Research & Reviews: Journal of Pure and Applied Physics

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1 Reserch & Reviews: Journl of Pure nd Applied Physics Choice of Origin, Loction of Group nd Uniqueness of Hermnn-Muguin Symols Vojtech Kopsky* Freelnce Reserch Scientist, Bjklsk 1170/8, Prh 10, Cech Repulic. Short Communiction Received dte: 8/10/015 Accepted dte: 8/03/016 Pulished dte: 30/03/016 *For Correspondence Vojtech Kopsky, Freelnce Reserch Scientist, Bjklsk 1170/8, Prh 10, Cech Repulic. E-mil: Keywords: Symmetry; Crystllogrphers; Trnsltion normlier; Eucliden spce. ABSTRACT It is shown tht widely ccepted opinion which sys tht Eucliden groups cnnot e locted in spce ecuse they re strct groups, is incorrect. Eucliden groups re quite certin groups of opertors in spce nd s the symmetry opertions hve its loction di ering y frctions of the trnsltion norml-ier of the group the group itself hs lso certin loction. We show, however tht derivtion of spce groups with the use of fctor systems indeed leds to groups which re not locted while derivtion with the use of systems of non-primitive trnsltions leds to set of groups di ering in loction y frction of their Eucliden normlier. Some exmples of possile use re given. FOREWORD At the time when the rest volume of the lst ut rest edition of the Interntionl Tles for Crystllogrphy (the one with lue cover) ws pulished, the uthor hs een sked to write review on it. On specil rrngement with Mthemticl reviews, he wrote some two pges nd eing enthusistic out the wide rnge of knowledge presented, he gve the tle nicknme "The Bile of Crystllogrphers" [1]. On the 16th Interntionl Congress in Beijing (1995) the uthor hd vigorous discussion with the editor of Vol. A, concerning the concept of group loction in connection with just prepred Vol E Su periodic groups []. It ppered tht mny crystllogrphers elieve tht spce nd generlly Eucliden groups cnnot e ssigned loction ecuse they re strct groups. To emphsie the con ict, the uthor presented n nlysis of group loction t Prgue Congress of IUCr [3] ut without response, though he used rther provoctive chnge of the nicknme to "The Old Testment". Recently, the uthor found on Wikipedi, under the title The Bile of Crystllogrphers, the reference to ook: Symmetry in crystllogrphy (Understnding the Interntionl Tles) [] y Polo G. Rdelli, Inter-ntionl Union of Crystllogrphy, Oxford University Press, 011. Neither this ook nor IT A contin even hint of group loction. Though it does not diminish the vlue of these pulictions, especilly for prctising crystllogrphers, we still elieve, tht the simple theory of loction should e included into these pulictions. THE CHOICE OF ORIGIN The uthor ws, for long time, not sure, how to interpret such trivil thing s the choice of origin in the Interntionl Tles Vol. A. Well, if physicist wnts to descrie the motion of rigid ody of nite sie, he hs to de ne two coordinte systems. First he hs to specify the coordinte system in spce y choosing point P (the origin) nd the three coordinte xes which re xed. Then he needs to specify the loction of certin point P within the ody (for exmple, s usul, the centre of grvity of the ody) nd lso the coordinte xes in the ody. Then you descrie the motion of the ody y the trck of its origin P 0 nd y rottion of its xes with reference to those of the coordinte system. Eucliden groups re represented y their digrms which were so nicely clled the "symmetry pttern" y Prof. Wondrtchek in Vol. A. It is ssumed tht the coordinte xes of the ody (symmetry pttern) in IT A re prllel with those of the spce. Then you descrie the loction of the symmetry pttern y vector from the RRJPAP Volume Issue June, 016 8

2 origin P to P. In IT A the origin is specified y the sentence: Origin t (or in) point in digrm of the group; s rule it is the left upper corner. We cn ssume, though it is nowhere explicitly sid, tht this is s well the origin of the coordinte system. This ssumption is not mndtory nd the origin of the group my e on distnce s from the origin of coordinte system. This is well known in cses of groups with two origin choices, where we cn write for the group type P /nmc=p /nmc (origin 1) tht the group P /nmc(origin ) equls the group P /nmc 3+c red the group P /nmc shifted y 3+c. This group is certinly different from the group P /nmc nd hence its Hermnn-Muguin symol contins lso the shift. We cn rgue tht it is the sme s if we write (origin ) ut the current nottion cn e used for ny shift nd it shows tht the spce group (generlly ech Eucliden group) hs loction, so tht the Hermnn-Muguin symol is not unique. It is unique up to shift in the fundmentl region of the group trnsltion normlier. We hve used such symols in Vol. E of the Interntionl Tles; for exmple: for the 1 1 group No 6, where the lyer group p/nmm(origin ) is denoted p/nmm (,,0). CLASSROOM PROOF OF THE CONDITION OF LOCATION COMPATIBILITY The conditions for spce group to e sugroup of nother spce group re rther severe. However, the reltions etween groups re given for the group types which mens in terms of Hermnn-Muguin symols. The ctul groups re then deduced y dding to these symols their prmeters, i.e. orienttion of point groups, with informtion out screw xes nd glide plnes nd orienttion, Brvis types nd metric prmeters of their trnsltion sugroups. If Hermnn-Muguin symols with these dt will de ne one certin group, then the reltions etween these symols will e vlid for ll specific groups. The prolem is tht this informtion does not provide unique group ut still n in nite set of groups with different loction. In ll editions of Vol A of the Interntionl Tles mximl non-isomorphic sugroups (type I, II nd II), mximl isomorphic sugroups of lowest index (IIc) nd miniml non-isomorphic super groups (I nd II) re listed just in terms of Hermnn-Muguin symols. The loction of groups is, however, completely ignored ecuse most of crystllogrphers refuse to ccept the concept of group loction. "How cn we locte such n strct oject s group?" is question which the uthor hd to fce mny times. It would e certinly rther strnge to ccept loction for n strct group. However, spce groups in this context nd in Vol. A re fr from eing strct. They re opertors on n Eucliden spce nd shift of spce y s moves n Eucliden group G to group {e s {e -s = (s),, where the lst symol mens the group G shifted in spce y s nd this shift my tke vlues in the fundmentl region of the group trnsltion normlier. We cn de ne certin loction of the group s the stndrd, ssign the Hermnn-Muguin symol to it nd write the shift s ehind this symol. Hence there is continuous set of groups with different loctions ut only one of them is the sugroup of the originl group. We shll now explin the use of the loction on n exmple. Thus, in the left hnd side of Figure 1 we hve the digrm of the group p/nm (origin 1) while the second digrm of the sme group, Figure 1 in nother loction (origin ) on the right is denoted y p/nm or p/nm 1 1 (,,0) where origin 1. Mens the origin of the spce coordintes, while origin.. Denoted the originl group moved y. There is further the digrm of the group P in red which is the digrm of tht group of this type which is the sugroup of the group p/nm, while neither the digrm of nor P tht of P 1 my correspond to sugroup of p/nm. However, if we move the group P y, we get group P which is sugroup of the group p/nm. The criterion for eing sugroup ccording to digrms is esy - the digrm of the sugroup must overlp with the digrm of the super group; we suggest using the term 1 1 "loction comptiility". Such group s P 1 cnnot e sugroup of the group p/nm (,,0) simply ecuse its Hermnn- Muguin symol contins the glide xis 1 which is not in the group P ecuse it is not loction comptile. It follows tht neither the equi trnsltionl sugroups listed y Neuuser nd Wondrtchek [5] nor the equi-clss sugroups y Boyle nd Lwrenson [6,7], nor the numerous issues of Vol. A. of the Interntionl Tles Vol. A () give exct informtion out the sugroups. Such informtion is only prtil. The Hermnn-Muguin symols which re considered SACRED COW y crystllogrphers re lso ctully not unique: To one symol G there corresponds set of groups G (s) with s from the fundmentl region of its trnsltion normlier. The symol G (Origin ) ctully mens G (s) is the shift of Origin from Origin 1. This nottion is so fr used only in Vol E of the Interntionl Tles 00 nd further editions). The digrm of the group P is loction comptile with the digrm od the group p/nm (overlps it nd is its sugroup) (Figure 1). Neither of the digrms of groups P nd P 1 overlps with tht of the group p/nm, ut the digrm of the group P does (Figure 1). We shll now consider four digrms of spce groups. The First two figures re the digrms of the group No 15 p/nm RRJPAP Volume Issue June, 016 9

