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X-ray crystallography Flashcards

1
Q

Why use crystallography?

A

Accurate and unambiguous result
Relatively cheap
Gives 3D representation of molecule/material
Bond lengths are accurate to 3 d.p. and bond angles are accurate to 2 d.p.

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2
Q

Experimental requirements for crystallography

A

A single crystal

Monochromatic (single wavelength) X-rays and a diffractometer

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3
Q

What is a single crystal composed of?

A

Small repeating units

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4
Q

Unit cell

A

The smallest volume portion of highest symmetry in a crystal

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5
Q

Number of unit cell types

A

7

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6
Q

What are the 7 unit cell types known as?

A

The 7 crystal systems

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7
Q

How is the crystal system determined?

A

By the relationships between the unit cell parameters

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8
Q

Cubic crystal system

A

a=b=c

alpha=beta=gamma=90

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9
Q

Hexagonal crystal system

A

a=b=/=c

alpha=beta=90, gamma=120

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10
Q

Trigonal (rhombohedral) crystal system

A

a=b=c

alpha=beta=gamma=/=90

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11
Q

Tetragonal crystal system

A

a=b=/=c

alpha=beta=gamma=90

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12
Q

Orthorhombic crystal system

A

a=/=b=/=c

alpha=beta=gamma=90

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13
Q

Monoclinic crystal system

A

a=/=b=/=c

alpha=gamma=90, beta=/=90

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14
Q

Triclinic crystal system

A

a=/=b=/=c

alpha=/=beta=/=gamma=/=90

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15
Q

a=b=c

alpha=beta=gamma=90

A

Cubic crystal system

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16
Q

a=b=/=c

alpha=beta=90, gamma=120

A

Hexagonal crystal system

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17
Q

a=b=c

alpha=beta=gamma=/=90

A

Trigonal (rhombohedral) crystal system

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18
Q

a=b=/c

alpha=beta=gamma=90

A

Tetragonal crystal system

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19
Q

a=/=b=/=c

alpha=beta=gamma=90

A

Orthorhombic crystal system

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20
Q

a=/=b=/=c

alpha=gamma=90, beta=/=90

A

Monoclinic crystal system

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21
Q

a=/=b=/=c

alpha=/=beta=/=gamma=/=90

A

Triclinic crystal system

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22
Q

How many lattice types are there?

A

4

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23
Q

Name the 4 lattice types

A

P (primitive)
I (body-centred)
F (face-centred)
C (centred on 2 opposing faces)

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24
Q

Number of lattice points in a primitive unit cell

A

1

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25
Q

Number of lattice points in a body-centred unit cell

A

2

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26
Q

Number of lattice points in a face-centred unit cell

A

4

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27
Q

Number of lattice points in a C-type unit cell

A

2

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28
Q

Order of symmetry of the crystal systems

A 
Cubic
Hexagonal
Trigonal (rhombohedral)
Tetragonal
Orthorhombic
Monoclinic
Triclinic
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29
Q

Combining the 4 lattice types with the 7 crystal systems gives…

A

…the 14 Bravais lattices

All crystal structures belong to 1 of these 14 lattices

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30
Q

How is the Bravais lattice type determined?

A

By the presence of lattice points in addition to those at the unit cell corners

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31
Q

Cubic lattice types

A

P, I, F

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32
Q

Hexagonal lattice types

A

P

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33
Q

Trigonal (rhombohedral) lattice types

A

P

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34
Q

Tetragonal lattice types

A

P, I

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35
Q

Orthorhombic lattice types

A

P, I, F, C

only crystal system with all 4 lattice types

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36
Q

Monoclinic lattice types

A

P, C

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37
Q

Triclinic lattice types

A

P

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38
Q

Why does the orthorhombic crystal system have all possible types of lattice?

A

Because this system has 3 different axial values (a, b, c) but all 3 axial angles (alpha, beta, gamma) equal to 90

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39
Q

Why do some crystal systems have ‘missing’ lattices?

A

The ‘missing’ lattices can be readily represented by a ‘listed’ Bravais lattice in the same crystal system
i.e. the missing lattices can be simplified into the other lattice types present in the crystal system

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40
Q

Method for determining the crystal system and Bravais lattice

A
  1. Identify the smallest motif in the XRD pattern
  2. Replace each motif with a dot
  3. Place the first lattice point anywhere, and the others at positions of identical environment
  4. Join the lattice points to form boxes (this is the unit cell)
  5. Select the smallest unit cell of highest symmetry - this will be one of the 7 crystal systems
  6. Check for the presence of additional lattice points at the centre of the unit cells faces or the centre of the unit cell - this will determine the lattice type
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41
Q

Notation for Bravais lattices

A

Crystal system followed by lattice type

e.g. cubic P

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42
Q

Important note about lattice points

A

They don’t necessarily tell you where atoms/ions/molecules are, just where the environments are the same

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43
Q

Joining of lattice points

A

Lattice points can be joined in 2D to give lattice lines and in 3D to give lattice places

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44
Q

Methods of indexing lattice planes

A
  1. Weiss indices

2. Miller indices

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45
Q

Weiss indices

A

= the intercepts of the line/plane with the axial system, so are different for every single line/plane

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46
Q

Miller indices

A

= the reciprocal of the Weiss indices, with the fractions cleared
Each Miller index corresponds to a family of parallel lines/planes with a characteristic ‘d’ spacing
X-rays interact with electron clouds in Miller indices

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47
Q

What is the axial system based on?

