Nuclear Weapon Effects Equation Sheet
Maj Jeremy Best
May 4, 2015
1
Fission Yield:
t
Y (t) = Σf vQn0 T (e T − 1)
(1)
where
t is time, Q is the average prompt energy from a fission, v is average 10 n velocity (1.08 × 107 m
s ), n0 is the
number of neutrons at initiation. ν is the average # of neutrons per fission. T is the period as defined
below and NA = 6.022 × 1023
Σf = N σf
N=
ρNA
M
T =
`
k−1
t = (#generations)(lif etime) = (g)(`) ` = `∞ L
Σa = N σa
k = k∞ L
k∞ = ν
(2)
`∞ =
σf
σa
Q = −∆mc2
Energy Form
Total Prompt Energy
Total Delayed Energy
Total Burnup
νσf
σa
ν
U-235
180 MeV
27 MeV
17.7 kT/kg
3.32 (barns)
1.44 (barns)
2.43
1kT = 2.613 × 1025 M eV
1
1
Σa v
(3)
(4)
(5)
Pu-239
185 MeV
27 MeV
17.8 kT/kg
5.61 (barns)
1.98 (barns)
2.98
(6)
2
Fusion Yield:
4
Y = πR3 Qt nt
(7)
3
where R is the radius of the assembly at the time of initiation, Qt is the energy per reaction, and nt is
the number of target atoms. For pure Deuterium ⇒ nt = 12 nD .
Ignition Energy
3
4σ
U = N (1 + Z) T k V ol + ( T 4 )V ol
(8)
2
c
where N is the atomic density of the fuel, T is the temperature throughout the volume [kelvin] (assumed
constant), Z is the average number of free electrons per atom (assumed to be atomic number) at keV temJ
peratures, k is Boltzmann constant (1.3807×10−23 K
), σ is the Stefan-Boltzmann constant (5.67×10−8 KJ4
) and c is the speed of light.
Fusion microscopic cross section Reaction Rates
2
1
2
1
(σv)DT = 3.68 × 10−12 T − 3 exp −19.94T − 3 [
(σv)DD = 2.33 × 10−14 T − 3 exp −18.76T − 3 [
cm3
]
s
(9)
cm3
]
s
(10)
where T is in keV.
Fusion Reaction Rates: D+T:
N = (σv)nt nb
(11)
For D+D
1
N = (σv)n2
2
where nt is the number of target atoms, and nb is the number of bombarding atoms.
5(21 D) = (11 H +32 He +42 He) + 2(10 n)
Where one 10 n is 2.5 MeV and one 10 n is 14.1 MeV and Qavg = 9.96 MeV
2
(12)
(13)
3
Thermal Effects
Fireball Radius at breakaway: R = 110W 0.4 ft
Q=
85.6f W τ cal
D2
cm2
(14)
where D is in kilofeet and W is kT. Below 180W 0.4 use Table 7.101 to get f , above 15,000 ft use Table
7.88 to get f .
For altitudes below 15,000 ft
Pmax ≈ 3.18W 0.56
kT
s
tmax ≈ 0.0417W 0.44 s
(15)
For altitudes above 15,000 ft
Pmax
4
3.56W 0.59
= h
i0.45
ρ(h)
ρ0
0.36
(16)
Blast and Shock
P
p = p1
P0
h = h1 W
1
3
d = d1 W
Burst Alt [ft]
40,000
60,000
90,000
120,000
150,000
Alt [ft]
40,000
60,000
90,000
100,000
120,000
150,000
5
tmax = 0.038W
ρ(h)
ρ0
0.44
1
3
P0
P
( 1 )
3
t = t1 W
Blast Efficiency Factor
Upper Limit
1.0
1.0
0.9
0.7
0.4
Alt [km]
12.192
18.288
27.432
30.48
36.576
45.72
T/To
0.75186
0.75186
0.77724
0.78773
0.835946
0.92366
1
Dr = Dr1 ∗ W 3
3
P0
P
Lower Limit
0.9
0.8
0.6
0.4
0.2
P/Po
0.18576
0.07136
0.01737
0.010996
0.0045368
0.001343
Underground / Underwater
1
3
ρ/ρ0
0.24707
0.09491
0.02236
0.01396
0.005427
0.001454
( 1 ) 3
T0
T
( 1 )
2
(17)
4
5
6
7
8
9