AE-226UDC 539.125.5.162.2

CNCM

LU Theoretical Time Dependent Thermal

Neutron Spectra and Reaction Rates

in H2O and D2O

S. N. Purohit

AKTIEBOLAGET ATOMENERGI

STOCKHOLM, SWEDEN 1966

AE-226

THEORETICAL TIME DEPENDENT THERMAL NEUTRON

SPECTRA AND REACTION RATES IN H.,0 and

by

S. N. Purohit

Aktiebolaget Atomenergi

Studsvik, Sweden

Printed and distributed in April, 1966,

LIST OF CONTENTS

Page

1. Introduction 3

2. Time dependent Boltzmann equation 3

2. 1 The time dependent thermal neutron source - S(E, t) 6

2.2 Thermal neutron scattering matrix 7

2.3 Numerical solution 10

3. Numerical results and discussion 11

3. 1 Light water (H2O) 11

3. 1. 1 Time dependent spectra 1 1

3. 1.2 Reaction rates 12

3.2 Heavy water (D2O) 14

3. 2. 1 Time dependent spectra 14

3.2.2 Reaction rates 14

3.3 Thermalisation parameters 15

3.3.1 Thermalisation time 16

3.3.2 Thermalisation time constant 16

3.4 Modification of the bound models of H?O and D~O 18

4. Moments method 19

4. 1 One dimensional Boltzmann equation 20

4.2 Time moments 22

. 24

25

26

30

35

5.

6.

7.

8.

9.

Conclusion

Acknowledgement

References

List of tables

List of figures

- 3 -

1. INTRODUCTION

The early theoretical and experimental time dependent neutron

thermalisation studies were limited to the study of the transient spec-

trum in the diffusion period. The recent experimental measurements

of the time dependent thermal neutron spectra and reaction rates, for

a number of moderators, have generated considerable interest in the

study of the time dependent Boltzmann equation.

In this paper we present detailed results for the time dependent

spectra and the reaction rates for resonance detectors using several

scattering models of H^O and D_O. This study has been undertaken

in order to interpret the integral time dependent neutron thermalisation

experiments in liquid moderators which have been performed at the

AB Atomenergi.

In addition to the above results we also present a short mathe-

matical formalism for studying the moments problem using the one

dimensional Boltzmann equation.

2. TIME DEPENDENT BOLTZMANN EQUATION

The time behaviour of a pulse of neutrons undergoing modera-

tion, thermalisation and diffusion in a moderating medium is described

by the time dependent Boltzmann equation. In the diffusion theory approxi-

mation, we write

00

so

Ss(E*-*E) <KE\£ , t) dE* +S(E,£, t) (1)

- 4 -

4>(E, r, t) is the neutron flux as a function of energy E (or velocity v),

space (r) and time t variables. 2 (E) and D(E) are the macroscopic total•- j

cross section and diffusion coefficient respectively. 2 (E —->E) is the scatte-s

ring kernel which governs the transfer of energy between neutrons and the

medium. S(E, r, t) is the external source.

We shall now expand <t>(E, r, t) and S(E, r, t) in a complete set of

orthonormal space modes.

• (E.r.t) = S^(r) yE.t) (2)

V2+p(r) ="B^ p ( r ) and +p(2) = 0 (3)

also

S(E, r, t) = 2 4;p(r) 6 ( E - E Q ) 6(t) (4)

The above expansion for §{&, r, t) implies that all the spatial modes

vanish at the extrapolated boundary (r) and the latter is independent of neu-

tron energy. In the case of source distribution we have assumed a pulse of

neutrons of energy E . Using the above expansions, one obtains

8* (E,t)r i 8<t> (E,t) r -i

(r) I -jrX = - i 2 (E) + 2 (E) U (E, t)

oo

(E '-> E) * (E*, t) dE* •+ 6(E - E ) 6(t) 1 (5)P ° J

where

2 (E) = 2 (E) + D(E) B2 (6)

- 5 -

In principle, one may construct the complete solution 4>(E, r, t)

by superposition of 4> (E, t) solutions. However, in practice the study

is usually limited to the fundamental spatial mode (1-5). This paper

is limited to the consideration of the infinite medium problem, B =0

for all p.

As is well known, the initial value problem may also be studied

with the Laplace transform method. If ^(E, s) is the solution of the

infinite medium Laplace transformed (in time) problem, then

b+ioo

4>(E, 0 = 2^T C 4>(E, S) exp fst] ds (7)J b-ioo L J

b is right of all the singularities of ^(E, s). The poles of ^(E, s)

contribute to the discrete modes in the form of exponential decays and

the other singularities give rise to continuous modes. It must be poin-

ted out that the knowledge of 4>(E, s) is intimately connected with the in-

finite medium solution of the energy dependent problem in the presence

of 1/v absorption. Only for very special cases - slowing down and

short collision time kernels - these solutions exist. For the latter case,

there is only an asymptotic solution of Corngold (6).

4>(E, t) solution for the slowing down kernel for hydrogen was gi-

ven by Ornstein and Uhlenbeck (7) and for a heavy element by Waller

(8) and others. The study of the initial value problem for the therma-

lisation kernel using the Laplace transformed or normal mode method

is infinitely complex, though highly desirable.

In this paper we shall only consider the determination of 4>(E, t)

by generating the numerical solution of the Boltzmann equation using

the source for thermal neutrons given by the slowing down theory. We

shall also assume that the scattering kernel for thermal neutrons is

given. Our treatment is an extension of the heavy gas case reported in

an earlier paper (1).

- 6 -

2. 1. The time dependent thermal neutron source - S(E, t)

The source for thermal neutrons is given by the following integral

E/or

S(E, t) = j * 2 8 ( E ! E) <fr(Ef , t) dE* (8)

E _ , the the rmal cut off energy is assumed to be equal to 1. 02 eV.A - 1 ̂a is equal to (.• • v) , with A equal to the rat io between the scat ter ing

atom and neutron m a s s e s . F o r hydrogenous mode ra to r s , the Ornstein

and Uhlenbeck solution (7) may be used . Fo r H ? O resu l t s p resen ted in

this study, we assumed

4>(E*. t) = S 2 t2N^E* exp - t 2 \ f l * (9)So So

S . is the free particle scattering cross section taken to be equalso *to 1. 33 cm , In order to take into account the effect of binding for ener-

Tgies above E T , the proton gas ke rne l with an effective t empera tu re

in units of thermodynamic t empera tu re has been used to calculate S(E, t)

in eq. (8).

35 ( E ' - * E ) « - £ erf E (10)s E1 kB Teff

for (E1 > E)

( " )

f(g) is the frequency spectrum of the dynamical modes, fi and k_, are

Planck's constant and Boltzmann's constant respectively. For the proton gas,

—^— = 1 and for the bound proton model of Nelkin (9) —?s— = 4. 66. Therefore,

- 7 -

S(E, t) = 2 2 2 erf J ^ r I t exp - t 2 NTE | (12)o B L o

For D_O studies, <j>(E ', t) given by von Dardel (10) has been

employed to obtain the thermal neutron source.

_J_ (i±a)

) exp-j-2 tsTE ^ , | (13)*• o s J

According to Eriksson (1 1), the empirical expression of von Dar-

del is in good agreement with Svartholm/s exact function (12) for A=2

and A=12. The value of b is equal to 0. 903 for A=2.

2. 2. Thermal neutron scattering matrix

In the neutron scattering formalism of Van Hove (13), the diffe-

rential scattering cross section, using the Gaussian incoherent approxi-

mation, is given by the following well known expression

+1 1/2 +co

J (^) J expicot X (k2, t) dtd^ (14)-1 -oo

21 is the bound atom scattering cross section, fiK and fito are the

momentum and energy transfers between a neutron and a moderating

atom. For the harmonic vibrations, the intermediate scattering func-

tion

t) = exp (15)

where X is the Debye-Waller integral.

- 8 -

(16)

In the expression of X, M is the mass of the scattering nucleus.

As noted earlier, f(£) is the generalised frequency distribution as

derived from the neutron scattering experiments. Theoretically, one may

consider f(|) as the frequency spectrum of the velocity auto correlation

function. In the case of harmonically vibrating Bravais lattice f(£) is the

frequency spectrum of phonons.

The dynamics of atomic motions in the bound proton model of Nel-

kin (9) for H?O and the Butler model (14) of D-O is represented by the

translational motion of the whole molecule plus the Ibiarmonic vibrations

of four oscillators representing three intr a-molecular vibrations and

hindered rotations. For these models,

f 4 ö (£-£.)gas . s ^ s,/

M ~ M ' ~ M.mol i=1 i

M i represents the mass of whole molecule and M. is the weight

of the i th harmonic vibration of frequency £..

In Table 1, we give weights of four vibrations for the original Nelkin

model and two modified Nelkin models I and II used in this. Intr a-molecular

vibrations are assumed to be of equal weight in all three cases. However,Mr

the relative weight (x = -r-r- ) between one intra-molecular vibration and hin-

dered rotations is different in each case. The basis of assigning the values

of x in these models will be discussed in a later section.

