THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING CALCULATION OF GAMMA RAY SCINTILLATION DETECTOR EFFICIENCIES AND PHOTOFRACTIONS BY MONTE CARLO METHODS Bernard J. Snyder A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Nuclear Engineering 1965 May, 1965 IP-706

Doctoral Committee: Assistant Professor Glenn F. Knoll, Chairman Assistant Professor Ziyaeddin Akcasu Professor Dietrich H. Vincent Associate Professor Franklin H. Westervelt

Dedicated to the Memory of my Father Jesse Snyder ii

ACKNOWLEDGEMENTS The subject of this investigation was first suggested to the author by Dr. Geza Lo Gyorey, who served as chairman of the doctoral committee in the initial stages of the work. It is a pleasure to acknowledge the assistance of Dr. Knoll, as chairman of doctoral committee throughout most of this investigation. The other members of the committee offered aid during various stages of the work, for which the author is appreciative. Many helpful conversations were held with K. Wainio who, as a colleague of the author, gave assistance in formulating part of the Monte Carlo programs. The cooperation of the staff of the University of Michigan Computing Center, in particular F. Little, contributed to the successful completion of this research. The author is grateful for financial assistance provided by three Atomic Energy Commission Pre-doctoral Fellowships; and the Industry Program, College of Engineering, The University of Michigan. I wish to express my appreciation for the assistance given by my family, in particular my mother. Finally, I will be forever indebted to my wife, Sondra, for her unselfish help and encouragement. iii

TABLE OF CONTENTS Page ACKNOWLEGEMENTS............................................... iii LIST OF TABLES...................................... viii LIST OF FIGURES..... o..............o................. xi CHAPTER I INTRODUCTION........................................ 1 A. Gamma Ray Scintillation Spectrometry............. 3 oB Absolute Total Efficiency... o 6 C. Absolute Peak Efficiency......................... 8 D. Photofraction.o.................................. 10 E. Intrinsic Total and Intrinsic Peak Efficiencies.. 11 Fo Calculation of the Absolute Total Efficiencies and Photofractions............................. 11 II CALCULATION OF THE ABSOLUTE TOTAL EFFICIENCY......... 13 Ao Definition of Absolute Total Efficiency ( E AT).0... 00....e...... 13 1o Isotropic Point Kernel for EAT0..*.*.....* 15 2. EAT for On-axis Point Sources............... 20 3. EAT for Disk and Cylindrical Volume Sources.................................... 22 4o Geometry Factors.......................,.... 23 Bo Calculation of CAT...................o......... 24 Co Checks Applied to Calculated EAT Results..... o. 24 Do Results of EAT Calculations.................... 30 1. On-axis Point Sources....................... 31 2o Off-axis Point Sources....................... 37 30 Disk Sources................................ 43 4, Cylindrical Volume Sources.................. 49 Eo Other Factors Effecting AT..................... 56 iv

TABLE OF CONTENTS (CONT'D) Page Fo Accuracy of EAT Results....................... 60 Go Extrapolation of Tabulated Results............... 65 1. Other Crystal Sizes......................... 63 2o Other Source Energies........................ 64 III APPLICATION OF MONTE CARLO TECHNIQUES TO CALCULATION OF PHOTOFRACTIONS..................... 68 A. Principles of Monte Carlo Calculations........... 68 1. Cumulative Probability Technique............. 72 2. Rejection Techniques......................... 74 B. Path Length Sampling.......................... 77 C, Gamma Interactions.............................. 78 1o Photoelectric Absorption................... 79 2. Compton Scattering........................... 80 3. Pair Production........................... 81 D. Secondary Gammas and Electrons.................. 81 E. Typical Histories............................ 83 IV DETAILS OF THE MONTE CARLO CALCULATIONS FOR PHOTOFRACTIONSo.................................. 86 A. Source Geometries Considered..................... 86 1. Isotropic Source Routines................... 87 a. Solid Crystal - Point Source............. 89 b. Well Crystal - Point Source,............. 91 c. Extension to Other Isotropic Sourceso.... 97 2. Collimated Beam Sources..................... 98 B. Gamma Interactions...................... 99 1. Interaction Processes Considered............. 99 2. Secondary Gamma Treatment................... 100 v

TABLE OF CONTENTS (CONT'D) Page 3. Photoelectric Events......................... 101 4. Compton Events............................... 102 5. Pair Production Events...................... 103 C. Approximation of Electron Transport.............. 105 D. Bremsstrahlung Production........................ 110 E. Escape Routines................................. 119 F. Termination Categories........................... 120 G. Photofraction Standard Deviation................ 121 Ho Absorption and Scattering in the Source.......... 123 V RESULTS FROM THE MONTE CARLO CALCULATIONS............ 129 A. Comparison of Experimental and Calculated Results.......................................... 129 1. Solid Crystals.............................. 150 2. Well Crystals............................... 135 B. Effects of Bremsstrahlung and Electron Escape on Photofraction.......................... 139 Co Effect of Geometry on Photofractions............. 140 1. Isotropic Sources........................... 140 2. Collimated Beam Sources..................... 144 3. Comparison of Solid and Well Crystal Photofraction................................ 147 D. Comparison of Monte Carlo and Integration Results.......................................... 147 E. Photofractions Calculated for NaI................ 150 F. Photofractions for Materials other than NaI..o... 151 VI CONCLUSIONS......................................... 155 A. Summary of Results............................... 155 1. Absolute Total Efficiencies (AT)........... 155 2. Photofractions........................... 156 vi

TABLE OF CONTENTS (CONT'D) Page Bo Application of Results........................... 157 C. Extension of Results............................. 157 APPEND ICES A DERIVATION OF EXPRESSIONS FOR THE ABSOLUTE TOTAL EFFICIENCY.......................................159 B SERIES SOLUTION FOR THE TOTAL EFFICIENCY............ 171 C DERIVATION OF KAHN'S METHOD FOR SAMPLING FROM THE KLEIN-NISHINA FORMULA......................... 175 D WELL CRYSTAL ESCAPE ROUTINE......................... 178 E INSTRUCTIONS FOR COMPUTER PROGRAMS FOR CALCULATION OF THE ABSOLUTE TOTAL EFFICIENCY AND PHOTOFRACTIONS,. 186 F METHOD OF SELECTING FROM A UNIFORM AZIMUTHAL DISTRIBUTION OF DIRECTIONS.......................... 200 G TABULATED VALUES FOR ABSOLUTE TOTAL EFFICIENCIES..... 204 REFERENCES..................................................... 226 vii

LIST OF TABLES Table Page II-1 Comparison of CIT Values for Point Sources on Centerline of Solid Crystals........................., 25 II-2 Comparison of CAT Values for Point Sources on Centerline of 2 x 2 Inch Solid Crystal................ 26 II-3 Comparison of CAT Values for Point Sources on Centerline of 3 x 3 Inch Well Crystal................. 26 I-4 Comparison of CAT Values for Disk Sources on Centerline of 3 x 3 Inch Solid Crystal................ 29 I1-5 Comparison of Solid Angles Subtended by Two Inch Diameter Solid Crystal at Off-axis Points............. 30 11-6 Isotropic Point Source Off Crystal Axis at Radial Distance = M 3 x 3 Inch Solid Crystal............... 36 II-7 Isotropic Point Source Off Crystal Axis at Radial Distance = M 2 x 2 Inch Solid Crystal............... 36 11-8 CAT(M)/ CAT(M=O) for 2 x 2 Inch Solid Crystal, at 1.0 MeVo........................................ o 37 11-9 Isotropic Point Source Off Crystal Axis at Radial Distance = M, 8F8 Well.......................... 42 II-10 Isotropic Point Source Off Crystal Axis at Radial Distance = M 3 x 3 Inch Well Crystal................. 42 II-11 Total Gamma Ray Sections in Pure NaI, From References 20, 21, Used in Evaluating CAT............... o 65 III-1 Fraction of Primary Interactions...................... 84 III-2 Fraction of the 1000 Interacting Gammas That Have Exactly X Scatterings Prior to Absorption............. 84 III-3 Fraction of the 1000 Interacting Gammas That Have Exactly X Scatterings Prior to Escape................ 85 viii

LIST OF TABLES (CONT'D) Table Page III-4 Typical Pair Production Events....................... 85 IV-1 Integration of the Differential Bremsstrahlung Spectra for NaI....1.............................. l14 IV-2 Absolute Total Efficiencies, Source Self Absorption and Scattering Included, 2 x 2 Inch Solid Crystal..... 123 IV-3 Absolute Total Efficiencies, Source Self Absorption and Scattering Included, 8F8 Well Crystal............. 128 V-l Broad Beam Photofractions, 2 x 2 Inch Solid Crystal... 135 V-2 Comparison for Point Source, TIp, 3 x 3 Inch Well Crystal................................. 136 V-3 Comparison for Point Source, CAP, 3 x 3 Inch Well Crystal......................................... 137 V-4 Comparison for Point Source, CAP, 5 x 5 Inch Well Crystalo..o..................................... 137 V-5 Comparison for Point Source, CAP, 8F8 and 7F8 Well Crystals....................................... 158 V-6 Effect of Electron and Bremsstrahlung Losses on Photofractions, Isotropic Point Sources............. 139 V-7 Source Geometry Effect on Photofractions, 3 x 3 Inch Solid Crystal................................... 143 V-8 Source Geometry Effect on Photofractions, 8F8 Well Crystal.e coee o oo.............................. 143 V-9 Solid Crystal Photofractions, Isotropic Point Source on Axis.................................... 150 V-10 Well Crystal Photofractions, Isotropic Point Source on Axis.............................,........ 151 ix

LIST OF FIGURES Figure Page I-1 Idealized Differential Spectrum for Gamma Ray Spectometer............................ 3 II-1 Solid Crystal Geometry............................... 16 II-2 Well Crystal Geometry.......................... 17 II-3 CAT Variation with Radial Distance from Solid Crystal Axis (.662 MeV),............................ 27 11-4 CAT Variation with Radial Distance from Solid Crystal Axis (.075 MeV)........................... 28 1I-5 CIT for 2 x 2 Inch Solid Crystal................... 32 1I-6 Effect of Moving Point Source Axially from 2 x 2 Inch Solid Crystal............................. 33 11-7 Effect of Moving Point Source Axially in 8F8 Well Crystal......................................... 34 1I-8 Increase in CAT of Well Crystal (8F8) for Noncentral Point Source............................. 40 11-9 CAT Variation with Disk Diameter of 8F8 Well Crystal..................................... 44 II-10 CAT Variation with Disk Diameter for 12AW12-W2 Well Crystal................................ 45 II-11 CAT Variation with Disk Position for 8F8 Well Crystalo..............o.............. 46 II-12 CAT Variation with Disk Radius...................... 48 II-13 CAT Variation with Cylindrical Volume Source Height for 8F8 Well Crystal..................... 50 II-14 CAT Variation with Volume Source Height for 8F8 Well Crystal.............o............................ 51 x

LIST OF FIGURES (CONT'D) Figure Page II-15 CAT Variation with Volume Source Height for 12AW12-W2 Well Crystal............................. 52 II-16 EAT Variation with Volume Source Height for 2 x 2 Inch Solid Crystal........................... 53 II-17 EAT Variation with Source Cylinder Radius........... 54 II-18 CAT Variation with Well Diameter................... 57 II-19 CAT Variation with Well Height...................... 58 II-20 CAT versus Well Size............................... 59 II-21 CAT Variation with Energy for 8F8 Well Crystal...... 65 1I-22 CAT Variation with Energy for 7F8 Well Crystal...... 66 11-23 CAT Variation with Energy for 2 x 2 Inch Solid Crystal.............................................. 67 III'l Flow Diagram for Monte Carlo Calculation of Photofractions..............................o 70 IV-1 Electron Transmission in NaI..o................0..... 107 IV-2 Bremsstrahlung Spectra from an Electron Slowing Down in Sodium Iodide............................. 111 V-l Photofraction versus Energy for 3 x 3 Inch Solid Crystal.............O....O..........o................ 132 V-2 Effect on Photofraction of Electron and Bremsstrahlung Escape from 3 x 3 Inch Solid Crystal.. 141 V-3 Effect on Photofraction of Electron and Bremsstrahlung Escape from 8F8 Well Crystal......................... 142 V-4 Variation of Solid Crystal Photofraction with Collimationo........................................ 145 xi

LIST OF FIGURES (CONT'D) Figure Page V-5 Photofraction (Relative to Narrow Beam) Variation with Collimation for 3 x 3 Inch Solid Crystal....,... 146 V-6 Absolute Peak Efficiencies for Collimated Sources Compared with Experiments of Jarczyk.................. 148 V-7 Photofractions for Solid and Well-Type Crystals....... 149 V-8 Ratio of NaI Photofraction to CsI or CaI2 Photofraction for 8F8 Well Crystal................... 152 xii

CHAPTER I INTRODUCTION The investigation described in this dissertation is concerned with the calculation of gamma ray scintillation crystal detector efficiencies. As gamma ray detectors, scintillation crystals are superior to gas-filled detectors in their ability to measure the total energy of incident gamma rays, and their much higher gamma detection efficiency. Hofstadter(1) first suggested the use of NaI(T~) as a gamma scintillation detector in 1948, and this material has remained the most important to date The high density of NaI (3.667 gm/cm3) and high atomic number (Z=53) of iodine account for its superior gamma ray detection characteristics over gas-filled detectorso The scintillation crystal can totally absorb the energy of an incident gamma ray, making possible a measurement of source energies, and use of the crystal detector in gamma ray spectrometry. Gas-filled detectors are not suitable for energy measurements, since total absorption of high energy gammas is negligible. Recent application of cooled germanium diode detectors in gamma ray spectroscopy show outstanding resolution characteristics, but are limited to much smaller volumes and efficiencies than readily available NaI crystals. Calculation of efficiencies for gas-filled detectors is considerably more difficult, and probably less accurate than efficiency calculations for large volume solid medium detectors. The difficulties arise because of the mechanism for gamma interactions with a gas-filled -1

-2detector The primary means of gamma detection is through the gas ionization produced by secondary electrons formed when a gamma interacts in the solid detector wall. The variations in range and energy of these electrons introduces considerable uncertainty in the calcula(2) tion of the gas-filled detector efficiency. Also, the sensitive volume of these detectors may be difficult to define accurately. However, the scintillation crystal efficiency is more readily calculated, as will be discussed subsequently. Once the efficiency of a given crystal has been obtained, another crystal of the same size will have exactly the same theoretical efficiency. The precise definition of the scintillation crystal sensitive volume insures reproducibility of the efficiency, independent of the electronic counting system. Since scintillation crystals are the most widely used gamma ray detectors, the present calculations of detection efficiencies are useful in a variety of applications. Right circular cylinders, with and without a coaxial cylindrical well, are the specific crystal shapes used almost exclusively in gamma ray counting. Therefore, this investigation has concentrated exclusively on these two crystal shapes. Knowledge of the detection efficiency is required to obtain a quantitative measurement of the absolute gamma ray source emission intensity (Iy). Determination of the absolute source strength is required in many applications such as quantitative activation analyses, and absolute neutron flux measurements The source strength may be related to the absolute disintegration rate of an isotope source if the decay scheme is known.

-3If the appropriate efficiency (6) of a gamma ray detector is known, experimental observation of the count rate (C.R.) will give the absolute source strength (Iy) I = C.R./E (1.1) A. Gamma Detector Scintillation Spectrometry Consider a monoenergetic point source of gamma rays of energy Eo, located such that some of the emissions interact with a scintillation crystal. If the differential count rate of the detector is observed with a pulse height spectrometer, it may appear idealized as H(E) in Figure I-1 below. The differential spectrum of emitted energy is as G(E). G(E) Back H(E) Scatter Compton m Peak P C P ositronnuum p Electronic Edge- F — WHM 5E I Noise o Both ~*^~ Annihilation \ Gammas Escape /11' a)| G as Escape Annihilation / 1 Crystal Gamma Escapes ~~~IS -.~ ~Crystal I A pp 0 Eb.51 E0-1.02 1.02 Eo-.51 Eo E,Energy Figure I-1. Idealized Differential Spectrum for Gamma Ray Spectrometero

-4The interactions which may contribute to the observed spectrum are depicted in the following sketch and are identified below. /Scintillation \^73 i 7~~~ Photo ctric absorption event 3. Compton scattering 7?,7 L. PhotomultiplierJ / < ~Tube / Scintillation Crystal Gamma rays from the source may have any one of three primary interactions in the crystal or surrounding media. Labeled rays from the source indicate these possible initial events: 1. Pair production event 2o Photoelectric absorption event 3. Compton scattering event.

-5These initial events give rise to the following secondaries, as labeled in the above sketch: 4. Compton scattered gammas 5. Positron annihilation photons (.51Mev) 6. Bremsstrahlung 7. Electrons 8. Iodine K x-ray 9. X-rays from surrounding media. Other radiation may be present from the source, or background, which further complicates the observed spectrum. The important features of the spectrum shown in Figure I-1 are as follows. The peak at E corresponds to total absorption of the source gamma within the crystal, and is conventionally defined as the photopeak. In addition to primary photoelectric events, the total absorption peak has contributions from successive events in the crystal which eventually result in total absorption of the source energyo Since these successive events occur within the crystal resolving time (0.25 x 10-6 sec. in NaI(2)), the energy deposited in each event is assumed to give a single output pulse. For energies below E, the observed continuum is due to gammas that deposit less than the total source energy in the crystal. For example, gammas which undergo an initial Compton event frequently are scattered from the crystal with only part of the source energy.in the form of electron kinetic energy, being absorbed in the crystal. This process is the major source of the observed continuum, which is conventionally called the Compton continuum.

-6Superimposed on the Compton continuum may be found a number of peaks which are identified on the figure. The backscatter peak at Eb is due to Compton events in the material surrounding the crystal which result in near 180~ scattering of the gamma into the crystal. The peaks which may be observed at Eo -.51 Mev and E - 1.02 Mev are due to initial pair production events within the crystal in which respectively one, or both positron annihilation gammas escape from the crystal. Also, positron annihilation in material surrounding the crystal may give rise to peaks at.51 and 1.02 Mev, if either, or both annihilation gammas are subsequently absorbed in the crystal. In some systems, peaks due to low energy x-rays may be seen near the region of electronic noise. The gamma emission intensity and observed total integral count rate are related to the differential spectra by: 1 -=f G (E)cdE (1.2) C.R. = H(E)dE E(dllsc.) where E(disc.) is the discrimination level of the counting system. Bo Absolute Total Efficiency Identifying the total area under the experimentally observed pulse height spectrum AT with the total integral count rate, C.R.,

-7at zero discrimination level we have: AT =J H (E)dE (1.3) and making further definitions: A* = ideal total area, i.e., the total area under the ideal T pulse height spectrum that would be obtained if no attenuating and scattering effects were present due to material surrounding the crystal (e.g. shielding) or due to gamma interactions within the source, 6AT = absolute total efficiency associated with the ideal total area, i.e., the fraction of gammas emitted from the source which interact with the crystal at least once and transfer energy to the crystal, Equation (1.1) becomes: iT= A/A (1.4) From the above equations, if A* AT and = can be calT AT culated, one should be able to determine the source intensity in gammas emitted per second from the observed integrated total count rate. In practice however, AT AT, i.eo, the ideal total area is not the observed total integrated count rate because of the perturbing effects of material surrounding the crystal and source self-absorption and scattering. As defined, EA is associated with A rather than AT, because interactions of the source gammas with anything other than the

-8scintillation crystal are ignored. For example, the backscatter peak usually observed in a pulse height spectrum is not taken into account. In addition, the observed total area may include pulses due to positron annihilation in the shield as well as pulses at low energy due to electronic noise. The observed total area is not a parameter uniquely related to the absolute source emission intensity, but it also depends on the presence of other materials and the electronic system. For a given experimental arrangement, it is concluded that calculation of IZ from AT is not feasible, and one must seek another measurable quantity that can be accurately reproduced for any experimental arrangement. C. Absolute Peak Efficiency Instead of AT, we will make use of the integrated pulse height under the photopeak (see Figure I-1). Making the definitions: Apy = photopeak area, i.e., the experimentally observed area of the pulse height spectrum under the photopeak, rAp = absolute peak efficiency, i.e., the fraction of gammas emitted from the source which interact with the crystal such that their total energy is transmitted to the crystal. Physically, App measures the number of source gammas incident on the crystal whose total energy is absorbed in the crystal. An implicit assumption made here is that the differential pulse height spectrum is proportional to the spectrum of energy deposition in the crystal. Investigations of this proportionality have been made by various workers, e.g Verheijke,(3) Engelkemeir(4) and for most applications the assumption has been shown to be justified.

-9When the photopeak is clearly defined A may be measured PP by direct summation of the appropriate portion of the differential spectrum. For cases when the spectrum is more complex and the photopeak is not as clearly defined, A is conventionally obtained by PP assuming the photopeak to be a Gaussian curve. Based on this assumption, it can be shown that App 2 2 Ct,0 where Cm = maximum of the photopeak (at E0) 5E = full width of the photopeak at half maximum (FWHM). When a source gamma ray has an interaction before entering the crystal (i.e., scattering in the detector shielding) some of its energy will be lost, preventing deposition of its total energy within the crystal and thus it will not be counted under the full energy peako The photopeak area serves the purpose of providing a readily measurable quantity which can be related uniquely to the absolute source intensity. For a given geometry, and source intensity and energy, A remains constant (assuming no change in the amount of PP shielding interposed directly between the source and the crystal) independent of changes in the counting system or the amount and position of external materials. For identical geometries and crystal sizes, the pulse heights may differ between two crystals and another photomultiplier may give different resolution, but the total number of counts under the photopeak does not depend upon pulse height or resolution.

-10The photopeak area is solely a measure of those source gammas which are totally absorbed in the crystal, and is reproducible since the experimental arrangement has essentially no effect Ap D. Photofraction From the above definitions of EAP and Ap, one may write: I = AP/c^ (1.5) Since Ap can be experimentally determined uniquely, 1 may be obtained if CAP can be calculated. Conventionally CAP is not calculated directly, but rather is determined by the following: -T F FAT (1-6) Ph AAT'AP cAT At (1.7),A T where App/AT = peak-to-total ratio, or photofraction. CAP is divided into these two factors primarily because EAT can be obtained from explicit analytical expressions (see Chapter II), while the photofraction must be calculated by simulation of the transport of particles through the crystal by Monte Carlo methods (see Chapter III).

-11E. Intrinsic Total and Intrinsic Peak Peak Efficiencies It will be convenient to define two other efficiencies, which have the solid angle dependence removed: CIT = intrinsic total efficiency, the fraction of source gammas incident upon the crystal that interact with the crystal at least once and transfer energy to the crystal, and jIP = intrinsic peak efficiency, the fraction of source gammas incident upon the crystal which interact with the crystal such that their total energy is transmitted to the crystal. Thus we have, CAT -4 IT C' C ~ where 4 where = mean solid angle subtended by the crystal at the source. Fo Calculation of the Absolute Total Efficiency and Photofraction In the present work CAT is calculated directly from analytic expressions (Chapter II). The quantity App/A, known as the peak to total ratio or photofraction, is calculated by Monte Carlo methods for practical cases of interest (Chapter III)o The photofraction

-12represents the ratio of the number of gammas that are completely absorbed, including all secondary radiation, to the number of source gammas that interact at least once. Calculation of the photofraction is performed by simulating with Monte Carlo methods the successive physical processes which an incident gamma undergoes. An analytical treatment involving solution of the problem is entirely impractical on account of the mathematical complexity of interactions, multiple secondary particles, and geometry. It should be noted that calculation of CAT can also be done by the Monte Carlo process, but to obtain results with an accuracy equivalent to that obtained by numerical integration of the analytic expressions would require excessive computer time.

CHAPTER II CALCULATION OF THE ABSOLUTE TOTAL EFFICIENCY A. Definition of Absolute Total Efficiency (CAT) As previously defined in Chapter I, the absolute total efficiency, CAT, is the probability that a gamma ray emitted from the source will interact with, and deposit some of its energy in the crystal. C AT is considered here to be the product of two independent probabilities. First, the probability that the source gammas are incident upon the crystal surface and second, the probability that those gammas will interact with the crystal. These two probabilities are respectively the geometry factor, 0/4jr, and the intrinsic total efficiency, C IT. Making the definitions: = solid angle subtended by the crystal at the source f (1-e do = probability that gammas incident upon the crystal will fo dn JdQ interact with the crystal = IT where, x(Q) = path length in the crystal extrapolated in the direction of emission of a source gamma T(E) = total gamma ray cross section (without coherent scattering) for the crystal material E = energy of the monoenergetic source gammas one obtains, A-T 4 r 4 TU t -1 -13

-14and finally, EAT (4-e )dS. (2.1) It should be noted that the coherent (Rayleigh) scattering cross section has been removed from the total cross section values used. This process occurs only at the lowest energies, where photoelectric absorption predominates. For gamma energies above about.20 MeV in NaI, the cross section for Rayleigh scattering is negligible. Below this energy, the photoelectric cross section is approximately 10-50 times greater than the Rayleigh cross section. Even when a coherent scattering event occurs, the photon is scattered by the atomic electron cloud and the entire atom recoils, so that only a slight change in direction and energy of the photon occurs. Equation (2.1) forms the basis for all calculations of 6AT For perfectly collimated monodirectional photon beams parallel to the cylinder axis, Equation (2.1) reduces to: e-TH E~AT~= C~T~ = I - eTH ~(2.2) where H = axial length traveled through the crystal cylinder. In Equation (2.1), the element of solid angle dO, can be written as: dn sin6d 9d0 = -a/ dfq

-15Referring to Figures II-1 and II-2, the definitions are made: Q = polar angle [1 = cos 9 cp = azimuthal angle and Equation (2.1) becomes: I d F,-e<X(/'.*)1d E' C AT~= re ( );d (2.3) For both the well and solid crystals, the extrapolated path length, x(,cp) is a complicated function, which is discontinuous at the edges of the crystal and well. Solution of Equation (2.3) for an isotropic point source a distance, m, from the crystal axis for either the well or solid crystal provides a point kernel which can, in principle, be integrated to determine EAT for any arbitrary source. The analysis here considered point sources, an isotropic disk source of zero thickness, or an isotropic cylindrical volume source of finite thickness. These point kernels are derived in Appendix A, using the definitions of Figures II-1 and 11-2. The results are summarized here. 1. Isotropic Point Kernel for EAT For a solid crystal: AT 2 T q o, _ s pt)~or0 F-,0)d.+, Fl,(/))df (2=.) 4,,... Z.

-16___.R _ m _I SOURCE Figure II-1. Solid Crystal Geometry.

-17R /^] ^ —— ~^\-S SOURCE YFge W Figure -2. Well Crystal Geometry. Figure 11-2. Well Crystal Geometry.

-18where 11= (H+b [(H+b) + IS A,= b/(b +s' ) -()) /- expG-T7 / ) S=-Ya sn14 + -V-T2COS For a well crystal with the point source within the well: E (oC-ax T s pt.)= q [ (/ F (,q) d4t + (/,4) 4 +fi( ) d4] (2.5) As discussed in Appendix A, the functions F2 i, 1 and L2 depend on the inequalities: Case 1: -- < b a+b Case 2: > S. b -c+b

-19where s, = -"m S-\ - ~f~t~_So S4 S,= -m Sin +VRt - m cos;/ The results are: F, =/ - exp( —ra/l) and for Case 1: b /X Vsc + SI /(ab) + S b - - 1 a+ b

-20or for Case 2: S. > St b O+b s V2+ b+ (o;4f~ Sb b Equations (2.4) and (2,5) are the desired point kernels for an off-axis point isotropic source for solid and well crystals, respectively. Any desired isotropic source geometry can be obtained from these equations, and in particular, the symmetrically located point, disk, and cylindrical volume sources are considered. 2. EAT for On-axis Point Sources The point source on the crystal axis equation for AT is obtained by letting m = 0 in Equations (2.4) and (2.5), and performing the integration over c. The results are: For a solid crystal: E (on-^oI. pt.) FK (t ) d + F (/)du (2.6) E,'acr 2~~~~~~~~~~~~~(~6

-21where V(H b)2 t R; U+- b IJ R b F, 1=l - exp(-T H/U) (l) =/- exp[-r(,. - - ) These results have been previously obtained.(5'6 For a well crystal, with the point source within the well: (~on-cx ps at.)=- ^0(daJ L+d i (Aj F(A)do (2.7) 2 r' 1 + i CC~)cL~ where4 J F. () =/ - exp(-r () FU'l)=I - +,expt( )j - -) Ati3 _ RC /)4 C "

-22and when t R b a+b _ ob+b Fa:)=, - exp- r(-, ] i to6 / -a b / R2+(<atb) Expressions for a point source on the crystal axis, but outside the well, are given in Appendix A. 3. EAT for Disk and Cylindrical Volume Sources The expression for a homogeneous isotropic disk source of radius g, located with its center on the crystal axis, is also derived in Appendix A and is given by F;()=/- expr ( a b )] yR'+(a+b) A ( Lsk) 2 M -8-n Expressions for a point source on the crystal axis, but outside the well, are given in Appendix A. 3. CAT for Disk and Cylindrical Volume Sources The expression for a homogeneous isotropic disk source of radius g, located with its center on the crystal axis, is also derived in Appendix A and is given by: C ( dask) = s(ofC-a<s pt.) d (2.8) So^

-25The expression is derived in Appendix A for a homogeneous isotropic cylindrical volume source of radius g and height (b2-bl), located with its axis along the crystal axis. The source cylinder axial distances are measured from the same plane as b, given in Figures II-1 and II-2 (b2 > bl). EAT (vol I) = 9 ( b ) db/ n-AT(~o -axis pt.)drn (2.9) For the point source on the crystal axis, it is easily shown that the geometry factor, Q- is given by: 4jt For solid crystal: _- _ / (A b \ (2.10) For well crystal, source inside well: L0. I t/+ C (2.11) 47r 2 f' +t ) For well crystal, source outside well: _7_ m -2 \ C ) (2.12) 47T 2, - C-+t-.

