2. THE SOLID STATE. Synopsis: INTRODUCTION:

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1 2. THE SOLID STATE Synopsis: INTRODUCTION: Solids ae chaacteized by thei high density and low compessibility as compaed to those of the gas phase. The popeties of solids indicate that the molecule (o ions) in them ae elatively close togethe. Solids can be boadly classified into two categoies, namely, cystalline and amophous solids. The main chaacteistics of these ae descibed in the following. Cystalline Solids: The outstanding chaacteistics of a cystal ae its shap melting point, its flat faces and shap edges. These popeties ae due to a high degee of intenal ode which extends thoughout the cystal (a definite patten constantly epeating in space). This is know as long-ange ode. Amophous Solids: Amophous solids do not have the long-ange ode but have a shot-ange ode. This chaacteistics may not be found aound a simila atom placed at a distance fom the othe atom. Examples of amophous solids ae glass, used silica, ubbe and polymes. Amophous solids do not have the chaacteistics as possessed by cystalline solids. In many ways, they ae moe closely elated to liquids than to the cystalline solids and ae, theefoe, egaded as supecooled liquids with high viscosity. A given mateial may be conveted into the amophous o glassy fom by apidly cooling the melt o feezing the vapou. CLASSIFICATION OF CRYSTALS BASED ON BOND TYPE The popeties of most of the cystals ae found to confom to one of the fou geneal types of chemical bonds, in tems of which it is possible to classify them into fou categoies as descibed in the following. Molecula Cystals (o van de Waals Cystals):Molecula cystals ae those in which the cystalline state is composed of an aggegate of discete molecules held togethe by van de Waals foces. Because of these weake foces, molecula cystals ae soft and posses compaatively low melting points. Examples ae CO 2, CCl 4, A and most of the oganic compounds. Ionic Cystals: Ionic cystals involve electostatic foces amongst thei stuctual units. Because of stonge foces, ionic cystals ae stong and likely to be bittle. The melting points ae high, which decease with inceasing size of the ions. In ionic cystals, some of the atoms may be held togethe by covalent bonds to fom ions having definite positions and oientations in cystal lattice. Covalent Cystals: Covalent cystals involve foces of chemical natue (covalent bonds) extended in thee dimensions. These foces ae stong, and consequently the cystals ae stong and had, with high melting points. Examples ae diamond, silicon, etc. Metallic cystals: Electons ae held loosely in these type of cystals. They ae god conductos o electicity. Metallic cystals ae stong and can be bent. Hydogen bonded cystals Molecules having H atom bonded to moe electonegative atoms like F, O, N exhibit stong hydogen bonding Cystals of H 2 O, HF, NH 3 and amino acids ae some examples of hydogen bonded cystals. CRYSTAL SYSTEMS : - Thee ae 230 cystal foms possible. These foms may be classified into 32 classes on the basis of thei symmety. 1

2 On the basis of inte facial angles and axes cystal systems ae 7 types. The cystal stuctues depends on the set of cystallogaphic paametes Ice may give hexagonal o tigonal cystals Quatz may give hexagonal o tigonal cystals BRAVAIS LATTICES : - Bavais showed that thee ae 14 diffeent possible kinds of thee dimensional lattices. The geometic shape of the cystal lattice must be same as that of solid cystal itself. Ex : If unit lattice is having cubical stuctue the cystal is also a cube. Fom fundamental laws of cystallogaphy thee ae 14 basic aangements known as Bavais Lattices. These 14 Bavais lattices ae gouped into seven cystal system based on unit cell symmety. These seven cystal systems ae 2

3 Bagg s Equation: If the waves ae in phase, then constuctive intefeence takes place. If the waves ae not in phase, they undego destuctive intefeence. If the two waves ae to be pesent in phase, the path diffeence must be equal to the wavelength(λ) o an integal whole multiple of wavelength (nλ). Bagg s equation is nλ = 2d sinθ n = ode of eflection (1, 2, 3,.) λ = wavelength of x-ay d = distance between planes θ = angle of incidence At the Bagg s expeiment, as the intensity of diffacted x-ay incease, the degee of ionisation also incease. It Bagg s angle (θ), intensity of the diffacted x-ay is maximum and maximum value is ecoded in electomete. The NaCl cystal has the d values in the atio 1 : : fom this it is confomed the cystal has fcc system. In the powde method, the holes and the ends of the photogaphic film ae inlet and the outlets of the x-ays. Packing of identical sphees: Thee ae two types of packing. 1) Close packing and 2) not close packing. Close packing of sphees leads two types of lactices a) hcp (hexagonal close packing) and b) fcc o ccp (cubic close packing) Not close packing type leads bcc (body cented cubic lattice) HCP: It has ABABAB.. type of layes. 3

