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NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
UNIT V
X-ray Diffraction and Defects in Crystals: Bragg’s law, X-ray diffraction
methods: Laue method, powder method; point defects: vacancies, substitutional,
interstitial, Frenkel and Schottky defects, line defects (qualitative) and Burger’s
vector, surface defects: stacking faults, twin, tilt and grain boundaries.
1. State and explain Bragg’s law. (Dec 2016)(2004,2006,2010,2011)
Statement: Bragg ’s law states that the X -rays reflected from different parallel
planes of a crystal interfere constructively when the path difference is integral
multiple of the wavelength of X-rays. The wavelength λ should not exceed
twice the inter planar spacing for diffraction to occur.
2. Derivation or explanation of Bragg’s law Let us consider a set of three parallel
planes MN, PQ and RS having inter
planar spacing ‘d’. Suppose two
parallel rays AB and DE are incident
at glancing angle θ. Suppose AB is
reflected from the plane MN and the
ray DE is reflected from the plane
PQ respectively. Suppose the
reflected rays are BC and EF. These
two rays interact constructively if the
path difference between them is nλ.
To find the phase difference draw BG and BH normals on the second ray. The
path difference between these rays is GE + EH. But GE = EH = d sin θ.
Hence the total path difference GE + EH = 2 d sin θ. For maximum intensity
2dsinθ = nλ. This is called Bragg’s law.
Problem 1: calculate the Bragg angle at which the electrons accelerated from
rest through a potential difference of 80 V will be diffracted from the (111)
planes of a FCC crystal of lattice parameter 0.35 nm.(J 2017)
3. Describe Bragg’s spectrometer
Schematic diagram of Bragg’s
spectrometer is shown in the
figure.S1 and S2 are two lead slits to
produce a narrow beam of X-rays. A
crystal D can be mounted at the
center of a turntable. The turntable
can be rotated about a vertical axis
passing through its center and the
position can be noted on a circular
2
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
scale C. The reflected X- rays can be detected with the help of an ionization
chamber E mounted on the arm. The arm is capable of rotation about the axis
of turntable. The turntable and the arm are so linked together such that when
the turntable rotates through an angle θ the arm turns through an angle 2θ.
4. Describe how do you determine the lattice type by using Bragg’s
spectrometer.
A narrow beam of monochromatic X- rays of
wavelength λ is made to be incident on the
crystal D. the angles for 1st order spectrum from
(100), (110) and (111) planes are noted. Suppose
they are θ1, θ2 and θ3 respectively.
Since all the spectra are obtained in the first order
we have
2d100 sinθ1 = 2d110 sinθ2 = 2d111 sinθ3= λ
From the above
321
111110100
θsin:θsin:θsin1
:1
:1
ddd
The ratios 111110100
1:
1:
1
ddd for different cubic lattices are given below
S.No. Cubic lattice
111110100
1:
1:
1
ddd
1 Simple cubic 3:2:1
2 Body centered cubic 3:
2
1:1
3 Face centered cubic 2/3:2:1
Problem 2: for a certain crystal first order reflection maxima occur at 5.9o,8.4o
and 5.2o for (100), (110) and (111) planes respectively. Find the type of the
lattice.
Hint: 321
111110100
θsin:θsin:θsin1
:1
:1
ddd
2.5sin:4.8sin:9.5sin1
:1
:1
111110100
ddd
= 0.1028 : 0.1461: 0.9
= 1.028 : 1.461 : 0.906 = 1.00: 1.421 : 0.881
= 2/3:2:1
The above crystal is face centered cubic crystal
Problem 3: A beam of X-rays is incident on an ionic crystal with lattice
spacing 0.313 nm. Calculate the wavelength of X-rays if the first order Bragg
3
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
reflection takes place at a glancing angle of 7°48’. (June 2011) [Answer: 0.085
nm]
Hint: 2d sin θ = n λ.