-24B. Calculation of CAT It is not possible to obtain solutions of Equations (2.4 - 2.9) in any closed-form analytical expressions, nor can these integrals, in general, be reduced to the form of any known tabulated functions. For a restricted number of special cases, series solutions have been found by Grosjean, Reference 7 and the unpublished derivation of N. McCormick given in Appendix B. Complete solution of these equations was carried out by numerical integration techniques on the Michigan IBM-7090 Digital Computer, using a Gaussian quadrature formula. Tabulations of calculated results are given in Appendix G. The computational algorithms used for the centerline point, disk and volume, and off-axis point sources are designated the BURP-1,2,3 Computer Programs, respectively. Operating instructions for use of these computer programs are given in Appendix E. C. Checks Applied to Calculated CAT Results Numerous independent methods are available to ascertain the proper operation of these computer programs. Since considerable computational data are available for solid crystals, with point and disk sources,5'56 13) direct comparisons are possible with other calculations for these source geometries. Only one published article(4) gives any results for right circular cylindrical crystals with a coaxial cylindrical well. Reference 14 gives data for only one crystal size. Comparisons were made with these references and agreement within all significant figures (usually three) was obtained, except for the calculations of Reference 11. These references generally used numerical integration

-25techniques based on the trapezoidal rule, using a small number of subintervals, since the computations were performed on older, smaller capacity machines than the IBM-7090. There are two reasons for the slight disagreement with the results of Reference 11. First, Reference 11 uses somewhat different cross sections, and second, the integration was performed by a different technique (Monte Carlo). However, since the maximum difference was less than 0.25%, no significant discrepency is indicated for the different approaches taken by Reference 11 and the present computer programs. Tables II-1, 11-2, and 11-3 give some of the comparisons made for CIT and CAT for centerline point sources. Figures 11-3 and 11-4 compare some of the results obtained for off-axis points with that of Reference 5. TABLE II-1 COMPARISON OF IT FOR POINT SOURCES ON CENTERLINE OF SOLID CRYSTALS EIT Crystal Size, cm. Source Height Energy Present Reference Dia. Ht. b, cm. MeV Calc. 0.2.5491.55 (1o)* 3.81.635 0.7 1.0.1157.12 (10) 3.81.635 3.0 1.0.0715.07 (10) 3.81.3175 1.5 2.0.0798.08 (lo) 3.81.3175 0.3 0.32.7793.7773 (11)** 7.62 7.62 11.0 0.661.6284.6277 (11) 7.62 7.62 11.0 4.43.3973.3976 (11) 7.62 7.62 9.3 7.48.3997.3995 (11) 7.62 7.62 9.3 * Plotted values **Tabulated values

-26TABLE II-2 COMPARISON OF EAT VALUES FOR POINT SOURCES ON CENTERLINE OF 2 x 2 IN. SOLID CRYSTAL CAT Energy Present Source Height MeV Calc. Reference(5) b, cm. 0.01.01539.01539 10.0 0.129.10605.10619 3.0 1.10.04118.04119 3.0 2.04.8233 x 10-4.823 x 10-4 100.0 2.04.5877 x 10-2.585 x 10-2 10.0 2.04.03343.03344 3.0 5.50.02940.02940 3.0 7.90.02994.02995 3.0 TABLE II-3 COMPARISON OF EAT VALUES FOR POINT SOURCES ON CENTERLINE OF 3 x 3 IN. WELL CRYSTAL CAT Energy Present Source Height MeV Calc. Reference(14) b, cm. 0.010.9800.980 0.0 0.10.9712.971 0.5 0.30.6989.699 2.0 0.60.6203.620 0.0 1.00.0831.0832 6.81 1.00.01737.0174 13.81 2.00.4048.405 0.0 3.00.3314.332 0.5 5.00.3169.317 0.5 5.00.3551.355 0.0 Well size: 7/8 in. dia. x 1-1/2 in. deep

1.0 0.9 z w 0.8 UL fL + EXPERIMENTAL (IDO-16370) Id E0 CALCULATED (IDO-16370) O- SOURCE ENERGY = 0.662 MeV _ q\ _J - |POINT SOURCE | 0 I 0.5 0 0.4 0 1D 2.0 3.0 4.0 m,DISTANCE FROM CRYSTAL AXIS, cm Figure II-3. EA' Variation with Radial Distance from Solid Crystal Axis (.662 MeV). Figure II-3. E'AT Variation with Radial Distance from Solid Crystal Axis (.662 MeV).

0.9 z U.m 0 0.7- + EXPERIMENTAL (IDO- 16370) o CALCULATED (IDO-16370) I ~0 CALCULATED (PRESENT WORKI ~J 0.6SOLID CRYSTAL SIZE 3x3 in Ms SOURCE ENERGY - 0.075 MeV SOURCE-DETECTOR DISTANCE, B = 1.0 cm + 0.5 POINT SOURCE 0.4 - I I I 0 1.0 2.0 3.0 4.0 m, DISTANCE FROM CRYSTAL AXIS, cm Figure II-4. eAT Variation with Radial Distance from Solid Crystal Axis (.075 MeV).

-29An independent check is provided for disk sources with solid crystals by the tabulations of Reference 5.* Table II-4 gives some of the comparisons made. In addition, Reference 15 provides some tables of the geometry factor for off-axis point sources. Since the present calculation of CAT reduces to calculations of -- for large values of 4ri the total cross section, T a direct comparison is possible. Table II-5 presents this comparison. Another check on the proper calculation of CAT for disk and volume sources is the asymptotic results obtained when the dijsk or cylindrical volume is made progressively smaller compared with the results obtained for the point source on the axis. As discussed later in this chapter, when the variation of C AT with the size of a distributed source is investigated, correct asymptotic values are obtained. TABLE II-4 COMPARISON OF CAT VALUES FOR DISK SOURCES ON CENTERLINE OF 3 x 3 IN. SOLID CRYSTAL CAT Energy Present Disk Radius Source Height MeV Calc. Reference(5) G, cm. b, cm. 0,0484.36105.36119 1.905 1.0 0.0484.03201.03206 1.905 10.0 0.081.35804.35954 1.905 1.0 0.129 -.34713.34856 1.905 1.0 0.212.31189.31250 1.905 1.0 0.566.21269.21291 1.905 1.0 0.129.08557.08591 2.857 5.0 7.9.03173.03194 3.086 5.0 *Reference 5 contains an error in Appendix III (Table VI) page 59. The heading at the top of this table specifies a value of R (disk radius), when actually this is r (crystal radius), in accordance with the authors' notation.

-30TABLE II-5 COMPARISON OF SOLID ANGLES SUBTENDED BY 2 IN. DIA. SOLID CRYSTAL AT OFF-AXIS POINTS Present Source Distance Source Height Calc. Reference(15) Off-axis m, cm. b, cm..082065.08206 o.635 3.81.002479.002479 0.635 25.4.024781.02479 1.27 7.62.052270 0.5 5.08.050758 1.0 5.08.049529.04956 1.27 5.o8.048364 1.50 5.08.045257 2.00 5.08.041650 2.50 5.08 D. Results of EAT Calculations Having performed all of these checks on the proper operation of the BURP- 1 2, and 3 computer programs, various detailed calculations were made, the results of which can be used for laboratory applications. In Chapter V, it is pointed out that changes in source geometry have only a small effect on the photofractions, whereas, the values of absolute total efficiency are strongly dependent upon these changes. In order to show the geometrical dependence of EAT, a number of different calculations have been performed as follows: (a) Solutions for on-axis point sources for well and solid crystals (b) Effect of moving a point source from the crystal axis (c) Determination when a disk source can be considered a point source

-31(d) Variations in 6AT with disk size and disk axial position (e) Determination when a thin cylindrical volume source can be considered a disk source (f) Variation in CAT with the size of cylindrical volume sources (g) Effects on CAT of crystal and well size, and thallium concentration. All of the data of this chapter are for NaI(Tl) crystals; some data for other scintillation materials used for gamma ray detection (i.e., CsI and CaI2) are given in Appendix G. 1. On-axis Point Sources As will be shown later in this chapter, the two and threedimensional sources of small enough extent can be well-represented by an on-axis point source. Therefore, the tabulation of results in Appendix G for on-axis point sources covers most of the commonly-used well crystals. The compilations are primarily for well crystals, since a considerable amount of data is available for solid crystals,(5)6'8-13) and except for Reference 14 no data is available for well crystals. Some calculations for 2 x 2 in. and 3 x 3 in. solid crystals, with on-axis point sources, are given to supplement those of the references. Assuming the source to be a point on the crystal axis, Figures II-5, 11-6, and II-7 show the variation in efficiency as the sourcecrystal distance is increased. Figure II-5 shows the intrinsic total efficiency, EIT for a solid crystal, while Figures II-6 and II-7 are

1.0 (I-e ) =0.9934 Total Cross Section, T = 1.0 cm 0.9I4) 0.8 0.7 - 0.0 0.01 0.1 1.0 10.0 100.0 DISTANCE FROM CRYSTAL, cm Figure II-5. E for 2 x 2 In. Solid Crystal. IT

0.6- ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 0.5 b 0.4 4I 0.3K E = 0.01 MeV. Q20 0.2 2.0 0.I 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 b, SOURCE DISTANCE FROM CRYSTAL FACE, cm Figure II-6. Effect of Moving Point Source Axially from 2 x 2'In. Solid Crystal.

-341.0 I 0 Q O~~~~~_ ^"^^^^9~ ~ISOTROPIC POINT 0.9 SOURCE ON CRYSTAL AXIS 0.8 E =0.01 MeV 0.7 0.5 - 5 _____-_ _ N0.20 MeV 4 I\ 0.4 0. 3 ~~ —-^^ O ^. 40 MeV Ad^ ^^^^^^0.60MeV^" 0.2 ^ ^ ^ ^ ^^ ~2.0 MeV 0.1 0 I, I 0 1.0 2.0 3.0 b, DISTANCE FROM WELL BOTTOM, cm Figure II-7. Effect of Moving Point Source Axially in 8F8 Well Crystal.

-35for the absolute total efficiency, AT for solid and well crystals, respectively. These figures are plotted for two commonly used crystals, the Harshaw Chemical Company Well Crystal No. 8F8 (2 in. dia. x 2 in. ht., with 1-1/8 in. dia. x 1-1/2 in. deep well)(16) and a 2 x 2 in, solid cylindrical crystal. Figure II-5 is interesting because with the solid angle dependence removed ( IT = fAT), the effect of variation of the extrapolated path length, x(>), with source-crystal distance is more clearly evident. The average path length in the crystal is longest for very large and very small source-crystal distances because few gammas escape out the cylindrical surface. As this distance increases from zero to infinity, the intrinsic efficiency decreases rapidly, passing through a minimum at a distance equal to approximately the crystal diameter, and then increasing again as the distance increases further. Other investigations have shown this same trend.(1217) Increasing the source distance from the crystal causes the divergence of the gamma rays from an isotropic point source to become a smaller effect, until finally the rays are essentially monodirectional. The latter effect can also be seen in Figure II-5 as the intrinsic efficiency asymptotically approaches the value calculated from Equation (2.2). Figures II-6 and 11-7 show that a significant change in CAT can result from only a slight axial movement of the point source,

-36TABLE II-6 ISOTROPIC POINT SOURCE OFF CRYSTAL AXIS AT RADIAL DISTANCE=M 3 x 3 IN. SOLID CRYSTAL D = 7.620000 H = 7.620000 B = 1.000000 ENERGY M=.CC.10.20.50 1.00 1.. ~C1 C.3731.S998.9994.961.9838.96j9.C15.3731.S998.9994.9961.9838.96.9.02.3731.9998S 9994.9961.9838.9609. 03C.3728.9998.9994.9960.9832.9595 ~C4G.3731.9998.9994.9961.9836.9608.05t,3730 C9998,9994.9960.9835.9601 C6G.3727.9998.9994.9959.9830.9592.C8C.3766.9998.9993.9958.9827.9583 * l.10. 3675.9998.9993.9956.9319.9563 15C.3550.9998.9992.9946.9778.9472 2CC.3341.9997. 9990.9937.9740.93'3 3C C.2887.9997.9989.9929.9714.9341.40 2 5.2571.9997.9989.9929.9712.9341 *5CC 251.S997. 9989.9929.9713.9346.6CC.22C 9.9997.9989.9929.9715.93,800.2C C1.9997.9989.9930.9719.936.CC.1857.997.9989.9931.9722.367 1. 5. 1622.5997.9989. 9932.9726.9376 2.CC.1496.~997. 989.933.9729.938D 3. CO3.1380.991.9989.9933.9732.93s1 4.C.1. 3 6.S9997.999.33.7 32 9 3 5.CCO.1325.9997.9989.9934.733.9394 6.C.1325.9997.9939. ) 34. 973.9394 8.CCO.1345.99979.9989.9933.932.933J 1C0.- CO.1377.9.97 9899.9933.-732.93-1 15.CC.1466.9997.99899.9-33.73.9386 20.CC. 1540.9997.9989. h932.9728.933 30. 0. 1656.9997.99.9932.92 67.9377 TABLE II-7 ISOTROPIC POINT SOURCE OFF CRYSTAL AXIS AT RADIAL DISTANCE=M 2 x 2 IN. SOLID CRYSTAL D = 5.080000 H 5.080000 = 1.000000 ENERGY M=.OC.1C.20.50 1.CC 1. 5.010.3168.9995.9980.9871.9450.8619.015.3168.9995.9980.9871.9450.8617.020.3168.9995.9980.9870.9444.8604.C30.3152.9995.9979.9867.9433.8585.040.3167.9995.9979.9869.9441.86CC.05C.3160.9995.9979.9867.9435.8589.C60.3146.9995.9979.9867.9432.8582.080.3111.9995.9978.9862.9413.8543.100.3C59.9994.9977.9854.9383.8487.15C.2833.9993.9974.9833.9311.8375.200.2528.9993.9972.9826.9292.8369.300.2C19.9993.9973.9831.9318.8454.400.1722.9993.9974.9836.9343.8516.5C0.1534.9994.9974.9840.9358.8554.6CC.1415.9994.9975.9842.9368.8578.8CO.1257.9994.9975.9846.9381.8669 1.C00.1150.9994,9976.9848.9390.8629 1.500.C983.9994.9976.9851.9403.8660 2.GCQ.C897.9994.9976.9853.9410.8675 3.CC0.C819.9994.9977.9854.9416.8689 4.00CC.C790.9994.9977.9854.9418.8694 5.000.0783.9994.9977.9855.9418.8695 6.CC0G.0783.9994.9977.9855.9418.8695

-372. Off-axis Point Sources No data have been previously available which indicate the effect on CAT of moving a point source radially from the axis of a well crystal. The BURP-3 Computer Program, described in Appendix E, was written to investigate the off-axis effect for both well and solid crystals. Reference 5 gives the results from a limited study of the off-axis effect on CAT for solid crystals only. Figures 11-3 and II-4 indicate the effect on the efficiency of moving an isotropic point source from the crystal axis. These figures are for a 3 x 3 in. solid NaI(Tl) crystal, with 0.662 and 0.075 MeV sources at b = 1.0 cm. from the crystal, and give a comparison of results from the present work and Reference 5. For solid crystals the trend of decreasing CAT, as a point source is moved radially from the crystal axis, has been observed in all calculations with this program. Figures 1I-3 and II-4 indicate such a trend, and as can be seen, agreement of the present work with experimental and independent calculations(5) is excellent. Table II-6 gives the results of calculation for a 3 x 3 in. crystal, with the point source located at b = 1.0 cm. from the crystal, over the energy range 0.01 - 6.00 MeV. Table II-7 gives a similar calculation for a 2 x 2 in. crystal. In both of these tables, the column labeled m = 0 gives the absolute total efficiency for the point source on the crystal axis. For other values of m, the efficiencies are given relative to the on-axis efficiencies i.e,, these are values of EAT(m)//CAT(m = 0). The relative decrease in AT with varying off-axis distance, m is not the same at all source heights, b measured from the face of the

-38solid crystal. In fact as the data of Table II-8 show, at a given off-axis distance, the ratio EAT(m)/EAT(m = O) goes through a minimum a short distance from the crystal and then approaches 1.0 as the source height, b is increased. Thus the magnitude of the effect is reduced further from the crystal. This is, of course, a very similar trend as that previously seen for calculated values of E IT for a solid crystal, Figure 11-5. The data of Table 11-8 are for CAT and thus the effect of solid angle decrease, as the source-crystal distance is increased, is included. TABLE II-8 CAT(m)/ AT(m = 0) FOR 2 x 2 IN. SOLID CRYSTAL, E = 1.0 MeV b, cm. m =.5 cm. m = 1.0 cm. m = 2.0 cm,.1.989.949.763.5.985.940.745 1.0.986.940.759 2.0.989.950.812 5.0 3996.976.913 10.0.999.992.970 Two conclusions from these data for solid crystals are evident. First, in order to use the tabulated values for point sources on the crystal axis, the sources must be located accurately on the crystal axis. Second, the accuracy of this location is more critical close to the solid crystal face. Quantitatively, a 1.0 MeV source located 0.1 cm. from a 2 x 2 in. crystal must be positioned within 0.5 cm. of the crystal axis

-39for an error of less than 1% if the centrally located point source values for EAT are used. If the source-crystal height is 10.0 cm., the radial location need be accurate only within 1.0 cm. of the crystal axis for the same error in applying centrally located point source values of CAT The above positioning requirements have been investigated at energies other than 1.0 MeV and are typical for energies in the range of laboratory sources. The effect of moving a point source radially from the axis of a well crystal is shown by Figure 11-8 for the Harshaw Chemical Company Crystal No. 8F8.(6) The effect for the well crystal is seen to be just opposite to that for the solid crystal, i.e., eAT(m)/ AT(m = 0) > 1.0, or m > 0. For the source within the crystal, two factors combine to cause the observed increase in efficiency as the point is moved from the crystal axis. First, the solid angle subtended at a point source by the top of the well decreases as the point moves off the axis(15) and as shown in Table 11-5, and thus fewer gammas escape the well without striking the crystal. Second, the average path length in the crystal increases as the source moves off the axis, for well crystal dimensions presently used in practice. Since the geometry factor for most well crystals with the source located near the well bottom is close to unity, the first factor does not contribute as significantly as the second, The second factor was observed by computer analysis of the four integrands which make up the multiple integrals of Equation (2.5), In general the rate of increase in efficiencyis less at the bottom of the well than in the upper portions (see Figure II-8)

I1.0 1.5 > 9.0' QLI I I I I I b /.0.75 5.0 b DISTANCE FROM / / / iu i WELL BOTTOM / / / M z[Energy 1.0 MV. W 7.0 - 0.5 i, 5.:- - b = DISTANCE FROM WELL BOTTOM z z 0 3.0 w I~~ b DISN0.1 cm 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 m,DISTANCE FROM CENTERLINE, cm Figure 11-8. Increase in CAT of Well Crystal (8F8) for Non-central Point Source [Energy = 1.0 MeV.]

-41because the geometry factor influences the results more near the top of the well where 2 deviates considerably from unity. From the results 4jr given in Figures II-3 through II-8, it is evident that axial position variation for a point source has a greater influence on AT than radial position variation. For comparison with the solid crystal, the requirements for accurate location of a 1 MeV point source in an 8F8 well crystal in order to use on-axis values for EAT are as follows, for an error of less than 1%. For b = 0.1 cm. (see Figure 11-2), m must be less than 0,7 cm. from the axis. For b = 3.0 cm., m <.3 cm, is required. Again, these positioning requirements are typical over the energy range of laboratory sources. Tables II-9 and II-10 give results for well crystals. The point source is located at various off-axis radial distances, m, and at a fixed axial distance from the bottom of the well, b = 0.2 cm. This value for b is typical of the crystal canning and reflector thickness of available detectors. 6) Table II-9 is for the Harshaw Crystal No. 8F8, and Table II-10 is for the 12AW(12)-W1 Harshaw Crystal, 3 x 3 in., with 0.791 in. dia. x 1.5 in. deep well. By. reference to these tables it is evident that fairly accurate radial location of a point source is essential if on-axis data for EAT are used. For counting measurements in which sources are just dropped into a well counter, the usually assumed reproducibility of source-detector geometry is not justified for precise work. A possible procedure, when an accurate radial positioning device is not available, would be to place the sources at the corner of the well where the location is more likely to be reproducible. Then by reference to data such as given in Tables II-9 and II-10, suitable correction can be applied to the on-axis values for EAT ~

-42TABLE II-9 ISOTROPIC POINT SOURCE OFF CRYSTAL AXIS AT RADIAL DISTANCE=M 8F8 WELL D = 5.080000 H = 5.080000 B =.200000 WD = 2.857500 WH = 3.810000 ENERGY M= 0.1.20.50 1.00 1.42.010.9649 1.O0000 1.COl 1.0008 1.0031 1.C058.015.9649 1.CCOO 1.CCl1 1.0C08 1.C030 1.0058.C2C.9649 1.OCCC 1.CC01 1.C008 1.CC30 1.0058.03C.9644 1.CO0 1.CC01 1.0008 1.0032 1.0060.040.9648 1.~OOG 1.CG01 1.0008 1.0 303 1.0058.C50.9646 1.0CC0 1.0001 1.0CC8 1.0031 1.0059.060.9642 1.CCCC 1.0001 1.CG09 1.C032 1.0061.C8C.9633 1.CCO 1.C001 1.0009 1.0034 1.0064,100.9612 1.CCOO 1.C002 1.0C10 1.C039 1.0073.150.9C65 1.C001 1.0004 1.0C24 1.0101 1.025d.200.769]3.C 1.01 10006 1.CC37 1.0175 1.0574.3C0.547: 1.C002 1.CC08 1.0050 1.0256 1.0973.400.438 1.C002 1.0C08 1.0C56 1.0291 1.1148.500.376. 1.0002 1.CC09 1.CC59 1.C309 1.1243.600.3395 1.CC02 1.C009 1.CC61 1.0320 1.1298.8CO.2932 1.C002 1.CCC9 1.0063 1.0333 1.1366 1.000.2633 1.C002 1.0010 1.0064 1.0342 1.1410.500.210.0002 1.2.010 1.0C66 1.C354 1.1473 2.0CC.1970 1.CC02 1.CC10 1.0C67 1.C360 1.1505 3.CCC.17'8 1.0003 1.0010 1.OC68 1.0365 1.1532 4.000.17)8 1.C003 1.C010 1.0068 1.0367 1.1542 5.000.161 1.C003 1.0010 1.CC69 1.0368 1.1544 6.COO.1691 L.C003 1.0010 1.C069 1.C368 1.1544 TABLE II-10 ISOTROIIC POINT SOURCE OFF CRYSTAL AXIS AT RADIAL DISTANCE-M 3 x 3 IN. WELL CRYSTAL D = 7.620000 H =7.620000 B =.200000 WD = 2.857500 WH= 3.810000 ENERGY A = 0.10.20.50 1.00 1.42.010.9649 1.0000 1.0CC1 1.0008 1.0031 1.0058.015.9649 1.0000 1.C001 1.0C08 1.0031 1.0058.02C.9649 1.CCoC 1.0001 1.0CC8 1.0030 1.0058.C30.9646 1.cCOC 1.0001 1.0008 1.0C30 1.0058.040.649 1.C000 1.C001 1.0008 1.CC030 1.0058.05C.9648 1.C000 1.0001 1.0008 1.0030 1.0057.060.9645 1.CCOC 1.0001 1.008 1.0031 1.0058.080.9634 1.CCOC 1.0002 1.0009 1.0035 1.0064.100.9618 1.0000 1.C002 1.0010 1.C039 1.0071.15C.9520 1.QO01 1.0003 1.0016 1.0061 1.0116. 200.9116 1.001 1.0004 1.00CC26 1.109 1.0249.300.7752 1.CC002 1.0006 1.0040 1.0181 1.0509.400.6733 1.0002 1.C007 1.0C45 1.0212 1.0640.500.6C41 1.0002 1.C007 1.0048 1.0229 1.0715.600.5595 1.C002 1.0008 1.oC50 1.0239 1.0758.800.4989 1.0C02 1.0008 1.0052 1.0251 1.0812 1.000.4572 1.0002 1.(008 1.0054 1.0259 1.0847

-433. Disk Sources The discussion so far has been restricted to sources which are small enough to be considered mathematically as a point. Of obvious interest then is determination of when a distributed, homogeneous source can be considered a point source. Considering the isotropic source is a disk, with its center located on the axis of the crystal, calculations were made for EAT as the disk diameter was decreased, at a fixed distance from the bottom of the well. Results for 8F8 and 12AW(12)-W2 crystals are shown in Figures II-9 and II-10, respectively. Tabulated results for these crystals, plus others, with disk sources are given in Appendix G. The 12AW(12)-W2 well crystal is 3 x 3 in., with 1-1/8 in. dia. x 1-1/2 in. deep well. Two facts are evident from these curves. First, as the disk diameter goes to zero, the results from the disk source computer program (BURP-2) reduce to those independently calculated by the point source program (BURP-1). This provides another check of the proper calculation with these programs. Second, over the range of energy.01 < E < 2.0 MeV, and for both the 2 x 2 in. and 3 x 3 in. well crystals, there is an insignificant difference between the point and disk source values of CAT for disks up to 0.8 cm. radius. Disks up to 1.0 cm. radius differ from on-axis point sources by less than 2% in CAT ~ Figure II-11 represents the variation in CAT as a disk of fixed radius (1.0 cm.) is moved axially from the well bottom of an 8F8 crystal.

-441.0, E=O.OIMeV 0.9 0.8 0.2 0.7.~d |~~~ ISOTROPIC DISK _a'~~^~ _ SOURCE ON CENTER 0.6 - LINE 0.5 cm. FROM WELL BOTTOM 0.5 ] Lo. cm 0.4 0.4:0 —-----------------------— g - -------— 0.6 0.3 0.2 2.0 0.1 0 l I I I,,, AQ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 C POINT SOURCE d, DISK DIAMETER, cm Figure II-9. EAT Variation with Disk Diameter for 8F8 Well Crystal.

-451.0 E = 0.01 MeV 0.9 0.2 0.8 ISOTROPIC DISK SOURCE ON CENTERLINE 0.5 cm FROM WELL BOTTOM 0.7 0.4 0.6 - 3, 0-.6 0.50.4 2.0 0.3 0.2 0. - 0 X I I I I I 0 0.2 0.4 0.6 0.8 1.0 1i.2 1.4 DISK DIAMETER, cm Figure 11-10. EAT Variation with Disk Diameter for 12AW12-W2 Well Crystal.

0.7 10, c,^^^~~ ISOTROPIC DISK r 1.0 cm SOURCE RADIUS 1.0Ocm 0.6 - ON CRYSTAL AXIS i-! b~~~ 0.5 - E.0.2 MeV 0.4 0.3 0.4 MeV 0. 2 0'2 ^^^-^^_^^ ^^^^~~~~~~~~~~~~~0.6 MeV ^^^-^^^~20 M e V 0.1 0 0 1.0 2.0 3.0 4.0 b, DISTANCE FROM WELL BOTTOM,cm Figure II-11. CA Variation with Disk Position for 8F8 Well Crystal.

-47Figure II-12 gives results for disk sources of various radii, all at.412 MeV. The 8F8 well crystal, 2 x 2 in., and 3 x 3 in. solid crystals are compared in this figure. It should be noted that the total efficiency increases with disk size at a fixed axial position for the well crystal, and decreases for the solid crystal. This is a manifestation of the same effects previously discussed for the point source moved radially from the crystal axis. Also, the advantage in absolute total efficiency of a well crystal over a solid crystal of the same outside dimensions can be seen from this figure. The data for the two solid crystals indicate that for disks located close to the crystal, little deviation from a point source is found for disk radii up to 0.8 cm. For a disk close to the crystal and a radius of 1.5 cm., the decrease in CAT is approximately 6% for the 2 x 2 in. solid crystal and 3% for the 3 x 3 in. solid crystal. At a disk-solid crystal distance of 10.Ocm.,it is seen that disks of 2.54 cm. radius cause no significant variation from the point source results. Thus for disks, as for off-axis points, the deviation from on-axis point source values of CAT decreases and becomes negligible as the source-solid crystal distance increases. Since it has been shown that only a small error results in using on-axis point source CAT values for disks of less than 0.8 cm. radius, it was considered sufficient to limit extensive calculations, to point sources only. Where greater accuracy is desired, some tables for disks are given for a limited number of well and solid crystals. All these tables will be found in Appendix G.

0.7 ENERGY=O.412 MeV 0.6 0.5 2x2 INCH (8F8) WELL X-TAL, b=0.2 cm7 0.4 J | 2X2 INCH SOLID X-TAL, b=0.2 cm.7 0.3,_ Q0_2 3X3 INCH SOLID X-TAL, b=1.O cm. 0.1 3x3 INCH SOLID X-TAL, b=1o.o cm 2X 2 INCH SOLID X-TAL, b=10. cm. 0 0-5 1.0 1.5 2.0 25cm. POINT SOURCE DISK RADIUS Figure 11-12. EAT Variation with Disk Radius.

-494. Cylindrical Volume Sources Verheijki(l8) has given the only previous results for cylindrical volume sources. His work was restricted to cylindrical sources and solid crystals of equal diameters. Again no data are available for well crystals, and therefore a computer program was developed to consider any sized cylindrical volume source located along the axis of either well or solid crystals. All calculations for homogeneous, isotropic disk sources assume that the source is two-dimensional. The effect of finite source thickness has also been investigated. Some results are shown in Figures II-13 through 11-17. Figures II-13 and II-14 give values of CAT for a cylindrical source which is 0.3 cm. from the well bottom, to allow for the combined thickness of a glass test tube, and crystal canning and reflector. Again these figures confirm one aspect of the computer programs because they indicate a smooth asymptotic behavior as a cylindrical volume source approaches zero thickness, finally converging to the independently calculated disk source results. Figures II-13 and II-14 are for the 8F8 well crystal, and show that for cylinder thicknesses up to.05 cm, (approximately 20 mils), insignificant variation from disk source CAT values results. For most neutron activation foils and other foil sources, this 20 mil foil thickness would not be exceeded. In fact, seldom are foils used which are greater than 1-2 mils in thickness. Thus, foils in general use can be considered to be two-dimensional sources, and if they are located at the well bottom and are well-approximated by a disk of radius less than 0.8 cm., they can be considered point sources.

-501.0,,. E=0.01 MeV 0.9 - -DISK SOURCE g = 1.13 cm b,= 0.3 cm 0.8 g9 0.5 -TEST TUBE, CANNING & REFLECTOR THICKNESS 0.4 - DISK SOURCE 0.661 MeV 0.3 0.2 - 2.75 MeV DISK SOURCE 0.1I -IOMILLS F 1/8 INCH 1/4 INCH FOIL THICKNESS,cm 0.0O 0.2 1 0.4 0.6 0.8 1.0 12 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 CYLINDER HEIGHT FROM WELL BOTTOM, b2,cm Figure II-13. EAT Variation with Cylindrical Volume Source Height for 8F8 Well Crystal.

g ~~0.6-~~ ---- ---- b, = 0.3 cm 0.I6 -t g = 1.28 cm SOURCE (b2;-b) ENERGY =0.661 MeV WELL b 0.5- CRYSTAL b 0.4 \i1 s- 0.3 0.2 0.10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 cm. CYLINDER HEIGHT, (b2.-b) Figure II-14. CAT Variation with Volume Source Height for 8F8 Well Crystal.

Te;SXID TTaM M-ZTM4VTZ.IoJ q.TaH aomos aunTOA %TFT uoryT.mA jV *'T-II ac- nLT. uwo q Wq'o 1lO8 113M WO- I1H913H 83N1A 3onos lls t'I'1 l0';1 8'0 9'0 t0o ZO - ro - I' -'0 AGW fL'Z - 9'O _9~0 -.'O -8'0 WO 0'q wo 89'1 =6 - 6'0 AW I0'0 0 — 3, _____ i _____ _________ ^ --------— l3 ~5$-~~~~~~~~~~~~~

0.35 I I I I EAT 0.3 0.30 ENERGY = 0.661 MeV 0.25 0~|,^~~ ~.~q~~ ~b, =0.0 cm 0.25 - DISK SOURCE 0.20.20 X SOURCE b 0.10 0.05 - C I I I I I I 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 SOURCE CYLINDER HEIGHT FROM CRYSTAL FACE, b2, cm Figure II-16. AT Variation with Volume Source Height for 2 x 2 In. Solid Crystal. ATS

-54ISOTROPIC CYLINDRICAL VOLUME SOURCE bl= 0. 3 b25.3-SOLID X-TAL b=1.755 -WELL X-TAL ENERGY =0.661 MeV 9t 0.5CE U b SOURCE b SOLID CRYSTAL 8F8 WELL CRYSTAL I — 0.20.1 - 2X2 in. SOLID CRYSTAL0.0 I I I l 0.0 0.5 1.0 1.5 2.0 2.5 cm. g, CYLINDER RADIUS Figure II-17. Variation in EAT with Source Cylinder Radius.