4 Co-odination numbe is 12 Packing faction is In one unit cell of HCP, on aveage 6 sphees ae pesent and 6 octahedal voids and 12 tetahedal voids ae pesent. Examples: Mg, Zn, Ce, Ti etc. CCP o FCC: It has ABCABC.. type of layes. Co-odination numbe is 12 Packing faction is In one unit of ccp o fcc, on aveage 4 sphees ae pesent and 4 octahedal voids and 8 tetahedal voids ae possible. Metals having this lattice ae moe meleable. Examples: Al, Cu, Ag, Au etc. BCC: It is not a close packing and it has ABAB type of layes. Co-odination numbe is 8 Packing faction is Examples: Li, Na, K etc. Two tetahedal voids ae pesent above and below each atom and the octahedal void is pesent midway between the two close-packed layes. Ionic cystals: In ionic cystals, it is not possible fo both cations and anions to have close-packed stuctues due to thei diffeent sizes. Howeve, if one of the ions is bigge than the othe, it is common fo the bigge ions (usually anions) alone to appoach a closed-packed stuctue and smalle ions to fit into holes (o voids) in this stuctue. The limiting sizes of cations that can fit into the voids without distubing the closest packing of anions ae as follows. c Octahedal void = Tetahedal void a c a = A few common ionic lattices ae descibed below. Rock-Salt Stuctue: In such a stuctue, anions fom a face-cented cubic unit cell and cations occupy octahedal voids. Thee ae fou anions and fou cations pe unit cell of this stuctue, and hence, the fomula of ionic compound is A 4 B 4 o simply AB. Examples ae NaCl, KCl, MgO, CaO and SO. Anti-Fluoite Stuctue: In this, anions fom a cubical-closest packing aangement and cations occupy tetahedal voids. Thee ae fou anions and eight cations pe unit cell of this stuctue and hence the fomula of the ionic compound is A 8 B 4 o simply A 2 B. Examples ae Li 2 O, Na 2 O, K 2 O and Rb 2 O. 4

5 Zinc-Blende Stuctue: In such a stuctue, anions fom cubical closest packed stuctue and cations ae pesent in the half of the tetahedal holes. Thee ae fou anionsand fou cations pe unit cell of this stuctue, and hence the fomula of an ionic compound is A 4 B 4 o simply AB. Examples ae BeO and ZnS. Body-Cented Cubic Stuctue: In this stuctue, the cation is located at the body cente of a cube of anions, and the anion is located at the body cente of a cube of cations. Thee is one cation and one anion pesent pe unit cell, and hence, the fomula of an ionic compound is AB. In this stuctue, each ion has eight positively chaged ions as its next neaest neighbous. Hence, the coodination numbe of each cation and anion is eight. Examples ae CsCl, CsB and CsI. Fluoite Stuctue: In this stuctue, cations fom cubical-closest packing and anions occupy tetahedal holes. Hee Coodination numbe of cations = 8 Coodination numbe of anions = 4 Geneal fomula: M 1/8 X 1/4 o MX 2. Examples include UO 2, ThO 2 and CaF 2. Coundum Stuctue: In this stuctue, anions fom hexagonal closest packing and cations ae pesent in 2/3 of the octahedal holes. The geneal fomula of the compound is M 2/3 X o M 2 X 3. Examples include Fe 2 O 3, Al 2 O 3 and C 2 O. Spinel stuctue: In this stuctue, oxides ae aanged in cubical-closest packing, one-eight of the tetahedal holes ae occupied by divalent metal ion (A 2+ ) and one-half of the octahedal holes ae occupied by tivalent metal ions (B 3+ ). In a unit cell, we have Numbe of divalent metal ions, A 2+ = 1 (8) = 1 8 Numbe of tivalent metal ions, B 3+ = 1 (4) = 2 2 Numbe of oxide ions, O 2 = 4 Hence, Geneal fomula of the compound: AB 2 O 4 Examples ae ZnAl 2 O 4, MgAl 2 O 4 and ZnFe 2 O 4. Stability of ionic Stuctue: Boadly, speaking the stability of ionic stuctues depends upon the adius atio of c and a. The stuctues pedicted fom the value of c / a ae as follows. c a Coodination Ratio Stuctue numbe > Body-cented 8 cube Face-cented c cubic i.e. 6 a octahedal c Tetahedal 4 a 5

