3.5 Gaseous (Barrier) Diffusion
141
N 2 = C
ρenter
stage 2
√
m
=
C
2
V
ρ o
√
m
2 .
(3.58)
Propagating this logic shows that after a total of n successive stages, the number
of atoms that emerge from the n’th stage will be
N n =
C
n
V n−1
ρ o
√
m
n .
(3.59)
If the gas consists of a mixture of two isotopes, say
235 U and
238 U, (3.59) will
apply to each according as the relevant values of ρ o and m. If we designate the two
isotopes with subscripts 5 and 8, then the final ratio of the number of 235 atoms to
the number of 238 atoms can be written as
N 5
N 8
=
ρ o5
ρ o8
m 8
m 5
n/2
.
(3.60)
Even if different stages of the cascade have different values of V, S, T or t, (3.60)
will still be correct as it is formulated as a ratio, and those quantities will cancel at
each stage since they apply equally to each isotope.
Since m 8 > m 5 , (3.60) indicates that the ratio N 5 /N 8 grows with each stage.
However, the amount of enrichment achieved at each stage is tiny: If we start with
uranium of natural isotopic composition and ignore the small natural abundance of
234 U, ρ o5 /ρ o8 = 0.0072/0.9928 = 7.25 × 10
−3 , and, with uranium hexafluoride,
m 8 /m 5 = 1.0086.
The extent of enrichment is usually quantified by the percentage of
235 U. If we
define x = N 5 /N 8 = ρ 5 /ρ 8 , then
% (235) = 100
x
x + 1
.
(3.61)
Figure 3.12 shows the run of percent
235 U as a function of the number of diffusion
stages, assuming that one starts with uranium of natural isotopic composition.
Bomb grade
235 U is usually considered to be reached at 90% enrichment (x = 9),
which requires n = 1665. In the case of 1000 stages, 34% enrichment can be realized,
whereas 50% enrichment requires n = 1151. The K-25 plant comprised 2892 stages,
which would theoretically have realized 99.94% enrichment, but in actuality the feed
material was input about one-third of the way along the cascade so that “depleted”
uranium hexafluoride could be recycled to preceding stages. Figure 3.13 shows a
schematic illustration of a section of a diffusion “cascade”.
At Oak Ridge, uranium went through various stages of enrichment in various
facilities as they were brought into service. When all enrichment methods had come
on-line by the spring of 1945, natural-abundance uranium hexafluoride was first
fed into the liquid thermal diffusion plant (code-named S-50), which enriched the
141
N 2 = C
ρenter
stage 2
√
m
=
C
2
V
ρ o
√
m
2 .
(3.58)
Propagating this logic shows that after a total of n successive stages, the number
of atoms that emerge from the n’th stage will be
N n =
C
n
V n−1
ρ o
√
m
n .
(3.59)
If the gas consists of a mixture of two isotopes, say
235 U and
238 U, (3.59) will
apply to each according as the relevant values of ρ o and m. If we designate the two
isotopes with subscripts 5 and 8, then the final ratio of the number of 235 atoms to
the number of 238 atoms can be written as
N 5
N 8
=
ρ o5
ρ o8
m 8
m 5
n/2
.
(3.60)
Even if different stages of the cascade have different values of V, S, T or t, (3.60)
will still be correct as it is formulated as a ratio, and those quantities will cancel at
each stage since they apply equally to each isotope.
Since m 8 > m 5 , (3.60) indicates that the ratio N 5 /N 8 grows with each stage.
However, the amount of enrichment achieved at each stage is tiny: If we start with
uranium of natural isotopic composition and ignore the small natural abundance of
234 U, ρ o5 /ρ o8 = 0.0072/0.9928 = 7.25 × 10
−3 , and, with uranium hexafluoride,
m 8 /m 5 = 1.0086.
The extent of enrichment is usually quantified by the percentage of
235 U. If we
define x = N 5 /N 8 = ρ 5 /ρ 8 , then
% (235) = 100
x
x + 1
.
(3.61)
Figure 3.12 shows the run of percent
235 U as a function of the number of diffusion
stages, assuming that one starts with uranium of natural isotopic composition.
Bomb grade
235 U is usually considered to be reached at 90% enrichment (x = 9),
which requires n = 1665. In the case of 1000 stages, 34% enrichment can be realized,
whereas 50% enrichment requires n = 1151. The K-25 plant comprised 2892 stages,
which would theoretically have realized 99.94% enrichment, but in actuality the feed
material was input about one-third of the way along the cascade so that “depleted”
uranium hexafluoride could be recycled to preceding stages. Figure 3.13 shows a
schematic illustration of a section of a diffusion “cascade”.
At Oak Ridge, uranium went through various stages of enrichment in various
facilities as they were brought into service. When all enrichment methods had come
on-line by the spring of 1945, natural-abundance uranium hexafluoride was first
fed into the liquid thermal diffusion plant (code-named S-50), which enriched the