Lecture V. Introduction to Spce Groups 2003. Chrles H. Lke Outline:. Introduction B. Trnsltionl symmetry C. Nomenclture nd symols used with spce groups D. The spce groups E. Derivtion nd discussion of spce group P2 /c F. Biliogrphy. Introduction Spce groups descrie the internl rrngement of crystl. There re only 230 symmetry rrngements tht crystl cn dopt to fill ll of spce. These symmetry rrngements re comintions of trnsltionl symmetry elements nd point groups. In this chpter, the focus will e entirely on conventionl, resonly ordered, three-dimensionl crystls. B. Trnsltionl symmetry Trnsltionl symmetry is when symmetry opertion moves ll points in such mnner tht spce itself is unchnged (rememer, in point symmetry, one or more points remin fixed under ll symmetry opertions). This leds to repet of some oject in n identicl orienttion (i.e., without ny rottion) t regulr intervls. Figure V-, shows simple 2-dimensionl pttern consisting of New Zelnd flgs. If the pttern is extended infinitely in ll twodimensionl directions then every flg will possess n identicl environment. lttice cn e constructed y plotting identicl points in the pttern (i.e. the center of the union jck or prticulr corner of the flg). This rry of points mkes up the lttice (ctully, 2-dimensionl lttice is clled net ). The unit cell is the smllest unit tht cn generte the entire pttern y simple trnsltions long unit cell edges (i.e. the nd directions). There re n infinite numer of possile unit cell choices, which cn descrie the entire pttern. In Figure V-, three possile unit cells, B nd C hve een drwn. Notice, in ll cses, one entire flg is contined inside the unit cell. Usully, the unit cell is defined to tke dvntge of symmetry or to enhnce some structurl feture. C Figure V-. 2-D lttice (net) with vrious possile unit cells. B
V-2 Introduction to Spce Groups There re three types of trnsltion symmetry elements tht re pplicle to three-dimensionl systems; the 4 Brvis lttices, screw xes nd glide plnes. If ny trnsltionl symmetry is present, the group must e n infinite one, since no finite numer of opertions cn return n oject to its originl position. ll spce groups re thus infinite groups. They my e pplied to crystl structures ecuse, with crystl of mcroscopic size, the unit cell is repeted thousnds of times effectively infinitely long ech of three independent directions. s pointed out in Lecture V, the four possile types of three-dimensionl lttice trnsltions re the primitive (P), C-centered (C), ody centered (I) nd fce centered (F) lttices s shown in Figure V-2. c c P primitive lttice C centered Lttice c c I ody centered Lttice F fce centered Lttice Figure V-2. Spce filling lttices. Unit cell in Figure V-3 is primitive unit cell. Notice, the four lttice points t the corners of the unit cell hve identicl environments nd tht ech corner is shred y four other unit cells, therefore ¼ of ech lttice point is inside unit cell. When ll four corners re dded, one complete lttice point is present inside the unit cell. This cn e esily oserved y exmining the str-figures. t ech corner, roughly ¼ of the str is inside the unit cell. When the pieces from ll four corners re dded, one complete str is generted. The entire pttern cn e generted y unit trnsltions long the primitive unit cell s edges trnsltions in the nd directions. But, unit cells re generlly chosen to tke dvntge of the symmetry of the pttern.
