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CHEMISTRY OF SOLID STATE Band theory of solids- non-stoichiometry-
point defects – linear defects- effects due to
dislocations-electrical properties of solids-conductor, insulator,
semiconductor-intrinsic-impurity semiconductors-optical
properties-lasers and phosphors-elementary study of liquid
crystals.

Difference between point group and space group – screw axis – glide plane - symmetry elements –relationship between molecular symmetry and crystallographic symmetry – The Concept of reciprocal lattice – X-ray diffraction by single crystal – rotating crystal – powder diffraction. Neutron diffraction: Elementary treatment – comparison with X-ray diffraction. Electron diffraction- Basic principle. Crystal Growth methods: From melt and solution (hydrothermal,

UNIT III: CHEMISTRY OF SOLID STATE II: DIFFRACTION METHODS

1. BAND THEORY OF SOLIDS

According to band theory the energy spectrum of materials contains conduction band and valence band. On the basis of distance between conduction band and valence band, the materials are classified in to three categories.

1. Conductors:

If there is no energy gap between conduction band and valence band, such materials are called conductors.

Examples: metals

2. Insulators:

Those materials in which the energy gap between conduction band and valence band is very high , are called insulators.

3. Semiconductors:

If the gap between conduction band and valence band is very low , the materials are called semiconductors.

Example: germanium and silicon

Compounds with non- integer values of atomic composition are called

non- stoichiometric compounds.

Example : Ni 0.999O

Impurities are the main reason

For example NaCl heated in Na vapour results Na 1.5 Cl

Stoichiometric Defects

The compounds in which the number of positive and negative ions are exactly in the ratios indicated by their chemical formulae are called stoichiometric compounds. The defects do not disturb the stoichiometry (the ratio of numbers of positive and negative ions) are called stoichiometric defects. These are of following types,

(a) Interstitial defect: This type of defect is caused due to the presence of ions in the normally vacant interstitial sites in the crystals.

(b) Schottky defect: This type of defect when equal number of cations and anions are missing from their lattice sites so that the electrical neutrality is maintained. This type of defect occurs in highly ionic compounds which have high co-ordination number and cations and anions of similar sizes. e.g., NaCl, KCl, CsCl and KBr etc.

(c) Frenkel defect: This type of defect arises when an ion is missing from its lattice site and occupies an interstitial position. The crystal as a whole remains electrically neutral because the number of anions and cations remain same. Since cations are usually smaller than anions, they occupy interstitial sites. This type of defect occurs in the compounds which have low co-ordination number and cations and anions of different sizes. e.g., ZnS, AgCl and AgI etc. Frenkel defect are not found in pure alkali metal halides because the cations due to larger size cannot get into the interstitial sites. In AgBr both Schottky and Frenkel defects occur simultaneously.

CRYSTAL IMPERFECTIONS( CRYSTAL DEFECTS)

Any deviation in a crystal from a perfect periodic lattice structure is called crystal defects. The three types of defects are

1. Point defects 2. Line defects( dislocations) 3. Surface defects(plane defects)

3. POINT DEFECTS

1. POINT DEFECTS

The deviation in a crystal,

from a perfect periodic lattice structure is localised in the vicinity of only few atoms, it is called point defects. The different point defects are

1.1 Vacancies

1.4 Schotky defects

1.5 Substitutional defects

1.3 FRENKEL DEFECTS (VACANCIY AND INTERSTITIAL DEFECTS):

When a missing atom, occupies the interstitial position, the defect caused is known as Frenkel defects. This is most common in ionic crystals in which the positive ions are smaller in size.

interstial

Fe 2+ O 2-Fe 2+ O 2-Fe 2+ O 2-

Fe 2+ vacancy

Fe 2+ O 2-Fe 2+ O 2-Fe 2+ O 2-

Number of Frenkel defects in a crystal can be calculated by the formula

N =

Where N total number of atoms and Ni number of interstitial positions

Derivation:

Let the energy required to displace an atom, from its proper position to an interstitial position be E1. If there are N atoms and Ni interstitial positions , then the number of ways in which ‘n’ Frenkel defects can be formed is given by

W = ×

The change in Helmholtz free energy by the creation of ‘n’ Frenkel defects is

A = nE – TS

= nE – T [ kln W]

= nE – T k ln [ × ]

= nE – T k [ ln N! + ln Ni! - 2 ln n! – ln( N-n)! – ln( Ni – n) !]

Using Sterling ‘s approximation ln N! = N ln N - N we get

A = nE – T k{[ N ln N – N] +[Ni ln Ni – Ni] - 2 [ nln n – n] –[ (N – n) ln( N-n)] –

(N – n) ] – [( Ni – n) ln( Ni – n) - ( Ni – n) ] }

= nE – T k{[ N ln N – N +Ni ln Ni - Ni - 2 nln n +2 n – (N – n) ln( N-n)+

(N – n) – ( Ni – n) ln( Ni – n) + ( Ni – n) }

= nE – T k{[ N ln N – N +Ni ln Ni - Ni - 2 nln n +2 n – (N – n) ln( N-n)+

(N – n) – ( Ni – n) ln( Ni – n) + ( Ni – n) }

= nE – T k{ N ln N +Ni ln Ni - 2 nln n – (N – n) ln( N-n)– ( Ni – n) ln( Ni – n) }

Differentiating with respect to ‘n’ at constant temperature,

) T = E - T k { -2 [n () + ln n] - [ ( N-n) ( + ln (N-n) (0-1)]

· [ ( Ni – n) × ( + ln ( Ni – n) ( 0-1)}

= E - T k { -2 -2 ln n] - [ -1 - ln (N-n) ] -[- 1- ln ( Ni – n) }

= E - T k { -2 -2 ln n] + 1 + ln (N-n) + 1+ ln ( Ni – n) }

= E - T k { -2 ln n + ln (N-n) + ln ( Ni – n) }

= E - T k { ln ( ) + ln [ (N-n) × ( Ni – n) ] }

= E - T k { ln }

When equilibrium is attained, the Helmholtz free energy is constant and its first derivative is equal to zero. i. e ) T = 0

∴0 = E - T k { ln }

E = T k { ln }

= { ln }

When N >>n, N- n ≈ N similarly, When Ni >>n, Ni- n ≈ Ni

Therefore the above equation becomes, = { ln }

Taking exponential on both sides,

=

∴ n = (NNi ) ½

This is the expression for the number of ways of forming the defects

SCHOTKY DEFECTS

When a positive as well as negative ions of a crystal are missing, the defect is known as Schotky defects.

