X-ray Diffraction- Structure Factor

Question 1

General strategy for seeing how crystals diffract X-rays

Answer 1

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

Question 2

How does scattering by an electron work?

Answer 2

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.

Question 3

What does the intensity of a scattered beam from an electron depend on?

Answer 3

The scattering angle 2θ

Question 4

Thompson equation intensity of scattered beam from electron

Answer 4

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

Question 5

What is the polarisation factor in the Thompson equation?

Answer 5

(1+cos(2θ)^2)/2

Question 6

Why doesn’t the atomic nucleus scatter x-rays?

Answer 6

It is much larger in mass than electrons

Question 7

Scattering in the forward direction

Answer 7

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

Question 8

Scattering in all other directions

Answer 8

Scattering angle 2θ>0. There is a phase difference between x-rays scattered by electrons in different parts of an atom

Question 9

Efficiency of scattering

Answer 9

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)

Question 10

Graph of atomic form factor vs (sinθ)/λ

Answer 10

Starts high on y-axis but decreases in sort of exponential decrease curve as x increases

Question 11

Path difference for scattering from adjacent atoms on different planes

Answer 11

δ=2dSinθ=λ
θ is incident angle to the plane
d is the plane spacing

Question 12

Difference in angular phase formula

Answer 12

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)

Question 13

Vector addition of waves

Answer 13

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

Question 14

Complex numbers for wave addition

Answer 14

Represent waves as complex numbers. Acosφ+iAsinφ. A is amplitude. Use a+bi form to add together. Can also use exponential form

Question 15

Exponential complex form of a scattered wave

Answer 15

Ae^iπ=fe^(i2π(hu+kv+lw))

Where f is atomic form (scattering) factor

Question 16

Structure factor

Answer 16

Fhkl=Σfne^(i2π(hu+kv+lw)
From 1 to N (atoms in unit cell)
hu, etc have subscript n as well

Question 17

How is intensity of diffracted x-ray beams related to the structure factor?

Answer 17

It is proportional to the square of the absolue amplitude of Fhkl.

Question 18

Physical significance of structure factor

Answer 18

Measure intensity- determine structure factor and hence atomic positions, know crystal structure.
From knowledge of structure, compute structure factor- predict intensity of diffracted beam

Question 19

Two possibilities for interference

Answer 19

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

Question 20

How to derive the systematic absence conditions for a Bravais lattice

Answer 20

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

Question 21

4 types of Bravais lattice

Answer 21

Primitive (P)
Body centred (B)
Face centred (F)
Side centred (A, B, C)

Question 22

Useful exponential form relations for systematic absences

Answer 22

e^iπn=1 when n is even

e^iπn=-1 when n is odd

Question 23

Deriving systematic absences for primitive

Answer 23

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

Question 24

Deriving systematic absences for primitive

Answer 24

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

Question 25

Deriving systematic absences for body centred

Answer 25

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

Question 26

Deriving systematic absences for face centred

Answer 26

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

Question 27

Systematic absences for side centred

Answer 27

k and l mixed (A)
h and l mixed (B)
h and k mixed (C)