Table 1: The generators of the icosahedral space groups (excluding those for lattice translations) employed in Tables 2-7. [R51≡ x,w,y,z,u,v, R31≡ y,z,x,w,−u,−v, R21≡ −x,−y,−w,−v,−u,−z, I≡ −x,−y,−z,−u,−v,−w, τ0=(0,0,0,0,0,0),τ1=(1,1,1,1,1,1)/2, τ2=(3,1,1,1,1,3)/4, τ3=(1,0,0,0,0,0)/5, τ4=(0,1,4,2,3,1)/5, τ5=(1,2,0,1,0,2)/10, τ6=(7,6,2,2,2,5)/10, τ7=(9,1,1,1,1,1,1)/10, τ8=(1,9,0,3,5,2)/10,τ9=(0,1,0,0,0,3)/4,τ10=(3,0,1,1,3,2)/4. τ11=(1,3,2,3,3,2)/4,τ12=(0,0,0,0,1,1)/2] Note that the origin is not at the inversion center in Fmxy35 and Pmxy35. The glide planes {σ21|τ} in Fmxy35 and Pmxy35 are given by {I|τ2}{R21|τ0} and {I|τ1}{R21|τ0}, respectively. (σ21≡ x,y,w,v,u,z.) When the inversion is not on the origin in centrosymmetric space groups, the second setting is given where the origin is at the inversion.
Space group generators Fm35(m352) {R51|τ0},{R31|τ0},{R21|τ0},{I|τ0} Fmxy35(m352) {R51|τ0},{R31|τ0},{R21|τ0},{I|τ2} (2nd setting) {R51|τ9},{R31|τ10},{R21|τ11},{I|τ0} Im35(m352) {R51|τ0},{R31|τ0},{R21|τ0},{I|τ0} Pm35(m352) {R51|τ0},{R31|τ0},{R21|τ0},{I|τ0} Pmxy35(m352) {R51|τ0},{R31|τ0},{R21|τ0},{I|τ1} (2nd setting) {R51|τ0},{R31|τ12},{R21|τ1},{I|τ0} F235(2352) {R51|τ0},{R31|τ0},{R21|τ0} F2351(2352) {R51|τ5},{R31|τ6},{R21|τ0} I235(2352) {R51|τ0},{R31|τ0},{R21|τ0} I2351(2352) {R51|τ7},{R31|τ8},{R21|τ0} P235(2352) {R51|τ0},{R31|τ0},{R21|τ0} P2351(2352) {R51|τ3},{R31|τ4},{R21|τ0}
Table 2: The Wyckoff positions of the space groups Fm35 and Fmxy35. The first column represents Wyckoff symbol (W.S.). The second and third columns denote the site symmetry and the representative coordinates, respectively. In the Wyckoff positions in the second setting in Fmxy35(m35), −τ2/2 shown in Table 1 should be added to the coordinates.
Fm35(m352) W.S. site symmetry coordinates 32· 1a m35(m352) (0,0,0,0,0,0) 32· 1b m35(m352) (1,0,0,0,0,0)/2 32· 1c m35(m352) (1,1,1,1,1,1)/4 32· 1d m35(m352) (3,1,1,1,1,1)/4 32· 15a mmm(mmm) (1,1,1,0,0,1)/4 32· 15b mmm(mmm) (2,0,1,1,1,1)/4 32· 15c mmm(mmm) (2,0,1,0,0,1)/4 32· 15d mmm(mmm) (1,1,0,0,0,2)/4 32· 12a 5m(52m) (x,y,y,y,y,y) 32· 20a 3m(3m) (x,x,y,−y,y,x) 32· 30a 2mm(2mm) (x,x,y,0,0,y) 32· 30b 2mm(2mm) (1/4+x,1/4+x,y,1/4,1/4,1/2+y) 32· 30c 2mm(2mm) (1/4+x,1/4+x,y,0,0,1/2+y) 32· 30d 2mm(2mm) (1/2+x,x,y,1/4,1/4,y) 32· 60a m(m) (x+y+z+u,x,y,u,0,z) 32· 120a 1(1) (x,y,z,u,v,w) Fmxy35(m352) W.S. site symmetry coordinates 32· 2a 235(235) (0,0,0,0,0,0) 32· 2b 235(235) (1,1,0,0,0,1)/2 32· 12a 5(52) (5,1,1,1,1,5)/8 32· 12b 5(52) (3,3,3,3,3,7)/8 32· 20a 3(3) (7,3,1,3,1,7)/8 32· 20b 3(3) (1,5,3,1,3,1)/8 32· 30a 222(222) (1,1,0,1,1,2)/4 32· 30b 222(222) (1,1,0,0,0,2)/4 32· 24a 5(52) (x,y,y,y,y,y) 32· 40a 3(3) (x,x,y,−y,y,x) 32· 60a 2(2) (0,0,x,−y,y,−x) 32· 60b 2(2) (1/4,1/4,x,−y,y,−x) 32· 120a 1(1) (x,y,z,u,v,w)
Table 3: The Wyckoff positions of the space group Im35(m352).