Introduction to Spce Groups V-3 In Figure V-3, it cn e seen tht different unit cell B cn e drwn with 90 0 ngles. This lrger cell is not primitive nd contins two lttice points. One complete lttice point is present in the center of the unit cell nd ¼ of lttice point is locted t ech corner. Using the str-figures, there re 2 complete strs inside the unit cell. This is clled douly primitive or centered cell. Notice tht even though there re corner lttice points nd centered lttice points, ll hve identicl environments in the overll pttern (i.e. LL LTTICE POINTS MUST HVE THE SME ENVIRONMENT). The centered lttice point is generted y the centering trnsltionl of ( )/2. B Figure IV-3. 2-D rectngulr net showing oth P- nd C- unit cell. Screw xes re symmetry elements, which comine rottion with trnsltion. The interntionl symol used to descrie screw xes is X n, X is the order of rottion nd n is the numer of unit cells needed to S, 2 2 complete the rottion in right hnded system. 2 -screw xis is shown in Figure V-4. Note, tht it tkes two symmetry opertions to complete /2 Figure V-4. Prllel nd perpendiculr views of 2-fold screw xis. S It is possile to replce 3-fold rottion xis with either 3 or 3 2 -screw xis ( 3 3 screw xis is in relity only proper 3-fold rottion). The 3 -screw xis descries counter-clockwise helicl screw trnsltion. The 3 2 -screw xis needs little more explntion. To generte this type of symmetry element, you need to complete 3-fold proper rottion during trnsltion tht is twice the unit cell length. This is shown in Figure V-5. Strting with n oject t position S, the first symmetry opertion is 20 0 rottion coupled with 2/3 trnsltion long the symmetry xis. This results in position. This is repeted to generte position 2 which is in the second unit cell nd finlly, the screw xis is complete with third opertion t position 3. Now, rememer, unit cell is the smllest prt of the structure tht cn mke 2-fold rottion. Therefore, 2 -screw xis involves ½ trnsltion long its length. Rememer, the right hnd rule gives rise to counter-clockwise rottion. If the screw xis is long the direction, the 2 -screw opertion will move the initil point x, y, z to -x, ½ y, -z. Unit Cell #2 Unit Cell # up the entire structure y simple trnsltions. With the oject only t positions S,, 2 nd 3, the unit cells cnnot possily e complete! We need to complete the unit cells y shifting the equipoints y unit cell (up or down) to get the contents of ll the unit cells to e identicl. This genertes points, 2 nd 3. 3 2 3 2 S Figure V-5. perpendiculr view of 3 2 screw xis.
V-4 Introduction to Spce Groups S, 3 S, 3 /3 2/3 2/3 2 /3 2 Note tht the overll effect is reversl of the direction of the screw. 3 2 -screw xis descries clockwise rottion! This cn e etter seen in the symmetry digrms provided in Figure V-6. Notice, the 3 nd 3 2 -screw xes re enntiomorphous with respect to ech other. The 4-fold proper rottion cn e replced with either 4, 4 2 or 4 3 screw xis nd the 6-fold proper rottion cn e replced with 6, 6 2, 6 3, 6 4 or 6 5 screw xes. Notice, tht there is no interconversion of hndedness in screw xis. Therefore, chirl molecules cn crystllize with these types of symmetry elements. Glide plnes re symmetry elements tht comine mirror plnes nd trnsltions. These symmetry elements cn replce mirror plnes in 3-dimensionl, One unit cell Figure V-7. Representtion of glide plne (/2 trnsltion followed y reflection). 3 3 2 Figure V-6. Symmetry digrms of the 3 nd 3 2 screw xes which, show the direction of rottion. spce groups. Figure V-7 shows simple glide plne. The motif is trnslted y ½ unit cell nd then is reflected cross the glide plne. The, represents the motif s hving the opposite chirlity. Chirl molecules cnnot crystllize with glide symmetry elements. The five types of glide plnes re collected in Tle V-. The n-glide is cross fce digonl, while the d-glide (clled dimond glide) is rre type of glide plne tht is only found in fce or ody centered unit cells. Tle V-. The five types of glide plces. Glide type c n d description trnslte /2 nd reflect cross the plne. trnslte /2 nd reflect cross the plne. trnslte c/2 nd reflect cross the plne. trnslte ( )/2 or ( c)/2 or ( c)/2 nd reflect cross the plne. trnslte ( )/4 or ( c)/4 or ( c)/4 nd reflect cross the plne