-55The rapid decrease in EAT with source cylinder height is as expected since the upper annular portion of the well crystal contributes less to the attenuation of source gammas than does the solid lower portion, and also the fraction that escape out the top of the well increases. The decreasing efficiency with distance from the bottom of the well has been a characteristic seen in calculations for all source geometries. There is no available data for calculated values of volume source CAT' except for Reference 18, so the results of some computations are tabulated for both solid and well crystals in Appendix G. These volume sources are assumed to be transparent to emergent gammas and no attenuation has been considered. For volume sources in which absorption and scattering of the source gammas is not negligible, a separate calculation has been made. This problem will be discussed in detail in Chapter IV, Section H. In order to take into account the combined thickness of the crystal canning, reflector, and source container, the tabulations for 6AT are made for various height cylinders which are placed 0.3 cm. from the crystal surface. Some calculations of the effect of varying the cylindrical source diameter have been made for a 2 x 2 in. (Harshaw No. 8F8) well crystal at 0.661 MeV, The results of these calculations are shown in Figure II-17. By comparison with Figure II-16, the decrease in EAT for a solid crystal is negligible as the source diameter increases. From Figures II-15 and II-17 for well crystals, a slightly greater change in EAT is seen with variations in source height than with increasing source diameter. Although not as pronounced for the well crystal, this trend for CAT with radial and axial dimensional changes is the same as for point and disk sources.

-56E. Other Factors Effecting CAT The final geometry variations were calculations to study the influence of crystal dimensions on CAT. In particular, the influence of well crystal overall dimensional tolerances, and the effect of various well sizes were determined, Also, since Reference 19 indicates the thallium concentration in the central portion of a typical Nal(Tl) crystal can vary from 0.11% -.25% by weight (w/o) calculations were performed to study this effect on the absolute total efficiencies for 8F8 and 12AW12 crystals. The results obtained indicated that 6AT varied at most by 0.2% for a variation of thallium concentration from zero to 0.25 w/o. The effect of well crystal dimensional tolerances on the total efficiency calculations has been determined for a typical well crystal. According to Reference 16, a manufacturing tolerance of +.005 in. is guaranteed. Using the worst combinations of this tolerance on all crystal dimensions causes a maximum deviation in the absolute total efficiency of 0.64% from a nominal sized crystal. The influence that well diameter and height have on EAT are shown in Figures II-18, II-19 and 11-20. Figures II-18 and II-20 show how EAT varies with well dimensions for an isotropic point source on the crystal axis, at the top of the well. These two figures are plotted from the same data, but using different coordinates, The source was located at the top of the well specifically to illustrate that the computer results for a well crystal converged to those for a solid crystal, as the well is made progressively smaller. This provides another independent check of the computer programs. Figure II-18 shows a rather

|~~~~~~~~~~~ ~ ~~0.39 - ~ ~~D = H = 3 in 0.39 ENERGY =0.661 Mev 0.38 SOURCE AT TOP OF WELL WELL HEIGHT 0.37 SOLID CRYSTALW'/^y __ ~~~~~~~~~~~~~~Hw (cm) 0.01 0.36 0.35 - 0.34 0.33 0.32 0.50 0,31 0.30- 1.00 0.29 - 2.00 3.00 0.28 - 0.278 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 WELL DIAMETER, cm Figure II-18. AT Variation with Well Diameter. AT

0.40,, D=H=3in 0.39 ~'~~~~~~~0.~391~ ENERGY =0.661MeV SOURCE AT TOP OF WELL 0.38 Dw= WELL DIAMETER 0.37 SOLID CRYSTAL (F —-----------------------------------------------------: —-----— 0.01 = DW -O.I cm 0.36 0.50 0.35 -50 0.34 -. ~~~~~~~~~~~~~0.31 ~~- 2\^^~~~~~~~~~~~~ 2 —-~ —1.00 0.30 - 0.29 0.28 - 2.8575 0.27i 0 1.0 2.0 3.0 4.0 5.0 WELL HEIGHT, cm Figure II-19. EAT Variation with Well Height.

-59Hw- WELL HEIGHT D = H = 3 in 0.9~~~~~~~0.9'~~~b = 0.5cm 0.8 0.7i ENERGY = 0.661 MeV 0.6 0.4 0.8 1.2 1.6 2.0 2.4 2.827 0. - WELL DIAMETER, cm Figure 11-20. EAT vs. Well Size. AT~~~~~~~~~~~50

-6osurprising linear variation of CAT with increasing well diameter, for various values of well height. Although comparison of Figures II-18 and II-19 indicates that increases in well height beyond about 2.0 cm. have no further influence on CAT, (for diameters < 2.0 cm.), it must be remembered that the source is located at the top of the well, and that the advantage of locating the source deeper within the crystal is not being considered. Figure 11-20, to be discussed subsequently, does show this effect. However, one can conclude from a comparison of Figures II-18 and II-19 that variations in well diameter have a much greater effect on (AT values than do variations in well height. That this is true, even when the source is located near the well bottom, can be seen in Figure 11-20. In this figure, the same general trend is seen for two values of gamma ray energy, 0.661 MeV, and 4.00 MeV. The decrease in efficiency as the well diameter is increased is seen to be much greater than the change caused by decreasing the well height the same amount. This trend would be expected since of course the well volume goes as the square of the diameter, and is linear with height. However, calculations have shown that the change in CAT does not correlate with well volume changes, i.e., for different diameter and height, but same volume wells, the efficiency is considerably different. EAT is seen to depend on both the diameter and height of the well, independently. F. Accuracy of EAT Results Correct and accurate application of all these calculations is dependent upon many things. Primarily they are: accurate definition of

-61crystal dimensions, accuracy of total cross sections used, and accuracy of the numerical integration technique used. The effect of manufacturing tolerances was seen to be small, of the order of 0.6%. The Gaussian quadrature numerical integration technique has been optimized, with respect to the number of subintervals into which each integral is divided and the order of the approximating polynomial, such that the results are accurate to four decimal places. The total gamma ray cross sections for NaI(Tl) included in the computer programs are taken from the National Bureau of Standards data given in References 20 and 21. References 20 and 21 give the most extensive summary and compilation of primary values of attenuation coefficients for photons on 29 materials. The summary of Reference 20 includes a detailed analysis of both theoretical and experimental results. These data were obtained primarily by theoretical calculations, with experimental data serving as a check. Reference 22 provides extrapolations and interpolations of older NBS data to include all elements up to atomic number 100. The NBS data are reported to be accurate within 3- 5% and this will cause some error in the application of any efficiency calculations to experimental results. Reference 5 presents some calculated results of the effect of an error in cross sections on detection efficiency. An estimate of the maximum percentage change in EIT for a given percentage change in cross section can be made. The intrinsic total efficiency can be written approximately as: -TX EIT / - e

-62Since the average path length, x, in the crystal is of the order of the crystal height (x < 10 cm. for most common crystals), two limiting cases are of interest. First, for low energy gammas (E < 0.2 MeV) and an average path of about 4 cm. or more, E IT is essentially unity, and changes in T will have no effect. Second, in limit of higher energy gammas (T generally decreases), the above expression can be expanded in a series and written approximately as: This case sets an upper limit to the effect of an error in cross section This case sets an upper limit to the effect of an error in cross section on the efficiency. The maximum effect is that a given percentage error in cross section causes, at most, the same percentage error in efficiency, Thus one can conclude that an error in cross section of 3% can cause no more than a 3% error in calculated efficiency. The EAT tabulations are presented with efficiency as a function of incident gamma ray energy, for convenience. As can be seen from Equation (2.1), the efficiency is a direct function of total cross section, and is related to energy only indirectly. If inthe future improved cross sections become available, the tabulated data in Appendix G can be plotted as a function of total cross section, using the data of Table Il-ll1 and the appropriate corrections made.

-63TABLE II-11 TOTAL GAMMA RAY CROSS SECTIONS IN PURE NaI(20,21) USED IN EVALUATING CAT (NO COHERENT SCATTERING) Energy, MeV T cm 1 Energy MeV T, cm1 0.01 564.718 1.0 0.212 0.015 178.950 1.5 0.171 0.02 81.041 2.0 0.151 0.03 26.806 3.0 0.135 0.03323 20.462 0 03323 111.477 4.0 0,129 5.0 0.127 0 04 66,739 6.0 0.127 0.05 37.037 8.0 0.128 0.06 22.625 0.08 10.488 10.0 0.134 15.0 0,147 0.10 5.757 20.0 0 158 0.15 2.083 30.0 0.176 0.20 1.118 0.30 0.568 0.40 0.407 0.50 0.330 0.60 0.289 0.80 0.241 G. Extrapolation of Tabulated Results 1. Other Crystal Sizes The tabulated calculations for the absolute total efficiency given in Appendix G have been made for scintillation crystals found to be in most general use for laboratory gamma ray detection. Interpolation and extrapolation of these tables is possible for other crystal and source dimensions which are in the proportions dictated by the following scheme, which has been given by References 6 and 10. Although this method was originally given by References 6 and 10 for on-axis point sources, it can

-64be extended to any source geometry. Inspection of the expressions for CAT given by Equations (2.4 - 2.9) shows that the crystal and source dimensions always occur as products with the total cross section, T or in ratio with one another. Thus the absolute total efficiency calculated for a given set of crystal and source dimensions and cross section equals that for N times each of the dimensions and at a cross section corresponding to T/N, where N is any number, 2. Other Source Energies The data tabulated in Appendix G may be readily interpolated for energy, as indicated in Figures 11-21, II-22 and II-23. Figures 11-21 and II-22 are for the 8F8 and 7F8 well crystals, respectively. Figure 1I-23 is for.a 2 x 2 in. solid crystal. These figures give the variation of 6AT with energy, for different source-crystal distances. At the lowest energies (highest cross section) the intrinsic total efficiency, CIT- is unity, so that CAT =. Any of the data given in Appendix G follows smooth curves such as these, so that accurately interpolated CAT values may be obtained for energies not specifically tabulated.

1.0 b = 0.0 cm I>1.*5 s Isotropic Point Source 3.5 \\\ \ - - I 0.9Figure -21. Variati on with Energy for 88 Well Crystal. Axis 20.A. y 0.83\ / 0.6 0. 3.81 4.0 I0.0 0:01 0.02 0.04 0.1 0.2 0.4 1.0 2.0 4.0 10.0 ENERGY, MEV Figure II-21. AT Variation with Energy for 8F8 Well Crystal.

1.0 b = 0.0 cm 1.5 T09 2.5 ~ ^\ ISOTROPIC POINT SOURCE ON CRYSTAL AXIS:.0 0.80.7- -.5 0.5- t3.81 0.5( 4.0 0.01 0.02 0.04 0.1 0.2 0.4 1.0 2.0 4.0 10.0 ENERGY, MEV Figure II-22. AT Variation with Energy for 7F8 Well Crystal. ~ATVaitowihEegfo78WelCyt.

ISOTROPIC POINT SOURCE |~b-=~ 0.0 cm ~ON CRYSTAL AXIS 0.50 0.5 0 0.4 1.0 0.30.2 2.5 0.1 5.0 0 \ 0.01 0.02 0.04 0.06 0.1 0.2 0.4 0.6 1.0 2.0 4.0 6.0 10.0 ENERGY, Mev Figure II-25. AT Variation with Energy for 2 x 2 In. Solid Crystal. AAT

CHAPTER III APPLICATION OF MONTE CARLO TECHNIQUES TO THE CALCULATION OF PHOTOFRACTIONS A. Principles of Monte Carlo Calculations The principles of Monte Carlo calculations have been discussed (23-29) elsewhere(2329) and only a brief mention of some general methods used in these calculations will be given here. Monte Carlo methods involve statistical estimates of the fraction of a given number of source particles that can be expected to terminate in preassigned categories, after having undergone various interactions in media of known geometry and composition. Use of the method is only limited to those problems in which the probabilities of each interaction and the relevant physical laws are known. Since solution of the coupled set of equations describing these processes can not be obtained in general, Monte Carlo methods provide a powerful tool for the solution of practical problems involving particle transport, The method consists of calculating on the digital computer the mathematical equivalent of a low intensity experiment. The basis of all Monte Carlo digital computer calculations is a set of random numbers, uniformly distributed between zero and one. In practice, however, such a set of numbers is not stored in the machine memory. Rather, upon demand, a so-called pseudo-random number is generated by any one of a number of techniques. The randomness of these generated numbers has been throughly tested,(25) such that the periodicity for their repetition is exceedingly large and that all digits have essentually an equal probability of appearing. The method of random number generation -68

-69employed in these calculations has a period of 235.(30) Henceforth when a random number is referred to, it will mean one of these machine-generated pseudo-random numbers. The discussion here is limited to a general outline of Monte Carlo techniques, as they are used in the present calculation of photofractions. The entire problem is treated here, without a detailed account of each of the physical processes and assumptions used, in order to provide an overall picture of primary photon and secondary particle transport through the crystal. Chapter IV will elaborate on methods of simulating each of these physical processes and source conditions. Two events are of primary interest in calculating the photofractions; the number (I) of incident source gammas that interact at least once in the crystal, and the number (A) of incident source gammas that deposit their entire energy in the crystal. By definition, the photofraction is given as the ratio of these two numbers, (A/I). Thus, the principal objective of the Monte Carlo calculation is to determine the number of incident source gammas that interact, and how many of these are totally absorbed. Figure III-1 gives the general flow diagram for the computer program. Termination of a given problem occurs when a pre-specified number of primary interactions (I) occur in the crystal. Starting with a monoenergetic source of specified geometry (e.g., isotropic point, collimated beam, etc.), a single gamma photon is chosen. If the photon interacts in the crystal, all the subsequent events involving the photon and its secondary particles constitute a history. The gamma is defined by seven variables: three rectangular coordinates (xyz),

-70START READ DATA REQUIRED NUMBER OF HISTORIES FOLLOWED YES, SELECT GAMMA SOURCE PARAMETERS GAMMA INCIDENT NO UPON CRYSTAL YES: YYES l COMPUTE TOTAL CROSS SECTION AND PATH LENGTH I NO E=I+i! COMPUTE DIRECTION AND COMPUTE PHOTOELECTRON 1ESC YES It PT CTION APDOSECNALONF FDARY GAMMATT ON E 1YH FOEYES i DETERMINE TYPE OF INTERACTION | -------- -.-V —STOPPED _ IN CRYSTAL NOIN CRYSTAL E SYES YES CTON STOPPEDGENERATE BREMSTRAHLUNG GENERATE BREMSSTRAHLUNG CODRYSTAL PHOTONS ALONG ELECTRON PATH PHOTONS ALONG ELECTION AIT YES o t ELECTRON PATH.NO N SECOND ELECTRON STOPPED IN CRYSTAL 5 ANY BREMSSTRAHLUNG - ESCAPE CRYSTAL YES YES GAlMMA~ |~GENERATE BREMSSTRAHLUNG F LLOW SECOND FOL SECONDARY GAMMA PHOTONS ALONG ELECTRON DATA ANNIHILATION TOTAL ENERGY (WITHIN.01 MeV) ENERGY <.01 MeV GAMMA N SORBED IN CRYSTAL (L)I~~~~~~~~~~~~~~~~ IY..lES ELECTRON S TOPPEGENEATEBANY BREMSSTRAHLUNG FOLLOW III-1. ESCAPE CRYSTAL SECONDARY PHOTON CR YES A-^ f qmnP PRINT YCALCULATE PHOTOFRACTION = A/I i oi~r K —RESULTS AND STANDARD DEVIATION Figure III-l. Flow Diagram for Monte Carlo Calculation of Eotofractions.

-71three direction cosines (uv,w), and energy (E). Rectangular coordinates are used, even though the medium geometry is cylindrical, since the direction cosines remain constant under linear displacements. Thus one has: X=X + uP Y y + v/ 2 =I + Wp (3,1). (=K,~;')) / C)S oS Vi COS- f l - Y a, e CBO The unprimed coordinates are initial values and the primed coordinates are final values after the gamma travels in a straight line radial distance p, in the direction u, v, w. Assuming that a source gamma is chosen by suitable sampling techniques to lie along a path which intersects the crystal, the distance traveled in the crystal will depend on the total macroscopic cross section,u. Calculation of this distance introduces

-72the first application of a fundamental principle of the Monte Carlo method, that of sampling from a properly normalized distribution function. 1. Cumulative Probability Technique To sample from a given distribution, the transformation must be obtained between that distribution, and the uniform distribution of the random numbers. One such transformation is obtained by the defined connection between a probability distribution function (P.D.F.), f(x) and its cumulative distribution function (C.D.F.)) F(X). The P.D.F. is defined as a function that yields the probability that the random variable will assume any particular value (within a small increment) over its range of definition. The C.D.F. is the probability that the random variable takes on values less than or equal to some specified value, Then, by definition: fr(X^ ) d| (8ClX/(3,2) -00 Consider now the P.D.F. and C.D.F. for the uniformly distributed random numbers over the range 0 - 1. Let g(r) = P.D.F., such that g(r)dr is the probability of obtaining a random variable in dr about r, Since the P.D.F. is a uniform (i.e., constant) distribution function, normalization requires that g(r) = 1. Thus we have: C.D.F. =G r) )dr r (3.3) o -. r i -/

-75G(r) 1.0/ /'~ -- - -71 /,0 C -- I I / I I I/. Consequently the C.D.F. for the random numbers is just the value of the random number. From Equation (3.2) it is seen that F(x) is a monotonically increasing, single-valued function of x for any f(x) > 0 and by definition f(x) is always positive. For a given value of the random number r, we wish to make the connection between Equations (3.2) and (3.3), such that we obtain a value of the variable x sampled from the distribution f(x). This connection is given by: r = F(x) = (x') dx' (34) It may be seen that Equation (3.4) determines x uniquely as a function of r, in such a way that since r is uniformily distributed on the interval 0 < r < 1, x falls with the frequency f(x) in the interval dx. It has been arbitrarily assumed here that the lower limit of the interval of definition of f(x) is x = 0. Therefore the so-called cumulative probability technique of Monte Carlo involves solving Equation (3.4) for x in terms of a random number r. Two objections to use of this method are immediately apparent.

-74First, no analytical evaluation of the integral may be possible, and second, even if the integral can be solved, the resulting expression may be so complicated that repetitive calculation with it may require too much computer time to be economical. For either case, other Monte Carlo techniques are available (e.go, rejection techniques). 2. Rejection Techniques To avoid the problems presented by use of Equation (3.4), the rejection technique is commonly used. Given the normalized probability distribution function f (x) where 0 < x < a, and that the maximum of f(x) is L, in the range definition of x what is required is a choice of the variable x which lies within the area denoted by fa f(x')dx'. The procedure is to choose two random numbers rl and r2. o Set xl = a.r1 and test if r2L < f(xl). If this inequality is satisfied, accept xl as a randomly obtained value for x from the function f(x). If r2L > f(xl), then reject rl, r2 and obtain two new random numbers, continuing the processes until the inequality r2L < f(rla) is satisfied. The probability of accepting xl is f(X)/L i eries fat the f (n which verifies that the frequency distribution from which the random variates are obtained is in fact the function f(x). Rejection of sets of random numbers causes an inefficiency in the method, and for a normalized f(x), the acceptance rate is 1/aL. The acceptance rate may be

-75considered to be ratio of the area of interest to the total rectangular area being sampled given by f (x) raL n r, Most frequently the cumulative probability method and rejection techniques are combined (e.g., sampling from the Klein-Nishina formula used for Compton scattering in these calculations, see Appendix C). The advantage of combining these techniques is that if a simple approximation, g(x) can be found to a complicated function, f(x), the cumulative probability method can be applied to g(x). Then g(x) and f(x) can be related by use of the rejection technique. For a properly chosen g(x), the acceptance rate in the rejection technique will be high and an efficient sampling procedure will be available for a given f(x). Reference 24 gives an extensive compilation of such techniques for many distribution functions encountered in physical and mathematical problems.

-76To select a random variable x from a given f(x) for 0 < x < a, one first obtains a value for x from an approximating g(x) by the cumulative probability technique. Thus one obtains x1 = G-l(rl), where rl is a random number, r,= Gc(,) = ^(x,)dx' and x is easier to obtain from g(x) than solving Equation (3.4), with f(x). The relationship between g(x) and f(x) is obtained from the rejection technique. Defining h(x) = f(x)/g(x), and L = maximum of h(x) in the range 0 < x < a, one obtains a second random number r2. Rejection of the set rl, r2 occurs until the inequality r2L < f(xl)/g(xl) is satisfied. When this is satisfied, x1 is accepted as the randomly chosen variable from the distribution f(x). The process is depicted below. o a o x, a ha(x L h(x,) /.O\ 0 ____ c -x, —-I —— x

-77The probability of accepting xl from the rejection technique is tr (X=.5X/L Dir h(X1), L and the probability of obtaining x1 is g(xl), from the definition of a probability distribution function. Thus the combined probability of choosing and accepting xl is the product of the independent probabilities (X.i) (X, = - L L which again verifies that one is sampling from the correct distribution function, f(x) B. Path Length Sampling Returning to the problem of determining the distance traveled by a gamma before any interaction occurs, we have the P.D.F., f(p) given by: f(p)dp = e-[Pe dp = probability that a photon travels a distance p without interaction and then has an interaction in the interval between p and p + dp 3 p = total cross section. Applying Equation (3.4) we obtain: p = _!un r (3.5) r =random number

-78Von Neumann(25) has given another method for obtaining the path length, p, without evaluating the logarithm of a random number. The incentive for using the von Neumann technique is to reduce the computer time, but a comparison between his method and Equation (3.5) with the Michigan computer give a negligible difference in computational time. Equation (3.5) was used throughout these calculations. C. Gamma Interactions Having determined the path length of the source gamma in the medium, a check is made to determine if the gamma interacts within the crystal boundaries, by using Equations (3.1) and the crystal geometry. If the interaction occurs outside the crystal, then the gamma is counted in the appropriate termination category as having passed through the crystal with no interaction. Another source gamma is selected and the process is repeated. When a gamma has an interaction within the crystal, the event is recorded, and the elementary probabilities for the possible gamma ray interactions dictate which interaction occurred. The possible interactions considered are photoelectric absorption, Compton scattering, and pair production. The following macroscopic cross sections, [cm] are defined: ~(E) = Compton scattering K(E) = photoelectric absorption T(E) = pair production 4(E) = 2 + K + T = total.

-79Then Z/1, K/[, T/I at a given energy are the probabilities per interaction of Compton scattering, photoelectric absorption, and pair production, respectively. The Monte Carlo procedure for deciding which interaction occured is shown by the following flow diagram, where r is another random number.:,... ^. L - Z+K -w pair producobtain r r-y r > y e [i ye s 1- yes tion event no no Compton Photoelectric event event 1. Photoelectric Absorption Depending on which primary gamma interaction occurs, as determined by the technique shown in the sketch above, a different set of calculations must be made. If a photoelectric absorption takes place, the following sequence is followed. The photon is assumed to transfer all of its energy to a single target electron. The electron originates at the point of interaction and is ejected in a specific (energy dependent)

-80polar angle with respect to the incident photon direction. The azimuthal angle of ejection is uniformly distributed, and a transformation is made to refer the electron direction back to the coordinate system fixed in the crystal. Calculations are then made to determine if the electron escapes the crystal. If this occurs, the history is terminated, If not, then data for bremsstrahlung generation are sampled. These bremsstrahlung photons are followed through the crystal, undergoing all the same processes that primary photons do, except that electron and subsequent bremsstrahlung generation is considered negligible. If the bremsstrahlung photons are all absorbed, the total source energy has been absorbed, a record is made in the appropriate category, and another source gamma is chosen. 2, Compton Scattering If a Compton scattering occurs, both the scattered gamma and free electron are followed. The polar angle for the scattered gamma ray is obtained by Monte Carlo sampling from the Klein-Nishina formula,(31) by the method given in Appendix C. The energy for the emergent gamma and electron, as well as the electron polar angle, are then calculated from the relationships given in Chapter IV. The scattered gamma and electron are assumed to be uniformly distributed in azimuthal direction. First the electron and subsequent bremsstrahlung are followed in an identical manner as described above. Then if the electron and bremsstrahlung are totally absorbed, the Compton scattered gamma is followed through the crystal in the same manner as for primary source gammas, with electrons being produced at every interaction. At each interaction in a

-81given history, a running summation is kept of the energy deposited in the crystal. Termination as a total absorption occurs when the difference between the source energy and the energy deposited is < 10 Kev. 3. Pair Production If a pair production event occurs, the two pair electrons are assumed to share equally the incident gamma kinetic energy excess over the rest mass of the two electrons. The electrons are ejected in an (energy dependent) forward direction, with a uniform azimuthal distribution given to one electron, the other one correlated in azimuth at 180~. The electrons and bremsstrahlung as followed, and if absorption of these secondaries occurs, one of the electrons is chosen at random to be the positron. The assumption is made that the positron comes to rest before annihilation, and this defines the position for emission of the two -. 51 MeV annihilation gammas. These gammas are assumed to be emitted isotropically in opposite directions, and they are followed through the crystal in the usual fashion, with electrons and subsequent bremsstrahlung being considered. D. Secondary Gammas and Electrons All of these gamma interactions give rise to secondary gammas and particles, i.e., scattered gammas and free electrons (positrons and negatrons). These gammas may interact further until they are absorbed, or escape from the crystal. In addition, the electrons produce positron annihilation gammas, bremsstrahlung gammas, and further electrons. Thus a sequence of gammas and charged particles is built up throughout the

-82crystal, which must be terminated at some point for any practical calculation. For usual applications of scintillation crystals, the sequence beyond the first few generations can be safely neglected with insignificant error. The termination of this sequence is discussed below. Gammas and particles from a given history are followed until either escape with more than 10 Kev energy, or the total source energy is absorbed. Whenever the former occurs, the history is immediately terminated since it is assumed that loss of 10 Kev energy removes the "count" from the total absorption peak. A count corresponds to the summation of the energy deposited in the individual events as discussed in Chapter I. The sequence of events from a primary source gamma is considered to be as follows: (a) Scattered gammas due to successive Compton events (b) Free negatrons from photoelectric and Compton events (c) Pair electrons (positrons and negatrons) from pair production events. Further interactions of the above give: (a) Annihilation gammas from positron annihilations (b) Bremsstrahlung photons from energetic electrons. It is assumed that electrons are slowed down by excitation and ionization of the target atoms, as well as by radiative losses (bremsstrahlung), but that the subsequent de-excitation photons and free electrons generated are totally absorbed in the crystal. Also it is assumed that the bremsstrahlung photons can undergo any of the three the primary gamma interactions, but no electrons are generated. The justification for these assumptions is that for the high atomic number materials used for gamma scintillation

-83crystals, absorption of these low energy gammas and electrons occurs with a very high probability. Thus the sequence is terminated at this point. E. Typical Histories The tables given below summarize the number of primary events found to occur in a typical history of gammas from an isotropic source. Reference 32 gives similar results obtained from Monte Carlo calculations with a narrow collimated beam source normally incident on a solid crystal. Reference 32 ignored all electrons, bremsstrahlung, and annihilation gammas, including only Compton scattered gammas. Calculations were limited to a maximum source energy of 1.5 MeV. All of the data given below were obtained from the Monte Carlo program for a 2 in. dia. x 2 in. high cylindrical solid NaI crystal. The calculations were made for 1000 primary interactions at each source energy and neglects energy loss by escaping electrons and bremsstrahlung. Table III-1 compares the observed fraction of primary interactions with the elementary probabilities per interaction given by the cross sections. Agreement is seen to be excellent, considering the relatively few histories considered. Table III-2 indicates that for gammas which are absorbed, most absorptions occur after zero, one, or two scatterings; while from Table III-3, one sees that most escaping particles suffer only one scattering prior to escape. In general, the results tabulated here indicate that as a gamma suffers more scattering the probability of absorption increases. Physically this corresponds to reducing the source gamma energy by successive scatters, until the gamma energy is in the range where photoelectric absorption predominates. Table III-4 gives a

-84sequence of typical events when pair production occurs in the crystal. It should be noted that relatively few pair production events occur for a gamma that suffers an initial Compton scattering. Also, the number of times both annihilation gammas are absorbed is significantly less than the number of times one annihilation gamma is absorbed, TABLE III-1 FRACTION OF PRIMARY INTERACTIONS Energy Photo- K Pair T (MeV) electric. Compton P Production P.51.182.174.818.826 0 0 2.0.021.025.938.915.041.059 3.0.016.017.800.808.184.174 4.0.020.013.691.703.289.285 5.0 0.009.614.620.386.380 TABLE III-2 FRACTION OF THE 1000 INTERACTING GAMMAS THAT HAVE EXACTLY X SCATTERINGS PRIOR TO ABSORPTION Energy (MeV) X=0 X = 1 X=2 X=3 X = 4 X= X =6 X = 7.51.182.212.105.033.011.004 0 0 2.0.021.076.087.033.015.003.002 0 3.0.016.044.063.028.002 0 0 0 4.0.020.041.041.033.009.002 0 0 5.0 0.032.032.013.007.002 0.001

-85TABLE III-3 FRACTION OF THE 1000 INTERACTING GAMMAS THAT HAVE EXACTLY X SCATTERINGS PRIOR TO ESCAPE Energy (MeV) X = 1 X = 2 X = 3 X = 4 X = 5.51.367.073.010.003 0 2.0.567.121.027.007 0 3.0.525.106.023.003.001 4.0.471.072.012.004.001 5.0.418.084.009.oo6 0 TABLE III-4 TYPICAL PAIR PRODUCTION EVENTS (For 1000 Primary Interactions) No. Pair Events No. Times X Annihilation Energy Total No. After Initial Gammas Totally Absorbed (MeV) Pair Events Compton X = 2 X = 1 X = 0 2.0 67 6 6 26 35 3.0 183 4 8 62 113 5.0 401 12 20 139 242

CHAPTER IV DETAILS OF THE MONTE CARLO CALCULATIONS FOR PHOTOFRACTIONS The Monte Carlo programs can be divided into sections consisting of: (a) Source selection routine (b) Particle interaction (c) Particle escape routine (d) Termination criteria and categories. A. Source Geometries Considered Two general types of monoenergetic gamma ray sources are considered. These are isotropic sources and monodirectional sources, the latter being collimated so that all rays are parallel to the crystal axis of symmetry. The isotropic sources are typical of small volume laboratory sources and are restricted to point sources (on and off the crystal axis), circular disk sources centered on the crystal axis, or right circular cylindrical sources located along the crystal axis. Self-absorption and scattering have been considered for the cylindrical volume source, and will be discussed in Section H of this chapter. The monodirectional source is the limiting case of a source at large distances from the detector, collimated to yield a parallel beam. Three degrees of collimation are considered: a narrow beam collimated to lie along the crystal axis, a beam collimated to any specified diameter, and a broad beam which provides full illumination of the crystal face. The -86

-87collimated beam sources are assumed to be normally incident upon the crystal face or well bottom. All of these seven sources have been considered for the solid cylindrical scintillation crystal, and for the well crystal except for the last two collimated sources. 1. Isotropic Source Routines The objective of any source routine is to obtain the direction of a source gamma and its coordinates at the point of intersection with the crystal surface. This point is determined by a line having the direction cosines uo g Vo, Wo from the source to the crystal, An isotropic source is defined as one in which the probability, P(s) of emission into any solid angle is a constant for all solid angles. Thus p(_l)ad -K = K d-a K= constant Expressing the solid angle in terms of an azimuthal angle cp and a polar angle. with respect to any one of the three rectangular coordinate axes, we have p(l) d- P( ) t) dcA d(t K dc( dc where [i - cos~, is the direction cosine relative to the arbitrary axis,

-88The probability is constant in azimuth about the arbitrarily chosen coordinate axis, and integrating over p gives p(ao) dn = 2T K dt Thus the probability of emission is a constant over all values of the direction cosine,. Since the reference axis was arbitrarily chosen, this argument can be repeated for the other two rectangular coordinate axes. One concludes that isotropic emission can be described by obtaining the three direction cosines uo, vo, wo, for a ray as shown below, from a constant distribution function such that all values from -1 to +1 have an equal probability of occurrence, Y to = COS o Vo = coS0 Wo = COS (4.1) These direction cosines may be obtained from a set of random numbers, uniformly distributed on the interval 0 - 1 by:,- 2r, -I w-=2 r-/ r r2 = /-ranm nu) (4.2) rid r2 = random numbers

-89The last equation is a consequence of the definition of the direction cosines; with the + or - being chosen at random. a. Solid Crystal - Point Source For the solid crystal consider a point source located off the crystal axis as indicated below. The origin, 0, of the coordinate system is fixed in the crystal face, in contrast to that defined in Chapter II, and the initial coordinates for this point source are: X = P Yo =0 = -b 0 ISOTROPIC SOURCE (xo,yozo) ~-~~~ —-- ~R ---- ~ ~p_. ~ /_ a /SELECTED A.- RA -b ZY | i SOLID CRYSTAL

-90Rather than sample over all directions, as indicated by Equations (4.2), it is more efficient to restrict the directions of emission to randomly chosen vectors, uniformly distributed within the cone defined by: w' by -b (4.3) b + (R+P)= and then to reject any rays which do not intersect the crystal face. Obviously as the distance p increases, the efficiency of this method will decrease since the fraction of the base of the cone which is covered by the crystal face decreases. In addition, it may appear that calculation of the square root may require more computer time than other methods, for example, sampling over the entire positive z direction and accepting only rays that intersect the crystal. However, for smaller distances p, the increase in efficiency due to the geometrical effect has been calculated to more than offset the time saved by use of more simple, but less efficient methods. In particular when p = 0 (point source on axis), one has the case of greatest interest, and the conditions of Equation (4.3) require that every gamma strike the crystal face. Applying the cumulative probability technique of Equation (3.4) one gets: wJ /- br + b _ Vb+(R+P) J - +P and to obtain -1 < (u0, vO) < 1 one uses:

-91u,=2 rf -I Vo = - -(o +w.) where rl, r2 are random numbers and the + or - is chosen at random. The coordinates in the plane of the crystal surface are: b x,=p + -4o u,,= 0 and for intersection with the crystal, the above process is repeated until: xyt+y <. b. Well Crystal - Point Source The procedure for the well crystal is somewhat more complicated, since gammas from an isotropic source can strike either the sides or the bottom of the well. Isotropic sources are limited to those contained within the well as shown below. (Again note that the origin of the coordinate system is different from that considered in Chapter II). The direction cosine wo (w = cos 7) is obtained from a uniform distribution 0 (-1 < wo < 1) by: Wo =2 r-I r = random number

"X 4~~~ Qf ~ ~~~~~ / cr 4^ —----- ^ -----— w C~~~~~j~

-95Next the sine and cosine of a uniformly distributed azimuthal angle 0 < cp < 2jr are obtained by the following technique first given by (25) von Neumann. (This procedure eliminates time-consuming series evaluation of the functions sincp, cosp, where cp = 2itr). Obtin x =2r; -I CoS~ x — rl,rA ^y:z yes St Y2XS no The other two direction cosines, u., vo, defined in the sketch above Equations(4.l) are obtained from: Uln.0 cs(/_W1Y 2Vz Then letting 4 = HW b Co= S -V^ C(Hv-bb and w, = -b = cos u. Vs2b^

-94where s = -p c os$ +0 t.- p sin the tests are made / -'/o -,,L (4.4) and vWL > Wo > - u (4.5) When (4.4) is satisfied the source gamma, labeled ray 1 in the sketch below, strikes the well bottom. The coordinates at the point of intersection with the bottom of the well are: x, -p+s' cosq =/p+(H,-b)(/ -vW ) cost =p + (Hw-b) -O Jw Wo %I,-!~ y,,4 = =,-

-95strikes the side of the well. SOURCE b Y ^ X Q \H So Z' Zs =Z y' strikes the side of the well.

