Bloch's theorem

Shmueli, U.

doi:10.1107/97809553602060000549

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.1, p. 9 | 1 | 2 |

Section 1.1.6.3. Bloch's theorem

U. Shmueli^a ^*

^aSchool of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel
Correspondence e-mail: ushmueli@post.tau.ac.il

1.1.6.3. Bloch's theorem

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It is in order to mention briefly the important role of reciprocal space and the reciprocal lattice in the field of the theory of solids . At the basis of these applications is the periodicity of the crystal structure and the effect it has on the dynamics (cf. Chapter 4.1 ) and electronic structure of the crystal. One of the earliest, and still most important, theorems of solid-state physics is due to Bloch (1928) and deals with the representation of the wavefunction of an electron which moves in a periodic potential. Bloch's theorem states that:

The eigenstates $[\psi]$ of the one-electron Hamiltonian $[ {\scr h} = (-\hbar^{2}/2m) \nabla^{2} + U({\bf r})]$ , where U(r) is the crystal potential and $[{\bf U}({\bf r} + {\bf r}_{L}) = U({\bf r})]$ for all $[{\bf r}_{L}]$ in the Bravais lattice, can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais lattice.

Thus $[\psi({\bf r}) = \exp (i{\bf k} \cdot {\bf r})u({\bf r}), \eqno(1.1.6.5)]$ where $[u({\bf r} + {\bf r}_{L}) = u({\bf r}) \eqno(1.1.6.6)]$ and k is the wavevector. The proof of Bloch's theorem can be found in most modern texts on solid-state physics (e.g. Ashcroft & Mermin, 1975). If we combine (1.1.6.5) with (1.1.6.6), an alternative form of the Bloch theorem results: $[\psi({\bf r} + {\bf r}_{L}) = \exp (i{\bf k} \cdot {\bf r}_{L}) \psi ({\bf r}). \eqno(1.1.6.7)]$ In the important case where the wavefunction $[\psi]$ is itself periodic, i.e. $[\psi({\bf r} + {\bf r}_{L}) = \psi({\bf r}),]$ we must have $[\exp (i{\bf k} \cdot {\bf r}_{L}) = 1]$ . Of course, this can be so only if the wavevector k equals $[2\pi]$ times a vector in the reciprocal lattice. It is also seen from equation (1.1.6.7) that the wavevector appearing in the phase factor can be reduced to a unit cell in the reciprocal lattice (the basis vectors of which contain the $[2\pi]$ factor), or to the equivalent polyhedron known as the Brillouin zone (e.g. Ziman, 1969). This periodicity in reciprocal space is of prime importance in the theory of solids. Some Brillouin zones are discussed in detail in Chapter 1.5.

References

Ashcroft, N. W. & Mermin, N. D. (1975). Solid state physics. Philadelphia: Saunders College.Google Scholar

Bloch, F. (1928). Über die Quantenmechanik der Elektronen in Kristallgittern. Z. Phys. 52, 555–600.Google Scholar

Ziman, J. M. (1969). Principles of the theory of solids. Cambridge University Press.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.1, p. 9