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Principles_Symmetry_PointGroups

Principles_Symmetry_PointGroups - PrincipIes of Symmetry...

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Unformatted text preview: PrincipIes of Symmetry K: [Fl REMENB 15-01111 e Ce 101011011 01 the 1% 13121111113 Lattices we have used. tenetiti 011 Operations C'ttlleeI x 1 11111110110215; namely the lattice trtmsluhons: the 'C' eated opei 1011011 01' three 011—0011100111 lattice 11‘1111511111011s 011 111101111 gave 11se '10 the space lattice. Othe1 1epetit1 1011 1J 1111511111011; other 1111111 lattice translations, are 1117112110115. reflections. inversions. for €31<111 1111 11e These are syiiimetty00e1‘z1tio11s. By 11131110111011 011:1 eses symmetry 0pe1‘at1ons an 01311321 is b1011ght 11110 coincidence 11:11111tse11' 0: 'L11‘01 11011 111301111111 0.111s (the 1011111011 11x15 such as 11101111115 11111115, triads. tetmtls 211111 hexads) 01 1etlec‘i 1011s in :1 plane (:eg. 13111101 planes) or lTlVEl‘SlOTl 111 1 p011 11C ((111 lflVETSlQH CBUECI} Definifignj LQCUS 0'1 11311111111311“; oueiutions is the 01111- line 01‘ lane let't LmCllLU'l‘TSC ' e _ _ 1 P p a 5/1: L 01'111e 53301111131111 Operation. This LOCUS is tet'erred 10 11s the symmettv element. app 11111111011 eg. some objects 1* l 1 , T- 7 1 ' t A. ‘1 .1 x... I‘ 1 1:41;. 1\ ‘1 . \n\“l‘!" \ i x ‘ . "1.1 — ~ .1 ~ ~ -_ 1A _‘ -4 - 1 UL 1116:: Cu 11111111113 nhu 1» 1 (JiJ JJt’LL: 8116 £11816 1 1:151} b 1 EL 1118:: V e1e1-11e11ts.s11tl1 15 1011111011 11xes.1‘e11ce 1111111311111es inversion 11111115 111 centms. 11s1111e 11:1 _\ 0111‘ \ oung and keen e} es ? NOTE: Point S mate.“ O 1e1at10n5‘ 1'1'11‘1112111ble 1,1,51th the 148111111115 Lattices are 1 Jv_ V 1. rotations. rettecti 1115 1:1,.11r1'0r planes), inversions (and combinations the 'enf. There al5<;1 are screw axes L 1nd gl1de Jpe1'5a‘tion ). .4 The 03511111 331151121115 and con'esponchng B1 1112115 Lattiee5 we most 511ccinctl 1,1 defined by the11 .5 5.1: l 1-'e;1qz111w111121115 1"1tl'1e1 than the Lt1n1L1ntrt11‘1 15 L), " l n 1m — (:11 a '\ 5 .2: lk. 'JLLl 411151511 LL. (I) .1513. r' ,. 1-1- FD 11-111'1111'111111 5'311'11111 @113? (1131.11 7 :11" } SETS 01: SYMRETRY ELENLENTS WETL H ARE VlUTU ALL‘: C ONSTSTENT ARE CALLED GROUPS. } Those groups of symmetry operations that leave at least one point unchanged. unmoved are calletl 1.101111 3111111175: } There are @1an _‘ " f3 101113511 at me comp LlllD le WI ith the t1a115 l Lt1 1Qnal perttlieitv in three dimensi 0115:. r 14 Brav £115 Lattices :De5eiibe all the p13551ble emmbi 1111110115 01 tran5latinnal ope1ator5. that. applied l‘Ep'titttedl}-'. will fill 3~D space with a set 01‘ points that have identical san'otmtlings. i.e. a spaee lattice ’r 3': POlUC Group 5‘: Point groups are matte an from point 5‘V'm1net1'y elements. that 15 point opei ratinn5 and. e13111b1nati110115‘ there 111 Tl1e1e ate 0 11l_\1 . - point groups. that are x 11l 1l1e 91111111651le L‘lelinetl b\' th 1: e1‘_\.-‘5‘t;1l5‘V'Stemsa11tl 1 B1.“ 1115 lattices. -5 11' ' -»~1'.;. 11' , » 1.71"; 1.11.. meme“ -.t -15151_.::e5 31;..5. S Yb’lt‘leTRY CONSIDERATIONS POINT GROUPS AND SPACE GROUPS Question: What is a group '? ...at least as 1311‘ as the eonte :{t of c1}sta11og1'ap1 151. 1 e Po111t tant1 Space 10111 s ate concemetl Answer: lt is an assemblv of all the s:- 111111net1'j operations and their combinations that me consistent 111th a crystal 01 C ij/stal lattice. Consider the set G which consists of Stine syfirneti 111' opetat- 113 U1, 1,1 Jere index 1 runs from 1 to infinity. for example. G is a g1oup if the 10110W1ng 1111133 apply to its elements: 1. The product of two operators or multiplicative operation of two element" “1 03' is 11:10 an element of the set G. ‘JL: For all Oi. Oj in G it is true that OR is also element of G if 01 . Oj : 0k "‘1 T113 m'xIV1f'1r11; 7fif i nnm 1mm 1 iq ace 3d 3_-ri\_r_\ LLALL 1.1:- L.