Periodicity, crystal systems and unit cells

Question 1

Define a lattice

Answer 1

An infinite array of points where each point has identical surroundings

Question 2

Define a lattice point

Answer 2

A dot in a 1-dimensional lattice. Note that a lattice point doesn’t need to be occupied by an atom or molecule. The separation between lattice points is a, and knowing a and a single point, the whole lattice can be generated.

Question 3

Define a unit cell

Answer 3

A block that when repeated in all directions gives the full lattice

Question 4

Define Bravais lattices

Answer 4

Combinations of the 7 unit cell shapes and 4 possible lattices give 14 Bravais lattices

Question 5

Define a motif/basis

Answer 5

An atom/molecule that when placed onto the lattice points generate a crystal structure.

Question 6

Describe how to count atoms in a 2D lattice

Answer 6

If an atom lies on a corner, it is shared between 4 cells and = 1/4. If it lies on a line, it is shared between 2 and = 1/2. If it lies in space it is not shared and = 1.

Question 7

Describe how to count atoms in a 3D lattice.

Answer 7

In an atom lies on a corner, it is shared between 8 cells and =1/8. If it lies on a line is it shared between 4 cells and = 1/4. If it lies on a face it is shared between 2, and = 1/2, and if it lies in space it is not shared and = 1.

Question 8

State the names of the 4 types of unit cell

Answer 8

Primitive, Body centred, face centred, C centred.

Question 9

Describe the 6 parameters of any unit cell

Answer 9

a b and c denote the sides of any cell, and α (alpha) β (beta) and γ (gamma) denote the angles between a, b and c. α describes the angle between b and c, β describes the angle between a and c and γ describes the angle between a and b.

Question 10

State the names of all 7 possible unit cell shapes

Answer 10

Cubic, triclinic, orthorhombic, rhombohedral, tetragonal, monoclinic, hexagonal.

Question 11

Describe the parameters of the cubic unit cell.

Answer 11

a=b=c

α=β=ɣ=90°

Question 12

Describe the parameters of the triclinic unit cell.

Answer 12

a≠b≠c

α≠β≠ɣ≠90°

Question 13

Describe the parameters of the orthorhombic unit cell.

Answer 13

a≠b≠c

α=β=ɣ=90

Question 14

Describe the parameters of the rhombohedral unit cell.

Answer 14

a=b=c

α=β=ɣ≠90°

Question 15

Describe the parameters of the tetragonal unit cell.

Answer 15

a=b≠c

α=β=ɣ=90°

Question 16

Describe the parameters of the monoclinic unit cell.

Answer 16

a≠b≠c

α=ɣ=90°β>90°

Question 17

Describe the parameters of the hexagonal unit cell.

Answer 17

a=b≠c

α=β=90°ɣ=120°

Question 18

Describe Bravais’ prediction regarding crystal structures

Answer 18

Crystal structure (a complete description of the atomic arrangement) = Bravais lattice(an arrangement of points) + basis (a group of atoms)

Question 19

Describe the parameters of the hexagonal primitive cell

Answer 19

a=b≠c
α=β=90
γ=120
Same parameters as the unit cell but each face is a quadrilateral. The unit cell contains primitive cells

Question 20

Describe miller indices

Answer 20

Miller indices are used to denote planes in crystals. A miller index has the form (h k l) where the three numbers are integers. The plane (h k l) intercepts the a, b and c axis at a/h, b/k, c/l. The separation between planes is referred to as dhkl. Note that a bar above a number means it is negative.