Lecture 9: Examples of Space Groups / Space Group Tables

February 6, 2007

LECTURE #09

Examples of Space Groups Space Group Tables ~'V, :; I,VJ ~ fc.u:~

&v ~ U If

. '--r' cons~sts of Cu at the 4a sites or just at the lattice points. For ~aCllit conStsts of Na at 4a and Cl at 4b, or vice versa. For CaF2 the basIs is Ca at 4a and F at Rc. Again we note that the general equivalent point is 192f, and there can be a crystal structure with this space group with atoms just at 192f (4 x 48 = 192). Of course, there are only 48 such atoms in the primitive unit eel!. For the structures shown in Fig. 3-5 the coordination numbers are: Cu-12; NaCI-6:6; CaF,-R:4. A number of elements have the faee-eent;red cubic (fee) s~rueture, and they are listed in Table 3-1. This means only the 4a sItes are occupied as in the Cu example. The fee structure is one of the two so-called close-packed structures. These are discussed in Secti~n 3-7, but suffLe it to say this structure can be obtained by the packlllg of hard spheres (billiard balls).

i' I I I I () S {} e , . -, (c) CaF (a) Copper (b) NaCl 2 Fig. 3-5 Several ,truclures wilh the 0h '(Fm3m) space group,

Zn at the 4a site at (0, 0, 0) and S at the 4c site at 0/4, 1/4, 1/4) plus the equivalent positions due to the F-Iattice. Figure 3-7e shows the structure. Unlike diamond, this structure does not have a screw axis or center of symmetry owing to the dissimilarity of the atoms. The lack of an 1(1) symmetry operation alIows materials with this crystal structure to be piezoelectric (Chapter 5), while diamond is not.

3-4 Point Group of a Spaee Group

So far in this chapter we have avoided mentioning the meaning of the term the point group of a space group and how it can be determined, altbough it was mentioned briefly in Chapter 1. Let us define it now. Consider all the symmetry operations of a space group. These operations include the lattice translations as in Eq. 2-1, point symme- r}, All try operations like {R Ia}, and glide and screw operations {R I t} where t is a three types may be written, in the Seitz notation, as {R I lattice translation (Eq. 2-1) or a fraction of a unit cell translation T as in Eq, 1-11. The point group of a spaee group is defined as the group of operations obtained if we take the space group symmetry operations and set all translations to zero. We are then left with a set of operations {R I O}, Note that for nonsymmorphic space groups some of these operations will not be symmetry operations of the crystal (as discussed in Section 1-3), so we might ask what is the use of the definition. The answer is that it is extremely useful and important. There is a one-to-one correspondence (an isomorphism) between the set of operations of the point group of the space group and the set of operations of one of the 32 crystallographic point groups; this leads to ways to classify the wave functions of the crystaL (This point will not be discussed here.) Also the macroscopic properties of crystal are determined by the operations !R I OJ because small translations such as ;. ~~$ IAcd. $e, 1I\c),tic.e&' 0\,0\ 4. lMa("oS<.o~ic:. ~Ie. (~ev 1.1II0..\11<0~'f l1l\t\\~IC!.) ,- \.-\ \ .2..

SIMPI.F CRYSTAL STRUCTURES 59

j //I 3 m Cubic No. 225 F4/1/1 2/m

Origin at centre (m3m)

Conditions limiting Co.ordinatcs of equivalent positions poSlibl1 reflections (0.0.0; 0.1.1; I.O.!; 1,1,0)+ General: hkl: h+k.k+I.(I+h)~2n I,X,y; y,r,x; x,z.)'; )',x,:; I,y,x; x,,.,z; hhl: (l+h-2n); 0 x,y,z; :,J/,p; y,f.J/; x,i,y; y.i,i; :.J','; .r,y,l; l,x,y; j,z,j; J/.:,P; j,x,i; t,y,'; Ok/: (k,l-2n); 0 J/.!,,:; f,J/,y; y,z,x; '.f.y; .v...f,z; t,y,x; x,j,t; f,J/,j; j,£,x; J/,f,y; p,J/,t; i,j.x; J/,y,:; t,x,y;' Y.z,x; x,l,y; ,,X,:; t..v,x; x,p,:; :,.t,y; y,i,x; x,i,y; y,J/,:; :",x; .,y,.f. x,y,l; "x,j; y,1.x; x,z,y; y,x,l;

