6 Reciprocal Latticeand Structure Factor

Amarjeet Singh

    Learning Outcomes

 

After studying this module, you shall be able to

  • Understand the need for reciprocal lattice to analyse diffraction phenomenon
  • Learn the relationship and interatomic planes
  • Construct the reciprocal lattice corresponding to various lattices for example, BCC, FCC, hexagonal lattice
  • Define structure factor
  • The relation of structure factor with the intensity of the diffraction peaks
  • Understand why certain diffraction peaks are absent in some XRD patterns.

    Introduction:

 

To determine a real space lattice structure, the most popular experimental technique is some form of diffraction. Methods include x-ray, electron, atom, and neutron diffraction. The particles in these techniques have dual properties and behave as matter waves through the de Broglie relationship, λ = h/p, where λ is wavelength, h is Planck’s constant ( = 6.63 × 10-34J · s), and p is the magnitude of momentum. The propagation of a wave is described by the advancement of a wave front. Assuming a plane wave propagation, the wavevector k is perpendicular to the plane wave front. The relationship between p and k is p = ħ k. From the de Broglie relationship (λ =h/p and p =[h/(2π)] k), one obtains |k| =  2π/λ. Note that the wavevector k in has a unit of inverse length. It is convenient to define a reciprocal space lattice in the momentum space that is related to the real space lattice. The symmetry of a real space lattice and the symmetry of its reciprocal space lattice are related. The unit vectors in the reciprocal space lattice have a reciprocal relationship with the unit vectors in the real space lattice. We have introduced real-space lattice points, basic unit vectors, the direction of a real-space plane, and interplanar spacing d. Reciprocal space also consists of reciprocal lattice points and reciprocal vectors. We can relate real space and reciprocal space using geometry in an actual diffraction experiment. The diffraction of a wave involves an incoming wavevector kin and a scattered wavevector kout. The direction of kout differs from the direction of kin (except in the forward scattering case). The difference in direction is the scattering angle 2θ.

 

Figure 5.1: Wave scattering from a sample a. The scattering angle is 2θ, and the outgoing wavevector kout has a momentum change K, relative to the incoming wavevector kin. b Specular scattering where θout = θin = θ

 

See Fig. 5.1. If the scattering is elastic then |kin| = |kout| = |k| consider specular scattering, θin =θout = θ. The change in wavevectors or the momentum transfer is defined as K=kout –kin. Applying trigonometry as shown in Fig. 5.1, one obtains

 

In a scattering experiment, one knows the wavelength λ and can measure the scattering angle 2θ, and then the magnitude of K change can be obtained from above Eq.

 

Bragg Condition

 

We will see that this change of momentum K is related to interplanar spacing d in a real-space crystal as shown in figure 5.2. The incoming wavevector k , and it is scattered as ray 1. The outgoing wavevector k incident on the first plane (plane 1) with an angle θ out in is scattered specularly with θ out = θ. The same wave scattering from the second plane is ray 2. The interplanar spacing d is the perpendicular distance between the first and second planes or a to b. The path length of ray 2 travels more than that of ray 1. The path length difference is in hkl in hkl is

 

Fig 5.2: Bragg scattering from two parallel planes. qin = θout = θ for specular diffraction. Wave 2 travels 2dhkl sinθ further than wave 1

 

If this path-length difference is an integer number n of wavelength λ, then a constructed interference occurs to give the maximum intensity. This is Bragg’s law.

 

2d hkl sin qB  = nl

 

Combining above equations, we obtain the reciprocal relationship between the change of wavevectors KB and the interplanar spacing dhkl at the Bragg condition,

 

Reciprocal Space Basic Vectors and their Relationship to Real Space Basic Vectors

 

In general, a diffraction experiment involves a 3D sample. The previous derivation is for one dimension. In the following, the real space and reciprocal space relationship in terms of vectors in three dimensions will be derived. We have defined the position vector of a lattice point in a 3D crystal in real space by

 

 

where n1, n2, and n3 are integers and a, b, and c are real space unit vectors. In Fig. 5.3, we sketched one unit cell in real space with basic vectors a, b, and c. Similarly, we can also define a reciprocal lattice vector G(hkl)

 

 

where h, k, and l are the Miller indices of a crystal plane (hkl) and a are reciprocal unit vectors. Mathematically, one can show that G(hkl) • r is always an integer, n*, b*, and c*

 

 

 

 

 

 

Fig. 5.3: Reciprocal basic vectors a*, b*, and c*andtheir relationship to thereal-space basic vectors a, b, and c

 

From vector algebra, when the previous equation is satisfied, one obtains the reciprocal lattice vectors

 

 

 

Where a·(b × c) is the volume V of a unit cell in real space. See Fig. 3.3. The previous relationships can be rewritten as

 

 

 

This means vector a* is the cross product of b and c or a* is perpendicular to the plane consisting of b and c. Using the right-hand rule, one obtains a Similarly, b* is obtained from the cross product of c and a, and c, shown in Fig. 5.3. c* is obtained from the cross product of a and b. Vectors a, b, and c are related to a *, b as*, and c*

 

 

 

 

The magnitude of a* is inversely proportional to the magnitude of a. The same relationship is true for b* and c*. This means the size of a reciprocal lattice unit cell is inversely proportional to the size of the real space unit cell.

