THE SOLID STATE MODULE - 3 OBJECTIVES. Notes

Transcription

1 The Solid Stte MODULE THE SOLID STATE You re wre tht the mtter exists in three different sttes viz., solid, liquid nd gs. In these, the constituent prticles (toms, molecules or ions) re held together by different forces of ttrction between them. However, the nture nd mgnitude of the forces vries. In the first two lessons of this module you hve lernt bout the gseous nd the liquid sttes of mtter. In this lesson you would lern bout solid stte- compct stte of mtter. The solids re distinguished from liquid or gs in terms of their rigidity which mkes them occupy definite volume nd hve well defined shpe. In solid stte, the constituent prticles re in close contct nd hve strong forces of ttrction between them. Here, you would lern bout structure, clssifiction nd properties of solids. OBJECTIVES After reding this lesson, you should be ble to explin the nture of solid stte; explin the properties of solids in terms of pcking of prticles nd intermoleculr ttrctions; differentite between crystlline nd morphous solids; explin the melting point of solid; clssify the crystlline solids ccording to the forces operting between the constituent prticles; define the terms crystl lttice nd unit cell; describe different types of two dimensionl nd three dimensionl unit cells; explin different types of pcking in the solids; define coordintion number; 175

2 MODULE - 3 The Solid Stte clculte the number of prticles in simple cubic, fce-centrd cubic nd body centered cubic unit cells; relte the density of solid to the properties of its unit cell; clculte the pcking efficiency of simple cubic body centrl cubic nd CCP/ HCP lttices; define rdius rtio; correlte the rdius rtio with the structure of solids; explin the structure of simple ionic compounds; explin Frenkel nd Schottky defects; clssify solids on the bsis of their electricl nd mgnetic properties; nd explin the effect of doping of semiconductors by electron deficit nd electron rich impurities. 6.1 NATURE OF SOLID STATE You hve lernt in lesson 6 tht ccording to Kinetic Moleculr Theory, the gses consist of lrge number of molecules, which re in constnt rndom motion in ll directions in the vilble spce. These molecules hve very wek or negligible forces of ttrction between them. A smple of gs cn be compressed, s there is lot of free spce between the molecules Fig. 6.1() In liquids Fig. 6.1(b) on the other hnd the molecules re lso in constnt motion but this motion is reltively restricted. Since there is very little free spce vilble between the molecules the liquids re reltively incompressible. () (b) (c) Fig. 6.1 A pictoril representtion of the three sttes of mtter () gs (b) liquid nd (c) solid stte. In solid stte the constituent prticles re rrnged in closely pcked ordered rrngement Fig. 6.1 (c) with lmost no free spce. They cn just vibrte bout their fixed positions. These re in close contct nd cnnot move round like the molecules of gs or liquid. As consequence, the solids re incompressible, rigid nd hve definite shpe. Like liquids, the volume of solid is independent of the size or the shpe of the continer in which it is kept. 176

3 The Solid Stte 6.2 CLASSIFICATION OF SOLIDS On the bsis of nture of rrngements of the constituent prticles the solids re clssified into morphous nd crystlline solids. MODULE Amorphous nd Crystlline Solids In crystlline solids the constituent prticles re rrnged in regulr nd periodic pttern nd give well defined shpe to it. The term crystl comes from the Greek word, krustllos mening ice. The regulr pttern extends throughout the solid nd such solids re sid to hve long rnge order. On the other hnd, some solids hve only short rnge of order. This mens tht the prticles re rrnged regulrly in only some regions of the solid nd re reltively disordered in other regions. Such solids re clled morphous solids. In Greek, mens without nd morph mens form. Thus the word morphous mens without form. Sodium chloride nd sucrose re common exmples of crystlline solids while glss, fused silic, rubber nd high moleculr mss polymers re some exmples of morphous solids. An importnt difference between the morphous nd crystlline solids is tht while morphous solids re isotropic in nture (i.e., these exhibit sme vlue of some physicl properties in ll directions) the crystlline solids re nisotropic (i.e., the vlues of some physicl properties re different in different directions). Refrctive index nd coefficient of therml expnsion re typicl physicl properties, which hve different vlues when mesured long different directions of given crystl. Another difference between morphous nd crystlline solids is tht while crystlline solids hve shrp or definite melting point, wheres the morphous solids do not hve definite melting point, these melt over rnge of temperture. The crystlline solids cn be further clssified on the bsis of nture of interction between the constituent prticles s discussed below Clssifiction of Crystlline Solids In crystlline solids the constituent prticles re rrnged in n ordered rrngement nd re held together by different types of ttrctive forces. These forces could be coulombic or electrosttic, covlent, metllic bonding or wek intermoleculr in nture. The differences in the observed properties of the solids re due to the differences in the type of forces between the constituting prticles. The types of forces binding the constituent prticles cn be used s bsis for clssifiction of crystlline solids. On this bsis, the crystlline solids cn be clssified into four different types- ionic, moleculr, covlent nd metllic solids. The chrcteristics nd the properties of different types of solids re compiled in Tble