Problem4: x-rays of wavelength 0.36 Å are diffracted in a Bragg’s spectrometer
at an angle 4o48`. Find the effective value of atomic spacing. (June 2017)
[2.15 Å]
Hint: 2d sin θ = nλ
Problem 5: Monochromatic X-rays of λ = 1.5 A.U are incident on a crystal
face having an interplanar spacing of 1.6 A.U. Find the highest order for which
Bragg’s reflection maximum can be seen. (Ans = 2) (June 2006)
Problem 6: The Bragg’s angle in the first order for (220) reflection from
nickel (FCC) is 38.2°C, when X-ray of wavelength 1.54Å are employed in a
diffraction experiment. Determine the lattice parameter of nickel. (Ans: 3.522
Å) (June 2008)
Problem 7: When a monochromatic X-ray beam of wavelength 0.1542 nm is
used, the first order reflection from (101) planes occurs at θ. If the lattice
parameter is 0.433 nm, find the value of θ. (Dec. 2010) [14o35`]
5.Describe the powder method of x-ray diffraction to calculate the lattice
constant of a cubic system(M 2003,2004,2006,Dec 2016)
6. Give a brief account of powder method for crystal structure analysis.(J
2017)
In most of the cases we do not single
crystals of sufficiently large size. So
we prefer powder diffraction method.
For this purpose we use a cylindrical
camera called Debye-Scherer camera.
It is a cylindrical camera with two
narrow openings for the entry and exit
of x-rays. A photographic film is
placed along the circumference from
inside. The crystalline sample is
powdered and taken in a capillary
glass tube. The specimen is positioned
at the center of the camera. A narrow
beam of monochromatic x-rays is
allowed to be incident on the specimen. Some of the x-rays are diffracted from
differ planes oriented at random. These rays spread in the form of cone. The
base of the cone is intercepted by the photographic film. On the film we get a
pair of arcs in diametrically opposite positions. Reflections from different
planes produce different sets of arcs. If S1 is the distance between two
respective arcs then the angle of reflection θ1 = S1/4R where R is the radius of
4
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
the camera. Similarly we can measure the angles θ2, θ3 , θ4 , θ5 etc. by
applying Bragg’s relation 2d sin θ = n λ we can find the inter planar distances.
From these interplanar spacings finally we can evaluate lattice parameters and
the type of the crystal.
7. Describe the Laue method to determine the crystal structure of a unit
cell. (M 2003, 2004, 2010, Dec 2016)
8. Describe the Laue method of determination of crystal structure
(J 2017)
This method is used to study the orientation of the crystal and verify crystal
symmetry. In this method a single crystal specimen is held fixed and it is
irradiated with white X-rays. (X-radiation with continuous wavelength is called
white X rays). The rays diffracted through the crystal are made to fall on a
photographic film placed on the other side as shown in Figure (a). Since the
crystal is fixed in position, the angles of diffraction θ are also fixed. If the
photographic film is at a distance D from the crystal, it will show reflection
spots at various distances R from the direct beam Fig.(b)
R = D tan 2θ
Each spot is due to all the orders of reflection n = 1, 2, 3. . . super imposed
from a single plane.
With the help of projection methods (such as the stereographic, gnomonic and
stereognomonic projections) we can find the crystal structure of unit cell. Laue
method can be used to study crystal orientation. The position of the Laue spot
on the photographic film depends on the orientation of the crystal relative to
the incident beam direction.
9. What are Laue spots? Explain (June 2017)
5
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
Laue spots: when a narrow beam of x-rays is passed through a crystal the
scattered x-rays emerge in specific directions. They are intercepted by a
photographic film placed in their path. When this film is developed we get
bright spots called Laue spots. The pattern of the Laue spots is the
characteristic of the crystal.
10. What are defects in crystal? Classify the defects.
In an ideal crystal there will be perfect periodic arrangement of atoms. Any
deviation from the periodic arrangement of atoms in real crystals is called
crystal defect or imperfection.
The defects are classified based on their dimensions as follows.