Im35(m352) W.S. site symmetry coordinates 2· 1a m35(m352) (0,0,0,0,0,0) 2· 6a 5m(52m) (1,0,0,0,0,0)/2 2· 6b 5m(52m) (1,1,1,1,1,1)/4 2· 6c 5m(52m) (3,1,1,1,1,1)/4 2· 10a 3m(3m) (1,1,1,0,0,0)/2 2· 10b 3m(3m) (1,1,1,3,1,3)/4 2· 10c 3m(3m) (1,1,1,1,3,1)/4 2· 12a 5m(52m) (x,y,y,y,y,y) 2· 15a mmm(mmm) (1,1,0,0,0,0)/2 2· 20a 3m(3m) (x,x,x,y,y,y) 2· 30a 222(222) (1,3,0,3,3,2)/4 2· 30b 222(222) (2,0,3,3,1,1)/4 2· 30c 2mm(2mm) (x,x,y,0,0,y) 2· 60a m(m) (x,x,0,y,y,0) 2· 120a 1(1) (x,y,z,u,v,w)
Table 4: The Wyckoff positions of the space groups Pm35(m352), and Pmxy35(m352). In the Wyckoff positions for the second setting in Pmxy35(m35), −τ1/2 in Table1 should be added to the coordinates.
Pm35(m352) W.S. site symmetry coordinates 1a m35(m352) (0,0,0,0,0,0) 1b m35(m352) (1,1,1,1,1,1)/2 6a 5m(52m) (0,1,1,1,1,1)/2 6b 5m(52m) (1,0,0,0,0,0)/2 10a 3m(3m) (1,1,0,0,0,1)/2 10b 3m (0,0,1,1,1,0)/2 12a 5m(52m) (x,y,y,y,y,y) 15a mmm(mmm) (0,0,1,0,0,1)/2 15b mmm(mmm) (0,0,1,1,1,1)/2 20a 3m(3m) (x,x,y,y,y,x) 30a 2mm(2mm) (x,x,y,0,0,y) 30b 2mm(2mm) (x,x,y,1/2,1/2,y) 60a 2(2) (0,1/2,x,y,−y,−x) 60b m(m) (x,y,z,u,u,z) 120a 1(1) (x,y,z,u,v,w) Pmxy35(m352) WS site symmetry coordinates 2a 235(2352) (0,0,0,0,0,0) 12a 52(522) (0,1,1,1,1,1)/2 12b 5(52) (3,3,3,3,3,3)/4 12c 5(52) (3,1,1,1,1,1)/4 20a 32(32) (1,1,0,0,0,1)/2 20b 3(3) (3,3,3,1,3,3)/4 20c 3(3) (1,1,3,1,3,1)/4 30a 222(222) (0,0,1,0,0,1)/2 24a 5(52) (x,y,y,y,y,y) 40a 3(3) (x,x,y,−y,y,−x) 60a 2(2) (0,0,x,y,−y,−x) 60b 2(2) (0,1/2,x,y,−y,−x) 120a 1(1) (x,y,z,u,v,w)
Table 5: The Wyckoff positions of the space groups F235(2352) and F2351(2352).
F235 W.S. site symmetry coordinates 32· 1a 235(2352) (0,0,0,0,0,0) 32· 1b 235(2352) (1,1,1,1,1,1)/4 32· 1c 235(2352) (3,1,1,1,1,1)/4 32· 6a 52(522) (1,0,0,0,0,0)/2 32· 12a 5(52) (x,y,y,y,y,y) 32· 15a 222(222) (0,0,0,1,1,2)/4 32· 15b 222(222) (1,1,0,1,1,0)/4 32· 15c 222(222) (1,1,0,0,0,0)/4 32· 15d 222(222) (1,1,0,1,1,2)/4 32· 20a 3(3) (x,x,y,−y,y,x) 32· 30a 2(2) (0,0,x,−y,y,−x) 32· 30b 2(2) (1/4,1/4,x,−y,+y,1/2−x) 32· 30c 2(2) (1/4,1/4,x,1/4−y,1/4+y,1/2−x) 32· 30d 2(2) (1/2,0,x,1/4−y,1/4+y,1/2−x) 32· 60a 1(1) (x,y,z,u,v,w) F2351(2352) W.S. site symmetry coordinates 32· 5a 23(23) (15,5,5,3,7,15)/20 32· 5b 23(23) (5,5,0,4,1,5)/10 32· 5c 23(23) (0,5,0,4,1,5)/10 32· 5d 23(23) (15,5,5,3,7,15)/20 32· 10a 32(32) (8,2,0,3,0,5)/10 32· 10b 32(32) (11,9,5,1,5,15)/20 32· 10c 32(32) (1,9,5,1,5,15)/20 32· 10d 32(32) (8,2,0,3,0,5)/10 32· 15a 222(222) (11,9,5,1,5,15)/20 32· 15b 222(222) (1,9,5,1,5,15)/20 32· 15c 222(222) (5,15,5,8,2,15)/20 32· 15d 222(222) (0,10,5,8,2,15)/20 32· 20a 3(3) (3/5+x,x,y,3/10−y,y,3/10+x) 32· 30a 2(2) (0,0,x,−y,y,−x) 32· 30b 2(2) (1/4,1/4,x,1/4−y,1/4+y,1/2−x) 32· 30c 2(2) (1/4,1/4,x,−y,+y,1/2−x) 32· 60a 1(1) (x,y,z,u,v,w)
Table 6: The Wyckoff positions of the space group I235(2352) and I2351(2352). (In the table the symbols ri=i/20, (i=1,5,18,19) are used.)
