Title: Determination of heterogreneous parameters by the neutron wave technique
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 Material Information
Title: Determination of heterogreneous parameters by the neutron wave technique
Alternate Title: Neutron wave technique
Physical Description: xiii, 137 leaves : illus. ; 28 cm.
Language: English
Creator: Bernard, Emile Anthony, 1936-
Publication Date: 1968
Copyright Date: 1968
 Subjects
Subject: Neutrons   ( lcsh )
Nuclear reactors   ( lcsh )
Nuclear Engineering Sciences thesis Ph. D   ( lcsh )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 135-136.
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098411
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000577304
oclc - 13957518
notis - ADA4999

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DETERMINATION
PARAMETERS
W AVE


OF HETEROGENEOUS
BY THE NEUTRON
TECHNIQUE


EMILE ANTHONY BERNARD












A DISSE-R'A iON I RSNL, ITED TO THIE GRADUATt COUNCIL 01"
1iLE UNIT-RSITrY or I LOrUDA
IN I'.AUTIALL I'LII-LL.MENT OL T1HE hL:LQUIREMNENTS FOR TILE
D)EGICE OF" DOCTOR 01 IPILOSOP'IY










UNIVERSITY OF FLORIDA
19e6
































To "B4i










TABLE OF CONTENTS


Page


ACKNtlOULEDGMHE TS . . . . . . . . .. iii

LIST OF TABLES . . . . . ... . . . . . vi

LIST OF FICURES . . . . . . . . .. . . vii

KEY TO S'Y- OLS . . . . . . . . . . . .ix

ABSTRACT . . . . . . . . .. . . . xi




CHAPTER I INTRODUCTION . . . . . . . . .

Background . . . . . . . .. . ... .1

Scope and Objectives . . . . . . . . .


CHAPTER II ITHEORY . . . . . . . .... .. 6

Introduction . . . . . . . . . . 6

The Theorecical Model . . . . . . . .7

Solution of Eqatrion For Fast Neutrons .. .. .

Solucton of equation For Thermal Neucrons .. .. 1

Combinacior. of Fast and Thermal Neutron Soluticns 16

Nu.erical Scluti on . . . . . . . ... .17

7 Analysis . . . . . . . . . .. 20

Ti-e Critical Frequency . . . . . . . . 21

CHAPTER III EX'PEDIETL-.L ?rTHOD. . . . . .... .26

Int ro jccion . . . . . . . . . ... .26

Prel imir ary procedures . . . . . . ... .51

Experimental Procedure . . . . . . ... .32











ACKNOWLEDGMENTS


The author wishes to express his thanks and appreciation to his

supervisory committee, especially to its chairman, Dr. M. J. Ohanian,

for their assistance and advice. Special thanks also go to

Dr. R. B. Perez who was chairman prior to his departure from the

University. Through his initial guidance and continued interest and

assistance he has remained closely associated with the work of this

dissertation. Appreciation is also extended to the Department of

Nuclear Engineering Sciences and to the U. S. Atomic Energy Commission

for their financial support of this dissertation.

Many other individuals rendered their time and services in support

of this dissertation and the author wishes to give recognition to

several of them. George Fogle gave invaluable assistance in the

electronic and technical problems encountered with the experimental

equipment. Jose Aldeanueva, Benny DuBois, Changmu Kang and Daniel

Meade were constantly available to assist in the data acquisition.

The competent assistance of Jim Burgess was very important in the

development of the computer programs used for data analysis.

Mrs. Edna Roberts was very helpful in the typing of the final manu-

script.

The author wishes to thank especially his wife, Betty, who was

always willing co do everything she could and assisted in many ways

in the preparation of this dissertation.










TABLE OF CONTENTS (cont'd)


Page

CHAPTER IV DATA ANALYSIS AND RESULTS . . . . ... 6.

General . . . . . . . . ... .. .. 6.

Continuous Mode Analysis . . . . . . . .. 40

Heavy later Pulsed Experiment . . . . . . . 42

One Fuel Rod Pulsed Experim2nt . . . . . . 53

A Lattice of Fuel Rods . . . . . . . ... .75


CHAPTER V CONCLUSIONS AND RECOC'--END.-.TIONS FOR FUTURE WORK 79

Conclusions . . . . . . . . ... . . 79

Reco mmenda t ns . . . . . . . . . ... . S.

APPENDICES

A MEASUPREENT OF RESOLUTION TILE: USING A PULSED
NEUTRON SOURCE . . . . . . . ... .. . 82

B DETERMINATION OF T-.RGET CUP.REN'T PULSE WIDTH
AND MULTICHANNEL ANALYZE ? CU;!;;EL WIDT4. ...... 9. 9

C EXPERI.AL CHECKS . . . . . . . .92

D COCMPUTER P FROGP...M.S . . . . . . . . . 93

E EXPERIMENTai L A.IMPLITUDES -ND FHK.'ES . . . ... 116

F NU'IERICAL CONSTAN;TS . . . . . . . . 13j

BIBLIOC RAPHY . . . . . . . . . . . . 155

BIOGRAPHIC.L SKETCH . . . . . . . . ... . 137











LIST OF TABLES


TABLE Page


1. COMPUTATION OF ALPHA AND XI AND CONTAMINATION
BY FIRST SPATIAL HARMONIC . . . . . ... 19

2. NUMERICAL ILLUSTRATION OF INTERSECTION OF
DISPERSION LAWS . . . . . . . . ... .24

3. CONTINUOUS MODE DECAY CONSTANTS . . . . .. 41

4. THEORETICAL VALUES OF ALPHA AND XI
FOR HEAVY WATER . . . . . . . . ... .46

5. EXPERIMENTAL VALUES OF ALPHA AND XI
FOR HEAVY WATER . . . . . . . . . 47

6. EXPANSION COEFFICIENTS OF RHO SQUARED AND
THERMALIZATION AND DIFFUSION PARAMETERS
FOR HEAVY WATER . . . . . . . .... .55

7. EXPERIMENTAL VALUES OF REAL AND IMAGINARY
COMPONENTS OF RHO SQUARED FOR HEAVY WATER . . .. 56

8. THEORETICAL VALUES OF ALPHA AND XI FOR
HEAVY WATER-ONE FUEL ROD SYSTEM . . . . .. 62

9. EXPERIMENTAL VALUES OF ALPHA AND XI FOR
HEAVY WATER-ONE FUEL ROD SYSTEM . . . . . 63

10. EXPANSION COEFFICIENTS OF RHO SCUA.ED FOR
HEAVY WATER-ONE FUEL ROD SYSTEM . . . . .. 67

11. EXPERIMENTAL VALUES OF REAL AND IMAGINARY COMPONENTS
OF RHO SQUARED FOR HEAVY WATER-ONE FUEL ROD SYSTEM 71

12. HETEROGENEOUS PARAMETERS BASED ON INTERSECTION
OF DISPERSION LAWS . . . . . . . ... .77

15. DATA FOR RESOLUTION TIME DETERMINATION . . .. 87











LIST OF FIGURES


FIGURE

1. A SCHEM-TIC VIEW OF THE HiETEROGENEOUS SYSTEM .

2. ILLUSTRATION OF DISPERSION LAW INTERSECTION .

5. EXPERIMENTAL ASSEMBLY . . . .

4. MOVABLE DETECTOR COUNTING SYSTEM . . . .

5. REFERENCE DETECTOR COUNTING SYSTEM . . .

6. METHOD OF ANA.LYSIS . . . . . . .

7. INTERRELATION OF COMPUTER PROGRAMS . . .

8. NORMA-LIZED NEUTROi; PULSES IN HEAVY WATER . .

9. EXPERLUEN FAL AM.PLITUDES FOR HEAVY WATER . .

10. EXPERILNTALI PH.'5ES FOR EEAVY AFTER . . .

11. EXPERIMENT.AL AND THEORETIC.-.L VALLES OF ALP?-,
FOR HEAVY '.'ATI.. . . . . . . . .

12. EXPERIIE:I.NL AND THEORETICAL VALUES OF XI
FOR HEAVY :ATER . . . . . . . .

15. EXPERiL-MENTAL .A:D THEORETICAL DISPERSIOI L.A.;S
IN RHO PLAkt: FO?. HEAY '.TER7E . . . ..

11. CC:P.RISO:I OF EX:?ERI::;T.L DIS3? AI' ON LA '15
IN FL'O PINE FOR HEA:.-Y .ATE. . . . . .

15. Ik.GL .ARY CCMPO;NNT CF RiHO SQCUAl:AD FOR HEAVY ri

16. REAL COMPO:;:NET OF RO1 SCL'UAED FOR HEAVY LATER

17. EXPERI-EN;TLL AND ThEOIrIC.AL DISPER-SIC: LA'.:S
IN RiHO SC'UA.-ED PL-.iNE FOP. HE.VY WATER . . .


1.5. :;NO,'L.LI?ED THE?'-t.L :N-UTC.O ?ULSS FOP. HL:.-'Y IATER-CO E
FUEL ROD S ST . . . . . . . . .


Page

4

. . 25

. . 28

. 29

. . 30









. . 35





. . 49



*. 49



. . 50


S51




54








LIST OF FIGURES (cont'd)


FIGURE Page

19. EXPERIMENTAL AMPLITUDES FOR HEAVY WATER-ONE
FUEL ROD SYSTEM . . . . . . . . . . 60

20. EXPERIMENTAL PHASES FOR HEAVY WATER-ONE
FUEL ROD SYSTEM.. ....... ........... . 61

21. EXPERIMENTAL AND THEORETICAL VALUES OF ALPHA
FOR HEAVY WATER-ONE FUEL ROD SYSTEM . . . . .. 64

22. EXPERIMENTAL AND THEORETICAL VALUES OF XI FOR
HEAVY WATER-ONE FUEL ROD SYSTEM ........... 65

25. EXPERIMENTAL AND THEORETICAL DISPERSION LAWS IN
RHO PLANE FOR HEAVY WATER-ONE FUEL ROD SYSTEM ..... 66

24. IMAGINARY COMPONENT OF RHO SQUARED FOR HEAVY
WATER-ONE FUEL ROD SYSTEM . . . . . . ... 69

25. REAL COMPONENT OF RHO SQUARED FOR HFAVY
WATER-ONE FUEL ROD SYSTEM . . . . . . ... 70

26. EXPERIMENTAL AND THEORETICAL DISPERSIO:; LAWS IN
RHO SQUARED PLANE FOR HEAVY WATER-ONE FUEL ROD SYSTEM 75

27. EXPERIMENTAL AND THEORETICAL INTERSECTION OF
DISPERSION LAWS IN RHO PLANE . . . . . . 74

28. EXPERIMENTAL AND THEORETICAL INTERSECTION OF
DISPERSION LAWS IN RHO SQUARED PLANE . . . .. 76

29. ILLUSTRATION OF RESOLUTION TIME AND TIME
BETWEEN BURSTS ON COUNTING RATE ... . ... . . 84

50. COUNT RATE VERSUS TIME BETWEEN BURSTS . . . ... 88

31. REPRESENTATIVE DISPLAY OF COUNTING PATTERN . . . 94


viii











KEY TO SYMBOLS


A . . TlHEIULAL NEUTRON ABSORPTION TERN;

B . TRANSVERSE BUCKLING OF nth SPATL'L NODE
n

c . EXPANSION COEFFICIENT OF THEl-'W'.L NEUTRON FLIU:
AT z 0

D .. DIFFUSION COEFFICIENT

J . ZERO OPDER BESSEL FUNC"LL)N
o

L~ .. DIFFUSION LENGTH SQUAI)

L . SLO:-ING DO'-.I TIM: FOP. FISSICN NEUTRONS

N . NOR .LIZATION CONSTANT

P . EXPANSION,, COEFFICIENT OF HiO SQU.A..RED IN PO'-.TE.
n
SERIES OF (tw)


p . RESON;ANCE; ESCAPE PROBABILITY

q . SLOWING DO'IN DENSITiY

R . EXTRAPOLATED RADIUS OF EXPERI;.ENLT-.L ASSEMBLY

r . RADLAL COORDINATE

r . RADIU C? FUEL ROD
o

S . THE-RM*.L ;EuTr?.N SOURCE YES.

