Difference between revisions of "Reciprocal lattice"

Latest revision as of 10:36, 15 December 2017

Réseau réciproque (Fr). Reziprokes Gitter (Ge). Reticolo reciproco (It). 逆格子 (Ja). Red recíproca (Sp).

Definition

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

Geometrical applications

Each vector OH = r*_hkl = h a* + k b* + l c* of the reciprocal lattice is associated with 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 = h₁x + 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 with these two families:

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

Centred lattices

Direct lattice			Reciprocal lattice
Bravais letter	Centring vectors	Unit-cell volume V_c	Bravais letter	Multiple unit cell	Unit cell volume V_c*
P	0	V	P	*a_c, b_c, c_c**	V*
A	½b_c+½c_c	2V	A	*a_c*, 2b_c, 2c*_c	½V*
B	½c_c+½a_c	2V	B	2a_c, b_c*, 2c_c	½V*
C	½a_c+½b_c	2V	C	2a_c, 2b_c, *c_c**	½V*
I	½a_c+ ½b_c +½c_c	2V	F	2a_c, 2b_c, 2c*_c	½V*
F	½a_c+ ½b_c	4V	I	2a_c, 2b_c, 2c*_c	¼V*
	½b_c+ ½c_c
	½c_c+ ½a_c
R	0	V	R	*a_c, b_c, c_c**	V*
(rhombohedral axes)	0	V	R	*a_c, b_c, c_c**	V*
R	⅔*a_c + ⅓b_c* + ⅓*c_c**	3V	R	3a_c, 3b_c, 3c*_c	⅓V*
(hexagonal axes)	⅓*a_c + ⅔b_c* + ⅔*c_c**	3V	R	3a_c, 3b_c, 3c*_c	⅓V*

where a_c, b_c, c_c are the basis vectors of the conventional multiple cell and a*_c, b*_c, c*_c the corresponding reciprocal lattice vectors.

An elementary proof that the reciprocal lattice of a face-centred lattice F is a body-centred lattice I and, reciprocally, is given in The Reciprocal Lattice (Teaching Pamphlet No. 4 of the International Union of Crystallography).

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, n. This condition is satisfied whatever r if the diffraction vector (OH = s_h/λ - s_o/λ) is of the form:

(s_h/λ - s_o/λ) = h a* + k b* + l c*,

where h, k, l are integers, namely the diffraction vector OH is a vector of the reciprocal lattice (Fig. 1).

A node of the reciprocal lattice is therefore associated with each Bragg reflection on the lattice planes of Miller indices (h/K, k/K, l/K). It is called the hkl reflection.

The relation s_h/λ - s_o/λ = OH generalizes the Laue equations. It is equivalent to Bragg's law, as can be seen in Fig. 2.

Consider the lattice plane passing through lattice point Q and perpendicular to reciprocal-lattice vector OH and let θ be the angle between the incident, s_o, or the reflected, s_h, directions and the lattice plane. It can be seen from the figure that

OH/2 = sin θ/λ,

and, since OH = n/d (d lattice spacing of the family of lattice planes associated with OH) and d_hkl = d/n:

2 d sin θ = n λ, or 2 d_hkl sin θ = λ,

which is Bragg's law. n is the order of the reflection.

Another way to express the diffraction condition in reciprocal space is to consider a sphere centred at a node Q of the direct lattice, of radius 1/λ and passing through the origin O of the reciprocal lattice (Fig. 3). If it passes through another node, H, of the reciprocal lattice, Bragg's law is satisfied for the family of direct lattice planes associated with that node and of lattice spacing d_hkl = n/OH if H is the nth node on the row OH (n = 2 in the example of Fig. 3). This sphere is called the Ewald sphere.

History

The notion of reciprocal vectors was introduced in vector analysis by J. W. Gibbs [(1881). Elements of Vector Analysis, arranged for the Use of Students in Physics. Yale University, New Haven; reprinted: Gibbs, J. W. and Wilson, E. B. (1902). Vector analysis, New York; 1960, Dover Publications]. 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 with a family of lattice planes.