Introduction to Spce Groups V-5 C. Nomenclture nd symols used for spce groups The Hermnn-Muguin nomenclture nd symols used to descrie the vrious spce groups ws developed in the lte nineteenth century. The first prt of the spce group nottion is the Brvis lttice type. This is lwys n itlicized cpitl letter (lower cse is used to descrie two-dimensionl plne groups). Note, tht the rhomohedrl lttice is designted R, not P, in the spce group tles. The lttice symol is followed y the point symmetry symol modified to tke into ccount dditionl trnsltionl symmetry (screw xes nd glide plnes). The point group ssocited with spce group cn e esily determined y replcing ll screw xes nd glide plnes with rottion xes nd mirror plnes, respectively. For exmple, the spce group P2 /c elongs to the point group 2/m (the 2 xis is replced with 2-fold xis nd the c-glide is replced with mirror plne). The symol P2 /c designtes monoclinic - P Brvis lttice with 2 screw xis long nd perpendiculr c-glide. Spce group Cmc elongs to the orthorhomic-c lttice (point symmetry mmm, which is short nottion for 2/m 2/m 2/m) nd hs mirror plne perpendiculr to, c-glide perpendiculr to nd n -glide perpendiculr to c (the long symol for this spce group would e C 2/m 2/c 2/). Tle V-2 shows which symmetry xes re represented y the sequence of symols in the spce group nottion. Tle V-2. The symmetry directions involved with the Hermnn-Mnguin symmetry nottion for the seven crystl systems. Crystl System symmetry symmetry symmetry direction direction direction Triclinic -fold - - Monoclinic -xis - - Orthorhomic -xis -xis c-xis Tetrgonl c-xis -xis [0] xis Cuic c-xis [] xis [0] xis Trigonl c-xis -xis [20] xis* (long cell xes)* Hexgonl c-xis -xis [20] xis * Note, for the trigonl-p lttice there re two permissile orienttions for 2-fold rottions. The 2-fold rottion xes my e long the unit cell edges (P32) or they my e long the unit cell edge digonls (P32) s shown in Figure V-8.
V-6 Introduction to Spce Groups P32 P32 Figure V-8. The orienttion of 2-fold xes in the spce groups P32 nd P32. Note, only the symmetry elements needed to descrie the differences re present. In the hexgonl spce group P6cc, the P- identifies the Brvis lttice, the 6 represents 6-fold proper rottion long the c-xis, the first c represents c-glide norml to the -xis nd the second c represents c-glide norml to the [20] direction. The cuic symol Pm3m, represents the Brvis lttice followed y symols descriing the symmetry ssocited with the c-xis, the unit cell ody digonl nd the unit cell fce digonl, respectively. The trigonl spce groups cn e represented s either rhomohedrl or hexgonl lttice. There re only seven unique wys to plce rrngements with 3-fold symmetry into primitive rhomohedrl lttice, nd thus only seven spce groups sed on primitive rhomohedrl lttice. These re customrily designted with the initil letter R in plce of the Brvis lttice symol. The other eighteen trigonl spce groups correspond to the plcing of rrngements with vrious 3-fold symmetry elements into primitive hexgonl lttice; they re now designted s P-lttice. ny structure sed on rhomohedrl cell my lso e descried s n lterntive, triply primitive hexgonl cell with the hexgonl c-xis coinciding with the ody-digonl of the primitive rhomohedrl cell. The Interntionl Tles for X-ry Crystllogrphy Vol. I (952) provides equivlent positions for ech of the seven R spce groups in oth the primitive rhomohedrl cell nd the triply primitive hexgonl cell. Structures with rhomohedrl lttice re usully descried more conveniently with hexgonl lttice. D. The Spce Groups Comintions of the trnsltionl symmetry elements with the thirty-two crystllogrphic point groups give rise to the different wys tht we cn pck mtter. If we comine the 4 Brvis lttices with the point groups we will generte 73 unique symmorphic spce groups in which the Brvis lttice is the only type of trnsltionl symmetry present. We cn lso replce mirror plnes with glide plnes nd rottion xes with screw xes to generte n dditionl 57 spce groups for totl of 230 unique wys to pck mtter.
