3 Brvis ltties nd rystl systems 3. Introdution The definitions of the motif, the repeting unit of pttern, nd the lttie, n rry of points in spe in whih eh point hs n identil environment, hold in three dimensions extly s they do in two dimensions. However, in three dimensions there re dditionl symmetry elements tht need to e onsidered: oth point symmetry elements to desrie the symmetry of the three-dimensionl motif (or indeed ny rystl or three-dimensionl ojet) nd lso trnsltionl symmetry elements, whih re required (like glide lines in the two-dimensionl se) to desrie ll the possile ptterns whih rise y omining motifs of different symmetries with their pproprite ltties. Clerly, these onsidertions suggest tht the sujet is going to e rther more omplited nd diffiult ; it is ovious tht there re going to e mny more threedimensionl ptterns (or spe groups) thn the seventeen two-dimensionl ptterns (or plne groups or the eighty two-sided ptterns Chpter ), nd to work through ll of these systemtilly would tke up mny pges! However, it is not neessry to do so; ll tht is required is n understnding of the priniples involved (Chpter ), the opertion nd signifine of the dditionl symmetry elements, nd the min results. These min results my e stted stright wy. The dditionl point symmetry elements required re entres of symmetry, mirror plnes (insted of lines) nd inversion xes; the dditionl trnsltionl symmetry elements re glide plnes (insted of lines) nd srew xes. The pplition nd permuttion of ll symmetry elements to ptterns in spe give rise to 30 spe groups (insted of seventeen plne groups) distriuted mong fourteen spe ltties (insted of five plne ltties) nd thirty-two point group symmetries (insted of ten plne point group symmetries). In this hpter the onept of spe (or Brvis) ltties nd their symmetries is disussed nd, deriving from this, the lssifition of rystls into seven systems. 3. The fourteen spe (Brvis) ltties The systemti work of desriing nd enumerting the spe ltties ws done initilly y Frnkenheim who, in 835, proposed tht there were fifteen in ll. Unfortuntely for Frnkenheim, two of his ltties were identil, ft first pointed out y Brvis in 848. It ws, to tke two-dimensionl nlogy, s if Frnkenheim hd filed to notie Denotes iogrphil notes ville in ppendix 3.
3. The fourteen spe (Brvis) ltties 85 (see Fig..4()) tht the rhomi or dimond nd the retngulr entred plne ltties were identil! Hene, to this dy, the fourteen spe ltties re usully, nd perhps unfirly, lled Brvis ltties. The unit ells of the Brvis ltties re shown in Fig. 3.. The different shpes nd sizes of these ells my e desried in terms of three ell edge lengths or xil distnes, ui Body-entred ui (I) Fe-entred ui (F) tetrgonl Body-entred tetrgonl (I) orthorhomi Body-entred orthorhomi (I) α α α 0 Bse-entred orthorhomi (C) Fe-entred orthorhomi (F) Rhomohedrl (R) Hexgonl β β β γ α monolini Bse-entred monolini (C) Trilini Fig. 3.. The fourteen Brvis ltties (from Elements of X-Ry Diffrtion, (nd edn), y B. D. Cullity, ddison-wesley, 978).
