COPYRIGHTED MATERIAL. Crystals and crystal structures. 1.1 Crystal families and crystal systems

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1 1 Crystls nd crystl structures Wht is crystl system? Wht re unit cells? Wht informtion is needed to specify crystl structure? Crystls re solids tht possess long-rnge order. The rrngement of the toms t one point in crystl is identicl, (excepting loclised mistkes or defects tht cn rise during crystl growth), to tht in ny other remote prt of the crystl. Crystllogrphy descries the wys in which the component toms re rrnged in crystls nd how the long-rnge order is chieved. Mny chemicl (including iochemicl) nd physicl properties depend on crystl structure nd knowledge of crystllogrphy is essentil if the properties of mterils re to e understood nd exploited. Crystllogrphy first developed s n oservtionl science; n djunct to the study of minerls. Minerls were, (nd still re), descried y their hit, the shpe tht minerl specimen my exhiit, which my vry from n morphous mss to well-formed crystl. Indeed, the regulr nd eutiful shpes of nturlly occurring crystls ttrcted ttention from the erliest times, nd the reltionship etween crystl shpe nd the disposition of crystl fces, the crystl morphology, ws soon used in clssifiction. At lter stge in the development of the suject, symmetry, treted mthemticlly, ecme importnt in the description of crystls. The ctul determintion of crystl structures, the positions of ll of the toms in the crystl, ws lter level of refinement tht ws dependent upon the discovery nd susequent use of X-rys. 1.1 Crystl fmilies nd crystl systems Creful mesurement of minerl specimens llowed crystls to e clssified in terms of six crystl fmilies, clled northic, monoclinic, orthorhomic, tetrgonl, hexgonl nd isometric. This clssifiction hs een expnded slightly y crystllogrphers into seven crystl systems. The crystl systems re sets of reference xes, which hve direction s well s mgnitude, nd hence re vectors 1. The crystl fmiliesndclssesregivenintle1.1. COPYRIGHTED MATERIAL 1 Vectors re set in old typefce throughout this ook. Crystls nd Crystl Structures. Richrd J. D. Tilley # 2006 John Wiley & Sons, Ltd ISBNs: (csed) (Pk)

2 2 CH1 CRYSTALS AND CRYSTAL STRUCTURES Tle 1.1 The seven crystl systems Crystl fmily Crystl system Axil reltionships Isometric Cuic ¼ ¼ c, ¼ ¼ g ¼ 90 ; Tetrgonl Tetrgonl ¼ 6¼ c, ¼ ¼ g ¼ 90 ; Orthorhomic Orthorhomic 6¼ 6¼ c, ¼ ¼ g ¼ 90 ; Monoclinic Monoclinic 6¼ 6¼ c, ¼ 90, 6¼ 90, g ¼ 90 ; Anorthic Triclinic 6¼ 6¼ c, 6¼ 90, 6¼ 90, g 6¼ 90 ; Hexgonl Hexgonl ¼ 6¼ c, ¼ ¼ 90, g ¼ 120 ; Trigonl or ¼ ¼ c, ¼ ¼ g; or Rhomohedrl 0 ¼ 0 6¼ c 0, 0 ¼ 0 ¼ 90, g 0 ¼ 120 ; (hexgonl xes) The three reference xes re lelled, nd c, nd the ngles etween the positive direction of the xes re,, ndg, where lies etween þ nd þc, lies etween þ nd þc, ndg lies etween þ nd þ, (Figure 1.1). The ngles re chosen to e greter or equl to 90 except for the trigonl system, s descried elow. In figures, the -xis is represented s projecting out of the plne of the pge, towrds the reder, the -xis points to the right nd the c-xis points towrds the top of the pge. This rrngement is right-hnded coordinte system. Mesurements on minerl specimens could give solute vlues for the inter-xil ngles, ut only reltive xil lengths could e derived. These lengths re written, nd c. β c α γ Figure 1.1 The reference xes used to chrcterise the seven crystl systems The seven crystl systems re nmed ccording to the reltionship etween the xes nd the inter-xil ngles. The most symmetric of the crystl clsses is the cuic or isometric system, in which the three xes re rrnged t 90 to ech other nd the xil lengths re identicl. These form the fmilir Crtesin xes. The tetrgonl system is similr, with mutully perpendiculr xes. Two of these, usully designted (¼ ), re of equl length, while the third, designted c, is longer or shorter thn the other two. The orthorhomic system hs three mutully perpendiculr xes of different lengths. The monoclinic system is lso defined y three unequl xes. Two of these, conventionlly chosen s nd c, re t n olique ngle,, while the third c, is norml to the plne contining nd. The lest symmetricl crystl system is the triclinic, which hs three unequl xes t olique ngles. The hexgonl crystl system hs two xes of equl length, designted (¼ ), t n ngle, g, of 120.The c-xis lies perpendiculr to the plne contining nd, nd lies prllel to six-fold xis of rottion symmetry, (see Chpter 4). The trigonl system hs three xes of equl length, ech enclosing equl ngles (¼ ¼ g), forming rhomohedron. The xes re clled rhomohedrl xes, while the system nme trigonl refers to the presence of three-fold xis of

