B M S INSTITUTE OF TECHNOLOGY [Approved by AICTE NEW DELHI, Affiliated to VTU BELGAUM] DEPARTMENT OF PHYSICS. Crystal Structure

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1 B M S INSTITUTE OF TECHNOLOGY [Approved by AICTE NEW DELHI, Affilited to VTU BELGAUM] DEPARTMENT OF PHYSICS COURSE MATERIAL SUBJECT: - Engineering Physics MODULE -IV SUBJECT CODE: - 14 PHY 1 / Crystl Structure The solid stte Atoms in solids my be rndomly positioned (s in liquid) Amorphous Solids (e.g. glsses) Arrnged in n orderly, repeting pttern within the mteril Crystlline Solids A crystl is one in which toms or molecules re in three dimensionl periodic rrngement. The periodicity my be sme or different in different directions. The periodic positions of the toms or molecules re clled spce lttice or crystl lttice. The geometricl representtion of crystl structure in terms of lttice points is clled spce lttice. How do we describe the rrngement of toms in crystls? There re two types of lttice 1) Brvis lttice ) Non Brvis lttice A Brvis lttice is one in which ll the toms t the lttice points re identicl or ll the lttice points re equivlent. A non-brvis lttice is one in which some of the lttice points re non equivlent. A non-brvis lttice is lso known s the lttice with bsis. Bsis is the set of toms locted ner to the Brvis lttice. Deprtment of Physics - 1 -

2 Non Brvis lttice is the superposition of two or more different Brvis lttice. In the digrm, points A, B, C etc., identicl points represents Brvis lttice wheres AA`, BB`, CC` etc., which re not identicl represents non- Brvis lttice. Spce Lttice + Bsis = Crystl Structure Bsis Vectors Consider two dimensionl rrys of points. Let O be the origin. be the coordinte vectors. nd The position or lttice vector R is R = n 1 + n b where n 1 nd n re integers whose vlue depends on lttice points. For B, (n 1, n ) = (1,0) Deprtment of Physics - -

3 For C, (n 1, n ) = (1,1) For D, (n 1, n ) = (0,1) For E, (n 1, n ) = (-1,0) Unit Cell nd Lttice Prmeters. A unit cell is the smllest portion of the spce lttice, which on repetition long the direction of three bsis vectors genertes the spce lttice itself. Unit cell hs bsis vectors, & nd interfcil ngles α, β & γ. Y-xis b O c X-xis Z-xis The bsis vectors nd interfcil ngles α, β & γ re the lttice prmeters. Ech unit cell hs only one lttice point becuse there re eight points t its corners nd ech point is shred by eight djcent cells. Primitive cell nd non-primitive cell Consider Brvis lttice in two dimensions s shown in figure. A unit cell with bsis vectors & b hs four lttice points t the vertex. It is 1 1 primitive cell. The other unit cell with bsis vectors nd hs n dditionl lttice point t the center long with four points t the corners, which is not the meeting point of the bsis vectors. This is non-primitive cell. 1 b1 A primitive cell is unit cell with ll the points in it re t the vertices. Deprtment of Physics - - b

4 A non-primitive cell is unit cell which incorportes n integrl multiple of primitive cells nd is imgined only for the ske of esy visuliztion of the symmetry in the rrngement of the lttice points. Crystl systems: There re seven crystl systems nmed on the bsis of geometricl shpe nd symmetry. The seven crystl systems re further divided into 14 Brvis lttice. The simple lttice hs points only t the corners. A body centered lttice hs n dditionl point t the centre of the cell nd fce centered lttice hs six dditionl points one on ech fce. The bse centered lttice hs two dditionl points one t the bottom fce nd other t the top fce. The seven crystl systems nd the 14 Brvis lttices re represented in the following tble: The 7 Crystl systems (From Most symmetric to lest symmetric) 1. Cubic =b=c ;α=β=γ=90 o Ex:- Au, Cu, NCl, CF, NclO simple (SC) Polonium The 14 Brvis Lttices bodycentered (BCC) Iron, Chrominium & Tungsten fcecentered (FCC) NCL(Hlite) or Rock Slt, luminium, copper, gold simple body-centered. Tetrgonl =b c ; α=β=γ=90 o Ex:-SnO, TiO,NiSO 4. Deprtment of Physics - 4 -