86 Brvis ltties nd rystl systems,,, or lttie vetors,, nd the ngles etween them, α, β, γ, where α is the ngle etween nd, β the ngle etween nd, nd γ the ngle etween nd. The xil distnes nd ngles re mesured from one orner to the ell, i.e. ommon origin. It does not mtter where we tke the origin ny orner will do ut, s pointed out in Chpter, it is useful onvention (nd helps to void onfusion) if the origin is tken s the k left-hnd orner of the ell, the -xis pointing forwrd (out of the pge), the -xis towrds the right nd the -xis upwrds. This onvention lso gives right-hnded xil system. If ny one of the xes is reversed (e.g. the -xis towrds the left insted of the right), then left-hnded xil system results. The distintion etween them is tht, like left nd right hnds, they re mirror imges of one nother nd nnot e rought into oinidene y rottion. The drwings of the unit ells of the Brvis ltties in Fig. 3. n e misleding euse, s shown in Chpter, it is the pttern of lttie points whih distinguishes the ltties. The unit ells simply represent ritrry, though onvenient, wys of joining up the lttie points. Consider, for exmple, the three ui ltties; ui P (for Primitive, one lttie point per ell, i.e. lttie points only t the orners of the ell), ui I (for Innenzentrierte, whih is Germn for ody-entred, n dditionl lttie point t the entre of the ell, giving two lttie points per ell) nd ui F (for Fe-entred, with dditionl lttie points t the entres of eh fe of the ell, giving four lttie points per ell). It is possile to outline lterntive primitive ells (i.e. lttie points only t the orners) for the ui I nd ui F ltties, s is shown in Fig. 3.. s mentioned in Chpter, these primitive ells re not often used () euse the inter-xil ngles re not the onvenient 90 (i.e. they re not orthogonl) nd () euse they do not revel very lerly the ui symmetry of the ui I nd ui F ltties. (The symmetry of the Brvis ltties, or rther the point group symmetries of their unit ells, will e desried in Setion 3.3.) Similr rguments onerning the use of primitive ells pply to ll the other entred ltties. Notie tht the unit ells of two of the ltties re entred on the top nd ottom fes. These re lled se-entred or C-entred euse these fes re interseted y the -xis. 09 60 () () Fig. 3.. () The ui I nd () the ui F ltties with the primitive rhomohedrl ells nd inter-xil ngles indited.
3. The fourteen spe (Brvis) ltties 87 The Brvis ltties my e thought of s eing uilt up y stking lyers of the five plne ltties, one on top of nother. The ui nd tetrgonl ltties re sed on the stking of squre lttie lyers; the orthorhomi P nd I ltties on the stking of retngulr lyers; the orthorhomi C nd F ltties on the stking of retngulr entred lyers; the rhomohedrl nd hexgonl lttie on the stking of hexgonl lyers nd the monolini nd trilini ltties on the stking of olique lyers. These reltionships etween the plne nd the Brvis ltties re esy to see, exept perhps for the rhomohedrl lttie. The rhomohedrl unit ell hs xes of equl length nd with equl ngles (α) etween them. Notie tht the lyers of lttie points, perpendiulr to the vertil diretion (shown dotted in Fig. 3.) form tringulr, or equivlently, hexgonl lyers. The hexgonl nd rhomohedrl ltties differ in the wys in whih the hexgonl lyers re stked. In the hexgonl lttie they re stked diretly one on top of the other (Fig. 3.3()) nd in the rhomohedrl lttie they re stked suh tht the next two lyers of points lie ove the tringulr hollows or intersties of the lyer elow, giving three lyer repet (Fig. 3.3()). These hexgonl nd rhomohedrl stking sequenes hve een met efore in the stking of lose-pked lyers (Chpter ); the hexgonl lttie orresponds to the simple hexgonl...sequene nd the rhomohedrl lttie orresponds to the f BCBC...sequene. Now oservnt reders will notie tht the rhomohedrl nd ui ltties re therefore relted. The primitive ells of the ui I nd ui F ltties (Fig. 3.) re rhomohedrl the xes re of equl length nd the ngles (α) etween them re equl. s in the two-dimensionl ses, wht distinguishes the ui ltties from the rhomohedrl is their symmetry. When the ngle α is 90 we hve ui P lttie, when it is 60 we hve ui F lttie nd when it is 09.47 we hve ui I lttie (Fig. 3.). Or, lterntively, when the hexgonl lyers of lttie points in the rhomohedrl lttie re sped prt in suh wy tht the ngle α is 90,60 or 09.47, then ui symmetry results. Finlly, ompre the orthorhomi ltties (ll sides of the unit ell of different lengths) with the tetrgonl ltties (two sides of the ell of equl length). Why re there C B () () Fig. 3.3. Stking of hexgonl lyers of lttie points in () the hexgonl lttie nd () the rhomohedrl lttie.
