74
2 Critical Mass, Efficiency, and Yield
geometry? From Fig. 2.8, it can be seen directly that the answer is no. To approach
this with Fig. 2.10, we determine what tamper outer radius is necessary to achieve
criticality, and then see if 150 kg of Al is sufficient to meet that radius. 12 kg of
239 Pu
would form a sphere of radius 5.68 cm. With d core = 2.985 cm, we are asking for
R thresh /d core = 5.68/2.99 = 1.90. Reading horizontally across Fig. 2.10 at R thresh /d core
= 1.90 shows that R tamp /R thresh ~ 5, or R tamp ~ 28.4 cm. This corresponds to a tamper
mass of ~ 250 kg, so the answer is again no. (More precisely, one needs ~ 230 kg of
Al to just achieve criticality for such a core.)
One further way of plotting Eq. (2.46) is shown in Fig. 2.11: creating curves of
threshold-critical core/tamper mass combinations for given compression ratios. The
limitation of a plot like this is that it becomes too cluttered if one attempts to display
data for more than one tamper material at once.
As for using Eqs. (2.47) and (2.48): Refer to the second way of solving (2.46)
above, where R thresh is determined for a given value of R tamp . Keep R tamp fixed to its
chosen value. Now choose a core radius R core > R thresh to use in (2.47) and (2.48).
Then solve (2.47) numerically for δ (= α), which enters the d’s and x’s of (2.47) and
(2.48) through (2.40) and (2.42). The value of knowing α will become clear when
the efficiency and yield calculations of Sects. 2.5 and 2.6 are developed.
A special-case application of (2.46) can be used to get a sense of how dramatically
the presence of a tamper decreases the threshold critical mass. Suppose that the tamper
is very thick, R tamp >> R thresh . In this case (2.46) reduces to
R thresh
d core
cot
R thresh
d core
= 1 −
λ
tamp
trans
λ
core
trans
.
(2.51)
0
20
40
60
80
100
120
5
1 0
1 5
2 0
2 5
3 0
3 5
4 0
4 5
Tamper mass (kg)
Core mass (kg)
U-235 + WC
C = 1
1.25
1.5
1.1
1.75
2
Fig. 2.11 Core/tamper mass combinations that are threshold critical for various compression ratios
(C = 1, 1.1, 1.25, 1.5, 1.75, 2) for a 235 U core and WC tamper
2 Critical Mass, Efficiency, and Yield
geometry? From Fig. 2.8, it can be seen directly that the answer is no. To approach
this with Fig. 2.10, we determine what tamper outer radius is necessary to achieve
criticality, and then see if 150 kg of Al is sufficient to meet that radius. 12 kg of
239 Pu
would form a sphere of radius 5.68 cm. With d core = 2.985 cm, we are asking for
R thresh /d core = 5.68/2.99 = 1.90. Reading horizontally across Fig. 2.10 at R thresh /d core
= 1.90 shows that R tamp /R thresh ~ 5, or R tamp ~ 28.4 cm. This corresponds to a tamper
mass of ~ 250 kg, so the answer is again no. (More precisely, one needs ~ 230 kg of
Al to just achieve criticality for such a core.)
One further way of plotting Eq. (2.46) is shown in Fig. 2.11: creating curves of
threshold-critical core/tamper mass combinations for given compression ratios. The
limitation of a plot like this is that it becomes too cluttered if one attempts to display
data for more than one tamper material at once.
As for using Eqs. (2.47) and (2.48): Refer to the second way of solving (2.46)
above, where R thresh is determined for a given value of R tamp . Keep R tamp fixed to its
chosen value. Now choose a core radius R core > R thresh to use in (2.47) and (2.48).
Then solve (2.47) numerically for δ (= α), which enters the d’s and x’s of (2.47) and
(2.48) through (2.40) and (2.42). The value of knowing α will become clear when
the efficiency and yield calculations of Sects. 2.5 and 2.6 are developed.
A special-case application of (2.46) can be used to get a sense of how dramatically
the presence of a tamper decreases the threshold critical mass. Suppose that the tamper
is very thick, R tamp >> R thresh . In this case (2.46) reduces to
R thresh
d core
cot
R thresh
d core
= 1 −
λ
tamp
trans
λ
core
trans
.
(2.51)
0
20
40
60
80
100
120
5
1 0
1 5
2 0
2 5
3 0
3 5
4 0
4 5
Tamper mass (kg)
Core mass (kg)
U-235 + WC
C = 1
1.25
1.5
1.1
1.75
2
Fig. 2.11 Core/tamper mass combinations that are threshold critical for various compression ratios
(C = 1, 1.1, 1.25, 1.5, 1.75, 2) for a 235 U core and WC tamper