4 MODULE - 3 Tble 6.1 Chrcteristics nd properties of different types of solids. The Solid Stte Type of Constituent Nture of Appernce Melting Exmples Solid Prticles interction Point between the prticles Ionic Ions Coulombic Hrd nd High Sodium chloride, brittle zinc sulphide, etc Moleculr Molecules Non polr vn der Wls Soft low Iodine, nphthlene Polr Dipole-dipole brittle wter crbon dioxide. Covlent Atoms Covlent bond- Hrd Very high Dimond, grphite, ing silic, etc. Metllic Atoms Metllic bonding Hrd nd Vrible Copper, silver, etc. mlleble Sodium chloride is n exmple of n ionic solid becuse in this cse the sodium ions nd chloride ions re ttrcted to ech other by electrosttic interctions. Iodine on the other hnd is n exmple of moleculr solid becuse in this the molecules re held together by wek vn der Wls forces. Dimond, with strong covlent bonds between the constituent crbon toms is n exmple of covlent solids while in metls lrge number of positive cores of the toms re held together by se of electrons. 6.3 PROPERTIES OF CRYSTALLINE SOLIDS You re fmilir with the following properties of solids on the bsis of hndling solids in dy to dy work. Solids re rigid in nture nd hve well defined shpes Solids hve definite volume irrespective of the size nd shpe of the continer in which they re plced Solids re lmost incompressible. You re fmilir with number of crystlline solids like sugr, rock slt, lum, gem stones, etc. You must hve noticed tht such solids hve smooth surfces. These re clled fces of the crystl. These fces re developed in the process of crystl formtion by ordered rrngements of the constituent prticles. It is generlly observed tht the fces of crystls re developed uneqully. The internl ngle between pir of fces is clled interfcil ngle nd is defined s the ngle between the normls to the intersecting fces. An importnt chrcteristic of crystlline solids is tht irrespective of the size nd shpe of the crystl of given substnce, the interfcil ngle between pir of fces is lwys the sme. 178

5 The Solid Stte This fct ws stted by Steno s the lw of constncy of interfcil ngles (Fig. 6.2). MODULE - 3 Fig. 6.2 The constncy of interfcil ngles Melting Point of Solid Wht is the effect of het on solid? You would hve observed tht when solid is heted it becomes hot nd eventully gets converted into liquid. This process of conversion of solid to liquid on heting is clled melting. You would lso hve observed tht different solids need to be heted to different extents to convert them to liquids. The temperture t which solid melts to give liquid is clled its melting point. Every solid is chrcterized by definite melting point. This in fct is test of the purity of the solid. The melting point of solid gives us n ide bout the nture of binding forces between constituent prticles of the solid. Solids like sodium chloride ( m.p = 1077 K ) hve very high melting points due to strong coulombic forces between the ions constituting it. On the other hnd moleculr solids like nphthlene ( m.p. = 353 K) hve low melting points. The effect of het on solid cn be understood in terms of energy nd motion of the constituent prticles. You re wre tht in solid the constituent prticles just vibrte bout their men position. As the het is supplied to the solid, the constituent prticles gin energy nd strt vibrting more vigorously bout their equilibrium positions. As more nd more het is supplied, the energy keeps on incresing nd eventully it becomes greter thn the binding forces between them. As consequence the solid is converted into liquid. INTEXT QUESTIONS 6.1 ) Differentite between solid, liquid nd gseous stte. b) How re solids clssified on the bsis of the intermoleculr forces. c) Wht is Steno s lw of constncy of interfcil ngles? 6.4 CRYSTAL LATTICE AND UNIT CELLS You know, the crystlline solids hve long-rnge order nd the closely pcked constituent prticles re rrnged in n ordered three dimensionl pttern. The 179

6 MODULE - 3 The Solid Stte structure of the crystlline solids cn be represented s n ordered three dimensionl rrngement of points. Here ech point represents the loction of constituent prticle nd is known s lttice point nd such n rrngement is clled crystl lttice or spce lttice or simply lttice Two Dimensionl Lttice nd Unit Cells To understnd the mening of the term lttice, let us first strt with two dimensionl lttice. A two dimensionl lttice hs set of points rrnged in regulr pttern on plne or surfce (in two dimensions). One such lttice is shown in Figure 6.4(). The rrngement of lttice points is s shown in Fig. 6.4(). When these lttice points re joined, the geometry of the lttice becomes cler (Fig. 6.4(b). The entire pttern cn be generted by repeting set of four points. On joining these points we get unit cell of the lttice (shown in drk). The unit cell of two dimensionl lttice is prllelogrm which is defined in terms of two sides nd b nd the ngle between them γ. Only five types of unit cells re possible in two dimensionl lttice. These re shown in Fig. 6.4(b). b y () (b) Fig. 6.3 () A two dimensionl lttice (b) nd its unit cell 90 =b y = 90 b 90 b y = 90 y =b y

7 The Solid Stte MODULE - 3 b y b y =b y = 60 Fig. 6.4 Five types of two dimensionl unit cells Three Dimensionl Lttice nd Unit Cells In three dimensions, the crystl structure of solid is represented s three dimensionl rry of lttice points. Remember tht the lttice points represent the positions of the constituent prticles of the solid (Fig. 6.5()). () (b) Fig. 6.5 Schemtic representtion of three dimensionl crystl lttice. In crystl lttice we cn select group of points which cn be used to generte the whole lttice. Such group is clled repet unit or the unit cell of the crystl lttice. The shded region in the Fig. 6.5() represents unit cell of the crystl lttice. The unit cell is chrcterized by three distnces long the three edges of the lttice (, b nd c) nd the ngles between them (α, β nd γ) s shown in the Fig. 6.5(b). We cn generte the whole crystl lttice by repeting the unit cell in the three directions. On the bsis of the externl ppernce the known crystls cn be clssified into seven types. These re clled crystl systems. In terms of the internl structure lso the crystl lttices contin only seven types of unit cells. The seven crystl systems nd the definition of their unit cells in terms of their unit distnces nd the ngles re compiled in Tble 6.2. The seven simple unit cells re given in Fig

8 MODULE - 3 Tble 6.2 The seven crystl systems nd their possible lttice types. The Solid Stte Systems Axes Angles Possible lttice types Cubic = b = c = β = γ = 90 P, F, I Tetrgonl = b c = β = γ = 90 P, I Orthorhombic b = c = β = γ = 90 P, F, I, C Rhombohedrl = b = c = β = γ 90 P Hexgonl = b c = β = 90 ; γ = 120 P Monoclinic b = c = γ = 90 ; β 90 P, I Triclinic b c β γ 90 P * P= primitive, I= body centered, F= fce centered nd C= side centered The unit cell shown in Fig. 6.5 nd the ones given in Fig..6.5 hve the lttice points t the corners only. c c Cubic b Orthorhombic 120 o Hexgonl c b c b c Triclinic Monoclinic Tetrgonl Rhombohedrl Fig. 6.6 The primitive unit cells; the reltive dimensions of the three repet distnces (, b nd c) nd the ngles between them (, β nd γ) re given in Tble 6.2. Such unit cells re clled primitive (P) unit cells. Sometimes, the unit cell of crystl contins lttice point(s) in ddition to the ones t the corners. A unit cell contining lttice point ech t the centers of its fces in ddition to the lttice points t the corners is clled fce centered (F) unit cell. On the other hnd unit cell with lttice points t the center of the unit cell nd t the corners is clled body centered unit cell (I). In some cses, in ddition to the lttice points t the corners there re two lttice points locted t the centers of ny two opposite fces. These re clled s end centered (C) unit cells. The possible lttice types in different crystl systems re lso indicted in Tble 6.2. The seven crystl systems when combined with these possibilities give rise to 14 lttice types. These re clled Brvis lttices. 182