Special: a. above, plus

z,x,x; ,Y.z,x; ',J/,f; f,x,J/; .f,t,.i; x,x,z: X,1,;(; x,x..fj z,.f,_i'i x,f,J/; .t',_".!; ',.T,Xj x.x.fj 1,.%,'£;.i.z,.f; x,x,z; It.!,.Y; x,1,x; .i.X,I; i..f.x; x,i,x; x,x,Z"; ZtxJ.fj .%,z,l. O,J,:; t,O.y; J,:,o; O,:,y; J,O,:; :,J,O; j>,i,0; O,i,y; y.O,!; t,y,O; O".t; t,O,y; O,y,J; f,O,y; y,f,O; O,l,y; y,O,f; t,J,O; no extra conditions O,p,'; J,O.p: p,J,O; O,"P; j,O,z; :",0. i.x,x; x,,,x; x,X,t; i,x,x; i,I,x; x,x,l; i..f.t; x,l,x; .i,x,ii V,x: x,iii; x,x,l. O.x,.x; x,O,x; x,x,O; O,x,i; x,O,x; x,.f,O; O,',f; .1',0";:; x..f,O; O,.f,x; x,O,.!; J/,x,O. x,U; 1,x,I; i,i.x; x.M: i,x,I: M,x; hkl: h,(k,I)-2n .t.M; !";:,I: i,I.J/; J/,M; 1,',1; M,i.

f 3m X,X,x; x,.f,.t'; .f,x..f; ~.x.x; x,x,x; x,x,x; x,x,x. i,x";:; no ..Ir. condilion. 4mm x,O,O; 0",.0; 0.0..<; i.O,O; 0,.i',0; 0,0";:. )

d mmm O,!,I; !.0.1; 1.1,0; O,M: 1,0,1; 1.1.0. 1 hkl: h,(k,l)-2n ~3m 1.1,1; 1,!,1. } m3m i.i.l. no exira condition" ~ m3m 0,0,0. J Fig.3-h 0h \(f'm3nl) T will not be noticed on a macroscopic scale (Neumann's principle, discussed in Chapter 5). Given a space group symbol it is very easy to determine the point group of the 5pace group. In the Schoenfljes notation the space group symbol is just a point group symbol with a superscript. If the superscript is ignored then we have the point group of the space group. In the International nolalion the space group symbol is a lattice type foJ1owed by a description of the poinl ~ymmetry operations as well as CHAPTER J SIMPLE CRYSTAL STRPCTCRES CHAPTER J 57

