TH E UN I V E R S ITY OF M IC H I GAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report NEUTRON DIFFRACTION AND INELASTIC SCATTERING BY VITREOUS BERYLLIUM FLUORIDE Charles A. Pelizzari DRDA Project l01111 supported by: NATIONAL SCIENCE FOUNDATION GRANT NO. GK-35901 WASHINGTON, D.C. administered through: DIVISION OF RESEARCH DEVELOPMENT AND ADMINISTRATION ANN ARBOR February 1974

TA3BLE O!' CONT,:ENTS ACKNOWLED)(fGEMEiNTS............ i LIST OF TABLES.......................................... i.v LIST OF FI'cl ES.. v CHAPTER ONEi: INTRODUCTION............. CHAPTER TO T THEORETICAL BACKGROUND FOR NEUTRON SCATTERING FROM DISORDERED SYSTMS................ 17 CHAPTER TIHRE.E: iEXPF'lT........... S 44 CHAPTErR FOUR: CALCILATION OF THE STRU CTURE FACTOR AND SCATTERING1 LAW FROM TIME-OF-FLIGHT DATA.. 69 CHAPTER FIVE: COMPUTATION' AND RE FIN.! M'rMNT OF THE RADIAL DENSITY FINCTIO T',............ 113 CHAPTER SIX: ANALYSIS OF THE TERIETAL RESULTS... 135 APPENDIX ONE: THE FUNCTIO.N ( (Q, ) AT CONST\ rT \NGLE FOiR VITREOUS BeF... 177 APPENDIX TWO: MtiLTIPLE SCATTERING IN AN I\COIERENTLY SCATTERING PLATE..r a....................... 213 APPENDIX THREE: SATMPLE ATTENUATION FACTOR........... 215 APPENDIX FOUR: ESTIMATION OF:RRORS IN lDF 1PAK PARAETERS.................... 216 LIST OF R,FEREF.NCES................. 218 i L

ABSTRACT NEU.TRON DIFFRACTTON AND INEL.ASTIC SCATTERING BY VITREOU S "BRYLLTI'F' FLIUORTI: by Charles Arthur Pelizzari Chairman: John,I. Carpenter The total neutron diffraction pattern and the scatterin:v law of vitreous BeF2 have been measured using time-of-fliht techniques. The diffraction pattern was measured at the University of bichi(-an time-of-flight diffractomcter for wave vector transfers in the range 0.9 1 2 Q<27X 1. The data were corrected for container scattering, multiple scatterin: in. the vandiilmr reference scatterer and multiple scatterin:- in the BcF, sample. Fourier transformation of the diffraction. data yieldedl a radial density function from which terin;iiation and normalization errors were partly removed using a novel computer:;raphic approach. The refined rdf was analyzed for neighbor distances, coordination numbers and root-mean-square variation in neig,hbor distances. for Be-F, F-F and Be-Be. pairs. Lon-,errange order was found to persist out to distances of?-10\. The structure parameters'and long-range rdf behavior were judged to be consistent with a tetrahedral random network structure proposed by Warren. The rms peak wi.dths were found to be smaller than those determined from X-ray di.ffraction bly Narten, and agree closely with rms vibration ampli tudes derived from a dynamical model of Bates. 1

2 The scattering law data were corrected for container scattering and multiple scattering in the vanadium reference scatterer. A generalized frequency distribution function was calculated from the measured scattering law. Systematic peaks in the generalized frequency distribution were found to correspond with previously observed Raman and infrared bands, and with maxima in the frequency distribution calculated by Bell and Dean for their tetrahedral random network model. Comparison between the nmeasured scattering law and a calculation including only contributions from long-wavelength acoustic modes showed the calculation to be valid only for small wave vector and energy transfer, exactly as expected.

ACiKNVO LEQLTG:, S The auitkho io id s illd.eb)ted to the- Arr:onn. National Lahoratory Center for i:dulcati onal Affai rs for i.ts support. (f1- t-he Thesi s Parts Fellowship durin w..li wch the i.nelasti.c scatteeri n portion of this work was completed. lie is also grateful to the staff of the Solid State Science Di vision at,ArP.onle, and in parti cular to Dr. J.'., Row r. L. rie, rce, r. J. R.Dl. Copley,!t. A. Rlahrman, r. 0.C. C Simpson, r. Ostrowski and,; r.:.. 1el.) for thei.r supplort, -duidance aind assisttace duclri nr, hi s stay at:Arr-onne.'lrhe. ti nle -of-fl irht (di ffracti.on work was supported in. part l)v the Na tiotial Scien.ce Foundactior uclder,-.rant nul_)ber K -35901. Finally,,.t is a r)leasure to ac.knowl edfe the con.tin'li:Isupport and gA ui.dance of LDr. Jack Carpe:nter, and( the many valuable conversa-tiolns with DTr. David:(il]d ner which contributed to this work; an(id the sterli; technical. suppr)rt of l's. Jalli ce''Tracht. i i i

LIST OF TABLES Table page 1 Warren's 1934 Random Metwork Model of Vitreous 6 BeF2 2 Summary of X-ray Diffraction Results for Vitreous BeF2 10 3 Scattering Cross Sections of Be and F 18 4 Bragg Scattering by Aluminum 76 5 Bragg Scattering by Vanadium 81 6 Summary of Structure Results for Vitreous BeF2 150 7 Coordination Peak Parameters of Bell and Dean 153 Model V 8 Peak Parameters in Rahman "Supercooled" Liquid Model 158 9 Infrared and Raman Bands of Vitreous BeF2 162 10 Systematic Maxima of G(Q,4) 167 iv

LIST OF FIGTURES Figure pae 1 Enerly - iavenumber Characteristi. cs of Vari.ous Radiati ons 1 i 2 Loci of Constant Angle for Monochromatic Neutrons 47 Loci of Constant Time of Arrival for Neutron Scattering at 20 and 90 Degrees 49 4 Comparison of Mleasured Structure Factor of Vitreous 3eF? with Tntegration of FMeasured Scatteri ng Law 51 5 Plan View of the Time-of-Flight Diffractometer 53 Integration Paths for the TNTOFS Experiment 5r 7 Plan View of the TNTOFS 10 E ncapsulation of the Vitreouls JB-eF' Granules,3 Target Arrangement for the Time-of-Flight Di ffractome te r,4 () Tar-,et Arrangement for the TNTOFS < 11 Geomletry of the TNT-OFS Experiment 67 12 Vitreous BeF2 Diffraction Data 74 13 Calibration of the Diffractometer 75 14 Diffraction Data with Errors Due to Vanadium Coherent Scattering 79 15 Vanadium Diffraction Data with Bragg Peaks O 16 Intensity Ratio for 140 A'z 20~Run 3 17 Intensity Ratio for 430 Hz 920Run 18 Intensity Ratio for 140 Hz 90 Run?5 19 Intensity Ratio for 480 Hz 90 Run 05

Figure page 20 Smoothing Spline Fit to 140 Hz 200 Data 88 21 Rahman Normalization Integral for Partially Corrected Data 93 22 Container Self-Shielding Factor 96 23 BeF Scattering.Cross Section from Partially Corrected Diffraction Data 98 24 Vanadium Multiple Scattering Correction Factors 101 25 BeF2 Multiple Scattering Correction Factors 108 26 Model BeF2 Structure Factor for Multiple Scattering Simulation 110 27 Fully Corrected BeF2 Structure Factor 111 28 Experimental BeF2 Radial Density Function 117 29 Refinement of Experimental Radial Density Function 132 30 Refined BeF2 Radial Density Function 136 31 Be-F Peak in the Refined RDF 142 32 F-F Peak in the Refined RDF 144 33 Be-Be Peak in the Refined RDF 146 34 Geometry for Calculation of Bond Angles 149 35 Radial Density Function for Bell and Dean Model V 154 36 Pair Correlation functions of Rahman BeF Liquid Model 2 157 37 Neutron RDF of Rahman Super cooled BeF Liquid Model eF 159 38 Pair Correlation Functions of Rahman Supercooled BeF2 Liquid Model 160 39 Determination of Systematic Maxima in G(Q, ) 168 40 Determination of Approximate Debye-WJaller Exponent 171

Fit gure page 41 Comparison of Aco ustLic (i(Q, )) wi.th Measured C(Q, c ) 172 42 Collpari-son of Acousti.c G(Q, A)) wi th Measured G(Q, I) 173 43 Comparison of Acoustic G(Q, W)) with Measured C(Q, )) 174 44 Comparison of Acoustic G(Q,A) with Measured G(Q,) 175

CHAPTER1 ONEINTRODUCTION 1.1 Preliminary Remarks In a time when condensed matter physicists and chemists have treated with remarkable success a large number of difficult and complex problems, even the most basic properties of the structure and dynamics of glassy solids seemingly remain beyond our theoretical grasp. Theorectical consideraations of glass structure consist largely of arguments as to which of two conceptual models postulated in the early to mid 1930's less offends physical and chemical sense. Attempts to adapt conventional solid-state (i.e. crystalline) dynamical analysis to glasses typically fall flat except under severly restrictive conditions (see Chapter Six below), and little else has been tried. This rather primitive theoretical state is quite natural, for glasses lie somewhere between the two extremes which have almost exclusively been calculated: on one hand, the highly ordered periodic world of crystals where translational invariance seemingly makes all things calculable; and on the other hand the stochastic world of liquids where kinetic theories can be used to advantage. A glass is not a liquid nor is it a crystal; it may display behavior characteristic of.either or both (or neither). In short, the glassy state constitutes a singularly difficult theoretical ground, one on which few are inclined to tread. We suspect that as understanding of 1

2 the liquid state catches up with that of the crystalline, the special problems of the intermediate, glassy state will begin to yield. We begin this report of an experimental investigation of some aspects of a single glass with a brief introduction to the history of diffraction studies of glass structure and the aforementioned structural models, which have their roots in the early diffraction results. 1.2 Early Histor of Glass Diffraction 1.2.1. Crystallite Model. Soon after the discovery by von Laule that x-rays are diffracted by crystalline solids, 2 3 both Debye and Ehrenfest speculated that other forms of condensed matter should also exhibit characteristic diffraction patterns. It was not until 1930, however, that the first quantitative analysis of silica (SiO2) glass diffrac4 tion was published by Randall, Rooksby and Cooper. Since the dominant broad band of their diffraction pattern was centered at 1.5 A1, near the position of the (111) reflection of the crystalline silica C -cristobalite, Randall et al. suspected a simple explanation of glass structure in terms of an assemblage of microcrystallites might be appropriate. Application of the Scherrer line-width formula to their data indicated an average crystallite dimension of 15-20A. An explanation of certain anomalous refractive and thermal properties of glass in terms of what might be

3 called a microcrystallite model based on quartz, another crystalline silica, had earlier been offered by A.A. Lebedev (about whom more later). 1.2.2. Continuous Rantdomn Network Hypothesis. -rior to 1930 diffraction studies of several crystalline polymorphs of silica had led to the conclusion that the basic unit of composition in these crystals was not the planar SiO2 molecule, but rather a silicon atom surrounded tetrahedrally by four oxygens. These tetrahedra were apparently linked at corners so that each oxygen was bonded to silicon atoms in_ each of two adjoining, tetrahedra. The various polymorphs of silica were found to liffer only in the particular relative orientation of neig'hbori.ng tetrahedra. (We quote many silica results here, because BeF2 and SiO2 have proven to be structurally analogous; thus isomorphs of B3eF are found to contain corner-linked 3eF2 tetrahedra, and one may speak of quartz-like, cristobalite-like, tridymite-like, etc., forms of BeF2 analogous to the various Si02 polymorphs. This structural analogy reflects the fact that the ratios Be2- a + 4 02of Be:F and Si:0 ionic radii are nearly the same, while BeF2 having ionic charges only half those of SiO2 might be expected to be a somewhat "weaker" structure, which turns out to be the case.) Zachariasen, noting these as well as similar regularities for other glass-forming oxides, postulated in 19326 that oxide glasses are formed by corner-linking of oxygen

4 polyhedra in a random manner, rather than in the regular manner of crystals. In a set of rules governing the formation of what has come to be known as a continuous random network, Zachariasen postulated that there be no sharing of edges or faces by neighboring polyhedra, and that at least three corners of each polyhedron be shared with neighbors. Of course a continuous random network can no more easily exist in a real glass than an assemblage of tiny crystallites with sharp boundaries, separated by voids. The relative orientation of SiO4 tetrahedra in silica glass cannot be truly random, since most of the oxygens must join two tetrahedra; the requirements of bonding impose restraints on the degree of randomness possible. If we are to avoid severly distorting a large fraction of the tetrahedra, in fact, the short-range structure of a realizable "random" network must probably look more or less like that in a crystal, with progressively less rigid ordering as one moves away from the starting point. A viable "random network" model of a glass will thus be characterized by random variation of relative polyhedron orientations only within a fairly small range about some mean. Warren made convincing use of a realizable continuous network in the analysis of his X-ray diffraction data for 7 8 SiO2 and BeF2 in 1934. Choosing a mean linking angle between neighboring tetrahedra (i.e. the Si-O-Si or Be-F-Be

5 bond angle) of 120~ and using distances and coordination numbers consistent with those in crystalline modifications, Warren calculated the scattering from a random network, which he found to be in substanti al agreement with experiment. Particulars of Warren's JBeF2 model are listed in table 1. 1.2.3. ~ Modern Crystallite Model. Not everyone was as convinced as Warren that his results had proven conclusively the veracity of the random network hypothesis. In 9 1936 N.N. Valenkov and E.A. Porai-Koshits postulated a crystallite theory of glass structure which was far more sophisticated than that of Randall et al. (It is worth noting that Randall himself had gone over to the random network hypothesis in 1934.) N\ioting, several inadequacies of the Zachariasen-Warren random network hypothesis, Valenkov and Porai-Koshits suggested a model of Flass structure based on cristobalite crystallites of 10-12A or more on a side, which are joined through severely distorted regions to neighboring crystallites. This avoids the obvious oversimplification of the Randall model where crystallites are sharply delineated and separated by voids, but still couches the description of glass in the comfortably familiar terminology of the crystalline state (it should be apparent that the random network model is in fact a description appropriate to a liquid). It is the opinion of this worker that the modern crystallite and random network hypotheses are not in essential conflict, that in fact a realizable

6 TABLE 1. Warren's 1934 Randomr Network Model of Vitreous BeF2 1 Be surrounded by 2 F each surrounded by 1 Be at r=0 1 F at r=0 4 F0A 2 r=l.60A 4 Be r=3.20A 6 F r=2.62A a 0 12 F r=4.00A 6 Be r=4.00A 12 Be r=5.20 9 F r=4.65+0.45A 18 F r=5.45A+0.45A Continuous distribution Continuous distribution beyond R1=5.45A beyond R2=4.65A

7 random network will as previously stated wind up looking, pretty much like the modern crystallite structure. The difference of course is only a matter of defree - whether one chooses to believe that it is the more or the _less strongly ordered regions in )glass which are more important. Clearly each type of regi. on will be more or less influential in determining, a particular property of glass - neither can alone provide a complete discription of all properties. (That the random network and modern crystallite hypotheses are not necessarily at odds has long ago been pointed out by A.A. Lebedev 1.) 1.2.4. Recent Developments Concerning the Crystallite Models. Both the original and the modern crystallite hypotheses were first advanced by Russians (Lebedev, Valenkov and Porai-Koshi ts). Accordingly we must impute quite some significance to a recently published letter of Evstropyev and Porai-Koshits, in which crystallite hypotheses were in essence declared not to have been supported by the sum of experimental evidence (both diffraction and spectroscopic). The letter described discussions and a resolution made by a group of 250 Soviet scientists in December 1971 on the fiftieth anniversary of Lebedev's first crystallite hypothesis.'We quote selected passages, hopefully not misreresenting thereby the authors' intent.

"In reports devoted to X-ray structural analysis (E.A. Porai-Koshits, V.N. Filipovich and others) and electron microscopy (F.K. Aleinikov) the speakers emphasized that... all the attempts to reveal the existence of crystallites...have failed so far. The analogous conclusion was made by V.A. Kolesova on the basis of... infrared spectroscopic investigations." From the Resolution approved by the participants, "The method of infrared spectroscopy... cannot be used at present for a quantitative evaluation of the dimensions, quantity and degree of geometrical ordering of the crystallites existing in glass...Other methods... (electron microscopy, X-ray structural analysis et al.) despite their extensive development in recent years, do not permit, for some reason or other, to reveal the regions of increased ordering (crystallites) in glass... the seminar on the crystallite hypothesis considers it expedient to recommend...to pay attention to the improvement of the structural methods which can produce quantitative or even semi-quantitative information about the regions of increased ordering in glass." One may well conclude on the basis of this letter and the Resolution it describes, that although the Soviets believe the crystallite hypothesis will ultimately be verified, they recognize that as yet there is precious little evidence to support this position. 1.3 Modern Diffraction Results on BeF2 Glass Several X-ray and neutron diffraction studies have

9 been made si.ncc the oriiinal work of Warren and li. ll, whose results have already been discussed. We will not dwell on the details of the measurements themselves except to note when experimental conditions lead us to disbelieve certain conclusions. The neiighbor distances, coordination numbers (when given) and bond angles (calculated from the distances, angles are rarely reported) are summarized in Table 2 for the X-ray measurements of Batsanova, Yur'ev and Doronina 2' 13 VU~ 1.4 Zarzvcki and':.Narten. Tn each case the diffraction data were Fourier analyzed to give a radial density function 15 (rdf) as first suggested by Zernike and Prins and first applied to liquids by Pebye and Menkel in the late 1920's, 1/ and first applied to glass data by Warren in 1936. (A reasonably complete discussion of the rdf will be given in Chapter Two. For now we note only that from it one can calculate coordination numbers and i nteratomic distances.);ie assign the greatest weigrht to the results of Narten, who took proper account of the termination effects in his Fourier transform. The Russian workers on the other hand, failed to use strictly monochromatic radiation in conjunction with a photographic recording. technique; to their result we can assipn but little weight. The 1.50A Be-F distance reported by Zarzycki seems a bit too low, and indeed indicates a mean F-Be-F bond angle significantly different from the expected tetrahedral angile of 109l 2'. The 1.43A Be-F distance found by Batsanova et al. is l.ow

10 Warren & Batsanova 1 Hill() et al.12 Zarzycki Narten re-F 0.43+.0 1.60 1.554+.004 rF-F 2.55 2.54 2.55 2.537+.004 re-Be 3.20 ---- 3.00 3.037+.005 4 (F-Be-F) 105~ 125 116~ 109.50 4(Be-F-Be) 180~ ---- 180~ 155.6~ NBe-F 4 4 4.4 3.8+.3 N 6 ---- 6.8 5.7+.3 F-F N eBe4 - ---- 3.8+.3 Be-Be Table 2. Summary of X-ray diffraction results for BeF2 glass. beyond all belief, and in fact is the same as the Be-F distance for isolated BeF2 molecules as determined by electron diffraction from BeF2 gas. Narten is the only author who has calculated bond angles from his interatomic distances; we again comment that his results are of the highest quality. The scheme for taking into account termination effects used by Narten is identical in principle with that of Konnert and Karle, about which we will have much to say in Chapter Five, and which we feel is the "state of the art" technique for extracting distances, coordination numbers etc. from rdf's and incidentally for removing the unavoidable termination errors. Narten's work must certainly be considered the benchmark at this time for BeF2 glass diffraction.

