Bravais lattice

A three-dimensional Bravais lattice iswhere the primitive vectors $a_1,a_2,a_3$ are linearly independent.

A tetragonal Bravais lattice represented by conventional-cell points $(0,0,0)$ and $(1/2,1/2,1/2)$, with $a=b\ne c$ and right angles. A motif attached to each point determines the atomic structure and can give more than two atoms per conventional cell.

The set $\{n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2:n_1,n_2\in \mathbb{Z}\}$ for two independent planar translation vectors. Its primitive-cell area is $|\det ({\mathbf{a}}_1,{\mathbf{a}}_2)|$.

A two-dimensional Bravais lattice with perpendicular primitive vectors of the same length $\ell$. An orthogonal matrix $R$ gives its orientation, and its primitive area is ${\ell }^2$. Its nearest-neighbor graph is a square lattice; the set of crystal translations and the graph with chosen adjacency are related but distinct objects.

A lattice represented by a rectangular conventional cell with corner points and one center point. For conventional sides $2a,2b$, primitive vectors can be $(a,b)$ and $(-a,b)$. The special case $a=b$ is a square lattice with a rotated primitive basis.

A direct lattice vector is an integer linear combination of primitive translation vectors ${\mathbf{a}}_i$ of a Bravais lattice. Translating by such a vector preserves the lattice. Its dot product with every reciprocal lattice vector is an integer multiple of $2\pi$ in the angular-wavevector convention.

For identical touching atoms represented by hard spheres of radius $r$, a face-centered cubic lattice has conventional side $a_F=2\sqrt{2}r$ and four atoms per conventional cell. A body-centered cubic lattice has side $a_B=4r/\sqrt{3}$ and two atoms per conventional cell. Their primitive unit cell volumes are respectivelyEach primitive cell contains one atom in the monatomic model, so its mass density is atomic mass divided by $\Omega$. These are contact-geometry estimates; a fixed hard-sphere radius does not encode thermal expansion or phase-dependent electronic bonding.

A region whose translates tile a lattice's ambient space, together with the lattice points represented in it. A primitive unit cell represents one lattice point; a conventional unit cell may represent several to display symmetry.

A crystallographic unit cell chosen to display the symmetry or a standard setting of a crystal structure. It tiles the periodic arrangement but need not be primitive. A conventional FCC cube contains four lattice points and a BCC cube contains two; a primitive unit cell contains one. These are lattice-point counts, not a universal count of atoms in a chemical motif.

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.

A tetragonal conventional cell with lattice points at its vertices and face centers describes a Bravais lattice. It is equivalent, after a rotation and a change of conventional cell, to a body-centered tetragonal description. The reciprocal lattice of a body-centered tetragonal lattice with conventional sides $a,a,b$ consists of $2\pi (h/a,k/a,l/b)$ with $h+k+l$ even; these points form a face-centered tetragonal conventional cell with sides $4\pi /a,4\pi /a,4\pi /b$.

A face-centered cubic lattice has lattice points at every corner and face center of a conventional cube. One primitive cell has one quarter of the conventional cube's volume.

For equivalent scatterers on the four conventional fcc sites, the centering factor is $1+e^{i\pi (h+k)}+e^{i\pi (h+l)}+e^{i\pi (k+l)}$. It is four when the integer indices are all even or all odd, and zero otherwise. An additional atomic basis of a crystal can impose further extinctions. This rule concerns conventional cubic indexing.

A lattice point is a vector $l=n_1{a}_1+n_2{a}_2+n_3{a}_3$ whose coefficients in a chosen primitive basis are integers.

For a simple polygon whose vertices are lattice points in ${\mathbb{Z}}^2$, its area is $A=I+B/2-1$, where $I$ counts interior lattice points and $B$ counts boundary lattice points.

The reciprocal lattice isIf $\Omega =a_1\cdot (a_2\times a_3)$, its primitive vectors areand satisfy $a_i\cdot b_j=2\pi {\delta }_{ij}$.

An elongated reciprocal-space reflection caused by finite extent in the corresponding real-space direction. A foil of thickness $t$ has reflection breadth of order $1/t$ along its normal, increasing the range over which a reflection can intersect an Ewald sphere. The exact profile depends on the specimen shape and scattering model.

A plane wave whose phase is unchanged by every direct lattice translation: $e^{i\mathbf{k}\cdot \mathbf{R}}=1$ for all $\mathbf{R}\in \Lambda$. Its wavevector is a reciprocal lattice vector. In one dimension $ka=2\pi m$.

A reciprocal lattice vector $\mathbf{G}$ has the displayed property for every direct lattice vector $\mathbf{R}$. Thus $e^{i\mathbf{G}\cdot \mathbf{R}}=1$. Fourier components of a lattice-periodic potential carry these wavevectors and couple plane-wave electronic states whose momenta differ by $\hslash \mathbf{G}$. This is the origin of the Bragg-plane mixing in a nearly-free electron model.

For free-electron dispersion proportional to $k^2$, states at $\mathbf{k}$ and $\mathbf{k}-\mathbf{G}$ are degenerate on this plane, obtained by equating their squared norms. A nonzero periodic-potential Fourier component can mix them and open an avoided crossing in the nearly-free electron model. First-zone boundaries are selected planes of this kind.

The reciprocal lattice of a body-centered cubic lattice is face-centred cubic, and that of a face-centered cubic lattice is body-centred cubic. In both cases its conventional cubic side is $4\pi /a$ if the direct conventional side is $a$. Thus a direct BCC-to-FCC phase transition reverses the reciprocal lattice types. The reciprocal primitive cell volume is $(2\pi {)}^3/\Omega$, so thermal expansion contracts reciprocal spacings, while a discontinuous change in direct primitive unit cell volume gives an inverse change in reciprocal cell volume.

A first-order structural transition need not continuously deform one perfect reciprocal array into another. Phase coexistence, new orientation variants, domains and loss of single-crystal alignment can produce two sets of diffraction peaks or many differently oriented sets. The two endpoint lattice types alone do not specify the transformation path or orientation relationship.

The Wigner--Seitz cell of a lattice point consists of points at least as close to it as to every other lattice point. Its faces lie on perpendicular bisectors to neighbouring lattice points.

For direct primitive vectors $(1,0)$ and $(-1/2,\sqrt{3}/2)$, reciprocal primitive vectors areThe reciprocal Wigner--Seitz cell is a regular hexagon of area $8{\pi }^2/\sqrt{3}$ and circumradius $4\pi /3$.

The cubic crystal system has cubic point symmetry. Its three cubic Bravais lattices are primitive, body-centered, and face-centered. The body-centered cubic lattice is one member of that classification.

A body-centered cubic lattice consists of the points of a cubic lattice together with their body centers. Its reciprocal lattice is face-centered cubic.

In a body-centred cubic lattice of conventional edge $a$, a lattice site has eight nearest neighbours along body half-diagonals at distance $\sqrt{3}a/2$, then six next-nearest neighbours along the coordinate axes at distance $a$. There are two sites per conventional cell, so identical atoms of mass $m$ and mass density $\rho$ obey $\rho a^3=2m$. Specifying a neighbour shell matters: the nearest and next-nearest distances differ.

Fora reciprocal basis isThus ${\Lambda }^{\ast }=(2\pi /a)\{(h,k,l)\in {\mathbb{Z}}^3:h+k+l\text{ even}\}$ and its shortest nonzero vectors have length $2\sqrt{2}\pi /a$.

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 lattice translation moves every lattice point of a Bravais lattice onto another lattice point. Its vector is an integer combination of primitive lattice translation vectors. The Laue equations require the wave phase factor of every such translation to be one.