screening charge and the effects of core polarization through antishielding were not accounted for in Window's model.

177 Recognizing its deficiencies, it is still instructive to apply the point charge model to estimate the quadrupole splittings that occur when small Fe clusters form in Cu. This has been done in Table B-1 for Fe57 with up to five Fe nearest-neighbors. Since the effects of Fe in Cu and Au are similar, the value a = 17 obtained by Window(16) for Au-Fe alloys is used here to calculate the effective charge from Equation (B.3). The EFG for a given near-neighbor configuration is calculated from Equation (B.1) and is expressed in terms of an effective charge qn, which is determined from Equation (B.3). The quadrupole splitting for a given configuration is determined from Equation (B.2) and is expressed in units of Q1 —see the third column in Table B-1. The fifth column of the Table lists the ratios of the quadrupole splittings resulting from n Fe nearest-neighbors to that for a single Fe neighbor. The values given in parentheses in the final two columns were obtained using the value a = 5 in Equation (B.3), which is more representative of (15) the density of states ratio.(15) On the basis of this simplified point charge model, the results of Table B-1 indicate that the cluster doublet for Fe in Cu should remain fairly well resolved for up to 5 Fe nearest-neighbors. It is interesting to compare the quadrupole splitting observed for the doublet in quenched Cu-Fe alloys (X.58 mm/sec —see Chapter IV and References 7 and 8) with that for a high spin ferrous compound. The splitting of the Fe57 excited state expected for a single d electron in a pure orbital state is about 3 mm/sec.(127) Assuming that the antishielding factors and covalency effects are about the same in both cases, a charge asymmetry of about 0.17e in the 3d state for an Fe pair in Cu is indicated.

178 TABLE B-1 ESTIMATED QUADRUPOLE SPLITTINGS FOR Fe CLUSTERS IN COPPER # Nearest Neighbors Configuration* AEn/Ql qn/ql AEn/AE1 (1) (1) 1 l-z axis ql1) 1 1 1 2 2-xy q2 (0.8) (0.8) 22-xy -q2 0.64 0.64 2 ~~~2-z axis ~(0.8) (1.6) 2 | 2 z axis 2q2 2q2 0.64 1.28 (0.67) (1.1) 3 | 3-xy 1.7q3 0.47 0.82 (0.67) (0.8) 3 2-xy, l-xz 1.2q3 0 47 4~ ~ 2y30.47 0.574x (0.57) (1.1) 4 4-xy 2q4 0.37 0.74 (0.57) (0.97) 4 3-xy, 1-xz, 1.7q4 0.37 0.64 (0.50 (1.0) 5 4-xy, l-xz l 2qQ Q(6'4-xy, l-xz 2q5 0.30 0.60 *Indicates the number of nearest neighbors in a given plane or on a given axis; e.g., 3-xy, l-xz indicates three nearest neighbors are in the xy plane of the test nucleus, while one Fe nearest-neighbor atom is located in the xz plane.

APPENDIX C MOSSBAUER SPECTROMETER DETAILS A. Drive System Design Considerations As discussed in Section B of Chapter III, modification of the original drive system was necessitated by the desire to (a) decrease equipment broadening of the resonant lines, (b) improve the velocity stability of the spectrometer, and (c) reduce the frequency of repairs and tuning operations required to keep the drive system on-line. To accomplish these objectives, an ultrastable function generator external to the MCA's time base system and a drive amplifier containing recent solid state components were designed and constructed. The function generator-drive amplifier combination was designed to provide a synthesis of square, triangular and parabolic waveforms which approximates the voltage signal required to give the drive transducer a linear velocity response. Details are given below. 1. The Drive Signal Although design details may differ between the many electro-mechanical Mossbauer drive systems presently in use, they all possess one feature in common —negative feedback. Basically, these systems employ a velocity transducer coupled to the drive's moving member which produces a voltage proportional to the instantaneous velocity. A reference signal of the desired form is derived from an external function generator111 or from the multichannel analyzer-time 179

180 base system(37109) and is compared with the signal from the velocity pick-up transducer. The resulting difference voltage, called the error signal, is fed to the driving transducer, thereby controlling the system's motion. Usually, such systems employ a single feedback loop and control the drive's motion with only the error signal. This method requires a relatively high loop gain which, in turn, leads to poor frequency stability. An improved approach to the problem is to derive a signal which approximates the exact waveform needed to give the desired velocity response (in this case, triangular) and employ negative feedback only to correct for small deviations from the desired motion. This concept has been used by M. B. Stearns* and was adopted by Ullrich(34) in the design of the Nuclear Engineering Department's original ME spectrometer. A similar approach has been described by Kankeleit.(1l2) In order to construct the required drive signal, it was necessary to determine the electro-magnetic and mechanical characteristics of the drive transducer. The transducer used consists of a drive coil rigidly coupled to the pick-up coil through a moving shaft whose motion is constrained at both ends by two linear leaf springs. Both coils move in their own permanent magnetic fields, which (a) for the drive coil produces a driving force proportional to its current; i.e., F(t) = cI(t) and (b) for the pick-up coil produces a voltage proportional to its velocity. The equation of motion for the drive is *Ford Scientific Laboratory, Ford Motor Company, Dearborn, Michigan.

181 md2 + Bx + kx = cI(t), (C.) + dt where k is the spring constant, 3 the damping factor, m the mass of the moving member (includes the source, both coils and the drive shaft), and I(t) the current supplied to the drive coil. The drive coil also possesses a resistance R and an inductance L. In addition, its motion in a constant magnetic field produces a back e.m.f. (Lenz's law) of dx cd, where the constant c is the same as that appearing in Equation (C.1). Thus, the voltage across the drive coil is given by V(t) = RI(t)+LdI(t) (Cdx Eliminating the current I(t) from Equations (C.1) and (C.2) gives the desired relation between the signal voltage and the drive's motion. (t) Lm d3x + (L )+Rm d2x Cdt3 c dt (Lk+R )+ d Rk (C.3) c dt c The drive coil's force constant c was given by the transducer's manufacturer as 28 Nt./amp. All of the other transducer characteristics were either measured directly or determined experimentally.* The values obtained are as follows: resistance, R = lOOa; inductance, L: 1.5 mH; mass, m =.306 kg; spring constant, k: 3.4(10)3 Nt./m; and damping *The spring constant k and damping factor A were obtained from the response of the drive to an initial displacement of the drive shaft as determined from the pick-up transducer's voltage.

