Crystal momentum

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Crystal momentum (usually denoted ) is a momentum-like vector quantum number associated with electrons in a crystal. It arises from the periodicity in the potential of a crystal lattice. In order to understand crystal momentum, it is important to consider the approximations that are made to describe the electrons in a crystal.

In a "real" crystal of finite size, all electrons have a "true" momentum of zero. If this were not the case, electrons would continuously be ejected from the edges of the crystal. However, for an infinitely large crystal, electrons may have a finite momentum without being ejected from the crystal.

To analytically model the behaviour of electrons in a crystal, the most important and commonly used approximation that can be made is to approximate the behaviour of electrons at the center of a large crystal to the behaviour of electrons in an infinitely large crystal. This results in a non-zero momentum arising in the analytic models. However, this momentum is not "real" momentum, and is termed crystal momentum.

Crystal momentum behaves in many ways like "real" momentum, and in many instances can be treated as if it were real momentum (for example, crystal momentum is conserved in interactions between electrons and phonons in a crystal). However, there are also many aspects in which crystal momentum differs from real momentum (for example, as described earlier, a net crystal momentum does not result in electrons shooting out from the edges of the crystal).

This may seem counter-intuitive from the point of view of classical physics, where it is reasonable for an electron to have a non-zero momentum in the center of the crystal and to be reflected from the edges of the crystal. In terms of quantum mechanics, however, the electron is in a wave state with a net momentum of zero. Interactions with the wave state in the center of the crystal may behave as if there is a non-zero momentum, but a full calculation of the wave state would reveal that the expection value of momentum in a finite crystal were zero.

[edit] References

• Kittel, Charles (1996). Introduction to Solid State Physics, Seventh Edition, John Wiley & Sons, Inc., 108-110,186. ISBN: 0-471-11181-3.

• Neil W. Ashcroft, N. David Mermin (1976). Solid State Physics. Brooks/Cole Thomson Learning, 482-486,784-785. ISBN: 0-03-083993-9.

• G. Rajagopal, R. J. Needs, A. James, S. D. Kenny and W. M. C. Foulkes (15 April 1995). "Variational and diffusion quantum Monte Carlo calculations at nonzero wave vectors: Theory and application to diamond-structure germanium". Phys. Rev. B 51 (16): 10591–10600. DOI:10.1103/PhysRevB.51.10591. PACS: 71.10.+x, 71.20.Ad.