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1-7: Alpha Decay of Atomic Nuclei

In the preceding section, we learned a surprising phenomenon, the tunnel effect, which is never seen in classical mechanics. If we could actually discover this phenomenon in the microscopic world, it would be strong evidence of the correctness of quantum mechanics and we could justify the probability interpretation of the wave function.
G. Gamow (Russia, USA, 1904 - 68) discovered that the alpha decay of atomic nuclei is well explained by the tunnel effect (1928). This was one of the strong indications of the success of quantum mechanics.

[Half Life of an Alpha Emitter]
An alpha particle is the nucleus of the helium atom. The radioactive decay of atoms was discussed on the page
2-1: The Discovery of Radioactivity
in the preceding Seminar (Microscopic World -1-).
When we deal with alpha decay, we sometimes use a concept named half life, which means the time in which the amount of a radioactive nuclide of a sample decays to half of its original value. The half life is very dependent on the kind of nuclide.
Geiger and others measured the half lives of various alpha radioactive atoms and discovered a close relation between the energy of the emitted alpha particles and the half life. The relation was written

where E is the energy of the emitted alpha particles and A and C are the constants to be adjusted to agree with experiment. It is clearly seen in Fig. (A) that the empirical formula (1) fits well to the experimental data.

Half lives of alpha decays
The black dots are experimental data. The solid curve shows the value of the empirical formula (1). (The constants, A and C , are adjusted to the experimental data.)

It should be noted from the experimental data in Fig. (A) that the half life of the alpha emitter changes to be shorter by a factor of about as the energy of the emitted alpha particles drops by a factor of 2.5 times. For example, the energy of the alpha particles emitted from is about 4 MeV and its half life is very long as . On the other hand, for the alpha particle energy is about 10 MeV which is about 2.5 times larger than the case of and the half life of is very short as .
Is it possible to explain this extreme difference?

[The Explanation of Alpha Decay by Gamow]
Gamow thought of the mechanism of the alpha decay as follows: The alpha decay of atoms is nothing else than the decay of an atomic nucleus by radiating alpha particles. The alpha particles in an atomic nucleus are considered to be bound in a nuclear potential as shown in the following Fig. (B).

The potential exerted to alpha particles in a nucleus.
The alpha particles in a level with negative energy cannot go out from the nucleus, but those in a positive energy level penetrate out with the tunnel effect.

The above figure is drawn in a one-dimensional space, but it must be three-dimensional in practice. However, it is somewhat troublesome to discuss in the three-dimensional space, so that we consider in a one-dimensional space (only on the x axis) here.
In Fig. (B), the region |x| < a is the inner part of a nucleus where alpha particles are bound by a strong nuclear potential. The region of |x| > a shows the Coulomb repulsive potential between the charge of an alpha particle, +2e, and the charge of the rest of the nucleus, +(Z - 2)e. This means that only the Coulomb repulsive potential works between the alpha particle and the rest of the nucleus at the region in which |x| is large enough.
The potential in Fig. (B) is extremely simplified. Although the precise potential must be more complicated, this simplified one seems sufficient to qualitatively understand the mechanism of the alpha decay.
In Fig. (B) three energy levels, , and , are drawn as examples. If the alpha particle is on the level , the energy is negative. Then it cannot go out of the region of the nucleus, and no alpha decay occurs. If the alpha particle is on the level ( > 0 ) or ( > 0 ), then the alpha particle would penetrate the potential barrier by the tunnel effect and alpha decay occurs. The question is what effect on the penetration coefficients results from the difference between the energies, and .
The half life is inversely proportional to the penetration coefficient. Because it is somewhat troublesome to estimate the penetration coefficient of the tunnel effect in such a case as the potential barrier shown in the following Fig. (C), let us substitute the following square potential for the potential barrier.

This figure is just the same as what is shown by Fig. (E) on the preceding page. For this square potential, the penetration coefficient has already been given by Eq. (16) on the last page. This is approximately proportional to

This exponential function is strongly dependent on the value aK , In this expression, a is the width of the barrier and the energy difference between the top of the barrier and the energy of the alpha particle under consideration. The penetration coefficients for the states of energies and therefore differ very much; i.e., the half life for each case varies considerably, and its energy dependence well matches the experimental data.
Carrying out a precise calculation of the penetration coefficient for such a potential as in Fig. (B), Gamow obtained the values of the constants A and C in Eq. (1) that are well suited to the experimental data. Gamow's result was one of the strong evidences of the correctness of the probability interpretation and contributed greatly to the development of quantum mechanics.
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