1]. 1.4 Comparison of`iffraction >Methods 1.4.1. Overview. We have in the previous sections referred to both X-ray and electron diffraction measurements. [n this section we will touch on some features of each method. We will have little to say abo-ut electron diffraction as a tool for investigation of glasses. Electrons interact so strongly with condensed matter that only a very thin layer of material can be "seen" by electron diffraction; this same property makes electrons an ideal probe for investigations of gaseous substances. Since many interesting amorphous semiconductor materials are most readily prepared by vapor deposition in thin films, electron diffraction is heavily used in work on such glasses. For macroscopic chunks of stuff such as a window pane of silica g-lass or a lethal dose of vitreous BeF2, however, electron diffraction can provide information o0nly about the immediate surface regions. Since we are interested in the bulk behavior of a glass which can be had in bulk, electron diffraction is not the most appropriate experimental technique. X-ray diffraction has been an indispensible tool in the study of condensed matter ever since von Laue discovered the phenomenon in 1912. X-ray crystallogjraphy was begun soon after Laue's discovery by W.H. Bragg and has since been refined to an incredible degree. it is fortunate that for many years X-ray diffractionists have also applied their

12 efforts to the study of glasses, although the analysis of glass diffraction is necessarily quite crude in comparison with what can be (and is) done with crystals. X-rays unlike electrons can probe the entire volume of an irradiated sample, and so give information characteristic of the bulk material. Their scattering by matter is comparable to that of neutrons; it is considerably simpler to increase the total amount of scattering by utilizing a more intense source of X-rays however, than to buy a similar increase in neutron intensity. X-ray sources are quite small in general, require only modest shielding and do not need licensing by the AEC. Neutron sources by contrast, are generally either nuclear reactors which have some rather obvious problems of large initial expense and frequently require an entire laboratory staff just to cope with the radiation problems they generate; or accelerator-based pulsed sources in which the accelerator and not the neutron source per se is a budgetwrecking expense. In short, an ordinary mortal, a company short of an industrial giant or even a minor university just cannot afford a neutron source useful for diffraction work, but can for a fairly modest capital expense have a respectable X-ray diffractometer. Why use neutrons at all then? The answer is that in a very real sense, neutron diffraction serves as a complement to X-ray diffraction, providing new and different information when applied to the same substances. Futhermore, there are several problems associated with X-ray

13 diffraction which do not ari se with neutrons ( and vice versa, as we shall see): a) Since X-rays scatter from the electrons in a material, andr since useful X-ray wavelengths are of the order of ionic radii, there is a diffraction effect due to scattering from different electrons in the same atom. This causes destructive interference which decreases the scattered intensity as Q increases. iNeutrons on the other hand scatter from nuclei in the sample which look like point scatterers, and so there is no destructive interference to diminish scattering at large Q. b) In addition to the elastically scattered or "tunmodified" X-rays, there is also a Compton scattered or "modified" component of scattered intensity which increases with Q. The removal of this modified component to get at the unmodified (diffraction) component is difficult, especially at large Q. c) There may be strong absorption of X-rays if the incident wavelength is near absorption edges of the sample constituents. Neutrons are also strorlnly absorbed by some nuclei, but on the whole X-ray absorption is a much more serious problem. 1.4.2. Complementarity of X-ray and Neutron Scattering. We have previously stated that X-ray and neutron diffraction are properly viewed as complementary techniques. One technique may be demonstrably superior in some applications, inferior in others; but on the whole both methods may be profitably applied to the same problems. The complementarity arises from the fact that X-ray and neutron scattering amplitudes differ for different atomic species. X-ray scattering factors increase with increasing atomic charge nlmber, while neutron scattering lengths do no vary simply

14 with increasing Z and are in fact different for various isotopes of the same atom. Since the various species in a heteroatomic system thus have different weights for scattering of X-rays and neutrons, the same measurement done by both techniques provides two different "views" of the system. This is most important in systems composed of both light and heavy atoms (the obvious example being biological systems containing H), since the high -Z atoms would completely dominate X-ray scattering, while most nuclei have neutron scattering lengths of the same order of magnitude so no one type in general can dominate. Also possible is isotopic substitution to change the neutron scattering properties of a substance, thereby increasing even more the amount of information available from diffraction. In principle one could (using SiO2 as an example) combine the results of one x-ray and two neutron diffraction measurements with isotopic substitution of, say, 018 for 016, and thereby be able to experimentally separate the scattering due to Si-Si pairs, 0-0 pairs and Si-0 pairs. This has never been successfully done, however. Finally we look at the possibility of investigating the dynamics of condensed matter. Suppose we wish to obtain information on the energy and momentum of collective excitations in a solid or liquid, typical energies of which may be.001 -.1 ev (1-100 mev) and typical momenta (expressed in wavenumber, k= p/h) several times 10B/cm (several A ).

15 In order to directly observe the momentum and energy of such excitations, we might wish to observe radiation scattered in inelastic processes whereby momentum and energy are exchanged with the excitations; there must be a detectable change in the momentum and/or energy of the radiation quanta, in order that we may interpret the results. Thus we would ideally like both the momentum and energy of the radiation used to be comparable to those of the excitations investigated. In figure 1 we plot energy vs. wavenumbers for neutrons, electrons and photons in free space. Obviously the neutron has the desirable momentum and energy combination while both e-m radiation (X-rays) and electrons do not. We will refrain from shouting too loudly about the ability of neutrons to scatter inelastically from collective excitations, because it is also the source of a problem when we try to do diffraction. As will be shown in Chapter Two, we would like to measure the scattering, at constant Q in a diffraction measurement; our ability to do this easily is adversely affected by the occurrence of significant energy transfer in scattering. Thus X-rays are in this respect better suited for diffraction applications than are neutrons.

104- Region of Collective Excitations in Solids - and Liquids 102 _ 1020 \ _____ - o r — _ 10 10-, IC-4 E,eV IO'4 Ie 10~ IO I... Figure 1. Energy- veFuer. Cll-ariacteristi es of Various: Radiatiorns

C IAPTI, TWO THEORETICAL BACKGROUND FOR N'EUTRON SCATTERING FROiN DISORDERED SYSTEMS 2.1 Scatterinp from Static Systems 2.1.1. The Structure Factor. For a system of N ri?,,idly bound nuclei, the neutron scattering amplitude iss *S (Sj~~~) = X, Fd~e J (2.1) where r. is the position of the j-th nucleus, a. its boundatom scattering length and Q = k - k is the change in the neutron wave vector. The differential cross section per nucleus associated with this scattering process is I thy) ^ /\/ | Wk($) | -.. ^ -,- i N= 4 > fa. e.- a r (2.2) The quantity I(Q2)/<a is known as the "static structure factor", or simply the "structure factor"; it contains all the structural information available from a diffraction experiment. We will denote it by Ss(Q). Thus I (2) = <a2> Ss(a) <a2> AN We can simplify somewhat the present analysis by noting that Be and F are almost completely coherent scatterers, as shown 19 in table 3. 17

19 isotope coh Tcoh+ inc coh/coh+inc abundance, % Be 7.53 7.54.999 100 F19 3.9 4.0.975 100 Table 3. Scattering cross sections of Be and F Accordingly we hereafter neglect incoherent scattering entirely, and ai will refer to the coherent bound-atom scattering length of the i th nucleus. We can separate (2.2) into "self" and "distinct" terms, -, -,C~ \ \ r <a> I- w' /-2(2.3) where the primed sum is restricted to terms with igj. 2.1.2. Static Pair Density g(r). By noting J,, r)toc) d3r =/; we can express (2.3) in the form iGt6;?) = } t A/ <& > Jg | - iP — ) e / I-,7 / \ -) where A I (d (2.5) We call g(r) the static pair density. The determination of this function is the ultimate aim of most diffraction

19 measurements on glasses and liquids. Its physical meaniny becomes clear on integrating (2.5) over some small region A: far) d 3~ X-, - I, X. jS. l r )'1 d' 4 (2.6) The integral on the right in (2.6) equals unity for each pair of indices (i,j) for which (ri-rj) AZ. Integrating g(r) over any region A gives the number of pairs of nuclei in the sample whose internuclear separation lies in A, weighted by their scattering lengths. It is convenient to rewrite (2.5) explicitly in terms of contributions involving nuclei of a given type in a polyisotopic system: 4 _iAl O~r) - N-' <y>"}L,22 V. Ag.- 4 r,) ^/ / t, i t (2.7) - < k~> X X t (t^) (2.8) where t denotes a nuclear type, nt runs over all nuclei of type t, Nt is the total number of type t nuclei, the factor (1- nn/ tt/) limits the double sum to distinct pairs as

20 before, and Ait ~44 o'1 vIl'./ (2.9) describes the distribution of nuclei of type t' about nuclei of type t. Since (r)= (-r) it is clear that "J /v^ S (2.10) The functions gt,t(r) are in fact what one might calculate from a model of the system under consideration; the function accessible to experiment, g(r), is simply a superposition of the gt, t(). In the case of a monoisotopic substance, one can experimentally determine a function even more closely connected to the structure: l- d-@ V, 3=. dZ', _c'r) (2.11) This function can be interpreted directly in terms of the structure, while the structural information in g(r) is obscured by the scattering-length weighting of the different isotopes' contributions. One would ideally like to extract the functions gt,t(r) from measurements of g(r), remove the scattering-length weighting and superimpose the resulting

21 functions to reproduce g(r). In order to do this, however, one would need n different measurements of g(r), varyinll in each the scattering len'gth of at least one nuclear type, where n is the number of distinct combinations (t't) (e..., 3 for BeF2: Be-F, F-F and Be-Be; one x-ray and two neutron measurements with isotopic substitution would suffice). In an isotropic meditim such as a liquid, gas, powder, polycrystal or glass, g(r) cannot depend on the direction of r, and we have for equation (2.4) - v1 -- f (2.12) where the integral has been extended over all space, and A3 S(/< I t fl rr^jr-:)j S< tr 4 ( G ) (2.13) Ignoring the last term which represents a forward scattering contribution (Q=0) which is not measured, we can write o } t"'= = 77 )Y-) ^,vr - ^(2.14) where 5~$ (2.15)

22 and S (Q) is assumed not to contain the forward scattering s contribution mentioned above. By way of evaluating g0, one can argue that as r becomes large, the distribution of internuclear distances will tend to look more and more uniform. Then (2.5) can be evaluated with ~ (r-ri+r.) replaced by its average value 1/V, /AJl. <a with <A>() I 4 k=J and p= V is the macroscopic number density of nuclei in the sample. Note that the structure factor for an isotropic system depends only on the magnitude of Q. Equation (2.14) is of course just a Fourier sine transform; the inverse tranc form is w sfQL(Q' $& a(2.17) This is the basic relationship by which g(r) is calculated from experimental S (Q) data.

23 2.2 Scatter in5 from Dynamic Systems 2.2.1. Partial diffefrenltial cross section; the scattering law. Here we abandon the restriction of the previous section to systems of rigidly bound nuclei, and proceed 1I,19 again from first principles. As shown in standard texts, the differential cross section per unit solid angle per unit interval of scattered neutron energy for coherent scatters is (within the framework of the first Born approximation, and using the Fermi pseudopotential) y r >s _\ </) 5( \)) (2.18) where C =,M) = = _k'['_) is the energy transfer to the neutron, and Li3) AI/V' -" I - (x) J-tS=c- i< (2.19) is the scattering law. The sum on n0 is a statistical average over initial states of weight p; the sum on n n. runs over all possible final states; the quantity in square brackets is a matrix element of the enclosed operator. The function restricts the sums to transitions with JI) (energy transfer to the neutron) equal E -l (energy transer from the satterin sy n fer from the scattering system). By introducing suitable

2 - time-dependent (Heisenberg) operators, r j(t)=e t/r.e t/ and the Fourier respresentation of the 6 function, one can rewrite (2.19) as S5,) l) l)' ^japwt) Fen,+) It (2.20) where _F, - )-<'c>-' ~ <SPo E,: —^..)(^ ) _ jd,"='~~t~ (2.21) and n- o In (2.21) we ignore the a. as dynamical variables, thus restricting the treatment to systems of noninteracting spins. 2.2.2. Space and Time Dependent Correlation Function ((r t). P(Q,t) is called the intermediate scattering function; it is intermediate in the sense of being the spatial part of a 4 dimensional space and time Fourier transform relation between the scattering law and the space and time 21 dependent correlation function of Van Hove21 R6ig ) = j fp(^-L<.) <?r&A)c1Fr;(2.22) _,'J+)~-()<a> ),US(2.23) (A) <BE[P -ty~l^ iL' t ) yp ir. ^)] d34 "I~~ d

25 Applying the convolution rule for the Fourier transform of a product and respecting the general noncommutativity of the operators r (o) and rj(t), 27T) /e ^ 2r) -fiJr < by) v - P^g( -')<<PW') 43~ G(r t) can alternatively be written S-t (r // ia -' A hSE ~L^R()^/ ) S(-'-d l > >' )2 I V fS i) (2.25) 2.2.3. Properties of the Correlation Function G(r t). Although Van Hove has named J(r_ t) the space and time dependent correlation function, it does not in general have the simple physical meaning that name seems to imply. For classical systems, the positions ri(t) are simply dynamical variables (c-numbers) and (2.25) may be simplified by performing the integration on r': -)' / re > (2.26) G (r t) has a simple physical interpretation: it directly measures the probability of finding a particle at position r at time t, given that some particle was located at the orgin at time o (again, there is a weighting by neutron scattering power of pairs of nuclei just as for g(r) ). Such a function is called a two-time conditional pair density. For quantum mechanical systems, the simplification

26. (2.26) (tantamount to replacing exp [iQ'r (o)] exp [-iQ.rj(t)] in (2.24) by exp | iQ ((rl(o)-rj(t) ) ) cannot be made since rl (o) and rj(t) are operators, in general noncommuting. Any pair of r. taken at the same instant of time however do commute, so one can write even in the quantum mechanical case for the equal-time conditional pair density GoCr, o) - /i,/- -<'<2t~s \ QAd~,'ir i -7) - r/''/ l 7"- (2.27a) =R =,<-;> a~)\22 + i Ad -; d r(2.27b) I -k__ - 6) 6rt) (2.27c) If the identification i m/ad. lerl dthe for o(2. 2) (2.28) is made. Clearly the form of (2.28) is the same as that of (2.5) where g(r) was defined for the static system. More than this, its physical content is the same. Aside from the statistical average in (2.28), which could as well be applied to (2.5) given that several configurations of the N-particle static system are possible, (2.28) is the exact analog of (2.5) for a static system whose N particles are at the positions r=r(o), r=r (o), etc. Equation (2.28) may be thought of as the evaluation of (2.5) for a system frozen in position at t=O. G(r o) thus has the character

27 of a "snapshot" of the dynamic N-particle system taken at the instant t=0. By exami-ning( the inverse transform of (2.23), Clr = ) ^ j L [ ^r- )t)] S(2 - )^ c^) (2.29) one finds a means of experimentally determining G(r o): G^g.- nA - ( ^f p^ ] S%3 ) J a "o J 4) 71-7'3 S,[,'a*r] S'&) P3J c< (2.30) with the identification SfL) - J S(; W) JLwO 2.31) Thus combining (2.28) and (2.30), io+ s ) = (21r rY) ^^.r] sti2) 5^ (2.32) or by analogy with (2.4), 5{&)= I t i fpL-t Jr g& )4Y (2.33) We w7ill call S(Q), the integral at constant Q of S(o W), the "structure factor". It is the analog of the static structure factor introduced previously, containing the same type of information but arrived at via a more realistic analysis. We have already discussed the information content of g(r) and its Fourier transform Ss(Q) in terms of the static system model. We have now found that by arranging to measure

28() S(Q) as defined in (2.31) we can collect the same sort of information for the more realistic dynamic model. If instead of S(Q) we measure only the "elastic diffraction" S(Q,o), we obtain different information as can be seen from equation (2.23): gS, o) ( fs) i'/p [-'. ] 6r, D d3r/ - W'^ - f'-'_L i Y- r) 3r, (2.34) Co t:r) - f ) &d e Thus by Fourier transformation of S(Q,o), one obtains r (r) which is somewhat difficult to interpret, and not g(r) as when S(Q) is transformed. It is clear then that to gain access to g(r) using neutron scattering, one must measure the total coherent scattering at each value of Q, ie, measure the structure factor S(Q). We postpone until later consideration of the feasability of arranging a neutron scattering experiment which will actually measure S(Q), a difficult question indeed. Finally, we note that for isotropic systems, g(E) =g(r) only and (2.33) can be orientation averaged to yield ao (2.35) or so

29( with i(Q)=S(Q)-l and the forward scattering term proportional to &(Q) has again been ignored. The analysis of glass diffraction data is based on equation (2.36). 2,2,4. Scattering from Systems of Harmonically Bound Nuclei. Here we derive expressions for the elastic and onequantum inelastic scatteringt cross sections for a large system of harmonically vibrating particles. We utilize the very general and elegant formalism of Zemach and 23 Glauber2 for scattering from molecules; we treat the entire glass sample as a single "molecule" in the sense of their work. The starting point here is equation (3.20) of the paper by Zemach and Glauber for the partial differential cross section of the harmonically vibrating system: LIy Atl t (2.37) where b is the position of the Vth nucleus, and (x) X;.j V'?z}/ A a) (-) (2.3a') (-x InV, /ldj (2.38)

30 wi th C = the polarization vector of the vth particle in the Ath normal mode, defined by lA,) = j' ~ ~ ) where U (t) is the displacement of particle' from equilibrium and q (t) is the Ath normal coordinate. )A = the frequency of the th normal mode MV = (myVm, ) I (x) = the modified Bessel function of the first kind of order n, = -/Boltzmann's constant / absolute temperature The continued producttT can be factored as AhA 4A=l CACO AL r% L —cv (^~) t ~AA S/)J (2.39) The first continued product of exponentials can be rewritten as the exponential of a sum,,' (2.40) 4,iA-' - S (2.40) which is the Debye-Waller factor expressing the diminution of scattered intensity arising from delocalization of particles in thermal vibration. Focusing attention now only on the continued product of infinite sums,

31 3 Al co A =7 X,^ ^^iA /) Z-AA/) -,tA C1Va SA U)'4/) 2 41 which can also be written 3^ " -- * 7T =p'nA ) Y /A^ ^ (X) - yn rL,~y S ^(,R,)J (2.42) where in? represents a (3N)-tuple of numbers (n1, n2..nN) and the sum on rfn) extends over all possible (3N)-tuples as n,n2,...n3N all vary from -0 to cc. Conmbining (2.37), (2.38), (2.40) and (2.42), A -.z ^>^ /-^^ —^')J -A,A?2 —(2'ri_., a,- &/ L-,"t't +i'.L(,-b _ -2,. (x ) 21 7it.....) Jf 5(-C 4)./2) (</V WA — J-p j^ (2.43) I — "2'a W (x a Les, -c, < ^1f^V) J Aj Ui.-4 — VA\ /I (X) TT "I', (2.44)

2 The sum on tAj has been integrated term by term over all t to produce the energy-conserving functions; this step is of course only justified if the sum is uniformly convergent throughout the ranige of integration. We ignore this question entirely (being a dilettante in theory saves considerable work at times such as this). There is one set f^ 2 with all elements zero; this corresponds to an event wherein no quanta are exchanged with any normal modes, i.e., elastic scattering. The contribution from this single elastic event is tcA C) (2.45) in order to approximate the I contributions, we simply assume that their arguments are small, *i^^yX').i<< so that the modified Bessel functions can be approximated to lowest order: or specifically I, tI.)',,/,;, (2.46) Then (2.45) becomes

).) c ~C") al,~ k/ T-,._' ~t~' (2.47) For "one-quantum" events, i.e. those for which the set,^A contains (3N-1) zeroes and a sinlirle one (2.44) is A:,!').t, L^ C _ G, ~ ~.._.~h\,, | )U. ~A=!k 2. ~ /' - 4 E r- ) /;w' 2.3 Scattering from Acoustic Mlodes in Disordered Systems of Harmonic Oscillators Here we attempt to calculate approximately the scattering law for an oversimple model of a glass, one which requires no more knowledge than we have of the real properties of a glass. To completely specify even the onequantum scatteringj as expressed in equation (2.42,), we must diagonalize a 3Nx3T\ matrix to obtain the normal modes

34 of the system -where N is of order 1023. Tn a crystal, symmetry considerations make it possible to consider a much smaller matrix as a function of the phonon wave vector %. However the size of this matrix is only 3n x 3n, where n is the number of atoms in a unit cell of the crystal. In a glass we have no corresponding symmetry, and must either investigate what other means can be used to reduce the size of the problem, or else set out to consider only a small part of the problem in the first place. What do we know about glasses that may be useful in calculating the scattering law? For one thing, glasses like all solids transmit sound waves. The wavelength of these waves however, must be sufficiently long that the glass appears to be a continuous medium. If the wavelength is too short, we will no longer have wavevector as a good characterization for the modes. We certainly know something about the glass structure; we have measured the neutron diffraction pattern and from it computed the neutron rdf. Although it is an integral property of the structure, still this information may prove useful. Suppose all we know about glass is its structure factor and that it transmits sound waves. Let us try to calculate the one-phonon scattering from sound waves. It is convenient to rearrange equation (2.48) somewhat c5u-bs Jk' c >'' (2.49)

35 where (X) — t (E'?.'),i`es (h _ -. / ) A, ~ — A v ) Al- - J y' [c^J2);iVi^ fl ^) ) ) / / ci a, L{ )&7'x~~l~a2i'Vp) S61cL^ / AI x\ 2.50) Note that )^ has been replaced by ) and brought out of the sum wherever it appeared in (2.42). The c (o'-qA ) factor allows us to do this. We call FA(Q) the "inelastic structure factor" and S(Q a)) the "symmetrized scattering law" since it has the property S(-Q -t)=S(Qa) while tlhei unsymmetrized scattering law apparently satisfies S(-Q -LO) = S -)= i~pc: S(Q,w) which is called the detailed balance condition. The factoring of S as a function of t times the inelasti c structure factor sum is made for convenience since FA(Q) contains all the effect of the A mode, while the factor depending on t is the same for all modes with frequency to. Now to incorporate the assumption of acoustic wave modes- we will treat modes for which nuclear displacements have the form ad wl auta=asre(2.51) and will assume that w = w q as is true for sound waves in a continuous medium. Furthermore, we will assume there

36 are only two values of 4, one for the longitudinal and one for the transverse modes; a mode is longitudinal or transverse if the polarization vector aA (9) is parallel or perpendicular to the propagation vector q. We expect one longitudinally and two (orthogonal) transversely polarized modes. Now let us explicitly write down the acousticAA mode portion of the scattering law, which we call S: 5 Y((,) = L2 X(fy)] X i ) Z, mFg ) r- dk~ [o?^^^f-^>" - fT^ 0Q-^^ )]"Z^ ^ ^e (,) j2a[pC^^+)~^V O] 6V-(L ) (2.52) We recognize the sum over atoms in (2.52) as the elastic structure factor, So T) - aa>J' J ^ d-lpr CL p >i *g ^h -Vb) (2.53) so (2.54) We transform the sum over a in (2.52) to an integral in the usual manner, where V is the sample volume.