88 Brvis ltties nd rystl systems P I C F () () Fig. 3.4. Plns of tetrgonl ltties showing () the tetrgonl P = C lttie nd () the tetrgonl I = F lttie. four orthorhomi ltties, P, C, I nd F, nd only two tetrgonl ltties, P nd I? Why re there not tetrgonl C nd F ltties s well? The nswer is tht there re tetrgonl C nd F ltties, ut y redrwing or outlining different unit ells, s shown in Fig. 3.4, it will e seen tht they re identil to the tetrgonl P nd I ltties, respetively. In short, they represent no new rrngements of lttie points. 3.3 The symmetry of the fourteen Brvis ltties: rystl systems The unit ells of the Brvis ltties my e thought of s the uilding loks of rystls, preisely s Hüy envisged (Fig..). Hene it follows tht the hit or externl shpe, or the oserved symmetry of rystls, will e sed upon the shpes nd symmetry of the Brvis ltties, nd we now hve to desrie the point symmetry of the unit ells of the Brvis ltties just s we desried the point symmetry of plne ptterns nd ltties. The sujet is fr more redily understood if simple models re used (ppendix ). First, mirror lines of symmetry eome mirror plnes in three dimensions. Seond, xes of symmetry (dids, trids, tetrds nd hexds) lso pply to three dimensions. The dditionl omplition is tht, wheres plne motif or ojet n only hve one suh xis (perpendiulr to its plne), three-dimensionl ojet n hve severl xes running in different diretions (ut lwys through point in the entre of the ojet). Consider, for exmple, ui unit ell (Fig. 3.5()). It ontins totl of nine mirror plnes, three prllel to the ue fes nd six prllel to the fe digonls. There re three tetrd (four-fold) xes perpendiulr to the three sets of ue fes, four trid (three-fold) xes running etween opposite ue orners, nd six did (two-fold) xes running etween the entres of opposite edges. This olletion of symmetry elements is lled the point group symmetry of the ue euse ll the elements plnes nd xes pss through point in the entre. Why should there e these prtiulr numers of mirror plnes nd xes? It is euse ll the vrious symmetry elements operting t or round the point must e onsistent with one nother. Self-onsisteny is fundmentl priniple, underlying ll the twodimensionl plne groups, ll the three-dimensionl point groups nd ll the spe groups
3.3 The symmetry of the fourteen Brvis ltties 89 () () Fig. 3.5. The point symmetry elements in () ue (ui unit ell) nd () n orthorhomi unit ell. tht will e disussed in Chpter 4. If there re two did xes, for exmple, then they hve to e mutully orthogonl, otherwise hos would result; y the sme token they lso must generte third did perpendiulr to oth of them. It is the neessity for selfonsisteny whih governs the onstrution of every one of the different omintions of symmetry, ontrolling the nture of eh omintion; it is this, lso, whih limits the totl numers of possile omintions to quite definite numers suh s thirty-two, in the se of the rystllogrphi point groups (the rystl lsses), the fourteen Brvis ltties, nd so on. The ui unit ell hs more symmetry elements thn ny other: its very simpliity mkes its symmetry diffiult to grsp. More esy to follow is the symmetry of n orthorhomi ell. Figure 3.5() shows the point group symmetry of n orthorhomi unit ell. It ontins, like the ue, three mirror plnes prllel to the fes of the ell ut no more mirror plnes do not exist prllel to the fe digonls. The only xes of symmetry re three dids perpendiulr to the three fes of the unit ell. In oth ses it n e seen tht the point group symmetry of these unit ells (Figs 3.5() nd 3.5()) is independent of whether the ells re entred or not. ll three ui ltties, P, I nd F, hve the sme point group symmetry; ll four orthorhomi ltties, P, I, F nd C, hve the sme point group symmetry nd so on. This simple oservtion leds to n importnt onlusion: it is not possile, from the oserved symmetry of rystl, to tell whether the underlying Brvis lttie is entred or not. Therefore, in terms of their point group symmetries, the Brvis ltties re grouped, ording to the shpes of their unit ells, into seven rystl