S . E. EXPM;NSION COC.FFLCIE:,',07 SOLCE AT z 0

s . LAPLACE T:hANSFOPRM PA,?, TR 'lim .RESPECT TO .-.GE

t .. TIME

u . L TH-.3,.

v . V.-LOC LT








KEY TO SYMBOLS (cont'd)


z . AXIAL COORDINATE

0 . REAL PART OF COMPLEX INVERSE RELAXATION LENGTH

6 . DIRAC DELTA FUNCTION

7 . FEINBERG-GALANIN CONSTANT

T . NEUTRONS PER THERMAL NEUTRON ABSORPTION IN FUEL

e . FAST NEUTRON SLOWING DOWN DENSITY

. . IMAGINA-RY PART OF COMPLEX INVERSE RELAXATION LENGTH

p . COMPLEX INVERSE RELAXATION LENGTH AND LAPLACE
TRANSFORMnTTION PARAMETER WITH RESPECT TO z

E . MACROSCOPIC CROSS SECTION

T . FERMI AGE

S. NEUTRON FLUX

. . FOURIER TRANSFORM PARnMETER WITH RESPECT TO TIME
AND THE RADLAN FREQUENCY








Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy



DETERIIINATION OF HETEROGENEOUS PARAMETERS
BY THE NEUTROI W.-,VE TECHNIQUE


By


Enile Anthony Bernard


March 1963


Chairrmn: Dr. M. J. Ohanian
Major Departn-nt: Nuclear Enginreerin Sciences

The neutron vvve technirqu has been used to study a heteroge-.eous

sysLtem conisLinsl of a nacur-! uraniun red surrounded by a r.oderator

of heavy "vater. By considering, onl-y simle unit cell confizuracion,

the theoretical derivations are siTplified and :one can perform a

relntively si.ile and inexpensive experit-ent. The experimental

technique applied per-it.s the. r-easurenent of several heterozenecus

parr.~aters and ca be used to study ocl-.er fu- l-,codrator s:s tes.

The Feinber-GC-l-nin Constrnct, 7, a measured. This r -e.s.jre-.en:

was acco:-;pl ished b: first obct inning th- e:-:perirM;-n ta disoersLon la-;.

for the unic cell. :Ie.tc, the theor'icical dispersion law is compuced

for various val-ues of y. The theoretical dispersion law best fictir.,

the experimental one det.rn.nine t.e v.lue of 7.

The theory: i,:ca -o..l us9ed ,s :h-: .A e-Di fus ion. odel. 3-y usin;

a series of tranCsform ion an.d an e:.::-nsion i r. transverse e i.gn-





functions, the two coupled equations are solved for the thermal flux.

Inversion of the solution yields the theoretical dispersion law. This

dispersion law is interesting in that a critical frequency is pre -

dicted at which the dispersion law for the unit cell is equal to that

of the pure moderator. This intersection allows one to measure two

other heterogeneous parameters of the group Ls, Ith' q and p.

The thermal neutron pulse, as a function of axial position, was

numerically transformed from the time to the frequency domain. The

frequency dependent attenuation constants and phase shifts were then

obtained and the experimental dispersion laws were determined.

The real and imaginary parts of the square of the complex inverse

relaxation length were expanded in even and odd powers of the radian

frequency. From the expansion coefficients 7 was determined. However,

it was found that more accurate results were obtainable when 7 was de-

termined from the dispersion law fit. The latter analysis yielded

the result:

7 = 0.25 + .02 cm.

The expansion coefficient analysis yielded the following results:


7(Po) = 0.29


and

7(P) = O.1 .



The dispersion law intersection was found to occur at


-i
a = -0.0532 + .0005 cm
-1
S= -0.0386 + .0034 cm

This corresponded to frequencies of 158 cps in the unit cell and

129 cps in the pure moderator dispersion laws. Evaluation of p and





Sth results in unrealistic values for their quantities indicating

that a more accurate model than Age-Diffusion Theory is required

for this analysis. Furthermore it is necessary to eliminate

variations in heavy afterr purity to improve the accuracy of the

experimental data.

The analyses of this work have been conducted primarily in

the p plane. It appears that more accurate and sensitive results
2
are obtainable when the analyses are conducted in the p plane.

Data to aid this type of future studies have been included.











CHAPTER I


INTRODUCTION



Background

The Department of Nuclear Engineering Sciences at the University

of Florida has been engaged in an extensive research program in the

field of neutron wave propagation since 1960. The original studies

dealt with the theoretical (1) and experimental (2, 16) determina-

tion of the diffusion and thermalization parameters of homogeneous

moderating media. Successful results prompted studies of two region

moderating media (3), moderating media with localized absorbers (4),

a "homogenized" subcritical assembly (5, 6) and a reflected subcritical

assembly (7, 8, 11). The continued success of the neutron wave

technique has been demonstrated in recent conferences held at Karlsruhe

(9), Gainesville (10), San Diego (11) and Ann Arbor (12). Since past

endeavors have been successful it is of interest to investigate the

application of this technique to other systems.

In this work a heterogeneous modIerator-fuel system will be investi-

gated using the neutron wave propagation technique. Cain (15) studied

the heterogeneous problem theoretically and computed the spatial and

frequency dependent neutron fluxes in various configurations. One of

the configurations consisted of two natural uranium fuel rods embedded

in a heavy w.'ater o-.oderating medium. Calculation of the flux pertu-rbation






showed no significant change due to the fuel rods. Experimental

measurements were predicted to be indeterminate unless enriched fuel

was used. Measurement of quantities other than the flux would have

to be made to obtain meaningful results with natural uranium fuel rods.

Corno (14) also studied heterogeneous systems theoretically and

proposed an experiment to study the heterogeneous characteristics of

a cylindrical fuel rod embedded in a moderating rediun. The ex.-peri-

ment involved the use of a highly enriched driving shell surrounding

che fuel rod. By measuring the exponential attenuation of the

fundamental mode for different driving shells, 7 and r can be determined.

7 is the Feir.berc-Galanin Constant (15) and q is the number of neutrons

produced per thermal neutron absorption in the fuel. The technical

problems involved in designing and using such a driving shell mak- a

simpler e;:--ri-ental configuration desirable.

Booth, Perez and Harcley (16) have d-veloped a procedure for

neutron wave proag3ation experir.2r.cs. This procedure is a zgeeral one,

applicable co alr-ost any system. The experimental quencity measured

is the cc- lIe:.: inverse rela.cation leng:h, p, as a function of crequen.- y

*-.he re



2 is the ia.licude a:te.-zaiicn per unit i.enr.:th nd is the j'ase

shift per unit length. By m-asJring p it is then pos.-ihle to obtain

the disperslc.n law', i.e., ; versus 2 with frequency as the parameter.

-nalysis of t.he co.n lex inverse rela::acion length allo-.s the e:x ri-

cental d e;r.-ir..tcioa of -ie FeinF er:-'C alanin Cfn.:ant for the feel rod

in chc s : e- cc.m ;ris-.on 'o.-: :n d1.-- j s redic r.e b'

theory and the e..' er-.-.na] dLs:-ersi:,. lUa. can a!so be -de This






method of analysis will be employedto study the heterogeneous system.



Scope and Objectives


The neutron wave technique has been used successfully to study

several systems. It has not been used experimentally to study

heterogeneous moderator-fuel systems and the question of its practi-

cality in such cases remains unresolved. This study will investigate

the use of the technique on a system other than those already con-

sidered. The scope of the neutron wave technique is to be extended

experimentally to include a method for investigating the heterogeneous

moderator-fuel system. The system is to be as simple as possible.

Heterogeneous parameters and their sensitivity to the neutron wave

parameters will be determined. In particular, the Feinberg-Galanin

Constant, 7, will be measured. One natural uranium fuel rod embedded

in a moderating medium of heavy water, as shown in Figure 1., will be

used to illustrate the method.

Two other heterogeneous parameters of the group T, p, Tth and L

can also be measured. Their measurements are made possible by the

intersection of the dispersion laws of the moderator only and the

moderator-fuel systems.

A review of the literature failed to show the existence of an

established experimental method to measure 7 directly. 7 is usually

computed but complex designs often call for approximations which lead

to uncertainties in the results. The method presented here allows the

measurement of 7 to be made in a simple and inexpensive way using

one fuel rod.


















130 cm


122 cm


c cm










122









FIGUirFL i.

A SC.i-i .LATIC '.'IE.'; CO THE
HETEROGENEOuLS S'iYTE--.




5

The effect on 7 when a lattice is considered needs to be examined.

Direct extrapolation from the one fuel rod case to that of a complex

lattice must be demonstrated experimentally. In the absence of such

data an extrapolation to the simple lattice configuration used by

Dunlap (5) will be investigated. If lattice effects are significant

the experimental method must be modified to make the measurement of

7 a valid one.

The potential of the method will be investigated and its exten-

sion to determine other heterogeneous effects, such as interaction

and shadowing effects, will be considered. Versatility of the method

will also be discussed.












CHAPTER II

THEORY

Introduction


The theoretical model used in this analysis is the Age-

Diffusion Model. Although this is a siinpliied approach it is

sufficient to exhibit the characteristics to be expected from a

more refined analysis. To be more accurate, thermalization

effects should be considered. However, Dunlap (6) found that ex-

peirnental results for the sucritical assembly agreed, within

several percent, with Age-Diffusion Theory for frequencies up to

530 cps. .As will be seen in the developm-.nt of this chapter,

the theoretical basis for the e.xerimental method is established

by cor.sidering treque-cies less than 350 cLs. There:cre the

Age-Ditrusion :-odel will be used.

The purpos- o: the theoretical model, in this case, is to

predict values for para.e cers -'hich are to be measured ex..:eri-

me.cally. .- ~-Dif:tsion The-ory, since it do-:s irn-ore nhe rm.lij -

cion effects, cnr. not be expex:.ed to preiic: accjraze values.

.Ho.--er :h2se :redictic..s are suf icient co .'eri'y tche .racricadity

of the experinent and do offer a comparison -vih the experimental

r-sults. E-ecter comparisons till be obtained 'iith more accurate

theoretical models.







An important number associated with heterogeneous calculations is

the Feinberg-Galanin Constant, 7 (15). This constant is defined as

the ratio of the net current of neutrons into a fuel rod divided by the

neutron flux on the surface of the rod. The value of 7 is determined

by two methods. The first method is to fit the experimental disper-

sion law, using 7 as the fitting parameter. The second method is
2
based on expanding p in the dispersion law in a power series of iw

and equating coefficients of like powers. The first two expansion

coefficients will be related to 7. The results of the two methods will

be compared.

In the theoretical development of this chapter it will be shown

that the dispersion laws for the moderator only and the moderator-fuel

rod intersect. The frequency of the moderator-fuel rod dispersion law

at which the intersection occurs is designated the critical frequency.

c Experimentally this is important. By setting the real and

imaginary parts of the moderator-fuel rod dispersion law equal to those

of the moderator dispersion law and substituting the value of the

measured critical frequency and a and : measured at the critical fre-

quency, two heterogeneous parameters can be determined. Further de-

tail is presented later in this chapter.


The Theoretical "cdel


Two equations make up the Age-Diffusion Model. The time dependent

fast flux, 4 (x, u, t), in terms of lethargy, u, is described by the

equation
-D(u) V- O(x, u, t) + (u) o(x, u, c) =
[2.1]
1 OiL.(x u S(x, u, t) 8s(x. u, t)
v(u) 3z +u




8


t h
and the associated equation for the thermal flux, a (x,t) is


th 2 th th th
D- c (::,t) + (x,t) =
a
th [2.2]
v a- t + q(x,t ,) A(x,t)
th



where

D = the diffusion constant [cm]

a = the macroscopic absorption cross section (cn1]
-I
v = velocity tcr. sec ]

q = slowing (own density [neutrons cm se ]
-5 -l
S = source of neutrons due to the fuel rod [neutrons cm sec

A = absorption of thermal neutrons in the fuel rod [neutrons cm sec 1

Sin-e cylindrical geomrry will be consiLdred,

x = x(r,z)

in both e atior.s.


SoluzLic- of EauAntion For Fast N-,eutrons The time dependence in

equLicon (2.1] is re-.oved with a Fourier Trans or-..aticn with respect:

to t. Since th6 El'-\ results :ro. -a he iJls-.L ne'Ctron source which is

assumed co sLrc J c zero tLia


I(x,u,t) = 0; t < 0.

Under this condition the Fourier Transform of equation [2.1j is


D(u) 72 c(x,u,' w) + (u,) -.(x,u,,) =2
a[2. ,]

ij ,, -.3';:I u ,'.,
vI) (:x,u,.) + S.x, u,) ----
v(') Cu


where n is che cransform parateter.







The source term, S(x,u ;1), represents the fission neutrons produced

by thermal neutron absorption in the fuel rod. It is assumed that

these source neutrons appear with lethargy zero, at the surface of

the fuel rod. The explicit form of the source term is then


S(x,u,co) = 6(u) s() 7 (r ro) th(x,h )
t (o) 2 T r



where

r = radius of the fuel rod [cm]


s (0) the probability of a neutron entering the slowing
Z (o) down process


S = the neutron yeild per thermal neutron absorption

7 = net therral neutron current into the rod [cm]
thermal neutron flux at the surface of the rod

5 = the Dirac Delta Function.




It is convenient to write equation [2.5] in terms of the slowing down

density, q(x,u,c). The infinite medium relation,


q(x,u,w) = ; t (u) (x,u,C),

is assumed since c-asuremrencs were made in heavy water which is a

weak absorber of neutrons. Substituting for f(x,u,c) and S(x,u,u)

equation [2.3] becomes

iu (x, u ,)__
( D(u) V- + E (u) +- ) (
a v(u) ( E(u)

[2.4]
_(.6( (o) 55(r r) h
u + 8(u) t (0) 1 7 2 r o h







A change of variables is now made from lethargy, u, to age, T, using

the definition


d "_ D(u) D(_)_
du = 5 Zt(u) t(


and the relations



t. (T)


6(u) -
Zt (.)


