Gamow factor

The Gamow factor or Gamow–Sommerfeld factor, named after its discoverer George Gamow, 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:
where formula_2 is the Gamow energy,
Here, formula_4 is the reduced mass of the two particles. The constant formula_5 is the fine structure constant, formula_6 is the speed of light, and formula_7 and formula_8 are the respective atomic numbers of each particle.
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.
Derivation.
Gamow 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 formula_9), and area 3 its other side, where the wave is arriving, partly transmitted and partly reflected. For a wave number "k" and energy "E" we get:
where formula_13 and formula_14.
This is solved for given "A" and "α" by taking the boundary conditions at the both barrier edges, at formula_15 and formula_16, where both formula_17 and its derivative must be equal on both sides.
For formula_18, this is easily solved by ignoring the time exponential and considering the real part alone (the imaginary part has the same behavior). We get, up to factors depending on the phases which are typically of order 1, and up to factors of the order of formula_19 (assumed not very large, since "V" is greater than "E" not marginally):
Next Gamow modeled the alpha decay as a symmetric one-dimensional problem, with a standing wave between two symmetric potential barriers at formula_22 and formula_23, and emitting waves at both outer sides of the barriers.
Solving this can in principle be done by taking the solution of the first problem, translating it by formula_24 and gluing it to an identical solution reflected around formula_15.
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 formula_15 requires two boundary conditions (for both the wave function and its derivative), so in general there is no solution. In particular, re-writing formula_27 (after translation by formula_24) as a sum of a cosine and a sine of formula_29, each having a different factor that depends on "k" and "α", the factor of the sine must vanish, so that the solution can be glued symmetrically to its reflection. Since the factor is in general complex (hence its vanishing imposes two constraints, representing the two boundary conditions), this can in general be solved by adding an imaginary part of "k", which gives the extra parameter needed. Thus "E" will have an imaginary part as well.
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 formula_30.
"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:
Note the factor of 2 is due to having two emitted waves.
Taking formula_32, this gives:
Since the quadratic dependence in formula_34 is negligible relative to its exponential dependence, we may write:
Remembering the imaginary part added to "k" is much smaller than the real part, we may now neglect it and get:
Note that formula_37 is the particle velocity, so the first factor is the classical rate by which the particle trapped between the barriers hits them.
Finally, moving to the three-dimensional problem, the spherically symmetric Schrödinger equation reads (expanding the wave function formula_38 in spherical harmonics and looking at the n-th term):
Since formula_40 amounts to enlarging the potential, and therefore substantially reducing the decay rate (given its exponential dependence on formula_41), we focus on formula_42, and get a very similar problem to the previous one with formula_43, except that now the potential as a function of "r" is not a step function.
The main effect of this on the amplitudes is that we must replace the argument in the exponent, taking an integral of formula_44 over the distance where formula_45 rather than multiplying by "l". We take the Coulomb potential:
where formula_47 is the Coulomb constant, "e" the electron charge, "z" = 2 is the charge number of the alpha particle and "Z" the charge number of the nucleus ("Z"-"z" after emitting the particle). The integration limits are then formula_48, where we assume the nuclear potential energy is still relatively small, and formula_49, which is where the nuclear negative potential energy is large enough so that the overall potential is smaller than "E". Thus, the argument of the exponent in λ is:
This can be solved by substituting formula_51 and then formula_52 and solving for θ, giving:
where formula_54.
Since "x" is small, the "x"-dependent factor is of order 1.
Gamow assumed formula_55, thus replacing the "x"-dependent factor by formula_56, giving:
formula_57
with:
which is the same as the formula given in the beginning of the article with formula_59, formula_60
and the fine structure constant formula_61.
For a radium alpha decay, "Z" = 88, "z" = 2 and "m" = 4"m""p", "E""G" is approximately 50 GeV. Gamow calculated the slope of formula_62 with respect to "E" at an energy of 5 MeV to be ~1014 joule−1, compared to the experimental value of formula_63 joule−1.