1. Point defects (zero dimensional defects).
2. Line defects or dislocations (one dimensional defects).
3. Planar defects (two dimensional defects).
4. Volume defects (three dimensional defects).
Explain point defects
Point defect is a defect localised to a point in a crystal. These can be further
divided in to
a) substitutional impurity b) interstitial impurity c) vacancy or Schottky defect
d) Frenkel defect.
a) substitutional impurity : if an tom of some other substance
occupies the lattice position in a crystal is substitutional
impurity as shown. substitutional impurity atom will be
of the order of atom of the crystal. E.g., p-type or n-type
crystals
b) interstitial impurity: if the impurity atom is very small
when compared with the atom of the crystal it will
occupy the space between the atoms
or interstice.E.g., carbon in steel
c) Vacancy: if an atom is missing from a lattice position
it is call vacancy.A vacancy decreases the density of the
material.
d) Schottky defect and Frenkel defect come under the category of point
defects.
Schottky defect : in this defect a pair of cation and a neighbouring anion will
be missing from their lattice positions. In this defect electric neutrality is not
disturbed whereas density decreases.
6
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
Frenkel defect: in this defect a cation will displaced from its lattice position to
a nearby interstitial space. This defect will not change electric neutrality as
well as density.
11. Explain edge and screw dislocations with neat diagrams (June 2017)
Line defects or dislocations: if a crystal defect spreads along a line it is called
line defect or dislocation. There are two kinds of dislocations.
1) edge dislocation 2) screw dislocation.
1) Edge dislocation: in this defect an
atomic plane starting from one edge fails
to reach the other edge. It gets terminated
in the middle as shown.
Screw dislocation: Screw dislocation results from a
displacement of the atoms in one part of a crystal
relative to the rest of the crystal, forming a spiral
ramp around the dislocation line as shown.
12. What is burgers circuit.
Burger’s circuit is generally a rectangular closed path surrounding a point. If
we get a perfect rectangle the crystal is perfect. Otherwise there is dislocation
present.
13. Draw the burgers circuits for an a) edge dislocation and b) screw
dislocation (June 2017)
a)
b)
7
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
14. What is the significance of Burger’s vector (June 2017)
Burger’s vector (b) describes dislocation quantitatively. It tells us the direction
and magnitude of the lattice distortion associated with a dislocation in a
crystal. It describes the slip which one part of the crystal undergoes relative to
the rest of the crystal.
The Burger’s vector of an edge dislocation is perpendicular to the dislocation
line and that of screw dislocation is parallel to the dislocation line.
15. What are grain boundaries (June 2017)
A grain boundary is a general planar defect that separates regions of different
crystalline orientations within a polycrystalline solid. Solids are formed when
the melt of the materials are cooled. Depending on the rate of cooling either
single crystal or polycrystalline solids form. Grain boundaries have extra
energy stored in them. Hence they are chemically more reactive than the grains
themselves.
16. What is meant by stacking faults?
Stacking faults are planar surface imperfections caused by fault in the stacking
sequence of atomic planes (or layers) in crystals. In FCC we have three
different stacking layers ABC.
In a perfect crystal we have ABCABCABCABC
with a staking fault we have ABCABCBCABC
17. Explain twin boundary in a crystal.
If the atomic arrangement on one side of twin boundary is a mirror reflection
of the arrangement on the other side is called twin boundary.The region
between the pair of boundaries is called twinned region.
Fill in the Blanks
l . All equally spaced parallel planes have the same ……(Miller indices)
2. Separation between adjacent lattice planes in a cubic crystal is given
by……….(d = 222 lkh
a
)
3. The ratio d100 : d110 : d111for a simple cubic structure is …….( 3/1:2/1:1 )
4. The ratio d100 : d110 : d111for a FCC structure is ………….( 3/2:2/1:1 )
5. The ratio d100 : d110 : d111for a BCC structure is ………….( 3/1:2:1 )
8
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
6. Crystal has to be considered as ……………….space grating. (3-D)
7. Bragg’s relation is ……………(2d sin θ = n λ)
8. If ‘a’ is the lattice parameter, the interplanar spacing for (100)
planes…….(a)
9. If ‘a’ is the lattice parameter, the interplanar spacing for (110) planes…
(a/√2)
10. If ‘a’ is the lattice parameter, the interplanar spacing for (111)
planes…(a/√3)
11.If ‘a’ is the lattice parameter, the interplanar spacing for (210) planes…(a/3)