5(T)


equation [2.4] becomes


(- V2 + 13(-,w)) q(x,t,w) = -


+ 56() s( TI
Lt(o)


5(r-r)
2 nr


[2.5]


th,
+ (x,> )


where
iw
) () + --
(B)= ul ve(T
D(B)



By multiplying equation [2.j] bv the integracing factor


f 1(r',W ) dt'
0


L 1
p(T,p) iL
p e s


where L is the slouing, doun tin- and p is the resonance escape

prob3bility and recognizing2 hat


( ____ ,-- :< (--, ) .+(-fL')
T p(,-u) p P'0,') T + p (T,) '


aT





11
2 s(0) o 7 5(r r th
(-v + ) e(xr,) = 6(T)) 8 r (x,W) [2.6]
e(x,-T) = (0) p(Tr,c) 2 n r



where
eo(x,,c = T(,O )
p(-r~w)


Equation [2.6] compares with equation [8.2.9a] of Beckurts and

Wirtz (17).

The next step is to remove the age dependence in equation [2.6]

using a Laplace Transformation with respect to T. The boundary

conditions
e(x,-=o,co) = 0

and p(T=o,c) = 1


apply since no neutrons slow down past age zero and the resonance

escape probability at age zero is one. The former condition holds be-

cause the source neutrons are taken into account by the source term

in equation [2.6]. The Laplace Transformation of equation [2.6] is

2 E (0) 5(r r ) th
V e(x,s,c) se(x,s,w) =- s o 7 (r ) (x,w) [2.7]
t (o) 2 x r




where s is the transform parameter.

At this point the dependent variables, e(x,s,,) and t(x,u),

are expanded in terms of the radial eigenfunctions, J (B r ), given

by the solution of the Helmholtz equation,


(7 + B2) J (B r) = 0.
r n o n


The orthogonality condition is







R
J (B r) J (5 r) 2 n r dr = N 5
o o n o p p np


where N is the normalization constant and 5
p
R, the extrapolated boundary, is such that

J (B R) = 0.
o n

The expansions are


O(X,s,,L) = Z b (z,s,a') J (B r)
n n o n


is the Kronecker delta.


and


th ch
o (x,,) = (z,'i) Jo (Br).
n n on


By noting that

V2 J (B r) = 2 ( r)
r o n n o n



multiplying by

2 r r J (B r),
o p

integrating and applying the orthogonality condition, equation [2.7]

is reduced to the following equation for the expansion coefficients:



262 (z,s, ) + (z,s,) =
SP P
6Oz.
[2.8]


Z(o) q 7
(0) (o
" p


th
th (z,F) J (nro o(Bpro )
nL 0 n0 0 p0


The lasc s:ep in the solution to the fast flux equation. is co re-

move the spatial dependency by a second Laplace Tran3sformation, this

time with respect to z. Th2 Laplace Transforniation of equation [2..]

is






2 2
p e (p,S,w) -(B + s) (p,s,) =


(o) 7 th
() Ti 4 (p,w) J (Br ) J (B r)
t (o) N p
t p


where p is the transform parameter and the boundary conditions are

taken to be
d9 (z=o,s,w)
ep(z=o,s,w) = dz = 0


which neglects the gradient at z=o because the dispersion law. is

independent of its presence. The solution for the fast flux expansion

coefficients is
[2.9]
1 E,(o) i 7 th
p (P,s, p s +(B2 2 E (o) N n no o p0
s + (B P ) t p
p


In order that the solution may be of use in solving for the thermal

flux, it is necessary to transform from the s domain back to the T

domain. The inverse transformation of equation [2.91 is

2 2
-(B p2)
S(p, T,) = e s (o) 7 th(p,) J(B r ) (pro)
pn n o nro 0 0
C (o) N


n,p = 1,2, . k.


Solution of Equation for Thermal Neutrons It is also necessary to

solve the thermal flux equation, equation [2.2]. The procedure is

similar to that used to solve the fast flux equation. The order of

transformation is changed so that the boundary conditions may be more

readily applied. Before solving equation [2.2] it is noted that








A(x,t) 7 6(r ro oth(x,t)
2 r

which represents the absorption of neutrons in the fuel rod as losses

occurring at the surface of tle fuel rod. By representing the absorp-

tions in this manner 'nd the source term in the manner described in

the previous section, the problem is reduced to a one region problem

with a source and absorber of neutrons.

Equation [2.2] is Laplace Transformed with respect to the z

variable:

Dth 2 ch 2 th th th th
(-D p I V + E ) (r,p,t) + D p o (r,z=o,t)
r a
th h [2.10)
+ th o. (r,z=o,t) = 1 d (r,p,t)t)
dt Vth at th'
th

th 6(r r )
7 th(r,p,t) r ro
2 r


where p is the transforrim parrmeter. In this case neither the thermal

flux nor it- derivative is zero at the origin. These terms are ex-

pressed as

.th (r,z-o,t) = S ( = S (t) J (B r)
dz 2


and
th
a (r,z=o,t) = c (c) J (B r).
n n an-


Making these substitutions, usiLrg the eigenfunction expansions and

invoking the orchogonality condition as was done in the solution of

the equajion for the fast neutrons equation [2.10] beco m r;









th 2 thB2 zth}t h Dth th
(-Dh + D B + h ) (p,t) + Dp (t) + D S (t)p
p a p p o p1




- __1 (p,t) + q (P,'tht)
th _t



E h n (p,t) Jo(Bnro) J (Br)
P


where


The time dependence is transformed out by a Fourier Transforma-

tion, again using w as the transform parameter. The result is


2 2
(-p + B +
P


th + iu
a
V th
th) h (p,O) + p c (C) + S (u) 5
Dth P p 0o
D


D th n n o no 0 p th p th
n n D h
P D


where


q (P,.thn,)) = P( th,") ep(p, ch, ).


q(r,p, th,t) = E q (p,T ,t) Jo(B r)
t n n th' Jo(Bnr








Combination of Fast and Therral ;Neutron Solutions The solution for

the thermal flux- expansion coefficients is



2-( p) -1 i-- teth(PW)
L p L2 Dth p p
th o
2 2
7 E S (o) -( P)rh
+ t L P(Tth') t e )- 1

P [2.11]



X Z Ot (p,,)J (B r )Jo ( r )= p c ('n) + S (mo)
n rL c, no J po p o pI




n,p = 1,2, . k


where the solution obtained for the fast flux expansion coefficients

has been substituted,


-th
L
a = I
th 2
D L
th



and

th th
cVh o


Since the moderator to fuel ratio considered is large, the roba ility

of a source neutron being absorbed on its fir.t .ollision is very

small. Therefore, the approximation



Z (o)
S(o)
t


is made










Numerical Solution

Two cases of the solution will now be considered, n,p = 1 and

n,p = 1 and 2. The former case represents the solution for the

fundamental spatial mode while the latter illustrates the case which

includes the first spatial harmonic as well as the fundamental solu-

tions. The latter was used in all the computations. The importance

of the first harmonic will be examined.

In both cases an inverse Laplace Transformation is required in

order to obtain the solution in z space. To simplify the inverse

transformation the poles of the transformed solution can be computed.

The solution in z space is then of the form


th(z,) = G.()e i()
p 1 i

where G.i()) is a constant, depending on the pole p (). The final

solution in terms of z will not be computed. Instead, the poles of

the transformed solution are conmuted, yielding the dispersion law.
2
For n,p = 1, p is computed from the equation



-(B{ 2 02) -
1 2 D thj
th o

2 -2) L
1(B t il s 2
+ pe 1 Jo (Bro) = 0 [2.12]
thN J
D N


For n,p = 1 and 5, o is com-uted by setting the determinant of a

2 X 2 matrix ecual to zero. The elements of the matrix are








2 2
B =- 2 + p -




-(
__2_ ~qpe
t h,





D thN







21 = 7- pe
DtI L


A = -3, + p
2 L


1 i4O
2 th
L D
th o

2 2
B1 p2)-,
B


t h L
th


s 0 2
11 Jo (B1r ),
0 o o


2 2
(B p2) th
1 h


(B2 p2)th
2 2
-< '^


1

th


s- 1 Jo(Bro)J (B2r o ),





iwL
s
1 oJ (B r )J (B r ),
S2 o 1


- to)
th
D
o


(B2 p2)
-(2 )ch iU


+ 7 .(ipe
Dt ,
"2


s N 2
1. J (B r )
S o 2 o


In each case soLutions *-aire obtained us in :_ie 'c YZVE.V Code (See

Appendi:,: D) .

Results of :he cc.jtadicr. :cr. selected frequean.ie are given in

Table 1. The fundamental eigenv',luees changed in the fourth signifi-

cant digit froi. t.he one ter.- (n,p = 1) to the two ter- (n,p = 1 and 2)

cas'c 3 Cc.:- icr.-.' ions -ere also -,d:- for a ten cerm c:.:pan.icr (21) d n -.

iunda.-.-rte- ei n.-alu s differed by. less :han 3.5 -rcr. :' ,0S of :he

one terry case. In the t 'o term cse the eien-'alue- of the first

harmonic code are of the same order of magnitude as the fundamental





19

r-4


U o\ o\0 o\ o\ oP c\0 o.
ZZ o O. 0 rcO aD a. a
0O 0 * *
-' 2 II --4 ,- ,-4 .-- .- 1-
H< N

ZI

-- O. o


Z II N N N N N N N
OU- N
u



0 -4 oo -4 r No
0O-- a rI k.0 0 Ie m

Z o C C C C 0 o
0 u. * * *
-C 0 0 0 0 0 0 0
Z Z I I I I I I
0 0 U

C. ce 7 :1c c4 Mri cs4
< < < (n -r cr SI -4 N N o 7N N o
H1 "Z 00 0 0 0 0 m0 a.


O H- - I I I I I




,'- L O -4o ,-4 O 0N L,.

zI < C
S -- * * *O


< 0>- C10. o o o o o o o

o E 0J0 C
OZ Z
< --' -J < N -1 C-

H- < X -4 N

S Z X o o ) o o C o
a-- D0 < *. .
L u. C o o o o o o o
0 < I I I I I I I

z
0 Z

Z 0 - 0 N -4.
S U 0 -4 N- 0 ) N Lf
2 0- C4 -4 "- co N%


Z X o o o o o o o ...--





0 I I I I I I I O
z -
LUJ cn r4 r-4 N '-4 LP, U- % ^
C) C) C) O C) C) C) C._


~^ . . <*
N'\ OT O O n O0 Ocl O



0 I 1 I I I I I




LU C) 0 C) 0 C C; 0

0 - CM- C- N-N
U-
Ct ^-








eigenvalues. However, the experimental source was found to contain a

low percentage of first harmonic contamination.1 These results are

included in Table 1. in this case only the fundamental mode is

significant.



7 Analysis


Using equation [2.12] it is possible to analyze 7. 7 is varied

parametrically and the resulting dispersion law is computed and com-

pared with the experimental one. The best comparison represents the

measured value of 7 within the limitations of the theoretical model.

If p in equation [2.12] is expanded in a power series of the

form

p2 n E) P
n n

and coefficients of like powers are equated, still another irezhod of

determinir- ;' is available. Tha first two coefficients of the- e:-

pansion are *.elated to 7 as follows


1 + 7 X(l q p)
2 L2
PO = B [2.15]
0 1
1 + r p X th
"th







1. Thes results were obtained fro:.1, the e:-.an..3 in ccefticients 0o
the cransverse ccnzinuous mode dacr at z = 10 ce. The ULFNLLS Code (20)
was used to Efi the data to the fundr.-cntal and fi.rs: h-rmnic r.:.des.
Con:2rLini tion a: z = 50 ca, rhe firs: d-ata Doint ainalyzed, ,as
deter .ined .- e*.* ua ir. t'.e- e:.-:cae.-.ial attc n'a.-ior. of eac. male and
cO,.putin r ;-, p 7r Cnc o.inatribactio.s. fhe- p terZ .1La3 are based on
mai. n -. 2- ol i tuda s and rpe rese.: r.-a:i'ir.j.:. co.:3.min tcion.






and D + X(qpL (1 B2 + ))
and -D t1 h th+ h
1 =2 [2.14]
1 + pyXTth( B th + P th)

where
J2 (B r)
X = 0 1 0
DthN
D 1
These expressions were obtained by separating the exponential term into

frequency dependent and independent terms, expressing the result as a

product of expcnentials and expanding each product in a MacLaurin Series.

The sensitivity of F0 and P to 7 was found to be

dP
0 -4
Po -4
6P 3.35 X 10

and
3P
S3.1 X 10 ,

both P and P changing about the same a-.ounL in the second significant

digit with respect to 7, yielding a 25% change in P and 7-/ change in P1.

