2.2 Critical Mass: Bare Core
65
function then allows one to solve (2.30) and (2.31) for x (assuming α = 0), from
which the bare critical radius and mass are computed.
In practice, having available only a single critical mass of fissile material will
not produce much of an explosion. The reason for this is that fissioning nuclei give
rise to fission products with tremendous kinetic energies. The core consequently
very rapidly—within microseconds—heats up and expands, causing its density to
drop below that necessary to maintain criticality. In a core comprising only a single
critical mass, this will happen at the moment fissions begin, so the chain reaction will
quickly fizzle as α falls below zero. To get an explosion of appreciable efficiency,
one must start with more than a single critical mass of fissile material, or implode an
initially subcritical mass to high density before initiating the explosion. The issue of
using more than one critical mass to enhance weapon efficiency is examined in more
detail in Sects. 2.5 and 2.6. The effect of using a tamper is examined analytically in
Sect. 2.3, and numerically in Sect. 2.6.
To determine the value of the exponential growth factor α for a core of more
than one critical mass, it is necessary to solve Eqs. (2.26), (2.30), and (2.31) for α
as described following (2.31) above. For the purpose of generating a seed value or
simply for making quick estimates, however, an approximate value can be obtained
as follows.
Equation (2.28) for the radial dependence of the neutron density appears as
N r (r ) = A
sin x
x
.
(2.32)
As a simplified boundary condition, assume that N r (R core ) = 0, that is, that the
neutron density falls to zero at the edge of the core. This is a more restrictive condition
that the true boundary condition, (2.29), and will lead to a larger bare threshold critical
radius. In this case, (2.32) indicates that we must have sin(x) = 0, or R/d = π. This
will be the case whether a core is supercritical or just threshold critical. If we use
subscripts “core” and “o” to designate a supercritical and bare-threshold-critical core,
respectively, then we must have
R core
d core
=
R o
d o
⇒
R o
R core
2
=
d o
d core
2
.
(2.33)
Substitute for d o and d core from (2.25), setting α = 0 in the expression for d o . The
result can then be solved for α core :
α core ∼ (ν − 1)
1 −
R o
R core
2
.
(2.34)
This result is expressed as an approximation as a reminder that it does not derive
from the true boundary condition for neutron diffusion. This simplified boundary
condition is explored further in Exercise 2.11.
65
function then allows one to solve (2.30) and (2.31) for x (assuming α = 0), from
which the bare critical radius and mass are computed.
In practice, having available only a single critical mass of fissile material will
not produce much of an explosion. The reason for this is that fissioning nuclei give
rise to fission products with tremendous kinetic energies. The core consequently
very rapidly—within microseconds—heats up and expands, causing its density to
drop below that necessary to maintain criticality. In a core comprising only a single
critical mass, this will happen at the moment fissions begin, so the chain reaction will
quickly fizzle as α falls below zero. To get an explosion of appreciable efficiency,
one must start with more than a single critical mass of fissile material, or implode an
initially subcritical mass to high density before initiating the explosion. The issue of
using more than one critical mass to enhance weapon efficiency is examined in more
detail in Sects. 2.5 and 2.6. The effect of using a tamper is examined analytically in
Sect. 2.3, and numerically in Sect. 2.6.
To determine the value of the exponential growth factor α for a core of more
than one critical mass, it is necessary to solve Eqs. (2.26), (2.30), and (2.31) for α
as described following (2.31) above. For the purpose of generating a seed value or
simply for making quick estimates, however, an approximate value can be obtained
as follows.
Equation (2.28) for the radial dependence of the neutron density appears as
N r (r ) = A
sin x
x
.
(2.32)
As a simplified boundary condition, assume that N r (R core ) = 0, that is, that the
neutron density falls to zero at the edge of the core. This is a more restrictive condition
that the true boundary condition, (2.29), and will lead to a larger bare threshold critical
radius. In this case, (2.32) indicates that we must have sin(x) = 0, or R/d = π. This
will be the case whether a core is supercritical or just threshold critical. If we use
subscripts “core” and “o” to designate a supercritical and bare-threshold-critical core,
respectively, then we must have
R core
d core
=
R o
d o
⇒
R o
R core
2
=
d o
d core
2
.
(2.33)
Substitute for d o and d core from (2.25), setting α = 0 in the expression for d o . The
result can then be solved for α core :
α core ∼ (ν − 1)
1 −
R o
R core
2
.
(2.34)
This result is expressed as an approximation as a reminder that it does not derive
from the true boundary condition for neutron diffusion. This simplified boundary
condition is explored further in Exercise 2.11.