6 Fo example, chloides of Li, K and Rb cystallize in the face-cented cubic lattice wheeas the chloide of Cs cystallize in the body-cented cubic lattice. CALCULATION OF THE CONTRIBUTION OF LATTICE POINTS PER UNIT CELL OF A SUBSTANCE. A thee dimensional cystal lattice is built up by the unit cells aanged in thee diections. The unit cells have diffeent shapes depending on the aangement of the component atom (o) ions in space lattice. Each lattice point and the component paticle contibute faction to the unit cell because they ae shaed by vicinal unit cells. The atoms, molecules (o) ions that ae making up the cystal ae pesent at the lattice points and ae epesented by points. A lattice point at the cone of a unit cell is shaed by 8 vicinal unit cells. A face centeed point is shaed by two adjacent unit cells. A edge centeed point is shaed by fou adjacent unit cells A body centeed lattice point belongs completely to a specific unit cell. Total contibution of lattice points can be calculated fom the unit cell.. CALCULATION OF DENSITY OF UNIT CELL Knowing the unit dimensions, the density of a cystalline substance can be calculated fom the fomula ZM ZM ρ = 1 o ρ = 3 Na NV 0 Whee ρ = density Z = No. of atoms (o) fomula units pe unit cell M = Molecula weight of the lattice paticle (Atomic wt / fomula wt ) unit cell length. v = volume 0 6

7 PACKING OF SOLIDS AND VOIDS In solids constituent paticles ae aanged as sphees in close stuctue. In close packed stuctue, maximum space is occupied by constituent paticles and leave minimum vacant space When the sphees ae closely packed the diffeent cystal systems ae geneated. As a consequence of this close packing in diffeent patten one gets the face cented cubic, body cented cubic and hexagonal closed packed stuctue etc. In such packings a cental sphee, pesent in two dimensional aay touches six neaest sphees in hexagonal close packing. In a squae packing, a cental sphee touches fou neighbouing sphees Hexagonal close packing is moe compact and occupies 74% volume and the emaining 26% space is void. In a cubic close packing it is 74% occupied space. INTERSTIAL VOIDS : The empty spaces between the thee dimensional layes ae known as holes o voids. The holes ae also efeed as intestices. Thee ae thee types of holes possible. (A) TETRAHEDRAL HOLES : A hole fomed by thee sphees in contact with each othe of a laye. The hole is capped by a sphee fom an uppe laye.(plana tiangle with vetex down wads) A hole fomed by thee sphees of a laye in contact with each othe and also with a sphee of a next lowe laye (plana tiangles with vetex upwad) Note : In the above types of holes the fou sphees ae aanged at the vetices of a egula tetahedon. If X sphees fom a solid thee ae a total of 2X tetahedal holes. Out of them X holes ae of the type (1) and the emaining X holes belong to type(2). (B) OCTAHEDRAL HOLE : It is the vacant space between a goup of thee sphees in a laye and anothe set of thee sphees of a next laye. These six sphees suounding the hole, lie at the vetices of a egula octahedon. If thee ae X atoms in a close packed stuctue, thee ae X octahedal holes pesent. Thus numbe of tetahedal holes is double the numbe of octahedal holes. The size of these holes depends on the size of the sphees poducing them. Octahedal holes ae lage than tetahedal holes. POINT DEFECTS IN CRYSTALS Stoichiometic compounds ae called Daltanides, non-stoichiometic compounds ae called Bethollides. Point defects ae the iegulaities o deviations fom ideal aangement aound a point o an atom in a cystalline substance Popeties like density, entopy and heat capacity ae influenced by cystal defects to a lesse extent. The defects in the cystal influence the popeties like mechanical stength, electical conductivity and chemical activity of the solids to a geate extent. Themodynamically all solids possess a tendency to acquie defects because defects esult in disode and hence incease the entopy of the system Electon mico scope is used to know the defects in a solid cystal The defects in the cystals can be devided in to diffeent kinds 7

