### SOLID STATE

SOLID STATE

How do you recognise a crystal lattice and a unit cell

Crystal lattice:– A crystal contains a structural unit called the “basis” or “structural motif”.

1. The basis is repeated in three dimensions to generate the crystal structure.
2. The basis may be a single atom or molecule or it may be small group of atoms   or ions. Each repeated group has the same structure and spatial orientation.

Ex:     i) In NaCl the basis is Na+ and Cl ion pair.                                                                                                     ii) For Cu metal the basis is an atom of Cu.

If the repeated basis group is represented by a point, a crystal lattice or space lattice is represented by a set of points.

Space lattice:-

1. It is a regular repeating arrangement of these points in space.
2. This forms the basis for classification of basic crystal structure.
3. In the space lattice each characteristic point has the same environment. More importantly the space lattice is not the same as the crystal structure If this space is represented with actual ions or atoms or molecules it is called crystal lattice.

Unit Cell:-

1. In a crystal lattice a specific fundamental structure appears repeatedly .This fundamental unit structure is a parallelogram .This parallelepiped has edges of lengths a, b, c and angles .
2. The three dimensional fundamental structure is known as “Unit cell”.
3. The arrangement of unit cell results in the crystal structure. The shape of a crystal depends on the shape and dimensions of the unit cell.

Q2) Place the atoms of an element A in the lattice points of face centered cubic structure.

Write short notes on the following giving suitable examples.

a) Schottky defect          b) Frenkel defect

a) Schottky defect:-

1. A Schottky defect consists of a pair of holes in the crystal lattice due to the absence of one positive ion and one negative ion. It is a point
2. Large number of Schottky defects lowers the density markedly but the overall composition of the solid is not usually affected because the number of vacancies of cat- ion and anion are equal.

b) Frenkel defect:-

A Frenkel defect is created when an ion occupies an interstitial site instead of occupying its correct Lattice points.  It is a point defect. It is favoured when there is large difference in the size between cation and anion and having low coordination number 4 or 6.

E.g.: AgCl, AgBr and AgI.Unlike Schottky defect Frankel defect do not change the density of the solid

What do you know about amorphous solids?

1. Amorphous substances are those in which the particles (atoms or ions or molecules) are arranged
2. Amorphous solids do not have sharp melting points. Generally they melt over a range of tempera­ture. In terms of the structures an amorphous solid is one which has no long range6rders that characterises crystalline solids. Instead they have a random disordered arrangement of atoms. Glass, rubber, many plastics form amorphous solid.
3. Any given material can be made amorphous or glassy by quenching a melt or by freezing the vapour of the specific structure. For example silica forms crystalline material quartz. If the quartz is melted and rapidly cooled amorphous solid results.
4. Amorphous solids can be blown or moulded into articles of different shapes, When this melts of amorphous solids are slowly cooled or annealed they become crystalline at a definite temperature that is characteristic of the material. But they themselves do not revert to the crystalline state at ordinary conditions.
5. Amorphous materials have many applications such as domestic constructions or appliances like photovoltaic cells, where the sun rays are transformed into electricity.

Giving suitable examples, explain the following.

a) Paramagnetic substances b) Ferromagnetic substances c) Piezoelectric effect.

a) Paramagnetic substances: .                                               ,                                   .

Paramagnetic solids contain unpaired electrons and are attracted into the applied external magnetic field. They lose their magnetism when the magnetic field is removed.

E.g.:O2 NO, Na atoms, Ti203, VO2·

b) Ferromagnetic substances:

A spontaneous alignment of magnetic moments in the same direction gives rise to ferromagnetism. In ferromagnetic solids there occurs magnetic interaction between the neighboring centers and the electrons in these centers interact in parallel direction. This interaction leads to an increase in magnetic moment.

Eg : Fe, Co, Ni,Cr02 etc.

C) Piezoelectric effect:

1. A dielectric substance is that which may not allow electric current through it, but charges are induced on its face when electric field is applied field.
2. When electric field is applied displacement of charges takes place and dipoles are created which results in polarisation.
3. Crystals in which dipoles may align to produce a net dipole moment are called piezoelectric. When piezoelectric crystals are subjected to pressure or mechanical stress electricity is produced, due to displacement of ion. This is known as Piezoelectricity and the effect is called piezoelectric effect.