In Schotky defect the displaced atom migrates in successive steps eventually settles at the surface. Since the number of missing positive ions and negative ions is same, the crystal remains as neutral

Na + Cl-

Na + Cl- Na + Na + Cl-

Na + Cl- Cl- Na + Cl-

Na + Cl- Na + Cl- Na + Cl-

Number of Schotky l defects in a crystal can be calculated by the formula

n = N×

Derivation:

Suppose a crystal contains N atoms and ‘n’Schotky defects are produced by removing ‘n’ cations and ‘n’ anions from the crystal. Let the energy required to displace an atom, from its proper position to an interstitial position be E1. The number of ways in which ‘n’ schotky defects can be formed is given by

W = ×

The change in Helmholtz free energy by the creation of ‘n’ Frenkel defects is

A = nE – TS

= nE – 2T k [ ln N! - ln n! – ln( N-n)! ]

Using Sterling ‘s approximation ln N! = N ln N - N we get

A = nE– 2T k{[ N ln N – N] - [ nln n – n] –[ (N – n) ln( N-n)] – (N – n) ]

= nE – 2T k{[ N ln N – N - nln n + n – (N – n) ln( N-n)+ (N – n)

= nE – 2T k{[ N ln N – N - nln n + n – (N – n) ln( N-n)+ (N – n) }

= nE – 2T k{ N ln N - nln n – (N – n) ln( N-n) }

Differentiating with respect to ‘n’ at constant temperature,

) T = E - 2T k { - [n () + ln n] - [ ( N-n) ( + ln (N-n) (0-1)]

= E - 2T k { -1 - ln n] - [ -1 - ln (N-n) ]

= E - 2 T k { -1 - ln n] + 1 + ln (N-n)

= E - 2T k { - ln n + ln (N-n) }

= E - 2T k { ln }

When equilibrium is attained, the Helmholtz free energy is constant and its first derivative is equal to zero. i. e ) T = 0

∴0 = E - 2T k ln

E = 2T k ln

Taking exponential on both sides,

=

∴ n = N ×

This is the expression for the number of ways of forming the defects

4. linear defects

7. CONDUCTOR, INSULATOR, SEMICONDUCTOR

1. Conductors:

If there is no energy gap between conduction band and valence band, such materials are called conductors.

Examples: metals

2. Insulators:

Those materials in which the energy gap between conduction band and valence band is very high , are called insulators.

3. Semiconductors:

If the gap between conduction band and valence band is very low , the materials are called semiconductors.

Example: germanium and silicon

8.INTRINSIC SEMICONDUCTORS:

A semi conductor which is pure and contains no impurity is known as intrinsic semiconductor.

9. IMPURITY SEMICONDUCTORS

Extrinsic Semiconductors:

A semiconducting material in which, the charge carriers originate from impurity atoms added to the material, is called extrinsic semiconductor or impurity semiconductor.

Theses are divided in to two types.

1 n- type semi conductor:

Pentavalent elements such as P, As, Sb , have five electrons in their outermost orbits. When any one such impurity is added to the intrinsic semi conductor, four electrons are engaged in covalent bonding with four neighbouring semi conductor atoms and the fifth electron is free. Free electron

2 p- type semi conductor:

Trivalent elements such as Al, Ga or In have three electrons in their outer most orbits. When such impurity is added to the intrinsic semi conductor, all the three electrons are engaged in covalent bonding with three neighbouring semi conductor atoms and creating a hole ( vacant electron site) on the semiconductor atom.

OPTICAL PROPERTIES

The optical properties of semiconductors have been studied extensively for their relevance to applications such as lasers, light-emitting diodes , and solar cells

LASERS AND PHOSPHORS

The term "laser" originated as an acronym for "Light Amplification by Stimulated Emission of Radiation"

A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation.

Types

Solid-state lasers

Solid-state lasers use a crystalline or glass rod which is "doped" with ions that provide the required energy states.

For example, the first working laser was a ruby laser , made from ruby ( chromium -doped corundum ).

The population inversion is actually maintained in the dopant.

Gas lasers

RUBY LASER

A ruby laser consists of a ruby rod that must be pumped with very high energy, usually from a flashtube , to achieve a population inversion .

The rod is often placed between two mirrors, forming an optical cavity , which oscillate the light produced by the ruby's fluorescence , causing stimulated emission .

SEMI CONDUCTOR LASERS:

GaAs LASER:

The gallium Arsenide laser is designed in such a way that a piece of N-type Gallium Arsenide material is taken and a layer of natural gallium aluminum arsenide material is pasted, The third layer of p-type gallium arsenide material is pasted over that.

Uses:

All these lasers can produce high powers in the infrared spectrum at 1064 nm. They are used for cutting, welding and marking of metals and other materials, and also in spectroscopy and for pumping dye lasers

PHOSPHOR

A phosphor, is a substance that exhibits the phenomenon of luminescence ;

it emits light when exposed to some type of radiant energy.

The term is used both for fluorescent or phosphorescent substances which glow on exposure to ultraviolet or visible light,

The energy from the lasers' light activates the phosphors, which emit photons, producing an image.

Phosphors are usually made from a suitable host material with an added activator . The best known type is a copper-activated zinc sulfide and the silver-activated zinc sulfide (zinc sulfide silver).

ELEMENTARY STUDY OF LIQUID CRYSTALS.

Solids yield a viscous cloudy liquids at a temperature known as transition point. If the temperature is increased beyond the transition point, the cloudiness disappear at the temperature called melting point

Between transition point and melting point the cloudy liquid shows double refaction. This state is called mesomorphic state. And the compounds in this state are called liquid crystals.

SMECTIC TYPE CRYSTALS WITH EXAMPLES

1.The word "smectic" originates from the Latin word having soap-like properties

2.There are two phases in smectic type. They are named as smectic A and smectic C

3. The smectic A phase has molecules organized into layers.

4. In the smectic C phase , the molecules are tilted inside the layers.

5. The layers can slide over one another .

Example:

. 1.The word nematic comes from the Greek which means "thread".

2. In a nematic phase, organic molecules have no positional order,

3. The molecules are free to flow

4. Nematics are uniaxial:

5. Nematics have fluidity similar to that of ordinary liquids

6. They can be easily aligned by an external magnetic or electric field.

Example:

CHOLESTERIC TYPE CRYSTALS WITH EXAMPLES

They exhibit the unique property that they reflect circularly polarized light when it is incident along the helical axis

Example: cholestryl benzoate

Point groups and space groups:

There can be 32 different combination of elements of symmetry of a crystal. These are called point groups. Some of the systems have been grouped together, so that we have only 7 different categories.

The 32 point groups, further produce 230 space groups.

SCREW AXIS

In crystallography, a screw axis symmetry is a combination of rotation about an axis and a translation parallel to that axis which leaves a crystal unchanged.

Diagram:

Figure 1 represents the normal 2-fold rotation and fig.2 represents a 2-fold screw axis in which rotation through 180 o , followed by t/2 transition, parallel to the axis. This is expressed as 2t screw axis.

Similarly, a 3-fold axis generate two screw axis namely 31 and 32 . The former represents rotation through 120 followed by translation t/3 and the latter corresponds to rotation through 240 o followed by translation through 2t/3. Similarly 4-fold axis generates three screw axis and 6- fold axis generates five screw axis.

GLIDE PLANE

A glide plane is defined as an operation which involves a translation t/2 parallel to the reflecting plane followed by reflection across the plane. Here t is the distance between the successive atoms.

or

In geometry and crystallography, a glide plane (or transflection) is a symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged. Glide planes are noted by a, b or c, depending on which axis the glide is along.

Diagram:

The glide planes are further classified in to three types

1.Axial glides: These are planes having glide component parallel to the crystallographic axis a,b and c and with length equal to a/2, b/2 and c/2. They are denoted as a-glide, b-glide and c-glide.

2.Diagonal glides: These correspond to the planes whose glide component is the vector sum of any two of the vectors a/2,b/2 and c/2. It is denoted by n.

3.Diamondglides:These are denoted by the symbol d and corresponds to the planes, whose glide component is the vector sum of any of the two vectors a/4, c/4 and d/4.

