# Reciprocal lattice

The reciprocal lattice is constituted by the set of all possible **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 *n*th node on the row *OH*, one has:

**OH** = *n* **OH _{1}** =

*n*(

*h*

_{1}

**a***+

*k*

_{1}

**b***+

*l*

_{1}

**c***),

where *H*_{1} is the first node on the row *OH* and *h*_{1} , *k*_{1} , *l*_{1} are relatively prime.

The generalization 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/

*OH*

_{1}=

*n*/

*OH*if

*H*is the

*n*th 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

**OH**

_{1}.

**OP**=

*K*where

*K*is a constant integer. Using the

**OH**

_{1}.

**OP**=

*h*x

_{1}*+*k

*y*

_{1}*+*l

*=*

_{1}z*K*. The Miller indices of the family are

*h*

_{1},

*k*

_{1},

*l*

_{1}. The

The Miller indices of the **family of lattice planes parallel to two direct space vectors**, **r _{1}** =

*u*

_{1}**a**+

*v*

_{1}**b**+

*w*

_{1}**c**and

**r**=

_{2}*u*

_{2}**a**+

*v*

_{2}**b**+

*w*

_{2}**c**are proportional to the coordinates in reciprocal space,

*h*,

*k*,

*l*, of the vector product of these two vectors:

*h*/(*v _{1}*

*w*-

_{2}*v*

_{2}*w*) =

_{1}*k*/(

*w*

_{1}*u*-

_{2}*w*

_{2}*u*) =

_{1}*l*/(

*u*

_{1}*v*-

_{2}*u*

_{2}*v*).

_{1}The coordinates *u*, *v*, *w* in direct space of the **zone axis intersection of two families of lattice planes** of Miller indices *h _{1}*,

*k*,

_{1}*l*and

_{1}*h*,

_{2}*k*,

_{2}*l*, respectively, are proportional to the coordinates of the vector product of the reciprocal lattice vectors associated to these two families:

_{2}*u*/(*k _{1}*

*l*-

_{2}*k*

_{2}*l*) =

_{1}*v*/(

*l*

_{1}*h*-

_{2}*l*

_{2}*h*) =

_{1}*w*/(

*h*

_{1}*k*-

_{2}*h*

_{2}*k*).

_{1}### Centered lattices

Direct lattice | Reciprocal lattice | ||||
---|---|---|---|---|---|

Bravais letter | Unit-cell volumeV_{c} | Bravais letter | Multiple unit cell | Unit cell volumeV*_{c} | |

P | 0 | V | P | a*, _{c}b*, _{c}c*_{c} | V* |

A | ½b+½_{c}c_{c} | 2V | A | a*, 2_{c}b*, 2_{c}c*_{c} | ½V* |

B | ½c+½_{c}a_{c} | 2V | B | 2a*, _{c}b*, 2_{c}c*_{c} | ½V* |

C | ½a+½_{c}b_{c} | 2V | C | 2a*, 2_{c}b*,_{c}c*_{c} | ½V* |

I | ½a+ ½_{c}b +½_{c}c_{c} | 2V | F | 2a*, 2_{c}b*, 2_{c}c*_{c} | ½V* |

F | ½a+ ½_{c}b_{c} | 4V | I | 2a*, 2_{c}b*, 2_{c}c*_{c} | ¼V* |

½b+ ½_{c}c_{c} | |||||

½c+ ½_{c}a_{c} | |||||

R | 0 | V | R | a*, _{c}b*, _{c}c*_{c} | V* |

(rhombohedral axes) | |||||

R | ⅔a* + ⅓_{c}b* + ⅓_{c}c*_{c} | 3V | R | 3a*, 3_{c}b*, 3_{c}c*_{c} | ⅓V* |

(hexagonal axes) | ⅓a* + ⅔_{c}b* + ⅔_{c}c*_{c} |

where **a _{c}**,

**b**,

_{c}**c**are the basis vectors of the conventional multiple cell and

_{c}**a***,

_{c}**b***,

_{c}**c***the corresponding reciprocal lattice vectors.

_{c}An elementary proof that the reciprocal lattice of a face-centered lattice *F* is a body-centered lattice *I* and, reciprocally, is given in The Reciprocal 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**/λ) .

_{o}**r**, where

**s**and

_{h}**s**are unit vectors in the scattered and incident

_{o}*n*. This condition is satisfied whatever

**r**if the

**OH**=

**s**/λ -

_{h}**s**/λ) is of the form:

_{o}(**s _{h}**/λ -

**s**/λ) =

_{o}*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 to 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**/λ =

_{o}**0H**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**, directions and the lattice plane. It can be seen from the figure that

_{h}*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 centered 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

*n*th 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 to a family of lattice planes.

### See also

- 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*