Describe Bragg’s method for determining the structure of a regular crystalline solid.

Bragg’s method:

1. The method is used to determine the crystal structures Bragg’s spectrometer is used for this purpose.
2. Beam of X-rays of definite wavelength, obtained froman X-ray tube, passes through a slit and falls on a
1. an X-ray tube, passes through a slit and falls on a
3.  known face of the crystal. The crystal is placed on  a round table which can be rotated. The position of the crystal can be read on a scale. The intensity of the diffracted X- radiations from the crystal is measured on a photographic plate or an ionization chamber. 4. An ionization chamber (com­monly used). The ionization chamber contains methyl bromide vapours. The      energy      of the X-rays causes ionization so that there is a flow of current. The current is measured on electrometer. The extent of ionization in the vapours is shown by the electrometer reading. As the intensity of the diffracted X-rays increase, the degree of ionization also increases. The angle of incidence X-rays on the crystal face or plane is gradually increased.

5.The extent of ionization at different angles is recorded in the electrometer. For a particular face there will be set of  values which satisfy Bragg’s equation and for which intensity will be maximum. These e values correspond to different values of ‘n’. A graph is drawn between the angle of incidence and electrometer reading. It is in the form shown in the diagram. The crests of the graphs reflect the mum diffraction. The angles corresponding to the crests are noted from the graphs. The angles are substituted for ‘ ‘ in Bragg’s equation and the values of’d’ are calculated. The values of’d’ in the direction of different co-ordination axes are compared and from their ratio the structure of the crystal is determined.

What is meant by constructive and destructive interference of waves? What sort of interference is necessary for the Bragg’s equation to be applied?

Electromagnetic radiations propagate in the form of waves in all directions from their origin. In the propagation of waves along a direction, the maxima and the minima of these waves mutually interfere with one another. If the waves are present in the same phase, they undergo constructive interference. If the waves are not in the same phase, they undergo destructive interference.

For the application of Bragg’s equation in the study   of structure of crystals, the constructive interference is necessary.

Derive Bragg’s equation for X – rays of wavelength (A) and a diffraction angle (0) for an nth order reflection.

1.  When X – rays are incident on the crystal plane, they undergo diffraction. The, atoms or ions are arranged in the regular planes in a crystal. When the waves are diffracted from the atoms or ions, they may have constructive interference or a destructive int`erference. 2. From the figure it can be seen that 1st and 2nd X – ray waves travel the same distance to the 1st layer of crystal. After crossing the 1st layer, the second X – ray waves travel more than the 1st ray: If the two waves are to be present in phase, the path difference must be equal to the wavelength A or an integral whole multiple of wavelength i.e., nλ where n = 1, 2, 3…any integral value. n is known as order of diffraction.
But AB= d and DB = BC = d sin θ
Hence nλ = (DB + BC) = 2dsin θ.This relation is known as Bragg’s equation. Knowing the values of θ andλ, the values of d can b calculated.

How many types of semiconductors are known? Explain the influence of doping on the conductivity of crystalline solids.

Semiconductors are the substances which at room temperature allow a portion of electric current to flow through them. Silicon and germanium are semiconductors. Doping is a process of mixing pure silicon or germanium with an impurity. Semiconductors are two types
1) n – type semiconductors (n – stands for negative)
2) p – type semiconductors (p – stands for positive)

ntype semi conductors: are obtained due to metal excess defect or by adding trace amounts of Vth or 15th group elements (P, As) to pure silicon or germanium. When P or As is added to silicon or germanium some of the Si or Ge atoms in the crystal are replaced by P or As atoms and 4 out of 5 electrons of P or As atoms will be used for bonding with Si or Ge atoms while the fifth electron serve to conduct electricity.

ptype semiconductors: are obtained due to metal deficiency defect or doping with impurity atoms  containing less electrons Le., atoms of III of 13th group elements. When B, Ga or In is added to silicon or germanium, some of the Si or Ge atoms in the crystal are replaced by B, Ga or In and only three valencies of Si or Ge are satisfied leaving an electron at Si or Ge because B, Ga or In have one electron less. So positive holes develop in p – type semi conductors. In the applied electric field current flow takes place due to the migration of positive holes due to the movement of electrons from adjacent site into positive holes.

