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ANP PROJECT QUARTERLY PROGRESS REPORT
where the square bracket in integral 8
vanishes, that is, at the points where
0 is a multiple of the period. At
these points dK/du may even go to 4
without violating the condition for
periodic oscillations.
Specific Examples. The previously
considered case of constant transit
time and constant power distribution
is obtained from the above general
case by giving K(D) the specific
values K(D) = 1/B for D< 0 and K(G) = 0
for 0 > 6 (compare Eq. 1 with Eq. 2
of ORNL-1227, p. 43,(’) or Eg. 2 of
Y-F10-109, p. 2(’)), dK/h = 0 at all
points except cr = 0 (where dK/do = a),
and periodic oscillations persist only
if 6 is a multiple of the period,
If the power distribution is only
constant in the direction of fuel
flow, but not necessarily in the
direction perpendicular to it, and if
there are a number of alternate fuel
paths with transit times 0,, e,, ...,
@ respectively, then the above
n ’
general theory applies with K(o) = ai
for B1-, < o < ol, where L = 1, 2,
... , n, 8, = 0, and the az are positive
constants, a, Periodic
oscillations are only possible if all
0 are multiples of the period, a
cAndition very unlikely to occur in
practice.
It should be mentioned that T. A.
Welton of the Physics Division has
made significant advances in reactor
kinetics by a somewhat different
technique.(3) The above results can
also be regarded as specialcasesof
We1 ton’s theorems,
REACTOR CALCULATPONS( )
R. K. Osborn, Physics Division
It was noticed that the method of
images can be used to solve some
specialized butnot unrealistic problems
of the slowing down of neutrons. The
(3)T. A. Welton, Phys. Quar. Prog. Rep. Sept.
20, 1952. ORNL-1421 (in press).
(4)B. K. Osborn, Sore Solutions of the Age
Equation, Y-F10- 111 (Nov. 17, 1952).
44
case considered involves two homo-
geneous, semi-infinite spaces of
different properties, joined along a
plane x = 0. A plane, isotropic source
of monoenergetic neutrons is placed in
one of t,he half spaces, say the right
half space. IJnder the conditions set
forth below, the left half space, the
“reflector,” can then be replaced, as
far as the slowing down density in the
right half space is concerned, by an
I1
image” source located symmetrically
to the given source with respect to
x = 8. The slowing-down density in the
reflector can be described by a
“refracted” source located to the
right of I: = 0. The image source and
the refracted source have the same
energy as the given source.
Properties of the right half space
are denoted by the index 1 and those
of the left half space by the index 2.
Is, 5, E, and q are, respectively, the
macroscopic scattering cross section,
the average lethargy increase per
collision, the average cosine of the
scattering angle, and the slowing-down
density; 7 is the “age” in the medium
that contains the source,
(1)
(2)
are constant with lethargy. Another
assumption is that the source plane is
parallel to x = 0, so its equation can
be written as x = X. The Fermi age
equation, with no absorption, is
assumed to describe the slowing-down
process :
and