THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING OME ASPECTS OF THERMAL NEUTRON DETECTORS ~~Geo~rge R~. Dalton A dissertation submitted. in partial fulfillment of the requirements for the d~egree of Doctor of Philosophy 'in the University-of Michigan 1960 May 196.~~~~~P 5

Nr, Doctoral Committee: Professor Richard K. Osborn, Chairman Assistant Professor Bernard A. Galler Professor Henry J. Gomberg Professor William Kerr Associate Professor Paul F. Zweifel

ACKNOWLEDGMENTS The writer wishes to express his sincere gratitude to the following persons: To Professors Richard K. Osborn, Chairman and Bernard A Galler for their adviceand interest in this thesis. To the staff of the Computer Center of the University of Michigan. To my wife for her assistance and patience in preparing this and other manuscripts. To the Industry Program of the College of Engineering at the University of Michigan for considerable assistance in the reproduction of this dissertation.

TABLE OF CONTENTS Page SLIST OF LES.4.....O............ o.. o a a aa 00a.........a... NOST NCFITURS a**. e * * o 0 00 aea o a* 0 0 o ea o G a a e 0 0 0 O a 0 * a o * a ~~~E..INTRODUCTION..e....O..,..*.~ 1...... I FORMULATION OF THE PROBLEM * a O a. * * a. a **. a O * 5 II UNIFORM ISOTROPIC INITIAL FLUX 1...........,....1.2 III NON-ISOTROPIC SCATTER AND NON-ISOTROPIC INITIAL FLUX (r, ) 19 IVRESULTS OF THE CALCULATION OF Ga(R) *.**..............24 V FLUX CALCULATIONS..... * 0 e *...... * O O O* O * VII COPARISON WITH EXPERIMENTS.3..6...-.............. 1a. Evaluation of G(r, gQ -r) a. a..a. la.,*.,..... a..,a..,a.l.a. 5, Spherical Be-ssel Functions. a a aaaa l Oa.................Is.- 92 4.I~ Spherical Harmonics a.. a. a. a a a a... 94 5. Numerical Integration......,*........ 97 APPENDIX II a a., a.0e 0 40 0 0 0 0@ 0a0 0 0 04;a0 00 0 0 0.0a0000000a 0. 0 0 101 APPENDIX III..o a...0,..4.ao0,.0.o0 0 a..* O04 0 0 aa0 a**0 * 0 a a.,0 a00 a a0.. a0 110 APPENDIX IV a.t.. a... a00 000a0a0 0a0 000a00a0a0000 a0 0 a 00 a a00a0 0 00aa0000000 117aa APPENDIX V..o..0000 00I a 0 000a000000 0 0 600aa0a0a04,0a 0 a0 a 00 0004 0 O0 0 0 a0 0 0 0 a 121 BPPIBLIOXAH VI000O0 * 00 00 0 000 00~.. 156

TABLE Table Pg ~I The Effect of Higher Order Correction8 on the Average Scalar Flux Within the Detector oo.6oo

LIST OF FIGURES Figure Page 1 The Function R2 Go (R) for Several Values of 6., _ 2 Zeroth Green's Function Coefficient for Water...8 3 Higher Order Green's Function Coefficient for Water29 4 Zeroth Green's Function Coefficient for Water for Two 0 Absorption Cross-Sections and Two Values of..... 5 Zeroth Green's Function Coefficient for Graphite.,1 6 First Order Green's Function Coefficient for Graphite 2 7 A Comparison of the Average Normalized Scalar Flux in a Coin Shaped Gold Detector of 0.5 cm. Radius in Water for the Integral Method and Skyrme's Method.... -............ 4 8 A Comparison of the Average Normalized Scalar Fluxin a Coin Shaped Gold Detector of 1.0 cm. Radius in Water for the Integral Method and Skyrme's Method....... 41 9 A Comparison of the Average Normalized Scalar Flux in a Coin Shaped Gold Detector of 1.5 cm. Radius in Water for the Integral Method and Skyrme Is Method...a0 *0..... W.. 6042 10 The Average Normalized Scalar Flux in a Coin Shaped Gold Detector in Water as Calculated by the Integral Method....4 11 The Average Normalized Scalar Flux in a Coin Shaped Indiu Detector in Water as Calculated by -the Integral Method...44 12 A Comparison of the Average Normalized Scalar Flux in a Coin Shaped Gold Detec tor in Graphite as Calculated by the Integral Method and by Skyrme's Method....45 15 A Comparison of the Average Normalized Scalar Flux in a Coin Shaped Indium Detector in Graphite as Calculated by the Integral Method and by Skyrme Is Method. a,....... 0 a46 14 A Comparison of the Average Normalized Scalar Flux in a Coin.Shaped Gold Detector in Water as Calculat~ed by the Integral Method and as Measured by Zobel. aaI a1 0.0. Da0.. 48

LIST OF FIGURES (CON 'T) Figure Page 15 A Comparison of the Average Normalized Scalar Flux in a Coin Shaped Indium Detector in Water as Calculated by the Integral Method and as Measured by Fitch and Drummond...... 49 16 A Comparison of the Average Normalized Scalar Flux in a Coin Shaped Indium Detector in Graphite as Calculated by the Integral Method and as Measured by Ritchie and Klema and by Gallagher...... 50 0 0. * a. * e * * e * * * 6 50 17 A Comparison of the Average Normalized Scalar Flux in a Wire Shaped Indium Detector in Water as Calculated by the Integral Method and as Measured by Fitch and Drummond *......1 18 A Comparison of the Average Normalized Scalar Flux in a Coin Shaped Gold Detector in Water as Calculated by the Integral Method and as Measured by Fitch and Drummond.. ~e 52 19 A Comparison of the Average Normalized Scalar Flux in a Coin Shaped Gold Detector in Graphite as Calculated by the Integral Method and as Measured by Ritchie and Klema 5 20 A Comparison of the Average Normalized Scalar Flux in a Coin Shaped Indium Detector in Graphite as Calculated by the Integral Method and as Measured by Thompson......,4 21 A Comparison of the Average Normalized Scalar Flux in a Coin.Shaped Indiu Detector in Graphite as Calculated by the Integral Method and as Measured by Gallagher.......... ooo55 22 A Map of the Normalized Scalar Flux Within a Coin Shaped Gold Detector in Water, Radius of 1.5 cm. and Thickness of 5 mils.......9 ' 0.457? 25 A Map of the Normalized Scalar Flux Within a Coin Shaped Gold Detector in Water, Radius 1.0 cm. and Thickness of~ 5 24 A Map of the Normalized Scalar Flux Within a Coin Shaped Gold Detector in Water, Radius of 0.5 cm, and Thickness of 5 nmils. *,......,....00,. 0. 0. 0..o 00.a.....a0 58 25 A Plot of the Normalized Scalar Fluxes Along the Central Axis Outside of a Set of Gold Coin Shaped Detectors in Graph-ite Thickness_ ofP 5Z mil............. 59

LIST OF FIGURES (CON'T) Figure Page 26 A Plot of the Average Normalized Scalar Flux Minus the Minimum Normalized Scalar Flux in a Coin Shaped Indium Detector in Water o................... oo oo a o o o oo e o 27 A Plot of the Average Normalized Scalar Flux Minus the Minimum Normalized Scalar Flux in a Coin Shaped Gold Detector in Water............................... 61 A Plot of the Average Normalized Scalar Flux Minus the Minumum Normalized Scalar Flux in a Coin Shaped Gold Detector in Graphite.. o.......... 6 o. o o a o o o 29 A Plot of the Normalized Scalar Fluxes Along the Central Axis Outside of a Set of Gold Coin Shaped Detectors in Water, Thickness of 5 mils o...........................e 6 30 A Map of the Normalized Scalar Flux Within a Coin ShapEd Gold Detector in Graphite Radius of loO cmo and Thickness 7 mils...... o o o o o a o o o a a 0 o o a o 0 a a o a o a o o a a o a o a a a o o 31 A Map of the Normalized Scalar Flux Within a Coin Shaped Gold Detector in Graphite, Radius of 1.0 cm. and Thickness of 7 mils.. e.o..,e.. o o o.o o o..o o o o o o oo. ao -o a o eo o o o o o e oo e64 52 A Map of the Normalized Scalar Flux Within a Coin Shaped Gold Detector in Graphite, Radius of 0.5 cm. and Thickness 35 A Map of the Normalized Scalar-Flux Within a Wire Shaped Indium Detector in Water., Radius -of 10 mils and Length of 2.54 cm.,,..0 0 80 0 0 0 0 0 0 IDD 0*,.a,,. 0 0 a aaa 0 0 88D 888*..,a0 0 66 54 A Map of the Normalize d Scalar Flux Within a Wire Shaped Indium Detect~or in Water, Radius of.20 mils and Length of 55 A Map of the Normalized Scalar Flux Within a Wire Shaped Gold Detector in Water, Radius of 5 mils, and Length of 1,27 36 A Map of the Normalized Scalar Flux Within a Wire Shaped Gold Detector in Water., Radius of 10 mils and Length of 1 7 mo- 0000 000 000 000a 0000 0a00 vii* aa 6

LIST OF FIGURES (CON'T) FiguePage 37 A Map of the Normalized Scalar Flux Within a Wire Shaped Gold Detector in Water, Radius of 20 mils and Length of 127 cm. * 0.o O ao..o0.......e..0aa... e 9........... 7Q 58 A Map of the Normalized Scalar Flux Within a Wire Shaped Gold Detector in Water, Radius of 5 mils and Length of 1.27,e eoo e o o- e o e oee o o o o eD o o e'o o o oe e*o e e oe e o * o oo e oeoo e oe o o o o e o - 39 A Plot of the Normalized Scalar Fluxes Along a Radius in the Mid-Plane of a Set of Indium Wire Shaped Detectors in Water, Length of 0.5 Inches o............................ * 72 4o A Plot of the Average Normalized Scalar Flux Minus the Minimum Normalized Scalar Flux in a Wire Shaped Indiu Detector in Water 7o5o............. 41 A Plot of the Normalized Scalar Fluxes Along a Radius in the Mid-Plane of a Set of Gold Wire Shaped Detectors in Water Length of 0.5 Inches. Q............oo........... 06.....6 0 74 42 A Plot of the Average Normalized Scalar Flux Minus the Minimum Normalized Scalar Flux in a Wire Shaped Gold Detector in Water.7....,.,.,.oa............,.... 7..5.. 453 A Graph of the FunctioncZ(x).*...*........,85 44 A Graph of the Functionc(x) *~.....,..,,..,.86 45 A Graph of theFunctionc'(x.)......,..,....,.87 46 A Graph of the Function (x) a..,.,..,,...,.,,.. 88 47 A Graph of the Functionlz~:-(x)......a.aa.a.......aa.a.....a..... 0 89 48 A Graph of the FunctioncN~.(x) go,....,,.,...,..9 49 Location of the Coordinate Ax.ies..............,...117 50 Location of the Grid Points.,,,,,.,,,...,118 51 Point-on the Same Z Axis Outside of Volume Element.......... 122 52 Point in the Same Z Outside of a Volume.Element..,.,........ 125

NOMENCLATURE ~~~~r ~A vector indicating a position in space. rA~~~ ~A vector of unit length with the direction of r. A unit vector indicating the direction of motion of a neutron. ~~~~~v ~The average speed of a neutron population in thermal equilibrium with its surrounding. ~N(r, Q) drd2 The number of neutrons in a volume dr about point r with directions of motion in solidangle d about Q before the detector is put in place. ~NI (Z ) drdQ2 The: number of neutrons in a volume dr about point r with directions of motion in solid angle dQ about Q after the detector is put in place. ~(r,, ) = vN(rI, ) This quantity will be called the angular flux, or simply flux, before the detector is put in place. D ~),3 = vNl(r 92) This quantity will be called the angular flux, or simply flux, after the detector is put in place. 'Y~r,, ~) = (r, ~)-O(r, Q) This quantity will be called the angular difference flux., or simply -the difference flux.cp(r) = (r, s2) d~2 This quantity will be called the scalar flux before the detector is put in place. y1(r = 'r,) d&2 This quantity will be called the scalar flux after the detector is put in place., 4r(r) = c(r) -pTI(r) This quantity will be called the scalar difference flux. D when the detector is considered. i = -~~~~~ 0 or not present when the surrounding medium is considered. Ea The probability of an absorption collision per centieterof path length,, for small paths.

