NEUTRON WAVE PROPAGATION
IN A HEAVY WATER, NATURAL URANIUM,
JOHNNY HALL .DUNLAP
A DISSTFhT iTlrJ 'l i .: I TrP TO TIHE GI C.\iT.r TE COULD CIL OF
THE L'rN..'EKilTY OjF LLOF.I.DA
TN P.AILL FTLLiPLU. i OF THE PL(,.*Li .EM.NTT IlUR TIL
bCEGFF OF DOcTOu Of FrHil..OOPLl
UNIVERSITY OF FLORIDA
....... ..iii .. .;; .
Ilany per -orln h-ave had a part in maki.:in thi disserta-
tion possible. The initial suggestion that the author take
a leave of absence from teaching at the School of Enin- eerriln
at '.'anderbll t Uriiver sity a in order to work to...'ar. the Ph. E.
degree came from Dean Robert 3. Rowe, His interest and
encouragement are greatly appreciated
Special thanks are due to Dr. Rafael E. Perez for hi:
overall guidance of this work during his service a; chair-
man of the superv.'isory committee until he resiqr,.ni from the
university to accept = position 'ith the Oak Ridge nationall
Laboratory. The author ia grateful to Frrfes- or .31en J
Schoessow for interest and assistrance while s-r-:.ing first
as a member of the committee, and later as the chairman
The author .-.'ihes to thank Dr. William F.. Hutcherson, Dr.
Billy S. Thomas, Dr. Billy 3. Duna-.ant and Dr- !lhran J.
Ohanian for their help as meiTbers of the comiTmittee The
author also thanks Dr. Robert E LTfrig for his helpful ad-
vice and service on the supervisory committee until his
leave of absence from the university in 1967.
Thanks ,re due also to the Ford Foundation for fijnan-
cia1 support through a fe 1iilc'ship for the first two .ears at
the (JUnr ersit: of Florida The author 1i also grateful to
the Uni.-ersit. of Florida and its nuclear Engiineering
DepartTment for firnncdial -supF.rt through a graduate aEssist-
.jrntship for the last half of the work at the uni,.ersity
.Juring which tim-e ih h.as been larqg-ll free t.-o prepare the
apparatus, conduct the experimental portion of this work,
analyze results and fincall '.-.rite this paper.
AcknCowiedrgenrt is also mride to the D'r.isiorn of Huclear
Education dnd Trainirnq of the Unlitd States Atomic Energy
Commission, for the loan of the he:'."' water and natural
ujranium necessary. for the subcritical assembly. Li1ewi.e,
thanks are .ue the Oak RidJae National Labrtotory for the
The author would like to thank the staff of the
Comruiunications Ecience Laboratory of the uni-.ersity for the
use of their IBM tape-to-card punch machine, and the staff
of the Uni'.-ersity Computing Center for the use of their
facilities without which this type of experiment and data
analysis would have been impossible.
For their aEssistance in the pre.-paratiorn for the e:.-
periment, the author thanks the foil:c.'ing: Rob-rt H. Hartley,
Fred A. Fiimo, and Henry H. Ioos .
earlyy all the data were taken ..ith the ass itarn.e of
John WthitfolJ, who rec.-:*rcJe.jd ,ta, mainait ne.i the le-ctronics,
and operated the neutron genir-retc-r. Hi s gooCd .:ol', che_,erful
spirit and willingness= to =ta_ through lunch or after hour-
for the good, of the e::xperim ent are appre:ciat-d.
The author thanks [ills J. Di L- f.-or hi a-sis-tance in
the iuse of the CEPTR and iBUCkSHOT conLpu.ter cO..jd- and for
sharing much of the literature and data he ha's a-cc umulatedi -
cornce-rnirJ heavy. water. Dr R'obert -:-. Coct:rell has al.so
been helpful in the area of computer codes
The author is Especially gratefull to his go-o frien-ri
and once fellow' student, Dr Ray Stur-ji- Booth It is .'eli
remembered that it was Bo:oth'sa anal tical appL-roach that
located the last "bug-?" in the electronic s-'tem, maakling
it possible finally to start taking good data earl:. in
April, 1965. Also, the heart o:f the analysis .a. per formed
through the use f computer ptr codes written e-arlier and later
modified for this experiment by Booth. In many -wa-y this
work is a refined extension of thie experimental and lata
.ana i,- is techniques in r eutLcon '.u..a-. prLc pl tion .aJ l.'.:1.'lop-.
b.' FE.rez, ecotrh and Hactl-.'.
Finaii .', rec .oni izin;g that tl-ieL i no reid -.-ithrout a
b-. : inn : n the authOiL .is-hes to th-ank his rrth. r for her
*.r.i inal ins i.stnc.: that her ,orlS ha'.e at le-s t orn y.ar
.f College? eiucat ion.
TABLE OF COL'In EliNT
ACK':IOWLEDGENT . . . ...
LIST -OF TABLES . . . . .
LIST OF FIGURES . . . .
AB TRACT . . . . .
I. I NC F ODUCTIOil . . 1
II. DERIVATI(,O COF THE DISPEP.EIO[i LAW BY
TUO-GP.CUIJ DIFFUSION THEORY . 11
III. AGE-THEORY TREATMEI T . . 14
IV. EX PEP. IME[TI' L APPARATUiiS . .. 28
V. EXPERIMEtlTAL FROCCEDUP.E AN[D DATA AiALISI .. F*
V"I. EXPERIMEliTAL RESULTS FOP HEA,"; WATER .. 79
VII. EXPERIMENTAL RESULTS FOR THE SUBCRiT'IC-AL
ASSEMrBLY . ..... .14
VIII. CC:MIPARISONi OF THEORY ArID EXPERIMENT 1 .
IX. CONCLUSIONS AI1D RECOr.MIEtDATIOJS . i8
TABLE OF CCONTEriTS ICcnr. inu.i])
A. SPATIAL DISTEIBUiTION OF [iEUTROLIl
FROM THE THERILALIZII]G A-FFARATlU .. 1
b. DETEPJ-IiIATIO[l OF RES'I'LUTIOIJ TIMlE
FOP IJEUTROCli DETECTIOli S'STEI1.S . 193
C ..A METHOD FOR DETEPJ.IIlINiG THE RESOLUTION
LOSS COP-RECTION FACTOR FOR \ DETECTOR
EXPOSED TO A FLI.-.: FIELD WHICH VARIES IN
A REPETITIVE FASHHIO Ill TIHE ... 199
D, A TEST OF THE TIC AIIALYZER FOR
ACCURACY OF TIME DIVISIONl .. 203
E. CALCULATION OF LATTICE P-ARAlIETERS 213
F. CO[i'EIJTIOIiAL DIE-AuAY MEASUREMENTS FOR DO 236
G. COlJVEIjlTIONtAL PULSE MEASUREMENTS Ill
THE SLBCRITICAL ASSEMBLY FOR SEVERAL
DIFFERENT WATER LEVELS .. . 239
H. E:,ACT SOLUTION TO THE AGE-THEORY
EQUATIONS BY A SEJ1IGRAPHICAL METHOD 243
LIST OF REFERENCES .... .... . 261
BIOGRAPHICAL SKETCH . ....... 265
LI T CF TABLES
Tab le FP
3-1 Values cf Parameters for the u-c.ritical
.*'m.s bl Used in t'r. Solution of the Afg' -
Theory Equa tions b: the tie'.*.'ton-Rapr.(sor
Ilethod . . . . 5
.-2 Comprison-i or RPsujlts from trh .ppFroirmate
..g rT- c r-. : 1 u i.o n w ith Thosri' from the
lewtocrn-Raphsor n CIomTpuIter S solution . 26'
5-1 Outlirn of Exp:-r im n r t r!a surer t-e rt rr.n
D,0 Moderator .'itho c Fu l . . .
5-2 Outline of Experiirenti i Meauruemrnts: with
Loaded Subcr tical A se i . . .
5-3 Equipment Secttinrg: and r-pical Int; tral
Count Information for the Wia-.-e P'r.op.ga-
tion Experiments . .. . . 75
6-1 Stand.ard De-.-iation for Figure 6-5 . 85
6-2 Amplitule V.ersu Position in Trars.-.er-se
Flans at Z = 5 cm for Certain Frsquencies 9;
6-3 Experimental '.'alues of in D., as a
Function of Frequ nc' .. .. . . 11l
6-4 Experimental valuess of .P In D.o as a
Function of Frequenc'. . . 121
6-5 Sensi tl.'i t of the Function a E to
Experimental Error? nr the 'Value of -i
in DO . . 1i '
6-6 Experlmental '.'alues for P Q D ,
and C for D.O 1 35
LIST OF TABLES (Continued)
.Ta ble Page
6-7 Colmpariso'n of E.-:xerimentall'1 Determined
valuess of C anJ D frmTi This Work .ith
Published values . ... 39
7-1 E:.:perimental 1V.'aljueZ of a and F In the
Subcritical Assembly . . 158
8-1 Co-mpariso-n o'f Calculated Results frL',,m Age
rTheory and from T'.-,o-qrrup Theory .,ithh the
Pesults from the E-:perirment .. . -. 164
6-2 Results of the r..'o-group Theor',' Calculation
ijsing the Parameters in Table 3-1 .. ... 169
6-3 Lattice C-nstants for ITwo Subcritical
Configuratio lrns . . . .. 176
8-4 The Dependence of the (in) C:oefficients
up:n Lattice Pitch for One-inch Diameter
tlatural Uranium C.'lindrical Slugs in D0 179
D-I Observ.ed Data and CalculateJ Results for
rest of IMC Multichannel Anaiyzer ... 211
E-l Basic iHoiminal Dimensions of Fuel Slugs 214
E-2 Calculate-d V.olumes and 'olume Ratios
Based on the Unit Cell .... . 215
E-3 Cross Section Calculation for 6061-T6
Aluminum for 2200 Meter,/sec Neutrons . 217
E-4 2200 Meter.'sec Cross Sections for 6061-T6
All.oy on the Assumption That Impurities
Are Present in Their Maximum Allowable
Quantities . . ... . 219
E-5 2200 Meter.sec Cross Sections for 6061-T6
Alloy on the Assumption That Impurities
Are Present in One third of Their Maximum
Allowable Quantities . . 221
LIST OF TABLES~ ICntinued.j
E-6 Cr.o, Section Caicu nation for Fuel Slug
Cladding Allo,.' for 22"00 Meter sec tieut,-,r, -
and for Nomiinal Alloy, Compo.sitior 22
E-7 Calculation of Fictitlious Cro.s 'Sect.ion
for .Aluminum for Use in CEFTP. and EUC:.iHOT
Code-s to Account for Pr- sern c of End
Ciaiding con Slug= . .
E-8 Cross Sections for Use in- CEPTR Coce 2
E-9 P.Re.ionr into i'F lch the Unit Cell Was
Divided for U=s in CEFTP Cod .
E-10 Data Cards Used ,-.'ith CEPTR Caiculation
for Subcritical Asse ibly . .. 2'9
E-11 Average Scalar Flux for Re'Iiors of the
Unit Cell As Calculated by CEPTR Coje .
E-12 Regions into i'Which the init Cell Was
Di'.'ided for Use in ELUCKSHOT Code .. ...
E-13 Identification of Informatior on Input
Cards for BUCKESHCT Code .. .... 2 4
E-14 Data Cards Used ..ith BUCKSHOT Calcu-
lation for Subcritical Assembly. 235
G-1 Data and Results for Pulse M.easurenimeints in
the .ubcritical Assembl.- As a Function
of Hoderator Level . . . 4i
H-l Values of Parameters for the Subcritical
Assembly Used in the Solution of the Age-
Theory Equations by the Semigraphical
Mlethod.. . .. . 24
LIST OF TA.BLE-S ICont inrued
H-2 Calculated FPesuiLt from ian E:act Jo iution
of the Farmi :-.g Equatio:ns b," the Semi-
graphicai M.etrihc.i U'sinfg tihe Farameters
Llitce in Table H-i
H-3 Structur-e of Fortran Frogqram Written to
Aid in the Solu.ition of the %ge-Theory
E.:ua t onS . . . 256
H-4 FCORTP.AI FroqrIm for the A.ie-The..ar
Equations . .. .. ... 258
LIST OF FIGURES
4-1 Overall 'ie.:- of the E.perimental Apparat',s 29
4-2 Front .'ie:.' of the Subcritical As embr: : :
4-3 End :ie'. of the Subcritical A sembl'.'
4-4 Catch Tank and Support Structure
for the Subcritical Assembi 3.
4-5 Thermalizing Apparatus . 42
4-6 Diagram of the Principal Detector System 47
4-1 Diayram of the Reference Derector S;'stem 4,
4-3 Oscilloscope Display of Signal at .Various
Points in the Principal Detector S.ste 49
4-9 Oscilloscope Display of Signal at various
Points in the Reference Detector S's tern .. .
