Scattering and tunnelling
Scattering and tunnelling

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Scattering and tunnelling

1 What are scattering and tunnelling?

The phenomenon of scattering was an important topic in physics long before the development of wave mechanics. In its most general sense, scattering is a process in which incident particles (or waves) are affected by interaction with some kind of target, quite possibly another particle (Figure 1). The interaction can affect an incident particle in a number of ways: it may change its speed, direction of motion or state of internal excitation. Particles can even be created, destroyed or absorbed.

Figure 1 The phenomenon of scattering
Figure 1 The phenomenon of scattering

It can be argued that almost everything we know about the world is learnt as a result of scattering. When we look at a non-luminous object such as a book or a building we see it because of the light that is scattered from its surface. The sky is blue because the particles in the Earth's atmosphere are more effective at scattering blue light (relatively short wavelengths) than yellow or red light (longer wavelengths). This is also the reason why sunsets are red (Figure 2). As the Sun approaches the horizon, its light has to traverse a lengthening path through the Earth's atmosphere; as a consequence, shorter wavelengths are increasingly scattered out of the beam until all that remains is red.

Figure 2 Red sunsets are a direct consequence of the scattering of sunlight
Jon Arnold Images/Alamy ©
Jon Arnold Images/Alamy
Figure 2 Red sunsets are a direct consequence of the scattering of sunlight

Much of what we know about the structure of matter has been derived from scattering experiments. For example, the scattering of alpha particles from a gold foil, observed by Geiger and Marsden in 1909, led Rutherford to propose the first nuclear model of an atom. More recent scattering experiments, involving giant particle accelerators, have provided insight into the fundamental constituents of matter such as the quarks and gluons found inside protons and neutrons. Even our knowledge of cosmology – the study of the Universe as a whole – is deeply dependent on scattering. One of the main sources of precise cosmological information is the study of the surface of last scattering observed all around us at microwave wavelengths (Figure 3).

Figure 3 Microwave image of the surface of last scattering
Bennett, C.L. et al. ‘First Year Wilkinson Microwave Anisotrophy Probe (WMAP) Observations; Preliminary Maps and Basic Results’, Astrophysical Journal (submitted) © 2003 The American Astronomical Society ©
Bennett, C.L. et al. ‘First Year Wilkinson Microwave Anisotrophy Probe (WMAP) Observations; Preliminary Maps and Basic Results’, Astrophysical Journal (submitted) © 2003 The American Astronomical Society
Figure 3 Microwave image of the surface of last scattering

In our detailed discussions of scattering we shall not consider cases where the scattering changes the number or nature of the scattered particles, since that requires the use of quantum field theory, a part of quantum physics beyond the scope of this course. Rather, we shall mainly restrict ourselves to one-dimensional problems in which an incident beam or particle is either transmitted (allowed to pass) or reflected (sent back the way it came) as a result of scattering from a target. Moreover, that target will generally be represented by a fixed potential energy function, typically a finite well or a finite barrier of the kind indicated in Figure 4. Despite these restrictions, our discussion of quantum-mechanical scattering will contain many surprises. For instance, you will see that a finite potential energy barrier of height V0 can reflect a particle of energy E0, even when E0 > V0. Perhaps even more amazingly, you will see that unbound particles can be reflected when they encounter a finite well.

Figure 4 (a) Particles with energy E0 > V0, encountering a finite barrier of height V0, have some probability of being reflected. (b) Similarly, unbound particles with energy E0 > 0 can be reflected by a finite well
Figure 4 (a) Particles with energy E0 > V0, encountering a finite barrier of height V0, have some probability of being reflected. (b) Similarly, unbound particles with energy E0 > 0 can be reflected by a finite well

The phenomenon of tunnelling is entirely quantum-mechanical with no analogue in classical physics. It is an extension of the phenomenon of barrier penetration, which may be familiar in the context of particles bound in potential energy wells. Barrier penetration involves the appearance of particles in classically forbidden regions. In cases of tunnelling, such as that shown in Figure 5, a particle with energy E0 < V0 can penetrate a potential energy barrier of height V0, pass through the classically forbidden region within the barrier, and have some finite probability of emerging into the classically allowed region on the far side.

Figure 5 Particles with energy E0 < V0, encountering a finite barrier of height V0, have some probability of being transmitted by tunnelling through the barrier. Such a process is forbidden in classical physics
Figure 5 Particles with energy E0 < V0, encountering a finite barrier of height V0, have some probability of being transmitted by tunnelling through the barrier. Such a process is forbidden in classical physics

Tunnelling phenomena are common in many areas of physics. In this course you will see how tunnelling provides an explanation of the alpha decay of radioactive nuclei, and is also an essential part of the nuclear fusion processes by which stars produce light. Finally you will see how quantum tunnelling has allowed the development of instruments called scanning tunnelling microscopes (STMs) that permit the positions of individual atoms on a surface to be mapped in stunning detail.

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