5 simple bse-centered bodycentered fcecentered. Orthorhombic b c α=β=γ=90 o Ex:- KNO,BSO 4, MgSO 4. simple bse-centered 4. Monoclinic b c α=γ=90 o β Ex:- CSO 4.H O, FeSO 4, N SO 4 5. Triclinic b c α β γ 90 o Ex:- K Cr O 7, CuSO 4.5H O 6. Rhombohedrl (trigonl) =b=c α=β=γ 90 Ex:- clcite, qurtz, Bi tourmline, As,Sb, 7. Hexgonl =b c α=β=90 o γ=10 o Ex:- SiO Zn,Mg,Cd,AgI MILLER INDICES OF DIRECTIONS AND PLANES: Miller hs introduced three numbers to explin the plnes nd directions in crystl. These numbers re clled Miller indices. A Miller indices of direction is bsiclly vector between two points in the crystl. Any direction cn be defined by following simple procedure: Deprtment of Physics - 5 -

6 1. Choose the position vector, so tht it is in convenient position within your chosen coordinte system.. Find the projection of the vector onto ech of the three xes in terms of the unit cell dimensions.. To get the Miller Index we express the vector s set of whole number, nd enclose in squre brckets. Exmple:- A cubic unit cell long with miller indices of directions. Procedure to find miller indices of plnes: 1. Find the intercepts of the plnes long x, y nd z xis. Express the intercepts s multiples of lttice constnts, b, c.. Find out the reciprocl of these numbers. 4. Find the lest common multiple (LCM) of the denomintor nd multiply ech term with LCM. 5. The result is in the form of h, k, l. re clled miller indices, denoted by (h k l) Intercepts Reciprocls Whole numbers Miller indices b c 1/ 1/1 1/1 1 (1 ) b ½ 1/1 1/ 1 0 (1 0) b c 1/1-1/1 ½ 1 ( 1) - b c -1/1 ½ -1/1-1 - ( 1 ) Few exmples for obtining the Miller indices Clcultion of miller indices Following points should be noted 1. The negtive digit indictes cutting of xis on the negtive side of the origin.. Zero index indictes tht the plne is prllel to corresponding xis.. A prllel set of plnes hve sme miller indices. Exmple: The plnes hve intercepts 4, 1, 1. x:y:z=4:1:. Reciprocls. LCM= : 4 1 : 4 The miller indices re 1, 4, i.e. 1:4: By knowing the Miller indices we cn drw the plnes within the unit cell. The two Figures shown below gives the ide of drwing the plnes with in the unit cell. Deprtment of Physics - 6 -

7 1) The Fig. shows the plne with Miller indices (110) Plne with Miller indices (110) ) The Fig. shows the plne with Miller indices (00) Plne with Miller indices (00) Due to the symmetry of crystl structures the spcing nd rrngement of toms my be the sme in severl directions. These re known s equivlent directions. A group of equivlent directions is known s fmily of directions denoted by < h k l>. Similrly the set of equivlent plnes re known s fmily of plnes is denoted by {h k l}. Deprtment of Physics - 7 -

8 Miller indices of directions Fmily of plnes is denoted by {h k l} nd Fmily of Directions is denoted by < h k l> Expression for inter-plnr spcing in terms of Miller indices: Consider plne ABC which belongs to fmily of plnes. h, k, l re the Miller indices of this plne, which represents the set of plnes. The perpendiculr ON from the origin O to the plne represents the inter-plnr spcing d=on of this fmily of plnes. Let ON mke n ngle α 1, β 1, γ 1 with the x, y, z xes respectively Deprtment of Physics - 8 -

9 The intercepts of the plne on the three xes re OA= /h ; OB= /k ; OC=/l Where is the length of the cube edge. Then from figure, we hve 1 d d cos OA / h 1 d d cos OB / k cos 1 d d OC / l From the figure ON d x y z 1 dh dk dl d cos d cos d cos But, cos cos cos 1 for the orthogonl coordintes Substituting the vlues of cos 1, cos 1, cos 1 in eqution (), We get, d OA d OB dh d OC dk 1 dl 1 Deprtment of Physics - 9 -