4-10 Plateau Curv.es for the Principal
and Reference Detector Systems . .. 51
5-1 Flow Diagram for the Data Accumulation and
Analysis for the Wav.e Propagation Experiments 66
6-1 Geometry for Transverse Flu:-: Measurements 5,
6-2 Trans-.'erse Flux in Hea'.' Water at Z = 5 cm
Measured with Cadmium Shutter up .
6-3 Transverse Flux in Heavy Water at Z = 5 cm
Measured :with Cadmium Shutter do'..n 2
6-4 Subtracted Transverse Flu:-: in Hear'.% Water
at Z = 5 cm Obtained by Subtracting Shutter-
down Flu:: from Shutter-up Flu : 83
LIST OF FIGiJR'ES I Ccnrt rinujed)
Fi iure Fa ,t
6-'5 Conti rnuc.uj Source Flux Distribut rn
in He-a- Wiatj er -.lornq the 2 -;i . .
6-6 Trar,n .erse Flux Distributiion at '.'arious
Tirrm Inter-als EA.fter the Beinr nninq of
tne- Tar;et Pui;e . .
6- .-Ljiplii tuJ:e cf Fourier Comporinents ''arcus
D tectcr FPo-it'lon Alo.rj a Trans- eCir
Axis for Se"ecal Frequencie . .
6-9 Shutter-up, Shutter-.jo.v?. arn: Subtracted
Pul -se '*'er uB Time in Hea ,' W.ater at
2 = cm . . . .
6-9 Shutter-up, z.nutter-d"o. n, ani Subtracted
Fule '.'er=us Time in- e Hea'- Water at
S = 4- m . . . .
6-10 Shutter-up, Shutti-d-o.:n, and Subtracted
Pulse \''er.us Time in Hea"y Waiter at
S = 70 cm . . .
6-11 Niormaired Subtracted Pulse '.ersus Time
at Several Detector Positionr Along the
....: . .. .
6-12 Width of the Subtracted Pulse at 10 per
cent and 1 per cent of the Feak Value as
a Functic)i cf Z Position in Hea''y Water .
6-13 Propaqation of the Pulse Peak in
Hea. W ter . . . .
6-]4 .ATiplitude of Fourier Componlents of the
Subtracted Pulse Versus Frequency for
Se eral positions . . .
6-15 Amplitude of Fourier Components- of the
Subtracted Pul-e Versus Po-sition for
Several Frequencies ..
LIST OF FIGURES i C':I-ntlnued.j
E-l16 Ph~=ie Angle :if Fourier CoIripon.C nts '..er;sj-
Fosition for Se-eral Frequencies.. In
Hea'-"/ Wate r. . .
6-17 Gain V.'erus Frequency in He"a'-:' Water for
3e"eral Detector PF.titic.ns 2
6-18 Phase 'Versus Frequency in Hea-.-y Water for
Several Dtetctcr Pos.itions 6
6-1'9 Graph of. a V.ersus Frequency for Therm1al
lieutron Wa"e Fr.paqgation in H-ea.' Water i.27
6-20 Graph of 5 Versus Frequency for Thermal
Ijeutr,.Dn Wav'e Propatat ic.ri in Hea.-: Water .. 1I2
6-21 The Furictic.n 20.-I Versus Frequency, f-or
Thernai Nleutron Wa'e Propa..ati.:.n in
Hea..'- Water . . 129
6-22 The Function a' '.'ersus Frequenc.
Squared for Thermarl Neutron jWa'--
Propagatio.n in He -.--- Water 1
7-1 Background Flux Distribution Alcng the
Z A:.:is in the Subcritical Assem.bl 142
7-2 Continuous Source Flu:: Distribution Alona
the Z Axis in the Subcritical Assetbly 1144
7-3 Shutter-up. Shutter-down, and Subtracted
Pulse at Z = 5 cri in the Subcritical
Assembly ........... 146
7-4 Corrected Subtracted Pulse at Se.'er.i
Detector Positions Along the Z A;-:is in
the Subcritical ssemblv ... .. 14
7-5 Propagation of the Peak of the Subtracted
Pulse in the Subcritical AF.semrbl. .. 49
7-6 Width of the Subtracted Pulse as a Function
of Position in the Subcritical Assembl,- 151
LIST OF FIGURES (Cc. ntinued1
Fi gure giqe
i7- AnrFpl itude of Fourier Compcrcnents Versus
Frequ'ncy for Se.'erl1 Pos it ions in the
5ub:riCical AsseiblTI .. .. 152
7-8 Aimpl itudje of Fourier C-omrponents Versus
P-sition for See''ral Frequencies in the
Subcritical Asscmbl... ... .... i154
7-9 Phase Angle of Fourier Comp:,nents Versus
F.osi.tion for t5.1eral Frequcncies in thl
Subcrti:,z Assembl'. ...... .155
7-10 The Functi.-.rn a and '.'ersus Fr-querncy
for the 3.ub;ritical Assembly .. .. . 157
7-ii Gain 'Versus Frequcncyi for Several Detector
Fositicn- in the S3b:ritical Assemrbl-, 161
7-12 Phase Versus Frequentc. for Se-eral Detector
Fositic.ns in the Subcrit:ical Assembly 162
8-i Gr.aFpI-i f a .'ersus Frequelncy for the Ex:pern-
ment, for IP:-rou..p Theory, and for Age theory 165
8-2 Graph of E '.ersus Frequency for the Experi-
ment, for T'wo-group Thecr', and for Age Theor'.' 166
2-3 Graph or a from Age Theory and from
the E:-:periment ... .. ..... 172
8-4 Graph of 23 from Age Theory and from
the Experiment . . . 173
A-i Subcadmium Flux Acrossc the Horizontal
Midline of the Graphite Face of the
Thermalizing Apparatus .. . 190
A-2 Epicadmium Flu:x: Across the Horizontal
Midline of the Graphite Face of the
Thermalizing Apparatus .. .. 191
LIST COF FIGLtPES IContinju.-J I
A-3 Total Flux Across the Horizontsl
Midline cf the Graphit.e Fac. ..f the
Therma lizing Apparatus . ... 192
B-1 Graph Used in tr.:- D.termrninat.in of the
Pcsolutic.n Tirrme f.r j-the R. ferernc-
Detector System ..m . .
B-2 Re.solution L.5ss Corriction Factors for
the R, fc-re- nc D.tc tcor 3 'st m . .. 1 7
C-1 Iilustration of the -comri:nistur-r f.or
the Deri.at.cion of the Equi"' i.:nt
r,-,ctani] uiar Pul e . .. 2:i1
C-2 TIhe Equii.alent Rectangular Puji.- and
the Actual FPuls f..r the Refereinc.
Detectcr System . . 204
C-3 The Equi-.-aent Rectangui.a r Puia.
Squared- iand the Actual Pulse Squared
for the Reference DcEtict..r System ..
E-1 SFlat-.'e Scalar Thermal Flu:: for thl-
Equiv'.alent Cylindrical Unit Cell
Calculat-ed by the CEPTR Code :1
0-1 Decay Constant .e'rsus Geometric Bucklinq
for the Subcritical Asserrbl '.' with Varius
Moderator Levels ... 242
H-1 Solutions to Equations 2-4 and -?-5 for
Frequencies from 0 cps to 300 cp 245
H-2 Solutions to Equations 3-4 and 3-, for
400, 600. and 800 cps .. . 246
H-3 Solutions to Equations 3-4 and 3-5 for
1000 cps : . 4.. 247
H-4 Solutions to Equation H-I for Zero
Frequency and for Various Values of BI2 250
LIST OF FIGURES (Corntinuedl
Fi gre Page
H-5 Solutions to Equation H-1 fo:r 100 cps
for "arious Values of I2 . .. 251
H-6 Soluticns to Equation H-i for 600 cps
fcr ''3 lous values s of e12 . . 252
H-7 Solutions to Equation H-2 for 50, 100,
200, and 300 cps for variousu s values s
of BR2 ... . . . 254
H-8 Solutions to Equation H-2 for 600 cps
for "aric.us V'alues of BR2 . .. 255
Abst-ract of DissectatlOn Prt-sented tO the Graduate -Cori.irl
in Facrtal Fulfiillment: of the P.equ'rimennLs for the D-Cree .-f
D' cto.r of Philos i phy
[lEUi'JRO WA'.E FROP.-RAG TIOL' Ii .A HEA'.' Wi.ATER.
[UATLrRAL URA.JIIUI.i. SUBCRPITICAL L AS'ErIELY
Jolhnn" Hall Dunlap
Ch irmTan: PcLf-of sor Gi-n J. Schoe.s --'
rIajor Department: [ucleac Enr.jineeri.n
The proCi-agatioo n of neutron ':-:'- in 3a ml.iplr:iying
medium .,as studied .:d:iperiri retaliily, JrId the re-su itirig dis-
per-sion '..' ;:as omiipared with that pr -iict.ed b t-:o
tleoretical mod.Ji: a tw.o-gr.,uip diffusion theory model.
anrd a Fermi age diffusion model
Thermal n utron '*.av.e pr:pri:ti on in a Zubcriti-:a
assembly. was studied ''a nLJui'merical Fourier naialys'is .:.f a
thermal fneutro'nf puls- of' lo.: spatial harmronic distortion
introduced through on- boundiary of th. assembly. COne
detector explocL- th-e a:-.:5 of the assrmbl' aan supplied -
description of the pulse in the time domain for input to a
:':." I 1
102-4 channel anal.ze.r A second.J dEtctccr sL'.-erd to monitor
the source str length for each run, prd."i.jing a means orf rzc'-
ma izing data taken- at differeFnt p.-ositionr. Correct i.on-s for
c*ujntin' rasoilutiO.n losses and backgr-.und were gi"en careftul
attention. A computer coJe .-.'as use.j t.. perfoTrmi the Four ier-
ana i_.si of the pul--se, givingg amplitude and phase iinf.ormat ion
s functions :f frequernc'. for e ach detector position. The
e:-'p Lirrnntal J isper-:ion law was determined by further analysis
of this o-utput t..- obtainn the real and imaginary parts of the
in-erse relaxation length of the F_.-.ricr c-.omrpnerts
Sin.c an aimos- perfect fundamental spatial mcde had
been oibtainrid for the soirce pulse in the experiment, the
cz.rrespondinq theoretical calculations w.ere specialized to''
this mode in the trans--erse directions. The source neutrons
were assumed to vary sinusoidaiily in time, and w:erLe intro-
duced into the assembi. through the X' plane at Z = 0
In the two-group treatment, the model led to an
equation for a bucklinq-like term front, W.hich the parameters
describing attenuation and phase shift as functions of fre-
qucnc', could be found. The Ferm-i age calculations were
based on a complex equation deri-ed b, P.. B. Perez, which
yielded two real, co.jpled, transcendental equations
The principal conclusions from the abov.'e work are:
(1) The experimentally determined dispersion law is a smooth
function of frequency and both the real and imaginary parts
of the inv.erse relaxation length increase *..'ith increasing
frequency in' the range of frequencies studied.
(2) The calculated disper-sion laws from both models aqree
in general trend with that of the experimnr t except for a
dramatic failure of the two-group theory for frequencies
over 353 cps.
(3) Effects of the a low'iri-dowrn process are important at
high frequencies. (That the Fermi age theory describ-ts
the high frequency behavior in a more correct fashion than
the two-group treatment is probably a con-sequenc- of the
more accurate description of Eilo;ing-dorwn in the aq- theor.
141 Since neither model gives entirely _ati factor agr3 -
ment with the experiment, it appears that the thermalization
effects should be included to describe the dispersion law of
(5) The dispersion law. poses a severe test to a theoretical
In addition to the measurements made with the loaded
assembly, the dispersion la.' for hea'"y water without fuel
was determined, and further analysis of the dispersion data
yielded the follo.Jing experimental values for the diffusion
coefficient and the diffusion cooling coefficient. respec-
5 2 -1
ti.'ely: D = (1.996 .002) x 10 cm sec and C =
5 4 -1
(3.0 .381) x 10 cm sec
Those in"oi.ed in the design and control of nuclear
reactors are necessarily interested in neutron kinetics_ in
.'arious arL rangqements of materials. Information in this area
ihas been sought by many persons through the t..'o principal
a'.'enues of scientific in'.'esctigation: ri-., studying the prob-
lem through a mathematical formulation based upon ar. assumed
ph/yical mojel, and by the performance of experimental ob-
serv.ations and measurements. An-_' genuine difference bet.;een-
the experimental results and the predictions based on a model
must be attributed to inaccuracies in the model or to errors
in the experiment.
