Reciprocal lattice

reciprocal lattice

Other languages

Réseau réciproque (Fr). Reziprokes Gitter (Ge). Red reciproca (Sp)

Definition

The reciprocal lattice is constituted by the set of all possible linear combinations of the basis vectors a*, b*, c* of the reciprocal space. A point (node), H, of the reciprocal lattice is defined by its position vector:

OH = r_hkl* = h a* + k b* + l c*.

If H is the nth node on the row OH, one has:

OH = n OH₁ = n (h₁ a* + k₁ b* + l₁ c*),

where H₁ is the first node on the row OH and h₁ , k₁ , l₁ are relatively prime.

The generalizaion of the reciprocal lattice in a four-dimensional space for incommensurate structures is described in Section 9.8 of International Tables of Crystallography, Volume C.

Geometrical applications

Each vector OH = r_hkl* = h a* + k b* + l c* of the reciprocal lattice is associated to a family of direct lattice planes. It is normal to the planes of the family, and the lattice spacing of the family is d = 1/OH₁ = n/OH if H is the nth node on the reciprocal lattice row OH. One usually sets d_hkl = d/n = 1/OH. If OP = x a + y b + z c is the position vector of a point of a lattice plane, the equation of the plane is given by OH₁ . OP = K where K is a constant integer. Using the properties of the scalar product of a reciprocal space vector and a direct space vector, this equation is OH₁ . OP = h22₁x + k₁y + l₁z = K. The Miller indices of the family are h₁, k₁, l₁. The subscripts of the Miller indices will be dropped hereafter.

The Miller indices of the family of lattice planes parallel to two direct space vectors, r₁ = u₁ a + v₁ b + w₁ c and r₂ = u₂ a + v₂ b + w₂ c are proportional to the coordinates in reciprocal space, h, k, l, of the vector product of these two vectors:

h/(v₁ w₂ - v₂ w₁) = k/(w₁ u₂ - w₂ u₁) = l/(u₁ v₂ - u₂ v₁).

The coordinates u, v, w in direct space of the zone axis intersection of two families of lattice planes of Miller indices h₁, k₁, l₁ and h₂, k₂, l₂, respectively, are proportional to the coordinates of the vector product of the reciprocal lattice vectors associated to these two families:

u/(k₁ l₂ - k₂ l₁) = v/(l₁ h₂ - l₂ h₁) = w/(h₁ k₂ - h₂ k₁).

Centred lattices

The reciprocal lattice of face-centred lattice F is a body-centred lattice I and, reciprocally, the reciprocal lattice of body-centred lattice I is a face-centred lattice F (see The Reciprocal Lattice). The reciprocal lattice of a C (or A, B) lattice is also a C (or A, B) lattice. The reciprocal lattice of an R lattice is also an R lattice

Diffraction condition in reciprocal space

The condition that the waves outgoing from two point scatterers separated by a lattice vector r = u a + v b + w c (u, v, w integers) be in phase is that the scalar product (s_h/λ - s_o/λ) . r, where s_h and s_o are unit vectors in the scattered and incident directions, respectively, be an integer, K. This condition is satisfied if (s_h/λ - s_o/λ) = h a* + k b* + l c*, where h, k, l are integers, namely if the diffraction vector (s_h/λ - s_o/λ) is a vector 0H of the reciprocal lattice. A node of the reciprocal lattice is therefore associated to each Bragg reflection on the lattice planes of Miller indices (h/K, k/K, l/K). It is called the hkl reflection.

History

The notion of reciprocal vectors was introduced in vector analysis by J. W. Gibbs (1981 - Elements of Vector Analysis, arranged for the Use of Students in Physics. Yale University, New Haven). The concept of reciprocal lattice was adapted by P. P. Ewald to interpret the diffraction pattern of an orthorhombic crystal (1913) in his famous paper where he introduced the sphere of diffraction. It was extended to lattices of any type of symmetry by M. von Laue (1914) and Ewald (1921). The first approach to that concept is that of the system of polar axes, introduced by Bravais in 1850, which associates the direction of its normal to a family of lattice planes.

See also

reciprocal space

The Reciprocal Lattice (Teaching Pamphlet of the International Union of Crystallography)

Section 5.1, International Tables of Crystallography, Volume A

Section 1.1, International Tables of Crystallography, Volume B

Section 1.1, International Tables of Crystallography, Volume C

Section 1.1.2, International Tables of Crystallography, Volume D