3 t origin choice 1 (1) nd t the origin (1), next is the digrm of the group No 89 P () nd the digrm of the group No 90 P 1 (3). These digrms show the symmetry elements of these groups in the unit cell t the plne, origin t the left upper corner. If we put the digrm () over the digrm (1) we see, thn ll the symmetry elements on () coincide with the elements on the digrm (1). The groups re loction comptile. The group P cnnot e therefore sugroup of p/nm, ut if we shift it y, it does. Figure 1. Continuous set of groups with different loctions ut only one of them is the sugroup of the originl group. PROOF WITH THE USE OF SYMMETRY OPERATIONS To ech group in IT A there is given list of representtive symmetry opertions. Insted of rther osolete nd clumsy nottion in IT A we list these opertors y Seit symols. Such symol hs the form {g u(g)p, where g is the element of the point group nd u ( g ) = uo ( g ) + s -gs, where u o (g) is the screw or gliding trnsltion nd P+s is the loction of the element. In the Tle elow, these symmetry opertions re listed for the group p/nm, p/nm, P nd P 1. The symmetry opertions of the group p/nm re the sme s for the group P nd hence the ltter group is sugroup of the former. For the group p/nm we first express the Seit symols with reference to the point P+. We need, however, to compre the opertions with reference to the sme point P. It is {g xp + s = {g x + s gs P,, clled the shift function, we cn red these symol either from the digrm or trnsforming those with reference to P (Tle 1). Tle 1. Compring the systems of non-primitive trnsltions p/nm {e { { { {y {x {xy {xy P {e { { { y { {x {xy {xy p/nm {e { P 1 {e { { { { { y { x {xy { xy { 1y { {1x {xy {xy Compring the systems of non-primitive trnsltions we see, tht the group P is sugroup of the group p/nm ecuse their Seit symols re identicl for ech Figure. We shll cll this property loction comptiility. Actully the groups re loction comptile, if three independent Seit symols re loclly comptile. Neither the group P nor the group P 1 re even loclly comptile with the group p/nm nd hence cnnot e its sugroups. However, the group P is loclly comptile with the group p/nm nd hence it is its sugroup. We cn find lot of exmples of wrong sugroups in Vol A itself without the Seit symols. Thus in The group P /mnm (No 138) there is listed the group Pn (No 118) s its sugroup. However, the list of symmetry opertions of this group on pge 69 contins opertions under the numers 7,8,11,1 which re listed s symmetry opertions 3,,7,8 of the "sugroup". Neither of c these coincides with the opertions of the super group - to correct it we hve to move the sugroup y s = +. Notice tht in this discussion we hve used only one principl criterion for group H eing sugroup of the group G: nmely, ll elements of the group H must e lso elements of G. RRJPAP Volume Issue June,

4 Figure. Two domins from phse trnsition to hlving equi trnsltionl sugroup. Quite generlly: If group H is sugroup of the group G, then the group H(s) is sugroup of G(s) nd vice vers. MATHEMATICAL APPROACH There re two lgeric pproches to the development nd systemtic of spce groups: 1. Theory of group extensions [8].. Theory of the sets of non-primitive trnsltions [9]. 1. Finding the extension G of the group Τ y the group G or solving the short exct sequence 0 Τ G G mens in lger tht of ending the group G, which hs norml sugroup Τ such, tht the quotient group G/Τ is isomorphic with the group G. This is typicl prolem of ending the spce group G with trnsltion sugroup nd point group G. Without going to intriccies of the cohomology theory we know the result. The pir (G; Τ) clled the rithmetic clss, where G cts s utomorphisms of the group Τ (in other words, the group G is group of opertors on T), induces set of groups G (α) which elong to this clss. Ech of these groups is defined y "fctor system" m T nd there is nite numer of different groups G (α) this property. All the groups G (α) thus defined re strct groups ecuse their derivtion uses only the strct groups. The fctor systems m (α) (g, h) Cohomology group whose order equls the numer of lgericlly different groups G (α) of the geometric clss (G, Τ).. In the next method we egin with the decomposition of the group G into cosets of its sugroup T. We use the fundmentl theorem on Eucliden groups which sys tht ech Eucliden group cn e uniquely expressed s o = { G, T, P, u ( g), Where G is the point group, T its G-invrint trnsltion sugroup, P the origin of coordinte system nd u ( α) (g) the so-clled system of non-primitive trnsltions, which stisfies so-clled Froenius congruences ( ) + ( ) ( ) T = 0 ( ) u g gu h u gh or mod T Where m g, h = u g + gu h u gh T, ( ) ( ) ( ) ( ) is the fctor system. If we dd the function t(g) + gt(h) t(gh) to