-96tV / ^^'*~P11 SOURCE Z( I ~L XIP ~XI VyI- Ir X a X / yl y"

-97The coordinates at the point of intersection with the side of the well are: x, =p + s cos, = S sin0 Z, = b+ W s (/- Wo2) If neither (4.4) nor (4.5) are satisfied, then the gamma escapes through the top of the well without hitting the crystal. For this case, the entire process is repeated. For most practical well crystal geometries the efficiency for selection of a source gamma which strikes the crystal is very high, especially if the source is located deep within the well. c. Extension to other Isotropic Sources The above methods are readily extended to cover the other isotropic sources. For on-axis points, p = 0 and the same procedures are used. For a disk source of radius RD the radial distance is sampled using the cumulative probability technique: p Re- - r = random number This is extended to a cylindrical volume source of height b2 - b1 by selecting b = r(b2 - bl) + b1, r = random number.

-98KII Rf IbI _____ ___Volume |, Sources Solid Crystal Well Crystal 2. Collimated Beam Sources The monodirectional sources have by definition, \o = / ldO = o = O For the narrow beam collimated along the crystal axis = Y1 = 0 Z1 = 0, solid crystal Zl = Hw, well crystal For a beam collimated to any diameter, Ds (solid crystal only) xi= Ds -F- r = random number = - t =0 and in particular for a broad beam, Ds = crystal diameter.

-99B. Gamma Interactions Having obtained the coordinates at the crystal surface xl, Yl, Z1 > given above for the appropriate source, the gamma travels into the crystal along a path in the direction (u0, Vo, wO). As discussed in the preceeding chapter, the radial distance traveled to the first interaction is obtained by sampling for the mean free path from the total cross section. The NaI cross section data of Grodstein and McGinnies(20)21) for Compton and photoelectric events, as well as the total cross section, are stored in the computer as an integral part of these programs, with interpolation for energy being performed when required. Table II-11 gives the energy increments used. As previously discussed, the type of interaction is determined by the relative probability for its occurrence. It is at this point that the various possible physical processes are simulated, based on assumptions made about the character of these processes. 1. Interaction Processes Considered For gamma rays the only important interaction processes in the crystal, for usual applications of scintillation crystals, are photoelectric, Compton scattering, and pair production. Rayleigh (coherent) scattering has been ignored, as discussed in Chapter II. The fluorescent radiation and Auger electrons that are emitted in filling the vacancy in the inner shell following a photoelectric absorption are neglected since the energy released (a 28 Kev for I) is so low. The photons will be absorbed essentially at the point of the photoelectric event for all crystals larger than % 1 cm., since the absorption cross section in NaI for these

-100low energy photons is so high, ~ 32 cm-1. Nelms(44) indicates that electrons of this energy have a range < 10-3 cm. 2. Secondary Gamma Treatment Gammas interacting in Compton events are assumed to scatter according to the differential Klein-Nishina formula for free electrons.(3 Experimental confirmation of this formula has been reported by References 34 and 35. Since the Compton process occurs with the outer electron shell, the binding energy of these electrons (of the order of a few ev.) is negilgible compared to the incident photon energy (of the order of 1.0 MeV.). Nelms(35) shows that the corrections required to this formula due to the electron binding energy are negligibly small. Polarization effects in Compton scattering were also ignored. Calculations made by Spencer and Wolff(36) indicate that when polarization effects are considered, the photon flux is only slightly increased (a 1%) over calculations made when polarization is neglected. Furthermore, these authors have shown the increase in photon flux occurs only for deep penetrations (i.e., many scatters), and photon energies in the range 0.1 -.2 MeV. Since typical histories for the present calculations, as given in the previous chapter, indicate that only 2 - 3 scatterings usually occur, and total penetration is only a few centimeters (typical crystal dimensions), polarization of Compton scattered gammas is negligible here. As a photon is degraded in energy by successive Compton events, a cut-off energy has been chosen, below which the photon is assumed to be completely absorbed. This is required for proper termination of some

-101histories in the computer program and is chosen to be.01 MeVo Since the absorption cross section is so large (- 564 cm.o in NaI), the probability for escape for a photon of energy < o01 MeV is negligible. In addition to Compton scattered gammas, the position created in the pair production events gives rise to secondary photons upon annihilation. These two annihilation photons each have a kinetic energy equal to the rest mass of an electron and are emitted from the location where the positron is essentially stopped. Heitler(37) has shown that for positron energies below 10 MeV, over 90% of the positrons will be reduced to essentially zero velocity before annihilationo The annihilation photons are assumed to be emitted isotropically, since there is no preferred orientation( ) To conserve momentum the two photons are emitted in opposite directionso 3, Photoelectric Events If a photoelectric event occurs, the gamma ray is totally absorbed and an electron is emitted with kinetic energy equal to the kinetic energy of the photono This neglects the electron binding energy which is about 30 Kev for K or L shell electrons in iodine, as being negligible in comparison to the incident photon energyo The polar angle of photoelectron emission, relativee to the incident photon direction, is assumed to be the average emission angle for the given electron energyo This average polar angle has been given by Davisson and Evans(39) as derived from the relativistic equation of Sauter(40)Use of this data takes into account the

-102variation of emission angle with energy. For low energies, emission tends to be at right angles to the direction of incidence while the angular distribution shifts to the forward direction at higher energies. The photoelectrons are emitted in a uniform azimuthal distribution. These polar and azimuthal angles are measured relative to the directed line of the photon incidence and a transformation must be performed to relate these angles back to the coordinate system fixed in the crystal. The usual formulas for performing this transformation(23 32)38) involve (41) computation with rather involved expressions. However Kleinecke(4 gives a highly efficient technique for performing the transformation. His method has been used throughout these calculations when it is desired to select a set of new direction cosines for a given set of original direction cosines and polar angle, knowing that the azimuthal angle is uniformly distributed. Kleinecke's method is given in Appendix F. Detailed treatment of the electron and bremsstrahlung will be given subsequently, in Sections C and D. 4. Compton Events If a Compton event occurs, the differential Klein-Nishina formula is sampled by the selection technique given by Kahn(24) Appendix C gives details of the sampling procedure used. This method selects the cosine of the polar scattering 0 and then the energy of the scattered gamma is obtained from the Compton relationship: EC pE' - _E o _ / +EO (/-cose)

-103where E' = scattered gamma energy, MeV E = incident gamma energy, MeV moc2 = electron rest mass energy, MeV The new direction cosines for the scattered gamma are obtained by the Kleinecke )technique, using cos 0 and a uniform azimuthal distribution. Prior to following the scattered gamma ray through the crystal, calculations are performed to determine if the scattered electron, with energy Ee and polar angle @' given by Ee = Eo-E' ton J~=El sine E0 -E'cose is stopped in the crystal. Also prior to following the scattered gamma, it is determined that any bremsstrahlung photons generated by this electron have all been absorbed in the crystal. As previously discussed, escape of any of these electrons or photons terminates any further consideration of that history. Assuming that the electron and generated bremsstrahlung are absorbed, the scattered gamma is followed through the crystal in an identical procedure as that used for source particles. (See flow diagram, Figure III-1, of previous chapter.) 5. Pair Production Events If a pair production event occurs, the incident gamma energy, less twice the electron rest mass energy, is assumed to be shared equally between the negatron and positron. This neglects the slight asymmetry in

-o04energy distribution between the position and negation. The mean polar angle for the emitted pair electrons is given by Bethe and Ashkin(42) as e Ee(4.6) where mOc = electron rest mass energy =.510976 MeV and Ee = electron kinetic energy in MeV The forward peaking predicted by Equation (4.6) for increasing electron energy, has been experimentally confirmed.(43) The azimuthal angles for the negatron-positron pair are uniformly distributed, 180~ apart. No differentiation is made between positrons and negatrons when electron ranges and bremsstrahlung generation are considered (see Sections C and D, below). If both electrons are absorbed within the crystal, as well as any generated bremsstrahlung, then the two annihilation photons are followed. Since the pair electrons are treated in an identical manner, one is chosen at random to represent the positron for purposes of generating annihilation radiation. The two annihilation gammas originate at the point of stopping the positron, which was calculated to determine if the positron remained within the crystal, and are emitted in an isotropic angular distribution correlated at 180~, with an energy = 0.510976 MeV. One annihilation gamma is followed through the crystal and if absorbed the second gamma is followed, All Compton scattered and annihilation gammas are considered by the same computational procedure which includes generation of electrons and bremsstrahlung.

-105C. Approximation of Electron Transport All of the three gamma interaction processes give rise to electrons. The pair production events produce negatrons and positrons. However, many more negatrons are produced in the crystal by successive (44) (45) Compton and photoelectric events, and since Nelms(44) and Seliger5) have indicated only a slightly greater range for positrons over negatrons, the assumption was made that both electrons would be treated as negatrons, as far as range calculations are concerned. Slowing down and eventual absorption of electrons involves many interactions. Treatment of these electrons by the same techniques as used for gamma rays, which undergo a relatively small number of large energy-loss interactions, is not feasible for individual electron tracks. For the calculation of photofractions such detailed treatment of the electrons is not necessary because the important determination is whether the electron is stopped in, or escapes from, the crystal. This can be readily calculated by the following approximate method. Nelms(44) gives calculated data for NaI for the energy loss per unit path length (stopping power, -dE/dx), and the integral: E d E R, (E) = id, (4d 7) (4.7) which represents an average of the actual path lengths the electron follows from its orgin to the end of its track. Nelms' data goes to E = 1.2 MeV and Zerby(38) gives values for R (E) for 2 < E < 10 MeV.

-106The maximum radial penetration of an electron determines whether it passes through a given thickness of material and this distance is approximately related to the Nelms and Zerby data by the intercept of the electron transmission curve. The transmission probability for electrons in sodium iodide is given in Figure IV-1. The data given in this figure has been calculated with the Monte Carlo program of K. Wainio(46) for an isotropic, monoenergetic electron source within sodium iodide. An isotropic source has been used to approximate the nearly random directions of electron incidence on the crystal boundary, after an electron has suffered many deflections. Also, by using an internal source within the medium, the effect of back scattered electrons is included in the transmission data of Figure IV-1. Conventional transmission curves, such as those of Seliger5) and Agu,(4) are obtained with a source of normally incident electrons external to the medium and do not include back scatter effects. In Figure IV-1 the percent transmission is plotted versus the normalized range (p/Ro), which is defined as the ratio of the mean penetration distance to the total path length traveled by the electron. This ratio is a measure of the range straggling that occurs. At p/Ro = 0.7, the electron transmission is less than 3%. Plotting the data from the computer program of Wainio(46) in this manner the transmission curve has been found to be nearly independent of energy for electron energies of interest. Other workers(45'47) have also reported that electron transmission data is nearly independent of energy, for low energy electrons, when plotted against the normalized range,

-1071.0 0.8I SOTROPIC ELECTRON DISTRIBUTION - 0.6 0 a. 0.4 z 0\ CO) Z 0.2 0 K 0 0.2 0.4 0.6 0.8 1.0 NORMALIZED RANGE, P/Ro Figure IV-1. Electron Transmission in NaI.(46)

-108Empirical equations have been fitted to the data of Nelms(44) (38) and Zerby as follows: RO(E)=.301 E cm., E.2 Mev. a.309 =.200 E cm.,.2' E 1.2 le v. =2/154 cmNL E >1.2 Mev. Also the transmission data of Figure IV-l are fitted by: P(e/) - -2.o05 - +.893 -P.277 EO R = 72 (- +.293 -.389 - - >.277 Let p equal the distance along the directed line from the point of the free electron emission to the nearest crystal surface and consider the following cases: (1) ~.... 7' R,(E) (2) M: <.7 Ro(E) For Case (1), the transmission probability from Figure IV-1 is essentially zero and the electron is assumed to be absorbed. The assumption is also made that the energy emitted by electron-induced excitation and ionization is absorbed in the crystal. For Case (2) the probability for electron escape is given by the function PE(p/Ro) which has the form

-109of the transmission curve, where 0 < PE < 1 Sampling from the transmission curve can be performed by selecting a random number r and saying that escape from the crystal occurs when r4 PE(L) otherwise absorption occurs. Thus one approximately takes into account the decrease in escape probability as the distance to the crystal surface increases, For those electrons which escape the crystal, a small fraction are returned to the crystal with nearly all their energy, due to small angle reflections from the canning of the crystal. Assuming the electrons are isotropically incident upon the canning, the number albedo and reflected energy spectrum data of Berger(47) have been used to determine the fraction of reflected electrons returned with approximately 98% of the incident energy, These electrons are assumed to deposit all their energy in the crystal. The number albedo data are stored in the program as a function of energy and the fraction of the returning spectrum that lies within about 98% of the incident energy has been determined to be approximately 0,08 by graphical integration of Berger's data for.51 MeV electrons isotropically incident upon aluminum. The shape of the energy spectrum for reflected electrons is expected to be nearly independent of incident electron energy, therefore this data of Berger has been used for all electron energies. The overall effect of electron reflection on the calculated photofractions is small enough so that these approximations are justified for all usual canning materials,

-110D. Bremsstrahlung Production If the electron is stopped within the crystal boundaries, then the bremsstrahlung photons generated by these electrons are considered, In calculating the bremsstrahlung, the average bremsstrahlung spectrum generated by a monoenergetic electron of known initial energy has been used. This average spectrum is obtained by integrating over the entire electron path as the electron slows down, and iseventually stopped. The average spectrum, differential in photon energy, has been calculated by Zerby and Moran4) for NaI and is given as Figure IV-2o This calculation is based on the Bethe-Heitler equation(51) for the bremsstrahlung cross section (includes screening effects of the atomic electrons), and the relativistic equation of Bethe(52) for energy loss by non-radiative collisions. No difference between the bremsstrahlung spectra generated by negatrons and positrons has been assumed, although a slight difference could be expected because of differences in energy loss by nonradiative collisions for positive and negative electrons. Zerby and Moran(49) have neglected any difference in their calculations, and Koch and Motz, in their review article,(50) point out that no data is available for positron bremsstrahlung spectra. The energy loss due to bremsstrahlung has assumed to be a small perturbation in comparison to the loss by all other mechanisms. The Bethe-Heitler equation is based on the validity of the Born approximation, Due to this approximation the Bethe-Heitler equation is expected to be less reliable for high Z materials, and at

-111403 402 -- - 5 - o='INITIAL ELECTRON ENERGY (my2) 2 -L —- k= PHOTON ENERGY (mc 2), 2 -__ \\\ — A- _- - 40 I- I y N ^__ __. C 5 EO::: == 2.0 x 2 - -0 4.56 < E0Q~ ~ Eo=6.-,,.N o 0-2 -, - W 5 I0-3 40-4 4O-a' ~ "EO-4 Slowing Down in Sodium Iodide. (Taken from Zerby and oran \\(49)) Fo-2. Be thlgSetafmnEecr Slowing Down in Soliu~ Ioclid-e. (Taken from Zerby and Moran. ())

-112lower initial or final electron energies. Koch and Motz ) make comparisons between the differential cross section calculated from the BetheHeitler equation and experimentalmeasurements in thin gold (Z = 79) targets, The greatest difference between theory and experiment exists for electron initial kinetic energies in the range 0.1 - 2.0 MeV, The difference is as much as a factor of two for kinetic energies close to the electron rest mass energy, and less than 10% in the energy region above 4 MeV. The theoretical values are consistantly lower than the experimental values and comparisons for aluminum indicate that the difference between theory and experiment is significantly reduced as Z decreases. These comparisons made by Koch and Motz indicate that in calculations where bremsstrahlung radiation represents a major portion of the energy released, a large error can result when the only available formula (Bethe-Heitler) is used. However, in the calculations being made here the bremsstrahlung contributes a second order effect. For example, for 5 MeV source energy gammas, the average energy of the free electron from the first Compton scattering is 3.1 MeV. Evans,(53) page 616 gives the following semi-empirical formula for monoenergetic electrons, the energy fraction, C, converted to bremsstrahlung in a thick target: E =.7 x 10O3 ZE E in MeV.

-135For 3.1 MeV electrons in NaI, C - 14%. As Figure IV-2 indicates, the most probable radiative energy loss occurs by emission of a large number of low energy photons, most of which will be totally absorbed in the crystal. Even though, 14% of the electron initial energy may be lost by bremsstrahlung emission, rarely will all this energy be concentrated in one or two high energy photons. Thus the error introduced by use of the Bethe-Heitler equation for NaI does not have a large effect on the overall calculation for the photofraction, for usual laboratory applications of scintillation detectors. The differential spectra given in Figure IV-2 were calculated (49) by Zerby and Moran, assuming that the initial monoenergetic electrons lose energy in a continuous manner due to Coulomb scattering, and ionization and excitation of the target atoms. This continuous slowing down model is consistant with the previously discussed method of approximating electron transport in the crystal. Using the Bethe-Heitler equation(51) Zerby and Moran(49) treated the bremsstrahlung events as a perturbation on the continuous energy loss model. Figure IV-2 gives their results for NaI, plotted as the average number of photons per unit photon energy versus normalized photon energy. Zerby(38) gives the total number of photons released, on the average, by integration of the differential 2 spectra from a minimum photon energy of.04 moc up to the maximum photon energy corresponding to the electron initial energy. The lower limit of integration is required because the spectrum has a singularity at zero photon energy. The assumption made here that all photons of energy less than,04 moc2 will be totally absorbed is justified because

114-the photoelectric cro the photoelectric cross section is greater than 80 cm,. for these low energy photons. The results of Zerby are given in Table IV-i1 below: TABLE IV-1 INTEGRATION OF THE DIFFERENTIAL BREMSSTRAHLUNG SPECTRA FOR NaI(3 ) Electron Initial Mean Number of Total (Rest + Photons Released, Kinetic; Energy, m(Eo) with Energy Eo moC units >.04 moc2 1.1.0009 1.5.0325 2.0.0972 6.o.8434 11.0 1.9770 16.0 3.1359 21.0 4.4144 The simulation of bremsstrahlung generation in the Monte Carlo programs is as follows. Of the electrons created in the three primary gamma interaction processes, only those stopped within the crystal are considered to generate some bremsstrahlung radiation along their path as they slow down in the crystal. (As previously discussed, if an electron escapes, the history is terminated.) For a given electron initial energy, the mean number of photons released is obtained by interpolation of the data of Table IV-,1 using a third order interpolation formula. Given this mean number, m, of photons with energies >.04 m c2 emitted over the electron path, the probability P(x) that exactly x photons will be emitted is assumed to be given by the Poisson distribution, since

-115the bremsstrahlung emission probability per electron interaction is small. p(x)= e x The probability that at most xo photons will be emitted is given by the cumulative probability, Xo P(x o) = p(x) (4.8) X=O which is a properly normalized cumulative distribution function (C.DF.). The number of emitted bremsstrahlung photons, Xo is obtained from this C.D.F. whenever the inequality p(xo )' r _ P(Xo+l) (4.9) is satisfied, where r is a random number. In these calculations, Equation (4.8) is evaluated for successive values of xO and the inequality (4.9) is tested. By reference to Table IV-1, it is apparent that for low energy electrons, the inequality (4.9) will usually be satisfied for Xo = 0 or 1. When Xo = 0, no bremsstrahlung of energy 2 >.04 moc are generated. For Xo > 1, a separate selection is made for the energy of each of the bremsstrahlung photons, by the following technique.

-1162 Making the definitions (energies in m c units): n(EoEX = mean number of photons with an energy between C and C +de, emitted by an electron of incident total (rest + kinetic) energy, E Thus, n(E50, ) versus E is the spectrum given in Figure IV-2, Table IV-1 above gives values for: mean number of photons with an ^ % 2 f EC"dC:energy >.04 moc released by m(E) n(E^6)de- an electron of total (rest +,04 2 kinetic) energy Eo; moc A normalized probability density function is obtained by: n o = E. _C E fraction of photons with an n(Eo) energy between C and +dC. Using the cumulative probability technique, Equation (3.4), one obtains a random sample of the probability density function f(Eo, C )d. No useful analytic expressions are available for f(Eo, 6 )dC, so that sampling must be performed from the function in a numerical form, Letting: jE ~ ) Jfo ) n(Eo, )dc E ~ ( i (Ea>)de E)o* (4.10) 4-4 (Eo.,~;dEE *o4

-117it is evident that: probability that an emitted photon will B( E ) =have an energy < ei e and o_ BL/. Thus Bi(Eo, Ci) is the appropriate cumulative distribution function (C.D.F.), defined as the probability that a random variable (photon energy in this case) is less than or equal to some specified value. Using the Monte Carlo technique of setting the C.D.F, equal to a random number, r, the selection of the bremsstrahlung energy is obtained from: B, (Eoo ) = r (4.11) Values of Bi versus Ci have been computed from Equation (4.10) by integration of the bremsstrahlung spectrum(54) and are stored in the computer programs in discrete steps. Equation (4.11) is used to obtain a value for Bi(Eo, Ci). A double linear interpolation scheme was devised to select the photon energy. First, an interpolation using the value of Bi(Eo, Ci) is made, and then an interpolation using the electron initial energy. The photons are uniformly distributed along the electron path in a random manner. This path is assumed to be a straight line starting from the creation of the electron (location of gamma interaction) to the greatest radial penetration. The radial penetration of some electrons goes beyond the crystal, yet sampling from the

-118transmission curve indicates that the electron actually does not escape. (See previous discussion of electron treatment.) For these electrons, the end of the path for purposes of bremsstrahlung emission, is considered to be at the nearest crystal boundary, along the direction of electron emission. For typical applications of gamma ray scintillation crystals, the electrons produced in the crystal will be of sufficiently low energy to justify the assumption that the primary mechanisms causing electron slowing down are ionization and excitation losses. This assumption has been made by Zerby and Moran(49)in the calculation of the bremsstrahlung spectrum as a perturbation on the excitation and ionization processes of electron energy loss. The electron energy loss by radiative processes is assumed small compared to other mechanisms. Under these conditions, momentum is essentially shared between the target nucleus and deflected electron, with the bremsstrahlung photons emitted uncorrelated in angular (53) distribution with respect to the scattered electron direction. Only for extreme relativistic energies is the photon emitted preferentially in the forward direction.(55) Thus, for all of these calculations, the photon from the bremsstrahlung spectrum has been assumed to emerge with an isotropic angular distribution. As discussed in the previous chapter, the bremsstrahlung photons are treated in the same manner as primary gammas, except that no electrons or further bremsstrahlung are generated. The escape of any bremsstrahlung generated by the above technique causes termination of that history, since total absorption of the source gamma energy is not possible.

-119E. Escape Routines Having considered the simulation of the interaction processes, one must next determine if a gamma or particle traveling a known distance in a given direction will escape the crystal. For the solid crystal, escape occurs whenever the coordinates at the end of the path satisfy any of the following inequalities: X+Jy R+, -0 0 or ZAH (4.12) where R = crystal radius H = crystal height. Although the same criteria applies for the well crystal, the problem is considerably more complicated. The derivation of the escape routine for the well crystal is given in Appendix D. Essentially it must be determined if the ray intersects the well, and if so, whether it reenters the crystal or escapes out the top of the well. Postponing considerations of absorption and scattering in a three-dimensional source to a later section, the well volume is assumed to be a vacuum. The effect of adding a cylindrical vacuum within the crystal is to increase gamma ray escape through the crystal exterior surfaces, as well as escape out through the top of the well. Reduction of the photofraction, in comparison with that of a solid crystal, will be the net result. It is also assumed for those gamma rays that pass through the well, attenuation in the crystal canning material is negligible. Aluminum is most frequently used as a canning material; typical thickness being about 0.8 mm. In general, gammas will not be normally incident upon the canning, but the

-120directions of incidence will be nearly isotropic. Except for those few gammas which are incident at very large angles from the canning surface normal, the path length in aluminum will not be more than - 1 mm, Attenuation through this thickness is appreciable only for that small fraction of gammas that cut through the well with energies less than about 80 Key. Since the absorption cross section in scintillation materials at this energy is so high ( 10 cm.-l in NaI), only a small fraction of the gammas near the well surface will be effected, most being absorbed before reaching the well. Thus if the gamma cuts through the well and re-enters the crystal the total distance traveled is increased by the length of path through the well. Making this increase in the path length for these gamma rays, the above escape criteria is applied. Allowance is also made for escape out the top of the well. F. Termination Categories A photon history is recorded according to the method of termination, For off-axis isotropic point sources, and disk or volume sources some of the selected source gammas may miss the crystal entirely, and these are recorded in the termination category, T. Those source gammas which are incident upon the crystal, but pass through without any interaction are recorded in the category, P, For any source gammas which interact with the crystal at least once, a count is made in I category. Of these I photons, the number that deposit the entire source energy (i.e., totally absorbed) are recorded in the category, A. The photofraction is given by A/I and the intrinsic total efficiency, IT is

-121given by I/(I + P). This later quantity has been calculated by numerical integration, as discussed in Chapter II, and thus provides a check on part of the Monte Carlo results. G. Photofraction Standard Deviation Estimates of the statistical significance of the Monte Carlo results for the photofraction may be obtained by calculation of the standard deviation. Statistical fluctuations occur in the number of source gammas totally absorbed in the crystal due to the random nature of the processes involved. For these calculations, the number of interacting source gammas, I, is a preassigned number. Of these I photons two events are of interest, either the total photon energy is absorbed, or not. The probabilities for either event are constant, and independent, thus satisfying the criteria for application of binomial statistics. It should be noted that the probability for total absorption is not always small, (the photofraction goes to 1.0 for low energies or very large crystals), so that application of the Poisson distribution is not justified, in general. One desires the variance of the ratio A/I, where I is a constant. It can be shown(56) that: V[ fA]= [A] Since in general the conditions for the binomial distribution are satisfied, the variance is given(56) by:

-122V[A] = I p - ) where p = probability of total absorption. Thus: V["^]= I]e'r Since the definition of the photofraction corresponds to that given above for p, A =p v[+-] = [ j' K ] (4.13) where p = standard deviation of the photofraction. As is generally p the case the standard deviation for an assumed constant photofraction varies as the inverse square root of the number of histories. Also one obtains the usual Poisson distribution result from: = ='^4; p <<l Equation (4.13), being the general case, has been used in the Monte Carlo programs which calculate the photofraction and its standard deviation.

