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PHYS3002: Introduction to Solid State Physics > 3. Diffraction & the reciprocal lattice > Flashcards

3. Diffraction & the reciprocal lattice Flashcards

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1
Q

What are the two categories of techniques to experimentally determine the structure of crystals?

A
  • Real space techniques
  • Reciprocal space techniques
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2
Q

Give an example of a real space technique to experimentally determine the structure of crystals

A

Scanning tunnelling microscopy (STM)

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3
Q

What does scanning tunnelling microscopy explain about a crystal?

A

It shows the structure of the crystal surface but little about the bulk.

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4
Q

Give an example of a reciprocal space technique to experimentally determine the structure of crystals

A

X-ray diffraction (XRD)

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5
Q

What does X-ray diffraction explain about a crystal?

A

It gives direct information about the reciprocal lattice and the bulk of the crystal.

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6
Q

Why can X-rays be diffracted by crystals?

A

X-ray photons have comparable wavelengths to the interplanar spacings in crystals so can be diffracted by them.

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7
Q

What does the Bragg law describe?

A

It describes the angle through which X-rays will be diffracted for a specific photon energy and set of planes.

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8
Q

State the Bragg law

A

d = interplanar spacing
n = integer number
λ = wavelength

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9
Q

Bragg reflection can only occur for wavelengths _______ than 2d, where d is the interplanar spacing on the order of Angstroms.

A

Smaller

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10
Q

What property must be known to complete Bragg analysis?

A

Interplanar spacing, d

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11
Q

Give the equation for interplanar spacing

A

d = interplanar spacing
a = lattice constant
hkl = Miller index planes

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12
Q

Give the equation for the energy of a photon (with the maximum wavelength for Bragg reflection)

A

E = hc/λ

E = energy
c = speed of light
λ = 2d = maximum wavelength

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13
Q

The Bragg law is a consequence of the __________ of the crystal lattice, not the basis.

A

Periodicity

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14
Q

In X-ray diffraction, what do the spot positions represent?

A

The direction of the beams scattered from the planes of lattice points.

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15
Q

Give the equation for the intensity of an elastically scattered wave

A

A_r = intensity
First term = incident wave
Second term = f = atomic scattering factor = form factor
Third term = amplitude decrease and phase change

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16
Q

What is the atomic form factor?

A

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.

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17
Q

What is the value of the atomic form factor, f, in the small angle limit?

A

Z

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18
Q

Give the approximated equation for the amplitude of an elastically scattered wave

A

Ar = amplitude
A0 = initial wave amplitude
R = distance to the detector from the atom
K = k’ - k
r = position of atom in the crystal

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19
Q

Give the equation for the amplitude of an elastically scattered wave in terms of the lattice and the basis of the diffraction crystal

A

A = amplitude
Term 1 = lattice (diffraction conditions)
Term 2 = basis (structure factor)

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20
Q

Give the expanded equation for the diffraction conditions

A

K = k’ - k
T = lattice translation vector
a, b, c = lattice vectors
u, v, w = integers

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21
Q

Give the equation for the structure factor

A

S = structure factor
hkl = Miller indices
f = form factor
K = k’ - k

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22
Q

What are the Laue conditions for diffraction?

A
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23
Q

There is a ______ scattering amplitude when the Laue conditions are true.

A

Large

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24
Q

What do scattering vectors, K, that satisfy the Laue conditions represent?

A

The directions of diffracted X-ray beams.

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25
Q

What is the recioprocal lattice?

A

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.

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26
Q

Give the equation for the reciprocal lattice translation vector

A

G = reciprocal lattice translation vector
a, b, c* = primitive reciprocal lattice vectors
h, k, l = Miller indices

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27
Q

When a scattering vector satisifies the diffraction condition what is it equal to?

A
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28
Q

Where does the Miller index plane intercept the axes?

A

At a/h, b/k, and c/l.

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29
Q

Give the equation that relates the three intercept points of the Miller index plane

A
30
Q

Give the equation for the vector perpendicular to the Miller index plane

A

d = vector perpendicular to hkl plane
d_hkl = Miller plane spacing
K = scattering vector
G_hkl = reciprocal lattice vector

31
Q

Give the equation for the length of the reciprocal lattice translation vector

A

G_hkl = reciprocal lattice vector
d_hkl = Miller plane spacing

32
Q

Describe the graphical representation of X-ray diffraction when elastic scattering occurs

A

Elastic scattering so |k| = |k’|=2π / λ

33
Q

Give the equation for the relationship between the initial and final wavevectors in the scattering process

A

k’ = final wavevector
k = initial wavevector
G = reciprocal lattice vector

34
Q

What is the Bragg angle?

A

Half the angle between the initial and final wavevectors for X-ray diffraction.

35
Q

What is an Ewald sphere?

A

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.

36
Q

How is an Ewald sphere constructed?