I235(2352) W.S. site symmetry coordinates 2· 1a 235(2352) (0,0,0,0,0,0) 2· 6a 52(522) (3,1,1,1,1,1)/4 2· 6b 52(522) (1,0,0,0,0,0)/2 2· 6c 52(522) (1,1,1,1,1,1)/4 2· 10a 32(32) (1,1,3,1,3,1)/4 2· 10b 32(32) (0,0,1,1,1,0)/2 2· 10c 32(32) (1,1,1,3,1,1)/4 2· 12a 5(52) (x,y,y,y,y,y) 2· 15a 222(222) (1,3,0,3,3,2)/4 2· 15b 222(222) (0,0,1,0,0,1)/2 2· 15c 222(222) (3,1,0,1,1,2)/4 2· 15d 222(222) (2,0,3,3,1,1)/4 2· 15e 222(222) (3,1,0,3,3,2)/4 2· 20a 3(3) (x,x,x,y,y,y) 2· 30a 2(2) (0,0,x,y,−y,−x) 2· 30b 2(2) (0,1/2,x,y,−y,−x) 2· 30c 2(2) (3/4,1/4,1/4+x,1/4+y,1/4−y,1/4−x) 2· 30d 2(2) (1/4,1/4,3/4+x,3/4+y,3/4−y,3/4−x) 2· 60a 1(1) (x,y,z,u,v,w) I2351(2352) W.S. site symmetry coordinates 2· 5a 23(23) (5,5,19,19,11,11)/20 2· 10a 32(32) (19,1,19,11,3,1)/20 2· 10b 32(32) (2,3,7,8,9,3)/10 2· 10c 32(32) (19,1,9,1,13,1)/20 2· 10d 32(32) (2,3,2,3,4,3)/10 2· 15a 222(222) (0,0,19,4,16,1)/20 2· 15b 222(222) (0,0,9,4,16,11)/20 2· 15c 222(222) (15,5,19,9,1,11)/20 2· 15d 222(222) (5,5,4,4,6,6)/20 2· 15e 222(222) (0,0,19,14,6,1)/20 2· 20a 3(3) (r19+x,r1+x,r1+y,r9−y,r5+y,r1+x) 2· 30a 2(2) (0,0,x,y,−y,−x) 2· 30b 2(2) (0,1/2,x,y,−y,−x) 2· 30c 2(2) (3/4,1/4,1/4+x,1/4+y,1/4−y,1/4−x) 2· 30d 2(2) (1/4,1/4,3/4+x,3/4+y,3/4−y,3/4−x) 2· 60a 1(1) (x,y,z,u,v,w)
Table 7: The Wyckoff positions of the space groups P235(2352), and P2351(2352).
P235(2352) W.S. site symmetry coordinates 1a 235(2352) (0,0,0,0,0,0) 1b 235(2352) (1,1,1,1,1,1)/2 6a 52(522) (0,1,1,1,1,1)/2 6b 52(522) (1,0,0,0,0,0)/2 10a 32(32) (1,1,0,0,0,1)/2 10b 32(32) (0,0,1,1,1,0)/2 12a 5(52) (x,y,y,y,y,y) 15a 222(222) (0,0,1,0,0,1)/2 15b 222(222) (0,0,1,1,1,1)/2 20a 3(3) (x,x,y,−y,y,x) 30a 2(2) (0,0,x,y,−y,−x) 30b 2(2) (0,1/2,x,y,−y,−x) 30c 2(2) (1/2,1/2,x,y,−y,−x) 60a 1(1) (x,y,z,u,v,w) P2351(2352) WS site symmetry coordinates 5a 23(23) (5,5,9,7,3,1)/10 5b 23(23) (0,0,2,1,4,3)/5 10a 32(32) (6,4,9,5,1,6)/10 10b 32(32) (1,9,9,5,1,1)/10 10c 32(32) (3,2,2,0,3,3)/5 10d 32(32) (1,9,4,0,6,1)/10 15a 222(222) (0,0,9,2,8,1)/10 15b 222(222) (0,0,9,7,3,1)/10 20a 3(3) (x,4/5+x,y,2/5−y,1/5+y,x) 30a 2(2) (0,0,x,y,−y,−x) 30b 2(2) (0,1/2,x,y,−y,−x) 30c 2(2) (1/2,1/2,x,y,−y,−x) 60a 1(1) (x,y,z,u,v,w)
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