-123H. Absorption and Scattering in the Source Up to this point in the present work, any interactions of the gamma rays within three-dimensional sources have been neglected. For some sources it will be desirable to consider these effects. In a review of the current literature, Reference 18 was found to give the only calculated results in which source interactions were considered. Reference 18 has given calculated results for the total efficiency of detecting, with a solid crystal detector, gammas from a homogeneous cylindrical volume source in which gamma attenuation occurs. The source was assumed to be an aqueous solution of a known energy gamma emitter. (A comparison is given in Table IV-2 between some of these results and those of the present work.) References 57 and 58 considered solution of a similar problem with an experimental approach. Unfortunately, the sources considered in References 57 and 58 were not homogeneous, and thus a comparison with the present work is not possible. TABLE IV-2 ABSOLUTE TOTAL EFFICIENCIES SOURCE SELF ABSORPTION AND SCATTERING INCLUDED 2 x 2 IN. SOLID CRYSTAL Energy Present Calc. Ref. 18 MeV CAT eAT' EAT 0.20.1490.1373.1380 1.00.0499.0453.0457

-124Since for some volume sources it may be necessary to consider attenuation effects within the source, the following analysis was made. At lower energies, where the gamma ray cross sections are the highest, attenuation in the source may cause an appreciable reduction in the photofraction and total efficiency, and thus in the absolute peak efficiency, EAP We will designate as EAP the absolute peak efficiency for an ideal source which has zero scattering and absorption cross sections, and EAP' for the same source geometry which has non-zero source cross sections, Analogous definitions are made for the photofraction, p and absolute total efficiency, CAT. Thus, by definition: AT? = fraction of source gammas that escape the source without interaction, and interact at least once in the crystal then, EAT = 7 f[/- e-XJe d d (4.14) where V = source volume, cm.3 x() extrapolated path length in crystal, cm. y(Q) = extrapolated path length in sourcerystal, cm. y(n) = extrapolated path length in source, cm. T(E) = total cross section of crystal, cm.>(E) = total cross section of source, cm. An experimental measurement of the pulse height spectrum would indicate a lower photopeak and increased Compton continuum for a lase in which gamma interactions occur in the source volume, as compared to a case

-125of identical geometry but the source being transparent to its radiation. This is because even with an initially monoenergetic source in the case of interactions in the source, the crystal has gammas of many different energies incident upon it since some variable fraction of the gamma energy can be lost in the source. Thus the experimental photofraction is reduced. Reference 59, page 87 gives a curve that shows this effect by comparing the pulse height spectra for a point source in air, and in water. This comparison approximates the situation considered here of a homogeneous source volume, with and without source absorption and scattering. However, if the calculation for the photofraction pt is made for only those gammas which escape from the source unattenuated in energy, then the desired absolute peak efficiency is given by: j / / P A, ~p (4. 15) where p! = photofraction, given by the ratio A'/I' A' = number of gammas that deposit the total source energy in the crystal I' = number of gammas that interact at least once in the crystal that could deposit the source energy in the crystal, i.e.,, those gammas that escape from the source without depositing any energy and interact with the crystal. Thus, Equation (4.15) is equivalent to writing I z' AI' IA' l -4 P A A (4.16) No T' No

-126where N = source absolute gamma intensity, This treatment of sources in which absorption and scattering occur is seen to be somewhat analogous to the treatment in Chapter I of the ideal total spectrum area AT* and the experimentally observed area AT. In both cases, use of the absolute peak efficiency eliminates consideration of the observed total spectrum, since this quantity cancels out in Equations (1.7) and (4.16). Thus since (AT' is obtained by numerical integration of Equation (4.14), the only parameter to be calculated is p'. The photofraction p' will differ from the photofraction p, which is calculated for the transparent source case. However, this difference is due only to a geometrical effect, based on the above definition of p'. If the volume source could be considered to be concentrated at a point, then this "source point" would be located nearer the crystal for the source with non-zero scattering and absorption cross sections than for the transparent source. This is because in the non-transparent source, gammas being emitted nearer the crystal will have a greater probability of escaping uncollided from the source and then striking the crystal. In Chapter V, the effect of geometry on the photofraction will be shown to be small for regular source shapes of not too large an extent. By restricting considerations to cylindrical volume sources with dimensions no greater than approximately the crystal dimensions, one can take p' = p, to a very good approximation. The above discussion is equally valid for solid and well crystals, even though the reentrant surface of the latter allows the possibility of a gamma which scatters in the crystal to reenter the source, depositing

-127some energy therein. The well crystal program was modified to study this effect, and for sources in an aqueous solution completely filling the well in less than 0.1% of the histories did such an event occur. Thus this possibility can be neglected, except for sources which are nearly "black" for their radiation. The computer program, BURP-2 evaluates Equation (4.14) for any specified source total gamma ray cross section (without coherent scattering). Details of using this program to include source scattering and attenuation are given in Appendix E. Table IV-2 above gives a comparison of the present calculations with some results of Reference 18 for a solid crystal detector and a homogeneous aqueous solution of the gamma source. The source diameter is the same as the crystal diameter, with different source heights. Agreement is seen to be excellent. Since Reference 18 provides extensive tabulations for solid crystals, no further calculations for solid crystals are given in the present work. However, no results have been previously published for well crystals, and therefore Table IV-3 is given below. This table gives calculated AT' values for the Harshaw 8F8 well crystal with various energy sources in a homogeneous aqueous solution totally filling the well. Values of CAT, in which source interactions are neglected, are also given in Table IV-3.

-128TABLE IV-3 ABSOLUTE TOTAL EFFICIENCIES SOURCE SELF ABSORPTION AND SCATTERING INCLUDED 8F8 WELL CRYSTAL, AQUEOUS SOURCE VOLUME EQUALS WELL SIZE Energy CAT' AT Energy CAT CAT MeV MeV.01.0697.8760.60.2589.2890.015.2388.8760.80.2261.2491.02.4158.8758 1.00.2049.2234.03.5908.8742 1.50.1728.1855.04.6539.8756 2.00.1567.1667.05.6774.8749 3.00.1429.1505.o6.6883.8737 4.00.1385.1444.08.7003.8705 5.00.1365.1429.10.7051.8652 6.oo.1371.1429.15.6703.8046 8.00.1404.1455,20.5698.6725 10.00.1452.1499.30.4081.4710 15.00.1581.1624.40.3284.3750 20.00.1689.1731.50.2849.3207 30.00.1864.1907

CHAPTER V RESULTS FROM THE MONTE CARLO CALCULATIONS A. Comparison of Experimental and Calculated Results Both calculational and experimental approaches have been previously used to obtain photofractions for scintillation crystals. Nearly all published results are for solid crystals, with only a limited amount of data reported for well crystals. Early calculations emphasized determination of detector response spectra, and accurate photofractions were not reported. Typical of these were the work of Campbell and Boyle, (6) (61) d (62 Foote and Koch, and Maeder, Muller, and Wintersteiger.() Experimental determination of photofractions for solid crystals have been given in References 63- 71. Calculations, using some type of Monte Carlo techniques with modern digital computers, have been reported in References 8, 9, 11, 32, 38, and 72 - 76. The recent papers of Miller and Snow,(9) and Zerby and Moran(38) are most noteworthy because their photofraction calculations for solid crystals cover a wide range of gamma energies. Trombka(77) has given some calculated values for photofractions (calculated by Miller and Snow) and absolute total efficiencies for a 2 in. spherical crystal. A limited amount of experimental data for well crystals have been reported in References 59 and 78 82. The only reported data for well crystals photofractions obtained by Monte Carlo calculations are (74) given by Wachter, but the geometry was restricted to spherical NaI crystals with empirical extrapolations used to approximately extend the data to cylindrical crystals. -129

-1301l Solid Crystals Results for solid crystals have generally indicated larger values for calculated photofractions than those measured in experimentso Since the photofraction is a ratio of the area under the total absorption peak to the total response spectrum, it can be influenced by any experimental effects which change either of these areaso Photons may undergo an initial Compton event in surrounding media, and can enter the crystalo These scattered photons may then be absorbed in the crystalo This produces the familiar back-scatter peak in the response spectrum, as well as contributing to the rest of the Compton continuumo Even if the detector is well-removed from shielding and other external material such as room walls, contributions from back-scattered gammas will unavoidably be present in the experimentally observed Compton continuum, since some scattering will occur from the reflector and canning surrounding the crystal, as well as from the photomultiplier tubeo For some isotopes that are difficult to prepare as point sources, absorption and scattering effects are present within the sourceo The above effects would tend to decrease the observed photofractions. In the careful experiments of Heath,(67,83) measurements were made for point source geometry with the source and detector located in the center of a 5 fto diameter plastic bag filled with helium to reduce air scattering. The experiments were done in a large open room, with minimum background presento Isotope sources were used in which electroncapture predominates over beta emission, so that no beta absorbers were needed, thus eliminating small-angle scattering from any beta absorbero

-131In References 67 and 83 a number of independent checks were made of the measured photofractions. By using the 4ir P-y coincidence counting methods given in Reference 84, a number of suitable sources were calibrated and used with these detectorso The peak efficiencies obtained agreed within better than 2% in all cases when compared to the product of the directly measured photofractions and the absolute total efficiencies of Vegors(5) Calibration of these sources was independently checked by the National Bureau of Standards and Chalk River Laboratories, and agreement within + 1% was found. Data from References 67 and 83 are given in Figure V-l. Photofractions obtained by Monte Carlo calculations are determined under idealized source-detector geometry, assuming no surrounding materialso Inherent errors are present in most of the previous Monte Carlo calculations primarly due to approximations made in treating secondary particleso At higher energies, deviation of calculated photofractions from experimental results, obtained under ideal conditions, has been about 10-20% probably because of these approximations. In addition, escape of secondary particles will have a greater influence on the photofractions for smaller crystals. In the previous Monte Carlo calculations only References 9 and 38 have considered electron production and subsequent bremsstrahlung generationo Miller and Snow(9) have treated electrons by simple range theory, which does not take into account the fractional leakage probability for electrons (see Chapter IV). On the other hand, Zerby and Moran(38) have neglected electron transport altogether. In data to be given in Section B below, it will be seen that electron escape has less of an effect on the calculated photofractions than does the bremsstrahlung escape~

0.8 + 3 HEATH (83),EXPERIMENTAL 0 PRESENT CALCULATION, MONTE CARLO + + ZERBY B MORAN (38), MONTE CARLO +S. ~ ~ ~ ~ L A MILLER B SNOW (9), MONTE CARLO + * GOSSETT a DAVISSON (75), MONTE CARLO 0.6- + + ISOTROPIC POINT SOURCE 10 cm FROM CRYSTAL z 0 + U0: \. 0.4 A ~ ~~~~~~~~~~*Y 0 C^^^ A+ A + + A 0 A 0.0 I___ 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 10.0 GAMMA ENERGY, Mev Figure V-l. Photofraction vs. Energy for 3 x 3 In. Solid Crystal.

-133Both References 9 and 38 have used the bremsstrahlung spectra calculated by Zerby and Moran (49) No elaboration on the technique of using this data in the Monte Carlo calculations has been given in Reference 9, thus a comparison with the methods of treating bremsstrahlung given in Reference 38 is not possibleo The method used in the present work to select bremsstrahlung photon energies (Chapter IV) is similar to that used in Reference 38~ However, the assumption has been in Reference 38 that all electrons will give rise to some bremsstrahlung 2 photons of energy > o04 m c o In Reference 38 two bremsstrahlung photons are selected for every electron produced, and a statistical weight of m(Eo)/2 is assigned to each, based on data given in Table IV-lo The photon is considered to be absorbed if its weight falls below a preassigned valueo The method used in the present investigation, described in Chapter IV, obtains the number of bremsstrahlung photons emitted based on the mean values of Table IV-1 and sampling according to the Poisson distributiono By this method a fixed number of photons is not emitted for all electronso The difference between these two approaches is that for low electron energies, the probability 2 for photon emission with energy > o04 moC is very low, and in most instances no bremsstrahlung will be generated in the present calculationso Bremsstrahlung effects on the calculated photofractions are nearly insignificant at these low energieso At higher energies, the methods used in Reference 38 and the present calculations give essentially the same number of bremsstrahlung photons and little difference is noted for the photofractions obtained (Figure V-l).

-134Figure V-1 gives a comparison made by Davisson and Gossett(75) of experimental and calculated results from References 9, 38 and 83 for photofractions of a 3 x 3 ino solid crystal, with an isotropic point source 10 cmo from the front crystal faceo Results from the present work have also been included. The comparison in Figure V-1 is especially valid due to the large amount of data available for this geometryo The data of Reference 75 given in Figure V-l is based on Monte Carlo calculations in which electrons and bremsstrahlung are neglected, and the divergence from the results of the more exact calculations can be seen at higher energies. However, the most significant aspect of this figure is the relatively poor agreement of References 9 and 38, especially at higher energies Fairly good agreement is seen among the results of Zerby and Moran, (38) the experimental values of Heath, (83) and the present calculationo Results from the present work are seen to be in best agreement with the experimental values from Reference 83. The relatively large difference between Reference 9 results and the others, even though rather exact (73) simulation methods were used, is unexplainedo Zerby has also mentioned this discrepancy and attributes it to the method of simulating bremsstrahlung emission Further doubt as to the validity of Reference 9 results, at (70) higher energies, is given by the experimental values of Jarczyk, some of which are listed in Table V-1 below. These values, obtained for collimated beam sources, were checked by three independent experimental methods which are discussed in detail in Reference 70. Considerable difference between References 9 and 70 is noted at these higher energies, at which the effects of

-135secondary particles would be expected to be more importanto Excellent agreement with Reference 70 has been obtained with the present calculationso At lower energies comparisons of experimental results with calculations from Reference 9 have been made by Roesch(85) and McCall(86) For energies < 133 MeV, the maximum difference between experiments and Reference 9 calculations was 9.3%, which is a considerable improvement over the large discrepancies seen at higher energies~ TABLE V-l BROAD BEAM PHOTOFRACTIONS, 2 x 2 INo SOLID NaI CRYSTAL Energy, MeV Miller and Snow() Jarczyk, et alo ( Present Calculation 2.68 0.189 0.156 0.152 7o10 0o056 00o42 oo046 8oo00 0o045 0.035 0~037 20 Well Crystals The data given in References 59 and 78-82 provide a basis of approximate comparison between experimental values of peak efficiencies for well crystals and the calculations of the present work. References 59, 78 and 79, based on some early measurements at ORNL do not clearly specify the source configuration and its location in the wello One can infer from these references that the source was probably a volume source, in some cases with the isotope in an aqueous solution. If this were the case, scattering and attenuation in the source would tend to increase the Compton spectrum at the expense of a decreased photopeak causing a reduction of the photofraction. This effect is discussed in Chapter IV,

-136Section H. (Reference 59, Po 87, gives experimental response curves that show the effect of a reduced photopeak and increased Compton spectrum for a source located in watero) Since the comparison of the present work and References 59, 78 and 79 is intended to show approximate agreement only, calculations were made at the three lowest energies neglecting the effects of electron and bremsstrahlung escape, The calculation at 0.661 MeV was made including these effects. Table V-2 below gives this comparison for a 3 x 3 in. well crystal, 1/2 ino well diao x 1-1/2 ino well height. For the calculations, the source was assumed to be a point isotropic source, 02 cm. from the well bottomo The values below from Reference 59 were taken from plotted data, and are only approximateo The standard deviation for the calculated results is that obtained from the variance in the Monte Carlo computations. TABLE V-2 COMPARISON FOR POINT SOURCE Ip, 3 x 3 IN. WELL CRYSTAL Energy, MeV References Reference 59 Present Calculations 78 and 79 Oo32 o067 0o75 0~762 + ~003 0o364 o6o o664 o0677 + o004 0o638 0o34 0O38 0o387 +,004 0o661 0o33 0o36 0o370 + o007 Gunnick and Stonner(2) have given values for experimentally determined absolute peak efficiencies for a 3 x 3 in. well crystal. The crystal dimensions for the well are not specifically given, but the well size including crystal canning and reflector has been given as

-13713/16 in. diao x 1-29/32 in. height. Information obtained from the manufacturer (Isotopes, Inc.) indicated a probable canning thickness of 00025 in. and reflector thickness of 1/16 in., which were used to determine the crystal well dimensions. Again the source position was unspecified and the assumption was made that it was an isotropic point at the well bottom, neglecting any source container. The comparison is given below for some of the values given in Reference 82. TABLE V-3 COMPARISON FOR POINT SOURCE, CAP 9 3 x 3 IN. WELL CRYSTAL Energy, MeV Reference 82 Present Calculations 0o412 0o465 o520 + oo006 1.17 0o168.180 +.005 2.75 00 72.079 + o003 Comparison between the experimental results of Colby(80) and those calculated in the present work are given in Table V-4 below. The crystal size is 5 x 5 in., with 5/8 in. well diao x 2-1/2 in. well height. For the calculations, the source was again assumed to be point isotropic, 0.2 cmo from the crystal well bottom. TABLE V-4 COMPARISON FOR POINT SOURCE, F, 5 x 5 IN. WELL CRYSTAL AP Energy, MeV Reference 80 Present Calculations 0,279 0.896 0o957 +.002 0.662 0o614 0o594 + o005 1o33 0~350 0.371 +.005

-138Finally, Table V-5 gives a comparison made with the experimental data of Nablo(81) for two well crystals, Harshaw Noso 7F8, and 8F8. In Reference 81 the experimental error is estimated to be about 3%, having used calibrated sourceso The values given in Reference 81 were for photofractions and absolute total efficiencies, EAT determined from a spectrometer response which included backscatter effects. As discussed in Chapter I, in the product of the photofraction and EAT to give CAP p the total spectrum area cancels. The absolute peak efficiencies, CAP of Reference 81 are comparable to the calculations of the present investigation. For the calculations, the point isotropic source was again assumed to be 0.2 cmo from the well bottom. TABLE V-5 COMPARISON FOR POINT SOURCE, P, 8F8 AND 7F8 WELL CRYSTALS Energy, MeV 7F8 8F8 Refo 81 Present Calco Refo81 Present Calco 0o80o 0o97 0o983 + o008 0o98 0.963 + o008 0.279 0,49 0o557 + 0005 0 43 0.515 +.005 0o412 0.24 0.294 +.005 0.23 0.268 +.005 0o662 0.12 00147 +.002 0.11 0o135 +.002 The following conclusions are made, based on the comparisons given in Tables V-2 through V-5. A direct experimental check on the well crystal photofractions calculated in the present work is not possible because no careful experiments such as those of Heath (for solid crystals)have been reported. However calculated values for AT have been shown to agree with independent calculations (Chapter II), and approximate agreement with

-139various experiments is seen for CAp or EIP values given in Tables V-2 through V-5. Thus an approximate check is given for the well crystal Monte Carlo calculated photofractions. B. Effects of Bremsstrahlung and Electron Escape on Photofraction In order to show the relative effects on the photofractions of the energy carried away by electrons and bremsstrahlung escaping the crystal, calculations were made in which these losses were ignored. (See Appendix E for the method of using the Monte Carlo programs without electrons and/or bremsstrahlung.) Table V-6 below gives some data for photofractions in which electrons and/or bremsstrahlung losses are neglected. This tabulation indicates that by neglecting electron losses only (Case 2), the photofractions for both the high and low energies considered here agree within one standard deviation with those obtained when electron losses are included (Case 1). However if only bremsstrahlung losses are neglected (Case 3), then a statistically significant increase in the photofraction occurs at the higher energyo TABLE V-6 EFFECT OF ELECTRON AND BREMSSTRAHLUNG LOSSES ON PHOTOFRACTIONS, ISOTROPIC POINT SOURCES Crystal Case 1 Case 2 Case 3 Case 4 EnergyMeV 8F8 Well o0515+.o049 o0530+O0050 oo829+00064,l000+o0042 6.00 8F8 Well 02650+.0099 o2820+. 0101 2945+,0102.2905+.0102 1,17 3x3 in Solid o1010+o0114.1055+.0069.1535+o0081 ol705+.0084 6.oo 3x3 in Solid.3245+.0105.3190+o0104 o3000+.0102.3200+.0104 1o78 Case 1 - Electron and bremsstrahlung losses considered Case 2 - No electron losses, bremsstrahlung only Case 3 - No bremsstrahlung losses, electrons only Case 4 - No electron or bremsstrahlung losseso

-14oAdditional calculations were made to determine at what energy bremsstrahlung and electron losses may be safely neglected for typical solid and well crystals. Figure V-2 shows the results for a 3 x 3 in. solid crystal. Below about 3 Mev the energy loss due to escaping electrons and bremsstrahlung are seen to have a negligible effecto Also (9) included are some data of Miller and Snow. It may be seen that the data of Reference 9 are in good agreement at low energies with the present calculations. However a significant divergence is noted at higher energies indicating that the probable difference from the present work is due to the methods of treating these secondary particles. Figure V-3 shows the results of similar calculations for the 8F8 well crystal. It may be seen that only for energies less than about 009 Mev may the electron and bremsstrahlung losses be neglectedo As anticipated, these losses become significant at a lower energy for the well crystal, since it has a greater surface area to volume ratio and electron and bremsstrahlung leakage is more probable than for the solid crystalo These results are useful for making calculations because by neglecting electron and bremsstrahlung losses, computer time can be significantly reducedo C, Effect of Geometry on Photofractions 1. Isotropic Sources The effect of various source geometries on the photofraction has been studied with both the solid and well crystal Monte Carlo programs. Reference 83 gives additional data on this effect for solid crystals onlyo

0.80.7ISOTROPIC POINT SOURCE, bl10 cm 0.6 -L.6\ QEl ALL PARTICLE LOSSES CONSIDERED o ELECTRON AND BREMMSTRAHLUNG LOSSES NEGLECTED A MILLER B SNOW REF.(9) 0.5z 0 - 0.4c,0 [ \ I -- 0.2 - N O..1~. 0.00.1 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 10.0 GAMMA ENERGY, Mev Figure V-2. Effect on Photofraction of Electron and Bremsstrahlung Escape from 3 x 3 In. Solid Crystal.

-1420.8 0.7 _8^~ ~ ISOTROPIC POINT SOURCE 0.2cm FROM WELL BOTTOM 0.6- \ ELECTRON AND BREMSSTRAHLUNG LOSSES NEGLECTED -~\ aE] ALL PARTICLE LOSSES CONSIDERED 0.5z 0 \ 0.4 0 I-, I D.310.20.1 - 0.1I 0.2 0.4 0.6 0.8 1.0 2.0' 4.0 6.0 8.0 10.0 GAMMA ENERGY, MEV Figure V-3. Effect on Photofraction of Electron and Bremsstrahlung Escape from 8F8 Well Crystal.

-143From the data given in Tables V-7 and V-8 below, most isotropic source geometries can be well-approximated by a point source located on the crystal axis, eliminating the need for different Monte Carlo calculations when the source geometry is varied within reasonable limits. The geometrical effect on the photofraction consequently is small when point source results are used for a cylindrical volume source. The data given in Tables V-7 and V-8 are intended to illustrate that the relative photofractions for two typical crystals at two different energies are not significantly different from unity when the standard deviation are consideredo For precise work, it would be preferable that photofractions be calculated for the exact experimental arrangement. TABLE V-7 SOURCE GEOMETRY EFFECT - 3 x 3 IN.o SOLID CRYSTAL Relative Photofraction* Isotropic Source o412 Mev 4,45 Mev Point, on-axis, b=lO0cm. 1.000 1.000 Point, on-axis, b=e2cmo 0o.969+.012 1l000+,078 Point, off-axis, m=3o81cmo, b=.2cmo 0o925+J012 0.850+.069 Point, off-axis, m=lo905cmo,b= o2cm 0.9703+.012.o014+.078 Disk, radius=3o81cm., b=o2cm, 0o939+.012 1o010T,078 Disk, radius=lo905cm,bb= 2cm. 0 o979+ 012 1.040+.079 Cylinder, radius=3o8lcm, bl=.2cm., b2=5oOcmo 0o985+.013 1.116+.084 Cylinder, radius=3.81cm, bl=.2cm., b2=10O0cmo 0.990+.013 1l055+.081 * Values relative to on-axis point, b = 10 cmo TABLE V-8 SOURCE GEOMETRY EFFECT - 8F8 WELL CRYSTAL Relative Photofraction* Isotropic Source.412 Mev 4.45 Mev Point, on-axis, b=.2cmo 1l000 1.000 Point, on-axis, b=lo2cm, 0o974+.015 1l000+.121 Point, off-axis, m=lo0cm,, b=o2cmo 1,l001+.016 1o142+.134 Point, off-axis, m=lo23cmo,b=.2cmo 0.979+.016 1o182+.138 Disk, radius=.5cmo, b=,o2cmo Oo988+,016 1.200+o139 Cylinder radius=l Ocm., bl=.2cmo, b2=lo2cmo 0o974+.015 1o,175+.137 * Values relative to on-axis point, b =.2 cm,

-14420 Collimated Beam Source An additional source geometry effect on the photofraction has been studied for the solid scintillation crystal. Varying degrees of collimation for monodirectional beams have been considered with the results given in Figure V-4o The photofraction for a narrow beam collimated along the crystal axis and incident on the crystal face is the largest possible value attainable for a given crystal size since leakage of scattered gammas through the crystal sides is minimizedo As the beam diameter increases, this leakage increases and the photofraction correspondingly decreaseso Thus the narrow beam source provides the most easily interpretable spectrum because the Compton continuum is minimizedo From Figure V-4, for energies less than 6ol3 Mev, less than 1% variation from narrow beam photofraction values are obtained for a beam diameter up to about one-half the crystal diametero Also, the percent variations are not significantly different between the 2 x 2 ino and 3 x 3 ino solid crystalso References 68, 70 and 72 have also considered collimation effectso Berger and Doggett(72) have calculated this effect at 0o662 Mev and state that the relative change in photofraction with collimation could be applied at other energies with only slight error However, as pointed out by Kreger and Brown their experimental data indicate a strong energy dependence for beam diameter/crystal diameter > o25 o Similar variation has been calculated in the present work and is given in Figure V-5o Since multiple scattering events are an increasingly important contribution to the total absorption peak as the tt eaenergy increases (see Chapter III), one would expect that the effect of beam collimation on the photofractions would depend upon the source energyo

ENERGY =0.279 Mev O09.- -- --- BEAM - - - 3 X 3 in CRYSTAL DIAMETER 0.8 — 2X2 in CRYSTAL 0.7 - Q 0.6 0.662 Mev 0 0.5 LI0 I 0.4 0. 2.62 Mev 0.2:_ -...____..0 Mev ____=_3.Mev| --..... 18.0 Mev O 0.:0 0.75 o.no - 0-025 0.500'75 SOURCE BEAM DIAMETER/CRYSTAL DIAMETER Figure V-4. Variation of Solid Crystal Photofraction with Collimation.

ENERGY =0.279 Mev 1.0 0.9- 0.662 ax 0.8 2.62 Mev OXo _ P| P - BROAD BEAM P B-NARROW BEAM 0 | BROAD. 8.00 0 Mev W 0.7- BEAM C,_ DIAMETER 0.6 0 0.25 0.50 0.75 1.0 SOURCE BEAM DIAMETER/CRYSTAL DIAMETER Figure V-5. Photofraction (Relative to Narrow Beam) Variation with Collimation for 5 x In. Solid Crystal.

-147Jarczyk,(70) presents experimental results for broad beam and narrow beam measurements for energies up to 10.83 Mev, which are in agreement with those of the present work, within the experimental error given as + 8-12%. Figure V-6, taken from Reference 70, includes the results of the present investigation. 3. Comparison of Solid and Well Crystal Photofractions Reference 79, p. 57, indicates that the experimental intrinsic peak efficiency for a 3 x 3 in. well crystal (.5 in. well diao x 1o5 in. well height) is nearly the same as that for a solid 3 x 3 in. crystal. Figure V-7 shows that the photofraction for well and solid crystals of the same outside dimension agree fairly well. A check has been made on the consistency of the separate Monte Carlo programs for well and solid crystals by calculating photofractions for progressively smaller well sizeo Convergence was obtained for the well crystal results, with sources at the top of the well, to the solid crystal photofraction for sources on the surface of the crystal. Do Comparison of Monte Carlo and Integration Results As another check on the Monte Carlo calculations, a comparison was made with nIT values obtained from direct calculation. The latter results for isotropic sources were obtained from EAT values calculated by the numerical integration program, BURP-1, described in Chapter II. For monodirectional beam sources, normally incident upon a solid crystal face, CIT = 1 - e o The comparison gave excellent agreement over a wide range of energies for both well and solid crystals, and confirms this portion of the Monte Carlo programs.

-1480.7 -- -. I 06x 4 " Nal 0.6-"1 -- 2 x 3" 23/4" CsI PRESENT G. 0.5 3 3 x 3 Nol ~ CALC., 0. - 5'Ill0. 4..2.Na. W O. Z 0 2 4 6 8 10 E (Mev) 0.7 _ _ - ] I 5"x 4" Nd 0.6 _ 2 3/4"x 23/4"c 3I 3"x3TM.oPRESENT l 3"x 3" Nol 0 CALC. o0.5 --- 4 4 2' x 2" Nol 0.4 LU 0.3 0 a 0.2 0._ 0 2 4 6 8 10 E (Mev) Figure V-6. Absolute Peak Efficiencies for Collimated Sources ompared with Experiments of Jarczyk.

0.8 0.7 - 0.6 - 0 2 IN.x 2IN. SOLID CRYSTAL \ 2 IN.x 21N. WELL CRYSTAL (8F8) Z 0.50 I 1/8 IN. DIA. x I IN. HT. WELL 0 0.3. 0.2- _ 0.1 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 10.0 GAMMA ENERGY, MeV Figure V-7. Photofractions for Solid and Well-Type Crystals.

-150Eo Photofractions Calculated for NaI Values of photofractions for solid and well-type NaI crystals are given in Tables V-9 and V-10, respectively. These tabulated values have been calculated with the computer programs developed in this investigation. Calculations have been restricted to point isotropic sources, since photofractions have been shown to be only slightly dependent on isotropic source geometry. The well crystal results are for the 7F8 and 8F8 crystalso The photofractions should be particularly useful since these two well crystals are the most commonly used, and no previous photofraction data have been available for well crystals. TABLE V-9 SOLID CRYSTAL PHOTOFRACTIONS* ISOTROPIC POINT SOURCE ON AXIS Energy, Mev 2 x 2 ino** 3 x 3 in,o*** 0o279.8315 + o0084 0.320.8150 + o0087 o0412 60o5 + o olo9 7230 +.0100 0o662.446o + o111 0.5560.0o111 1o17.3055 +.0103 lo78 o3245 + o0105 2.14 02685 +.0099 2o75.1565 +,0081 3.57 1790 + o0086 4o45 ooo0900 + oo0064.1425 +.0078 6oo0.0760 +.0059 0olo0 + o0114 7.48.0875 +.0063 8.00 00365 + o0042 10oO o.0180 + 0030.0530 +.0050 * For 2000 primary interactions ** Source-crystal distance = 0.0 cmo ~** Source-crystal distance =10.0 cmo

-151TABLE V-10 WELL CRYSTAL PHOTOFRACTIONS* ISOTROPIC POINT SOURCE ON AXIS Energy, Mev 7F8 8F8 0.279 08590 +,0078.8365 + o0083 0o412.6460 +.0107.6225 + o0108 0o662.4305 +.oOlll 4520 +.0111 1.17.2775 +.0100.2650 + o0099 2.75.1420 + 0078 o1210 +.0073 4o45 00785 + o060o o0815 + oo6i 60oo o0445 + o0046 0o515 + o0049 8o00 0235 +.0034.0215 +.0032 10o00.0185 + o0030.0125 +.0025 * For 2000 primary interactions, source 0.2cm. from crystal well bottom. F. Photofraction for Materials other than NaI Since scintillation materials other than NaI have been used for gamma ray detection, a limited investigation was made of the total absorption characteristics of CsI(Tl) and CaI2(Eu) relative to NaI(T1), for a well crystal corresponding to the Harshaw 8F8 dimensions. Figure V-8 shows the NaI photofraction relative to the photofractions for CsI and CaI2o As previously mentioned, the Monte Carlo programs contain electron and bremsstrahlung data for NaI only and some error is introduced in the results for any other scintillation material at the higher energies. It can be seen that CsI has a significant advantage over NaI, in that the photofraction for the latter is only 40-65% of the former, over the energy range of most laboratory interest. This gives a better defined pulse height spectra

2Q 2|-O CaI2 0 o 0.8 0 I0 0.6c? ro Q: F,z z CsI 0 0.4 0. 0 0.2 0.4 0.6 0.8 1.0 2.0 4.0 ENERGY, Mev Figure V-8. Ratio of NaI Photofraction to CsI or CaI2 Photofraction for 8F8 Well Crystal.

