164
4 Complicating Factors
formula for predicting the neutron-generation rate for some impurity. For sake of
definiteness, I have in mind beryllium as the impurity.
The yield y of a reaction can be understood as follows. Suppose that one has
a well-mixed sample of Be and some alpha emitter such as uranium, plutonium,
radium, or polonium. Not all of the emitted alphas will find a Be nucleus to react
with; atoms are mostly empty space. The yield of the reaction is the number of
neutrons produced per alpha emitted. From figures given by Fermi (1950, p. 179), 1
Curie (Ci) of radium well-mixed with beryllium yields about 10–15 × 10
6 neutrons
per second, whereas 1 Ci of polonium well-mixed with Be yields some 2.8 × 10
6
neutrons per second. (Both Ra and Po are alpha-emitters.) With 1 Ci = 3.7 × 10
10
s
−1 , these figures correspond to yields of 2.7–4.1 × 10
–4 and 7.6 × 10
–5 , respectively.
Radium and polonium alphas have energies of about 4.8 and 5.3 MeV. More energetic
alphas actually give lower yield due to the fact that a higher-energy particle will have
a longer range of travel in some material before being consumed in a reaction; this
is discussed further in the next paragraph. Plutonium alphas have energies of about
5.2 MeV, so we might expect a yield for Pu-alphas on Be somewhere between these
two results, say y ~ 10
–4 . This is in the ballpark: West and Sherwood (1982) give the
neutron yield of 5.2-MeV alphas on
9 Be as 6.47 × 10
–5 .
As the terminology suggests, the stopping power S of a material is a measure of
how effective the material is at stopping particles that are traveling through it. As
you might infer, the range of a particle in some material is inversely proportional to
its stopping power. Empirically, the Bragg-Kleeman rule (Evans 1955, p. 652) states
that stopping power is proportional to the mass density of the material and inversely
proportional to the square root of its atomic weight:
S ∝
ρ
√
A
.
(4.28)
Suppose that one has a mixture of two materials, A and B, each with their own
stopping power for alpha particles, S A and S B . If S B > S A , an alpha will have a
greater probability of reacting with a nucleus of material B than one of material A,
presumably in the proportion S B /S A . We will use stopping power as a measure of
relative amounts of “reactivity” of the two materials. For the impurity, the density to
be used will not be its “normal” density, but rather that given by its hopefully small
mass distributed throughout the volume of the core.
Now consider a bomb core made of a heavy fissile material of atomic weight A H
and density ρ H along with an admixture of some light-element impurity of atomic
weight A L and density ρ L as defined above. We presume that the amount of impurity
is so slight that ρ H will be essentially the “normal” density for the core material. Also,
let the nuclear number densities of the two materials be n H and n L , respectively; the
goal here is to get an expression for the tolerable limit on n L /n H . If V is the volume
of the core, the mass of the impurity will be n L A L V /N A , and its mass density will be
n L A L /N A (N A = Avogadro’s number). This will give a stopping power S L according
as
4 Complicating Factors
formula for predicting the neutron-generation rate for some impurity. For sake of
definiteness, I have in mind beryllium as the impurity.
The yield y of a reaction can be understood as follows. Suppose that one has
a well-mixed sample of Be and some alpha emitter such as uranium, plutonium,
radium, or polonium. Not all of the emitted alphas will find a Be nucleus to react
with; atoms are mostly empty space. The yield of the reaction is the number of
neutrons produced per alpha emitted. From figures given by Fermi (1950, p. 179), 1
Curie (Ci) of radium well-mixed with beryllium yields about 10–15 × 10
6 neutrons
per second, whereas 1 Ci of polonium well-mixed with Be yields some 2.8 × 10
6
neutrons per second. (Both Ra and Po are alpha-emitters.) With 1 Ci = 3.7 × 10
10
s
−1 , these figures correspond to yields of 2.7–4.1 × 10
–4 and 7.6 × 10
–5 , respectively.
Radium and polonium alphas have energies of about 4.8 and 5.3 MeV. More energetic
alphas actually give lower yield due to the fact that a higher-energy particle will have
a longer range of travel in some material before being consumed in a reaction; this
is discussed further in the next paragraph. Plutonium alphas have energies of about
5.2 MeV, so we might expect a yield for Pu-alphas on Be somewhere between these
two results, say y ~ 10
–4 . This is in the ballpark: West and Sherwood (1982) give the
neutron yield of 5.2-MeV alphas on
9 Be as 6.47 × 10
–5 .
As the terminology suggests, the stopping power S of a material is a measure of
how effective the material is at stopping particles that are traveling through it. As
you might infer, the range of a particle in some material is inversely proportional to
its stopping power. Empirically, the Bragg-Kleeman rule (Evans 1955, p. 652) states
that stopping power is proportional to the mass density of the material and inversely
proportional to the square root of its atomic weight:
S ∝
ρ
√
A
.
(4.28)
Suppose that one has a mixture of two materials, A and B, each with their own
stopping power for alpha particles, S A and S B . If S B > S A , an alpha will have a
greater probability of reacting with a nucleus of material B than one of material A,
presumably in the proportion S B /S A . We will use stopping power as a measure of
relative amounts of “reactivity” of the two materials. For the impurity, the density to
be used will not be its “normal” density, but rather that given by its hopefully small
mass distributed throughout the volume of the core.
Now consider a bomb core made of a heavy fissile material of atomic weight A H
and density ρ H along with an admixture of some light-element impurity of atomic
weight A L and density ρ L as defined above. We presume that the amount of impurity
is so slight that ρ H will be essentially the “normal” density for the core material. Also,
let the nuclear number densities of the two materials be n H and n L , respectively; the
goal here is to get an expression for the tolerable limit on n L /n H . If V is the volume
of the core, the mass of the impurity will be n L A L V /N A , and its mass density will be
n L A L /N A (N A = Avogadro’s number). This will give a stopping power S L according
as