5-14 Engineering Physics-I

The logarithmic term in the above equation can be simplified by applying Stirling’s approximation
log x ! = x log x – x.

 N! Ni !  N! Ni !
∴ log  ×  = log
 + log
 ( N − n) ! n ! ( N i − n) ! n !  ( N − n) ! n ! ( N i − n) ! n !

≅ N log N + Ni log Ni − (N − n) log (N − n) − (Ni − n) log (Ni − n) − 2n log n ___________ (5.32)

Substituting Equation (5.32) in (5.31), we have:

F = nEi − KBT [N log N + Ni log Ni − (N − n) log (N − n) − (Ni − n)

log (Ni − n) − 2 n log n] ___________ (5.33)

At thermal equilibrium, the change in free energy is minimum w.r.t. ‘n’, so we have:

 ∂F  ( N − n) ( N i − n) ___________ (5.34)
  = 0 = Ei − K BT log
 ∂n  T n2

( N − n) ( N i − n) Ei ( N − n) ( N i − n)
∴ Ei = K BT log (or) = log
n 2
K BT n2

Taking exponential on both sides, we get:

 E  ( N − n) ( N i − n)
exp  i  =
 K BT  n2

 − Ei 
n 2 = ( N − n) ( N i − n) exp   if n << Ni
 K BT 

 −Ei  1  −Ei 
n 2 ≈ N N i exp   (or) n = ( N N i ) 2 exp   ___________ (5.35)
 K BT   2 K BT 
 

The above equation shows that n is proportional to (NNi)½

5.5 Diffraction of X-rays by crystal planes

and Bragg’s law
The visible light rays when pass through a sharp edge of an object can form some bright regions inside
the geometrical shadow of the object. This is due to the bending nature of light, called diff raction.
Diff raction of visible light rays can also be produced using plane-ruled grating. This grating consists of
about 6000 lines/cm; so that the spacing between any two consecutive lines in the grating is of the order
of the wavelength of visible light used to produce diff raction. The wavelength of X-rays is of the order of
 X-ray Diffraction 5-15

an angstrom, so X-rays are unable to produce diff raction with plane optical grating. To produce diff raction
with X-rays, the spacing between the consecutive lines of grating should be of the order of few angstroms.
Practically, it is not possible to construct such a grating. In the year 1912, a German physicist Laue
suggested that the three-dimensional arrangement of atoms in a crystal can serve as a three-dimensional
grating. Inside the crystal, the spacing between the crystal planes can work as the transparent regions as
between lines in a ruled grating. Laue’s associates Friedrich and Knipping succeeded in diff racting X-rays
by passing through a thin crystal.
In 1913, W.L. Bragg and his son W.H. Bragg gave a simple interpretation of the diff raction pattern.
According to Bragg, the diff raction spots produced are due to the reflection of some of the incident X-rays
by various sets of parallel crystal planes. These planes are called Bragg’s planes. The Bragg’s interpretation is
explained in the following topic.
Bragg’s law: W.L. Bragg and W.H. Bragg considered the X-ray diff raction as the process of reflection of
X-rays by crystal planes as shown in Fig. 5.11. A monochromatic X-ray beam of wavelength λ is incident

Figure 5.11 Bragg’s law

A C

D
θ θ B θ θ F
Plane 1

θ θ d
P Q
Plane 2
E

Plane 3

Plane 4

at an angle θ to a family of Bragg planes. Let the interplanar spacing of crystal planes is ‘d ’. The dots in the
planes represent positions of atoms in the crystal. Every atom in the crystal is a source of scatterer of X-rays
incident on it. A part of the incident X-ray beam AB, incident on an atom at B in plane l, is scattered along
the direction BC. Similarly, a part of incident X-ray DE [in parallel to AB] falls on atom at E in plane 2 and
is scattered in the direction EF and it is parallel to BC. Let the beams AB and DE make an angle θ with the
Bragg’s planes. This angle θ is called the angle of diff raction or glancing angle.
If the path difference between the rays ABC and DEF is equal to λ, 2λ, 3λ… etc. or nλ, i.e., inte-
gral multiples of wavelength, where n = 1, 2, 3,… etc. are called first-order, second-order, third-order
… etc. maxima, respectively. As path difference is equal to nλ, then the rays reflected from consecutive
planes are in phase; so, constructive interference takes place among the reflected rays BC and EF, hence
the resulting diff racted ray is intense. On the other hand, if the path difference between the rays ABC and
DEF is λ/ 2, 3λ/ 2, 5λ/ 2, … etc., then the scattered rays BC and EF are out of phase so that destructive
 5-16 Engineering Physics-I

interference takes place and hence the resulting ray intensity is minimum. To find the path difference between
these rays, drop perpendiculars from B on DE and EF. The intersecting points of perpendiculars are P and
Q as shown in Fig. 5.11. The path difference between the rays is PE + QE. From the figure, we know that
BE is perpendicular to plane 1 and BP is perpendicular to AB. So, as the angle between ray AB and plane
1 is θ, then ∠PBE = ∠QBE = θ. In the triangle PBE, sin θ = PE/BE = PE/d or PE = d sin θ. Similarly,
EQ = d sin θ.

∴ For constructive interference, PE + EQ = nλ or d sin θ + d sin θ = nλ

i.e., 2d sin θ = nλ
The above equation is called Bragg’s law.