NOMENCLATURE (CONT) The probability of any type of a collision occurring per centimeter of path length1, for small paths. S(r~, s) drd~ The number of neutrons being supplied to volume element dr about r with direction of motion in solid angle d2 about Q due to scattering from higher energies per second %, (Lu * Q)2 dThe probability per centimeter per nucleus per square cubic centimeter that a neutron having suffered a scattering collision while traveling in direction X will be scattered into soli angle d2 about Q. W. -s(- - )x Atomic density. The following Green's functions denoted by G are all for an infinite homogeneous scattering and absorbing medium. G(rI, Q r ) drdQ The speed v times the number of neutrons in dr about r with directions of motion in about Q due to a source of neutrons of spee v at point r' which emits one neutron per second in direction W'. G(rl I r, s2) = G(r',J~ -r.j~) dS2 The speed v times the numberlof neutrons in dr about r with direction of motion in da2 about Q2 d~ue to a source of neutrons of speed vat point rT isotropically emitting 4it neutrons per second. G(r', QT I r) a92lI-r,)d The speed v times the number of neutrons in dr about r due to a source of neutrons of speed v a~t point r' emitting-one neutron per ~second in direction W. G(rI -r) = G (r'A? -r,Q92) dQ21 d2 The q spee v times the C+nume fnetosi

NOMENCLATURE (CON tT) The following, denoted by a script * f is the Gree for an infinite homogeneous medium where all collisions result in remova from the population..r, Q)drdQ The neutron speed v times the number of neutrons in dr with directions of motion in d2 about due to a source emitting one neutron per second at point r' in direction Qtin an infinite purely absorbing medium. F(a;b~cIZ) The Hypergeometric function, discussed Appendix I, Part 2. 00 x=0 A special function discussed in Appendix I Part 2. ~~ja(Z) ~The spherical Bessel function, discussed in Appendix I, Part 3. b~a (:~2) The spherical harmonic function, discussed in Appendix I, Part 4.

INTRODUCTION When a thermal'neutron absorber is used to. measure a thermal neutron density, the extent to which the detector disturbs the neutron density mst be considered. One method for investigating this problem is to derive an analytical expression which relates the neutron population which exists when a detector is present to the population when a detector is not present. Then the effect on this expression of such quantities a detector composition, detector geometry, composition of the surrounding medium, d many others may be investigated. In deriving the expression relating the steady state detector absorption and the unperturbed neutron population there are no restrictive assumptions. In adapting the problem for a numerical solution the nuber of assumptions has been k~ept as small as possible while still allowing the prob~o lem to be handled on a large digital computer. There are tWo general re-P strictive 'assumptions which are made throughout this paper. First, the neutron energy spectrum iA and around the detector is assumed to be indaependent of position and to be the same energy spectrum that exists when the detector is not present. Second, it is assumed that the detector is located in a large homogeneous medium Further, the detector is assumed to be several mean free paths from any boundaries of the medium, There are severa features which are built into the analytical relationship between the detector activation and the unperturbed neutron

-2 -1. The initial unperturbed neutron density is allowed to be nonsotropic in angular distribution. This implies that the initial neutron density is non-uniform in space. 2. The scatter of thermal neutrons is allowed to be non-isotrop in the laboratory coordinate system, 5. The detector may have up to three independernt dimensions and it may be of anay arbitrary size and shape,, 4. Isotropic' scatter in the laboratory coordinate system of thermal neutrons by the detector is allowed, 5,The neutron~ density at points inside and outside the detector is available as a result of the calculation..

CHAPTER I FORMULATION OF THE PROBLEM The time independent neutron transport equation isassumed to be applicable to the neutron population both before and after the neutron detector is in'place. It will be assumed that the thermal neutron energy spectrum 'is independent of position both before and after the detector is,ut in place. The energy dependence of the transport equation can now be nte grated out and all cross-sections will be average cross sections. averaged over the neutron spectrum. Define N(r _2)drd_ = number of neutrons in unit volume dr about r with directions of motion in dQ about Q. Let v = average thermal neutron speed in centimeters per second, Define O(r, 2) = v N(r, and call it angular flux or simply flux, Let the neutron flux before the detetor -is put in place be called T(L ~) and the neutron flux after the detector is put in place be called t(r,, Finally define a difference flux as This difference flux is not necessarily small compared to 0 (r, s).The transport equat~ion_:vhich 0 (r:, s0) must satisfy before the detector -is Sput in place is:. - (r Q) (1) -5-~~~~~~

(rw _) = source of thermal neutrons due to slowing don of neutrons. The transport equation which -O (I _2) must satisfy after the detector is present is: rV ' (, -) +[ tr (s )] r (r (2) [, - ) A F,...).(W ) d,i(row Q)4 (r ) rwQ Where the superscript D indicates detector and no superscript indicates surrounding miedium.0 0 when r is outside Detector 1when r is inside Detector By subtracting one transport equation from the other -and introducing the difference flux notation one obtains: ~ * ~ (~&) +~jv ~, ~ - ~(r,, 9) Z5(~-Q) do. -A (Z T~ (r, )-A (Z- (r, 2) -(5 - ~(E,)Z -l (co Q)J dco

-5 -A S (r, _) L sD (r_, ) Now efe the Green's function G(r', ' 2r ) as the solution to Equation). G(r', Q' r, 2) + t G(r, Q' r, Q)_,_ -r_ _,_ -_r _) f)~~~~~~~~~~~~~~~4 - G(r', ' r,) Es (w- Q) dc = 5(r'-r) ( ) Where is the Dirac delta function.;The function G(r', Q - r, Q) can be given a physical meaning. G(r',~ Q~ - r~, Q) drJl is the number of neutrons in dr about r with directions of motion in dQ about SQ times the average speed v. clue to a source located at point rt emitting one neutron per second in direction Q' into an infinite homogeneous medium of total cross-section Etand scattering cross-section 7.(w Q) dQ2. Where s(wn- ) d is the differential scattering cross-section in the laboratory system and ZS(ct -- Q) = o(cn -4 Q) x the atomic density. Using the general properties of Green's functions Equation (2) can be converted into an integral equation: T (r, -2) (E - (r, T)G (r', SV --- r, ~2) drtdQ.1 r t () + (~t - z~) T (r, ~2) G (rt I -1 -r, Q~ dr dQ' _r',21

-6 -grations over vectors r, rtetc. are limited toS integrationoverthedeteco.r volume, Conversion of the diff erential-integrcql Equation (2) to the integral Equation 5 by means of the Greens function has effecti-tely separated the problem into two separate problems, The first problem is that of formulating the Greenis fu point o n t will bsome useablnderstoodblem is that oall tcarryingout th re spatiallimited to integrations over thedetectorvolume. Equation 5 will now be converted to an iterati~ve equation by mer~ely appending subscripts to the ~ (rQ) terms. Tn( )= ~~~ ~(rt, S7t)G(r'j Q -erp Q) dr dS (Z ~fl4J2~) Q) G (L, Ql*j> Q2) dr'dQ'

-7 -7~~~~~~~~ - -2-) J n-lr 9) dc5)[s(Ct dglt (a G~r' W-4,r, ~Q) ly (2) doQ) dD cl) + G(r-_, W 4 2) S(r't Q')-SD(rT, 2')) dr'dQ' Any initial guess for 'O(r, Q) can be used in Equation (6) to of functions In (r, _) for n > 0. Furtner as n goes Qto infinity 1n(,)1 approach (rT ) te soluti of Equation The proof of tihis convergence is discussed in Appendix III. Apparently no analytic expression for the Greents function G (rt -r~ is known, however the somewhat less complex ~Greents function G (r-* 2) has been -expressed i'n analytical form, Where G (r' ~,~ G (r', d~ ~ ) Q' (7) G (r *. )satisfies the equation which is Qbtained by integrating Equation (4) over-all Sn In appendix I it is shown that G (rt,-4 r, can be expressed as ~~~~~~~b ==a a=oo b=-a ~ ~ A t~(r~(

-8 -(IRI) = 00 f a,(Kx) eZt dx x 00 2 Ja (KR) x = o K2 Jt K 1KEs K=0; tan l K K K Et The function Ga (JR I) will clearly depend upon IR and upon the J-;. ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.. _ -.properties of the external medium. Except for the assymptotic limits the funchas ot beeion expressed in a simple analytical form, however a numerical evaluation is coded for the IBM 704. Details of the numerical evaluation are given in Appendix I.o Thus Ga (IR) may be considereda known anction of R once the external medium has been specified. Using the reciprocity theorem it is easily shown that G (r V-r G (rlo~,Qt 1) - ~~ -b A b a~b (I rI) Y(r -r) Y (Q') Due t o the f act that only the Gileen s funct ion G (ri, V-r) is available rather than the complete G (r? Q? ->r, s2) only a few iterations can be carried out using Equation (6). Thus it is desirable to.use the best possible guess for To (r, s7) the starting function in the series Tn (,~ In order to obtain a good starting guess for To (r, sQ) expand T (r T b (r) -Yb ( (11 a -ab

-9 -b Assume tat all a (r) for any r and for "a"- > 0 are small compared to.0 (r) and tus can be neglected. This is equivalent to assuming that the differenceflux is nearly isotropic, Similarly expand ac S(DQ) = Sa (r) Yb (2) (2 a,b D'a Db b a (r) Ya (Q) "a = 0 (rm, / a + 4rr) G0 dr,(15) lt it~~~~~~~1i

-10 -Where cp (r) = r n) 12 4f @ (r) (r) = /S (_ 2) dcQ =-~/-S~ (r) 2 (r) = ji (r, ) d2 =N 4/ (r) atsli In tnesensethat v (~,~) isa searable fucto ofradr2adi is oi its assumed that the sperical harmonic expantia form Aagna ierati ofo of E ( an b writtentas Tn (r) art (2.tl r

-11 -gnri (3)O crge t 4r (ra) the s t of E ar r t as n goes to infinity. Since the function G rha I) is completely knom -as many iterations of Equation (15) as are necessary may be carried out. The integrations and iterations of. Equation (15) are actually carried out on a digital computer. The details of this numerical work are presented in Appendix V. Now that j,(r) has been calculated it may. be ~used for the initial gu~ess To (r 2) in Equation (6).

CHAPTER II UNIFORM) ISOTROPIC INITIAL FLUX The effects outlined earlier will be considered oneat a time. The first case considered is that of an initial flux which is uniform AX space and isotropic- in angular distribution, First r (r) solution f Equation (1) will be obtained by the iterative solution of Equation (15) This function (r) will then be used as an initial guess in Equation (6) and (, ( ) will be calculated using Equation (6). Integration of the lequation fo (r Q) over _2 will show that r (r) = l (rs Q) d__ ot. d if it is assumed that the source due to slowing-down of neutrons in the detector is zero and that the source due to slowing*down of neutrons in the sur~rounding medium 1krt Ths't is seen that the first iterationo qain()poue nua information about T (r, ) The function Ql(',s) is not calculated explicitjy lbut it is implicit in the expression for 'J2(, T Although the function Tl (r), Q) is not explicitlyrj-evaluated numaerically this calculation could be carried out to give the detailed r and ~2 dependence of Tl, ) Equation (6) is applied a second time to produce a function

-13 -will show that it contains the Greent's function G (r -r 2) which i r not available. Since the absorption rate is relatedD l'Y (r, 2) d 2 the detailed dependence of ' (r.2) upon 2 is not necessary in order to calculate absorption rate. Thus the equation for 2 (ay be integrated) over to produce:.fY2 ( Q)dQO = a G (rt9, Q' -r) dr'dQ' ~2 r>Q2t + (4I ZA Za qi G (r '- -,Q )) G (r", Q' _) d"dr ~+ 4 ~ 6G -r(r" r) G (rt -.r) drIdr 4kc r",r' GZr (r") G (r_" -e r '")G....a t(_r' G -er') G (r'r r d it~~~~~~~~~i Note that G (r' -r) = Gr, ~2' -r.) d~2 It is seen that Equation (16) now contains only the Green's functions of the form G (, ~2' -r) or G (" -4 r' Q2'),both of which are known.

