Learning outcomes

By the end of this section you should:• understand the derivation of the reciprocal lattice

• be able to derive the Laue condition for the reciprocal lattice

• understand how reciprocal space relates to the diffraction experiment

• be able to use reciprocal space to make calculations

Reciprocal Space

The reciprocal lattice

• A diffraction pattern is not a direct representation of the crystal lattice

• The diffraction pattern is a representation of the reciprocal lattice

We have already considered some reciprocal features -

Miller indices were derived as the reciprocal (or inverse) of unit cell intercepts.

Reciprocal Lattice vectors

Any set of planes can be defined by:

(1) their orientation in the crystal (hkl)

(2) their d-spacing

The orientation of a plane is defined by the direction of a normal (vector product)

Defining the reciprocal

Take two sets of planes:

Draw directions normal:

These lines define the orientation but not the length

We use 1

dto define the lengths

These are called reciprocal lattice vectors G1 and G2

G1

G2

Dimensions = 1/length

Reciprocal Lattice/Unit Cells

We will use a monoclinic unit cell to avoid orthogonal axes

We define a plane and consider some lattice planes

(001)

(100)

(002)

(101)

(101)

(102)

Reciprocal lattice vectorsLook at the reciprocal lattice vectors as defined above:

G100

G001

G002

G101

G102

O

These vectors give the outline of the reciprocal unit cell

a*

c*

* So a* = G100 and c* = G001

and |a*| = 1/d100 and |c*| = 1/d001

a* and c* are not parallel to a and c - this only happens in orthogonal systems

* is the complement of

Vectors and the reciprocal unit cell

The cross product (b x c) defines a vector parallel to a* with modulus of the area defined by b and c

The volume of the unit cell is thus given by a.(bxc)

We can define the reciprocal lattice, thus, as follows:

From the definitions, it should be obvious that:

a.a* = 1 a*.b = 0 a*.c = 0 etc. i.e. a* is perpendicular to both b and c

V

cba

*

V

acb

*

V

bac

*

Reciprocal vs real

Reciprocal lattice vectors can be expressed in terms of the reciprocal unit cell a* b* c*

For hkl planes: Ghkl = ha* + kb* + lc*

compared with real lattice: uvw = ua + vb + wc

where uvw represents a complete unit cell translation and uvw are necessarily integers

x u=1 u=2v, w = 0

The K vector

We define incident and reflected X-rays as ko and k respectively, with moduli 1/

Then we define vector K = k - ko

K vector

As k and ko are of equal length, 1/, the triangle O, O’, O’’ is isosceles.

The angle between k and -ko is 2hkland the hkl plane bisects it.

The length of K is given by:

K k hklhkl 2

2| |sin

sin

The Laue conditionK is perpendicular to the (hkl) plane, so can be defined as:

K hkl

2n

sin

where n is a vector of unit length

G is also perpendicular to (hkl) so n GG

hkl

hkl

KG

Ghkl

hkl hkl

2

sin and Gd

from previoushklhkl

1

Kd

Ghkl hklhkl

2 sin

But Bragg: 2dsin =

So K = Ghkl the Laue condition

What does this mean?!

Laue assumed that each set of atoms could radiate the incident radiation in all directions

Constructive interference only occurs when the scattering vector, K, coincides with a reciprocal lattice vector, G

This naturally leads to the Ewald Sphere construction

Ewald SphereWe superimpose the imaginary “sphere” of radiated radiation upon the reciprocal latticeFor a fixed direction and wavelength of incident radiation, we draw -ko (=1/) e.g. along a*

Draw sphere of radius 1/ centred on end of ko

Reflection is only observed if sphere intersects a point

i.e. where K=G

What does this actually mean?!

Relate to a real diffraction experiment with crystal at O

K=G so scattered beam at angle 2

Geometry:

tan tan2 2 1 xR

xR

but K hkl2sin

so KxR

2 12

1

sin tan

Practicalities

This allows us to convert distances on the film to lengths of reciprocal lattice vectors.Indexing the pattern (I.e. assigning (hkl) values to each spot) allows us to deduce the dimensions of the reciprocal lattice (and hence real lattice)In single crystal methods, the crystal is rotated or moved so that each G is brought to the surface of the Ewald sphereIn powder methods, we assume that the random orientation of the crystallites means that G takes up all orientations at once.

Examples

Q1 In handout:

Indexing - Primitive

Each “spot” represents a set of planes.

A primitive lattice with no absences is straightforward to index - merely count out from the origin

Indexing - Body Centred

In this case we have absences

Remember, h+k+l=2nl =0l =1

Indexing - Face Centred

Again we have absences

Remember, h,k,l all odd or all evenl =0l =1

Notes on indexing

Absences mean that it’s not so straightforward - need to take many images

The real, body-centred lattice gives a face-centred reciprocal lattice

The real, face-centred lattice gives a body-centred recoprocal lattice

Summary

The observed diffraction pattern is a view of the reciprocal lattice

The reciprocal lattice is related to the real lattice and a*=1/a, a*.b=0, a*.c=0 etc.

By considering the Bragg construction in terms of the reciprocal lattice, we can show that K = G for constructive interference - the Laue condition

This leads naturally to the imaginary Ewald sphere, which allows us to make calculations from a measured diffraction pattern.