16.SYMMETRY ELEMENTS

1. Plane of symmetry:

If an imaginary plane, which divides the crystal into two parts, such that one is the exact mirror image of the other, exists in a crystal , it is said to have plane of symmetry.

a. Rectangular(vertical or horizontal) plane of symmetry

b. Diagonal plane of symmetry

2. Axis of symmetry:

If a crystal possesses an imaginary line, about which the crystal may be rotated such that it presents similar appearance, then , it is said to have axis of symmetry.

If the similar appearance is repeated after an angle of 180 o , the axis is called 2- fold axis of symmetry. If it appears after 120 ,90, 60 o, it is called 3- fold axis of symmetry, 4-fold and 6-fold axis of symmetry respectively. In general if a rotation through an angle of , brings the molecule to similar appearance, then the crystal is said to have n – fold axis of symmetry

3. Centre of symmetry

Centre of symmetry of a crystal is such a point that any line drawn through it intersects the surface of the crystal at equal distances in both directions.

Elements of symmetry

The total number of planes, axes and centre of symmetries possessed by a crystal is termed as elements of symmetry.

Elements of symmetry in a cube:

Rectangular planes of symmetry = 3

Diagonal planes of symmetry = 6

2- fold axis of symmetry = 6

3- fold axis of symmetry = 4

4- fold axis of symmetry = 3

Centre of symmetry = 1

18. THE CONCEPT OF RECIPROCAL LATTICE

RECIPROCAL LATTICE:

, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice).

In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the direct lattice. ?

There are 14 space lattices belonging to all the seven crystal systems( cubic ., orthorhombic …) These 14 lattices are called Bravais lattice ( primitive ,FC,BC- cubic, etc)

For each Bravaislattice , there is a corresponding reciprocal lattice of the same symmetry which may be derived geometrically.

From the origin O lines are constructed normal to the families of the plane (hkl). Points are marked off along each of these lines such that the distance d of any first point from O is inversely proportional to the corresponding interplanar spacing d

d(hkl) = 1 / d ( hkl)

Thus the first point along OP the normal to the (100) family of plane in real space is labelled 100 in the reciprocal space.

001 101

O

100

The particular reciprocal lattice points 100, 010, 001 define the reciprocal unit cell.

A = K bc sin / V 001

100 002 003

20. ROTATING CRYSTAL METHOD

This method is used to determine the structure of crystals using diffraction of X- rays The technique makes use of Bragg’s X-ray spectrometer, where crystal is used as reflecting grating .

X- rays generated in the tube T are passed through a slit so as to obtain a narrow beam. This narrow beam is allowed to strike the crystal C mounted on the turn table. The reflected rays are sent to ionisation chamber where the intensities are recorded.

The crystal is rotated gradually by means of the turn table , so as to increase the incident angle at the exposed face of the crystal. The process is carried out for each plane of the crystal. The lowest angle at which , maximum reflection occurs is , called first order reflection which corresponds to n= 1. The next higher angle , at which maximum reflection occurs again is called second order reflection.

Diagram:

The lattice constant d is found out using different planes of the crystal as reflecting surface for the same known wavelength of X – rays.

Applying Bragg’s equation

2dsinθ = n λ

For first order spectrum n= 1, hence the above equation becomes

2dsinθ = λ

=

If the ratio : : = 1 : √2 : √3 the crystal is simple cubic. If it is 1 : : √3 then the crystal is body centred cubic whereas it is 1 : √2 : the crystal is face centred cubic

Problem1: The values of θ for the first order reflection from the three faces of sodium chloride are 5.9 o, 8.4 o and 5.2 o .Find the crystal lattice.

Solution:

It has FCC structure.

Problem2: Find the crystal structure of potassium chloride .The values of θ for the first order reflection from the three faces are 5.22 o, 7.30 o and 9.05 o .

Solution:

= 0.0910 : 0.1272 : 0.1570

= 1 : 1.4 : 1.73

= 1 : √2 : √3

21. POWDER DIFFRACTION.

Powder method( Debye- Scherrer method)

The substance to be examined is finely powdered and is kept in the form of cylinder inside a thin glass tube. This is placed at the centre of Debye Scherer camera which consists of a cylindrical cassette,

X- rays are generated and allowed to fall on the powder specimen. The X- ray beam enters through a small hole, passes through the sample and the unused part of the beam leaves through the hole at the opposite end. The powder consists of many small crystals which are oriented in all possible directions. So the reflected radiation is not like a beam ; instead, it lies on the surface of a cone whose apex is at the point of contact of the incident radiation with the specimen.

Diagram:

For each combination of d and θ, one cone of reflection must result. Therefore, many cones of reflection are emitted by the powder specimen. The recorded lines from any cone are, a pair of arcs. The first arc on either side of the exit point corresponds to the smallest angle of reflection.

The distance between any two corresponding arcs on the film ( S) is related to the radius of the powder camera R

S = 4Rθ where θ is the Bragg angle in radians( 1 rad = 57.3 o ) . ---------1

Combining d(hkl) = with Bragg equation, we get

nλ = 2 sinθ

∴sin2 θ = ( h2 + k2 + l2 ) [ for first order reflection n = 1]

Θ values can be obtained from the powder pattern using equation 1 The values of sin2θ are compared with the below mentioned extinction rules.

1:2:3:4:5:6:8 SC [ 7 cannot be written in the form h2 + k2 + l2 ]

2:4:6:8 BCC [ odd integer for h + k+l are absent]

3:4:8:11:12 FCC [ h,k,l are either all odd or all even 111, 200, 220,311,222]

3:8:11:16 DC

Problem. From a powder camera of diameter 114.6 mm, using an X – ray beam of wavelength 1.54 Ao , the following S values in mm are obtained for a material: 86,100,148,180,188,232,and 272.determine the structure and the lattice parameter of the material.

Solution:

The Bragg angles in degrees = S/4

21.5,25,37,45,47,58 and 68

Sin2 θ values are , 0.1346 : 0.1788 : 0.362 : 0.5003 : 0.5352 : 0.7195 : 0.8596

These values can be expressed in the ratio of integral numbers

3:4:8:11:12: 16: 19

From the extinction rules, the structure is FCC.

The lattice parameter calculated from the highest Bragg angle is 3.62 A.

22. NEUTRON DIFFRACTION: ELEMENTARY TREATMENT

Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material.

A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of the structure of the material.

Instrumental and sample requirements

The technique requires a source of neutrons.

Neutrons are usually produced in a nuclear reactor or spallation source .

At a research reactor , other components are needed, including a crystal monochromator , as well as filters to select the desired neutron wavelength.

Some parts of the setup may also be movable.

At a spallation source, the time of flight technique is used to sort the energies of the incident neutrons (higher energy neutrons are faster), so no monochromator is needed, but rather a series of aperture elements synchronized to filter neutron pulses with the desired wavelength.

The technique is most commonly performed as powder diffraction , which only requires a polycrystalline powder. Single crystal work is also possible, but the crystals must be much larger than those that are used in single-crystal X-ray crystallography . It is common to use crystals that are about 1 mm3.

23. COMPARISON WITH X-RAY DIFFRACTION.

Neutron diffraction technique is similar to X-ray diffraction but due to their different scattering properties, neutrons and X-rays provide complementary information:

X-Rays are suited for superficial analysis, strong x-rays from synchrotron radiation are suited for shallow depths or thin specimens, while neutrons having high penetration depth are suited for bulk samples.

