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UNITED STATES ATOMIC ENERGY COMMISSION
CALCULATION OF NEUTRON DISTRIBUTIONS IN HETEROGENEOUS PILES
A. M. Weinberg
Date of Manuscript:
May 27, 1947
May 29, 1947
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CALCULATION OF NEUTRON DISTRIBUTIONS IN HETEROGENEOUS PILES
By A. M. Weinberg
The calculation of the slow neutron density in a pile which consists of lumps of uranium disposed
in a lattice is a problem of fundamental importance in elementary pile theory. The method used for
such a calculation is rather similar to the method used for calculating the distribution around control
rods; it is therefore doubly worthwhile to examine the calculation in detail.
In a thermal neutron heterogeneous pile, fast neutrons produced as the result of fission in the
uranium lumps are moderated by, for example, the graphite matrix in which the rods are embedded
until they become slow. The neutrons then diffuse in accordance with laws of diffusion theory until
they are finally captured in the moderator, or in the uranium, or until they escape from the boundaries
of the pile.
We consider an infinite prototype of an actual pile. Our problem is to calculate the fraction of the
thermal neutrons absorbed inside this infinite pile which are absorbed in the uranium. We call the
neutrons absorbed in uranium
neutrons absorbed in uranium and moderator
the thermal utilization; in accordance with usual practice we denote this quantity of f. or simply f.
The ratio of slow neutrons absorbed in uranium to neutrons produced in a finite pile, called the ef-
fective thermal utilization, and denoted by feff, is evidently less than the ratio of slow neutrons ab-
sorbed in uranium to slow neutrons produced as a result of slowing down-the reason being that not
all slow neutrons produced are absorbed inside the pile. However, in pile theory, it is customary to
first examine the infinite prototype of a pile, then to deal with the corrections introduced by the finite-
ness of the pile, i.e., it is customary to separate microscopic pile theory from macroscopic pile theory.
It will be our purpose here to consider only the calculation of the thermal utilization in an infinite
lattice. The thermal utilization in the finite lattice depends on the fraction of neutrons which escape
from the pile, and therefore upon the size of the pile in which the lattice is placed. Its calculation is
very much more difficult than the calculation of f.
The calculation of f according to Plementary diffusion theory is very straightforward. The method
and the assumptions are quite analogous to those used for the calculation of wave functions in crystal
lattices. We list the assumptions as follows:
1) Elementary diffusion theory is applicable.
2) The production of thermal neutrons is uniform throughout the moderator, and is zero in the
3) The units cell of a lattice can be replaced by a sphere in case the lattice is a point lattice, e.g.,
simple cubic, or a cylinder in case the lattice is a rod lattice.
We now discuss the validity of these assumptions:
1) The assumption of elementary diffusion theory as is well known, is accurate first, if the
dimensions of the system are large compared to a mean free path; second, if the system does not
absorb neutrons very heavily and third, if the system contains no sources. None of these assumptions
are strictly correct in a uranium lattice. The errors which are involved because elementary diffusion
theory is used, can be estimated. It turns out that they are sizeable in lattices of interest but the labor
involved in extending calculations to include the more accurate theory is prohibitive if a large number
of calculations, such as are needed in calculating optimum lattice dimensions, are to be performed.
2) The assumption that thermal neutrons are produced uniformly throughout a cell is accurate
if the cell size, i.e., the separation between lumps, is not large compared to a slowing down distance.
It is easy to compute the slowing down density in an infinite lattice at each of whose points neutrons,
which slow down according to a Gaussian picture, are produced. It is rather remarkable how uniform
the slow neutron production rate is in such a lattice.
3) This procedure of replacing e.g., a cubic cell by a spherical one is called sphericizing of
the unit cell; it is entirely analogous to the Wigner-Seitz method for calculating the first approximation
to the wave functions in a crystal lattice. The sphericizing assumption is made essential in order to
simplify the calculational procedure. Thus if the lattice is a simple cubic lattice the unit cell is a
cube and the neutron distribution involves all spherical harmonics of order 4n. The sphericizing
approximation implies that only the spherical harmonic of zero order is considered. Evidently the
nearer the actual cell is to the "sphericized" cell, the closer is the approximate neutron density to
the actual one. In case the uranium is disposed in parallel plates, the "sphericized" cel and the actual
cell are identical, both being defined by the planes of symmetry between the uranium plates.
