Special Cases: Si and S,(b)S(a)(1)Rotate(1)Rotate(2)Reflect(2)ReflectS, =O,C =0S2 =O,C2 =iNeither S, nor S, axis is necessary!
Special Cases: S1 and S2 S1 sh C1 s S C i 2 s h 2 • Neither S1 nor S2 axis is necessary!
StereographicProjectionsReflectionInversionWe will use stereographic projections to plot the perpendicularto a general face and its symmetry equivalences, to displaycrystal morphologyoforupperhemisphere; xforlower
Stereographic Projections o x We will use stereographic projections to plot the perpendicular to a general face and its symmetry equivalences, to display crystal morphology o for upper hemisphere; x for lower o Reflection Inversion x
S3S!COhx中S.E orS,6S3=S3S31?.S3OOS3S3S3S3SXSx3X3SOh二塑:. S3 = C3 +O hE> S, is not an independent symmetry element! (no need!
1 3 1 S3 s h C S3 S3 C3 s h S3 is not an independent symmetry element! (no need!) S3 1 S3 2 =S3 1S3 1 S3 3 S3 4 S3 5 E S3 1= sh C3 1 S3 1 S3 1 S3 1 S3 1 or S3 6 2 3 5 S3 s h C 2 3 2 S3 C S s h 3 3 1 3 4 S3 C S E 6 { 3 } =C3
Example: H,C-CHstaggered formS6S?=C!S6S= 0.C1=(iC) C =iC3S2=C3S4=C?S3=0n·C3 =iS=ES =0nCS =(iC).C? =i.C3→ S = C3 + iS6 is not independent at all
staggered form S6 Example: H3C-CH3 S6 S6 1 S6 1 S6 2 =C3 1 S6 = C3 + i * S6 is not independent at all! S6 1 = sh C6 1 = (iC6 3 )C6 1 = iC3 2 S6 2 = C3 1 S6 3 = sh C6 3 = i S6 4 = C3 2 S6 5 = sh C6 5 = (iC6 3 ) C6 2 = i C3 1 S6 6 = E
S4l = oC4;S = CxsSS43 = oC;S4 = EC S, is an independent sym. element!SSs =Cs+OhNot independent at all!Possible operations pertaining to a S, axis:S, = oC,; S5? =C, ; S53 = oC,3; S54 =C,4; S5=0S5° =Cg;S, =oC2;Ss -C.3;Ss = oC,4;S,1 = EIt demands the coexistence of a C, and a or, which readilyproduce all symmetry operations arising from S
S C S E S C S C 4 4 3 4 3 4 1 2 2 4 1 4 1 4 ; ; s s S C S C S C S C S E S C S C S C S C S 10 5 4 5 9 5 3 5 8 5 2 5 7 5 1 5 6 5 5 5 4 5 4 5 3 5 3 5 2 5 2 5 1 5 1 5 ; ; ; ; ; ; ; ; ; s s s s s x S4 3 x S4 1 S4 2 5 S • S4 is an independent sym. element! Not independent at all! Possible operations pertaining to a S5 axis: C5 s h • It demands the coexistence of a C5 and a sh , which readily produce all symmetry operations arising from S5