Comparison of the two 7 analyses with the experimental results

will be made in Chapter IV.


The Critical Freauenvc

To investigate the critical frequency equation [2.12] is again

considered, both for the moderator only and the moderator-fuel rod

systems. The right hand sides of both equations are zero so the left
2 tn
hand sides can be ecua:ed. L and D are constants of the moderator
th
th th
but D differs between the two systems because v is different.
o
2
However, this difference is small and they are assumed to be equal.

It is now assumed that the dispersion laws intersect

th
2. Subsequent analysis showed the difference in v to be less
than 2.








so that the two p's are equal. The relation is then obtained:

2 2
hc yJ (B-C r ) P1 c th L
- c + o 1 o fpe I [2.15]
Dt thction a
o 1
iu
m
D th
where ishe frequency of the moderator only dispersion law asso-

ciated with the intersection and


p = + i2
















C C tC
ib LIt vlue uf p at the intersection. J is the critical frequency



dispersion law'.s intersect at the same values of an and but the

associated frequencies are different. By equating real and imaginary




;' 2 2
[BI (2 c -th

r p cos (22 Tth L )e = 1





and








r) 3;- v( e L-( r a S0 1 0LI o S '--Ca31 -3 1 ) t h- q anci
both e_ -onh aa O D
p s!n(2.-c -,'.a L "e = c m 1
D 7J (Br )



atc obtair.ed-. Compiuca ions performed for many conditions of inter-

sec iton S3oJ t'3ht .. > .0 Therefore, the m.ne and cosine must be

Dcs3ii,.ve for a soluc t.ioi to e::L: because all che other quancities in

both eq-atiors are positive. The angle








2a cT t Wc L
c th cs

must be in the first quadrant. If these two equations are divided,

one by the other, the result is




S- DthN 2.16]
tan (2C T o L ) c m 1 (2.16]
s Dth J 2(Br )
o o 1 0o

By determining the experimental intersection and computing N1 and

J 2(Bro) the right hand side of equation [2.16] can be evaluated,
th
if v and 7 are known. The argument of the left hand side can then

be determined. This allows either T h or L to be evaluated in
th s
terms of the remaining parameters. Once the argument is found, either

of the two original equations can be used to solve for n or p in

terms of the other. The experimental results will be discussed in

Chapter IV.

The intersection of the dispersion laws is illustrated in

Table 2. and shown in Figure 2., where they are plotted for 7= 0.0

(pure moderator) and 7 = 0.6. These values were selected to give

a qualitative description of the intersection. Based on Cain's

results (13), the value of 7 was expected to be in this range.

The sensitivity of c to 7 was investigated by using the two

term expansion of equation [2.12]. It was found that a change of

0.1 in y produced a 2 cps change in w .
c























TABLE 2.

NUM-ERICAL ILLUSTPATIO;N OF
INTERSECTION OF DISPERSIO;J LA.-S


GA;-:.A = 0.0

FREQUENCY ALPHA XI
CCPS) (Cc1) (-

294.0 -0.07326 -0.06260

295.0 -0.07337 -0.05272

296.0 -3.07347 -0.06284

297.0 -0.07357 -0.06235

298.0 -0.07363 -0.06309

::: INTc 'SZCTI


GA.'MMA = 0.6

FREQUENCY ALPHf XI,
(CPS) (C;.,,- ) C c.-1 )

305.0 -0.07331 -0.00626

306.0 -0.07341 -0.0627S

3: 06.5 -0.07347 -0.06264

307.0 -0.07352 -0.06239

305.0 -0.07363 -0.06301






0 100-







o--o-











I-I-
xi

0-040-







0,020








0


+ = 0.0

V 7 = 0.6


. 350 (

300 CPS


250 CPS


- 300 CPS

0 CPS


200 CPS


150 CPS-- -
\--o150 CPS
xa-
g-
100 cps--


--100 CPS


50 CPS <

-50 CPS

V+


o-'02


0d0A 4 O-o/
ALPHA I/CM


FIGURE 2.
ILLUSTRATION OF DISPERSION
LAW INTERSECTION


50 CPS


0G08


0.10


_1_ _L *













CHAPTER III


EXPERIMENTAL METHOD


Introduction


Two experiments were conducted. The first was in a pure

moderating medium of heavy water. The second was in the same system

with one fuel rod inserted as shown in Figure 1. The experimental

dispersion laws for both systems were determined using the pulse

propagation Lechnique (16) and converting the data into the frequency

domain in the u sal way:. A Texas Nuclear Corporation Neutron

Generator (Mode! 9505) was used as the source. Tne system in which

the measuremnarns were r'tade and the neutron ge.neracor which .:as used

have been descr ibed In deta il by Dunlap (5, 6). The 17 inch graphite

stack between the light water ther.alizing rank and the heavy a,, ter

tank -'as removed. This was done :o reduce the w-idth of the therral

neutron :ulze and thus increase the hih frequency cencnr..

NIo data acouisit i on sys .e.T:z '.'-re u'sed, one h'vinQ a r.o'.'able

drrtecor 'which was uc-ad for measure.emnt.s alon- the axi:l lengci of

the assembly ani :he o'thn: beinj the reference detector whichc h .wa

used for nor.i.iizacion purposes. The reference d-eector was moved

to a position adjac-en to the 2 rachite in Lhe chnernaizing assemrbi.y.







This was done to reduce the effects of source anisotropy. A brief

sketch of the set up is shown in Figure 3. Each detector system

contains basically the same components with the following exceptions:

(a) The movable detector system has a 12 inch He3

detector and the reference detector system has

a 6 inch He3 detector

(b) The movable detection system has additional

components for data acquisition and analysis

of the neutron pulse in time.

The-two detection systems are shown in Figures 4. and 5.

Considerable time was spent in becoming familiar with each

and every component. This step can not be overemphasized in im-

portance for it proved invaluable in the immediate identification

of equipment failures. Several failures occurred and data acquisi-

tion was terminated with a minimum collection of faulty data. The

use of oscilloscopes was likewise important in the verification of

proper pulses, monitoring signals and general trouble shooting.

Light water contamination in the heavy water moderator changed

between the two experiments. The first experiment was conducted at

99.5% purity while the second one was conducted at 99.0% purity.

The decrease in purity was caused by the several transfers of the

moderator that took place between the performance of the two

experiments.



1. The angular distribution of neutrons was found to be de-
pendent on the am.pli:ad of the target current. Experin.ental data
showed tha: the variation of anisotropy for the range of target
current amplicudes used 'wa red,,ced to less than 1% when the reference
detector was positioned adjacent to the graphite.









































Graphite





Light
"Wa cer


Experimental
System










Cadmium
"Shutter






Re ference
Detecto-
Therma iz ing
Assemb l'.


FIGURE 3.


F :; F' E: RI;I L.-.L ~-.L.








Atomic Instrument Co.
Regulated High Voltage
Power Supply Model 318


Texas Nuclear Corp.
12 in., 1 atm., He3
Texilium Detector


Hamner Electronics
Co., Inc.
Pre-Amplifier
Model 102


Hamner Electronics
Co., Inc.
Linear Amplifier
Model N-318


Hamner Electronics
Co., Inc.
Scaler
Model NS-11


Tektronix, Inc.
Pre-Amplifier
Power Supply
Type 127


IType CA -
Plug-in Unit]


Technical Measurements
Corp.
Data Output Unit
Model 220C


Tally 420
Binary Tape
Perforator


Hewlett-Packard Corp.
Digital Recorder
Model J44-561B


FIGURE 4.


MOVABLE DETECTOR COUNTING SYSTEM


Technical Measurements
Corp., Multichannel
Analyzer Digital Computer
Unit, Model CN-1024/Model
212 Pulsed Neutron Logic
Unit


~--------C~-----------
















Atomic Instrument Co.
Super Stable High
Voltage Po'...'er Suppl''
Model 312


Texas Nuclear CorD.
6 in., 1 atm., He3
Texilium Detector


Hamner Electronics
Co., Inc.
Pre-Ampli ier
Model 1 102


Hamner Electronics
Co., Inc.
Linear ."-.molifier
Model N318


Hamner Electronic.
Co., Inc.
Scaler, ;i=,-del TS-11


FIGURE 5.


REFERENCE DE SECTOR COUNT'IliG SYSTL':.







Preliminary Procedures

Before any data were taken the following pre-operational checks

and tests were performed on the counting systems:

(a) compatability of output-input signals between
components

(b) detector plateau determination

(c) chi-squared checks

(d) resolution time determination

(e) gamma ray discrimination checks

(f) minimizing of internal noise

(g) reverification of plateaus, chi-squared check
and resolution time.


Step (g) is necessary to assure that changes made in the system in

the preceding steps have not affected the counting characteristics

of the systems. Adequate guidance in the completion of most steps

is easily obtained from the appropriate technical manuals and any

Nuclear Engineering laboratory manual. Resolution time determina-

tion is described in Appendix A.

The last task to be accomplished before the actual experiment

is begun is the selection of the target current pulse width and re-

petition rate for the neutron generator. Coincident with this is the

selection of a compatible channel width for the multichannel

analyzer. This procedure is described in Appendix B.

In the course of checking out the detector systems a limiting

count rate was observed. Increased source intensity produced no in-

creased scaler count rate at approximately 200,000 and 250,000

counts per second for the reference and movable detector amplifiers







respectively. All data were checked to insure that the inximum

count rate during the neutron pulse peak did nor approach the

saturation rates.


Experimencai Procedure


In order to obtain the thermal neutron pulse propagation charac-

ceristics through the assembly the cadmium difference Lechnique !as

employed (16), using the cadmium shutter located at the forward edge

of the assembly. The epicadmium contribution of the source vas

thereby eliminated.

Norrmally a completed neutron wave experiment consu;.es days ar.d

even v-eeks. During this tine components ray change enough to affe:-

che data significantly. In order that one may have a standard of

comparison for reproducibility purposes a continuous mode run was

made at all the a:ial positions. These data can be acquired in

several hours, a period in which variations are n.?:ligible. If

necessary, normalii.ciion for minor variation: of che syster.,s can

then be made.

The conrinuous mode E::peirirent correspcnds to Ohna z-ro frecuenc'.

copone-rL: of che r.eurron pulse in chat the amnplliude atcenucation per

unit Lenlrh are the same (16). It minor anronlies occur in the zelo

frequc'.ie : cc.-.onent o cthe pulsed da;c.a :he dj:: car be assily

normalized to the continuous m.ode dact, In chis manner, small changes

due co d;y-to-day variations of the source, e.>., angular dL:cribu-

cions, and counting sysrens, e.g., shifts in detector plateaus, can

be eli.:inaced. In this series of ex::perirer:ts, no normr-aliita io. jas

required.





33

The continuous mode data which were recorded at each position for

both the cadmium shutter up and down runs included:

(a) run time

(b) reference detector scaler counts

(c) movable detector scaler counts.

After the continuous mode experiments were completed, the pulsed

experiments were conducted. At this point the multichannel analyzer

which was set up in accordance with Appendix B was used. Once the

pulsed mode has been established, one proceeds to examine the propaga-

tion of the thermal neutron pulses through the assembly, employing

the same cadmium difference technique used in the continuous mode case.

The pulsed mode data which were acquired for each run included:

(a) run time

(b) number of triggers

(c) reference detector scaler counts

(d) mcvable detector scaler counts

(e) multichannel analyzer printed output

(f) multichannel analyzer binary tape output.

Verification checks which were performed as soon as the data were re-

corded are described in Appendix C.

During each run, two signals were continuously monitored with

oscilloscopes. One was the target current of the neutron gun. The

other was the input to the amplifiers of the movable detector systems.

Thus, two visual presentations were available to assure proper pulsed

operation. A secondary purpose of monitoring the amplifier input

signal was to measure the effective countinZ tite of the movable

detector. The effective counting time is discussed in Appendix C.








In both cadmium covered and bare pulsed runs, a minimum of

64,000 counts were accumulated in the peak channel of the multichannel

analyzer. This sets the time required for each run. Data were ac-

quired every 5 cms from 50 through 90 ems. These limits are based on

the continuous mode analysis which showed definite deviations from a

simple exponential decay with distance for positions outside this

region. Deviation at the lower end was caused by the source while

end effects were significant at the higher end. The procedure em-

ployed in determining the appropriate heterogeneous parameters is

outlined in Figure 6. The data analysis is described in Chapter IV.












0
0 C)
W co

E--a n
S.o o


c I 4 1S


Q












CHAPTER IV

DATA ANALYSIS AND RESULTS

General


Data analysis was accomplished with a modified version of the

Moore Moments Code (6) and a series of supplemental programs

written for the IBM-1800. By using the 1800 the direct interface

capability bec:een the paper tape daca and the computer could be

utilized. A flow chart of the computer programs is shown in

Figure 7., while more detailed descriptions are given in Appendix D.