,7 Thus'S t,^ - bLLst%/e;)&' @ft6) t6tc-4_ ) J (2.55) (x) [ I t Se(R 4) ] bta~J Q) d g (2.56) Now to evaluate the _ integral, we need expressions for the aj(4)- Since we have only longitudinal and transverse modes this is simple: for the L modes, ) = g) ~cqL ) (2.57) for the T modes, - 9, Xg 4'1(Lt)= 6L<(^ (2.5Q) and b.b2=0 to insure orthogonality of the T and T2 modes. We can evaluate the numerical value of a and the aT using the normalization condition for the c, expressed in terms of the a's: 1;=I - Z c < - Z^-(2.59)(2.59)

for L modes and 14V- 41.T- = 7 r2, _- _V = A / (2.60) for T modes. Using (2.57) and (2.59) we find Al — M is apparently the total mass of the system. Using (2.58) and (2.60), VY- I and analogously for ya. (q). In toto then, the normalizations are 71-I (2.61) Now to evaluate the integral in (2.56). We first note that for systems of interest here, Se is a function only of the magnitude of its argument. Also noting &(ax) = a-l(x), we write for the a- integral J3T/ f) a-f - A ] ^ ( J] at) S14:() d.^6 (2.62)

39 with $1 = (Q<.) Q To evaluate (2.62) we use a spherical polar coordinate system with polar axis along Q, - c 7YcjLZ'<o\yj f[i L i~t s( (C < O') I /. r27, OZ y;trg3 + 5 +.i ) J I — I I (2.63) Similarly we find /~r l i ) -;T, )? L) — I j -"iL 2' noting that ('a.T )2+(. aT )2 is just the squared projection ~ 2 2 of Q on a plane normal to a, and using the fact that a T, 2 T Thus 2 1 i l~' -- ) t $- q L ei)_<r )y -I (2.64) Substituting JL and JT into equation (2.56) for the acousticmode scattering law we have finally

40 - iL ^^eA statYe)] -n i^ ) )- 4 LC, _4 -3,mT 3"[ ) L 3 <) + - 3 # t t2,,r )14 (2.65) We have separated the total system mass M into the product of a "unit of composition" mass MI, and the number N of such units. A unit of composition can be whatever is convenient; in our case one Be and 2 F's is an obvious choice. The ratio V/N is just the reciprocal density of such units, u u. The quantity h Q2/2M may be thought of as a recoil e u u energy ( it is actually an angular frequency) associated with the transfer of momentum tQ to a single unit of composition. For convenience we have also introduced the moments <W) " A4 ) ( ( +itt< V + 2t- - ) d (2.66) 2.4 Generalized Frequenc Distribution Function, G(C,t) As an aid in interpretation of the inelastic scattering data, we will reduce them to a functional form in which the underlying information about the system's normal mode frequencies will be made more clearly visible. We define

41 (r4 ) Sr,4 o2/s4 A):, 7 "l ( 2.,7) with S(Q,4o) as in equation (2.50). This function is easily calculable from S(Q,A>), which we;,et out of the inelastic scattering data reduction proceduire (note that equation (2.50) represents only one-quantum scatterin,; we must assume multiquantum contributions are small for the present analysis to be valid.) Now consider the frequency dependence of G(Q,4)), which we first write explicitly: iQ= L>>7- re f: ^ c> *~ Tl ^ C yC )..*j-QVP)L ]-'vMSV- <2v) IS 2.r -%3 (M2.,) We will not attempt to evaluate the interference terms, b]ut will focus attention on the self terms which we separate and call G. In the simplest case of a monatomic system ~v/e*.= 5/ ~ and -2U i VJ (2. 9) _- L__ ^Q'.. (2.70) where we have defined the "polarization weighted directional

42 frequency distribution" according to C A (2.71) A th and C, is the ot component of C. The determination of the functions GV (w) is as close as we can come using neutron scattering to a measurement of the actual normal mode frequency distribution of the system, 3,J (2.72) Note that the normalization of the CA implies ~v sL~ v Z< — = 21 J - ) 6L- - -L) Unfortunately, in the present case we have a polyisotopic system, and in reality one cannot ignore the interference terms in (2.68). We can however, still evaluate G(Q I) using (2.67) and we expect that (treated as a function of w) it will at the very least exhibit the same singularities as g(w). Just for the record, we write down a polyatomic analogue of (2.70) including interference terms: wha 6 =L.4ereLJ <']>j'' ___ 4Ac where.- (kPLK( L CL d-Lb97 C/L V J~l^;- irv-^,)j Adi

by analogy with (C (w). There is little need to point out that the dynamical information contained in C(r,hl ) for a polyatomic system is considerably obscured by the various weighting factors (scattering length, scatterer mass, Debye-Waller factor, polarization and interference factors) appearing in (2.68), as well as by the additional spherical averaging over L which occurs in a glass. As we shall see later, however, the G(Q w) function computed from glass scattering law data can still provide some insight on the nature of g(W) for the glass.

CHAPTER THREE EXPERI MENTS 3.1 Time of Flight D!iffraction 3.1.1. Method. We begin by writing down a very general expression for the result of a neutron scattering measurement in which the scattered beam is not explicitly energy analyzed. For the counting rate per unit solid angle per unit time at angle 4 and time t, we may write (, ) S(r- - _ d E (3.1) with /4 E' = = the energy of a scattered neutron E ='^m'= the energy of an incident neutron 7(E') = the detector efficiency at neutron energy E' p(E,E'$) = the probability of single scattering through angle 4 with energy change from E to E' for an infinitesimally thin target I(t-T E) = the intensity of the neutron source at time t-T and energy E. F(EE',-) = a correction for nonideal scattering in the sample (this includes both "attenuation" and "multiple scattering" effects, based on the assumption that the neutron interacts instantaneously with the sample) 44

/~ 5 L,L' = flight path lengths between source and sample, and lbetween sample and detector In general, if Ni scattering units are illumi nated by the beam, N Z We will consider cases of the above general relation, which we classify according to the nature of the source term in equation (3.1): a) conventional diffraction T(t-r, i) (E) (E ( (-_) independent of t-t where i0 is the fixed incident energy and.(E ) is the monochromatic flux at E. b) time-of-flight diffraction Ilt-' Jl) (.4. ) (t-rt ) where S(E) is the energy-dependent intensity, in neutrons/ unit area, of a source pulse. In the conventional diffraction case, (3.1) becomes 4 (+t, = tr2,?, s)^- IK t >, z tE) d'/ E" c-~~, >~~~ (3.2) In the time-of-flight diffraction case, P r'1t?) =f 3v')E LV Fv (')s ) (*) ~(T- _-L')e-' (3.3)

4t6 Each type of diffractometer may be thought of as measuring the integral of the partial differential cross section along some path in (Q,E) space. In order to understand exactly what function one measures in a given experiment, it is necessary to examine the integration paths of the various types of diffractometers. For the conventional diffractometer at a scattering angle $, the instrumental integration path is determined by conditions E-= E' s IL /ih k I- Gl (E Jo~-^ m ) (3.4) and is shown in figure 2 for a variety of E: and. For the time-of-flight diffractometer, the integration path is determined by (3.5) or (2.6) which depends on the flight path lengths L and L'. Examination of (2.5) shows that for L>> L', t is determined mainly by the incident energy ~~, while for L<< L', t is determined mainly by the scattered energy A'd. Loci of constant time of arrival in the (Q,) plane are shown in

1 7 00^ - cI0 O OO O 6 - Cd X rN -, o. c 0 ep'iC) (d o (rN,3 -.I - 01. 0 -4 C(

48 0 figure 3 for 90 scattering and for various choices of L and L', all with the total path length L+L'=5 meters. When one measures an experimental point at Qe=a 2k using either diffraction method, what one gets is thus not S(Qe) but the integral of 2'E (Q,e) (weighted by detector efficiency, and in the time of flight case by the incident intensity) along a path such as those shown in figures 2 and 3. Recalling that one wishes to measure 5^)= JS(Q^U)) diu f(i& )- d< it appears that the most favorable arrangement is that whose integration path is most nearly vertical, and figures 2 and 3 show that this condition is best fulfilled by the time of flight diffractometer with L=L'. Furthermore, for the case L=L' the integration path is more nearly vertical for smaller times of arrival, i.e., for higher energy at a given scattering angle. 24 Carpenter and Sutton have examined the effects of the detector efficiency and incident spectrum in "weighting" the integration performed by diffractometers: their conclusion is that the equal flight path (L=L') time of flight configuration provides a good approximation to the structure factor. Powles, Sinclair and Wright and others however 27 prefer to apply what are called Placzek corrections to

nowrr~~~~~~~~~~~/, ("uf CO C r - -------------- * i lliliXc... "ft* G-) ~-' ^ (., 4, U.,,.. ~~~'~-.,,, t._,.,. —-4 ~~~~~~~~I eee~ t~~~~~~~~~~~~s 11-i., -;'V, -: ~ N....eee e ei ee-' r0 r (,, -1^0't N'N.^ —~...^~~~~~~~~~~ ~_ I L_,, m.; -.

50 extract S(Q) from the diffraction data. A series of correc22 tions was calculated by Placzek and by Wick in terms of the energy moments of the scattering law; these corrections of course can only be calculated insofar as the energy moments are known. We consider the subject of whether or not Placzek corrections are preferable to an equal flight path arrangement as unsatisfactorily resolved at the present time; we show here some experimental results supporting the equal flight path method. Figure 4 compares measured structure factor data for glassy BeF2 from the Michigan equal-flightpath diffractometer, with integrated scattering law data from the Argonne time-of-flight spectrometer. We have made the comparison at an equivalent level of data reduction in both cases; multiple scattering and resolution effects are not corrected for. It is emphasized that these are totally independent measurements and that no normalization of one to the other has been made. With the exception of the immediate vicinity of the first two peaks in S(Q), where differences in multiple scattering and resolution effects are expected to be greatest ( as will be explained later, the configuration of the target was different for the two experiments; multiple scattering and resolution effects are expected to be more severe for the time-of flight diffractometer, consistent with the trend of figure 4) the agreement is within several percent throughout the region

21 2.5.. 2.0- l~ 0* * ~,.0 * 0 O0 * 0 9o o0. o 0 0 a'; ~ 4Tr S(Q,w) dw from ANNL TOF Spectrometer Independently Normalized, Neither Corrected for Multiple Scattering 0 2 4 6 8 * Go 0 ~* 0 *...Ii c~0 A 0 *. Sructure Factor from U-M Diffrctometer o-ifS(Q~w) dw from ANL TOF Spectrometer Vj Lu-i; 1- Vi- ii 1. n- t - v-i t i- ScatLteringI

52 1.0<Q< 6A where the scattering law integration is reliable. This is well within the combined statistical error of the two measurements; we thus assert that the equal-flightpath arrangement provides a reasonably good measurement of S(Q) even for a sample such as BeF2 composed of light nuclei where Placzek corrections might be expected to be severe. This may be due at least in part to a fortuitous cancellation of errors due to omitted Placzek corrections in the sample and reference; but this seems improbable at best. 3.1.2. Time of Flight Diffractometer. The equalpath time-of-flight diffractometer at the University of 29 Michigan has been described in detail elsewhere. We repeat only pertinent details here. The diffractometer operates at beam port J of the 30 Ford Nuclear Reactor, viewing the D20 source tank0 (The general layout is shown in figure 5.) The beam is coarsely defined and shielded by a 103" long plug in the beam port, with an exit beam size of 1.97" x.54"(height x width). Between the rotor and sample is a 95" long, 1" x 3/8" collimator which together with the chopping rotor collimates the beam to.0092 x.0079 radian (vertical x horizontal). Beam size at the target is 1.5" x.5". A rotating collimator (10" diameter fiberglass resin rotor, denoted "rotor 1" in figure 5) in the beam upstream from the chopping rotor cuts down the unwanted fast component of the beam, thereby reducing

53 C 0 w ILLIZ U.~. Q 0~~~~~~~~~~~~~~ ar~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C Co K: v w; U \o. m E 34 w cr C Z _ < to ow I v CO~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C X C) o D z Z -J~~~~~~~~~~~~~~~~~~~~~~~~~~~~cr cCO - V <- < zz (j co wxo LLV) W LL. W C 04 W.~~~~~~~~~~~~~~~~~0 ZW TC w Oco d-i~~~~~~~~~~~~~~~~~~.0 U) cr.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r W ~ ~~dI; ~ ~_O ~~~~~~? 0 F w.e ~~~~~~~~~~0w O rl" o- 0~,[r Ws [x 3 _ u uLL I Ir.. Q wcro~~~~~~~~~~~~r O~~~~~~~~~~~~~~~~~~ cr 3 r w ~-C-~~-1~L < tI II wiQ v, C~~~~C U -Jk z~~~~~~~~~~~~~~ F' PT~~~~~~~~~~~~~~~~~~~~~~~~~r

54 background counting rates. Distance from rotor 2 to the sample position is nominally 2.37 meters. Banks of 5" x 5" scintillator type neutron detectors are placed at nominally 2.47 meters from the sample, at scattering angles of 20 and 90. The detector faces are 31 situated tangent to time-focusing loci at the respective angles. Borax, borated water, B4C, concrete and cadmium in various amounts shield the detectors from room returned neutrons. The TMC 1024-channel time-of-flight analyzer is normally used to analyze four scattered intensities in 256-channel segments: sample scattering at 900 and 20, and reference scattering (see below) at 90~ apd 20. A homemade instru32 ment, the "Quadapter", routes detector signals into the appropriate memory segment after pulse shape discrimination 33 against gamma-ray induced pulses. A variable delay of from 1 to 2047 microseconds precedes the start of analysis after each rotor opening. 3.1.3. Vanadium Reference Method. Writing for the partial differential cross section Poet~ J&) - <'c> 5YQe) S6() (3.7) which we will call the "static approximation", Equation (3.3) become s

55 F'or a s i. mi la r me sll trel' t- on an.r i rcoherFent lst. t i f. ('e rence scatterer, for which we have C (~,) = 7S) N 6 r) st) 55- 4Lt ) (3.9) Dividing (3.9) by (3.9), _tr_ _ p< d> SQ) F Now i.f the properti es of the reference scatterer are known and F can be calculated, S(Q eO).(,) can,be extracted from (.10), without knowledge of either the detector efficiency or the source intensi ty. Furthermore, if th e mieasuremesnts are n-ade more or less simultanecusly tby alternati ng the sample arcl reference scatterers in the tar!get posi tion, any fluctuations in'(E) or S(i) on a time scale lonA compared to the target cycling time will average out in forming the ratio (3.10) A useful reference scatterer is vanadium, whose coherent 34 and incoherent cross sections are.03 and 5.13 barns re35 spectively, constant in energy to excellent approximation and predominantly elastic.

56 The reference scatterer used in the present measurement was a V plate of dimensions 2"x4"x.125", with density NR=.0720 atoms/b-cm. The sample and reference scatterers were alternated in the target position each 5 minutes; the total counting time at each of two rotor speeds was about two weeks. 3.2 Time of Flight Inelastic Scattering Spectroscopy Returning to equation (3.1) we treat the case of a pulsed monochromatic neutron source: i(-6, E) = (Eo) &Co) s&-r) (x) t- J (3.11) If we displace the time origin by L/c so that time is measured from the instant at which the burst reaches the target,'5- ) = lt) yo ) $4) =t ~ E -) Z_ LIe (3.12) with E = C tL/,) ~ Thus each time of arrival t is uniquely associated with a particular final energy. Furthermore, since _=- Eo - t: - L/J ) -L and Q- 3i i^ L-D t L=, - J3. LI-, L CK I )

57 each different combination of v and t corresponds to a unique point in the (Q,v) plane. The optimum way to perform such ameasurement then, is to time analyze simultaneously the output of several detectors at different scatterin, angles; thereby measuring, S(Q W) at a set of points (one for each time channel) along a locus such as that shown in figure 2. Loci for the present measurement are shown in figure 6. It is then a simple matter of interpolation between angles to arrive at S(Q w) at constant Q, probably the most generally useful presentation of scattering law data. Wle pause to remark that the TOF spectrometer is most useful for cases in which S(Q w) information is desired at all Q and w, since all this information is simultaneously gathered by the TOF instrument. This is in contrast to the constant -Q scans possible with triple-axis spectrometers, where the scan is limited to a small range of Q and w of interest. For this reason the triple axis systems are more useful for phonon measurements in crystals where the positions of peaks are more important than the overall shape of S(Q,w); conversely the TOF spectrometer is most useful in measurements on isotropic systems such as liquids and PMlasses. 3.2.1. Thermal Neutron Time of Flight Spectrometer (TNTOFS). The time-of-flight spectrometer (T',TFOFS) at the Argonne National Laboratory C: -5 reactor has been fully

aI) OD o, 0 0 0 44 1* 0 C l-^ - o.9) ^-Q) ~} C./ F-r 0 _0 c: 0000000 00. 000000 0 oof.00t 1') - - N rO~t gI / I L 0 3 -, 1 I3 o - I ~ I

36~ 5 described elsewhere. Ve present only the briefest description here. A plan view of the TNTOIFS is shown in figure 7. A monochromatic component of the reactor neutron beam is selected by the double-crystal monochromator. Ilse of the double monochromator effectively eliminates gamma-ray and fast neutron contamination i.n the monochromated beam with their associated shielding and background problems, as well as allowing a more compact layout of the spectrometer with constant monochromator take-off angle. Tn the present experiment, copper crystals cut for (220) reflection were used to produce a monochromatic beam of nominal 1.5A wavelength. The 1.5A beam was fil.tered throuph 6" of NigO single crystals held at 77 K, to remove any short wavelength contamination. A 3" diameter Fermi chopper was used to pulse the beam at 1,320 pulses/minute. Nrinety-five 1" diameter le3 proportional coun-ters situated Twith axes vertical on an arc 2.5 meters from the target position were used to monitor the scattered intensity. These were divided into 31 subgroups each consisting- of two, three or four adjacent detectors; the signal responses of all detectors in a given subgroup were analyzed as a unit. The,roupin?- of detectors entails a relaxing') of Q resolution due to increased angular spread vis a vis a single detector, wi.th a concomitant i ncrease i.n counti.ng rate. Most of the detectors used had an active length of 12", with a few 9"

60 no Monochromator Crystals FC2 /FC I Fiure 7. Plan View of the TNTOFS Sample Pickup Coil - Chopper C! Detectors Reactor Face Fi_,rure 7, Plan View of the'T.?<'?OFS

or 41." longl detectors at the smaller scatterinig angles to preserve Q resoluhtion. The range of scattering angles monitored in the present experiment was 13.2 ~ 4 120. The entire 40196 channel mnemory of the Nuclear rData 50/50 time of flight analyzer was utilized i-n a confi.?guration of 32 subgroups of 122- channels each (the 32nd subgroup was used to analyze the signal of the two beam monitors. ) Vanadium was used as a reference scatterer in the TNTOFS measurement, but the details of the method used are quite different fronm that described above for the time of flight diffractometer. At TNTOFS a separate run was made with a thin V sample as target, and the calculated detector efficiency was adjusted for each subgroup to reproduce the theoretical value of the V elastic scattering in the data analysis. The adjusted detector efficiencies were ther used in the analysis of the sample data. Such a procedure depends quite heavily on the detector efficiencies remaining constant throughout the V, sample and empty container runs; this may not be a particularly bad assumption in the case of the He detectors, but the fission chamber beam monitors used to measure the integrated incident intensity for each run tended to vary quite a bit during these measurements. This is overcome to some extent by use of a supplementary 3 monitor, a He' detector which measures scattering of the incident beam by the chopper. It is this peripheral monitor

and not the beam monitors per se which was taken in the present experiment to provide a measure of integrated beam intensity at the target position. 3.3. BeF2 Sample The vitreous BeF2 was prepared by C. Bamberger of ORNL. It is in granular form, granule dimensions being typically 1-2 mm. Immediately on preparation the BeF2 was placed into.25" O.D. aluminum tubes which were sealed at both ends with 0-ringed plugs, crimped and filled with epoxy cement (see figure S). Each of the 14 tubes contains 1 gram of BeF2, filled with approximately 62% void fraction. The effective BeF2 mass density averaged over the sample volume is 0.75 g/cm3. For use at the time of flight diffractometer, six tubes were arranged with axes vertical as shown in figure 9. The choice of 45 inclination between the beam and the target face is made on the basis of time focusing considerations for the 90 detector bank. The effective thickness of BeF2 along the beam (averaged across the target) for this arrangement is 1.085 cm yielding an effective area density in the beam of.0104 BeF2/b. It is worth pointing out that the choice of 45 rotation of the target with respect to the incident beam, which optimizes time focusing for 90 scattering in reflection geometry, results in suboptimal focusing at 20~ and in fact causes the 20~ scattering

EPOXY I ji C Scale I" I/2" C "O" RING Fure. he L A0 C) Scale I":I/2" "0" RING

o9 / 4- E / a)I /20~ Scattered m / Direction' 45~ 90~ Scattered Direction 0.58" I Fitgure 9. Tar.!.t Arran.!eu.ont for tile' TIi nl. -of -Flight Dit ffractometer

"5 to be in transmission geometry. This introduces some complications into the calculation of the various;-eometric correction factors, but is deelmed worthwhile solely on the basis of improvement in resolution at 90. Making separate measurements at 90~ and 20, each with the target inr the respective optimum focused geometry, was riot considlered a vi.able alternative; since the 20 bank measures at.ow ( where the resolution is adequate even i r nonrfocused;reometry, the superior efficiency of making- both measuremen,(Its simultaneously was opted for. For the T'NTOFS measurement, all 14 sample tubes were assembled into a single layer with axes horizontal (fifgure 10). Cadmium leaves.020" thick were placed between adjacent tubes to prevent intertube multiple scattering,. Thi s lecoupling of the tubes not only reduces the sevxeri ty of multiple scattering, contributions, but also amikes the calculation of a multiple scattering correction somewnhat simpler (see next chapter). The single layer of tubes was placed in the TNTOFS beam with the sample face rotated 45 clockwi se from the beam direction. As shown in figure 11, this resulted in transmission geometry at all scattering angles examined in this experiment. A cadmium mask with a 3.7f" x.8'" rectangular aperture placed normal to the beam direction, upstream from the target, was used to accurately define the beam

r*6 A ~~~~~ II. _ __' — I.- l==d _ i-,i I I _~~ ~._....... rilled _.. BeF,.-.-2_ __.. lu bes l...Li~uc~ I. _~r t~ —,t T'~_rO.. ~_ _ i ii,Fiure 10. -arget Arran~onienL for the TNTOFS _: - ~_ _ ELI:..,..