To which type semiconductors do the following belong?

a)Germanium doped with indium b) Boron doped with silicon

a) Germanium doped with indium belong to p – type semi conductor. When indium is added to germanium some of the germanium atoms in the crystal are replaced by indium and only three valencies of germanium are satisfied leaving an electron. at germanium because indium has one electron less. So positive holes develop in a p-type semiconductor. In the applied electric field current flow takes place due to the migration of positive holes due to the movement of electrons from adjacent site into positive holes.

b) Boron doped with Silicon is also p-type semiconductor and works in a similar way as explained above.

What do you mean by imperfections in solids? Write an essay on crystal defects.

All solids have imperfections or defects in structure or composition even though they were prepared with utmost care. The defects in the crystals can be divided into different kinds.

1) Intrinsic defects 2) Extrinsic defects 3) Point defects 4) Extended defects

1) Intrinsic defects: It is seen in pure crystals. These are difficult to detect directly.
2) Extrinsic defects: These are due to the impurities in the solid.
3) Point defects: These arise due to the irregularity in the arrangement of atoms or ions. The atomic defects caused by missing or misplaced ions are called point defects. These are two types

1. i) Stoichiometric defects and ii) non stoichiometric defects.
2. i) Stoichiometric defects are again two types a) Schottky defects and· b) Frenkel defects
3. a) Schottky defects: It consists of a pair of holes in the crystal lattice due to the absence of one positive ion and one negative ion. Schottky defects occur mainly in the ionic compounds which contain smaller ion of similar size which have high coordination number.

Eg: NaCl, CsCl. Large number of vacancies in the lattice lowers the density markedly but overall composition of the solid is not usually affected because the number of vacancies of cation and anion are equal.

b) Frenkel defect: It is created when an ion occupies an interstitial site instead occupying its correct lattice site. The small cations (compared to anions) occupy the interstitial positions causing Frenkel defect. Frenkel defect takes place when there is large difference in the sizes of cation and anion and having low coordination number 4 or 6. There is no change in the density of the solid due to Frenkel defect.

ii) Non stoichiometric defects: In these defects the ratio of positive ion to negative ion is not exactly one .These are again two types a) metal excess defect b)metal deficiency defect.

1. a) Metal excess defect: It is due to the absence of negative ion from its lattice points leaving a hole which is occupied by an electron, there by maintaining the electrical balance. The anion site occupied by an electron is called F – The solids having F – centers are coloured and the intensity of the colour increases with increase in the number of F -centers. Solids containing F – centers are paramagnetic since the electrons occupying the vacant site are unpaired. When materials with F – centers are irradiated with light they become photo conductors.

Eg: i) Excess potassium in KCl gives violet colour.

ii) Zinc oxide on heating loses oxygen leaving excess metal accommodated in interstitial spaces.

b) Metal deficiency defect: It is due to the absence of a positive ion from  its lattice point and the charge can be balanced ‘by an adjacent metal ion having an extra positive charge.

Eg : When SrCl2 is added to NaCl, the  Sr2+ ion replaces Na+ ions resulting in  cationic vacancies

4) Extended defects: These may present in one, two or three dimensions.

Name the crystal systems.

i) Cubic   ii) tetragonal   iii) orthorhombic or rhombic   iv) hexagonal v) trigonal or rhombohedral vi) monoclinic and   vii) Triclinic.

How many unit cells share each of the following lattice points in a cubic lattice?

a) A lattice point at the corner of cell b) A lattice point at the face center c) A lattice point at the centre of the body

a) A point that is located at the corner of a unit cell is shared together by eight such unit cells.
b) A face centered point is shared between two unit cells.
c) A body centered point lying entirely within a unit cell, belongs completely to this unit cell.

Explain the crystal structures formed, when first and third layers in a three dimensional packing look alike. In this type of arrangement every sphere is in contact with six other spheres as shown in the figure.