A. lieutron Wav.es
In this dissertation, neutron behavior was investigated
in a multiplying medium by studying the propagation of neutron
wa.a'es in a subcricical assembly of natural uranium and hea-v.
water. The investigation .-:as conducted by theory and experi-
ment followed by a comparison of the results from the two
It should be under t.ood at the -.ut set that the neutron
wa.'ves spoken of are not iE Br.,glie w'a-es. but are temprFor a
.*:riationrs in the neutronr population density per unit 'olume,
which ma,' be observed at an' position in a moJeratingq or
multipl-ing medium, whenr ar. aFppr.'prlate time-de-F.ndent Csojur.e
of neutrons is placeJ at the b':undary .of the rmediuim.
It is also important to note that when one ipe-ak. of
a sinusoidal neutron w.a-'e, ,h d is rn.:t imply a ph-ysiczl im-
porssibility. In practice, the sinu-soial si3nl rides on
top of a background of population density siuci t-hat the
algebraic sum of this backirouni plus the sinu.soidal siqnal
is never negaCtr.'e. Thus, one is not forc-ed to find 5 supply
of negative neutrons to compacse half of the signal in order
to perform the ..:ave experiment. The bottom half of thef
sinusoidal pattern represents riegatie neutrons no more
than the troughs between the crests of ocean c:a.es represent
negative w.'ater. Of course, in this analogy, as-suinr that
the air-w.ater interface is spatially s-inusoidsal, one could
represent the depth of the water as the sum of the average
depth plus a sinusoidal term which is negat.'e half of the
tire. Such a representation worksk s .'ell for neutrons also,
and is not a source of Jifficultl' in the mathematics.
M. [1. Moore II) draws an 11lustrat.ive analogy between
r.eutron wa"es and elastic wave'. in a v.iscous dilute gas.
He pictures a '.omijrrme ojf gas in a aonr.tainer at rone end Jo
whichc h is a fle:-~ibie diaphragm and points to the Similarity
bett- eeRn neutron a..ea-.Se a and the a.co st ic wa'.'- that can bn e
eneiirated within this container if the .iaphrag.T is caused
to 'ibrat_- in a harmonic fashion. Moor;e 5use tlho framework
of tciis analog; to illus-trate various s other points nrt und'ir
cons icderation here For our pre-sent purposes, one may
obser'.' that cthe acoustic wa'- is one in whichh th- popu-
iation d ensity- of the gas molecules could be detected by
.uitable pressure, measurinr d,'.'ices. In the case of neutron
'a.'a.es, the population density of the neutrons may' be d.tectcd
by con'-entionia n,-utron Jd-t_.ctors which report a count rate
proportional to the neutron flux. variationsn s in count rate
as a function of time or position ri.-c information concern-
ing the neutron ,wave. .An important difference between the
neutron a..a.e and the acoustic wa'.'-, ho.we.ver, is that the
neutron wa.e is studied in a material medium in which the
neutrons scatter about, colliding with nucl.i of the medium,
but not with other neutrons. With the acoustic w'av.e, there
is no medium other than the gas molecules themsel'.-es, and
their collisions arc with each other. The absorption and
leakage of the neutrons are not described by the analogy,'.
B. Historical tiotes
Although the propaiat ion of neutron w.a -e wa= inr'.-ti-
gated in 1i'94 b' Weinberg and 3ch.-wenier 1 2 wh..ho u-sed .an
oscillating neutron ab-scrber in muliriyirng mi-edium, the
subject has not recei.-ed as much atte t io.rn since thn'iri a
has been gil.'en to related techniques such as e:-xp..nri, tlai an.
pulse die-aw..'a. measuremlents Ten years. later, in th-ir weii-
kno.r, text on nuclear reactor the. .r.' 3.1 We irbert and
Winner included de ri'.aticns for the amp.itude attenuatic.n,
propagation :.eloci!t, san.nd .w length of neutron ..e.'es in
inFinite media. In 1955 Raie-.-ski and iirc.r'itz reported
the determination of the mean transfer free path o.f therm.li
neutrons b' measurement of the cormplip:e diffuse icrn length.
The measurements .-'ere made suiccesstfully it, both graphite
and heavy .-;ter, and a mechanical source was used- 14 1.
In thet field of Nuclear E inineering, o:ne turn ultimate-
ly to the Boltzmann equation. writingn g it in a form which
describes the neutrn populuticn denrsi~' as a function cf time,
position, speed (or energy), and direction. The solution of
the Boltzmrann equation as applied to a nuclear reactor is to.
difficult to perfo-rm without resorting to simpiFlifying ass'ump-
tions ..,hich result usually' in an attendant loss of detail in
the description of the neutrons being studied. The detailed
Information most frequently surrenJdered2 in or Jer to obtain
a formulation amenable to sEolution is that of the detailed
diirectio.n of travel of the neutrons. Such a surrender is
characteristic of the often-use diffusion theory., in whichh
isoctro-pic scattering is assumrred.
Cne-group dJifrfusi.o theory vas used by R. E. Uthrig
I.,61 In 1959 and 1961 to relate phase shift and attenuation
to diffusion parameters. The results were compared with
measurements in both ..cater-moderated anj graphite-imc.derated
R. 5. Booth 17) extended the diffulsion treatment to
include t..o groups, and concluded that t.o-gro.up theory
predicts the presence of tw.:o thermal wa,.'es that propagate
in the fundamental spatial mode.
Multigroup Jiffusion theory was applied to a subcriti-
cal assembly by. Perez, Booth, Denning, and Hartley (8), and
showed promise of relating resonance wave effects to the
paranreters of the system.
Both the experimental methods and theoretical treat-
ments employed in the study of neutron vwa'.es have undergone
considerable refinement. The energy dependence was given
additional attention by Perez and Uhrig 19) b.' use of an
expansion in associated Laguerre polynomials. A power series
in i ',.vD was obtained which related the complex inverse
relax-ation len~rth to them-iializat cn and diffusion .prop.' rtle
Pefinement: in experimental -.ior- .'ith neutron '.'a'-s
have appeared in two principal areas: the source ua-ed, and
the method of data accuulation and anr'lyis The earlier
rotating absorbersa or Lr.tatinr Zourc-s hae a been largel'q
replaced b, the use ;.f sinusoidal ly modulated neutron gener-
itors using the I'D,TI reaction .:r the iD,D) r eacti n.
Higher source strengths and 3 wilder frequency range ha'-e
been obtained. The use *-f s'.. itchin circuits and systems
of scalers to record different inter- 31 of the det-cted
signal 110' has been replaced by the use of mulichatnnel
time analyzers and computer ccdes.
The use of numerical methods to Fourier analyze the
pulses represented an impro,.,emrent c'er the data anal _is
performed by Raie'.ski and Horc.oitz, who did not ha'.' time
analyzers, and hence 'ere restricted to less sophisticated
Moore I11') extended the Fourier analysis b,' the in-
clusion of the Fourier widths, which allow the calculation
of not only the dispersion law but also of its higher deri-a-
ti.es. An application cf Moore's method '*.'.as performed by
Perez, Booth, and Hartley (12) in the pulse propagation of
neutrons in graphite.
Since it is not possible to review here all the liter-
ature no, in print ori the subject of neutron 'a'.'es, many
good apers, both theoretical and experimental, ..ill not
be mentioned. However, the rea*.'ier .-,ouid find the work of
P.. L. Bremhi (131 helpful, as he expires the use cf both
on-e-e.eloci ty diffusion theory and ige theory in the P1
approximation as tools for anal/'zing neutron .'ave experi-
iments, and deals .itih the dispersion parameters of the
system as functions of frequency.
There experiment described in the body of this work
is of such a nature as to afford an opportunity to seek
experimental e'.vidence of the existence of the "exceptional
frequencies" first discussed by Moore 114). He spoke of the
interesting possibility that there may exist preferred
frequencies of excitation which produce a wae ha'.ving either
no spatial attenuation, or no phase change while traversing
the me.diuim, or both. This phenomenon is alsc discussed by
Brehm 113 and 15), but it has not yet been found experi-
C. A DescriptiLn of This Study
The principal emphas is in this '.-ork is directed to;:ard
the experimental appro.cn to the study of neutron '..'a-e prcpa-
gat.i.on. The theoretical portrrions of the '.:ork make use .:,f
available theories, e'.'aiuatinq the results fraom these fcr
system parameters appropriate for the nuclear system in :hicl
the experiments were performed, so that a critical comparison
of theor'r and exp-eriment car be made. The Fr incip i a il
of this '..:ork ma.' be listed as fc..llows:
1. To determine the dispersion law. fr:.r a multiply-
ing system by experimental means.
2. To evaluate the performance of vario-us theoretical
models in describing the dispersion la...
The dispersion la'..' spoken of above is the quantita-
ti'.e relationship Letreen the complex inerse relaxation
length of the neutron '.'aves as a function of frequency The
terminology stems from the use of the term in elecctroma rnetic
theory and the fact that the relacicnship bet'..'een the complex.
inverse relaxation length and frequency: for neutrons is anal-
oqous to that between .:ave number and frequency for electro-
magnetic :-.oa.'es. Some secondary, although essential, aims of
the 'work are:
1. To develop a tnermalizing apparatus which yill effec-
tivelv thermalize neutrons from the (D.T) reaction to
provide a suitable supply of primarily thermal neutrons
in the fundamental trans'.erse spatial mode to ser.'e as
a source for ..ave or pulse propagation experiments.
2. To further explore the ad. antages. and disad-antages of
the Fourier analysis= of pulse FroFpaiation data .a a
method of studying nc.iutron wav propagation (161.
? To determine diffusion and thermalization parameters
of the moderating media by further analysis of the
experimentally determined dispersion' law.
4. ro gi"e attention to an, excep:tional frequencies which
may be di.s:o ered ha3'.in; unusual properties of propa-
Wa-ve propagation was studied b'.' ,nian of Fourier analy-
sis of pulse propaqati-on data because:
1. If a rmo Jlated source were used, a new s-et of data
would be necessary, for each frequenc-y, whereas one
set of pulse jata can be Fourier analyzed aE a
function of many different frequencies.
The use of the method in .raphite was su-cessful I16),
and the a"ailabillty of the MOORE 1MOMENTS code made
the method further attractive.
-. The Fourier analysis of pulse propagation data is a
method which itself was considered worthy of further
use and e'.'aiuation.
In Chapters II and III. two different physical models
are adopted and each is adhered to through the mathematical
manipulations necessary to arrivee at a useful prediction of
neutron behavior. Both formulations emplo, a simplified
treatment of the neutron energy variable e in addition to using
the diffusion approximation. In each, the neutron population
density is left as a function of only time and space,
In Chapter II, two-group diffusion theory is used, in
which the neutron energy v.ariable is treated by assuming that
the neutrons are in two energy groups Chapter III is devoted
primarily to the solution of equations tased cn Fermi aQe
diFfusion theory In both of these chapters, a mat emia ticia
prediction of the dispersion la., for a nmult iplying i|-med im is
obtained and s solved specifically for the sur.critical
assembly used in the e:-:perimerital measurements, thus making
possible a com-parison of theory and ep'Feriment.
Chapters I'' through '.'II describe the e::periment--the
physical conditions imposed upon the neutron population under
study, the e:-:perimental procedure and data analysis, and the
results, including the dispersion lia.
Throughout this -.rk-', the experiment itself is under
study. Measurements are made in order to determine how'' :..ell
the experimental apparatus performs its task of imp.csirn the
assumed physical conditions upon the neutrons under stud;.
Remaining chapters compare the results of e::periment
and the predictions of theory .i.th the dispersion la': being
the principal point of comparison. Some clear disagreements
are found the merits of each theory are evaluated, and
suggestions for future '.ork are made
DEPI'ATIOI OF TiiE DISPERSIOLU LAW
BY TWO-GROUP DIFFUSIOII THEORY
The ti..o-group diffusio.n approx-im.ation has already been
applied by others to describe neutron '.av'e propagation. The
rrief deri'.'ation gi'ren here is based on the work of Booth
() The t'.-. '.,ell-kno.'n differential equations for this
Di. l r'r t) rR l( ,ti + F W0 (rt; r i t) .2-
1Po 1 t 1
_2 1 -
D '2: (r,'t E 'r,t) : -: (r t) = (r t) 2-
0 0 00 0 '. 0
The gecmetry of interest in this problem is that of
a long, box-shaped medium, into .&.-hich sinusoidally varied
thermal source neutrons enter the boundary surface at Z = 0.
The trans'.'erse a:xos are X and Y. Propagation of the neutron
..a.'e do,..T the Z a:-:is is to be studied. Both fast and ther-
mal flu:xes are everyw-.here zero at times t = O, and are zero
at the boundaries, except for the source neutrons entering
the surface Z = 0, .here the diffusion theory relationship
bet.'een current and flu:: gradient is used.