systems. For exmple, rystls with ui P, I or F ltties elong to the ui system, rystls with orthorhomi P, I, F or C ltties elong to the orthorhomi system, nd so on. However, omplition rises in the se of rystls with hexgonl lttie. One might expet tht ll rystls with hexgonl lttie should elong to the hexgonl system, ut, s shown in Chpter 4, the externl symmetry of rystls my not e identil (nd usully is not identil) to the symmetry of the underlying Brvis lttie. Some rystls with hexgonl lttie, e.g. α-qurtz, do not show hexgonl (hexd) symmetry ut hve trid symmetry. (see Fig..33, Setion..5) Suh rystls re ssigned to the trigonl system rther thn to the hexgonl system. Hene the trigonl system inludes rystls with oth hexgonl nd rhomohedrl Brvis ltties. There is yet nother prolem whih is prtiulrly ssoited with the trigonl system, whih is tht the rhomohedrl unit ell outlined
90 Brvis ltties nd rystl systems in Figs 3. nd 3.3 is not lwys used lrger (non-primitive) unit ell of three times the size is sometimes more onvenient. The prolem of trnsforming xes from one unit ell to nother is ddressed in Chpter 5. The rystl systems nd their orresponding Brvis ltties re shown in Tle 3.. Notie tht there re no xes or plnes of symmetry in the trilini system. The only symmetry tht the trilini lttie possesses (nd whih is possessed y ll the other ltties) is entre of symmetry. This point symmetry element nd inversion xes of symmetry re explined in Chpter 4. 3.4 The oordintion or environments of Brvis lttie points: spe-filling polyhedr So fr we hve onsidered ltties s ptterns of points in spe in whih eh lttie point hs the sme environment in the sme orienttion. This pproh is omplete nd suffiient, ut it fils to stress, or even mke ler, the ft tht eh of these environments is distint nd hrteristi of the ltties themselves. We need therefore method of lerly nd unmiguously defining wht we men y the environment of lttie point. One pproh (whih we hve used lredy in working out the sizes of interstitil sites) is to stte this in terms of oordintion the numers nd distnes of nerest neighours. For exmple, in the simple ui (ui P) lttie eh lttie point is surrounded y six other equidistnt lttie points; in the (ui I) lttie eh lttie point is surrounded y eight equidistnt lttie points nd so on. This is stisftory, ut n lterntive nd muh more fruitful pproh is to onsider the environment or domin of eh lttie point in terms of polyhedron whose fes, edges nd verties re equidistnt etween eh lttie point nd its nerest neighours. The onstrution of suh polyhedron is illustrted in two dimensions for simpliity in Fig. 3.6. This is pln view of simple monolini (monolini P) lttie with the xis perpendiulr to the pge. The line lelled represents the edge or tre of plne perpendiulr to the pge nd hlf wy etween the entrl lttie point 0 nd its neighour. ll points lying in this plne (oth in the plne of the pper nd ove nd elow) re therefore equidistnt etween the two lttie points 0 nd. We now repet the proess for the other lttie points, 3, 4, et., surrounding the entrl lttie point. The plnes,, 3 et. form the six vertil fes of the polyhedron nd in three dimensions, onsidering the lttie points ove nd elow the entrl lttie point 0, the polyhedron for the monolini P lttie is losed prism, shown shded in pln in Fig. 3.6. Eh lttie point is surrounded y n identil polyhedron nd they ll fit together to ompletely fill spe with no gps in etween. In this exmple (of monolini P lttie) the edges of the polyhedron re where the fes interset nd represent points whih re equidistnt etween the entrl lttie point nd two other surrounding lttie points. Similrly, the verties of the polyhedron represent points whih re equidistnt etween the entrl lttie point nd three other surrounding points. However, for ltties of higher symmetry this orrespondene does not hold. If, for exmple, we onsider ui P lttie, squre in pln, nd follow the proedure outlined ove, we find tht the polyhedron is (s expeted) ue, ut the edges of whih re equidistnt etween the entrl lttie point nd three surrounding