C12T (Solid State Physics) Topic Crystal Structure (Part 4)

Dr. Avradip Pradhan, Assistant Professor,

Department of Physics, Narajole Raj College, Narajole.

PAPER: C12T (Solid State Physics) TOPIC(s): Crystal Structure (Part – 4)

C12T (Solid State Physics)

Topic – Crystal Structure (Part – 4)

We have already discussed part 3 of this e-report.

Now let us continue part 4 of it.

Laue Equations:

The outcome of diffraction theory expressed as , may be written in

another way to give what are called the Laue Equations. These are important

because of their geometrical representation. We need to take the scalar product

of the given equation successively with , and (using the fact that

and can be expressed as

)

to obtain the Laue Equations, written as

with , and being three positive or negative integers.

All these equations have a simple geometrical interpretation. The first equation

tells us that has a few specific values of projections along

the direction of the first crystal axis . This is possible only when lies on a

certain cone about the direction of . Similarly the second equation tells us that

lies on a cone about as well, and the third equation requires that lies

on a cone about . Thus, at a Bragg reflection must satisfy all three

equations. So, it must lie at the common line of intersection of the three cones,

which is a severe condition that can be satisfied only by systematic sweeping or

searching in wavelength or crystal orientation.




The Ewald Construction:

Fig. 1

P. P. Ewald gave the following simple geometrical construction that guided

experimentalists to develop various methods for deducing the crystal structure

from the observed diffraction pattern. A sphere of radius is drawn in the

reciprocal space with centre at A (as shown in Fig. 1). Let

represent the

wave vector of the incident X-ray beam terminating at the origin of the

reciprocal lattice C. The direction and magnitude of the wave vector of reflected

X-rays (say given by ) will be such that change in wave vector

equals a reciprocal lattice vector as per the Laue condition for diffraction.

Therefore, is a vector connecting C to another appropriate reciprocal lattice

point B. To satisfy the above geometrical requirements, B must lie on the

sphere’s surface and be given by AB. The whole construction is shown in

Fig. 1 and known as the Ewald construction. The corresponding sphere is called

as Ewald Sphere.

A

B

C




The Ewald construction implies the elastic X-ray scattering, since we know

the radius of the Ewald Sphere. With this, the vector relationships

shown in the sphere give the Laue condition that has been shown to

be equivalent to the Bragg’s Law. The angle is the Bragg angle. The

construction clearly establishes how rare it is to find the reciprocal lattice points

on the surface of the Ewald Sphere to get Bragg reflections from various planes

of a crystal. Different methods are used to create situations that render the

recording of these reflections feasible.

Brillouin Zone:

Fig. 2

Brillouin provided us with the statement of the diffraction condition that is most

widely used in Solid State Physics, which means in the description of electron

energy band theory and of the elementary excitations of other kinds. A Brillouin

Zone (to be precise, first Brillouin Zone) or in short, BZ is defined as a Wigner-

Seitz primitive cell in the reciprocal lattice. The construction method for such

type of cell in the direct lattice has been shown in previous e-report. The

Brillouin zone gives a vivid geometrical interpretation of the diffraction

condition . We divide both sides of this equation by 4 to obtain




or

We now work in reciprocal space, the space of the s and s. We select a

vector from the origin to a reciprocal lattice point. Then a plane is built

normal to this vector at its midpoint. This plane forms a part of a zone

boundary (shown in Fig. 2). An X-ray beam in the crystal will be diffracted if

its wave vector has the magnitude and direction required by the condition

given by

. For example, in the given figure any vector from the origin

to the plane 1, such as will satisfy the diffraction condition

. The

diffracted beam will then be in the direction , as we see that . Thus

the Brillouin construction exhibits all the wave vectors which can be Bragg

reflected by the crystal.

The set of planes that are the perpendicular bisectors of the reciprocal lattice

vectors is of general importance in the theory of wave propagation in crystals. A

wave whose wave vector drawn from the origin terminates on any of these

planes will satisfy the condition for diffraction. These planes divide the Fourier

space of the crystal into fragments. The central cell in the reciprocal lattice is of

special importance in the theory of solids, and we call it the first Brillouin Zone.

The first Brillouin Zone is the smallest volume entirely enclosed by planes that

are the perpendicular bisectors of the reciprocal lattice vectors drawn from the

origin.

Here we take an example of a one dimensional infinite crystal lattice and its

reciprocal lattices (as shown in Fig. 3). The crystal axis in the direct lattice is

given as . The basis vector in the reciprocal lattice is , of length equal

to

. The shortest reciprocal lattice vectors from the origin are and .