The fast and thermal fluxc:s '..11i hrave the form:
S --. + ) t
:1 .r ti = 7x.; ), "i e
S ---z + eut
:. r, ti = .:..:-:,'i A e-
Inserting, Eqs 2-3 and 2--4 into the initial se1 t ,ye-ids*
DiB2 -" 1 = 0
i i B F. A + F A = o
S, D B2 + J = 0
R L ao v J
2 2 2
where B = B
The Eqs, 2-5 and 2-6 for-i a homc-oeneous set f.r ..h-ichj
the compatibility condition is:
D1B2 Jzo + ]
D [DB 11 + ]
By developing the determinantr one obtains an equation
4 1 1
B L-"Z + -
e L + I
1 1 '. 1 2-
+ + -. 3j
+ 0 1
Ld D T L
i-eg3lectinr3 the term with ],j D.. the roots of Eq. 2-8 are:
B = 2
L = -L I v
S -+ + I + i +
1 2 L L' D rT
Whr c = B + B = B. + B
03 *-; 1 1i 1
The quantity., is the squared iner e complex relaxation
length ,of the thermal neutron wa'.a.e of frequency w. The
fcllrowinq relaticOnships and definition-s will be used to
obtain a arnd the real and imaginary parts of Lo the
in 'erse complex rela:.:ati,,n length.
2 2 2
k = + +
0, = a + ]r
0 = -+
..= [ I r + ,.-
S 1 [ 21-
"'= Ir *** )
r + ,+
With the abo0-e relationships and Eq. 2-9, a and can be
calculated for a nuclear system pror'.ided the nuclear param-
eters In Eq 2-9 are kno.,'n. The results of such a calcu-
lation, performed through the use of a computer program based
on these equations, are presented in Chapter VIII.
A.GE-THEOR' r TIREAr'T'IEllT
The disperiion ia.-.' for th.e subcriticail ai-e-mbl .-.'as
calculated uSinq a Fe-rmL aqe- diffusion treatment. ir this
treatment the fission siectrum wa.jsdecr ibi by .a Dirnc
.eit a function in lethargy, anJ the frequent./ dep:n.iren.ce of
the fast fission factor .J.a not taTken into account. Unnder
these conditions, and. for the fundam.-en-ital pactial mci., the
resulting equation for this m,,ei is:
2 2 1 i..
B 0 e:p -B T + -T iL u i 13-1)
L L- D
A simTilar result '.:was obtained ini.ependent'ly by P. L. Brehm
in Ref. 13.
The Jispc-rsioni la-.. is contained .wi.thin Eq. 3-1, but
its solution is not entirely si mple. The imethcJ by --''hich
the e>:act solution -'as obtained will be- outlined first then
a useful series appro-:imation '..i ll be deri'.'ej.
R B. Perez, personal communication.
A. Exact Sr-.iut in -f the Aq Te-Th-.-.r.- Eq'uati .rin
First c:xpr-ess the c.rrnple:.:x in-1"r" e rel axi ti -.n llnrgth,
S, in tr -ms of its r.e l and irmaqinar- parts:
S= *. + 7- Hence c = e )+ ilZa
Lettini = and 0 = 20, .. next
s'-ubs t tute th- expre -~=i, n I. + 1' ) nf r c in Eq. 3-1
tn ,t tain:
B \ 1 = --7 :-:p
B = --- exp
2 2 -,
- I.B IT + i 3 T L Ul
IB .' I T CcS 1,7 T L ,J)
I I 1
+ 1 S11-n i 0 T- L J I -
Equatinq real aind irniaglinar partsof Eq. 3-3,
B = B -
B = -,
2 e;.:p (-B2 T
Cos IB T + L JI -
1 3 LL
Sin iB T + L ) -, -
1 5 D
From these two equations, B and B were computed for
selected values s of 'j, and from the B and E *'..lue so ob-
stained. ,a *and ? '.w.ere calculated as functions of i', thus
'ieldinq the dispersion ia.: as predicted b.- tnhe iowinq-down
Sonm 1 difficult: as esncounte-re, 3 in Sol irnq for & andr
BE since Equations 3-4 and ?-5 are coupled in botn ujn'Krn'..- n
and are both transcendental in either BI or E ..sc.., the
periodic nature of the trtgonr,-metric functions make possible
the existence of solutions related to a mujitiplicit.- of sicv-
ing-down modes. It was found, how''eer, that the additional
solutions lie far outside of the range of valuess .of B' and-
B encountered in the experimental results. Thereforre,
attention was directed toward the finding of solutions
related to the experiment.
The exact solution of Eqs. 3-4 and 3-5 ;as first accm-
plished by the use of a lengthy se.migraphical method. Brieff'
in this method, values of B arnd BL which 'satisf.' both
Eqs. 3-4 and 3-5 for a particular value of a were obtained
by plotting B v.ersu B- for each of the t:.wo equations for
various .values of ju, and observing the points of intersection.
For a glien w, the coordinates of the pcint of intersection
are solutions to both equations. The reader is referred to
Appendix H, 'where the method is described more full',, and a
sarp-plI calculation is given with the associated graphs and
results based upon the initial set of parameters used. The
semigraphical solution is a 'valuable check for alternate
methods, and can be used to find the solutions related to
higher sl.wing-d o -.rn modes. The semigraphical method is also
useful because the g:raphs rev.eal the manner in which the
functions .ariy, and provide a basis for an estimate of the
region of values s which should be examined in the search for
tlh higher mode solutions.
Coupled transcendental equations may also be solved
by the liewton-Raphson method (17). .-. computer program for
solving Eqs. 3-4 and 3-5 by this iterative technique recently
became available, making parameter studies feasible.
B. Approximate Solution to the Age-Theor.. Equation
The usefulness of the age-theory model would be greatly
enhanced if one could obtain from Eq. 3-1 a simple approxi-
mation to the dispersion law which would give answers of
reasonable accuracy over a useful frequency range. This was
IThe semigraphical solution was performed in late 1965.
In 1966, R. G. Cockrell adapted the Newton-Raphson method to
the transcendental equations encountered in this work, and Nils
J. Diaz wrote a computer program employing this technique in the
solution of Eqs. 3-4 and 3-5 and studied the dependence of the
results upon the .'alues of various parameters. (tU. J. Diaz,
four, to be poTsibie, and the deiri,. at ion 12 fnr-,: pre- -nte:D', in
jhich the rsal and imaelirnary parts ,cf the -uanrtit IE 0
are expr-2'ss-d ias a po'eLser s-eris in frequency.
Defninng F = k Lg ansd E = B :, q 3-1 bi:-o.:mes
2 2 1 ",'
flultiplyilng throi.qgh by T .gi 's:
B T = FT e:.:p'-B T iL ,)
S L D
Add iL 'J to both -ile;; anJ let
S(,) = B T + iL u
'' I.J,'I = FT e:.:p[- ': lu) ] -
S ij, I L j
De'.*?lop f l'1j) in a rpo.wer serle-s:
' I ) = i Io + 1 I
+ 171' )
+ "T jl^
' (o) =
lit)~11~ + ) 7
F; expF-i 0) 3 -
--- = -FT ---- ex-: ,- ) ---- L
3 1 L ,' b f i ) D,
FT expi- --- ---- .
. ,( i ) '.) '1.1 ')
F ex. F.1
U')F -. [ I
a 1 ii
'i ar its' d-eci.'.at.-e at j.r = 0 :an no- be obt3 ini .
f io) = FT exp[-' (c.l -
Using FT e:.p[-'i ( ] = l i .) +
' r I 7 S./
i 11 (
, I = i I 0 -
I' 1 = 1t"), +
, 1 ,,) LU
( I P -
- I Ii
L I 2- ,t ai 'C,
0, I 1.1 i i 0 I I' J ) ,'j '
I J I
1 ) / 0
i1 + '!t ,) + -
' ( J. ) "' ,
i) + T 1 7 ., I
1 + '1' 1) + -
1 1 + r I ,) + -
.\. ) ____*
\ 0'r 1 i *)/;-, -si ) I .*}.- J
1 ia) *
The last two equatlonr abo.e rua',' ,b rewritten:
[ ( ], +- L ]
+1 + '() +
- 3., 5 -
K 1[ = 1 C+ T T ]L + ]r i -
1 + 'i1 + 1 + *Ic + --
!-Je:t, -:.pre-S. B am thl- h um *f- its re-al .an- ima inak;:. paL'rt.
and 3subsi ti tu the abo. '. 'es-lc ts irt-nt tch p:.'ec s3eri f:-c '. I.I
I -- L i A1.l
E 5 + E B I + i L = '10) -
1 + c.) + -
1,1 + L iL
[ + Y ,l, + --
3[ ,,, + L]-
S:i + ,, .> + -L
SFpaLratio- n of rcal and im&ginarl F:parts -ields:
L --l [ i L ]
B + -- +
S2T[ 1 --T
+ +B.:,) T --
1 ) + ] T
[ o. 7) ] -. [ L ]
L L D
2Ti +': Ic + --
[,.. T. ] _] 1
[T i + 'oi + L---_
Since B lc = ? 'o can bE reFlad by B'. (1 7 in Eq-
3-6 and 3-' to yiell the fioli-o.in;g:
B lul = B (ol
_- ] [ ]- I '
B 0l) + D L
1 + B Cl) T + -
I .3- ,
-- + L B I +
D s r
B- 1 ) = L]-1
1 + B i ol +
[- ---- L ---- -- --
E.1 b I T + = -
B i-i + c.',c -I-.j ,. +
6 1 + E 10) T +
Br l.1 for the ab,_:-,e equati.ris can be obtained by s.l"ing
Eq. 3-1 for '. = 0.
B 1 = 7 ~ 5 1 17J 7 lT?-10
From th, aboc.'e results, B I, i and B ii,) can be comput.-d if
L L D and r ." are krno-.'.'n. Then, f'-,ri E i and
B I' and krino-.inq. ncr can calculate 1 an rj frun.:rctir
ofr thr, cu.h the iius.- ,-f the. relartic.cnsr.h p :
-' 2 -'
5 = B -
B = B + iB
= E (E2 + IB 2
S + 1
The coe fficients in Eqs. 3- an -9 '.-. re i i"ai'aluat. f-r.r
the parameters in Table 3-1, yil- Idng rthe follow. 1. nr appro:xiima-
t Ic1n s :
'2 -4 -in 1
B l-.) = 7 .1 1 1.044 Nx i a" + -
From these a.lues of EB and E .. arnj 2a .-.'.re ca sil
obtained and are listed in Tabl' 3-2 for comparisons irth nr,
results from thr e':act solution of Eqs. ---4 anrd ---5 b' the
Ne,'t.ron-P.ap'horn rrmethod, using the -same input piarameterr iate.d
in Table --1.
An examination of Table 3-2 sho..s; that the seriess
approximation obtained by keeping only the first t'.-.'o termff
yields results differing from the exact soluti.ron by less tan
VALUES OF PARA:.fMETERS FOR TiLE SLUCRiTICAL ASSElIBLY
USED IH THE SOLUTION OF THE A.CE-THtEORY EQUA TIOtS
BY THE [(EijTOIJ-P-.PiSSOl[i IELETHOD
P ra meter V 'ue Remarks
T 0 2 012 .3 crr,
1 *i 2.229 cm
L ?1. S8 cm
L 5.0 :.: 10 sec Slc.. in -do.-n time
D 2 .224 10 cm D = D
2.,7 x 10 c ". rc
2 .' ,7 x 10 cm;,*s
0_ X r-co
E,3 -. -1 n C r G -r o
o o i i" i *" '.. r- .., ':' -i .* I u"1 I"~ --
.I I -t"
.t < ,' .
Wr 1 ..
i C, C -r r- '
S- Oi j CO T u . *
U II : [ LZ .* J. UI Ll I U-l Ur U L :' .U'
L,,., L i- -, -
. i- ri -
'm ,C,. 0_ -,
V' Z --
i 4 C, > :, ', ,J ", r- t 'X', ,r", (c --, ., ,: .-- '-,,
H -Cz..z C ,
r. -r r,
:-- u I I
*,,, r E- .- C ,. ,2 ,-
H 3" *-. ..:.-. -
< O- r- r-1 L -, '. i ..
