by the end of this section you should: understand the derivation of the reciprocal lattice be able...
TRANSCRIPT
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
Objectives
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)
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
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
The K vectorWe 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 condition
K 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 lattice
For 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 sphere
In powder methods, we assume that the random orientation of the crystallites means that G takes up all orientations at once.
ExamplesQ1 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
SummarySummary 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.