1m 3m III .~ III Cubic Im3m-Oh9 0: Originat centre 1'",Jm) po~iti,)n~ Co.ordinatci Qrequi ~'ale"1 Condilion.limiting 6b sites l'I}~~ible reftcctioo$ . (0.0,0; 1.1.1)+ 0 12d sites General: A X,)',l, I,X,y; )',:.x; X,Z.y; )I,X,.!; t,)',Xj hkl: I,J.k+l-p, 24h si tes x,p,l; I,il,y; )'.i,~; .~,!,ji; f,x,i; z,y,x; MI. (1-0,); G .i,y,l; i,_t,y; y,.!,Xj R,l,,r; p,.T,I; I.y,.i; Okl: (k+/-p,); G I,.f,y; i,i,xj x,!,y; ,,j,l; Fig. 3-3 Some of the .t,.P,I; I".x; t,J,I; l,.t,p; r,t,.f; x,t,y; j,~,:; I,M; y,:,;(; .i,z,y; l,y..Y; positions of the -',y,I; l""y; f,X,'; :,.i,y; y,i.~; )',l,t; l,y,x; 0h9(1m3m) space x",:; ''',i,y; x",l; .,x,,; Y,l,j; X,t,'; ".t,:: .,y,.f. groups. The very Specinl: os above, plu, large circles repre- 48 k m X,x,'; ',X,x; X,Z,x; .f,.f,t; l,i,.i; x.i,f; 'elll 'pher" at the x,i,l; z,',.fj .t,i,.i; x,x.:; i,.t,x; ;i.z,x; .\;L1 .f,r,t; x,l,x; 2a site at (0. 0, 0) t,.t,.f; .f.z,.t; x..f,1; l',.f,.~; ,-v\7 x,j..; l,j,x; .i,I",; x,x,f; l,X,.f; .t,l~. ... and(1/2, 1/2, 1/2). _ . , f 48 m O,y,:; I,O,y; y,I,O; O.l,,; y,O.:; z,.v.O; O,p,l; I,O,J; J.t,O; O,I,p; P.O.l; I.P,O; O,y,t; I,O,y; y.t,O; O,t,y; y,O,t; l,y,O; O,p,l; I,O.P; ;;,:,0; O,l,r; P,O,I; 1..V.O.

l,x,l-x; I,I.,+.<; 1'%.1u; 1,j.I-x; ,-x.I",; 1+x.I,.t; l~x,l,x; l-x,I,i; .<,1-".1; f,i+x,l; x,l+x,l; t,l-x,l; /.I-x,%; 1,1+X,t; 1.1 +X,x; I,I-.<"!; x,/.I-x; t,I,I.x; .....i.ll.t; .I,I,I-x; no eJltrRcondirions l-x,x,l; I! -',f,l; 1+x.x,l; I--'..t./. h mm O.x..t; X,O,.Y;x,.'t.O; O,x,Jj -f,O.,'t; x,x,O; O,l.t; .t.O..f; .t,t,O; O,x,x; x,D,.!; .t,x,O.

, mm ...0,1; 1....0; O,I,x; x,I.O; 0,.<,1; I,O,x; .t.O,I; l.t,O; O,!..t; ..q,0; O..f.l; i.O,x. f 3m X,X,X; .'f..i..f; .t,x..f; .t,..f,.tj .f,x,i; .f.X,A; x..f,.tj x,x,J.

4mm .<,0,0; O,x,O; 0.0,..: .t,O,O; O,.t,O; O,O.t.

d ~2m 1,0.1; 1,1,0; 0,1,1; 1,0,1; 1,1,0: 0.1,1.

3m /.i,I; 1,I,t: 1,1.1; !.i.I. hkl; h,k,(I)-p, b 4/mmm 0,1,1; 1.0.1; /.1,0. nu condition' a ",3m 0,0,0. }

Fig. 3-4 0h9(lm3m). CHAPTER 3 S..... LE CR YST AL STRUCTURES 53 't.Q..\M..~~~ ~a..\.i03 E m3m Cubic Pm3m No. 221 P 4jm 321m O~ Origin at centre (m3m) Conditions limiting Number of positions. CCH)rdinates of equivalent positions Wyckotrnotation, possible refto;tions and point symmetry General: hkl: x,y,z; z~,y; y,z,x; X,z,y; y,x,z; z,y,x; 48 n hhl: No conditions x,y.i; z,x,y; y,i,x; x,i,y; y,x,!; z,ji~; .i,y,:; i,x,y; y,z,x; x,z,y; y,x,i; i,y,x; Okl: 1, x..ji,z; i,x,y; y,i,x; j~,y; y,x,z; z,y.x; x,y,z; i~,y; j,i~; x,i,y; y,x,i; i,j,x; x,y,z; :,x,y; j,z,x; x,z,y; joX,Z; i,y,x; x,y,z; z,x,y; r,i,x; x,i,y; y.x,z; Z,1,x; x,y,i; z,x,j; y,z,x; x,z,y; YrX,i; :,y,x. Special: i,i,x; No conditions 24 m m X,X,:; :,x,x; x,z.x; i,x,i; z,x,x; x,x,i; z,x,x; x,i.x; x,x,z; i.;c,."t; x,z,x; i,x,i; i,x,x; i,z,.f; x,.f~z; z,x,x; x.i,x; x,x,:; i,x,x; x,i,x; x,x,i; z,x..x; x,z,x. z,y,l; 24 m ~,y,z; z,!,y; y,z,!; l,z,y; y,!,:; ~,1,i; i,l,y; y,i,I; VJi; yM; i,y,j; t,y,i; i,i.y; y,i,l; V,y; y,!,i; i,y,l; !Ji,:; z,l,y; y,z,l; l,z,y; y,!,:; zJi,l. z,y,O; 24 k m O,y,z; z,O,y; y,z,O; O,r,y; y,O,z; O,y,%'; i,O,y; y,i,O; O,iJi; y,O,i; iJi,O; O,y,i; i,O,y; y,i.O; O,i,y; y,O,i; i,y,O; O,y,:; z,OJi; ji,z,O; O,z,y; y,O,z; z,j,O.

