Crystallography- Point Groups

Question 1

How many point groups are there in 3D?

Answer 1

32

Question 2

What is a point group?

Answer 2

A set of symmetry elements and symbols associated to any object, mutually compatible and compatible with infinite lattices. Often a synonym is crystal class

Question 3

How to derive the 32 point groups

Answer 3

Start with one symmetry direction. Take the 3 alternatives: pire rotation, inversion-rotation, rotation and inversion. There are 5 possible rotations (1,2,3,4,6-fold).
Then consider objects with symmetry along all 3 coordinate directions. Permutate pure rotation, inversion-rotation and separate inversion. 30 extra possible groups.
Among the 45 identified, eliminate any double occurrence of same overall symmetry to get 32

Question 4

Order of crystal systems used for order of point groups

Answer 4

Triclinic, monoclinic, orthorhombic, trigonal, hexagonal, tetragonal, cubic

Question 5

In the symbol for point groups what does / mean?

Answer 5

2/m would mean 2 and m perpendicular to each other.

If was 2m they would be parallel

Question 6

What point groups are the Bravais lattices for each crystal system?

Answer 6

The ones with the most symmetry elements for each crystal system (last number for each system). They are all holosymmetric (highest symmetry within system

Question 7

What does subscript p mean on symmetry element symbols?

Answer 7

Means the axis is polar (has a direction of charge)

Question 8

What type of symmetry does a symbol within a symbol indicate?

Answer 8

Inversion axis

Question 9

What is the symbol for an inversion centre?

Answer 9

Small circle

Question 10

What does the inversion centre column show in the point groups table?

Answer 10

If the point groups has (+) or doesn’t have (-) an inversion centre

Question 11

Chriality

Answer 11

Aka enantiomorphy. Enantiomorphous groups must have absence of rotation-inversion axes and inversion centre

Question 12

What is optical activity?

Answer 12

Rotating polarised light. Similar to chirality apart from some weak effects in some non-enentiomorphous groups

Question 13

Pyroelectric effect

Answer 13

Change the temperature and find a polar charge dipole forming on two opposite ends of the material. Needs a unique polar axis

Question 14

Piezoelectric effect

Answer 14

Apply stress (tension or compression) and find a polar charge dipole forming on two opposite ends of material. The opposite effect is to apply a voltage and the crystal reacts by expansion/compression (used for actuators)

Question 15

Ferroelectricity

Answer 15

Spontaneous alignment of polar dipoles within domains.

Question 16

What does the lowest symmetry point group of a crystal system show?

Answer 16

The essential or defining symmetry elements of that crystal system