Crystallography- Symmetry

Question 1

What are symmetry elements?

Answer 1

Instructions how to mirror/turn/translate an object relative to a coordinate system. The element applied if the entire object with all its point and all its internal details/properties remain unchanged upon applying the operation

Question 2

Two divisions of symmetry

Answer 2

Translational, point symmetry

Question 3

What mirror planes and how are they represented?

Answer 3

2D objects oriented in 3D through which a 3D shape can be reflected across and remain unchanged. Represented by their traces (bold line) on a 2D face or cross-section of the object. Symbol m

Question 4

What are rotation axes?

Answer 4

For symmetry upon rotation by a central axis. They are 1D objects oriented in 3D

Question 5

How are rotation axes represented?

Answer 5

Thin lines if in plane of page. Has symbol at ends of the axis. Oval for 2-fold, triangle for 3-fold, square for 4-fold, hexagon for 6-fold (filled in). Symbol is a number, n

Question 6

How to determine n off rotation axes

Answer 6

n is the number of times the objects repeats as it is rotated in a circle around the axis. If it repeats every 180°, n=2 (360/180=2) and the rotation axis is 2-fold

Question 7

What is an inversion centre?

Answer 7

A point (normally at origin) upon which the entire object can be reflected so that every point finds a copy at the opposite side of the centre (in 3D). The lines of reflection all point towards the origin. Point (x,y,z) goes to (-x,-y,-z). Symbol i or 1bar

Question 8

Rule for how many axes/planes count

Answer 8

Only count once per pairs of faces penetrating

Question 9

How many counts of symmetry are in a cubic cell?

Answer 9

6x 2-fold (edges)
3x 4-fold (faces)
4x 3-fold (corners)
6x m diagonal
3x m central
Has inversion centre

Question 10

(Rotation) inversion axes

Answer 10

An operation for which rotation and inversion must be carried out simultaneously. Symbol derived from underlying rotation but with bar on top. Doesn’t matter order done.

Question 11

How are inversion axes drawn?

Answer 11

1D line through the object with symbols at either end. 2 bar is filled oval with empty circle in. 3 bar is filled triangle with empty circle in. 4 bar is empty square with filled oval in. 6 bar is empty hexagon with filled triangle in

Question 12

Which inversion axes include/equal other symmetry elements?

Answer 12

1 bar = i 
2 bar = m
3 bar includes 3 and i 
4 bar includes 2
6 bar includes 3 and m

Question 13

When is a structure symmetric?

Answer 13

If a mathematical transformation such as rotation, mirror, inversion exists which converts the lattice and unit cell content onto itself

Question 14

What are the only order rotations allowed for crystallography?

Answer 14

2, 3, 4, 6 fold because these are the only ones compatible with translation symmetry (can fill a 3D lattice in infinite space)

Question 15

Greek names for rotation symmetry orders

Answer 15

2 is diad
3 is triad
4 is tetrad
6 is hexad

Question 16

How can quasi-crystals show 5-fold rotational symmetry?

Answer 16

Every origin point is surrounded by a neighbourhood of 5-food symmetry but there is is no long range repeat symmetry. You need two differing unit cells (broad and narrow rhombus) to fill the entire space (against the rule)

Question 17

The relationship between crystal symmetry and lattice symmetry

Answer 17

Crystal symmetry is equal or lower than lattice symmetry, never higher. The fill in of the unit cell gradually reduces symmetry