24. Electron diffraction- Basic principle.

Electron diffraction refers to the technique used to study matter by firing electrons at a sample and observing the resulting interference pattern.

This phenomenon is commonly known as wave–particle duality , which states that a particle of matter (in this case the incident electron) can be described as a wave. For this reason, an electron can be regarded as a wave much like sound or water waves. This technique is similar to X-ray and neutron diffraction .

Electron diffraction is most frequently used in solid state physics and chemistry to study the crystal structure of solids.

In these instruments, electrons are accelerated by an electrostatic potential in order to gain the desired energy and determine their wavelength before they interact with the sample to be studied.

The periodic structure of a crystalline solid acts as a diffraction grating , scattering the electrons in a predictable manner. Working back from the observed diffraction pattern , it may be possible to deduce the structure of the crystal producing the diffraction pattern. However, the technique is limited by phase problem .

25. CRYSTAL GROWTH METHODS: FROM MELT AND SOLUTION (HYDROTHERMAL,

there are three general categories of crystal growth methods, viz.,

(1) growth from melt,

(2) growth from solution,

GROWTH FROM MELT

Melt growth is the most widely applied method, especially for the growth of not too high melting point substances.

CZOCHRALSKI CRYSTAL PULLING TECHNIQUE

The process involved in this method is termed as ‘crystal pulling’, since it involves relative motion between a seed and the melt so that crystal is literally pulled out from the melt. The crystal pulling is applicable only to materials that melt congruently. The melt is first raised to a temperature a few degrees above melting point. Then the seed crystal, rotating slowly, is brought slowly into contact with the melt surface, and then lowering is stopped. After getting the desired length, the seeded crystal is slowly and carefully pulled out from the melt The crystal can be observed as it grows and adjustment in both temperature and the growth rate can be made as needed. With suitable precautions, the material withdrawn from the melt solidifies as a large cylindrical crystal. The practical aspects of the method have been discussed at length by Draper36). Fig. 2.1 illustrates schematically the basic principle of the technique.

BRIDGEMANN - STOCKBARGER TECHNIQUE

The material to be crystallized is placed in a cylindrical, conical shaped crucible, which can be lowered through a twozone vertical furnace where the temperatures of upper and lower zones are respectively above and below the melting point of the eventual material. The temperature profile of the growth chamber is shown in Fig. 2.2(b). In some cases the Ch.2.Crystal growth methods 22 crucible is raised through a furnace. The basic requirement for this procedure is that the freezing isotherm should move systematically through the molten charge, and this can be satisfied by moving the crucible or the furnace, or by changing the furnace temperature. The tip of cone allows restricted nucleation and therefore, under favorable conditions, the material is almost entirely transformed into a large single crystal whose diameter is equal to the internal diameter of the conical crucible. The method is useful in preparation of crystals of metals and semiconductors, alkali and alkaline earth halides, and complex ternary fluorides of alkali and transition metals. This method is, however, not appropriate to materials, which expand on solidification, e.g. aluminium tungstate.

VERNEUIL FLAME FUSION TECHNIQUE

This technique, developed by Verneuil in 190216,37), is mainly used to grow crystals with high melting point, like ZrO2 (2700o C), SrO (2400o C) etc. An oxyhydrogen or oxy-acetylene flame is established and is used for heating purpose. The feed powder of the material to be crystallized is shaken mechanically or electrically from the hopper through a sieve, using a small vibrator with a low amplitude capacity. The flame is made to impinge on a pedestal where a small pile of partly fused alumina quickly builds up. As the pile rises, it reaches into the hotter part of the flame so that the tip becomes completely molten. The molten region increases in size and starts to solidify at the lower end. As more and more powder arrives, the solidifying region broadens into a crystal growing in length. Such a crystal is called boule. The method has been schematically illustrated in Fig. 2. 3. The largest use of this method has been for the growth of gem - quality ruby and emeralds with high melting point and for which no suitable crucible is found. Keck and Gulay18) introduced floating zone variant to produce ultra pure silicon.

ZONE-MELTING TECHNIQUE

This technique, discovered by Pfann38) in 1852 was originally used for the purification of semiconductor materials. But since the product is usually crystalline, the technique is also used for growing single crystals. Zone refining technique is the most important zone melting method, where numbers of molten zones are passed along the charge in one direction either horizontally or vertically. This technique is illustrated in Fig. 2.4(a). By moving either the boat or the coil, the molten zone is moved along the boat, thus melting the material in the front portion and solidifying at the back to form the crystalline material. If the conditions are suitable, then the resultant material will be single crystalline. Fig. 2.4(b) shows a modification of the float zone technique, devised by Keck and Gulay18). In this method the material to be Ch.2.Crystal growth methods 23 purified or grown is arranged in a vertical compacted rod. The molten zone floats below the two solid parts of the rod held in place by surface tension. Each zone carries a fraction of impurities to the end of the charge, thereby purifying the remainder. This technique is used for growing crystals as well, in addition to purifying several metals and compounds.

GROWTH FROM SOLUTION

This is the simplest and one of the oldest methods42) of growing crystals in which the material to be crystallized is dissolved in a solvent to the desired degree of Ch.2.Crystal growth methods 24 supersaturation. The solution is then slowly cooled or evaporated. If a suitable solvent is found, crystals can be grown at temperatures much below the melting point of the eventual crystal. The low temperatures involved here indeed relieve demand on expensive furnaces and power supplies. Crystal growth from aqueous solutions has been extensively and phenomenologically studied by measuring the concentration and temperature gradient around crystals growing in two-dimensional cell at the growth interface. The growth rate of the crystals is mostly found to be proportional to the normal component of the gradients43).

GROWTH FROM WATER SOLUTION

This method is extensively used for obtaining single crystals of organic and inorganic materials. Two basic methods (cooling and evaporation) are used to grow large crystals from water solution. In both the cases, a saturated solution is prepared and the seed crystal is inserted. In one of the methods, temperature is lowered slowly so as to reduce the solubility and produce crystallization, while in the other method, the temperature is held constant and the solvent is made to evaporate isothermally to induce crystallization. Crystals like alkali halides44), sodium borate45), barium strontium nitrate46), Rochelle salt47), potassium and ammonium dihydrogen phosphate48-50), Ammonium Oxalate51,52), Potassium Hydrogen tartrate53), potash alum54), oxalic acid have been grown from water solution.

HYDROTHERMAL METHOD

This method of crystal growth, schematically illustrated in Fig. 2.5, using aqueous solution at high temperature and pressure, was first used by Spezia55) to grow quartz hydrothermally, and quartz is still the prime material grown commercially hydrothermally on a large scale. To obtain even a low solubility of quartz in water, the temperature of water well above boiling point is necessary. To prevent the water from the boiling away, necessary pressure is applied. As this solubility is not sufficient for satisfactory growth, a mineralizer is added to the system. The method is carried out using a sealed high pressure vessel known as autoclave or bomb. Special, strong, corrosion-resistant and chemically inert material is used for the construction of an autoclave to withstand high pressure and temperature. It is kept at two different temperature regions. In the upper cooler part, seed material is supported while in the lower hotter part, feed material is used. The rate of growth depends on the temperature difference between top and bottom of the autoclave, pressure and the amount of mineralizer present. When hot solution from the bottom rises into the cooler part of the autoclave on account of convection, excess material gets deposited on the seed, which then grows in size.