For the Butler model of D~O, the weights of different modes for

studying the contribution to the scattering cross section by deuteron and

oxygen atoms separately are also given.

Based upon the Zemach-Glauber formalism (15) using the above fre-

quency spectrum, one obtains

- 9 -

—J exp -

-1

In the above expression, 2W, I (Z.) and S (k, (J ) represent

the Debye-Waller factor (for four oscillators), the modified Bessel

function of the first kind and the scattering law for the gas model re-

spectively. We define

2M. g . sinh ^

=

09)

and

0 ) ' = CO - Of

(21)

The scattering matrices for the Nelkin and Butler models with

the parameters listed in Table 1 were obtained from the GAKER code*)

of Honeck (16) . This code calculates the incoherent scattering ma-

trix for the model of free translational mass and two harmonic oscilla-

' The scattering matrices for H?O and D?O models with the parame-

ters of Table 1 were reproduced by Mr. M. Lindberg using Honeck's

GAKER code with the 7090 IBM machine.

- 10 -

tors based upon the simplified version of the above expression eq. (18).

In the case of H-O, the contribution of oxygen atoms is neglected. How-

ever, for D_O separate matrices for oxygen and deuterons have been

obtained and added together.

2. 3. Numerical solution

The Boltzmann equation for thermal neutrons, eq. (5), is trans-

formed into a set of linear algebraic equations using the multigroup for-

malism. This transformation is based upon the replacement of cross

sections by the group cross sections and the derivatives by the finite

differences. Thermal neutrons between energies zero and E_ (thermal

cut off) are divided into a number (G) of energy groups. Due to the

Ferranti Mercury computer storage limitation, G is less than or equal

to twenty-two.

Let the flux distribution for g th group at time t be represented

by 4>n. The suffix g stands for the g th energy group and the prefix n

for the n th time point. For the g th energy group from eq. (5), one

(22)

obtains

We

A g

g

define

2At vo

g" B g (

ETT"

* g + 1 + • * -? * (Qgn+1

+ Q g

e } { ( s S } (24)

s 2 *?Vs8 4 > j | (25)j = 1 s.^g J . s g j

S(E(tnJ)dE (26)

The group g lies between E + h and E - h .

- 11 -

The numerical solution of the transformed equations is obtained

by two multigroup programes" - NEFLUDI and NEFLUDI TDCS pre-

pared by L. Persson in collaboration with K. Nyman for the Ferranti

computer. Both programes require the thermal group cross

sections (2 , D B and S ) and the thermal neutron source asN a g s.

the input data and handle twentyetwo thermal groups.

The NEFLUDI generates <t>(E, t) distribution and calculates reac-

tion rates for 1/v absorber and three resonance detectors Cd, Sm and

Gd for a given scattering matrix and thermal neutron source at each

time point. In the NEFLUDI TDCS, the modified version of NEFLUDI,

the group cross sections are recalculated during the running of the

problem using the calculated time dependent flux. This procedure is

adopted so as to obtain an effective increase in the number of thermal

groups by having better group cross sections. Moreover, it is con-

sistent with the multigroup approach. For details about NEFLUDI and

NEFLUDI TDCS programes see (17) and (18) respectively.

3. NUMERICAL RESULTS AND DISCUSSION

In this study we present detailed results for several scattering

models of H9O and D_O obtained with the NEFLUDI TDCS. Results

obtained with NEFLUDI for the light water case were given in an earlier

report (1 7).

3. 1. Light water (H-,O)

3.1.1. Time dependent spectra

Time dependent thermal neutron spectra generated by the nume-

rical solution of the Boltzmann equation are presented in Figs. 1 (a),

(b), (c), 2 and 3. As described in an earlier section (2. 1), these spectra

have been calculated using the Ornstein - Uhlenbeck solution as the ther-

mal neutron source starting from 10 |asec and are for zero absorption.

Results for the proton gas and the bound proton model of Nelkin are gi-

ven. Groups above 0. 1 eV (from 1 to 1 0) reach a distinct maximum be-

fore decaying because of the preferential downward scattering for these

- 12 -

groups. On the other hand, each group with energy below 0. 1 eV has a

broad maximum (which disappears with the decrease in the energy of

the group) followed by an asymptotic level. We plot in Fig. 4 t (the

time at which the energy group has a maximum) as a function of energy.

Qualitatively, the two models give similar spectra. For illustration, the

development of time dependent spectrum for the Nelkin model is shown

in Fig. 3. After 25 usec the spectrum is found to have attained the asymp-

totic distribution. A comparison between the calculated asymptotic spec-

trum at 81. 9 |J.sec and the Maxwellian distribution at the room temperature

is given in Fig. 1 5.

• 1 •From the above spectra reaction rates, for 1/v and three resonance

(Cd, Sm and Gd) detectors, have been calculated. To demonstrate the con-

sistency of these calculations with the physical considerations, we plot in

Fig. 5 unnormalised reaction rates for Cd for the hydrogen gas and bound

proton models, using the identical group structure and source at 1 0 jasec.

We observe that for times smaller than 2 fxsec and larger than 20 |j.sec the

reaction rates are almost identical. At small times the binding effect is

small and at long times the results are independent of models due to the

detailed balance theorem.

In Fig. 6, we present theoretical reaction rates along with the ex-

perimental results of Möller and Sjöstrand (19) for three detectors. For

comparison, all the curves have been normalised so as to have a level

value of 50 and correspond to the zero absorption case. These curves de-

monstrate the importance of each detector in studying the spectral effects

in different energy regions.

The proton gas model is inadequate to explain the measured reaction

rates for all three detectors. -=r*- ( -: --*- —) values forR-level reaction rate at level '

Cd and Sm are equal to 1. 84 and 1. 25 respectively compared to the expe-

rimental values of 1.91 and 1.29. The gas model underestimates the reac-

tion rates for Cd and Sm and overestimates for Gd. We attribute the diffe-

rence between the proton gas and the experimental results to the chemical

binding effect as the difference is too large to be attributed to numerical

factors.

- 13 -

Comparing the Nelkin model reaction rates with the experi-

mental results we find better agreement than the gas case discussed

above. Nevertheless, there still exists some disagreement as evi-

dent from Fig. 6. An independent study of Ghatak and Krieger (4) has

given a similar conclusion. The disagreement between the Nelkin mo-

del and experimental reaction rates can be attributed to several fac-

tors - the group structure, the source condition and the limitations

o£ the Nelkin model.

It is well known that the application of the Sachs-Teller mass

tensor approximation in the Nelkin model has introduced uncertainty

in the assignment of the relative weight (x) between one intra-molecular

vibration and hindered rotations. We, therefore, repeated the calcu-

lations for, two modified Nelkin models I (x = 0. 222) and II (x = 0. 674)

in order to investigate the effect of varying x on the reaction rate. In

Fig. 7, we present reaction rates for the above two cases along with

the Nelkin model (x = 0. 397) and experimental results. We note that

an increase in the contribution of the intra-molecular vibrations in

model II enlarges the disagreement between theory and experiment.

On the other hand, an increase in the contribution of the hindered ro-

tations in model I slightly diminishes this disagreement.

We also present a preliminary comparison between the Nelkin

model and the Haywood kernel reaction rates for Cd in Fig. 8. The

former is from this study and the latter from Poolers calculation. '

The Haywood kernel has been constructed from the Chalk River neu-

tron scattering experiments, following the treatment of Egelstaff and

Shofield (20). No attempt is made to discuss the theoretical results

of Fig. 8, as they have been obtained by different multigroup program.es,

source conditions and group structures. If we set aside these differences

then it appears that the Haywood kernel gives better agreement with ex-

perimental results beyond 7 [asec than the Nelkin model. The peak of

Cd reaction rate occurs at 4. 2 (j-sec according to the Haywood kernel.

' The Haywood kernel curve calculated by Dr. Poole of Harwell has

been communicated to the author by Dr. Möller. See also L. G. Larsson,

E. Möller and S. N. Purohit; Neutron scattering in hydrogenous moderators,

studied by Time dependent reaction rate method. AE-223 (1966).

- 14 -

This is in good agreement with the experimental peak position at 4. 1 - 0. 2

p-sec. The Nelkin model, according to this study, gives 3.49 |J.sec. On the

other hand, there exist differences between experimental and Haywood ker-

nel results at small times and also in the amplitude of the Cd peak.

In Table 2, we list a set of parameters obtained from reaction rate

studies for the proton gas and the Nelkin model for H9O along with the

experimental values.