10 d h k l 1 d h k l This is the reltion between inter-plnr spcing d nd the edge of the cube. It should be noted tht this formul is pplicble only to primitive lttices in cubic, orthorhombic nd tetrgonl systems. d hkl Spcing between the plnes 100, 110 nd 111 if h k l= 100, d 100 h k l If h k l=110, d If h k l=111, d Expression for spce lttice constnt for cube lttice Let be the lttice constnt, ρ be the density of the mteril nd n be the number of molecules in unit cell. Totlmssof molecules in unitcell Volume of the unit cell The totl mss of the molecules in one kilo mole of substnce is equl to the moleculr weight of the molecule expressed in kg. Therefore the mss of ech molecule = M/N A Where N A is Avgdro number The totl mss of the molecule = For cubic lttice =b=c... the volume of the unit cell = Hence, the density (ρ)is = nm N A nm N A nm N A Deprtment of Physics

11 nm N A 1. Simple cubic structure (SC): 1. Body centered cubic structure (BCC):. Fce centered cubic structure (FCC): Properties of the Unitcell (Cubic) 1. Volume of unit cell: The generl expression for finding the volume unit cell is V bc 1 1 cos cos cos cos cos cos Where,b,c,, nd re clled lttice prmeters. For cubic system =b=c nd ===90 0. Therefore V=. Deprtment of Physics

12 . Co-ordintion number: It is the number of nerest neighbors directly surrounding given tom well within crystl. The co-ordintion number for n tom in simple cubic structure is = 6 The co-ordintion number for n tom in body centered cubic structure is =8 The co-ordintion number for n tom in fce centered cubic structure is =1. Number of toms per unit cell: In unit cell toms re t the corners, t the center of the fces nd t the center of the body. An tom situted t the corner shre 1/8 th prt to unit cell. An tom situted t the fce shre 1/ prt to unit cell. An tom situted t the center of the body shre one full prt to unit cell. 1. In simple cubic structure there re 8 corner toms.... Totl shre of ll the corner toms/unit cell = (1/8) 8 =1... The number of toms/unit cell in simple cube = 1. In body centered cubic structure there re 8 corner toms nd n tom t the center of the unit cell.... Totl shre of ll the corner toms per unit cell = (1/8) 8 = 1 The shre of n tom t the center of the body =1... The number of toms per unit cell in body centered cube = 1+1=. In fce centered cubic structure there re 8 corner toms nd 6 fce centered toms in unit cell....totl shre of toms t the corner/unit cell= (1/8) 8=1 Totl shre of toms t ll the fces/unit cell = (1/) 6=... The number of toms/unit cell in fce centered cube = 1+ = 4 4. Atomic pcking fctor (APF) The frction of the spce occupied by toms in unit cell is known s tomic pcking fctor. It is the rtio of the totl volume occupied by the toms in the unit cell to the totl vilble volume of the unit cell. 1. Simple cubic structure (SC): There is only one lttice point t ech of the eight corners of the unit cell. In simple cubic structure n tom is surrounded by six equidistnt neighbors. Hence the co-ordintion number is 6. Since ech tom in the corner is shred by 8 unit cells, the totl number of toms in one unit cell is (1/8) 8 = 1 The nerest neighbor distnce r is the distnce between the centers of two nerest neighboring toms. Deprtment of Physics - 1 -

13 The nerest neighbor distnce r = The number of lttice points per unit cell = 1 Volume of ll the toms in unit cell v = πr Volume of unit cell = V = = (r) v 4r Pcking fctor is P.F = 0.5 5% V 8r 6. Body centered cubic structure (BCC): In BCC structure eight toms re present t eight corners nd one tom is t the center. The co-ordintion number is 8. The number of toms per unit cell is = [(1/8) 8]+1 = The lttice constnt is (AB) = + = (AC) = (AB) + (BC) (4r) = + = Lttice constnt 4r Volume of ll the toms per unit cell v = Volume of the unit cell V = = 64r 4 r v 8r Atomic pcking fctor = % V 64r 8. Fce centered cubic structure (FCC): In FCC structure eight toms re t the corners of the unit cell nd six toms re present t the center of the six fces. The center tom is surrounded by 1 points. The co-ordintion number is 1. Ech corner tom is shred by 8 unit cells nd the fce centered tom is shred by surrounding unit cells. Deprtment of Physics - 1 -

