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Study of the process of neutron moderation in beryllium and beryllium oxide by the impulse method
Journal
of Nuclear
Energy
Parts A/B,
1966, Vol. 20, pp. 683 to 695.
Pergamon
Press Ltd.
Printed
in Northern
Ireland
STUDY OF THE PROCESS OF NEUTRON MODERATION IN BERYLLIUM AND BERYLLIUM OXIDE BY THE IMPULSE METHOD* I. F. ZHEZHERUN (First received
30 November
1963 and injnalform
29 February
1964)
Abstract-The neutron slowing down time to energies 1.46, 0.3, 0.178 and 0.0976 eV and also the thermalization time below ~0.1 eV have been measured. The time dependence of slowing down neutrons at 0.3 eV was obtained. The measurements made it possible to compute the correction for neutron moderation in the energy range below 1.46 eV and to obtain the square of the fission neutron slowing down length to various energies close to the thermal region. 1. INTRODUCTION FOR the majority of moderators used in nuclear engineering the square of the slowing down length Lf2has been measured down to an energy of 1.46 eV (resonance of indium). A correction for neutron moderation to thermal energy is introduced by calculation, and usually contains substantial indeterminacy due mainly to not knowing the average logarithmic energy loss 5 in this region. Measurement of Lf2to lower energies has been performed only for beryllium oxide, down to 0.3 eV [resonance of 239pu](1).
Neutron slowing down time and its fluctuation have been studied theoretically by a number of authors.(2-6) Interesting results have been given by DYAD’KIN and BATALINA”), who examined the time dependence of the space-energy distribution of neutrons N(r, u, t) having an initial velocity u0 and coming from a pulsed source. They found that at distances r G h/3 (;I is the scattering length, u the lethargy, /? a constant of the order of unity) from the source
i.e. it is the product of the space-energy energy-time distribution iV,,(u, r), where N(
t)=
04
(steady-state)
distribution
NSt(r, u) and the
(F)“‘exp(-!!)
(1 -!L)“;“~’
(2)
E
I? I+? ( t)
at t > [A/(2+ - v)]. Thus, the time distribution of neutrons having velocity coincides with the Poisson probability of the occurrence of 2/E neutrons in a time t on condition that the probability of the occurrence of neutrons at any moment is identical and equal to (u/L) dt. It should be noted that N,(u, t) is almost independent of the initial velocity of the neutron u0 and reaches a maximum at instant
* Translated by J. J.
CORNISH
from Atomnaya 653
Energiya
18, 127 (1965).
684
I. F. ZHEZHERUN
which is slightly dependent on r (LXis a very slowly varying function of u). An additional slight dependence of N(r, u, t) on time is contained in term [l + E(r, u, t)], which however, differs little from unity for t w tmax $ At, where At is the dispersion at tmax, tmax is the mean slowing down time of neutrons having a velocity U. The slowing down time to velocity u will obviously be t, = tmax - A/v. One paper ~3)and two notes@~rO)are known on the measurement oft, in water down to thermal energies; two further papers give t, SW160 psec for graphiteol) and t, = 230 h 30 ,usec for beryllium oxide (r2). The shortcoming of these papers is that there is no indication of the energies to which the aforesaid t, values refer. The study of the last stage of moderation-the establishment of an equilibrium spectrum-is dealt with in several papers, describing work in which a pulsed neutron source was used. Of the theoretical works we shall refer only to PUROHIT’S paper(13), in which the relationship between the diffusion cooling coefficient C and the thermalization time was found without using the concept of neutron temperature, and which indicated the possibility of experimentally determining t,, by measuring the decrement of attenuation of the first harmonic, similar to the way in which the diffusion coefficient is determined from the decrement of attenuation of the zero harmonic. Of the experimental work, attention should be drawn to that described in papers,(12F14-16) in which t,, was measured in beryllium oxide of density 2.96 g/cm3 by various methods (passing through a filter made from a l/v absorber measuring the diffusion cooling coefficient and so on) and a value t,,l = 165 + 10 psec obtained. For beryllium (density 1.79 g/cm3) ZHEZHERUN(~‘) gives an approximate value tt,, = 172 ,usec, found from measurements of the diffusion cooling coefficient. In the present work, the slowing down time was measured to various energies E < 1.46 eV; the thermalization time was also measured in beryllium and sintered beryllium oxide. The measurements provided data on the neutron slowing down length in Be and Be0 below 1.46 eV. 2.
MEASURING
PROCEDURE
The neutron slowing down time was determined by the transmission r(t) of filters with strong resonance in the absorption cross section as a function of the time t elapsed since the instant of the neutron pulse. One of the beams of the Kurchatov Institute of Atomic Energy linear accelerator was used as the pulsed neutron sourceo6). The pulse duration was 0.5-l ,usec, the pulse repetition rate 50-100 c/s. The filters were cylindrical and were fitted over a small cylindrical BF, counter placed in the moderator block. The filters used were of 0.073 g/cm2 indium, 0.086 g/cm2 cadmium and 0.047 g/cm2 samarium oxide, having resonances at 1.46, 0.178 and 0.0976 eV respectively. In addition to the filters, use was made of a pulsed 0.3 eV neutron detector [a plutonium chamber enclosed in a shield consisting of a mixture of samarium and gadolinium oxides] which proved to be very convenient for measuring not only t, but also the time distribution No(u, t). The detectors were placed on the axis of the moderator block coinciding with the beam axis. By changing the position of the detector along the axis, it was also possible to study t, as a function of the distance r to the neutron source. The mean density of the material in the beryllium and beryllium oxide blocks was I.79 and 2.79 g/cm3 respectively.
Neutron moderation
in beryllium and beryllium oxide by the impulse method
685
The thermalization time was determined from the measurement of the transmission r(t) of plane boron filters containing O-012 and O-023 g/cm2 boron. The filters, measuring 12 x 30 cm, were placed in a slit under a neutron collimator located on the moderator block (Fig. 1). The collimator was made from cadmium and boron carbide and had apertures in it 35 mm in diameter and 120 mm high. Above the collimator there were four lOBF,_ counters 20 mm in diameter and 25 cm long connected in parallel. The counters were shielded with cadmium, boron carbide and paraffin wax, in such a way that neutrons from the moderator block could only
FIG. I.-Experiment geometry for measuring the transmission of boron filters (cross section): 1: filter, 2: boron carbide of collimator, 3 : cadmium can, 4: protective shield for counters, 5 : BF, counters, 6: collimator apertures, 7 : collimator support, 8 : moderator block. The shield of the block (Cd and B,C) is not shown.
