Chapter 2: Crystal Structures and Symmetry

Transcription

1 hpter 2: rystl Structures nd Symmetry Lue, rvis Jnury 30, 2017 ontents 1 Lttice Types nd Symmetry Two-Dimensionl Lttices Three-Dimensionl Lttices Point-Group Symmetry Reduction of Quntum omplexity Symmetry in Lttice Summtions Group designtions Simple rystl Structures F HP

2 theory of the physicl properties of solids would e prcticlly impossile if the most stle elements were not regulr crystl lttices. The N-ody prolem is reduced to mngele proportions y the existence of trnsltionl symmetry. This mens tht there exist set of sis vectors (,,c) such tht the tomic structure remins invrint under trnsltions through ny vector which is the sum of integrl multiples of these vectors. s shown in Fig. 1 this mens tht one my go from ny loction in the lttice to n identicl loction y following pth composed of integrl multiples of these vectors. α Figure 1: ne my go from ny loction in the lttice to n identicl loction y following pth composed of integrl multiples of the vectors nd. Thus, one my lel the loctions of the toms 1. which compose the lttice with r n = n 1 + n 2 + n 3 c (1) where n 1, n 2, n 3 re integers. In this wy we my construct ny periodic structure. 1 we will see tht the sic uilding locks of periodic structures cn e more complicted thn single tom. For exmple in Nl, the sic uilding lock is composed of one N nd one l ion which is repeted in cuic pttern to mke the Nl structure 2

3 1 Lttice Types nd Symmetry 1.1 Two-Dimensionl Lttices These structures re clssified ccording to their symmetry. For exmple, in 2d there re 5 distinct types. The lowest symmetry is n olique lttice, of which the lttice shown in Fig. 1 is n exmple if nd α is not rtionl frction of π. Notice tht it is invri- Squre Rectngulr =, γ = π/2 =, γ = π/2 Hexngonl entered =, γ = π/3 Figure 2: Two dimensionl lttice types of higher symmetry. These hve higher symmetry since some re invrint under rottions of 2π/3, or 2π/6, or 2π/4, etc. The centered lttice is specil since it my lso e considered s lttice composed of two-component sis, nd rectngulr unit cell (shown with dshed rectngle). nt only under rottion of π nd 2π. Four other lttices, shown in Fig. 2 of higher symmetry re lso possile, nd clled specil lttice types (squre, rectngulr, centered, hexgonl). rvis lttice is 3

4 the common nme for distinct lttice type. The primitive cell is the prllel piped (in 3d) formed y the primitive lttice vectors which re defined s the lttice vectors which produce the primitive cell with the smllest volume ( (c c)). Notice tht the primitive cell does not lwys cpture the symmetry s well s lrger cell, s is the cse with the centered lttice type. The centered lttice is specil since it my lso e considered s lttice composed of two-component sis on rectngulr unit cell (shown with dshed rectngle). sis Primitive cell nd lttice vectors Figure 3: squre lttice with complex sis composed of one nd two toms (c.f. cuprte high-temperture superconductors). To ccount for more complex structures like moleculr solids, slts, etc., one lso llows ech lttice point to hve structure in the form of sis. good exmple of this in two dimensions is the 2 plnes 4

5 which chrcterize the cuprte high temperture superconductors (cf. Fig. 3). Here the sis is composed of two oxygens nd one copper tom lid down on simple squre lttice with the tom centered on the lttice points. 1.2 Three-Dimensionl Lttices ic ody entered ic Fce entered ic c c = x = y c = z = (x+y-z)/2 = (-x+y+z)/2 c = (x-y+z)/2 = (x+y)/2 = (x+z)/2 c = (y+z)/2 Figure 4: Three-dimensionl cuic lttices. The primitive lttice vectors (,,c) re lso indicted. Note tht the primitive cells of the centered lttice is not the unit cell commonly drwn. The sitution in three-dimensionl lttices cn e more complicted. Here there re 14 lttice types (or rvis lttices). For exmple there re 3 cuic structures, shown in Fig. 4. Note tht the primitive cells of the centered lttice is not the unit cell commonly drwn. In ddition, there re triclinic, 2 monoclinic, 4 orthorhomic... rvis lttices, for 5