Purity of the heavy water was estimated to be 99.5 4 .2% for

the moderator only experiment and 99.0 + .2% for the one fuel rod

experiment. The 99.5. purity was based on the theoretical model

and the experimental data. This purity yielded the experi,.entally

observed values of Ci and s at near zero frequencies, where better

agreement exists betcw.en theory and experiment, when .: and were

computed from equation [2.12]. The 99.0 purity is based on the

difference of 0.5,: purity in samples from the two experiments w'.hich

were analyzed by the Departr.ent of Chemiscry. Purities were re-

ported to be 99. +. .05, and 93.8 ~ .05, for the two experiments (1s).

In both experiments the heavy wacer temperacure was 2,1 + --C.

Experimental errorsin measuring a and E verecompuced based on

the mean square deviation of the data points from the computed least

squares fit. E-rors rani.ed from 0.;'. to 2.0' and the latter 'w:s

taken as the experimental error for all determinations of _- and

Resolution time measurements were conducted as described in

Appendix A. The subsequent analysis showed this method to be in












-000
*HOj
0c Acn A-i
E-P~ 0) C
00 0 X (n
H >L 0 )
0 -4 x Q
04 a4P


-H1 4J


w (n (n
rdcfn
**CQ) d) j

0 O
0 0 V0 0
< *H I
cr1 0
OPOEI L j~

CHmum


SI
o e
0- 0 E E
'-1 0 ,









S!- o.. Q ) '--4 )-
30
14 0c 0 3 0 C-4










r:Ir
4 .l 0 r. ^










H = ^ 0
2 0 3 ^ _j


C,

'JO
S 0


04 0 22 0n
F ^- w a -i75


D 7
Q. -


C1)-




,4

x
&1

Mq


Q)


(U7


44
H-
















c,
Q) 0)



H Q) OrO
HO) O4J
** 0 l


I !








,-4




E-4 0-

SO









-H J O
4J 0 -H
-0 0 7
00 0

H :: 0 0:










-r1
CO J



cd30







error and a discussion of the error is also included in Appendix A.

DuBois (7) measured the resolution times of the two detector systems

just prior to this series of experiments. Resolutions times were

determined to be 1.42 and 1.77 microseconds for the movable and

reference detector systems respectively. Since the components of the

systems had not been changed, these resolutions times were used in

this analysis.

As was mentioned in Chapter Ill, the 17 incl. graphite stack

that had been located between the thermalizing and subcritical

assemblies was removed to obtain higher frequency content in the data.

There being less moderation, the asymptotic region of the assembly was

smaller. Dunlap (5) found the region to begin at 25 cms. Without

the graphite stack the asymptotic region began at 50 cns.

The asympLotic region was established by scanning the data.

The undesirable characteristics of higher spatial and energy modes at

axial positions near the source and end effects at positions far fro,,

the source tend to increase the computed values of a and . In region.

where neither effect is significant ninimun values of a and I are

comp.i ed. These cini.nu- .'aLJes '..:er de-ernied by compluting and :

usin, a four pote: scan of the dai.. Then a five point scan w'as used

and thz rtiLi-a ;wi re ccnpared '-i'h the previous o;es. if they differed

by lesa than I, the additional data point was considered to be in the

asymptotic region. Progressively higher point scans were used until

the minimal differences e:-.ceeded 1. The daca point produce irg this




1. See the speciall Controis Section of The .ALP?-. proran des-
cription in Appendi:: D.







deviation was excluded. The 1% criterion is based on the results of

Booth, Perez and Hartley (16) who found a 1% tolerance in the measure-

ment of a and . A consistent asymptotic range, for all frequencies

in both experiments, was found to be 50-75 cms. The end point was

the same as that determined by Dunlap (6).

In some cases deviating trends in the amplitudes and phases were

obviously present. Deviations at a given axial position increased or

decreased with increased frequency. In these cases the data points

were eliminated, beginning at certain frequencies. The criteria for

elimination were:

a) the obvious deviating trend at maximum frequency

b) the existence of a 0.25%2 difference in a and t between

the least squares computations which included and

excluded the data point in question.

Once a 0.25% deviation was observed the data for higher frequencies at

the given position were also eliminated.

In both experiments an expansion of the real and imaginary com-
2
ponents of the experimental p in powers of o(rad/sec) was made. The

expansions are:


2 2 2 +
0 2 4









2. Daviations greater than. 0.25% produced significant dis-
continuities in a and : 'hen subsecuenc data points were eliminated.







and

20( = P u) + Pgn5.



The P's were determined by using the UFNLLS Code (20). Data above

500 cps were excluded in the polynomial fitting because statistical

fluctuations were deemed too significant.

Perez, Ohanian and Dunlap (19) have found that greater sensitivity

of the dispersion law to ther;ralization and diffusion parameters

exists when it is considered in the p plane rather than in the p plane.

No analysis will be conducted in this manner because the experimental

errors appear to be more significant when the p dispersion law is

considered. The dispersion laws in the p plane will be shoun to

aid any future studies that may be conAlicted along these lines.

The remainder of this chapter will be divided into four parts.

The first part will deal with the continuous mode analysis. The pure

moderator and one fuel rod pulsed experic.nts will then be analyzed

and finally a lattice of fuel rods will be considered.



Continous Mode Ar.a!'.'sts


The continuous ro-e data serve, three purpoass:

1) it oFfers a st-rdard for conoirison with rhe zero

frequency component of the ther mal neutron pulse

2) it provides a standard for norm-lization of minor

ano rallies in the pulsed data due to source or count-

ing s.ste;- '.'ariat ion

5) it establish-s a sta3nl. rd for co-.oirison ~rith pref.io..s

and subsequent e:peririntt .







It becomes apparent that the continuous mode data are the key to the

reproducibility of the experiment.

At each data position the thermal ratio, TR, is computed from

the relation

N N
TR = mov 1 mov
Nref bare ref cd


where


N = the resolution corrected scaler counts

mov = the movable detector system

ref = the reference detector system

bare = the bare run data

cd = the cadmium run data.


TR, as a function of distance was fitted to an exponential function

using the least squares technique. The logarithmic decay constants

for each experiment are given in Table 3.



TABLE 5.


CONTINUOUS MODE DECAY CONSTANTS


Experiment Asymptotic Range Alpha (cm )



Heavy Water 50-75 cms 0.0571


Heavy Water- 50-75 cms 0.0585
One Fuel Rod


As will be seen, there was no normalization required for anomalies

in the data. In this case the continuous mode data are used only for

comparative purposes.








Heavy 1 water Pulsed Experimnnt


The thermal neutron pulses obtained in the heavy water only

experiment are shown in Figure 8. The last 512 channels contain few

or no counts. These are insignificant compared to the peak counts

of appro:-:imately 45,000 counts and are not plotted. The pulses are

normalized to the 50 cn pulse. Each channel represents 50 micro-

seconds. These pulses are the ones that were numerically Fourier

Transformed.

The Fourier Transformed flux:es and the least squares fit to the

anplitudes and phases are shown in Figures 9. and 10. The maximum

deviations of points included in the least squares fit Erom the least

squares fit are h", in 0 and 1. in ;. Theoretical values of 0. and -

computed fror. equation (2.12] are listed in Table b. while the result-

ing experiTental values are listed in Table 5. Theoretical and ex-

perir.ental -vlu s of a are illustrated in Figure 11. A raaxinun devia-

tion of 2.1"2 occurs at 300 cps. The smaller deviations at higher

frequencies is fortuicous. Statistical deviatio~is at these higher

freue.ncies are too significant to 3llow valid alaiLyses.

An illustration of -, si.il-r :o che one- of is given in

Fieure 12. In this case :he exo.eri-antal data acree .i:h t heor:

within 1. 5'.

The heavy after dis-ersion laws are shoin in Figures 1i. and 14.

The former shc:..s :r.e experiranta and theoretical discersion las.s of

this ''crk a-.d the later suo's a co-.c-rison uf chis ..'or'.- with r.'.a

of 'niap ,. 3cLh Co-:.arisoas i ire re;. .c:.ably co The better 'i;.r

freque-ncy cort.cnt of che-se data over cha: of Dcala.?'s (6. is decon-














40,000















20;000-
20 / 000Y


__J
-
LLI
LU
H-


109000-







0


50 CiMi






55 CM




-.2 C.1;


5 CM

?3' C..


CHANNEL


iNME


FIGURE 9.


NORMALIZED NEUTRON PULSES IU HEAVY WATER


F-
F-I.

Li_
EL]


?Si


_, C i l





1000


100


0 CPS
50 CPS


100 CPS


150 CPS


200 CPS


250 CPS


300 CPS


2350 CPS


10+


0


X DAT-', POiiT-, OMITTED FRO;M- LEAST SQUARES FIT

- LEAST SQL'UAES FIT TO FE:-AAINIIG DATA POINTS
I I I i t


50


so 70 B08
AXIAL POSITION9 CMS
FIGUF'E 9.
E:*.PERI-tE[ITA..L APLI TUIDES
FOPR HEA',L WATER


SO 100






1' I I

I L.---------


0-







-B-


DATA POINTS
X OMITTED FROM'LEAST
SQUARES FIT


100 CPS




150 CPS



200 CPS



250 CPS


300 CPS


350 CPS


- LEAST SQUARES FIT TO REMAINING DATA POINTS
I- ----- - --II I -L


60 70 SO
AXIAL POSITION CMS


50


100


EXPERIMENTAL PHASES FOR HEAVY WATER


4+ 50 CPS


-4


LL
n-
I--
in
LJ

C-
31


--h~
t-~ :


FIGURE 10.


-i(40
;9T 0.









TABLE 4

TIEORET ICAL VALUES OF ALPHA A;:U XI
FOR HEAVY ,lATER


FREQUEIICY (CPS)
0.
10.
20.
'o.
40.
50.


GO.
70.
80.
90.
100.

110.
120.
130.
140.
150.

150.
170.
180.
190.
200.

210.
2, 0 0 .


220.
230.
2140.
250.

2G0).
270.
30.
290.
300.

310.
32 0.
33 J.
3 i 0.
350'.


ALPHA(I/Cil)
-0.0375
-0.0377
-0.03 S
-0.0393
-0.0; 05
-0.0410


-0.0433
-0. 0 4 47
-0.01462
-0.0477
-0.0491

-0.0506
-0.0520
-0.0534
-0.0547
-0.0561

-0.0574
-0.05"7
-0.0500
-0.0512
-0.06214

-0. 0636
-3. 03 4 36
-0. 0650
- 0 6 7 1


-0.063

-0. 07 16
-0.0726
-0.0737


-C. 0 75
-0.0767
-0.07 7 "
- 0. T ,: .2


XI (1/Cl)
0.0000
-0. 0041
-0.00.81
-0.011S
-0.0153
-0.01SG

-0.0215
-0.0243
-0.0269
-0.0294
-0.0317

-0.0339
-0.0359
-0.0379
-0.039S
-0.01415

-0.0434
-0.0451
-0.0467
-0.0433
-3.0 099

-0.05i0
-0. 0 5

-0.0543
-0.0557
-0.0570

-0.05 c,'
-0.0507
-0.0 6 09
-0.0622
-0.0534

-0O. r; 6
-0 C 35
-0.0570
-0. 0 3 2 1
-0. 0j n2









TABLE 5.

EXPERIMENTAL VALUES OF ALPHA AND XI
FOR HEAVY WATER


FREQUENICY(CPS)
0.
10.
20.
30.
40.
50.


60.
70.
80.
90.
100.

110.
120.
130.
140.
150.

160.
170.
180.
190.
200.

210.
220.
230.
240.
250.

250.
270.
280.
290.
300.

310.
320.
330.
340.
350.


ALPHA(1/CIH)
-0.0376
-0.0378
-0.0383
-0.0391
-0.0402
-0.0414


-0.0428
-0.0443
-0.04158
-0.0473
-0.0483

-0.0503
-0.0519
-0.0534
-0.0549
-0.0564

-0.0573
-0.0593
-0.0606
-0.0619
-0.0631

-0.0643
-0.06555
-0.0669
-0.0682
-0.0696

-0.0708
-0.0719
-0.0730
-0.0741
-0.0753

-0.0762
-0.0759
-0.0776
-0.0782
-0.0726


XI (1/CM)
0.0000
-0.0040
-0.0079
-0.0117
-0.0153
-0.0187

-0.0218
-0.0247
-0.0275
-0.0300
-0.0324

-0.0347
-0.0368
-0.0388
-0.0407
-0.0425

-0.0442
-0.0458
-0.0473
-0.048S
-0.0503

-0.051S
-0.0533
-0.0547
-0.0561
-0.0572

-0.0582
-0.0592
-0.0603
-0.0613
-0.0623

-0.0631
-0.0541
-0.0652
-0.0664
-0.0677





0 100-


0' 080







Li




1<04G
I_
0_

<0o040






0 020


350 CPS


30C CPS


50 CPS


CPS


150 CPS


100 CPS


50 CPS


E:: PERIP.'E'iT-.L ERROR


-- A.E-DIFFUSION THEORY

-I- E.:PERIUIENTAL DATA POIflTS


- -- I I


100 G00
FREOUEINCY s


300
CPS


FIGURE 11.
EXPERIrI-ENTAL .AD THEORETICAL
VALTrES CP AL.PHA. FOR FEA'-Y WATERR


400









- EXPERIMENTAL DATA POINTS

-- AGE-DIFFUSION THEORY


I EXPERIMENTAL ERROR


350 CPS


300 CPS


50 CPS


CPS


150 CPS


100 CPS


CPS


.00 200 300
FREQUENCY CPS
FIGURE 12.
EXPERIMENTAL AND THEORETICAL
VALUES OF XI FOR HEAVY WATER


0 10(


0


0 010
LJ
N


0 042

0 -04
0QCE~





00-10-


0o080











0-o04
L00
OOCY





0 0P0


1- EXPERIMEiNTAL DATA POINTS

- AGE-DIFFUSION THEORY

- EXPERIMENTAL ERROR


4t--350 CPS
CPS
y 300 cPS


--200 CPS

/-150 CPS


---100 CPS



50 CPS


ALPHA


0 .5
i/CM


0O


FIGURE 13.