impinge on them through the collimator apertures at an angle SO” to the block surface. The distance from the block to the counters was 15 cm. The moderator blocks were in the shape of a cube or a parallelepiped measuring 60 x 60 x 60 and 50 x 50 x 50 cm for beryllium, and 80 x 70 x 75 and 60 x 60 x 60 cm for beryllium oxide; they were enclosed on all sides by a layer of cadmium (086 g/cm? and b oron carbide (5 g/cm2). Data on the diffusion parameters of these materials are given by ZHEZHERUN.(~‘>~~) The counting rates of a counter with filter and without filter as functions of time, from which the transmission n(t) was determined, were measured with the time analyser used previously by the author(17p20)and a 1lo-channel analyser, the minimum channel width in which was 1 ,usec. The source strength was monitored with two monitors. The background (which did not exceed 1 per cent of the effect) was determined from the counting rate at instants immediately prior to a neutron pulse. 3. RESULTS
OF
THE
MEASUREMENTS
AND
DISCUSSION
3.1 Slowing down time The results of measuring the slowing down time are given on a semilogarithmic scale in Fig. 2. For beryllium these measurements were made in a block measuring 50 x 50 x 50 cm, for beryllium oxide in blocks 80 x 70 x 75 and 60 x 60 x 60 cm. The time elapsed since the instant of the neutron pulse is plotted on the x-axis; on the y-axis is plotted the counting rate N(t) of the 0.3 eV neutron resonance detector and the inverse transmission rr-l (t) for the resonance filters. The time corresponding to the maxima is obviously equal to the slowing down time t, to the given energy, plus the time of flight of the neutron from the position of the last collision till absorption
686
I. F. ZHEZHERUN
in the detector 1, = {[A(v) + +d]/u} (dis the mean transverse dimension of the detector allowing for the voids at the place where it is located) and the time of flight from the accelerator target to the block (~0.5 psec). The t, values obtained are given in Table 1. The indicated errors (from 1 to 10 ,usec in the different measurements) are due to the finite width of the channel and inaccuracy in calculating t,. For comparison, Table 1 also gives t, values obtained in
FIG. 2.-Results of measurements of the slowing down time: 1,2: counting rate N(t) of 0.3 eV neutron detector in a block of Be and Be0 respectively; 3,4: inverse transmission +(t) of a cadmium filter in Be and Be0 ; 5,6 : same for samarium oxide filter ; 7,s : same for indium filter. All the curves at the maximum are normalized to unity.
accordance with equation (3), assuming that the term in the brace which allows for tmax as a function of the distance r from the source is unity. Under our conditions, the sources are obviously where the first collisions occur between the beam neutrons and the moderator nuclei, which are located in a plane layer of the front face of the moderator block of the order of 240,) = 5-7 cm thick. Measurements in the blocks with an 0.3 eV neutron detector at different distances (r < 30 cm) from this layer showed no dependence between t, and r, which justifies the assumption made above. The t, values given in Table 1 will be valid for fission neutrons as well, as the neutron spectrum of the accelerator is close to the fission neutron spectrum. It can be TABLE I.--MEASURED
Resonance energy (ev) 1.46 (indium) 0.3 (plutonium) O-178 (cadmium) 0.0976 (samarium)
AND CALCULATED VALUES OF THE SLOWING DOWN ENERGIES (SW)
In beryllium Calculation Experiment 7.2 15.7 20.4 27.6
7.5 17.5 40 73
+ + + *
1 1 3 5
TIME TO VARIOUS
In beryllium oxide Experiment Calculation
26.3 34.3
9.5 26 51 88
+ 1 h 2 &3 + 5
Neutron moderation
in beryllium and beryllium oxide by the impulse method
687
seen from Table 1 that for energies less than 1.46 eV the experimental value of t, is greater than the calculated value. This is evidently connected with the fact that 6 in this region is lower than the value of 5 for collision with a free atom. Measurements made with the O-3 eV neutron detector make it possible to compare the experimental distribution NO(u,t) with the theoretical [see equation (2)]. Figure 3 gives experimental data for beryllium and beryllium oxide, and Poisson distributions
0
IO
20
30 1,
40 pisec
50
60
70
FIG. 3.-Comparison
of the experimental time distribution of 0.3 eV neutrons with the Poisson distribution, 0 : beryllium, 0: beryllium oxide.
N,(u, t) w (vt/11)“5 exp (- d/l) with 2/t = 12 for beryllium (solid line) and 2/E = 18 for beryllium oxide (broken line) selected according to the experimental data. It can be seen that they are in good agreement. Discrepancy is noted only at high t values, which is apparently due to the influence of the term in square brackets in equation (1) [see DYAD'KIN et cd.(‘)]. Making use of the fact that in the Poisson distribution the dispersion is equal to the mean value, from the relations (ut,,,,/jl)Be = 12 = 2/(&J and (%,Xl%eO = 18 = 2/(4&o) it is also possible to obtain the slowing down time of neutrons t, to an energy 0.3 eV for beryllium and beryllium oxide. For this, the dispersion caused by the width of the energy sensitivity range of the detector must be computed from tmax(lg), and is At = 4 . (AE/E)tZ (2.9 and 4.3 psec for beryllium and beryllium oxide respectively); and the time of flight at 0.3 eV must be computed as well. As a result we get t, values of 17.3 psec for beryllium and 28 psec for beryllium oxide, which agree with the values found from the position of the maximum. The average logarithmic energy losses at 0.3 eV are:
C$ne= 0.19,
5&o = 0.12.
3.2 Thermalization time The results of measuring the n-(t) of boron filters for blocks of beryllium (60 x 60 x 60 cm) and beryllium oxide (80 x 70 x 75 cm) are given in Fig. 4. It follows from the figures that for time t 2 1200 psec in beryllium and t > 1400 psec in Be0 the transmission reaches the minimum asymptotic value corresponding to the steadystate neutron spectrum. The asymptotic values were measured with great accuracy and, as was to be expected, they were identical for beryllium and beryllium oxide: 0.577-0.003 and 0*355-0.002 for filters containing 0.012 and 0.023 g/cm2 boron respectively.
I. F. ZHEZHERUN
688 0.9
0.8
0.7
5
06
c
05
o-4
0.5
160
320
460
I 800
1
,Ll
t,
PC
960
II
set
FIG. 4(a).-Transmission of boron filters for a beryllium block. The curves are drawn visually from the experimental points of various series of measurements: 1,2: for filters containing 0.012 and 0.023 g/cm* boron respectively.
-
i.
-
r
-
-
-
-
2 -
$8 -
-
-
: :;;:57
,-
p;1
‘1
5. I28
t, FIG. 4(b).-Transmission
psec
l6(
1760
of boron filters for a beryllium oxide block. for explanation.]