3 MORPHOLOGY AND CRYSTAL CLASSES 3 rottion symmetry in the crystl (see Chpter 4). Crystls descried in terms of rhomohedrl xes re often more conveniently descried in terms of hexgonl set of xes. In this cse, the hexgonl c-xis is prllel to the rhomohedrl ody digonl, which is three-fold xis of symmetry (Figure 1.2). The reltionship etween the two sets of xes is given y the vector equtions: () (c) c R R ¼ Ù H þ ˆÙ H þ ˆÙ c H R ¼ ˆÙ H þ ˆÙ H þ ˆÙ c H c R ¼ ˆÙ H Ù H þ ˆÙ c H H ¼ R R H ¼ R c R c H ¼ R þ R þ c R Three-fold xis Rhomohedrl H c R R R R H Three-fold xis c Hexgonl Figure 1.2 Rhomohedrl nd hexgonl xes: (, ), xes in equivlent orienttions with the trigonl 3-fold xis prllel to the hexgonl c-xis; (c) superposition of oth sets of xes, projected down the hexgonl c-xis (¼ the rhomohedrl 3-fold xis) () R γ H H where the suscripts R nd H stnd for rhomohedrl nd hexgonl respectively. [Note tht in these equtions the vectors, nd c re dded vectorilly, not rithmeticlly, (see Appendix 1)]. The rithmeticl reltionships etween the xil lengths is given y: H ¼ 2 R sin 2 R ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 H 3 þ c2 H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c H ¼ R 3 þ 6 cos sin 2 ¼ 3 H p 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 H þ c2 H where the suscripts R nd H stnd for rhomohedrl nd hexgonl respectively. 1.2 Morphology nd crystl clsses Oservtions of the shpes of crystls, the crystl morphology, suggested tht the regulr externl form of crystl ws n expression of internl order. Among other oservtions, the clevge of crystls, tht is, the wy in which they could e frctured long certin directions in such mnner tht the two resultnt frgments hd perfect fces, suggested tht ll crystls might e uilt up y stcking of infinitesimlly smll regulr rick-like elementry volumes, ech unique to the crystl under considertion. These elementry volumes, the edges of which could e considered to e prllel to the xil vectors, nd c, of the seven crystl systems, eventully cme to e termed morphologicl unit cells. The reltive xil lengths,, nd c were tken s equl to the reltive lengths of the unit cell sides. The vlues,, c,, nd g re termed the morphologicl unit cell prmeters. [The solute lengths of the xes, lso written, nd c or 0, 0 nd c 0, determined y diffrction techniques, descried elow, yield the structurl unit cell of the mteril. Unit cell prmeters now refer only to these structurl vlues.]

4 4 CH1 CRYSTALS AND CRYSTAL STRUCTURES A centrl concept in crystllogrphy is tht the whole of crystl cn e uilt up y stcking identicl copies of the unit cell in exctly the sme orienttion. Tht is to sy, crystl is chrcterisedyothtrnsltionl nd orienttionl long-rnge order. The unit cells re displced repetedly in three dimensions, (trnsltionl long-rnge order), without ny rottion or reflection, (orienttionl long-rnge order). This restriction leds to severe restrictions upon the shpe (strictly speking the symmetry) of unit cell; sttement which will e plced on firm footing in lter chpters. The fct tht some unit cell shpes re not llowed is esily demonstrted in two dimensions, s it is pprent tht regulr pentgons, for exmple, cnnot pck together without leving gps (Figure 1.3). [A regulr pentgon is plne figure with five equl sides nd five equl internl ngles.] Not only could unit cells e stcked y trnsltion lone to yield the internl structure of the crystl, ut, depending on the rte t which the unit cells were stcked in different directions, (i.e. the rte t which the crystl grew in three dimensions), different fcets of the crystl ecme emphsised, while others were suppressed, so producing vriety of externl shpes, or hits, (Figure 1.4), () () (c) (d) (e) Figure 1.3 Irrespective of how they re rrnged, regulr pentgons cnnot fill plne completely; spces lwys pper etween some of the pentgons. Just one rrngement is drwn, others re possile Figure 1.4 () schemtic depiction of crystl uilt of rectngulr (orthorhomic) unit cells. The unit cells must e imgined to e much smller thn depicted, thus producing smooth fcets. ( e) different crystl hits derived y differing rtes of crystl growth in vrious directions