E E LL a- C -1 *" -
E-1 2^ > c I
" 1' .'-
:- I rr c1X1
C l1 'J
,a, 3', ,, ., ._ .. 3 L 3 -' ", "
D rIl -'p ,-, Ld" r, iT. ,,, ,. ,- ,
1 peL cient f?)r flrt,.-':uecrc is. urnder -4' :i Cips. 'The or nl.' -i ntrl in
the table ..'ith an -rrrer exc:eeJing 1 per cent is the- 50' cps
alu ", .'hi ch .-i ffers from the. ex,_act rzulit b. 1.3
F'er cent. T'lerefore, it is ser-n th3t th*i --ries a ppr:: .:imat l.ron
as: s.t rorth in Eqs. 3-8 and --9 intro.iuce little additional
error compaieJ w:ith that inherent with the 3e-.iiffuslior-tlheory
model, and is useful ou.-ver a re r, nabpi wide fr.-queLncy. range.
Four digits are used to preserit the .'alues in Table
-2 in or.ier to -sow the if ference between- the approximat-
solutionr an.1 th]-i exact solution. The rea.--:r must not assume
from this that the the-ry' it-alif, =ern '.hen sold ..'ith no
further .apiproxim at ions, is qood to four-figure accuracy Ii
Chapter ''III, tl-~ accuracy and,. usefulness of the age-theory
model in this application is e'.-aiu.ated by,' comparing the
results of the e;-:.act solution of the a.g-theory equations
'.:.ith the exparlmentali results.
E:XPER IMEIITAL APPARATUS
A. in t ro.:uc t ion
The appa r.9tus ass-errbledj for this .ar k co.nsisted ..f
four units or sE'stems: the EuIbcritical a.sserrbl within
*.nich the principal measurements '.*.re perf r rmed1. trh lliutr:.on
generator o'.hich e-:tern all supplied th-: in[ut neutr:'ns fr.:.
the subcritical assembly, the thermaliLn4 apparatus inter-
posed bet.'eer the neutron g-enerirat iand the Fubcriti:al
assembly, and the deCect.or sys teams. F.,.ire- 4-1 sh.:-.s the
subcritical assembly, neutron generator, and thermavlizin'g
apparatus in their relativ.e operation. posi tons. Each of
these systems '*..11ll be discussed in turn.
B. Subcritica Assembi'-
A rectangular subcritical assembly usinq heavy -..ter
as moderator and natu. ra uranium as fuel ;.as designed and
constructed at the Uni'.'ersit' of Florida for this arid future
i -dil.d n~Jo
experimental '.:ork F.igures 4-2 and -2 p reent fronr ani1d
end :.ie'.:s of the assembly. hbo.ve the lo..er tank, ...hic:h con-
taiin the core is a separate iry.erted tank ..ith a '..rd..
light, and t..o gl.oes on both sides. These features permit
iimanal positionin.q of the rlc tL-:nr detector (L.or detectors 'i
.-ilthout opening the assembly and e:pfostring the hea.". ..ater
to e'.' portion CL:. tr,: c:nt ai ina tion. by light ..at r present
in the air. A small posit .'Ve pressure is ra initairn ed irnsidi
the upper tank by ieakinic in dr_, nitrogen at a rate of
appro:- imately 1 mi.'sec. Thus an-' .' Fapcr leakage through
seals or 'elds above moderator le"el is outwardly directed.
For ioadinq or unloading the core, or '.hene'-er complete
access to the inside of the assembly is required, the upper
tank may be rolled aside b: means .:f four attached *.-heels
and an elevated track structur-
The assembly rests or a heav.'y steel table inside an
emergency D20 catch tank ..;hich also *onitains the necessary
pump, -.'alv.es, and plumbing for filling anrd em't, ing the
assembly. See Fig: 4-4. The entire assembly, support table,
and catch tank rest on a rigid steel frame equipped .'ith
leveling screws and 9" casters for a moderate degree of
Je::t, the detector positionir i apparatus and the core
will be described in greater detail, including? not only
pC, s -t -
;::: ::.r_-~------J--...-- : ---1--'---'-~-~---- --" -
S -I ii
-. L -
I:L : --_ -- ::.:, -iii3II- ----_ < ;
_ : -- --- --, -- . . -'i . ... ; 3- i '
_-_---_ ----r----- -.--- --"-. ------ ] <:
I_ __iI-__ I~_ ~L --: l .,",' :
;: i: -r ::" :::: r^ ,, -
r.---- ._~~--I _1__-l- --J]
l,2-- -l _-" .-_-- --- --E- -J---------'L--- -"-
I ~ :^ '---'^ 'i^^ --^ ^ ^ ^ j
7 s.'- s I e'aC t ,' tjb e
5/ -r' ub e a-' e .
52- 51 3f3l >?Sj a'e 4 7e5 07
5 C 24
Fig. 4-2. Front View of the Subcritical Assembly.
3 i.... r
*'~, ~c.; t 1 -
,. 3 5
r_ ll r
- -- '-' r"- -n-r- l-4 r--re- -r "r e- nfr--Tr' T 1
o o 0 O OI,O 0 OO c
o 0 O 0 01,0 0 0 0 o
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0',0 0 0 0 0 0
0 0 0 0 00 0 0 0 0 0
S0 o0 'o o 0 o 0
Ir. Li -b4 ..35r---
00 0 0 000 0 0 0
0 0 0 0 0 0 0 0 0 000
0 0 0 0 0 0 0 0 0 0 0
O OOO OO OO OO O
..- !r, :."j
5t.'eC 'j 7"ra
-, a o ald C'as
".ns .5' a' 5 .
Fig. 4-3. End View of the Subcritical Assembl','.
,./ r r ,
..a ..s.mbl .pprt t :be ,
,'t ,A .att ',- -', ;, ,. ,, ,,,,
, o-, t. d -....... ', '.
'' -' > ,r ~
,P__$on '-). 'a taft 4-/
Fig. 4-4. Catch Tar.k and Support Structure for
the Subcritical -ssembly.
information relative to the use of the equipment, but also
those dimensions, deflections and tolerances ..hich are ex-
pected to be important in core caiculatiCrons or in the con-
sideration of experimental errors stemmin.i frcm imperfect
core geometry' and inaccuracies in detect,-r placement
Posit-ioninq apparatus for detector
Filure 4-3 sho.cs the manually operated de te&t.or
positioning apparatus -..hich rolls the length or the a semrrinly
in the Z direction parallel to. the fuel tubes. The tracks
are above the after r line. screed to inside :aiii orf the
core-containmnent tank. An aluTinum reasuring rod ..ith
drilled holes spaced e;',ry 2.50 0.025 cm engages an
alignment pin on the center line of the detector holder,
pro'.'ding accurate positioning of the Z detector coordinate
from Z = 5 cm to Z = 125 cm Scre'..,-operated friction brake-
clamp against the tracks to prevent t accidental mov.'emient
after the Z position has been set.
A nominal 1" dia.reter, 12" long He-3 detector may be
wrapped with a small amount of tape at each end to impro'
centering, and lo..ered into a 125" OLD aluminum tube sealed
at the bottom. This tube slides in a vertical guide and is
locked at the desired Y position by a thumb screw-. mcst
measurements are made with the center of the detector active
length at Y = 0, the mid-hei-ght of the core. Because
the lernth of the detector ]lies '.n th'e direction, trans-
verse flux mapping is best done along the :: direction,
for then no part of the detector active volume is more
than i0.5" from the nominal detector position. Furthermore,
since the assembly core and the thermalizing apparatus are
both square in the ..' plane, a true flux traverse in the '
direction should differ little from that in the :: direction
if the influence of epicadmium neutrons scattered from the
laboratory floor is negligible.
Since i full fuel loading consists of 121 fuel tubes
in a square array of 11 columns containing 11 fuel tubes
each, there are only 10 vertical YZ planes which are midw..ay
between adjacent IZ columns of fuel tubes. These 10 planes
provide the most useful and natural detector positions for
flu. mapping across the core if thermal flu:-: depression by
the fuel is not the object of study. For convenient and
accurate detector placement, 10 alignment holes '.:ere drilled
into the transverse beams of the detector positioning appa-
ratus for an alignment pin.
Since one column of fuel tubes lies in the central
plane of the core, at :: = 0, axial measurements which would
ordinarily be made '.ith the detector placed on the true Z
axis must be made in a plane offset from the center plane.
If the 10 planes desc-ribed abo'.ve are numbered fro-m one side
of the core to the other, planes 5 and 6 are cl.oest to the
Z a::is, and are at :: = t 2 .175", ..here the dimension is half
the lattice spacing of 4 35". .ll of the c-called Z a:-.is
measurements in this work ..ere made in plane number L6 Li
problems should arise from this offset because if the funda-
mental spatial mode is well established, measurerrient made
along a line parallel to the i a:i.s are related to true -
a:is measurements through a multiplicativ.'e constant, '...hich
in this case is almost unity.
Figure 4-3 sho...' the nearly opt imum lattice spaciing
of approximately 4.35" for a core using DC and Mark I
solid natural uranium slugs in a square lattice This
spacing produces 11 unit cells across the inside tank dimen-
slion of 4 .87'', Other lattice spacings are obtainable by
leaving some fuel positions '.'acant or by constructing
additional pairs of core support plates. The full loading
requires 726 slugs in 121 horizontal fuel tubes 51" loni.
each containing six slugs with an accumulated length of
50.5", Moderator (and possibly a fe'.' tenacious gas bubblesa
occupy the small remaining ;volume of the tubes. The machined
diameter of the uranium metal is 1 '00" Aluminum cladding
adds 1/16" to the diameter. gi ing a nominal outside diameter
.f 1 0625" manufact urging tolerances. The length of each
slug, includinq- end cladding m'eau ures approximately 8.41".
The fuel tubes hav.e an IOD of i.2c5", a ..all thickness
of 0.049" (stubs cauuge no, 1i8). and are of ALC.OA alloy
'061-T6, the SaTie mriaterial used in the construction of the
.aubcritical asserbl'. This full" tempered alloy has the
following specifications: tension field d = 40,000 pia, tension
ultimate = 45,000 psi, shear = 30,000 psi. Its allowing ele-
ments have reluti.-eli lo.. absoLption cross sections and are:
0.25:: copper, 0 6-. silicon, 1.0-. magnesium, and 0.251: chromium
Aluminum and "normal impurities" constitute the remainder-
A calculation of the thermal neutron cross section for this
alloy is included in Appendix E. The resulting alloy is
also unusually resistant to corrosion in vater. Each fuel
tube constitutes a uniformly loaded horizontal beam supported
near the ends and has a measured na.ximum deflection of 1.'"
at the middle. It is likely that deflections up to 1/4"
could be tolerated, particularly since the deflections are
in the Y direction and since most trans.'erse flu:-: studies
'..ith this assembly are done in the X direction for reasons
already gi.ren. The design calculations had selected stubs
no. 20 (0.025" wall), but 1.25" OD tubing was not a-.'ailable
in wall thickness less than no, 18, Since the 18 gauqe
tubing used weighs 0 210 pounds per foot 121 tubes put 108
pounds of structural TaLterial into the core. This figure
could be reduced to 77 pounds safely if 20 gauge 6061i-T6
tubing were a.'ailable
Iqgnorng deflections caused by rihdrcstatic forces, and
provided that the correct amount of moderator is present, core
dimensions are 47>,7" :: 47.87" :.. L, '.here L = 50.5" if the
core length is taken to be the accumulated length of 6 fuel
slugs. L = 51.12" if the core length is taken to be the
moderator-filled dimension between fuel tube -suppcrt plates.
(The moderator-filled dimension exceeds the fuel length be-
cause the fuel tubes e:.-tend beyond the slugs by about i' .25"
on each end, and because of clearances; required for loading
and unloading the core.) If one w..ishes to take into account
the additional small thickness of moderator present in the
0.38" space pro'.'ded for movement of the cadmiuiim shutter,
L = 51.50" with shutter up, and about 51.4' .with shutter
do-.n (The shutter is a 0 030" sheet of cadmium cemented
to a 0.0625" aluminum backing sheet.) When the shutter is
lowered, it displaces moderator, Therefore. if the tank is
filled to a depth of 47 87" '.hen the shutter is do-'..n the
level 'ill be less by about 0.i8" '.hen the shutter is raised
(The 47.87" dimension corresponds to the lower surface of the
cadmium sheets which cover the upper moderator boundary.)
The out.ward deflection cf the tank ..alls mertiioned aro',.oe is
a maximum of approximate ly 0. 1" on each of the two largest
eide panels, at a depth of about 30" Therefore, the maximum
inside tank- '. dth wouldl d exceed the rno-iinal .alue of 47.27"
...hen the tank i-s filled ..ith moderator: The bottom of the
.cre tank rests on a steel table of such riqidity that de-
fiections in the bottom core boundar- are negligible.
Cadmium sheet, 0 030" thick, is attached to all .outer
surfaces c.f the core containment tank except for the end
that has the mo"ablc cadmiumi shutter. Cadmium sheet in
smaller, moa-able pieces aisolies just above the water-le-.-el
surface Therefore, '.,ith the shutter down, the core has
cadmium on all si: boundaries.
