3. Diffraction & the reciprocal lattice Flashcards
What are the two categories of techniques to experimentally determine the structure of crystals?
- Real space techniques
- Reciprocal space techniques
Give an example of a real space technique to experimentally determine the structure of crystals
Scanning tunnelling microscopy (STM)
What does scanning tunnelling microscopy explain about a crystal?
It shows the structure of the crystal surface but little about the bulk.
Give an example of a reciprocal space technique to experimentally determine the structure of crystals
X-ray diffraction (XRD)
What does X-ray diffraction explain about a crystal?
It gives direct information about the reciprocal lattice and the bulk of the crystal.
Why can X-rays be diffracted by crystals?
X-ray photons have comparable wavelengths to the interplanar spacings in crystals so can be diffracted by them.
What does the Bragg law describe?
It describes the angle through which X-rays will be diffracted for a specific photon energy and set of planes.
State the Bragg law
d = interplanar spacing
n = integer number
λ = wavelength
Bragg reflection can only occur for wavelengths _______ than 2d, where d is the interplanar spacing on the order of Angstroms.
Smaller
What property must be known to complete Bragg analysis?
Interplanar spacing, d
Give the equation for interplanar spacing
d = interplanar spacing
a = lattice constant
hkl = Miller index planes
Give the equation for the energy of a photon (with the maximum wavelength for Bragg reflection)
E = hc/λ
E = energy
c = speed of light
λ = 2d = maximum wavelength
The Bragg law is a consequence of the __________ of the crystal lattice, not the basis.
Periodicity
In X-ray diffraction, what do the spot positions represent?
The direction of the beams scattered from the planes of lattice points.
Give the equation for the intensity of an elastically scattered wave
A_r = intensity
First term = incident wave
Second term = f = atomic scattering factor = form factor
Third term = amplitude decrease and phase change
What is the atomic form factor?
A reflection of the strength of an interaction between radiation and an atom. It increases with an increasing atomic number, Z, but decreases with increasing angle, 2θ.
What is the value of the atomic form factor, f, in the small angle limit?
Z
Give the approximated equation for the amplitude of an elastically scattered wave
Ar = amplitude
A0 = initial wave amplitude
R = distance to the detector from the atom
K = k’ - k
r = position of atom in the crystal
Give the equation for the amplitude of an elastically scattered wave in terms of the lattice and the basis of the diffraction crystal
A = amplitude
Term 1 = lattice (diffraction conditions)
Term 2 = basis (structure factor)
Give the expanded equation for the diffraction conditions
K = k’ - k
T = lattice translation vector
a, b, c = lattice vectors
u, v, w = integers
Give the equation for the structure factor
S = structure factor
hkl = Miller indices
f = form factor
K = k’ - k
What are the Laue conditions for diffraction?
There is a ______ scattering amplitude when the Laue conditions are true.
Large
What do scattering vectors, K, that satisfy the Laue conditions represent?
The directions of diffracted X-ray beams.
What is the recioprocal lattice?
A 3-D grid of all the possible scattering vectors, K, in k-space that satisfy the Laue conditions. It is derived from the crystal structure.
Give the equation for the reciprocal lattice translation vector
G = reciprocal lattice translation vector
a, b, c* = primitive reciprocal lattice vectors
h, k, l = Miller indices
When a scattering vector satisifies the diffraction condition what is it equal to?
Where does the Miller index plane intercept the axes?
At a/h, b/k, and c/l.
Give the equation that relates the three intercept points of the Miller index plane
Give the equation for the vector perpendicular to the Miller index plane
d = vector perpendicular to hkl plane
d_hkl = Miller plane spacing
K = scattering vector
G_hkl = reciprocal lattice vector
Give the equation for the length of the reciprocal lattice translation vector
G_hkl = reciprocal lattice vector
d_hkl = Miller plane spacing
Describe the graphical representation of X-ray diffraction when elastic scattering occurs
Elastic scattering so |k| = |k’|=2π / λ
Give the equation for the relationship between the initial and final wavevectors in the scattering process
k’ = final wavevector
k = initial wavevector
G = reciprocal lattice vector
What is the Bragg angle?
Half the angle between the initial and final wavevectors for X-ray diffraction.
What is an Ewald sphere?
A geometric construction used in X-ray diffraction to visualise the diffraction condition and to show the relationship between the initial and final wavevectors. It identifies that the X-ray diffraction condition is true when the Ewald sphere intercepts more than one point.
How is an Ewald sphere constructed?