-153for CsI crystals. The comparison given here has been made on the basis of equal crystal dimensions and the greater density of CsI (4.510 gm/cm3) over NaI (3.667 gm/cm3) accounts for the major part of (73) this advantage. Moran(73) has made a similar comparison between Nal and CsI, in which the two crystals were assumed to have linear dimensions inversely proportional to the crystal densities. On this basis the CsI volume is 0.665 that of the NaI, with the CsI weight 0.816 that of the NaI. On this basis Reference 73 found that CsI photofractions were slightly greater than those for NaI, while the intrinsic efficiencies were about the same, for the limited source energies and solid crystal sizes considered. A further advantage of CsI over NaI is that the former is non-hygroscopic, thus decreasing the manufacturing difficulties and allowing its use with no window between the source and crystal, for limited periods of time. CsI crystals are useful in applications where high shock-resistance is necessary since the material is not subject to cleavage. However, the material is relatively plastic and must be supported to prevent distortion during long periods of (87) stress. Special shapes can be bent from thin crystals for particular applications(16) Resolution of CsI crystals is comparable to NaI, but the light yield is considerably less. More recently Hofstadter(88) has developed a new scintillation material, CaI2(Eu) Glos9) has also reported information on this material. Measurements by Hofstadter have given promising results. Pulse heights 1.5-1.8 times those of NaI(Tl) were obtained, with resolution as good as the best NaI(Tl) values. To date only small-sized

-154crystals have been produced, 1-2cm2 x.3 -.6 cm. thick. CaI crystals 2 have been found to cleave very easily when subjected to thermal or mechanical shock, and the material is highly deliquescent. Figure V-8 includes photofraction data calculated for comparison with Nal, and indicates less than a 10% advantage for CaI2 over the energy range considered. Appendix G contains some calculations of EAT for CsI and. CaI2 crystals.

CHAPTER VI CONCLUSIONS A. Summary of Results 1o Absolute Total Efficiencies ( CAT) The analytical expressions for CAT of solid and well crystals have been derived in the form of general point kernels. Integration of these kernels can be performed to reproduce any source geometry. Solutions have been obtained for point, disk, and cylindrical volume sources by numerical integration, and the accuracy of the results have been verified. Previously, solid crystal EAT values had been avail(5,6) (14) able, but except for the limited calculations of Verheijki, no >AT values for well crystals had been available. Using the computer programs developed in the present work, numerous quantitative studies have been carried out to investigate typical laboratory conditions for gamma ray detection by solid or well type scintillation detectors. Calculations have been made to determine the accuracy with which point sources can approximate disk or cylindrical volume sources. The errors introduced by applying on-axis point source CA values for points located off the crystal axis have been calculatedo Also, the effect on EAT values for variations in crystal dimensions and materials have been investigatedo The problem of calculating absolute peak efficiencies for scintillation detectors when absorption and scattering are present within a source has been considered. The treatment has been shown to be a -155

-156logical extension of CAT calculations developed in the present investigation. Well crystal results are given in Chapter IV for homogeneous, monoenergetic sources in an aqueous solution. The tabulated results for CAT of different commercially available scintillation crystals for various source conditions given in Appendix G provide an extensive reference for experiments. The EAT values are tabulated over a range of energies. If the CAT values are plotted versus cross section (using the energy-cross section data of Table II-11), interpolation may be made for any future improvements in cross sections. 20 Photofractions Monte Carlo methods have been used to simulate the transport of gamma rays, and the production of secondaries (electrons and bremsstrahlung) in solid and well type scintillation crystals. The general method of calculating photofractions by Monte Carlo techniques has been used previously for solid crystalso However, the present investigation has extended the calculations to cover well type crystals, for which no calculated photofraction values have previously been available in the literature. The present work also includes an improved simulation of electron transmission and bremsstrahlung generation through the crystal material. When compared with calculated data,(938) which were limited to solid crystals only, photofractions obtained in this investigation are in closer agreement with experiment. (83 Isotropic and collimated sources used here cover the usual applications of gammy ray scintillation detectorso Numerous studies

-157were made using these Monte Carlo programso Photofractions have been shown to be only mildly dependent on isotropic source position and thus the studies made of CAT variations with source position are also valid for peak efficiencies, CP ~ Chapter V contains tabulation of calculated photofraction values for selected well and solid crystals which are commonly used in the laboratory~ Bo Application of Results Application of the results of this work for practical gamma ray scintillation detectors lie in two general areas. First, the absolute peak efficiencies obtained here may be used to calculate the absolute emission rate for gamma ray sources. Knowledge of the absolute source activity is required in many experimental studies, including absolute neutron flux measurements, activation analyses, etco The second general area of application of the present work is in the choice of a gamma ray detection arrangement to serve a particular purposeo For example, one could obtain the minimum crystal size for a given efficiency or peak-to-total ratio in the detector pulse height spectrum. Co Extension of Results Since no tabulation of calculated results can ever cover all possible experimental arrangements that may arise in the laboratory, the values given here have been limited to typical situationso To permit extensions of these results, the computer programs are being made available through the Code Center of Argonne National Laboratoryo

-158Quantitative information for specific experimental situations not included in these tabulations may be calculated by obtaining the programs from the Code Center. The ability to perform calculations for any scintillation materials make these computer programs useful as new materials are developed.

APPENDIX A DERIVATION OF EXPRESSIONS FOR THE ABSOLUTE TOTAL EFFICIENCY Io Isotropic Point Source Kernel Using the notation previously defined in Chapter II, the general expression for the absolute total efficiency is AT4 d L - (A.l) Solid Cylindrical Scintillation Crystal Referring to the sketch below it is seen that integration over G must be performed in two steps, due to a discontinuity in the function x(cp,G) when 9 = 91. All gamma rays from the source enter the nearest face of the crystal, and as the polar angle, G, increases, the rays first leave the crystal through its lower face. For G > 91, the rays leave the crystal through its cylindrical sides. Making the transformation sin 9 dG = -dt (p = cosg), and referring to the sketch below, it is seen that: 2 z 1, (H+6)/ [(H+b) + s ] l2- b/( b2 S ) The distance s(m,cp) may be determined from the sketch showing the lower crystal face. s (m,) =-m sinp + R- m0cos^0 This expression is restricted to -jr/2 < cp < it/2, i.eo, positive values of Y, in order to have a single functional form for s. Since the problem is symmetric about the X-axis, the integration is performed over -159

-16o-ar/2 < cp < _c/2 and a factor of 2 is introduced. The positive square root is required to insure s > 0. SOURCE — R v —-~ - b Yox m I H a',X x Y I,

-161The path lengths x(cp,L) may be determined from the following sketches: I'. ^ \ ~~~~pR I I' I 1 > > >1 11 _> _> L2 x = H /X = S - For an isotropic point source located off the solid crystal axis a radial distance m, IT -2 I -TH/% ] /' b - AT -XiS Pt.)= 2f(d,/-e -dU-e Yd^>l whi~~~~ch ~ 3 which is Equation (2.4).

-162Well Type Scintillation Crystal This crystal geometry is considerably more complicated because of the cylindrical well. The sketches below define the parameters to be used, and three discontinuities exist in the path length function x(cp,.) These discontinuities occur when the polar angle, i, equals 91, 92 and 93, respectively. Again transforming to i we have:' = b/(so+ ) L- =(ob)/[S +(a +b) ], _= -c/(2 + St1 ) i, = -c/(cc t S') Using the sketches showing the well bottom and the crystal lower face we have sl and s2, given as: S,(m0) = -m sinq +Vt cos S*(m,4)=-m sinq5 +vR -m 0cos7 with the restriction -t/2 < cp < i/2 Noting that p. and k2 depend on the source-well bottom distance, b, two cases are possible, dependent upon whether p1 > p2 or pi < p. Physically this means that as the integration proceeds over p., for i > 4,2 rays from the source first strike the well bottom and emerge from the lower crystal face. Second, the rays strike the side of the well and emerge from the lower crystal face. Third, the rays strike the side of the well and emerge from the crystal side. However, when p > [1, the rays for the second path

-163m m Hw II SOURCE y i ( ~~e m I — x (^ -\ -- x 4 i, Y ---- X Y Well Bottom Y Lower Crystal Face

-164strike the bottom of the well, and then the side of the crystal. Writing the integrand in Equation (A.1) as a sum of four different functions, I T At. 1_2 Io p0 (0 ) f (P (i02 1)?A3 4l)d4j (A.2) The form of F2(cp,p) depends on the inequalities ~i > * or tl < t. The derived path lengths, x(cp,pt) are shown in the following sketches. x = a/4

-165b m t~~ b c_>_'\ -- --------— K 1 c When 41 > 42 41~~~~~~~~~ I ~CI a+b Si S2 ( L 4 N/ 1. J.2j2 ~~~I I I; I Icos x.L _ \ I ~ ~~~~~~~ rnS -,~ ~~~~~~~~owhen ^>^ ~~~~~~or when < P2 ~~~~C - — ~~~~~~~~~~~~~~~~0 Cr > CI C 3 > II > C4 K ~' 32 K 4~ S2 S 2 l X c - Jx=%-l _ 2 f2 fC'""''^ -^~ ~ ~~~ ~~ *^ —- *%ICO-... ^'2 ^ ^ ~'^3 13 >. ^' >.^ _ - Si S_ c.fi- n2 I- 4j-n

-166Thus we have F, (, l)= - exp (-Ta/P) and when * > (>.e. < ) /x,- /, = b/(s,+ b ) 2=P2= (a+b)/[s +(a+b) ] (<,)=/-exp[-r( b _,)] or when i < (i-e- > sb ) b o-*b ~,=/,=- (a+b)/[~^ +(a+b)Y] P<= b/b+ S. ((q0,uL()= / - _exp-T(_ and for both cases F(h)g= / - ex PL-Ta(, s)] (gi)= / - exp[ -(- - -- (,) This gives Equation (2,5).

-167II. Isotropic Point Source On-axis The expressions for on-axis points are readily obtained by letting m = 0 in the previously derived results and performing the integration over p. In the above expressions for a solid crystal s = R; and for a well crystal sl = t, s2 = Ro It should be noted that the above derivation for the well crystal was for point sources within the crystal well. In order to check the results given in Reference 14, the solution for an on-axis isotropic point source was obtained for the point outside the well. Using arguments similar to those above, the results for the functions F(t), and t to be substituted into Equation (Ao2) are best summarized by the following table. (Note that Fi = 1 - e-Txi for i = 1, 2, 3, 4) This table includes all the limits of integration, pi, and path lengths xi, for on-axis points both inside (CASE 1) and outside the well (CASE 2), as well as the solid crystal (CASE 3)0 IIIo Two and Three-Dimensional Sources Making the assumption that these sources are homogeneous and isotropic, the point kernel expressions can be integrated to cover any two or three-dimensional source. In particular, the disk and cylindrical volume sources were considered in this investigation. For a disk source consider first a ring coaxial with the crystal axis, with a width dm. The absolute total efficiency for this differential area is: 2TT m dm[CAT(off- is pt.)] 2T( m dm

Case* Conditions 1i Il I X 2Xi I2 X 4 1 b < A-a _________ _________ ____ ____ ___ a ( a+b b a+b a+b -C -c a R-t -C t ((l)2 R )11 t (1- 2, ____ ((o__)2+__) 2 ls= (b2+t2)1I2 ((a+b)2+ R2)"/2 ((a+b)24+ R2)112 (c2+R2)112 (c2+t2)12 p..&2)Il p (1.2 2)i lba+b b b R b lb >,,, 1~ ((a+b) +R2) 112 (b2+t2)11i2 (b2+t2)2 ______________ _________ ____ (12)1 ______ ___________ a+b b b a+b a+b t le < a' ab ((a+b)2 +R2)/t2 (b2+t2)1/2 (b2+t2)112 ((a +b)2+R2) 1/2_________ ________ (1 L2)_ i 2 b >k-a 2_ fa___b -&b a+b c a+b t R c 2a c- a+b (b2+t2)/112 ((a+b)2+R2)l12 (c2+ R2)l2 (1 )a (1p2)l _____t R_______________ ____ _____ _____________ ______________________ —--- H ((a+b)f+R2)112 (c2+tl)l! a+b c c 2 a~b t R-t1 2b and ___ __ __ _ _ __ __ _ _ _ _ _ bb a.b aad.. < ((a+b)2+RL)1l2 (C2+t2)U2 (C2+R2) 2 p ^ (1-L2)1/2 (1 - L2)1/2 __________ (b2+t2) 1/2 ((04b)2+R2)1I! _______________ _______________ _______________ ___________ _______________ ___________________b a+b b g 2c a t -R _____2_ (b2+t2)l/2 t b a+b a+b 1 b a t 2d _ _ _ _ _ _ - - _________ (b2+t2)12 ((a)22) ((b)22)12 _______________ _______________ ________ (2)12 ________ __ c a+b b c +b +b t A 2e _ _ _ _ _ _ _ _ _II _ (c2+t2)l/2 ((a+b)2+R2)lI (b2+t42)lI (c2+t2)12 ((a+)2 ) ______ ___ (1.2)1 M I b aab ab b 1 R b 3 Solid crystal ( a+b b_ —R - ((a+b)2i+Rs2)2 (b2+R'" (,ir) In * All cases of same number must satisfy conditions b 5 h - a, as well as specific condition for eases a, b, etc. t Integral is zero.

=169and for a disk of radius g JEA T (disk) =277 m EAT ( off- Ox s pt)d w27T f mdm simplifying: 2 rf AT (dlsk)- 2 m CET (ff-o-xis pt) cd (A.3) For a cylindrical volume source, consider a differential volume of radius g, and height db located coaxially with the crystal axis. The absolute total efficiency for this volume, assuming the source is transparent to the source gammas is: nT' dCb, (is k) T 92 db and performing the integration for a right circular cylinder extending between the axial dimensions bl and b2 (b1, b2 being measured from the same plane as b we have: C v ) T21CA(s(disk) db C (vol)-) - Q b-a —f- I

-170which gives ^T (voL) = (b,- b,) b or E^AT (vo.)= I b b mCAT ^ (off-ais p6t. Am (A.4) Equations (A.3) and (A.4) are the same as Equations (2.8) and (2.9), respectively.

APPENDIX B SERIES SOLUTION FOR THE TOTAL EFFICIENCY The following derivation, obtained by No McCormick, is given for an on-axis isotropic point source located on the face of a solid crystal. Noting that Q/4r = 1/2 for a solid crystal, with the source located at b = 0, the derivation will be given for the intrinsic total efficiency. By previous definition of parameters. -a/4 11 and for this case, H J -XH/M W1/R )1Rl _/; ) ~>2~^ = / -f e d f - e d We introduce I1, I2 for the first and second integrals, respectively, Using the definition of the exponential integrals: E.( ) a'e ze dZ we find I, TH e-'z- dz - ( — T(H^+R) )ez-'d: t+ (2 + ) -171

-172or I, = E2 (T) - (H^i~RY Ez(jrH2+) To obtain 12, let f2 1 - 2 and rewrite 12 I2- r e-R/< C dC J R I_ e ^ (H R) Y —^since 0 < 1, then 0 < f < 1, and -- can be expanded in a binomial series: ~1/_~,2 2! 3! I2=-d e-^f + 3+(-)() - + (- (f+)()-).. 73,.. (H2+ RR)Y let p thus -. (.+ R2) T R e-P R 2T( 2) TR "CTrR2R L 31(2 - ) +... pC 3 1

-173This can be written as a summation:.co, 2M+z2 t(HN +R) Xl(2!! (rR) ne- dp.,=2 m 2+3 T*R iP where (2m-I)!! = (2m —)(2m-3)(2m-5)...(3,)(1) (-)!!-= l 0=! -I This summation can be rearranged mo 00 2^+2 ^^42, -, 7.pi. -+3 2 t2n +2 m+Z R r, r (NH R1) / e (H n +R pZ ( j Again using the exponential integrals M=- / [ ( (H2- RR1 +2.

-174Combining we finally obtain the desired solution. cT= / - E (r H) + (HLR)/ E2 (V) f^ E ~(2 -R) 2- r2 + r 1! EZ Vf% +3 3RVH'+R m:o 2 m (H[ R *' The above equation has been used to check the results of some calculations made with the BURP-1 Computer Program. Excellent agreement was found between results of the above equation and the computer program value s

APPENDIX C DERIVATION OF KAHN'S METHOD FOR SAMPLING FROM THE KLEIN-NISHINA FORMULA He Kahn(24) has given a number of selection techniques for sampling from the Klein-Nishina formula, differential in polar angle, for Compton scattering events. The method used in these calculations has been shown by Kahn to be most efficient for photons below about 4 MeV, and even though source particles of greater energy are considered, most will be either degraded in energy below 4 MeV in the first or second scatterings, or will undergo a pair production event. Thus the technique used is the most efficient in the energy range where multiple Compton scatterings are the principal interaction mechanism. The differential Klein-Nishina formula(31) for Compton collisions of incident unpolarized radiation is: co t' j (/+ cose ), (- cos e) 2dL,[/A(/(-cose)]2L (I/coSze)[I+(-CO0s) (C.I) where ac = incident gamma energy, moc2 units 2 ro = classical electron radius moc2 The Compton energy-angle relationship is: = K/-/ + /-cose)] (C.2) or X- / + e( (/-Cose) where a' = scattered gamma energy, moc2 units x = ac/a. -175

-176Integration of Equation (Col) over the azimuthal angle, and rearrangement using Equation (Co2) gives: dLcr(X,e o)= i Cr dQx (x- + -/)( ),-)- x —I xC Noting that in a rejection method, the normalization constant of the probability distribution need not be specified, one can define the PoDoFo for x as: / / + (x,~,o )= (cose -/ +x + X ) (C.4) The P.D.Fo of Equation (C.4) is split into the sum of two other PoD.Fo 2x) = +9 20 ( 4 X 8 20(t 1 (/coS' +.) 5 2-(+9 20-x- 2 x let 2o( 20'(X2 The procedure is to sample from fl(x) = gl(x)hl(x) and f2(x) = g2(x)h2(x) with relative frequencies + and, respectively. Sampling 2a + 9 2a + 9 is accomplished by applying the cumulative distribution function method to gl(x) and g2(x) and then applying the rejection method to hl(x) and h2(x)o The functions gl(x) and g2(x) are properly normalized for 1 < x < 1 + 2a and one obtains: for gl(x): x = 2crl + 1 for g2(x): x =c + 1 2acr2 + 1 where r1, r2 are random numbers.

-177The method is given in the following flow diagram: l —--------- Ob.tain r,,r29 | 2,L ~2,~ +' 9 yesn -= la=,''es _r SW=/'3^ ^+ no yes I —1 I ~ —jkS y -s cose=t ose =I x-/ cK -

APPENDIX D WELL CRYSTAL ESCAPE ROUTINE In the Monte Carlo calculation of photofractions for scintillation crystals with a coaxial cylindrical well, consideration must be given to those gamma rays which intersect the surface of the well. If a secondary gamma ray has (for any step in the calculation) its location in the crystal at the coordinates x, y, z, and then travels a radial distance, t, in the direction defined by the direction cosines u, v, w, its final coordinates are given by: X' x + at y= y +t Zi- = 4- \c t (Dl) If the gamma ray path cuts the surface of an inner, coaxial cylinder of infinite height and radius ro, then the radial distance to this surface is obtained by simultaneous solution of Equations (D1l) and a 2 2 X' +' = r (D.2) The radial distances to this inner cylindrical surface are given by: t+_ -(a- +vc) (,ax, -(l-w )(x +" -~r ) (-o7) -178

-179The following sketch illustrates these distances. A ( X "'^y tUV- (x y z) Let: S L = X v A - (/-w)(X2 y 2 thus atL (/-w and note that for the coordinates (x,y,z) assumed in the above sketch x +y ^rO (D.4) For a given point of departure (x,y,z) and direction (u,v,w), the gamma cuts through the inner cylinder only if both solutions to Equation (D.3) give different, positive, and real values for t+ and t_. (Requiring different values for t+ and t_ excludes the case of a path tangent to the inner cylinder.) Thus for: noO,or s O nner cylin |5) no intersection occurs with the infinite height inner cylinder.

-180The actual well crystal geometry, in which the well depth is less than the crystal height, introduces a further consideration. Now a point of departure may be located below the well so that: X f t l i Except when w + 1, all paths from this location intersect the extended cylindrical well surface, and the condition A > 0 is automatically satisfied. In addition the restrictions for intersection, 8 < 0 and I|b > /- which insure that t > 0, are no longer applicable because t+ and t_ are measured in opposite directions and one of the roots of of Equation (Do3) must be negativeo For points within the well crystal, three locations are possible and must be considered separately. These locations are characterized by the following conditions for a point of departure x,y,z, well radius, ro and well height, Hw~ Location A: x2 + y2 < r2 and z > Hw o > Location B~o x2 + y2 >_ r2 and z > Hw Location C~ x2 + y2 > r2 and z < Hw Some preliminary considerations are applicable to all locationso If w = + 1, then no intersection with the well cylindrical surface is possible, and a test is made first for this condition, Next, if the path length obtained from Equation (3~5) is less than the distance to the nearest well surface, in the direction u,v,w, then the particle does not enter the wello In addition, when any of the conditions (Do5) are satisfied for Locations B and C, no intersection is made with the well cylinrical surface, extended. Assuming that a vacuum exists in the well, when a gamma does enter the well, no interactions occur. Under this assumption,

-181the gamma path length is increased by the distance traveled through the well. (Chapter IV, Section H, discusses the effect of absorption and scattering by a three dimensional source, considerations of which are ignored here.) Assuming all these conditions have been checked, then the calculation of the particle's escape through the well, or its increase in path length due to re-entry into the crystal after passing through the well, are summarized below. Defining: Zt+ = Z +t+w = Z ~ Q =, +-, 2 where al = path length in crystal from Equation (355) 2 = distance traveled through well Thus the location of end point of the gamma path is: X -X + )(-y +- x= z + VW i?'^ -^^Z~ = &A (6)(Do6) and the criteria given in Equation (4.12) are applied. In the sketch below showing Location A, four different rays are possible and are identified as Cases 1-4, below:

-182LOCATION A CASE CONDITIONS 22 REMARKS 1 zt < 0 Escape through well 2 0 < zt H w t-+ - (Hw-z)/w Re-enters crystal 3 zt > Hw 0 Does not enter well 4 w > 0 0 Does not enter well In the following sketch which shows Location B, 5 different rays are identified as Cases 1-5 below~ LOCATION B CASE CONDITIONS 2 REMARKS 1 zt < 0 0 Does not enter well 2 0 z t < Hw and zt < 0 Escape through well 3 0 < zt < Hw and 0 < zt+ < Hw t+ - t Re-enters crystal 4 zt > Hw and 0 < zt < Hw t+ - (Hw-z)/w Re-enters crystal 5 w > O 0 Does not enter well In the following sketch which shows Location C, 5 different rays are identified as Cases 1-5 below: LOCATION C CASE CONDITIONS REMARKS 1 zt <0 0 Does not enter well 2 0 < zt < Hw and zt+ 0 - Escape through well 3 0 < zt_ < Hw and 0 < zt+ < Hw t+ - t_ Re-enters crystal 4 0 < zt < Hw and zt > Hw (Hw-z)/w-t. Re-enters crystal 5 zt > H 0 Does not enter well -w

-183z Location A Hw /z - i ~ x ly I Location A

-184ro yX,~ I I, - (x,y,z) ~, i —ocaion B Location B

-185W Y ~ - I I z Location C

APPENDIX E INSTRUCTIONS FOR COMPUTER PROGRAMS FOR CALCULATION OF THE ABSOLUTE TOTAL EFFICIENCY AND PHOTOFRACTIONS* Absolute Total Efficiency - BURP 1, 2, and 3 Three separate computer programs have been written to compute the absolute total efficiency, 6AT' of scintillation crystal detectorsO Each of these programs will calculate EAT for either solid or well type cylindrical crystals for various source geometries (see Figures II-1, II-2)o The programs, and the equations that are solved are: BURP-I: Isotropic Point Source on Crystal Axis - Equations (2,6), (2~7)o BURP-2: Isotropic, Homogeneous Disk, or Cylindrical Volume Sources Centered on Crystal Axis - Equations (2o8), (2o9)o BURP-35 Isotropic Point Source Located Radially off the Crystal Axis - Equations (2,4), (2,5)o The programs are written in the Michigan Algorithm Decoder (MAD) language(90%91) for the University of Michigan IBM-7090 computer~ The programs use a numerical integration technique to evaluate the integrals for (AT ~ This technique, which forms a part of the available library system (Michigan Executive System Subroutines-MESS) 30) is a Guassian quadrature formula. The subroutine allows selection of the power of the approximating polynomial and the number of interval sub-divisions for each integral, with no limit being placed on the multiplicity of integrals. For the numerical integration the order of the polynomial approximation and the number of subintervals have been optimized for maximum accuracy in a reasonable computational time. The procedure used was to sequentially * All computer programs are available from the Argonne National Laboratory Code Center, Argonne, Illo -186

-187increase the order of the polynomial approximation and the number of sub-intervals until the results showed no change in the first four decimal places. These optimized values are included in the programs. In addition, the BURP programs use another MESS library routine for interpolation of the total cross section data at a specified energy. This subroutine is based on the iterative interpolation formula of Aitkeno(92) The general instructions are the same for use of each of these programs, although due to different source geometries there are some differences in the details of required input data. The required data is put in on punched cards and is in the so-called "simplified input" form, as opposed to format input. (9091) Use of this simplified input provides greater flexibility in these programs and insures that these programs can be used with relative ease by those not familiar with digital computerso All data must have dimensions in cm., energies in MeV, and cross sections in cm. For all programs, by proper choice of certain input parameters, various options are available. These options area solid or well type crystal; evaluation of 6AT for NaI at a single energy or any number of energies; or calculation for any crystal material at a single energy. For all three programs the data may be given in any order, being punched in columns 1-72 on standard "IBM-Cards." The only requirements are that each value punched must be of the form, PARAMETER = value, separated from the next value by a comma, with the last value followed only by an asterisk. With this simplified input form, the first set of data must specify all numerical values required by the program for a given problem. However, subsequently run problems

-188for which only some of the values are changed, need contain only those values which are different. This saves considerable time when a large amount of input data is to be prepared. Since the most common material used for gamma ray scintillation crystals is NaI, the cross sections of Table II-11 have been incorporated in all these programs. However, provision has been made to read in any values of cross sections so that calculations can be performed for any material, or to take. into account future improvements of the data given in Table II-11. To read in different values, one uses the form TAU(1) = xxox, y.yy, etc. E(l) = zozz, zw.x, etc. where the corresponding numerical values of TAU and E are respectively the total cross section (without coherent scattering), cm- and energy, MeVo In addition, the integer number of cross section values read in must be specified as KK = a, (no decimal point)o When total cross sections are read in this manner, the NaI values stored in the program are automatically deletedo If subsequent calculations for NaI are desired in the same computer run, the NaI cross sections must be read in. All three programs require input data of: D = crystal diameter, cm H = crystal height, cm For well crystal: CASE = 1 (no decimal point) and WD = well diameter, cm WH = well height, cm or for solid crystal: CASE = 2 (no decimal point)

-189Specified integer values of J (no decimal point), depending on the three options: 1) J = 0 calculation for NaI (or any other material if cross sections are supplied) at a single energy, which must be specified as: E = incident gamma ray energy in MeV, where.01 < E < 30 MeV 2) J = 1 calculation for NaI (or any other material) over all energies given in Table II-11, except at Kedge of iodine (E =.03323 MeV), starting at E = o01 MeV. The calculation is repeated for every energy in sequence, up to and including the value specified by NUMBER, where: NUMBER = number of gamma energies for which calculations will be performed (eog., if NUMBER = 23, output will give calculations for.01 < E < 60 MeV)o Other requirements for source geometry data will be discussed below. 3) J = 2 single calculation for any scintillation material for which the total cross section (without coherent scattering) must be specified as: TAU = total cross section of crystal, cm. By specifying TAU = some large number (e.g. 500), one obtains the mean solid angle subtended by crystal at the sourceo

-190The output for J = 0 or 2 is a single value for EAT' designated as EFFAT, while for J = 1, the output is in a tabular form similar to the tables given in Appendix G. For BURP-1 only, the output for the solid crystal, when J = 0 or 2 includes EFFIT = 6IT, and OMEGA = solid angle. Each program also requires the following additional input data: BURP-1 When J = 0, or J = 2 options are specified, must also specify: B = source distance, cmo (see Figures II-1, II-2), Or when J = 1, must also specify: B(l) = xxox, y.yy,...., where the xx.x and y.yy are symbolic representation of the numerical values in cm. for the desired source distanceso Also specify NN = integer number (no decimal point) of these source values read in. Typical data for a 2 x 2 in. NaI well crystal (Harshaw No. 8F8) and solid crystal are: 1) D = 5o08, H = 5,08, CASE = 1, WD = 2~8575, WH = 3581, J = 0, E =.661, B =.2* 2) J = 1, B(1) =.5, 10, 2.0, NUMBER = 20, NN = 3* 3) J - 2 B = o5, TAU = 500* 4) CASE = 2, TAU = o519* The problems to be calculated by this data are: 1) EAT of a NaI well crystal for an isotropic point source of 0.661 MeV located 0.2 cmo from the well bottom. 2) CAT for the same sized well crystal, for point sources located at 0.5, 1.0, 2.0 cm. from the well bottom having all energies,01 < E < 350 MeV, tabular output.

-1913) CAT for the same sized well crystal, with a total cross section TAU = some large number, (e.g., 500). For such a large value of TAU, the calculation will give the value for the geometry factor, Q/4c, for a point source 0.5 cm. from the well bottom. 4) CAT for a 2 x 2 in. solid crystal, for energy corresponding to the total cross section TAU =.519 cm1. BURP-2 When J = 0, 2 must specify either: a) K = 3 (no decimal point), disk source with: G = disk radius, cm. B = distance of disk from crystal, cm. b) K = 4 (no decimal point), cylindrical volume source with: G = cylinder radius, cm. Bl = distance from crystal to bottom of cylinder, cm. B2 = distance from crystal to top of cylinder, cm. When J = 1, must specify either: a) K = 3 (no decimal point), disk sources of same radii, different source-crystal distances G = disk radii, cm. B(1) = Xoxx, yyoy, etc. numerical values, in cm., for desired disk source-crystal distances. NN = integer number (no decimal point) of these source distances.

-192b) K = 4 (no decimal point), volume sources of same radii, different heights. G = cylinder radius, cm. B1 = distance from crystal to bottom of source cylinder, cm. B2(1) = x.xx, y.yy,.o,, etc. numerical values, in cm., for desired source cylinder heights. NN = integer number (no decimal point) of these source cylinder heights. Source absorption and scattering can be taken into account by evaluation of Equation (4o14), for the well or solid crystal. The source volume dimensions for the well crystal must be numerically equal to the well volume dimensions or for the solid crystal the source must have the same radius as the solid crystal with any desired values for Bl and B2 allowed. The additional input data required is: SIGMA represents the source material total gamma cross section values cm-1 corresponding to the energy values, Eo (1) For CASE = 1, well crystal ABS = 1 G = WD/2 (numerical value) B = 0 (a) When J = 0 B2 = WH (numerical value) SIGMA(l) = XoX, y.y, oo., etc.

-193(b) When J = 1 B2(l) = WH (numerical value) SIGMA(l) = x.x, y.y,..., etc. NN = 1 (c) When J = 2 B2 = WH (numerical value) SIGMA = x.x (2) For CASE = 2, solid crystal ABS = 1 G = D/2 (numerical value) (a) When J = 0, 1 SIGMA(l) = Xox, y.y,..., etc. (b) When J = 2 SIGMA = x.x BURP-3 When J = 0, 2 specify: B = source-crystal axial distance, cm. M = radial distance of source from crystal axis, cm. When J = 1, specify: B = source-crystal axial distance, cm. M(1) = xoxx, yyoy,..o, numerical values, in cm., for desired off-axis radial distances. NN = integer number (no decimal point) of these off-axis distances.