8 TYPES OF CRYSTAL DEFECTS INTRINSIC DEFECTS : These ae seen in pue cystals EXTRINSIC DEFECTS : These ae due to the impuities in the solids POINT DEFECTS : These occu at the lattice points o sites in the cystals EXTENDED DEFECTS : These ae pesent in one o moe dimensions In stoichiometic compounds two types of defects ae obseved Solid State (A) SCHOTTKY DEFECT(SCHOTTKY-WAGNER DEFECTS) This defect aises due to a vacancy at cation sites and equal numbe of vacancies at anion sites In ionic cystals electical neutality is to be maintained This type of defect occus in ionic compounds with (a) High co-odination numbes and + (b) Whee the positive and negative ions ae of simila size i.e., 1 Ex : NaCl, CsCl Schottky defect is found in highly ionic compounds Lage numbe of vacancies in the lattice lowe the density (B) FRENKEL DEFECT This type of defect aises due to a vacancy at a cation site. Actually, The cation moves to anothe position between two layes (intesticial position) and it is suounded by a geate numbe of anions Fenkel defect is favoued by a lage diffeence in sizes between cation and anion + Compounds having ions of diffeent sizes i.e is low since positive ions ae smalle than negative ions, geneally the positive ions ae found in intestitial positions In these compounds the coodination numbe is low(usually 4 o 6) Ex : ZnS, AgB, AgI, AgCl. Some ionic cystals have both the Schottky as well as Fenkel defects Ex : AgB ELECTRICAL AND MAGNETIC PROPERTIES Popeties of solids depending on the natue of bonding o cohesive foces pesent in them In ionic solids cohesive foces ae due to the lattice enegy and they oiginated fom Coulombic foces of attactions between the two ions In metals electons ae essentially mobile and so the popeties of metals can be explained in tems of electon bands fomed by molecula obitals. Cystals can possess vaious stuctues, these stuctues can be studied by X-ay diffaction methods. Pue cystalline solids ae also impefect aangements. These defects in solids cause changes in the popeties of the solids. Cystalline solids exhibit a wide ange of popeties namely, Electical, Mechanical, Magnetic and othe popeties. The physical popeties of solid, its stuctue and its chemical composition ae all closely inteelated. ELECTRICAL PROPERTIES : Based on electical conductivity, solids can be boadly classified into thee types 8

9 They ae Metals, Semi-conductos and Insulatos o Non-conductos Metals ae conductos and have conductivity of the ode of 10 3 to 10 8 ohm 1 cm 1 Solid State Insulatos have vey low conductivity of the ode of to ohm 1 cm 1 The solids whose conductivity lies between those of Metallic conductos and insulatos ae called Semiconductos. The ode of semic conductos 10 6 to 10 4 ohm 1 cm 1 The electical conductivity that is obseved is usually elated to defects in the cystal. In such defective cystal the migation of ions o chage impefections(in case of Ionic conductivity) o the motion of electons and shifting of holes(incease of electonic conductivity) elate the conductivities. The mechanism of electical conductivity may be given in tems of a) Vacancy Mechanism b) Intestitial Mechanism and c) Intestitialcy Mechanism The Magnitude of electical conductivity stongly depends on the numbe of electons available to paticipate in the conduction pocess. In Metals the conductivity depends on the numbe of valence electons pesent pe atom. when a lage numbe of atoms unite thei wave functions inteact. This esults in a lage numbe of molecula obitals o enegy levels, some Bonding and some antibonding. These enegy levels ae closely spaced enegy levels and ae known as bands. The numbe of enegy levels is equal to the numbe of Atomic obitals inteacting. One half of these esulting obitals have loweed enegy levels and ae known as bonding obitals. They ae of simila enegies and fom a continuous band of obitals. The othe half of esultant obitals have aised enegy levels ae known as Antibonding enegy levels o obitals. At all eal tempeatues the band has a lage numbe of half filled enegy levels. Electons can flow eadily though these bands only when the band is incompletely filled with electons o when electonic band(filled enegy levels) is vey close to vacant molecula obitals. At laboatoy tempeatues the conductivity of the metals is nealy independent of impuities and the lattice defects. At low tempeatues the mobility due to vibations in the lattice is quite negligible and as a esult the conductivity becomes infinitely lage. The conductivity of metals geneally decease with an incease in tempeatue. This is pobably due to inceased vibations in the lattice points the esistance to the flow of electons inceases. The impuities in the metal and the impefections in the lattice also contibute to the conductivity. Measuement of the esistance of the metal as esistance atio explain the puity of the metal. The conductivity of semi conductos vaies completely in the opposite way to that of the metals. Semi conductos conductivities inceases with incease in tempeatue. This is due to the fact that the electons fom the valence band jump to the conduction band. Pue semi conductos which exhibit this popety ae known as intinsic semi conductos. The tempeatue zone whee the conductivity depends on the themal electons and the holes in the lattice of the semi conductos is known as intinsic egion. At low tempeatue the conductivity is mainly detemined by the concentations of the electon donos and the acceptos. This egion is known as extinsic egion. Doping : - Addition of B ( IIIAGoup ) ( o ) P ( o ) As ( VA Goup ) element to alte the conductivity of ( ) Si Ge o is called as doping. 9