Page 21 of 56

Difference between point group and space group – screw axis – glide plane - symmetry elements –relationship between molecular symmetry and crystallographic symmetry – The Concept of reciprocal lattice – X-ray diffraction by single crystal – rotating crystal – powder diffraction. Neutron diffraction: Elementary treatment – comparison with X-ray diffraction. Electron diffraction- Basic principle. Crystal Growth methods: From melt and solution (hydrothermal,

UNIT III: CHEMISTRY OF SOLID STATE II: DIFFRACTION METHODS

1. BAND THEORY OF SOLIDS

According to band theory the energy spectrum of materials contains conduction band and valence band. On the basis of distance between conduction band and valence band, the materials are classified in to three categories.

1. Conductors:

If there is no energy gap between conduction band and valence band, such materials are called conductors.

Examples: metals

2. Insulators:

Those materials in which the energy gap between conduction band and valence band is very high , are called insulators.

3. Semiconductors:

If the gap between conduction band and valence band is very low , the materials are called semiconductors.

Example: germanium and silicon

Compounds with non- integer values of atomic composition are called

non- stoichiometric compounds.

Example : Ni 0.999O

Impurities are the main reason

For example NaCl heated in Na vapour results Na 1.5 Cl

Stoichiometric Defects

The compounds in which the number of positive and negative ions are exactly in the ratios indicated by their chemical formulae are called stoichiometric compounds. The defects do not disturb the stoichiometry (the ratio of numbers of positive and negative ions) are called stoichiometric defects. These are of following types,

(a) Interstitial defect: This type of defect is caused due to the presence of ions in the normally vacant interstitial sites in the crystals.

(b) Schottky defect: This type of defect when equal number of cations and anions are missing from their lattice sites so that the electrical neutrality is maintained. This type of defect occurs in highly ionic compounds which have high co-ordination number and cations and anions of similar sizes. e.g., NaCl, KCl, CsCl and KBr etc.

(c) Frenkel defect: This type of defect arises when an ion is missing from its lattice site and occupies an interstitial position. The crystal as a whole remains electrically neutral because the number of anions and cations remain same. Since cations are usually smaller than anions, they occupy interstitial sites. This type of defect occurs in the compounds which have low co-ordination number and cations and anions of different sizes. e.g., ZnS, AgCl and AgI etc. Frenkel defect are not found in pure alkali metal halides because the cations due to larger size cannot get into the interstitial sites. In AgBr both Schottky and Frenkel defects occur simultaneously.

CRYSTAL IMPERFECTIONS( CRYSTAL DEFECTS)

Any deviation in a crystal from a perfect periodic lattice structure is called crystal defects. The three types of defects are

1. Point defects 2. Line defects( dislocations) 3. Surface defects(plane defects)

3. POINT DEFECTS

1. POINT DEFECTS

The deviation in a crystal,

from a perfect periodic lattice structure is localised in the vicinity of only few atoms, it is called point defects. The different point defects are

1.1 Vacancies

1.4 Schotky defects

1.5 Substitutional defects

1.3 FRENKEL DEFECTS (VACANCIY AND INTERSTITIAL DEFECTS):

When a missing atom, occupies the interstitial position, the defect caused is known as Frenkel defects. This is most common in ionic crystals in which the positive ions are smaller in size.

interstial

Fe 2+ O 2-Fe 2+ O 2-Fe 2+ O 2-

Fe 2+ vacancy

Fe 2+ O 2-Fe 2+ O 2-Fe 2+ O 2-

Number of Frenkel defects in a crystal can be calculated by the formula

N =

Where N total number of atoms and Ni number of interstitial positions

Derivation:

Let the energy required to displace an atom, from its proper position to an interstitial position be E1. If there are N atoms and Ni interstitial positions , then the number of ways in which ‘n’ Frenkel defects can be formed is given by

W = ×

The change in Helmholtz free energy by the creation of ‘n’ Frenkel defects is

A = nE – TS

= nE – T [ kln W]

= nE – T k ln [ × ]

= nE – T k [ ln N! + ln Ni! - 2 ln n! – ln( N-n)! – ln( Ni – n) !]

Using Sterling ‘s approximation ln N! = N ln N - N we get

A = nE – T k{[ N ln N – N] +[Ni ln Ni – Ni] - 2 [ nln n – n] –[ (N – n) ln( N-n)] –

(N – n) ] – [( Ni – n) ln( Ni – n) - ( Ni – n) ] }

= nE – T k{[ N ln N – N +Ni ln Ni - Ni - 2 nln n +2 n – (N – n) ln( N-n)+

(N – n) – ( Ni – n) ln( Ni – n) + ( Ni – n) }

= nE – T k{[ N ln N – N +Ni ln Ni - Ni - 2 nln n +2 n – (N – n) ln( N-n)+

(N – n) – ( Ni – n) ln( Ni – n) + ( Ni – n) }

= nE – T k{ N ln N +Ni ln Ni - 2 nln n – (N – n) ln( N-n)– ( Ni – n) ln( Ni – n) }

Differentiating with respect to ‘n’ at constant temperature,

) T = E - T k { -2 [n () + ln n] - [ ( N-n) ( + ln (N-n) (0-1)]

· [ ( Ni – n) × ( + ln ( Ni – n) ( 0-1)}

= E - T k { -2 -2 ln n] - [ -1 - ln (N-n) ] -[- 1- ln ( Ni – n) }

= E - T k { -2 -2 ln n] + 1 + ln (N-n) + 1+ ln ( Ni – n) }

= E - T k { -2 ln n + ln (N-n) + ln ( Ni – n) }

= E - T k { ln ( ) + ln [ (N-n) × ( Ni – n) ] }

= E - T k { ln }

When equilibrium is attained, the Helmholtz free energy is constant and its first derivative is equal to zero. i. e ) T = 0

∴0 = E - T k { ln }

E = T k { ln }

= { ln }

When N >>n, N- n ≈ N similarly, When Ni >>n, Ni- n ≈ Ni

Therefore the above equation becomes, = { ln }

Taking exponential on both sides,

=

∴ n = (NNi ) ½

This is the expression for the number of ways of forming the defects

SCHOTKY DEFECTS

When a positive as well as negative ions of a crystal are missing, the defect is known as Schotky defects.