 

One can obtain the reciprocal unit vectors a*, b*, and c**, and c* from a, b, and c in the previous relationships. A reciprocal lattice can be generated by G(hkl) =ha*+kb*+lc*, where h, k, and l are integers. We illustrate the relationship between the reciprocal unit vectors and the real space unit vectors in a two-dimensional lattice shown in Fig. 3.4. A two-dimensional real space unit mesh consists of unit vectors a and b that are parallel to the page. The aisperpendicular to b, and the b* is perpendicular to a. Also, the length of a Projected on a is 2π/a and is the inverse of the length of a. Also, b * and b are related in a similar way. For example, if one has the (001) plane in real space, the reciprocal lattice direction in the reciprocal space can be determined as G(hkl) =G(001) =c * since h =0, k =0, and l =1. This means G(001) is perpendicular to a and b (because of the cross product of a × b) and its magnitude is inversely proportional to the magnitude of c.**

 

Fig. 5.4 Relationship between real-space basic vectors a and b and reciprocal-space basic vectors a* and b*

 

space unit mesh consists of unit vectors a and b that are parallel to the page. The a is perpendicular to b, and the b * is perpendicular to a. Also, the length of a projected on a is 2p/a and is the inverse of the length of a. Also, b * and b are related in a similar way. For example, if one has the (001) plane in real space, the reciprocal lattice direction in the reciprocal space can be determined as G(hkl) =G(001) =c * since h =0, k =0, and l =1. This means G(001) is perpendicular to a and b (because of the cross product of a × b) and its magnitude is inversely proportional to the magnitude of c.*

 

Reciprocal Lattice Vector and its Relationship

to Interplanar Spacing

 

We can further examine the direction and magnitude of the reciprocal lattice vector G(hkl) for a general case. The (hkl) plane is defined as a plane intercepting the a, b, and c axes at a/h,b/k, and c/l, respectively. The plane ABC shown in Fig. 3.5 represents the (hkl) plane. The vectors AB and AC equal b/k –a/h and c/l –a/h, respectively. The cross product AB × AC is a vector G(hkl) perpendicular to the (hkl) plane or parallel to the normal of the (hkl) plane

 

 

 

The unit vector

 

 

 

To obtain the shortest distance dhkl between a family of (hkl) planes, that is, the distance from point O to the (hkl) plane, one can take the dot product of OA or any vector in the (hkl) plane and n.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 5.5 Reciprocal latticevector G(hkl) is perpendicular to the (hkl) plane consisting of vectors AB and AC with interception a/h,b/k, and c/l on x-, y-, and z-axes

 

Structure Factor

 

In condensed matter physics and crystallography, the static structure factor (or structure factor) is a mathematical description of how a material scatters incident radiation. The structure factor is a particularly useful tool in the interpretation of interference patterns obtained in X-ray, electron and neutron diffraction experiments. The static structure factor is measured without resolving the energy of scattered photons/electrons/neutrons.

 

In last chapter we learned about x-ray diffraction in reciprocal space. Structure factor is the signature of the real structure expressed in reciprocal space. In other words, the structure factor depends on the arrangement of atoms in a unit cell. In x-ray diffraction we perform measurements in reciprocal space, therefore it is convenient to work in reciprocal space. Here we define a new parameter called structure factor which can predict the intensity of peaks and their extinction from the diffraction spectrum. The structure factor is defined as

 

 

 

 

 

 

 

Structure Factor for BCC lattice

 

Fig 5.6: A unit cell of BCC lattice

 

The bcc basis referred to the cubic cell has identical atoms at x1 = y1 = z1 = 0 and x2 = y2 = z2 = ½. Thus above equation becomes

 

 

 

 

 

 

The idea is look for atoms per unit cell and carry the sum over all atoms in a unit cell. For example BCC structure has 2 atoms per unit cell and FCC structure has 4 atoms per unit cell. In BCC the sum will be carried over 2 atoms whereas in FCC it will be carried over 4 atoms.

 

Structure Factor for FCC lattice

 

Fig: 5.7: A unit cell of FCC lattice

 

 

The basis of the FCC structure referred to the cubic cell has identical atoms at (000); (0, 0.5, 0.5); (0.5, 0, 0.5); (0.5, 0.5, 0), Thus structure factor becomes

 

 

 

 

SG = 4f if h, k, l are either all odd or all even

 

SG = 0, otherwise

 

When structure factor becomes zero that means the diffraction peak corresponding to that plane (h k l) will become extinct.

 

Value Addition:

 

Do You Know?

  1. The concept of reciprocal lattice is very useful in x-ray diffraction measurements. For a measurement in a real space we can do it with meter scale, microscope and electron microscopy depending on the length scale which is in question. In real space measurement we scan the location in real space and measure the property. Similarly in x-ray diffraction measurement we scan the reciprocal space ( | K |= 2 | kin | sin q ) when we scan angle q and measure the property (Intensity). Therefore X-ray measurement is a measurement of reciprocal space. Thus we obtain the reciprocal lattice associated with the crystalline structure. The Fourier transform of the reciprocal lattice gives us the knowledge of real space lattice.
  2. Structure factors are very useful in x-ray crystallography. Structure factors in solids are intimately related to the atomic arrangement of solids. Looking at the structure factor it is possible to determine if the material is crystalline or amorphous. Also the intensity of the diffraction peak can be explained from structure factor. Using several diffraction patterns from various angles, the measured structure factors are used to determine the structure of very complex molecules such as proteins and DNA etc.

    1. Suggested Reading

 

For More Details (on this topic and other topics discussed in Text Module) See

  1. Neil W. Ashcroft and N. David Mermin, Solid State Physics, Thomson Brooks/Cole, Eastern Press Bangalore (India) 2005
  2. Charles Kittel, Introduction to Solid State Physics, John Wiley & Sons, Singapore 1999
  3. Jens Als-Nielsen and Des McMorrow, John Wiley & Sons, UK, 2011
  4. Wikipedia