3. 2. Heavy water (D^O)

3.2.1. Time dependent spe_ctra

For three scattering models of TJ2O - the Butler model with parame-

ters of Table 1 and two gas models of masses equal to 3. 6 (Brown and

St. John (20) ) and 2, with the contribution of oxygen as a gas model of

mass 16 for both the gas models - time dependent spectra have been ge-

nerated along similar lines as for H?O. As mentioned in section (2. 1),

the von Dardel distribution function eq. (13) has been employed to start

the calculations at the initial time of 1 |xsec. We have assumed 2 equalM S O

to 0. 346 cm" for D~O and set 2 = 0 in this study.c a

The development of thermal neutron spectrum for the Butler model

of D9O at different times is shown in Fig. 9. We plot the time dependent

spectra for three scattering models of D?O in Figs. 1 0 to 14 at 21, 41. 8,

64.2, 102. 6 and 166. 6 fjisec respectively. A comparison between the

asymptotic spectrum for D_O at 409. 8 fisec and the Maxwellian distri-

bution at room temperature is shown in Fig. 15. In Figs. 1 6 to 19, we

also present the time behaviour of four energy groups (0, 85 to 0. 625;

0. 1 8 to 0. 14; 0. 1 to 0. 08 and 0. 02 to 0. 025 eV) for four scattering mo-

dels - the Butler model with and without the contribution of oxygen and two

gas models.

3. 2. 2. Reaction_rate_s_

Reaction rates for three resonance detectors for different scattering

models of D_O are shown in Figs, 20 to 23. All the curves have been nor-

malised to a level value of 50 and are for zero absorption. The Butler mo-

- 15 -

del reaction rates have been used as a standard for comparison. In

Table 3, we list a set of reaction rate parameters for the Butler mo-

del of D-,O and the gas model of mass 2.

We present in Fig, 24 a comparison between theoretical and ex-

perimental reaction rates for Cd. Theoretical results are for two

scattering models of D_O - the Butler and the deuteron gas models.

The experimental curve is essentially an infinite medium curve. It

has been obtained by Möller (21) from a series of spatial measure-

ments in a D^O assembly.

From the time dependent spectrum and reaction rate studies

presented here we observe the following facts.

1. The difference between the results given by the Butler

model of D?O and the gas model of mass 2 is significant.

2. Butler and Brown and St. John models give similar spectra

and reaction rates. The Butler model also gives reaction

rates for Cd and Gd which are in good agreement with ex-

perimental results as shown in Fig. 24. This observation

was reported earlier (22).

3. The contribution of oxygen is significant and should not be

neglected in D?O studies.

4. The gas model of mass 2 is inadequate to explain experi-

mental results.

According to the studies of the RPI group (23) it also appears

that the Honeck model of D_O (24) and the effective mass model of

Brown and St, John give similar spectra.

3. 3. Thermalisation parameters

We shall describe the relaxation process leading to the establish-

ment of the Maxwellian distribution by two parameters - the therma-

lisation time and the thermalisation time constant. As there is no un-

animity in the literature in defining these concepts, therefore, we

shall define them again in this study.

- 16 -

3. 3. 1. Thermalisation time

The total time required to establish the Maxwellian distribution

after the introduction of a pulse of neutrons is defined as the therma-

lisation time or t . There is some arbitrariness involved in thisasympdefinition, as it is difficult to determine the state of complete therma-lisation precisely. In this study, we take t to be the time at

* y y asymp

which the reaction rate differs by 0.4 % from its asymptotic value, at

81. 9 usec for H?O and 409. 8 |j.sec for D?O. For the proton gas and the

Nelkin model of H00 t values are equal to 22 and 28 usec respec-2 asymp ^ t~ v

tively compared to 25 - 30 usec from the experimental data (19). For

the Butler model of D^O and the mass 2 gas model we obtain 230 and 160

usec respectively. Measurements on D?O give about 200 usec. See

Tables 2 and 3.

3. 3. 2. Thermalisation time constant

As already defined in the literature by several authors, it is the

time constant with which the Maxwellian distribution is established in

an infinite and non-absorbing medium. This definition is rigorously true

as long as the Maxwellian is attained via an exponential decay.

To estimate the time constant from experimental and theoretical

studies one usually fits the data for a quantity A(t) (for example v(t),

E(t) or R(t) for a resonance detector) during the last stage of the ther-

malisation process by a single exponential.

A(t) = AQ +A1 exp - \ j t (27)

A determines the asymptotic level and X... determines the time con-

stant as X = -— .

In Tables 2 and 3, we present the values of X. - | A\.| for H O and

D9O respectively. Theoretical |A\ | is the mean deviation from the average

value of \ . , obtained from a large number of fittings using the data bet-

ween 10 and 81. 9 usec for H?O. This range has been chosen as it has also

been used in the analysis of the experimental data.

- 17 -

For the proton gas model t , is estimated to be equal to 3. 5 -

- 0. 1 |asec (an average of values obtained from Cd and Sm data). For

the Nelkin model this value is found to be equal to 4. 7 - 0. 1 5 (isec.

The experimental value of Möller and Sjöstrand (19) is equal to 4. 1 -

_-_CL4 pisec. The proton gas model underestimates the time constant

compared to the experimental results and the Nelkin model overesti-

mates it.

The t , values from the fittings for the mass 2 gas and the Butler

model are equal to 22, 8 - 0 . 9 (isec and 36. 0 - 4. 3 [isec respectively.

The latter value has an unusually large deviation. It has been obtained

by taking an average of several fittings involving the data from 70 to

409 H-sec and dropping points up to 1 66 |isec. If we drop the points up

to 96 p.sec only and take the average of fittings we obtain 33. 7 - 1 . 5

|i.sec. The average of fittings using the data beyond 115 [isec gives

41. 5 - 1.9 fJ.sec. One therefore suspects 36. 0 |j.sec to be a composite

of two values.

A number of papers (23, 25, 26) presented at the IAEA pulsed

neutron symposium at Karlsruhe have also dealt with the D_O studies.

Theoretically, one may also estimate t , , from the first discrete

eigenvalue assuming it to be dominant at long times, from the following

expression (27).

fft~v

2 o

In Table 4, we list M2 (second energy transfer moment weighted

by the Maxwellian distribution) and t , for H~O and D_O models. The

values of M2 are from a paper of Purohit and Sjöstrand (28). M2 esti-

mates for the bound models of H?O and D-O are based upon the experi-

mentally derived frequency spectrum for hindered rotations as given

by Larsson and Dahlborg (29). f. (the correction factor to lJ1!approxi-t , l j

mation) values used in estimating t , correspond to proton and deuteron

gases. These are from the studies of Shapiro and Corngold (30). We

also give in Table 4 t , values from the eigenvalue studies of Ghatak

and Honeck (3) and Ohanian and Daitch (5).

- 18 -

3.4. Modification of the bound models of H^O and D,O

The presenif bound models (of Nelkin and Butler) need improvement

along two lines - (i) the replacement of the delta function representation

of vibrations by the structure, especially for the hindered rotations and

(ii) the assignment of realistic weight to different dynamical modes. In

addition, one would also like to include the small energy transfer modes

(diffusive and hindered translational). At present these are not very well

understood.

The two factors mentioned above depend upon a detailed understan-

ding of the hindered rotations which are the least understood of all dy-

namical modes. At present a heavy reliance is placed upon the neutron

scattering experiments. The cold neutron scattering experiments demon-

strate a temperature dependent structure for the hindered rotations but

fail to cover energy transfers of the magnitude of intra-molecular vibra-

tions. Therefore, these experiments do not provide direct information

about the relative weights between hindered rotations and intr a-molecular

vibrations. Attempts have been made to obtain the relative weight (x)

between one intr a-molecular vibration and hindered rotations from the

neutron scattering law data, see Beyster, J. R. et al. (31) and Egel-

staff, P. A. et al. (32). In the latter study the value of x= 0. 674 has

been estimated. This value is the basis of the modified Nelkin model II

used in this study.

In a theoretical study Yip and Osborn (33) treated hindered rota-

tions as small angle rotations of a molecule having the permanent electric

dipole moment |a in the local electric field of intensity e generated by

the neighbouring molecules. The energy associated with the torsional

oscillator has been predicted to be equal to N/2^B, where X. = fJ. x e and_ 3

B is the rotational constant. Using X = 0. 825 eV and B = 2. 2 x 10 eV

one obtains the oscillator energy equal to 0. 06 eV which is consistent

with the neutron scattering and Raman spectra experiments. By comparing2

the coefficient of the K term in the intermediate scattering function one

may assign an effective mass for hindered rotations in the Yip-Osborn

case. It is found to be equal to 3/4 B b , with b equal to the nuclear di--9 -3

stance from the center of mass. For b = 0. 9 x 10 cm and B = 2. 2 x 10Mr

eV one obtains = 1. 764 and x = 0. 222. This is the basis of the modifiedm

Nelkin model I.

- 19 -

The determination of the relative weight from the analysis of

two integral experiments has also been proposed in a study of the

integral parameters of the neutron scattering law (34). From this

study one also obtains x of the same order as in Nelkin model I.

