X-ray Diffraction- Structure Factor Flashcards
General strategy for seeing how crystals diffract X-rays
First consider how x-rays are scattered by electrons. Then how the x-rays are scattered by atoms. Finally how the x-rays are scattered by the arrangement of atoms in the unit cell
How does scattering by an electron work?
The oscillating electric field of an x-ray will cause a nearby electron to oscillate. An oscillating (accelerating/decelerating) electron emits an EM wave. This is scattering. The scattered beam has the same frequency and wavelength as the incident beam.
What does the intensity of a scattered beam from an electron depend on?
The scattering angle 2θ
Thompson equation intensity of scattered beam from electron
I=(I0/r^2)(e^2/mec^2)^2(1+cos(2θ)^2)/2 Where I0 is intensity of incident beam I Si intensity of scattered beam at distance r from electron. e is charge of electron me is mass of electron c is speed of light
What is the polarisation factor in the Thompson equation?
(1+cos(2θ)^2)/2
Why doesn’t the atomic nucleus scatter x-rays?
It is much larger in mass than electrons
Scattering in the forward direction
Scattering angle 2θ=0. All scattered x-rays are in phase. Resultant amplitude is the simple sum of all the scattered amplitudes. Amplitude of scattered wave by an element atomic number Z is Z times that scattered by a single electron
Scattering in all other directions
Scattering angle 2θ>0. There is a phase difference between x-rays scattered by electrons in different parts of an atom
Efficiency of scattering
In a given direction is described by the atomic form factor, f
f=(amplitude of wave scattered by atom)/(amplitude of wave scattered by single electron)
Graph of atomic form factor vs (sinθ)/λ
Starts high on y-axis but decreases in sort of exponential decrease curve as x increases
Path difference for scattering from adjacent atoms on different planes
δ=2dSinθ=λ
θ is incident angle to the plane
d is the plane spacing
Difference in angular phase formula
φ=2πhx/a=2πhu
x is distance between atoms on different plane
a/h is from definition of Miller indices
u=x/a
In 3D, φ=2π(hu+kv+lw)
Vector addition of waves
Represent waves as vectors on a graph. Add together graphically to get final vector. Length of this is amplitude and angle from positive x-axis is phase.
Useful but cumbersome
Complex numbers for wave addition
Represent waves as complex numbers. Acosφ+iAsinφ. A is amplitude. Use a+bi form to add together. Can also use exponential form
Exponential complex form of a scattered wave
Ae^iπ=fe^(i2π(hu+kv+lw))
Where f is atomic form (scattering) factor
Structure factor
Fhkl=Σfne^(i2π(hu+kv+lw)
From 1 to N (atoms in unit cell)
hu, etc have subscript n as well
How is intensity of diffracted x-ray beams related to the structure factor?
It is proportional to the square of the absolue amplitude of Fhkl.
Physical significance of structure factor
Measure intensity- determine structure factor and hence atomic positions, know crystal structure.
From knowledge of structure, compute structure factor- predict intensity of diffracted beam
Two possibilities for interference
Constructive or partially destructive: Fhkl has finite value so diffracted beam has measurable intensity.
Completely destructive: Fhkl has a value of 0 and diffracted beam has no measurable intensity and the the hkl reflection is systematically absent
How to derive the systematic absence conditions for a Bravais lattice
Write down structure factor expression. Identify Bravais lattice type. Write down fractional coordinates (u,v,w) of unique atoms in unit cell. Insert coordinates into SF expression (1 term per atom). Factorise expression in f. Does exponential term resolve to +1 or -1 for particular combinations of hkl? Does Fhkl^2 have non-zero value? If yes reflections corresponding to conditions on hkl are systematically absent. If no, reflections have measurable intensity so no systematically absent
4 types of Bravais lattice
Primitive (P) Body centred (B) Face centred (F) Side centred (A, B, C)
Useful exponential form relations for systematic absences
e^iπn=1 when n is even
e^iπn=-1 when n is odd
Deriving systematic absences for primitive
Contains one atom at origin (0, 0, 0) Sub into SF gives Fhkl=fe^i2π(0) so =f Means |Fhkl|^2=f0^2 SF same for all reflections and no systematic absences
Deriving systematic absences for primitive
Contains one atom at origin (0, 0, 0) Sub into SF gives Fhkl=fe^i2π(0) so =f Means |Fhkl|^2=f0^2 SF same for all reflections and no systematic absences
Deriving systematic absences for body centred
Contains two atoms at (0,0,0) and (1/2,1/2,1/2). Sub into SF for (u,v,w) and add. Get to
Fhkl=f(1+e^iπ(h+k+l))
If h+k+l even Fhkl=2f so |Fhkl|^2=4f^2. If h+k+l odd Fhkl=0 so |Fhkl|^2=0. So only reflections with h+k+l being even are observed
Deriving systematic absences for face centred
Contains 4 atoms at (0,0,0), (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0). Sub into SF for (u,v,w) and add together. Get to
Fhkl=f(1+e^iπ(k+l)+e^iπ(h+l)+e^iπ(h+k)). If h,k,l are all even or all odd (unmixed) Fhkl=4f. If not (mixed) Fhkl=0. So only reflections with hkl unmixed are observed
Systematic absences for side centred
k and l mixed (A)
h and l mixed (B)
h and k mixed (C)
