Measurement of neutron spectra in moderators and reactor lattices—I Aqueous moderators

I. Nuclear Energy, 1957. Vol. 5. pp. 325 to 341. Pergamon Press Ltd., London

MEASUREMENT OF NEUTRON MODERATORS AND REACTOR AQUEOUS

MODERATORS

M. J.

POOLE

Atomic Energy Research Establishment, (Received

SPECTRA IN LATTICES-I

12 November

Harwell, Berks

1956)

Abstract-The application of pulsed source time-of-flight ‘techniques to the measurement of reactor lattice and moderator spectra is described. Results are given for a series of measurements on the spectrum in water loaded with boric acid, and compared with the predictions of theory. INTRODUCTION

effective use for reactor design of the available nuclear data, it is necessary to know the energy spectrum of neutrons at all places in the reactor under consideration, In principle, once all the cross-sections are known, this may be calculated, but in practice even then such a calculation is usually too complex to be feasible, and recourse must be made to experiment. The most generally used experimental method has relied upon observing response of energy-sensitive detectors. However, the interpretation of such results usually requires the prior assumption of the mathematical form of the spectrum, and it does not give explicit verification of this form. This paper describes a method of measuring the spectra of neutrons in moderators and in lattice tanks, using the time-of-flight method, by which means it has proved possible to plot out the detailed shape of the spectrum. A series of results is given for moderation in water poisoned by varying amounts of boron. To MAKE

METHOD

An electron linear accelerator bombarding uranium is used as a pulsed neutron source.* Round this source a section of lattice (or moderator) is built up, and the spectrum of neutrons from a selected region in this assembly is observed by a time-offlight experiment. The set-up is schematically illustrated in Fig. 1. Tube A contains the target of the accelerator, and B is a probe tube placed above A and at right angles to it. A carefully designed collimator C is arranged so that it transmits only neutrons originating from a definite area in the end of A. These neutrons then pass along a 12m flight path and are detected by a bank of thirty-six proportional counters filled with l”BF,. The signals from the neutron detector are timed electronically relative to the neutron pulse at the source. It is now necessary to see how the energy distribution should be calculated from the observed time distribution. The experiment differs from the ordinary time-of-flight energy measurement in the following respects : (a) The neutron density (and so flux) at any point in the assembly dies away approximately exponentially (l) after the end of the pulse. The mean time depends primarily on the neutron-capturing properties of materials in the assembly, but is modified at any particular point by the effects of diffusion of the neutrons. For each moderator or lattice used, the die-away time was measured at a point corresponding * 50 pulses

per second,

and 1’~s~ long each containing about lo* neutrons. 325

326

M. J.

POOLE

to the end of the probe tube (B, Fig. l), by inserting a BF, chamber small enough to have a l/u response and connecting the output from this to a set of 10 psec gates. Measured die-away times lay between about 20 psec for the strong boric-acid solutions to 250 ,usec for pure water at some distance from the neutron source. These times are

? ?Boric oxide shielding ? ?Concrete ? ?Collimator C ? ?Tank E FIG. l.-Layout

Linear accelerator

of apparatus.

all very large compared to the original pulse length, and so exert a controlling influence on the resolution of the experiment. The gate length used in any run is made at least equal to twice the die-away time. A correction to the mean delay corresponding to a given gate is also needed. This correction is, to a first approximation, equal to the die-away time. A more detailed calculation by COATES@) of the effect on the shape of an originally Maxwellian energy spectrum showed that, even when this correction was made, the temperature observed for the neutrons would be a little lower than the true neutron temperature. For a d&-away time of 230 psec (the measured value 10 cm from the source in water) this amounts to 7°C. (b) The pulse lengthening mentioned above is caused by diffusion of the neutrons after thermalization and before capture. The actual value of r is energy-dependent. Qualitatively it may be seen that in a non-multiplying assembly any neutrons observed with energies well above thermal (say above 0.25 eV) cannot have spent much time diffusing. Thus, as the slowing-down time in hydrogen is very small (~10 psec) the die-away time for these neutrons is also very small, and there is no need either to use long gates, or to make any correction to the mean delay for a gate. This property has been used to measure the high-energy end of the spectrum with better resolution. Under these circumstances the gate length used is limited only by intensity considerations.’ There is an intermediate region between the case of the neutron which has gained thermal equilibrium, and the fast neutron which emerges with no delay, and the delays in this case cannot easily be calculated. In the case of a multiplying lattice, a thermal neutron may be re-born as a fast neutron by virtue of the fission process. This largely destroys the correlation between