The neutron scattering data, if available, provides the best

source to construct the generalised frequency spectrum. The Hay-

wood kernel for H^O developed at Harwell is an effort in this di-

rection. For the sake of further improvement, one has also to keep

in mind that the weights assigned to various dynamical modes in

the Haywood kernel have been obtained from the best fit to the avail-

able neutron scattering data.

4. MOMENTS METHOD

Alternative to the numerical solution of the Boltzmann equation

one may attempt to construct the complete distribution from the know-

ledge of its moments. For example, let a function 4>(x) be represented

by a complete set of orthogonal functions.

where 41 (x) form the orthonormal set between given limits, a are

the coefficients and W(x) is the weight function. As i|> (x) is a linear(

combination of terms of powers of x, therefore, one may represent

a in terms of the moments of ^(x).

4i (x) = S b x m (30)m=o

:) dx (31)

a = Ä D \ x l ) (32)m=o

- 20 -

We define

/ ir

Therefore

4>(x)

the m th moment as

b

a

00

= 2 W(x)n=o

dx

n

m=o

(33)

(34)

We present a general formalism to obtain the distribution function

using one dimensional Boltzmann equation.

4. 1. One dimensional Boltzmann equation

00 1

+0

\ U J E U E , ^ ) ^(E 'V.X, t) dE^n1 + S(E, x, fJi, t)

(35)

See Weinberg and Wigner (35).

We take the' t_

integrate over all jx.

We take the' t th angular moment. Multiply eq. (35) by P (|a) andV

> t+1(E,x,t)

84> (E,x,th p , ,+ I & *••• >• = \ d E S Q (E - » - E ) <t>. ( E , x , t ) d E

o x J J s I

+ S, (E,x,t) (36)V

- 21 -

where

+ 1(E, x, t) = J <f>(E, x, t.'ix) P^ (jt) d|i (37)

, » v C C i •(E —HE) = \ \ S (E —H£; (J. ) djj. P, ta) dfx (38)J J s o ^

-1

1 1

1

S^ (E, x, t) = \ S(E , x, t, |x) P t (|x) dft (39)_ i

Now we take the n th spatial moment by multiplying with x11 and

integrating between limits a and b (a = 0 for semiinfinite medium case,

a = -oo for infinite medium and b = oo for both cases).

, a* (E.t)

(40)

where

b

and

b

- 22 -

4. 2. Time moments

Finally, we take the r th time moment of eq. (40).

(E)

* t,n, • t+ , t E J-1 , r

= f z .(E—>E)4>n,

(E) (43)

where

o

oo

Therefore

4>(E, x, t,[i)= Sv» n, r

n (x) C r (t) 4>̂ ; n> r (E) (46)

A (x) and C (t) are suitably chosen orthonormal functions. The cosine

functions for x variable and the Laguerre functions for t variable.

The symmetry property of the distribution function

, t,x, , t, -x, - | (47)

implies that

> (E) = 01, n, rv '

For 1 and n , ,even odd

(48)

1 , , and nodd even

- 23 -

4» , (E) ^ 0 when n has the same parity (49)

as 1.

Also for n < 1

+1. n, r<E> = ° <5°>

Eq. (43) represents a set of interlinked integral equations. To

proceed further the steady state infinite medium energy dependent

problem must be solved. Only in very special cases can this problem

be exactly solved analytically. Time dependent solutions for these

cases - the Ornstein-Uhlenbeck solution for hydrogen and the Waller

solution for a heavy element - exist. One may also employ Corngold's

asymptotic solution to calculate the time moments, as has been under-

taken by Williams (36). For a realistic thermalisation kernel, the

problem is not tractable. However, one may employ the modal ex-

pansion used extensively in the study of the interpretation of pulsed

neutron experiments to obtain time moments in terms of eigenfunctions

and eigenvalues of the normal modes (discrete and continuous). Koppel

(31) attempted the modal expansion to obtain the time moments for the

heavy gas problem. A brief discussion of two methods, asymptotic and

modal expansions has been given in (38).

If we represent the distribution in a set of discrete and continuous

modes, that is

<x>

<t>(E, t) = S exp - \ t 4> (E) + \ 4>(E, X) exp - \ t d \ (51)n xJ*

then r th time moment is given by

d\A > A

n=1

( 5 2 )

- 24 -

We define

00,. r i

it (53)

The first time moment may be approximately represented by

M r «, 4> (E ) X Xt(E) =^ l ' + ^ <>M

00

X,(54)

Further discussion of the moment problem v/ould require the know-

ledge of the eigenfunctions and eigenvalues of the discrete and continuous

modes which are given in terms of the matrix elements of the scattering

operator of a given scattering model.

5. CONCLUSION

The proton gas and the deuteron gas models are inadequate to ex-

plain the measured reaction rates in H~O and D?O. The bound models

of Nelkin for H?O and of Butler for D?O give much better agreement with

the experimental results than the gas models. Nevertheless, some disagree-

ment between theoretical and experimental results still persists. This study

also indicates that the bound model of Butler and the effective mass 3. 6 gas

model of Brown and St. John give almost identical reaction rates. It is also

surprising to note that the calculated reaction rate for Cd for the Butler

model appears to be in better agreement with the experimental results

of D?O than of the Nelkin model with H?O experiments.

The present reaction rate studies are sensitive enough so as to

distinguish between the gas model and the bound model of a moderator.

However, to investigate the details of a scattering law (such as the effect of the

hindered rotations in H?O and D?O and the weights of different dynamical

- 25 -

modes) with the help of these studies would require further theoreti-

cal as well as experimental investigations. Theoretical results can

be further improved by improving the source for thermal neutrons,

the group structure and the scattering model.

It is easy to calculate and measure spectra and reaction rates

for a resonance detector in a moderator. At the same time it is diffi-

cult to discuss them in terms of the details of the dynamics of atomic

motions. Only with the help of well defined parameters one can hope

to undertake a systematic study of different scattering laws and under-

stand the importance of different dynamical modes. We have employed

in this study the thermalisation parameters (thermalisation time con-

stant and thermalisation time), the amplitude of the reaction rate peak,

and the time of occurrence of the peak to undertake a quantitative com-

parison of theoretical and experimental results. In addition to this list,

time moments also need to be studied.

Finally, we wish to emphasize that the cadmium reaction rate

studies may be sensitive enough for investigating the details of dy-

namical modes (especially hindered rotations) in hydrogenous liquids

at several temperatures. In that respect these studies would supple-

ment neutron scattering investigations in liquids.

6. ACKNOWLEDGEMENT

The author expresses his sincere thanks to Dr. E. Möller for

communicating his experimental results and for stimulating discussions.

During the course of this study, he held several fruitful discussions

with Prof. N. G. Sjöstrand, Chalmers Technical University, which are

gratefully acknowledged. He expresses his thanks to Drs. R. Pauli

and A. Claesson for their interest in this study. To the latter he is in-

depted for the critical review of this manuscript. The. assistance of

Mr. L. Persson and other members of the reactor physics division

in the numerical computations of this study is also appreciated.

- 26 -

7. REFERENCES

1. PUROHIT, S NTime-dependent thermal-neutron energy spectra in a monatomicheavy gasNucl. Sci. Eng. 2(1961) 305

2. BARARD, E et al.Time dependent neutron spectra in graphiteNucl. Sci. Eng. JT7 (1963) 513

3. GHAT AK, A K and HONECK, H COn the feasibility of measuring higher time decay constantsNucl. Sci. Eng. 2M_(1965) 227

4. GHATAK, A K and KRIEGER, TNeutron slowing-down times and chemical binding in •waterNucl. Sci. Eng. _21_ (1965) 304

5. OHANIAN, M J and DAITCH, P BEigenfunction analysis of thermal-neutron spectraNucl. Sci. Eng. j_9_(1964) 343

6. CORNGOLD, NThermalization of neutrons in infinite homogeneous systemsAnn. of Phys. _6 (1959) 368Chemical binding effects in the thermalization of neutronsIbid JJ_ (1960) 338

7. ORNSTEIN, L S and UHLENBECK, G ESome kinetic problems regarding the motion of neutrons throughparaffinePhysica 4 (1937) 478

8. WALLER, IOn the time-energy distribution of slowed-down neutronsInt. Conf. on the peaceful uses of atomic energy, Geneva 2. 1958.Vol. 16 Geneva 1958 p. 450. (Also to be published in Arkiv förFysik)

9. NELKIN, MScattering of slow neutrons by waterPhys. Rev. _M_9 (1 960) 741

10. Von DARDEL, G FThe interaction of neutrons with matter studied with a pulsedneutron sourceTrans. Roy. Inst. Techn. Stockholm No. 75(1954)

11. ERIKSSON, K-EStatistical time moments and an asymptotic formula for the time-energy distribution of slowed-down neutronsArkiv för Fysik 1_6_ (1959/60) 1

- 27 -

12. SVARTHOLM, NTwo problems in the theory of the slowing down of neutrons bycollision with atomic nucleiActa Polytech. 177 (1955) Phys. Nucl. ser. 3_: No 1