EXPERIMENTAL AND THEORPETICAL DISPEFRSION;
LAWS I1E RHO PLACE FOR HEAVY WATER





0 oC00


0 o080


L0o080
Lj


+- EXPERIMENTAL DATA POINTS

X DUNLAP'S DATA (6)
- EXPERIMENTAL ERROR
EXPERIMENTAL ERROR


350 CPS

300 CPS-

250 CPS--\


200 CPS


50 CPS


300 CPS


250 CPS


200 CPS


150 CPS-


0.040


0 020


K '---150 CPS

100 CPS00 C

-4-- 100 CPS


50 CPS C
\-N-50 CPS


AL'04 0 O/0
ALPHA I/CM


0-08


FIGURE 14.
COMPARISON OF EXPERIMENTAL DISPERSION LAWS
IN RHO PLANE FOR HEAVY WATER


0oi0


1 LII~ _


! 0 -02







strated by the more consistent results in the 2!0-510 cps range.

Results of the two experiments differ by 3.21 in 0, a phenomenon

attributable to a difference in heavy water purity.

2
The imaginary and real cciponents of p are shown in Figures 15.

and 16. respectively. The polynomial fits are included in each ploc.

Values obtained for the expansion coefficients are listed in Table 6.

Resultant values for the thermalizstion and diffusion pararmieters are

included. Values obtained by Dunlap (6) are also listeJ. PO and P1

agree within '(.1L and 1.27. respectively but the other coefficients

differ radically in magnitude and sign. An cx-aninatiion of Figure
2
16. shows that the real component of p is erratic, e::hibiting

oscillations and increasing less rapidly than Dunlap's results (6).

The expcrinental values of the real and imaginary components of p2

are listed in Table 7. The e:-:perir.r.tal and Lheoretical dispersion

laws for heavy water in the p plane are sho.'ni in Figure 17. The

vertical line result of .iAe-Diffusioa Theory is due Zo the neglect

of chermalizatior defects. The experimental dispersion 'aw obtained

by Dunlap (6) is also sho-:n. The theoretical dispersion law' obzained

by Perez, Ohantin and Dunlap (1?) is also included. The la-ter in-

cludes thernali zai:J L effects. .art front :he shifC in 2 due to. a

difference in heav-.'y ..: r purity, the t-o sets acree relativel-y ?!ll

at loit frecquenccie buc pronounced differerces occur at high fre-

quencies. The anomalous behavior of tie heavy '..ater data in this

experiment is more apparent h.-r.

In the hecercgeneous analysis the disperston la' in the p plane

is reqir. Thiss aree sc na' lv .'ell with :oth t.t.-or: an

Dunlap's data (6).










0 -0i



0o010


Li
[U




O-0
c 1





00(







00(


+ EXPERIMENTAL DATA POINTS

- LEAST SQUARES FIT TO DATA


I EXPERIMENTAL ERROR


++\J 50
+ CPS

-300 CPS


250 CPS


200 CPS


150 CPS


100 CPS


50 CPS


DO 1000 1500
OMEGAW RAO/SEC
FIGURE 15.
IMAGINARY COMPONENT OF RHO
SQUARED FOR HEAVY WATER




54
0 005


-- EXPERIMENTAL DATA POINTS

LEAST SQUARES FIT TO DATA POINTS
0-004--
SEXPERIIIENTAL ERROR

I--I
x

F-I
O003-



CL





100 P- -250 CPS 350 CPS
--1ii03 CPS

0o001

&00200




00 2 ,00 '000 4 00 000
O!,'LEG\A QUnJAREO (RAOE)(RAD) / (SEC) (L
FIGU RE 16.
REAL CO[iPO,]EiT OF R-IH
SQUARED FOR HEAVY W'.ER


Er)


/














TABLE 6.

EXPANSION COEFFICIENTS OF RHO SQUARED AND THERMALIZATION AND DIFFUSION
PARAMETERS FOR HEAVY WATER


QUANTITY


PO [cm-2]

P [cm-2 secj

P2 [cm- sec ]

P3 [cm- sec ]

-2 4-.
P4 [cm sec J

a0 Lsec I


Do [cm sec ]

C [cm4 -sec
CO [cm sec _7


EXPER I MENTAL
VALUE

1.3746 x 10-3
-6
5.0613 x 106

-II
-3.1374 x 10

-15
8.443 x 10 15

4.3259 x 10 17

20


1.976 x 10

*


EXPERIMENTAL VALUE
OF DUNLAP (16, 19)

1.4713 x 10-3

5.0083 x 10-6

3.9684 x 10- 1

1.8026 x 10 14

-17
2.5045 x 10 1

20


1.996 x 105

3.73 x 105


*** Lack of high frequency content of the heavy water data prevented
a suiTable determination of C .
0








TABLE 7.

EXPERPIENTAL VALUES OF REAL AMID
IMAGIN'ARY COI1PO1JE;TS OF R;IO SQUARED
FOR HEAVY WATER


OMEGA(RPAD/SEC)
0.000
62.3 1
125.663
130.1195
25] .327
314.159

375.990
4 39 .322
502.65 4
515.43G
623.312

691.143
753.931
815.313
879.61 5
942.476

1005.30C
10 6:3. 14 0
1130.972
1193.304
1256. 636

1313 53
1332.29 9
11 5 13 1
1507.9u3
1570. 795

1633 62 L,
1696 4 5 3
1759 2 ) 0
1322.122
134 953

1947.785
2010 17
2073.U49
2136.231
219 113


OIE'lGA SQUARE)
0.0
3947 .3
15791.3
35530.5
63165. 3
93695.

14 2122.0
193443.3
252661.
319771. 6
394 733. 4

477637.9
5 G 438 .1
6671 3.9
773775.4
383262.5

1010645.14
11 04 92 4.2
1273 0908.4
1425163.1
1579133.7

1740994 7
1910751.7
203340L. 1
2273952.7
24G739G.0

2 6 7 35.
2877971. 1
3095191.9
3320123.14
3553050.7

37933 G .0
404251.6
4299190. 4
45G360..3
4 3 39 35 9


REAL
0.0014139
C.0011 159
0. 00110 3
0.0015970
0.00138S 1
0.0013714

0.00136)0
0. 001350()
0. 0013421
0.0013362
0.0013325

0.0013315
0.00134101
0. 0013';! '
0. 00135 3
0.0 61 746

0.0013?32
0.001416u
0. 0 9144 0 3
0.0')15514
0.0014597


0.0016553
0. 90174ESJ
O.OG14535
0.0014557
0.0014764
0. 00 I51
0. 001535J9

0.0016259
0. 0 0 1 G1 2 C 2
0. 00 160 2
0.0016927
0.0017350
0.0017 777

0.0013256
C.001.01 3I
C. 0, 1777 .
0. 0017 11'3
0 0016 C 7 3


I MAG I NARY
0.0000000
0.0003043
0.0006109
0.0009211
0.0012354
0.0015536

0.001274h6
0.0021977
0.0025221
0.002 '-471
0.003172E;

0.0034932
0.0033271
0. 0041511)
0.0044734
0.0047949

0.0051153
0.005L360
0.0057505
0.0060573
0.0063002

0.0006659
0. 0 9 6 '^ 3
0.007327;';
0. 7606 2
0.00 79 712

0. 9i23253 :
0.090 5311
0.0038 207
0. 0 0 3 9 2 0 7
0.0091095
0.0093906

0.009622 :1
0.0090174 2'
0. 1001351
0.0103992
0. '1: c u







THERMALIZATION THEORY (19)


0






00









; a





CL





10,

ru





0o
0


150
CPS


+

+
100
CPS


+
4-


300
CPS


0


--250 CPS
7


200
CPS


150 CPS


+ EXPERIMENTAL DATA POINTS


100
CPS


V DUNLAP'S DATA (6)


l- EXPERIMENTAL ERROR


50 CPS


0-0010 0-'0015 0o0020 0-0025 0o003
(ALPHA) (ALPHA) (XI) (XI) 5 i/(CM) (CM)
FIGURE 17.

EXPERIMENTAL AND THEORETICAL DISPERSION
LAWS IN RHO SQUARED PLANE FOR HEAVY WATER


AGE-DIF-











One Fuel Rod Pulred Experiment


In Figure 1.. the thermal neutron pulses are shown Lor the various

axial positions. Again, the last 512 channels are not shown and each

channel represents 50 microseconds. All pulses are normalized to the

pulse at 50 cm.

Figures 19. and 20 show the Fourier transformed flux.es. At each

position considered in the final analysis, the amplitude and phase of

the frequency components are show.n. Included are the least squares

fits to the data. Deviations between the least squares fit and the

data points considered in the least square fic are small, the largest

being less than 5, in amplitude and less than I, in phase. The

theoretical results for 2 and are given in Table 8.and the e::peri-

mrental results a 3e given in Table 9. The results of both are illus-

trated in Figures 21.anl 22. The deviations at high frequencies

caused by the neglect of theroalizacion effec:s, are seen in the

illus :ra .ons. Taery under estimates and over estimates c at

high freq.encies. Msaxi-n-j e'.'Lacicns, hw:.,-. er, are less than 5.53 in

2 and 6.." in ;. These devia-ions occur at 1,10 cps and 320 cps

respc:tive ly. The closer agreem-nt bet':a-n theory and e:-peri .-nt 2a

higher reque.ncies is a:ain fortuitous. In the subsequent analyses

data above 500 cps, where m-n:-:iirum deviations :.er 5.1", in Ct and

V.97 iL: r, were excluded.

Both the expe-ir:i:.1 aid theoretical dispersion lawJs :re so:-'n

in Fi'ure .. TLe theoretical dispersicn .a.;: is ob:ained front

equ-tL.on [2.12] wich 7 = 0.25. This value of :- ga.'ve agreem-enc a:

zero frequency with the expert irental data aad the resulting





50s






40,


30 s00


20,00


10o


-55 CM


60 CM


5 CM

70 CM


>-


Z


LJ
a
22
En




EL
F-


<-


L,
F-


5 CM
.90 CM


12 256 384
CHANNEL NUMBER
FIGURE 18.
NORMALIZED THERMAL NEUTRON PULSES FOR
HEAVY' WATER-ONE FUEL ROD SYSTEM


75 CM

- 80 CM
















--0o cPs


-50 CPS

-100 CPS


-150 CPS


-200 CPs


250 CPS



300 CPs


350 CPs


-+- F.PEP.IIE.T..L DATA POI[;TS


X DATA POINiTS OMITTED FROP.. LEAST SQUAFREL FIT


0i -- LELAT SQUARES FIT TO PRE11.,IIIItIG POIDiT-

1 o F 0 70 b sb 10o
AXIAL POSITION,9 /MS
F IGUFE 19.
EXPEP.I-MElTAL A-IIPLITULDES FOR
HEAVY WA.-TER-OC.E FUEL ROD SYSTErI


ioo10


10


LUQ

I -
-l
I


L.
_j


10






\__0 CPS


I I I I- --t

t--~i I ~t--3


--100CPS



150 CPS
+-.I












200 CPS


---250 CPS


CPS


X DATA POINTS OMITTED FROM LEAST SQUARES FIT

LEAST SQUARES FIT TO REMAINING DATA POINTS
I I I I I


3 S6 70 O
AXIAL POSITION CMS
FIGURE 20.
EXPERIMENTAL PHASES FOR HEAVY
WATER-ONE FUEL ROD SYSTEM


30 100


U3
m
E<
0-i
[ 3


-
Or:
LL
C-fl

C-n
1-
EL


-4


-E


-%1


- --~








TABLE 3.

THEORETICAL VALUES OF ALPHA AND XI
FOR HEAVY WATER-ONE FUEL ROD SYSTEM


FREQU .IlCY(CPS)
0.
10.
20.
30.
40.
50.


