R3m C3v5 3m Trigonal info
No. 160 R3m Patterson symmetry R-3m
HEXAGONAL AXES

symmetry group diagram

Origin on 3 m

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

Symmetry operations

For (0, 0, 0)+ set

(1)  1   (2)  3+   0, 0, z(3)  3-   0, 0, z
(4)  m   x-xz(5)  m   x, 2xz(6)  m   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, -1/61/3)   x + 1/2-xz(5)  g(1/61/31/3)   x + 1/4, 2xz(6)  g(2/31/31/3)   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(-1/61/62/3)   x + 1/2-xz(5)  g(1/32/32/3)   x, 2xz(6)  g(1/31/62/3)   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 c 1
(1) xyz(2) -yx - yz(3) -x + y-xz
(4) -y-xz(5) -x + yyz(6) xx - yz
hkil:  -h + k + l = 3n
hki0:  -h + k = 3n
hh(-2h)l: l = 3n
h-h0l: h + l = 3n
000l: l = 3n
h-h00: h = 3n
    Special: as above, plus
9 b  . m 
x-xzx, 2xz (-2x), -xz
no extra conditions
3 a  3 m 
0, 0, z
no extra conditions

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' = 1/2(a + 2b)   b' = 1/3(-a - 2b + c)   
Origin at x, 0, 0
Along [210]   p1m1
a' = 1/2b   b' = 1/3c   
Origin at x1/2x, 0





R3m C3v5 3m Trigonal info
No. 160 R3m Patterson symmetry R-3m
RHOMBOHEDRAL AXES

symmetry group diagram

Origin on 3 m

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)  m   xyx(5)  m   xxz(6)  m   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 c 1
(1) xyz(2) zxy(3) yzx
(4) zyx(5) yxz(6) xzy
no conditions
    Special: as above, plus
3 b  . m 
xyxxxyyxx
no extra conditions
1 a  3 m 
xxx
no extra conditions

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' = c   
Origin at x-x, 0
Along [2-1-1]   p1m1
a' = 1/2(b - c)   b' = 1/3(a + b + c)   
Origin at 2x-x-x








































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