Crystallography- Mathematical Description of Symmetry

Question 1

Unit vectors in crystallography

Answer 1

The 3 unit vectors (bold a, b, c) are the 3 edges of the unit cell (not necessarily in Cartesian axis directions). The lengths of the edges are the absolute vector values (lattice parameters)

Question 2

Lattice vector

Answer 2

A general vector connecting origin with any other lattice point in another unit cell. Constructed as L=Ua+Vb+Wc where U, V, W are integers

Question 3

Lattice direction

Answer 3

Independent of length. Is the direction along the lattice vector, L, and all directions parallel to it. Labelled [UVW] (the direction indices). Negative values represented with bar above and spoken ‘bar one’. Use smallest common factor of UVW so [202] is [101]

Question 4

What is a lattice plane?

Answer 4

A plane that passes through three non-colinear lattice points. There can be infinitely many parallel to each other

Question 5

When do lattice planes have the same Miller indices?

Answer 5

When they are parallel and equally spaced. Have same Miller indices (hkl)

Question 6

What can Miller indices be used for?

Answer 6

Assign an orientation symbol to any planes even if they don’t pass through lattice points as long as they are parallel to a lattice plane. Can describe the orientation of mirror planes relative to a unit cell.

Question 7

What happens if the origin of the unit cell sits on a lattice plane?

Answer 7

The next adjacent parallel plane cuts the reference axes x, y, z at a/h, b/k, c/l respectively

Question 8

What happens if the plane cuts the axes on the other side of the origin?

Answer 8

The Miller indices are negative and written with bars

Question 9

What does it mean if h, k or l are 0?

Answer 9

The planes are parallel to (don’t intersect) the reference axis that corresponds to 0. (110) is parallel to z-axis

Question 10

How are the Miller indices related to the unit vectors?

Answer 10

h=a/A
k=b/B
l=c/C
Where A, B, C are points of intersection of plane with reference axes

Question 11

General plane equation

Answer 11

(h/a)x+(k/b)y+(l/c)z=n

n is vector normal to plane

Question 12

Reaction between density of atoms in plane to lattice plane distance

Answer 12

Density of atoms in plane is proportional to reciprocal of lattice plane distance. This is so there is a constant volume density.

Question 13

Formula for d-spacing of lattice planes in orthorhombic

Answer 13

1/dhkl^2=h^2/a^2+k^2/b^2+l^2/c^2
For tetragonal a=b
For cubic a=b=c

Question 14

Process for finding Miller indices

Answer 14

Find values where the plane nearest to the origin crosses the base line. Calculate the reciprocal intersect value and bring to smallest full integer

Question 15

How to apply a symmetry operation to a vector u

Answer 15

Use a matrix representation of the symmetry element and multiply u by it

Question 16

Matrix for inversion

Answer 16

Like identity 3x3 matrix but all -1 along diagonal

Question 17

Matrix for mirror

Answer 17

Like identity 3x3 matrix but if one is negative, the mirror is perpendicular to that. Top left -1 means perpendicular to a

Question 18

Rule for direction [UVW] lying parallel to the plane system (hkl)

Answer 18

hU+kV+lW=0

Question 19

Zone axis [UVW]

Answer 19

A preferred crystallographic direction sharing many planes (h1k1l1), (h2k2l2), showing typically high symmetry, ideal for oriented diffraction experiments

Question 20

For what crystal system does the normal vector N low parallel to the direction index vector [UVW]?

Answer 20

Cubic