60.
70.
0O.
90.
100.

110.
120.
130.
140.
150.

160.
170.
ISO.
190.
200.

210.
220.
230.
24 1 .
250.

260.
270.
280.
290.
300.

310.
320.
330.
340.
350.


ALPHA( 1/Cl l)
-0.0379
-0.0381
-0.C387
-0.03'J7
-0.0400
-0.01121


XI ( /CM)
0.0000
-0.0040
-0.0078
-0.0115
-0.0149
-0.0121

-0.0211
-0.023S
-0.0264
-0.0232
-0.0311

-0.0332
-0.0353
-0.0373
-0.0391
-0.0400

-0.0427
-0.0444
-0.04000
-0.0475
-0.0401

-0.0506
-0.0520
-0. 0534
-0. 0 '. :
-0. 0562

-0.0575
-0.0523
-0.0600
-0.0513
-0.0625

-0.0537
-0. 0643
-0. 0600
-0.01571
-0.0 0 2


-0.00434
-0.00449
-0.0463
-0.0477
-0. 0492

-0.0500
-0.0520
-0.0533
-0.0547
-0.0560

-0.0573
-0.052G
-0.0599
-0.0611
-0. 06 4

-0.0536
-0.0643
-0o.n0653
-0.0671
-0.0332

-0.0603
-0.0704
-0.0715
-0.0725
-0.0737

-0.0747
-0.0757
-0.0762C
-0.0772
-0.0783








TABLE 9.

EXPERIMENTAL VALUES OF ALPHA AND XI
FOR HEAVY WATER-ONE FUEL ROD SYSTEM


FREQUENiCY(CPS)
0.
10.
20.
30.
40.
50.


GO.
70.
80.
90.
100.

110.
120.
130.
140.
150.

160.
170.
180.
190.
200.

210.
220.
230.
240.
250.

260.
270.
2S0.
290.
300.

310.
320.
330.
340.
350.


ALPHA(1/CN)
-0.0380
-0.0382
-0.0387
-0.0395
-0.0405
-0.0418


-0.0431
-0.0446
-0.0461
-0.0476
-0. 0491

-0.0505
-0.0520
-0.0535
-0.0549
-0.0564

-0.0579
-0. 0593
-0.0606
-0.0619
-0.0532

-0.0646
-0.0660
-0.0674
-0.06S9
-0.0701

-0.0714
-0.0723
-0.0741
-0.0755
-0.0771

-0.0783
-0.0791
-0.0796
-0.0799
-0.0S00


XI (1/CM)
0.0000
-0.0038
-0.0076
-0.0112
-0.0147
-0.0179

-0.0209
-0.0236
-0.0262
-0.0286
-0.0309

-0.0330
-0.0351
-0.0370
-0.0388
-0.0406

-0.0422
-0.0437
-0.0452
-0.0467
-0.0482

-0.0496
-0.0510
-0.0522
-0.0534
-0.0544

-0.0555
-0.0566
-0.0576
-0.0586
-0.0532

-0.0599
-0.0604
-0.0616
-0.0G28
-0.0551





0 10
0-10














0 06


CL



<0 04


0


50
SCPS
300 CPS


250 CPS


00 CPS


50 CPS


100 CPS


50 CPS


- EX:PERIMENT.-.L D..TA. POINTS

.- .GE-DIFFUSIO:; THEOR'Y


I


EXPERII-.EiITAL ERROR


20'0
FREQUJENCYE CPS


FIGURE 21 .
EXPERItLE[ITAL AND THEORETICAL VALUES OF
ALPHA FOR HEAVY. WATER-ONE FUEL ROD SYSTEM






0 10







008,









1-d


0G0


OaG2(


+ EXPERIMENTAL DA'A POINTS

AGE-DIFFUSION THEORY


-I


EXPERIMENTAL ERROR


\_3 50
-t1
CPS

300 CPS


50 CPS


CPS


150 CPS


CPS


50 CPS


FREQUENCY,


CPS


FIGURE 22.
EXPERIMENTAL AND THEORETICAL VALUES OF
XI FOR HEAVY WATER-ONE FUEL ROD SYSTEM





0-1(






0 O.







Li





0 o
01


4- EXPERIMENTAL DATA POINTS

- AGE-DIFFUSION THEORY


I


EXPERIMENTAL ERROR


350


CPS


200 CPS


o50 CP


100 CPS


50 CPS


O-'04 0 '05
ALPHA 1/CM


FIGURE 23.
EXPERIMENTAL .MID THEOCF'.ETICA'L DISPERSIGCi LA-JS IN
RHO FLAITE FOR E.AVY W.ATER-ONE FUEL ROD SYSTEM-1







dispersion law is shown. Again, the deviation at high frequency is

seen. Since the determination of 7 is based on the zero frequency

data where theory and experiment better agree, the high frequency

deviation has no significant effect on the experimental value.

The sensitivity of the fit to 7 was determined by computing the

dispersion law for several values of 7. It was found that


6 7 = 22 6 a'

where a' is the zero frequency Q. Assuming that a is accurate to

+ 0.0008 (2.1% error) this sensitivity allows the determination of 7

to + 0.02. The feasibility of using the neutron wave technique to

measure 7 based on the dispersion law is demonstrated.

Attention is now turned to the polynomial expansion of p in

powers a). The expansion coefficients obtained from the UFNLLS Code (20)

are listed in Table 10.




TABLE 10.

EXPANSION COEFFICIENTS C'OF HO SCARED
FOR HEAVY WATER ONE FUEL ROD SYSTEM'

CO EFFI~ :. _:TS UFNLLS ?JSL1T

PO [cm-2] 1.4285 X 10o

P1 [cm sec] L.354 X 10 '

P [cm-2sec] 6.6175 X 10i

P, [cm- sec ] 1.2051 X 10-1

P [cm 2sec ] 5.8555 X 1017








Figures 2h. and 25. showi the e::perirmental data and the leasr squares fit

for the imaginary and real parts of p2 respectively. The 23D fit has

a maximum deviation of ) .5, frotn the experimental data at j500 cps while
3 2
r' E, has a ma-'imum deviation of 2.51 at 300 cps. Experimental

2 2
results for 2a; and a - are listed in Table 11.

In Chapter LI PO and P were related to 7. Using these relations,

equations (2.13] and [2.11], 7 vas computed to be 0.29 in terms of P0

and 0.l1 in terms of P The difference of these values from the 0.25

value obtained in the dispersion 12W fit are now discussed individually.

The P value of 7 is based principally on zero frequency data, P
0 0
being the zero frequency intercept of L' -. It follows that fairly

good agreement should exist between the .two values of 7, 0.25 and 0.29,

both bein de:ermin?d from zero frequency data. Nonetheless, a 16''

difference e::ists in '. This d.s:repancyv s e.cplained by recalling

that P,, the zero frecuen'cy intercept was 2 cer.iF uced p rarareter in t'ae

least squares fit co the e:xperi-ental daJta. This value is less chan

the observed i-tercept, a phenor.encn w'hch is caused by the minimum
2 '.
occurring .n the 2 nta at apprcxirately 50 cps. The miniu.-.

has :,et to be e:-: lined, :h.re be -g. no es:ablisrhed tCeoretical basis

for tC e::i.Ste .ca. it m.Tiy oe che result of sc-.Se s.sce-.aj ic error. Zo-.-

seGq.n:l, che e.-.er:.-i en:.l .val; of ?P is in question. It ito note

that if the value of the obser.'ed intercept is used, y is ccrmputed to

be 0.24 and this agrees 'ithin L', with the dispersion law result.

The greater :ev'.'itl o of e ? value of t has C':o c .auses. First,

the rela:ion deri-.a:' for 7 n c ?r- of ?, is dEj: e.' .1: on- te fuil fra-

quency raJn e, ? aei-n i e f:rst derL.ar.3 L" of the 2 ; fun :ion.

mere accurate model would predict a lo'.er v..'lue of P. l Lo:er -'alj:
-l














+ EXPERIMENTAL DATA POINTS


LEAST SQUARES FIT TO DATA POINTS


EXPERIMENTAL ERROR


+ \__350
+ CPS


300 CPS


Li



-0







=0
Li



O
F--I






<-






0


200 CPS


150 CPS


100 CPS


10 CPS


1000 15 00
OMEGA s RAD/SEC


FIGURE 24.
IMAGINARY COMPONENT OF RHO SQUARED
FOR HEAVY WATER-ONE FUEL ROD SYSTEM


Ic


0 02


50 CPS





0-005-


0-004+


1-j
)o .003



n_

50 002-
-<-
0F
<


0 001-


0'
OMEGA


+ EXPERIMENTAL DATA POINTS

- LEAST SQUARES FIT TO DATA POINTS


I


EXPERIMENTAL ERROR


300 CPS


350 CPS---


250 CPS


200 C


N-- 150 CPS
-100 CPS


000
SQUARE[],


A 4000 000
(RAO) /(SEC)(SEC)


FIGURE 25.
RE.L COMPONENT OF RHO SQUARED FOR
HE.'AVY WATER-OHIE FUEL ROD SYSTEM


0030
(RAO)


- -~-' ~L









TABLE 11.

EXPERIMENTAL VALUES OF REAL AND
IMAGINARY COMPONENTS OF RHO SQUARED
FOR HEAVY WATER-ONE FUEL ROD SYSTEM


OMEGA(RAD/SEC)
0.000
62.831
125.663
188.495
251.327
314.159

376.990
439.022
502.654
565.486
628.318

691.149
753.981
816.813
879.645
942.476

1005.308
1063. 140
1130.972
1193.804
1256.636


1319.468
1332.299
1445.131
1507.963
1570.795

1533.626
1696.450
1759.290
1822.122
1384.953

1947.735
2010.617
2073.449
2136.281
2193.113


OMEGA SQUARED
0.0
3947.8
15791.3
35530.5
63165.3
98695.8

142122.0
193443.8
252661.3
319774. 6
394783.4

477687.9
568488.1
667183.9
773775.4
338262.6

1010645.4
1140924.2
1279093.4
1425168.1
1579133.7


1740994.7
1910751.7
2033404.1
2273952.7
2467396.0

2663735.9
2877971.1
3095101.9
3320123.4
3553050.7

3793868.9
4042531.6
4299190.4
4563G96.8
4336095.9


REAL
0.001446G
0.0014455
0.0014415
0.0014363
0.0014315
0.0014232

0.0014231
0.0014304
0.0014354
0.0014433
0.0014526

0.0014643
0.0014774
0.0014921
0.0015112
0.0015343

0.0015676
0.0016016
0.0016230
0.0016544
0.001G773

0.0017161
0.0017604
0.0012203
0.0013927
0.0019571

0.0020113
0.0020363
0.0021691
0.0022590
0.0024522

0.0025514
0.0026108
0.0025451
0.00244G0
0.C021629


I MAG I NARY
0.0000000
0.0002949
0.0005913
0.0008903
0.0011934
0.0014990

0.0013065
0.0021147
0.0024241
0.0027320
0.0030399

0.0033485
0.0036562
0.0039662
0.0042771
0.0045890

0.0043936
0.0051967
0.0054947
0.0057953
0.0061051

0.0064234
0.0067373
0.0070574
0.0073555
0.0076470

0.0079417
0.0082553
0.0035456
0.0088723
0.0091399

0.0094011
0.0095803
0.0093191
0.01C0510
0.0104173







of P1 produces a lower value of 7. y computed from the PF relation

was found to be very sensitive to the value used for v th. 2,h80 m/sec

and 2,490 n/sec yielded results of 7 = 0.27 and 7 = 0,22 respectively.

The 2,470 m/sec is the average thermal neutron velocity in heavy water.

Insertion of the fuel rod would tend to increase the average velocity

because of preferential absorption of lower energy neutrons and pro-

duction of fast neutrons in the fuel rod. Consequently, the correct

value of vth should be soiaewhat larger than that for the moderator,

resulting in a more consistent value of 7.

For the present, 7 = 0.25 .02 obtained from the dispersion law:,

is considered the r:ost accurate value. Assuming that Diffusion Theory

holds in the rod, 7 was computed to be 0.3-27 by Cain (13).

2
The experimen-al and theoretical dispersion laws in the p2 plane

are sho'.'n in Figure 26. Tne high frequency difference is am:li.Lied.

The next analysis deals with the intersection of the dispersion

laws of the iieav,' -'ter outly and heavy water-one fuel rod systems.