9 The Solid Stte 6.5 NUMBER OF ATOMS IN CUBIC UNIT CELLS Of the seven crystl systems, let us discuss unit cells belonging to the cubic crystl system in somewht detils. As you cn see from Tble 6.2 tht in the cubic crystl system the three repet distnces re equl nd ll the three ngles re right ngles. The unit cells of three possible lttice types viz., primitive or simple cubic, body centered cubic nd the fce centered cubic, belonging to cubic crystl system re shown in Figure 6.7. MODULE - 3 () (b) (c) Fig. 6.7 () primitive or simple (b) body centered nd (c) fce centered cubic unit cells. Number of toms per unit cell As you know tht in unit cells the toms cn be on the corners, in the body center nd on fce centers. All the toms do not belong to single unit cell. These re shred mongst different unit cells. It is importnt to know the number of toms per unit cell. Let us lern how to compute these for different cubic unit cells. () Simple Cubic Unit Cell The simple or primitive unit cell hs the toms t the corners of the cube (Fig. 6.7 ()). A lttice point t the corner of the unit cell is shred by eight unit cells s you cn see from the encircled tom in the Fig Therefore, the contribution of n tom t the corner to the unit cell will be 1/8. The number of toms per unit cell cn be clculted s follows Number of corner toms = 8 Fig. 6.8 A corner lttice point is shred by eight unit cells Contribution of ech corner tom = 1 8 The number of toms in simple cubic unit cell = = 1 (b) Body Centered Cubic Unit Cell A body centered cubic (bcc) unit cell hs lttice points not only t the corners but 183

10 MODULE - 3 The Solid Stte lso t the center of the cube( Fig. 6.7 (b)). The tom in the center of the cube belongs entirely to the unit cell, i.e., it is not shred by other unit cells. Ech corner tom, on the other hnd, s in the cse of simple cubic unit cell, is shred by eight unit cells. Thus the number of toms per unit cell cn be clculted s Number of corner toms = 8 Contribution of ech corner tom = 1 8 Contribution of ll the corner toms to the unit cell = = 1 Number of toms t the center of the cube = 1 Contribution to the unit cell = 1 ( s it is not shred) The number of toms in body centered cubic unit cell = = 2 (c) Fce Centered Cubic Unit Cell A fce centered cubic ( fcc) unit cell hs toms not only t the corners but lso t the center of ech fce. Thus it hs eight lttice points t the corners nd six t the fce centers (Fig. 6.7 (c)). A fce centered lttice point is shred by two unit cells, s shown in Fig Fig. 6.9 A fce centered lttice point is shred by two unit cells Number of corner toms = 8 Contribution of ech corner tom = 1 8 Contribution of ll the corner toms to the unit cell = = 1 Number of toms t the fce center = 6 Contribution of ech tom t the fce centre = 1 2 Contribution of ll the fce centered toms to the unit cell = = 3 The number of toms points in fce centered cubic unit cell = = 4 184

11 The Solid Stte The number of toms per unit cell in different types of cubic unit cells is given in Tble 6.3. MODULE - 3 Tble. 6.3 Number of toms per unit cell S.No. Type of unit cell No. of toms per unit cell 1. Simple cubic 1 2. Body centered cubic 2 3. Fce centered cubic CALCULATION OF DENSITY OF UNIT CELL Mss Density = Volume (i) Volume of Unit cell If the edge length of the cubic unit cell is its volume is 3 (ii) Mss of the Unit cell Let the molr mss of substnce = M Mss of one tom or molecule = M N A Where N A = Avogdro s constnt. Let the number of toms or molecules of the substnce present per unit cell = z (See tble 6.3 for vlues of z for different type of cubic unit cells) Mss of unit cell = ( number of toms/molecules present per unit cell) mss of one tom/molecule ( ) z M = N A (iii) Density Mss of one unit cell Density = Volume of one unit cell z M d = 3 NA Density of the unit of cell of substnce is the sme s the density of the substnce. In cse of ionic substnces, the formul unit is tken s one prticle. (For exmple, formul unit of NCl = 1N + + 1Cl ions; of N 2 SO 4 = 2N + + 1SO 2 4 ions). 185

12 MODULE - 3 The Solid Stte The bove reltion cn be used to clculte z nd hence the nture of cubic lttice if d nd M re known or M if d nd z re known Exmple. The unit cell of metllic element is fce centred cubic nd the side of the cube is pm. Clculte the density of the metl in gcm 3 if its reltive tomic mss is Solution Given z M d = 3 N A Side of the cube = = pm = cm Number of toms per unit cell of fce-centred cubic unit cell = z = 4 Molr mss of the element, M = g mol 1 Putting these vlues in the bove reltion g mol d = ( cm) ( mol ) = 8.53 g cm 6.7 CLOSE PACKED STRUCTURES OF SOLIDS In the process of the formtion of crystl the constituent prticles get pcked quite closely. The crystl structures of the solids cn be described in terms of close pcking of identicl spheres s shown in Fig These re held together by forces of ttrction. Let us lern bout the possible close pcked structures of solids nd their significnce. Fig Arrngement of identicl spheres in one dimension A liner horizontl rrngement of identicl spheres in one dimension forms row (Fig. 6.10). A two dimensionl close pcked structure cn be obtined by rrnging number of such rows to form lyer. This cn be done in two possible wys. In one of these, we cn plce these rows in such wy tht these re ligned s shown in (Fig ()). In such n rrngement ech sphere is in contct with four other spheres. This rrngement in two dimensions is clled squre close pcking. 186