182 factor, s -.74 kg/sec. The system's natural frequency is then f k1= L - 17 Hz o 2 V m Because of the coil's relatively small inductance value, terms in Equation (C.3) containing L can be ignored without causing significant error in the design. Thus, the drive system's response to the voltage signal V(t) can be written as Rm d2x (R dx Rk V(t)- c 2 + c) + x. (C.4) dt Since the desired velocity, dx, is to be triangular in shape, Equation (C.4) requires the driving voltage to be the sum of square, triangular and parabolic waveforms. One method of determining the relative amplitudes of these waveforms as a function of frequency and maximum velocity is to expand the parabolic displacement x in a Fourier series and then substitute the expansion into Equation (C.4); i.e., x(t) = xmP(t) = 32xsin(2n+l)t m 7 2 n (2n+l) V(t) -Rm 2 32 sin(2n+l)wt - ^ 32 cos(2 n+n)rt +( +c)wx(321 cos(2n t c +c m n (2n+1)2 + Rk 32) sin(2n+l)wt (C.5) c a n (2n+l)

183 4 Noting that the maximum velocity is vm =- xm or 8fxm, and that the terms T(t) 8 cos(2n+l)wt TM 2 1 2 T n (2n+1) and S(t) = 4 sin(2n+l)t - 2n+l are the Fourier expansions of the triangular and square waveforms, Equation (C.5) can be written as 4Rm fS(t)+(-Rk VV(t) - fS)+( +)vT(t) + p (tm (C.6) C ml c c f (t). In this expression the functions S(t), T(t) and P(t) represent the square, triangular and parabolic waveforms with unit amplitudes. Their coefficients given in Equation (C.6) relate the respective amplitudes of the individual waveforms to the transducer's characteristics. The amplitude coefficients are: 4Rm Rk Vm A-s v f A(@ +c)v A (C.7) s c m t c m p c f t m' p (Cf 7) The ratios A/vm are plotted against the drive frequency in Figure C-1 along with their measured values obtained at two frequencies. In the design of the drive system, Figure C-1 was used to determine the proper resistive feedback ratios at the summing amplifier for the three waveforms.

184 2. The Drive System Figures C-2 and C-3 show the schematic diagrams of the function generator and drive amplifier, respectively. Two of the three required waveforms are provided by the function generator (Figure C-2), while the third, the parabolic wave, is obtained by integration of the triangular wave at OA-5* (operational amplifier #5, Figure C-3). The synthesis of the three waveforms occurs at the summing junction of the drive amplifier, just prior to the driver or booster amplifier stage. The relative amplitudes of the triangular and parabolic waves are determined by the variable resistors Rt and R in the drive amplifier, while the square wave's amplitude is controlled by the variable feedback resistor Rs in the function generator's output circuit (OA-4). Negative feedback is provided by the output of OA-7 in which the triangular reference signal is compared to the amplified (x50) velocity pick-up signal. The relative weight of the resulting error or correction signal is controlled by the variable resistor Rc. The function generator of Figure C-2 is a modified version of Cohen's design (3) and produces precise, ultra-stable triangular and square waves. (The triangular wave's linearity and stability are better than.04%, while the square wave's halves are equal to within.01%). This is accomplished by the operational amplifiers OA-l and OA-2, which act as an integrator and comparator, respectively. Because of positive feedback, any positive input voltage tends to drive OA-2 to saturation. Its output, however, is clipped at about +9v by the Zener *All operational amplifiers were obtained from Philbrick/Nexus Research of Dedham, Massachusetts. Their + 15v DC bias supply was provided by two Sorensen Model QSA 18-1.1 power supplies.

185 diodes, Z1 and Z2, and the resulting constant 9v signal is integrated (and inverted) by OA-1 to produce a negative ramp function. When the comparator's input is driven negative by OA-l's output, the output of OA-2 reverses polarity and is clipped at a -9v by the Zener diodes. Thus, the output of the comparator (OA-2) is a square wave, and the output of the integrator (OA-1) is a triangular wave. The signal frequency is determined by the product of C and (Rfl+Rf2) and can be varied by changing the setting of Rf2. Both the triangular and square waves are fed to the drive amplifier through two high stability, low drift operational amplifiers, OA-3 and OA-4. The ratio of the variable feedback resistor Rt to the 11 kQ input resistor of OA-3 determines the amplitude of the triangular reference signal, and thus the maximum velocity of the source's motion. The trigger generator provides a synchronizing pulse a few microseconds long once per cycle to initiate the MCS sweep. Since the operational amplifiers are DC coupled in the drive system, care was taken to minimize DC drifts. All of the operational amplifiers at the output of the function generator and in the signal shaping and input stages of the drive amplifier were chosen for their high precision, high stability and low drift characteristics. Initially, it was believed that chopper-stabilized amplifiers would be required at the inputs to the drive amplifier to eliminate DC voltage drifts. However, the use of recent low drift operational amplifiers in these key locations gave satisfactory results. Shifts in the zero velocity channel positions were within.015 mm/sec per month, while trimming to eliminate the DC offset drifts had to be performed only once every one to two months.

186 3.0 0 0 u\ > \ _~ \\~ A /v p m in \P'\ 1.0- \ A s /Vm AT/Vm 4 8 12 16 20 24 Drive frequency (cps) Figure C-1. Waveform amplitudes required to give drive transducer a triangular velocity response versus frequency.

Rfl Rf2 200K 50K 100K R 0.l f Is 5K ~~1 _20 3.OOK 2N55 1 K 100K A A Wave Output -lN3155A Z2 ~ 1~200pf 10K Trigger OAl,OA2........P/N Q102A =r P Output OA3,OA4../..... P/N SQ1OA *All operational amplifiers from Philbrick/Nexux Research __ _ 2N706 of Dedham, Mass. \ lOOK slN100O Figure C-2. Triangular and Square Wave Generator. 11fis. -15v

Square Wave ~Input Rs Rf 49.9K 24. 99K I 5M A~I~ \ ~ ^ ~1 "Parabolic + A B Triangular Wave" "Booster Amplifier" Drive Wave Input 100K Rp Coil 1100^ f _ "Triangular | Operational Amplifier Key* Wave" OA5............ P/N Q102A 249K RT OA6,OA7,OA9....P/N 1020 ~-lS(C6> Iv -A OA8............P/N 1008 - 64K 50K Bl.......B.... P/N 2001 ---— _"~~~ ~~*All operational amplifiers from Philbrick/Nexus Research of i.0011 f Dedham, Mass. 200K "Correction Signal1" Figure C-3. Mbssbauer Drive Amplifier IOOK RC 10K, 10T 100 kH f.5M 100lK 200K 10K Transducer Pickup _-?~ _ _^ ~ ~V^V^~ Input A9

189 B. Commercial Components Used in Mossbauer Spectrometer System Pickup Calibrator/Exciter Model: C31, MB Electronics, New Haven, Connecticut. Multichannel Analyzer Model: ND-180FM, Nuclear Data, Inc., Palatine, Illinois. Time Base Generator Model: ND-180ITB, Nuclear Data, Inc., Palatine, Illinois. Oscilloscope Type 503, Tektronix, Inc., Beaverton, Oregon. Teletype Model: 33TC, Teletype Corporation, Skokie, Illinois. Kr-C02 Proportional Counter Model: RSG-61-M1, Reuther-Stokes Electronic Components, Inc., Cleveland, Ohio. High Voltage Supply Model: HV-1544, Power Designs Pacific, Inc., Palo Alto, California. Preamplifier Model: 109PC, Ortec, Incorporated, Oak Ridge, Tennessee. Linear Amplifier Model 451, Ortec, Incorporated, Oak Ridge, Tennessee. Single Channel Analyzer Model: NC-15, Hamner Electronics Corporation, Princeton, New Jersey.

