Diffraction

Question 1

What is the main feature of a crystal?

Answer 1

Array of atoms/molecules which have periodic translational symmetry

Question 2

What is a lattice?

Answer 2

Array of points in space which describe translational symm
Mathematical

Question 3

What is the basis of a crystal?

Answer 3

Atoms or molecules which are attached to each lattice point to give structure

Question 4

How can points in a lattice relate to eachother?

Answer 4

r’ = r + n₁a + n₂b = n₃c

Where these a,b,c are along x,y,z axis respectively

Question 5

What is a primitive lattice?

Answer 5

One lattice point per unit cell

All pairs of lattice points r and r’ related by integer values of cell units

Question 6

How are lattices defined in 3D?

Answer 6

Question 7

What is a cubic cell?

Answer 7

4x 3-fold rotation axis
Seen as a cube with a point on each vertex

Question 8

What are the 7 possible primitive lattices?

Answer 8

Question 9

What is a body-centered cell?

Answer 9

Question 10

What is a face-centered lattice?

Answer 10

Question 11

What is a face-centered lattice?

Answer 11

Question 12

How many lattice points are in a body-centered cell?

Answer 12

2 points

so 2 basis units

Question 13

How many lattice points are in a face-centered-cell?

Answer 13

4 lattice points

So 4 basis units

Question 14

How many lattice points and atoms are in the unit cell of hcp?

Answer 14

2 atoms but 1 lattice point

Question 15

What is the range of lattice spacing in a crystal?

Answer 15

1-20 angstrom

where 1 angstrom = 10^-10 m

Question 16

How do X-rays interact with a 3D lattice?

Answer 16

Reflected by planes of the crystal lattice

Difference in phase of waves reflected gives separation of planes of the lattice

Question 17

What is Bragg’s Law and how is it used?

Answer 17

nλ = 2dsinθ

2 waves in phase must be a whole wavenumber difference

d is distance between planes of lattice

Question 18

What is the method for understanding sets of lattice planes?

Answer 18

Miller Indicies

Question 19

How are the miller indicies done in 2d?

Answer 19

Number of times intercept x and y axis

(n₁, n₂)

Where n₁ is number of times planes intercept x axis per cell
And n₂ is number of times planes intercept y axis per cell

Question 20

Describe these using Miller Indicies

Answer 20

LHS: (1,2) plane

RHS: (2,-1) plane

Question 21

What does a negative in miller indicies indicate?

Answer 21

Different orientation of the plane
Runs in opposite direction

Question 22

How are miller indicies in 3d worked out?

Answer 22

Planes cut x,y,z axes at (1/h, 1/k, 1/l)

These are then reffered to as a (h,k,l) plane

Question 23

Describe these 3d planes using Millers indicies

Answer 23

LHS: parallel to z-axis and cuts through others once so is a (1,1,0) plane

RHS: (2,2,3) plane

Question 24

What is the relation of distance between lattice planes and h,k,l for a cubic cell?

Answer 24

d = a/Sqrt[h² + k² + l²]
where a is the cubic lattice constant

(1/d²) = [h² + k² + l²]/a²

Question 25

What is the relation between d and h,k,l for a tetragonal cell?

Answer 25

(1/d²) = [h² + k²]/a² + l²/a²

Question 26

What is the relation between d and h,k,l for a orthorombic cell?

Answer 26

(1/d²) = h²/a² + k²/b² + l²/c²

Question 27

How many diffraction peaks are seen for a primitive lattice?

Answer 27

Peak for each hkl lattice plane
No systematic absences

Question 28

What is the equation relating a to h,k,l?

Answer 28

a² = (λ²/4sin²θ) (h²+k²+l²)

Question 29

How are crystals measured in a powder?

Answer 29

Measure powder with large number of crystals and assume some are correctly orientated

Those at bragg angle will diffract and many so is likely

Question 30

How are crystals measured in a powder?