5 THE DESCRIPTION OF CRYSTAL STRUCTURES 5 thus explining the oservtion tht single minerl could occur in differing crystl morphologies. The fces of crystl, irrespective of the overll shpe of the crystl, could lwys e lelled with respect to the crystl xes. Ech fce ws given set of three integers, (h kl), clled Miller indices. These re such tht the crystl fce in question mde intercepts on the three xes of =h, =k nd c=l. A crystl fce tht intersected the xes in exctly the xil rtios ws given importnce s the prmetrl plne, with indices (111). [Miller indices re now used to lel ny plne, internl or externl, in crystl, s descried in Chpter 2, nd the nomenclture is not just confinedtotheexternl fces of crystl.] The ppliction of Miller indices llowed crystl fces to e lelled in consistent fshion. This, together with ccurte mesurements of the ngles etween crystl fces, llowed the morphology of crystls to e descried in reproducile wy, which, in itself, led to n pprecition of the symmetry of crystls. Symmetrywsrokendownintocomintionof symmetry elements. These were descried s mirror plnes, xes of rottion, nd so on, tht, when tken in comintion, ccounted for the externl shpe of the crystl. The crystls of prticulr minerl, regrdless of its precise morphology, were lwys found to possess the sme symmetry elements. Symmetry elements re opertors. Tht is, ech one descries n opertion, such s reflection. When these opertions re pplied to the crystl, the externl form is reproduced. It ws found tht ll crystls fell into one or nother of 32 different groups of symmetry opertions. These were clled crystl clsses. Ech crystl clss could e llocted to one of the six crystl fmilies. These symmetry elements nd the resulting crystl clsses re descried in detil in Chpters 3 nd The determintion of crystl structures The descriptions ove were mde using opticl techniques, especilly opticl microscopy. However, the solute rrngement of the toms in crystl cnnot e determined in this wy. This limittion ws overcome in the erly yers of the 20 th century, when it ws discovered tht X-rys were scttered, or diffrcted, y crystls in wy tht could e interpreted to yield the solute rrngement of the toms in crystl, the crystl structure. X-ry diffrction remins the most widespred technique used for structure determintion, ut diffrction of electrons nd neutrons is lso of gret importnce, s these revel fetures tht re complementry to those oserved with X-rys. The physics of diffrction y crystls hs een worked out in detil. It is found tht the incident rdition is scttered in chrcteristic wy, clled diffrction pttern. The positions nd intensities of the diffrcted ems re function of the rrngements of the toms in spce nd some other tomic properties, such s the tomic numer of the toms. Thus, if the positions nd the intensities of the diffrcted ems re recorded, it is possile to deduce the rrngement of the toms in the crystl nd their chemicl nture. The determintion of crystl structures y use of the diffrction of rdition is outlined in Chpter The description of crystl structures The minimum mount of informtion needed to specify crystl structure is the unit cell type, i.e., cuic, tetrgonl, etc, the unit cell prmeters nd the positions of ll of the toms in the unit cell. The tomic contents of the unit cell re simple multiple, Z, of the composition of the mteril.

