Gamov formula

From the Gamov formula, calculate the probabilities for an electron and for a proton to tunnel through barriers of 1 and 10 A thickness with a height of 1 eV. 

In the deep tunneling limit E Vmax, i-e-> kl > this becomes the same as the Gamov formula. On the other hand, at > Vmax, e(= e ) is positive [see Equation (2.17)1 and the transmission probability becomes... 

This is the well-known Gamov formula of tunneling probability. It is now well understood that the connection formulas of Equations (2.49) and (2.50) are crucial. These formulas can be obtained from the Airy function, since the potential in the vicinity of the turning point can be approximated by a linear function of x for which the Airy function gives the exact analytical solution. 

The Gamov formula derived above is a deep tunneling approximation at Emax. where is the potential barrier top. Actually, Equation (2.55) gives 1.0 at = Vmax, but the correct transmission probability at A Enax is 1 /2. This error is obvious, since the linear potential approximation does not hold in the vicinity of the potential barrier top. It is better to use a quadratic potential approximation. Instead... 

This formula, aside from the prefactor, is simply a one-dimensional Gamov factor for tunneling in the barrier shown in fig. 12. The temperature dependence of k, being Arrhenius at high temperatures, levels off to near the cross-over temperature which, for A = 0, is equal to ... 

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