-194The output for the J=l option will give a column headed M=o00, which are the on-axis values of EAT ) and other columns headed M=xoxx > M=yyoy, etco which are values of EAT(M) normalized to the on-axis value, at a given energyo For all of these programs, the source dimensions must be no greater than the well dimensions for CASE=1- or the solid crystal diameter for CASE=2o The only exception is that BURP=1 will calculate for point sources located outside the well on the crystal axiso PHOTOFRACTION - BURP 4 5 PROGRAMS The input data for the Monte Carlo Programs are similar to that required for the previously described BURP 1-3 programso The solid, and well crystals are considered in two separate programs, designated BURP 4),5 respectively, but certain data are common to botho Again, both programs contain cross section data for NaI crystals (Table 1I-11), with provision being made for other materialso For other scintillation materials the following cross section (units of cm ) and corresponding energy values must be read inMU(1) = Xoxx, yyoy, etco (total cross section) KAPPA(1) = Zozz, wwow, etco (photoelectric cross section) SIGMA(l) = Xoyx, yoxx, etco (Compton cross section) E(l) = xoxy, xoyy, etc (energy in MeV) KK = B (no decimal point) - integer number of energy values read ino

-195As in the BURP 1-3 Programs, the NaI cross sections stored in the BURP 4,5 Programs are automatically deleted when cross sections for another material are read in in this manner. If subsequent calculations for NaI are desired in the same computer run, the NaI cross sections must be read in. When obtaining photofractions for scintillation crystal materials other than NaI, some error is introduced in the result because these programs use electron penetration data and bremsstrahlung spectra for NaI. The magnitude of these effects is reduced for lower energy incident gamma rays. Termination of the calculation is based on a specified number of primary interactions occuring in the crystal. This number is read in as the integer value, N=ca (no decimal point)O The incident gamma ray energy is specified by a single value EO=z xx, (MeV). The crystal dimensions in cmo are specified by D, H, WD, and WH as previously discussedo The various source configurations are given by reading in a numerical value for the parameter, SOURCE and other parameters, as summarized below~ SOURCE=1, B=xoxx~ Isotropic point source on crystal axis, located B cm. from coordinate axes origin which is at the top of either the solid or well crystal. (Note difference from Figure 11-2, for well crystal.) SOURCE=2, B=xoxx, RHO=y.y: Isotropic point source off crystal axis a distance equal to RHO cmo, located B cmo from coordinate axes origin. SOURCE=3, B=xoxx, DD=yoy: Isotropic disk source of diameter equal to DD cmo, located B cmo from coordinate axes origino

-196SOURCE=4, o Monodirectional beam source, collimated to solid crystal diametero SOURCE=5, DS=xoxy: Monodirectional beam source, collimated to diameter equal to DS cmo, where DS < Do SOURCE=69 ~ Monodirectional narrow beam source of zero divergence collimated to crystal axiso SOURCE=7, G=xox, Bl=yoy, B2=zoZo Isotropic, homogeneous cylindrical volume source with radius equal to G cmo, and distances Bl, B2 measured to the source surfaces as shown belowo 8I lII ~B2 ~II _________ L 8 Well Crystal Solid Crystal The solid crystal program BURP-4 considers all seven of the above sources, while the well crystal program BURP-5 includes all sources except 4 and 5o All sources must be located within the crystal outside diameter for the solid crystal, or within the well dimensions for the well crystal,

-197In addition to specifying a value for SOURCE, a value must be read in for the parameter FORM. One must specify FORM=P and where: P = X = SOURCE value for first problem run P = 0 for each subsequent problem run for which only the energy is changed 3 = XX, where X=SOURCE value for each subsequent problem run in which changes are made in the crystal or source dimensions, but retaining the same type of sourceo Various options are provided by specifying values for the parameters BR and ELECo If no values for BR or ELEC are specified in the input data, all electron and bremsstrahlung effects are consideredo For low incident gamma energies, or when less accurate photofraction values are acceptable, computer time can be reduced by specifying any of the following values for these parameters: ELEC=O: Neither electrons, nor bremsstrahlung considered ELEC=1: No electron escape, bremsstrahlung considered to be distributed along electron path BR=O Electrons produced and allowed to escape, no bremsstrahlung consideredo Finally, by specifying ALPHA=1, one will obtain an unbroadened energy deposition spectrum presented as a table of 128 values in the form of an 8x16 matrix. These values represent the number of histories which result in energy deposition equal to n x EO/128, where n is the position in the 8x16 matrix. Thus, the lowest deposited energy is given by the first row, first column position, and the last row, last

-198column position corresponds to the unbroadened photopeako This spectrum is obtained, neglecting electrons and bremsstrahlung, and requires a relatively large amount of computer timeo The output from the Monte Carlo programs consists of columns headed by ENERGY, PHOTOFRACTION, A, P, Io For input data of SOURCE=2, 3, or 7 an additional column headed by the parameter T is giveno The value for ENERGY is the incident gamma energy in MeV (read in as input data, EO), and the corresponding PHOTOFRACTION is given with its standard deviation. A is the number of totally absorbed source particles; P is the number of source particles incident upon the crystal that pass through with no interaction; I is the number of source particles that interact at least once in the crystal (I=N, where N is part of the input data); T is the number of source particles selected that are not incident upon the crystalo Typical input data for the well crystal program might consist of~ 1) N=5000, E0=o661, D=5o08, H=5o08, WD=208575, WH=3o81, SOURCE=l, FORM=-1, B=3 o61 2) FORM=O, EO=1o17* 3) EO=2075* 4) FORM=11, B=3o51, EO=o661* 5) SOURCE=3, FORM=3, B=3061, DD=205* 6) ELEC=O* The photofractions calculated by this data are (all calculations are for 5000 primary interactions in crystal)

-1991) 2 x 2 in. well crystal (8F8) for a.661 MeV point isotropic source located on the axis 002 cm. from well bottom. 2) Same crystal and source, for 1.17 MeV gammas. 3) Same as 2) for 2 75 MeV gammas. 4) Same crystal and source, but now located 0.3 cmo from well bottom for.661 MeV gammas. 5) Same crystal,.661 MeV disk source of 2,5 cmo diao, located.2 cm. from well bottom. 6) Same as 5), but electrons and bremsstrahlung are not considered.

APPENDIX F METHOD OF SELECTING FROM A UNIFORM AZIMUTHAL DISTRIBUTION OF DIRECTIONS The following technique is due to Kleineckeo (4) In the transport of particles through a medium, the various physical laws give the cosine of the angle between the paths of the incident and emergent particleso This polar angle,, defines a cone of possible emergent directionso When the assumption of azimuthal symmetry is made, one must select a direction from a uniformr distribution over this coneo In the Monte Carlo calculation the direction cosines, relative to a fixed rectangular coordinate system are known for the incident path; cos 9 is obtained from the physical laws, and the direction cosines for the emergent path are desiredo We will denote the incident direction cosines by Aly B 1 C1 and the desired emergent ones by A2, B29 C2 with -= cos o Now suppose U, V, W are three numbers selected such that (U, V, W) is a point distributed uniformly within the unit sphereo The vector U from the origin to the point (U,V,W) will be distributed with respect to the vector A to (Al, B1, C1) in a uniform azimuthal direction Moreover, the vector J+kA = (U+kAi, V+kB1, W+kCl), for any k, is also uniformly distributed in azimuth, with respect to A -200

-201^A + y /?-u+k_ A A The length of the vector U+kA is S=[+V +V +2k(uA,+VB,+WC,)+k 2 2 2 since A + B1 + C =1. Let Y = UA1 + VB1 + WC1, and then the condition that the angle between the vectors U + kA and A has the cosine, p. = cos 0 gives y2 + 2kY +K2'=2,(U +V \+2KY + W) Solving, we obtain K = -Y +-[ u-iv +W -Y and also S= Y+k =. +V +W -Y P I -/II

-202Thus the vector U+kA is a vector uniformly distributed in azimuth about the vector A, at the polar angle G o The direction vector, i (U+kA) defines the desired direction, and has the components (U+kA1)/s, (V+kB1)/s, (W+kC1)/s o The desired direction cosines are thus A = (u+kA,)Z = A,(Y - (YA-u) B,= (v +'B,)z = GB, -Z(Y B,-V) C, _(w+kC,)z C, -Z(YC, -w) where / /1 S V In summary, the procedure followed is~ (1) Select U,V,W uniformly distributed in the unit sphere, au +\- +s I \

-203(2) Let y = LA, +VB, +WC, z —.= [+v2 \/ y (3) Then A,= A, - z(YA,-a) C= B,- z (YB,-V) C2,= C,1- z (YC,-)

APPENDIX G TABULATED VALUES FOR ABSOLUTE TOTAL EFFICIENCIES* * All calculations for NaI crystals, with dimensions in centimeters, and. energies in MeV, unless otherwise notedo Well crystal numbers refer to Harshaw Chemical Coo Catalog(l6) numberso -204

ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 2 x 2 IN. SOLID CRYSTAL 0 = 5.080000v H 5.080000 ENERGY B=.03.20.50 1.00 2.50 5.00 10.00 20.00 50.00 100.00.010.5000.4608.4034.3168.1493.0542.0154.0040.0006.0002.015.5000.4608.4034.3168.1493.0542.0154.0040.0006.0002.020.5000.4608.4034.3168.1493.0542.0154.0040.0006.0002.030.5000.4608.4034.3168.1493.0542.0154.0040.0006.0002.040.5000.4608.4034.3168.1493.0542.0154.0040.0006.0002.050.5000.4608.4034.3168.1493.0542.0154.0040.0006.0002.060.5000.4608.4034.3168.1493.0542.0154.0040.0006.0002.080.5000.4608.4034.3168.1493.0542.0154.0040.0006 0002 100.5000.4608.4034.3168.1493.0542.0154.0040.0006.0002 150.4988.4471.3776.2833.1276.0475.0142.0038.0006.0002 200.4832.4187.3445.2528.1131.0431.0133.0037.0006.0002.300.4172.3479.2794.2019.0909.0359.0116.0033.0006.0001,400.3649.3001.2393.1722.0780.0314.0103.0030.0005.0001.500.3287.2686.2134.1534.0698.0283.0094.0028.0005.0001.600.3051.2485.1971.1415.0646.0263.0088.0026.0005.0001.800.2729.2213.1752.1257.0576.0236.0080.0024.C004.0001 1.000.2506.2027.1603.1150.0528.0218.0074.0022.0004.0001 1.500.2153.1737.1371.0983.0453.0188.0064.0019.0003.0001 2.000.1968.1585.1250.0897.0414.0172.0059.0018.0003.0001 3.000.1801.1448.1142.0819.0378.0158.0054.0016.0003.0001 4.000.1739.1398.1102.0790.0365.0153.0052.0016.0003.0001 5.000.1724.1385.1092.0783.0362.0151.0052.0016.0003.0001 6.000.1724.1385.1092.0783.0362.0151.0052.0016.0003.0001

ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 3 x 5 IN. SOLID CRYSTAL 0 = 7.620000, H = 7620000 ENERGY 8=.00.20.50 1.00 2.50 5.00 10.00 20.00 50.00 100.00.010.5000.4738.4349.3731.2257.1023.0328.0088.0014.0004.015.5000.4738.4349.3731.2257.1023.0328.0088.0014.0004.020.5000.4738.4349.3731.2257.1023.0328.0088.0014.0004.030.5000.4738.4349.3731.2257.1023.0328.0088.0014.0004.040.5000.4738.4349.3731.2257.1023.0328.0088.0014.0004 r.050.5000.4738.4349.3731.2257.1023.0328.0088.0014.0004 0.060.5000.4738.4349.3731.2257.1023.0328.0088.0014.0004.080.5000.4738.4349.3731.2257.1023.0328.0088.0014.0004.100.5000.4738.4349.3731.2257.1023.0328.0088.0014.0004.150.4999.4690.4240.3550.2051.0922.0304.0085.0014.0004.200.4965.4581.4071.3341.1884.0850.0286.0082.0014.0004.300.4641.4153.3601.2887.1597.0731.0256.0076.0013.0003.400.4270.3767.3234.2571.1417.0655.0234.0071.0013.0003.500.3969.3475.2969.2351.1294.0603.0218.0067.0012.0003.600.3756.3275.2790.2205.1214.0568.0207.0064.0012.0003.800.3444.2989.2538.2001.1102.0519.0191.0059.0011.0003 1.000.3216.2782.2358.1857.1023.0483.0179.0056.0010.0003 1.500.2833.2441.2064.1622.0895.0425.0159.0050.0009.0002 2.000.2623.2256.1905.1496.0826.0394.0148.0047.0009.0002 3.000.2427.2084.1758.1380.0762.0364.0137.0044.0008.0002 4.000.2353.2019.1703.1336.0738.0353.0133.0043.0008.0002 5.000.2334.2003.1689.1325.0732.0350.0132.0042.0008.0002 6.000.2334.2003.1689.1325.0732 0350.0132..0042.0008.0002

ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 7F8 WELL CRYSTAL D = 4.445000, H = 5.080000 WO = 1.905000, WH = 3.810000 ENERGY =.00.10.20.30.40.50.75 1.00 1.25.010.9851.9843.9835.9825.9816.9805.9774.9735.9686.015.9851.9843.9835.9825.9816.9805.9774.9735.9686.020.9851.9843.9835.9825.9816.9805.9774.9735.9686.030.9851.9843.9835.9825.9816.9805.9774.9735.9686.040.9851 9843 9835.9825.9816.9805.9774.9735 9686 ~.050.9851.9843.9835.9825.9816.9805.9774.9735.9686.060.9851.9843.9835.9825.9816.9805.9774.9735.9686.080.9851.9843.9835.9825.9816.9805.9760.9718.9664.100.9835.9825.9815.9805.9793.9781.9744.9699.9640.150.9492.9423.9375.9336.9303.9275.9214.9151.9076.200.8355.8167.8054.7971.7908.7858.7765.7690.7612 300.6186.5926.5781.5679.5604.5547.5447.5377.5313.400.5031.4779.4641.4546.4476.4423.4333.4272.4218.500.4351.4114.3986.3898.3834.3785.3703.3649.3601.600.3944.3720.3600.3518.3457.3412.3336.3285.3242.800.3424.3220.3111.3036.2982.2941.2872.2828.2790 1.000.3086.2895.2795.2726.2676.2638.2575.2534.2500 1.500 -2579.2413.2326.2266.2223.2190.2137.2102.2073 2.000.2325.2173.2093.2038.1998.1969.1920.1888.1862 3.000.2102.1962.1889.1839.1802.1775.1730.1701.1678 4.000.2021.1886.1815.1767.1732.1705.1662.1634.1611 5.000.2001.1866.1796.1748.1714.1688.1645.1617.1595 6.000.2001.1866.1796.1748.1714.1688.1645.1617.1595

ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 8F8 WELL CRYSTAL D = 5.080000, H = 5.080000 WD = 2.85750C, WH = 3.810000 ENERGY B=.00.10.2C.30.40.50.75 1.00 1.25.010.9682.9666.9649.9631.9612.9591.9530.9457.9366.015.9682.9666.9649.9631.9612.9591.9530.9457.9366.020.9682.9666.9649.9631.9612.9591.9530.9457.9366.030.9682.9666.9649.9631.9612.9591.9530.9457.9366.040.9682.9o66.9649.9631.9612.9591.953C.9457.9366.050.9682.9666.9649.9631.9612.9591.9530.9457.9366.060.9682.9666.9649.9631.9612.9591.9530.9457.9366.080.9667.9651.9633.9613.9593.9570.9506.9428.9330.100.9650.9632.9612.9592.9569.9545.9477.9393.9290.15-0.9205.9130.9065.9006.8951.8900.8781.8664.8537.203.8017.7828.7691.7578.7482.7398.7225.7081.6945.300.5916.5646.5473.5339.5229.5137.4960.4825.4710.400.4813.4548.4384.4258.4157.4074.3915.3798.3701.500.4165.3914.3761.3646.3553.3477.3333.3229.3143.600.3778.3539.3395.3287.3200.3129.2996.2900.2821.800.3262.3062.2932.2834.2756.2693.2574.2488.2419 1.000.2959.2754.2633.2543.2471.2413.2304.2226.2163 1.500.2475.2295.2190.2112.2050.2000.1907.1841.1788 2.000.2232.2067.1970.1899.1842.1796.1711.1651.1604 3.000.2C19.1866.1778.1712.1661.1619.1541.1487.1443 4.000.1942.1794.170.1645.1595.1555.1480.1427.1386 5 00.1922.1776.1691.1628.1579.1539.1464.1412.1371 6.000.1922.1776.1691.1628.1579.1539.1464.1412.1371 8.000.1956.1807.1721.1658.1601.1,67.1491.1438.1396 10.000.2014.1862.1773.1708.1657.1615.1537.1483.1440 15.000.2176.2014.1919.1849.1794.1749.1666.1608.1561 20.000.2316.2145.2046.1972.1914.1866.1778.1716.1667 30.000.2542.2359.2251.2171.2108.2057.1961.1894.1840 (CONTINUED ON NEXT PAGE)

-209ISOTROPIC POINT SOURCE ON CRYSTAL AXIS (CONT'D) 8F8 WELL CRYSTAL 0 = 5.08000, H = 5.080000 WO = 2.857500, WH = 3.8100C0 ENERGY 8=1.50 2.00 2.50 3.00 3.50 3.81.OC1.9252.8925.8379.7466.6060.5000.015.9252.8925.8379.7466.6060.5000.020.9252.8925.8379.7466.6060.5COO.030.9252.8925.8379.7466.6060.5000.040.9252.8925.8379.7466.6060.500o.050.9252.8925.8379.7466.6060.5060.060.9252.8925.8379.7466.6060.5000.080.9209.8860.8288.7363.6001.500V.1CO.9162.8796.8207.7280.5955.4996.15C.8390.8C01.7423.6582.5462.4676.200.6804.6461.5986.5327.4477.3890.30C.4600.4355.4036.3610.3072.2703.400.3610.3415.3166.2837.2426.2144.500.3064.2897.2687.2411.2C66.1830.600.2750.2599.2411.2164.1857.1647.800.2357.2227.2066.1857.1596.1418 1.0C0.2107.1991.1847.1660.1429.1270 1.500.1741.1645.1526.1373.1183.10!3 2.000.1561.1474.1368.1231.1062.0946 3.000.1405.1327.1231.1109.0956.0853 4.000.1349.1273.1182.1064.0918.0819 5.000.1334.1260.1170.1053.0909.0810 6.C00.1334.1260.1170.1053.0909.0810 ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 2 x 2 IN. WELL CRYSTAL 0 = 5.080000, H = 5.080000 WO = 3.492500, WH = 4.127500 ENERGY 8=.00.10.20.30.40.50.010.9605.9587.9569.9549.9528.9505.015.9605.9587.9569.9549.9528.9505.020.9605.9587.9569.9549.9528.9505.030.9605.9587.9569.9549.9528.9505.040.9605.9587.9569.9549.9528.9505.050.9605.9587.9569.9549.9528.9505.060.9605.9587.9569.9549.9528.9505.080.9589.9570.9551.9530.9507.9483.100.9547.9525.9501.9476.9450.9423.150.8693.8590.8498.8413.8334.8261.200.7201.6982.6818.6678.6557.6451.300.5087.4802.4615.4467.4345.4241.400.4078.3804.3632.3498.3390.3299.500.3503.3246.3088.2967.2870.2788.600.3164.2922.2774.2662.2571.2496.800.2736.2514.2381.2281.2200.2134 1.000.2459.2253.2131.2039.1965.1904 1.500.2048.1869.1763.1684.1622.1570 2.000.1844.1679.1582.1510.1453.1406 3.000.1665.1513.1425.1359.1307.1264 4.000.1600.1453.1368.1304.1254.1213 5.000.1584.1438.1354.1291.1241.1200 6.000.1584.1438.1354.1291.1241.1200

-210ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 12AW(10)-W3 WELL CRYSTAL D = 7.620000, H = 6.350000 WD = 2.009140, WH = 5.080000 ENERGY B=.00.10 20.30.40.50.010.9905.9901.9897.9893.9889.9884.015.9905.9901.9897.9893.9889.9884.020.9905.9901.9897.9893.9889.9884.030.9905.9901.9897.9893.9889.9884.040.9905.9901.9897,9893.9889.9884.050.9905.9901.9897.9893.9889.9884.060.9905.9901.9897.9893.9889.9884.080.9905.9901.9897.9893.9889.9884.100.9905.9901.9897.9893.9888.9884.150.9792.9784.9778.9771.9764.9757.200.9298.9263.9241.9225.9213.9203.300.7804.7704.7652.7619.7597.7582.400.6732.6610.6549.6511.6487.6471.500.6016.5888.5825.5786.5761.5745.600.5559.5430.5367.5328.5304.5288.800.4941.4814.4753.4716.4693.4678 1.000.4519.4396.4338.4302.4279.4265 1.500.3861.3747.3693.3660.3639.3627 2.000.3518.3410.3360.3329.3310.3298 3.000.3211.3109.3061.3033.3015.3004 4.000.3098.2998.2952.2924.2906.2896 5.000.3069.2970.2924.2896.2879.2868 6.000.3069.2970.2924.2896.2879.2868 ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 12AW(10)-W4 WELL CRYSTAL D0 7.620000, H = 6.350000 WD = 2.857500, WH = 5.080000 ENERGY 8=.00.10.20.30.40.50.010.9813.9806.9799.9791.9782.9773.01.5.9813.9806.9799.9791.9782.9773.020.9813.9806.9799.9791.9782.9773.030.9813.9806.9799.9791.9782.9773.040.9813.980f.9799.9791.9782.9773.050.9813.980d.9799.9791.9782.9773.060.9813.9806.9799.9791.9782.9773.080.9813.9806.9799.9791.9782.9773.100.9801.9792.9784.9775.9766.9756.150.9671.9658.9645.9632.9619.9606.200.9088.9038.9001.8971.8944.8920.300.7459.7333.7255.7197.7152.7115.400.6360.6211.6124.6060.6011.5972.500.5646.5491.5403.5339.5291.5253.600.5196.5042.4955.4892.4845.4809.800.4597.4447.4363.4303.4259.4225 1.000.4191.4046.3966.3910.3867.3835 1.500.3564.3431.3358.3307.3269.3240 2.000.3241.3115.3047.2999.2963.2936 3.000.2952.2834.2770.2725.2692.2667 4.000.2846.2731.2668.2625.2593.2568 5.000.2819.2705.2643.2600.2568.2544 6.000.2819.2705.2643.2600.2568.2544

-211ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 3 x 3 IN. WELL CRYSTAL 0 = 7.620000, H = 7.620000 WD = 1.270000, WH = 3.810000 ENERGY B=.00.10.20.30.40.50.010.9932.9928.9924.9920.9915.9910.015.9932.9928.9924.9920.9915.9910.020.9932.9928.9924.9920.9915.9910.030.9932.9928.9924.9920.9915.9910.040.9932.9928.9924.9920.9915.9910.050.9932.9928.9924.9920.9915.9910.060.9932.9928.9924.9920.9915.9910.080.9932.9928.9924.9920.9915.9910.100.9932.9928.9924.9920.9915.9910.150.9900.9893.9886.9878.9870.9860.200.9725.9692.9671.9653.9636.9619.300.8721.8619.8562.8520.8486.8455.400.7778.7648.7577.7526.7486.7451.500.7081.6942.6867.6813.6771.6735.600.6614.6473.6397.6343.6300.6264.800.5959.5819.5743.5690.5648.5614 1.000.5498.5360.5287.5235.5195.5161 1.500.4756.4628.4559.4510.4473.4442 2.000.4362.4239.4173.4127.4092.4062 3.000.4002.3886.3824.3780.3747.3719 4.000.3868.3755.3694.3652.3619.3592 5.000.3834.3722.3661.3619.3587.3560 6.000.3834.3722.3661.3619.3587.3560 ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 12AW(12)-W1 WELL CRYSTAL D = 7.620000, H = 7.620000 WD = 2.009140, WH = 3.810000 ENERGY 8=.00.10.20.30.40.50.010.9835.9826.9817.9807.9796.9785.015.9835.9826.9817.9807.9796.9785.020.9835.9826.9817.9807.9796.9785.030.9835.9826.9817.9807.9796.9785.040.9835.9826.9817.9807.9796.9785.050.9835.9826.9817.9807.9796.9785.060.9835.9826.9817.9807.9796.9785.080.9835.9826.9817.9807.9796.9785.100.9819.9809.9799.9787.9775.9761.150.9770.9755.9739.9722.9705.9686.200.9528.9478.9441.9407.9376.9345.300.8418.8287.8202.8134.8077.8025.400.7453.7292.7191.7113.7047.6991.500.6758.6588.6484.6403.6336.6279.600.6298.6126.6022.5941.5875.5818.800.5658.5489.5387.5309.5244.5190 1.000.5211.5047.4947.4872.4810.4758 1.500.4496.4344.4252.4182.4125.4077 2.CO.4118.3973.3886.3820.3767.3722 3.000.3775.3638.3556.3494.3443.3401 4.000.3647.3513.3434.3373.3324.3283 5.000.3615.3482.3403.3342.3294.3253 6.000.3615.3482.3403.3342.3294.3253

ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 12AW(12)-W2 WELL CRYSTAL D = 7.620000, H = 7.620000 WD = 2.857500, WH = 3.810000 ENERGY B=.00.10.20.30.40.50.75 1.00 1.25.010.9682.9666.9649.9631.9612.9591.9530.9457.9366.015.9682.9666.9649.9631.9612.9591.9530.9457.9366.020.9682.9666.9649.9631.9612.9591.9530.9457.9366.030.9682.9666.9649.9631.9612.9591.953C.9457.9366.040.9682 o9666.9649.9631.9612.9591.9530.9457.9366.050.9682.9666.9649.9631.9612.9591.953C.9457.9366.060.9682.9666.9549.9631.9612.9591.9530.9457.9366 6.080.9682.9666.9649.9631.9612.9591.9530.9457.9366 H.100.9654.9637.9618.9597.9575.9552.9484.9401.9298.150.9572.9546.952J.9492.9463.9432.9346.9243.9118.200.9240.9172.9116.9064.9014.8966.8845.8717.8572. 300.8023.7864.77 52 7659.7577.7533.7339.7187.7034. 400.7049.6861.6733.6629.6540.6460.6290.6140.5996.500.6367.6171.6041.5936.5846.5767.560C.5456.5321.60).5920.5724.5595.5491.5403.5326.5163.5025.4896.800.5305. 5114.4989.4889.4804.4 731.4577.4449.4330 1. 00.4878.4692.4572.4476.4395.4325.4180.4058.3948 1.500.4200.4028.3918.3830.3757.3693.3562.3454.3357 2.000.3842.3630.3576.3494.3425.3365.3242.3142.3052 3.000.3519.3366.3268.3191.3126.3070.2956.2863.2780 4.000.3399.3249.3154.3379.3016.2962.2851.2760.2680 5.000.3363.3220.3126.3051.2988.2934.2824.2735.2655 6.000.3368.3220.3126.3051.2988.2935.2824.2735.2655 8.000.3421.3271.3176.3100.3037.2982.2871.2780.2699 10. 00C.3511.3358.3261.3184.3119.3064.2950.2857.2774 15.000.3758. 3598.3496.3414.3347.328 8.3167.3069.2981 20.000.3967.3801.3695.3610.3540.3479.3353.3250.3158 30.000.4297.4123.4011.3922.3847.3783.3649.3539.3440

-213ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 12AW(12)-W3 WELL CRYSTAL D = 7.620000, H = 7.620000 WD = 2.009140, WH = 5.080000 ENERGY B=.OU.10.20.30.40.50.010.9905.9901.9897.9893.9889.9884.015.9905.9901.9897.9893.9889.9884.020.9905.9901.9897.9893.9889.9884.030.9905.9901.9897.9893.9889.9884.040.9905.9901.9897.9893.9889.9884.050.9905.9901.9897.9893.9889.9884.060.9905.9901.9897.9893.9889.9884.080.9905.9901.9897.9893.9889.9884.100.9905.9901.9897.9893.9889.9884.150.9870.9863.9856.9849.9842.9835.200.9630.9592.9567.9547.9531.9516.300.8444.8330.8263.8214.8176.8146.400.7429.7287.7205.7147.7102.7067.500.6710.6558.6473.6412.6366.6329.600.6237.6084.5998.5937.5891.5855.800.5585.5434.5350.5290.5245.5210 1.0 OO.5132.4985.4904.4846.4802.4768 1.500.4414.4277.4202.4148.4108.4076 2.COO.4036.3906.3834.3784.3745.3716 3.000.3693.3571.3503.3456.3420.3392 4.000.3567.3447.3381.3335.3300.3272 5.C00.3535.3416.3350.3304.3269.3242 6.000.3535.3416.3350.3304.3269.3242 ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 12AW(12)-W4 WELL CRYSTAL D = 7.62C000, H = 7.620000 WD = 2.857500, WH = 5.080000 ENERGY B=.00.10.20.30.40.50.010.9813.9806.9799.9791.9782.9773.015.9813.9806.9799.9791.9782.9773.020.9813.9806.9799.9791.9782.9773.030.9813.9806.9799.9791.9782.9773.040.9813.9806.9799.9791.9782.9773.050.9813.9806.9799.9791.9782.9773.060.9813.9806.9799.9791.9782.9773.080.9813.9806.9799.9791.9782.9773.100.9801.9793.9784.9776.9766.9756.150.9750.9736.9723.9710.9697.9683.200.9420.9367.9327.9293.9262.9234.300.8099.7958.7866.7792.7731.7679.400.7057.6888.6780.6695.6626.6568.500.6339.6162.6051.5965.5896.5837.600.5875.5697.5587.5502.5433.5375.800.5241.5067.4960.4878.4811.4756 1.000.4804.4636.4533.4454.4390.4338 1.500.4118.3961.3867.3795.3737.3689 2.000.3758.3611.3521.3454.3399.3354 3.000.3435.3295.3212.3148.3097.3055 4.000.3315.3179.3098.3036.2986.2945 5.000.3285.3150.3069.3007.2958.2918 6.000.3285.3150.3069.3007.2958.2918