6 6 CH1 CRYSTALS AND CRYSTAL STRUCTURES Origin y c r z x The vlue of Z is equl to the numer of formul units of the solid in the unit cell. Atom positions re expressed in terms of three coordintes, x, y, nd z. These re tken s frctions of, nd c, the unit cell sides, sy ˆÙ, ˆÙ, ˆÙ. The x, y nd z coordintes re plotted with respect to the unit cell xes, not to Crtesin set of xes, (Figure 1.5). The position of n tom cn lso e expressed s vector, r: r ¼ x þ y þ zc where, nd c re the unit cell xes, (Figure 1.5). An tom t cell corner is given the coordintes (0, 0, 0). An tom t the centre of the fce of unit cell is given the coordintes (ˆÙ, ˆÙ, 0) if it lies etween the - nd -xes, (ˆÙ, 0,ˆÙ )if etween the - nd c-xes, nd (0, ˆÙ, ˆÙ ) if etween the - nd c-xes. An tom t the centre of unit cell would hve position specified s (ˆÙ, ˆÙ, ˆÙ ), irrespective of the type of unit cell. Atoms t the centres of the cell edges re specified t positions (ˆÙ, 0, 0), (0, ˆÙ, 0) or (0, 0, ˆÙ ), for toms on the -, - nd c-xis, (Figure 1.6). Stcking of the unit cells to uild structure will ensure tht n tom t the unit cell origin will pper t every corner, nd toms on unit cell Atom t x, y, z Figure 1.5 The position of n tom in unit cell, x, y nd z, isdefined with respect to the directions of the unit cell edges. The numericl vlues of x, y nd z re specified s frctions, (ˆÙ, ˆÙ, etc.) of the unit cell edges, nd c Atom t (0, 0, 0) c edges or fces will pper on ll of the cell edges nd fces. In figures, the conventionl origin is plced t the left rer corner of the unit cell. The - or x-xis is represented s projecting out of the plne of the pge, towrds the reder, the - or y-xis points to the right nd the c- orz-xis points towrds the top of the pge. In projections, the origin is set t the upper left corner of the unit cell projection. A frequently encountered projection is tht perpendiculr to the c-xis. In this cse, the - orx-xis is drwn pointing down, (from top to ottom of the pge), nd the - ory-xis pointing to the right. In projections the x nd y coordintes cn e determined from the figure. The z position is usully given on the figure s frction. A vst numer of structures hve een determined, nd it is very convenient to group those with topologiclly identicl structures together. On going from one memer of the group to nother, the toms in the unit cell differ, reflecting chnge in chemicl compound, nd the tomic coordintes nd unit cell dimensions chnge slightly, reflecting the difference in tomic size, ut reltive tom positions re identicl or very similr. Frequently, the group nme is tken from the nme of minerl, s minerl crystls were the first solids used for structure determintion. Thus, ll crystls with structure similr to tht of sodium chloride, NCl, (the minerl hlite), re sid to dopt the hlite structure. These mterils then ll hve generl Atom t (½,½,0) Atom t (½,½,½) Atom t (0,½,0) Figure 1.6 Atoms t positions 0, 0, 0; 0, ˆÙ, 0; ˆÙ, ˆÙ, 0; nd ˆÙ, ˆÙ, ˆÙ in unit cell

7 THE CUBIC CLOSE-PACKED (A1) STRUCTURE OF COPPER 7 Tle 1.2 Strukturericht symols nd nmes for simple structure types Symol nd nme Exmples Symol nd nme Exmples A1, cuic close-pcked, copper Cu, Ag, Au, Al A2, ody-centred cuic, iron Fe, Mo, W, N A3, hexgonl close-pcked, Mg, Be, Zn, Cd A4, dimond C, Si, Ge, Sn mgnesium B1, hlite, rock slt NCl, KCl, NiO, MgO B2, cesium chloride CsCl, CsBr, AgMg, BCd B3, zinc lende ZnS, ZnSe, BeS, CdS B4, wurtzite ZnS, ZnO, BeO, CdS, GN C1, fluorite CF 2,BF 2,UO 2,ThO 2 C4, rutile TiO 2,SnO 2,MgF 2,FeF 2 formul MX, wherem is metl tom nd X non-metl tom, for exmple, MgO. Similrly, crystls with structure similr to the rutile form of titnium dioxide, TiO 2, re grouped with the rutile structure. These ll hve generl formul MX 2,forexmpleFeF 2.Asfinl exmple, compounds with similr structure to the minerl fluorite, (sometimes clled fluorspr), CF 2, re sid to dopt the fluorite structure. These lso hve generl formul MX 2,nexmpleeing UO 2. Exmples of these three structures follow. Crystllogrphic detils of numer of simple inorgnic structures re given in Appendix 2. Some minerl nmes of common structures re found in Tle 1.2 nd Appendix 2. A system of nomenclture tht is useful for descriing reltively simple structures is tht originlly set out in 1920, in Volume 1 of the Germn puliction Strukturericht. It ssigned letter code to ech structure; A for mterils with only one tom type, B for two different toms, nd so on. Ech new structure ws chrcterised further y the lloction of numerl, so tht the crystl structures of elements were given symols A1, A2, A3 nd so on. Simple inry compounds were given symols B1, B2 nd so on, nd inry compounds with more complex structures C1, C2, D1, D2 nd so on. As the numer of crystl structures nd the complexity displyed incresed, the system ecme unworkle. Nevertheless, the terminology is still used, nd is useful for the description of simple structures. Some Strukturericht symols re given in Tle The cuic close-pcked (A1) structure of copper A numer of elementl metls crystllise with the cuic A1 structure, lso clled the copper structure. Unit cell: cuic Lttice prmeter for copper 2, ¼ 0:3610 nm. Z ¼ 4Cu Atom positions: 0, 0, 0; ˆÙ, ˆÙ, 0; 0, ˆÙ, ˆÙ ; ˆÙ, 0,ˆÙ ; There re four copper toms in the unit cell, (Figure 1.7). Besides some metls, the nole gses, Ne(s), Ar(s), Kr(s), Xe(s), lso dopt this structure in the solid stte. This structure is often clled the fce-centred cuic (fcc) structure or the cuic close-pcked (ccp) structure, ut the Strukturericht symol, A1 is the most compct nottion. Ech tom hs 12 nerest neighours, nd if the toms re supposed to e hrd touching spheres, the frction of the volume occupied is 2 Lttice prmeters nd intertomic distnces in crystl structures re usully reported in Ångström units, Å, in crystllogrphic literture. 1 Å is equl to m, tht is, 10 Å ¼ 1 nm. In this ook, the SI unit of length, nm, will e used most often, ut Å will e used on occsion, to conform with crystllogrphic prctice.