C neutron Generator
The neutron generator used was model no. 9505, serial
no. 52, rianufactured by' the Texas tluclear Corporation of
.Austin, Texas The neutron generator itself is shown in
Fig. 4-1 along .'ith the other maIor items of experimental
.quipimrent. The associated high .oltage supply and control
console are not shown. F gure 4-1 shows the generator
mounted on a ..heeled apparatus .with elevating screw's and an
eight t-foot-long track on which the generator could be rolled
without disturbing alignment ..ith the reentrant tube of the
thermalizer apparatus. This '..heeled hard..are was cons tructed
at the Uni.ersit: of Florida and is not a standard part of
the generator as supplied by the manufacturer
The original outside dimensions of the target end of
the generator exceeded the inside diameter :of the reentrant
tube of the thermalizing apparatus, and *-.as replaced a
redesigned assembly of dsa c dimensions, ith no glass
parts and no suppressor ring This prc.ed satiefactor..-
The neutron generator is basically' a Cockcroft-Uialton
linear accelerator with a maximum accelerating .oltage of
150 KV. Neutrons ma,' be produced b. either the (D, n)
reaction .,itlh a tritium target, :jelding neutrons cf i4 74
lex.', or the iD,ni reaction .vith a deuterium target, ..'hich
gives 2 .86 Me'' neutrons. The lo..er energy is desirable.
but the tritium target -was chosen because of its much
greater yield During this :aork, the ma.-imum obtainable
deuteron beam current as indicated b' a microamme ter circuit
bet'..,een target and ground w..as about iSOuamp for continuous
beam operation. During pulsing, with a typical duty cycle
of 5 per cent, the time-a..'eraged team current w:as less than
tno further description of the neutron generator w..ill be
given here as it is described in the manufacturer's literature
The reader is also referred to the work of Hartley
(18) for a description of the electronic equipment required
to determine the pulse duration and frequency and to synchro-
nize the data-recording equipment with the erinssion of the
pulse from the generator,
D- Thermalizing apparatuss
In order to stud,. the propagation of a thermal neutron
pulse or wave in the subcritical assembly, it ;as considered
necessary to assemble a thermalizing apparatus for the pur-
pcse of receiving 14 Me'; neutrons emitted from the small
target of the neutron generator and deli'-ering these to
the four-foot-square source boundary of the subcritical
assembly as thermal neutrons in the fundamental spatial
mode having a cosine shape in the X and Y directions.
After considerable experimentation :.'th various
thicknesses of graphite, water, lead and steel., an almost
optimum arrangement was determined, and is shown in Fig.
4-5, Graphite ser-.es as the final thermalizing and flux-
shaping medium. A 48" x 48" x 17" stack of machined
graphite is located outside the :ater tank. An additional
48" : 47.5" x: 12" stack of rough, unmachined graphite was
added inside the 49" x 48" x 31" ..ater tank, bringing the
graphite to a total thickness of 29', and a total weight of
sr-- 14 '"i, I/ "-
Fig. 4-5. Thermalizing Apparatus.
.bout 3800 lbs
Four inches of steel in the form of steel bricks, and
about t'.-.o inches of lead sheet surround the reentrant tute
housing the target end of the neutron generator as shown in
Fig 4-5. The relatively large inelastic scattering cross
sections of these healer materials serve to reduce 14 let.'
neutrons to lo-.er energies at .-.hich graphite : is an efficient
moderator: The 2" :-: 4" x teel bricks, machined and i.elded
..here ne.-essar.. to reduce streaming and to improve stacking,
,ere a.'ailable from earlier neutron .ave w'..ork.
Ordinary ..ater occupies the remaiining volume of the
thermalizing tank, usefully filling v.'oids and cracks,
I!easurements showed that the presence of ..'ater reduced the
total ne-utron intensity at the output face of the graphite
stack by a factor of 116, '..'hilc reducing the epicadmium
intensity by a greater factor of approximately 3.4. These
t:wo effects resulted in an increase in the cadmium ratio
from 40 to 90 upon filling the dry tank v..ith water, These
measurements ..'ere made ..ith the thermalizing apparatus aw.ay
from the subcritical assembly, The detector w.as a 12" He-3
detector, The cadmium measurements were made ..ith a cylinder
of cadmium '.:rapped around the detector
water should not be regarded as merely an inexpensive
substitute for graphite in its use here, A measurement '.as
made after approximately 4 cubic feet of additional graphite
-.as placed adjacent to the stack already in the tank- dis-
placing an equal v.olume of water. There -as no measurable
improv.erment in intern it' or cadmium-i ratic at the output t face.
Therefore that additional graphite 'as remov'.ed in order to
avoid unnecessary broadening of the iqrgna pulse .'n effort
..'as made to use no more graphite than needed to produce a
useful cadmium ratic and to pro.'ide a sufficient '.-olume f or
adequate development of the fundamental spatial mode at the
output face. Graphs to be presented later sh,'.-' that the
thermalized pulse is much Droader than the input iddth of
two mEec, and associated with this broadening is an undesir-
able attenuation of the high frequency' content of the source
pulse which presents a limitation in the e::periment.
It was found that the pulse does not broaden nearly
so much in passing through light w..ater as it does in graphite,
A detector placed in the corner of the -;ater tank about 27"
from the target, sees a nearly rectangular pulse with a ver'
rapid rise to saturation, and a rapid decay,. Graphs of the
pulse are included in Appendi:x C. The difference between the
two materials is largely related to the absorption cross
sections. Therefore it is concluded that the '.-ater used in
the thermalizing tank is far less detrimental in w-idening
the output pulse than an equal '.-olume of graphite '-ould be
At onr- time, heav.'y water ','as somi..hat ish fuil ; considered
for use as the best Dpssible thermalizin? medium in place
of part or all of the graphite, t o.. in retrospect it
appears fortunate that this ..as not done because irn "ew
of the abo.e considerations., it appeared likely that the
'w'y lo.w absorption cross section of D.O would ha'-e resulted
in an e"ern c.ider output pulse than an equal volume of graphite
ii- this application.
The after r also serves as shielding for the target, and
really diminishes leakage of epicadmium neutrons from the
back and side surfaces of the tank. It is important that
such leakage be small L.ecause epicadmium neutrons can
scatter into the subcritical assembly and become an e::traneous
signal entering the '-rong surface and out of phase ",ith the
Flli-: maps at the face of the thermalizing apparatus
are presented in Appendix A. Further information about the
performance of the thermalliinc apparatus is included in
E. Detector Systems
Tr.o detector systems ..ere employed. One system used
a 12" He-3 detector and terminated with equipment which pro-
duced binary and digital records of the neutron pulse shape
in the subcritical a.ss.embinly. Figure 4-6 des.Crires this
system. A second system. employing a 6" He-3 detector, ..as
used to monitor the source strength fromn the neutr:-.n genera-
tor, and is shown in Fig 4-7 isjLi 1 l.'Ecope displays of
the signal pulse Idetector system response to irndi-idual
neutrons) at "aricus points in each system are reproduced
in Figs 4-8 and 4-9 for the 12" and 6" systems respecti',ely
In both systems, the signal rise time .as 0 3 peiec or less,
and the pulse fell tc. 1'e of its ma:-:iI'I.T, -.alue i in 2 usec or
less. Figure 4-10 presents graphs of count rate versuS high
voltage for the t.o E'ssteims The operating voltages used
w.'re higher than ordinarily encountered with these detectors
because of the length of cables required.
Additional information related to detector system
resol'.'ing time, count rates, and resolution loss corrections
is presented in Chapter v and in the appendices.
Atom i Ins rum ient C o. -
Pea. H.V. Power C. ckrn 1 He- Dete7tor
SuppS/ M.rdel aJ i
Te. u rr I .oi de! 3 1 9
Serial -l i
Si-anil + and Filament
Hamnin. Ir El.-ccrcnic: Co.. P.H.S. E= 1 .'0
Linear Amplifier A fr A Otput O
ouu -= O OSig.nl
,..utt ,.'-=., 0 0 F=
-5^1 .oIls Signal
H smrier Scaler
l'cd-.. I [.i-295
+2.-' Si nnall
Technical l na u rerrimerr t
Co.rp. Data Ourput i.init
.1odel ?2?U C"
Tektroni:. Input A
Out put A
S T:pe CA
"- I Plug-in l.init
I+.-- n3Ian l
+?_. -'':'. Signal
Technical Measureme ntr
Corp. CiQital Ccmpputer
Unit I.lodel C ,i-102-1 with
Model 212 'ulsed !jeutron
Fig. 4-6. Diagram oE the Principal Detector System.
Aormnic insrun,,:ir Co.
Super Stdble H V.
Power SuppIy' Mold. 31i
I-Ice H.V. Se~tin-i "0"
1e*:Iurn NIC-dci ?I
,Z : rial P:I.qelI:
'C.I 5.3a S
Flq. 4-7. Didargrar of '-he Reference 'etectc~r System.
1 /e- 2 3
2 U t -
,3 1 Usec 2
Fig. 4-8. Oscillc.scope Display of Signal at Various
Pcints in the Principal Detector System.
Fig. 4-9. Oscillasc.-pe Display of Siinal at \'arIo-us
PointS in the Reference Detector System.
Plateau Cur'.es for the Principal
and Reference Detector Systems.
EX:PERI!iEiiTAL PROCCEDURE AIJD DATAi .'.I'.LY SIS
A. Int rod-D.U cti cn
Thro.ighout the e:peFrmerntal .'ork, the temperatuLr or
the moderator was 2-3 1 C, and. the purity w:.as :':.'. 0. 1
per cent D.O.
Fcr measurements '..ith the ic..jaded asseiTil' l, a faLll
lattice of 121 fuel tubes ,'as used, witi a m.,Jioderato.r -ldepth
of 47.8". The same full water ie'ei was used during the
measurements with the moderator c.nly.. Also, *.hni the
t]hrmaiizing apparatus was used, the- H20 le .ei in the ther-
malizing taii' w.as maintained at the normal full i o- f "1 .f
Cert ain necessary pcLelimiiar measurem.i cnct related
to the construction, adjustment, and performance of the
thermalizing apparatus and the ,detectc- systems are not
described in this chapter, but the r--ults :.f the moriCe-
important of these are gi'.-en in s'.-eral of the appendices.
Tables 5-1 and 5-2 divide the emaninin g experimental
measurements into six mr-io-'.r parts In Parts 1, 2, 5, an.d
.1 ,ll "1,
0 1 L4
4-1 1 -
1 'I U
'. .. 'lW
Li C1 C-
,' M :I
4J 3 .t -.
E *- r-
4-1 4 3 .
*, C. 44 \
'. -j- .0r
4-' 0 I'2
E-l 01 $S Ii
C -. i'.
- -' 1' 4 4-.
.1 1' I 4 .t
-.T C T-
V, -- L '
d. 10 *
u i .. c iC 0
, -. E -n D .,-
'4. ,4 rj -. ,
-4 .74 '
the r-ir 1 a p.osition-s of the neuttr.nn iener ator, thermal-
iz'inr app 'aratus, an, Subcr i L ctlc a a--ieblZ w. ere- E jE jecribe
in Fig. 4-1 of crh p re i'us c apt-er. Because e th.- b-ha -"10r
or thermia neutrons ..;as of rrii3mar: interest, the effect .of
tLri-: .F pcdmiuim neutrons entering the core tank" frcm the
thermfal.in: ir aFppar.tus ..as largely: remonr .-ed from conr.. 2dera-
tion by performing one set of measur-ements .'Lth the cadmium
shutter .Jo.n tthus permitting ornly epicamiuim riutrons to
enter the core tank. and a second, other'.:ise identical set
of measuremnt s i-ith the shu-tter up, permit ting both fast and
lo'.-: neutrons to enter. After corrections h.ad. been appied
to both _ets of dLta for resolution losses. back iroun.- and
'-aiations in source strength and run times, the shutter-down
counts '.-'re subtracted rrcm the shutter-up counts, lea.'ing
counts related to the subcadmium content of the source.
The reflection of neutrons across the boundary between
the thermalizing apparatus and the assembly has an undesir-
abie effect in tne experiment, ..'hich no'..' ..'ill be pointed out.