-14 -G (r t - r' ') are now inserted in equation and the integration over is carried out to produce: ~/( ~) d~=Z~ 2 % (ir I) dr' + _Z ) p Ga (jrt - r) b ) A (r at r'l Ga (:Kt r |)yb (r_ rA) _drldrt + ) __ (. t rIo (t ) % - 'rI) dr ' + s~~~~r 0 -,-c) (- ) a _,+,-" Z)- a) _ (>I") Ga (lrt r") x a 4< 4 ~~~r' r_' * z_, Ya (r G r ) rt) Y (r rt) drf'dr' (ZD~Z (rS (ra rI ( rj dr"dr' Due to the conplexity of the numnerical problenm it is not practical to obtain ~(,)i,but ~ rs) drdO is obtained. This

-15 -integral is all that is necessary to calculate detector activation. Terminating the summation over "a" in Equation (17) at some vaue A is equivalent to termination of the spherical harmonic expansion for ) at a = A, When ' 2 (r, 2) drd_ is calculated numerially, A will be usedas a parameter and varied in order to determine how many terms in the spherical harmonic expansion must be retained in a given case so that Lthe neglected higher order terms will have only small effect upon.% (o r Q) dridQ The results:of the numerical calculations show that for small detectors, i,e. thin coins or small diameter wires, the value of T{ (r, 2) drdS2 calculated from Equation (17) will be independent of the value. of A, the cut,off point for the spherical harmonic expansions, For detectors of this small size the spherical harmonic expansions may be terminated at the zeroth term, thus in this case When the value-of ~ (r2) drd92 is insens itive to "aa" for some "a, A but r rS

-16 -it is clear that including higher order spherical harmonics will not effect the value2 (rof Q) d~drv But since the value -of (, ) drd ( c as J fr(r)dr) may be different from 2 drd r r _ then the question arises of how much would E3 (r, _) differ from 2 (.Since T ( )c'L Lnot be calculated due to lack of information about G (r, ->)Q)the magnitude of E3 (r_ _) will have to be estimated In Appendix I it is shown that 1ITn+l(r 9) T 2)1 [ < Kll[n(Ll Q>n(r, Q) where K < 1 II'~( ~ Max, for any r of fD911 1) (r.JMax. for any 9 fr 2)[(9 For the cases where T (r, Q is not a sharply peaked function of r or of Q the maximum differences may be replaced approximately with differences of averages-, K! 'if2 (r, QTrd - (r, O)drdQ I drd~2 For the case of a nearly isotropic ~ (I Q) it was noted above that Eqatio rn (6). q-I- reduce to t' r I- Equatio n (15) +and thus

-17 -n+l(r —) -rn(r) I< K1I ~n(r-) -n-(1) I1 ~The nors)- 'rn.l(r) 11 for all iterations of Equation able and can be used to get a good estimate of the value of K o given the difference between (r ) drdy2 and /j ( ) drd which results from the calJculations, an estimate of the difference between (r_ _)drdQ and /2 (r_ _)drdO can be obtained. All of the f ormalism is now set up to evaluate the average neutron flux within the detector compared to the flux which existed before the detector was put in place. Since all of the- above integrations over the d~etector —volume are carried out numerically a completely arbitrary three dimensional detector geometry could be treated, Practical limitations dictated the choice of a right circular cylindrical geometry for consideration here. This cylinder can be in -one extreme case a finite length wire or in the other extreme a thin coin$ depending upon the dimensions chosen. Thus the effect-of finite length of a wire detector and of finite radius of a coin detector are built into this model, The function 4r (r) will be available as a result of the numerical cacuaton at a lag ubro Jonstruhu h eetrvlm n

at a few select points outside the detector. For those detectors thatare small enough so that 4r (r) and, Q2)d are reasonably close to dependence of T (r, 2)dQ

CHAPTER III NON-ISOTROPIC SCATTER AND NON-ISOTROPIC INITIAL FLUX $ (r, _) Consider the problem of a detector placed in a medium which has nonisotropic scatter in the laboratory system. Assume that the initial flux (r, ) is uniform in space and isotropic in direction. The Greens function G(r r) for the case of non-isotropic scattering is available in analyt ical form which is similar to but more complex than the form of G(r ->r) for the isotropic scattering case. A complete discussion of the derivation of the Green's function for j $ 0 is given in Appendix II. Here is defined as: 4 = Zs (w - Q) w * d/ Z (w- ) dQ Q f it is assumed that terms can be neglected compared to terms in _~ then the zeroth term in the spherical harmonic expansion for the Green's function can be calculated for the case of non-isotropic scatter, i~e. 00 G Pt Qtr) dPQ =o (Ir t rI)- 2 — (21) (2it)2 [F0 (K Zb) -p — F1 (K, Zt)]~ Kr-rJ 2d 4TC 0 (4it) 21 This function Go (R)7 is evaluated in a numerical manner very -1 A ~-1 9- T%

-20 -Equation (15) may now be rewritten replacing Go ( with (Jr I)- The solution to this equation will be called to denote the pressence of non-isotropic scatter. ar: - p o (_- r'| ' 4= rt + ' (r') Go (IL - r')I dr_ rt + o ( r - rtl-Dl (r') _ 'D (r') d_ In Chapter II the procedure was discussed for determining how small a detector must be in order that the solution to Equation (13),4 (r),be essentially the same as the solution to Equation (5,' (r, Q2) For small detectors.Equation (22) can now be evaluated numerically using the -'non-isotropic Go (Ir -r'j) and a comparison of ~ (r) with ~r() will show the effect-of non-isotropic scattering in the external medium upon the average flux in the detector. Next consider the case where scattering is isotropic and the departure of the initial flux o (r,. from isotropy is small enough so that ai b

-21 -Putting thisexpansion in the transport equation yields the usual diffusion relationship: -'V f O ( Q, dQ2= 0 ( (r25 Q)dQ t f (r) = (r, _Q) diQ = _ (24) 0 Dividing Equation (23) into rectangular components and evaluating yields: o To (r),\ = i1 If te spericl hrmoi exaso6fteiiil lxi eZntda ~ (r, ~) dS anoh gain of((2 2 Q hchaeasue specified.~~~~~~~~~)XC

-22 -Equation (15) can be solved for this case of a spatially dependent flux. The r (r) which results from this solution will again be used as -an initial guess for a single iteration of Equation (6). Carryingout a single iteration and integrating 1 (r over all ) l(r5 Q) dQ = (Z - Y n-~' Q:)G(rt, Q' -4r) drtd rt ~' r~~ ~ J 1t + (a- ar) ) (r (rt, - r) ' dr'dQ ' +(26 r7 Q + s s G9G(.r t r ) r) (r_ w) d(or_ r, _ + G(rt Q r) S(r, Q) - S&(rt', ft j dr it d Assume the source in the detector S (r, Q) due to the slowing down of neutrons is zero and the s-ource due to slowing down of neutrons outside the dete ctor., S (.) = (r ) d ~ Putting the spherical harmonic expansion for 0 (r, ) and G (r', QI.->) into Equation (26), and integrating lover all SW? gives:,

-23 -b A (r 4 — ii~ (4t) AjO(r')Y (r - r? (Jr')a ammo i x + a n (r) ar_ (r ) dr (27) s4i s 0o(I tl); 0(' r r' + ___( - do | ) t i O' (r t) d r t. 0 rr fortheIn-i c or dr t eal h average flux within r thfeetr Itsolbentdtain the case of anui r stoi initial lyuia none-unrifor non-sotopi flu only onea iteraioentofEqation (6) canb carrihedu dueotropthifct i that thxae genea Greenmprts oh function G(t (-r, d)i ot avaiabl~ bt thsshouldb noteta present caserious handiiicaplf n-nior rm)wl be very close to- ' (r Q) for a large range of small detectors'.

CHAPTER IV RESULTS OF THE CALCULATION OF a (R) a Rather than presenting the function Ga (R) which has a l/B dependence forsmall R the graphs and tables all present R x a (R) The upper limit for error specified in the calculation f Ga () is 10 per cent, but an examination of the convergence of the numerical calculations of Ga (R) shows that in all cases the actual errorin a () is much less than 1.0 per cent, A more realistic error estimate is about 01 per centr 02 per cent. value of Ga (R) was calculated at points R = n/6 cm where n runs from 0 to 60. The function R2 x G-a (R) is then stored in tabular form for later use. The spacing of 1/6 cm. between points is used so that alinear interpolation scheme can be used in the table lookup process while rcstricting the interpolation error to 0,1 per cent. The results of the calculations show that the functions B2 x da,(B) for wmater and graphite are smooth enough so that alarger spacing can be used and still -maintain the interpolation error limit at about 0,1 per cent, It is of interest to note that for B greater than several mean free paths the function -G (B) agrees very well with the conventional diffusion app-roximation eB/L Go (B) DR

-25 -The usual factor of 4t is missing from the denominator due to the fact that the (R) has a source of 4c neutrons per second. it is also observed that the Ga (R) curves falloff more rapidly for increasing "a, "t The terms in the spherical harmonic expansion G(r,.r) for ttatt greater than zero correspond to higher order anisotropy in the source. This more rapid fall off corresponds to the physically observed phenomenon that the nonisotropic components of a point source are less important at a given observation point than the isotropic component. 2 sets of calculations are presented for Rx (R) for water. The only difference between the two calculations is the sed. In one case = 0.0196 cm1 and in the other 0.0184 cm_1 As expected) increasing the absorption cross* —section ~causes the neutron density to decrease at any given observation point. Another set of calculations are presented for R2 x G. (R) for water, In one calculation -jT is 0.5, in the other. one ~'FLis,0.0. In this case the neutron density near the source is reduced for = 0.5. This is due to the fact that for 0.3 scattered neutrons tend to be scattered forward rather than isotropically, which is the case when ji = 0. This preferential -scattering allows the neutrons to penetrate greater -distances into the medium. Since the source strength is unchanged the number of neutrons absorbed is unchanged: r r

-26 -Although the = > Go (R)= for small R the inequality must eventually reverse for some larger R. The change of the shape of the R2 x Go (R) curve with change in the value of suggests a method for measuring i directly. Assume that several curves R2 o (R) are available for-several valuesof T The maxium or each curve will occur at a different point R depending upon the value-of _ used, B2 x Go(R)-~ ~ <2 5 0 B (cm.) Figure 1. The Function B G 0 (B) for Several Values ofjT

-27 -Since (R) is merely the neutron speed v times the neutron density at a distance R centimeters from an isotropic thermal neutron emitting c neutrons per second, G-o (R) can be me activation techniques. From this type of measurement a plotof G () can be obtained and the position of the maximum determined. The position of the maximum should then determine the proper value of ~ I~t should be recalled that for the calculations carried out here Go (R~j was evaluated. assuming that terms. in ~t can be neglected compared to terms in jT*For the type of analysis outlined in the previous paragraph 2 the terms in -T can not be neglected, and a more careful calculation taking into account higher-order terms in ~Tshould be made,

-28 -10 8~~~~~~~~~~~ XT= 3.14 I~~~~~~~6 X ~~0.o= 0.0196 c 0 I0 0 2 4 6 8 10 R (cm) Figure 2. Zeroth Greents Function Coefficient for Water,

J.0 0.8 0.8~~~~~~~~~~~ 3.14 cm o=0. 0196 cm-1 ~ 0.6 0 2C or 0.4~ 0.2 0 0 0.25 0.5 0.75 1.0 R (Cm) Figure 3. Higher Order Green's Function Coefficients for Water.

