Periodic boundary conditions

Schwarz, K.

doi:10.1107/97809553602060000639

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 296

Section 2.2.4.2. Periodic boundary conditions

K. Schwarz^a ^*

^a Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.4.2. Periodic boundary conditions

| top | pdf |

We can expect the bulk properties of a crystal to be insensitive to the surface and also to the boundary conditions imposed, which we therefore may choose to be of the most convenient form. Symmetry operations are covering transformations and thus we have an infinite number of translations in T, which is most inconvenient. A way of avoiding this is provided by periodic boundary conditions (Born–von Karman). In the present one-dimensional case this means that the wavefunction $[\psi(x)]$ becomes periodic in a domain [L=Na] (with integer N number of lattice constants a), i.e. $[\psi(x+Na)=\psi(x+L)=\psi(x).\eqno(2.2.4.9)]$ According to our operator notation (2.2.4.6), we have the following situation when the translation t is applied n times: $[\widetilde{t}^{n}\psi(x)=\psi(x-na)=\mu^{n}\psi(x).\eqno(2.2.4.10)]$ It follows immediately from the periodic boundary condition (2.2.4.9) that $[\mu^{N}=1\eqno(2.2.4.11)]$ with the obvious solution $[\mu=\exp[2\pi i({n}/{N})]\quad\hbox{with}\quad n=0\pm1,\pm2,\ldots.\eqno(2.2.4.12)]$ Here it is convenient to introduce a notation $[k={{2\pi}\over{a}}{{n}\over{N}}\eqno(2.2.4.13)]$ so that we can write $[\mu=\exp(ika)]$ . Note that k is quantized due to the periodic boundary conditions according to (2.2.4.13). Summarizing, we have the Bloch condition (for the one-dimensional case): $[\psi(x+a)=\exp(ika)\psi(x),\eqno(2.2.4.14)]$ i.e. when we change x by one lattice constant a the wavefunction at x is multiplied by a phase factor $[\exp(ika)]$ . At the moment (2.2.4.13) suggests the use of k as label for the wavefunction $[\psi_{k}(x)]$ .

Generalization to three dimensions leads to the exponential $[\exp(i{\bf kT})]$ with $[\textstyle\sum\limits_{i=1}^{3}k_{i}n_{i}={\bf k}\cdot{\bf T}\quad\hbox{using (2.2.2.6) and (2.2.2.1)}\eqno(2.2.4.15)]$ and thus to the Bloch condition $[\psi_{{\bf k}}({\bf r+T})=\exp({i{\bf kT}})\psi_{{\bf k}} ({\bf r}), \eqno(2.2.4.16)]$ or written in terms of the translational operator $[\{E|{\bf T}\}]$ [see (2.2.3.15)] $[\{E|{\bf T}\}\psi_{{\bf k}}({\bf r})=\psi_{{\bf k}}({\bf r-T})=\exp({-i{\bf kT}})\psi_{{\bf k}}({\bf r}).\eqno(2.2.4.17)]$ The eigenfunctions that satisfy (2.2.4.17) are called Bloch functions and have the form $[\psi_{{\bf k}}({\bf r})=\exp({i{\bf kr}})u_{{\bf k}}({\bf r}), \eqno(2.2.4.18)]$ where $[u_{{\bf k}}({\bf r})]$ is a periodic function in the lattice, $[u_{{\bf k}}({\bf r})=u_{{\bf k}}({\bf r+T})\quad\hbox{for all }{\bf T},\eqno(2.2.4.19)]$ and $[{\bf k}]$ is a vector in the reciprocal lattice [see (2.2.2.6)] that plays the role of the quantum number in solids. The $[{\bf k}]$ vector can be chosen in the first BZ, because any $[{\bf k}^{\prime}]$ that differs from $[{\bf k}]$ by just a lattice vector $[{\bf K}]$ of the reciprocal lattice has the same Bloch factor and the corresponding wavefunction $[\psi_{{\bf k+K}}({\bf r})]$ satisfies the Bloch condition again, since $[\exp[{i({\bf k+K)T}}]=\exp({i{\bf kT}})\exp({i{\bf KT}})=\exp({i{\bf kT}}),\eqno(2.2.4.20)]$ where the factor $[\exp({i{\bf KT}})]$ is unity according to (2.2.2.7). Since these two functions, $[\psi_{{\bf k+K}}({\bf r})]$ and $[\psi_{{\bf k} }({\bf r})]$ , belong to the same Bloch factor $[\exp({i{\bf kT}})]$ they are equivalent. A physical interpretation of the Bloch states will be given in Section 2.2.8.