Crystal Structure and Band Theory

Bragg condition and band gaps

The Bragg condition 2d\sin\theta = n\lambda states that a family of lattice planes with spacing d scatters strongly only when the path difference between waves reflected from successive planes equals a whole number of wavelengths. In reciprocal-space language the same content is the Laue condition \mathbf{k}'-\mathbf{k}=\mathbf{G}, and for elastic scattering |\mathbf{k}|=|\mathbf{k}'|. Because \theta is the glancing angle measured from the planes, the detector records 2\theta, so a \theta value is only half of the angle that is actually measured; this distinction is easy to miss. For a fixed d, a reflection can be observed only if \lambda \le 2d, since \sin\theta \le 1, so a longer wavelength cuts off the high-order peaks of a finely spaced set.

Within a unit cell the scattered amplitude is governed by the structure factor F=\sum_j f_j e^{i\mathbf{G}\cdot\mathbf{r}_j}, so the relative positions of atoms in the basis decide which orders are strong. For a diatomic basis whose two atoms are half a period apart, the phase between them is \pi n for order n, giving |F|^2 \propto f_P^2+f_Q^2+2f_Pf_Q(-1)^n. Even orders then reach (f_P+f_Q)^2 while odd orders fall to (f_P-f_Q)^2, which is weaker but nonzero unless the two form factors are equal.

In the nearly-free-electron picture, energy gaps open where |\mathbf{k}|=|\mathbf{k}-\mathbf{G}|, that is at Brillouin-zone boundaries. For a square lattice of constant a, the first-zone edges are the lines k_x=\pm\pi/a and k_y=\pm\pi/a, so a gap can occur at any point where either coordinate takes one of these values, whereas the zone centre never satisfies the condition.

The same geometry applies to neutron diffraction. A particle of kinetic energy E has k=\sqrt{2mE}/\hbar, and an elastic peak with reciprocal vector \mathbf{G} satisfies |\mathbf{G}|=2k\sin\theta, so the glancing angle follows directly once k is computed in consistent units.

Lattice type decides which reflections exist. For a face-centred cubic lattice, (hkl) reflections appear only when h, k, l are all odd or all even, so (111) is the lowest-angle allowed peak; a body-centred cubic lattice forbids (111) because h+k+l is odd. For touching rigid spheres of radius r, the cube edges are a=2r (simple cubic), a=4r/\sqrt{3} (BCC) and a=2\sqrt{2}\,r (FCC).

Since \sin\theta=\lambda/(2d), a change in lattice parameter moves all peaks in a definite direction. If every lattice constant is multiplied by a factor f at fixed wavelength, each d is multiplied by f and \sin\theta is divided by f; for a cubic cell the spacing of (hkl) planes is a/\sqrt{h^2+k^2+l^2}.

Phonon density of states

The phonon density of states g(\omega) counts vibrational modes per unit frequency, and its frequency dependence depends on the dimension and on the dispersion. The number of allowed wavevectors with magnitude below k grows as k^d in d dimensions. For a linear dispersion \omega=ck, the density of states therefore scales as \omega^{d-1}: it is a constant in one dimension, proportional to \omega in two dimensions, and proportional to \omega^2 in three dimensions. The long-wavelength acoustic modes of the Debye model follow this linear dispersion.

For a one-dimensional monatomic chain of length L and sound speed v, both the +k and -k modes count, giving g(\omega)=L/(\pi v), a constant that does not depend on frequency. Because this constant is fixed by L and v, a measured value of g gives the sound velocity directly through v=L/(\pi g). Since g is a count per unit frequency, its unit is seconds, and the final velocity must be expressed consistently in SI units.

Electronic heat capacity of metals

At low temperature the heat capacity of a metal contains an electronic term C_{el}=\gamma T, because only electrons within about k_BT of the Fermi level can be thermally excited. The Sommerfeld coefficient is \gamma=\frac{\pi^2}{3}k_B^2N(E_F), so it is set by the density of states at E_F. For a band whose electrons behave like free particles with an effective mass, N(E_F)\propto m^*, which means \gamma\propto m^*; a heavier thermal mass gives a larger electronic specific heat. The linear law holds for a degenerate Fermi gas in any dimension: dimensionality changes N(E_F), and hence the size of \gamma, but not the power of T.

The lattice term follows the Debye T^3 law, so at low temperature C=\gamma T+\beta T^3. Dividing by T gives C/T=\gamma+\beta T^2, a straight line in T^2 with intercept \gamma and slope \beta. Plotting C/T against T^2 is the standard way to read \gamma from data, and the linear electronic term is visible only well below the Debye temperature, where it dominates the phonon contribution.

Effective mass from band dispersion

The effective mass describes how sharply a band curves in k-space: \frac{1}{m^*}=\frac{1}{\hbar^2}\frac{d^2E}{dk^2}. It is not a fixed property of the electron but depends on the band, so m^* can be either larger or smaller than the free-electron mass m_e. For a parabolic band E=\hbar^2k^2/(2m^*), a steep, narrow parabola corresponds to a small m^*, and a flat band corresponds to a large m^*. When dispersion curves are compared, the one with the greatest curvature has the smallest effective mass.

For a tight-binding band E(k)=E_0-2\gamma\cos(ka), the second derivative is 2\gamma a^2\cos(ka), so at the band minimum k=0 one obtains m^*=\hbar^2/(2\gamma a^2). The ratio m^*/m_e is therefore set by the hopping energy and the lattice constant, and a wide band with large \gamma gives a light carrier. The usual calculation step is to evaluate d^2E/dk^2 at the extremum with all energies converted to joules.

BCS superconductor ground state

In BCS theory, electrons near the Fermi level form Cooper pairs, and the single-particle spectrum acquires an energy gap \Delta. The ground-state occupation of a state with \xi_k=E_k-E_F at T=0 is v_k^2=\frac{1}{2}\left(1-\frac{\xi_k}{\sqrt{\xi_k^2+\Delta^2}}\right). This function equals 1 well below E_F, 0 well above it, and \frac{1}{2} exactly at E_F, so the transition is smoothed over an energy width of order \Delta.

A normal metal at T=0 would instead show a sharp Fermi-Dirac step with a discontinuity at E_F. The smooth crossover is the fingerprint of pairing and should not be confused with thermal broadening, which is absent at zero temperature.