5 selected space group in crystal structure

These are the 5 selected space groups with detailed points.

1. 5 Space Group -Eugene gumatab

2. CONTENTS P23 P 432 F23 I23 01 02 03 04 05 F432

3. 01 P23

4. P23 Basic Info International Number 195 Crystal system Cubic Bravais Lattice Primitive (P) Point group 23 Space Group type Symmorphic a space group is considered " symmorphic " if all its generating symmetry operations, excluding lattice translations, leave at least one point fixed.

5. Symmetry Operations Design Business 3 two-fold rotations along [xyz] 8 three-fold rotations along the body diagonals such as [111], [1̅11], [11̅1], [111̅] 1 identity

6. Vertices asymmetric unit 1,0,0 0 ≤ x ≤ 1 0,1,0 0 ≤ y ≤ 1 1/2,1/2,1/2 0 ≤ z ≤ 1/2 y ≤ 1 - x z ≤ min(x, y) asymmetric unit is the smallest portion of a crystal structure that, when repeated through symmetry operations (like translation, rotation, and screw operations), generates the entire repeating unit cell.

7. 3 + Symmetry

8. X Z Y 2 2 2 2-Fold Rotation ( –x, –y, z ) (–x, y, –z) (x, –y, –z)

9. 02 P 432

10. Basic Info International Number 207 Crystal system Cubic Bravais Lattice Primitive (P) Point group 432 Space Group type Symmorphic indicate the symmetry elements: a screw axis along one direction and a 3-fold rotational axis along the diagonal. P 432

11. Vertices asymmetric unit 1,0,0 0 ≤ x ≤ 1 1/2,1/2,0 0 ≤ y ≤ 1 /2 1/2,1/2,1/2 0 ≤ z ≤ 1/2 y ≤ min(x, 1 - x) z ≤ y

12. Symmetry Operations Design Business 2 -fold rotations parallel to a 3 -fold rotations along the body diagonals such as [111], [1̅11], [11̅1], [111̅] 1 identity 4-fold rotations along the diagonal screw axis at the center of two 4-folds

13. Design Business 3-fold rotations along the body diagonals

14. Design Business 4-fold rotations 4-fold rotations parallel to a 4-fold rotations parallel to b with 1/2 away from the projections 2-fold rotations 2-fold rotations parallel to a 2 1 screw axis parallel to a between the projections

15. 03 F23

16. Basic Info International Number 19 6 Crystal system Cubic Bravais Lattice Face-centered Point group 23 Space Group type Symmorphic F23 Even though F23 is symmorphic (strictly speaking), fractional translations arise from centering and symmetry operations that reproduce the full lattice by moving atoms by fractions of the lattice constants.

17. Vertices asymmetric unit 1/2,0,0 0 ≤ x ≤ 1 /2 1/2,1/2,0 0 ≤ y ≤ 1 /2 1/4,1/4,1/4 -1/4 ≤ z ≤ 1/ 4 1/4, 1/4, -1/4 y ≤ x max(x - 1/2, -y) ≤ z ≤ min(1/2 - x, y)

18. Symmetry Operations Design Business 2 -fold rotations along the projections 3 -fold rotations along the body diagonals such as [111], [1̅11], [11̅1], [111̅] and such 1 identity screw axis (2 1 ) parallel to a //

19. Design Business 2-fold rotations

20. Design Business 2 1 screw axis

21. Design Business 3-fold rotations

22. Axis Position: (0, y, ¼) This defines the line along which the screw axis operates.It's parallel to the y-axis at a fixed x=0 and z=¼. Translation Part: [0, ½, 0] After the 180° rotation, the atom is translated by ½ unit cell along the y-axis.

23. 04 I23

24. I23 Basic Info International Number 19 7 Crystal system Cubic Bravais Lattice Body-centered ( I ) Point group 23 Space Group type Symmorphic

25. Vertices asymmetric unit 1/2,0,0 0 ≤ x ≤ 1 1/2,1/2,0 0 ≤ y ≤ 1 /2 1/4,1/4,1/4 ≤ z ≤ 1 /2 1/4, 1/4, -1/4 y ≤ min(x, 1 - x) z ≤ y

26. Symmetry Operations Design Business 2 -fold rotations parallel to the projections 3 -fold rotations along the body diagonals such as [111], [1̅11], [11̅1], [111̅] and such 1 identity screw axis (2 1 ) parallel to the projections in between the rotation

27. Design Business 2-fold Rotation 2 1 -Screw axis 3-fold Rotation along the diagonals 3 1 -Screw axis

28. Axis Position: ( 1/4, y , ¼) This defines the line along which the screw axis operates.It's parallel to the y-axis at a fixed x= 1/4 and z=¼. Translation Part: [0, ½, 0] After the 180° rotation, the atom is translated by ½ unit cell along the y-axis.

29. 5 F432

30. F432 Basic Info International Number 209 Crystal system Cubic Bravais Lattice Face-centered ( F ) Point group 432 Space Group type Symmorphic

31. Vertices asymmetric unit 1/2,0,0 0 ≤ x ≤ 1 /2 0 ≤ y ≤ 1 /4 1/4,1/4,1/4 -1/4 ≤ z ≤ 1 /4 1/4, 1/4, -1/4 y ≤ min(x, 1/2 - x) -y ≤ z ≤ y

32. Symmetry Operations Design Business 2 -fold rotations parallel to the projections 3 -fold rotations along the body diagonals such as [111], [1̅11], [11̅1], [111̅] and such 1 identity screw axis parallel to the projections in between the regular rotation

33. Design Business 4-fold rotations 2-fold rotations 3-fold rotations

34. Design Business 4-fold screw axis 2 1 screw axis 3-fold screw axis

35. Axis Position: ( 1/3 + x, 1/6 + x , x ) This defines the line along which the screw axis operates.It's parallel to the y-axis at a fixed x= 1/3 + x , y= 1/6 + x Translation Part: [ 1/3 , 1/3 , 1/3 ] After the 1 2 0° rotation, the atom is translated by 1/3 unit cell along the axis.

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