In Schotky defect the displaced atom migrates in successive steps eventually settles at the surface. Since the number of missing positive ions and negative ions is same, the crystal remains as neutral

Na + Cl-

Na + Cl- Na + Na + Cl-

Na + Cl- Cl- Na + Cl-

Na + Cl- Na + Cl- Na + Cl-

Number of Schotky l defects in a crystal can be calculated by the formula

n = N×

Derivation:

Suppose a crystal contains N atoms and ‘n’Schotky defects are produced by removing ‘n’ cations and ‘n’ anions from the crystal. Let the energy required to displace an atom, from its proper position to an interstitial position be E1. The number of ways in which ‘n’ schotky defects can be formed is given by

W = ×

The change in Helmholtz free energy by the creation of ‘n’ Frenkel defects is

A = nE – TS

= nE – 2T k [ ln N! - ln n! – ln( N-n)! ]

Using Sterling ‘s approximation ln N! = N ln N - N we get

A = nE– 2T k{[ N ln N – N] - [ nln n – n] –[ (N – n) ln( N-n)] – (N – n) ]

= nE – 2T k{[ N ln N – N - nln n + n – (N – n) ln( N-n)+ (N – n)

= nE – 2T k{[ N ln N – N - nln n + n – (N – n) ln( N-n)+ (N – n) }

= nE – 2T k{ N ln N - nln n – (N – n) ln( N-n) }

Differentiating with respect to ‘n’ at constant temperature,

) T = E - 2T k { - [n () + ln n] - [ ( N-n) ( + ln (N-n) (0-1)]

= E - 2T k { -1 - ln n] - [ -1 - ln (N-n) ]

= E - 2 T k { -1 - ln n] + 1 + ln (N-n)

= E - 2T k { - ln n + ln (N-n) }

= E - 2T k { ln }

When equilibrium is attained, the Helmholtz free energy is constant and its first derivative is equal to zero. i. e ) T = 0

∴0 = E - 2T k ln

E = 2T k ln

Taking exponential on both sides,

=

∴ n = N ×

This is the expression for the number of ways of forming the defects

4. linear defects

7. CONDUCTOR, INSULATOR, SEMICONDUCTOR

1. Conductors:

If there is no energy gap between conduction band and valence band, such materials are called conductors.

Examples: metals

2. Insulators:

Those materials in which the energy gap between conduction band and valence band is very high , are called insulators.

3. Semiconductors:

If the gap between conduction band and valence band is very low , the materials are called semiconductors.

Example: germanium and silicon

8.INTRINSIC SEMICONDUCTORS:

A semi conductor which is pure and contains no impurity is known as intrinsic semiconductor.

9. IMPURITY SEMICONDUCTORS

Extrinsic Semiconductors:

A semiconducting material in which, the charge carriers originate from impurity atoms added to the material, is called extrinsic semiconductor or impurity semiconductor.

Theses are divided in to two types.

1 n- type semi conductor:

Pentavalent elements such as P, As, Sb , have five electrons in their outermost orbits. When any one such impurity is added to the intrinsic semi conductor, four electrons are engaged in covalent bonding with four neighbouring semi conductor atoms and the fifth electron is free. Free electron

2 p- type semi conductor:

Trivalent elements such as Al, Ga or In have three electrons in their outer most orbits. When such impurity is added to the intrinsic semi conductor, all the three electrons are engaged in covalent bonding with three neighbouring semi conductor atoms and creating a hole ( vacant electron site) on the semiconductor atom.

OPTICAL PROPERTIES

The optical properties of semiconductors have been studied extensively for their relevance to applications such as lasers, light-emitting diodes , and solar cells

LASERS AND PHOSPHORS

The term "laser" originated as an acronym for "Light Amplification by Stimulated Emission of Radiation"

A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation.

Types

Solid-state lasers

Solid-state lasers use a crystalline or glass rod which is "doped" with ions that provide the required energy states.

For example, the first working laser was a ruby laser , made from ruby ( chromium -doped corundum ).

The population inversion is actually maintained in the dopant.

Gas lasers

RUBY LASER

A ruby laser consists of a ruby rod that must be pumped with very high energy, usually from a flashtube , to achieve a population inversion .

The rod is often placed between two mirrors, forming an optical cavity , which oscillate the light produced by the ruby's fluorescence , causing stimulated emission .

SEMI CONDUCTOR LASERS:

GaAs LASER:

The gallium Arsenide laser is designed in such a way that a piece of N-type Gallium Arsenide material is taken and a layer of natural gallium aluminum arsenide material is pasted, The third layer of p-type gallium arsenide material is pasted over that.

Uses:

All these lasers can produce high powers in the infrared spectrum at 1064 nm. They are used for cutting, welding and marking of metals and other materials, and also in spectroscopy and for pumping dye lasers

PHOSPHOR

A phosphor, is a substance that exhibits the phenomenon of luminescence ;

it emits light when exposed to some type of radiant energy.

The term is used both for fluorescent or phosphorescent substances which glow on exposure to ultraviolet or visible light,

The energy from the lasers' light activates the phosphors, which emit photons, producing an image.

Phosphors are usually made from a suitable host material with an added activator . The best known type is a copper-activated zinc sulfide and the silver-activated zinc sulfide (zinc sulfide silver).

ELEMENTARY STUDY OF LIQUID CRYSTALS.

Solids yield a viscous cloudy liquids at a temperature known as transition point. If the temperature is increased beyond the transition point, the cloudiness disappear at the temperature called melting point

Between transition point and melting point the cloudy liquid shows double refaction. This state is called mesomorphic state. And the compounds in this state are called liquid crystals.

SMECTIC TYPE CRYSTALS WITH EXAMPLES

1.The word "smectic" originates from the Latin word having soap-like properties

2.There are two phases in smectic type. They are named as smectic A and smectic C

3. The smectic A phase has molecules organized into layers.

4. In the smectic C phase , the molecules are tilted inside the layers.

5. The layers can slide over one another .

Example:

. 1.The word nematic comes from the Greek which means "thread".

2. In a nematic phase, organic molecules have no positional order,

3. The molecules are free to flow

4. Nematics are uniaxial:

5. Nematics have fluidity similar to that of ordinary liquids

6. They can be easily aligned by an external magnetic or electric field.

Example:

CHOLESTERIC TYPE CRYSTALS WITH EXAMPLES

They exhibit the unique property that they reflect circularly polarized light when it is incident along the helical axis

Example: cholestryl benzoate

Point groups and space groups:

There can be 32 different combination of elements of symmetry of a crystal. These are called point groups. Some of the systems have been grouped together, so that we have only 7 different categories.

The 32 point groups, further produce 230 space groups.

SCREW AXIS

In crystallography, a screw axis symmetry is a combination of rotation about an axis and a translation parallel to that axis which leaves a crystal unchanged.

Diagram:

Figure 1 represents the normal 2-fold rotation and fig.2 represents a 2-fold screw axis in which rotation through 180 o , followed by t/2 transition, parallel to the axis. This is expressed as 2t screw axis.

Similarly, a 3-fold axis generate two screw axis namely 31 and 32 . The former represents rotation through 120 followed by translation t/3 and the latter corresponds to rotation through 240 o followed by translation through 2t/3. Similarly 4-fold axis generates three screw axis and 6- fold axis generates five screw axis.

GLIDE PLANE

A glide plane is defined as an operation which involves a translation t/2 parallel to the reflecting plane followed by reflection across the plane. Here t is the distance between the successive atoms.

or

In geometry and crystallography, a glide plane (or transflection) is a symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged. Glide planes are noted by a, b or c, depending on which axis the glide is along.

Diagram:

The glide planes are further classified in to three types

1.Axial glides: These are planes having glide component parallel to the crystallographic axis a,b and c and with length equal to a/2, b/2 and c/2. They are denoted as a-glide, b-glide and c-glide.

2.Diagonal glides: These correspond to the planes whose glide component is the vector sum of any two of the vectors a/2,b/2 and c/2. It is denoted by n.

3.Diamondglides:These are denoted by the symbol d and corresponds to the planes, whose glide component is the vector sum of any of the two vectors a/4, c/4 and d/4.

