The Gamow factor, Sommerfeld factor or Gamow–Sommerfeld factor,[1] named after its discoverer George Gamow or after 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. When George Gamow instead applied quantum mechanics to the problem, he found that there was a significant chance for the fusion due to tunneling.
The probability of two nuclear particles overcoming their electrostatic barriers is given by the following equation:[2]
PG(E)=
-\sqrt{{EG | |
e |
/{E}}}
EG
EG\equiv2mrc2(\pi\alphaZa
2 | |
Z | |
b) |
Here,
mr=
mamb | |
ma+mb |
\alpha
c
Za
Zb
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.
Gamow[3] first solved the one-dimensional case of quantum tunneling using the WKB approximation. Considering a wave function of a particle of mass m, we take area 1 to be where a wave is emitted, area 2 the potential barrier which has height V and width l (at
0<x<l
\Psi1=Aei(kx+\alpha)e-i{Et/{\hbar}}
\Psi2=B1e-k'x+B2ek'x
\Psi3=(C1e-i(kx+\beta)+C2ei(kx+\beta'))e-i{Et/{\hbar}}
k=\sqrt{2mE}
x=0
x=l
\Psi
k'l\gg1
B1,B2 ≈ A
C1,C2 ≈
1 | A ⋅ | |
2 |
k' | |
k |
⋅ ek'l
Next Gamow modeled the alpha decay as a symmetric one-dimensional problem, with a standing wave between two symmetric potential barriers at
q0<x<q0+l
-(q0+l)<x<-q0
q0
x=0
Due to the symmetry of the problem, the emitting waves on both sides must have equal amplitudes (A), but their phases (α) may be different. This gives a single extra parameter; however, gluing the two solutions at
x=0
\Psi3
q0
kx
The physical meaning of this is that the standing wave in the middle decays; the emitted waves newly emitted have therefore smaller amplitudes, so that their amplitude decays in time but grows with distance. The decay constant, denoted λ, is assumed small compared to
E/\hbar
λ can be estimated without solving explicitly, by noting its effect on the probability current conservation law. Since the probability flows from the middle to the sides, we have:
\partial | |
\partialt |
(q0+l) | |
\int | |
-(q0+l) |
\Psi*\Psi dx=2 ⋅
\hbar | |
2mi |
* | |
\left(\Psi | |
1 |
\partial\Psi1 | |
\partialx |
-\Psi1
| |||||||||
\partialx |
\right),
Taking
\Psi\sime-λ
λ ⋅
1 | |
4 |
⋅ 2(q0+l)A2
k'2 | |
k2 |
⋅ e2k'l ≈ 2
\hbar | |
m |
A2k,
k'l
λ ≈
\hbark | |
m(q0+l) |
k2 | |
k'2 |
⋅ e-2k'l
λ ≈
\hbark | |
m2(q0+l) |
⋅ 8
E | |
V-E |
⋅ e-2\sqrt{2m(V-E)l/\hbar}
\hbark | |
m |
Finally, moving to the three-dimensional problem, the spherically symmetric Schrödinger equation reads (expanding the wave function
\psi(r,\theta,\phi)=\chi(r)u(\theta,\phi)
\hbar2 | \left( | |
2m |
d2\chi | |
dr2 |
+
2 | |
r |
d\chi | |
dr |
\right)=\left(V(r)+
\hbar2 | |
2m |
n(n+1) | |
r2 |
-E\right)\chi
n>0
\sqrt{V-E}
n=0
\chi(r)=\Psi(r)/r
The main effect of this on the amplitudes is that we must replace the argument in the exponent, taking an integral of over the distance where
V(r)>E
V(r)=
z(Z-z)e2 | |
4\pi\varepsilon0r |
\varepsilon0
r2=
z(Z-z)e2 | |
4\pi\varepsilon0E |
r1
2 | \sqrt{2mE |
t=\sqrt{r/r2}
t=cos(\theta)
2 ⋅
r | ||||
|
x=r1/r2
Gamow assumed
x\ll1
\pi/2
λ\sim
-\sqrt{{Eg | |
e |
/{E}}}
Eg=
2\pi2m\left[z(Z-z)e2\right]2 | |
4\pi\varepsilon0\hbar2 |
Za=z
Zb=Z-z
\alpha=
e2 | |
4\pi\varepsilon0\hbarc |
For a radium alpha decay, Z = 88, z = 2 and m = 4mp, EG is approximately 50 GeV. Gamow calculated the slope of
log(λ)