Aqueous moderators

321

die-away time and energy, and in this case all neutrons have been assumed to die away at the same rate. The gates used were divided into two groups, twelve which could be either 100 psec or 50 ,usec long, followed by twenty-eight which were either 400 psec or 200 psec long. The resolutions (AZ/(E)) which could be obtained at different energies (E) are tabulated in Table 1. (c) If the method is used for measurements in reactor lattices, then it is necessary to be satisfied that the spectrum of neutrons within a small section of the lattice is the same as it would be were the lattice critical. The evidence for this will be discussed in a later paper when the results on reactor lattices are presented. (d) The neutrons observed emerge in a particular well-defined direction. It is necessarily true that their spectrum is the same as the spectrum of neutrons averaged over all directions. NELKIN(~)has shown that in a homogeneous medium, the spectrum emerging is’related to the spectrum in the tank by N(E) = N,(E) + 2 2 N,(N) + A2-& N,(E) - ilDA2No(E) TABLE 1

-

!

Boric acid Energy

Water (100 ysec)

_

(50,usec)

(100 psec)

8eV

Lattice

(200psec)

1

1

1

i7

i3

5

1

1

1

2eV

3

3

6

0.5 ev

1

1

1

6

6

i-i

_:_

400psec 0.125eV 0.02eV 0.005 eV

400,usec

200ysec

400,usec

1

1

1

1

2

5

6

52

1

1

1

1

J

i-3

Is

6

1

1

1

1

lo

Is

30

iz

-

-

when N(E) is the observed spectrum, N,(E) is the true spectrum, 1 is the mean free path for a neutron of energy E, and D is the diffusion coefficient. The differentials are all to be taken at the position of the end of the probe tube, and the x direction is the direction of the flight path. To a first approximation, the true spectrum only

328

M. J.

POOLE

differs from the observed spectrum when dNldx # 0. Now, the’diffusion of neutrons is approximately radially away from the target, except near to the walls of the tank, and so at the position of the end of the probe tube 6N/6x + 0. More exactly, COATES (unpublished) has calculated that in this case N(E) = N,(E)( 1 + 0%X)741- 0.0112P) Over the energy range 0.01 to 0.1 eV, 3,varies from -0.2 cm to 4.4 cm, leading to a maximum difference (N)E and N,,(E) of $ per cent. The foregoing only applies to a homogeneous medium. In the case of a lattice, the question is complicated by the possibility of large local variations due to the fuel rods, and it has been shown experimentally by the author that there is pronounced anisotropy in the spectrum of neutrons in a uranium rod. (e) Systems can only be investigated if their die-away time is not much greater than 200 psec (permitting gates of 400 ,csec). This arises because experience has shown that the present source intensity is not enough for the use of a flight path longer than 12 m, making it undesirable to increase the gate length above 400 psec. This means that, except for some systems very heavily loaded with absorber, the method is at present limited to ordinary water-moderated systems. Even so, because of the small collimator aperture, extraordinary amounts of shielding are needed to keep the background count small relative to the wanted count. The whole flight tube is lined with boric oxide to a minimum thickness of 4 in., the only opening into this shielding being through the collimator. NORMALIZATION

OF RESULTS

In practically all spectra measured so far, a Maxwellian component could be identified. In each case the spectra have been so normalized that the integral of the flux in this Maxwellian component is equal to unity. This is equivalent to writing for the low-energy end of the spectrum,

where E, = kT. T = effective neutron temperature. The value of E, (and of T) was always determined from the least-squares fit of a straight line between log N(&)/E and E. The intercept of this straight line gives the logarithm of the normalization factor. Where a “il value” has been quoted for a A/E tail, it comes from a similar linear fit between the normalized value of N(E) and l/E. SPECTRA