The details of the calculation are now easy to write down. We must solve a simple diffusion equa -
tion for the thermal neutron density in the moderator and a similar diffusion equation in the uranium
and must apply the appropriate boundary conditions at the boundary of the sphericized unit cell and at
the interface between the uranium and the metal. Thus in the moderator we have the equation
S A #1N1,al 1l+q=0 (1)
where O1 is the sum of the speeds of all the thermal neutrons in a cubic centimeter, NI oai is the
macroscopic absorption cross section of the moderator, KI is the reciprocal of .he diffusion length
in the moderator, and q is the number of thermal neutrons produced per cubic centimeter per second.
The quantity N1lal is recognized as the diffusion coefficient according to the elementary diffusion
theory. In the uranium a similar equation is written down, but since practically no neutrons are pro-
duced there, the uranium atomic weight being so large, it is appropriate to set q = 0. Thus in the uran-
ium we have
No2ao A 0o-No rao o =0 (2)
where the subscript zero refers to the uranium region. The cell we take to be either a composite
sphere, a composite cylinder, or a composite slab (see Figure 1); in all cases we denote the one-half
diameter of the uranium by ro, the one-half thickness of the ceUll by r1.
The boundary conditions at the interface between the uranium and the moderator, i.e., at r = ro,
must be continuity of flux and continuity of net current. (This corresponds to ensuring continuity of
the first two harmonics in a spherical harmonic expansion of the slow neutron angular distribution.)
The boundary conditions can therefore be written:
Nj=ai or l at r = ro (3)
2 2 2 C1 V
at r = ro (4)
*1 = o0
where v represents the outward normal. Finally, at the outer boundary of the cell there can be no net
flow of neutrons, i.e.,
6 i 0 at r = rI. (5)
To solve these equations is straightforward. We set
00o = A Zol(Kor) (6)
01 a 1 B K1r1 Zl2(1 r) Z (ll r) Z 2( rlI] (7)1
where A and B are arbitrary constants, Zij( KIr) is a function which solves the equation
SZj KZi = 0 (8)
in the appropriate geometry and denotes derivative.
Since (8) is a second order differential equation it has two solutions which we distinguish by the
subscript j = 1 or 2. One of these solutions will be singular at the origin (in cylindrical or spherical
geometry), or will be nonsymmetric about the origin (plane geometry). This function we denote by j = 2.
The other function will be symmetric and nonsingular at the origin; this we denote by j = 1.
We tabulate the functions Zij and their derivatives for the three geometries in the following table:
The Functions Zij
Geometry Zil(Kir) Z 1(l ir Zi2(Kir) Zi2(Kir)
Plane Cosh Kir sinh Kir sinh Kir cosh Kir
Cylinder lo(Kir) I1 (Ocir) Ko( Klr) -Kl( ir)
Sphere sinh Kir Kir cosh Kir -sinh Kir cosh Kir *r cosh Kir -coshwir
Kir ( Kir)2 Kir (Kgr)2
The linear combination of Z11 and Z12 chosen in (7) automatically satisfies boundary condition (5)
i.e., no net current at the edge of the unit cell. There are still the two boundary conditions (3) and (4)
to be satisfied, and these two conditions are just enough to determine the arbitrary constants A and B.
Substituting (6) and (7) into (3) and (4) we get the following pair of equations for A and B:
-B N a [Z11( Kirl) Zi2( xlro) + Z'll Ilro) Z'12( Krl)]
+ A No ao Zo1(~o,ro) = 0 (8)
B [Zil( Klrl) Z12( Kro) Zl1( lro) Z'12( Kirl)]
+ A Z01(Koro)= N (9)
from which we obtain I/A:
No ao Z0'1(Koro) C(r r 1, ro) + a Zo(Koro) C'( K rl, K ro)
A 0 1(10)
C'( Kirl, Kiro) q/ K1
C(x,y) = Z'l(x) Z12() Z' I(y) Z12(x) (11)
C'(x,y) = Z11(x) Z'2(y) Z'I(y) Z'12(x) (12)
From the expression for A and the corresponding expression for B, the nuetron fluxes (6) and (7)
can be calculated. The general shape of the flux distributions are given in Figure 2. It is seen there
that the density is highest in the moderator where the slow neutrons are produced, and dips in the
uranium where the neutrons are absorbed but are not produced.