13 The Solid Stte MODULE - 3 () Fig () Squre close pcking nd (b) hexgonl close pcking of identicl spheres in two dimensions (b) In the other wy we cn plce the spheres of the second row in the depressions of the first row nd so on nd so forth (Fig. 6.11(b)). You my notice tht in such n rrngement ech sphere is in contct with six other spheres. Such n rrngement in two dimensions is clled hexgonl close pcking. In such pcking, the spheres of the third row re ligned with the first row. You my lso hve noticed tht in the hexgonl close pcked the spheres re more efficiently pcked. In Fig n equl number of identicl spheres re rrnged in two different types of pcking. A three dimensionl structure cn be generted by plcing such two dimensionl lyers on top of ech other. Before we move on to the three dimensionl pcking let us look t the hexgonl close pcked lyer some wht more closely (Fig. 6.12). You my note from Fig tht in hexgonl close pcked lyer there re some unoccupied spces or voids. These re tringulr in shpe nd re clled trigonl voids. You cn further note tht there re two types of tringulr voids, one with the pex pointing upwrds nd the other with the pex pointing downwrds. Let us cll these s X type nd Y type voids respectively s mrked in the Fig IIIrd row IInd row Ist row Fig A hexgonl Close Pcked lyer showing two types of tringulr voids. Close Pcked Structures in three dimensions Let us tke hexgonl close pcked lyer nd cll it A lyer nd plce nother hexgonl close-pcked lyer (clled the B lyer) on it. There re two possibilities. 187

14 MODULE - 3 The Solid Stte 1. In one, we cn plce the second lyer in such wy tht the spheres of the second lyer come exctly on top of the first lyer. 2. In other, the spheres of the second lyer re in such wy tht these re on the depressions of the first lyer. The first possibility is similr to squre close pcking discussed bove nd is ccompnied by wstge of spce. In the second possibility when we plce the second lyer into the voids of the first lyer, the spheres of the second lyer cn occupy either the X or Y type trigonl voids but not both. You my verify this by using coins of sme denomintion. You would observe tht when you plce coin on the trigonl void of given type, the other type of void becomes unvilble for plcing the next coin (Fig. 6.13). Fig Two lyers of close pcked spheres, the second lyer occupies only one type (either X or Y ) of tringulr voids in the first lyer. In this process, the sphere of second lyer covers the trigonl voids of the first lyer. It results into voids with four spheres round it, s shown in Fig. 6.14(). Such void is clled tetrhedrl void since the four spheres surrounding it re rrnged on the corners of regulr tetrhedron, Fig. 6.14(b). Similrly, the trigonl voids of the second lyer will be plced over the spheres of the first lyer nd give rise to tetrhedrl voids. () (b) Fig A tetrhedrl void In yet nother possibility, the trigonl voids of the first lyer hve nother trigonl void of the opposite type (X type over Y nd Y type over X type) from the second lyer over it. This genertes void which is surrounded by six spheres, Fig (). Such void is clled n octhedrl void becuse the six spheres 188

15 The Solid Stte surrounding the void lie t the corners of regulr octhedron, Fig (b). MODULE - 3 () Fig An octhedrl void (b) A closer look t the second lyer revels tht it hs series of regulrly plced tetrhedrl nd octhedrl voids mrked s t nd o respectively in Fig o t Fig The top view of the second lyer showing the tetrhedrl nd octhedrl voids. Now, when we plce the third lyer over the second lyer, gin there re two possibilities i.e., either the tetrhedrl or the octhedrl voids of the second lyer re occupied. Let us tke these two possibilities. If the tetrhedrl voids of the second lyer re occupied then the spheres in the third lyer would be exctly on top (i.e., verticlly ligned) of the first or A lyer The next lyer ( 4 th lyer) which is then plced would lign with the B lyer. In other words, every lternte lyer will be verticlly ligned. This is clled AB AB. pttern or AB AB. repet. On the other hnd if the octhedrl voids of the second lyer re occupied, the third lyer is different from both the first s well s the second lyer. It is clled the C lyer. In this cse the next lyer, i.e., the fourth lyer, howsoever it is plced will be ligned with the first lyer. This is clled ABC ABC. pttern or ABC ABC... repet. In three dimensionl set up the AB AB. pttern or repet is clled hexgonl closed pcking ( hcp) (Fig (c) ) while the ABC ABC. pttern or repet is clled cubic closed pcking (ccp) (Fig ()). 189

16 MODULE - 3 The Solid Stte A Lyer A-Lyer C Lyer B Lyer A Lyer A Lyer B Lyer C Lyer B-Lyer A-Lyer A Lyer () (b) (c) Fig () Cubic closed pcking (ccp) s result of ABC pttern of close pcked spheres; (b) the lyers in () tilted nd brought closer to show fcc rrngement (c) hexgonl closed pcking (hcp) s result of ABAB pttern of close pcked spheres. This process continues to generte the overll three dimensionl pcked structure. These three dimensionl structures contin lrge number of tetrhedrl nd octhedrl voids. In generl there is one octhedrl nd two tetrhedrl voids per tom in the close pcked structure. These voids re lso clled s interstices. As mentioned erlier, the identicl spheres represent the positions of only one kind of toms or ions in crystl structure. Other kind of toms or ions occupy these interstices or voids. In the close pcked structures (hcp nd ccp) discussed bove, ech sphere is in contct with six spheres in its own lyer ( s shown in Fig. 6.12) nd is in contct with three spheres ech of the lyer immeditely bove nd immeditely below it. Tht is, ech sphere is in contct with totl of twelve spheres. This number of nerest neighbor is clled its coordintion number. The prticles occupying the interstices or the voids will hve coordintion number depending on the nture of the void. For exmple n ion in tetrhedrl void will be in contct with four neighbors i.e., would hve coordintion number of four. Similrly the tom or ion in n octhedrl void would hve coordintion number of six. INTEXT QUESTIONS 6.2 () Wht is the difference between the squre close pcked nd hexgonl close pcked structures? (b) Which of the bove two, is more efficient wy of pcking? (c) Clerly differentite between, trigonl, tetrhedrl nd octhedrl voids. 190