The experiment i. one in which a purely thermal neutron
source is desirable, but since none is available, the thermal-
izi.nJ, apparatus and cadmiumi, shutter are used to permit a sub-
traction of the effect of the epicadmiun content in the
source from the effect of the subcadmium plus epicalmium con-
tent of the source to obtain the effect of only the neutrons
of subcadmrium energies, This subtraction mray b- writtenen :
iA + B) B=
The problem in this is that the two B' are not equal in the
actual experiment, because the flu:-: in the asembl1 caused
by epicadmium neutrDns ent-erin g the assembly from the ther-
malizing apparatus Is greater ..henr the shutter is3 jp than
it is for the same sc--.u-rce strength of e icadmium neztrc.r,-
.*.:hen the shutter 13 doj'..n. This is true becau:s:- '.Thern thr,
shutter is down, it estabisihes a bounJarv condition on
the thermal neutrons in the ass-iembl', bu..hen it he i up,
thermal neutrons are free to employ the thermal iz in appa-
ratus as a reflector, thus increase in the flux in the
assembly, especially, in the vicinity. of the shuLtter p's 'ition
The subtraction of shutter-do..'n counts from shutter-up
counts is more accurately represented b. the equatc.in,
(A + B) B' = A' or A' = -. + IB B
Since B > B', as explained above, the subtr.acton .ield.
an A', which though supposed to be the response of the
detector to the subcadmirum content rf the source '..'ith the
shutter up, is actually the sum of that quantity plus the
difference between the response to the epicad3mium content
with shutter up minus the response to the epicadriLum con-
tent with shutter down Therefore, not all of the detector
response to the epicadmium coenent in the source is
subtracted i r the data analysis prc.cedure f-oll '..ed in this
w..ork. 13! real remerdy. for thi. Problem w.as found, but by
using data: from onr, the middle thirJ of the core tank,
.vhern possible, for the dcteraminat ion rf D i, the error in
the results is ex-pected to be small, since the quantity
B B' is small at distances far from the shutter. More
.ill be said about this matter in c-nnecti.niDrn with some
o.f the graphs in the next t;. chapters.
Correct ions fort background counts were made in differ-
ent wa.'s, depending upon whether the assembly had fuel in
it or not, and upon the mode of operation of the neutron
generator. For the continuous s-ource measurements, the
background rate was determined with the neutron generator
turned orff and w.as found to be negligible (about 2 to 5 cpa.
for the unloaded assembly. With fuel, however, the back-
ground was a ma-:imumn of approximately 5,0:,j) cpm in the
center of the assembly. as measured with the 12' detector
system. Therefore for continuous source measurements on the
loaded assembly, the appropriate background rate correspond-
in3 to the correct detector po-sition and shutter position
,wa taken from Fig. 7-1 for each run These correct ion
were applied to the shutter-up and shutter-down data after
the resolution loss corrections were made
With the eneriat,'r operate.:J as a pulled .source the
situation c-oncernin: bac-k rrund r aB sa'-*i.e..hat more coimp licated
because the data '.,cre handled b' the computer anj because the
neutron generator iwas not -completi1r : "o.ff" betweer rulie;-
During th 'ff" part :f the pulsi'3 ] .equernce, the Leam n'.. a
deflected away from the target, but stray i:.nri c. -aueid a fe -.
nei.itr.-nri t.c be emitte3. The "r:.n-offr neutron r ield. rat.:. foro
the Generatc.r w;as =oi.metimes go-- d j as 2000'D: to. i. This Lrati,
:as not constant, an.jd as dep-enent upon man-y lar-l: unr-:.n-
tr.ollable factors a sociated with the adj],ustmient and c_.n.i-
tion c.f the generato-r arnd target. With the piuled -o:ur-ce
measurements, the MOORE rMOMETlr5 co-.e per .frme- ._crre :_t:.n=
for resolution los., anj --ariatilonF in s-.urce Btrrenrth first.
then subtracted the shutter--d-.wn data ,.-hic h included the
background for the shutter-do.:.n data) fr::m the shutter-up
data plus background. In the subtractir..n the back.l-rounl
tended to cancel out, but =.some residual error usually
remained. As a later step, the -ode .ailed fcr the sub-
traction of a residual background correction .Ahich had to be
calculated b-. hand and entered as input information ror
each detector positio-n.
In practice, it was found that the background in the
pulse runs on D.O2 was about 10 counts per channel, and the
residual background was approximately 2 or 3 counts per
channel in some test data for .-hich it -*as -omputed There-
fore, for the measurements on D. ::i tho'.it fuel no correction
:'*as made for the residual background, because of its small
magnitude. With the leaded assemrblY. ho..e.ver, the ra:. back-
ground and residual baci;qround '.*.'ere of sufficient magnitude
to .'arrant the full correction procedure. The hand caic.u-
latior, of the residual correcticns were based on the average
channel content of the last 1i00 channels for both the shutter-
up and shutter-dc.-.n runs.
Section B, ..hich follo-'s, describes the measurements
in Parts 3 and 4 of the tables, in -lhich some conventional
die-a--.'a techniques were performed usinq a source pulse of
fas-t ne-utrons Parts I and 5 pertain to continuous source
ex perimenrts along the Z axis, and are described in Section C
of this chapter. Section D describes the measurements in
Parts 2 and ';., which h constitute the heart of the experl-
mental w'ork, as these lead to the determination of the
dispersion li.. and .various parameters for the assembly both
'.ith and ..ithout uranium.
B- Con'.enti.onal Pulse Dec-a Measuremen ta
Fcr the conr.entional Fulse die-a'a', i meaureri'ents cut-
lined in Parts 3 and 4 .cf Tables 5-1 and 5-2. the equipment
'was arranged to admit during the Fulse a large number of
fast neutrons .hich would become thermalized throughout the
tank, Frov.idin a distributed source cf thermal n'rutrons
whose decay .-.ould then be studied. F-r these measurermnt;
the thermalizing apparatus .'.as roiled aside and the target
end of the neutron generator '.-as placed "-ery near the sur-
face of the core tank, at Z = 0. The cadmi um s-hutter ',. a
in place, co.erinq the input surface of the core tank.
For Part 3 of Table 5-1, the deca; constant for the
heavy water moderator was determined '.ith no fuel present.
Appendix F describes this measurement in greater detail,
In the measurements outlined in Part 4 of Table 5-2,
the decay of the neutron population ..as obser-ed for several
different moderator le'.els during the initial filling; of
the subcritical with moderator after the full load of fuel
had been loaded into the dry core tank. The seria seuremenrts
were pF-rforrmed principally for safety reasons during the
initial loading in order to foresee and avoid any possible
C Zero Frequenc flea -urements
FP'art 1 and S of Table, 5-1 and 5-2 w.ere continuous
:.source measurements perf. orrmed ..1ith m oderator .on- an: d with
the loaded assembly, respecti--el". Only the e:-:per mental
proce .irue wi I be de;tcribed here. Th.e graphs and o.--ther
results from these meisurrerir-nts are gi.'en in Chapter '.'I for
the mriderat-or. only, and in Chapter 'V II for the loaded assae.bli
For the zerC frequency' rinea urements, the neutron gercr-
ator '.as operated in the continutouSL mode twithc.ut pulsing)
an.d at a near' czonscant beam current ohile counts '...ere
accumulated for a particular detect-or position. The bear,
current was increased for detector posliti-o a.a" from the
s-.Lurce boundaryx in order co maintain a satii-actDor' counting
rate from the 12" detector s'"stem.
H!ormall., a counting time of 5 minutes at each detector
position was sufficient to accumulate several million counts,
making the statistical error in the initial count data ap-
pro.-: i:mately" 0.1 per cent or less. Since the resolution
error was usually' several times this amount, corrections for
resolution loss were necessary for both of the detector
systems. ) The counting time ,was short enough to permit a
set of data to be gathered in a period of se.'eral hours or
less: a fact w-hich later prov'.ed to be especially useful
because thea data were found to be largely. free o-.f error Adue
to.. drift in the detector s .stems, and' .'ere used as a ba1is f.or
a drift correction to. the pulse data obtained later o-'er 5
period of several das'. The detector sYstem setup for th.se
measurements -.as aa shcow.n in Fi's. 4-6 through 4-10 in ,7rapter
I''. Ilote, however, that the time anal.--er and data c:,,tput
equipment with the 12' detectc.r ','stenm .r.ere not in use be-
causc only integral count data 'fere needed in continuous
source measurements. Data recorded included detector posi-
tion, run time, and scale counts for both 12" and 6"
detector systems for both Ehutter-up and shutter-do..rn r'.inL.
-. full list of the column headings on the data and calcu-
lation sheets used is given below., along with e:-planator'.
1. Detector Position.
2. Shutter Position. (Designate up o.r do.n.)
3. Neutron Generator Beam Current in uamp. (The
beam current used was from 5 to S6 uamp. For Z
positions greater than 70 cm, a stronger source
would have been useful to increase the count rate
of the 12" detector, but would hae e:.ceeded a
practical limit of 1.5 x 10 cpm for the 6' system
monitoring the source strength.)
4. Run Time in Minutes. IUsuall .' 5 minutes for either
shutter-up or shutter-down run at each position.)
5. Observed 12" Detector Counts on Scaler. IFrom 5 x 10
for Z = 125 cm to 6 x 106 cpm. Less for shutter-
6. 'Ober-.'ed 12" Detector Count Rate. (Calculated,
in c[:.m. Typically approximately& 1 x 10' but
less for iarge Z and shutter-doc'.'n run's.
12" Detector Resolution Correction Factor, 1-F.
.This is read from a graph prepared for the
purpose, and. is based on the observe.'d count rate.)
3. CObs-er.ed 12" Detector Count Rate Di.-ided by 1-F.
IThis gies the resciution-corrected count rate )
9. Background Count Rate for 12" Detector. IThis .'a
determined in a separate measurement, and was er.
mall for the moderator only, but quite significant
in thie loaded assembly.
1i. Resoiution-Corrected Count Pate for 12" Detector
Minus Back-round. iBackground was a function of
detector positi in and of the position of the cad-
mium shutter. This quantity will be referred to
-impi' as the corrected 12" count rate.)
11. Observed 6" Detector Counts on Scaler.
12. Obser'.ved 6" Detector Count Rate. iCalculated, in
cpm. From 10i cpm for low beam current to 1.5 x 10'
cpm at 80 wamp.)
13. 6" Detector Resolution Correction Factor, 1-F.
14. Obser-.ed 6" Detector Count Rate Di'.'ded by 1-F.
15. Background Count Rate for 6" Detector. (This quan-
tity .,.as alway-i found to be negligible, and is
listed here only for clarity. The result in item
14 above will be referred to as the corrected 6"
detector count rate.)
16. Corrected 12" Count Rate Divided by Corrected 6"
Count .ate. (This quantity is in a sense output
divided by input. Recall that the 6" detection
monitors the source strength.)
17. TiherIn l 1 Flux. (T.iis quantity i; fcund-I b- su.,b-
tracrtinrg itemr 16 ab.ove fcr the 'hu tt.er-down run
from 1tem 16 fDr the shutter-up .run. For a
particular detectz.-or p.ea i ir.i r, thiz qu.anrity ;i
the ccr-ected a-ndr, norma rjli-d ze res-a orine to =ub-
cadmniuim' r neutronr: er t1rrg Ithe ?r'urce boun.da ry
of the :?re tank. The in'.-rse r-laxit: tin length
for this qu3nt t', 3a a functrin .:-f Z rposititc .ri
the "a i uc o- f for -ero ueienc. I in Fart 1,
for their tr.ins.er= e flu:-: mapping, this quantity
is to be-- co-.mfre a t'l th the fundiamenal 'tcor inr.
d tr ibuticn. I
D. Frocedure and Data .ial.a- is for the
fleutron Wa-e Propaqatiorn E:-:pr imernti
For Parts 2 and 6 of Tabl.-- 5-i and 5-2, the fi...e-
ment of apparatus .'as as illustrated in F i. 4-i r.ad th.
detector s'stemrs ..ere .:-s dc.e ribed in Fig-. 4-6 thbrcuqh --i1i.
The neutron generator was. ,-.peracred as .a pushed. source .of the
subcritical assenmbl- after passing throiuLigh- th- tlhernriializin
Binary and diic.l data from rt 1024 ch-arinel anal-.zer
described the time behav.'or of the r.ulse as dJEt-cted at the
position occupied by the 12" detector .placed in rthe sub-
critical assembly. This pulse at cacht detector position
was Fourier analyzed to yield phase and amplitude informa-
tion as a function %f frequency and position. Fo.r the 3
axis measurements, this information .,'as further analzc-d toc
g1'.e -- and as functions of frceque-rc'y. Thus, in a sense,
a thermal pulse propagation experiment .as used to obtain
wav.0e prop agation information.