8 8 CH1 CRYSTALS AND CRYSTAL STRUCTURES c c Figure 1.7 structure The cuic unit cell of the A1, copper, Figure 1.8 structure The cuic unit cell of the A2, tungsten, More informtion on this structure is given in Chpter The ody-centred cuic (A2) structure of tungsten A second common structure dopted y metllic elements is tht of the cuic structure of tungsten, W. Unit cell: cuic Lttice prmeter for tungsten, ¼ 0:316 nm. Z ¼ 2W Atom positions: 0, 0, 0; ˆÙ, ˆÙ, ˆÙ ; There re two tungsten toms in the odycentred unit cell, one t (0, 0, 0) nd one t the cell centre, (ˆÙ, ˆÙ, ˆÙ ), (Figure 1.8). This structure is often clled the ody-centred cuic (cc) structure, ut the Strukturericht symol, A2, is more compct designtion. In this structure, ech tom hs 8 nerest neighours nd 6 next nerest neighours t only 15% greter distnce. If the toms re supposed to e hrd touching spheres, the frction of the volume occupied is This is less thn either the A1 structure ove or the A3 structure tht follows, oth of which hve volume frction of occupied spce of The A2 structure is often the high temperture structure of metl tht hs n A1 close-pcked structure t lower tempertures. 1.7 The hexgonl (A3) structure of mgnesium The third common structure dopted y elementl metls is the hexgonl mgnesium, Mg, structure. Unit cell: hexgonl Lttice prmeters for mgnesium, ¼ 0:321 nm; c ¼ 0:521 nm Z ¼ 2Mg Atom positions: 0, 0, 0; ˆÙ, Ù, ˆÙ ; There re two toms in the unit cell, one t (0, 0, 0) nd one t (ˆÙ, Ù, ˆÙ ). [The toms cn lso e plced t Ù, ˆÙ, ˆÙ ; ˆÙ, Ù, Ù, y chnging the unit cell origin. This is preferred for some

9 THE RUTILE STRUCTURE 9 c More informtion on this structure, nd the reltionship etween the A1 nd A3 structures, is given in Chpter 7. () 1.8 The hlite structure The generl formul of crystls with the hlite structure is MX. The minerl hlite, which nmes the group, is sodium chloride, NCl, lso clled rock slt. Unit cell: cuic. () Figure 1.9 The hexgonl unit cell of the A3, mgnesium, structure: () perspective view; () projection down the c-xis purposes]. The structure is shown in perspective, (Figure 1.9) nd projected down the c-xis, (Figure 1.9). This structure is often referred to s the hexgonl close-pcked (hcp) structure. If the toms re supposed to e hrd touching spheres, the frction of the volume occupied is , equl to tht in the A1 structure of copper, nd the rtio of the lttice prmeters, c =, is equl to H8 =H3, ¼ The idel volume, V, of the unit cell, equl to the re of the se of the unit cell multiplied y the height of the unit cell, is: pffiffiffi 3 V ¼ 2 2 c ¼ 0: c Lttice prmeter for hlite, ¼ nm. Z ¼ 4 {NCl} Atom positions: N: ˆÙ, 0,0; 0,0,ˆÙ ; 0, ˆÙ, 0; ˆÙ, ˆÙ, ˆÙ Cl: 0, 0, 0; ˆÙ, ˆÙ, 0; ˆÙ, 0,ˆÙ ; 0, ˆÙ, ˆÙ There re four sodium nd four chlorine toms in the unit cell. For ll mterils with the hlite structure, Z ¼ 4. In this structure, ech tom is surrounded y six toms of the opposite type t the corners of regulr octhedron (see Chpter 7). A perspective view of the hlite structure is shown in Figure 1.10, nd projection, down the c-xis, in Figure This structure is dopted y mny oxides, sulphides, hlides nd nitrides with formul MX. 1.9 The rutile structure The generl formul of crystls with the rutile structure is MX 2. The minerl rutile, which nmes the group, is one of the structures dopted y titnium dioxide, TiO 2. [The other common form