13. Van HOVE, LCorrelations in space and time and born approximation scatteringin systems of interacting particlesPhys. Rev. 9_5_(1954) 249

14. BUTLER, DThe scattering of slow neutrons by heavy water. 1. Intramole-cular scattering.Proc. Phys. Soc. _81_(1963) 276

1 5. ZEMACH, A C and GLAUBER, R JDynamics of neutron scattering by moleculesPhys. Rev. J_0I_(1956) 118

16. HONECK, H CTHERMOS . A thermalization transport theory code for reactorlattice calculations1961 (BNL-5826)

1 7. PUROHIT, S N and PERSSON, LTime dependent thermal neutron spectra and reaction ratesfor scattering models of water (NEFLUDI)1964 (AB Atomenergi, Sweden, Internal report RFR-283. Part 1)

18. PUROHIT, S N, NYMAN, K and PERSSON, LTime dependent reaction rates with multigroup programmenefludi TDCS1965 (AB Atomenergi, Sweden, Internal report FFT-6, RFN-216;RFR-283. Part 2)

19. MÖLLER, E and SJÖSTRAND, NGMeasurement of the slowing-down and thermalization time ofneutrons in waterArkiv för Fysik Zl_ (1964/65) 501

20. BROWN, H D and St. JOHN, D SNeutron energy spectrum in D?O1954 (DP-33)

21. MÖLLER, ENeutron moderation studied by the time-dependent reaction ratemethod.Pulsed neutron research, IAEA symposium Karlsruhe 1965.Vol. 1, Vienna 1965 p. 155.

- 28 -

22. PUROHIT, S NTime-dependent neutron thermalization in liquid moderatorsH2O and B'OPulsed neutron research. IAEA symposium Karlsruhe 1965.Vol. 1, Vienna 1965 p. 273

23. KRYTER, R C et al.Time-dependent thermal neutron spectra in D?OPulsed neutron research. IAEA symposium Karlsruhe 1965.Vol. 1, Vienna 1965 p. 465

24. HONECK, H CAn incoherent thermal scattering model for heavy waterTrans. Am. Nucl. Soc. J5 (1962) 47

25. DAUGHTRY, J W and WALTNER, A WThe diffusion parameters of heavy waterPulsed neutron research. IAEA symposium Karlsruhe 1965.Vol. 1, Vienna 1965 p. 65

26. POOLE, M J and WYDLER, PMeasurement of the time-dependent spectrum in heavy waterPulsed neutron research. IAEA symposium Karlsruhe 1965.Vol. 1, Vienna 1965 p. 535

27. PUROHIT, S NNeutron thermalization and diffusion in pulsed mediaNucl. Sci. Eng. _9 (1961) 157

28. PUROHIT, S N and SJÖSTRAND, N GNeutron thermalization parametersPulsed neutron research. IAEA symposium Karlsruhe 1965.Vol. 1, Vienna 1965, p. 289

29. LARSSON, K E and DAHLBORG, USome vibrational properties of solid and liquid H-,O and D^Oderived from differential cross-section measurementsReactor Sci. Technol., J. Nucl. Energy: Pt A & B 1_6_(1 962) 81

30. SHAPIRO, C S and CORNGOLD, NApproach to equilibrium of a neutron gasPhys. Rev. 137A (1965) 1686

31. BEYSTER, J R et al.Integral neutron thermalization. Annual summary report,October 1, 1961 through September 30, 19621963 (GA-3642)Integral neutron thermalization. Quarterly progress reportfor the period ending December 31, 19621963 (GA-3853)

- 29 -

32. EGELSTAFF, P A et al.The motion of hydrogen in waterInelastic scattering of neutrons in solids and liquids. IAEA sym-posium Chalk River 1962. Vol. 1, Vienna 1963, p. 343

33. YIP, S and OSBORN, R KSlow-neutron scattering by hindered rotatorsPhys. Rev. J_30 (1963) 1860

34. PUROHIT, S NIntegral parameters of the thermal neutron scattering law.1964 (AE-154)

35. WEINBERG, A M and WIGNER, E PThe physical theory of neutron chain reactorsChicago Univ. Chic. Pr . , 1958

36. WILLIAMS, M M RThe effect of chemical binding and thermal motion on a pulseof slowing down neutronsNucl. Sci. Eng. _T9 (1964) 221

37. KOPPEL, J UTime dependent, space independent neutron thermalizationNucl. Sci. Eng. _T2 (1962) 532

3 8. PUROHIT, S NNeutron scattering and thermalization lectures1965 (AB Atomenergi, Sweden, Internal report S-333)

- 30 -

8. LIST OF TABLES

1. Parameters of the scattering models of H?O and D?O,

2. Reaction rate parameters from Cd and Sm results for H?O models.

3. Reaction rate parameters from Cd results for models of D2O.

4. M2 and t , values for H-,0 and D_O models.

- 3 1 -

Table 1. Parameters of the scattering models of H~O and D_O

Parameter

X

M /m

M /mv'

MT\Tnfp> • -v — — —

Nelkin

0.397

2.32

5.84

(Ef f e ctive

H2O

Nelkin I

0.222

1. 764

7.94

mass for

Nelkin

0.674

3. 195

4.74

hindered

D 2

II Deuteron

0.325

4.39

13.52 —>

rotations)

O (Butler)

Oxygen

0.424

181.8

428.6

M (Effective mass for each intra-molecular vibration)

Table 2. Reaction rate parameters from Cd and Sm results for H?O models.

Cd Sm

Proton Nelkin Exp. Proton Nelkin Exp.

R.

R,> e a k 1.84 1.96 1.91 1.25 1.31 . 1.29level

t p e a k ( ^ s ) 3.49 3.49 4. i t 0.2 5.73 5.73 6.1

t a g m (|JLS) 22 28 25 - 30 22 28

X1 + A\ (2.95-0.14) (2.22-0.08) - (2. 81 t o . 04) (2. 051 0.07)

((Xs)"1 x iO" 1 10"1 - ~ x iO" 1 x iO" 1

t - 0 1 6 4 5 1 0 1 5 4 1 t o 4 3 5 6 0 0 5 4 8 8(|J.S) 3 . 3 9 - 0 . 1 6 4 . 5 1 - 0 . 1 5 4 . 1 t o . 4 3 . 5 6 - 0 . 0 5 4 . 8 8 - 0 . 1 7

Note: Average t from Cd and Sm data for the proton gas and the Nelkin model is equal to

3.48 - 0. 1 and 4. 69 - 0. 1 5 fJisec respectively.

- 33 -

Table 3. Reaction rate parameters from Cd results for models of D9O

Model Deuteron gas Butler model of Experimentplus oxygen D2°

Rpeak

R level

2.28 2.49 2.54

tas ymp

22. 6

160

(4.38-0. 18)

xiO"2

22. 8^0.9

(a)

Cb)

(c)

(a)

(b)

(c)

24.2

230

(2.78-0.

xiO"2

(3.02^0,

x10" 2

(2.41-0,

x10" 2

36. 0-4.

33. 1 - 1.

41.5-1.

.33)

.14)

.11)

3

5

9

26.6

200

3. 16^0.04 x10" 2

(Möller)

32-4 (Möller)

Note: (a) From data (70 - 409. 8) |J.sec dropped to (166 - 409. 8) (Jisec.

(b) From data (70 - 409. 8) (isec dropped to (96. 2 - 409. 8) |isec.

(c) From data (1 1 5. 4 - 409. 8) fxsec.

- 34 -

Table 4. M_ and t., values for H_O and D_O models2 th 2 2

Moderator

A.

(a)

(b)

B .

(a)

(b)

H2O

Nelkin model

Proton gas

D2O

Butler model

Gas modelMass 2

My (barns)

47.31

57.56

10. 71

28. 78

My (cm )

3. 16

3.85

0.355

0.95

f t

1. 152

1. 152

1.244

1.244

tft0».O

4. 4 usec

3. 6 usec

42.3

15. 8

Note: (1) M9 values for the Nelkin and Butler models of H9O and

D?O respectively have been obtained using the Larsson

and Dahlborg frequency spectra for the hindered rotations

in these moderators.

(2) f values are for the gas model from Shapiro (BNL.-8433),

1964.

(3) From the eigenvalue studies of Ghatak and Honeck (3) and

Ohanian and Daitch (5) t , is equal to 5. 64 and 5. 84 [isec

respectively.

- 35 -

9. LIST OF FIGURES

1. (a) Neutron flux versus time for different energy groups (1 to 7)

for the proton gas and the Nelkin model of HyO by NEFLUDI

TDCS.

1. (b) Neutror flux versus time for different energy groups (8 to 14)

for the proton gas and the Nelkin model of H_O by NEFLUDI

TDCS.

1. (c) Neutron flux versus time for different energy groups (15 to 21)

for the proton gas and the Nelkin model of H5O by NEFLUDI

TDCS.

2. Neutron flux versus mean energies of groups at different times

for the proton gas and the Nelkin model of H~O by NEFLUDI

TDCS.