14 ... The number of toms per unit cell is Atomic rdius of fce centered cube AB = 4r (AB) = + (4r) = Lttice Constnt = 4r Nerest neighbor distnce r = Volume of ll the toms in unit cell v = 4 4 r Volume of unit cell V = 64r =.. v 16r. The pcking fctor = % V 64r 6 Prmeters SC BCC FCC Co-ordintion number Atomic Rdius (r) 4 4 Atoms per unit cell 1 4 Atomic pcking fctor Brgg s Lw: Consider set of prllel plnes clled Brgg s plnes. Ech tom is cting s scttering center. The intensity of the reflected bem t certin ngles will be mximum when the pth difference between two reflected wves from two djcent plnes is n integrl multiple of λ. Deprtment of Physics

15 Let d be the distnce between two djcent plnes, 'λ be the wvelength of the incident x-ry, θ be the glncing ngle. The pth difference between the rys reflected t A & B is given by = CB + BD = d sinθ + d sinθ = dsinθ For the reflected light intensity to be mximum, the pth difference dsinθ = nλ, where n is the order of scttering. This is clled Brgg s lw. Brgg s x-ry spectrometer: The schemtic digrm of Brgg s x-ry spectrometer is shown in fig. It is used to determine lttice constnt nd inter-plnr distnce d. It hs 1) x-ry source ) A Crystl fixed on circulr tble provided with scle nd vernier. ) Ioniztion chmber. Deprtment of Physics

16 A collimted bem of x-rys fter pssing the slits S 1 nd S is llowed to fll on crystl C mounted on circulr tble. The tble cn be rotted bout verticl xis. Its position cn be mesured by vernier V 1. An ioniztion chmber is fixed to the longer rm ttched to the tble. The position of which is mesured by vernier v. An electrometer is connected to the ioniztion chmber to mesure the ioniztion current produced by diffrcted x-rys from the crystl. S nd S 4 re the led slits to limit the width of the diffrcted bem. Here we cn mesure the intensity of the diffrcted bem. If x-rys incident t n ngle θ on the crystl, then reflected bem mkes n ngle θ with the incident bem. Hence the ioniztion chmber cn be djusted to get the reflected bem till the ioniztion current becomes mximum. A plot of ioniztion current for different incident ngles to study the x-ry diffrction spectrum is shown in fig. The rise in Ioniztion current for different vlues of θ shows tht Brgg s lw is stisfied for vrious vlues of n. i.e. dsinθ = λ or λ or λ etc. Peks re observed t θ 1, θ, θ etc. with intensities of P 1, P,P etc. i.e. dsinθ 1 : dsinθ : dsinθ = λ : λ : λ The crystl inter-plner spcing d cn be mesured using dsinθ = nλ If d 1, d, d be the inter-plnr spcing for the plnes (100), (110) & (111) respectively. It cn be shown - Deprtment of Physics

17 For cubic crystl: d 1 :d :d = For FCC : d 1 :d :d = For BCC : d 1 :d :d = Allotropy nd Polymorphism The property possessed by certin elements to exist in two or more distinct forms tht re chemiclly identicl but hve different physicl properties is clled llotropy. Two or more distinct crystl structures for the sme mteril (llotropy/polymorphism). Allotropy (Gr. llos, other, nd tropos, mnner) or llotropism is behvior exhibited by certin chemicl elements tht cn exist in two or more different forms, known s llotropes of tht element. In ech llotrope, the element's toms re bonded together in different mnner. Note tht llotropy refers only to different forms of n element within the sme phse or stte of mtter (i.e. different solid, liquid or gs forms) - the chnges of stte between solid, liquid nd gs in themselves re not considered llotropy. In ech different llotrope, the element's toms re bonded together in different mnner. For exmple, the element crbon hs two common llotropes: Dimond, in which the crbon toms re bonded together in tetrhedrl lttice rrngement, nd grphite, in which the crbon toms re bonded together in sheets of hexgonl lttice. Dimond An extremely hrd, trnsprent crystl with tetrhedrl bonding of crbon very high therml conductivity very low electric conductivity. The lrge single crystls re typiclly used s gem stones The smll crystls re used to grind/cut other mterils dimond thin films hrd surfce cotings used for cutting tools, medicl devices. Grphite soft, blck, flky solid, with lyered structure prllel hexgonl rrys of crbon toms wek vn der Wl s forces between lyers plnes slide esily over one nother The chnge between different llotropic forms of n element is often triggered by pressure nd temperture, nd mny llotropes re only stble in the correct conditions. For instnce, iron only chnges from ferrite to ustenite bove 1, F (7 C), nd tin undergoes process known s tin pest t 56 F (1. C) nd below. Other exmples of llotropes include: Phosphorus: Red Phosphorus polymeric solid White Phosphorus crystlline solid Blck Phosphorus semiconductor, nlogous to grphite Oxygen: dioxygen, O colorless ozone, O blue tetroxygen, O 4 red Deprtment of Physics