APPENDIX D DATA ANALYSIS DETAILS A. Description of Programs As summarized in Chapter III, the experimental Mossbauer data are fit to a sum of distribution functions by a non-linear least squares fitting program similar to other published programs.(37'120) The method used for finding the set of parameters which minimizes chi(119) square is the Gauss-Newton iteration method.^ The resonant lines are fit to a combination of Lorentzian functions or Gaussian-distributed Lorentzians, while the non-resonant background is fit to a parabola. These functions, as well as theirpartial derivatives with respect to the fitting parameters, are supplied to the main program, which performs the least squares analysis, by special sub-programs named THEORY. The partial derivatives are used to set up the "normal equations" whose solutions are the parameter increments for the next iteration. The functions performed by the main program and its supportive subroutines include the following: the main program reads in the control and fitting parameters, sets up the normal equations, determines convergence, calculates and prints out the fitting parameters and their standard deviations, and also provides the input data for the Calcomp plotting program. The subroutine DREAD reads the input data (usually from a file where the MCA paper-tape output is stored) and makes background corrections if required. At present a maximum of 25 parameters 190

191 may be fitted to a spectrum consisting of 512 data points. The normal equations are solved by a linear systems subprogram called DPINVS. The inverted coefficient matrix of the normal equations, called the error matrix, is also supplied by DPINVS. In addition to THEORY, there exists a unique subprogram TAREA for each function being fit. TAREA calculates the area of each line or pair of lines and its standard deviation, and also determines the ratios of these areas and the ratios' errors. The printed output includes the fitting parameters and the normalized x2 after each iteration; the final parameter values and their standard deviations; the total and individual resonant line areas; the distribution in units of standard deviation of errors between the measured and fitted spectrum; and an area ratio matrix containing the ratios of individual line areas as off-diagonal elements and individual to total areas along the diagonal. As an option, the error correlation matrix can also be provided. The Calcomp program plots the normalized experimental data and fitted function on the same graph as well as a distribution of their differences on a separate graph. The main data analysis program, its subroutines and the Calcomp plotting program are listed in Section C of this Appendix. All programs were written in the Fortran IV language. Only the two THEORY subprograms (plus TAREA) used most often in fitting the data are listed. Other THEORY subprograms used to provide special constraints on the fitting parameters involved only minor modifications of these two general subprograms, and therefore, are not listed.

192 B. The Fitting of Gaussian-Lorentzian Convolution Functions Gaussian-distributed Lorentzians are used to account for the small variations in isomer shifts and quadrupoles splittings that occur when the Fe5 nuclei experience a distribution of local Fe configurations. For solid solution Fe these configurations involve further than nearest neighbors, while for Fe in clusters both nearest and distant neighbors are involved. The fitting of the G-L convolution function by the Gauss-Newton iteration scheme requires that its value and the value of its partial derivatives be calculated for each velocity or channel number. Unlike the Lorentzian function, the distribution function of Equation (111.4) can not be obtained in closed form. The function and its derivatives must be calculated numerically for each data point. How this has been accomplished is described below. The first step is to recognize that the G-L convolution dj(v), in Equation (111.4) can be expressed in terms of the real part of the complex error function;(24) i.e., the real part of w(z) = ()j Z-SL s (D.1) Z-S _00 or, its more common form o00 w(z) = e-z 1 + 2 et dt (D.2) where z = x+iy. To accomplish this consider the following variable transformations in Equation (111.4): v'-v. V-V s =, x J 2 =. (D.3) 3 J 3

193 Making these transformations, the G-L convolution becomes d.(v) = - |~ 2~Is. (D.4) dj(v): ~/-ej _(xj-s) 2+y To see the relationship between dj(v) and Re[w(z)], Equation (D.1) can be rewritten as w. = 2 2 w(z)= s+ i,A s s (D.5) T (x-s) 2+y J (x-s) 2+s -00 o00 or, w(z) = R(x,y) + iI(x,y) (D.6) where R(x,y) and I(x,y) are the real and imaginary parts of w(z). Comparison of Equations (D.4), (D.5) and (D.6) then leads to A. dj(v) -- (xjy) (D.7) which is the desired result. Thus, the required resonant line function, dj(v), is directly proportional to the real part of the complex error function which can readily be evaluated by existing numerical techniques. For this purpose, the subroutine COMER* is used. COMER evaluates both the real and imaginary parts of w(z) for arguments z = x+iy in the first quadrant of the complex plane. Using the symmetry relations R(x,y) = R(-x,y) and I(-x,y) = -I(x,y), the functions can be evaluated in all four *Obtained from the Computing Technology Center, Union Carbide Corporation, Oak Ridge, Tennessee.

194 quadrants. The recursive procedure used to calculate w(z) in COMER is described by Gautschi.(125) In addition to the function d.(v), the partial derivatives — ad. ad. dd. " ___J j Iand J — are also required in the fitting procedure. These aAj' av j aej partial derivatives can be expressed in terms of R(x,y) and I(x,y), and thus can also be evaluated using COMER. The expressions for the derivatives can be obtained either directly from Equation (D.4) or by using Equation (D.7) and the expression for the total derivative of w(z). The latter approach involves less algebra and is given below. From Equations (D.7) and (D.3) the required partials are ad*(v) --— R(x.,.), (D.8a) aAj /e adj(v)R. A. v. A(v) Aj aR Aj aR ax; av-,40 avj /e axj avj i / j e ij J J -j DR (D.8b) 2 ax. -/r7G. 3 and d.(v) Aj ^ J. J= J A. 1 R1 DRax' aR D aDe 0 3 2 ej axj aej ayj ae -A. f R R (D.8c) - [R+xj- y. - 2 J 0+ Yj - + j] 0' tef2 texesoyax j Only the first expression, (D.8a) above, is in final form; the others involveR aR involve the partial derivatives ~and,~which can be determined from