6 totl of 14 in three dimensions. 2 Point-Group Symmetry The use of symmetry cn gretly simplify prolem. 2.1 Reduction of Quntum omplexity If Hmiltonin is invrint under certin symmetry opertions, then we my choose to clssify the eigensttes s sttes of the symmetry opertion nd H will not connect sttes of different symmetry. s n exmple, imgine tht symmetry opertion R leves H invrint, so tht RHR 1 = H then [H, R] = 0 (2) Then if j > re the eigensttes of R, then j j >< j is representtion of the identity, nd we expnd HR = RH, nd exmine its elements < i R k >< k H j >= < i H k >< k R j >. (3) k k If we recll tht R ik =< i R k >= R ii δ ik since k > re eigensttes of R, then Eq. 3 ecomes (R ii R jj ) H ij = 0. (4) So, H ij = 0 if R i nd R j re different eigenvlues of R. Thus, when the 6

7 sttes re clssified y their symmetry, the Hmiltonin mtrix ecomes lock digonl, so tht ech lock my e seprtely digonlized. 2.2 Symmetry in Lttice Summtions s nother exmple, consider Mdelung sum in two-dimensionl squre centered lttice (i.e. 2d nlog of Nl). Here we wnt to clculte ij ± p ij. (5) This my e done y rute force sum over the lttice, i.e. lim n i= n,n j= n,n ( 1) i+j p ij. (6) r, we my relize tht the lttice hs some well defined opertions which leve it invrint. For exmple, this lttice in invrint under inversion (x, y) ( x, y), nd reflections out the x (x, y) (x, y) nd y (x, y) ( x, y) xes, etc. For these resons, the eight points highlighted in Fig. 5() ll contriute n identicl mount to the sum in Eq. 5. In fct ll such interior points hve degenercy of 8. nly specil points like the point t the origin (which is unique) nd points long the symmetry xes (the xy nd x xis, ech with degenercy of four) hve lower degenercies. Thus, the sum my e restricted to the irreducile wedge, so long s the corresponding terms in the sum re multiplied y the pproprite degenercy fctors, shown in Fig. 5(). n pproprite lgorithm to clculte oth the degenercy tle, nd 7

8 () () Figure 5: Equivlent points nd irreducile wedge for the 2-d squre lttice. Due to the symmetry of the 2-d squre lttice, the eight ptterned lttice sites ll contriute n identicl mount to the Mdelung sum clculted round the solid lck site. Due to this symmetry, the sum cn e reduced to the irreducile wedge () if the result t ech point is multiplied y the degenercy fctors indicted. the sum 5 itself re: c First clculte the degenercy tle c do i=1,n do j=0,i if(i.eq.j.or.j.eq.0) then deg(i,j)=4 else deg(i,j)=8 end if end do 8

9 end do deg(0,0)=1 c c Now clculte the Mdelung sum c sum=0.0 do i=1,n do j=0,i p=sqrt(i**2+j**2) sum=sum+((-1)**(i+j))*deg(i,j)/p end do end do y performing the sum in this wy, we sved fctor of 8! In fct, in three- dimensions, the svings is much greter, nd rel nd structure clcultions lwys mke use of the point group symmetry to ccelerte the clcultions. The next question is then, could we do the sme thing for more complicted system (fcc in 3d?). To do this, we need some wy of clssifying the symmetries of the system tht we wnt to pply. Group theory llows us to lern the consequences of the symmetry in much more complicted systems. group S is defined s set {E,,, } which is closed under inry opertion (ie. S) nd: 9