12. In Laue X-ray diffraction method the specimen used is……….. (single)
13. In Laue X-ray diffraction method the x-radiation used is………..
(continuous)
14. In Debye Scherrer camera the specimen used is in …….form.(powder)
15. For powder diffraction method we use ……………..camera.(Debye
Scherrer)
16.. A foreign atom that substitutes for or replaces a parent atom in the crystal
is referred to as………….. impurity. (substitutional)
17. A small sized atom occupying the void space in the parent crystal without
disturbing the parent atoms from their regular sites is an………..impurity.
(interstitial)
18. In ionic crystals, an ion displaced from a regular site to an interstitial site is
called ……………imperfection. (Frenkel defect)
19. A pair of one cation and one onion can be missing from an ionic crystals.
Such a pair of vacant ion sites is called……… imperfection. (Schottky defect)
20. Errors in charge distribution in solids are called……defects.(electronic)
9
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
21. Schottky defects and Frenkel defects come under ……defects.(point
defect)
22. Line defects are………….. dimensional imperfections. (one dimensional)
23. Line imperfections are also called ……………. (dislocation)
24. The magnitude and the direction of the displacement are defined by a
vector called the …………vector. (Burger’s vector)
25. A displacement of the atoms in one part of a crystal relative to the rest of
the crystal, forming a spiral ramp around the dislocation line is called ……….
dislocation. (screw dislocation)
26. Regions of distortions that lie about a surface having a thickness of a few
atomic diameters are referred to …………….. imperfections.(surface defect)
27. Planar surface imperfections caused by fault in the stacking sequence of
atomic planes (or layers) in crystals is referred to as …………(stacking fault)
28. In semiconductors, electrical conductivity …….. with increase of
pentavalent or trivalent impurity concentration. (increases)
29. The presence of impurity atoms in the crystals lattice results in
characteristic colours to the crystals. There are called …………..(colour
centers)
30. Copper added to gold increases………….. of gold so that it can be drawn
into wires.(ductility)
31. In Bragg’s spectrometer the specimen used is …………..(single crystal)
MULTIPLE CHOICE QUESTIONS
1. Choose the correct statement
a) In Debye Scherrer method of X-ray diffraction sample is used in
powder form
b) In Debye Scherrer method of X-ray diffraction sample is used in the form of
a single crystal
c) Debye Scherrer method of X-ray diffraction is used for studying the crystal
orientation and symmetry
d) In Debye Scherrer method white X-radiation is used
2. If a foreign atom replaces a parent atom in the crystal, it is called
10
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
a) Frenkel defect b) Schottky defect
c) substitutional impurity d) electronic defect
3. In ionic crystals, an ion displaced from a regular site to an interstitial site is
called
a) Frenkel defect b) Schottky defect
c) substitutional defect d) electronic defect
4. A pair of cation and anion missing from an ionic crystal leaving vacant ion
sites is called
a) Frenkel defect b) Schottky defect
c) substitutional defect d) electronic defect .
5. Frenkel defect is a
a) point defect b) line defect
c) surface defect d) volume defect
6. Schottky defect is a
a) point defect b) line defect
c) surface defect d) volume defect
7. When the composition of an ionic crystal does not correspond to the exact
stoichiometric formula, then ----------arise.
a) Frenkel defects b) Schottky defects
c) substitutional defects d) electronic defects
8. Burgers vector is to indicate the magnitude of
a) point defects b) line defects
c) surface defects d) volume defects
9. Stacking fault is a
a) point defect b) line defect
c) surface defect d) volume defect
10. Edge dislocation is a
a) point defect b) line defect
c) surface defect d) volume defect
11. Screw dislocation is a
a) point defect b) line defect
c) surface defect d) volume defect
12. Twin boundary is a
11
NAWAB SHAH ALAM KHAN COLLEGE OF ENGINEERING & TECHNOLOGY
Prof. S.M.ASADULLAH 9290233597 NSAKCET
a) point defect b) line defect
c) surface defect d) volume defect