3.4 Electromagnetic Separation of Isotopes
137
of TNT—some 100 times the yield of Little Boy itself! In the summer of 1945, Oak
Ridge was consuming close to 1% of all the electricity being generated in the United
States.
3.5 Gaseous (Barrier) Diffusion
Like electromagnetic separation, gaseous diffusion played a central role in enriching
uranium for the Little Boy fission bomb. The physical principle utilized in this facility,
which was code-named K-25, was that when a gas of mixed isotopic composition
is pumped against a barrier made of a mesh of millions of tiny holes, atoms of the
lighter isotope will tend to diffuse through the barrier slightly more readily than those
of the heavier one (Strictly, the correct name for this process is effusion). The gas
on the other side of the barrier, slightly enriched in the lighter isotope, is collected
with a vacuum pump. However, the enrichment realizable through any one stage of
barrier is limited by the relative masses of the two isotopes; the process must be
repeated hundreds or thousands of times to achieve significant overall enrichment.
In the case of uranium this is particularly so as the isotopes differ in mass by only
about 1.3%. In fact, the input material to the K-25 plant was uranium hexafluoride
gas, for which the isotopes differ by <1% in mass:
235 UF 6 has atomic weight 349,
while that of
238 UF 6 is 352.
In view of the importance of gaseous diffusion to the success of the Manhattan
Project, the physics of this process is derived here from first principles.
We begin with a result from classical thermodynamics. Suppose that we are
dealing with a gas of atoms, each of mass m trapped in a container at absolute
temperature T. According to the Maxwell-Boltzmann distribution, the mean atomic
speed is given by
v =
8 k B T
π m
.
(3.48)
We can imagine all atoms to have this speed, racing about in all possible directions.
As shown in Fig. 3.9, imagine an abstract three-dimensional space where the axes
are the (x, y, z) components of an atom’s velocity. The magnitude of the velocity
vector v shown in the diagram is v and its direction is given by spherical coordinates
(θ, φ).
If there is no preferred direction of motion, then any direction of travel (θ, φ) must
be as probable as any other. The solid angle subtended by angular limits θ to θ + dθ
and φ to φ + dφ is dΩ = sinθ dθ dφ; if (θ, φ) are measured in radians then the solid
angle is said to be measured in steradians. Integrating overall all possible directions
[θ = (0, π ); φ = (0, 2π )] shows that the total available solid angle is 4π steradians.
The probability that any atom chosen at random is moving in the direction of a
particular solid angle dΩ is then given by P(dΩ) = dΩ/4π, that is,
137
of TNT—some 100 times the yield of Little Boy itself! In the summer of 1945, Oak
Ridge was consuming close to 1% of all the electricity being generated in the United
States.
3.5 Gaseous (Barrier) Diffusion
Like electromagnetic separation, gaseous diffusion played a central role in enriching
uranium for the Little Boy fission bomb. The physical principle utilized in this facility,
which was code-named K-25, was that when a gas of mixed isotopic composition
is pumped against a barrier made of a mesh of millions of tiny holes, atoms of the
lighter isotope will tend to diffuse through the barrier slightly more readily than those
of the heavier one (Strictly, the correct name for this process is effusion). The gas
on the other side of the barrier, slightly enriched in the lighter isotope, is collected
with a vacuum pump. However, the enrichment realizable through any one stage of
barrier is limited by the relative masses of the two isotopes; the process must be
repeated hundreds or thousands of times to achieve significant overall enrichment.
In the case of uranium this is particularly so as the isotopes differ in mass by only
about 1.3%. In fact, the input material to the K-25 plant was uranium hexafluoride
gas, for which the isotopes differ by <1% in mass:
235 UF 6 has atomic weight 349,
while that of
238 UF 6 is 352.
In view of the importance of gaseous diffusion to the success of the Manhattan
Project, the physics of this process is derived here from first principles.
We begin with a result from classical thermodynamics. Suppose that we are
dealing with a gas of atoms, each of mass m trapped in a container at absolute
temperature T. According to the Maxwell-Boltzmann distribution, the mean atomic
speed is given by
v =
8 k B T
π m
.
(3.48)
We can imagine all atoms to have this speed, racing about in all possible directions.
As shown in Fig. 3.9, imagine an abstract three-dimensional space where the axes
are the (x, y, z) components of an atom’s velocity. The magnitude of the velocity
vector v shown in the diagram is v and its direction is given by spherical coordinates
(θ, φ).
If there is no preferred direction of motion, then any direction of travel (θ, φ) must
be as probable as any other. The solid angle subtended by angular limits θ to θ + dθ
and φ to φ + dφ is dΩ = sinθ dθ dφ; if (θ, φ) are measured in radians then the solid
angle is said to be measured in steradians. Integrating overall all possible directions
[θ = (0, π ); φ = (0, 2π )] shows that the total available solid angle is 4π steradians.
The probability that any atom chosen at random is moving in the direction of a
particular solid angle dΩ is then given by P(dΩ) = dΩ/4π, that is,