195 the derivative of w(z). From the second form for w(z) given by Equation (D.2) w'(z) = 2-2zw(z) = 2[yI(x,y)-xR(x,y)] +2i [ -xI(x,y)-yR(x,y) (D.9) where use has been made of Equation (D.6). Making use of the fact that w(z) is analytic and applying the Cauchy-Riemann conditions (i.e., AR =I and a = I), the partial derivatives R and can be obtained 3x ay ay ax aD ay directly from Equation (D.9) and are given by.= 2[yjI(xj,yj)-xjR(x,y)] (.10) and a -2[ 1 -xjI(xj,yj)-yjR(xj,yj)]. (D.11) Combining the results of Equations (D.8b) and (D.8c) with Equations ad. (D.l0)and (D.11) above, leads to the desired expressions for aJ and ad. aej ad.(v) 2A. -v -=- 2 [xjR(xj,yj)-yjI(xj,yj)] (D.12) J 9d.(v) 2A~, J 2 djv = 2Ai L ( 2-y - )R(xj' yj)-2xYjI(xy 3) + (D.13) ad, The expression for aA. is given by Equation (D.8a). 3

196 The proportionality between the G-L convolution integral and its partial derivatives with either the real or imaginary parts of the complex error function allows their evaluation by existing numerical methods. The subroutine COMER, which is listed in Section C of this Appendix, is used to evaluate these functions in conjunction with the THEORY subroutine.

197 C. Listin of Mbssbauer Computer Programs 1. Main Program C MOSFIT PROGRAM INTEGER SP,P3S,P~5, PiS,POS, IP () REAL*4 INCR REAL*d SETUP(i8),P(25),REF,X(51),AM(2525,25),DP 25),WT,ULIFFZ, iAZ OF (25),AV(25,25),C( 5),PS(25),FYI,Q DIMENSION FYJ(512,t-),FY(512),YiRM(5i2)FIT( 51Z ),XV(512), ICVKR(2225),YS(512),SIG(25) fVR(25),DIFF(512),N(I8) CJMMON/CONTR /M,NS, NP;,, NL,NK, IUP,ILP, IS CCM MON/I)ATA/YS ADDC, MM, IM, IDT,JDT COMMON/L I N/AM P,AV,UP COMMON/PAR/FYI, P X I,P X2, PX3, PX4,VCAL, RZP CCMMON/AREAS/P,CVR,VR, AREA ARS IGNORE dLANCS GN INPUT CALL FCVTHb(1) I PR=O IRD)=5 C REAL) AND PRINT DA A 952 FORMAT(' 1',7Ab/7Ab//i4HVELOCITY CAL=,F8.5, 1 1OH MM/SEC/CH,3X,bH ZERO=,F8,3,7H CH VS,Ad/14,9H CHAN\NELS 23X, I3, I1H PARAtl tEI'RS,3X, 13,13H SINGLE LINESt3XI3t,4H PRS) 125 READ( IRD,901 tEND=i00) (SETUP( l) I 1l, 14),REFMM, INDECKS,M, NS, NP,N, IiT TLPL,KSC,ICAL,IDIF, ILP, IUP, IMIN, IMAX, 2YM, YMX SFXS FY,FF, C ON 901 FORMAT( 7A /7A8/A,2 3/91 3/414/4F 53/2F6. ) JDT=0 I READ( IRD,90 ) RP,VCALPX,PX2,PX3,PX4 902 FORMAT(6F8.0) READ(IRD,905 }(PI), I=l,M) 905 FORMAT(7Fd6.0 2 CALL DREAD C CHECK FOR PRLGPER CUN CITIONS FOR ANALYSIS IF(M.GT.U.ANJ..LtL. 5)GO TO 4 WRITE( IPR,907)M 907 FJRMAT('O' 0,20X,4dttuNLY I TO 25 FITTING PARAMETERS ARE ALLOWED, NOT 1,I3) GO TO 125 4 IF( IM.LE.512)GO TU 5 WRITE(IPR,98) IM 908 FORMAT(' o,2OiX,42hUNLY 1 TO 512 DATA PCINTS ARL ALLOWED, NUT,13) GO TO 125 5 CONTINUE C SETTING UP NURMAL Lt'UATIDNS DO 500 1=1,I M 500 X(I)=DFLOAT( ) WRITE ( IPP R952) (SETUP ( I ), I 1,14,) VCAL,RZP,REF,I M,M,NS,P WRITE IPR,951) 951 FORMAT('O',24X,'IiNTRMRJIArE RESULTS'/IX,'ITERATlUN',2X,'CHIS'/NN 1' t 19X,' PARAMftERS' )

198 J I T =0 K IT=0 N K= Q=:6LE ( FF) 7 WVAR=O. L) 8 K=1,M C (K) =0.0 D)O L=I,M 3 AM(KL)=O.G )0 9 I=ILP,IIUP CALL THEURYl X( I ),P,AZoZ,FDF) WT =.O/DBLE (YS(I ) ) )IF=DBLE( YS( I)-F) VAR =VAR+WT *D IF*2 DO 9 K=l,M C (K) =C(Ki+WT*D0IFDF (K) DO 9 L=1,M 9 AM(K, L) =AM(K,L) +'vT*Uf- (K)*DF (L) CHISQ=WVAR/( IUP-ILP-M) NK= WRITE ( IPR 9i 1) JI rT,LH CI SQ, (P ( I ) I1 M) 91.1 FiKMAAT (' 0, 3X, 1 3,4X t E4.O 2 X,3014.6/( 27X, 30D4.6 ) C TEST FUR CGISDill ONS OF ITERATIO N IF(CHISQ.GT.500. ) GC TO 99 iFIJIT.LQ.O) GO Tu 1C ABV=ABS (WVAR-VC) IF(ABV. LE.CGN GU TO i2 IF(KIT.E<.O.ANOD. WVAP.bGT.VO) GO TO 18 IF(KIT.LQ.I.ANU. WVAk.GT.VO) GO TO Zl IF(JIT-20.E..0) Gu TrO 1 C SOLUTION OF EQUAl IUNS 10 CONTINUE 00 11 K=1,M 11 DP(K)=C(K) CALL DPINVS I(,-1 IS I C) IF(ISIG.NE.O) GO TU luO C LCHECK VALUES UF ITtRATION C(D=0.0 DO 14 K=1,M 14 CD=CO+C (K) DP ( K) IF(CD.GT.O) GO TC 1l DO 15 K=1,M 15 DP(K)=-DP(K) WRITE (I PR, 925) 923 FOURKAT('O',3X, VLCTOK PRODUCT NEGA IVE ) 16 CONTINUE DO 17 K=1,M PS(K)=P(K) DP( ) =Q:DP(I K) 17 P(K) =P(K) +DP K) VO=WVAR JI T=JIT+1 KIT=O GO TO 7 1d CONTINUE DO 180 K-I,MPi P(K)=PS (K ) 180 PS K) =PS (K) + P(K)/2.

