nawab shah alam khan college of engineering &...

1

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



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



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



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



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



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



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



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



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



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


c) substitutional defect d) electronic defect

4. A pair of cation and anion missing from an ionic crystal leaving vacant ion

sites is called


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



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



10. Edge dislocation is a



11. Screw dislocation is a



12. Twin boundary is a

11





nawab shah alam khan college of engineering &...

Documents