IN PURE

WATER

Spectra have been measured in ,pure water at distances of 2 cm, 5 cm, 10 cm, 12.5 cm from the neutron source, with the water at room temperature, and at 10 cm only with the water at 98°C. The spectrum at 2 cm is almost certainly distorted by capture of neutrons by the uranium target of the linear accelerator. These spectra have been normalized as described above and neutron temperatures and 1 values obtained. Typical spectra are plotted in Fig. 2 and all the results are tabulated in Table 2. In all cases a Maxwellian component was observed corresponding to a temperature equal, within the error, to the moderator temperature. At energies greater than 0.2 eV the spectrum goes over into a ;I/E.slowing-down spectrum. The

Aqueous

329

moderators

\

x\x , . ?+

(

x-

.

\ \,

% :\

x, L

0.10

1.0

Neutron

energy

FIG.2.-Flux spectrum in water. X Spectrum 2 cm from source. 0 Spectrum A Spectrum 0 Spectrum

5 cm from source. 10 cm from source. 12.5 cm from source.

IO .O

1 3

ev

12.5

291°K

0.3

10 10

291°K

291°K

291°K

2.9

3.13

4.55

3.03

430 & 40 432 Zk 40 501 f 50 515 f 50

10 10 10 10

369°K

370°K

371°K

371’K

4.54

7.6

9.9

340 * 30

310 -+ 30

306 f 30

314 f 30 i

283 f 30 297 f 30 304 * 30

308 * 30 314 f 30

368 + 35

-

-

(Z&”

Measured

-

--

T

299 f 15

Averages

-

--

353

345

332

313

( :ovEYou

_

T-

-

-_

466

348

342

330

313

BROWN

Calculated Tneut

-

452

417

341

320

303

AMSTER

c TABLE2.-SUMMARIZED RESULTS ON Tm AND ,I

10

-

10

291°K

1.5

-

10 12.5 10

291°K 291°K 291°K

0.3 0.3 0.3

Tap water Tap water Demineralized water Demineralized water

Boric-acid solution Boric-acid solution Boric-acid solution Boric-acid solution Boric-acid solution Boric-acid solution Boric-acid solution Boric-Acid solution

2. 5

(CT)

29l’K291°K

--

T Position ,of probe (,cm fron 1 source)

0.3 0.3

-

OO

(barns)

T

-

Tap water Tap water

Moderator

7--

I

044

0.328

0.236

0.121

0.214

0.13

0.067 1 0.130

0.016 J

0.048 0.033 Average 0.016 0.015 f 0.016 0.018 1 -f oxlO

(All ilO%)

Measured

L

--

_

-i-

0.356

0.19

0.125

0.214

0.19

0.137

0.069

POOLE

-

-_

-

0.34

0.215

0.180

0.139

0,072

AMSTER

-

--

0.35

0.240

BROWN

- ~-

Calculated II

Aqueous moderators

331

value of il is at first a function of distance from the source, and then reaches an equilibrium value of 0.015 at -10 cm. The way in which the l/E portion joins the Maxwellian cannot be very accurately determined in pure water, because this is the

??

I

I

I

Resolution

f I

0.01

0.1 Neutron

FIG. f.-Flux

)

spectrum in boric-acid solution.

a, = 1.5 b.

1 0

10.0

energy

.

eV

T, = 291’K.

region in which the die-away time is a rapidly varying function of energy. As it is only a correction factor, an approximation has been made by arbitrarily assuming a linear variation in die-away time from zero O-2 eV to full value at 0.1 eV. SPECTRA

IN BORIC-ACID

SOLUTION

Spectra have been measured for concentrations of boric acid corresponding to values of co varying from 1.5 to 7.6 barns, measurements being taken both at room temperature and at 370°K (c,, is the effective capture cross-section per H atom at 2200 m/set). These spectra are plotted in Figs. 3 to 9. The spectra again consist of Maxwellian components going over into l/E tails, and to show the form of the

332

M.