199 V 1=0. 0 DO 185 =I LP,IUP CALL THE RY( X I ), S, AL,,F,OF) WT=I./DBLE(YS( I)) )iF-=D)LE(YS( )-F) 185 VI=V1+WT*)IF**2 V2=( VAR+V( ) /2. lF(V1.GT.Vz) GC 1'0 2O ALPr-A=. b5 (A VAR-4. VI +J. *V0) /( 2. V AR-4. V 1+2 *VO) WRITE(I1PK,912) ALPHA 91Z FORMAT(lOX,' ALPHA=', IX,Fu.' J LO 19 K=l,M 19 P(K )=P ( K+ALPHA*P (K) KI T=1 J lT-J IT+ NK=0 GO TO 7 C TERMINATION OF ITERATION 20 WRITEIIPR,913) 913 FORMAT( O',5X.' FURTHER ITERATI N UNSUCCESSFUL') GO TO 80 21 WRITE(IPR,914) 914 FORMATC'O',1OX,'CONVERGENCE CRITERIA NCT MET') IF(JIT.GT.4) GO TO 22 GO TO 80 22 WRITE(IPR,915) 915 FORMAT 0 O',I OX,'F INAL RESULTS') C NORMALIZATION OF DATA & CALCULATION OF AREAS ARE A= O.O ARS=O.O DO 25 I=ILP,IUP CALL THEURY ( X I ),P, AL, Z F, DF) FIT(I )=8B YNRM(I)=YS( I /AZ i IFF(I)=YNRM (I-FIT( I AREA=AREA-YNRM(I )+10 ARS=ARS-FIT( I)+.1O DO 24 J=1,NL 24 FYJ(I,J)=BN(J) 25 FY(I)=FYI. C CALCULATION OF EtRRORS WRITE(IPR,9gO) CHISQ 930 FURMAT('v1',X,'FINAL CHISQ/NN='t2X,E14.6) CALL DPINVS(M,O,ISIG) IF ISIG.NE.O) GO TO 100 DO 26 K=1,M VR(K)=CHI SQ*DABS(AV{KK)) SIG(K)=SQRT(VR(K)) DO 26 L=1,M CVRl,L)=CHISQ*AVK, L) 26 CVRIL,K)=CVR(K,L)

200 C ERROR D[STRIBUTION SU=DSQRT(AL) /AL P3S=0 P2S=0 PIS=O POS=O NOS=O N1S:O N2S=O N3S=0 00 27 I=ILPIUP IF (DIFF i) }.GT.SD*3.0)P3S=P3S+1 IF (D IFF ( ).GT.SLb*2. AND.O IFF( I ).LT.SD*30.) P2S=P2S+1 IF (DIFF( I ).GT.S*.O.AND. DIFF( I) LT.SD*2.0)P1S=PlIS+1 IF (D OIF I ).GT.O.. AND. i)FF( I ).LT.SD*l C)POS=POS+I IF (DIFF(1).GT.-SU*i.O.AND. DFF(I ).LT.O.O)NOS=NOS+1 IF { If F( 1) GT.-S*2.O. AND.DIFF( I ).LT.-SD*1.O) N 1S=NIS+I IF {O IFF( I).GT.-SD*3.0. ANO DIFF{ I.LT.-SD*2.0)N2S=N2S+1 27 IF (DIFF(I) LT.-S* 3.0) iN3S=N3 S+1 C DATA PRINT OUT WRITE( I PR,91 7) 917 FORMATI'0,2O X,*FITTING PARAMETERS') DO 30K=i,M 30 WRLIi(IPR,918)K,P(K),SIG(K) 918 FORMAT(', 3X,13I 10H PARAMETER,2X, I15.6,.4H +-,EI15.U WRITE(IPR,919) P3S,P2S,PIS,POS,NOSNISN2S,N3S 919 FORMAT('',5X,lbHERRCR DISTRIBUTION,8I5) C CALCULATION CF AREAS THRU SUB TAREA CALL TAREA IFILPL.NE.1) GC TO 37 WRITE( IPR,920) 920 FORMAT(I O', 5X, COVARIANCE ELEMENTS FRCM ERROR 1MATRI X//) DO 35 K=1,M KPI=K+1 WRITE(IPR,922) (CVR(K,L),L=KPIM) 922 FORMAT('',2X,6(EO. 3,1X)) WRITE IPR,923) 923 FORMAT('') 35 CONTINUE 37 CONTINUE IF LPL.EQ.-1.AND. JUT.EQ.O) GO TO 60 C PRINT OUT FOR CALCOMP PLOT PROGRAM IF(JDT.EQ. ) GO TO 125 970 FORMAT(' 4F7.3,2F9.4/513,414) 971 FORMAT(', 1OF8.4) WRITE (7,970) SFX SFY, YI,YMXt RZP,VCAL,NL,KSC, ICAL, I DIF I1M, IMIN, I MAX, I UP, ILP WRITE(7,971)(FY(I}, I=,IM) WRITE(7,971)(YNRM(I),I=,IM) WRITE(7,971) (t-T(I),I=,IM) WRITE(7,971) (IFF( I),1=1,IM)

201 DO 320 K=1,NL 320 WR I T(,971) (f-YJ(JK),J=1, IM) 80 I F IDT) 12 5,1 25,1 81 JOT= 1 GO TO 1 99 WRITE(IPR,921) 921 FORMAT('3t,' CHISQ GREATER THAN 500 _THEORY INCORRECT OR 1ESTIMATES VERY BAD' ) 100 CALL SYSTEM END SUBROUTINE DREAD C REAL) AND ARRANGE MCA DATA DIMENSION Y(512),YS( 512) YZ(512) CUMMUN/DATA/YS,ADDC,MM, IMIDT,JDT 100 FORMAT(F8.0) 101 FORMAT(8(f6 b t2XX) IF(JDT.EQ.1I GC TO 50 IF(IDT)3,3,5 3 IM=MM GO TO 10 5 IM=MM/2 10 DO 15 I=1,MM 15 YZ(I)=O.O REAL( 5,100) ADUC D0 20 J=I,NDECKS READ(4,101) (Y( ),I=1,MM) DO 20 1=1,MM 20 YL( I=-Y((IY (I )+ADCC IF(IDT)30,25,25 25 DO 26 I=1,IM LM=1+ IM-I 26 YS(I)=YL(LM) GO TO 80 30 DO 35 I=1,IM 35 YS( I )=YZ I ) Gd TO 80 50 DO 55 I=i,IM KM= M+ I 55 YS(I)=YZ(KMJ 80 CONTINUE RETURN END