67 Q) E ~/ \ -^~m 4/ \ aC Range of Scattering Angles Covered in Present Work 120 Target' \. 0 Fi.'AS 1 f ] 1.. eo.'OicL He y lo r th.L 1 11e r NT o v, l- s!.x i r',.. 1-.

A area. The same mask was used in the V reference scatterer normalization run. Effective thickness of the filled region of the target for this configuration is Tr/2Acp; r the inside radius of a tube and S the rotation angle away from the beam direction. For the present case, this is.634 cm. The sample area density in the beam for this configuration is.6077 BeF2/b.

CT AI T:R t O i R' CALCI).LAT TON OF THri STiR CTI UR lE FACTOR FtRO' DrI FFRlAC' TION IATA 4.1 Introduction In the discussion of the V reference method in Chapter Three, we asserted that by forming the ratio of sample to reference scattering we could,et at the sample structure factor:'Z~~(4.1) In this chapter we will deal with two problems: first, to move from raw data to the sample and reference scattering. intensi-ti es C" and C; and second, to calculate the factors F(E I ) and FI(E ( ), and thereby become able to extract S(Q() from (4.1). Before beginning. a detailed description! of the data reduction manipulations, we briefly outline the entire rrocedure. Tt is summarized in the following( steps: (1) Remove the background contribution from both sample and vanadium raw data. (2) Form the ratio of sample to vanadium counttint rates. (3) >iake a prelimi.nary removal of the container contribution from the samnple scat t ir'in-r,. (4):sstimnate the sample scatterinrg cross secti on, and utilize it in effecting a fi nal remnoval of container scattering. (5) Compute the factor F" to correct the i ntensity ratio.,. n

70 (6) Compute an improved scattering cross section estimate and differential cross section. (7) Compute the factor F and correct the differential cross section estimate for multiple scattering. (8) Iterate if necessary, steps (6) and (7). The factors F(E,4) and F (E ~ ) will be called "multiple scattering corrections." We take this term to mean the following - suppose we have an infinitesimally thin slab sample, such that the incident beam has no chance of being attenuated before scattering, and the scattered neutrons have no chance of interacting after scattering. Then we will see scattering proportional to t <a2> S(Q), which we call P, the probability of scattering for an "ideally behaved" sample (e is the number density and t the thickness of the sample). In this ideal case we detect only oncescattered neutrons. If the target is somewhat thicker, the sample is not uniformly illuminated due to the attenuation of the incident beam by scattering; the front of the target "shadows" the back. Furthermore, once-scattered neutrons heading for a detector may interact (i.e. be scattered or absorbed) with the sample and be removed from the scattered beam. These two effects taken together reduce the once-scattered intensity seen at the detector, and are simply termed "attenuation." We may define an "attenuation correction" FA(E,)=P1(E )/ P (w 9 ) where P1(E,f) is the probability of observing once-scat

71 tered neutrons for the thick target. It is obvious, however, that we cannot have attenuation effects going on witihout also havin:g some probability of a twice-scattered, thricescattered, etc., neutron happening alontg into the detector (except possibly for strongly absorbi-ing sample materials). We can define a hierarchy of probabilities of single scattering, double scattering, etc. resulting, i.n our detection of a scattered neutron: -l One may choose to call FA=PI P1 an "attenuation correction"'1 and l+P PF, a "multiple scattering correction", which is applied to make up for the inadequacy of the "attenuation correction." This is no more than an exercise in algebra and reflects the fact that one must really know F=F'I lT to get at the interpretable quantity PI from the observed quantity PT. In point of fact FA is generally quite simple -1 to calculate while 1+P1 P, is generally quite difficult. WV'e will utilize a Monte Carlo approach to compute FI, P1, P4 and hence F; and we will refer to F as a multiple scattering correction. 4.2 Calibration of the Diffractometer Before proceedi.ng with the data analysis, it is useful

72 to establish a calibration of the instrument in terms of Q vs. observed time of arrival. This must be done for each different rotor speed used to accumulate data. We define an overall timing error, terr, by the following relation between observed time of arrival and actual time of flight from rotor to detector for channel n: i-(n-I^A~~ - +td t,+ f ^ +^, *err =ots +e~rr- (4.3) with t the true time of flight, n the observed channel number, At the channel width, tdel the preset delay between rotor opening and analyzer start signal and terr the overall timing error. The fixed contribution of -1.5 A t is due to the fact that the time analyzer requires some time to store a count presented at its signal input; thus a count stored in channel n actually arrived in the interval (n-1.5) At + ~At. We allow t to absorb all the possible sources of timing errors, principally misalignment of the "rotor open" signal device and delays introduced by the detector electronics. In order to establish a calibration, we take advantage of the fact that the diffraction pattern of a powder or polycrystal consists of sharp Bragg peaks at well-defined values of Q. Using the Bragg law nA =24 sin i9/2 for lattice planes with spacing d, and the flight time for neutrons of wavelength \ over a path of length L+L',

7'". t(A) — (t+-t )/A-) -- - 1* - the -- gLtL') > SChB L = -- t L )A^ k F (4.4) for Bragg scatteri n from planes (hkl). if a polycrystalline sample produIces rraggr peaks in channels n. which can be identified as dule to Bra-g planes (hkl), one has then combining (4.4) with (4.3), j (t L')X\ 0-A 1 5)4-t t-4 + t-r t (4.5) The adjustable parameters i.n this relation are (L+L')sin -/2 and t. Tf the indexi.rn of Brapgr peaks has been done err correctly, all the points (thklhkl) should lie on a straight line whose slope and intercept at A =0 fix the adjustable parameters (L+Il')sin7/2 and trr, respectively. In practice we assume a value for A4, which fixes the X associated with each plane. Only the path length L+L' is treated as unknown. For example, the Bragg peaks from the aluminum container seen superimposed on the 90 BeF2 data of fi;gure 12 can be indexed as shown*, and the resultin-,g vs.t plot (figure 13) yields t =-t(X =0)=44 $ sec and L+L' = 4 =.90 meters. The accuracy of our estimate for 4 is irrelevant in determini.ng Q(t) since Q= 21F/d and the calibration of d vs. t is independent of our chosen.';Information on aluminum Bragg scattering is presented in table 4.

74 (OOZ) - O c'J o-J q C; ----''g: O __1 I —: "O Z (11~)- - - - m 0 E L(O(O Z2b -- J6' - - 0 L C n 0 0 Lr) C b, oo o- o o oo I-II II I I C co- 0 N ^ ^t- rC) rO) N - - "13NNVH3 NI SINnlO

0 0 *(OOz) 0 O 0 r ) C\ OleO \ N - 0 C (O) tI \ \ /~I v0 j (:\ <^ 0 1,, c (ZZZ,\ 1 0 0 0o 0 0 0 00 0 \) LOIK 0 0 m 10 O ) N 0 — CM CM c\i ( ) 0M

76 TABLE 4. Br>,; Scat torLi, by Ahlnvnrlnun Plane Multiplici.t' d(A) Q((A1) 111 4 2.33803 2.68738 200 3 2.02479 3.10313 220 6 1./3174 4.,388499 311 12 1.22099 5.14597 222 4 1.1(901 5.37479 400 3 1.01240 6.20622 331 12 0.92904 6.76299 420 12 0.90551 6.93883 422 12 0.82662 7.60105 333 4 0.77934 8.06218 511 12 440 6 0.71587 8.77698 531 24 0.68450 9.17923 442 12) 0.67493 9.30938 600 3 620 12 0.64029 9.81302 533 12 0.61756 10.17420 622 12 0.61050 10.29186 444 4 0.52471 10.74581 551 12 0.56705 11.08047 711 12) 640 12 0.56158 11.18840 642 24 0.54115 11.61079 553 12) 0.52721 11.91779 731 24) 800 3 0.50620 12.41245 733 12 0.49473 12.70022

j:ach scatterinn-, ant, le of course ive:s its own value for (I,+L') sin,/2, but tr should be the samne for both lanles sirlce only minor el-.ictrorlic del.acys dli fer' i,)(tLwee - l t le tw- o. 4. 3 Background Removal ard( (-TReducti on to Inlensity Refeerred to Vanadium 4.3.1. Procedure. One has two options regarding the determination background: measure it separately (i.e. with no sample in the beam) or obtain it from the data. V'e have chosen the latter alternative,. In a typical set of raw data such as that shown in figure 12, one sees a b-road central peak due to sample scatterin,, superimposed on a background inrte-nsity which can be approximately represented by a straight line in the refion where sample scattering is appreciab)le. If one arranes the analyzer channel width and scattering delay to ensure that a region of no sample scattering is observed both earlier and later than the central peak, a lirle can be fitted to those regions and subtracted as background. This is shown in. figiure 12. The same method of background subtraction may be applied to the V data. A program, DATRSJDI, written to perform the first data reduction steps, does the following:

7,8 a. Read in various parameters including analysis timing information, calibration parameters, duration of runs, etc. b. Read in sample, V and (optionally) background data c. If specified, fit and subtract a linear background contribution from sample and V data d. Perform a 5-point weighted smoothing of the V data e. Compute point-by-point the ratio of sample to V scattering, and the associated statistical error. 4.3.2. Coherent Scattering from Vanadium. At this point in the analysis a totally unexpected problem was detected. Several distinctly unphysical features in the intensity ratios (see e.g. figure 14) prompted a reexamination of the raw data and were traced to the apparently unphysical features shown in figure 15. These sharp peaks in the V scattering were, however, identified as Bragg reflections from that almost totally incoherent scatterer. Recall that V has a coherent cross section of.03 barns compared to 5.13 barns incoherent; the small coherent contribution is manifested in figure 15. Peaks were observed to values of Q corresponding to the (110),(200), (222) and (211) reflections in V. Since they are rather small and narrow, it was a simple matter to interpolate the incoherent scattering under them. 4.4. Matching and Interpolation of Individual Data Sets MInformation on vanadium Bragg scattering is summarized in table 5.

70 T C, Co (002) -....-................ _...............................L L- j |( e 0,.~~~~~~~~~~~~-~~~~~~~~~J, 5( -}_) L, _a ).... —' i (OZZ)- _'':'::''::':::::::.::::.::::::.-:::::::::::::::: _::::; (ZZZ~ - - _ 0 E' -~ii. - - 1-2 (OOr) - _ - | (zzt)) -_- - >, z p, _-I_ ~0 i ~ -'- ~- -- T I 0I ---— oC\io', N~~1 ~ I S —~L":0 0 o^OLd AISIIN: Un 0 n Oi!.Vd AllSN31NI

00 NI. s N 0 _ I (OOZ'' — - 13 _>-....' -- 0 0 (II Z) — -- ~ J 0r zo -- I O 0 0 0 0 0 0 0 0 0:5 (ZZZ) — -'3 N" C 0 --'~i ) o - -N N, _ -> "3NIS)-4- -_Z 0 N -- Z' gO I.L) OJ 00 - O D o ao t ro to N c( - j - - aO3NNVHO Nl SJLnOD cLJS o- - - -

"1. LA~j~i7.. ua c ri 2 1, )y V naca] i.i t Plane i,'ltipli.ci ty d(A) (1 110 I 2.1 32> 2.9,3 200, 1. 512 4.1 5 211 24 1.235 5.0q9?20 12 1.069 5. 77 310 A 24.9 5,. 570 222 7. Q73 7.19' 321 4(.839 7.4992 4 1006.756:.311 330 12.713 8.815 420 24.676 9.292 332 24.64 5 9.74( 422 24.617 10.179 431 4 ).593 10.595 501 24I W 521..552 11.320 440 12.535 11.753 530 24.519 12.115 433 24 422 24.504 12.467!00

a2 There are several purposes for which the entire experimental structure factor is needed. The actual data are in the form of four sets, each measured with a different combination of rotor speed and scattering angle, and each extending only over a limited Q range. Furthermore the data are tabulated on an equally spaced time-of-arrival mesh, which transforms to an unequally spaced Q mesh. It is then necessary to patch the several data sets together and sometimes to interpolate onto an equally spaced Q mesh; calculated corrections are applied to the individual data sets since most corrections are scattering angle dependent. The range over which each data set proves useful is determined by statistical accuracy. We have chosen the criterion in establishing ranges for use of each data set, that fractional statistical error (as calculated by DATRED, and which is unchanged by VANCOR but affected by container scattering removal and also by the multiple scattering correction) be everywhere minimized. As seen from plots of raw data or intensity ratios (figures 16-19) the statistical error is smallest in the middle portion of a data set and smoothly worsens toward either end. This makes application of the above criterion quite simple. The ranges of Q covered by the present four runs are established by the minimum error criterion to be: 140 Hz at 20, 0.9A~-1 Q<2.42A1'; Q~<9 ~01 o _ _ 480 Hz at 20~, 2.45A1 Q.< 2.95A; 140 HZ at 90, 2.98A Q < 9.0A'1; 480 Hz at 90, 9.05A 1 Q 27.A1. The overall

O U) (O - 0 T o<3 0 __~_ 0 --- -O L c Z. C * o__~~i —- N ~ - 0 X - b - N= "-: 0 4 LO ~~I~~~~~-n o ^t <D do 0r tD *' 0 O01/1V~ AISN31NI

84 0 0 C) o< _ 0 o zcJ <r C' O o o IC) ~~o~~~~~~_ r O' Co N I I |,.S - I I Lr,, lO l in.0 o D. OI.LV21 AIISN31NI N

35 0 LO TQ N o< (D N aI' — -C9- 0::............................................................. (OZZ) --—:: ——.: ——:__ Z 0'-_ <["""" x ^(OZZ). (OO) - *^ u ILC) *r o zz - -_- - -_ o — -_ (OOr) _ O o ((z -— __ o r( ~LO —. -0 i _ o- _ - - Ni ~I) I II I(* 0 0'.IJ

<f 0 ( 0Ic) LO- 0: S_- )- Mlrbs) LO = (Io~S) - _ - 0 (009)(Ztb) - -_ (OZ9) - (~~) - (ZZ9) - (IIZ)(IS) --- _ (1~Z)(S-S) - (~~Z) -- (OZ8)(tt9) 0: (ZZ~X099). _.'__- _ rr') ~j L) 0 JI uI Co ~00. lOI o Il S~ 011V8 AllSN31NI i.

, 7 -, / upper and lower limits of 0.9A and 27A are chosen so that no data are used with fractional statistical error of more than 10% in the count from an individual time channel. having chosen the range for each data set there mnay or may not be a mi smatch between any two data sets at a joining point. For intermediate steps in the data reduction process, we eliminate mismatches by scaling the individual data sets with respect to each other; the fully corrected data will be expected to match acceptably well without scaling. To interpolate onto an equally spaced Q mesh, we use a cubic spli. ne smoothing polynomial computed by a program obtained from Argonne's Applied Nathematics Division.37 This polynomial can optionally smooth the data, taking into account their statisti cal uncertainty; we smooth to a limited extent i.n sonme instances, not at all in others. A typical smoothing spline through part of the 140 Hz 200 data is shown in figure 20; note particularly its inability to adequately fit the peakl reg:io)n. This failure is most bothersome in the small KQ region where peaks are relatively sharp; we overcome it by hand interpolation in the peak regions. 4.5 Normalization of Diffraction Data The problem of normalizing the diffraction data may be treated in several ways. Tn doing the measurement by a V reference method we expect that the data will be correctly normalized with respect to the V incoherent scattering cross

2.11 --- 140 Hz 20~ 201.8 1.7 1.3 Qe Fi1gure 20. Smoothi.n spline fit to 14. 0 data.

section. However, various sources of error may cause the data to be improperly normalized. We can check on the shape of the measured structure factor (but not its magnitude) by applying one of several integral normalization conditions derived from the relation between structure factor and radial density function: - r)() -cT~i 7jja;(i 31C~k C7r S First, note that for small r the function g(r) must be zero since no two nuclei can approach each other closer than a fixed distance of order several Angstroms due to interatomic repulsion. For very small r, we also approximate sin Qr by Qr and so — js\-r'/ z~n-j (4. 7) ~~~~~~~or~~3q uii> / (kL(6) (4.8,) This relation was noted by Krogh-Noe. Since i(Q)=S(Q)-l which is calculated from the observed intensity T(Q) by ci(2) [( [x/^ (~~5 a /z('^ (4.9) this normalization procedure can only help us to choose an appropriate I(oo), and not to scale the entire structure factor. Hence the statement above, that such normalizations

90 are sensitive only to the shape of the structure factor. We have made use of the Krogh-Moe normalization by adjusting the upper Q limit of data used so that equation (4.8) reproduces the known value of g, which is,He =c Ph: R~C> it] /Z>R(4.10) with the unit cell number density. For BeF2 at 1.96 g/cm o3 A-1 mass density, g =.0736/A. An upper limit of 26.8A for the data of figure 26 reproduces this value via equation (4.8) and yields a value of I(O ) I(26.8A-1)=1.085 b/ster/ BeF2. This does not compare well with the expected value of <a2) =1.238 b/ster/BeF2, about which more later. 39 Another normalization condition due to Rahman 9is obtained from (4.6) by a further sine transform over a small region: o o o j 4rr 3^ ) ^ r = f l Q) 2 r ] 3 r d rQ 0 ~ (4.11) if R is less than the first neighbor distance, o3i)~ ^~O (4.12) cooR -ifQALTy g C i4 h 7 i, I t I - if= i / L- ( - ( l^X JL< 39 D9 ~ ~ (4.13) I4,,i r^9\ A,(s e} -dy^< / ^ l'(@ t e -jo[^tL1 2J{?(4.14) - 7 (4.14)

()1 -9 where j,(,i R)=(I R) 2(, R cosLR-sin,/iR) is the first-order spherical Bessel function. For any value of [R less than the first eoigrhbor distance, we must have the rig.!,ht side of equation (4.14) which depends on the diffraction pattern of the substance -under consideration, equal (as a function of,/) to the left side which is simply a multiple of the fixed functionr j,(jR). The transform of Qi(Q) defined by equation (4.14) must have the same j1 shape for every isotropic substance provided only that R is smaller than any internuclear distance. It is interesting to note that one can derive any number of normalization conditions similar to (4.14) by multiplying both sides of (4. ) by some function of r and integrating over a range 0 r< R. The simplest of all is obtained by intetgrating (4.6) as it stands: - ^nr r -, / o.^' ) st's. Cn ^,-'idr 6,t J (14.15) (4.1 ) Furthermore, all these integral normalizations reduce to the Krogh-Moe condition for very small R: witness the Rahman condition:' L - J~rg s f( i (j'~ = ) s ts';>-,.e. i' lde d O~~~~f

92 or the simple condition (4.16), a functont o = (e figure 2o V^ X a -27rXdoT = Ai <e C L3 Obviously, there is much information in the various normalization conditions. The interpretation of a plot of both sides of (4.14), for instance, for some value of R as a function of A (see figure 21) is not clear, which diminishes somewhat the practical utility of such conditions. They are quite useful in comparing data from different measurements on the same substance (as in the original paper of Rahman), but one is hard pressed to decide what data changes would improve the none-too-satisfying agreement of figure 21. Indeed, since the left side of (4.14) has the same shape for every substance, one can obviously modify i(Q) to a form which bears no relation to the structure factor of the substance under study, but which satisfies (4.14) as well- as, or better than, one's experimental i(Q). We have thus not

<3 3 0.8 0.7 0.60.5 0 Q40 S(1.35,1 ) 0.30 0.2 O.I - o \ -0.1 0 -0.2 o ~ SN(I.35.,/L) -0.3 -, 0 -0.4- eee I I...I I..I I 0 1 2 3 4 5 6 0 ~,A-l Fi. pt ire 21. Rahl,.-iran >iorinal i zati.on T-rte'.ri]. for Partli a t1 y Corrected rData. Solid ci.rve, theoreticcal. value; open ci.rcl.es, data- withl It].ti l.e scaltteri:''?:. cortrecti on;'dots data wlthout iut] ul. scatteri er corre ct i.o l

94 attempted to make corrections to the data based on the Rahman normalization, but we note with pleasure that the multiple scattering corrected data satisfy (4.14) better than the uncorrected data ( see figure 21). 4.6 Removal of Container Scattering At this point the intensity ratio contributions from both the BeF2 and the aluminum container and can be expressed ct ) gFi ) + M, F+( )J /C Abe) from which we wish to obtain CS(t, )/C (tj,) before proceeding to multiple scattering corrections. The factor Fs expresses the effect of the container on the observed sample scattering intensity, and FAithe effect of the sample on the observed container scattering intensity. We dispose of Fs as follows: the fraction of the beam -2z unattenuated by the tube walls is of order & - where AL - t is the wall thickness,.0254 cm, and.098 cm. w r Thus accounting for attenuation of both incident and scattered beams, one expects to see an attenuation effect of about 1-e- (' ) -.005. Multiple scattering contributions will be even less; we thus set FS=1 and proceed. FAL expresses the effect of the contents of the container tubes on the observed tube scattering. It is simply the ratio of the observed scattering of full to empty tubes. We illustrate (Appendix) by calculating FAL for a simple