[See Fig. 4(a)
Neutron moderation
in beryllium and beryllium oxide by the impulse method
689
Similar transmission curves were also obtained for smaller blocks with a slightly smaller asymptotic transmission. Thus, for a beryllium block measuring 50 x 50 x 50 cm it was 0.56%0.008 and 0.346-0.002 respectively. In order to obtain data on neutron energy from the transmission n(t) the neutron spectrum was assumed to have a Maxwellian distribution with temperature T(t) = T, + A
exp (- ~0,
(4)
where y-l is the thermalization time in the given moderator block, T, is the equilibrium temperature under room conditions. Taking into consideration the deformation of the spectrum due to the flight length Z,for the transmission we get expression
$$ sow
{T, + [T(t) - TJ exp
tr
x exp
4Wl
E
-
(1 - exp (- [b/&Z])) exp (-
T, + [T(t) - T,l exp(a/&j
= s
(a/~~~>}-1 [c/&3)
om-f-f$ (T, + [T(r) - T,l exp (Q/&)}-~ tr x exp
i
E
-
T, + [T(t) - T,l exp@/v%
(1 -
exp (-[@I)). (5)
Here u I yl(JKTo/qJ; ubis the most probable velocity corresponding to To = 293°K; and c are constants determined by the design of the detector and the thickness of the boron filter respectively; o,,(E) is the transport cross section of the moderator material. Function n-(T) was calculated on an electronic computer in the temperature range from T, to 3000°K; the value of T, in the block (allowing for diffusion cooling) was takenas 285°K. The thermalizationtime y-l entering equation(5), whichwas unknown earlier, was taken as 120 and 170 psec. Figures 5 and 6 give the results of calculating r(T) at y-l = 170 ,usec using calculated o,,(E)(~~) and experimental a,(E) data.(22y23) The graphs in these figures and measured values of the transmission (smooth curves Figs. 4a,b) were used to obtain the temperature as a function of the time elapsed since neutron pulse T(t) (Figs. 7 and 8). The equilibrium temperature T, for the neutrons escaping from the block was found to be slightly different exceeding T, inside the block by 20-40°K [in accordance with theoretical predictions(24)]. The slope of the linear part of the graphs, determining the thermalization time y-l, is practically independent of whether o,,(E) or u,(E) is used for calculating n(T), and gives a y-l value of 178 i 25 and 213 f 30 ,usec respectively for a beryllium block (60 x 60 x 60 cm) and a beryllium oxide block (80 x 70 x 75 cm). For smaller size blocks the following y-l values were obtained: 170 f 30 psec for Be (50 x 50 x 50 cm) and 177 f 40 psec for Be0 (60 x 60 x 60 cm). The value oft,, for an infinite medium was computed from equation b
t,,l
= y -
$DB2
(6)
where D is the diffusion coefficient and B2 the geometric buckling of the block allowing
I. F.
690
ZHEZHERUN
0.60 i= y 050
0.40
0.30
O-20,
fi
4
I
6 7 II 910’ T,
8
2
1
4
‘K
FIG.
5.-Transmission of boron filters as a function of neutron temperature for beryllium : la, 2~: for filters containing 0.012 and 0.023 g/cm2 boron respectively, using o,,(E) from the calculated data of SINGWP ; lb, 2b: when at,(E) is replaced by experimental data for a,(E)‘*2’.
T,
OK
FIG. 6.-Transmission of boron filters as a function of neutron temperature for beryllium oxide: la, 2~7: for filters containing 0.012 and 0.023 g/cm2 boron respectively using u,,(E) from calculated data of HUGHES tzr’; lb, 26: when utr(E) is replaced by experimental data for a,(E) for a specimen with 13 ,LA grain siz.cP’.
Neutron moderation
in beryllium and beryllium oxide by the impulse method
691
for a slight correction to y-l in those cases when this quantity does not coincide with those used in the calculations of n(T). It can be evaluated from n(T) for y-l equal to 120 and 170 ,usec. Using D for Be and Be0 from the author’s previous data(17*20) and averaging agreeing values of tt,, over measurements in blocks of different sizes we get t,!, = 185 & 20 ,usec for beryllium and ttl, = 204 fr 25 pusec for beryllium oxide. The tth value for Be accords with that obtained from measurements of the diffusion parameters, viz. 172 psec (l’). There is no such accordance for BeO. Converting t,, for Be0 to a density 2.96 g/cm3, we get 200 & 25 psec; according to earlier measurementS(12J4*l.i) f,, = 165 f 10 psec. That this divergence (which is within the limits of the measurement errors) is due to the difference in the crystalline structure of the materials cannot be excluded. We did not study the crystalline structure of the beryllium and beryllium oxide which we used. It was therefore unclear which of the available data for o,(E) for different grain sizes should be used in computing r(T). Calculations for the minimum (8 ~1) and maximum (29 p) grain sizes showed, however, that t,, can differ by approximately 5 per cent, increasing with increasing grain size. Thus, inaccuracy in calculating n(T) from equation (5) has a slight influence on the value of ttll measured by the transmission method, but has a marked effect on T, and the value of A [see equation Tmsx = T, + A, at which a neutron distri(4)], i.e. on the maximum temperature bution close to the Maxwellian is established. As can be seen from Figs. 7 and 8, quantity A also depends on the thickness of the boron filter. This can serve as an indication that there is not yet an exactly Maxwellian distribution. On all the graphs, however, T,,, does not exceed 700°K. For graphs 2a,b (see Figs. 7 and 8) it can be approximately established that Tmax = T, + A m 520°K (-0.045 eV) and that this temperature is reached at 190 ,usec for Be and 215 psec for BeO. Subtracting from these values the time of flight of 0.045 eV neutrons from the position of the last collision to the detector, which is 55 psec, we can get an approximate t, value to this energy: 135 ,usec for Be and 160 psec for BeO. Measured ttl, values for Be and Be0 are close to experimental values oft,, for other materials: 185 & 45 psec(25) and 207 & 23 psec(ll) for graphite, 130 5 16 psec and 194 & 32 psec for zirconium hydride [from measurements by different methods](26). 3.3 Square of the neutron slowing down length in the energy range below 1.46 eV It has been noted earlier that at energies below 1.46 eV experimental values of t, are greater than calculated values. This is evidently caused by the reduction in the average logarithmic energy loss due to the atomic bonding in the crystal lattice. We shall attempt to use measured t, values for evaluating E and the slowing down area below I.46 eV. The slowing down time of a neutron from energy El to E, can obviously be represented as the sum of the collisions (7’) where ;li is the scattering range of a neutron having a velocity vi. In the 1.46-0.1 eV range li for Be and Be0 are constant and equal to 1.39 and 1.46 cm respectively. We shall assume that 6 does not vary in the small range El-E2 either, i.e. that vi+Jvi =
692
I. F.
ZHEZHERLJN
0
0 t,
psec
FIG. 7.-Neutron
lium block.
temperature as a function of the time past since the pulse for a berylCurves la, lb, 2a, 26 are obtained from the appropriate graphs in Fig. 5.
4”
la 2b 2a 0
FIG.
1
I
I
I
I
I
I
I
I
160 320 480 640 800 960 1120 I280 1, psec
8.- -Neutron temperature as a function of time for a beryllium oxide block. Curves la, lb, 2u, 26 are obtained from the appropriate
graphs of Fig. 6.