The perpendicular bisectors of these vectors form the boundaries of the first

Brillouin Zone (shown as the shaded area). The zone boundaries are located at




with the zone centre at . The next zones are called as second

zone, third zone etc.

Fig. 3

Construction of Reciprocal Lattice of a few Popular Lattices:

(a) Simple Cubic (SC) Lattice. The primitive translation vectors of a simple

cubic lattice may be taken as the set given by , and .

The volume of the cell is .

The primitive translation vectors of the reciprocal lattice are found from the

standard prescription given earlier as

We therefore obtain

,

and

. Looking at the values

and mutual direction of the primitive vectors, it can be concluded that the




reciprocal lattice is itself a simple cubic lattice, now with a lattice constant of

.

The boundaries of the first Brillouin Zones are the planes normal to the six

reciprocal lattice vectors taken at their midpoints, at

,

, at

,

and at

,

. These six planes bound a

cube of edge

and of volume

. This cube forms the first BZ of the SC

crystal lattice.

Fig. 4

(b) Body Centred Cubic (BCC) Lattice. The primitive translation vectors of

the BCC lattice are given by

,

and

, where is the side of the conventional cube.

The primitive translations of the reciprocal lattice, defined previously can be

calculated as

,

and

. By a

comparison we see that these are just the primitive vectors of an FCC lattice of

lattice constant

. So an FCC lattice is the reciprocal lattice of the BCC lattice.

The volume of the primitive cell of the reciprocal lattice is

.




The first Brillouin Zone is the Wigner-Seitz cell of the reciprocal lattice which

contains one lattice point at the central point of the cell. Here this zone is

bounded by the planes normal to the 12 vectors (nearest neighbour vectors) at

their midpoints. Therefore the BZ is a regular 12-faced solid, a rhombic

dodecahedron, as shown in Fig. 4.

Fig. 5

(c) Face Centred Cubic (FCC) Lattice. The primitive translation vectors of

the FCC lattice are given by

,

and

The primitive translations of the reciprocal lattice, defined previously can be

calculated as

,

and

These are primitive translation vectors of a BCC lattice, so that the BCC lattice

is reciprocal to the FCC lattice. The volume of the primitive cell of the

reciprocal lattice is

.




The boundaries of the central cell in the reciprocal lattice are determined for the

most part by the eight planes normal to these vectors at their midpoints. But the

corners of the octahedron thus formed are cut by the planes that are the

perpendicular bisectors of six other reciprocal lattice vectors. Therefore the BZ

looks like a truncated octahedron as shown in Fig. 5.

Structure Factor & Atomic Form Factor:

When the diffraction condition is satisfied, the ·scattering amplitude for

a crystal of cells may be written as

The quantity

is called the structure factor and is

defined as an integral over a single cell, with at one corner. Often it is

useful to write the electron concentration as the superposition of electron

concentration functions associated with each atom of the cell. If is the

vector to the centre of atom , then the function defines the

contribution of that atom to the electron concentration at . The total electron

concentration at due to all atoms in the single cell is the sum

The structure factor may now be written as integrals over the atoms of a

cell

or

or

Here

is called as the

atomic form factor of the lattice. If is an atomic property, will also be

an atomic property.




So, the structure factor can be written as

. Using the

expressions of as

and , we

finally get

We will take an example of a BCC lattice. The BCC basis referred to the cubic

cell has two identical atoms at ( ) and at (

). Therefore, can be

written as

Here is the atomic form factor is a single atom. Here is same for both the

atoms as they are identical. The previous expression of shows the conditions

for maximum and minimum of it.

when is an odd integer: minimum condition

when is an even integer: maximum condition

Metallic sodium has a BCC structure. The previous discussion clearly explains

why the X-ray diffraction pattern of sodium does not contain lines such as

( ), ( ), ( ) or ( ), but lines such as ( ), ( ) and ( ) will be

present.

Out next example will be of an FCC lattice, whose basis consists of 4 identical

atoms present at ( ), (

), (

) and (

). Therefore, we get as

The conditions for maximum and minimum of can be obtained as

when two of , and are odd or even integer: minimum condition

when all of , and are odd or even integer: maximum condition

KBr has an FCC structure. Its X-ray diffraction pattern contains the peaks from

( ), ( ), ( ) whereas peaks from ( ), ( ), ( ) are missing.




Reference(s):

Introduction to Solid State Physics, Charles Kittel, Wiley

Elements of Solid State Physics, J.P. Srivastava, PHI Learning

(All the figures have been collected from the above mentioned references)

C12T (Solid State Physics) Topic Crystal Structure (Part 4)

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