3. Time dependent thermal neutron spectra for the Nelkin model

of H2O.

4. t versus mean energy of group for the proton gas and the

Nelkin model of H2O.

5. Unnormalised reaction rate curves for Cadmium for the proton

gas and the Nelkin model of H2O by NEFLUDI TDCS.

6. Reaction rate curves for Cd, Sm and Gd, Corrected to zero

absorption for H2O. All curves are normalised at the final

level of 50.

7. Reaction rate curves for Cd and Sm corrected to zero absorption,

for H?O obtained by varying the relative weight between one

intra-molecular vibration and hindered rotations. See Table 1.

8. Reaction rate curves for Cadmium for Haywood kernel and

Nelkin model.

9. Time dependent thermal neutron spectra for D-O (Butler model),

10. Time dependent thermal neutron spectra for D~O at 21 fisec.

1 1. Time dependent thermal neutron spectra for D-O at 41. 8 [xsec.

12. Time dependent thermal neutron spectra for D»O at 64.2 |isec.

13. Time dependent thermal neutron spectra for D?O at 102. 6 usec.

14. Time dependent thermal neutron spectra for D-O at 166. 6 |asec.

15. Comparison between the theoretical Maxwellian distribution

and asymptotic spectra calculated by the NEFLUDI TDCS.

36 -

16. Time behaviour of different energy groups for D_O (Butler model).

17. Time behaviour of different energy groups for two cases of Butler

model for D~O (Butler and mass 3. 6 gas).CM

18. Time behaviour of different energy groups for two scattering mo-

dels of D_O (Butler and mass 2 gas).

19. Time behaviour of different energy groups for two scattering mo-

dels of D2O.

20. Reaction rate curves for Cd, Sm and Gd. Corrected to zero ab-

sorption and normalised to the level of 50 (Butler model of D-O).

21. Reaction rate curves for Cd, Sm and Gd. Corrected to zero ab-

sorption and normalised to the level of 50 for Butler and mass

3. 6 gas models.

22. Reaction rate curves for Cd, Sm and Gd. Corrected to zero ab-

sorption and normalised to the level of 50 for Butler and mass

2 gas models.

23. Reaction rate curves for Cd, Sm and Gd. Corrected to zero ab-

sorption and normalised to the level of 50 for two models of

D2O.

24. Comparison between theoretical and experimental reaction rate

curves for Cd and Gd corrected to zero absorption for D2O and

normalised to the level of 50.

FIG. 1 (a). Neutron flux versus time for different energygroups for the proton gas and the Nelkinmodel by NEFLUDI TDCS.

Group No

11

9 -

7 -

5 -

3 -

\

\

Energy (e»v) » 10i

(102-85)

(85 -62.5)

(62.5-40)

(40 -32)

(32 -28)

(28 -25)

(25 -18)

PROTON GAS

NELKIN MODEL

Time (jjsecs)

FIG. 1 (b). Neutron flux versus time for different energygroups for the proton gas and the Nelkinmodel by NEFLUDI TDCS.

Energy (e-v)-IO

( 1 8 - U )

( U - 1 0 )

( 1 0 - 8)

(8 - 6.7)

(6.7- 5.8)

(5.8- 5.0)

(5.0- 4.2)

PROTON GAS

NELKIN MODEL

3 5 Time (psecs)-

40-

x£ 35

30-

25-

20-

15 -

13-

FIG. 1 (c). Neutron flux versus time for different energy groups for the pronton gas and theNelkin model by NEFLUDI TDCS.

25

Group No

15

16

17

18

19

20

21

22

— —

Energy (e-v)-10'

(42-35)

(35-30)

(30 -25)

(25-20)

(20-15)

(15 -10)

(10 - 5)

( 5 - 0.065)

PROTON 6AS

NELKIN MODEL

1

30 Time

c

3

FIG. 2. Neutron f lux versus mean energies of groups at differenttimes for the proton gas and the Nelkin model by NEFLUDITDCS.

0.02 0.04 0.06 0.08 0.10 0.12

PROTON GAS

NELKIN MODEL

0.UEnergy ( e - v )

0.16

-510

X

co

OJ

1Ö6

i57

-

-

7.65 p s / /

3.49ps/

1.01 ps /

i i i i i i i i i

F

\ i i i i i i 111

G. 3. TimeneutNelk

\ \

\i i i i

dependentron spectran model of

• — .

\

i i 111 i

thermalfor theH2O.

i i i i i 11

0.001 0.01 0.1 1.0Enery

to

70 -

60 -

m a x ve rsus mean energy of group for the protongas and the Nelkin model.

10 •

0.8

Mean energy of g roup (e^v)

19

FIG. 5. Un normalised reaction rate curvesfor the proton gas and the Nelkinmodel by NEFLUDI TDCS.

15-

-4->

O

trco

uo

OC 1 0 -

5-

— NELKIN MODEL

PROTON GAS

T5 10 15

T"20 Time (fjsecs)

25

t« 90-

O

.Q

« 70O

enco

oov

50-

30

10

FIG. 6. Reaction rate curves for Cd,Sm and Gdcorrected to zero absorption for H20.All the curves are normalised at thefinal level of 50.

/A.W

NELKIN MODEL

PROTON GAS

EXPERIMENT

10 15 20 TIME

100-

o•*->

'3

aEC

oX>ooQ)

a:

Fl G. 7. Reaction rate curves for Cd and Smcorrected to zero absorption forfor different scattering models.See table 1.

o o— Nelkin U

x x — Experiment

10Time (psecs)

FIG.8. Reaction rate curves for Cadmiumfor Haywood kernel and Nelkinmodel.

100-

c3

O

co

-t->uoat

a:

50-

35-

2 0 -

Haywood kernel (Poole)

Möl ler Expt

Nelkin model

i

2 8 10i

20 30

Time (psec)

-1010

co

-1110

1012

409.8 M S / / / /

102.6 M S / / // / /

64.2 MS/ / /

41.8 M S / /

21 M S /

FIG. 9.

\ \ \ \

i

Time dependentneutron spectra(Butler model).'

\

\

\

thermalfor D2O

0.001 0.01 0.1 1.0Energy(e-V)

CO

-1110

-1210

-1310

FIG. 10. Time dependent thermalneutron spectra for D2Oat '21 psec.

D20 (Butler Model)

Mass 2 gas

Mass 3,6 gas(Brown and St. John)

—r~0.10.001 0.01

— i —

1.0Energy ( e * V )

-10

x•3

CO

10 -

-1210 -

— i -j

10 -

/ /

/ /

/

I I 1

FIG. 11.

\

\

\

\

\%\

Time dependent thermal

neutron spectra for DoO

at 41.8 usec.

— — Mass 2 gas

— — ——« —— Mass 3*6 gas

(Brown and St.John)

»

i

0.001 0.01 0.1 1.0Energy (e-V)

-1010 -

CO

v -11z 10

ro12 .

-1310 H

FIG. 12.Time dependent thermalneutron spectra for D2Oat 64.2 usec.

D20 (But ler Model)

Mass 2 gas

Mass 3.6 gas(Brown and St . John)

0.001 0.01 0.1 1.0Energy (e-V)

-10

i

X

coc

•*->

Z3V

z

1 U -

1 Ö 1 1 -

io 1 2 -

-1310 -

f\\

FIG.

i

\

13.Time dependent thermal

neutron spectra for D«O

at 102.6 (jsec.

——— Mass 2 gas

Mass 3-6 gas(Brown and St. John)

0.001 0.01 0.1 1.0 Energy ( e«V)

-1010

XZ3

CO

-1110

-1210

-1310 -I

FIG. K. Time dependent thermal

neutron spectra for D2O

at 166.6 jjsec.

D20 (Butler Model)

Mass 2 gas

Mass 3*6 gas

(Brown and St.John)

—r~0.10.001 0.01 1.0 Energy (e-V)

FIG. 15. Comparison between the theoreticalMaxwellian distribution and asymptoticspectra calculated by the NEFLUDI TDCS.

-410

tco

Hi

z

-510 •

X

Maxwellian distribution(T = 293°K).

Asymptotic spectrum at 409409 psec from DoO matrix.

Asymptotic spectrum at81.9 Jjsec from HjO matrix.

~ 610

0.05 0.10 0.15

Energy (e-V)

FIG. 16. Time behaviour of different energy groups forD20 (Butler Model).

A (0.85

B (0.18

C (0.10

D (0.02

0.625 e-V)

0 . U e-V)

0.08 e-V)

0.025 e-V)

100 150

Time (|Jsec)

FIG.17. Time behaviour of different energy groups fortwo cases of Butler Model for

A (0.85

B (0.18

C (0.10

D (0.02

0.625 e-V)

0. U e-V

0.08 e-V)

0.025 e-V)

— D20 (But le r Model)

Model neglect ingoxygen)

150Time (jjsec)

c3

O

•»-•

.Q

XZ3

CO

Dl

-111 0 -I

FIG. 18. Time behaviour of different energy groups fortwo scattering models of D2O (Butler and3*6 mass gas).