I0 8 =.00196 cm 0o -I 6T" 3.14 cm le~ X,.U) 40 Z 0 0196 Cm' 0. a 2 0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 R (cm) Figure 4.Zeroth Green's Function Coefficient ~for Water f~or Two Absorption Cross-Sectos

8 8 r j: N [ ~~~~~~-II 6 | _ > ~~a = 0.385325 cm | / -- 0.00032 5 cm 0 go 2 0 _ _ _ _ _ _ _ i 0 I 2 3 4 5 6 7 R (cm) Figure 5. Zeroth Green's Function Coefficient for Graphite.

1.0 Za 0. 3853 25 cm 0.9 IT= 0A00325 cm Cr~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r 0.8 0.7 0 0.25 0.5 0.75 1.0 1.25 1.5 R (cm) Figu~re 6.First Ord~er Green's Fiuncti~on Coeff~icient for Graphite.

CHAPTER V FLUX CALCULATIONS In the calculation of the neutron population after the detector Is ut in place there are several assumptions made which are open to question First it is assumed that the energy spectrum of the neutron population is independent of position before and &fter the detector is put in place and tat it is thesame before and after the detector is put in place. The validity of this assumption has not been investigated in this paper. Second it is assume that the detector is placed in an infinite homogeneous npdum. Third it is assumed that scatter is isotropic in the external'medium, Fourth it is assumed that the integrals in Equations (6) and (15) can be replaced with the summations over a finite set of points as given in Appen isassumed that two iterations of the integral equation..Equation will be suffccient for reasonable.accuracy. Sixth it is assumed in setting up the computer calculations that the sourcelof thermal neutrons due to 4lowing-m down of neutrons from higher energies within the detector is zero. Finally it is assumed that the source of thermal neutrons outs~ide the detector is isotropic in angular distribution and equal to the capture rate before the detector is. in -place., The asmtion of an infinite homogeneous medium has the effect- of liiin h application of this type of a calculation to detectors plae at least several mean free paths from any boundaries in the system..Since the mode'rating material with the largest jTis wter it is

efficient for the Greents function is calculated for and for 0. Both of these Greens f unctions are then used to calculate the flux within a detector. For the case of a gold foil of 0.5 cm. radius 0.0127 cm, (5 mil.) thickness the ratio of average flux in the detector to the flux before the detector is in place is 0,815 for 0 and CO for = 3 The effect is essentially the same for indium in water.Thus it is seen that the effect of non-isotropic scattering in the moderating material will effect the flux within the detector by 3 per cent in the case considered here, The next assumption considered is the one that the integrals may be replaced by a summation over a finite set of points. One test of the.validity of such an assumption is to repeat the summation, each time in cluding a larger number of p~oints, A given detector 'was calculated three different times. The only difference between calcu at ions was that the.axies were divided into four, five and six subdivisions, The greatest difference between the calculations was the order of 0,1 per cent. This indicates, that the summations -are essentially 'Independent of subdivision size and that the summLations are good approximations to the integrals when there are about five subdivisions along each-axis. It is not necessary to assume that a certain number of iterations of 'Equation (6) will be necessary for -a given -accuracy. For the quantities T,(~ &Q) drd~Q and '12( Q) drd~2 are both avai~lable as the result of the numeri al calc-cuati-on,. An estimate -of the, quantity K*

-35 -which is the maximum difference between n+l (r Q) and relative to the maximum difference for the previous iteration, is also available. From these quantities it will be possible to recognize that range of small detector dimensions, thickness for coins and radius for wires, where r, will be sufficiently accurate. It will also be possible to rec ognize that range of larOirdi-dAmensions where the second iteration will be sufficiently accurate. In the case of the 0.5 cmo radius gold coin in water it is seen that the difference between (r ) drd. and Y2(r _ 2) drd2 is 1.,5 per cent for a 5 il thickness and1 per cent for a 10 mil thickness. It is further seen that the third. iteration would. d~iffer~2from the second. it eration by about 0.15 per cent and. 1.2?_ per cent for -the 5,mil and. the 10 nil thick coins respectively, In the case of the 0,5 erm, rad~ius gold. coin in graphitq it is seen that the second. iteration d~iff ers from the first by 1 per cent~and. 12 per cent for the 7 mil and. the 15 mil thickness respectively and. that the third. iteration would. d~iffer from the second. by- 0.1 per cent and. 1,.2 per -cent for the 7 mil and. 15 mil thickness respectively. Inspection of Equation (13) shows that ~ (r) is ind epend~ent -of the scattering-cross seetion of the detector~, but ~2 (r 2) is d~epend~ent on it. Results of the calculation T2 (r, Q drd.Q shows that for 0.5 r, cm. rad~ius and. 5 mil thick gold. coin -variation of the scattering crosssection from zero to 0.549 cm, change~s the value of the flux in the d~~etector by les tFha_ n0. pe nc-r ce~nt. fohr bonth water and. graphite.

-36 -TABLE I THE EFFECT OF HIGHER ORDER CORRECTIONS ON THE AVERAGE SCALAR FLUX WITHIN THE DETECTOR Change Due To cp ifr (r) (p 0. 3 =.33 Gold Coin in Water t 5 mils, = 0.5 cm..813 +.027 -.013 +.001 t =l ilsR = 0.5 cm.707 -.026 +.003 Gold Coin in Graphite t=7nils, R 0,5 cm..827 * -.024 +.002 t = 15 ils, R 0.5 cm..730 * -.118 +.011 L=1.27 onm, R =20 mils *The quantity pr is assumed to be effectively zero for graphite, As discussed in Chapter III the case of a non-uniform nonisotropic initial flux can be calculated. But due to the assumption of axial symmetry and symmetry across the midplane which were necessary only due to the computer time and space limitations the gradient problem was not carried to a numerical calculation,

-37 -Theassmption of a zero source due to slowing-down of neutrons within the detector is not necessary due to the theory, merely convenient It has the effect of limiting the application of the calculations to detectors which have high enough atomic weight so that the detector thermalizes very few neutrons As the numerical problem now stands one could not calculate a~reh detector made up of a solution of gold salt in plastic for example But there is no reason in principle why the calculation could not be modified to take a non-zero source within the detector into account0 The assumption of a source of neutrons in the external mediu which, Is isotropic and equal to the capture rate before the detector is introduced is -again not nece~,sary. The theory will permit the use of a completely arbitrary source0 For the cases considered here) where the initial flux has a constant gradient in space., there is no net flow into any unit voIlum-e in the space0 Thus thb supply mrust equal the rate of loss) i~e. the absorption rate. In the absence of any reliable information about the anisotropy of neutrons as they slow down., the source in the external medium due to the slowing-~down of neutrons is assumed isotropic0

CHAPTER VI COMPARISON WITH OTHER THEORIES It seems generally agreed that for finite foil detectors Skyrmes method () is the most adequate method available for calculating the average flux in the detector. It is of interest to see how the results 1of the integral technique developed in Chapter I compare with the results of Skyrmes method. Let the quantity |' (r, 2) dQ be called the scalar flux. A comparison of final results shows that for gold and indium in ~giwater mthe integral method consistantly gives average scalar fluxes in the detector which are higher than those calculated by Skyrme's method. The comparison of the integral method with Skyrme's method is presented graphically for gold in water only, the comparison for indium in water shows the same trends. For the case of gold and indium foil detectors in graphite the integral method is seen to agree within 1 per cent with Skyrmets method. It would be desirable to show how the integral method reduces to the method used by Sky-rme. But Skyrme superimposes two independent calculations, one for "self-s'hieldin-f" within the detector and the other for "flux-depression" in the surrounding medium, -This approach of separating the problem is basically inconsistant with the unlified approach of the integral method. Skyrrne also makes several other approximations which could not be included in the integral method,

-39 -was found which could consistantly account for the differences between the integral method and Skyrme's method. There is one other method of calculating foils with]which comparison ight be made The method is the PL Legendre polynomial method for infinite slab geometries. This method is discussed at some length by Bengston(l) and nuerical solutions were carried out by him and are quotd in the report There is a basic difficulty with using the inte~gral-method discussed here to calculate infinite foils or foils with very large radii. The radius must be divided into a finite number of intervals. As -larger and larger radii are considered the number of radial divisions must be increased t~o maintain accuracy. But the time required to carry out the computation on the computer goes as the third power- of the number -of radial divisions. Due t~o the limited amount of computer time available it does not se'en desirable to push the calculations along this line,

1.0 I. INTEGRAL METHOD ---;SKYRME 0.9 R 0.5 cm. %%~~~~~~I 0.8 0. 8~~~~~~~~~~~~~~~~~~~~ 0.7 0.6 0 I 2 3 4 5 6 THICKNESS (MILS) Figure 7. A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Gold Detector of 0.5 cm. Radius In Water As Calculated By The Integral Method and Skyrme 's.

1.0 -.o.......... '..., INTEGRAL METHOD.. SKYRME 0.9 R: 0 cm. 0.8 0.7 0.6.. 0 I 2 3 4 5 6 THICKNESS (MILS) Figure 8 A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Gold Detector of 1.0 cm. Radius In Water As Calculated By The Integral Method and Skyrme's Method.

1.0 INTEGRAL METHOD.... SKYRMVE 0.9 R = 1.5 cm. 0.9 0.7 0.6 0 I 2 3 4 5 6 THICKNESS (MILS) Figure 9. A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Gold Detector of 1.5 cm. Radius In Water As Calculated By The Integral Method and Skyrme's Method.

1.0 0.9 0.8 0 0.7 0.6 0 I 2 3 4 5 6 THICKNESS (MILS) Figure 10. The Average Normalized Scalar Flux In A Coin Shaped Gold Detector In Water As Calculated By The Integral Method.

1.0 0.9 0.8 0.7 0.6 0 I2 3 4 5 6 THICKNESS (MILS) Figure 11. The Average Normalized Scalar Flux In A Coin shaped Indium Detector In Water As Calculated By The Integral Method.

1.0melT II...._ INTEGRAL METHOD 0 SKYRME 0.9 0.8 0.7 0.6 0 I 2 3 4 5 6 7 8 THICKNESS (MILS) Figure 12. A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Gold Detector In Graphite As Calculated By the Integral Method and By Skyrme's Method.

1.0 I NTEGRAL METHOD 0 SKYRME 0.9 0. 8 0.6 0 I 2 3 4 5 6 7 8 THICKNESS (MILS) Figure 13. A Comparison of The Average Normalized Scalar Flux In a Coin Shaped Indium Detector In Graphite As Calculated By the Integral Method and By Skyrme's Method.

