View source for Nuclear Bomb Physics

== Nuclear Bomb Physics ==

'''Nuclear bomb physics''' deals with the principles, mathematics, and engineering behind nuclear weapons, which unleash enormous energy by triggering extremely rapid, uncontrolled nuclear chain reactions. These bombs rely primarily on nuclear '''fission''' and '''fusion''' to release energy measured in kilotons to megatons of TNT equivalent.

== Fundamental Concepts ==

=== Energy from Fission ===

Each fission event of <math>{}^{235}U</math> or <math>{}^{239}Pu</math> releases about 200 MeV of energy:

<cmath>
E_{fission} \approx 200 \text{ MeV} = 3.2 \times 10^{-11} \text{ J}
</cmath>

If <math>N_f</math> nuclei undergo fission, total energy released is:

<cmath>
E_{total} = N_f \times E_{fission}
</cmath>

For example, 1 gram of <math>{}^{235}U</math> contains approximately

<cmath>
N = \frac{6.022 \times 10^{23}}{235} \approx 2.56 \times 10^{21} \text{ atoms}
</cmath>

If 1% fissions:

<cmath>
E = 0.01 \times N \times 3.2 \times 10^{-11} \approx 8.2 \times 10^{8} \text{ J}
</cmath>

This is equivalent to about 200 tons of TNT (1 ton TNT = <math>4.184 \times 10^{9}</math> J).

=== Chain Reaction Growth ===

The number of neutrons (and fissions) grows exponentially during detonation:

<cmath>
N(t) = N_0 e^{\alpha t}
</cmath>

where <math>\alpha</math> is the neutron population growth rate, related to neutron generation time <math>\Lambda</math> and multiplication factor <math>k</math> by:

<cmath>
\alpha = \frac{k - 1}{\Lambda}
</cmath>

Typical neutron generation time <math>\Lambda</math> in a bomb core is on the order of <math>10^{-8}</math> seconds, and <math>k</math> can be <math>>2</math>, so <math>\alpha</math> can be <math>\sim 10^{8} \text{ s}^{-1}</math>, meaning neutron numbers can double every nanosecond.

=== Critical and Supercritical Mass ===

The '''critical mass''' <math>M_c</math> is the minimum fissile material mass for sustaining a chain reaction:

<cmath>
k = 1 \quad \Rightarrow \quad \text{steady neutron population}
</cmath>

Above critical mass, the system becomes supercritical (<math>k > 1</math>) and reaction grows explosively.

The multiplication factor depends on geometry, density <math>\rho</math>, neutron reflection, and fissile purity:

<cmath>
k = \eta f p \epsilon \times \text{(geometry factor)}
</cmath>

Increasing density via implosion reduces critical mass:

<cmath>
M_c \propto \frac{1}{\rho^2}
</cmath>

=== Assembly Methods ===

'''Gun-type assembly''': Rapidly shoots one subcritical mass into another, forming a supercritical assembly. This method requires less precision but can only be used with uranium-235 due to spontaneous fission constraints.

'''Implosion-type assembly''': Uses shaped conventional explosives to symmetrically compress a subcritical plutonium or uranium core, increasing density and making it supercritical. The compression factor <math>C</math> increases density:

<cmath>
\rho_{compressed} = C \times \rho_0,
</cmath>

reducing critical mass by

<cmath>
M_c \propto \frac{1}{C^2}.
</cmath>

=== Fusion Boosting and Thermonuclear Stages ===

Fusion reactions add tremendous yield:

<cmath>
{}^{2}H + {}^{3}H \rightarrow {}^{4}He + n + 17.6 \text{ MeV}
</cmath>

Neutrons from fusion can cause additional fission in a uranium tamper:

<cmath>
E_{fusion} \sim 10 \times E_{fission} \text{ per reaction chain}
</cmath>

=== Neutron Diffusion and Leakage ===

Neutron leakage reduces efficiency. The neutron diffusion length <math>L</math> affects neutron retention:

<cmath>
L = \sqrt{\frac{D}{\Sigma_a}},
</cmath>

where

* <math>D</math> = diffusion coefficient,
* <math>\Sigma_a</math> = macroscopic absorption cross-section.

Optimizing geometry minimizes leakage and maximizes <math>k</math>.

=== Time Scale of Explosion ===

With neutron generation time <math>\Lambda \approx 10^{-8}</math> s and initial neutrons <math>N_0 \sim 1</math>, after <math>n</math> generations over time <math>t = n \Lambda</math>:

<cmath>
N(t) = e^{\alpha t} = 2^{t / t_d}
</cmath>

where doubling time

<cmath>
t_d = \frac{\ln 2}{\alpha} \sim 10^{-9} \text{ s}.
</cmath>

The chain reaction completes in microseconds, releasing energy almost instantaneously.

=== Yield Estimates ===

Energy yield <math>Y</math> in kilotons of TNT:

<cmath>
Y \text{ (kt)} = \frac{E_{total}}{4.184 \times 10^{12} \text{ J/kt}}
</cmath>

For a bomb with 10 kg of fissioned material:

<cmath>
N_f = \frac{6.022 \times 10^{23}}{235} \times 10 = 2.56 \times 10^{25}
</cmath>

Energy:

<cmath>
E = N_f \times 3.2 \times 10^{-11} = 8.2 \times 10^{14} \text{ J}
</cmath>

Yield:

<cmath>
Y = \frac{8.2 \times 10^{14}}{4.184 \times 10^{12}} \approx 196 \text{ kt}
</cmath>

== See Also ==

* [[Nuclear Fission]]  
* [[Nuclear Fusion]]  
* [[Critical Mass]]  
* [[Nuclear Chain Reaction]]  
* [[Nuclear Reactor Physics]]  
* [[Hydrodynamics of Explosions]]

[[Category:Physics]]
[[Category:Science]]