16.SYMMETRY ELEMENTS

1. Plane of symmetry:

If an imaginary plane, which divides the crystal into two parts, such that one is the exact mirror image of the other, exists in a crystal , it is said to have plane of symmetry.

a. Rectangular(vertical or horizontal) plane of symmetry

b. Diagonal plane of symmetry

2. Axis of symmetry:

If a crystal possesses an imaginary line, about which the crystal may be rotated such that it presents similar appearance, then , it is said to have axis of symmetry.

If the similar appearance is repeated after an angle of 180 o , the axis is called 2- fold axis of symmetry. If it appears after 120 ,90, 60 o, it is called 3- fold axis of symmetry, 4-fold and 6-fold axis of symmetry respectively. In general if a rotation through an angle of , brings the molecule to similar appearance, then the crystal is said to have n – fold axis of symmetry

3. Centre of symmetry

Centre of symmetry of a crystal is such a point that any line drawn through it intersects the surface of the crystal at equal distances in both directions.

Elements of symmetry

The total number of planes, axes and centre of symmetries possessed by a crystal is termed as elements of symmetry.

Elements of symmetry in a cube:

Rectangular planes of symmetry = 3

Diagonal planes of symmetry = 6

2- fold axis of symmetry = 6

3- fold axis of symmetry = 4

4- fold axis of symmetry = 3

Centre of symmetry = 1

18. THE CONCEPT OF RECIPROCAL LATTICE

RECIPROCAL LATTICE:

, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice).

In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the direct lattice. ?

There are 14 space lattices belonging to all the seven crystal systems( cubic ., orthorhombic …) These 14 lattices are called Bravais lattice ( primitive ,FC,BC- cubic, etc)

For each Bravaislattice , there is a corresponding reciprocal lattice of the same symmetry which may be derived geometrically.

From the origin O lines are constructed normal to the families of the plane (hkl). Points are marked off along each of these lines such that the distance d of any first point from O is inversely proportional to the corresponding interplanar spacing d

d(hkl) = 1 / d ( hkl)

Thus the first point along OP the normal to the (100) family of plane in real space is labelled 100 in the reciprocal space.

001 101

O

100

The particular reciprocal lattice points 100, 010, 001 define the reciprocal unit cell.

A = K bc sin / V 001

100 002 003

20. ROTATING CRYSTAL METHOD

This method is used to determine the structure of crystals using diffraction of X- rays The technique makes use of Bragg’s X-ray spectrometer, where crystal is used as reflecting grating .

X- rays generated in the tube T are passed through a slit so as to obtain a narrow beam. This narrow beam is allowed to strike the crystal C mounted on the turn table. The reflected rays are sent to ionisation chamber where the intensities are recorded.

The crystal is rotated gradually by means of the turn table , so as to increase the incident angle at the exposed face of the crystal. The process is carried out for each plane of the crystal. The lowest angle at which , maximum reflection occurs is , called first order reflection which corresponds to n= 1. The next higher angle , at which maximum reflection occurs again is called second order reflection.

Diagram:

The lattice constant d is found out using different planes of the crystal as reflecting surface for the same known wavelength of X – rays.

Applying Bragg’s equation

2dsinθ = n λ

For first order spectrum n= 1, hence the above equation becomes

2dsinθ = λ

=

If the ratio : : = 1 : √2 : √3 the crystal is simple cubic. If it is 1 : : √3 then the crystal is body centred cubic whereas it is 1 : √2 : the crystal is face centred cubic

Problem1: The values of θ for the first order reflection from the three faces of sodium chloride are 5.9 o, 8.4 o and 5.2 o .Find the crystal lattice.

Solution:

It has FCC structure.

Problem2: Find the crystal structure of potassium chloride .The values of θ for the first order reflection from the three faces are 5.22 o, 7.30 o and 9.05 o .

Solution:

= 0.0910 : 0.1272 : 0.1570

= 1 : 1.4 : 1.73

= 1 : √2 : √3

21. POWDER DIFFRACTION.

Powder method( Debye- Scherrer method)

The substance to be examined is finely powdered and is kept in the form of cylinder inside a thin glass tube. This is placed at the centre of Debye Scherer camera which consists of a cylindrical cassette,

X- rays are generated and allowed to fall on the powder specimen. The X- ray beam enters through a small hole, passes through the sample and the unused part of the beam leaves through the hole at the opposite end. The powder consists of many small crystals which are oriented in all possible directions. So the reflected radiation is not like a beam ; instead, it lies on the surface of a cone whose apex is at the point of contact of the incident radiation with the specimen.

Diagram:

For each combination of d and θ, one cone of reflection must result. Therefore, many cones of reflection are emitted by the powder specimen. The recorded lines from any cone are, a pair of arcs. The first arc on either side of the exit point corresponds to the smallest angle of reflection.

The distance between any two corresponding arcs on the film ( S) is related to the radius of the powder camera R

S = 4Rθ where θ is the Bragg angle in radians( 1 rad = 57.3 o ) . ---------1

Combining d(hkl) = with Bragg equation, we get

nλ = 2 sinθ

∴sin2 θ = ( h2 + k2 + l2 ) [ for first order reflection n = 1]

Θ values can be obtained from the powder pattern using equation 1 The values of sin2θ are compared with the below mentioned extinction rules.

1:2:3:4:5:6:8 SC [ 7 cannot be written in the form h2 + k2 + l2 ]

2:4:6:8 BCC [ odd integer for h + k+l are absent]

3:4:8:11:12 FCC [ h,k,l are either all odd or all even 111, 200, 220,311,222]

3:8:11:16 DC

Problem. From a powder camera of diameter 114.6 mm, using an X – ray beam of wavelength 1.54 Ao , the following S values in mm are obtained for a material: 86,100,148,180,188,232,and 272.determine the structure and the lattice parameter of the material.

Solution:

The Bragg angles in degrees = S/4

21.5,25,37,45,47,58 and 68

Sin2 θ values are , 0.1346 : 0.1788 : 0.362 : 0.5003 : 0.5352 : 0.7195 : 0.8596

These values can be expressed in the ratio of integral numbers

3:4:8:11:12: 16: 19

From the extinction rules, the structure is FCC.

The lattice parameter calculated from the highest Bragg angle is 3.62 A.

22. NEUTRON DIFFRACTION: ELEMENTARY TREATMENT

Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material.

A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of the structure of the material.

Instrumental and sample requirements

The technique requires a source of neutrons.

Neutrons are usually produced in a nuclear reactor or spallation source .

At a research reactor , other components are needed, including a crystal monochromator , as well as filters to select the desired neutron wavelength.

Some parts of the setup may also be movable.

At a spallation source, the time of flight technique is used to sort the energies of the incident neutrons (higher energy neutrons are faster), so no monochromator is needed, but rather a series of aperture elements synchronized to filter neutron pulses with the desired wavelength.

The technique is most commonly performed as powder diffraction , which only requires a polycrystalline powder. Single crystal work is also possible, but the crystals must be much larger than those that are used in single-crystal X-ray crystallography . It is common to use crystals that are about 1 mm3.

23. COMPARISON WITH X-RAY DIFFRACTION.

Neutron diffraction technique is similar to X-ray diffraction but due to their different scattering properties, neutrons and X-rays provide complementary information:

X-Rays are suited for superficial analysis, strong x-rays from synchrotron radiation are suited for shallow depths or thin specimens, while neutrons having high penetration depth are suited for bulk samples.