Introduction to Spce Groups V-7 Tle V-3. The monoclinic spce groups (C2, C2 /m nd C2 /c re lterntive descriptions of C2, C2/m nd C2/c respectively). point group 2 point group m point group 2/m P Lttice P2 Pm P2/m C Lttice C2 Cm C2/m P Lttice w/ screw xes replcing rottion xes P Lttice w/ glide plnes replcing mirror plnes P Lttice w/ glides replcing mirrors nd screws replcing rottions C Lttice w/ screw xes replcing rottion xes C Lttice w/ glide plnes replcing mirror plnes. C Lttice w/ glides replcing mirrors nd screws replcing rottions P2 - P2 /m - Pc P2/c - - P2 /c C2 - C2 /m - Cc C2/c - - C2 /c In the monoclinic system, we hve three point groups 2, m nd 2/m nd two possile lttice types P nd C. These cn e comined in vrious wys s shown in Tle V-3. Comining the monoclinic Brvis lttices nd point groups forms six symmorphic spce groups P2, Pm, P2/m, C2, Cm nd C2/m. Figure V-9 is spce group digrm for C2. By convention, spce group digrm is lwys drwn so tht is downwrd, is to the right nd c is towrd the viewer. In the first setting of the monoclinic system, the 2-fold xis is long the c direction, which is consistent with the hexgonl nd tetrgonl systems. Historiclly, crystllogrphers hve preferred the second setting with the symmetry xis long the direction. - - - - Figure V-9. The spce group C2. Notice the formtion of the screw xes when point group 2 is comined with C-centering opertion. -
V-8 Introduction to Spce Groups Both settings re shown in the Interntionl Tles for X-ry Crystllogrphy. The two equipoints in the upper left-hnd corner re relted y 2-fold rottion xis long. This symmetry element is represented y doule-heded rrow. The equipoints in the center of the digrm re relted y the C-centering opertion (½ trnsltion long nd ½ trnsltion long ). Notice, the comintion of the symmetry elements C-centering nd 2-fold rottion genertes new symmetry element, 2 -screw xis t ¼. These screw xes re represented y single red rrows. If we shift the entire system y ¼, we chnge the spce group from C2 to spce group C2. Therefore, spce groups C2 nd C2 re relly only lterntive descriptions of the sme spce group nd re relted y shift of ¼ long. Replcing mirror plnes nd rottion xes with glide plnes nd screw xes, respectively will generte nother 0 possile choices, ut C2, C2 /m nd C2 /c re redundnt. comintion of 2-fold rottion with C- centering opertion will generte 2-fold screw xes nd comintion of mirror plne with C-centering opertion genertes glide plnes. Therefore, C2, C2 /m nd C2 /c re shown in the spce group tles s lterntive settings of C2, C2/m nd C2/c, respectively. This gives totl of 3 different monoclinic spce groups. E. Deriving the spce group P2 /c P2 /c is very specil spce group. Over 30 % of ll known crystl structures dopt this spce group! Spce group P2 /c is sed upon the point group 2/m with the 2-fold rottion xis replced with 2-fold screw xis (y convention, the xis is the xis in the monoclinic system). nd the ½ -, - ½, ½, 2 3 4 Figure V-0. Deriving P2 /c, step. mirror plne replced with glide plne. We strt the derivtion y drwing the equivlent points (equipoints) ssocited with the symol 2 /c. First dd the c-glide perpendiculr to the screw xis. If we strt with point t position x, y, z in the unit cell, the c-glide will trnslte ½ long c nd reflect cross the c plne. This genertes point 2 t position x, -y, ½ z with the opposite chirlity. The symol for c-glide is dotted line. Next, use the screw opertion to generte points 3-4. Notice, the symol for the screw xis is two singly-red rrows. This is shown in Figure V-0. Relizing tht ech unit cell must hve the exct sme