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6 Appendices
Fig. 6.10 Spherical shell of
material of inner radius r and
thickness dr
r
r + dr
⎛
⎜
⎝
net rate
o f neutrons
entering shell
⎞
⎟
⎠ = 2 π v r
(r + dr)
2
∂ N
∂r
r +dr
− r
2
∂ N
∂r
r
.
(6.100)
On expanding out the factor of (r + dr)
2 and writing
∂ N
∂r
r +dr
=
∂ N
∂r
r
+
∂
2 N
∂r 2
r
dr,
(6.101)
one arrives at
net rate o f neutrons
entering shell
= 2 π v r
r
2
∂
2 N
∂r 2
r
+ 2r
∂ N
∂r
r
dr
+2r
∂
2 N
∂r 2
r
dr
2
+
∂ N
∂r
r
dr
2
+
∂
2 N
∂r 2
r
dr
3
.
(6.102)
Now recall the Laplacian operator in spherical coordinates:
∇
2 N
r
=
1
r 2
r
2
∂
2 N
∂r 2
r
+ 2r
∂ N
∂r
r
.
(6.103)
But for a factor of 1/r
2 , this is exactly the square-bracketed term in (6.102). Hence
we can write