Neutron moderation
exp (-[t/2]).
in beryllium and beryllium oxide by the impulse method
693
Then equation (7’) will take the form
t 22
(7”) -
tzl-
<
1 -
&p([/2)*
Substituting measured t, values in this equation, we get the following values for moderation below 1.46 eV: (a) in the 1.46-0.3 eV range: O-193 f 0.004 for Be and O-109 & 0.012 for BeO; (b) in the 0.3-0.178 eV range : 0,048 & 0.008 for Be and 0,047 & 0.007 for Be0 ; (c) in the 0.178-0.0976 eV range: 0.050 * 0,009 for Be and 0.047 f 0.007 for BeO; (d) in the 0.0976-0.045 eV range (approximate evaluation) : 0.048 for Be and 0.044 for BeO. Thus, [ for Be and Be0 at energies below 0.3 eV does not in fact vary. In the 0+30.0976 eV range we therefore get more accurate [ values: 0.049 & 0.005 for Be and 0.047 f 0.005 for BeO. In the 1.46-0.3 eV range equation (7”) gives only approximate values of 6, particularly for BeO. In the region where equation (4) is valid, it is easy to find that
From here it can be seen that as E approaches E,, 6 decreases, and when E = E, it vanishes. Some authors describe the last stage of slowing down by the quantity A/ (us. t,,), which is O-034 for Be and 0.031 for Be0 (~3~corresponds to an energy E, = kT,). Now, making use of the 5 values obtained, the correction to the experimentally measured value of Lf2 down to an energy of 1.46 eV can be calculated from the known
relation Lf2(Er -
~72)
(9) (D is equal to 0.50 and 0.54 cm for Be and Be0 respectively). Thus, for example, values BeLf2(1.46 + O-3eV) = 5.7 * 1.2 cm2; ueoLfz (1.46 -+ O-3 eV) = Il.4 * 1.2 cm2, are in good agreement with the value 12.5 + 2.5 cm2 obtained for Be0 from direct measurements of Lf2 to 1.46 and 0.3 eV(r’, which indicates the acceptable accuracy of equation (7”) in the 1.46-0.3 eV range as well; u,L12(O*3-+ 0.178 eV) = 7.4 6 0.8 cm2; ueoLY2(0.3+ 0.178 eV) = 8.7 & 0.9 cm2; ueLY2(0.3-+ 0.13 eV) = 11.9 f 1.2 cm2; BeOLf2(0.3 -+ 0.13 eV) = 14.3 f 1.4 cm2 and so on. In the region where equation (4) is valid it is easy, using equations (8) and (9), to get
(10)
I. F. ZHEZHERUN
694
i.e. when E, -+ E,Lf2 -+ 00. The latter result, as has been pointed out several times [see, for example, KOEN t2’)] testifies to the fact that it is incorrect to take value E = kT, as the lower limit of intekral(9) when computing the moderation length to thermal
energy. 4. CONCLUSIONS The measurements performed show that the process of neutron moderation in beryllium and beryllium oxide to 1.46 eV occurs in collisions with free atoms and lasts a comparatively short time not exceeding 10 psec (Table 2). In the 1.46-0.3 eV range
the effect of the atomic bonding in the crystal lattice is already marked, particularly for BeO. The logarithmic energy loss E decreases on the average by 10 per cent in Be TABLE2.-SLOWING DOWNTIMEAND THE SQUARE OF THE NEUTRON SLOWING DOWN Lfa TO ENERGIES E 1.46eV1~
Energy range -2 MeV-1.46 eV -2 MeV-0.3 eV -2 MeV-0.078 eV -2 MeV-0.13 eV -2 MeV-0.0976 eV -2 MeV-0.07-0.045 eV -007-0~045 eV-O.025 eV
Be (density 1.79 g/cm”) L,2* (cm2) t, (set) 7.5 * 17.5 * 40 * 56 * 73 + -135 185 *
1 1 3 5t 5 20
LENGTH
BERYLLIUMAND BERYLLIUM OXIDE
85.8 + 91.5 * 98.9 * 103.4 & 107.5 & -
2.1’28’ 2.4 2.6 2.7 2.9
Be0 (density 2.79 g/cm”) t, (set)
9.5 * 1 27 + 2 51 * 3 69 i 4t 88 $ 5 -160 204 ??25
L,2 (cm2)
92 k 103.4 6 112.1 * 117.7 k 122.5 h -
1.5’” 1.9 2.1 2.4 2.9
* L,2 data given with corrections computed in the present work. t Calculated from equation (7”) using the values obtained in the 0.3-0.0976 eV range.
and by 60 per cent in Be0 compared with its value for a free atom. The slowing down time in this range is 1.5-2 times greater than t, to 1.46 eV. In the range from 0.3 to approximately 0.07-0.08 eV (below which a spectrum close to the Maxwellian is established) 5 for Be and Be0 decreases to approximately 25 per cent of its value for a free atom. The slowing down time in this range is six to seven times greater than t, to 1.46 eV. The energy-time distribution of neutrons in the region to ~0.07 eV can be obtained from equation (2) if the 5 values given above are substituted in it. Below -0~07-0~045 eV moderation proceeds very slowly and lasts on the average 185 & 20 ,usec for beryllium and 204 & 25 ,usec for beryllium oxide. The Lf2 value to 0.13 eV (5.2 kT,) for beryllium given in Table 2 is slightly higher (when converted to a density of 1.84 g/cm3) than the value of the Fermi age given by WEINBERG and WIGNER(~~) in which the lower limit in integral (9) is taken as 5.2 kT, in accordance with thermalization theory. The value of Lf2 to 0.3 eV for Be0 agrees well with the value 104.5 & 2 cm2 obtained by direct measurements of Lf2 to O-3 eV for the same beryllium oxide. Acknowledgments-The author expresses his gratitude to the operating personnel of the accelerator on the beam of which the measurements were performed, to M. P. SHIJSTOVA for the numerical calculations, A. A. OSOCHNIKOV and G. V. YAKOVLEVfor their help in servicing the analysers and Yu. D. KURDYUMOVand G. P. PEROVfor help in the measurements. REFERENCES 1. ZHEZHERUNI. F. et al. Atomn. Energ. 13, 258 (1962). 2. SYKES J. J. nucl. Energy 2, 31 (1955). 3. HAYNEMAN G.and CROUCHM.NUCI. Sci.Engng2,626(1957).