contents, we must dd numer of equipoints nd symmetry elements (Figure V-). If you py close ttention, you will note tht this genertes numer of other symmetry elements in the unit cell. First, new screw xis is generted t /2, y, 0 nd new c-glide plne is generted t x, ½, z. lso, series of inversion centers re generted t ¼ z (smll open circles, see Figure V-B). Rememer, nything tht occurs t ¼ z lso occurs t ¾ z! Next, we wish to shift the origin of the unit cell to one of the inversion centers. This llows us to mke etter use of the symmetry in the spce
Introduction to Spce Groups V-9 ½ -, - ½, ½, ½ -, - ½, ½, ½ -, - ½, ½, ½ -, - ½, ½, B Figure V-. The dditionl equipoints nd symmetry elements needed to complete the unit cells nd the extr symmetry elements generted. group. In order to do this, remove the equipoints (Figure V-2). Then shift ll the symmetry elements down y ½ c. This plces the inversion centers t 0 z nd the screw xes t z. Rememer, in order to keep our unit cells consistent, - z is equivlent to ¾ z, which in turn is equivlent to ¼ z! Then we shift the symmetry elements down y ¼. This will generte Figure V-2B. Next regenerte the equipoints with the symmetry elements. This results in Figure V-2C, which is the exct sme digrm presented in the Interntionl Tles for X-ry Crystllogrphy volume I. -, ½, ½ - -, B -, ½ ½ - -,, C Figure V-2. The derivtion of P2 /c.
V-0 Introduction to Spce Groups The symmetry tle for P2 /c is shown in Tle V-4. This contins tremendous mount of useful informtion, which includes the multiplicity (numer of positions), the Wyckoff nottion, point symmetry nd the numer of equivlent positions. Numer of positions The numer of positions (or multiplicity) is the numer of equivlent positions inside the unit cell. If the motif is n tom nd the tom resides t site with multiplicity of four, then there MUST e four equivlent toms present in the unit cell. The Wyckoff nottion is useful wy to reference the vrious symmetry positions in the unit cell. The point symmetry is the symmetry of the equivlent positions. There re two types of equivlent positions, the generl positions, which re sites of -fold point symmetry nd the specil positions, which re sites with greter thn -fold symmetry. In P2 /c, the Wyckoff sites e re the generl positions. These sites hve multiplicity of four. This mens tht the unit cell must contin t lest four of ny feture (i.e. n tom), which resides in one of the generl positions. For every tom t generl position x, y, z there must MUST e three other equivlent toms t positions -x, -y, -z; -x, ½ y, ½ - z nd x, ½ -y, ½ z. If n tom resides t specil position 0, 0, ½ (multiplicity of two), then the unit cell MUST contin two equivlent toms t positions 0, 0, ½ nd 0, ½, 0. Both of these positions re locted t inversion centers. The equivlent positions re generted y symmetry inside the unit cell. For exmple, in the generl condition the point x, -y, -z is relted to point x, y, z y the inversion center locted t 0, 0, 0. The point x, ½ y, ½ z is relted to x, y, z y the 2-fold screw xis locted t 0, y, ¼ nd the point x, ½ -y, ½ z is relted to x, y, z y the c-glide plne locted t x, ¼, z. F. Biliogrphy Tle V-4. Symmetry tle for spce group P2 /c. Wyckoff nottion Point symmetry Equivlent positions (equipoints) 4 e x, y, z; -x, -y, -z; -x, ½ y, ½ - z; x, ½ -y, ½ z 2 d ½, 0, ½; ½, ½, 0 2 c 0, 0, ½; 0, ½, 0 2 ½. 0, 0; ½, ½, ½ 2 0,0,0; 0, ½, ½ ) Interntionl Tles for X-ry Crystllogrphy, Volume I (952), edited y N.F.M. Henry nd K. Lonsdle, The Kynoch Press: Birminghm. 2) Buerger, M.J. (954). Elementry Crystllogrphy, John Wiley &Sons: New York.