24. Electron diffraction- Basic principle.

Electron diffraction refers to the technique used to study matter by firing electrons at a sample and observing the resulting interference pattern.

This phenomenon is commonly known as wave–particle duality , which states that a particle of matter (in this case the incident electron) can be described as a wave. For this reason, an electron can be regarded as a wave much like sound or water waves. This technique is similar to X-ray and neutron diffraction .

Electron diffraction is most frequently used in solid state physics and chemistry to study the crystal structure of solids.

In these instruments, electrons are accelerated by an electrostatic potential in order to gain the desired energy and determine their wavelength before they interact with the sample to be studied.

The periodic structure of a crystalline solid acts as a diffraction grating , scattering the electrons in a predictable manner. Working back from the observed diffraction pattern , it may be possible to deduce the structure of the crystal producing the diffraction pattern. However, the technique is limited by phase problem .

25. CRYSTAL GROWTH METHODS: FROM MELT AND SOLUTION (HYDROTHERMAL,

there are three general categories of crystal growth methods, viz.,

(1) growth from melt,

(2) growth from solution,

GROWTH FROM MELT

Melt growth is the most widely applied method, especially for the growth of not too high melting point substances.

CZOCHRALSKI CRYSTAL PULLING TECHNIQUE

The process involved in this method is termed as ‘crystal pulling’, since it involves relative motion between a seed and the melt so that crystal is literally pulled out from the melt. The crystal pulling is applicable only to materials that melt congruently. The melt is first raised to a temperature a few degrees above melting point. Then the seed crystal, rotating slowly, is brought slowly into contact with the melt surface, and then lowering is stopped. After getting the desired length, the seeded crystal is slowly and carefully pulled out from the melt The crystal can be observed as it grows and adjustment in both temperature and the growth rate can be made as needed. With suitable precautions, the material withdrawn from the melt solidifies as a large cylindrical crystal. The practical aspects of the method have been discussed at length by Draper36). Fig. 2.1 illustrates schematically the basic principle of the technique.

BRIDGEMANN - STOCKBARGER TECHNIQUE

The material to be crystallized is placed in a cylindrical, conical shaped crucible, which can be lowered through a twozone vertical furnace where the temperatures of upper and lower zones are respectively above and below the melting point of the eventual material. The temperature profile of the growth chamber is shown in Fig. 2.2(b). In some cases the Ch.2.Crystal growth methods 22 crucible is raised through a furnace. The basic requirement for this procedure is that the freezing isotherm should move systematically through the molten charge, and this can be satisfied by moving the crucible or the furnace, or by changing the furnace temperature. The tip of cone allows restricted nucleation and therefore, under favorable conditions, the material is almost entirely transformed into a large single crystal whose diameter is equal to the internal diameter of the conical crucible. The method is useful in preparation of crystals of metals and semiconductors, alkali and alkaline earth halides, and complex ternary fluorides of alkali and transition metals. This method is, however, not appropriate to materials, which expand on solidification, e.g. aluminium tungstate.

VERNEUIL FLAME FUSION TECHNIQUE

This technique, developed by Verneuil in 190216,37), is mainly used to grow crystals with high melting point, like ZrO2 (2700o C), SrO (2400o C) etc. An oxyhydrogen or oxy-acetylene flame is established and is used for heating purpose. The feed powder of the material to be crystallized is shaken mechanically or electrically from the hopper through a sieve, using a small vibrator with a low amplitude capacity. The flame is made to impinge on a pedestal where a small pile of partly fused alumina quickly builds up. As the pile rises, it reaches into the hotter part of the flame so that the tip becomes completely molten. The molten region increases in size and starts to solidify at the lower end. As more and more powder arrives, the solidifying region broadens into a crystal growing in length. Such a crystal is called boule. The method has been schematically illustrated in Fig. 2. 3. The largest use of this method has been for the growth of gem - quality ruby and emeralds with high melting point and for which no suitable crucible is found. Keck and Gulay18) introduced floating zone variant to produce ultra pure silicon.

ZONE-MELTING TECHNIQUE

This technique, discovered by Pfann38) in 1852 was originally used for the purification of semiconductor materials. But since the product is usually crystalline, the technique is also used for growing single crystals. Zone refining technique is the most important zone melting method, where numbers of molten zones are passed along the charge in one direction either horizontally or vertically. This technique is illustrated in Fig. 2.4(a). By moving either the boat or the coil, the molten zone is moved along the boat, thus melting the material in the front portion and solidifying at the back to form the crystalline material. If the conditions are suitable, then the resultant material will be single crystalline. Fig. 2.4(b) shows a modification of the float zone technique, devised by Keck and Gulay18). In this method the material to be Ch.2.Crystal growth methods 23 purified or grown is arranged in a vertical compacted rod. The molten zone floats below the two solid parts of the rod held in place by surface tension. Each zone carries a fraction of impurities to the end of the charge, thereby purifying the remainder. This technique is used for growing crystals as well, in addition to purifying several metals and compounds.

GROWTH FROM SOLUTION

This is the simplest and one of the oldest methods42) of growing crystals in which the material to be crystallized is dissolved in a solvent to the desired degree of Ch.2.Crystal growth methods 24 supersaturation. The solution is then slowly cooled or evaporated. If a suitable solvent is found, crystals can be grown at temperatures much below the melting point of the eventual crystal. The low temperatures involved here indeed relieve demand on expensive furnaces and power supplies. Crystal growth from aqueous solutions has been extensively and phenomenologically studied by measuring the concentration and temperature gradient around crystals growing in two-dimensional cell at the growth interface. The growth rate of the crystals is mostly found to be proportional to the normal component of the gradients43).

GROWTH FROM WATER SOLUTION

This method is extensively used for obtaining single crystals of organic and inorganic materials. Two basic methods (cooling and evaporation) are used to grow large crystals from water solution. In both the cases, a saturated solution is prepared and the seed crystal is inserted. In one of the methods, temperature is lowered slowly so as to reduce the solubility and produce crystallization, while in the other method, the temperature is held constant and the solvent is made to evaporate isothermally to induce crystallization. Crystals like alkali halides44), sodium borate45), barium strontium nitrate46), Rochelle salt47), potassium and ammonium dihydrogen phosphate48-50), Ammonium Oxalate51,52), Potassium Hydrogen tartrate53), potash alum54), oxalic acid have been grown from water solution.

HYDROTHERMAL METHOD

This method of crystal growth, schematically illustrated in Fig. 2.5, using aqueous solution at high temperature and pressure, was first used by Spezia55) to grow quartz hydrothermally, and quartz is still the prime material grown commercially hydrothermally on a large scale. To obtain even a low solubility of quartz in water, the temperature of water well above boiling point is necessary. To prevent the water from the boiling away, necessary pressure is applied. As this solubility is not sufficient for satisfactory growth, a mineralizer is added to the system. The method is carried out using a sealed high pressure vessel known as autoclave or bomb. Special, strong, corrosion-resistant and chemically inert material is used for the construction of an autoclave to withstand high pressure and temperature. It is kept at two different temperature regions. In the upper cooler part, seed material is supported while in the lower hotter part, feed material is used. The rate of growth depends on the temperature difference between top and bottom of the autoclave, pressure and the amount of mineralizer present. When hot solution from the bottom rises into the cooler part of the autoclave on account of convection, excess material gets deposited on the seed, which then grows in size.

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