18 Polymorphism is the phenomenon where compound cn exist or precipitte to form numerous crystl structures. The different crystlline structures ech hve different physicl properties, which cn chnge the use of the chemicl. The physicl properties tht my differ from one polymorphism to nother include: solubility, density, melting point nd even color. One of the vribles tht ffect the crystlliztion process is the solvent tht is used in the precipittion. Different polymorphisms cn lso be formed by mnipulting the solute concentrtions, flows rtes, nd equipment configurtions. Crystls re used in mny res of science, phrmceuticls, nd mterils engineering. Crystls differ from mny other orgnic nd inorgnic mterils becuse of their bility to form polymorphisms. There re mny physicl nd chemicl properties tht cn be ffected by chnge in the polymorph. When crystl trnsforms from one of its polymorphs to nother, both the chemicl nd physicl properties of the substnce re ltered. For exmple, solubility, hrdness, shpe, melting point, dissolution rte, density, opticl nd electricl properties, nd electromgnetic spectr re some properties tht re ffected by chnge in the crystl structure. In the cse of iron the crystl structure hs one form t room temperture nd nother t high temperture. When heted bove C the tomic structure chnges from body centered cubic to fce centered cubic but reverts gin when cooled. The llotropy of iron modifies the solubility of crbon, nd it is becuse of this tht steel cn be hrdened. Crystl Structure of Dimond nd Perovskites The dimond lttice cn be considered s the superposition of fcc sublttices one of which is displced from the other long the body digonl of the unit cell by (1/4) th the length of the digonl. Thus crbon tom will be present t the center of tetrhedron, with four crbon toms of the other sublttice s its nerest neighbours locted t four corners of the sme tetrhedron. Thus the coordintion number of dimond is 4 nd there will be 8 crbon toms present per unit cell. Tht is ech crbon tom is t the center of tetrhedron, 4 crbon toms re t digonlly opposite in the plnes. The lttice constnt is.5a o nd the bond length is 6.56A o. The semiconductors like Si nd Ge re hving dimond structure. An FCC lttice hs 4 toms/unit cell. But the dimond lttice comprises of two interpenetrting FCC sub-lttice. Therefore, Number of toms per unit cell in the dimond lttice is x4 = r And Atomic Pcking Fctor = r r 4% 16 The prt of the unit cell considered is s below: Deprtment of Physics

19 The coordintes of the centrl C tom re t the corner positions re respectively 1 1,, ,0,0, 1 1, 0,, 1 1, 1 1,0,,,0 1 nd the ocordintes of C toms Perovskite crystl structure (Qulittive) The structurl fmily of perovskites is lrge fmily of compounds hving crystl structures relted to the minerl perovskite CTiO. In the idel form the crystl structure of cubic ABO perovskite cn be described s consisting of corner shring [BO 6 ] octhedr with the A ction occupying the 1-fold coordintion site formed in the middle of the cube of eight such octhedr. The idel cubic perovskite structure is not very common nd lso the minerl perovskite itself is slightly distorted. Distorted perovskites hve reduced symmetry, which is importnt for their mgnetic nd electric properties. Due to these properties, perovskites hve gret industril importnce, especilly the ferroelectric tetrgonl form of BTiO. Deprtment of Physics

20 Idel perovskite: exmple, SrTiO ( Perovskite is the minerl nme of CTiO, whose ctul structure, though it pproximtes to tht of SrTiO, is more complicted nd of lower symmetry.) Cubic; primitive lttice P = 90.5 pm 1 Ti t 0, 0, 0 1 Sr t O t,, 1, 1, 1,0,0 0,,0 0,0, Ech Ti tom hs 6 neighbouring O toms, t distnce /, forming regulr octhedron. Ech octhedron shres ech corner with one similr octhedron to form three-dimensionl frmework, with cvities holding the lrger Sr toms. Ech Sr tom hs 1 neighbouring O toms t distnce ()/. Ech O tom is linked to Ti toms nd 4 Sr toms. The idel perovskite structure imposes prticulr reltion between ionic rdii s condition of ction nion contct for both ctions, nd only occurs when this is nerly stisfied, s in SrTiO. With moderte misfit, relted structures of lower symmetry re found, collectively described s structures of the perovskite fmily. Mny mterils possessing such structures t room temperture hve high-temperture form with the idel perovskite structure. Deprtment of Physics - 0 -