5.6 Powder method

X-ray powder method is usually carried for polycrystalline materials. The powder photograph is obtained in
the following way. The given polycrystalline material is ground to fine powder and this powder can be taken
either in a capillary tube made up of non-diff racting material or is just struck on a hair with small quantity of
binding material and fixed at the centre of cylindrical Debye-Scherrer camera as shown in Fig. 5.12(a).

(a) Debye-Scherrer cylindrical camera; (b) film mounted in camera and

Figure 5.12 (c) film on stretchout

Specimen

Collimator

Luminescent
screen
X-rays

Cones of diffracted rays

Film
(a)

Incident
beam

S
(c)

(b)
 X-ray Diffraction 5-17

A stripe of X-ray photographic film is arranged along the inner periphery of the camera. A beam
of monochromatic X-rays is passed through the collimator to obtain a narrow fine beam of X-rays. This beam
falls on the polycrystalline specimen and gets diff racted. The specimen contains very large number of small
crystallites oriented in random directions. So, all possible diff raction planes will be available for Bragg reflec-
tion to take place. Such reflections will take place from many sets of parallel planes lying at different angles
to the incident X-ray beam. Also, each set of planes gives not only first-order reflections but also of higher
orders as well. Since all orientations are equally likely, the reflected rays will form a cone whose axis lies along
the direction of the incident beam and whose semi-vertical angle is equal to twice the glancing angle (θ),
for that particular set of planes. For each set of planes and for each order, there will be such a cone of reflected
X-rays. There intersections with a photographic film sets with its plane normal to the incident beam, form
a series of concentric circular rings. In this case, a part of the reflected cone is recorded on the film and it is
a pair of arcs, the resulting pattern is shown in Fig. 5.12(c). Diameter of these rings or corresponding arcs is
recorded on the film, and using this the glancing angle and interplanar spacing of the crystalline substance can
be determined. Figure 5.12(b) shows the film mounted in the camera and the X-ray powder pattern obtained.
The film on spread-out is shown in Fig 5.12(c). The distance between any two corresponding arcs on the film
is indicated by the symbol S.
In case of cylindrical camera, the diff raction angle θ is proportional to S. Then,
S
θ= where R represents the radius of the camera.
4R
If S1, S2, S3 … etc. are the distances between symmetrical lines on the stretched film, then,
S1 S S
θ1 = , θ2 = 2 , θ3 = 3 ...
4R 4R 4R
Using these values of θn in Bragg’s equation nλ = 2 dhkl sin θn
where n = 1, 2, 3, … etc = order of diff raction
dhkl = interplanar spacing
θn = angle of diff raction for nth order
The interplanar spacing dhkl can be calculated.

5.7 Laue method

In Laue method, a narrow beam of white X-rays [usually in the wavelength range, 0.2 to 2.0 Å] is obtained by
passing X-rays through a collimator ‘C’. This beam is allowed to fall on a stationary single crystal ‘S’ as shown
in Fig. 5.13(a). The crystal act as a 3-dimensional diff raction grating to the incident beam. The processes of
reflection of X-rays by crystal planes is considered as X-ray diff raction. The diff raction phenomenon satisfies
Bragg’s law, nλ = 2d sin θ. where n = 1, 2, 3, … represent the order of diff raction, λ = wavelength of diff racted
X-rays from a system of crystal planes with interplanar spacing ‘d ’ and θ = glancing angle i.e., the angle made
by X-rays with a crystal plane. As the crystal is not rotated, so, the angle ‘θ ’ is fixed for a set of planes having
separation ‘d ’. Different sets of crystal planes satisfy Bragg’s law with different wavelengths of X-rays and
produce diff raction. The diff racted X-rays from a set of planes produce constructive interference, if they are in
phase and form an intense beam, and this produces dark spots on photographic film. If the diff racted rays are
out of phase, they produce destructive interference so that photographic film is unaffected.
 5-18 Engineering Physics-I

Figure 5.13 (a) X-ray diffraction by crystal plane and (b) Laue pattern for NaCl crystal

C
S
X-rays

Crystal
Transmission Laue film
(b)

Crystal
C

X-rays

Back-reflection
Laue film

(a)

Laue photograph is obtained either by allowing the transmitted diff racted rays or by back-reflected
diff racted rays on photographic film as shown in Fig. 5.13(b).
As we observe the diff racted film, the diff racted spots lie on certain curves. These curves are either ellipses
or hyperbolas on transmission Laue photograph and hyperbolas on back-reflection Laue photograph. The way
of arrangement of spots on a film is a characteristic property of the crystal. Laue method is useful to decide the
crystal symmetry and orientation of the internal arrangement of atoms/molecules in the crystal. Cell parameters
of a crystal cannot be determined using Laue method. For transmission Laue method, the crystal should be thin.
Laue method can be used to study imperfections or strains in the crystal. The presence of above defects
forms streaks instead of spots in the Laue photograph.

Formulae
1 a
1. d= 2. d=
2 2 2
h k l h + k2 + l 2
2

2
+ 2+ 2
a b c

 − EV   −E 
3. 4.  P 
n ≈ N exp   ← Metallic crystal n ≈ N exp   ← Schottky defect
 K BT  2 K
 B  T
   

1  − Ei 
5. n ≈ ( NN i ) 2 exp   ← Frenkel defect 6. 2d sin θ = nλ
 2 K BT 
 