17 The Solid Stte 6.8 PACKING EFFICIENCY In ll closed pcked structures there re lwys some voids or empty spces. The percentge of the totl spce tht is filled by the constituent prticles is clled the pcking efficiency. It is clculted from the following reltion filled spce Pcking efficiency = 100% totl spce It depends upon the nture of crystl lttice. For the three types of cubic lttices, the clcultions re given below MODULE - 3 (1) Pcking Efficiency of Simple Cubic Lttice In simple cubic lttice, the constituent prticles occupy only the corner positions of the cubic unit cell. These prticles touch one nother long the edge of the cube s shown in the Figure Let be the edge length of the cube nd r the rdius of the prticle then = 2r C B H A F C E D Fig Simple cubic unit cell The volume of the cube = (edge length/side) 3 = 3 = (2r) 3 = 8r 3 No. of constituent prticles present in ech unit cell = 1 The volume of the filled spce = volume of 1 prticle = filled spce Pcking efficiency = 100 totl spce 4 3 πr 3 π = 100 = r 6 = 52.36% = 52.4% 4 3 πr 3 191

18 MODULE - 3 (2) Pcking Efficiency of Body Centred Cubic Lttice The Solid Stte G B H A c c F b C E D Fig Body centred cubic until cell. The body centred cubic unit cell hs constitutent prticles present t ll its corners s well s t its body centre. The prticle t the body-centre touches the prticle t corner positions. Thus, the prticles re in contct long the bodydigonl of the cube. Consider the body-digonl AF long which the contct between the centrl prticles with those t corners A nd F hs been shown in Fig Length of body-digonl AF = c = 4 r (6.1) (Whole of the centrl prticle (2r) nd one-hlf of ech of the two corner prticle (r + r) occupy the body-digonl) Length of the body-digonl In the tringle EFD on the fce CDEF, EF ED FD = EF + ED b = + = 2 Where is the edge-length or side of the cube. In the tringle AFD, AD FD AF = AD + FD c = + b = + 2 = 3 c = 3 (6.2) But c = 4r 192

19 The Solid Stte 4r = (6.3) 3 MODULE - 3 The number of constituent prticles in body-centred cubic unit cell = Volume of filled spce = 2 π r (6.4) 3 3 4r Volume of the cubic unit cell = = π r filled spce Pcking efficiency = 100= totl spce 3 4r π r 3 3π = 100 = 100 = 68.0% 64 3 r (3) Pcking Efficiency of CCP nd HCP Lttices 3 (6.5) Cubic close pcked or fce centred cubic (FCC) nd hexgonl close pcked lttices hve equl pcking efficiency. Pcking efficiency of fce centred cubic (FCC) or cubic close pcking (CCP) unit cell cn be clculted with the help of Figure Let the edge length or the side of the cubic unit cell be nd its fce digonl AC be b. In the tringle ABC, AB BC, therefore, AC 2 = AB 2 + BC b = + = 2 or b= 2. (6.6) H G A B b F C E D Fig Fce centred cubic or cubic close pcking unit cell 193

20 MODULE - 3 The Solid Stte Unit cell of fcc (or ccp) lttice hs constituent prticles t ll the corners nd t the centre of ech fce. The prticle of fce-centre touches the prticles occupying the corner positions of the sme fce. Length of the fce digonl AC = b= 4r (6.7) Where r is the rdius of the constitutent prticle (whole of centrl prticle, 2r, nd one-hlf of ech of the two corner prticles, r + r) From equtions 1 nd 2 or b= 4r = 2 4r = = 2 2r (6.8) 2 The number of constituent prticles in fce centred cubic unit cell = 4 4 The Volume of filled spce = 4 π r Volume of the cubic unit cell = (side) = = (2 2 r) π r filled spce Pcking efficiency = 100 = totl spce 3 (2 2 r) 16 π 3 π = 100 = = 74% Pcking efficiencies of the three types of cubic lttices re summrized in the tble 6.4. Tble 6.4 Pcking efficiency of cubic unit cells S.No. Lttice Pcking Efficiency % 1. Simple Cubic 52.4% 2. Body Centred Cubic 68.0% 3. Fce Centred Cubic or 74.0% Cubic close pcking 194

21 The Solid Stte 6.9 STRUCTURES OF IONIC SOLIDS In cse of ionic solids tht consist of ions of different sizes, we need to specify the positions of both the ctions s well s the nions in the crystl lttice. Therefore, structure dopted by n ionic solid depends on the reltive sizes of the two ions. In fct it depends on the rtios of their rdii (r+/r-) clled rdius rtio. Here r+ is the rdius of the ction nd r- is tht of the nion. The rdius rtios nd the corresponding structures re compiled in Tble 6.5. MODULE - 3 Tble 6.5 The rdius rtios (r + /r ) nd the corresponding structures Rdius rtio (r+/r ) Coordintion number Structure dopted Tetrhedrl Octhedrl Body centered cubic >= Cubic Close Pcked structure The common ionic compounds hve the generl formule s MX, MX 2, nd MX 3, where M represents the metl ion nd X denotes the nion. We would discuss the structures of some ionic compounds of MX nd MX 2 types Structures of the Ionic Compounds of MX Type For the MX type of ionic compounds three types of structures re commonly observed. These re sodium chloride, zinc sulphide nd cesium chloride structures. Let us discuss these in some detils. () Cesium Chloride Structure In CsCl the ction nd the nions re of comprble sizes (the rdius rtio = 0.93) nd hs bcc structure in which ech ion is surrounded by 8 ions of opposite type. The Cs + ions is in the body center position nd eight Cl ions re locted t the corners (Fig. 6.21) of the cube. Thus it hs coordintion number of 8. Cesium Ion Chloride Ions Fig Cesium chloride structure (b) Sodium Chloride Structure In cse of NCl the nion (Cl ) is much lrger thn the ction (N + ). It hs rdius rtio of According to Tble 3.3 it should hve n octhedrl rrngement. In sodium chloride the( Cl ) form ccp (or fcc) structure nd the sodium ion occupy the octhedrl voids. You my visulise the structure hving 195