This andly;,is is ha-.pered sor.e,'hat by, the difference in heavy water

purity that existed between the two sets of data. This difference

Wjs :2:.er. into account: b'j reducing the 2 of the c .e fuel rod dat.,

0.00157 to cc,--pns2-:e for absorptions of the a.di-:ional 0.5' li.it

water. This corre:cion was ob:.-tin'ed by comoput ir. the zero frequency

a at the two concentrations of heavy water frori equation [2.12). The

intersection is then based on a 99.5' heavy water purity. The dis-

persion law intersection is shio'n in Figure 27.'where both the ex.eri-

r.in al and t;ec.:r ic ii intersections are sho:.-r,. Tril or e ical '.L the

intersectio;-. occurs t lO :ps, a value --ch higher than the

e::perimencal inc.--rsection at 1 .&, c's which was obtained by plotting






0 012(





0o010(


0r











O0a00(
u M
I--










0000


0C 0010 0-0015
(ALPHA) (ALPHA)


--350 CPS
/-


\-- 300 CPS


+. \-250 CPS


+ --200 CPS


-150 CPS


100 CPS


+ EXPERIMENTAL DATA POINTS

AGE-DIFFUSION THEORY
10 CPS-


EXPERIMENTAL ERROR


0 a0020 0 O25 0- 0030
- (XI) (XI) 9 I/(CM) (CM)


FIGURE 26.
EXPERIMENTAL AND THEORETICAL DISPERSION LAWS IN RHO
SQUARED PLANE FOR HEAVY WATER-ONE FUEL ROD SYSTEM


.+





0-10







OoS
0 08











---0


0 a04







0 02(


- MODERATOR ONLY DATA POINTS

V MODERATOR-FUEL ROD DATA POINTS


-- AGE-DIFFUSION THEORY

Li EXPERIMENTAL ERROR
I


300 CPS


250 CP


2003 C


'00 CPS


150 CPS


5' CPS


100 CPS


100 CPS


50 CPS


MODERATOR ONiLY


o04


0-05
ALPHA L


u uo
1/C1NI


FIGURE 27.
EXPERIMENTAL AIID THEORETICAL INTERSECTIONi
OF DISPERSION LA'.-:S IN PJHO PLANT


2503 CPS








the observed dispersion laws for the heavy water only and heavy

water-one fuel rod systems. This discrepancy is due to the experi-

mental error and the inadequacy of the Age-Diffusion Model. The

dominant experimental error is caused by the uncertainty of the heavy

water purity for it was found that


&U 7,000 5C


where C is the correction term applied for the purity difference be-

tween the two experiments. C is approximate and introduces signifi-

cant deviations. More accurate results would have been obtained if

the experiments had been conducted at the same heavy water purity.

On the other hand, an improved theoretical model would predict a

smaller value of we thereby producing better agreement between the

theoretical and experimental intersection. The intersection in the

p2 plane is shown in Figure 28. Uncertainty in the heavy water

results obscure this analysis but a more distinct intersection may

occur.

The combination of theoretical and experimental uncertainties

prevent the accurate dete:'mination of heterogenectus parameters based

on the intersection of the disper3ion laws. This is readily apparent

in Table ]2. where the experimental and predicted parameters are given.

A more accurate experiment and theoretical model are needed for

proper analysis of the dispersion law intersection.


A Lattice of Fuel Rods

7 has been measured for a one fuel rod system Will the value of

7 change when a lattice of fuel rods is considered? It is not possible

at this time to answer this question resolutely but che work of






0o0120





0 -0100





0o 0080





'0 OOG







r-




Oo0020


200
CPS


150
CPS


100
CPS



50.
CPS


(ALPHA)


300 +
CPS

+

+ v
+

+ v -250 CPS
+
+ V


_--300 CPS


-- 200 CPS




150 CPS


100 CPS


+ 'IODEPRTOR OL iL' D.-T.-. POI'-iTS


'50 C PS


(0'-0015
(ALPHA)


V7 :MODERATOR-FULEL
POI ITS


ROD DATA


-- AGE-DIFFLITC.il THF.TOR

-1E::PERIilEiTTAL ERROR

0o 20 00025 0 ---00t30
- (XI) (XI) I/ (CM) (EM)


F'IG'UJE 2,3.
EXPERILMEN[TA.L .liD THEORETICAL INTERSECTION
OF DISPERSION LAWS IN RHO SQUARED PL.-iE














TABLE 12.

HETEROGENEOUS PARAMETERS BASED ON
INTERSECTION OF DISPERSION LAWS


EXPERIMENTAL VALUE


-0.05515

-0.03855

150.8

120.9

552.3

>. I


PREDICTED VALUE


1
-0.07478

-0.06375
31o.o1
310.01

510.141

118.82

.9992


These values were cb:ained from the Age-Diffusion Model.
These values .:ere obtained from N. J. Diaz by private comuni-
cation.


VARIABLE


-1
a (cm"I]

~c[cm-l]

c [cps]

r[ (cps]

P [cm2]

p




78
Dunlap (6) does offer a clue to the answer. His study included the

calculation of cell parameters of a lattice of fuel rods of the

same type as the one considered in this experiment. The calculated

parameters yielded a theoretical dispersion law which agreed reasonably

well With the experimental one. The agreement leads one to believe

that the calculated and true values of the parameters are nearly the

same. The value of L2 for the lattice will now be considered.
2 2
L of the lattice is related to 7 and L of the moderator (15)

and is very sensitive to 7 in that


L2 1.6 X 10 7.


Using Dunlap's data and = 0.25, L2 was computed to be 92.2 cm2

This compares within 0.6. of the 91.6.2 cm value co.iputed by Dunlap

(6). The close a[ree.ant in L and the agree.mer.t obtained by

Dunl=p between the theoretical and experir.ental dispecjion law in-

dicate that :he 7 measured fcr one fuel rod jay be applicable to

a simple lattice of fuel rods.















CHAPTER V

CONCLUSIONS AND RECO:MiENDATIONS FOR FUTURE WORK

Conclusions

The neutron wave technique has been used to study a heterogeneous

system. 7 has been determined using the experimental and theoretical

dispersion laws and the expansion coefficients of 2 in a power series

of (iw). The intersection of the experimental dispersion laws of the

heavy water only and the heavy water-one fuel rod systems made possible

the determination of two other heterogeneous para-maters. Heterogeneous

parameters have been extracted from the neutron wave data. These para-

meters are sufficiently sensitive to neutron wave parameters to allow

their determination by this technique. Accuracy of this technique has

been partially established in that Y can be determined through the dis-

persion law fit to two significant digits. Further investigation of

accuracy was precluded because of the theoretical model and/or experimental

phencr.ena.

The exie:.sicn of the results obtained for v to a lattice has beer

investi-tel for the case cf a simple lattice. The diffusion length

squared for the lattice was co=puted based on the measured value of 7

and it was found to agree very well with previous results. The fact that

the correct value of the diffusion length squared can be computed from

the measured value of y is very sign ficant.

Tne exoer-ienta.l zechnicu has 'esn tested under the most adverse







conditions in that a .inimum-L of fuel was used. Vith larger amounts

of natural uranium or with enriched uranium more pronounced character-

istics will be exhibited. In these systems improved results should be

obtained.

Reccc-mendat ions

'Tne most imnor-arnt reco-r-mendation is the development of a more

accurate theorelical model. Better accuracy at high frequencies would

improve the technique, predicting more accurate values for the appropri-

ate parameters.

Future studies should be conducted to determine interaction and

shado-in; effect's on the heterogocneous rara.-.etcrs. These studies can

be accomplished by a study: equivalent to this one wh'here two, three or

more fuel rods are ir.serten at various locations in the assembly. AAn

intercstin- an.-d useful study.- would be the jdternination of heter:--enous

oarans.eter s as function of -ne nu.'.ber of fuel rods in the asse.bi.'.

Still another a:.roz:h wsu'ld be the sta:." of t.te heterocenreous para-

meters a: a functi on of the :enter lir-e se- r'tio of the fuel rods.

.ne ar.aly:is of :'uJre e::eri.-.eants :.?. be nodifiedi in the follo'--

in -:':. Let t.-e car-::ric f: in ; cf the th-ore:'cal to 'the e::eri-

centa_ UQi3_-.-i---... ... '."-:f i .t -e: .--, .2 : f...t.r.-".i;." 'Ihen; in the

ev'alu'icn of ticE e;. .: s -.-. c.-ef-fi : ts,~; .(? .13) .n (Z2.1 ), -

is kr.'-r. and -:'c .t.- r t:sr.x-s ters can c cet -r.-.ne..

The disperse or. l-aw- ir,tersettion aj not sta-iied in the lanse.

It ma- be -.rtr.-'hle to do so in the f,:.-ure. -. artears that a imore

dirstinc t ii n :.::..O _.. r iclut-:r.









time of the counting system is needed. The method described in Appen-

dix A may serve as a starting point for the development of a suitable

method. It is important to determine the resolution time accurately

to 0.01 seconds because the experimental value of a and C are dependent

on the resolution time used. A more accurate resolution time yields

a more accurate experimental values of a and C and the other neutron

wave parameters.

Determination of heavy water purity proved to be a difficulty in

this experiment. Should heavy water or some other moderator with vari-

able properties be used in future experiments, the physical properties

must be accurately determined and verified. Efforts should be made to

conduct the moderator only and moderator-fuel experiments under the

same conditions when such moderators are used. The error in determin-

ing the physical properties must be minimized to improve the validity

of the data analysis.


































APPE.1DIX A

1.EASU .ILJ:LiT 0?7 RSOLUJTIO; Ti ':3 USI;;G A PULSED

i;EUT3:I f SOURCE







The method presented here for determination of the resolution

time was found to be inadequate when performed in the manner described

herein. The method, however, is believed to be sound and with one

improvement good results should be obtained.

The difficulty concerns the rectangular shape of the neutron

pulse at the detector. The asymptotic decay constant of the thermalizing

apparatus was such that the individual pulses at the detector are not

sharp and the decay time consumes a great percentage of the time between

pulses. Consequently, an "effective" time between pulses exists rather

than the time between the end of one pulse and the beginning of the

subsequent one which was measured at the source. An improved procedure

to overcome this difficulty is discussed at the end of this Appendix.

Resolution time is defined as the minimum time between successive

neutron interaction events such that both events are registered by the

counting system. The definition is not changed if, instead of a single

neutron, a burst of neutrons, as shown in Figure 29. is used. Assuming

that the neutron bursts from the neutron generator are:

(a) the same

and

(b) near a square wave at the detector

a method for determiiing the resolution tir.e of the counting syste:rs

was developed which used the neutron generator. The major advantage

of this method is that the resolution time of the counting system in

its operating condition is determined.

The method is illustrated grachically in Figure 29. It is seen









(A) EFFECTIVE ;;U;-SER OF CCONTS
REGISTE.PIED ?ER BURS'J
,I r--


/IF


(B) LES3 -:; --FECTIV- :I. '.-. OF COL.;T
RiEGISTD.ED ?.' 3U.1"ST


ILL'S-..'. IO.,:; -l !S ,L'5 O:7 : 1 T.l : A 3 TI-
BET.Z- :i L..S S '.; CC.LC'::T ?.'TZ


I/


I


II I


11111








that when the time between the end of one burst and the beginning of

the next is less than the resolution time, -, of the system the effec-

tive number of neutrons counted per burst is less, because some of the

leading neutrons of the burst are not counted. Otherwise the effective

number of counts per burst is constant.

To verify assumption (a) the target current was monitored with

a Type 5h7 Tecktronix Oscilloscope with a 1Al Plug-in PreAmplifier

unit. This PreAmplifier unit permitted double amplification of the

input signal by cascading the two channels of the PreAmplifier. This

monitoring method presented a trace of the target current which could

be monitored accurately enough for assurance of constant target current

and hence a constant burst of neutrons.

Since moderation will spread the neutron burst, the detector was

placed as near the source as practical. In this case, the detector

was placed in the reference detector position shown in Figure 3.

This arrangement was the optimum position for obtaining nearly square

pulses of thermal neutrons at the detector.

Tne resolution time of the system was believed to be of the order

of several microseconds. For this reason data were taken using a 10

microsecond target pulse initially separated by an 8 microsecond time

interval. Tne time intervals were measured by observing tne terminating

and initiating signals for the neutron pulses on an oscilloscope.

Tne time interval of separation was decreased until a nonlinear devia-

tion was observed.

The integral number of counts were recorded by the detector system.

If the Li:e interval of pulse separation is greater than the resolution

time, the counts should increase linearly with decreasing time separa-







Lion (increo-sins i-epetition rate). '.Then the tire separation is less

than the resolution ti.me there should be a nonlinear deviation in t-he

scaler counts. Reprcsertative data are siven in Table 33. ani illustrated

in Figure 30.

In order to ovcrcome the difficulty of pulse spreading the genera-

tor should be withdra'-'n from the ther,.alizing. asse bl, and a small

amount, for example several inches, of rodcrator placed between the

target and the detector. Although the sharpness of the detector pulse

has not been cc:.ipletely verified, a later experiment indicates that

this procedure 'ill yield suitable results for the resolution time.



























TABLE 13.

DATA FOR RESOLUTION TIME DETERMIIN'ATION


TIME BETWEEN
BURSTS (-ICROSECONDS)


AVERAGE COURTS
FOR ONE MINUTE


340,792

344,665

343,661

318,507

301,070




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