B

A (0.85

B ( 0 . 1 8

C (0 .10

D (0 .02

0 .625 e-V)

0 . U e-V)

0.08 e-V)

0.025 e-V)

Mass 3-6 gas model (Brown & St.John)

— D20 (ButLer Model)

50 100 150Time (psec)

FIG.19. Time behaviour of different energy groups fortwo scattering models of D2O (Butler andmass 2 gas).

D

A

B

C

D

(0.85 —

(0.18 —

(0.10 —

(0.02 —

0.625

0.H

0.08

0.025

rt/ic m 1

e-V)

e -V)

e-V)

e-V)

nrlai

D20 (Butler Model)

100 150Time (psec)

t150 -

o

FIG. 20. Reaction rate curves for Cd.Sm and Gdcorrected to zero absorption and normalisedto the level of 50 (Butler Model of D00).

01

oa:co

oo0i

a:

100 -

50 -

100 150 200 250

Time ( jjsec)

FIG. 21150 -

O1_

•*->

n

<

a»-̂*oo:

I 100uoa:

5 0 -

Reaction rate curves for Cd,Sm and Gdcorrected to zero absorption and normalisedto the level of 50 for Butler and mass 3*6gas models.

D20 (Butler Model)— — — — Mass 3*6 gas model

(Brown and St John)

100 150 200 250

Time (psec)

150-

I 100o

50 -

0

FIG. 22. Reaction rate curves for Cd,Sm and Gdcorrected to zero absorption and normalisedto the level of 50 for Butler and mass 2 gasmodels.

DO (Butter Model)

— ~~ Mass 2 gas model

150 200 250Time (usec)

t3 150

Dl_

•4->

"is

ocn

•r. 100 -uoO)

a:

50 -

FIG. 23 Reaction rate curves for Cd, Sm and Gd

corrected to zero absorption and normalised

to the level of 50 for two models of D2O.

Do0 (Butler Model)

— — — — D20 (Butler Model neglectingoxygen)

100 150 200 250Time ( psec)

Reactionrate,arbitraryunits

100

50

0

i i

f:

ri

i-

/

i— *

/

•l/L.* i i

1 ' 1 ' '

\VC d

'V.1 1 i 1 1

1 1 • 'D2O (Butler

Mass 2 gas

• • Experiment

i 1 i i

i i i i i i i

model )

model

J

—

i i I i i * «

50 150 200100Time , JJS

FIG. 2k. Com parison between theoretical and ex peri mental reaction rate curves for Cd and Gdcorrected to zero absorption for D2O and normalised to the level of 50.

LIST OF PUBLISHED AE-REPORTS

1—145. (See the back cover earlier reports.)146. Concentration of 24 trace elements in human heart tissue determined

by neutron activation analysis. By P. O. Wester. 1964. 33 p. Sw. cr. 8;—.147. Report on the personnel Dosimetry at AB Atomenergi during 1963. By

K.-A. Edvardsson and S. Hagsgård. 1964. 16 p. Sw. cr. 8:—.148. A calculation of the angular moments of the kernel for a monatomic gas

scatterer. By R. Håkansson. 1964. 16 p. Sw. cr. 8:—.149. An anion-exchange method for the separation of P-32 activity in neu-

tron-irraditecl biological material. By K. Samsahl. 1964. 10 p. Sw. cr8:—.

150. Inelastic neutron scattering cross sections of Cu6' and Cu65 in the energyregion 0.7 to 1.4 MeV. By B. Holmqvist and T. Wiedl ing. 1964. 30 p.Sw. cr. 8:—.

151. Determination of magnesium in needle biopsy samples of muscle tissueby means of neutron activation analysis. By D. Brune and H. E. Sjöberg.1964. 8 p. Sw. cr. 8:—.

152. Absolute El transition probabilities in the deformed nuclei Yb"7 andHf"». By Sven G. Malmskog. 1964. 21 p. Sw. cr. 8:—.

153. Measurements of burnout conditions for f low of boil ing water in vertical3-rod and 7-rod clusters. By K. M. Becker, G. Hernborg and J. E. Flinta.1964. 54 p. Sw. ct. 8:—.

154. Integral parameters of the thermal neutron scattering law. By S. N.Purohit. 1964. 48 p. Sw. cr. 8:—.

155. Tests of neutron spectrum calculations with the help of foi l measurementsin a D2O and in an hhO-moderated reactor and in reactor shields ofconcrete and iron. By R. Nilsson and E. Aal to. 1964. 23 p. Sw. cr. 8:—.

156. Hydrodynamic instability and dynamic burnout in natural circulationtwo-phase f low. An experimental and theoretical study. By K. M. Beck-er. S. Jahnberg, I. Haga, P. T. Hansson and R. P. Mathisen. 1964. 41 p.Sw. cr. 8 i—.

157. Measurements of neutron and gamma attenuation in massive laminatedshields of concrete and a study of the accuracy of some methods ofcalculation. By E. Aalto and R. Nilsson. 1964. 110 p. Sw. cr. 10:—.

158. A study of the angular distributions of neutrons from the Be9 (p,n) B'reaction at low proton energies. By. B. Antolkovic', B. Holmqvist andT. Wiedl ing. 1964. 19 p. Sw. cr. 8:—.

159. A simple apparatus for fast ion exchange separations. By K. Samsahl.1964. 15 p. Sw. cr. 8:—

160. Measurements of the FeH (n, p) Mn H reaction cross section in the neutronenergy range 2.3—3.8 MeV. By A. Lauber and S. Malmskog. 1964. 13 p.Sw. cr. 8:—.

161. Comparisons of measured and calculated neutron fluxes in laminatediron and heavy water. By. E. Aal to. 1964. 15 p. Sw. cr. 81—.

162. A needle-type p-i-n junction semiconductor detector for in-vivo measure-ment of beta tracer activity. By A. Lauber and B. Rosencrantz. 1964. 12 p.Sw. cr. 8:—.

163. Flame spectro photometric determination of strontium in water andbiological material. By G. Jönsson. 1964. 12 p. Sw. cr. 8;—.

164. The solution of a velocity-dependent slowing-down problem using case'seigenfunction expansion. By A. Claesson. 1964. 16 p. Sw. cr. 8:—.

165. Measurements of the effects of spacers on the burnout conditions forflow of boil ing water in a vertical annulus and a vertical 7-rod cluster.By K. M. Becker and G. Hemberg. 1964. 15 p. Sw. cr. 8:—.

166. The transmission of thermal and fast neutrons in air f i l led annular ductsthrough slabs of iron and heavy water. By J. Nilsson and R. Sandlin.1964. 33 p. Sw. cr. 8:—.

167. The radio-thermoluminescense of CaSCUs Sm and its use in dosimetry.By B. Bjärngard. 1964. 31 p. Sw. cr. 8:—.

168. A fast radiochemical method for the determination of some essentialtrace elements in biology and medicine. By K. Samsahl. 1964. 12 p. Sw.cr. 8:—.

169. Concentration of 17 elements in subcellular fractions of beef heart tissuedetermined by neutron activation analysis. By P. O. Wester. 1964. 29 p.Sw. cr. 8:—.

170. Formation of nitrogen-13, fluorine-17, and fluorine-18 in reactor-irradiatedH2O and D2O and applications to activation analysis and fast neutronflux monitoring. By L. Hammar and S. Forsen. 1964. 25 p. Sw. cr. 8:—.

171. Measurements on background and fall-out radioactivity in samples fromthe Baltic bay of Tvären, 1957—1963. By P. O. Agnedal. 1965. 48 p. Sw.cr. 8:—

172. Recoil reactions in neutron-activation analysis. By D. Brune. 1965. 24 p.Sw. cr. 8:—.

173. A parametric study of a constant-Mach-number MHD generator withnuclear ionization. By J. Braun. 1965. 23 p. Sw. cr. 8:—.

174. Improvements in applied gamma-ray spectrometry with germanium semi-conductor detector. By D. Brune, J. Dubois and S. Hellström. 1965. 17 p.Sw. cr. 8:—.

175. Analysis of linear MHD power generators. By E. A. Wital is. 1965. 37 p.Sw. cr. 8:—.

176. Effect of buoyancy on forced convection heat transfer in vertical chann-els — a literature survey. By A. Bhattacharyya. 1965. 27 p. Sw. cr. 8:—.

177. Burnout data for flow of boiling water in vertical round ducts, annuliand rod clusters. By K. M. Becker, G. Hernborg, M. Bode and O. Erik-son. 1965. 109 p. Sw. cr. 8 : - .

178. An analytical and experimental study of burnout conditions in verticalround ducts. By K. M. Becker. 1965. 161 p. Sw. cr. 8:—.