10 10 CH1 CRYSTALS AND CRYSTAL STRUCTURES c () Ti O () Figure 1.11 The tetrgonl unit cell of the rutile structure: () perspective view; () projection down the c-xis Figure 1.10 The cuic unit cell of the B1, hlite, structure: () perspective view; () projection down the c-xis of TiO 2 encountered is clled ntse. Other structures for TiO 2 re lso known.] Unit cell: tetrgonl. Lttice prmeters for rutile, ¼ 0:4594 nm; c ¼ 0:2959 nm: Z ¼ 2 ftio 2 g Atom positions: Ti: 0, 0, 0; ˆÙ, ˆÙ, ˆÙ O: /ˆ, /ˆ,0; Ù, ˆÙ, ˆÙ ; É/ˆ, É/ˆ,0; ˆÙ, Ù, ˆÙ There re two molecules of TiO 2 in the unit cell, tht is, for ll mterils tht dopt the rutile structure, Z ¼ 2. In this structure, ech titnium tom is surrounded y six oxygen toms t the corners of n octhedron. A perspective view of the rutile structure is shown in Figure 1.11, nd projection, down the c-xis, in Figure This structure is reltively common nd dopted y numer of oxides nd fluorides with formul MX The fluorite structure The generl formul of crystls with the fluorite structure is MX 2. The minerl fluorite, clcium fluoride, CF 2, which nmes the group, is sometimes lso clled fluorspr. Unit cell: cuic. Lttice prmeter for fluorite, ¼ 0:5463 nm.

11 THE STRUCTURE OF UREA 11 Z ¼ 4 fcf 2 g Atom positions: C: 0, 0, 0; ˆÙ, ˆÙ, 0; 0,ˆÙ, ˆÙ ; ˆÙ, 0,ˆÙ F: ˆÙ, Ù, ˆÙ ; ˆÙ, Ù, Ù ; ˆÙ, ˆÙ, ˆÙ ; ˆÙ, ˆÙ, Ù ; Ù, ˆÙ, ˆÙ ; Ù, ˆÙ, Ù ; Ù, Ù, ˆÙ ; Ù, Ù, Ù There re four clcium nd eight fluorine toms in the unit cell. The numer of molecules of CF 2 in the unit cell is four, so tht, for ll fluorite structure compounds, Z ¼ 4. In this structure, ech clcium tom is surrounded y eight fluorine toms t the corners of cue. Ech fluorine tom is surrounded y four clcium toms t the vertices of tetrhedron (see lso Chpter 7). A perspective view of the structure is shown in Figure 1.12, nd projection of the structure down the c-xis in Figure This structure is dopted y numer of oxides nd hlides of lrge divlent ctions of formul MX The structure of ure The structures of moleculr crystls tend to hve different significnce to those of inorgnic nd minerl structures. Frequently, the informtion of most importnce is the moleculr geometry, nd how the molecules re rrnged in the crystllogrphic unit cell is often of less interest. To introduce the chnged emphsis when deling with moleculr mterils, the crystl structure of the orgnic compound ure is descried. Ure is very simple molecule, with formul CH 4 N 2 O. The unit cell is smll nd of high symmetry. It ws one of the erliest orgnic structures to e investigted using the methods of X-ry crystllogrphy, nd in these initil investigtions the dt ws not precise enough to locte the hydrogen toms. [The loction of hydrogen toms in structure remins prolem to present times, see lso Chpters 6 nd 7.] The crystllogrphic dt for ure is 3 Unit cell: tetrgonl. Lttice prmeters for ure, ¼ 0:5589 nm; c ¼ 0:46947 nm. Z ¼ 2 fch 4 N 2 Og Atom positions: C1: 0, , C2: , 0, O1: , 0, N1: , , N2: , , N3: , , N4: , , Figure 1.12 The cuic unit cell of the fluorite structure: () perspective view; () projection down the c-xis 3 Dt dpted from: V. Zvodnik, A. Stsh, V. Tsirelson, R. de Vries nd D. Feil, Act Crystllogr., B55, 45 (1999).