J.

POOLE

.

1oc

1o.c Maxwellian

_ u

calculated

1x

s

r=

E b

-5 \3 O.lC : c c

2

3 $ 0.01

0.0

001

.

0.10 Neutron

Fro. 4.-Flux

1.0

10.0

energy

spectrum in boric-acid solution.

q, = 29 b.

T,,, = 291’K.

100 eV

Aqueous moderators

333

I

0.001

0.01

0.10 Neutron

1.0

10.0

energy

FIG.5.-Fluxspectrum in boric-acid solution.

crO= 3.05b. T, = 369°K.

100 eV

M. J. POOLE

334

I 0.001

Resolution

m

001

??

I

10.0

0.1 Neutron

FIG. 6.-Flux spectrum

in boric-acid

eV

energy

solution.

IT,,= 3.8 b.

i 100

T, = 291°K.

Aqueous moderators

Observed spectrum tioxwellion

335

4 minus component

11

.

I

.

Resolution

31I 0. 01

C

1

1

FIG. 7.-Flux

3 energy

eV

spectrum in bori&acid solution.

uO = 4.55 b. T,,, = 291°K.

Neutron

336

M. J. POOLE

lO(

I

10X Maxwellian

calculated

-0 S z E b d >’ *

1.c

2 r 0.K

minus Maxwellian component

E *; z”

0.0

??

I Resolution o.oc

31

0.1

1.0 Neutron

FIG. 8.-Flux

energy

spectrum in boric-acid solutidn.

10-O

100 eV

CT,, = 4.54 b. 7’, = 370’K.

Aqueous moderators

331

transition more clearly, the Maxwellian has in each case been subtracted off (dotted curves). Now, both the temperature of the Maxwellian and the value of 3, vary with oo, the values obtained being shown in Table 2. To facilitate comparison with theory, LutP’moa and also 1 are plotted against o0 in Fig. 10. I~$Maxwellian

cakated for T,=50°K

t

)-

Observed

spectrum

minus

component

Maxwellian l-

T

11-

.

.

.

Resolution

1

I1-

ox11

C1.

3 Neutron

FIG.

9.-Flux

energy

eV

spectrum in boric-acid solution. COMPARISON

WITH

a, =

7.6 b.

T,,, = 371°K.

THEORY

It can easily be shown that for moderation in monatomic gaseous hydrogen the slowing-down spectrum is given by W%(E)

= 0,/E

provided that the capture of neutrons before they come into thermal equilibrium is neglected. Here (T,is the cross-section at energy E for n-p scattering, cr, is the capture cross-section for thermal neutrons by hydrogen, and N(E) is the normalized neutron flux. This will also be the spectrum in water as long as E is larger than the molecular binding energy, and as long as the effect of moderation by oxygen can be neglected. Due to the effect of binding, Gorises from its almost constant value of 20 barns at a few tens of eV to 80 barns at thermal energy (4.025 eV). Considering only the region where o, = 20 barns, and putting in oc = 0.3 barns, then N(E) = 0*015/E

in good agreement with the measured equilibrium value of 1 = 0.015.

Strictly,

338

M. J. POOLE

1

1.:3-

>_ 1.: I-

ElI=

il

k

1.1I -

oveyous formula fcat- T,= 291 OK (Brown’s formula is -3% lower )

I

l*( .

T,=291°H

x

T

0

Cmolculated

pointL

at ?9l 0

Q

Calculated

points

at 369’K

=369’K

0.2

x

0.:

.d

0.