Neutron moderation
in beryllium and beryllium oxide by the impulse method
695
4. WAXER J. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, Vol. 16, p. 450. United Nations, N.Y. (1958). 5. POLL. and NEMETHG. Nukleonik 1, 165 (1959). 6. KOSALY G. and NEMETHG. Nukleonik 1, 225 (1959). 7. DYAD’KIN I. G. and BATALINAE. P. Atomn. Energ. 10, 5 (1961). 8. KROUGH M. F. Nucl. Sci. Engng 2, 631 (1957). 9. JUNERJ. DE Nucl. Sci. Engng 9,408 (1961). 10. MOLLER E. and SJOSTRANDN. Nucl. Sci. Engng 15, 2 (1963). 11. ANTONOVA. V. Trud Fiz. Inst. Lebedeva 14, 147 (1962). 12. IYENGERS. et al. Proc. Indian Acad. Sci. A45,215 (1957). 13. PUROHITS. N. Nucl. Sci. Engng 9, 157 (1961). 14. RAMANNA R. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, Vol. 16, p. 314. United Nations, N.Y. (1958). 15. COUHALLV. A. et al. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, Vol. 16, p. 319. United Nations, N.Y. (1958). 16. IYENGERS. et al. Proc. Indian Acad. Sci. A45, 224 (1957). 17. ZHEZHERUNI. F. Atomn. Energ. 16, 224 (1964). 18. VORONKOVR. M. et al. Atomn. Energ. 13, 327 (1962). 19. ZHEZHERUNI. F., SADIKOV I. P. and CHERNYSHOVA. A. PriborJ tekh. Eksp. No. 3, 43 (1962). 20. ZHEZHERUNI. F. Atomn. Energ. 14, 193 (1963). 21. SINGWI K. and KOTHAR~L. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, Vol. 16, p. 325. United Nations, N.Y. (1958). 22. HUGHESJ. and SCHWARTZ R. Neutron Cross Sections, 2nd edn. Report BNL-325. 23. ZHEZHERUNI. F., SADIKOVI. P. and CHERNYSHOVA. A. Atomn. Energ. 13,250 (1962). 24. SINGWI K. Ark. Fys. 16, 385 (1959). 25. BECKURS~K. Nucl. Sci. Engng 2, 516 (1957). 26. MEADOWSJ. and WHALEN J. Nucl. Sci. Engng 13,230 (1961). 27. KOEN E. Experimental Reactors and Reactor Physics (Reports of Foreign Scientists at the International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1955), p. 557, Gostekhteorizdat, Moscow (1956). 28. HUGHESJ. Neutron Studies on Nuclear Boilers, p. 161, Izd. Inostran. Lit., Moscow (1954). 29. WEINBERGL. and WIGNER E. Physical Theory of Nuclear Reactors, p. 310, Izd. Inostran. Lit., Moscow (1961).
of Nuclear
Energy
Parts A/B,
1966, Vol. 20, pp. 683 to 695.
Pergamon
Press Ltd.
Printed
in Northern
Ireland
STUDY OF THE PROCESS OF NEUTRON MODERATION IN BERYLLIUM AND BERYLLIUM OXIDE BY THE IMPULSE METHOD* I. F. ZHEZHERUN (First received
30 November
1963 and injnalform
29 February
1964)
Abstract-The neutron slowing down time to energies 1.46, 0.3, 0.178 and 0.0976 eV and also the thermalization time below ~0.1 eV have been measured. The time dependence of slowing down neutrons at 0.3 eV was obtained. The measurements made it possible to compute the correction for neutron moderation in the energy range below 1.46 eV and to obtain the square of the fission neutron slowing down length to various energies close to the thermal region. 1. INTRODUCTION FOR the majority of moderators used in nuclear engineering the square of the slowing down length Lf2has been measured down to an energy of 1.46 eV (resonance of indium). A correction for neutron moderation to thermal energy is introduced by calculation, and usually contains substantial indeterminacy due mainly to not knowing the average logarithmic energy loss 5 in this region. Measurement of Lf2to lower energies has been performed only for beryllium oxide, down to 0.3 eV [resonance of 239pu](1).
Neutron slowing down time and its fluctuation have been studied theoretically by a number of authors.(2-6) Interesting results have been given by DYAD’KIN and BATALINA”), who examined the time dependence of the space-energy distribution of neutrons N(r, u, t) having an initial velocity u0 and coming from a pulsed source. They found that at distances r G h/3 (;I is the scattering length, u the lethargy, /? a constant of the order of unity) from the source
i.e. it is the product of the space-energy energy-time distribution iV,,(u, r), where N(
t)=
04
(steady-state)
distribution
NSt(r, u) and the
(F)“‘exp(-!!)
(1 -!L)“;“~’
(2)
E
I? I+? ( t)
at t > [A/(2+ - v)]. Thus, the time distribution of neutrons having velocity coincides with the Poisson probability of the occurrence of 2/E neutrons in a time t on condition that the probability of the occurrence of neutrons at any moment is identical and equal to (u/L) dt. It should be noted that N,(u, t) is almost independent of the initial velocity of the neutron u0 and reaches a maximum at instant
* Translated by J. J.
CORNISH
from Atomnaya 653
Energiya
18, 127 (1965).
684
I. F. ZHEZHERUN
which is slightly dependent on r (LXis a very slowly varying function of u). An additional slight dependence of N(r, u, t) on time is contained in term [l + E(r, u, t)], which however, differs little from unity for t w tmax $ At, where At is the dispersion at tmax, tmax is the mean slowing down time of neutrons having a velocity U. The slowing down time to velocity u will obviously be t, = tmax - A/v. One paper ~3)and two notes@~rO)are known on the measurement oft, in water down to thermal energies; two further papers give t, SW160 psec for graphiteol) and t, = 230 h 30 ,usec for beryllium oxide (r2). The shortcoming of these papers is that there is no indication of the energies to which the aforesaid t, values refer. The study of the last stage of moderation-the establishment of an equilibrium spectrum-is dealt with in several papers, describing work in which a pulsed neutron source was used. Of the theoretical works we shall refer only to PUROHIT’S paper(13), in which the relationship between the diffusion cooling coefficient C and the thermalization time was found without using the concept of neutron temperature, and which indicated the possibility of experimentally determining t,, by measuring the decrement of attenuation of the first harmonic, similar to the way in which the diffusion coefficient is determined from the decrement of attenuation of the zero harmonic. Of the experimental work, attention should be drawn to that described in papers,(12F14-16) in which t,, was measured in beryllium oxide of density 2.96 g/cm3 by various methods (passing through a filter made from a l/v absorber measuring the diffusion cooling coefficient and so on) and a value t,,l = 165 + 10 psec obtained. For beryllium (density 1.79 g/cm3) ZHEZHERUN(~‘) gives an approximate value tt,, = 172 ,usec, found from measurements of the diffusion cooling coefficient. In the present work, the slowing down time was measured to various energies E < 1.46 eV; the thermalization time was also measured in beryllium and sintered beryllium oxide. The measurements provided data on the neutron slowing down length in Be and Be0 below 1.46 eV. 2.
MEASURING
PROCEDURE
The neutron slowing down time was determined by the transmission r(t) of filters with strong resonance in the absorption cross section as a function of the time t elapsed since the instant of the neutron pulse. One of the beams of the Kurchatov Institute of Atomic Energy linear accelerator was used as the pulsed neutron sourceo6). The pulse duration was 0.5-l ,usec, the pulse repetition rate 50-100 c/s. The filters were cylindrical and were fitted over a small cylindrical BF, counter placed in the moderator block. The filters used were of 0.073 g/cm2 indium, 0.086 g/cm2 cadmium and 0.047 g/cm2 samarium oxide, having resonances at 1.46, 0.178 and 0.0976 eV respectively. In addition to the filters, use was made of a pulsed 0.3 eV neutron detector [a plutonium chamber enclosed in a shield consisting of a mixture of samarium and gadolinium oxides] which proved to be very convenient for measuring not only t, but also the time distribution No(u, t). The detectors were placed on the axis of the moderator block coinciding with the beam axis. By changing the position of the detector along the axis, it was also possible to study t, as a function of the distance r to the neutron source. The mean density of the material in the beryllium and beryllium oxide blocks was I.79 and 2.79 g/cm3 respectively.