12 j mm !,x,x; x,i,x; x,x,!; l,x,x; x,i,x; x,x,!; i,x,x; i,l,x; .>:.x,l; !,x,x; x,I,.f; X,x,I.

12 milt O,x,x; x,O,x; x,x,O; O,x,.f; x,O.x; x,.f,O; O,x,.f; .f,O,.i; i,x,O; O,f,x; x,O,x; x,x,O.

12 h mm x,l,O; O.x,l; l.o.x; x,O,l; 1.x,O; O,l,x; x,!.O; O,.f,i; l,O,x; i,O,l; l,x,O; O,l,.f.

8 g 3m x,x,x; x,f.x; x,x,x; x,i,x; i,i,x; i,x,x; x,x,x; x.x,.i.

6 f 4mm x,l,l; t,x,t: i.l,x; i,M; i.x,i; i.l.x.

6 e 4mm x,O,O; O,x,O; 0.0,x; x,O,O; O,x,O; O,O,x.

3 d 4/mmm i,O,O; O,i,O; O,O,i.

3 c 4/mmm 0,1,1: !'O,i; 1,1,0.

b m3m i,M.

Q m3m 0,0,0.

Fig. 3-2 °hl(Pm3m\.

,..... "- )(. O-..'M ~\~ ~Q.\l 6~

o Fa..C€ 0-- Ce.I.\.-\e.Ve& ~\~e~ c.. cf 3-3 Diamond and Zinc Blende Structures

In this section we discuss the last two simple, important, cubic crystal structures. Figures 3-7a and 3-7b show the diamond structure, space group 0,/ (Fd3m). This is the first nonsymmorphic structure discussed in this chapter. It has an F-Iattice with atoms at the 8a positions (0, 0, 0), (1/4, 1/4, 1/4) and the face-centering positions, Le., the coordinates of equivalent positions arc the same for all F- lattices as in Fig. 3-6. The full space group symbol is F4j/d32/m and 1/4. Note that there Fig. 3-7b shows the 41 axis at x = 1/2 and y = is a center of inversion at (l/8, 1/8. 1/:'5) and (l/8, 3/8, 3/8). He-

sides diamond and gray tin, the technologically important clements . sUieon (Si) and germanium (Ge) have this crystal structure. ..i' The zinc bien de structure is closely related to that of diamond. ZnS, GaAs, and many other binary compounds have the zinc blende structure. The symmorphic space group is T i(F43m) and the basis is

c -c - '!If'I I I . 0 10 . I "'" I I . 2 0 .I.. 0 . . b . b a I a 2 a (c) ZnS or GaAs (a) Diamond (b) Diamond (T/)F~3m (Oh7)Fd3m (Projection) S"( WI.1MC~ \,\111' c.. . Fig. 3-7 On page 14H these structures are shown with an emphasis on the tetrah~dral bonding.