13 Ewald Construction and vector form of Bragg equation

Prof. P. N. Kotru

13.1 Ewald Construction.

The methods of analysing x- ray diffraction data are based on the shape and size of the direct cell of crystalline matter. However, the data, particularly single crystal diffraction data, are more conveniently handled in terms of the reciprocal lattice and Ewald sphere concepts introduced in 1921 by P. Ewald. The reciprocal lattice, as already explained, is an extension of Miller Index notation for crystallographic planes which is constructed by erecting vectors, from an arbitrary origin within the crystal, perpendicular to the various crystallographic planes in the crystal lattice of length σhkl given by σhkl = k/dhkl, where k is a positive constant, usually taken an unity, but sometimes the x-ray wavelength λ being used. It is, therefore, very essential and important too to understand the significance of reciprocal lattice in x-ray diffraction.

During our discussion on reciprocal lattice, it has been established that crystal lattices and reciprocal lattices are related through equations {(12.9-12.11 of section 12.2 and 12.38-12.40 of section 12.4 in module XII)}. While the crystal lattice is a lattice in the real space, the reciprocal lattice is a lattice in Fourier space which is governed by the equation: exp (I π G r ) = 1,……………13.1 where , G is known as reciprocal lattice vector.

In the direct crystal lattice, the points are given by:

where, h, k and l are integers.

The reciprocal lattice points or reciprocal lattice vector G in Fourier space is given by:

where h/ , k/ and l/ are integers

The scalar product of direct lattice vector and the reciprocal lattice vector:

Referring to equation 7.4 of module VII, we learn that it is only when three equations are satisfied for integral multiples of h/, k/ and l/ that we can obtain strong diffracted beam. One can summarily put the significant points as follows:

(i) The x – ray diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal in the same way

as we look at microscopic image as a map of real crystal structure.

(ii) In a crystal there is a periodic arrangement of atoms and the atoms are associated with electrons.

Reciprocal lattice is a concept that makes us to understand the wave mechanical behaviour of electrons in a

periodic crystal lattice.

Consider a diffraction of x-rays from hkl planes with interplanar spacing as dhkl incident at a glancing angle Өhkl, then Bragg equation can be written as:

We may now imagine construction of a right angled triangle with one side λ/dhkl and 2 as its hypotenuse and inscribe such a triangle inside a circle of unit radius as shown in figure 13.1(a)

Figure 13.1(a,b) : Construction of right angled triangle with one side λ/d hkl and 2 as its hypotenuse inscribed inside a circle of unit radius

The operation can be explained through figure 13.1(b) as follows:

a) The diameter of the circle is the direction of incident beam of x-rays.

b) Through the origin of the circle draw a line parallel to LQ This line makes an angle incident  beam  and  as 

such  represents  a  crystallographic  plane  hkl Өhkl with that follows the Bragg condition of diffraction and

giving Өhkl as the Bragg angle.

c) Now, connect points P and Q through a line PQ which makes an angle Өhkl with the crystal plane .PQ,

therefore, represents the direction of diffracted beam. The line OQ represents the reciprocal lattice vector

to the reciprocal lattice point Q which falls on the circumference of the circle. The reciprocal lattice vector

σhkl originates from a point on the circle where the direct beam leaves the circle.

In the three dimensional case the circle will turn out to be a sphere. If the sphere, known as Ewalds sphere, does not pass through any points it will suggest that the particular wavelength in question will not get diffracted by the crystal in that orientation and readjustment of orientation will have to be carried out for diffraction to occur. No x-ray diffraction can occur if λ> 2a because a sphere with radius PO < 1/2a cannot pass through any point on the lattice and as such the above said construction cannot be done.

The relation of the reciprocal lattice to the direct space lattice is such that the planes in the one are perpendicular to the rows of points in the other and the plane spacing in one is 2π times the reciprocal point spacing in the other. The general reciprocal lattice vector G has magnitude 2π/d, where d is the spacing of planes of Miller indices (hkl) of the direct space lattice. This is known as Ewald’s construction.

On rotating a crystal, both the direct lattice as well as reciprocal lattice get rotated. It is also learnt that while vectors in the direct lattice have the dimensions of length, they have the dimensions of length─1 in the reciprocal lattice. Considering direct lattice, the interplanar spacing d is related to the glancing angle Ө and the wave length of x-rays through Bragg’s law given as 2d sin Ө = λ. This law can be expressed in reciprocal lattice as well which is of great significance so far as crystallography is concerned.