9'5 tube - the extension to multiple-tube geometry is obvious. The integration involved is by necessity done numerical l, and the results for the six-t:ube array used in the diffraction measurement is shown i.n fi.ure 22 as a function of the sample cross section 1T' In order to apply these corrections to extract CS(t, ) from the data, we must have an approximate sample cross section, since FA varies with. S We T are at least nearly in a position to compute S from the data; for in the static approximation, E9) -' <4,> 5Q) and by definition the microscopic total scattering cross section is the integral of the differential cross section, i.e. for w-(&) = p 1 s(x) we can write (S - J (S (S.17) ~7T /t <<RJs&Q I ('/,1') in the static approximation. R?,ecallin; Q=2k sin l/2 0 or - ( i) ~. ~-. so at fixed k, C = - jT-'' /k (4.19)

9r; 0.90 0.88 0.8 6 6"z.FAL: 0.9755-0.0092 aTT FAL 0.84 0.82 0.80 i i i 12 13 14 15:rT Barns Figure 22. Container Self-S-li::'ir., Factor

97 By effecting a rotiugh removal of the container scatterin_!, we can arrive at a crude estinmate for <a'-) S(:); whichi yields an approximate cross s(ctli.on vi a (4.19) for use( in a nmo(re careful removal of the container scatteri n-. Si.nce the( cross section at thi-s point can only be con-sidered approximate, a technique allowing some variation in the cross section used for removal of container scattering, was adopted. By using a constant value for > (), the factor FL(E ) becomes a function only of S(E:) and the scattering an:.le, as in figure 22; since FA is nearly linear as a function of 1., it can be interpolated rather than calculated C3 for each value of r. For the sample cross section >2 we use (1+C ) times the approximate cross section shown in figure 23 which was obtained by interpolating a smooth curve,undler the contain er 3raig peaks in the intensity ratio data. We allow the normalizationl parameter O to vary within the range -,1i- c i.1 and choose O~ so that the function S&~~~R S lt is made most smooth (R, R' are the intensity ratios for the sample and empty runs). It happens that this particular bit of parameter fiddlin,g was done using a graphic computer terminal to display the data with peaks substracted, and to adjust O& (and hence FAL) so RS-FAl' is acceptably smooth. 4.7 Correction for':ul]tiple Scatter;in: in the Vanadium Reference Scatterer

98 c L, O 0 c OD C C i? I I ^ I C U GD 9_ - o | c f) U 4 cD / -<D O >, Z009/q (8)q)4. L - L4

n Calculation of the multiple scattering correcti on factor FR (Ej ) for the reference scatterer is considerably simplified by the fact that scattering, from V is nearly isotropic, and predominantly elastic. Again in voking,., the static approximation, we write the total scattered intensity from V as V(Q) = V (Q)+V2(Q)+3(Q)+... where V1(Q) contains contributions only from once-scattered neutrons, V2(Q) from twice-scattered, etc. As shown by Vineyard4 for an isotropic scatterer Vn+ (Q)/Vr(Q) changes little with n if n >2, and so Vc( - -,(& V ^&2) ) i-t,?Wi)a/ -,lV V1 ) For an infinitely large but thin plate the f irst and second For an infinitely large but thin plate the first and second orders of scatterin/g can readily be calculated (Appendix) and so also F (E^ -) = V6\)I). We have written a protram, VANCOR, which reads as input the intensity ratios computed by DATRED (or in the present case, intensity ratios with container scattering removed) and computes an appropriate FR(Et) for each data point. Microscopic scattering and absorption cross sections of V are taken to be 5.13 barns and 5.06(.0253/E)~ barns, respectively (E in eV). The angular integrations (Appendix) are performed numerically

100 on a mesh of, (azimuthal) by 2/10 (polar, expressed in A/ =cos ). The computed correction factor is shown in figure 24. 4.2 Multiple Scatterin Correction 4.8.1. Preliminaries. The multiple scattering correction F(E,9) was computed using, a Monte Carlo simulation technique. This approach to the problem of multiple scattering corrections was developed by Bischoff 42,43 for use in the time-of-flight inelastic scattering program at RPI44 Adaptation to slow-neutron measurements, and in particular to the Argonne time of flight spectrometer, was made by 45 Copley5 who introduced many innovations into the method. 46 We use a condensed version of the Copley code, treating the scattering as though entirely elastic; i.e., no inelastic data are used in the correction of the diffraction data. The Argonne inelastic scattering code is called MSCAT; our version is titled MSCATD. Ideally, one might like to include the effects of the sample container in a Monte Carlo simulation such as this, and thereby eliminate the need for a separate step of container scattering removal. Unfortunately this is quite complicated, and we have not yet built it into MSCATD. 4.8.2. Method. We present briefly here the details of the Monte Carlo simulation, beginning with the general

101 cu \Jc 0 0 0 *c'0.1o c4 - 0 CO E 4-..C) L^~~~~~~~ I~5~~~~~~~~ I^I - -j 0 0) C) c 1'I'< 0 0 ~~c o~ ~Ci~~0 CJ 0 o<::)DC) Cj "CL-. 0 1 C r~ Y ^ 1 ro c'-: 0 < co 1 CD. 0- -0 d d' SdOI.OVJ N0113OI.d03 VAllONVAA

102 approach. For a uniform monoenergetic neutron beam incident on a sample of known geometry, we can compute an average contribution to the response of a bank of detectors at scattering angle 9 by simulating histories of many neutrons in the beam, which enter the sample at various points. This response can be separated into first order, second order, third order, etc. so we can compute PI/(Pi+P2P3+...), I1/P, P /(P2+P+3...) and whatever other ratios we may wish to examine. The sample material must be characterized in terms of macroscopic scattering and total cross section is and iT; and differential cross section used in the simulation should be a reasonable model of that of the actual sample, in order that the corrections calculated be a good approximation to those for the real sample. In practice, we have no better differential cross section model for use in the simulation than the experimental data themselves. We may wish to initially estimate multiple scattering corrections on some other basis and use a provisionally corrected experimental differential cross section as the model. There is but one way to test the acceptability of a model differential cross section: to calculate multiple scattering corrections, apply them to the uncorrected experimental data, and see if the model is reproduced acceptably well. If the match is good, we can say with confidence that the model differential cross section is a good approximation to <a>2 S(Q ) in equation (4.1). In principle

one could adopt an iterative approach, whereby i f the model is not acceptably reproduced by the corrected experi ll(etal data one recalculates corrections isinp, the corrected ldata as model. It is not entirely clear if such an approach would eventually convere (and if so, to what). We have not followed this iterative approach and have computed corrections only once, based on an initial model. Before displaying the model differential cross sectior and the resulting correction factors, we briefly describe the simulation process. 4..3. Simulation. The calculation be.ins with a sinrle neutron with enerwg EF traveling along the incident beam A direction -QL. We choose at random a point on the entrance face of the sample, rO=(x,y,z ) (constrained to lie within the beami area, of course) and say that thoe lnutron enters the sample there. Associated with any point on the entrance surface (or any point in the sample for that matter) and with any direction is a unique distance along that particular direction to the point where the neutron would leave the sample if it continued unmolested on its way.'e call this distance t(r L ) the "effective thickness." The probability of an interaction occuring alongR the incident neutron's path is then tc(-I-e) - - We choose a random number C from a uniform distribution between 0 and 1, and then solve for 10 such that p(1 )/p(t )- [.

104 This fixes the next interaction point via the relation A r=r +1,. We require the interaction to occur at r. Notice that there is no chance for l > t, i.e. no chance 0 O for the neutron to escape the target. This is efficient since we keep the neutron in use, but unphysical; we take the effect of this device into account by lowering the "statistical weight" of the neutron from its initial value of 1 down to p(t ). This correctly accounts for the fact that we have forced an interaction to occur which in reality would only occur with probability p(t ). At the point r we require a scattering event to occur; the probability of this is ZS/ZT' given that some event occurs. We therefore multiply the statistical weight by -S /-T' To establish a new direction of travel, we choose a value of Q for the scattering event at r as follows: again pick a random [ number 0 4 i 1, and choose Q such that fjrE'A,jsa, = i? oo This fixes the scattering angle ~ by Q =2k sin-/2. Too gether with a random azimuthal angle (eg, 2 T ) this establishes a new direction of travel -_L. Before proceeding to the next interaction however, we stop to do some scoring. To each detector in the bank at scattering angle -, can be assigned a probability of detection for a neutron scattered at r_ simply on the basis of the effective sample thickness t between r and the D -1

105 th detector i. The "score" for this event for the i detector is just e t' times the current statistical weight of the neutron times the relative cross section for scattering into the detector; S1- e gr ge \R s tQ <^ d A ) We have included the subscript 1 to indicate that this score is for a once scattered neutron; later collisions will contribute to higher orders of scattering. Having seen the neutron through its first scatter, the rest is repetition. At each collision point r=(xn, Yn' zn) with the neutron moving in direction Q n1 we score each detector for nth order scattering,. We then choose a new direction -Q n, compute tn(rS-,n) change the neutron's statistical weight to w,= W,, U ) and choose the next interaction point just as the first was chosen. The score for nth order scattering into the ith detector is, e 2 E (t -A and the score for nt order scattering into the s detector bank is Z s~i~= w.^ - e^ r Ds d (X < _-,OQD) The sum runs over all the detectors in the ~ bank, each with weight ui, We allow this neutron to collide along its merry way until its statistical weight drops below a cutoff value

we; at which point we either (a) double its weight, or (b) terminate its history, with equal probability. This "Russian Roulette" tends to concentrate the calculation of low-weight events in fewer neutrons without biasing the simulation. Since the simulated neutrons cannot "leak out" or be "absorbed" under the rules we have set forth, it is necessary to provide such a means for ending their histories. After simulating the histories of a reasonably large number of neutrons, we compute the quantities of interest, the correction factors, by comparing the total scores (normalized to a "per incident neutron" basis) for various orders of scattering with each other and with the normalized score which would result from single scattering in an "ideally behaved" sample, with N the total number of nuclei illuminated by the beam, and A the beam area. The normalized scores are in fact probabilities, and the quantities P1/ PI, P/ etc. can be computed by using appropriate scores for the probabilities. We repeat the simulation for each scattering angle, and for incident energies such that Qe=2' sin -/2 covers the desired range. In practice, every collision can be scored for all the different scattering angles of interest, so the consideration of numerous scattering angles does not require much additional work over that for one angle. The results of simulations for various values of Qe

1.07 are shown in figure 25. These were interpolated and smoothed to some extent usin-g a cubic spline smoothing- polynomial. The error flags indicate a standard deviation ill tl!he calculated correction factors, due to the statistical natulr' of the simulation process. The number of histories traced in any simulation is of course determined by the required level of statistical accuracy in the correction factors. We have compiled 200 hi stories at each incident neutron energy represented in figure 25. The model differential cross section used in these simulations was simply the four sets of experimental data, patched together by scaling each set and renormali zed to make the Krogh-iioe integral normalization re3 produce the bulk density of BeF2 glass, 1.96 g/cm (Note that this is for us a microscopic density only, since in our sample there are voids on a macroscopic scale.) These model data are shown in fi gure 26( along with the scaling, factor by which each set was multiplied. The experimental. data with smoothed correction factors appliecd are shown in figulre 27. 4.9 Inelastic iScatteri.n g D)ata Reductior Data from the time of flight spectrometer were reduced 40 to scatteriqng law form as described by Copley..ve will not elaborate here on the data reduction procedures. As previously mentioned, a vanadium reference scatterer was used to cali.brate the'e neutron detectors. Correction was.made for the following? effects:

108 rD --- - (D i — ------ (_ - 1~ _ — - h- o-I C___^ I I, * — -,.. ^frl o L0 o-S^~ 0 u~s~0 L^ o Id / Od td/ld U -I:-Ii 0 ~ - ---— 4. I —-Sp ~ 1 I I (IN'Ftd-.-l — i CM__- I-.-4 - 1 0 - 0 0 Id/-Ld Zd/- Ld.~ 0f0

1 (5() kg l e L et —— * —-- C(j Gus AL~~~n I HO0 IF"loV o H O o o I — - Id' — H - ~,D d/1dC/.)d ~ r" I -- J,, —I -.:d/'Ld

el WN0 =0 (!3 I L 1~^t;C: CY 0 -' 0~'" (:D.- (-,,j~~~~ 0~~~~~\

?1L (\j I/(~~VJ 00 C3l LO /-^~~~~~0 0^, I C r~ c X L 4 <5 d^__> LL r^ r I ^ ^8/oO-^^ ^

112 a) background counting rates in sample and empty runs b) container contribution to the sample scattering c) multiple scattering in the reference scatterer, calculated as described for the diffraction measurement. We made no attempt at a multiple scattering correction; the main problem was a lack of a suitable model to extend the data to large Q. Since we are not particularly interested in the shape of the elastic peak and require knowledge only of the location of other features in S(Q,)), we have also omitted a resolution correction. Using the reduced S(Q,Q)) data we have also computed the generalized frequency distribution function G(Q,&) introduced in section 2.4, namely

C.i!:\~i'I:iK FT' VK CO`lPUTATION AND REF]"N"-'"iNT O THlL RADIAL -'S-TTY 7l NCTIO' 5. 1 Tntrodulction -In this chapter -we t-ouich- briefly on some of the: corputational and analyti c probl.ems which mutst -be doealt with i.n doinr-, Fourier transforms i n real life. WVe say "i.n real life" because in. principle there need be no hitches whatever in the evaluation of eq.(2.3':), which we repeat here for convenienIce: 0 47TrV, )-j.) _ f-fJ 6CQ) S1 Si d( (5.1) with ~ i - S3cS - 1 (5. la) In practice there are several problems we must face which bear on the evaluation of (5.1). These we will discuss here are: a) the representation of the integral in (5.1) by a sum b) the effect of our lack of i(Q) data in the regions Qma< Q and Or < i (Qnlin c) the effect of our imprecise knowled;ge of i(Q:;) in the region of Qmin Q Q due to statman M ma istical imprecision and to possible normalization errors. 5.2 Discretization of the Fourier Transform 5.2.1. Available;ethods. Here we may make any of several choices, opting for more or less computationally efficient and sophisticated techniques. The most efficient techniques are the so-called "fast Foulrier Transforms" (FFT) of which the algorithm of Cooley and Tukey 7 is the prototype. These techniques take advantage of a priori knowledge of phase 113

114 relationships between various components in a Fourier series to reduce the amount of computation required. They represent the known function and its transform as vectors of length n, and constitute optimized algorithms for evaluating the nxn matrix equation T = MF with T the vector of transform values, F the vector of known function values, and (M)k = e /n. Such techniques work best for highly composite n, and in particular for n = 2P they are spectacularly efficient, resulting in reduction of computing labor by a factor roughly p/n. We have chosen a less efficient but simpler alternative, that of approximating the integral in (5.1) by a sum. In this approximation: with This is computationally a terribly crude and inefficient process, but has the advantage for our purposes of eliminating the relationship implicit in FFT between the Q mesh and the r mesh!, = 2itr/^ IaQ. We can also save at least some computation by tabulating the sine function and interpolating, rather than computing every sine needed. Most important for our purposes, we can calculate as few or as many transform values as needed at whatever values of r we wish; the FFT techniques requiring the computation of exactly n transform values on a mesh of 2 71/Qmax, given n values of Qi(Q) on a max

mesh of Qna/n. One can artificially extend the data to a larger value of Q, say Q mamax' by setting s(Q) = S((nax); (max < Q Q' ax and thus become able to calculate the rdf on a denser mesh of spacing 2 7r/Q'm. The extension to Q'max however, max max obviously entails increasing the number of data points, and in the square matrix FFT techniques, the number of transform points which must be calculated. There is still no possibility of computing only a small number of points in the region of peaks in the rdf on a very fine mesh, and the rest on a relatively coarse mesh. 5.2.2. Computation. A program DATINV has been written which does the following: a) reads in I(Q) values on an equally spaced mesh b) forms the function Qi(Q) using as 1(0c) either the last value of I(Q), or another value specified by the user c) optionally weights Qi(Q) by eK with oC controlled by the user (the use of such modification functions will be discussed later) d) evaluates the transform (5.2) on an equally spaced r mesh specified by the user e) evaluates the Krogh-?Ioe normalization integral f) optionally computes the first, second and third r moments of g(r)-gfor use in finding coordination numbers, neighbor distances, etc. g) optionally prints and/or stores all the computed functions. Typically, one eva].uates 47Tr(.(r)-gjo on a relatively coarse mesh over a wide range of r (eg, O.OA <r <10.OA by.05A) and

11U then on a relatively fine mesh (eg..01A) in the region of peaks. The transforms of the fully corrected S(Q) of figure 27 is shown in figure 22. 4~ 5.3 Errors in the Transform 5.3.1. Termination Errors. The limits of int-gration in Q space in equation (5.1) are 0 to o. we of course cannot hope to measure S(Q) for all Q,O< Q<O; we must live with Qminx Q Qmax' we will not be concerned with the region OQ<Q( min since in the transform (5.1) there is a weighting of i(Q) by Q; the small Q region is thus relatively unimportant in its contribution to (5.1). We extrapolate S(Q) _-1 smoothly to zero below Qmin(Qin.9A in the present case). There is a small error due to the extrapolation to S(O)=O; the correct value of S(O) is related to the isothermal compressibility of the sample, but does not differ significantly enough from zero to give trouble (especially when weighted by Q near Q=0). The lack of data in the region beyond Qmax is a far more serious problem with which we must come to grips, since we will have no way around it. It will be useful first to derive the mathematical form of the spurious oscillations in 4 7Tr(g(r)-go) which we will call "termination errors." We begin by substituting the inverse transform of (5.1), &.QtBW Jca-C,)-S) we dr- (5 *3)'V

117 I "'00 O) < ~~~~~~~~~~~~.5 >~~~~~t -* o> —'t cc - 4~~~~~~~~~~~~~4 /~~~r (\J~- /0 er) 0 LC) (Nj~~~~~~~~~~~~~~~~~~~~~~~~~~,

11~ into (5.1) which then becomes co W W(Nt#9)o) _ =-E J,//^^"r^-S>'<p Q.^S ^Q.^ dt ( 5.4) which is just a statement of the Fourier integral theorem, appropriate to this problem. Now we define an "experimental" g (r) by the truncated transform e q reA d) - = f OJ OQ) r d (5.5) o This is the function we arrive at by transforming the experimental Qi(Q). Substituting (5.3) into (5.5), we obtain an expression which is analogous to (5.4): ^-^.^-eLc')o -t )-'3) 3'Q QL't i (5. 6) This relates the experimentally observed g-(r) to the true g(r) of the sample. We may perform the Q integral in (5.6), JkJ qcoCQr Qj ddC = Li (),9.=.S klA- - ^) so (5.6) is finally 00oa v 5/ ~~~(5.7) The experimental rdf of fPigure 2~ is thus a convolution of the true rdf with an oscillatory function. We have only (!) to solve the singular integral equation (5.7) to unfold the desired true rdf from the experimental rdf. Being practical, we will seek only an approximate solution; but first we will mention another possible source of error in ge(r).

1.19 5.3. 2. Normalization i.rror:. II we ohave incorrectly norlal - izced tlie structurc factor Cdat;la, the funlction tranlsfornmed i!, (5.5) will be not,'a) = Gl [ zr/rix>)- ibut 4i tc Q) C& ^[ rQ?)rW)rL>) -i where f is the fractional error in our determination of {(~?). Then'() = 6: () - a 6&)/C:6) + Q a()/(~). r ) = 6aca) + s zr< /r6.) i /4,-) _~ cX ^ (G) jHi^ t 4 (5.0) Dernotiing by:, (r) the erroncous experim'en tal rdf oltain-ed by transformiing Qi (Q), which Cye.et b)y sulbsti.tutini ( 5.:') into (5.5),, /i-F iThlat we have plotted in figut.re 28 is in fact 4 lr(.e(r)-, ) 20 and not t/ ITr( (r)-, ) since we hlave not yet made sure that no normialization error is present. If we know f, it is a simple matter to:et 4-rr(c (r) -0o) by T4.^-pL ) =6jt) iWr - )-+ Q ) 4 0 ) and then proceed to treat equation (5.7). In practice we do not know f; if we did, we would have done the normalization correctly in the first place rather than leave i-t until this point.!ie therefore have to g.et at -r(r) throullh an equtation

120 incorporating the effects expressed in both (5.7) and (5.9): g-(A)-p-=r /-) ) C ^. Cr)) 0 (5.11) Equation (5.11) relates the rdf derived from an improperly normalized experimental structure factor to the true rdf, and reduces to (5.7) in the case of proper normalization (f=O). 5.3.3. Resolution in the Experimental rdf. The broadening of peaks in the experimental rdf, implicit in equation (5.7), bears on our ability to resolve contributions arising from closely spaced peaks in the true rdf. An infinitely sharp peak at r=r0 in the true rdf would appear in the experimental rdf as ic (ignin e second term, which is ne ))c which (ignoring the second term, which is negligible compared to the first for r-ro) has height Qmax /7 and FWHM. 3.7/Qax o ax max We may thus expect to be unable to resolve contributiouns from peaks in the true rdf separated by less that about 4/Q ax, which is rmin =.1A for the present case with Qmax=25.' 1 ife do not expect (or see) ay peak separations in he vireous do not expect (or see) any peak separations in the vitreous BeF2 rdf as small as this, so the broadening effect does not present a resolution problem for us. In any case, the broadening can be removed to some extent by applying a competent termination error correction (section 5.4 below).