Answer 30

Measure powder with large number of crystals and assume some are correctly orientated

Those at bragg angle will diffract and many so is likely

Question 31

What kind of beam is required for x-ray diffraction?

Answer 31

Monochromatic

Use a metal filter

Question 32

What is the format of a diffraction pattern?

Answer 32

2θ is on x-axis (where θ is from detection), this gives d-spacing

Intensity is on y-axis

Question 33

How can you find h,k,l from d in diffraction pattern?

Answer 33

1/d² = [h²+k²+l²]/a²

find 1/d² for each signal

Find h²+k²+l² from this

Then can derive (h,k,l)

Question 34

How can x-ray diffraction pattern show the average particle size?

Answer 34

The broader the diffraction peak then the smaller the particle size

Due to destructive interference

Question 35

What is the formula for intensity of h,k,l of a reflection?

Answer 35

I = K x Lp x J x F² x exp[-2W] x A

where I is i not L

K = scaling constant, depends on instrument
L = lorentz polarisation factor
J = multiplicity of {hkl} plane
W = debyte temp correction factor
A = absorption factor
F = structure factor

Question 36

What is the Lorentz polarization factor?

Answer 36

Tabluated correction factor

Diffraction polarises incoming X-ray
Not all polarisation orientations reflected equally so scattered beam is partially polarised

Means intensity of reflected radiation depends on Bragg angle and diffractometer

Question 37

What is J for intensity of a reflected wave?

Answer 37

Multiplicity of {hkl} - equivalent lattice planes are present

Squares mean (1,1,1), (-1,1,1), (1,-1,-1), etc all have the same d-spacing (8 with the same 2θ value)

Question 38

What is W, the debye temperature correction factor?

Answer 38

Correction for debye temp

Vib motion in solid means diffraction grating not static and decreases intensity

This is noticeable at small lattice spacings

Question 39

What is A, the absorption factor?

Answer 39

Generally a standard value for intensity

Ignored unless samples have strongly absorbing elements

Most affects signals at low 2θ

Question 40

How can you avoid problems with absorption?

Answer 40

Changing wavelength of radiation used

Question 41

What is F, the structure factor of the intensity?

Answer 41

Structural factor relating to structure of crystal to rel intensity

F = Σ f_q cos2π(hx + ky + lz) + isin2π(hx + ky + lz)

where f_q is the scattering power of each atom q
sin term 0 for cells with inversion symmetry - generally ignored

Question 42

What does f_q, the scattering factor, depend on?

Answer 42

Depends on # of electrons of an atom/ion
Allows for e- density in a crystal to be calculated

But hard to tell isoelectronic apart, e.g. Na+, F-, O2-

Question 43

What is the angular dependence of scattering factor, f_q?

Answer 43

X-rays scattered by e- as radiation siimilar size to atom
Scattered waves leads to partial destructive interference

More e- atoms have larger scattering form factor

Question 44

What diffraction peaks are present for face-centered sturctures?

Answer 44

Diffraction peaks where h,k, and l are either all even or all odd
(0 counts as even)

Due to structure factor, F

Question 45

What diffraction peaks are present for a body-centered cubic stuctures?

Answer 45

Diffraction peaks when h+k+l = 2n

Question 46

Why are there systematic absences in diffraction patterns?

Answer 46

Arise from additional translational symmetry of centred lattices

Question 47

What are the diffraction peaks present in a primitive lattice?

Answer 47

All present, no absences

100,110,111,200,210, etc.

Question 48

How do you find he lattice parameters from d-spacing?

Answer 48

1/d² then divide each from initial 1/d², and then can multiply to get full numbers

Then find h²+k²+l² and plot on x-axis against 1/d² on y-axis

Slope = 1/a²

Question 49

What happens if h²+k²+l² = 7 ?

Answer 49

Must multiply by a factor as impossible to sum three square integers
If first value is 1 then must multiply all by 2 (body-centered)