\_‘—._yL r y:\.a-t-.vt Oi. (‘Oj .Ok1:(fOi.Oj‘1.0lt 3.11‘1e1‘e exists a unit element. or unity. e e.Oi=Oi "‘ Each eleinen has its 111\-’C‘31‘5€ element: ‘1'. 1" 1_O1:‘1:1_’01\:.(".. :- » w .,_ . ~_ '- ‘, _, ' 1— » 1111\3‘13‘5‘1 O1» 101 (1-. “.1311 \111:L\_:1111e: 1~—r1131. s‘:" I ’ 1 . I, —‘ 1. 1. ‘1 s .1 r fir ~.- » 1‘1 1‘ t CHL [11‘ 11. .1 1o: 141 ‘1.» 1: t ;‘ 1 ‘ PonrrGRoors A) proper rotation axes, cyclic point groups: 2 3 4 6 B) dihedral point groups - combination of proper rotations with two—fold rotations: 222 32 422 522 V— ? X C) rotoinversion axes: ~11 or ~X point groups - combinations of proper rotations with inversion centers: D) The n/m point groups — combination proper rotation with perpendicular reflection elements: - - g z ‘- 9 21m 4/ m 6/ m E) The nm point groups — combination of proper rotations with reflections: .4 6mm 2mm o ‘5 m F) The wan-m point groups — combining rotoinversions with coinciding reflections: 3m G) The 111111 In point groups - combining proper rotations with coinciding perpendicular reflections mmm 4/ mm lmmm nun!!! ”an!!!” I- U) H) The n1n2 point groups - combining proper rotations 23 432 EEmE NEE E Em m: m 325 E E..: EcE . E 51.11;; EC: CF: «I Em . :E _ [I r5922 11 V . . E . x c t . . m 952 E; E: :5: EE E c mEm Em . EEE E m Emw mE EEw :Em , NEE E 4 . m wa‘ 1 A Q A 1. o: 5. «NE cw ANN . N .EEEEEEEEEEEE ..E.E.EEEEEE EEEEEEEEEEEEEEOEEEEEEEEEo>EEEEEEEEEfEE£EEEE1.EBEEEEEEEEcmE.EEEEEEEoEEEEmmanEEE: EEESEEEEEEEEE: E magi. MEG—.EEKE E<ZCE22>§EE .I liinlr’lE ECEMEEEEQ E3 ExEmElEEEEcE E: 2 EEEEEEEEEcu EEEEEE muEEEE fl EcEEEEEm 2:. E . EE .rEEpE .EEEEQEE _ UEEE :33 HEA CONE HA com HE Geo HA _ :co NR NEE”: scans”: u n: ”and aeonEnc Egg: .E.EE.E..E.E.EE.EEE u H EE H 2 .E 3 EE H EE . u R E H z E a» Q «E a E w «E v u w E «E m ..E<ZCE.E.E.EE.E>ICE,E C5152: EEEE >~E5 EEZEQ EEZEEZEEE EEE EFEEMErm E< EESELE; 2:2,: v 938; E EEEEE E EEEEEEE E . mEEEEEEE m \SEZOOIEEEEEE. UEZCEEOEEdO EEEEEEE _EEE:cE E<ZOC<XE=E EE/EEZOGEME. UEZZUCZCZ UEZ_EU_:.E. EC:<~ECCAEE<EmE>~EmE 1E <EEZHE<CZDEE Problems for exercise by individual students l. For each of the ohiects sketched below mark the location of the rotational svmmetrv axes. etrads 1md hexads. Pay attention to l‘nllow the inte nationallv accented v " L' g.“ r _ .4 A r—r such as dlads. triads, conventions for nomenclature. a) a hexagonal prism b) a tetrahedron of equilateral triangles b) :1 cube _...M.*..- . w w— m.~ d) a regular octahedron _‘.h J 4 am many square base pynmlids me [has in the regular ocmhedmn of figure d) shown L. ‘ , n AJO\C; Consider again the hexagonal Dnsm and the cube as shown below. and mark in the g ‘— L drawings {he surface traces of at 1 the mirror planes. LEM LHC LLC::[H{1,1.LL1.LI;LLLLL.>LLL£H L. L31u\_il§ HILLLCE. .«.~L\XY 1:.» V~>\>1Y '1 LHLL—rLLIILpLLHL heir unitce .3 -._-. c, 5) On the next page. six primitive crystal structures or elements are i- Fresco r~ 3 r—r ( 3 n ,._- G“ / (L1) to H), a) Identify the orthorhombic and the tetrugonul crystal structures and describe their respective motif. b) For each of the unit cells mark the direction [ll 1] and the plane (l 10,). c) Indicate t Use directional indices forrotation axesl Miller indices for mirror planes and fractional coordinates for points of inversion. Note that a mirror plane ranks higher than a dl‘dd. he location of the highest order point symmetry element for each c‘ysttil structure. (/7 “ .7 ~—/ "1 a; L ‘/‘f” b a ‘i’ ”'1"- 30 = b0 —- CO azflzy:%° %¢b0¢%:a) a: y=90° fl>90° a) C d) C {1 . \l v I Co \ 'fi‘ ‘r'wj/K‘T} 1‘ f}: /JTK‘_{ \— ‘g_j [ ¥ /.J R « L" _ r” 4 [\le / { ’ v b ‘ On 5 "\ Ga—lg o 4:» x QM {J ~f’,)\‘{ b a/Y’ <F, b 0‘ 4r %¢%¢%¢% aifliyqfia -a,[i,y;‘-90° F) C C I \4 ) l d—_.\/‘v I , \ .1 , x/v/ , 3— , -z A —‘_/ T ugly gig, a par} 9 Q _/ v be ! ...
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