10 the inry opertion is ssocitive ( ) = ( ) there exists n identity E S : E = E = For ech S, there exists n 1 S : 1 = 1 = E In the point group context, the opertions re inversions, reflections, rottions, nd improper rottions (inversion rottions). The inry opertion is ny comintion of these; i.e. inversion followed y rottion. In the exmple we just considered we my clssify the opertions tht we hve lredy used. lerly we need 2!2 2 of these (ie we cn choose to tke (x,y) to ny permuttion of (x,y) nd choose either ± for ech, in D-dimensions, there would e D!2 D opertions). In tle. 1, ll of these opertions re identified The reflections re self inverting s pertion (x, y) (x, y) (x, y) (x, y) (x, y) ( x, y) (x, y) ( x, y) (x, y) (y, x) (x, y) (y, x) (x, y) ( y, x) (x, y) ( y, x) Identifiction Identity reflection out x xis reflection out y xis inversion reflection out x = y rottion y π/2 out z inversion-reflection inversion-rottion Tle 1: Point group symmetry opertions for the two-dimensionl squre lttice. ll of the group elements re self-inverting except for the sixth nd eight, which re inverses of ech other. is the inversion nd one of the rottions nd inversion rottions. The 10

11 set is clerly lso closed. lso, since their re 8 opertions, clerly the interior points in the irreducile wedge re 8-fold degenerte (w.r.t. the Mdelung sum). This is lwys the cse. Using the group opertions one my lwys reduce the clcultion to n irreducile wedge. They the degenercy of ech point in the wedge my e determined: Since group opertion tkes point in the wedge to either itself or n equivlent point in the lttice, nd the former (ltter) does (does not) contriute the the degenercy, the degenercy of ech point times the numer of opertions which leve the point invrint must equl the numer of symmetry opertions in the group. Thus, points with the lowest symmetry (invrint only under the identity) hve degenercy of the group size. 2.3 Group designtions Point groups re usully designted y their Schönflies point group symol descried in tle. 2 s n exmple, consider the previous exmple of squre lttice. It is invrint under rottions to the pge y π/2 mirror plnes in the horizontl nd verticl (x nd y xes) mirror plnes long the digonl (x=y, x=-y). The mirror plnes re prllel to the min rottion xis which is itself 4-fold xis nd thus the group for the squre lttice is 4v. 11

12 Symol j S j D j T i s Mening (j=2,3,4, 6) j-fold rottion xis j-fold rottion-inversion xis j 2-fold rottion xes to j-fold principle rottion xis 4 three-nd 3 two-fold rottion xes, s in tetrhedron 4 three-nd 3 four-fold rottion xes, s in octhedron center of inversion mirror plne Tle 2: The Schönflies point group symols. These give the clssifiction ccording to rottion xes nd principle mirror plnes. In ddition, their re suffixes for mirror plnes (h: horizontl=perpendiculr to the rottion xis, v: verticl=prllel to the min rottion xis in the plne, d: digonl=prllel to the min rottion xis in the plne isecting the two-fold rottion xes). 3 Simple rystl Structures 3.1 F The fcc structure is one of the close pcked structures, pproprite for metls, with 12 nerest neighors to ech site (i.e., coordintion numer of 12). The rvis lttice for the fcc structure is shown in Fig. 6 It is composed of prllel plnes of nerest neighors (with six nerest neighors to ech site in the plne) Metls often form into n fcc structure. There re two resons for this. First, s discussed efore, the s nd p onding is typiclly very long-rnged nd therefore rther non-directionl. (In fct, when the 12

13 Fce entered ic (F) lose-pcked plnes Principle lttice vectors z x c = (x+y)/2 = (x+z)/2 c = (y+z)/2 y 3-fold xes 4-fold xes Figure 6: The rvis lttice of fce-centered cuic (F) structure. s shown on the left, the fcc structure is composed of prllel plnes of toms, with ech tom surrounded y 6 others in the plne. The totl coordintion numer (the numer of nerest neighors) is 12. The principle lttice vectors (center) ech hve length 1/ 2 of the unit cell length. The lttice hs four 3-fold xes, nd three 4-fold xes s shown on the right. In ddition, ech plne shown on the left hs the principle 6-fold rottion xis to it, ut since the plnes re shifted reltive to one nother, they do not shre 6-fold xes. Thus, four-fold xes re the principle xes, nd since they ech hve perpendiculr mirror plne, the point group for the fcc lttice is h. p-onding is short rnged, the cc structure is fvored.) This nturlly leds to close pcked structure. Second, to whtever degree there is d-electron overlp in the trnsition metls, they prefer the fcc structure. To see this, consider the d-oritls shown in Fig. 7 centered on one of the fce centers with the fce the xy plne. Ech loe of the d xy, d yz, nd d xz oritls points to ner neighor. The xz,xy,yz triplet form rther strong onds. The d x2 y 2 nd d 3z 2 r2 oritls do not since they point wy from the nerest neighors. Thus the triplet of sttes form 13