given system of non-primitive trnsltions, the resulting function will e gin the system of non-primitive trnsltions leding to the sme fctor system, so tht we cn restrict the function u () (g) to vlues in the unit cell of T. Such Systems of non-primitive trnsltions re clled normlied. The proof of the theorem is rther esy. We perform the coset resolution of the group G of the trnsltion sugroup T : = T T {gg u (g ) p T {gg u (g ) p. Multipliction of elements from two cosets then gives: {g u (g)p{h u (h)p ={gh u (gh)p = {gh u (g) + gu (h) - u (gh) + tp, which leds to Froenius congruences. Systems of non-primitive trnsltions hve the property, tht the sum of two systems is gin the system of non-primitive trnsltions ecuse it lso stisfies Froenius congruence nd the shift function ϕ(g; s) is lso system of non-primitive trnsltions, ecuse ϕ(g, s) + g ϕ(h, s) ϕ(gh, s) = s gs+ gs ghs+ ghs s = 0 The shift y ms leds to the group {e s {e -s = (s) = {G,T,P + s,u (g) = G(s) = {G,T,P,u (g) + s - gs Which is the group G shifted in spce y s? The set of the systems of non-primitive trnsltions u (α) (g) + s - gs is therefore n elin group U nd the set of the of shifts RRJPAP Volume Issue June,

5 ϕ(g; s). All groups of rithmetic clss (G; T) re expressed y nd the set of shifts ϕ(g; s) is group G (ms), where distinguishes n group types nd s the vectors of the fundmentl region of the trnsltion normlier Τ N (G) which is identicl for ll groups of the rithmetic clss. These re the ctul groups which re listed in Vol A or Vol E. Let us note tht equi trnsltionl sugroups of spce group with respective shifts re given lredy in [10] nd tht ll previous dt in Vol. A re lso completed y shifts in [11]. Unfortuntely, the lst puliction though it provides very vlule informtion out reltions mong spce groups interprets the shifts s the shifts of the origins ut not of sugroups. Exmples of possile use Prcticl crystllogrphers re usully influenced y the fct, tht they re nlying single crystls. But there exists undoutedly crystl structures, which contin two or more monocrystls, like domins, twins nd we shll mke even some modest ttempt t in commensurtion [1-1]. 1. Antiphse domins. In Figure, we hve two domins from phse trnsition to hlving equi trnsltionl sugroup. There will pper two domins of different structures. As n exmple we consider the cse when in one domin toms re moved in one direction with reference to originl structure, in the other they re moved in the other directions. The domins will hve the sme symmetry G ut it will not e necessrily the sme groups - they my well differ in their loction. This will e cler if we perform the scnning for lyer groups. These my follow even in different positions in the individul domins Thn the positions, predicted y scnning theory for homogeneous crystl. The oundry etween the two domins will therefore interrupt the homogeneity, the structure corresponding to lyer groups my fde in the direction to the centre, losing some toms nd respective symmetry elements ( process which cn e nmed symmetry dilution, till, in the centre, there will remin only trivil lyer symmetry. We do not use here the term plne symmetry, ecuse it is the symmetry of two dimensionl ojects in two dimensionl spces while trivil lyer symmetry is two dimensionl symmetry in three dimensions. Holser (11) uses the term symmetry of two sided plne which is ctully n oxymoron, ecuse ll plnes in three dimensions hve two sides, nd the groups he hs in mind re now stndrdly clled the lyer groups. The trivil lyer groups re not the plne group; the difference is lso emphsied y symols nd nmes in Vol. A. s compred with Vol.E. In chpter. y Bertut in Vol. A re used the nmes for symmetry elements: rottion points nd ordinry or glide lines.. Lmellr twins. Agin we hve two components in the left nd right twins, generlly with different symmetries. The respective groups my now differ not only in their loction ut lso in their orienttion. Agin we ssume different scnning from left nd right. In the middle there will gin remin region with t symmetry of some of the trivil lyer groups (Figure ). In Figure 3, we illustrte the twin of symmetry p/nm with -xis horiontl. You my now consult Vol. E, pges 38, 85 nd 336. We consider the scnning direction (001) where the group p/nm is lso the scnning group. There re the lyer groups p/n of two orits on distnces d/ - different orits mens tht though the lyer groups re identicl, their environments re different (their loction differ y hlf of unit cell in this direction not