121 5. *} Siupressi on or RClemv\'al of Transfornl errors 5.4.1. I'iodi ficationl Fu'.inctions. Havi. nri at least soIme quantitative',rasp of the termination effects, we mulst proceed to deal with them. The crudlest and least effectiv e method available, I)tt the one which has been most used( for many years, is to try t-o m minmize the prol. em throu rh wei i-thting of the diffraction data. If we multi.ply the Qi(Q) f:Inction by a modification function —let us use a Gaussian, for example —before transforming, the termination satelli tes will be altered somewhat, and hopefully made less serious. The analysis of IVaser and Schomaker4 however shows that every common choice of modification function not only fails to si gnificantly reduce the termination errors, but also tends to broaden the peaks in the rdf. [-i the context of the analytic methods presented later, the use of any mrodification function is entirely redundant and in. fact quite equivalent to the use of no modificati on fiunction at all. As a device for suppressing termination errors, we must conclude that data modification is none too effective. (In fairness we must point out that the use of a Gaussian modification function —or "artificial temperature factor" or "convergence factor" - -does have the salutory effect of decreasin!g the weig;ht given to the large Q region where statistics tend to be worst in X-ray and neutron diffraction measurements.) We will require that a useful procedure remove nearly all the termination effects (and ideally also remove the effects of the unknown normalization error.)

122 5.4.2. Method of Kaplow, Strong and Averbach49 The method used by Kaplow, Strong, and Averbach to suppress termination errors in their study of liquid lead and mercury 49 has been applied by several workers with considerable success. This method does not require knowledge of the analytical form of the termination errors as expressed by equations (5.7) and (5.11). We will use the notation of Kaplow et al. in which F(Q)- Qi(Q) and G(r)- 4 Tr(g(r)-g ). The first step is to transorm the experimental data Fe using several different termination points. Those features in the resulting CG which change position with termination point are deemed to be spuriouls. The spurious features thus identified are removed from CJ as are all the features below the first coordination peak (where g(r) must be zero). The rdf thus corrected, Ic, is then transformed to yield a function Fc which is calculated for 0 Q, Qmax and also beyond Qmax yond Qmax' At this point one begins to iterate the following steps: a) truncate FC at Qm and transform b) compare the resulting rdf with Gc from which Fc was calculated, to identify spurious features c) correct G using the identified spurious features, and alter Gc based on a comparison with d) compute a new Fc from the corrected (,c and compare with Fe.

Steops (a)-((d) ar~e1 repatt ti FC tc Stteps (") -(d) are rp'et e(1 i.1F match(.,s i n, th1( ret;i oln 0 Q $ Qmax and (G chanres -li ttle from one i terationl to the next. At this point FC is used to extend Fe beyond QTnax' and the transform of the tresultinp. extendedl Fe is the final G(r) with termination errors removed. W'e have chosen not to use this procedure, not 1because it fails to adequately remove the termirnation effects —it can in fact do quite well —but because the.steps taken in the correction procedure are lar-,ely arbitrary.,vcen the initial identification of spurious features is not foolproof —real features may chang-e shape and thus appear to move, and the movement of spurious features is not so %reat as to preclude the possibility of a small thoui.h real feature lyin, underneath a spurious one bein, wiped out i n the very first step. Furthermore, it does not admi.t inclusion of the effects of a normiali zation error. In spite of our re.servati ons concernin..this procedure we do not denigrate i ts effectiveness whten properly used; Yarnell et al. 5 effected atn almost total removal of termination etffects fromn their liquid ar-on rdf in a si.nfle iteration usin? a method very sinmilar to this one, which they attribute to Verlet. 5.4.3. Method of Narten and of Konnert and Karle. ionnert and Karle of the?,ava] Research Laboratory have developed a very powerful method1 for removal of termination effects and normalization error (and, indeed, several other sources of error we have not worried about, such as errors in measured backrround or i.mprecise knowledsge of X[-ray scat

124 tering factors). The technique can be extended to include nearly any source of error one can postulate; it operates via a least-squares refinement procedure. The basic assumption made by Konnert et al. is that the unmeasured diffraction pattern beyond Qmax would, if it could be measured, affect only the shape of the first few peaks in the rdf. In addition, it is assumed that the dominant contribution in i(Q) to an rdf peak at ri (one of the first few peaks) is (5.12) The shape thus forced on the (ij) coordination peak in the rdf is _ 0'^N t -. *d (5.13) The second exponential term is negligible compared to the first except near r=O, so the overall shape of the coordination peak is approximately Gaussian; this is certainly not a particularly bad assumption on its face. Characterizing a small number of short-range peaks (three in the case of vitreous SiO2, the subject of their work: Si-O, 0-0, and Si-Si) by expressions such as (5.12) and (5.13), a "short-range diffraction pattern" and a "short-range rdf" are parametrized. When the short-range diffraction pattern and the unmeasured portion beyond Qax are subtracted from the

12S infinitec-ra:ilue diffractiont pattern, the remai-nder i s a sui)set of ttr:h ex:peri mental data1 in the ran:',e (O,<( (.'This "resi(lual diffractioTn pattern:" i (Q) is the tranlslform of( n "resi dual rdlf" r) (r) (i r t:lh notation of hKonnert and ka!r.e, D(r) 4 41 r(. (r)-.-.)) completely determined by di ffraction 0 data below Q ax; and furthermore -) (r) should be completely maxmax free of termination errors, since Qmax is as m;ood as t for an upper limit on the transform of i' (Q), which hopefully dies away below Qnax' The way i.s clear now to proceed with refinecmeint of the rdf. All one need do is adjust the parameters Nij, r. and li to make the transform of SC -Q.a) -Cd a JS. 1 i-2i) I) (5.14) free of any feature in the short-distance region (note the sum over "sd" in equation (5.14) runs over the short distances bein, removed from the rdf). This should be possi ble if we have correctly assumied that or.ly trhe re;:gion beyond Qax and the postulated contri butions (5.12) -rive rise to short-range structure in the rdf. The adjustment of parameters is accomplished using a standard nonlinear least-squares refinement technique such as the Causs-N ewton iteration or the method of steepest descent. We feel there is much to recommend the procedure of Konnert and Karlo. One attractive feature is the ease with which one can account for the other sources of error- -in our

126( case, it would be desirable to include the effects of a narmalization error, as previously stated. This would simply add another parameter (the fractional normalization error f) to the least-squares fitting problem. Perhaps more sig.nificant on physical,rounds is the fact that one does not try to artificially extend the diffraction data, as i:l the procedure of Kaplow et al. In this method, one makes up one's mind to do without the data beyond Q max and see what refinement can be done on the basis of what one actually knows. The fundamental approximation of a damped jo shape for peaks in the short-range diffraction pattern with resulting Gaussian peaks in the short-range rdf, is really the only thing one must buy; after that the method proceeds free of approximation. As to the acceptability of the Gaussian shape for short-range rdf peaks, it is probably safe to say that any deviation from the Gaussian peak shape at short distances in the true rdf is reflected in the diffraction pattern only in the region beyond Qmx' and thus cannot be calculated from our diffraction data in any case. It seems a reasonable statement then, that the Konnert and Karle procedure when properly executed yields a refined rdf which is as close to the true rdf of the sample substance as we can possibly come on the basis of a limited range diffraction measurement. We would be remiss not to g:..ive credit in this discussion to A.H. Narten, whose X-ray diffraction study of glassy BeF2 was mentioned in Chapter One. As early as 1969 (three years before published application of the Konnert and Karle refine

127 ment procedure) Narten and Levy52 utilized a procedure identical in principle in their X-ray diffraction study of water. This procedure was alluded to in Chapter One, having also been applied by Narten to his BeF2 glass data13 Narton utilizes a least-squares variation of a model ij(Q) which is a sum of short-distance terms such as (5.12), plus a term accounting for continuous pair density at large r, against the experimental i(Q). Refinement at the level of i(Q) rather than the level of g(r) offers the advantage of reducing the amount of computation required, but introduces the additional assumption that there be no contributions to the diffraction pattern other than those from short-distance and continuous terms. The Konnert and Karle procedure admits the possibility of structure in g(r) not confined to short distances but not in the continuum region either. On this basis we would be hard pressed to declare the slightly more general but considerably more costly procedure of Kornnert and Karle superior to that of Narten. Having, so highly praised this method for suppression of termination and normalization errors, it becomes imperative to explain why we have not utilized it in the present work. The answer, simply stated, is time and money. Time: as we became adequately informed of this method in late 1973 when there was not sufficient time to seriously develop a program applying it. Fioney: as even though we began work on the aforementioned program, charges for computation became prohibitive. To illustrate the large amount of computation

12c implicit in the method, consider that any competent nonlinear least-squares fitting algorithm requires knowledge at each iteration of not only the total squared deviation, but also its partial derivative with respect to each parameter in the fitting function. The fitting function in the present case is the Fourier transform of equation (5.14); each of its partial derivatives requires as much work in the evaluation as the function itself, and there are 3n of these (n the number of short-range peaks modeled). This adds up to a considerable amount of computing labor per iteration, and numerous iterations are in general necessary to converge on a satisfactory solution. All of this boils down to the fact that in order to apply this method at a reasonable cost, one must have access to a computer of the number-crunching variety —and we do not. 5.4.4. Graphic Removal of Termination Errors. The method described in this section was developed with the intent to take advantage of the particular computing environment of The University of Nichigan; that environment being one of a medium power computer, certainly not a number-cruncher but with features (e.g. a large virtual memory) which make it ideally suited to a time-sharing system. In addition a large variety of terminal devices are available, and the Computing Center Staff provide the user with much software support for his use of these devices. We have found invaluable in our treatment of the termination error problem the Tektronix 4010 storage tube (graphic) terminals available at no charge to the user in the Computing Center, and the Integrated Graphics software

12.9 packageo (developed:by':-. James Bl.i.rnn of the Computin;- Center) which llas proven the si.nLe qu(a non for onur profitablle use othe;,raplhi.c tormi;nal. in:t'an i ll teractiv lmodc. iv'e will. proceed to (lescri.be our "rapt!l-hic tri a.l -anr-ld-crror solution of equiatioin (5.11). Wve will i. lclulde the niormalizatrion error f in the eqltuations, since it has provecn ulseful]. to have the effects of f i. ncluded in the - Graphi c procedure. First we note that if tile experimental rdf i47rr(p (r)-i, ) and a mnodel rdf 4Ti r(^,.(r)-p,o ) are both taulilated on the same mesh, arnd if the meslh is sufficiently fi:n(e that we can approximate the ta.bulated function ^Yr<, )-r) cF. g ) 8; = - c )4r (5.1 5) then? equatilon (5. 11) has a very simple forlm.whten (5.15) is sub,!sti tuted: r- (< rK) -) = 75v) coT,r) [C, r {)) - <~~~~~ >^i&4.^ ^~~~~~~~~~~~~~~~4 e r —))-^ jb)) 4' d~9~P)')... ~~ I(5.16) Equation (5.16) predicts the experime-r.tal rdf we would obtain by invertinw diffraction da.ta, out to max' from a substance whose true rdf is 4 7r( 7:(r) -, ). This prediction is of course based on the assumplilntion that the only errors possible inr the transform are the termi nation arid normalization effects

130 as expressed in equation (5.11). The simple principle of our graphic method is to adjust the G. (and f) so that (5.15) reproduces as closely as possible the experimentally determined rdf. There is of course no guarantee that the model arrived at by this procedure is the only solution to equation (5.11); but it is certainly a solution, and we assert that any siG nificantly different solution would have some physically uniacceptable features by which we could rule it out. In practice, the -raphic error removal program (titled GRAFIK) proceeds as follows: a) reads the experimental rdf b) prompts the user for a an d Q c) displays the experimental rdf on the screen as a solid curve d) allows the user to set,(r)-O in any reeion he chooses e) initializes the model by making-t g t(r)=,e(r) in regions not required to be zero f) displays the model as a histoygram superimposed on the experimental rdf g) folds the model into equation (5.16) with f=O initially h) displays the folding as a series of crosses i) prompts the user to make adjustments to the model or to exercise one of several other options, then returns to step (f) if iteration is to continue j) iterate steps (f-i) until the crosses lie upon the solid curve acceptably well. The "other options" at step (i) include giving a nonzero value to the normalization error f, integrating peaks in the model, changing the scale of the display or blow ingi up a specified

131 regi orn of the picture, andl terminatJing the fitting procedure with optional di sk aril/or print a-nd/or plotter Oult2t)llt. A typi.cal reflnementL of an expeY. 1mental rdf,'i;t 200 miesh points takes 15 to 20 nmirutes at the terTnminal; most of tlhis time is spent pondelri lng o-ne's iext move, so the computilngj time is far less thant the elapsed tiime.'Thi s is cideal since one is char,-ged relatively little in the Viichig;ari Terminal System for "i.dli.ng" as opposed to actual computin-. Results of a refin-ement of the rdf in fi.gure 2' are shown in figure 29. Olr failure to accurately reproduce the oscillations near r=0 can be traced in part to the resolution correction we failed to apply.''he results otr 53 Sutton' s work on vitreous SiO show that the effect of the resolition correction is to brilng, the low-r oscillatino's in the experimental rdf closer to their pre(li cted forml; i.e. to i.mprove the lmatch at low r in fi.,gure 29.:Sutton0's res.ults show thlat the real fe.atures in the rd Ff are essentially u.niffected by the resolution correction. Sin-ce we are interested in the rdf and not the structure factor per se, we can thus g et away without: the resolultion correction. In the case of diffraction from liquids, the structure factor is itself of considerable interest, ar.nd in particular, the detai ls of its shape at low Q. A resoluition correction must clearly be applied i n such cases. 0

132 0 JB K^~~-( - C).^ — J~~r / 4 "-~TI~ZIII~ — SX I - 4 0 = C "K~" —" - —... C'" 0\ ~~ c~ ~.~" ( ),),,,... C) rr, r~~,

1 33 5.5 Implications of the Termination Errors for Lar:e-Q Diffraction We close this chapter on a note of reflection,- colcernini.the terminatiorn errors. Since theo prollel arises F-rnom oulr i nability to measure the structure factor.)beyonld a faii-ly small value of Qmax' one might expect at first that by g:oingr to a larger and larfer Q ax' we could eventually overcome the ~ax termi nation errors. This unfortunately turns oult not to be the case. That the problem of teriniation (and, incidentally, norlalization) errors i.s in fact exacerbated by increasingf Qmax can be seerl from equation (5.11). Immediately obvious is the fact that the amplitude of the termination ripples is proportional to Qmax' and of the normali.zation ripples to the square of Qma Furthermore the frequency of the ripples is proportional to X' x, so they become more numerous as well as larfger with increasi.nig Qmax In fact, reflection on the form of equation (5.11) indicates that only for Q so large that 2IrQ- is small compared to the width of the narrowest feature in g(r) will we )begin to get away from the ternlirnation error problem. This may correspond to Q ma in the range of 50 to 100A or more, depending on the exact nature of the material under investigation. The implications of the foregoingr considerations for work of the sort described here are clear. If we expect to gain anythilng by extending the Q ran;ge of our measurements, we had better be prepared to deal effecti vely vifith the increasingly severe termination effects. Only if we can push Q.ax far maxt

134 beyond presently attainable limi ts will we be able to avoid entirely the termination error problem.

CHAPTER1 SIX ANALYSIS OF EXPERIMENTAL RESULTS 6.1 Analysis of the RDF 6.1.1. Information in the rdf. We have previously noted that the rdf contains information on the relative arrangement of nuclei in the sample. In particular, the function 47Tr g(r) r represents the normalized number of nuclear pairs with spacing in 4r at r, weighted by their scattering lengths. From such a function, which may correctly be called a pair distribution (g(r) itself is not a pair distribution in the literal sense), we may deduce certain average properties of the sample structure, such as coordination numbers, neighbor distances and near-square thermal displacements for the various pairs of nuclear species. 6.1.2. Preliminary comments on the fully corrected experimental rdf. The rdf with graphic termination effect removal applied (figure 30) is our best estimate of the true rdf of vitreous BeF2. That we have not been completely successful in removing the nonphysical features has already been mentioned. In particular, the feature centered about 2.8A is almost certainly spurious, as is the small feature at about 3.3Ao We may note further that the first coordination peak at 1.56A rises and falls too sharply; we would expect a more gradual variation in pair density both above i135

136.... i1 i l 1/~~.^~r -~~~j~~.~ 4 C) r-4 CDIW o C3, I 6() JcO 0N - cu O cor ( D b -) c O J (~~- m(.~) j.t

137 OO *ra~o.~. 5J 0 C.) cZ N-~~~~~~~c'q 00 o<: Lo r~ -.. C. C-, IC) 0 cO -0 N 0 N (05_ b).J~,u~.-; j^ c.o Cr-~~~~~~~~~~~~1 "* (~6-^6y) J1L? t'.T.<

138 and below the peak. Without trying to explain away these and other shortcomings of this final rdf model, we feel it necessary to point out the very simple nature of the modeling procedure used, and to stress that there may well be sources of spurious structure in the rdf which we did not take into account (for example, the possibility of spurious structure in I(Q) contributing to the rdf was not considered). 6.1.3. Identification of features in the rdf. With reference to previous work on vitreous BeF2, we can immediately determine which atom pairs contrubute to the first few rdf peaks. Warren's 1934 model (table 1), for example, predicts peaks due to Be-F pairs (with coordination number n=4) at 1.60A, due to F-F pairs (n=6) at 2.62A, due to Be-Be pairs (n=4) at 3.20A, due to second-neighbor Be-F pairs (n=12) at 4.00A, due to second neighbor F-F pairs (n=9) at 4.65+0.45A, and due to second-neighbor Be-Be pairs (ni=12) at 6 5.20A. We can thus with some confidence (subject to further a verification) label the peak at 156A in figure 30 as due to first-neighbor Be-F coordination; at 2.52A, first-neighbor F-F; at 3.06A, first-neighbor Be-Be; at roughly 4.0A, secondneighbor Be-F (which we abbreviate Be-(2)F); at 4.80A, secondneighbor F-F (F-(2)F); and at 5o06A, second-neighbor Be-Be (Be-(2)Be). It is ridiculous to expect that any features beyond the first can be explained completely in terms of only one pair type; it is apparent from figure 30 that the features beyond the F-F peak are not resolved, and we must

139 expect significant overlap of the peaks at larger r. The assignment of atom pair types to the peaks beyond Be-Be, although qualitatively reasonable on the basis of the various structural models, cannot be pushed too hard in a quantitative sense; we will not even attempt to analyze the peaks beyond Be-Be. 6.1.4. Analysis of the Be-F peak. We expect the Be-F peak to reflect a coordination number of 4, i.e. to show four fluorines bonded to each beryllium. We must take into account the scattering length weighting; we can see from equation (2.8), b t' that in the vicinity of the Be-F peak (6.1) where (repeating equation (2.9)) degie ti o t i describes the distribution of type t' nuclei about an average type t nucleus. Furthermore we know from equation (2.10) that in the isotropic case, Sinc)= wa k jw N.i'w) Since we also know that NF=2Be and that NNNF+NBe, we can write for the first peak in g(r)

140 (6.2) and using equation (2.10), (6.3) so that (6.4) If we redefine a2 on the basis of a unit of composition, -? Z which is just three times the value used in Chapter Two, then Jq _24<cZ~5 (6.5) We know that gFBe(r) describes the coordination of fluorines around berylliums, so we expect its integral over the first peak (properly weighted) to yield the coordination number: Ji~Tpae - r = e AC P-4 ( 45-X. 774) Ye -.437 Le (6.6)

141 and Jf-T. Wr-.70 -Fe If nFBe is 4.0 as we expect, then the inttegtral of,(r) over the first peak should be 2.84. If we sum up the area of the first peak in 4 7r2 g(r), figure 31, we find 7.c 1 l) r.4 which implies nFBe=4.14. We are thus comfortably close to the expected coordination number of 4.0 (3.5% high). On the basis of our finding nBe=4.14, we have no reason to conjecture that any coordination other than fourfold is present. 2Since 4 Tr g(r) represents a distribution of pairs, we can also extract the mean pair separation, Again from the model rdf, we find L l 3f^ ^} r = Ad -73 _ (.4 8 so that _ _ 4,1S 3/7 3l =-,-w A Knowing the centroid of the peak we can compute its variance, </i\ -L e r/ 7 We find from the model rdf that r.z4 v l^;) =.=4 w= ivr

142 35 30 25 r, 1.2 1.4 15 3 1.6 1BF 18 r,A Figure 31. Be-F Peak in t~he Refji.ed RDF

so the variance is a ~ (<^rZ _.-oof/.00= I-A 49 and the rms width is <ar. =.o'43 We may interpret this rms width of 4i r2 g(r) directly as an estimate of the rms deviation of the Be-F pair spacing from its mean value in vitreous BeF2. For comparison, we have superimposed on the peak in figure 31 a Gaussian with centroid, variance and area equal to those of the model peak. This Gaussian is a reasonable fit to the model histogram; we mentioned in connection with the Konnert and Karle termination error procedure that an assumption of the correctness of such Gaussian peaks is basic to that procedure. 6.1.5. Analysis of the F-F peak. If the F-F peak were isolated, we could evaluate the indicated coordination number: fwlIrN6Pe4r" f7z. <v f 2i^ ) TheF pep 02 t2 C F a 6 a IF_ The F-F peak in 4 r2gM (r), figure 32, is not isolated, however; contributions fron the Be-Be peak and from the apparently spurious peak between the F-F and Be-Be peaks overlap with the F-F peak on the high-r side. We have chosen to integrate only as far as the minimum of 4'1r2gM(r), and to accept whatever inaccuracies that may introduce. On this