179. Hindered El transitions in Eu"5 and Tb>". By S. G. Malmskog. 1965. 19 p.Sw. cr. 8:—. v

180. Photomultiplier tubes for low level Cerenkov detectors. By O. Strinde-hag. 1965. 25 p. Sw. cr. 8:—.

181. Studies of the fission integrals of U235 and Pu239 with cadmium andboron filters. By E. Hellstrand. 1965. 32 p. Sw. cr. 8:—.

182. The handling of liquid waste at the research station of Studsvik,Sweden.By S. Lindhe and P. Linder. 1965. 18 p. Sw. cr. 8:—.

183. Mechanical and instrumental experiences from the erection, commis-sioning and operation of a small pilot plant for development work onaqueous reprocessing of nuclear fuels. By K. Jönsson. 1965. 21 p. Sw.cr. 8:—.

184. Energy dependent removal cross-sections in fast neutron shieldingtheory. By H. Grönroos. 1965. 75 p. Sw. cr. 8:—.

185. A new method for predicting the penetration and slowing-down ofneutrons in reactor shields. By L. Hjärne and M. Leimdörfer. 1965. 21 p.Sw. cr. 8:—.

186. An electron microscope study of the thermal neutron induced loss inhigh temperature tensile ductility of Nb stabilized austenitic steels.By R. B. Roy. 1965. 15 p. Sw. cr. 8:—.

201.202.

187. The non-destructive determination of burn-up means of the Pr-144 2.18MeV gamma activity. By R. S. Forsyth and W. H. Blackadder. 1965.22 p. Sw. cr. 8:—.

188. Trace elements in human myocardial infarction determined by neutronactivation analysis. By P. O. Wester. 1965. 34 p. Sw. cr. 8:—.

189. An electromagnet for precession of the polarization of fast-neutrons.By O. Aspelund, J. Björkman and G. Trumpy. 1965. 28 p. Sw. cr. 8:—.

190. On the use of importance sampling in particle transport problems. ByB. Eriksson. 1965. 27 p. Sw. cr. 8:—.

191. Trace elements in the conductive tissue of beef heart determined byneutron activation analysis. By P. O. Wester. 1965. 19 p. Sw. cr. 8:—.

192. Radiolysis of aqueous benzene solutions in the presence of inorganicoxides. By H. Christensen. 12 p. 1965. Sw. cr. 8:—.

193. Radiolysis of aqueous benzene solutions at higher temperatures. ByH. Christensen. 1965. 14 p. Sw. cr. 8:—.

194. Theoretical work for the fast zero-power reactor FR-0. By H. Häggblom.1965. 46 p. Sw. cr. 8:—.

195. Experimental studies on assemblies 1 and 2 of the fast reactor FRO.Part 1. By T. L. Andersson, E. Hellstrand, S-O. Londen and L. I. Tirén.1965. 45 p. Sw. cr. 8:—.

196. Measured and predicted variations in fast neutron spectrum when pene-trating laminated Fe-DjO. By E. Aalto, R. Sandlin and R. Fräki. 1965.20 p. Sw. cr. 8:—.

197. Measured and predicted variations in fast neutron spectrum in massiveshields of water and concrete. By E. Aalto, R. Fräki and R. Sandlin. 1965.27 p. Sw. cr. 8 : - .

198. Measured and predicted neutron fluxes in, and leakage through, a con-figuration of perforated Fe plates in D2O. By E. Aal to. 1965. 23 p. Sw.cr. 8:—.

199. Mixed convection heat transfer on the outside of a vertical cylinder.By A. Bhattacharyya. 1965. 42 p. Sw. cr. 8:—.

200. An experimental study of natural circulation in a loop with parallelflow test sections. By R. P. Mathisen and O. Eklind. 1965. 47 p. Sw.cr. 8:—.Heat transfer analogies. By A. Bhattacharyya. 1965. 55 p. Sw. cr. 8:—.A study of the "384" KeV complex gamma emission from plutonium-239.By R. S. Forsyth and N. Ronqvist. 1965. 14 p. Sw. cr. 8:—.

203. A scintillometer assembly for geological survey. By E. Dissing and O.Landström. 1965. 16 p. Sw. cr. 8:—.

204. Neutron-activation analysis of natural water applied to hydrogeology.By O. Landström and C. G. Wenner. 1965. 28 p. Sw. cr. 8:—.

205. Systematics of absolute gamma ray transition probabilities in deformedodd-A nuclei. By S. G. Malmskog. 1965. 60 p. Sw. cr. 8:—.

206. Radiation induced removal of stacking faults in quenched aluminium.By U. Bergenlid. 1965. 11 p. Sw. cr. 8:—.

207. Experimental studies on assemblies 1 and 2 of the fast reactor FRO.Part 2. By E. Hellstrand, T. L. Andersson, B. Brunfelter, J. Kockum, S-O.Londen and L. I. Tirén. 1965. 50 p. Sw. cr. 8:—.

208. Measurement of the neutron slowing-down time distribution at 1.46 eVand its space dependence in water. By E. Möller. 1965. 29. p.Sw.cr.8:—.

209. Incompressible steady flow with tensor conductivity leaving a transversemagnetic f ie ld. By E. A. Witalis. 1965. 17 p. Sw. cr. 8:—.

210. Methods for the determination of currents and fields in steady two-dimensional MHD flow with tensor conductivity. By E. A. Witalis. 1965.13 p. Sw. cr. 8:—.

211. Report on the personnel dosimetry at AB Atomenergi during 1964. ByK. A. Edvardsson. 1966. 15 p. Sw. cr. 8:—.

212. Central reactivity measurements on assemblies 1 and 3 of the fastreactor FRO. By S-O. Londen. 1966. 58 p. Sw. cr. 8:—.

213. Low temperature irradiation applied to neutron activation analysis ofmercury in human whole blood. By D. Brune. 1966. 7 p. Sw. cr. 8:—.

214. Characteristics of linear MHD generators with one or a few loads. ByE. A. Witalis. 1966. 16 p. Sw. cr. 8:—.

215. An automated anion-exchange method for the selective sqrption of fivegroups of trace elements in neutron-irradiated biological material.By K. Samsahl. 1966. 14 p. Sw. cr. 8:—.

216. Measurement of the time dependence of neutron slowing-down andthermalization in heavy water. By E. Möller. 1966. 34 p. Sw. cr. 8:—.

217. Electrodeposition of actinide and lanthanide elements. By N-E. Barring.1966. 21 p. Sw. cr. 8:—.

218. Measurement of the electrical conductivity of He3 plasma induced byneutron irradiation. By J. Braun and K. Nygaard. 1966. 37 p. Sw. cr. 8:—.

219. Phytoplankton from Lake Magelungen, Central Sweden 1960—1963. By T.Wi l lén. 1966. 44 p. Sw. cr. 8:—.

220. Measured and predicted neutron flux distributions in a material sur-rounding av cylindrical duct. By J. Nilsson and R. Sandlin. 1966. 37 p.Sw. cr. 8:—.

221. Swedish work on brittle-fracture problems in nuclear reactor pressurevessels. By M. Grounes. 1966. 34 p. Sw. cr. 8:—.

222. Total cross-sections of U, UO2 and ThO2 for thermal and subthermalneutrons. By S. F. Beshai. 1966. 14 p. Sw. cr. 8:—.

223. Neutron scattering in hydrogenous moderators, studied by the timedependent reaction rate method. By L. G. Larsson, E. Möller and S. N.Purohit. 1966. 26 p. Sw. cr. 8:—.

224. Calcium and strontium in Swedish waters and fish, and accumulation ofstrontium-90. By P-O. Agnedal. 1966. 34 p. Sw. cr. 8:—.

225. The radioactive waste management at Studsvik. By R. Hedlund and A.Lindskog. 1966. 14 p. Sw. cr. 8:—.

226. Theoretical time dependent thermal neutron spectra and reaction ratesin H2O and D2O. By S. N. Purohit. 1966. 62 p. Sw. cr. 8:—.

Förteckning över publicerade AES-rapporter

1. Analys medelst gamma-spektromelri. Av D. Brune. 1961. 10 s. Kr 6:—2. Bestrålningsförändringar och neutronatmosfär i reaktortrycktankar —

några synpunkter. Av M. Grounes. 1962. 33 s. Kr 6:—.3. Studium av sträckgränsen i mjukt stål. Av G. Östberg och R. Attermo.

1963. 17 s. Kr 6:—.4. Teknisk upphandling inom reaktorområdet. Av Erik Jonson. 1963. 64 s.

Kr 8:—.5. Ågesta Kraftvärmeverk. Sammanställning av tekniska data, beskrivningor

m. m. för reaktordelen. Av B. Lilliehöök. 1964. 336 s. Kr 15:—.6. Atomdagen 1965. Sammanställning av föredrag och diskussioner. Av S.

Sandström. 1966. 321 s. Kr 15:—.Additional copies available at the l ibrary of AB Atomenergi, Studsvik,Nyköping, Sweden. Transparent microcards of the reports are obtainablethrough the International Documentation Center, Tumba, Sweden.

EOS-tryckerierna, Stockholm 1966