CHAPTER VII COMPARISON WITH EXPERIMENTS InFigures 14 through 21 a comparison of the results of the calculations using the integral method with various experiments is presented. It should be noted that the error bounds indicated on the experimental points vary considerably in meaning from one set of results to another. At one extreme Zobel (13) attempts to estimate the total uncertainty from all causes in his measurements and he quotes a very large range of uncertainty. At the other extreme is the work of Fitch and Drummond (4)where no indication is given of the possible errors. In between the two extremes is the work of Klema and Ritchie (8) who specifically state that their error estimates reflect only the uncertainty of the corunting process and nothing else. An examination of the comparison for indium and gold detectors in graphite shows that most of the experimental results fall slightly above the theoretical curve. But the most complete and well documented -work, that of Thompson,(2 consistantly falls below the theoretical curve, Even with the somewhat uncertain error bounds available a minor renormalization of the experimentall-results within these bounds produces in almost every case an excellent fit with the theoretical curves. For the case of gold and indium in water the meager amount of experimental data shows a qualitative agreement with the theoretical calculations. The largeness, or complete lack, of error estimates -will not permit any qualitative comments about the agreement-with the theory,

1.0 INTEGRAL METHOD 0 ZOBEL (EXP.) 0.9, 0.8 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ 0.7 0.6 0 2 3 4 5 6 THICKNESS (MILS) Figure 14. A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Gold Detector In Water As Calculated By the Integral Method and As Measured By Zobel.

1.0 INTEGRAL METHOD. FITCH aS DRUMMOND(EXP 0.9 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~0. 635 CM. 0.8 ~ 0.7 0.6 J.. 0 2 3 4 5 6 THICKNESS (MILS) Figure 15. A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Indium Detector In Water As Calculated By the Integral Method and As Measured By Fitch and Drummond.

E] GALLAGHER R=1.9 Cm. o KLEMA & RITCHIE R 1.9 Cm. 1.01 2~- INTEGRAL METHOD R 2.O Cm. 0.9 0.8 0.7 0.6 0 I 2 3 4 5 6 7 8 THICKNESS (MILS) Figure 16. A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Indium Detector In Graphite As Calculated By the Integral Method and As Measured By Ritchie and Klema and By Gallagher.

1.0 I INTEGRAL METHOD FITCH a DRUMMOND (EXP) 0.8 > 0.6 0.4 0.2 0 5 10 15 20 25 30 35 RADIUS (MILS) Figure 17. A Comparison of The Average Normalized Scalar Flux In A Wire Shaped Indium Detector -In Water As Calculated By The Integral Method And As Measured By Fitch and Dryum

1.0 0.9 LENGTH =0.635 cm. 0.7 _____LENGTH= 1.27 cm. 0.6 0 5 10 15 20 25 30 RADIUS (MILS) Figure 18. A Comparison of The Average Normalized Scalar Flux In A Wire Shaped Gold Detector In Water As Calculated By the Integral Method and As Measured By Fitch and Drummond.

INTEGRAL METHOD {] RITCHIE & KLEMA R 1.9cm 0.9 0.8 cm 0.7 0.6 0 2 3 4 5 6 7 THICKNESS (MILS) FigLre 19. A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Gold Detector In Graphite As Calculated By the Integral Method and As Measured By Ritchie and Klema.

1.0 INTEGRAL METHOD a THOMPSON 2.54 x 1.9 cm. 0.9 0.8 0.7. 0.6 0 2 3 4 5 6 7 THICKNESS (MILS) Figure 20. A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Indium Detector Inz Graphite As Calculated By the Integral Method and As Measured By Thompson,

1.0 INTEGRAL METHOD 09 sGALLAGHER R.954 cm. 0.9 0.8 0.7 0.6 0 I 2 3 4 5 6 7 THICKNESS (MILS) Figure 21B A Comparison of The Average Normalized Scalar Flux In A Coin Shaped Indium Detector In Graphite As Calculated By the Integral Method and As Measured By Gallagher.

-56 -estimates will allow no more tthan the comment that the qualitative.agreement is good. Maps of the scalar flux are presented for the 5 mil thick gold coin shaped detectors in water and in graphite. The maps for the coins of 005 and 2.0 mil thickness are not presented due to their great similarity to the plots for the 5 mil coins, The only essential difference between the 5 mil coins and the thinner ones is in the average value of the scalar flux within the coins and in the quantity the average scalar flux less the minimum scalar flux, Both of these quantities are presented for all coins, Similarly the maps of the indium coins are not presented due to their great similarity to the maps for the gold coins. Graphs of the normalized scalar flux for points along the central Z axis outside the detector are presented for the 5 mil thick gold detectors in water and in graphite. Graphs of the normalized scalar flux for points along a radius in the mid-plane outside the detector are presented for the 0.5 inch long indium wire in water, and for the 0,5 inch long gold wire in water. A question which is frequently discussed in the literature is the relation of the scalar flux at the surface of a detector to the average scalar flux in the detector, When examined in detail it is found that the scalar flux on the surface of the detector is not a single number but is a strong function of the position on the surface. In fact the scalar flux on the surface varies from several per cent below the average scalar neutron flux to several per cent above. Since there is no obvious method for defining a "surface flux" no attempt is made to present the relation of surface to average scalar flux,

-57 -0.76 0.74 N N 0.69 0.70 0.72 M 0.68 4-(L )/#=0.673 MID-PLANE _P Figure 22. A Map of The Normalized Scalar Flux Within A Coin Shaped Gold Detector In Water, Radius of l.5 cm. and Thickness C,) b- 1.0 x )/J0.7 3 MID-PLANE_ P Figure 23. A Map of The Normalized Scalar Flux Within A Coin Shaped Gold Detector In Water, Radius of 1.0 cm. and Thickness of 5 mils.

-58 -~~~~- ~ ~ ~ ~ 0 - 0.8 - ____N____ ____ ____ _ 0.81 0.82 N 0.80 4 0.795 Z MID_ PLANE P Figure 24. A Map of The Normalized Scalar Flux Within A Coin Shaped Gold Detector In Water, Radius 0.5 cm. and Thickness of 5 mils,

1.0 R = 0.5 cm. 0.9 R=1.5cm R= 1.0 cm. 0.8 0.7 0.6 0 0.4 0.8 1.2 1.6 2.0 AXIAL POSITION Z(cm.) Figure 25. A Plot of The Normalized Scalar Fluxes Along the Central Axis Outside of A Set of Gold Coin Shaped Detectors In Graphite, Thickness of 5 rils.

0.5 0.4 z 0.3 0.2 0 0.I 0 0 i 2 3 4 5 6 THICKNESS (MILS) Figure 26. A Plot of the Average Normalized Scalar Flux Minus the Minimum Normalized Scalar Flux In A Coin Shaped. Indium Detector In Water.

0.05 0.04 zi 0.03 s-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~01 0.02 H 0.01 0f 0 2 3 4 5 6 THICKNESS (MILS) Figure 27. A Plot of the Average Normalized Scalar Flux Minus the Minimum Normalized Scalar Flux In A Coin Shaped Gold Detector In Water.

0.03 R=1.5 cm. R=I.O cm. 0.02 z 0 [ R= 0.5 cm. Ie0.01 0 12 3 4 5 6 7 8 THICKNESS (MILS) Figure 283 A Plot of the Average Normalized Scalar Flux Minus the Minimum Noomalized Scalar Flux In A Coin Shaped Gold Detector In Graphite.

1.0 0.9 0.8 0.7 0.6 0 0.4 0.8 1.2 1.6 2.0 AXIAL POSITION Z (cm.) Figure 29. A Plot of The Normalized Scalar Fluxes Along the Central Axis Outside of A Set of Gold Coin Shaped Detectors In Water, Thickness of 5 mils.

-64 -0.80 z 0.76 0/-'0 0755 5 MIDPLANE_ p Figure 30. A Map of The Noxmalized Scalar Flux Within A Coin Shaped Gold Detector In Graphite, Radius of 1.5 cm. and Thickness of 7 mils. 0~~~~~~~~~~~~~~~~~~0. 1 cci II I I MID PLANE _ p Figure 31. A Map of The Normalized Scalar Flux Within A Coin Shaped Gold Detector In Graphite, Radius of 1.0 cm, and Thickness of 7 mils,.

-65 - MID-PLANE-P Figure 32. A Map of The Normalized Scalar Flux Within A Coin Shaped Gold Detector In Graphite, Radius of 0.5 cm. and Thickness of 7 mils.

-80.78.77.76 Co 1 x z ) MID- PLANE-P Figure 33. A Map of The Normalized Scalar Flux Within A Wire Shaped Indium Detector In Water, Radius of 10 mils and Length of 2.54 cmo

76.68 64.62.59 =575 MID-PLANE-P Figure 34. A Map of The Normalized Scalar Flux Within A Wire Shaped Indium Detector In Water, Radius of 20 mils and Length of 2.54 cm.

.91.92.905.90 -r w (r =.897 XX MID -PLANE-P Figure 35. A Map of The Normalized Scalar Flux Within A Wire Shaped Gold Detector In Water, Radius of 5 mils and Length of 1,27 cm.

.86.84.82.8.805 N MID- PLANE-P Figure 36. A Map of The Normalized Scalar Flux Within A Wire Shaped Gold Detector In Water, Radius of 10 mils and Length of 1.27 cm.

72.70.68.66,c L.65 IM-N- 643 MID- PLANE- P Figure 37. A Map of The Normalized Scalar Flux Within A Wire Shaped Gold Detector In Water, Radius of 20 mils and Iength of 1.27 cm.

-71-.90.89.88.875.87 () z Qi MID- PLANE- P Figure 38. A Map of The Normalized Scalar Flux Within A Wire Shaped Indium Detector In Water, Radius of 5 mils and Length of 2.54 cm.

1.0 R = 20 MILS.9 __ I R = 10 MILS R=5 MILS r.8.7.6 0 A.8 12 1.6 2.0 RADIAL POSITION R, Cm Figure 39, A Plot of The Normalized Scalar Fluxes Along A Radius In The Mid-Plane of A Set of Indium Wire Shaped Detectors In Water, Length of 0.5 Inches.

.08.06.0 '4.04 LENGTH 2.54Cm |LENGTH = 1.27 Cm.02 0 5 10 15 20 25 30 RADIUS (MILS) Figure 40. A Plot of the Average Normalized Scalar Flux Minus the Minimum Normalized Scalar Flux In A Wire Shaped Indium Detector In Water.

1.0 R = 20 MILS.9 R = 10 MILS.8 R= 5 MILS -.7.6 0.4.8 1.2 1.6 2.0 RADIAL POSITION R, Cm Figure 41. A Plot of the Normali~zed Scalar Fluxes Along a Radius In the Mid-Plane of A Set of Gold Wire Shaped Detectors in Water, Length of 0.5 Inches.

.06.04 ENGTH = 1.27 Cm.-.02 LENGTH:0.635 Cm 0 ' ' 0 5 10 15 20 25 RADIUS(MILS) Figure 42. A Plot of the Average Normalized Scalar Flux Minus the Mimium Normalized Scalar Flux in A Wire Shaped Gold Detector In Water.

CHAPTER VIII CONCLUSIONS In conclusion it can be observed that for neutron absorbers commonly used in neutron population measurements the conversion of the transport equation, Equation (2), to an integral equation, Equation (5), is a very practical method of treating the thermal neutron population and around a neutron detector. For a large range of detector sizes it is, in fact, necessary to consider only the zeroth spherical harmonic of the neutron population, Equation (13). For these cases such problems as:.1, Initially anisotropic neutron population. 2, Non-isotropic scattering by the surrounding material. 5e Detailed spatial dependence of the neutron population. 4. Arbitrary three dimensional detector geometry, can be calculated with an accuracy of 1 per cent using only the zeroth spherical harmonic of the neutron population. For the larger detector sizes the calculation of two iterations of Equation (6) must be carried out, where the higher spherical harmonics are included, in order to achieve an accuracy of 1 per cent. For very large absorbers, e.go reactor control rods, the iterations of Equation (6) should be continuted beyond the second iteration, but as noted earlier this is not possible due to the limited amount of information available about the Greents function G (r':, r_ _r ). In the case of water there is still another limiting factor. Even for the small detectors where the higher spherical harmonics need not be

-77 -considered the variation of p from zero to 0,3 causes the average scalar flux in the thickest detectors to increase by as much as 5 per cent. Thus, the limiting factor for the calculations presented here for the case of water is the value of I used, The value of j was taken as zero for all of the calculations except the one case of T of 0.3. So, the average scalar flux in water may be low by as much as 3 per cent in the case.of thick coins in watero Aside from the uncertainty caused by the two factors mentioned above there are no other known sources of error which could contribute a correction of I.-more than about 0,1 per cent.

