Crystallography- Space Groups

Question 1

What is different for space groups?

Answer 1

It considers extra symmetry elements extended by translation elements (glide and screw axes). Symmetry is no longer defined around one point (origin)

Question 2

How do space groups relate to point group and Bravais lattices?

Answer 2

There are 230 space groups which are subgroups to the 32 point groups. They are compatible with the 14 Bravais lattices

Question 3

Why are there many more space groups per crystal system than Bravais lattices?

Answer 3

Symmetry axes and planes can pass through the unit cell at any fraction not all just through origin. Maximum of 68 space groups for one system

Question 4

Describe glide planes

Answer 4

Like mirror planes but all atoms on other side of plane are shifted by some fraction of the unit cell. The direction of translation determines name of glide plane.

Question 5

How do screw axes work?

Answer 5

Like a rotation axis but the new atom position is always shifted by the same fraction of a unit cell. Motion follows the shale of a screw

Question 6

What does it mean if a screw axis is 2-fold?

Answer 6

After 2 operations the atom should be one full unit cell away from where it started

Question 7

What fold rotations are allowed for screw symmetry?

Answer 7

1, 2, 3, 4, 6

Question 8

Symbol for screw symmetry

Answer 8

Large number indicates angle of rotation for each operation (360/n). Subscript number is pitch which indicates translation as fraction of unit cell in screw direction. E.g 6 subscript 5 means rotation by 60° and translation by 5/6 along axis

Question 9

Representing screw symmetry diagrammatically

Answer 9

Have straight line(s). In centre is normal shape symbol for order of rotation. There are hooks that show direction of rotation. Number of hooks tells you how many operations until whole number of unit cells away

Question 10

Representing a glide plane

Answer 10

Give symbol, give plane normal direction, indicate on a drawing through which point of a unit cell the glide plane runs

Question 11

Axial glide symbol and translation

Answer 11

a: a/2
b: b/2
c: c/2

Question 12

Diagonal glide symbol and translation

Answer 12

Symbol n
(a+b)/2 
Or any other combination of a, b, c
(a+b+c)/2 
This one only for cubic or tetragonal systems

Question 13

Diamond glide symbol and translation

Answer 13

Symbol d
(a+/-b)/4
Or any other combination of a, b, c
(a+/-b+/-c)/4
This one only for cubic or tetragonal systems

Question 14

Why can’t glide planes be represented on stereograms?

Answer 14

Stereograms use two angles describing orientations through the origin only

Question 15

Symbol changes from point groups (stereograms) to space groups (unit cell 2D projections)

Answer 15

Below plane: point uses hollow circle, space uses hollow circle with - sign next to it.
Above plane: point uses filled circle, space uses hollow circle with + sign next to it.
Below and above: point uses filled circle within circle, space uses circle with central vertical line and + and - either side

Question 16

What does a comma mean in circles for projection drawings of space groups?

Answer 16

Indicates alternating symmetry (left-right-left-right). Not always necessary for atoms (circles) which are the same in all orientations

Question 17

General equivalent positions

Answer 17

Used in the orthogonal projections of space groups. Display all equivalent locations starting from one position near a corner. Not to be confused with atoms which are unlikely to sit so asymmetrically. Represented as open circles.

Question 18

Special equivalent positions

Answer 18

Used in orthogonal projections of space groups. They are at edges or corners where there is reduced multiplicity. These are more likely to be occupied by real atoms. Often shown as shaded circles

Question 19

Converting side view of mirror plane to plan view

Answer 19

Side view has straight horizontal line with mirrored atoms either side which are either both above (+) or both below (-) plane of paper. Plan view shows the plane as a rectangle viewing down from above the previous horizontal line. The symbols will be for above and below plane (vertical line in circle with + and - either side)

Question 20

Converting side view of a, b or c glide plane into plan view

Answer 20

Starts as horizontal line with one atom above or below it. The next is on the other side and shifted and so on. All remain either above or below plane of paper. For plan view the circles lie on the same straight line across the rectangle and alternate between + and -

Question 21

Converting side view of diagonal glide to plan view

Answer 21

Side view has horizontal line with one atom above or below it. This goes onto the other side and is shifted but will have a 1/2 next to it (remains same side of paper). For plan view start with one circle above plane of paper (for example). The next will be below plane (-) but will also be shifted vertically on the page relative to the first

Question 22

Converting point group to space group symbols

Answer 22

Add Bravais lattice letter to point group symbol (P, F, I, C…). Expand mirror planes into possible glide planes (m becomes a, b, c, n, d). Expand rotation axes into possible screw axes (4 goes to 4 sub 1)

Question 23

Converting space group to point group symbols

Answer 23

Remove the Bravais lattice letter. Replace all different glide planes with letter m (ba becomes mm). Remove any subscripts for the screw axes so only rotation axes