22 MODULE - 3 The Solid Stte chloride ions t the corners nd the fce centers nd the sodium ions t the edge centers nd in the middle of the cube (Fig. 6.22). (c) Zinc Sulphide Structure Fig Sodium chloride structure. In cse of zinc sulphide the rdius rtio is just = According to Tble 3.3 it should hve n tetrhedrl rrngement. In Zinc sulphide structure, the sulphide ions re rrnged in ccp structure. The zinc ions re locted t the corners of tetrhedron, which lies inside the cube s shown in the Fig These occupy lternte tetrhedrl voids. Fig Zinc Sulphide structure Structure of Ionic Compounds of MX 2 type () Clcium fluoride or fluorite structure In this structure the C 2+ ions form fcc rrngement nd the fluoride ions re locted in the tetrhedrl voids (Fig. 6.24). 196

23 The Solid Stte MODULE - 3 Fig Clcium fluoride or Fluorite structure; clcium ions occupy the corners of the cube nd fce centers The F - ions re on the corners of the smller cube which dipict the positions of tetrhedrl void. (b) Antifluorite Structure Some of the ionic compounds like N 2 O hve ntifluorite structure. In this structure the positions of ctions nd the nions in fluorite structures re interchnged. Tht is why it is clled ntifluorite structure. In N 2 O the oxide ions form the ccp nd the sodium ions occupy the tetrhedrl voids (Fig. 6.25). Fig Antifluorite structure dopted by N 2 O; The oxide ions occupy the corners of the cube nd fce centers nd the N + ions ( shown in blck ) re on the corners of the smller cube DEFECTS IN IONIC CRYSTALS You hve lernt tht in crystlline solid the constituent prticles re rrnged in ordered three dimensionl network. However, in ctul crystls such perfect order is not there. Every crystl hs some devitions from the perfect order. These devitions re clled imperfections or defects. These defects cn be brodly grouped into two types. These re stoichiometric nd 197

24 MODULE - 3 The Solid Stte non-stoichiometric defects depending on whether or not these disturb the stoichiometry of the crystlline mteril. Here, we would del only with stoichiometric defects. In such compounds the number of positive nd negtive ions re in stoichiometric proportions. There re two kinds of stoichiometric defects, these re Schottky defects Frenkel defects () Schottky defects This type of defect re due to the bsence of some positive nd negtive ions from their positions. These unoccupied lttice sites re clled holes. Such defects re found in ionic compounds in which the positive nd negtive ions re of similr size e.g., NCl nd CsCl. The number of missing positive nd negtive ions is equl. The presence of Schottky defects decreses the density of the crystl [Fig. 6.26()]. (b) Frenkel defects This type of defect rise when some ions move from their lttice positions nd occupy interstitil sites. The interstitil sites refer to the positions in between the ions. When the ion leves its lttice site hole is creted there. ZnS nd AgBr re exmples of ionic compounds showing Frenkel defects. In these ionic compounds the positive nd negtive ions re of quite different sizes. Generlly the positive ions leve their lttice positions, s these re smller nd cn ccommodte themselves in the interstitil sites. The Frenkel defects do not chnge the density of the solids [Fig. 6.26(b)]. A B A B A B A B B A B A B A A B A B B A A B A A B A B A B A B A B B A B A B () (b) Fig Stoichiometric defects ) Schottky nd b) Frenkel defects These defects cuse the crystl to conduct electricity to some extent. The conduction is due to the movement of ions into the holes. When n ion moves into hole it cretes new hole, which in turn is occupied by nother ion, nd the process continues. 198

25 The Solid Stte MODULE - 3 INTEXT QUESTIONS 6.3 () Wht do you understnd by crystl lttice? b) Wht is unit cell? c) How mny toms re there in fcc unit cell? 6.11 ELECTRICAL PROPERTIES Conductnce of electricity is n importnt property of substnce. Solids show very wide rnge of conductivities from high of 10 7 to low of sm 1, thus spnning 27 orders of mgnitude. Bsed upon their bility to conduct electricity, solids my be clssified into three ctegories; conductors, insultors nd semiconductors Conductors' Insultors nd Semiconductors (i) Conductors These re the solids with conductivities rnging from 10 4 to 10 7 S m 1. Metls conduct electricity through movement of their electrons nd re clled electronic conductors. Ionic solids conduct electricity when in molten stte or dissolved in wter, through movement of their ions. They re clled electrolytic conductors. (ii) Insultors These re the solids with extremely low conductivities rnging from to S m 1. Insultors re used to provide protective covering on conductors. (iii) Semiconductors These re the solids with intermedite conductivities rnging from 10 6 to 10 4 S m 1. Although semiconductors hve low conductivities, they find vide pplictions in solid stte devices like diodes nd trnsistors. Their conductivities cn be modified by introduction of suitble impurity Conduction of Electricity in Metls Metls conduct electricity through movement of their electrons. In unit 5 you hve lernt tht when two toms come closer, their tomic orbitls overlp nd they form n equl number of moleculr orbitls. One-hlf of these hve lower energy while the other hlf hve higher energy thn the energy of the tomic orbitls. As the number of moleculr orbitls increses, the energy-seprtion between them decreses. 199

