Absences Flashcards
What is an absence?
When F(hkl) = 0
Reasons why absences arise
Lattice centring (i.e. anything other than a P lattice) Presence of translational symmetry elements in the space group
Why are all absences referred to as systematic absences?
Because they are governed by patterns/rules
How is the presence of symmetry elements in a unit cell/crystal lattice resolved?
By examination of diffraction spot intensities
Symmetry elements can be determined by looking at reflections for which I(hkl) = 0
The Miller indices for which I(hkl) = 0 often form patterns, called absences
What does space group determination involve?
Relating the absences to the symmetry elements that govern the spatial relationships between molecules in the unit cell to each other
Lattice-centring absence conditions
Result from any lattice type other than P
Destructive interference of the X-rays will take place according to a selection rule that affects the entire set of intensities (i.e. all h, k, l values)
Systematically absent reflections for P lattice type
None
Systematically absent reflections for I lattice type
When h+k+l = 2n+1
i.e. sum of indices is odd
Systematically absent reflections for F lattice type
Mixed odd/even values for h, k, and l are not allowed
i.e. the reflection conditions is that h, k, l must be all odd or all even
Systematically absent reflections for A lattice type
When k+l = 2n+1
i.e. sum of k and l is odd
Systematically absent reflections for B lattice type
When h+l = 2n+1
i.e. sum of h and l is odd
Systematically absent reflections for C lattice type
When h+k = 2n+1
i.e. sum of h and k is odd
Reflection condition
Opposite of a systematic absence
Absences due to translational symmetry elements
Result from screw axes and glide planes present in the space group symmetry
Affects only small groups of reflections (subsets) within the data set
Systematically absent reflections for a 2(1) screw axis along a
For (h, 0, 0) when h = 2n+1
i.e. h odd
Systematically absent reflections for a 2(1) screw axis along b
For (0, k, 0) when k = 2n+1
i.e. k odd
Systematically absent reflections for a 2(1) screw axis along c
For (0, 0, l) when l = 2n+1
i.e. l odd
Systematically absent reflections for c glide perpendicular to b
For (h, 0, l) when l = 2n+1
i.e. l odd
Systematically absent reflections for a glide perpendicular to b
For (h, 0, l) when h = 2n+1
i.e. h odd
Systematically absent reflections for a glide perpendicular to c
For (h, k, 0) when h = 2n+1
i.e. h odd
Why do mirror planes, inversion centres, rotation axes and rotary-inversion axes not give rise to absent reflections?
They are non-translational symmetry elements
Systematically absent reflections for a 4(1) screw axis along c
For (0, 0, l) when l=/=4n
i.e. reflection condition is l = 4n
When does an absence occur?
When the I(hkl) of a ‘reflection’ from a set of Miller planes has a mathematical value of 0
(This is the same as destructive interference)
This means the F(hkl) for a given Miller index is also 0
When does the imaginary part of the structure factor expression = 0?
When the space group is centrosymmetric (has a centre of symmetry)
Cos(pi x odd number)
= -1
Cos(pi x even number)
= 1
d-spacing equation for cubic
1/d^2 = (h^2+k^2+l^2)/a^2
d-spacing equation for tetragonal
1/d^2 = (h^2+k^2)/a^2 + l^2/c^2
d-spacing equation for orthorhombic
1/d^2 = h^2/a^2 + k^2/b^2 + l^2/c^2
Method for calculating the structure factor for a given Miller set when given scattering factors (fj) as a function of sin(theta)/lambda
- Determine sin(theta)/lambda for the given Miller set using the d-spacing equation for that particular unit cell type
- Calculate the atomic scattering factors for the atoms in the unit cell at the sin(theta)/lambda value for the given Miller set (read off graph)
- Sub all terms into structure factor equation
