R3c C3v6 3m Trigonal info
No. 161 R3c Patterson symmetry R-3m
HEXAGONAL AXES

symmetry group diagram

Origin on 3 c

Asymmetric unit 0 ≤ x ≤ 2/3; 0 ≤ y ≤ 2/3; 0 ≤ z ≤ 1/6; x ≤ (1 + y)/2; y ≤ min(1 - x, (1 + x)/2)
Vertices
0, 0, 0  1/2, 0, 0  2/31/3, 0  1/32/3, 0  0, 1/2, 0  
0, 0, 1/6  1/2, 0, 1/6  2/31/31/6  1/32/31/6  0, 1/21/6  

Symmetry operations

For (0, 0, 0)+ set

(1)  1   (2)  3+   0, 0, z(3)  3-   0, 0, z
(4)  c   x-xz(5)  c   x, 2xz(6)  c   2xxz

For (2/31/31/3)+ set

(1)  t(2/31/31/3)   (2)  3+(0, 0, 1/3)   1/31/3z(3)  3-(0, 0, 1/3)   1/3, 0, z
(4)  g(1/6, -1/65/6)   x + 1/2-xz(5)  g(1/61/35/6)   x + 1/4, 2xz(6)  g(2/31/35/6)   2xxz

For (1/32/32/3)+ set

(1)  t(1/32/32/3)   (2)  3+(0, 0, 2/3)   0, 1/3z(3)  3-(0, 0, 2/3)   1/31/3z
(4)  g(-1/61/61/6)   x + 1/2-xz(5)  g(1/32/31/6)   x, 2xz(6)  g(1/31/61/6)   2x - 1/2xz

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(2/31/31/3); (2); (4)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
 (0, 0, 0)+  (2/31/31/3)+  (1/32/32/3)+  General:
18 b 1
(1) xyz(2) -yx - yz(3) -x + y-xz
(4) -y-xz + 1/2(5) -x + yyz + 1/2(6) xx - yz + 1/2
hkil:  -h + k + l = 3n
hki0:  -h + k = 3n
hh(-2h)l: l = 3n
h-h0l: h + l = 3nl = 2n
000l: l = 6n
h-h00: h = 3n
    Special: as above, plus
6 a  3 . 
0, 0, z 0, 0, z + 1/2
hkil: l  =  2n

Symmetry of special projections

Along [001]   p31m
a' = 1/3(2a + b)   b' = 1/3(-a + b)   
Origin at 0, 0, z
Along [100]   p1
a' = 1/6(2a + 4b + c)   b' = 1/6(-a - 2b + c)   
Origin at x, 0, 0
Along [210]   p1g1
a' = 1/2b   b' = 1/3c   
Origin at x1/2x, 0





R3c C3v6 3m Trigonal info
No. 161 R3c Patterson symmetry R-3m
RHOMBOHEDRAL AXES

symmetry group diagram

Origin on 3 c

Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1; yx; zy
Vertices
0, 0, 0  1, 0, 0  1, 1, 0  1, 1, 1  

Symmetry operations

(1)  1   (2)  3+   xxx(3)  3-   xxx
(4)  n(1/21/21/2)   xyx(5)  n(1/21/21/2)   xxz(6)  n(1/21/21/2)   xyy

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (4)

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
 General:
6 b 1
(1) xyz(2) zxy(3) yzx
(4) z + 1/2y + 1/2x + 1/2(5) y + 1/2x + 1/2z + 1/2(6) x + 1/2z + 1/2y + 1/2
hhl: l = 2n
hhh: h = 2n
    Special: as above, plus
2 a  3 . 
xxxx + 1/2x + 1/2x + 1/2
hkl: h  +  k  +  l  =  2n

Symmetry of special projections

Along [111]   p31m
a' = 1/3(2a - b - c)   b' = 1/3(-a + 2b - c)   
Origin at xxx
Along [1-10]   p1
a' = 1/2(a + b - 2c)   b' = 1/2c   
Origin at x-x, 0
Along [2-1-1]   p1g1
a' = 1/2(b - c)   b' = 1/3(a + b + c)   
Origin at 2x-x-x








































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