Neutron moderation
in beryllium and beryllium oxide by the impulse method
685
The thermalization time was determined from the measurement of the transmission r(t) of plane boron filters containing O-012 and O-023 g/cm2 boron. The filters, measuring 12 x 30 cm, were placed in a slit under a neutron collimator located on the moderator block (Fig. 1). The collimator was made from cadmium and boron carbide and had apertures in it 35 mm in diameter and 120 mm high. Above the collimator there were four lOBF,_ counters 20 mm in diameter and 25 cm long connected in parallel. The counters were shielded with cadmium, boron carbide and paraffin wax, in such a way that neutrons from the moderator block could only
FIG. I.-Experiment geometry for measuring the transmission of boron filters (cross section): 1: filter, 2: boron carbide of collimator, 3 : cadmium can, 4: protective shield for counters, 5 : BF, counters, 6: collimator apertures, 7 : collimator support, 8 : moderator block. The shield of the block (Cd and B,C) is not shown.
impinge on them through the collimator apertures at an angle SO” to the block surface. The distance from the block to the counters was 15 cm. The moderator blocks were in the shape of a cube or a parallelepiped measuring 60 x 60 x 60 and 50 x 50 x 50 cm for beryllium, and 80 x 70 x 75 and 60 x 60 x 60 cm for beryllium oxide; they were enclosed on all sides by a layer of cadmium (086 g/cm? and b oron carbide (5 g/cm2). Data on the diffusion parameters of these materials are given by ZHEZHERUN.(~‘>~~) The counting rates of a counter with filter and without filter as functions of time, from which the transmission n(t) was determined, were measured with the time analyser used previously by the author(17p20)and a 1lo-channel analyser, the minimum channel width in which was 1 ,usec. The source strength was monitored with two monitors. The background (which did not exceed 1 per cent of the effect) was determined from the counting rate at instants immediately prior to a neutron pulse. 3. RESULTS
OF
THE
MEASUREMENTS
AND
DISCUSSION
3.1 Slowing down time The results of measuring the slowing down time are given on a semilogarithmic scale in Fig. 2. For beryllium these measurements were made in a block measuring 50 x 50 x 50 cm, for beryllium oxide in blocks 80 x 70 x 75 and 60 x 60 x 60 cm. The time elapsed since the instant of the neutron pulse is plotted on the x-axis; on the y-axis is plotted the counting rate N(t) of the 0.3 eV neutron resonance detector and the inverse transmission rr-l (t) for the resonance filters. The time corresponding to the maxima is obviously equal to the slowing down time t, to the given energy, plus the time of flight of the neutron from the position of the last collision till absorption
686
I. F. ZHEZHERUN
in the detector 1, = {[A(v) + +d]/u} (dis the mean transverse dimension of the detector allowing for the voids at the place where it is located) and the time of flight from the accelerator target to the block (~0.5 psec). The t, values obtained are given in Table 1. The indicated errors (from 1 to 10 ,usec in the different measurements) are due to the finite width of the channel and inaccuracy in calculating t,. For comparison, Table 1 also gives t, values obtained in
FIG. 2.-Results of measurements of the slowing down time: 1,2: counting rate N(t) of 0.3 eV neutron detector in a block of Be and Be0 respectively; 3,4: inverse transmission +(t) of a cadmium filter in Be and Be0 ; 5,6 : same for samarium oxide filter ; 7,s : same for indium filter. All the curves at the maximum are normalized to unity.
accordance with equation (3), assuming that the term in the brace which allows for tmax as a function of the distance r from the source is unity. Under our conditions, the sources are obviously where the first collisions occur between the beam neutrons and the moderator nuclei, which are located in a plane layer of the front face of the moderator block of the order of 240,) = 5-7 cm thick. Measurements in the blocks with an 0.3 eV neutron detector at different distances (r < 30 cm) from this layer showed no dependence between t, and r, which justifies the assumption made above. The t, values given in Table 1 will be valid for fission neutrons as well, as the neutron spectrum of the accelerator is close to the fission neutron spectrum. It can be TABLE I.--MEASURED
Resonance energy (ev) 1.46 (indium) 0.3 (plutonium) O-178 (cadmium) 0.0976 (samarium)
AND CALCULATED VALUES OF THE SLOWING DOWN ENERGIES (SW)
In beryllium Calculation Experiment 7.2 15.7 20.4 27.6
7.5 17.5 40 73
+ + + *
1 1 3 5
TIME TO VARIOUS
In beryllium oxide Experiment Calculation
26.3 34.3
9.5 26 51 88
+ 1 h 2 &3 + 5
Neutron moderation
in beryllium and beryllium oxide by the impulse method
687
seen from Table 1 that for energies less than 1.46 eV the experimental value of t, is greater than the calculated value. This is evidently connected with the fact that 6 in this region is lower than the value of 5 for collision with a free atom. Measurements made with the O-3 eV neutron detector make it possible to compare the experimental distribution NO(u,t) with the theoretical [see equation (2)]. Figure 3 gives experimental data for beryllium and beryllium oxide, and Poisson distributions
0
IO
20
30 1,
40 pisec
50
60
70
FIG. 3.-Comparison
of the experimental time distribution of 0.3 eV neutrons with the Poisson distribution, 0 : beryllium, 0: beryllium oxide.
N,(u, t) w (vt/11)“5 exp (- d/l) with 2/t = 12 for beryllium (solid line) and 2/E = 18 for beryllium oxide (broken line) selected according to the experimental data. It can be seen that they are in good agreement. Discrepancy is noted only at high t values, which is apparently due to the influence of the term in square brackets in equation (1) [see DYAD'KIN et cd.(‘)]. Making use of the fact that in the Poisson distribution the dispersion is equal to the mean value, from the relations (ut,,,,/jl)Be = 12 = 2/(&J and (%,Xl%eO = 18 = 2/(4&o) it is also possible to obtain the slowing down time of neutrons t, to an energy 0.3 eV for beryllium and beryllium oxide. For this, the dispersion caused by the width of the energy sensitivity range of the detector must be computed from tmax(lg), and is At = 4 . (AE/E)tZ (2.9 and 4.3 psec for beryllium and beryllium oxide respectively); and the time of flight at 0.3 eV must be computed as well. As a result we get t, values of 17.3 psec for beryllium and 28 psec for beryllium oxide, which agree with the values found from the position of the maximum. The average logarithmic energy losses at 0.3 eV are:
C$ne= 0.19,
5&o = 0.12.
3.2 Thermalization time The results of measuring the n-(t) of boron filters for blocks of beryllium (60 x 60 x 60 cm) and beryllium oxide (80 x 70 x 75 cm) are given in Fig. 4. It follows from the figures that for time t 2 1200 psec in beryllium and t > 1400 psec in Be0 the transmission reaches the minimum asymptotic value corresponding to the steadystate neutron spectrum. The asymptotic values were measured with great accuracy and, as was to be expected, they were identical for beryllium and beryllium oxide: 0.577-0.003 and 0*355-0.002 for filters containing 0.012 and 0.023 g/cm2 boron respectively.
I. F. ZHEZHERUN
688 0.9
0.8
0.7
5
06
c
05
o-4
0.5
160
320
460
I 800
1
,Ll
t,
PC
960
II
set
FIG. 4(a).-Transmission of boron filters for a beryllium block. The curves are drawn visually from the experimental points of various series of measurements: 1,2: for filters containing 0.012 and 0.023 g/cm* boron respectively.