12 12 CH1 CRYSTALS AND CRYSTAL STRUCTURES H1: , , H2: , , H3: , , H4: , , H5: , , H6: , , H7: , , H8: , , Notice tht toms of the sme chemicl type re numered sequentilly. The numer of molecules of ure in the unit cell is two, so tht Z ¼ 2. The toms in unit cell, (including hydrogen), re shown in Figure This turns out to e not very helpful, nd n orgnic chemist would hve difficulty in recognising it s ure. This is ecuse the molecules lie long the unit cell sides, so tht whole molecule is not displyed in the unit cell, only moleculr frgments. [The unit cell is chosen in this wy ecuse of symmetry constrints, descried in the following chpters.] The chemicl structurl formul for ure is drwn in Figure 1.13, nd this is compred to molecule of ure viewed front on (Figure 1.13c), nd edge on (Figure 1.13d) extrcted from the crystllogrphic dt. The crystl structure is redrwn in Figures 1.13e, f with the toms linked to form molecules. This ltter depiction now grees with chemicl intuition, nd shows how the molecules re rrnged in spce. Note tht the list of toms in the unit cell is ecoming lengthy, leit tht this is n extremely simple structure. The wys used y crystllogrphers to reduce these lists to mngele proportions, y using the symmetry of the crystl, is explined in lter chpters The density of crystl The theoreticl density of crystl cn e found y clculting the mss of ll the toms in the unit Figure 1.13 The structure of ure: () perspective view of the tetrgonl unit cell of ure; () structurl formul of ure; (c) ll nd stick representtion of ure fce on, s in (); (d) ll nd stick representtion of ure sidewys on ; (e) projection of the structure long the c-xis; (f) projection of the structure down the -xis

13 ANSWERS TO INTRODUCTORY QUESTIONS 13 cell. The mss of n tom, m A, is its molr mss (grms mol 1 ) divided y the Avogdro constnt, N A ; ð6: mol 1 Þ m A ¼ molr mss=n A ðgrmsþ ¼ molr mss=ð1000 N A ÞðkilogrmsÞ The totl mss of ll of the toms in the unit cell is then n 1 m 1 þ n 2 m 2 þ n 3 m 3...=ð1000 N A Þ where n 1 is the numer of toms of type 1, with molr mss of m 1, nd so on. This is written in more compct form s X q i¼1 n i m i =ð1000 N A Þ where there re q different tom types in the unit cell. The density, r, is simply the totl mss is divided y the unit cell volume, V: ( ) r ¼ Xq n i m i =ð1000 N A Þ =V i¼1 For exmple, the theoreticl density of hlite is clculted in the following wy. First count the numer of different tom types in the unit cell. To count the numer of toms in unit cell, use the informtion: n tom within the cell counts s 1 n tom in fce counts s ˆÙ n tom on n edge counts s ˆÙ n tom on corner counts s ˆÙ A quick method to count the numer of toms in unit cell is to displce the unit cell outline to remove ll toms from corners, edges nd fces. The toms remining, which represent the unit cell contents, re ll within the oundry of the unit cell nd count s 1. The unit cell of the hlite structure contins 4 sodium (N) nd 4 chlorine (Cl) toms. The mss of the unit cell, m, is then given y: m ¼½ð4 22:99Þþð4 35:453ÞŠ=1000 N A ¼ 3: kg Where g mol 1 is the molr mss of sodium, g mol 1 is the molr mss of chlorine, nd N A is the Avogdro constnt, mol 1. The volume, V, of the cuic unit cell is given y 3, thus: V ¼ð0: Þ 3 m 3 ¼ 1: m 3 : The density, r, is given y the mss m divided y the volume, V: r ¼ 3: kg=1: m 3 ¼ 2164 kg m 3 The mesured density is 2165 kg m 3. The theoreticl density is lmost lwys slightly greter thn the mesured density ecuse rel crystls contin defects tht ct so s to reduce the totl mss per unit volume. Answers to introductory questions Wht is crystl system? A crystl system is set of reference xes, used to define the geometry of crystls nd crystl

14 14 CH1 CRYSTALS AND CRYSTAL STRUCTURES structures. There re seven crystl systems, cuic, tetrgonl, orthorhomic, monoclinic, triclinic, hexgonl nd trigonl. As the crystl systems re sets of reference xes, they hve direction s well s mgnitude, nd hence re vectors. They must e specified y length nd interxil ngles. The three reference xes re lelled, nd c, nd the ngles etween the positive direction of the xes s,, nd g, where lies etween þ nd þc, lies etween þ nd þc, nd g lies etween þ nd þ. The ngles re chosen to e greter or equl to 90 except for the trigonl system. In figures, the -xis is represented s projecting out of the plne of the pge, towrds the reder, the -xis points to the right nd the c-xis points towrds the top of the pge. Wht re unit cells? All crystls cn e uilt y the regulr stcking of smll volume of mteril clled the unit cell. The edges of the unit cell re generlly tken to e prllel to the xil vectors, nd c, of the seven crystl systems. The lengths of the unit cell sides re written, nd c, nd the ngles etween the unit cell edges re written,, nd g. The collected vlues,, c,, nd g for crystl structure re termed the unit cell or lttice prmeters. Wht informtion is needed to specify crystl structure? The minimum mount of informtion needed to specify crystl structure is the unit cell type, i.e., cuic, tetrgonl, etc, the unit cell prmeters nd the positions of ll of the toms in the unit cell. The tomic contents of the unit cell re simple multiple, Z, of the composition of the mteril. The vlue of Z is equl to the numer of formul units of the solid in the unit cell. Atom positions re expressed in terms of three coordintes, x, y, ndz. These re tken s frctions of, nd c, the unit cell sides, for exmple, ˆÙ, ˆÙ, ˆÙ. Prolems nd exercises Quick quiz 1 The numer of crystl systems is: () 5 () 6 (c) 7 2 The ngle etween the - nd c-xes in unit cell is lelled: () () (c) g 3 A tetrgonl unit cell is defined y: () ¼ ¼ c, ¼ ¼ g ¼ 90 () ¼ 6¼ c, ¼ ¼ g ¼ 90 (c) 6¼ 6¼ c, ¼ ¼ g ¼ 90 4 A crystl is uilt y the stcking of unit cells with: () Orienttionl nd trnsltionl long-rnge order () Orienttionl long-rnge order (c) Trnsltionl long-rnge order 5 Miller indices re used to lel () Crystl shpes