P 0123456

00 FIG. lo.--Plot

of TJT,,, and of 1 VS.0,.

barns

339

Aqueous moderators

the comparison should not be with the equilibrium value measured at large r, but with the mean value for all neutrons. This will differ little from the equilibrium value, because in water, equilibrium is established very close to the source. A similar calculation has been carried out for the boric-acid solutions, but now account has to be taken of neutrons captured in the epithermal region. Assuming only l]v capture, and assuming that us is a constant, it can be shown that the slowingdown spectrum takes the form 1

which for all practical purposes may be written 0.886 d N(E) = E (c, - 0.7~) where co is the effective capture cross-section per hydrogen atom, measured at energy E,,, (O-0253 eV), cS is the scattering cross-section of hydrogen, Es the initial source the effective neutron temperature (referred to the Maxwellian energy, and Tneut component only). Values of jz calculated from this formula (using experimental values are included in Table 2. of Tneut) It is also of interest to compare the results of various more complicated theoretical calculations with the experimental spectra. COVEYOU has made Monte Carlo calculations for monatomic hydrogen gas loaded with a l/v absorber, and has derived the formula for the neutron temperature

Tneut = T,

1 + 0.91 %’ o,

where A is the atomic weight of the moderator. BROWN has carried out a more elaborate calculation which attempts to take into account the molecular binding of the water, and produces a similar formula to COVEYOU, but with Tneut/Tm dependent Finally, AMSTERhas made a numerical integration of the Wigner-Wilkins also on T,,,. equation for the case of monatomic hydrogen loaded with l/u absorber. Values for Tneut and 1 taken from their work are quoted in Table 2 wherever. relevant and also included in Fig. 10. A comparison between the detailed shape of the experimental and theoretical curves is given in Fig. 11. It will be seen that in the example shown the slowing-down spectrum is well reproduced by the theory and the Maxwellian part well reproduced in shape (possibly shifted a little in energy); but in the region O-1 to 0.3 eV, where chemical binding is important, the experimental values of N(E) lie significantly above the theoretical curves. This appears true in all the examples so far compared, the divergences getting greater as co increases. However, the interesting fact is that these elementary theories give results which are in fact so close to the experimental curves; certainly close enough for reactor calculations. As will be seen for Fig. 10, the data for effective neutron temperatures are consistent with theory, although not really accurate enough to distinguish BROWN’S and COVEYOU’ calculations. S

340

M. J. POOLE

The closeness of agreement between theory and experiment is in part explained by BROWN, who points out that the shape of the spectrum obtained is extremely insensitive to the scattering law for slow neutrons in moderators, and so the present lack of knowledge is not very important. Points Line

0.01

0.10 Neutron

FIG.

il.-Comparison

1

I

are

experiment

is theory

10.0

energy

of experimental and theoretical

100 ev.

(AMSTER) spectra.

Acknowledgements-The author wishes to state his particular indebtedness to Mrs. D. RUSSELLand to Mr. W. D. HOWE,whose skill in setting up the apparatus and patience in taking the results contributed largely to the success of the experiment; to Mr. M. COATESfor calculating the effect of resolution function on neutron temperature and assisting in the preparation of the diagrams; to Mr. G. DEANE,who collaborated in the design and construction of the electronic apparatus; and to Dr. and Mrs. HAILSTONE,who, together with Miss D. FIELDS,used the I.B.M. machinery to compute the experimental results. Thanks must also be given to the whole of the linearaccelerator group for the use of the facilities of the machine, and to Dr. J. TAIT for many helpful discussions.

341

Aqueous moderators REFERENCES 1. 2. 3. 4. 5.

SYKESJ. B. A.E.R.E. T/R 1367 and J. Nucl. Energy 2, 31 (1956). ~OATESM. S. private communication. NELKINM. S. (Knolls Atomic Power Laboratory, Schenectady) private communication. COVEYOUR. R., BATESR. R., and OSBORNE R. K. J. Nucl. Energy 2, 3 (1956). BROWNH. D. Report DP-64, E.Z. Du Pont de Nemours and Co. (Available from Office of Technical

Services, Department 6. AMSTER H. Report Communication.

of Commerce, Washington 25, DC., U.S.A.) WAPD-T-379 Westinghouse Atomic Power

Laboratory,

and

Private