144 25 20 CI I05 2.2 2.3 2.4 2.5 p2.6 2.7 2.8 2.9 r,A Fisure 32. F-F Peak i.n the Refined F

basis, we find =5.4A )^4^ t3<E4A t - 7.~77 so nFF=5.43 rFF=2.524A. Using this value for r, we also find X TrI-a.)C- r =. O/ so the variance is E\ A>~ =-.~ oo4oI A and the rms width is <A^>^ =.0o63 A Again we have shown a Gaussian with the experimental area, centroid and variance superimposed on the model histogram; the fit is not as good as for the Be-F peak, as the F-F model peak is markedly asymmetric. 6.1.6. Analysis of the Be-Be peak. The Be-Be peak in 247r gM(r), figure 33, is even less well defined than the F-F peak. Again we have chosen to integrate between the minima of 4'fr2gM(r) on each side of the peak. Were the peak isolated, we would obtain #ITrr, dr Ja iie f2Crd <i r = < >'= f-d = Seae &&a. <"E4~~~ ~

146 20 E CP 15 5 0~ 23 ^ 0 ——. 2.8 2.9 3.0 3.1 p3.2 3.3 3.4 3.5 rA Figure 33. Be-Be Peak in the Refined RDF

147 Evaluating the area of the model rdf and moments between the minima of 41 r2 g, (r), we find 3,2e Y^T~ry^r3< O..5v-5Z. from which we have nBeB=4'44, BeBe3050' Recognizing that the estimate for nBeBe is not very good, we may still have confidence in the centroid value. If we leave off one histogram bar on each end of the sum, we have a smaller estimate of the coordination number: 3,(3 y.l(~; — =2. - from which n eBe=3.65 and r e=3,056. This coordination Be Be I3 BelIe number estimate is just as far from what we expect as the original one; the true value undoubtedly lies somewhere be2tween. From examining 4cr 2gM(r), we see that there is considerable overlap from nearby features, so we cannot really expect to get a clean estimate of the Be-Be coordination number. Likewise we will have some difficulty establishing the Be-Be peak variance. Adopting the first set of values above for the centroid and coordination number (i.e., rBeBe=3.050A, nBeBe4.44) we find E. ^ 6r-) Cr-. 4 r-.31 6 rc=2.8s

14~ so 0 2A <o ) *=*. <^t^/2.?48 _.O c 48fA idae <'4t e = * o93 d Again we have plotted a Gaussian with the experimental Be-Be centroid, area and variance on figure 33. The fit to the model histogram is not particularly striking; but this comes as no surprise since the model peak itself is not very well defined. 6.1.7. Evaluation of mean bond angles. The mean F-Be-F and Be-F-Be bond angles can be evaluated in a straight forward manner from the peak centroid positions. To do this we simply consider a triangle (figure 34) with (taking the F-Be-F angle first) fluorines at two vertices and a beryllium at the third. Two sides then have length equal to the Be-F distance we have determined, and the third side is equal to the F-F distance. The F-Be-F bond angle is then @ e- P-F) =2 S-L- o 7. Similarly we find @ (,^-F-S) = ^^-' L[J _ ] =o We summarize our analysis of the first three coordination peaks in our model rdf in table 6, and also repeat the results presented in table 2.

2b x X Y s: (X-Y-X) = 28=2 Sin-l(b/a) 2Sin-' (Fxx/2Fy ) Fi gure 34. -eometrnv i-c) c.tlc:i-.ation of bond angles. X~i= 3e or'"i

150 U) 0t - 0O 0 0 CO ~ ~ ~ ~ L C) C) CDSrZ i <d <r <^ r~r* t 0 0 0 0 I ~'~~~~~~~~~I -l) 0: -tr N- r- 0c *n +1N it' c' t * * o i~~~~~~t' it' 0 t ( Nnc; O t 0 4) * * * * U) U) 4 c * * * r - 4n en en * - + 4 $H en en s o k Ln Ln o C 0 L c r- o oo0 0r Y5 (N C y- y- %'J* * * *.fQ)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 -r4 C"', *r-i >NJ 4J U 4-4~~~~~~CN - N 0. I0n 0 n =3 k m n o o co ".- CZ C * I pQ) ~ - ^Z -t * -~ -r f L 4 ^ ~ ~ ~ ~ ~ - (NJ 0 *.i * ~ 4 D4Q 0 0 0 40 CO 4c - * 4 O* J 0 it' 00 *c *f U) P4rA \-< i L (N i L 0 * C.r.4 CZ P -4 0 C p4- pQT- r r-( ~ <r ^ ~U4'-4 ( CY'),~ - ~ ~ * * *C 4r-4 C) C~~~~~~~~~~~~~~~l ~~~~ ~ s 0) C^ L_0 Ci (N 0 CD 0 4^ A4 KI 0 - v4 -zi 0 c~~~~~~ ~- - -H -H 4-0 LfI 1e O IN C,) 00a,~~~~~~~~~~~~% Ls r =-r c C:~~~~~~~~~~~~~~~~~~~~~~C CDC 0 D kO - (ICOCD ^r-^ ^CLn c~i rI ^^ O ~~~~t~^o c'n o co co^ \s, rs~ t- < cn <^ < cn cn E H^~cnc r r- < ^ t' 03~~..* 3 % cJ " O c?~~ t u S^<^' "- "- -^ 1 ~3. m ^, <?*r mC

151 6.2 Comparison of RDF Results with Previous Work 6.2.1. Comparison with diffraction results. The comparison of individual peak positions, etc., among the various works does not require much comment; we are pleased to note that our results agree best with those we indicated in Chapter One as being the most reliable. We will take time here to compare our results with those of Narten, which we feel are the highest quality of all the published diffraction results on vitreous BeF2. Our peak positions agree reasonably well with Narten's with the possible exception of the F-F distance. Our experimental coordination numbers for F-F and Be-Be are somewhat far off, but still respectably close. (On reflection, it seems that all the problems just noted could have been overcome had we used a somewhat finer r-mesh in the computation of gM(r)). Most encouraging is the fact that our experimental values for the rms peak widths in 47Tr2g(r), while considerably lower than those reported by Narten, are in quite good agreement with the rms vibration amplitudes calculated by Bates5 for his dynamical model of vitreous BeF2 based on the Bquartz structure. Except for the Be-F peak, where Bates' amplitude value lies midway between our rms peak width and that of Narten, our values are considerably closer to those of Bates than are Narten's. The discrepancy between his values for the rms amplitudes and those of Narten was noted by Bates, who conjectured that low-frequency acoustic modes

152 not included in his dynamical model might be responsible for broadening the X-ray rdf peaks. This seems unlikely, since the wavelengths of low-frequency acoustic modes in vitreous BeF2 are quite large compared to the first few neighbor distances (we will have more to say about this in our discussion of the inelastic scattering results below) so there should not be appreciable broadening of the coordination peaks from this source. In any case, our results certainly support Narten s assertion that the breadths of the vitreous BeF2 rdf peaks are sufficiently small to be due entirely to thermal vibration. 6.2.2. Comparison with structural models. There are several structural models with which we may compare our refined experimental rdf. The previously mentioned work of Narten consisted in part of comparison of his X-ray rdf with a model based on the 3 -quartz structure with a number of random vacancies introduced. We have already noted our agreement or disagreement with several features of Narten's model (since the model fits his X-ray rdf quite acceptably, our comparison with the diffraction results constitutes a comparison with the model). We will make comparisons with two other models of a disordered BeF2 structure —one a traditional random-network model (a "ball and stick" model), and one, the by-product of a molecular dynamics calculation (computer simulation) of BeF2 liquid.

153 6.2.3. Bell and Dean vitreous SiO2/BeF2 "Model V." R.J. Bell of the National Physical Laboratory (U.K.) has generously provided us with a full set of coordinates of the atom sites in the largiest of a series of models constructed by Dr. Bell and P. Dean. The model represents 182 Be and 426 F atoms, and was constructed according to a set of rules conlsistent with the Zachariasen random network hypothesi s (there have been a number of such models built, by Bell and Dean and others ). From the model coordinates we have computed [g(r) according to equation (2.7), utilizing scattering lengths appropriate to BeF (the model can also be considered 2 to represent SiO2 or GeO2). The function 47r 2g(r) for the PBell and Dean model is shown in fi.gure 35. We have computed peak areas, centroids and variances from the model rdf; these are displayed in table 7.'ABLE 7. Peak Analysis of "eutron RDF from Bell and Dean Mlodel V Peak n r <'1^>1 Be-F 4.00 1.561A. 054A F-F 6.12 2.546A.112A Be-Be 4.45 3.051A.086A (F-Be-F) = 109..3 (Be-F-Be) = 155.5~ IUnfortunately the statistics in the model rdf are rather bad at larger values of r. It is apparent that one cannot hope to build such a model. which represents more than a small

1 54 04~~ - r04',-'*'* I_, S -,' *0'-C ~ ly,,_hs~~~~ hW' -- o Nul 0 00 ^) 1 CM C C~ ~ _ * 0r(~OmIJ-..OI 0 o Q CD 2 i-) Nr C) H C. r ~ N3 Cd Od - --

155 region in an actual material; even the (physically) very large M:odlel V represents only a region of radius roughly 13A (we arrive((d at this "equi valent radil.s" by computing the average distan;-ce betweei aztom site-s on the surface, of the model, noting that for a splere the avera.ge chord length is 4/3 times the radius).. We note that the variations in iBe-F and Be-'Be first neighb)or distances for the Bjell and Dean model are quite close to our observed values; the F-F width on the other hand, i s even larg:er than that observed by TNarten, which we feel to be an overestimate of the true value. 17e also note that the Be-Be coordination number is overestimated from the neuItron rdf even though the coordi antion number calculated from the individual pair distribution 4 7Tr2l g (r) f- eBe (r) is 3.9"','% below the expected value of 4. This explains to some extent our own overestimate of ne; the effect is B. e Lb e sinplyl due to overlap of the Be-Be and F-F' pair distributions above and below the Be-Be peak, and! additionally, to overlap of thel Be-F and fBe-Be distributions above the Be-Be peak. This is of course no different than what we expected when we first noted that the F-F and Be-Be peaks were not sufficiently isolated to provide clean estimates of nFand nS Be. The FF^ 13BBee Bell and Dean model neutron rdf overestimates nFF, however, and our experimental value was a considerable underestiniate (again, the coordination number calculated from gFF(r) is less than 1% different from the expected value of t). In the larger-r re:gion, experimental rdf in general agrees reasonable well with the model rdf. The presence of

156 some residual spurious detail in the experimental rdf, and the statistical variations in the model rdf together make a detailed comparison in the region beyond the Be-Be peak of dubious value. We can certainly conclude, however, that the Bell and Dean Model V is not in substantial disagreement with our experimental rdf. 6.2.4. Rahman "frozen liquid" BeF2 model. Molecular dynamics calculations have had widespread application in the study of liquid structure and dynamics. Aneesur Rahman of 57 Argonne has kindly provided us with a set of pair correlation functions derived from his molecular dynamics work on BeF2. Simply stated, a molecular dynamics calculation is a computer simulation of the motion of a reasonably large number of particles interacting via some postulated force law. Knowing the positions of all the particles, the force on each can be calculated. One can then allow the particles to move under the influence of the calculated forces for a short time; after which new forces can be calculated and the simulation continued. The suitability of this technique for the simulation of liquid motion is readily apparent; to our knowledge, there has been no previous attempt to identify a "frozen" (or perhaps more properly, a "supercooled") molecular dynamics liquid structure with a glass. In the present case, Be+ and F ions were simulated; the resulting liquid (figure 36) shows that the ions arranged themselves into the familiar tetrahedral BeF2 structure.

1'57 1'! ~ c l0 JC I; o< I 0 0I 1\ \ 0 0 IL oM<.***^ ~~~~r~~L L —,^^-^-"^^'^ ^^~~~~~~~~~~~~~~- uE / co~~~~~~~~~c i^06 coc LL~~~~~~~~~~~, IL ~.......... CC* cOD I~~~~~~~~.~.... 6)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~il eCl ('",'~ (-t,..,rsC) r —. (. " q'

158 The BeF2 liquid was "cooled" by reducing the total system energy, until the mean particle energy was brought to a level corresponding to a temperature of 480 K, at which point the structure became static. Pair correlation functions for the 480K 3eF2 system are shown in figure 37. Comparison with the liquid pair functions of figure 36 shows that the coordination peaks are considerably sharper and higher, with the areas of course unchanged. The fact that the peaks are only mari.nally narrower in the 480K case indicates that crystallization has been avoided; the structure at 480K is remarkably similar to that of the liquid. We have also computed the neutron rdf, g(r), for the supercooled liquid model. This is shown in figure 38. Extracting the usual parameters from the coordination peaks in 4Tr 2g(r), we find the values given in table 8. TABLE 8. Peak Parameters in Rahman "Supercooled" Liquid Model 47fr g,(r) Peak n r <~r)> Be-F 4.01 1.575A.081A F-F 5.94 2.577A.193A ~ o Be-Be 5.61 3.061A.118A Y (F-Be-F) = 109.8 (B1e-F-Be) = 152.7' The overlap of the coordination-peaks is so severe in this case that we cannot get a good estimate of nBeBe from the neutron rdf (but note that the integration of 4T7r2gTAR(r)

1. 5') IIl I II i I C) gI (0 01,* - o 1Ct.40'-, w-. r^) c' r2; Ct 4) CP ) Lt,* C 00~~, F. ED S t s ^ i CT0'. ~: 0., ~ o " o...,'.-c CT;'~~............;.., k ^ (1:- V'j.5 O~~~~~~.* ^

1( C) 160.r4 0r~~~~~~~~r 0 0 "0 r-u C3~~~~~~~ oC ( () 0 cLL (Q) 0 ~~~~m 00) -.D \V c c c (\ J.2 C.C~~~a I.Q A L I..~.~,Q, 0~~~~~~~~~~~~~~~~~ 1.. 0 -< 0 4-.- 0- f'-j CO ^P 3'I*0'-( ^ M Q..~o~~J c C o ( OD 2 LL ~~~~~ —)~b

161 yields the expected value of 4 as indicated on figure 37). With the exception of the Be-Be peak, all the coordination peaks for this model are significantly broader than in our observed rdf. In addition, the peak centroid positions differ appreciably from those we (and other workers) have determined. We conclude that the supercooled molecular dynamics liquid does not represent a model of the BeF2 glass structure. 6.3 Interpretation of Inelastic Scattering Results 6.3.1. Overview. There is a wealth of information concerning the dynamics of vitreous BeF2 contained in the scattering law we have measured. As we indicated in Chapter Two, however, this information is obscured to a large extent due in part to the polyatomic nature of BeF2, and in part due to interference effects which can complicate the frequency dependence of the scattering law. We must rely heavily on various assumptions, which may or may not be valid in various regions of momentum and energy transfer space in order to extract from the measured scattering law any information directly interpretable in terms of the dynamics of vitreous BeF2. We will attempt only the simplest analysis of the measured scattering law; first comparing features in the "generalized frequency distribution function" G(Q, 4), (section 2.4) derived from the experiment with bands observed in infrared absorption and Raman scattering measure

1 62 ments on vitreous BeF2, and then comparing our calculation of the "acoustic mode scattering law" (section 2.3) with the measured scattering law. 6.3.2. Spectroscopic results on vitreous BeF2. We will analyze the experimental G(Q, )) in terms of three spectroscopic studies of vitreous BeF2: the infrared absorption 58 work of Zarzycki and Naudin5, the Raman spectroscopy of Batsanova et al., and the infrared and Raman studies of 54 Bates. We summarize the findings of these authors in table 9. TABLE 9. Infrared and Raman Bands of Vitreous BeF2 Zarzycki58 Batsangva 54 59 Author & Naudin et al.1 Bates' observed 800 900-910 805 frequenfrequen 715 760 410 eies in cm-l* 435 392 280 246 *1 cm1 optical wave number is equivalent to an energy transfer of.124 meV, or an angular frequency of.189 ps-l. 6.3.3 Other information on the vitreous BeF2 frequency distribution. Unfortunately most of the spectroscopic data is at higher frequencies than we can reach using thermal neutron inelastic scattering. We are fortunate to have some other frequency distribution information. Bell, Bird and Dean have calculated the vibrational spectrum of several of the Bell and Dean random network

l63 models. We quote the results for their Model II (503 atoms) considered as BeF2: Bell et al. have also calculated spectra for the model taken as SiO2 and GeO2. The calculated spectrum for the BeF2 model contains four main peaks, at 195, 335, 630 and 715 cm - The lower two of these four frequencies are accessible to us via thermal neutron scattering. Bell et al. suggest that the 195 cm1 peak is probably due to bond-bending and bond-rocking vibrations of F atoms, and the 335 cm peak to bond-stretching F vibrations. The only published inelastic neutron scattering results on vitreous BeF2 are those of Leadbetter and Wright.61 By utilizing very coarse time-of-flight resolution in their cold neutron experiment (4.24 meV incident energy), Leadbetter and Wright were able to collect statistically meaningful data over a very wide range of energy transfer, out to 500 cm= and beyond. They were of course unable to resolve much fine detail in the scattering law, but the glass scattering law is not expected to show particularly sharp structure so this inability may be of small importance. Utilizing a function analogous to our G(Q,6 ), Leadbetter and Wright concluded that there are peaks in the glass frequency spectrum g( ) at frequencies of 38 + 4 cml, 60 + 5 cm-, 80 + 5 cm, and 105 + 5 cm * They also reached the important conclusion that at least for low Q and 4, the positions of peaks in the scattering law can be plotted as "average dispersion curves" as for polycrystals.62 This is not particularly surprising if we truly believe in the importance of

164 long-wavelength acoustic modes in determining one-quantum scattering at low Q and (, but it is certainly interesting that wave-vector conservation conditions appear to operate much as for polycrystals, near the first peak of the glass structure factor (the region in Q space near the first peak in the structure factor is analogous to the second Brillouin zone in a crystal). This is a clear manifestation of the importance at low Q and 4) of the interference terms we were so anxious to ignore in equation (2.68). 6.3.4. Interpretation of the generalized frequency distribution G(Q,&)). As stated in section 4.9, we have transformed the measured scattering law to the function G(Q s3) defined by equation (2.67).'.e have included plots of G(QW) for each detector subgroup in the TNTOFS experiment as an appendix. Each subgroup, it will be recalled, represents the scattering at fixed scattering angle with approximately 1.29 angular resolution. We have only displayed the upscattering (neutron energy gain) portion of the data in the Appendix. The downscattering (energy loss) data are much more densely spaced due to the time-of-flight mesh, but typically extend only over a limited range of energy transfer (w<30 ps ). The upscattering data seem quite adequate for the present purposes. The curve drawn through the data points are cubic spline smoothing functions (see section 4.4) with the smoothing

165 parameter chosen for each subgroup to make the fit pleasing to the author (we would assert that the spline function fits shown are if anything conservatively drawn). The splines used here are considerably less stiff than those used for the structure factor data. In light of the discussion of Chapter Two and of the results of Leadbetter and Wright, we have chosen to divide the data further into what we will call "low-Q" and "high-Q" regions. The low-Q region encompasses data subgroups 2 through 8, the high-Q region subgroups 9 through 32 (this corresponds to 2.90< Qe 6 6.90 in the high-Q region, where Q. is the value of Q at 03=0 for each subgroup). Recall that we expect the self terms in equation (2.68) to become relatively more important as Q increases; thus we may expect the one-quantum scattering in the high-Q region to be influenced to a lesser extent by interference terms than that in the low-Q region. On the other hand, if we wish to map out "average dispersion curves" by plotting the positions of maxima in G(Qk0), the data in the low-Q region will probably be more useful.* There is a caveat concerning the interpretation of G(Q,4)) in the high-Q region, however. The G function has a simple interpretation only for one-quantum scattering; it was derived on the assumption that one-quantum scattering is dominant. This assumption unfortunately becomes worse *The high-Q region as we have defined it corresponds roughly to the third and higher Brillouin zones in a crystal; the low-Q region to the second zone.

as Q increases, so that as interference effects die out, mul-tiquantum scattering, processes. come incroas;i n,!ly i nto the picture and do their ownl. job of obscuring; the desired information in G(Q,.)). Since we have no easy handle on the magnitude of the multiquantum terms in the scattering law, we will continue to iTgnore their presence and turn to the problem of extracting the information on g( ) contained in G(Q, )). DeWvette and Rahman have shown that for a polycrystal G(Q, 3) may have much more structure than g( ). ]t seems a fair statement on the basis of their detailed calculations for noble gas polycrystals, that ((Q,0 ) show peaks for all values of 4) at which the derivative of g(s ) is discontinuous. This includes Van Hove singularities ("corners") as well as peaks in g(c3), so it is misleading to say simply that peaks in G(Q,)) correspond to peaks in g(( ). G(Q,m)) may also have peaks at values of u not corresponding to pronounced features in g( )). This is because of the influence of the polarization factors Q2c in equation (2.68), 7which may cause G(Q, 1) to vary sharply in regions where the character of the atomic motions changes. We have utilized the following simple procedure to identify the values of a) for which ((Q, ) has systematic maxima (ie, those appearing in a number of detector subgroups) in the high-Q region: first, associate with each peak in G(Q, >) at a given angle, the value of e correspond

