Gamow factor

The Gamow factor, Sommerfeld factor or Gamow–Sommerfeld factor, named after physicists George Gamow and Arnold Sommerfeld, is a probability factor for two nuclear particles' chance of overcoming the Coulomb barrier in order to undergo nuclear reactions, for example in nuclear fusion. By classical physics, there is almost no possibility for protons to fuse by crossing each other's Coulomb barrier at temperatures commonly observed to cause fusion, such as those found in the Sun. In 1927 it was discovered that there is a significant chance for nuclear fusion due to quantum tunnelling.

While the probability of overcoming the Coulomb barrier increases rapidly with increasing particle energy, for a given temperature, the probability of a particle having such an energy falls off very fast, as described by the Maxwell–Boltzmann distribution. Gamow found that, taken together, these effects mean that for any given temperature, the particles that fuse are mostly in a temperature-dependent narrow range of energies known as the Gamow window. The maximum of the distribution is called the Gamow peak.

Description

When two positively charged nuclei approach each other, they are repulsed by the strong electric field between them – the Coulomb barrier. In order to undergo a nuclear reaction, the nuclei must quantum-tunnel through the barrier. The probability of this happening is proportional to the following factor:

P

G

(

E

)

∝

e

−

E

G

/

E

,

{\displaystyle P_{\text{G}}(E)\propto e^{-{\sqrt {{E_{\text{G}}}/{E}}}},}

where

E

G

{\displaystyle E_{\text{G}}}

is the Gamow energy

E

G

≡

2

μ

c

2

(

π

α

Z

a

Z

b

)

2

,

{\displaystyle E_{\text{G}}\equiv 2\mu c^{2}(\pi \alpha Z_{\text{a}}Z_{\text{b}})^{2},}

where

μ

=

m

a

m

b

m

a

+

m

b

{\displaystyle \mu ={\frac {m_{\text{a}}m_{\text{b}}}{m_{\text{a}}+m_{\text{b}}}}}

is the reduced mass of the two particles. The constant

α

{\displaystyle \alpha }

is the fine-structure constant,

c

{\displaystyle c}

is the speed of light, and

Z

a

{\displaystyle Z_{\text{a}}}

and

Z

b

{\displaystyle Z_{\text{b}}}

are the respective atomic numbers of each particle.