21 Objective Type Questions 1) A crystl of hexgonl lttice hs unit cell with sides ) Four types of Brvis lttice re observed in orthorhombic system ) The configurtion for the unit cell with sides =b=c nd interfcil ngles 0 90 defines rhombohedrl systems 4) For cubic unit cell the plne ( ) will be prllel to Y Z plne 5) The interplnr spcing in crystl is 1 nd the glncing ngle is 5. For the first order Brgg reflection to tke plce the pth difference between the rys reflected between two successive plnes should be ) The crystl structure for NCl is fcc 7) The intertomic distnce between sodium nd chlorine tom in sodium chloride crystl is ) The coordintion number in cse of simple cubic crystl structure is ) Which of the mteril crystllizes in fcc structure luminum. 10) The number of molecules present in the unit cell of sodium chloride is ) The miller indices for the plne prllel to the x nd y xes re (0 0 1) 1) In Brgg spectrometer for every rottion of the turn tble the detector turns by n ngle ) The coordintion number in cse of the body centered cubic structure is ) Coordintion number for fcc is ) The number of toms per unit cell ) For sc structure is (8 )= 1 b) For bcc structure (8 )+ 1 = c) For fcc structure(8 )+ (6 )= 4 16) Reltion between tomic rdius nd the lttice constnt ) For sc structure is = r 4 b) For bcc structure is = r c) For fcc structure is = r 17) Atomic pcking fctor (APF) ) For sc structure APF is= = 0.5 b) For bcc structure APF is = % 8 c) For fcc structure APF is = % Deprtment of Physics - 1 -

22 APF for dimond crystl structure is 0.4=4%. 18) Brgg s lw of diffrction is d = n 19) Expression for spce lttice constnt for cubic lttice is nm N A 0) The crystl of tetrgonl lttice is =b c. 1) The nerest neighbor distnce between two toms in cse of bcc structure is [()/4]. ) A plne intercepts t,b/,c in simple cubic unit cell. The Miller indices of the plne re (14). Numericls 1. An x-ry bem of wvelength 0.7A 0 undergoes minimum order Brgg reflection from the plne (0) of cubic crystl t glncing ngle 5 0. Clculte the lttice constnt. ( Jn 008). X-rys re diffrcted in the first order from crystl with d-spcing.8x10-10 m t glncing ngle 6 0. Clculte the wvelength of x-rys. (July 008). Clculte the glncing ngle for incidence of x-rys of wvelength 0.58A 0 on the plne (1) of NCl which results in second order diffrction mxim tking the lttice s.81a 0 (Jn 007, Jn 009). 4. Clculte the glncing ngle of the (110) plne of simple cubic crystl (=.814A 0 ) corresponding to second order diffrction mximum for the x- rys of wvelength 0.710A 0. ( July 011) 5. Interplnr distnce for crystl is A 0 nd the glncing ngle for second order spectrum ws observed to be equl to `. Find the wvelength of the x-rys? (Jn 011) Descriptive type questions: 1. Define lttice points, Brvis lttice nd primitive cell. Explin in brief the seven crystl systems with net digrms (Jn 007, July 007, Jn 010).. Explin how Miller indices re derived. Derive n expression for inter plnr spcing of crystl in terms of Miller indices.( July 008, Jn 009, Dec 010, Jn 011). Define coordintion number nd tomic pcking frction. Clculte pcking frction for SC, BCC nd FCC structures. (Jn 007,Jn 008, jn 010, Dec 010, Jn 011) Deprtment of Physics - - 1

23 4. Determine the coordintion number, number of lttice points per unit cell nd tomic pcking fctor for the FCC lttice? 5. Describe how Brgg s spectrometer is used for determintion of crystl structure? (July 007, Jn 008) 6. Explin the structure of NCl? ( July 008) 7. Explin with net sketch the dimond crystl nd show tht tomic pcking fctor of dimond is 0.4. (July 007, July 011) 8. Drw the following plnes in the unit cube i) ii) Deprtment of Physics - -