y the cell sie. Between lyer groups we hve the symmetry p/m of trivil lyer group where the structures lso elong to two orits, depending on the lyer group in vicinity. Now we ssume tht towrds the centre, the lyer groups lose some of their symmetry elements till there remin only the elements of the trivil lyer group pm in the cnter. The distnces on which the distorted lyer groups s well s the resulting centrl group pper my e of severl d which mkes the model plusile. Figure 3. The twin of symmetry p/nm with -xis horiontl. These two hypotheticl cses my e experimentlly verified. If we cut thin plte in the cnter of this rrngement, the plte will llow properties which will not e llowed in the ulk. Theoreticlly, ll properties which re llowed y trivil lyer or mgnetic lyer groups my e exhiited. Notice gin, tht we use the term trivil lyer groups, not the plne groups. 3. This cse is highly hypotheticl. Let us ssume regulr rrngements of the units of group G. A unit with slightly higher length of unit cell in certin direction my e descried s G(d), the next y G ( d),... G (nd), up to mximum of n nd then diminish the cell ck. If n is irrtionl, we otin some structure which cn e clled "incommensurte" (Figure ); such incommensurility my pper in three independent directions. Mye this is rther nive, ut such structures relly pper in RRJPAP Volume Issue June, 016 3

6 the nture. If you trvel y plne nd your tke o is directed towrd the se, you cn oserve set of ridges on the surfce, which is, how you see the set of wves. After the plne height rises, these ridges will vnish ut nother increse in height leds to the ppernce of new ridges, which evidently correspond to the wves of igger wvelength (Figure 5). If you re lucky, the wider wves will pper while the nrrower re still recognile then you will see specil wve structure. Figure. The units of group G. A unit with slightly higher length of unit cell. RRJPAP Volume Issue June, 016 Figure 5. The wider wves will pper while the nrrower re still recognile. CONCLUSION The whole prolem with group loction is in the smll region in which the loction cn chnge. It is even frction, out one hlf, of the sie of unit cell nd hence the justifiction my pper only in very exct experiments, if it is not cry drem. The width of the centrl region my e however much igger thn d which mkes the model plusile. The uthor is neither specilied neither in twin theory nor experienced with in commensurte sttes. However, he thinks tht the loction of groups should e t lest recognied in such pulictions s the Interntionl Tles of Crystllogrphy. The systems of non-primitive trnsltions re given for clssicl groups in ook on crystllogrphy in higher dimensions [15]. ACKNOWLEDGEMENT The uthor expresses his sincere thnks to Prof. Prochk for the nice lectures of cohomology he gve us yers ck, s to Prof. Litvin nd Jnovec for discussions nd encourgement. REFERENCES 1. Interntionl Tles for Crystllogrphy. Vol. A: The Spce Group Symmetry. Editor T. Hhn. Springer. 1983: 00: Interntionl Tles for Crystllogrphy. Vol. E: Superiodic groups. Editors V. Kopsky nd D. G. Litvin. Kluwer

7 3. Kopsky V. Is revision of Vol. A of the Interntionl Tles for Crystllogrphy desirle or necessry? In Advnces in Structure Anlysis. Proceedings of the Europen Crystllogrphic Meeting, Prgue 1998: Polo G. Rdelli. Symmetry in crystllogrphy (Understnding the Interntionl Tles). Interntionl Union of Crystllogrphy, Oxford University Press. UK Neuuser J, et l. Tech ; Boyle LL nd Lwrenson JE. The ellengleichen Supergroups of the Spce Groups. Act Cryst. A. 197;8: Boyle LL nd Lwrenson JE. Klssengleichen Supergroup-Sugroup Reltionships etween the Spce Groups. Act Cryst. A. 197;8: Ascher E nd Jnner A. Algeric spects of crystllogrphy. I. Spce groups s extensions. Helv. Phys. Act. 1965;38: Ascher E nd Jnner A. Algeric spects of crystllogrphy. II. Non-primitive trnsltions in spce groups. Commun. Mth. Phys. 1968;11: Interntionl Tles for Crystllogrphy. Vol. D: Physicl properties of crystls. Editor A. Authier Kluwer Interntionl Tles for Crystllogrphy. Vol. A1: Symmetry reltions etween spce groups. Editors H. Wondrtschek nd U. Muller. Kluwer Holser WT. The Reltion of Structure to Symmetry in Twinning. Z. Kristllogr. 1958;110: Holser WT. Point Groups nd Plne Groups in Two-sided Plne nd Their Sugroups. Z. Kristllogr. 1958;110: Holser WT. Clssi ction of symmetry groups. Act Cryst. 1961;1: Brown H, et l. Crystllogrphic groups of four-dimensionl spce; Wiley nd Sons: Toronto RRJPAP Volume Issue June, 016 3