-214ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 12AW(14)-W3 WELL CRYSTAL D = 7.620000, H = 8.890000 WO = 2.009140, WH = 5.080000 ENERGY B=.00.10.20.30.40.50.010.9905.9921.9897.9893.9889.9884.015.9905.9901.9897.9893.9889.9884.020.9905.9901.9897.9893.9889.9884.030.9905.9901.9897.9893.9889.9884.040.9905.9901.9897.9893.9889.9884.050.9905.9901.9897.9893.9889.9884.060.9905.9901.9897.9893.9889.9884.080.9905.9901.9897.9893.9889.9884.100.9905.99J1.9897.9893.9889.9884 0150.9874.9866.9859.9853.9846.9838.200.9678.9642.9614.9594.9577.9561.300.8627.8509.8438.8385.8343.8308.400.7671.7522.7434.7369.7318.7211 500.6972.6813.6721.6652.6599.6555.600.6506.6345.6251.6182.6128.6086.800.5855.5696.56C3.5535.5482.5439 1.000.5398.5242.5152.5086.5034.4993 1.500.4665.4520. 4436.4374.4326.4287 2. 00.4276.4138.4C58.3999.3954.3917 3.000.3922.3791.3715.3660.3617.3582 4. 00.3790. 3662.3589.3535.3493.3459 5.0S0.3757.3630.3557.3503.3461.3427 6.000.3757.3630.3557.3503.3461.3427 ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 12AW(14)-W4 WELL CRYSTAL 0 = 7.62UCO, H = 8.890000 WO = 2.857500, WH = 5.C83C00 ENEKGY B=.0 ~4'..3;.54.010.9813.9826.9799.9791.9782.9713.015.9813.98.6.979)9.9791.9782.9713.020.9813.98.66.9799.9791.9782.9713.030.9813.98.6.9799.9791.9782.9773.040.9813.98 6.9799.9191.9782.9773.350.9813.98,6.9799.9191.9782.9773.060.9813.98.;6.9799.9791.9782.9773 080.9813.98Co.9799.9791.9782.9773.100.9801.9793.9784.9776.9766.9756.150.9753.974.9727.9714.970C.9687.2 0.9469.9415.9374.9339.9308.9279.300.8282.8138.82'41.7963.7898.7841.400.7299.7122.7.8.6917.6842.6778 500.6602.6417.6299.6205.6128.6C63.600.6144.5958.5839.5747.5670.5605.800.5511.5328.5213.5123.5048.4986 1.000.5070.4893.4781.4694.4623.4562 1.500.4369.42.4.4121.4021.3955.3900 2. 000.3999.3842.3 745.3669.3608.355S 3.000.3663.3515.3424.3353.3295.3240 4.300.3538.3395.33C5.3236.3179.3132 5.000.3507.3364.3275.3206.3150.3103 6.000.3507.3364.3275.3206.3150.3103

-215ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 2 x 2 IN. SOLID CESIUM IODIDE CRYSTAL D0 5.080000, = 5.080000 ENERGY Bs.00.20.50 1.00 2.00 5.00 10.00 20.00.010.5000.4608.4034.3168.1907.0542.0154.0040.015.5000.4608.4034.3168.1907.0542.0154.0040.020.5000.4608.4034.3168.1907.0542.0154.0040.030.5000.4608.4034.3168.1907.0542.0154.0040.04n.5000.4608.4034.3168.1907.C542.0154.0040.050.5000.4608.4034.3168.1907.0542.0154.0040.060.5000.4608.4034.3168.1907.0542.0154.0040.080.5000.4608.4034.3168.1907.0542.0154.0040.100.5000.4608.4034.3168.1907.0542.0154.0040.150.4999.4535.3878.2948.1717.0494.0145.0039.200.4954.4382.3660.2719.1562.0458.0138.0038.300.4531.3842.3114.2262.1293.0394.0125.o035.400.4047.3361.2694.1944.1114.0348.0113.0032.500.3682.3031.2417.1740.0999.0316.0104.0030.600.3433.2812.2237.1609.0925.0295.0098.0029.800.3090.2518.1997.1435.0827.0267.0089.0026 1.000.2847.2312.1832.1315.0759.0246.0081 -0024 1.500.2469.1997.1579.1133.0655.0215.OC73.uOZ0 2.000.2276.1838.1452.1041.0603.3198.0067.0020 3.000.2120.1709.1349.0967.0561.0185.0063.0019 4.000.2082.1678.1324.0950.0551.0182.0062.0019 5.000.2092.1687.1331.0955.0554.0183.0062.0019 6.000.2112 -1702.1344.0964 j.0559.0184.0063.0019 ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 8F8 CESIUM IODIDE WELL CRYSTAL 0 ) 5O08CC00, H 5.080000 WD r 2.857500, WH = 3.810000 ENERGY 8=.20.40.60 1.00 2.00 3.QO 3.81.010.9649.9612.9568.9457.8925.7466.5000.015.9649.9612.9568.9457.8925.7466.5000.020.9649.9612.9568 i9457.8925.7466.5000.030.9649.9612.9568.9457.8925.7466.5000.040.9649.9612.9568.9457.8925.7466.5000.050.9649.9612.9568.9457.8925.7466.5000.060.9649 i9612.9568.9457.8925.7466.5000.080.9649.9612.9568.9457.8882.7396.5000.100.9629.9588.9541.9421.8845.7342.5000.150.9442.9373.9304.9151.8507.7007.4901.200.8578.8421.8293.8077.7420.6106.4391.300.6482.6240.606&.5818.5274.4361.3233.400.5186.4944.4777.4550.4102.3402.2554.500.4445.4217.4062.3855.3466.2880.2175.600.4003.3788. 3642.3449.3097.2575.1952.800.3454.3257.3124.2952.2646.22203. 16i76 1.000.3097.2915.2793.2635.2359.1966.:1500 1.500.2585.2425.2320.2184.1953.1629,1247 2,000.2341.2193.2096 i 1971.11762.1470.,1127 3.00C0.2149.2012.1921.1806.1613.1347.1034 4.C00.2104.1969.;1880.1766.1578.1317.1011 54000.2117.1981.1892.1778.1588.13Z6.1018 6.000.2140,2003 i1912.1797.1605.1340.1029

-216ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 2 x 2 IN. SOLID CALCIUM IODIDE CRYSTAL D 5.080000, H - 5.080000 ENERGY B=.00.20.50 1.00 2.00 5.00 10.00 20.00.010.5000.4608.4034.3168.1907.0542.0154.0040.015.500C.4608.4034.3168.1907.0542.0154.0040.020.5000.4608.4034.3168.1907.0542.0154.0040.030.5000.4608.4034.3168.1907.0542.0154.0040.040.5000.4608.4034.3168.1907.0542.0154.0040.050.5000.4608.4034.3168.1907.0542.0154.0040.060.5000.4608.4034.3168.1907.0542.0154.0040.080.5000.4608.4034.3168.1907.0542.0154.0040.100.5000.4608.4034.3168.1907.0542.0154.0040.150.4993.4492.3808.2867.1660.0481.0143.0038.200.4878.4217.3512.2586.1481.0439.n135.0037.300.4296.359.2897.2096.1199.0371.U119.0034.400.3785.3122.2493.1796.1030.0325.0106.0031.500.3434.2813.2238.1609.0925.0295.0098.0029.600.3198.2610.2072.1489.0857.0276.0092.0027.800.2871.2332.1848.1326.0766.0248.0083.0025 1.000.2643.2141.1694.1216.0703.0229.0077.0023 i.500.2284.1844.1457.1045.0605.0199.06bb.0020 2.000.2093.1687.1332.0955.0554.0183.0062.0019 3.000.1925.1549.1222.0876.0509.0168.0058.0017 4.000.1870.1504.1186.0851.0494.0164.0056.0017 5.000.1861.1497.1181.0847.0492.0163.0056.0017 6.000.1867.1502.1185.0850.0493.0164.0056.0017 ISOTROPIC POINT SOURCE ON CRYSTAL AXIS 8F8 CALCIUM IODIDE WELL CRYSTAL D = 5.08CCCC, H = 5.080000 WD = 2.8575CC, WH = 3.810000 ENERGY B=.2C.40.60 l.CC 2.00 3.00 3.81.010.9649.9612.9568.9457.8925.7466.5C0O.015.9649.9612.9568.9457.8925.7466.5C00.020.9649.9612.9568.9457.8925.7466.5000.030.9649.9612.9568.9457.8925.7466.5COC.C40.9649.9612.9568.9457.8925.7466.5CCO.050.9649.9612.9568.9457.8925.7466.5000.060.9649.9612.9568.9457.8925.7466.5000.080.96334.9595.9548.9430.8866.7372.5COO.1CO.9619.9576.9527.9403.8812.7299.4998.150.19Y4.9094.9003.8825.8165.6718.4753.200.7961.7764.7613.7376.6742.5555.4C40.300.5782 55536.5364.5124.4631.3835.2863.400 40.4408.4249.4036.3632.3016.2274.500.4004.3788.3642.3450.3098.2576.1952.600.36,0i).3417.3280.3101.2782.2315.1759.800.3131.2947.2824.2665.2386.1988.1516 ifo00.2815.2645.2531.2385.2135.1780.1360 1.500.2351.2203.2105.1980.1769.1477.1132 200.2118.1982.1893.1779.1589.1327.;1018 3.000.I92<J.1794.1712.1608 *1435.1199.0922 4,C00,1856 -1734.1654.1553.1386.1158.0A9n 5.COO.1846.1725.1646.1545.1379.1152.0886 6.C00.1853.1732.1652.1551.1385.1157.0889

ISOTROPIC DISK SOURCE ON CRYSTAL AXIS 3 x 3 IN. SOLID CRYSTAL D = 7.62CCOO, H = 7.620000, G = 2.000000 ENERGY B=.00.20.50 1.00 1.50 2.00 3.00 5.00 10.00 30.00.010.5000.4705.4272.3598.3011.2519.1784.0969.0320.0040.015.5000.4705.4272.3598.3011.2519.1784.0969.0320.0040.020.5000.4705.4272.3598.3011.2519.1784.0969.0320.0040.030.50CO.4705.4272.3598.301.0.2518.1783.0968.0320.0040.040.5CCO.4705.4272.3598.3011.2519.1784.0969.0320.0040.050.5000.4705.4272.3598.3011.2519.1784.0969.0320.0040.060.5000.4705.4272.3597.3010.2517.1782.0968.0320.0040.080.5000.4705.4270.3588.2994.2498.1763.0957.0317.0040.100.5000.4704.4258.3557.2953.2456.1727.0936.0312.0039.150.4994.4660.4148.3394.2786.2302.1612.0877.0297.0039.200.4920.4502.3930.3163.2575.2119.1482.0811.0280.0038.300.4521.4014.3432.2719.2198.1804.1264.0701.0251.0035.400.4130.3621.3072.2420.1953.1603.1126.0630.0230.0034.500.3827.3332.2817.2214.1785.1466.1032.0581.0215.0032.600.3616.3137.2647.2078.1675.1376.0970.0548.0204.0031.800.3310.2859.2407.1887.1521.1250.0883.0501.0188.0029 1.COO.3087.2660.2236.1752.1412.1161.0821.0467.0176.0027 1.500.2718.2333.1958.1532.1235.1016.0720.0412.0157.0024 2.CCO.2515.2155.1807.1414.1140.0938.0665.0381.0146.0023 3.000.2329.1993.1670.1306.1053.0867.0616.0353.0136.0021 4.000.2255.1929.1616.1264.1019.0839.0596.0342.0132.0021 5.C00.2237.1913.1603.1253.1011.0832.0591.0340.0131.0020 6.COO.2237.1913.1603.1253.1011.0832.0591.0340.0131.0020 8.CO0.2269.1941.1626.1271.1025.0844.0600.0344.0132.0021 10.CO0.2322.1987.1665.1302.1050.0865.0614.0352.0135.0021 15.C00.2466.2112.1771.1385.1117.0920.0652.0374.0143.0022 20.C00O.2586.2217.1860.1455.1173.0966.0685.0392.0150.0023

ISOTROPIC DISK SOURCE ON CRYSTAL AXIS ISOTROPIC DISK SOURCE ON CRYSTAL AXIS 2 x 2 IN. SOLID CRYSTAL 2 x 2 IN. SOLID CRYSTAL D = 5.080000H = 5.080000 G = 2.000000 D = 5.080000 H = 5.080000 G = 1.000000 ENERGY B=.20 1.00 3.00 10.00 20.00 ENERGY R=.20 1.CO 3.00 10.00 20.00.01C.4457.2767.1037.0150.0040.C10.4583.3C84.1146.0153.0040.015.4457.2767.1037.0150.0040.015.4583.3084.1146.0153.0040.02C.4457.2767.1037.0150.0040.C2C.4583.3084.1146.0153.CC40.03C.4455.2758.1034.0150.0040.30.4583.3082.1142.0153.04.040.4457.2766.1037.0150.C040.040.4583.3084.1146.0153.0040.050.4456.2763.1036.0150.040.050.4583.3C83.1145.0153.0040.060.4454.2754.1032.0149.0040.060.4583.3080.1140.0153.040 D.080.4437.2716.1012.0148.0039.080.4582.3048.1115.0151.C040.1 0.4396.2653.0985.0145.0039.100.4575.2983.1083.0148.0039.150.4156.2423.0901.0138.0038.15.4446.274.0984.0141.0038.200.3779.2160.0809.0130.0036.20C.4123.2440.0878.0132.CC37.300.3C80.1740.0664.0113.0033.300.3400.1950.0713.0115.0033.400.2650.1493.0576.0101.0030.400.2S29.1666.0615.0103.0030.500.2370.1335.0518.0092.0027.5C0.2619.1484.0552.0094.0027.600.2193.1235.0481.0086.0026.6C0.2422.1371.0512.0088.0026.800.1954.1100.0430.0078.0023.800.2157.1218.0457.0079.0024 1.0CC.1791.1008.0396.0072.0022 1.CCO.1976.1115.0420.0073.0022 1.500.1536.C865.0341.0063.0019 1.5CC.1692. 954.0361.OC64.0019 2.000.1402.0790.0312.0058.0018 2.C00.1545.0870.0330.C0059.0018 3.000.1284.0723.0286.0053.CC16 3.CCO.1413.0796.C302.0C54.0016 4.CCO.1238.0697.0276.0051.0016 4.CG00.1362.0767.0292.0052.0016 5.000.1227.0691.0274.0051.0015 5.COC.1350.0760.0289.0052.0016 6.000.1227.0691.0274.0051.0015 6.CC.1350.0760.0289.052.0016

-219ISOTROPIC DISK SOURCE ON CRYSTAL AXIS 7F8 WELL CRYSTAL D = 4.445000 H = 5.080000 G = 1.000000 WD = 1.905000 WH = 3.810000 ENERGY R=.20.50.80 1.50 2.00 2.52 3.C; 3.81.,l:.9843.9817.9783.9662.95C6.9227.8676.50C^.(175.99943.9817.9783.9662.9506.9227.8676.5020.'2?.9843.9811.9783.9662.9506.9227.8675.5C'0.3'.9843.9817.9783.9660.9502.9217.8649.5rC, *~ 4r ~9843.9817.9783.9662.9505.9226.8673.5COO.,5 4.39843.9817.9783.9661.9504.9223.8662.52'0.J.9843.9816.9783.9659.9499.9212.8640.5030.'8.9840.9812.9777.9647.9479.9175.858C. 50C 1'1:.9831.98l1.9763.9623.9442.9117.8487.4999.15-' 9446.9385.9332.9161.8942.8557.7847.4820 w.C.:8213.8099.8331.7857.7639.7266.6615.42?2.3G.5985.5845.5776.5632.5463.5179.4703.3120.4-^.4836.47'J3.464-.4520.4381.4151.3771.2539.50'.4169.4045.3987.3882.3761.3563.3238.219P.6~3'.V.3772.3655.3602.35,5.3396.3217.2924.1994.3L. 3268.3161.3113.3029.2934.2779.2527.1733 i1.C'.2940.2842.2797.2721.2636.2496.2270.1563 1.53^.249?.2367.2329.2264.2193.2077.1890.1308 2.,:'.2209.2130.2J95.2037.1973.1869.17j1.118C 3.^]'.1998.1926.1894.1841.1783.1689.1531.1C69 4.C90j.1 C18.1848.1f18.176*.1711.1621.1475.1026 5. )UC.1899.1829.1799.1748.1693.16)4.1460.1. 16 5.CC-'.1899.1829.1799.1748.1693.16)4.146C.1716 8.2OS. <133.1362.1831.1780.1724.1633.1486.1r34 1^.:C:.1991.1919.1887.1834.1776.1682.1531.1065 ISOTROPIC DISK SOURCE ON CRYSTAL AXIS 8F8 WELL CRYSTAL D = 5.080000 H = 5.080000 G = 1.000000 WD = 2.857500 WH = 3.810000 ENERGY B=.20.80 1.50 2.00 2.50 3.00 3.80.010.9665.9544.9310.9027.8560.7749.5044.015.9665.9544.9310.9027.8560.7749.5044.020.9665.9544.9310.90Q27.8559.7747.5044.030.9664.9542.9305.9016.8539.7712.5043.040.9665.9544.9310.9026.8558.7744.5044.050.9664.9544.9308.9022.8550.7728.5043.060.9663.9541.9302.9012.8532.7702.5043.080.9655.9528.9278.8976.8482.7639.5041.100.9635.9499.9236.8919.8405.7547.5035.150.9112.8854.8535.8190.7661.6844.4737.200.7755.7340.7003.6691.6241.5580.3976.300.5537.5075.4790.4563.4253.3815.2793.400.4441.4017.3775.3593.3349.3009.2225.500.3813.3425.3211.3055.2848.2561.1904.600.3443.3081.2885.2744.2558.2302.1716.800.2975.2649.2477.2355.2196.1977.1479 1.000.2673.2372.2216.2107.1965.1770.1327 1.500.2224.1965.1834.1743.1625.1465.1102 2.000.2001.1764.1645.1563.1458.1315.0991 3.000.1809.1592.1483.1409.1315.1186.0895 4.000.1736.1526.1422.1351.1261.1137.0858 5.000.1718.1510.1407.1337.1247.1125.0849 6.000.1718.1511.1407.1337.1247.1125.0849 8.000.1749.1538.1433.1362.1270.1146.0865 10.000.1802.1586.1478.1404.1310.1181.0891 15.000.1950.1718.1602.1522.1420.1280.0965 20.000.2078.1833.1710.1625.1516.1366.1029 30.000.2286.2021.1886.1793.1672.1507.1133

ISOTROPIC DISK SOURCE ON CRYSTAL AXIS 8F8 WELL CRYSTAL D = 5.080000, H = 5.080000 B=.50 B =.50 WD = 2.857500, WH = 3.810000 ENERGY G =.00.10.15.25.50.75 1.00 1.4.010.9591.9591.9591.9592.9596.902.9611.9632.015.9591.9591.9591.9592.9596.9602.9611.9632.020.9591.9591.9591.9592.9596.9602.9611.9632.030.9590.9590.9590.9591.9595.9601.9610.9630.040.9591.9591.9591.9592.9596.9602.9611.9632.050.9590.9591.9591.9592.9596.9602.9611.9631.060.9589.9589.9589.9590.9594.9600.9609.9629.080.9575.9576.9576.9577.9581.9588.9598.9620.100.9547.9547.9548.9549.9554.9563.9574.9600.150.8900.8901.8902.8905.8917.8940.8973.9061.200.7398.7399.740n.7404.7422.7455.7507.7668.300.5137.5138.5139.5143.5161.5194.524R.5429.400.4074.4075.4076.4079.4095.4124.4172.4338.500.3477.3478.3479.3482.3496.3522.3565.371.600.3129.3130.3131.3133.3147.3171.3211.3353.800.2692.2693.2694.2696.2708.2729.2765.2893 1.000.2412.2413.2414.2416.2427.2446.2479.2597 1.500.20C0.2001.2001.2003.2012.2029.2057.2158 2.000.1796.1797.1797.1799.1807.1823.1848.1940 3.000.1621.1621.1622.1623.1631.1645.1668.1753 4*000.1555.1555.1555.1557.1564.1578.1600.1682 5.000.1538.1539.1539.1541.1548.1561.1584.1664 6.000.1538.1539.1539.1541.1548.1561.1584.1664 8.000.1567.1567.1567.1569.1576.1590.1612.1695 10.000.1615.1615.1616.1617.1625.1639.1662.1746 15.000.1749.1750.1750.1752.176J.1775.1800.1893 20.000.1866.1867.1867.1869.1878.1893.1920.2015 30.000.2057.2057.2056.2060.2069.2386.2115.2218

ISOTROPIC DISK SOURCE ON CRYSTAL AXIS 12AW(12)-W2 WELL CRYSTAL 0 = 7.620030, H = 7.620000, B =.50 WD= 2.857500, WH = 3.810000 ENERGY G =.00.15.25.50.75 1.00l.010.9591.9591.959?.9596.9602.961 962.015.9591.9591.9592.9596.9602.9611.9632.020 *9591.9591.9592.9596.9602.9611 9632.030.9591.9591.9592.9596.9602.9611.9631:^:9^:^;:.9:^9t:9959; *9 *s^*9^;'9591t~~~~~9592.9596 9602.':.040.9591.9591.9592.9596.9602-.9611.962.050.9591.9591.9592.9596.9602.9611.9632.060.9591.9591.9592.9596.9602.9611.9631.080.9585.9586.9586.9590.9596.96V5.9625.100.9564.9564.9565.9570.9577.9588 9611.150.9435.9436.9437.9445.9458.9476.9517.200.8967.8968.8971.8983.9005.9037.9-I8.300.7504.7505.7539.7526.7555.7601 7733.400.6468.6462.6466.6483.6513.6559.6700.500.5768.5769.5772.5788.5817.5862.6001.600.5326.5327.5331.5346.5314.5417 5552.800.4731.4732.4735.4750.4715.4815 944 1.00.432694.4327.4330.4343.4367.4405.4527 1.500.3694.3694.3697.3709.3730.3764.3874 2.000.3306.3366.3368.3380.3399.3431.3533 3.000.3075.3075.3078.3088.3106.3135.3231 4.000.2962.2963...2965.2975.2993.3021.31 5.000.2935.2936.2938.2947.2965.2993 3085 6.000.2935.2936.2936.2947.2965.2993.3085 8.000.2983.2984.2986.2996.3013.3042 3135 10.000.3064.3065.3067.3077.3095.3124 3220 15.000.3283.3289.3291.3302 3322.3352.3453

ISOTROPIC DISK SOURCE ON CRYSTAL AXIS 5 x 3 IN. SOLID CRYSTAL ISOTROPIC CYLINDRICAL VOLUME SOURCE ON CRYSTAL AXIS D = 7.62 B =.20 2 x 2 IN. SOLID CRYSTAL H = 7.62 G = 1.0 D = 5.080000, H = 5.080000 ENERGY WD=2.009 WD=2.8575 WD=2.009 WD=2.8575 WH=3.81 WH=5.81 WH=5.0 8.010.9826.9665.9900.9804 ENERGY B2=1.30 3.30 6.30 10.30 15.30.015.9826.9665.9900.9804.020.9826.9665.9900.9804.030.9826.9665.9900.9804.010.3400.2218.1408.0928.0630.040.9826.9665.9900.9804.015.3400.2218.1408.0928.0630.050.982 6.9665.99u0.9804.020.3400.2218.1408.0928.0630.060.9826.9664.9900.9804.030.3394.2210.1402.0923.0626.080.9825.9661.9900.9803.040.3400.2218.1408.0927.0630.100.9820.9647.9899.9799.050.3398.2215.1405.0926.0628 r.150.9769.9554.9870.9740.060.3391.2206.1399.0921 0625.200.9504.9167.9605.9357.080.3358.2174.1377.0906.0614.300.8323.7820.8357.7917.100.3302.2128.1346.0885.0599.400.7327.6801.7320.6836 50 307 958.12376836.150.30710812.1958.1237.0812.0549.500.6621.6106.6594.6108.200.2759.1749.1106.0727.0493.600.6158.5658.612.5642.300.5622242.30 0.2220 1408.0895.0591.0403.800.5518.5047.5470.5013.400.1900.1207.0770.0510.0348 1.000.5073.4627.5020.4584.500.1695.1078.0689.045 7 313 1.500.4367.3967.4310.3914.600.1566.0997.0638.0424.0290 2.000.3994.3622.3937.3565.800.1393.0888.1393.080569.0379.0260 3.000.3661.3314.3605.3257 1.000. 0813.0522.0347.0238 4.000.3532.3195.3476.3138 1.500.1091.0697.0448.0299 0205 5.000.3500.3166.3444.3109 2.000.0996.0636. 0273.0273 0188 6.000.3500.3166.3444.3109 3.000.0911.0582.0375.0250.0172 8.000.3556.3217.3499.3160 4.00.0878.0562.0362.0241.0166 10.000.3649.3303.3592.3246 5.000.0870.0556.0358.0239.0165 15.000.3906.3540.3849.3484 6.000.0870.0556.0358.0239.0165 20.000.4124.3741.4066.3686 30.000.4467.4061.4411.4008

ISOTROPIC CYLINDRICAL VOLUME SOURCE ON CRYSTAL AXIS ISCTRGPIC CYLINDRICAL VOLUME SOURCE ON CRYSTAL AXIS 2 x 2 IN. SOLID CRYSTAL 3 x 3 IN. SOLID CRYSTAL D =' 7.62GCG0, H = 7?62CO00 0 = 5.080000, H = 5.080000 7.62CC, H= 7.62 G6.CCCC 8 = 1.CCCCC, 1 = 1.CC-O G = 2.540000, B1.300000 ENERGY B2=1.30 3.30 6.30 10.30 15.30 ENERGY 821.1C 1.2C 1.50.clo.3641.3582.3410.C15.3641.3582.341C.010.2801.1824.1173.0778.0533 C20.3641.3582.341.015.2800.1824.1173.0778.0533.C3C.3641.3582.3410.020.2798.1823.1172.0778.0533.C4C.3641.3582.3410 030.2785.1813.1166.0774.0530 C50.641.3582.3410.040.2797.1823.1172.0778.0533.060.3641.3581.3409.050.2791.1818.1170.0776.0531.C80.363.3570 3394.060.2780.1810.1164.0772.0529.100.3596.3534.3356.080.2743.1783.1146.0760.0521.150.3449.3386.3204.100.2686.1743.1121 0744 0509.0.3233.3169.2990.150.2477.1604.1033.0687.0471,30C 2786 2728 2565.200.2227.1443.0932.0622.0428.4C0.2479.2426.2280.300.1815.1180.0767.0514.0356.500.2266.2218.2083.400.1567.1021.0666.0448.0310.600.2126.2C80 1953.500.1405.0917.0599.0403.0280,800.1929.1888.1772.600.1302.0851.0556.0375.0261 1.CO0.1790.1752.1644.800.1163.0760.0498.0336.0234 1.5C.1565.1531 1437 1.000.1068.0699.0458.0309.0215 2.COO.1443.1412.1325 1.500.0918.0601.0394.0267.0186 3.CCO.1333.1304.1224 2.000.0839.0550.0361.0244.0170 4.C.1289.1261.1184 3.000.0769.0504.0331.0224.0156 5.CCC.1279.1251.1174 4.000.0742.0486.0320.0216.0151 6.CCO.1279.1251.1174 5.000.0735.0482.0317.0214.0150 8.CCO.1297.1269.119 6.000.0735.0482.0317.0214. 0150 10C0.132 130 12 15.C00.1414.1383.1299 20.CO. 1486.1453.1364 30.CCO.1597.1563.1467

ISOTROPIC CYLINDRICAL VOLUME SOURCE ON CRYSTAL AXIS ISOTROPIC CYLINDRICAL VOLUME SOURCE ON CRYSTAL AXIS 3 x 3 IN. SOLID CRYSTAL 3 x 3 IN. SOLID CRYSTAL D = 7.620000, H = 7.620000 D = 7.620000, H = 7.620000 G = 1.9050009Bl a.300000 G = 3.810000, 81.300000 ENERGY 82=1.30 3.30 6.30 10.30 15.30 ENERGY 82=1.30 3.30 6.30 10.30 15.30.010.3883.2850.1962.1362.0972.010.3272.2347.1626.1137.0814.015.3883.2850.1962.1362.0972.015.3272.2347 1626.1137.0814.020.3883.2850.1962.1362.0972.020.3271.2347.1626.1137.0814 rO.030.3882.2847.1960.1360.0970.030.3261.2340.1622.1134.0812.040.3883.2850.1962.1362.0972.040.3270.2346.1626.1137.0814.050.3883.2849.1962.1362.0971.050.3266.2344.1624.1136.0813.060.3881.2846.1958.1359.0969.060.3258.2338.1620.1133.0811.080.3870.2826.1941.1346.0959.080.3234.2317.1604.1122.0803.100.3843.2791.1913.1326.0943.100.3192.2283.1580.1105.0790.150.3705.2655.1812.1253.0890.150.3032.2160.1494.1045.0748.200.3489.2472.1682.1163.0825.200.2826.2008.1390.0973.0698 300.3028.2124.1445.1001.0711.300.2450.1739.1206.0848.0610.400.2704.1892.1289.0895.0637.400 2198 1561.1085.0765.0551.500.2477.1732.1182.0821.0586.500.2022.1436.1000.0706.0509.600.2326.1626.1111.0773.0551.600.1904.1353.0943.0666.0481.800.2114.1478.1011.0704.0503.800.1738.1236.0862.0610.0441 1.000.1964.1373 0940.0655.0468 1.000.1619.1152.0804.0569.0412 1.500.1718.1202.0824.0575.0412 1.500.1424.1013.0708.0502.0364 2.000.1586.1109.0761.0532.0381 2.000.1317.0938.0656.0465.0337 3.000.1465.1025.0704.0492.0353 3.000.1220.0869.0608.0431.0313 4.000.1418.0992 0681.0476.0341 4.000.1181.0841 0589.0418.0303 5.000.1406.0984.0676.0472.0339 5.000.1172.0835.0584.0415.0301 6.000.1406.0984.0676.0472.0339 6.000.1172.0835.0584.0415.0301

ISOTROPIC CYLINDRICAL VOLUME SOURCE ON CRYSTAL AXIS ISOTROPIC CYLINDRICAL VOLUME SOURCE ON CRYSTAL AXIS ISOTROPIC CYLINDRICAL VOLUME SOURCE ON CRYSTAL AXIS 7F8 WELL CRYSTAL 8F8 WELL CRYSTAL 12AW(12)-W2 WELL CRYSTAL o = 4.445CCC, H = 5.8008C0 H = 5.080000 0 = 5.080000, D = 7.6200CO, H = 7.620000 G =.952500, 61 =.3CCOCO Bl =.300000 G = 1.428750, G = 1.428750, Bk =.3CCOCO Wo = 1.9050C', WH = 3.810000 WH = 3.810000 WO = 2.857500, WC = 2.8575CC, WH = 3.81000C ENERGY 82=1.3C 2.30 3.81 ENERGY B2=1.30 2.30 3.80 ENERGY 82=1.3C 2.30 3.81.010.9777.9667.9054.010.9565.9390.8688.010.9565.9390.8678.C15.9777.9667.9654.015.9565.9390.8687.015.9565.9390.8678.020.9777.9667.905-.020.9565.9390.8684.020.9565.9390.8676.030.9776.9664.9033.030.9561.9383.8668.030.9563.9386.8661.040.9777.9667.9049.040.9565.9389.8683.040.9565.9390.8675.050.9777.9666.9040.050.9563.9386.8675.050.9564.9388.8668.060.9775.9662.9028.060.9559.9380.8663.060.9562.9384.8657.080.9768.9649.8991.080.9547.9363.8628.080.9551.9366.8622.100.9753.9626.8932.100.9524.9331.8572.100.9532.9338.8569.150.9314.9157.8401.150.8961.8735.7956.150.9424.9193.8358.200.8C0k.7339.7154.200.7547.7315.6632.200.8999.8727.7885.300.5741.5609.5109.300.5314.5121.4632.300.7586.7298.6564.400.4608.4498.4.97.400.4236.4074.3683.400.6551.6279.5643.500.3958.3861.3518.500.3626.3483.3148.500.5857.5603.5633.600.3575.3486.3177.600.3268.3138.2836.600.5413.5172.4646.800.0C89.3011.2745.800.2817.2703.2443.800.4813.4592.4124 1.000.2775.2705.2460 1.000.2528.2424.2191 1.000.4403.4198.3770 1.500.2309.225:.2052 1.500.2099.2011.1818 1.5C0.3763.3583.3217 2.C00.2C78.2024.1846 2.000.1887.1807.1634 2.OOC.3430.3263.2931 3.000.1878.1829.1669 3.000.1704.1632.1475 3.000.3135.2981.2677 4.C00.1802.1755.1601 4.000.1635.1565.1415 5.CCO.1783.1737.1585 5.000.1618.1549.1400 6.000.1783.1737.1585 6.000.1618.1549.1400

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