The thermal utilization is defined as the ratio of slow neutrons absorbed in the uranium to slow
neutrons produced (by moderation) in the moderator. Since the number of neutrons absorbed in the
uranium must equal the number which flow into the uranium
1 qlV1 = qlV 1
N So 4ao 01 r No a SZ'01( oro) A
where So is the surface of the uranium lump and Vi is the volume of the moderator. We substitute (10)
into (13), and use Gauss's theorem:
f 1 f So Z0((or)o
ZoVo = Z01( or)dVo = AZo01 or)dV So z
f I f AC(lr)dV SoCi (Klro)
CIVI V1 CI( cKr)dVI = -- AC( 1r)dVI
where bar denotes average. Thus
= N alVIF + E =+N al F+E-1 (15)
f No vaoVo No oaoVo
F oVo Z01( Koro) Z01(Koro) (16)
So Z01( or ) Z01( Koro)
E = i Cl( Kro, cirl) = Cl( Kr, Kr) (17)
So C l( lro, Kgrl) ( lro-, 1rl()
The actual formulas for F and C depend upon the geometry of the cell. We list these formulas in the
Table of F and E
Plane Koro coth Koro K1(rl ro) coth al(r1 ro)
Coro lo( oro) 2(ri-ro) II( Kirl)Ko( Kro) Io( lro)Ki(xiri)
2 Il( Koro) 2K iro Ii(Klrl)Kl(Kiro) + Ii( Kro)Kl(clrl)
3 2 3
Sphere Koro tanh Koro Kl(r r3) 0lro coth Ki(rl ro)
3 Koro tanh 10ro 3 K2(r1 r2) K1ro + coth Ki(rl ro)
The structure of the thermal utilization formulas, as pointed out by J. A. Wheeler, is easily under-
stood. If the diffusion coefficient in the moderator were infinite (Figure 3) then there could be no
gradient of neutrons in the moderator, i.e., the neutron density there would be uniform. In this case
K1 = 0, and as may be easily verified, E = 1.
Thus the quantity I + N Ia iV F represents the reciprocal of the thermal utilization if the neutron
density throughout the moderator were just the same as its value at the interface between moderator
and metal. This can also be seen if we observe that the quantity F according to (16) is the ratio of the
neutron density at the interface to the average inside the lump. The quantity Noa-V F is therefore
the ratio of the absorptions in moderator and uranium if the neutron density in the moderator is uni-
form and equal to its value at the uranium surface. This quantity is often called the "relative absorption".
Actually the diffusion coefficient is finite and therefore the neutron density in the moderator is
higher than it is in the uranium. This causes an additional absorption of neutrons in the moderator and
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the magnitude of this so-called excess absorption is measured by the quantity (E 1). For K1(r1 ro)
=0,E 1 = 0. For finite but small Kl,E 1 can be expanded in a series which involves powers of
K 1(rl ro). The reciprocal thermal utilization can therefore be written
-- = 1 t relative absorption plus excess absorption.
It is worthy to note, as a matter of computation that E involves only the constants of the moderator
and F only the constants of the moderator region. It is feasible therefore to prepare tables of F and E
which can be used for many lattice calculations. Such tables were prepared under the supervision of
J. A. Wheeler.
From the behavior of the functions E and F, it is easy to make some general statements about the
behavior of the therm..l utilization. Evidently, as the volume ratio of moderator to uranium increase,
the thermal utilization becomes smaller. Furthermore, for a given volume ratio the thermal utilization
is smaller in a system containing large uranium lumps than in a system containing small uranium lumps.
This is so because the function F, being the ratio of surface density to average density, is larger in a
large lump than in a small one.
One might prepare a series of curves in which the quantity 1/f is plotted as a function of V1/Vo
for various ro. The general trend of these curves is given in the accompanying figure. (See Figure 4).
Since the resonance absorption goes in just the reverse way, i.e., the resonance absorption is large
when the volume ratio is small, the multiplication constant, having the product of p and f as factors,
must have a highest value at some point which represents a compromise between the value of p and the
value of f.
The choice of this compromise in any actual case is of course complicated by questions of heat
transfer, fabricability of lattice units, and other matters of an engineering nature; the solution of these
conflicting demands between engineering and nuclear physics is a perpetual problem which always
arises in the design of a lattice. 'o,
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