- Plot the reciprocal lattice of the crystal.
- Draw the incident beam wavevector, k, such that it terminates at a reciprocal lattice point.
- Draw a sphere of radius k = 2π / λ about the origin of the incident wavevector.
- If the Ewald sphere intersects any other points in the reciprocal lattice, a diffracted beam will therefore be formed.
What is the Brillouin zone?
The Wigner-Seitz unit cell. It is the smallest volume enclosed by the perpendicular bisectors of the surrounding reciprocal lattice vectors.
If a wave has a wavevector of suitable direction and magnitude that, drawn from the origin, it terminates on the edge of the Brillouin zone it will meet the __________ _________.
Diffraction condition
State the conversion equation
a* = reciprocal lattice vector
b* = reciprocal lattice vector
c* = reciprocal lattice vector
a, b, c = lattice vectors
What is the purpose of the conversion equation?
To convert real space lattice vectors to reciprocal lattice vectors.
Give the equation for the volume of a simple cubic unit cell
V = volume
a, b, c = lattice vectors
What are the primitive translation vectors of a simple cubic lattice?
What are the reciprocal lattice vectors of a simple cubic lattice?
What are the boundaries for the first Brillouin zone of a simple cubic lattice?
The planes normal to the 6 shortest reciprocal lattice vectors, at their midpoints.
Give the equation for the volume of a body-centred cubic unit cell
V = volume
a, b, c = lattice vectors
What are the primitive translation vectors of a body-centred cubic lattice?
What are the reciprocal lattice vectors of a body-centred cubic lattice?
What are the boundaries for the first Brillouin zone of a body-centred cubic lattice?
The planes normal to the 12 shortest reciprocal lattice vectors.
Give the equation for the volume of a face-centred cubic unit cell
What are the primitive translation vectors of a face-centred cubic lattice?
What are the reciprocal lattice vectors of a face-centred cubic lattice?
What are the boundaries for the first Brillouin zone of a face-centred cubic lattice?
The planes normal to the 8 reciprocal lattice vectors and the six next-shortest vectors, forming a truncated octahedron
For fcc and bcc, if a lattice constant is ‘a’ in real space then in reciprocal space it is ____.
4π / a
Give the equation for the general structure factor solution for cubic crystals
S = structure factor
f = form factor
K = scattering vector
r = basis vector
What is the structure factor for a simple cubic lattice with a basis of one atom?
r1 = 0
What is the structure factor for a body-centred cubic lattice with a basis of 2 atoms?
r1 = 0
r2 = 1/2 (a + b + c)
State the condition to determine the structure factor of a body-centred cubic lattice
For bcc, the sum of h + k + l must be even to observe a diffracted beam.
What is the structure factor for a face-centred cubic lattice with a basis of 4 atoms?
r1 = 0
r2 = 1/2 (a + b)
r3 = 1/2 (b + c)
r4 = 1/2 (a + c)
State the condition to determine the structure factor of a face-centred cubic lattice
For fcc, h, k, and l must be all odd or all even to observe a diffracted beam.
The reciprocal lattice of bcc is a ___ _____ in reciprocal space with a lattice constant of ____.
fcc lattice
4π / a
The reciprocal lattice of fcc is a ___ _____ in reciprocal space with a lattice constant of ____.
bcc lattice
4π / a
For simple cubic, ___ combinations of h, k and l are allowed.
All
For body-centred cubic, the sum of h+k+l must be _____.
Even
For face-centred cubic, h, k, and l must be ____ ___ or ____ ____.
All odd
All even
Electrons with energies around 10-200 eV have de Broglie wavelengths on the same scale as __________ ________.
Interatomic distances
What is LEED?
Low energy electron diffraction. It is a probe of surface structure that forms a LEED pattern (a direct representation of the 2D reciprocal lattice of the surface).
What can be used to derive the LEED pattern?
- Reciprocal lattice
- Ewald sphere
How are the following related:
a
a*
b
b*
a* is perpendicular to b.
b* is perpendicular to a.
Give the equation for the modulus of a* in relation to |a|
|a*| = reciprocal lattice vector modulus
|a| = translational lattice vector modulus
α = angle between a and b
Give the equation for the modulus of b* in relation to |b|
|b*| = reciprocal lattice vector modulus
|b| = translational lattice vector modulus
α = angle between a and b
Which waves meet the diffraction condition? (In terms of the Brillouin zone)
Waves whose energy is large enough to allow the wavevector to touch the edge of the Brillouin zone.