-78 -APPENDIX I Part 1 EVALUATION OF G(r ' ' r) By definition G(r', Qt -r, Q) satisfies the equation (1) a v G(r', Q' r, Q) + CZ G(r', Q' r, Q)Zs (co Q) G(r', 2' r_, c) dco = 6(r'- r) (Q2'-Q) Again by definition IQ r) G(r', Q' r ) dQ Assume isotropic scattering in the laboratory coordinate system. Then.Zs ( ) (2) Next letjS(r", -> r, Q) denote the solution to the following equation: Q ~ v(r=", Q" - r, Q) + Zt (=(r, " Q ~ (2) = (rt"-r) (E"-aQ) It is well known that (2) _,_r -Qtt, _ Q) = e_=__=121 (4) h (i-rQ) b (Ge -[fnr "i ]) The functiona-(r", Q" -r, Q) is a Green 's function for Equation (1) thus

-79 -G(r', ' r, ~) = G(r", r, Q) r (5) fG(r', r ) dc) + (rl-r") 65(0'-Q") dr"dQ2" Spatial integrations are over all r space. Making the indicated substitutions yields: e-Gt I r-r I G(r',, Irr 12 (6) r fG/(rt', ' -r", ") dw b( - L-Ar"]) dr" + Now integrating over all ~ yields: G(r' I'V4 4 rt. G(r' Qd (7) I4n~ I r Ir-rl2 b(~2Krrfl) Expanld G(r', Qt -'r) as 2'~ Qt r) = ) 6(r~, r) Ybr) (8) alb Thus ) Gb(ra Y) b() = YQ) x (9)r a, b

-8o~Gb (r~l 'dr" dr r e- I _A_ 1r _ r (Q'- I-2A~' I) Now multiply by Yg (Q') and integrate over all Q'. G (/, )' r") e-t, e dr" (10) r -z I_-=_' ym (TAr) e n(K=| 1 ) - r2; (12) For U thee sake of convenience let r = O Now take the Fourier transform of Equation ( _ r e-Z r-iriKr G () G (r) e 'd (11) r ~r~~ jC n..-Ce r - K o S ~;(), ~di.A~

-81 -00 2Ztr * r e -iKr A A e r Y (r) drdr= r A r=o r 00 ~r~= 0 ~ ~~ *A a- b rL' 00 = f4 (-i)1 jl (Kr) dr Y (K) 00 Deine F1 (K ) _ 4i (1i) jl (Kr) e dr; 1 = 0O 1, 2..o (14) r=0 Thus! At () dr = Y () F1 (K, Zt) (15) Note that in the special case of 1 = m = 0 the integral in Equation L 4):can be done directly yielding Fo (KK, t) = Arctangent( ) (16) Next examine the integral: Let x = r"-r e.~- I rr [| iKor -iK-r t eJ:-t ~ ij2- - dr = e -- x (17) r

-82 -Ex- iK.x e -= x -iK-r"t r e FO (Kt Et) Combining Equations 11 and 15 gives: G1 (K) = G(1 (r e Fo (K= 1) dr." (2T/2 4 j 1(d *. (18) + (2)521 ` (K) F1 (K Zt) Using the definition in Equation 12 and solving for G1 (K) m I~1, (K) FY1 (K) F (K, t ) () ) e Y1 (K) F1 (K, ) (20) 1(2X)EK s FO (K, Z) + iKor Again using the expansion for e it is easily shown that '/ iK "r ASn ln — (r) dA er)27 Y dK = 4 () (Kr) ) (21) K Agi bn-h xaso oreIti aiysonta

-83 -Thus G () 2(i)l () jl (Kr) F1 (KI Zr) K2 4 Fo(KtZ) In order to regain the dependance on r r eplace r with r-r'. 1 * Gm (r-r') =.2i y (rAr') x (22) 1 -- (2as)3 f j~ (KJrrl) F1 (K, Zt),jv,; KdK -4 Fo (K, Zt) Define a1 (K-r' J) 3F) ( Zt) K 23) 4G Thus G(r',.r) yb r Ga( r-r) (24) alb

APPENDIX I Part 2 THE FUNCTION Fa (K, Et) In Equation (14) of Part 1 of this Appendix Fa(K, Z) was defined as 00 F a(K, Zt) = 4 (-i)a / ja (Kx) e-tX dx (25) x=O 4T (-i)a (~)l(a+) Ka 2 r=(a + 3/2) (K2+E 2ta)~(a + 1) x F (a+. a + 1 2a + j 3 2 2 +22 2-() (9) The function F(A; BICIZ) is the hypergeometric function. AB A(A+1)B(B+) Z F(A;BICIZ) = 1 + B Z + Ac(AcB() z2 +,.,.. (26) For Z near 1 it is useful to note F(A;Bjcjz) r(c P C-A-B) F (27) = r(C) (C-B) F(A;BIA+B-C+111-Z) + (27) rPC)r(A+B-C) (lz))C-A-B F(C-A; C-BIC-A-B+111-Z) r (A) r(B) * Note that Equation (25) should contain the term r(~).r (A+l) O r(~)r,(a+N (3)T(A+1) NOT 2r(-)r(a+,) as given in Reference (9), 2aflrF('aS+/2) 2a-lF (a+~-)8 -84 -

12 X W.0 4 -I 0 I 2 10 10 10 10 x Figure 43. A Graph of The Functionci(x). W t | | [ i l m~~~~~~~~~~~~~~

4 I I I I I I I I I I I I I I -I 0 I 2 10 10 10 G0 Figure 44. A Graph of The Functio4L-(x).

1.4 1.2 r; 0.8 0.4 -I 0 I 2 10 10 10 10 X Figure 45. A Graph of the Functionc 2(x).

0.7 0.6 0.4 Co 0.2 0 1 2 3 00 10 10 10 Figure 46. A Graph of the Function.94(x)o

0.9 0.8 0.6 0.4 4 0.2 10 10 10 10 Figure 47. A Graph of The Functionc3(x).

0.5 0.4 0.3 0.2 0.I 100 10' 102 x Figure 48 A Graph of~ The Function ~(X).

-91 -For the large K, i.e. Z near 1, it will be more convenient to use Equation (3) with its power series expansions in (1-Z) rather than the slowly converging power series in Z given by Equation (2). From the power series expansion it is obvious that F(A;BICIZ) -1 Z \ 00 Thus for large K F(K,.t) 4~l-i)aP2(_)F(a+l) 1 -F....x — (28) K - oo 2a+l2(. 1) K ( Define c a(K/t) = Zt x Fa(K, Zt) (29) The function a(K/t) is dimensionless and depends only upon a and K/4. The functions. a(K/Zt) are displayed graphically in Figures 43 through 48.

-92 -APPENDIX I Part 3 SPHERICAL BESSEL FUNCTIONS jo(Z) inn Sin Z (5') =z.Sin Z jO(z) = z n (31).5(z) Cos Cos Z Sin ( jl(Z) = -Sin Z CoZ Sin (32) j2(Z) = 3 Z + 3- (33) c- os Z z Sin z js(z) = CosZ +15z +15 F (34) z 2..... -420 Sin Z Cos-945 Z Sin Z -105 +S 105 z =_(Z) +15 Z2 +105 '3 (36) -420 YE >95 +945 Z

Z)0-(932n+1) as Z - 0 Zn in(Z) ` CosEZ (nil)] as Z - co (38)

APPENDIX I Part 4 SPHERICAL HARMONICS m - 2 +1 (1 - m)' eime Pm rn 1+1 _ _ ell (39) Z cos cpa= / I N /G Em (1) = Legendre's associated function of the first kin,(7) b* Y1 (_Q) Yb (_) d2 = 6(a - 1) 6(b-m) (40) Y 21+1 ) = 2 1m) e (41) -(1 +m) -94 -

-95 -Note that P P (1 P1 (M) (42) 11 ~ ( + m21~+1 1i m) enm Pim (43) ' (i + m)' e - () Thus y1 (~) _e 2 1+ 1ime p Another useful formula is: 00 (2n+1) A A Piker =: (2 n + ) 2i Pn (k ) ()45 Pn (k _) 2__ (n + m) pm (Cos k) x (46) -n (k ~ _) 2n. +1 2n (4 ) n=o 2n+ 1 n - m' pm (Cos c) eim(Ik Or) Thus A A 4 A A m=- n Thus Equation (7) becomes: +ik-r n m A m A e- - 4(i) Yn (k) Y () (kr) (48) n,m

Using the definitions and results of the above work it is easily shown that: b =a a ) (9) a f(b,.....) Yb (_ Yb (50) a * b =-a if f(b,...) = f(-b,.....)

APPENDIX I Part 5 NUMERICAL INTEGRATION Consider the integral of Equation (23) of Appendix I, Part 1. 00 a() (2K ia (KR)Fa(K ) K (51) Examination of the integrand in Equation (51) will show that for large K it becomes ia j (KR) Fa(K, Zt) K2.... =...... - - (52) 2 lo S Fo(K, 1t) K (K R x 2a+l ~2(a + 1) The value of the integral will oscillate with constant amplitude about some mean value as K goes to infinity,. As Davison (3) notes when the Fourier transform integral oscillates, the mean value about which the oscillations take place is the appropriate value for the integral, The upper limit for the integral (51) will then be changed from infinity to some number T. The number T must be large enough so that all of the terms in the integrand of Equation (51) are near to

-98 -their assymptotic form. Further T must be selected so that the mean value requirement can be satisfied, For small K 1 _ 0Fo(K, Zt) (53) 4n K -e0 2 Due to the denominator the integrand will be changing very rapidly near K = 0. On the other hand for large K the integrand becomes a cosine of period R * The integral is evaluated using Simpson's rule in four separate ranges of K. Region one is 0 < K < Z, region two is ~ < K < 100i/4R, region three is 100/R4 < K < (602 + 2a) A/4R, region four is (602 + 2a)A/4R < K < (642 + 2a)A/4R. In each of these regions the step size is subdivided until two successive values of the integral over the region are within epsilon of each other. The epsilon is specified as input data and is usually taken as 0.01 The end point K = (642 + 2a) r/4R is chosen so that the mean value requirement will be satisfied. If the value of the integral over region four is greater than epsilon times the sum of all the other three integrals then a xfegion five is added, Region five is the same length as region four. If the end point of region five is large enough so that all of the terms in the integrand have reached their assymptotic value then the integral over region five will be very small. Successive regions are integrated until the K is large enough so that the integral over the last region is very small. Let us now examine the assymptotic form of Ga(lr - r'l) for Ir - r'I going to zero, Rename Ir - r'j =R

Rewriting Equation (51) 00 — G (R) 2i2a / ja(KR) Fa(K, t) K2 Ga) dK (54) K=O 1- 4it FO (K, t) Define Z =KR; dK = dZ/R 2 a 00 Z 2ia Ja(Z) Fa(Z/R, Zt) R dZ (55) Ga (R) -(2a)2 dZ Z=O 1 - 4t FoRt lim Ga (R) = R-O 0 2ia flim ja(Z) Fa(R, E0,2' —:k (2it)2 0 dZ (56) z=o - Fo( ) Recalling Equation (28) urn 4a(~' F2 ) - r 2 r)(a+l)(-i) aR Rlim F a(l(t) =a + 1) Z R - 0 Combining equations 00 im Gai (R) ra'2(~)rI(a+l) 2a+1 r2(a + 1)2tR2 a(Z)ZdZ58) 00 Z= a 2a+lp(a+3/2) 2D Where F is the Hypergeometric Function and equal to

