Lectures co-financed by the European Union in scope of the European Social Fund

Crystallography and Diffraction Theory and Modern Methods of

Analysis

Lectures 3-4Lattice Planes and the Reciprocal

Lattice

Dr. I. AbrahamsQueen Mary University of London

Lectures co-financed by the European Union in scope of the European Social Fund


Early in the history of crystallography and diffraction WL Bragg showed that diffraction of X-rays by crystals could be explained in terms of reflections from semi-transparent mirrors or planes within the crystal lattice. These planes are termed Lattice Planes.

Consider a 2D-lattice

Lattice Planes

On the right hand side a set of parallel lines have been drawn on this 2D-lattice.

How do we distinguish this set of lines from other sets of lines ?

If we consider the spaces between lattice points, then travelling along vector a two spaces are crossed between lattice points and along b one space is crossed. We can therefore index this set of lines as 2,1


Now consider the following lines on the same 2D-lattice

2,1 1,2 1,2or

In these cases, travelling along the axial vectors between lattice points, the same number of spaces are crossed. However, whereas in the first case the lines are crossed from the same side, in the second case travelling along a, lines are crossed on the reverse side to that when travelling along b. Hence, one of the indices is given a negative value.

In this example, the same lattice can be described by either a primitive or centred unit cell. The set of lattice lines shown is given different indices for the two cells.

For the primitive cell the lines are indexed as 2,1 while for the centred cell the same lines are indexed as 2,0.


The same rules apply to 3D-lattices which require three indices designated h, k, l

These are known as the Miller indices, where

h is the index corresponding to the a-axisk is the index corresponding to the b-axisl is the index corresponding to the c-axis

The directions of the lattice vectors are normally chosen according to the right hand rule. i.e. the thumb, first finger and second finger correspond to +x (a), +y (b) and +z (c) directions respectively.

Right hand convention for crystal lattices.Ref: X-ray Structure Determination. A practical Guide. G.H. Stout and L.H. jensen 2nd ed. 1989 Wiley New York.



When trying to sketch lattice planes for more complicated cases it is best to first mark out the cell edges.

Then, starting from the marks nearest to the origin join the marks

Continue joining up the marks until all are used. Remember what you are drawing are parallel planes in 3D.


Finally, shade the planes to make them clearer


The perpendicular distance between parallel planes is known as the d-spacing and in normally quoted in Å

e.g. 0,2,0 planes in an orthorhombic cell

d020 = b/2

For a simple orthogonal system the d-spacing is easily calculated where two of the Miller indices are zero. Generally for an orthogonal crystal system (i.e. orthorhombic, tetragonal or cubic)

2

2

2

2

2

2

2

1

c

l

b

k

a

h

d

d-Spacings


e.g. The following diffraction peaks were collected for a tetragonal cell using Cu-K radiation ( = 1.5418 Å). Calculate the unit cell parameters a and c.

h k l 2 /2 0 0 40.0731 1 1 30.942

First use Bragg's Law = 2 dhkl sinhkl to convert the 2 values to d-spacings in Å. i.e

250.2

2

073.40sin2

5418.1200

d

890.2

2

942.30sin2

5418.1111

d

Since the unit cell is tetragonal (i.e. the axes are orthogonal) a = b = 2 d200 = 4.500 Å. Now we know a and

b we can use the 111 reflection to calculate c.

2

2

2

2

2

2

2

1

500.4

1

500.4

1

890.2

1

c

c = 6.906 Å

Å

Å

Lectures co-financed by the European Union in scope of the European Social FundLectures co-financed by the European Union in scope of the European Social Fund

Direct Cell Equations for d-Spacings

coscoscos2

coscoscos2coscoscos2

sinsinsin11

2

22

222222222222

22

chlab

bcklahkabc

balcakcbh

Vd

Triclinic

ac

hl

c

l

b

k

a

h

d

cos2sin

sin

112

2

2

22

2

2

22Monoclinic

2

2

2

2

2

2

2

1

c

l

b

k

a

h

dOrthorhombic

2

2

2

22

2

1

c

l

a

kh

d

Tetragonal

2

2

2

22

2 3

41

c

l

a

khkh

d

Hexagonal

2

222

2

1

a

lkh

d

Cubic

2

1222 coscoscos2coscoscos1 abcV

where


The Reciprocal Lattice

The reciprocal lattice is a theoretical concept that makes the interpretation of X-ray diffraction data easier.

Consider a 2D optical diffraction grating

If we shine light through this grating we get a diffraction pattern

a

b


These spots can be indexed as follows:

So 0 2 is twice as far away from the centre as 0 1 etc.

The spacings of the points are inversely proportional to the lattice spacings.

X-ray patterns are analogous in 3D. Interpretation of these patterns is easier if we redefine the lattice in terms of the direction and spacings of the lattice planes This new lattice is called the Reciprocal Lattice.


Consider a simple orthogonal lattice. A projection down the c-axis of the unit cell is shown below.

Starting at the origin in the real (direct) lattice, lines are drawn perpendicular to the lattice planes.010 and 100

These lines are marked at points d* where d* = 1/d.

Using this convention, d* has units Å-1. However, it is more convenient to use d* = /d which has dimensionless units known as reciprocal lattice units (r.l.u.).


Note that because in this case the direct lattice is orthogonal a* is parallel to a and b* is parallel to b. The magnitude of a* is inversely proportional to a and similarly b* to b.

Additional layers of the reciprocal lattice can be built up in the c* direction to give a sphere of reciprocal space.

This is continued to give a layer of the reciprocal lattice


Consider now the case of a non-orthogonal direct lattice such as in a monoclinic system.

a = 6 Åc = 4 Å = 110(b = 10 Å)

100

1

sin

1*

daa

001

1

sin

1*

dcc

180*

Therefore

a* = 0.177c* = 0.266* = 70


Triclinic symmetry represents the general case:


If we assign an intensity to each reciprocal lattice point corresponding to the intensity of the observed X-ray reflection then we obtain an intensity weighted reciprocal lattice.


The diffraction pattern shown by the weighted reciprocal lattice will show the point symmetry and the centre of symmetry of the crystal structure (if it is centrosymmetric). This is known as Laue symmetry.

Simple X-ray photographs can be taken of the undistorted reciprocal lattice using a variety of cameras.

Max von Laue

de Jong Bouman photograph of the hk0 layer of a crystal of ammonium oxalate monohydrate (space group P21212)

Documents

Lectures co-financed by the European Union in scope of the European Social Fund