202 2. Theory Subprograms for Lorentzian Functions. SU83ROUT I NL THECtY(XT, PT,AT, BT FT,DFT) C THEORY FOR NS INDIV LINES + NP PAIRS OF EQUAL LINES CENTERED AT C SAME CENTROID. MAX. NO. LINES + PAIRS = NL C J=INDIV. LINE NO, K=PAIR NO. JMAX=NS,KMAX=NP+NS, KMIN=NS+i C INDIV LINE PARAMS. P(3*J-2)=PUS., P(3*J-1)=INT,9 P(3*J)=GAM/2 L PAIR PARAMETERS. P(3*NS+ )=CENT., P( 3K-i )SPLIT, P(3*K)I NT. C P 3*K+1) =GAM/2 C BKGRD. PARAM. NP=O. P(3*NS I)=CCNST, P(3*NS+2)-=LIN. COEF., C P(3*NS+3)=QUAD. COEF. C dKK RD PARAM. r\P.NE.O P(3*NL+2)=CGNST, P(3*NL+3)=LIN. COEF. C P(3*NL+4)=QUAD.COEF. C TYPE OF 6KGRU: NB=-lI(LIN); Nb=OICCNST); Ni=1(QUADRATIC). IMPLICIT REAL*S (A-H,O-L) REAL*4 FT,BI (8),PX1,PX2,PX3,PX4,VCALRZP DIMENSION PT (5), DFT(25) COMMON/PAR/BSi,I,PX1 PX2 PX3 tPX4,VCAL RZP COMMON/CONTR/MNS,NP,NB,NL,NK IUP,ILP,IS BS=I.O IF(NK.EQ.I) GOi TO 6 NL=NS+NP NL3=3*NL NS3=3 *NS 4 AT=PT(NL3+1) IF(NP.NE. O) AT=PT(NL3+2) 6 CONTINUE IF(NS.EQ.U) GO TO 30 C INDIVIDUAL LINES' FUNCTIONS AND UERIVATIVES DO 20 J=1tNS JJ=3*J XO=PT(JJ-2) G=PT JJ) G2=G*G T=PT(JJ-1) C=T*G2 L=XT-XO D=Z Z +G2 31 J)=1.O-C/ L BS=BS-C/D DFT (JJ-2)=-2.0OAT*C*Z/(D*D) DFT (JJ-1 ) =-AT*G2/D 20 DFT(JJ)=-2.0 OAT*G({(T*O-C)/(D*D)) IF(NP.EQ.O) GO T0 50 30 DFT(NS3+1)=0.O NSS=NS+i C PAIR FUNCTICNS AND DORIVATIVES DO 40 K=NSS,NL KK= 3*K XL=PT NS3+1 -.5*PT (KK-1 ) XR=PT NS.3+1) + 5*PT( KK- ) T=PT(KK) G=PTIKK+1 ) G2=G*G C=T G2

203 ZL=XT-XL LR=XT-XR DL= L LL L G2 DR=ZR*ZR+G2 I (KK) =1. O-C/DL-C/DR BS=BS-C/DL-C/DR OFT(NS3+i)=DFT(NS3+l)-2.0*AT*C*(ZL/(CL*DL)+ZK/DRDR) OFT(KK-1)=AT*C*(LL/(DL*DL)-LR/(DR*DR) ) DFT (KK)=-AT* (G2/DL+G2/DR ) 40 DFT(KK+LI=-Z.O*G*I(T*DL-C)/(DLC *DL)+(T*DR-C)/ DR*DR)*AT GO TJ 80 C BACKGROUND 50 CONTINUE IF(NB)55,65,60 5) BTTBS+PT NS3 +2)*XT DFT (NS3+ 2)=A TXXT GU TO 70 60 dT=BS+PT(NS3+2)*XT +PTINS3+3)*XT*XT DFT ( NS3+2)=A T*XT OFT (NS3+3 )=AT*XT*XT GO TO 70 65 rT=BS 70 OFT(NS3+1)=BT GO TO 99 80 CONTINUE IF(NB)8595, 90 85 dT=BS+PT(NLi +3)*XT DFT(NL3+3 )=AT*XT GO TO 96 90 BT=BS+PT( NL3 +J3)*XT+PT(NL3+4)*XT*XT DFT (NL3+3 )=A T*XT DFT (NL.3+4 )AT*XT*XT GO TO 96 95 BT=BS 96 DFT(NL3+2)=BT 99 FT=AT*BT NK=1 RETURN END SUBROUTINE TAREA C AREAS, RATIGS OF AREAS AND ERRORS C FOR LORENTZIAN SINGLETS AND/OR DOUBLETS REAL*8 P(25) REAL*4 AA(8,8),RSG(J,, )CVR(25,25)tVR(25)tAR(8),SGA(8),VRA(8) DIMENSION IP(8) CGMMON/CONTR/M,NS, "P,Nd, NLtNK, IUPILP IS COMMON/AREAS/P,CVRVR, AREA,ARS VRT=O.0 IPR=6 IF(NS.EQ.O) GO TO 72 DO 271 I=1,NS I 1=3*I 11=11-1 AR( I )=3.14159P( 1 1) *P I 1 ) ARE=ARE+ARE I )

204 VRA(I)=AR(I)*AR i)*(VR(I1)/(P(I1)*P(I1))+ IVR(1I)/(P(II )*P(II l)+2CVR( I,Il)/(P(Ill)*P(IJ )) SGA I )=SQRT (VRA( I)) Z71 VRT=VRT+VRA( ) IF(NP.EQ.O) GO TO 280 272 NSL=NS+l DO 273 K=NS1,NL KK=3*K KI=KK+l AR(K)=6.2831 8*PKK)*P(KI) ARE=ARE+AR( K ) VRA(K)=AR(K)*AR(K)*(VR(KKJ/(P(KK)*P(KK))+ IVR(K)/(P(K1 )*P(KI} )+2*CVR(K1,KK)/(P(KI)*P(KK))) SGA(K)=SQRT(VRA(K)) 273 VRT=VRT+VRA(K) 280 CONTINUE IF(NS.EQ.0) GO TO 292 DO 291 I=1,NS 1 1=3*1 lI=II-1 IZ=I+I IF(IZ.GT.NS) GC TO 292 DO 291 J=IZLNS JJ=3*J Jl=JJ-1 VRT=VRT+2*AR I )*AR.(J);(CVR( 1,JI)/(P( 11)*P(J1))+ ICVR(I l,JJ)/( P(i)P(PIJJ) )+CVR( IJJ )/(P(I )*PJJ) ) ) AA(I,J)=AR(I )/AR(J) AA(J,1)=AR(J)/AR(I ) RV=VRA I )/ ( AR ( I ) *Af< (I) ) +VRA (J )/ ( AR ( J ) *A J) ) i-2*CVR( I Ji)/(P(1 )*P(Ji) -2*CVR ( Il,JJ)/(P( 1)*P(JJ ) ) 2-2CVR( I I,J1 )/(P( [1) P( JI))-2*CVR( I,JJ )/(P( I)*P(JJ} RSG(1,J)=AA( I,J)*SQRT(RV) 291 RSG(J,I)=AA( J, I) wRT (RV) 292 iNS=NS+1 IF(NP.EQ.O) GO TU L97 00 296 K=NSI,NL KK=3*K KL=KK+I KL=K+I IF(KZ.GT.NL) GU TO 294 DO 293 L=KZ,NL LL=3*L L 1LL+I VRT=VRT+2*AR ( )*A (L )*(CVR(KI,L1)/ (P(Ki)*P(L1))+ ICVR(Kl1LL)/( P K )P LL) )+CVR(KK,LL)/(P(KK)*P(L ) )) AA(K,L)=AR(K )/AR(L) AA( L,K)=AR(L )/AR(K) RV=VRA(K)/(AR(K)*AK(K) +VRA(L)/(AR(L)*AR(L))-2* 1CVR{(K1LL)/( P(K 1 ) PLL))-2*CVR(K1,L )/(P(K1 *P( L1 ) 2-2*CVR(KK,LL) / (P(KK) *P(LL)) -2*CVR KK, L1 )/(P(KK) *P(L I) ) RSS (KL) =AA K L)*SRT(RV) 29-3 RSG L,K))=AA (L, K)*SRT (V) 294 CONTINUE