The procedure for tne data accumulation and analysis
for the wa"e propagation experiments is outlined in a flow
diagram in Fig. 5-1. The complete diagram ion three pages
describes the procedure for the Z ax:is D C measurements of
Part 2. All but the last two blocks of the diagram also
apply to the analysis of Z axis measurerrents on the loaded
assembly in Part 6. The analysis of the transverse data for
the moderator in Part 2 ends with the output from the MOORE
MOMENTS code and some hand calculations necessary to put
the code output in a form to be graphed. This FORTRAN. code
is the principal tool in the data analysis. Based on a
m-ethod suggested by M. N. Moore, and written b;y R. S. Booth
116), it was used here to perform the Fourier transform of
the time analyzer data descriptive of the thermal neutron
pulse in order to yield as output the amplitude and phase
of frequencies present in the pulse.
A list of column headings on the data and calculation
sheets used for the wave. propagation experiments is given
below,a along with explanatory remarks. Some of these quanti-
ties were obtained as preliminary information for input into
the MOORE MOC-1E1TS code for background and resolution cor-
rection of the count data, and for initial plotting of graphs
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to see if the data wvere ;.orth putting through the Fourier
1. Run Identification [Number.
2. Detector Position.
3. Shutter Fosition. (Designate up or down.
4. Neutron Generator Beam Current in wamp. IThis is
the time-averaged current of deuterium ions received
by the target of the Cockcroft Walton accelerator
and was recorded principally as a guide during
experimentation. It .;as not relied upon for use
5. Run Time. iGenerally the time necessary to complete
some chosen number of 'seeps of the analyzer.)
6. TRIG or TRIGC. (TRIG is the number of triggers or
s..'eeps for the shutter-up run. TRIGC is the number
of triggers for the shutter-down run. These were
7. Observed 12" Detector Counts on Scaler.
8. 12" Detector Integral Counts From Analyzer. Read
from the digital print-out from the analyzer.)
9. item 8 Driided by Item 7. (This was calculated as
a running check on the electronics. Since the
resolution times for the scaler and the analyzer
were about the same, and since the signal from the
linear amplifier and discriminator for the 12"
system went to both the scaler and the analyzer,
any drift in the value of this constant indicated
a drift in the trigger frequency, in the channel
width, in the pulse shape or resolution time of the
scaler or of the analyzer, or almost any other mal-
function of either the scaler or the analyzer and
its digital print-out system.)
10. Background Counts Per Channel. IThis quantity
was the average counts per channel for the last
100 channels, calculated frcm the digital print-
out. Note that the ch anel width useJi must be
long enough for the pulse to decayo o backgoroiand
in the first 900 channels.)
11. Observed 6" Detector Count.s on Scaler.
12. Observed 6' Detector Count Rate. (Calculated,
in cpm, by dividing total counts b. run time.
tlote that the 6" detector accumul i tes nearly
all its counts in the duration .-f the target
pulse. This count rate therefore is a timie-
averaged quantity and is far less than the
actual count rate during the pule. See Ap-
13 Bunching Factor for 6" Detector. (This is that
factor by which one. must multiply the average
count rate in item 12 in order to obtain the
effective count rate upcn wnich the resoluticn
correct ion for the 6" system can be based. See
14. Effecti'.e Count Rate for 6" Detector. IItem 12 x
15. Resolution Correction Factor, 1-F. (Taken from
Fig. B-2 in Appendix B for the effective count
rate in item 14).
16. Observed 6" Detector Counts on Scaler Dv.'ided by
1-F. (This resolution loss correct ion was usually
le-s than 2 oer cent.
17. RATIO. (For input to cede. PATIO =is item 16 for
the shutter-up run divided by item 16 for the shutter-
18. CORPFAC. (For input t' code. CORFAC i1 the factor
whizh takes into account the different source
strengths used for runs at different Z positions,
and is item 16 for th- shutter-up run at Z = 5 cm
divided by item 16 for the shutter-up run at the
detector position for which COP.RAC is beinq calcu-
lat.:d. Note that 5 cm is the first, or smallest
Z position at ..hich the detector could be placed.
19. 12" Detector Inteqral Counts From Analy.zerm D1.'ided
b" Obsere'-d 6" Detector Scaler Counts. (This 1s
item i divided by item ii, and is in a sense,
outF.put di'.id-ed by input. Although neither quantity
is corrected for background or 1ire out ion loss,
the logarithm of the quotient may be plotted .ersus
detector position to re-.eal any large errors in the
integral data prior to performing the Fourier
analy1 sis. )
20. itemTi 19 for the ihutter-up Run Mlinus Item 19 for the
Shutter-do-..n Run at the Same Detector Position.
'Although this quantity contains resolution and
background errors, the logarithm .-ersus position
may be plotted to reveal bad data or mistakes prior
to further analysis. The slope of this graph must
agree closely with that from the --ro frequency
measurements described earlier.)
In the analysis procedure outlined above, no provision
has been made for applying an end-effects correction to take
into account the fact that the medium is not of infinite
extent in the Z direction. The possibility of taking end-
effects into account in this work .'as studied, but it wa.
concluded that to do so correctly would add considerable com-
plication to the analysis, and, furthermore, no suitable
approximate method was found. It was concluded that, if
possible, the end-effects would be made negligible through
the simple expediency of not using data taken too close to
the end of the tank Some of the findings concerning the
nature of the end-effects error i...11 now be presented,
however, in the event that some reader may' be interested
in tackling the problem. The end-effects correctioDn spoken
of here is that complex quantity (hav.inq amplitude and phase'
which one would have tc add v.*ectoriall to the ex:Feriimentally
determined Fourier component from the MOORE mOMlENtTS code for
each detector position and frequency analyzed in order to
obtain the same amplitude and phase at that detector position
and frequency that could be obtained from measurements in a
tank of infinite length. The only case that is easy, to
calculate is the zero frequency correction .herein the cor-
rection has the same phase angle as the ex:perimentally
measured quantity. For other frequencies, transcendental
equations are encountered because the correction needed is
a function of the inverse relaxation length, w-.hich in turn
is dependent upon the experimental quantity and the correction
term. It was found that the amplitude of the end-effects
correction term at a given distance from the end of the
assembly is a maximum for the zero frequency case, and
decreases with increasing frequency, because the relaxation
length decreases with frequency. Therefore, if the cor-
rection is negligible for the zero frequency case at, say,
55 cm from the end of the tank, the correction is surely
neq liq ble fcr ;il other frequencies This statement is
further upheld by the fact that the correction term is
usually out of phase ~.ith the measured quantity, and this
reduces its effect upon the amplitude data For higher
frequencies, the correction sometimes adds, and sometimes
subtracts from the measured amplitude, tending to show up
as point-scatter rather than as a recoqriizable, *systematic
Before data .ere taken for the wa'*e propagation experi-
ment, it .was clear that good results from the Fourier analy-
sis of the thermal pulse .-.ould be impossible without the
accumulation of a large number .of counts, Also it w.'as
desirable that the time spent in accumulating counts at a
particular detector position .'ith cadmium shutter up or
down be less than an hour on the a'.erage to permit 40 runs
for obtaining shutter-up and shutter-down pulse data at
some 20 detector positions in a period of three or four
working days, assuming no equipment failures in that interval.
The relationships among run time, counts accumulated,
count rates, and resolution errors recoi.-ed further atten-
tion. Preliminiar' measurements showed that a 20 usec
channel counting width and a pulse repetition rate of 25
cps ..ere appropriate settings of the 1024 channel analyzer
for stud''inq neutron propagation in the assembly filled with
heavy water w'.ithout fuel. 1ith this pulse repetition rate
one pulse and one s..eep of the analyzer '.-.ould occur every
40 msec. Therefore. with a 20 jusi-c channel *idth each
channel o.ould be open to rece1.ve counts only one second out
of every, 2000C' seconds of run time Furthermore, if f:irm
statistical considerations, 30 O00C counts is taken to: be a
reasonable and desirable number to accumulate per channel
in the peak of the pulse, one finds that for a count rate ro
10" cpm in the peak. each channel must be open a total of
1 a sec during a run. and the run time required ..'ould be
ex:actl,' one hour. To meet the -amr statistical accuracy
with run times shorter than an hour would d require short-term
counting rates exceeding 10O cpm during th e plse peak
Table 5-3 presents information concerning run time, counts
accumulated, beam current, pulse width, pulse rates. and
channel .widths used for the .*.av.e propagation experiments.
A calculation of the counts lost because of the finite
resolution time of the counting equipment yields an error
of 2.83 per cent based on 10 cpm and a resolution time of
1 7 psec per neutron counted. If this error '.'ere ignored.
it would likely be the largest source of e::perimental error
in the initial data for the ..ave propagation experiments.
If the experimenter does not plan resolution corrections,
*-. I .
C J** i
,- C r"
,, .4 r
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-4 3 -4 -
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'4 C. '4
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he should use Ic:.er count rates and c~cmpririis, e bet'.-e rn sta-
tistical errors and resolution errors.
The effect of unco.rrected resolution errors in the
sort cf data encountered in this *'*rk is ..orth:'.' :f comi:ment.
The fractional error in the c-urnts stored in each cihann~l f
the analyzer is proportional to the instantantarou-s c-harnel
count rate, and hence pr:,port onal to: the channel c-ontent
This error \-.culd reach a ma:.imum orf 2 .83 per cent in the
peak channels for the resolution tiinm and cco.unt rate assumed
in the abo'.e example, but is propo'rtiona'-itelv le~s in the
channels describing the rise or the fall *:of the pulse in
time. Thus the ra.. data from the analyzer .i'-e a disto-rted,
some-:.hat flattened picture of the pulse. without resjluJtion
corrections, the flattened pulse .;,oulid contain a s :me'..hat
smaller amount of the higher frequency Fourier coimp.:onents
than the true pulse, and this error, beinq itself a smcr.th
function of frequency, -..ould probably propagate through all
the ensuing calculations '.--'th-out giving ..arning thr u.,ir h
The analyzer data require the applicatiL.n of a differ-
ent correction factor for each channel. Such a correction
'..ould be tedious to. do :.'itr manual calculators, but ..as
easily built into the MOORE .MOMELTS .:codJe by E.. S. Etooth, in
the following manner: For each channel, F was computed first
aind then th,;, channel content ..as divided by 1-F.
F = observed channel counts, T
(channel counting .idtth) (nurrmer of s..:eepl
Corrected channel counts = o d channel counts
T is the resolution time in usec per observed count. The
channel counting :-idth .is also expressed in usec and does
not include the 1'0 usec that the TMIC logic unit uses as
storage time between channels. [cte that F 15 the fraction
of counting time lost because of finite resolution time and
is proportional to the average count rate during the counting
time at each particular channel. The correction is quite
simple and straightfor..a rd.
The resolution time of the detector system beginning
:;ith the 12" detector and ending ..ith the analy'zer as sho''n
in Fig. 4-6 '.as found to be 1.7 ,..1 usec for count rates
up to 2.3 x 10 cpm. Also the resolution time of that por-
tion of the system in Fig. 4-6 beginning 'with the 12" detec-
tor and ending -.ith the scale ..-as 1.6 = 0.1 usec for count
rates up to 2.3 x: ir, cpm. The resolution time of this
second combination of components w.as needed because count
data from this scale -.ere used during constant source flux
mapping, for ..,iich the time analyzer ...'as
the mrronitor detectccr system in Fig. 4-7,
through scale, the resol,.'ing tim. '-.as 1
obser'.ve: cou i.h t rattea ip to. 4 :6 i ci cpr
played to determine the resolution times
is explained in ,Appendi.. C.
not needed. for
from '. I" de tector
.47 0.1 1u.c ftor
The- mretihod em-
EXPERirLE[TAL RESULTS FOR HE.'Y W.-.TER
A. Crntin'r .u Source lea ureme,&tslt in
Tra-ns erse Direct ion
In this criapter the results of the riea suirements out-
lined in Parts 1 and 2 of Table 5-1 in Chapter '. are
presented. The aeometr; for the trans.erse flux. mieasuremrents
is sho'.wn in Fi?. 6-1. :.here the 10 detector positions sh,.*.',i
lie in the 10C planes described in Chapter I.' under "Position-
inr appar.j~tu5 for detector." [jate that the cadmium ehutter
*.as not placed around the detector, or e'.en- located at the
detector position, but "'as lowered into the moderator
adjacent to the end .-f the tank, near 2 = 0.
Graphs cf the transverse fluxi inr the core tank along
a horizontal line at mid-height and in the plane Z = 5 cm
are gi.en in Fiqs. 6-2, 6-3. and 6-4. Figure E-2 was
obtained '..ith the cadmium shutter raised. Figure 6-3 is the
shutter-do.n curve, and Fig. 6-4 presents the difference
bet:.een the shutter-up and the shutter-dow.;n flux. The
graphs are normalized to unity at the center line to
Thinria lizing Apparatus
o--o--o- Oe -0-- -- --
Po it on
Fig. 6-1. Geometry for Transverse Flux Measurements.