R3 C34 3 Trigonal info
No. 146 R3 Patterson symmetry R-3
HEXAGONAL AXES

symmetry group diagram

Origin on 3

Asymmetric unit 0 ≤ x ≤ 2/3; 0 ≤ y ≤ 2/3; 0 ≤ z ≤ 1/3; 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, 1/3  1/2, 0, 1/3  2/31/31/3  1/32/31/3  0, 1/21/3  

Symmetry operations

For (0, 0, 0)+ set

(1)  1   (2)  3+   0, 0, z(3)  3-   0, 0, z

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

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

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
 (0, 0, 0)+  (2/31/31/3)+  (1/32/32/3)+  General:
9 b 1
(1) xyz(2) -yx - yz(3) -x + y-xz
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
3 a  3 . 
0, 0, z
no extra conditions

Symmetry of special projections

Along [001]   p3
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]   p1
a' = 1/2b   b' = 1/3c   
Origin at x1/2x, 0





R3 C34 3 Trigonal info
No. 146 R3 Patterson symmetry R-3
RHOMBOHEDRAL AXES

symmetry group diagram

Origin on 3

Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1; z ≤ min(xy)
Vertices
0, 0, 0  1, 0, 0  1, 1, 0  0, 1, 0  1, 1, 1  

Symmetry operations

(1)  1   (2)  3+   xxx(3)  3-   xxx

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
 General:
3 b 1
(1) xyz(2) zxy(3) yzx
no conditions
    Special: as above, plus
1 a  3 . 
xxx
no extra conditions

Symmetry of special projections

Along [111]   p3
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]   p1
a' = 1/2(b - c)   b' = 1/3(a + b + c)   
Origin at 2x-x-x








































to end of page
to top of page