Diffraction Flashcards
What is the main feature of a crystal?
Array of atoms/molecules which have periodic translational symmetry
What is a lattice?
Array of points in space which describe translational symm
Mathematical
What is the basis of a crystal?
Atoms or molecules which are attached to each lattice point to give structure
How can points in a lattice relate to eachother?
r’ = r + n1a + n2b = n3c
Where these a,b,c are along x,y,z axis respectively
What is a primitive lattice?
One lattice point per unit cell
All pairs of lattice points r and r’ related by integer values of cell units
How are lattices defined in 3D?
What is a cubic cell?
4x 3-fold rotation axis
Seen as a cube with a point on each vertex
What are the 7 possible primitive lattices?
What is a body-centered cell?
What is a face-centered lattice?
What is a face-centered lattice?
How many lattice points are in a body-centered cell?
2 points
so 2 basis units
How many lattice points are in a face-centered-cell?
4 lattice points
So 4 basis units
How many lattice points and atoms are in the unit cell of hcp?
2 atoms but 1 lattice point
What is the range of lattice spacing in a crystal?
1-20 angstrom
where 1 angstrom = 10-10 m
How do X-rays interact with a 3D lattice?
Reflected by planes of the crystal lattice
Difference in phase of waves reflected gives separation of planes of the lattice
What is Bragg’s Law and how is it used?
nλ = 2dsinθ
2 waves in phase must be a whole wavenumber difference
d is distance between planes of lattice
What is the method for understanding sets of lattice planes?
Miller Indicies
How are the miller indicies done in 2d?
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
Describe these using Miller Indicies
LHS: (1,2) plane
RHS: (2,-1) plane
What does a negative in miller indicies indicate?
Different orientation of the plane
Runs in opposite direction
How are miller indicies in 3d worked out?
Planes cut x,y,z axes at (1/h, 1/k, 1/l)
These are then reffered to as a (h,k,l) plane
Describe these 3d planes using Millers indicies
LHS: parallel to z-axis and cuts through others once so is a (1,1,0) plane
RHS: (2,2,3) plane
What is the relation of distance between lattice planes and h,k,l for a cubic cell?
d = a/Sqrt[h2 + k2 + l2]
where a is the cubic lattice constant
(1/d2) = [h2 + k2 + l2]/a2
What is the relation between d and h,k,l for a tetragonal cell?
(1/d2) = [h2 + k2]/a2 + l2/a2
What is the relation between d and h,k,l for a orthorombic cell?
(1/d2) = h2/a2 + k2/b2 + l2/c2
How many diffraction peaks are seen for a primitive lattice?
Peak for each hkl lattice plane
No systematic absences
What is the equation relating a to h,k,l?
a2 = (λ2/4sin2θ) (h2+k2+l2)
How are crystals measured in a powder?
Measure powder with large number of crystals and assume some are correctly orientated
Those at bragg angle will diffract and many so is likely
How are crystals measured in a powder?
Measure powder with large number of crystals and assume some are correctly orientated
Those at bragg angle will diffract and many so is likely
What kind of beam is required for x-ray diffraction?
Monochromatic
Use a metal filter
What is the format of a diffraction pattern?
2θ is on x-axis (where θ is from detection), this gives d-spacing
Intensity is on y-axis
How can you find h,k,l from d in diffraction pattern?
1/d2 = [h2+k2+l2]/a2
find 1/d2 for each signal
Find h2+k2+l2 from this
Then can derive (h,k,l)
How can x-ray diffraction pattern show the average particle size?
The broader the diffraction peak then the smaller the particle size
Due to destructive interference
What is the formula for intensity of h,k,l of a reflection?
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
What is the Lorentz polarization factor?
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
What is J for intensity of a reflected wave?
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)
What is W, the debye temperature correction factor?
Correction for debye temp
Vib motion in solid means diffraction grating not static and decreases intensity
This is noticeable at small lattice spacings
What is A, the absorption factor?
Generally a standard value for intensity
Ignored unless samples have strongly absorbing elements
Most affects signals at low 2θ
How can you avoid problems with absorption?
Changing wavelength of radiation used
What is F, the structure factor of the intensity?
Structural factor relating to structure of crystal to rel intensity
F = Σ fq cos2π(hx + ky + lz) + isin2π(hx + ky + lz)
where fq is the scattering power of each atom q
sin term 0 for cells with inversion symmetry - generally ignored
What does fq, the scattering factor, depend on?
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-
What is the angular dependence of scattering factor, fq?
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
What diffraction peaks are present for face-centered sturctures?
Diffraction peaks where h,k, and l are either all even or all odd
(0 counts as even)
Due to structure factor, F
What diffraction peaks are present for a body-centered cubic stuctures?
Diffraction peaks when h+k+l = 2n
Why are there systematic absences in diffraction patterns?
Arise from additional translational symmetry of centred lattices
What are the diffraction peaks present in a primitive lattice?
All present, no absences
100,110,111,200,210, etc.
How do you find he lattice parameters from d-spacing?
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
What happens if h2+k2+l2 = 7 ?
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)