-
i.
-
r
-
-
-
-
2 -
$8 -
-
-
: :;;:57
,-
p;1
‘1
5. I28
t, FIG. 4(b).-Transmission
psec
l6(
1760
of boron filters for a beryllium oxide block. for explanation.]
[See Fig. 4(a)
Neutron moderation
in beryllium and beryllium oxide by the impulse method
689
Similar transmission curves were also obtained for smaller blocks with a slightly smaller asymptotic transmission. Thus, for a beryllium block measuring 50 x 50 x 50 cm it was 0.56%0.008 and 0.346-0.002 respectively. In order to obtain data on neutron energy from the transmission n(t) the neutron spectrum was assumed to have a Maxwellian distribution with temperature T(t) = T, + A
exp (- ~0,
(4)
where y-l is the thermalization time in the given moderator block, T, is the equilibrium temperature under room conditions. Taking into consideration the deformation of the spectrum due to the flight length Z,for the transmission we get expression
$$ sow
{T, + [T(t) - TJ exp
tr
x exp
4Wl
E
-
(1 - exp (- [b/&Z])) exp (-
T, + [T(t) - T,l exp(a/&j
= s
(a/~~~>}-1 [c/&3)
om-f-f$ (T, + [T(r) - T,l exp (Q/&)}-~ tr x exp
i
E
-
T, + [T(t) - T,l exp@/v%
(1 -
exp (-[@I)). (5)
Here u I yl(JKTo/qJ; ubis the most probable velocity corresponding to To = 293°K; and c are constants determined by the design of the detector and the thickness of the boron filter respectively; o,,(E) is the transport cross section of the moderator material. Function n-(T) was calculated on an electronic computer in the temperature range from T, to 3000°K; the value of T, in the block (allowing for diffusion cooling) was takenas 285°K. The thermalizationtime y-l entering equation(5), whichwas unknown earlier, was taken as 120 and 170 psec. Figures 5 and 6 give the results of calculating r(T) at y-l = 170 ,usec using calculated o,,(E)(~~) and experimental a,(E) data.(22y23) The graphs in these figures and measured values of the transmission (smooth curves Figs. 4a,b) were used to obtain the temperature as a function of the time elapsed since neutron pulse T(t) (Figs. 7 and 8). The equilibrium temperature T, for the neutrons escaping from the block was found to be slightly different exceeding T, inside the block by 20-40°K [in accordance with theoretical predictions(24)]. The slope of the linear part of the graphs, determining the thermalization time y-l, is practically independent of whether o,,(E) or u,(E) is used for calculating n(T), and gives a y-l value of 178 i 25 and 213 f 30 ,usec respectively for a beryllium block (60 x 60 x 60 cm) and a beryllium oxide block (80 x 70 x 75 cm). For smaller size blocks the following y-l values were obtained: 170 f 30 psec for Be (50 x 50 x 50 cm) and 177 f 40 psec for Be0 (60 x 60 x 60 cm). The value oft,, for an infinite medium was computed from equation b
t,,l
= y -
$DB2
(6)
where D is the diffusion coefficient and B2 the geometric buckling of the block allowing
I. F.
690
ZHEZHERUN
0.60 i= y 050
0.40
0.30
O-20,
fi
4
I
6 7 II 910’ T,
8
2
1
4
‘K
FIG.
5.-Transmission of boron filters as a function of neutron temperature for beryllium : la, 2~: for filters containing 0.012 and 0.023 g/cm2 boron respectively, using o,,(E) from the calculated data of SINGWP ; lb, 2b: when at,(E) is replaced by experimental data for a,(E)‘*2’.
T,
OK
FIG. 6.-Transmission of boron filters as a function of neutron temperature for beryllium oxide: la, 2~7: for filters containing 0.012 and 0.023 g/cm2 boron respectively using u,,(E) from calculated data of HUGHES tzr’; lb, 26: when utr(E) is replaced by experimental data for a,(E) for a specimen with 13 ,LA grain siz.cP’.
Neutron moderation
in beryllium and beryllium oxide by the impulse method
691
for a slight correction to y-l in those cases when this quantity does not coincide with those used in the calculations of n(T). It can be evaluated from n(T) for y-l equal to 120 and 170 ,usec. Using D for Be and Be0 from the author’s previous data(17*20) and averaging agreeing values of tt,, over measurements in blocks of different sizes we get t,!, = 185 & 20 ,usec for beryllium and ttl, = 204 fr 25 pusec for beryllium oxide. The tth value for Be accords with that obtained from measurements of the diffusion parameters, viz. 172 psec (l’). There is no such accordance for BeO. Converting t,, for Be0 to a density 2.96 g/cm3, we get 200 & 25 psec; according to earlier measurementS(12J4*l.i) f,, = 165 f 10 psec. That this divergence (which is within the limits of the measurement errors) is due to the difference in the crystalline structure of the materials cannot be excluded. We did not study the crystalline structure of the beryllium and beryllium oxide which we used. It was therefore unclear which of the available data for o,(E) for different grain sizes should be used in computing r(T). Calculations for the minimum (8 ~1) and maximum (29 p) grain sizes showed, however, that t,, can differ by approximately 5 per cent, increasing with increasing grain size. Thus, inaccuracy in calculating n(T) from equation (5) has a slight influence on the value of ttll measured by the transmission method, but has a marked effect on T, and the value of A [see equation Tmsx = T, + A, at which a neutron distri(4)], i.e. on the maximum temperature bution close to the Maxwellian is established. As can be seen from Figs. 7 and 8, quantity A also depends on the thickness of the boron filter. This can serve as an indication that there is not yet an exactly Maxwellian distribution. On all the graphs, however, T,,, does not exceed 700°K. For graphs 2a,b (see Figs. 7 and 8) it can be approximately established that Tmax = T, + A m 520°K (-0.045 eV) and that this temperature is reached at 190 ,usec for Be and 215 psec for BeO. Subtracting from these values the time of flight of 0.045 eV neutrons from the position of the last collision to the detector, which is 55 psec, we can get an approximate t, value to this energy: 135 ,usec for Be and 160 psec for BeO. Measured ttl, values for Be and Be0 are close to experimental values oft,, for other materials: 185 & 45 psec(25) and 207 & 23 psec(ll) for graphite, 130 5 16 psec and 194 & 32 psec for zirconium hydride [from measurements by different methods](26). 3.3 Square of the neutron slowing down length in the energy range below 1.46 eV It has been noted earlier that at energies below 1.46 eV experimental values of t, are greater than calculated values. This is evidently caused by the reduction in the average logarithmic energy loss due to the atomic bonding in the crystal lattice. We shall attempt to use measured t, values for evaluating E and the slowing down area below I.46 eV. The slowing down time of a neutron from energy El to E, can obviously be represented as the sum of the collisions (7’) where ;li is the scattering range of a neutron having a velocity vi. In the 1.46-0.1 eV range li for Be and Be0 are constant and equal to 1.39 and 1.46 cm respectively. We shall assume that 6 does not vary in the small range El-E2 either, i.e. that vi+Jvi =
692
I. F.
ZHEZHERLJN
0
0 t,
psec
FIG. 7.-Neutron
lium block.
temperature as a function of the time past since the pulse for a berylCurves la, lb, 2a, 26 are obtained from the appropriate graphs in Fig. 5.