205 IF(NS.EQ.0) GU TO 296 DO 295 I-lNS I I=3*I I l= I I-l VRT=VRT+2*AR ( I )*AR (K)*(CVR( Il,Ki )/ (P( I )*P(KI))+ ICVR 1 KK)/(Pi i)*P (KK) +CVR( II,Kl )/ ( P i *P KI ))+ 2CVR(I IKK )/Il F( II )*PKK) AA I( K)=AR ( I )/AR(K) AA(K, I)=AR ( )/AR I ) RV=VRA( I)/AR( AR I)A( I))+VkA (K)/ (AR(K)*AR K))-2* I CVR( I1, KI)/( P i l )P (K ))-2*CVR( I 1,KK)/ P ( I*P ( KK)) 2-Z*CVR( II,Ki)/(P( )*P (K1) )-2*CVR(I1,KK)/P(11 )*P(KK)) RS,(,K)=AA( I, K) SQRT(RV) 295 RSGK, I )=AA(K, I)SQRiT(RV) 296 CONTINUE 297 SGAT=SQRT(VRT) DO 298 J=1,NL AA(JJ)=AR(J)/ARE RVD=.5*(VAA (J )/(AR(J) AR J) +VRT/ ARE*ARE)) 298 RSGJ J)=AA( J,J)*SQRT( VD) WR I TE ( IPR 96 0) AREA, A PS, ARE, SGAT 960 FORMAT( 0,5X,'EXPER. AREA=',F12.4//6X, FIT AREA=, I F 10.4//b6X,' KGRU COURECTED AREA=',F10.4, 2X'+-',F7.4 WRITE (IPR,96 ) 961 FORMATI'O', X,'INDIVIDUAL LINE AND OK PAIR AREAS'//IOX,'LINE NUMBE iR') DO 962 J=1INL 962 WRITE(IPR,963)JAR(J ),SGA(JJ 9o3 FORMATI'',13X, 3, 3X,E 156,4H +-,E15.6) DO 302 K=1,NL 302 IP(K)=K WRIlTE(IPR,964)(IP(K),K=1,NL) 964 FURMAT(0',5X,'AREA FRACTION MATRIX'//7X,'I)RATIO OF AREA OF ONE P 1EAK TO TOTAL AREA CN DIAGONAL'//7X,' 2)RATIO OF AREAS OF ANY PAIR 0 2F PEAKS OFF DIAGONAL'///2X,'PEAK(S', 10 I7) tU 303 K=1,NL 303 WRITE IPR,96 K, (AA(K,L),L=1tNL) 965 FORMAT(,', X, 13,8X 8F7.3) ^RITE(IPR,966J(IP(t),K=lNL) 96o FORMATI('05,X,'AREA FRACTION ERRORS'//2X,'PEAK(S)', 11017) DO 305 K=1,NL 305 WRITE(IPR,965)K,(RSG(K,L),L=1,NL) KET RN END

206 3. Theory Subprograms for Gaussian-Distributed Lorentzian Functions. SUBROUTINE THtUKY(XT,PT,ATBTFTDFT) C NN GAUSIAN-LURENTIIAN CONVOLUTED FITTING FCNS. PLUS C (NdB=) ONE LURENTZIAN CJNSTR. IN POSITICN (PX ) ONLY C OR (NB=2) ONE CONVOLUTED GAUSSIAN-LORENTZIAN C CONSTR. IN POS. (PXi). FIXED LORENTLIAN WIDTHt C IN CONVOLUTION FCN. IS W/2=PX2. PARAMETERS FOR C UNCONSTR. G-L LINES: POS=PT(3*J-2), AREA=PT(3*J-1), C GAUSSIAN WITH=PT( 3*J) BACKGROUND EQUATION: C PT(M-2)+PT(M-1)*X+PT(M)*X**2. IMPLICIT RLAL*8(A-H, Q-L) REAL*4 VCAL,RZP,FT,I (8) REAL*8 IMPXPYPZt,PT(25),UFT(25) COMMON/CUNTR/M,NS, NP,N3, NLt NK, IUP, ILP t I S COMMON/PAR/ S, f3I,PXi,PX2,PX3,PX4,VCAL,ZP BS=1.0 IF(NK.EQ.1) GO TO 10 NL=NS NS3=3*NS PX=DBLE (PX1/VCAL +KP) PZ=DBLE[ PX2/VCAL ) B'=PT ( M-2) NN=NS IF(NB.NE.0) NN=NS-1 10 AT=1.0 DO 20 J=I,NN JJ=3*J X=(XT-PT(JJ —) )/PT (JJ) XA=DABS( X) Y=PZ/PT(JJ) CALL COMER(XA,Y,RE I.) FX=PT(JJ-1)*RE/t(.77245*PT(JJ)) IF(X.LT.O.) IM=-IM BI (J=1.-FX/B dS=BS-FX/B DFT( JJ-2)=-2.*AT*P' (JJ-1 )*( XORE-Y*IM) /( 1.77245*PT(JJ )*Z2) OFT(JJ-L)=-AT*X/PT (JJ-1) 20 OFT(JJ)=-2.*AT*PT(JJ-1)*((X*X-Y*Y-0.5)*RE-2.*X*Y*IM 1+Y/1. 77245) / (1.71/45*PT ( JJ) *2) IFiNB-1) 90t 50,0 50 Z=XT-PX G=PT(NS3-1) T=PT(NS3-2) D=*ZL+G*G FX=T*G/( 3. 14 15 9* ) B[ ( NS )= 1.-FX/B BS=BS-FX/B DFT ( NS3-2 )=-AT*F X/T L)FT ( NS3- l )=-AT*FX* 1./G-2.*G/ ) GO TO 90 80 X=(XT-PX)/PT(NS3-1) XA=DABS (X) Y=P /PT NS3-1) CALL COMER(XA,Y,E, IM)