Vacant spaces are called holes are formed when three spheres in a  layer are in contact with each other. If the second layer of spheres is arranged by placing these spheres over the holes in the first layer three of the six holes formed only are covered by the spheres.

A third layer of spheres can be super imposed on the spheres of this layer directly over  the spheres of the first layer i.e., the third layer is a repetition of the first, similarly the fourth layer is the repetition of the second layer. This type of arrangement is called AB AB AB … type or hexagonal close packed (or hcp) arrangement.

Write a note on tetrahedral hole.

A void created when four spherical particles are in contact with each other is called tetrahedral void or tetrahedral hole. In other words a tetrahedral hole ·is formed by a planar triangle of spheres capped by another sphere in the dip produced by three spheres in contact with one another. What are octahedral holes? How are they formed? The void created when six spherical particles are in contact with each other is called octahedral void or octahedral hole. In other words an octahedral hole is the vacant space that lies between two oppositely directed· planar triangles of spheres in adjacent layers.

Derive a relation between the density of a crystalline substance and the unit cell length.

The density of the unit cell The mass of ‘z· lattice points =ZM/N_0

Where, Z = No. of atoms present per unit cell.
M = Molecular weight of crystalline substance.
No = Avogadro’s number.

If the unit cell length is ‘a’, the volume of the unit cell is “a3” (=v) Write a brief account of VBT of metal bonding.
1. VBT theory of metals was proposed by Pauling. This theory is based on resonance theory.
2. According to this theory the metallic bond is essentially a polar or a non-polar covalent bond. This covalent bond involves resonance between a number of structures having one electrons and electron pair bonds.
3. As there is a possibility of insufficient valence electrons for the formation of electron pair bonds with each atom of the metal, it is assumed that resonance takes place through out the solid metal. This resonance not only involves covalent bonds but also ionic linkages. The atoms undergo hybridization. For example, the resonance structure of sodium metal is represented taking four Na atoms only.
4. This theory does not explain the conduction of heat in solids or their base or the retention of the metallic properties in either the liquid state or in the solution.

Calculate the number of particles present in fcc structure.
In the face centered cube.

1. A point that is located at the corner of a unit cell is shared together by eight such unit cells. That is each unit cell has a share of 1/8 of such points.
Total points contributed from eight corners atoms = 8 x 1/8 =1 atom or point.
2. A face centered point is shared between two unit cells and therefore only half of each point lies within anyone cell. Total points contributed by face – centered points
i.e., 6 faces of a cube = 6 x 1/2 = 3
3. Total contribution of points per unit cell in fcc arrangement = 4 atoms.

Explain schottky defect in solids.

1. Schottky defect is a point defect in which an atom or ion is missing from its normal site in the lattice. To maintain electrical neutrality in ionic crystals if a vacancy is created by one ion, it is accompanied by the vacancy creation by the loss of another kind of ion. 1. Schottky defects occur mainly in the ionic compounds which contain smaller ion of similar size which have high coordination number. E.g.: NaCl, CsCl.
2. Large number of vacancies in the lattice lowers the density markedly but overall composition of the solid is not usually affected because the number of vacancies of cation and anion are equal.

Atoms ‘A’ and ‘B’ which do not differ greatly in their electro negativities form an ionic compound. Explain its crystal defect.

Crystals formed by atoms ‘A’ and, ‘B’ which do not differ greatly in their electro negativities will show   Frenkel defect. Frenkel defect is also a point defect. The ion or the atom is shifted from its normal lattice point and occupies an interstitial position in the lattice.

It is more common that positive ions occupy   the interstitial position. This type of defect is favoured by a large difference in sizes between cation and anion. In these compounds the coordination numbers are low. Ex: Silver halides, ZnS etc. exhibit this type of defect. Unlike Schottky defect, Frenkel defects  do not change the density.

Define coordination number in crystalline solid. What is the coordination number in a body centered cubic lattice?

The number of atoms or oppositely charged ions or molecules that present around a particular atom or ion or molecule respectively in a crystal lattice is called coordination number.

The coordination number in a body centered cubic lattice is 8.

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