68
2 Critical Mass, Efficiency, and Yield
Superscripts and subscripts tamp and core will be used liberally here as it will be
necessary to join tamper physics to core physics via suitable boundary conditions.
As was done in Sect. 2.2, take a trial solution for N tamp of the form N tamp (t, r ) =
N
tamp
t
(t) N
tamp
r
(r ), where N
tamp
t
(t) and N
tamp
r
(r ) are respectively the time-and space
dependences of N tamp ; r is the usual spherical radial coordinate measured from the
center of the core. Upon substituting this into (2.35) we find, in analogy to (2.19),
1
N
tamp
t
∂ N
tamp
t
∂t
=
λ
tamp
trans v neut
3
1
N
tamp
r
1
r 2
∂
∂r
r
2 ∂ N
tamp
r
∂r
.
(2.36)
Define the separation constant here to be δ/τ, where τ is the mean time that a
neutron will travel in the core before causing a fission, that is, as defined in (2.21):
τ =
λ
core
f iss
v neut
.
(2.37)
While it may seem strange to invoke a core quantity when dealing with diffusion
in the tamper, this choice is advantageous in that the neutron velocity v neut , which
we assume to be the same in both materials, will cancel out in later algebra. This
choice is not equivalent to assuming at the outset that the core and tamper separation
constants are the same, as δ may be different from the exponential factor α of Sect. 2.2.
However, we will find that boundary conditions demand that they too must be equal.
This choice of separation constant renders (2.36) as
1
N
tamp
t
∂ N
tamp
t
∂t
=
λ
tamp
trans v neut
3
1
N
tamp
r
1
r 2
∂
∂r
r
2 ∂ N
tamp
r
∂r
=
δ
τ
. (2.38)
The solution of (2.38) depends on whether δ is positive, negative, or zero; the
latter choice corresponds to threshold criticality in analogy to α = 0 in Sect. 2.2. The
situations of practical interest will be δ > 0, in which case the solutions have the
form
N tamp =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
A
r
+ B
(δ = 0)
e
(δ/τ )t
A
e
r/dtamp
r
+ B
e
−r/dtamp
r
(δ > 0),
(2.39)
where A and B are constants of integration (different for the two cases), and where
d tamp =
λ
tamp
trans λ
core
f iss
3 δ
.
(2.40)
The situation we now have is that the neutron density in the core is described by
(2.22) and (2.28) as
2 Critical Mass, Efficiency, and Yield
Superscripts and subscripts tamp and core will be used liberally here as it will be
necessary to join tamper physics to core physics via suitable boundary conditions.
As was done in Sect. 2.2, take a trial solution for N tamp of the form N tamp (t, r ) =
N
tamp
t
(t) N
tamp
r
(r ), where N
tamp
t
(t) and N
tamp
r
(r ) are respectively the time-and space
dependences of N tamp ; r is the usual spherical radial coordinate measured from the
center of the core. Upon substituting this into (2.35) we find, in analogy to (2.19),
1
N
tamp
t
∂ N
tamp
t
∂t
=
λ
tamp
trans v neut
3
1
N
tamp
r
1
r 2
∂
∂r
r
2 ∂ N
tamp
r
∂r
.
(2.36)
Define the separation constant here to be δ/τ, where τ is the mean time that a
neutron will travel in the core before causing a fission, that is, as defined in (2.21):
τ =
λ
core
f iss
v neut
.
(2.37)
While it may seem strange to invoke a core quantity when dealing with diffusion
in the tamper, this choice is advantageous in that the neutron velocity v neut , which
we assume to be the same in both materials, will cancel out in later algebra. This
choice is not equivalent to assuming at the outset that the core and tamper separation
constants are the same, as δ may be different from the exponential factor α of Sect. 2.2.
However, we will find that boundary conditions demand that they too must be equal.
This choice of separation constant renders (2.36) as
1
N
tamp
t
∂ N
tamp
t
∂t
=
λ
tamp
trans v neut
3
1
N
tamp
r
1
r 2
∂
∂r
r
2 ∂ N
tamp
r
∂r
=
δ
τ
. (2.38)
The solution of (2.38) depends on whether δ is positive, negative, or zero; the
latter choice corresponds to threshold criticality in analogy to α = 0 in Sect. 2.2. The
situations of practical interest will be δ > 0, in which case the solutions have the
form
N tamp =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
A
r
+ B
(δ = 0)
e
(δ/τ )t
A
e
r/dtamp
r
+ B
e
−r/dtamp
r
(δ > 0),
(2.39)
where A and B are constants of integration (different for the two cases), and where
d tamp =
λ
tamp
trans λ
core
f iss
3 δ
.
(2.40)
The situation we now have is that the neutron density in the core is described by
(2.22) and (2.28) as