207 FX=PT(NS3-2) *RE/(i.77'45*PT(NS3-1)) IF( X.LT..) IM=-IM b I ( NS)= 1.0-FX/ dS=BS-FX/ DFT(NS3-2 =- AT*-X/P r (NS3-2) OFT NS3- 1 )=-2*AT*PT (NS3-2)*( (X X-Y*Y-.5) RE 1-2'.*XY* M+Y/i.7724 )/ (.7'245*PT NS3-i)**2) 90 AT=PT(M-2) FT=AT*BS+PT ( M-1)*T+PT ( M)*XT*2 bT=FT/AT UFT (M-2)=. OFT (M-1)=XT DFT(M)=XT**2 iK= 1 RETURN iND -j# J L)UT I NEo CMER t X, Y,t,i I M)';JrtROUTINE CCMER(X,Y,R.,IMj C PROGRAM TU COMPUTE THE COMPLEX EKROR FUNCTION IMPLICIT REAL*( A-H O-Z) REAL*8 IM,LAMBDA LUGICAL B IF(Y.LT 4.29.AND X.LT.5.33) GO TO 10 H=O. N.CAP=O NU=8 GO TO 20 1 0 S=(. 1 -Y/4.2 9)CS T (l.-X ( X/2 4 1 ) H=1.oS H2=2.*H NCAP=6+23*S NU=9+21*S 20 LAMbDA=O. I F (H.GT.0) LAMBDA=H2~**NCAP b=.FALSE. IF(H.EQ.0..OR.LAMBA.EQ.0. ) B=.TRUE. S1=0. SI:O. S2=0. R 1= O R2=0. N NU 30 NPL=N+1 T 1=Y+H+DFLOAT( NPL) 3K 1 T2=X-DFLOAT NPL ) RL CL. 5/ lTI**2+ T2**2) R =C4*T1 R2=T2*C IF(H.GT.O..AND.N.LL.NCAPj GO TO 40 GU TO 50 40 T1=LAMBDA+SI SI=RI*Tl-R2Z S2 SZ=R2*T i+RI*S2 LAM8DA=LAMBD A/H2 50 N=N- 1 I F(N.LT.0) GO TU t)

208 GO TO 3D 60 RS=S1 IF(B) RS=RI RE=i 12183791 671095 I1RS IF(Y.EQO.O.) RE=tUXP(-X*X) SR=S2 IF(B) SR=R2 IM=1. 1283 7916709551 SR RETURN END SUBROUTINE TAREA C CALCULATION GF INDIVIOUAL AREAS, RATIOS AND ERRORS C FOR NN GAUSSIAN-LORENTZIANS PLUS CNE CCNSTR LOR. REAL*4 AA {d, 8) RSG (8 t) t CVR( 25 t 25) VR (25) VA(8) REAL*8 P(25) DIMENSION IP(8) CUMMON/AREAS / PCV r, VR K AREA, AR S COMMON/CONTR/M,NS, NP,NL,NL NKI, UP, ILP, IS IPR=6 ARE=O. VRT=O. NN=NS IF(NB.NE.0) NN=NS-I S3=3 * NS DO 10 1=1,NN 1 1=3*1-1 ARE=ARE+P{ II ) VRT=VRT+VR( II) VA I )=VR( I)/P(I ) IZ=I+1 IF IZ.GT.NN) GO TL 15 DO 10 J=IZNN JJ=3*J-1 AA(I,J)=P(1 )/P(J ) AA(J,I)=i./AA(I,J) RV=VR ( I )/(P( I I)P( I I) )+VR(JJ)/( P(JJ) P(JJ) ) 1-2.*CVR( I1,JJ)/ (P( I1 )*P(JJ) ) RV=ABS(RV) RSG( I,J)=AA( I,J)SiR T (R V) RSG(J, I)=AA( J, I)*SRT(RV VRT=VRT+2.*CVR(I,JJ ) 10 VA I)=VA I)+CVR( I 1,JJ)/P( I 15 CONTINUE IF(NB.EQ.O) GO TC 50 ANS=P (NS3-2) VNS=VR NSJ-2) VA (NS) VNS/ANS ARE=ARE+ANS VRT= VRT+VNS

209 DO 40 =11,NN I I=3*j- 1 AA( INS)=P(I I)/ANS AA(NS,I )=I./AA( I,NS) RV=VR I I I )/(P (I 1)p( I I) )+VNS/(ANS ANS) 1-2.*CVR( I I t -2/P(I I )*ANS) VA(NS)=VA(NNS)+CVR I I,NS3-2)/ANS VRT=VRT+2.*CVR( 1 II, S 3-2) RV=ABS( RV) RSG(I,NS)=AA I,NS)SCRT(RV) 40 RSG(NS, I )=AA (NS, I )*SRT (RV) 50 SGA T=SQRT (VR T) /P ( M-2 ) DO 60 1=1,NN I [=3*1-1 AA(I,I)=P(II)/ARE RV=VR(II)/lP( I)*P(II)) VRT/(ARE*ARE)-2.*VA(I)/ARE RV=AbS(RV) 60 RSG( I,I)=AA( I, I)*SSRT(RV) IF(NB.EQ 0) GO TO 80 AA( NS, NS )=ANS/ AR R V= VNS/ ANS ANS) +VRT / A RE*ARE )-2.*VA (NS/ARE RV=AbS(RV) RSG ( NS NS ) =AA ( hSj S) *SORT (RV) 80 ARC-ARE/P(M-2) WR ITE IPR, 960) AREA, ARS, AR E, SGAT 960 FORMAT(', X,' EXPLR. AREA=' F12.4//6X'F IT AREA=' 1FI0.4//6X,'tKGRD CORRECTED AREA =,F12.4,2X,' +-F7.4) DO 302 K=iNL 302 IP(K)=K WRITE(IPR,964)(IP(K),K=I,NL) 964 FORMAT(tO',SX,'AREA FRACTION MATRIX'//7X, 1)RATIO OF AREA OF ONE P 1EAK TO TOTAL AREA ON DIAGONAL'//7X,'2)RATIO OF AREAS OF ANY PAIR U 2F PEAKS OFF DIAGGOAL'///2X,PEAK(S)',10I7) DO 303 K=1,NL 303 WRITE(IPR,965)K,(AA(KL),L=1,NL) 965 FORMAT (0', X, 13,8X,8F7.3) ARITE(IPR,9bb) (IP(K),K=l,NL) 966 FORMAT('0',5X,tAREA FRACTION ERRORS'//2Xt'PEAK(S', 11017) DO 305 <=1,NL 305 WRITE(iPR,965)K, (SGO(K,L,L=1,NL) RETURN END

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