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A charge density wave is a spatial modulation of the electron charge density, often accompanied by a periodic lattice distortion. Write a weak modulation as $\rho (x)={\rho }_0+{\rho }_1\cos (Qx+\phi )$. In a one-dimensional metal, the Peierls distortion couples states near opposite Fermi points most strongly for $Q=2k_F$. Opening a gap near the Fermi energy lowers occupied electronic energies, while the lattice deformation has a positive elastic energy cost. Their competition can make a periodic state favorable below a transition temperature.

The modulation can be commensurate or incommensurate with the underlying Bravais lattice. Diffraction detects additional satellite peaks, while transport can show reduced low-energy carrier density. Impurities can pin its phase; above a depinning field a collectively sliding modulation can contribute nonlinear current. These effects distinguish collective density modulation from simply assigning independent static charges to isolated lattice sites.

Consider a chosen primitive cubic lattice $a{\mathbb{Z}}^3$ with $a=0.400\mathrm{nm}$ and one identical point scatterer per primitive unit cell. Retain the incident wavelength $\lambda =0.154\mathrm{nm}$. This is an ideal point model. Its reciprocal lattice vectors are $q=(2\pi /a)(h,k,l)$ for integer triples other than $(0,0,0)$. Write $S=h^2+k^2+l^2$. The associated interplanar spacing and first-order Bragg angle satisfyThe reflection labels $(hkl)$ use the Miller index convention, keeping unreduced triples to encode diffraction order. With that convention, set $n=1$ in Bragg's law for each label. The $(200)$ reflection is equivalently the second diffraction order of $(100)$: $d_{200}=a/2$ does not imply extra scatterers halfway between the occupied $(100)$ planes. Counting both descriptions as distinct reflections would count the same direction twice.

For $(100)$, $(110)$, $(111)$ and $(200)$, respectively, $S=1,2,3,4$ and the calculated interplanar spacings are $0.400000$, $0.282843$, $0.230940$ and $0.200000\mathrm{nm}$. Their calculated deflections $2\theta$ are $22.197435^{\circ }$, $31.594744^{\circ }$, $38.952948^{\circ }$ and $45.287481^{\circ }$. Equal steps in $S$ give equal steps in ${\sin }^2\theta$, not equal steps in $2\theta$. This regularity makes an angular pattern a test of a proposed crystal structure.

A real crystal structure has a Bravais lattice together with an atomic basis of a crystal. The crystal lattice structure factor selects the periodic directions; the crystal basis structure factor combines the scattering amplitudes of atoms within a unit cell and can cancel a reflection. Consequently Bragg's law alone predicts neither allowed reflections nor their wave intensities. ^[ref] Systematic absences further restrict observed families; for example, face-centered cubic lattice centering permits only indices all even or all odd in conventional cubic indexing. The four strips above apply to the stated primitive cubic lattice, not to sodium chloride or a face-centered cubic lattice. ^[ref]

A member of a set of linearly independent vectors whose integer combinations generate every lattice point of a Bravais lattice. Such basis vectors specify translations; an atomic basis of a crystal specifies the chemical motif repeated under them.

A Bravais lattice with mutually perpendicular primitive vectors of lengths $a,a,c$, with $c\ne a$. A chemically ordered L10 crystal structure has this translation lattice even when drawn using a larger four-site cell that resembles a distorted FCC conventional cell.

A fundamental cell of a Bravais lattice, of volume equal to the absolute determinant of any primitive lattice basis. Direct and reciprocal lattice primitive cell volumes multiply to $(2\pi {)}^d$ in dimension $d$ with the angular-wavenumber convention.

For a divalent square Bravais lattice, let $E_X={\hslash }^2{\pi }^2/(2m_e{a}^2)$ and real periodic potential coefficients $V_{10}<V_{11}<0$. The leading two-wave edge levels are $E_X\pm V_{10}$. The four-wave corner levels are $2E_X+V_{11}\pm 2V_{10}$ and a doublet at $2E_X-V_{11}$. Comparing the lowest corner level with the upper edge level gives the displayed indirect band gap. The first spin-degenerate energy band can hold the two electrons per primitive unit cell; it becomes isolated when $-E_X-V_{11}-3V_{10}>0$ in this truncation. This follows from resonant degenerate perturbation theory, and higher plane waves can shift the threshold.

The lattice translations commute with one another and with a lattice-periodic Hamiltonian. They can therefore be diagonalized simultaneously within each energy eigenspace. Every unitary character of a Bravais lattice has the form $r\mapsto e^{-ik\cdot r}$, with $k$ defined modulo the reciprocal lattice. Hence $T_r{\psi }_k=e^{-ik\cdot r}{\psi }_k$, so $u_k(x)=e^{-ik\cdot x}{\psi }_k(x)$ is lattice-periodic.

X-ray diffraction is the angular redistribution of X-ray wave intensity through wave interference between scattered waves. In a crystal structure, repeated lattice translations select discrete outgoing directions by the Laue equations. A photographic plate can therefore turn invisible atomic periodicity into a visible constellation of spots. Their geometry constrains the Bravais lattice and its orientation; their wave intensities also depend on the crystal basis structure factor. Determining atomic positions belongs to X-ray crystallography and requires more information than recognizing a spot pattern.

The Laue X-ray diffraction experiment of 1912 established this phenomenon experimentally. The Laue method uses a fixed crystal structure with a range of incident wavelengths; the Bragg crystal reflection experiment of 1913 instead measured angularly resolved signals with a moving detector. These are different ways of finding the same coherent elastic scattering directions.