-100 -2+a a + I1) - '(a + (6o) 2; p)=r(a + 1 )r(Lt+ 3) 2 2 Thus lim Ga (R) = Re 0 4 r (~)r(a+l)r (a+2) 1 22a2, r2 iC.)T a -)lr (a) v Evaluation of the term on the right of equation shows that -!lii Ga (R) 2= (62) R- 0

-101 -APPENDIX II NON-ISOTROPIC SCATTERING Consider a point source of thermal neutrons located at point r' in an infinite homogeneous medium. This source is emitting 4c neutrons per second isotropiQallyo Scattering of neutrons in the medium is allowed to be non-isotropic ~n the laboratory coordinate system, Assume that the spherical harmonic expansion for Fs(_ e _) is known. eXbb (cU)) Y (b ) (1) a} b The usual representation for Zs (_c -* ) is 1 %s (_ ~ 0) ~_,4P= ( Q).1 (2) Where the P1 (c * ' ) are the Legendre polynomials. In this case EL ( Q) d~ (5) so = is (u < a) do The average of-the cosaine of the scattering angle 4 is defined Using the results) sobecomes: 1 Using the results of Appendix I Part 4 Equation-(2) becomes: (ceQ) j Zsa Ya (') Y (_) (5) agb Where sl = Zso x A and Zso is the atomic density.times the isotropic

-102 -component of the scattering cross-section which is found tabulated in Nuclear Cross Sections. Note that all spatial integrations in this appendix extend over all space, contrary to the convention used in the other parts of this paper. The 1:rsapart:: equation for the system outlined above may be obtained by integrating Equation (4) of Chapter I over::'all ~2 to obtain: _ o V G(r' -r, 2) + Zt G(r' - r, 2) - (6) - Zs ((U)W ) G(r' -r, w) d = b(r' - r) Using the same conversion as is used in Appendix I, Part 1 G(r' e r, ~)/,.6(_1 _ -.R= -r - r"t Z (_U) ~2_") G(r' -4 r_ ) dcw + 6(r' - r")] dr"d,," G(r' -r, ) = ( [r - r]) x

-103 -/ ~ ( -_) G(r — r, r n) dr" + Et Ir' -r + s~2_ - [r r Ir' I-j+ ------ — S(Q - [r _ rP]) Now expand G(r' -r, Q) Gb(r', r) (b ) (8) a - ab Combine Equations (5), (8), and (9) _ om~b _m e=tII -A r^,. r 5- [r - r ]) Gj Y1 (r() Y( Ga (r () ddr" () W lm a,b Irb I —r 1 / YG (r) r) Y (n2) (2slr r 1br) x a "G (r rtt Ya ( ) dra" +( eaZt r..' |-_ + 6(Q - [r - r']) r - =t 12

-104 -Now multiply by YD (Q) and integrate over all.2 r" Ir" - rl2 lm GC (r' r") r") (r - r")) + _,r' -r r') For convenience let r' =0 Now take the Fourier transform of the above equation and define D <) 2)3/2 GD re- eiK~r dr (12) r InD Appenir 1 et Ioe-d r"-r1-iKr r,r 1,m Gaj () < (- r") ) r - r) )d"atd_ + +;s ebtrL- *K. rn A In Appendix I, Part 1 it is noted that

J f"t r iK ( Pr() drP = Ya (K) Fc (K, Zt) (14) r2 - _ The function Fc (K, Z) is discussed in Appendix I, Part 2. Using Equation (13) G (K) e-t I' -iK x - (2it)3/2.... 2 1 m A *0 A G (r) Y1 ( r t- r") Y (r r") dr"dr 1 err t - -- Y (),mdr + ~ (231~ j e.-Etr-iK'r. 'o A (2ir)3/.= r"- Yo (=-2 2- r Let x = r - r" Now examine the integral r e " Y —y (r- r") y (r- r") dr = (16) -iK'r" -txiKox.- A 1 x e iKr" Y1 (K) Fe (K,Z ) x '

-106 -Thus 1 o 1 % (K_) - ( / - - (17) G ( G1 (2) Y1) F (KF (t) d 1 m m lm (18) 1 oA + 13~ YO (K) F (K, Zt) (2t)3/52 o Assume that terms that make a contribution of the order of (,Sl)2 are small and may be neglected compared to terms of the order of sl in the calculation of Go (K) Using the same procedure as was used above it can be shown that G (K) =,so o () Y1 (K) F1 (K, t) + (19) 1 m A (2E)/2 KY1 (K) F1 (K,,t) O~)/

-107 -Combining equations to solve for Go (K) Go () = so Go (K) Fo (K, >t) + (20) so o '1 ynIA * A so Z Go (K) F12 (K, Zt) Ym(K) Ym(K) m=-l 1 m=-_ Fo (K t) + F12 (K, Z) - F0 (K, 2k) 2o Zsl Z(K1 Ft ) 1 0 A Go (K) - 1 (K) x (21) "K (2-)3/2 0 Fo (K, Zt) + Z1 F12 9(K, 4 ) 1- h5o 2 1oFo (K, )t) - F., - ) 4 (r41c

-108 -Now use the results of Appendix I, Part 4. 00 Go~2) (r) J3 (23) K=O F (K, ) + 7 sl F1 (K, Z)j K2 x OF (K, )(.3 2 so Zs1 F1 (K, t) /o (A 1*m A m A i Yo (K) 4T (i)1 Y1 (K) Y1 (r) x A 1l,m A j (Kr) dKdK 00 G (r)- 2 ~ (24) K=0 IF (K t) + 4 Zsl F1 (K, ) K jo (Kr) dK SO FO (K, Zt) - (4)2 so Zsi F1 (K, Zt) To regain the r' dependance set r r - r' 00 G (r- () o r2') Jo (Kr - rj) K X (25) K=O

-109 -Fo (K, Zt) + 3 F 2 (K t) F ( 3 2 o (K, Zt) (4)2 so sl

APPENDIX III CONVERGENCE Consider two elements T (r_, Q) and Y (r, Q) of the series n m generated by successive application of Equation (6) Chapter I _(rt) _ (rt 2) G (r, G(r ' Q) dr'd" (1) + G(r_ I' '-r, _Q) (r,' w) dwir'dd' + rr' By changing the n's to m's in Equation (1) an equation in m (r Q) is obtained. Now subtracting the equation for Ym (r, _) from the equation for Yn (r, _) one obtains: ~ = (r - -QI) -Zm- (r- -) - ( i- r) x (2) -110 -

-111 -(-1 _ (r', ) 'm-1 (r', Q') G(r' ' r, Q) dr.'d' - r' t Q - _ ' ' Max' any r Define || A (r, Q) II = Max. any Q Of A (r ) (3) IK1 n(5&) '- -+ (r )-1 )(r ) -(m) _ x. v ' S n G 1 - r ' -o) ('. _ Q)) G)arta' - r'fiQ' ~ ~~~~((1) _ _ II n-l (r_, _)- m-l (r, _) x | (E -+) x r' 2 ' I GS(r", t ' rr, (,) dr(dr aQ So the mapping of elements (or functions) ~n-l (r, _) into elements Yn (r, Q) which Equation (1) defines will be a contraction mapping if: -Za- | x lx G(r', Q' r, ) r'dQ' Qt 1 (5)

-112 -or if: | fG(rj Q'-r ) dr'dQ | (6) Using the reciprocity theorem Z ~ G(r' — r, 2) dr = a G(rZ -Q - r') r' = (7 a. a) (7) r' rt the integral over the detector volume of the rate of capture of neutrons by the moderator due to a source of neutrons at point r emitting one neutron per second in direction -Q' Clearly the integral over a finite volume of the capture rate is less than one for a source emitting one neutron per second, Thus f G(r' i r, _) dr' K Z (8) rt and Equation (8) holds for all detector sizes and compositions, thus G(r'.- r, ) dr' < < (9) ac( rNext consider two elements Next consider to elements n (r) and m (r) ofChapt the series generated by successive application of Equation (15) Chapter I

-113 -D Fa) (lO) ~n (r) -= _pa ~ G(r' or) r' - () r...... Z J) (r') G(rt' - r) dr' and mP G(r 'g cPr) a _ _ _(ll) r. ' 4n4 f~fi (rt) G(r? '>r) dr' Subtracting Equation (11) from (10) 'n (r) - *m (r_) (h - Ha) x a (12) I (nl (r_') - _m-1 (r ) G(r_' e-r) dr ) rl - 4G (r -4_- (r)I mx G(r'- r) dr

Thus.Equation (10) will define a contraction mapping if Ma G(rt er) drt < 1 (15) r T Using the reciprocity theorem 1it a G(r. tr) dr' - a G(r - r') dr' = (16) 4i- a 4 — r' r' ~the integral over the detector volume of the rate of capture of neutrons by the surrounding medium due to a source of neutrons at point r emitting isotropically one neutron per second at any point r in the surrounding medium. Clearly the integral over a finite volume of the capture rate is less than- one, Thus 1 G(r r) dr' 1 (17) Y-a r' and 1 G (r r) r. 1 for all (18) r' a a detector sizes and compositions, Both Equation (1) and (10) are of the form: f(x) = Pf(x) where P is an operator. The iterative equation is: fn (x) P l (x) (20) n ) ~ni-l

-115 -Define || fn (x)11 = Max. any x of fn (x) j (21) Subtracting Equation (19) from (20) and taking norms one obtains: I fn (x) - f (x) = P (fn-1 (X) f (x)) I| further (22) ii fn (x) fm (x) 11 = 1J P (fn-f (x) (- () 1 I| P. =K < 1 (24) It has been shown that for the operators P used here: |I fn (x) - f (x) i K I< fn-i (X) - f (x) (25) for all detector sizes and compositions where K is some number less than one, With successive applications of equation one finds: II f= (x) - f (x) ||Kn jj f (x) - f (x) (26) Where f (x) is some arbitrary initial guess at the solution to Equation (20) and f (x) is the solution to Equation (19). Clearly as n goes to infinity Equation (26) requires that the quantity |I fn (x) - f (x) || go to zero. Thus as n goes to infinity the solution of the iterative equation, Equation (,20), approaches the solution f (x) of the original Equation (19).

Thus it has been demonstrated that the series generated by Equation (1) and (10) converge, and that they converge respectively to the solutions of Equations (5) and (13) in Chapter I

APPENDIX IV GEOMETRY AND GRID POINT NETWORK The basic shape of the detector will be a right circular cylinder with radius RG and height 2 x ZE Cylinderical grid surfaces will be located around the central axis and will be spaced a distance MGR from one another. Plane grid surfaces will be located perpendicular to the central axis and will be spaces at distances ~ZE from one another. A A ZE Z A IR -117-I -117 -

-118 -A ZE ~~~~~I ~~~~~~~~outletr surface IA1 I IRG i=o l i=2 i= i=I Mid-plane Figure 50. Location of the Grid Points. Note that a half interval is placed at the outer surface so that a full unit volume may be associated with those points next to the outer surface. No points are located on the mid-plane so that reflection across the mid-plane can be carried out without regard to points on the mid-plane.