14 strong onding nd nti-onding nds, while the doulet sttes do not split. The system cn gin energy y occupying the triplet onding z z z d y d y d xz xy yz y x x x z d x 2 - y 2 y z d 3z 2 - r 2 y x x Figure 7: The d-oritls. In n fcc structure, the triplet of oritls shown on top ll point towrds nerest neighors; wheres, the ottom doulet point wy. Thus the triplet cn form onding nd ntionding sttes. sttes, thus mny metls form fcc structures. ccording to shcroft nd Mermin, these include, Sr, Rh, Ir, Ni, Pd, Pt,, g, u, l, nd P. The fcc structure lso explins why metls re ductile since djcent plnes cn slide pst one nother. In ddition ech plne hs 6-fold rottion xis perpendiculr to it, ut since 2 djcent plnes re shifted reltive to nother, the rottion xes perpendiculr to the plnes re 3-fold, with one long the ech min digonl of the unit cell. There re lso 4-fold xes through ech center of the cue with mirror plnes perpendiculr to it. Thus the fcc point group is h. In fct, this sme rgument lso pplies to the cc nd sc lttices, so h is the pproprite 14

15 group for ll cuic rvis lttices nd is often clled the cuic group. 3.2 HP s shown in Fig. 8 the Hexgonl lose Pcked (HP) structure is descried y the D 3h point group. The HP structure (cf. Fig. 9) is 3-fold xis mirror plne three 2-fold xes in plne Figure 8: The symmetry of the HP lttice. The principle rottion xis is perpendiculr to the two-dimensionl hexgonl lttices which re stcked to form the hcp structure. In ddition, there is mirror plne centered within one of these hexgonl 2d structures, which contins three 2-fold xes. Thus the point group is D 3h. similr to the F structure, ut it does not correspond to rvis lttice (in fct there re five cuic point groups, ut only three cuic rvis lttices). s with fcc its coordintion numer is 12. The simplest wy to construct it is to form one hexgonl plne nd then dd two identicl ones top nd ottom. Thus its stcking is... of the plnes. This shifting of the plnes clerly disrupts the d-oritl onding dvntge gined in fcc, nevertheless mny metls form this structure including e, Mg, Sc, Y, L, Ti, Zr, Hf, Tc, Re, Ru, s, o, 15

16 F HP These spces unfilled Figure 9: comprison of the F (left) nd HP (right) close pcked structures. The HP structure does not hve simple rvis unit cell, ut my e constructed y lterntely stcking two-dimensionl hexgonl lttices. In contrct, the F structure my e constructed y sequentilly stcking three shifted hexgonl twodimensionl lttices. Zn, d, nd Tl. 3.3 Just like the simple cuic nd fcc lttices, the ody-centered cuic () lttice (cf. Fig. 4) hs four 3-fold xes, 3 4-fold xes, with mirror plnes perpendiculr to the 4-fold xes, nd therefore elongs to the h point group. The ody centered cuic structure only hs coordintion numer of 8. Nevertheless some metls form into lttice ( V N, T W M, in ddition r nd Fe hve cc phses.) onding of p-oritls is idel in lttice since the nnn lttice is simply composed of two interpenetrting cuic lttices. This structure llows the next-nerest 16

17 neighor p-oritls to overlp more significntly thn n fcc (or hcp) structure would. This increses the effective coordintion numer y including the next nerest neighor shell in the onding (cf. Fig. 10). R(r) 2 1 1s fcc cc 2s,2p o r() Figure 10: solute squre of the rdil prt of the electronic wvefunction. For the cc lttice, oth the 8 nerest, nd 6 next nerest neighors lie in region of reltively high electronic density. This fvors the formtion of cc over fcc lttice for some elementl metls (This figure ws lifted from I&L). 17