Let us consider a set of parallel and equidistant lattice planes around O ( the origin) as shown in figure 13.2 (a).The x-ray beam is incident say along LO making an angle , say γ,with a set of parallel planes represented by XX/ and lines parallel to it. The interplanar distance is d as shown in the figure.

Figure 13.2

To construct reciprocal lattice points for these planes we choose the constant of proportionality (or magnification factor), for the sake of convenience, as λ so that the reciprocal lattice point Q along OR which is perpendicular to the set of planes XX/ is at a distance λ/d from the origin O. Therefore, OQ = λ/d. Now, a point P which is at a unit distance from the origin O is taken as centre of a circle of radius OP. Therefore, LP= PO = 1.

Let us assume that the reciprocal lattice point Q is located on the circle of reflection. In that case the lattice planes represented by the reciprocal lattice point Q are appropriately oriented planes for Bragg reflection.

It suggests that the set of planes which are parallel to XX/ and identified by the reciprocal lattice point Q satisfy Bragg’s law.

Now, P is the centre of the circle and PQ is the line joining the reciprocal lattice point to the centre of the circle. PQ is parallel to the direction of reflection OM.It means that the lattice planes are appropriately oriented for Bragg diffraction if their reciprocal lattice point lies on the circle of reflection .The direction of diffraction is given by the line joining the origin to the reciprocal lattice point. This circle is, therefore, known as circle of reflection which, if considered in three dimensions, will be substituted by a sphere. The above said construction by Ewald and the above described sphere of reflection is named after him as Ewald Sphere.

Figure 13.2(b) represents the geometrical construction for Bragg’s law as shown in a different way. Considering the triangle OPQ of figure 13.2 (b) let the incident x-ray beam PO be represented by vector S and the diffracted beam PQ by vector S/.

For the sake of convenience 1/λ is used instead of 1 as the radius of the reflection circle and constant of proportionality M as 1.

13.2 Vector form of Bragg Equation

Crystal, as we know, has a regular arrangement of atoms and there are a large number of free electrons, particularly more so in metallic crystals. The motion of free electrons more or less resemble the motion of molecules of gas. The periodicity of lattice, however, affects the dynamics of free electrons. When atoms are brought together to make a crystal we have allowed energy bands separated by forbidden bands. The forbidden energy gaps arise because of the Bragg diffraction of electrons from the lattice planes of the crystal. Electrons are associated with wave like character and get diffracted from crystal planes in almost the same way as x-rays do. The reciprocal lattice concept can be easily applied to interpret diffraction effects of electron waves. In the discussion on application of reciprocal lattice concept to solid state theory, it is convenient to take the proportionality constant (or the magnification factor) of the reciprocal lattice to be equal to 2π instead of 1. Therefore, we can use expressions of earlier section regarding reciprocal lattice with the magnification that a factor of 2π is used.

Having explained the geometrical construction of Bragg’s law, using Ewald sphere in the theory of solid state is discussed in a way similar to the one as already described.

K is taken as wave vector of the incident wave and K/ as the wave vector of the Bragg diffracted wave as shown in figure 13.2(b)

Radius of Ewald sphere is taken as 2π/λ

References.,

1. Kittel, C. “ Introduction to Solid State Physics”, 4th Ed., Wiley, N.Y.1971.
2. Verma ,A.R.&Srivastava,O.N. “ Crystallography For Solid State Physics”Wiley Eastern Ltd.,
3. Guinier,A: “ X-ray Diffraction”, Freeman,San Francisco,1983.
4. Bacon,G.E. “ X-ray & Neutron Diffraction” Pergamon,N.Y.,1966.

Suggested Reading for more information on the subject.

1. Cohen, J.B. “ Diffraction Methods in Materials Science” ,Macmillan,N.Y,1966.
2. Zachariasen,W.H.” Theory of X-ray Diffraction in Crystals “, Wiley,N.Y.,1945.
3. Woolfson,M.M. “ X-ray Crystallography”,CambridgeVikas
4. Phillips.F.C. “ An Introduction to Crystallography”,Longmans,London.