1(07 ing to the local maximum in G(Q, N ). Next, plot as a function of U) the total number of peaks (in all subgroups of the high-Q region) appearing in 0D aboutS. We illustrate such a plot for D) =2.5 ps-1 (=13.3 cm"1) in figure 39. The positions of peaks in figure 39 are listed in table 10. TABLE 10. Systematic Maxima of G(Q,k) 57 + 8 cm'1 194 + 5 cm-1 88 + 10 cm'1 263 + 15 cm'1 126 + 10 cm'1 354 + 15 cm-1 164 + 10 cm Our peaks at 57 + 8 and 88 + 10 cm- undoubtedly correspond to those observed by Leadbetter at 60 + 5 and 80 + 5 cm-1 We do not see a peak corresponding to that observed at -1 -1 105 + 5 cm by Leadbetter, though our peak at 126 + 10 cm may be the same. Our peak at 194 + 5 cm,1 probably corresponds to the 195 cm- peak calculated by Bell et al., and the peak at 354 + 15 cm1 to that calculated at 335 cm. Our peak at 263 + 15 cm1 lies between the IR band of Batsanova et al. at 246 cm, and the strong Raman band of Bates et al. at 280 cm; there is considerable structure in the plot of figure 39 in this region, and we are hard pressed to make an unequivocal statement concerning this peak. There remains our observed peak at 164 + 10 cm-; it corresponds to no other reported band, yet there is no question of its presence in our G(Q,k) ). We recognize the very simple nature of our algorithm

16~S 0 -OD Ile) Q o 00 tI I I Gi 0)~ X _ rt * * 0 cx ~ * C. 10 (Drr() * <^.* 3 n T' -' J 0~^ -^ 00 I O0 E' o.SV d ~ 3 o0) Ao mqOwNII I I 1 1 mLj S>43d JO ~381IfnNl g~

169 for identifying systematic peaks in (G(Q,O). Certainly there is a great deal of subjective judgment involved, at every step of the process beginning with the selection of smoothing spline parameters, and finally with picking peaks off a plot such as figure 39. We do not intend to apologize for using such a simple procedure; we feel that despite its simplicity it has worked admirably well. 6.3.5. Comparision with the calculated acoustic-mode scattering law. We have calculated the acoustic-mode scattering law derived in section 2.3 for comparison with the experimental results. We have used the bulk density of 3 vitreous eF2, 1.96 g/cm, and the sound velocities quoted by Leadbetter and Wycherley4 which were measured by Kurkjian, v( = 4.231 km/sec and vr = 2.814 km/sec. The structure factor which appears in the expressions for the acoustic-mode scattering law is not the total, but the elastic structure factor. In principle our measured scattering law for vitreous BeF2 can be integrated to give the elastic structure factor. We have chosen instead to use an approximate elastic structure factor which is just out measured total structure factor weighted by an approximate Debye-Waller factor. We have proceeded thus because it is the total structure factor we have in detail; to get the elastic structure factor on a sufficiently fine Q-mesh would require considerable interpolation rendering the result

170 approximate anyway. To g.et the approximate Detye-WValler exponent, we have plotted the ratio of elastic to total scattering as determined by integration of the measured 2 scattering law, vs. Q- as shown in figure 40. This plot -.02Q2 shows that the ratio can be roughly approximated by e which is the approximate Debye-Waller factor we have used in evaluation of the acoustic-mode scattering law. The results of evaluation of the acoustic-mode scattering law for several angles is shown in figures 41-44. We have chosen to plot C(Q, ) instead of S(Q, )). We see from the figures that the acoustic G(Q,4)) underestimates the measured G at the smallest angle, fits pretty well in subgroup 4, and overestimates the measured G at the larger angles, if we confine attention to the region -8ps < 0 < 8ps The failure to fit at larger ~ is no surprise, since the picture of purely longitudinal and purely transverse modes with linear dispersion curves cannot be trusted at large It is very interesting that the agreement is best for the angles (e.g. subgroup 4 at 21.60) which correspond to an elastic Q near the first diffraction 0-l peak at 1.60A. A possible explanation is that in this region (where Leadbetter has shown an average dispersion curve exists) our assumption of simple longitudinal and transverse acoustic modes with no dispersion is at its best; and in particular the no-dispersion assumption (i.e.4)= g ) is best quite close to the first diffraction peak where the

171 9 - g Sel/S = e-02Q2 8'_* 75-i S CZ S:) 0. r-4.r3 5 7 9 I 13 I5 1 5 I OQ 0 C 4-) 0 ~J Q2.r

172 0 0~~~~~~- 0 o a 0 - o — 4o 0 O lI --- ~ J' 3 0 3 0 Cl o0 ____ 0 I0 _________1.4 ----- o — ) D -' D01 xI 0O Olw^m~gl~ Cs (0 in dr C ) - N.0 OI|x(m'0)9,

173 Subgroup 4 21.60 0 6 Cr 5^. -0:3 CD c3 0 0 0 4r) 0~~~~~~~~~~~~~~~~ K~~~~~~~~~~~~~~~ i o oo~~~~~~~~~~o r4 C~5 3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C 6 (2 ~(93~ ~ ~~ 0 Ol~~~~~~~~~~~~~~~~~~~z ~20 -10 010 20 0 I 0~~~~~~~~ 0 4t3 I~~~ ~ Co CO 1 (-~~~~~~~~~~~~~ii 05 -20 -10 0 10 20 W, PS1 0'?O *r-< C4

174 -8 0 N / ic 0~~~~~~~~~~~ 0 LC) O~~~~~~~~~~~ -I (^ O~~~~~L 4Ji Q. - > 0 s o 00) LO 0..0 I o~~~~~~~~-i ICM 0 O ~~~~~~~~~~~~~~~~~~~~~~C) oO~~~~~~~~~~~~~~~~~~r co CO ~ ~ ~ ~ _ C 0~J (C.' i 0 ^-~~~~~~ o ( M - 0 r') NU 0)j oix Im'o~o ( w fO

175 N O _o C - ct O0~ — O _ _0 D _(0 cn o ook 0 - _ I' 0' cn- - ~ Oo.., -S, L. If 00 (M *~~~ o 0r Cf)V) ~ 0 m')9. _ _ _~ IsI xl(o' )9 a ____ix wo Cl

17 average dispersion curve has not yet broken over from its initial linear rise. We may estimate the wavelength of our "long-wavelength" acoustic modes from the frequency at which the calculation begins to fail. Taking this value as ) = 8 ps, we find for L modes 8 so L.2e^^ L. 7 -I or and for T modes, The at wh These would appear to indicate the minimum wavelength at which the simple modes we have assumed can provide a decent picture of the glass dynamics, at least as reflected in the scattering law.

APPENDIX ONE THE IFUNCTION G(Q, ) ) FOR VITRIIOUS BeF2 We present plots of the nleasured scattering law interpolated at several values of Q, and of the function G(Q,4 ) at constant angle (for positive energy transfers) from the 31 detector subgroups. 177

17~ (Xl ~~~CM~J~~~~~~~,C ~~0 ~ ~ ~ ~ ~ ~ 0 1 O e- 0 i- - I _I - o Ji - _t c =__ 0I.l a,-~- 0. = - s"i q C5 5 0 0 8

179 We - 10 O Ct) - C 3 t - -.- - I -cY,, ~C CQ 0 0 0 0 0 0 080 00 0 0000 0 0 0 0 0 CC 0,C 0D( 0 >5 0 0 o o o bdddddd0 0 =3 ~r~~~~~~~'-. ~~~~~~~~~~~~~~~c 0 0 0 0 0 Q 0 ~ d d d d d d d o

120 o< ~ 0 of - N0 CI 0 3.3'4r C;, 00 __ 0 (0 X 0 ~ ~ ~ ~ ~ - Q or 0 _ 00 CI -^r" fg C;.. II I, I I I I - 0 0 0 0 0 0 0 0 jD c, c0 d 0 0 0 0 00 0

1 1 00 (t o- _0 N 0 -- (I O - N - w(vi 0 00 I ~ ~~ ~~ ~~ ~ ~ ~~~ ~~I I I I I I I lo II 0"0 CO *ri (0 OD 0 0 D CM ~ S Oo d o o o' o( o o o 3. ^ 0

182 I0 ~=^~~~~~~~~~c cc i "s~.:~~~ ~~a: CZ,\.4 \\ i — o o OQ~' 080' 090' 0 ~' 00' 000' C1:t93W0'0)9 ~ ~ ~ ~,

0. 01 (n Cccr4 o 0 0 4J C) C'8 o * / " ~T- ~ ~~~~~~~~ ci\~~ -\ 2J3 ^^>1^ OQC O 0 001 00' 090-Oo-~~ 000' (U9JWO'O) 9 -^~~ 3 ^ * 0~~~~~~~~~~~~~ -- ^^ "2 5~ "7 - ^~ -/- ~ ^~~~~lu J~ -2 3~~ ^c-~ ~ (y 001 * 080- 090- oOftO OZO- 000' (yOBNOi' OlO

184 o 0 CC.4 u ii ~i D 2a: C (D ~ QC a cr <.~~ -~o *o CC V 0 r.,., 090' 8O' 90' bO' 810' 000' (U9O3W0O' 9

1? 5 o \- t4~~~~~~ 4 s o. /^~~~~~ Z \ <. o,; 7 - o ~ o I'' I' I' I'I 090' 9h0' 9~O' hO' 0' 00' t(93J'O"O) 09 (U9~3WO0)91

186 Oli I I 0 ~~~~~/ ~~~~ a: o Q) \ +~~~~~ o *SU~') OSO QO' 0~0 OZO 010' ~~ L *(\9WOO) 0 (O3WO''0 O

1a 7 oc CS o t c Iz C CY C to (b93W'O) 9 "\ ___ ^.5~~~~~~ir \ R ^~~~~~~ / ^~\~~~~~~~~~~~~~~~~~~~~~~ri <~~~~~ 0 3 OSO' OfiO' 060' 020' OfOA^.^ ^ [t1~~V- ^ o'

188 o 0 0 N^ 1 -o t 0 r _i =I \ 0 *h0' Z~0* hZ0' 9* 0* 4~4 (89W ( I 0 9 (U9JWO\~~~~~~)- 9c4'~iO';'_ f-''J....J......

189 (N o CC cc 0 - Q) (*c *r-c VI' 4i. U) rr =E4 C~ --— *C oso. OfiO' OCO' O~~~~~~~~O' 0 —-0' 0S (UOJWO ) )~~~~V ri OSO' one. 026. or_ 0. 5 t133WO'"s^ *r-'

190 o 0=1 C C,): -4 C (I;;.a S, 0! (2 090' ObO' 9W0' hZO0 ~''' 000' ~(93W0'0 9 _ O~ 090' 81t0' 9E0' htZO' ZIO' 000' (WDRNWO' 0)

191 o OM CN,- i I~,. I C O (l0 090oso ohO' 090' O 00O' (U93WO'01)

192 CL C) 0 (3 I w o s ^ ~ ~~~~~~~'0 C1-1 = T s~~~~~~~~~~~~~~~~~~~~~el In LC) a \ _ ~~~~~~~~~~~~~~~~~~~~~~~~i 4~~~~~~~~~~L oso.\ OW GO i' 1' o. (UOJWOO) 9 2^ ----..t^Or~c -d \. 1"~~~~~a -\ --- 5 C~~ JL.~~~~~~~C' ^ \~~~~~ C3 \ m~~~~ -\ ~) 3 ^- **2 -~~~~C -L ~ ~~V <y I ^^^~~~p T~~~ 1 OSO' OhO' 0~0' OZO' OIO1 m in (t133WO'DY. "

193 a. o cc /L 0.4 co rl!!'!W \ uJ c I:O3H7'0) O ^ *s-^~~~~~( \ ~~~ s ^ (U~3O W'013

194 o c, 10 o I) ~. _\~ \^ ~ U ~o,'t, t.., ID c O0O',~O' bh0' 910 00' 000' (U93W'O) 0 OtrO'~~~~~~~~~~~ o EO d r~ o~`od'L It133W0'0 ^" s^

C)C 5 ctt a. \ one C I -^.~~,,,t-,.^ c^ (1:U93W0'0) 9 (uoa3Wo'0)

196 V) 0 o'p-I cr 0 3 OCO'Z~ Yi~~~0 810' ~~ 1O 900' 00 0 -..(W^')o 9 ^ \ -II \ ci O0~0' h;ZO' 810' 900' 00' ^r 0 ^s~~~~~~~~~~~~~~~~~~~~~~~~~ \-..10~ 3 X w4 -~~~~~~~~~~~~~~~~~i1 ^^...m ^~~~~~V ^^^^ P* 1-1~~~C ^^*1^**^^rC cl (U933WO'0)9

197 0 o/ hi =_o _ _____ TiU ri ~~~~~~~~~~~~~r!,,,,,, L a o.0, r Q \ 0 0E0R 9ZO'-810 Z10- 900.o c (U933WO'01)

198 An 0 0 CC o ff f a CY )4.. o 2= Cy (2 0 Q Cj) /~~~~~~~~~~~~~~~~~v O^ -— *0~ ~ lO' BoO' --- (U HO') 9 ^______\ r-<~'^-^s ^~~~~~~~~~~~~~~~~~~~~t -7-^ ^~~~~~~~~~~~~~~~~~~~~~~~~~/~ ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~i -— f-0. 4^~~~~.<(~~~~~~~~3 \-5 ^~~~~~~~~~~~~~~~~~~Z'^S_0'~~~~~~~~~I ^"^.-s ^~~~ ^>^*~,._*> r^ 0~0' tiZO* 810' 210' 900' 000*' LI (U03lWO( D)9

199 co o.r' -E I/) U/ /' *. 9 In i 0 2 X?- ^ IO sz. S. Sa J - - |.~.H 0 r — SZO OZO' S10' 010' SO0' 000* (UO3WO'O1 9

200 U An0 I ~ -l V) Cr) (V) U) W 0; r. /.:. O r C),/ U / - - -r *s 3, 0930< 0'L (U93NO'0) en ~ (U193N.' 0)9

201 0 le ro (n a -- 1 — ~~~~~~~~~~~~~~~~~~~~~~~~~CCY 04 p, -^~~~~~~~~~~~~~~~v j & CT) C) o _ In - c:3 O sO SW 010 50' 00 (U9JNOO) ~ ~ C 9~ 7^- u* ~o~~~~~~~~~~~~~~~~~~~~~~~~~~ ^ —-^w JK~ * *~~~~~~~~~~v \~~~~~~~~C "arszo* ozo' sio*~3WO ow so* o

202 * w -. I 0 / -3 \ V> -~~~~~~~~~~~~~~~~~~~~C, 00 Vo..! uzg t 0 a _ _ _ _ _ S0o OZO' SIO' 010' S0' 000' (U93WO'0) 9

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204 0 ) ) oa: -tl ^ U.C /SO 0OZ S1' 010' ~0-' _(B9JWO O 0

205 0 -— ii * ^ - 1-:~~~ n ^ i ia i C) CN \ - y *: ~ oSOON S -' t' SOO (BW'OO) 9' (t~~3W0' > 0

206 V) co U.. /0 oa ~I~~,~ ~ ~ ~ 1 I ~ i \ OR ~ 1_. 91 8o C' * CN 00 910 ~tO 900' hOO' 00'00 (UW3O''0)9

207 8 0i N o.. /.,c M - --- r oc CY O0 t ____ i 0 \ —~~~~ Cc o) CDt" * 0 0 OZO' 910' 10O':800' iOO' 000' (U933WO'0)

208 o lo CC,c II fo r) 1/3W' 0 7-., ^ E m o ^ i 1 E U o * ^ - o^r U 3 Sco' 0o:0' ~10' 010' SOO' 000' (U93W0'0)9

209 u) < lo^r~~~~~~~c o o. YC 01 co I I " 00' 910' ZO' 800' bOO' 000' X (93WO 0)9

210 Co CZ o z a^ X Z C) / o J C;I. / ~ " Q (UOJWQO))~~ ~. 0. / — uj f x o~~~~~~~~~~~~~~~~~~r X o -p~~~~~~~ ^_ o u~~~~~~~~~~~~~\ ""T ^~~~~~~~0 ^ /-^\~~~~~~~~~~~~~~~~~~~~J5 T-~~~~ ~ ^ -L ~ ~ ~ ~ ~ C - *2 \ ^ ~~~~~~~~Cy \" ^"^~~~~~~~~~~Q

211 0.aR 0 T~ cr CZ 0 0,_v, ~ C) o. S. wr- (93W0'O) 9 ---— ( ~ ~ ~ ~ ~ ~ ~ O (^ ^^- os ^~~~~~~~~~~~~~~~~~~~~~ -^- **~ ^~~~~~~~~~~~~~~~~~~~r \~~~~~~~ ~ S~~~~~O`~~~ ~ o1'60 00 0'0 [t^ - ^01

212 U, 0. U}.\.. 0 g X ol \...s s..0o j ='UI) I I I I I I I I I i..OZX I 9T0' Zy BOO- hioot 000.6- - (.93W00) C.4 \ w' OZO' 910' 210' 800' bOO' 000' a, (U.93WO'O)9

APPENDIX TWO MULTIPLE SCATTERING IN AN INCOHERENTLY SCATTERING PLATE Consider a plate of thickness t and of infinite lateral extent. The incident beam (figure A36 ) makes angle $. with Figure A36. the plate normal. The scattered beam makes angle G1 with the normal, and the scattering angle < = (o-!)., 9<o) The first scattering is simply V/1gn) 2,(/,) e ) I e) where T63~') - ~r tscs e- Sc49,) and g(/ - e c- 49, < /z (transmission i?/(g~~'y~~ )tS - )case) I; )i >. (reflection case) Considering only elastic scattering, we have for the second scattering M/ zQ\-l8, -?./. ~;~rgl -e ) s ts9, ~ Stece 9,/5ec i(1 - sec ~ 7s~ec ) ( - sec~ @Cr s) e 1 213

214 where 2 k are the wave vector transfers for the intermediate scattering events, and the intermediate scattering angles are C2: COO )>AAA SA C03 ~ c + Cw eo o ) (- tS.o ^^ ^^ X~ + 40d e ] The indicated imtegrations are necessarily done numerically. The total scattering in Vineyard's approximation is expressed in terms of the first and second scattering as vca) = -6a) ~ Vp ) --! ^ /^ ] It is not obvious that this is a valid approximation for thin plates. The results of a calculation of PT/ I using this approximation, however, agree with results of Monte Carlo simulations for incoherently scattering; plates. The Monte Carlo results are expected to be valid for a slowly varying cross section such as that of vanadium, so we can reasonably conclude that Vineyard's approximation is adequate for our reference scatterer.

APPENDIX THREE SAMPLE ATTENUATION FACTOR For a sample with macroscopic total cross section AZ contained in a holder of macroscopic total cross section CT we define the following distances: dC = distance traveled through container before scattering d = distance traveled through sample before scattering dc = distance traveled through container after scattering = distance traveled through sample after scattering. All these distances depend on the scattering point within the container, and the exit distances also depend on the scattering angle. The sample attenuation factor is the ratio of tube scattering with and without the sample, namely SA= f LpC ET (doC<) Z5(4.4 k)JV -fjr pc[-W EC )] C with the integration taken over the container volume. In the case of a multiple tube sample container, one need only generalize the entry and exit distances as the total distance traveled through sample or container in all the tubes traversed, and extend the integration over the volume of all the tubes. 215

APPfN iD' pTX' FO UR 4J'STTM.ATTON' OF ERRORS TIN RiF P:TAK PARAMi'..TPE.RS In order to arrive at an estimate of the uncertainity in our determination of the area, centroid and variance of the Be-F, F-F and Be-Be peaks in the refined experimental rdf, we have utilized the familiar error-propagation formula f [8(s0) )] - 0)(b~3 )X + r(T^ 8)(^/ z which is valid provided the variances of x,y,z...are independent. For our determination of coordination numbers, we used the relation where K accounts for the scatterin, lengi:th wei -htinf in the rdf, and 4A is the mesh spacing. For the variance in fi due to errors in 4rzT2-(r) then, we have rY3 i): - K-t C 7 6(^f irt7<vc))(4v Our determination of r used the relation _ 4 3 -<Al ) W/ _ so that And finally we calculated the variances usingr, <4A I> I TFL&6C) (_f /C from whi ch + rG 4&- r) } 216

217 and We calculated the bond angles from the distances by r(X-/'-x) x <. -- (a-o ) This gives for the variances -4(s,^Btv-)) - i^) Y) &6(r)& A^/ vY )A r O(~(-~-x)) ^= <r A.- X+.-r-x))/ L (.r-Y- ) We have calculated variances in our determinations of the various parameters by first assuming that there are inde2. pendent errors in the values of 4Tr2 (r) in the refined rdf. As an estimate of these errors, we used the mean squared deviation of the refined rdf with termination errors folded in, from the experimental rdf, in the peak regions (ie., the mean squared deviation of the crosses in figure 29 from the solid curve). This is probably an overestimate of the fitting errors in the refined rdf, but ignores the statistical imprecision of the diffraction data. In this way we can associate an estimated uncertainity with each of the peak parameters derived from the refined rdf. These have been included in table 6.

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