15 PROBLEMS AND EXERCISES 15 () Crystl fces (c) Crystl sizes 6 Crystl structures re often determined y the scttering of: () Light () Microwves (c) X-rys 7 In crystllogrphy the letter Z specifies: () The numer of toms in unit cell () The numer of formul units in unit cell (c) The numer of molecules in unit cell 8 The position of n tom t the corner of monoclinic unit cell is specified s: () 1, 0, 0 () 1, 1, 1 (c) 0, 0, 0 9 The numer of toms in the unit cell of the hlite structure is: () 2 () 4 (c) 8 10 When determining the numer of toms in unit cell, n tom in fce counts s: () ˆÙ () ˆÙ (c) ˆÙ Clcultions nd Questions 1.1 The rhomohedrl unit cell of rsenic, As, hs unit cell prmeters R ¼ nm, ¼ Use grphicl vector ddition (Appendix 1) to determine the equivlent hexgonl lttice prmeter H. Check your nswer rithmeticlly, nd clculte vlue for the hexgonl lttice prmeter c H. 1.2 Cssiterite, tin dioxide, SnO 2, dopts the rutile structure, with tetrgonl unit cell, lttice prmeters, ¼ nm, c ¼ nm, with Sn toms t: 0, 0, 0; ˆÙ, ˆÙ, ˆÙ ; nd O toms t: /ˆ, /ˆ,0; Ù, ˆÙ, ˆÙ ; É/ˆ, É/ˆ,0;ˆÙ, Ù, ˆÙ. Tking one corner of the unit cell s origin, determine the tom positions in nm nd clculte the unit cell volume in nm 3. Drw projection of the structure down the c-xis, with scle of 1cm¼ 0.1 nm. 1.3 The structure of SrTiO 3, is cuic, with lttice prmeter ¼ nm. The toms re t the positions: Sr: ˆÙ, ˆÙ, ˆÙ ; Ti: 0, 0, 0; O: ˆÙ, 0,0;0,ˆÙ, 0;0,0,ˆÙ. Sketch the unit cell. Wht is the numer of formul units in the unit cell? [This structure type is n importnt one, nd elongs to the perovskite fmily.] 1.4 () Ferrous fluoride, FeF 2, dopts the tetrgonl rutile structure, with lttice prmeters ¼ nm, c ¼ nm. The molr msses re Fe, g mol 1, F, g mol 1. Clculte the density of this compound. () Brium fluoride, BF 2, dopts the cuic fluorite structure, with lttice prmeter ¼ nm. The molr msses re B, g mol 1,F,18.998gmol 1. Clculte the density of this compound. 1.5 Strontium chloride, SrCl 2, dopts the fluorite structure, nd hs density of 3052 kg m 3. The molr msses of the toms re Sr, g mol 1, Cl, g mol 1. Estimte the lttice prmeter,, of this compound. 1.6 Molydenum, Mo, dopts the A2 (tungsten) structure. The density of the metl is

16 16 CH1 CRYSTALS AND CRYSTAL STRUCTURES kg mol 1 nd the cuic lttice prmeter is ¼ nm. Estimte the molr mss of molydenum. 1.7 Ametldifluoride, MF 2, dopts the tetrgonl rutile structure, with lttice prmeters ¼ nm, c ¼ nm nd density 3148 kg m 3. The molr mss of fluorine, F, is g mol 1.Estimtethe molr mss of the metl nd hence ttempt to identify it. 1.8 The density of nthrcene, C 14 H 10, is 1250 kg m 3 nd the unit cell volume is m 3. Determine the numer of nthrcene molecules, Z, which occur in unit cell. The molr msses re: C, g mol 1, H, g mol 1.