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Scattering and tunnelling

Keith Hannabuss

Published Mar 20, 1997• 2 citations
Publisher

Abstract

Abstract Ever since Rutherford’s pioneering work, passing alpha rays through a metal foil target, beams of particles have provided a means of probing the fine details of subatomic structure. Modern accelerators hurl particles through a target at velocities only slightly less than that of light. The wavelength of each particle shrinks as its momentum increases and it becomes possible to distinguish features far beyond the resolving power of any microscope. In the simplest cases (such as Rutherford’s sedate Edwardian experiments) the particles emerge from their encounter unscathed, and it is by analysing the changes to their momentum that one must build up a picture of the target. To develop some feeling for what happens we shall look at the simplest case in which the particle suffers no deflection. We therefore consider particles moving along a line on which there is some kind of potential barrier.

DOI
10.1093/oso/9780198537946.003.0005

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Figure 1.1 represents schematically a typical scattering experiment, in fact this is the sketch of E. Rutherford experiment in 1910 [ 1 ]. Projectiles (here, α -particles) from a source go through a collimator and collide with a target (gold foil in Rutherford’s experiment). Some projectiles are scattered by the target and reach the detector (here, a fluorescent screen). Rutherford expected them to go straight through with perhaps a minor deflection. Most did go straight through, but to his surprise some particles bounced directly off the gold sheet! What did this mean? Rutherford hypothesized that the positive alpha particles had hit a concentrated mass of positive particles, which he termed the nucleus. We shall describe the Rutherford scattering theoretically later in this chapter.
There are several ways of gathering experimental evidence about microscopic systems. Spectral analysis gives information about the bound states of the system, while scattering experiments probe their inner structure. Rutherford's pioneering alpha scattering experiments yielded the first glimpse into the structure of atoms. Such experiments opened the door to modern collider experiments, allowing us to peer deeper into the structure of matter. In this chapter, we will study only elastic collisions, in which the nature of the scattered particles does not change. This is expected for low energies in the non-relativistic regime. We first introduce the concept of cross-section and then study two complementary approaches to scattering: the Born approximation, which is suitable when the energy of the incoming particles is high in comparison with the potential energy of the target, and the partial waves method, which applies best when the two energies are comparable.
Abstract Having examined free particles and particles that are confined in space by a potential energy term, we now consider the impact of a disturbance in the flat energy landscape for a free particle. By disturbance we means some kind of fixed “obstacle” which is either a positive (repelling) or negative (attractive) potential. We are interested in determining the impact on the free particle. Continuing to work mostly in one dimension, the particle described by a plane wave corresponding to momentum moving in the positive direction (a positive k−vector in the x−direction), we study elastic scattering. In one dimension, this means that we determine the probability that the particle is transmitted (continuing in the forward direction) or reflected (now moving in the backward direction.) We will also determine the nature of the solution inside the potential and in the case that the potential energy maximum is greater than the kinetic energy of the particle, we will show that the particle tunnels through the barrier. Interestingly, when we have two barriers, we can find conditions where the probability that the particle is transmitted is unity. This is the result of resonance, a feature of the wave-like nature of the particle’s wave function.