-119 -The circle defined by the intersection of cylinder i with plane j will be called circle (i,j). The point where this circle (i,j) cuts the A A ZE - RG plane will be called point (i,j,O) i = 0,1,2,3....I where I = the number of points along the Rs axis excluding the point on the centeral axis. j = 1,2,3,4.....J where J = the number of points along the ZE axis. Coordinates of the point (i,j,O) are X =i x BG$ Y = Z= (j - )x ZE The circle (i,j) will be divided by a series of equally spaced points. The circumferential spacing between points will be equal and as near as possible to AR9 in length. The number of points on circle (ij) = 6xI. The angular spacing between points on the circle (i,j) will be 2T/6i The points spaced around the circle (i,j) will be denoted point (i,j,k), k = 0. The volume associated with point (i,j,k) = i (Li\2R) ZE, ((i + ~)2 - (i - )2)/ 6i except where i-~ is negative, then the volume around point (itj,k) = E (aRt /2)2 ZE Coordinates of the point (i,j,k) are X = izRG x Cos (2jrK/6i) Y = izRG x Sin (2TCK/6i) Z = jtzZE

-120 -Due to the cylindrical symmnty and the s eatry across the midplane it is necessary to calculate unit volumes and fluxes only for points in the RG - ZE plane. Associated with each of these points (iO,,0) in the RG - ZE plane there will be several other points having the same flux and unit volumes: 1. The point (i, -j,0) 2. The points around the circle (i,k) excluding the point (i,j,0) 3. The points around the circle (i, -j) excluding the point (i, -j;;)

APPENDIX V NUMERICAL INTEGRATIONS In order to solve Equation (15) of Chapter I one must evaluate 'n-1 (r-') do (Jr - r'j) dr' (1) rt and G_ (Ir - r'I) dr' (2) rt Where 4n-1 (r) is known function. Let the detector volume be divided into a series of small volumes vi surrounding a series of points ri. Thus:! n-l (r') o (Jr - r'|) dr' (3) r ' G (Jr - r') dr (r ) vi r' i Examination of the numerical results shows that to a good approximation -121 -

-122 -G-a ( I) - - r or __ir,2 for 1-2i Cmo (5) Ca dir -r [K i- r Ga ( - rl)] (6) Due to the 1/Ir - r'| dependence of Ga (Ir - r') difficulty will occur in the summation process when r = ri Further difficulty will occur in the case of a coin shaped detector, p Zo r' R > Cl V.1 Figure 51. Point on the Same Z Axis Outside of Volume Element. For a point r located at the origin in the above figure a (Ir-r'|) dr' r Ga (Ir- ri) 1) (7) vi

-123 -when R >> h and R >> Zo. For a coin shaped detector it will be necessary to evaluate the integral Ga (lJr ril) dri (8) Vi analytically when the points r and r' have the same x and y coordinates. In the case of a wire detector a similar difficulty will occur. X dA 0 V. Az Figure 52. Point in the Same Z Plane Outside of a Volume Element, Ga (Ir - r- i) dr. =Ga (Ir -ri) vi when LZ~>> NdA and L_>Z Xo For a wire shaped detector it will be necessary to evaluate the integral Here again Ga (r - i I) dri (10)

-124 -analytically for the case where the points r and ri have the same Z coordinate o If the point r is inside the volume vi p / —111 1 t1 oz Figure 53, Point Inside a Volume Element, then: Ga (Jr - r) dr' - (11) V. -1 h 2l2 4+ R tan aR + 2 n (1 + (h)2 ) ~+ 2 Ca 2 '-p2 R in (2 h h) +R2 ) + 2ltCaL (Z~BB l(~ R

-125 -R2 ln (R) - ()2 1 If the point r is outside the volume vi but r and rt have the same x and y coordinates as in. Figure 51 then / Ga (r - rl) drt = (12) V. 1 2 L(Zo+)h ln (- -R - (Z, 2) n (Z R 9 h 2 -h 21 h 2 + C (Z +) R2 + (zo + )h2 + R +(Z ) -(Z 2 R2 +(Z )2 R2 ln(ZO h 2 R2 +(ZO 2) 2 2 2 -( + 2 )2 + (z h)2 If r is outside the volume v. but r and r have the same Z coordinate as Figure 52 t L.'-w;. 2dA -1 Dn Ga (r-.r'tj) dr' tan1 (2 ) rt (13) Vi. 2c A Ca - (.___~ + Ca 2x. 2x'

From this point on in this appendix a convention will be observed. X Ga (r - 1il) X f (ri.) vi will be understood to mean i the sum for all points except r = r. In the case of a coin shaped detector all points for which the x and y coordinated of (r - ri) = O will also be excluded from the sum. In the case of a wire detector all points for which the Z coordinateL of (r - r.) = 0 will be excluded from the sum. For all of the points excluded from the summation the analytical calculation of the integral p j Ga (Ir r'i) O_' V. 1 will be carried out as outlined above, Thus: rt (14) and S o(Ir - t_1 dr (1) ri r~.

-127 -In the limit as vi goes to zero the integral will be exactly equal to the summation. In the numerical problem it will not be practical to let vi go to zero. But the same detector will be calculated using several different sub-division sizes in order to see the effect of sub-division size upon ~ (r) o Next consider the integral (r_") G(r"' r' "r l where r (r) is a known function. Using the spherical harmonic expansion for G(r', Q' -er) the integral becomes, after integration over Q' rT (r _ ) b (Ir' r) x tr'tr a a b A b b Ya (r- r_') Ga (Lr r'j) dr"dr'dr Denote the real part Ya (s2) as RY (Qa) and the imaginary part as X Ya(2) Next consider two terms, for b - b' and b = -b', in the above summation. = r' - r" r r'

-128 -yb (.) Ga (P) x Ya () Ga () + (18) *-bf A -b ' - Ya (cp) Ga (P) x Ya ) Ga (9) = (-) -- F b A b'A - G ()a (9) RYa (cp) RYa (9) + i 92 Ya (T) Ya (9) + Rya a a A a (9 *bA bIA *.b? A * bA b (cp) RY (e) +RY () RYa (P),' K.)Y - i cS Ya (g) tQ Yab (A) + yb t yb t ( } bfA...b'A 1 a ( ) RYa (A) Note that RYa () = RYab(2) (19) Ya ()=-Y (Q) (20) Thus Y () G (cp) Y_ () Ga (9)+ (21) Ya (P) Ga (P) Ya () Ga (9 = Ga (cp) a (9) x Eb x RYa( ) Rba (9)

-129 -a a Where Eb = 2 if b' (22) b =a b =a a=o a=oo So the summations cahn mow be replaced with a=O a=O b=-a b=0 Thus (r") G(r"-r, ) G(r', Q -r) dr'"drd'dr = (23) r",r ',Q',r = (r") ),b yb (r'- r") Ga (Irt- r"l) x,r ',r ab b A Ya (r - r') Ga (Ir - r'j) dr"drdr Next consider (r") G ( ) dr r + a (' - r) () + i Fa (r') (24) r Where b RY (r - r.) V. a 1 Fa F(r') = Z b ( ri) Ga (Ir' - ri) x (26) a~~ ~ ~ _ ri ( 6

*b A Ya (r'- r) v. Next consider da (Ir - r ) dr (ra')+ iEb (r' ) (27) r Where b V b A REb (r') = Fb Ga (|ri r r) RYb (ri A( RE' fb Ga I rtl r. - r') vi (28) Ea(r') = G rt) (r ) v (29) Thus ~ (r") a_(jr' - r"l) Ya - r") - a( - r 31) x () r",r r yb (r r) dr"dr'.dr a a a ' - - - a1 [R k " b () a ar Eba dr' k Finally v (rt") G(r" r Ig Q") G(r ' ' - r) dr"dr'dQdr = (31) r"t r S ' r ab k {RF RE adb k

-131 -Fa (r XE_ (rk) } Vk Now consider the integral G(r" - r't Q') G(r' Qt r) dr"dr'dQ'dr= (32) r" r',Q.' r - ZJ Y(r rb A G (r' Y-b rl) Yx r_,r,r a,b Ga (Ir - r'|) dr"dr'dr Consider b H\b (r' - r") Ga(It' - r"l) dr" RHa (r') + iHa (r') (33) r -a ta Rb (rt) = 7 e RYa (r' A ri) Ga (Jr' r"tl) Vi (34) Ha (r') = Rb r b -Ab A Thus RH (r Eb r) (r-) r") v( (3) 3 a - - -- Thus b Thus(r" ( ' d"ddQ (38) rtl rt rt r

= X f(-' [{RE (r)}%-{ Ea(rt)} ] dr t abb b (a { L) ( i; (k)} { a (k a,b K

APPENDIX VI: NUCLEAR CONSTANTS The nuclear constants used are all taken from the Tables of (6) Neutron Cross-Sections in BNL - 325 In the energy range which the room temperature thermal neutron distribution the cross sections are either constant or have a 1/v dependence. For the case of a constant cross-section the average of the cross-section over a thermal neutron distribution presents no problem. For the 1/v cross-section it is well known that the cross-section averaged over a thermal neutron distribution satisfies the equation: CY= a- (2200 m) For water a = x 106 b/molecule = 940 b/mol. t 2 Since the absorption cross-section for oxygen is very small compared to hydrogen aa for H20 = 2 x x12..588 b/mol. 94.0o x 602 997 14 cm.588 x.6023 x 997 - a = 18.0196 cm1 -133 -

-134 -For j the classic value of 2/3A is not valid for bound hydrogen. Radkowsky ) has suggested the prescription M (E) = -a(E) >2 This formula gives values of T of.2 to.4 for energy around thermal. A value of = 0.3 was selected. For graphite C, 4.8 x.6023 x.6 12.1.385 cm-1 s X,..0033 x,6023 x 1.6.00023 cm a =.385235 cm t oi-:: For gold E I.. 98.8 x.6023 x 19.32 1 cm Ia,1.:2,62 = 5.17 cm a X,9.3x,6023 x 19.32 = s192

-135 -= 5.719 cm 1 t For indium =; 196. x.6023 x 7.28 = 664 cm La= 2 - - 114076 = 66 =2 x.6023 x 7.28 C11476.0842 cmi X = 6o7242 cm t

BIBLIOGRAPHY 1. Bengston, Joel, Neutron Self-Shielding of A Plane Absorbing Foil, Oak Ridge National Laboratory, OF-56-3-170, Microdard, United States Government Printing Office, Washington 25, Do C., 1956. 2. Case, K. M,, F. de Hoffman and G. Placzek, Introduction To The Theory of Neutron Diffusion, Volume 1, United States Government Printing Office, Washington 25, D. C., 1953. 30 Davison, B. and J, B, Sykes, Neutron Transport Theory, Oxford University Press, Oxford England: 1957. 4o Fitch, S. H. and J. E. Drummond, Neutron Detector Perturbations, Livermore Research Laboratory, LRL-95, United States Government Printing Office, Washington 25, D. C., 1954. 5. Gallagher, Tom Lo, "Flux Depression Factors for Indium Disk Detectors, Nuclear Science and Engineering (January 1958), 3:110-112. 6. Hughes, Donald J. and Robert B. Schwartz, Neutron Cross Sections, Brookhaven National Laboratory, BNL-325, Second Edition, United States Government Printing Office, Washington 25, D. C,, 1958. 7. Jahnke, Eugene and Fritz Emde, tTables of Functions, Dover Publications, New York, New York, 1945. 8. Klema, E. D. and R, H, Ritchie, "Thermal Neutron Flux Measurements In Graphite Using Gold and Indium Foils," Physical Review (July 1952), 87:167. 9. Morse, Philip M. and Herman Feshbach, Methods of Theoretical Physics, McGraw Hill, New York, New York, 1953. 10o Petrie, C, Do, M. L, Storm and PO F. Zweifel, "Calculation of Thermal Group Constants for Mixtures Containing Hydrogen," Nuclear Science and Engineering (November 1957), 2:728-7440 11. Skyrme, T. Ho R,, Reduction In Neutron Density Caused By An Absorbing Disc, British Atomic Energy Research Establishment, M. S. 91. 12, Thompson, M. W. "Some Effects of The Self-Absorption of Neutron Detecting Foils," Journal of Nuclear Engineering (British Journal, 1955), 2:286-290. 130 Zobel, W,, "Determination of Experimental Flux Depression and Other Corrections for Gold Foils Exposed In Water," Section 8.8 in Neutron Physics Division Annual Progress Report, Oak Ridge National Laboratory, ORNL 2842, ppo 202-203, United States Government Printing Office, Washington 25 D. Co,, 1959. -136 -

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