Crystal Lattice, Reciprocal Lattice, Diffraction

We review the crystal lattice and reciprocal lattice, and the Bragg's law.

What is Crystal?

Materials are usually classified into gas, liquid, and solid from the viewpoint of mechanical properties. However, from the viewpoint of the arrangement of atoms, materials are classified as follows:

Most of materials with three-dimensional long-range order have periodicity. This is the traditional definition of crystal.
A material with three-dimensional long-range order but no periodicity is called a quasicrystal.

CrystalFThe state of substances in which the same atomic arrangement is repeated three dimensionally

Unit Cell, Primitive Vectors, and Crystal Lattice

First we focus on the periodicity of the crystal before thinking about the arrangement of atoms.

The repeating unit of atomic arrangement is called "unit cell". In other words, in a crystal, unit cells with the same shape, size and orientation are not overlapped and lined up with no gaps. The periodicity with which the unit cells are repeated can be represented by three independent vectors. These are called primitive lattice vectors for the lattice. Hereafter, we express the primitive vectors as a, b, and c.

The atomic arrangement of the crystal is invariant under the translation by a primitive vector.
If one focuses on this periodicity, each unit cell may be represented by a certain point instead of considering the arrangement of all the atoms. The point is called lattice point. The lattice points are also regularly arranged with a periodicity determined by the primitive vectors. The geometric arrangement of the lattice points is called "crystal lattice".

Bragg's Law

When x-rays are scattered from a crystal lattice, special patterns of intensity distribution are observed. Lawrence Bragg and his father William Henry Bragg explained this phenomenon as follows:
The crystal lattice is considered as a set of parallel planes separating a constant distance d. The planes are called lattice planes. Each lattice point is located on one of the lattice planes. Lawrence Bragg proposed that the x-ray wave should be reflected by the lattice planes.
Waves reflected by a periodic array of planes interfere with each other. The path difference between the waves reflected on the adjacent planes is 2d sin θ. Here one should note that the angle θ is measured from the plane in the research field of x-ray, in contrast to the optics research field. The condition for constructive interference of reflected x-rays is expressed as
2d sin θ=nλ .
This is called the Bragg condition.

Reciprocal Lattice

Bragg condition can be expressed also by using the momentum of the x-ray photon.
In a diffraction process, the wavelengths of the incident X-ray and diffracted x-ray are the same. Therefore, the magnitude p = h/λ of the momentum of the x-ray photon does not change either. Only the direction of motion changes by 2θ, and hence the change in the momentum is 2 h sin θ/λ.
From the Bragg condition, the change is represented as
nh / d.
The differencial momentum is oriented normal to the lattice plane that reflects the x-ray.

Since the change in momentum is a vector, we can express it (divided by hbar) as a point in three dimensional space. Because the Bragg's law for a set of parallel lattice planes contains integer n, a series of points are aligned with a spacing 2π/d along the line normal to the lattice plane.

Considering the x-ray reflection by another set of parallel planes, the points representing the change in momentum are aligned in a different direction by a different spacing. Many sets of points will result in a grid with periodicity in three dimensions. This is called "reciprocal lattice".

Reciprocal Lattice Vectors

Here we discuss the relationship between the crystal and reciprocal lattices.

Consider a lattice plane formed by two primitive vectors. The corresponding interplanar spacing "d" is calculated as the quotient of the unit-cell volume divided by the absolute value of the outer product of the two primitive vectors. The normal vector of the plane is orthogonal to the two vectors.
Therefore, the primitive vectors of the reciprocal lattice can be defined as
Here, the volume Vcell of the unit cell is calculated as the scalar product of the primitive vectors a, b, and c.

Miller Index

One reciprocal lattice point corresponds to a set of parallel lattice planes and the order of reflection. Therefore, it is possible to express the set of lattice planes using three integers h, k, and l. This is called Miller index. The set of lattice planes is written by using parentheses as (h k l). The lattice planes are parallel to a plane including three points a/h, b/k, and c/l.
Let's find a vector Ghkl that is perpendicular to this plane and as long as 2π/dhkl.
Consider a triangular pyramid with the coordinate origin as the vertex with the triangle connecting the above three points as the bottom. Its volume is Vcell/(6hkl). The area of the triangle on the bottom surface is (a/h-c/l) × (b/k-c/l)/2. Therefore, the height of the pyramid, which is the same as dhkl, is written as
dhkl = 3 * Vcell/(6hkl) / [|(a/h-c/l) × (b/k-c/l)|/2].
Because Ghkl is orthogonal both to a/h-c/l and to b/k-c/l,
Ghkl = 2πhkl (a/h-c/l)×(b/k-c/l) /Vcell
= 2π (la×b-ka×c -hc×b)/Vcell =ha*+kb*+lc* .

Bragg condition represented by Reciprocal Lattice Vectors

The change in the momentum of x-ray photons for diffraction is determined by the set of parallel planes and the order n. Here we can re-define the Miller index so that the order of reflection is always one, by just multiplying all the indices by n. Then the orientation of the planes does not change and the spacing becomes 1/n.
Then the change in wavevector with the reflection by the plane (hkl) including three points a/h, b/k, and c/l is expressed as
kiks =Ghkl =ha*+kb*+lc* .
Here, ki and ks are the wavevectors of incident and scattered x-rays, respectively. (Note that there are several different expressions about the sign and the factor 2π.)

Since the length of G is 2π/d, the diffraction angle θ can be easily calculated from h, k, l, a*, b*, and c*.


Taka-hisa Arima