A
  1. Plot the reciprocal lattice of the crystal.
  2. Draw the incident beam wavevector, k, such that it terminates at a reciprocal lattice point.
  3. Draw a sphere of radius k = 2π / λ about the origin of the incident wavevector.
  4. If the Ewald sphere intersects any other points in the reciprocal lattice, a diffracted beam will therefore be formed.
37
Q

What is the Brillouin zone?

A

The Wigner-Seitz unit cell. It is the smallest volume enclosed by the perpendicular bisectors of the surrounding reciprocal lattice vectors.

38
Q

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 __________ _________.

A

Diffraction condition

39
Q

State the conversion equation

A

a* = reciprocal lattice vector
b* = reciprocal lattice vector
c* = reciprocal lattice vector
a, b, c = lattice vectors

40
Q

What is the purpose of the conversion equation?

A

To convert real space lattice vectors to reciprocal lattice vectors.

41
Q

Give the equation for the volume of a simple cubic unit cell

A

V = volume
a, b, c = lattice vectors

42
Q

What are the primitive translation vectors of a simple cubic lattice?

A
43
Q

What are the reciprocal lattice vectors of a simple cubic lattice?

A
44
Q

What are the boundaries for the first Brillouin zone of a simple cubic lattice?

A

The planes normal to the 6 shortest reciprocal lattice vectors, at their midpoints.

45
Q

Give the equation for the volume of a body-centred cubic unit cell

A

V = volume
a, b, c = lattice vectors

46
Q

What are the primitive translation vectors of a body-centred cubic lattice?

A
47
Q

What are the reciprocal lattice vectors of a body-centred cubic lattice?

A
48
Q

What are the boundaries for the first Brillouin zone of a body-centred cubic lattice?

A

The planes normal to the 12 shortest reciprocal lattice vectors.

49
Q

Give the equation for the volume of a face-centred cubic unit cell

A
50
Q

What are the primitive translation vectors of a face-centred cubic lattice?

A
51
Q

What are the reciprocal lattice vectors of a face-centred cubic lattice?

A
52
Q

What are the boundaries for the first Brillouin zone of a face-centred cubic lattice?

A

The planes normal to the 8 reciprocal lattice vectors and the six next-shortest vectors, forming a truncated octahedron

53
Q

For fcc and bcc, if a lattice constant is ‘a’ in real space then in reciprocal space it is ____.

A

4π / a

54
Q

Give the equation for the general structure factor solution for cubic crystals

A

S = structure factor
f = form factor
K = scattering vector
r = basis vector

55
Q

What is the structure factor for a simple cubic lattice with a basis of one atom?

A

r1 = 0

56
Q

What is the structure factor for a body-centred cubic lattice with a basis of 2 atoms?

A

r1 = 0
r2 = 1/2 (a + b + c)

57
Q

State the condition to determine the structure factor of a body-centred cubic lattice

A

For bcc, the sum of h + k + l must be even to observe a diffracted beam.

58
Q

What is the structure factor for a face-centred cubic lattice with a basis of 4 atoms?

A

r1 = 0
r2 = 1/2 (a + b)
r3 = 1/2 (b + c)
r4 = 1/2 (a + c)

59
Q

State the condition to determine the structure factor of a face-centred cubic lattice

A

For fcc, h, k, and l must be all odd or all even to observe a diffracted beam.

60
Q

The reciprocal lattice of bcc is a ___ _____ in reciprocal space with a lattice constant of ____.

A

fcc lattice
4π / a

61
Q

The reciprocal lattice of fcc is a ___ _____ in reciprocal space with a lattice constant of ____.

A

bcc lattice
4π / a

62
Q

For simple cubic, ___ combinations of h, k and l are allowed.

A

All

63
Q

For body-centred cubic, the sum of h+k+l must be _____.

A

Even

64
Q

For face-centred cubic, h, k, and l must be ____ ___ or ____ ____.

A

All odd
All even

65
Q

Electrons with energies around 10-200 eV have de Broglie wavelengths on the same scale as __________ ________.

A

Interatomic distances

66
Q

What is LEED?

A

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).

67
Q

What can be used to derive the LEED pattern?

A
  • Reciprocal lattice
  • Ewald sphere
68
Q

How are the following related:
a
a*
b
b*

A

a* is perpendicular to b.
b* is perpendicular to a.

69
Q

Give the equation for the modulus of a* in relation to |a|

A

|a*| = reciprocal lattice vector modulus
|a| = translational lattice vector modulus
α = angle between a and b

70
Q

Give the equation for the modulus of b* in relation to |b|

A

|b*| = reciprocal lattice vector modulus
|b| = translational lattice vector modulus
α = angle between a and b

71
Q

Which waves meet the diffraction condition? (In terms of the Brillouin zone)

A

Waves whose energy is large enough to allow the wavevector to touch the edge of the Brillouin zone.

PHYS3002: Introduction to Solid State Physics (21 decks)

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