4”
la 2b 2a 0
FIG.
1
I
I
I
I
I
I
I
I
160 320 480 640 800 960 1120 I280 1, psec
8.- -Neutron temperature as a function of time for a beryllium oxide block. Curves la, lb, 2u, 26 are obtained from the appropriate
graphs of Fig. 6.
Neutron moderation
exp (-[t/2]).
in beryllium and beryllium oxide by the impulse method
693
Then equation (7’) will take the form
t 22
(7”) -
tzl-
<
1 -
&p([/2)*
Substituting measured t, values in this equation, we get the following values for moderation below 1.46 eV: (a) in the 1.46-0.3 eV range: O-193 f 0.004 for Be and O-109 & 0.012 for BeO; (b) in the 0.3-0.178 eV range : 0,048 & 0.008 for Be and 0,047 & 0.007 for Be0 ; (c) in the 0.178-0.0976 eV range: 0.050 * 0,009 for Be and 0.047 f 0.007 for BeO; (d) in the 0.0976-0.045 eV range (approximate evaluation) : 0.048 for Be and 0.044 for BeO. Thus, [ for Be and Be0 at energies below 0.3 eV does not in fact vary. In the 0+30.0976 eV range we therefore get more accurate [ values: 0.049 & 0.005 for Be and 0.047 f 0.005 for BeO. In the 1.46-0.3 eV range equation (7”) gives only approximate values of 6, particularly for BeO. In the region where equation (4) is valid, it is easy to find that
From here it can be seen that as E approaches E,, 6 decreases, and when E = E, it vanishes. Some authors describe the last stage of slowing down by the quantity A/ (us. t,,), which is O-034 for Be and 0.031 for Be0 (~3~corresponds to an energy E, = kT,). Now, making use of the 5 values obtained, the correction to the experimentally measured value of Lf2 down to an energy of 1.46 eV can be calculated from the known
relation Lf2(Er -
~72)
(9) (D is equal to 0.50 and 0.54 cm for Be and Be0 respectively). Thus, for example, values BeLf2(1.46 + O-3eV) = 5.7 * 1.2 cm2; ueoLfz (1.46 -+ O-3 eV) = Il.4 * 1.2 cm2, are in good agreement with the value 12.5 + 2.5 cm2 obtained for Be0 from direct measurements of Lf2 to 1.46 and 0.3 eV(r’, which indicates the acceptable accuracy of equation (7”) in the 1.46-0.3 eV range as well; u,L12(O*3-+ 0.178 eV) = 7.4 6 0.8 cm2; ueoLY2(0.3+ 0.178 eV) = 8.7 & 0.9 cm2; ueLY2(0.3-+ 0.13 eV) = 11.9 f 1.2 cm2; BeOLf2(0.3 -+ 0.13 eV) = 14.3 f 1.4 cm2 and so on. In the region where equation (4) is valid it is easy, using equations (8) and (9), to get
(10)
I. F. ZHEZHERUN
694
i.e. when E, -+ E,Lf2 -+ 00. The latter result, as has been pointed out several times [see, for example, KOEN t2’)] testifies to the fact that it is incorrect to take value E = kT, as the lower limit of intekral(9) when computing the moderation length to thermal
energy. 4. CONCLUSIONS The measurements performed show that the process of neutron moderation in beryllium and beryllium oxide to 1.46 eV occurs in collisions with free atoms and lasts a comparatively short time not exceeding 10 psec (Table 2). In the 1.46-0.3 eV range
the effect of the atomic bonding in the crystal lattice is already marked, particularly for BeO. The logarithmic energy loss E decreases on the average by 10 per cent in Be TABLE2.-SLOWING DOWNTIMEAND THE SQUARE OF THE NEUTRON SLOWING DOWN Lfa TO ENERGIES E 1.46eV1~
Energy range -2 MeV-1.46 eV -2 MeV-0.3 eV -2 MeV-0.078 eV -2 MeV-0.13 eV -2 MeV-0.0976 eV -2 MeV-0.07-0.045 eV -007-0~045 eV-O.025 eV
Be (density 1.79 g/cm”) L,2* (cm2) t, (set) 7.5 * 17.5 * 40 * 56 * 73 + -135 185 *
1 1 3 5t 5 20
LENGTH
BERYLLIUMAND BERYLLIUM OXIDE
85.8 + 91.5 * 98.9 * 103.4 & 107.5 & -
2.1’28’ 2.4 2.6 2.7 2.9
Be0 (density 2.79 g/cm”) t, (set)
9.5 * 1 27 + 2 51 * 3 69 i 4t 88 $ 5 -160 204 ??25
L,2 (cm2)
92 k 103.4 6 112.1 * 117.7 k 122.5 h -
1.5’” 1.9 2.1 2.4 2.9
* L,2 data given with corrections computed in the present work. t Calculated from equation (7”) using the values obtained in the 0.3-0.0976 eV range.
and by 60 per cent in Be0 compared with its value for a free atom. The slowing down time in this range is 1.5-2 times greater than t, to 1.46 eV. In the range from 0.3 to approximately 0.07-0.08 eV (below which a spectrum close to the Maxwellian is established) 5 for Be and Be0 decreases to approximately 25 per cent of its value for a free atom. The slowing down time in this range is six to seven times greater than t, to 1.46 eV. The energy-time distribution of neutrons in the region to ~0.07 eV can be obtained from equation (2) if the 5 values given above are substituted in it. Below -0~07-0~045 eV moderation proceeds very slowly and lasts on the average 185 & 20 ,usec for beryllium and 204 & 25 ,usec for beryllium oxide. The Lf2 value to 0.13 eV (5.2 kT,) for beryllium given in Table 2 is slightly higher (when converted to a density of 1.84 g/cm3) than the value of the Fermi age given by WEINBERG and WIGNER(~~) in which the lower limit in integral (9) is taken as 5.2 kT, in accordance with thermalization theory. The value of Lf2 to 0.3 eV for Be0 agrees well with the value 104.5 & 2 cm2 obtained by direct measurements of Lf2 to O-3 eV for the same beryllium oxide. Acknowledgments-The author expresses his gratitude to the operating personnel of the accelerator on the beam of which the measurements were performed, to M. P. SHIJSTOVA for the numerical calculations, A. A. OSOCHNIKOV and G. V. YAKOVLEVfor their help in servicing the analysers and Yu. D. KURDYUMOVand G. P. PEROVfor help in the measurements. REFERENCES 1. ZHEZHERUNI. F. et al. Atomn. Energ. 13, 258 (1962). 2. SYKES J. J. nucl. Energy 2, 31 (1955). 3. HAYNEMAN G.and CROUCHM.NUCI. Sci.Engng2,626(1957).
Neutron moderation
in beryllium and beryllium oxide by the impulse method
695
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