214
6 Appendices
z
dV
r
R
L
x
y
θ
Fig. 6.6 Neutrons escaping from a small volume dV within a bomb core. A neutron begins at
position r. R is its position when it reaches the surface of the core. Vector L, the neutron’s straightline path, goes from the volume element to the edge of the sphere in a direction defined by sphericalcoordinate angles (θ, φ) as shown in Fig. 6.7
dV
L
θ
φ
x
y
z
Fig. 6.7 Detailed view of vector L of Fig. 6.6
will be e
−σ nL times P(L)dL. The second step is to integrate this product over all
possible values of L to determine the average escape probability.
To begin the first step, apply the law of cosines to the triangle r-L-R:
R
2
= r
2
+ L
2
+ 2r L cos θ.
(6.72)
It is convenient to develop this derivation in terms of dimensionless variables. To
this end, define
6 Appendices
z
dV
r
R
L
x
y
θ
Fig. 6.6 Neutrons escaping from a small volume dV within a bomb core. A neutron begins at
position r. R is its position when it reaches the surface of the core. Vector L, the neutron’s straightline path, goes from the volume element to the edge of the sphere in a direction defined by sphericalcoordinate angles (θ, φ) as shown in Fig. 6.7
dV
L
θ
φ
x
y
z
Fig. 6.7 Detailed view of vector L of Fig. 6.6
will be e
−σ nL times P(L)dL. The second step is to integrate this product over all
possible values of L to determine the average escape probability.
To begin the first step, apply the law of cosines to the triangle r-L-R:
R
2
= r
2
+ L
2
+ 2r L cos θ.
(6.72)
It is convenient to develop this derivation in terms of dimensionless variables. To
this end, define