What is tunnel effect? Discuss the Alpha decay as an example of tunnel effect. Prove that tunneling increases on decreasing the height and width of the barrier. Explain what do you mean by mean by the terms potential well and potential barrier. Obtain an expression for transmission coefficient of a rectangular potential barrier.
Where U_{0} = height of the rectangular potential barrier. Draw graphs showing variation of T with particle energy E and barrier width a.
What is quantum mechanical "tunneling" ? Give one example.
Potential Step
If a particle having energy less than V_{0} i.e. E0 approaches this barrier form the left i.e. from 1^{st} region, classically the particle will always be reflected and hence will not penetrate the barrier. However, ware mechanics predicts that the particle has some probability of penetrating to region , the probability of penetration being greater if and a are smaller. Morever, if E>V_{0} classical mechanics predicts that the particle will always be transmitted; while according to wave-mechanics, the particle has a finite probability of transmission and hence it is not certain that, the particle will penetrate the barrier. If a particle with energy E is incident on a thin energy on a thin energy barrier of height greater than E, there is a finite probability of the particle penetrating the barrier. This phenomenon is called the tunnel effect. This effect was used by George Gamow in 1928 to explain the process of decay exhibited by radioactive nuclei.
Applications of Tunnel Effect
Emission of particles from radioactive nuclei of Alpha-decay. The average energy of an particle formed within the nucleus is less than the height of the potential barrier around the nucleus which is formed by the nuclear binding forces. Classically, the particle cannot escape from the nucleus, but Quantum mechanically it tunnels the barrier. This tunneling constitutes radioactive decay. The decay of nuclei by emission of an alpha-particle can be regarded as a tunneling process. A radioactive nucleus can be thought of having an alpha particle (a helium nucleus) trapped in a spherical potential well arising out of extremely strong nuclear attractive forces between the nucleons. These attractive forces are short range. They operate on the particle so long as it is inside the nucleus. Once the alpha-particle is out of the nucleus Coulomb repulsive force operates between the positive charge of alpha-particle and the positive charge of residual nucleus. This electrical repulsion is negligible when the alpha-particle is under the effect of a strong attractive nuclear force up to a certain distance (nuclear size) after which long range coulomb repulsive force operates on it. As the exact form of nuclear forces is still unknown, the potential as seen by the alpha-particle is generally represented as shown in fig. Thus there appears a potential barrier height E < V (coulomb repulsive barrier), then according to classical mechanics such alpha-particles cannot come out of the nucleus. But because of wave nature they actually have small probability of tunneling through the barrier. The tunneling probability per unit time is equal to the number of bounces per unit time multiplied by the tunneling coefficient per of the natural radioactive nuclei is of the order of 10^{7} m/sec. and the nuclear size is of the order of 10^{-14} m, the alpha particle strikes the barrier about 10^{21 }times per second. Each time it bounces the barrier the probability that it penetrates the barrier is equal to the transmission coefficient T. Hence the tunneling probability per unit time is for a barrier much higher than the energy of the particle and the barrier width a = R _{c} -R from fig. the decay rate is thus determined predominantly by the exponential factor in T. Its value is very sensitive to actual shape of potential curve and can vary significantly in order of magnitude from nucleus to nucleus. Thus qualitatively it is possible to explain decay as an example of quantum mechanical Tunnel effect.