Wednesday, 7 May 2014

Quantum tunnelling and Gamow Energy

Quantum tunnelling:
Quantum tunnelling also known as the barrier penetration, is the key to the occurrence of fusion reactions in stars. If the particle's total energy is less than the potential energy of some barrier it cannot pass that barrier. Think about throwing a ball over a wall. Unless there is sufficient kinetic energy to attain a height greater than the wall, there is no way the ball will reach the other side. Kinetic energy of the ball, once transformed to gravitational potential energy should exceed the gravitational potential energy of the wall.
However this restriction does not apply to quantum mechanics. We have to consider the wave properties of the particle in quantum mechanics. The amplitude of particle's wave function goes to zero only for an infinitely high barrier although the wave function associated with a particle is attenuated by a potential barrier.

As we know that the finite potential barriers exist in reality, the wave function inside and beyond the barrier is non-zero. Hence the position probability density is also non-zero. (position probability density).  Therefore there is a small but non-zero probability that a particle will make its way through a barrier, in such a case and even though that may appear impenetrable or insurmountable in normal circumstances. If we think about the ball again, if a ball and the wall were small enough for quantum properties to dominate over the classical scenario, it would be possible for a ball to reach the other side. That is even if it did not have enough kinetic energy to overcome the gravitational potential energy at the wall's top. This seems strange but this is precisely what happens during collisions of nuclei.

For the nucleus to be stable, the nucleons in a nucleus sit in a potential well surrounded by a Coulomb potential barrier of finite height and width. There is a non-zero probability of a particle with energy less than the height of the barrier would make its way from outside the barrier to inside due to barrier penetration to reach the nucleus.

The Gamow energy
The  barrier potentials do vary with separation, i.e. V is a function of r. For the Coulomb barrier, the penetration probability may be expressed in terms of the particle energy E, and the Gamow energy EG which depends on the atomic number of the interacting nuclei, and hence the size of the Coulomb barrier:

Therefor Gamow Energy:

where α is the fine structure constant ≈ 1/137.0.  and and C is Speed of light.
The rate of nuclear fusion therefore depends on the penetration probability of the Coulomb barrier. This penetration probability in turn is described by its Gamow energy.
A higher EG reduces the probability that the barrier will be penetrated. The Gamow energy measures the strength of the Coulomb repulsion, which determines the height of the Coulomb barrier.

The Gamow energy is named for George Gamow (1904–1968), a Russian physicist and cosmologist who escaped to the USA in 1934. His paper ‘The origin of chemical elements’ (1948) by Alpher, Bethe & Gamow attempted to explain much of astrophysical nucleosynthesis in the big bang and also predicted the existence of the cosmic microwave background radiation.

The Gamow Peak: 
The fusion probability as a function of energy for nuclei.  The energy at which the fusion rate is a maximum is called the Gamow peak. The region on either side of the peak in which the fusion probability is significant is called the Gamow window.

Ref: Stellar Evolution and Nucleosynthesis - Sean G Ryan, Andrew J Norton

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