26 MODULE - 3 The Solid Stte In piece of metl, the number of metl toms is very lrge nd so is the number of their vlence orbitls. This results in formtion of n eqully lrge number of moleculr orbitls which re so close to one-nother tht they form continuous bnd. The bnd in which the vlence electrons re present is clled vlence bnd. Electrons present in this bnd re strongly bound to the nucleus nd cnnot conduct electricity. The bnd formed by vcnt moleculr orbitls of higher energy is clled conduction bnd. When electrons rech conduction bnd from vlence bnd on excittion, they become loosely bound to the nucleus nd cn conduct electricity by moving under the influence of n electric field. Such electrons re lso clled free electrons. Conductivity of solid depends upon how esy or difficult it is for the vlence electrons to jump to the conduction bnd. See figure Conduction bnd Empty bnd Forbidden bnd (Lrge energy gp) Empty bnd Smll energy gp Emergy Filled bnd Prtilly filled bnd Metl () Overlpping bnds Insultor (b) Semiconductor (c) (i) Fig Vlence nd conduction bnds in () metls, (b) insultors nd (c) semi conductors. In conductors either the vlence bnd is only prtilly filled or it overlps vcnt conduction bnd of slightly higher energy. In both the cses its electrons cn esily flow under the influence of electric field nd the solid behves s conductor (Fig. 6.27()) (ii) In insultors the gp between the vlence bnd nd conduction bnd is lrge. Due to this the vlence electrons cnnot jump to the conduction bnd nd conduct electricity (Fig. 6.27(b)) (iii) In semiconductors the gp between the vlence bnd nd nerest conduction bnd is smll (Fig. 6.27(c)). On pplying the electric field, some electrons cn jump to the conduction bnd nd provide low conductivity. On incresing the temperture more electrons cn jump to the conduction bnd nd the conductivity increses. Silicon nd germnium show this type of behviour. They re clled intrinsic semiconductors. 200

27 The Solid Stte Doping of Semiconductors Conductivities of silicon nd germnium re too low to be put to ny prcticl use. Their conductivities cn be improved by introduction of controlled quntities of impurities which re either electron-rich or electron-deficit with respect to these elements. This process is known s doping. () Doping with electron rich impurities Silicon nd germnium both belong to group 14 of the periodic tble nd hve 4 vlence electrons ech. In their crystl lttice ech silicon (or germnium) tom forms four covlent bonds with its neighbours (Fig. 6.23()) when it is doped with group 15 element like As or P, whose ech tom crries 5 vlence electrons which is one more thn Si. After shring its four electrons with four neighbouring Si toms, it hs 9 electrons in its vlence shell (Fig. 6.28(b)) which is highly unstble electronic configurtion. The ninth electron being highly unstble roms freely in the whole of the crystl lttice rndomly. In the presence of n electric field this electron (mobile electron) moves from negtive to positive terminl nd increses the conductivity of Si (or Ge). Since the incresed conductivity of Si (or Ge) is due to negtively chrged mobile electrons, the Si doped with electron rich impurity becomes n-type semiconductor. Silicon tom Mobile electron Positive hole (no electron) MODULE - 3 As B Perfect crystl () n-type (b) Fig Cretion of n-type nd p-type semiconductors p-type (c) (b) Doping with electron-deficit impurities. When Si (or Ge) is doped with group 13 element like B or Al contining 3 vlence electrons (1 electron less thn Si/Ge), this results in cretion of one-electron vcncy in the structure which is clled n electron hole (Fig. 6.28(c)). An electron from its neightbouring tom cn come nd occupy it, leving hole t its originl position. Electrons cn rndomly occupy holes nd the hole would pper to move rndomly. On pplying n electric field, the electrons move from negtive to positive terminl nd the hole would pper move in the opposite direction, i.e., from positive to negtive terminl nd would behve s if it is positively chrged. Such semiconductors re clled p-type semiconductors. The movement of electrons nd the hole increses the conductivity of Si. Semiconductors like Si, doped 201

28 MODULE - 3 The Solid Stte with electron-deficit or electron-rich impurities re clled extrinsic semiconductors. Applictions of n-type nd p-type semiconductors Due to their specil properties, n-nd p-type semiconductors find severl pplictions. (i) Diodes Diode is combintion of n-type nd p-type semiconductor. Diodes re used s rectifier to convert n AC signl to DC signl. (ii) Trnsistors. Trnsistors re mde by combintion of 3 lyers of semiconductors. Trnsistor of n-p-n type is mde by sndwiching lyer of p-type semiconductor between two lyers of n-type semiconductors nd of p-n-p type by sndwiching lyer of n-type semiconductors between two lyers of p-type semiconductor. Trnsistors re used s detectors nd mplifiers of rdio or udio frequency signls. They re used in circuits of solid stte devices. (iii) Solr cells. Solr cells re photo diodes which hve specil property of emitting electrons when sunlight flls on them. They re used to convert solr energy into electricity MAGNETIC PROPERTIES All substnces re ffected (ttrcted or repelled) by mgnetic field. Mgnetic properties present in ny substnce re due to the electrons present in it. Ech electron in n tom behves like tiny mgnet. Electrons re negtively chrged prticles. When n electron revolves round nucleus nd spins bout its own xis, two types of mgnetic moments re creted orbitl mgnetic moment due to its revolution round the nucleus nd spin mgnetic moment due to its spin. Overll mgnetic properties of substnce depend upon the orienttion of these tiny mgnets. On the bsis of mgnetic properties, ll substnces cn be clssified into five ctegories (i) prmgnetic, (ii) dimgnetic, (iii) ferromgnetic, (iv) ntiferromgnetic nd (v) ferrimgnetic Prmgnetic Substnces On plcing these substnces in mgnetic field, these re wekly ttrcted by it. O 2, Cu 2+, Fe 3+ nd Cr 3t re some exmples of prmgnetic substnces Prmgnetism is due to the presence of one or more unpired electrons in n tom, molecule or ion. 202