10 Pue Si ( o) Ge ae intinsic semiconductes Solid State In doping, goup IIIA element behaves as electon accepto and goup VA( 15goup elements) element behaves as dono. When VA (o) 15 th goup element is added to, the cystal lattice does not change, but few atoms ae eplaced by goup VA (o) 15 th goup elements, it foms covalent bonds with the fifth electon is delocalised, theefoe becomes electical conducto. Silicon doped with VA (o) 15 th goup element is known as n type semi conducto ( n = negatively chaged electons ae esponsible fo conductivity) In doping VA (o) IIIA goup element is called as dopant When IIIA (o) 13th goup ;element like (o) o is added to the substitution of few atoms by goup IIIA takes place IIIA goup element has only 3 valence electons. One moe electon is equied, it is left as a vacant place on the atom. This is called as electon vacancy o a hole. This electon vacancy in the cystal stuctue migates fom one atom to anothe. This is esponsible fo electical conductivity of Si. Si doped with IIIA (o) 13 th goup element is called p type semiconducto n type and p type semi conductos both ae used in electonic devices Example : Diode, pnp o npn type semiconductos ae used in tansistos. These tansistos ae used to detect (o) amplify adio o audo signals In the convesion of adiant enegy into electical enegy sola cell (p n type ) is used Many solid state semiconductos ae pepaed by combining IIIA o 13 th goup elements with VA o 15 th goup elements and IIB o 12 th goup elements with VIA o 16 th goup In this type of conductos bond is not pue covalent natue of the bond depends on electonegativities of two elements. Tansition metal oxides shows wide vaiation in this electical popeties. Metal sulphides of tansition elements also exhibits wide vaiation in thei electical popeties. MAGNETIC PROPERTIES : The substances can be classified into five types depending on thei esponse to an applied magnetic field. (A) DIAMAGNETIC MATERIALS : The substances which ae weakly epelled by magnetic field examples NaCl, KCl, TiO 2 ZnO 2 Benzene etc. Alignment of magnetic moments in diamagnetic substances (B) PARAMAGNETIC MATERIALS: The substances which ae attacted by magnetic field due to the pesence lone electons on atoms, ions o molecules. They lose thei magnetic popety when the applied filed is emoved Ex : O 2, Cu ++, Fe ++, Na, Ti 2 O 3, VO 2, NO etc (C) FERROMAGNETIC MATERIALS : The substances which ae stongly attaacted by magnetic field. 10

11 These substances contain domains of magnetization. All of them ae oiented in the same diection. They etain thei magnetism even afte withdawal of applied field. Ex : Fe, Co, Ni, CO 2 etc. Fe, Co, Ni, ae the only thee elements which show feomagnetism at oom tempeatue. The alignment of magnetic moments in these substances is (D) Anti Feomagnetism : It is due to compensatoy alignment of moments. The net moment is Zeo. The alignment in these substance is Ex : MnO 2, V 2 O 3, NiO etc. (E )Feimagnetism : It is due to the unequal numbe of alignement of moments in paallel and antipallel diections. It esults in the net moment. Fe 3 O 4, MgFe 2 O 4, CuFe 2 O 4, ZnFe 2 O 4 etc. All the magnetically odeed solids tansfoms to the paamagnetic state at elevated tempeatues due to the andomization of spins Ex : V 2 O 3 changes to paamagnetic at 150 K and NiO at 523 K Evey substance has some type of magnetic popety, these magnetic popeties ae measued in tems of magnetic susceptibility and magnetic moment. Magnetic subsceptibility is expessed in gam susceptibility, volume susceptibility o mola susceptibility. BRAIN TWISTING CONCEPTS Length of face diagonal - 2a Length of body diagonal - 3a Fo fcc lattice of an ionic cystal of NaCl type edge length, a = ( + ) Packing faction = volume of atoms in the cube volume of the cube 2 c a Pecentage of volume occupied by aotms in a packing = packing faction

The geometric construction of Ewald sphere and Bragg condition:

The geometric construction of Ewald sphere and Bragg condition: The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in