Past exam of the quantum condensed matter course of the University of Cambridge / 2009 / Section B / B1

The attractive part of the Lennard-Jones potential represents the London dispersion force between fluctuating atomic electric dipoles. The dipole-dipole interaction is proportional to $r^{-3}$; for nonpolar ground-state atoms its first-order expectation vanishes. In second-order perturbation theory, virtual excitations give a negative energy shift proportional to the square of that interaction, hence $-A/r^6$ with $A>0$. This applies at separations large compared with atomic size but small enough that electromagnetic retardation is negligible.

The repulsive term models the steep energy increase when closed electron shells overlap, involving the Pauli exclusion principle and increased electronic energy. The $r^{-12}$ power is an empirical convenient approximation, not a universal power derived from the exclusion principle; its being the square of $r^{-6}$ makes computation convenient. At still larger separations retardation changes the attractive asymptotics, outside this model.

A wave packet in an energy band has group velocity $v={\hslash }^{-1}dE/dk$. The semiclassical force law $\hslash \dot{k}=F$ then gives $\dot{v}={\hslash }^{-2}(d^{2}E/d{k}^2)F$. The curvature therefore defines the one-dimensional effective mass by $1/m^{\ast }=E^{\prime \prime }/{\hslash }^2$. Here $E^{\prime \prime }(k)=A{a}^2\cos (ka)$, so at the band minimum $k=0$,It measures acceleration within the band, rather than a change in the electron's rest mass. At a band maximum the electron curvature mass can be negative; a missing electron is then described by a positively charged hole with a positive curvature mass.

For elastic scattering, $|{\mathbf{k}}_{\mathrm{in}}|=|{\mathbf{k}}_{\mathrm{out}}|=2\pi /\lambda$. Contributions from sites differing by any Bravais lattice translation $\mathbf{R}$ add in phase exactly whenfor every $\mathbf{R}$, namely when the momentum-transfer wavevector is a reciprocal lattice vector $\mathbf{G}$. For a family of planes of spacing $d$, the $n$th reciprocal vector normal to those planes has magnitude $2\pi n/d$. The scattering geometry gives $|\mathbf{G}|=2k\sin \theta$. Combining the two expressions provesConversely, reflection from that plane family satisfying Bragg's law supplies the corresponding normal reciprocal vector. The scalar angle condition alone needs this plane-normal geometry; it does not specify an arbitrary outgoing direction. A nontrivial atomic basis can extinguish an otherwise allowed reciprocal-lattice reflection through its structure factor.