29 The Solid Stte Dimgnetic Substnces When plced in mgnetic field, dimgnetic substnces re wekly repelled by it. H 2 O, NCl nd C 6 H 6 re some exmples of such substnces. Dimgnetism is shown by substnces in which ll the electrons re pired. MODULE Ferromgnetic Substnces When plced in mgnetic field, ferromgnetic substnces re strongly ttrcted by it. Fe, Ni, Co, Gd, MnAs, CrBr 3 nd CrO 2 re such substnces. These substnces cn be permnently mgnetized. In solid stte, the metl ions of ferromgnetic substnces re grouped together into smll regions clled domins. In ech domin, the individul mgnetic moments of the metl ions re directed in the sme direction nd they dd up. As result, ech domin cts s tiny mgnet. Ordinrily, these domins re rndomly oriented which cncels out their mgnetic moment. When () (b) (c) Fig Arrngement of mgnetic moments of domins in () ferromgnetic, (b) nti ferromgnetic nd (c) ferrimgnetic substnces plced in mgnetic field, ll the domins get oriented in the direction of the mgnetic field. This dds up their mgnetic moments nd mkes them strong mgnets (Fig. 6.29()). They lose their ferromgnetism on being given mechnicl jerks or on heting bove certin temperture, clled Curie temperture they become prmgnetic Antiferromgnetic Substnces Some substnce tht hve domins in them like ferromgnetic substnces but their domins re oppositely oriented nd cncel out the mgnetic moments of ech other (Fig (b)) re cled ntiferro mgnetic substnces. FeO, MnCl 2, MnO, Mn 2 O 3 nd MnO 2 re exmples of ntiferromgnetic substnces. They lso become prmgnetic on heting bove certin temperture Ferrimgnetic Substnces. Some substnces like Fe 3 O 4 (mgnetite) nd ferrites (MFe 2 O 4 where M is bivlent ction like Cu 2+, Zn 2+ ) show n intermedite behvior between tht 203

30 MODULE - 3 The Solid Stte of ferromgnetic nd ntiferromgnetic substnces. These substnces re quite strongly ttrcted by mgnetic field s compred to prmgnetic substnces but wekly s compred to ferromgnetic substnces. Their domins re ligned in prllel nd ntiprllel directions in unequl numbers (Fig. 6.29(c)). They lso become prmgnetic on heting bove certin temperture. WHAT YOU HAVE LEARNT In solid stte the constituent prticles re rrnged in closely pcked ordered rrngement with lmost no free spce. These re held together by strong forces of ttrction nd vibrte bout their fixed positions. Solids re incompressible nd rigid nd hve definite shpes. Solids re clssified into morphous nd crystlline solids. The crystlline solids hve long rnge order while morphous solids hve only short rnge order. The crystlline solids cn be clssified into four different types- ionic, moleculr, covlent nd metllic solids on the bsis of nture of forces of ttrction between the constituent prticles. The temperture t which solid melts to give liquid is clled its melting point. The three dimensionl internl structure of crystlline solid cn be represented in terms of crystl lttice in which the loction of ech constituent prticle is indicted by point. The whole crystl lttice cn be generted by moving the unit cell in the three directions. The crystl structures of the solids cn be described in terms of closepcking of identicl spheres. In three dimensions there re two wys of pcking identicl spheres. These re hexgonl closed pcking (hcp) nd cubic closed pcking (ccp). The hcp rrngement is obtined by ABAB repet of the two dimensionl lyers wheres the ccp rrngement is obtined by ABCABC repet. On the bsis of the externl ppernce the known crystls cn be clssified into seven types clled crystl systems. The unit cells of cubic crystl system hs three possible lttice types. These re simple cubic, body centered cubic nd the fce centered cubic. The toms t the corner of cubic unit cell is shred by eight unit cells while fce centered tom is shred by two unit cells. The tom t the body center, on the other hnd is exclusive to the unit cell s it is not shred. 204

31 The Solid Stte The number of toms per unit cell for the simple cubic, bcc nd fcc unit cells re 1,2 nd 4 respectively. The structure dopted by n ionic solid depends on the rtios of their rdii (r+/r-), clled rdius rtio. The structures of some simple ionic solids cn be described in terms of ccp of one type of ions nd the other ions occupying the voids. Actul crystls hve some kind of imperfections in their internl structure. These re clled defects. There re two types of defects clled stoichiometric nd non-stoichiometric defects depending on whether or not these disturb the stoichiometry of the crystlline mteril. There re two kinds of stoichiometric defects, these re clled Schottky defects nd Frenkel defects. Solid cn be clssified s conductors, insultors nd semiconductors on the bsis of their electricl conductivities. Electricl properties of solids cn be explined with the help of bnd theory. On the bsis of their interction with externl mgnetic field, solids cn be clssified s prmgnetic, dimgnetic, ferromgnetic, ferrimgnetic nd ntiferromgnetic substnces. MODULE - 3 TERMINAL EXERCISES 1. Outline the differences between crystlline nd n morphous solid. 2. How cn you clssify solids on the bsis of the nture of the forces between the constituent prticles? 3. Wht do you understnd by the melting point of solid? Wht informtion does it provide bout the nture of interction between the constituent prticles of the solids? 4. Wht do you understnd by coordintion number? Wht would be the coordintion number of n ion occupying n octhedrl void.? 5. Explin the following with the help of suitble exmples. () Schottky defect (b) Frnkel defect 6. Explin why prticulr solid behves s conductor or semiconductor or insultor on the bsis of bnd theory. 7. Wht re (i) prmgnetic (ii) dimgnetic nd (iii) ferromgnetic substnces? 205

32 MODULE - 3 The Solid Stte ANSWERS TO INTEXT QUESTIONS Solids hve definite shpe nd definite volume. Liquids hve indefinite shpe but define volume. Gses hve indefinite shpe nd indefinite volume. 2. Coulombic forces, dipole-dipole ttrctions, covlent bonding nd metllic bonding. 3. Irrespective of the size nd shpe of the crystl of substnce, the interfcil ngle between pir of fces is lwys the sme Refer to section Hexgonl close pcked. 3. Refer to sections Ordered three dimensionl rrngement of points representing the loction of constituent prticles. 2. A select group of points which cn be used generte the whole lttice. Unit cell is chrcterised by three edges of the lttice nd ngles between them. 3. Four. 206