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# QUANTUM TUNNELING

In quantum mechanics particles are treated as waves. Out of that idea come some of the most surprising results in all of science. One of them is the ability of particles to pass through barriers that would be completely impenetrable according to pre-quantum physics. For example, a ball thrown against a brick wall will rebound every time. An electron striking the atomic equivalent of a brick wall can pass through without even slowing down.

An atomic particle approaching a barrier is like a ball rolling toward a hill. If the ball doesn't have enough energy, it will not make it over the hill; the hill is a barrier. In quantum mechanics what makes the difference is that the particle can also be treated as a wave. A wave might be reflected from a barrier, for example, light reflected from a mirror, but waves never stop abruptly when they strike barriers. Light waves penetrate a short distance into the mirror before being totally reflected.

Similarly, particle waves penetrate into barriers. If the barrier is not too thick, some of the wave gets through to the far side even though the wave amplitude decreases rapidly (Figure 1). Since the wave represents the particle, it seems that some part of the particle has gotten through the barrier. But atomic particles such as electrons cannot be partly reflected: either the particle is reflected, or it is not. The proportion of the wave that survives at the far side of the barrier represents the probability that the particle will get through. Thus, some particles are reflected, and the others pass through the barrier, but there is no way to tell what will happen to a particular particle.

Since quantum mechanics applies in principle to everyday objects, there is even a slight possibility that a BB could pass through a steel plate (without damaging the plate or leaving a hole). The BB does not even have to be going very fast. Still, calculations show that the probability that a BB would actually pass through a steel plate is unimaginably small. Thus, we never observe such behavior in everyday processes.

However, the tunneling probability increases as the mass of the particle and the thickness of the barrier decrease. As we approach atomic dimensions

FIGURE 1

and masses, the probability can become substantial. It is common to use electron tunneling in semiconductors as part of the design of common electronic devices. It is also used in the scanning tunneling microscope, which is able to image individual atoms.

## Alpha Decay

One of the first applications of quantum tunneling was by the physicist George Gamow in 1928, soon after the development of quantum mechanics. Alpha particles, which consist of two protons and two neutrons, are emitted by some nuclei. For example, ordinary uranium, 238U, with a lifetime of 4.5 billion years, decays by emitting an alpha particle.

For decades alpha decay had presented a problem: the emitted alpha particles seemed to have too little energy to get out of the nucleus. The Coulomb barrier arises from the combined effect of the Coulomb repulsion between the alpha particle and the nucleus (both positively charged) and the nuclear force that attracts the two particles. The energy of the emitted alpha particle is less than the top of this barrier. Classically, the particle would be unable to get out of the nucleus, but it obviously does.

Gamow suggested that alpha particles tunnel through the barrier. If so, the half-life of the decay should depend on the width and height of the barrier, and it does: the lower and thinner the barrier, the greater the chance of penetrating it. As the alpha particle's energy increases, the particle sees both a lower and thinner barrier so the probability of getting through increases extremely rapidly. For example, the energies of the alphas emitted by 232Th and 212Po are 4.05 MeV and 8.95 MeV, respectively, while their respective half-lives are 14 billion years and 0.3 millionth of a second. Thus, a factor of about two in energy produces a difference in half-lives of sixteen orders of magnitude (that is, sixteen powers of ten)!

## Hydrogen fusion in the Sun

The Sun is mostly hydrogen. Its energy arises from combining hydrogen nuclei (protons) to form helium in a process called hydrogen fusion. In order for the protons to react with each other, they must get close enough for the strong nuclear force to hold them together. But the strong force does not reach very far, which means that the protons must come close enough to touch.

Since protons are positively charged, they repel each other and must approach with a lot of energy in order to get close enough to react. The higher the temperature of a gas, the more energetic the particles. At fifteen million degrees Celsius the center of the Sun is hot enough to give the protons the required energy. The reactions are able to take place, however, because the protons tunnel through the barrier, thereby producing the energy that sustains life on Earth.

## Bibliography

Binnig, G., and Rohrer, H. "The Scanning Tunneling Microscope." Scientific American253 , 50–56 (1985).

Feinberg, G. What Is the World Made of: Atoms, Leptons, Quarks and Other Tantalizing Particles (Anchor, Garden City, NY,1977).

Lawrence A. Coleman