A

The unit cell parameters

i.e. a, b, c

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48
Q

Indexation in 2D

A

Parallel lines will have the same (h, k) and will have the same d-spacing between adjacent pairs

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49
Q

Indexation in 3D

A

Miller indices refer to sets of parallel planes

The spacing between the planes that make up a Miller index is known as the ‘d’ spacing

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50
Q

What planes cannot have a Weiss index?

A

Planes that contain the axis

But these planes can be Miller indexed by looking at the Miller indices of a parallel plane

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51
Q

Bragg’s law

A

n(lambda) = 2dsin(theta)

n=1

52
Q

Assumption of Bragg’s law

A

Treats ‘diffraction’ as ‘reflection’

53
Q

What does Bragg’s law denote?

A

The conditions needed to observe a ‘reflection’ from a given set of Miller planes (i.e. for one given Miller index h, k, l)
Provides NO information on the intensity of the reflection (but this is measured in the experiment)

54
Q

Derive Bragg’s law

A

Draw

55
Q

What information is needed to determine a crystal structure?

A

Need to measure the intensity of diffraction for each Miller index (each h, k, l) (Ihkl)

56
Q

Crystals are 3D, which means…

A

…they produce a 3D diffraction pattern

57
Q

What is the diffraction pattern of a crystal?

A

The Fourier transform of the crystal

and vice versa - the crystal is the Fourier transform of the diffraction pattern

58
Q

What do we get from the X-ray diffraction pattern of a single crystal?

A

‘Frames’ of data
A ‘frame’ of data is a 2D snapshot (‘photograph’) of part of the diffraction pattern from a crystal
Because crystals are 3D, they have 3D diffraction patterns
Therefore lots of frames need to be taken of a crystal to measure its diffraction pattern
This is done by moving the crystal in increments and recording a frame of data at each orientation

59
Q

What does each ‘spot’ on the frame of data correspond to?

A

Diffraction from 1 set of Miller planes
i.e. from 1 set of (h, k, l)s
Different spots have different intensities

60
Q

Low theta

A

Low (h, k, l) values

Stronger reflection intensities

61
Q

High theta

A

High (h, k, l) values

Weaker reflection intensities

62
Q

Wavelength of radiation used in X-ray diffraction experiments and how this relates to resolution

A

Mo(Kalpha) radation, lambda = 0.71073 A
For this wavelength, the entire diffraction pattern is collected up to theta = 27.5degrees
Subbing these values into Bragg’s law gives d = 0.77 A
This is the resolution of the experiment - can distinguish between atoms >= 0.77 A apart

63
Q

What does the position of a spot in the diffraction pattern tell us?

A

It is related to the unit cell and associated h, k, l

64
Q

What does the intensity (brightness) of each spot in the diffraction pattern tell us?

A

It contains embedded information on every atom type and its location (x, y, z) in the unit cell

65
Q

What are (x, y, z)?

A

Fractions along a, b, and c, respectively,

= fractional coordinates

66
Q

How to relate the positions of the spots in the diffraction pattern to the unit cell/Miller planes

A

Using reciprocal space/the reciprocal lattice

67
Q

How to relate the diffraction spots to the contents of the unit cell (i.e. the atom types present in molecules)

A

Using their intensities (Ihkl) and how they relate to electron density at any location (x, y, z) in the unit cell

68
Q

The reciprocal lattice

A

Consists of points on a grid that represent diffraction possibilities
Each of these points can be labelled with a Miller index, which corresponds to the planes from which diffraction could occur

69
Q

Why is a diffraction pattern said to be in reciprocal space?

A

It is based on units of 1/d (i.e A^-1)
There is an inverse relationship between sin(theta) and d (sin(theta) is proportional to 1/d, from Bragg equation)
This relationship is observed when examining a diffraction pattern - the distance of each spot from the centre of the frame is proportional to sin(theta) and is therefore also proportional to 1/d

70
Q

How is the position of a spot in the diffraction pattern determined?

A

By the 1/d value of one set of Miller planes (one h, k, l)
Start at the origin and draw a line perpendicular to a Miller set, terminating the line at a distance of 1/d from the origin
At precisely this point, the intensity of the spot in the diffraction pattern can be observed for this particular Miller set

71
Q

What does a set of planes in direct space appear as in reciprocal space?

A

A spot

These spots in reciprocal space are then referred to as the reciprocal lattice

72
Q

Data collection process

A

Involves measuring the intensities of the diffraction spots in reciprocal space
This is done by processing the frames of data that make up the complete diffraction pattern from a single crystal
The diffraction from each Miller set is ‘integrated’ across the frames to which it contributes

73
Q

What can intensity (Ihkl) data be used for?

A

To produce an electron density map for a portion of the unit cell in direct space, for a given crystal
From this, the atom positions and types in the structure can be determined

74
Q

Why is X-ray diffraction so successful at structure determination?

A

Due to the fact that the arrangement of molecules in a crystal is highly ordered
Allows you to identify locations within the 3D array that are identical (= lattice points)

75
Q

What are the necessary conditions for observing diffraction, according to Bragg’s law?

A

Wavelength of incident radiation
Angle of incidence between this X-ray beam and a Miller set
d-spacing of the Miller index

76
Q

Reciprocal space

A

Allows us to relate diffraction patterns to unit cells, Bravais lattices and Miller planes
It relates Miller planes in direct space to the location of the diffraction spots from each set of planes as they appear in the diffraction pattern

77
Q

What information is contained in the intensity of each diffraction spot?

A

Information about all the atoms in the unit cell
Crystallographers can extract this information from the intensities in order to work out where molecules are located within the unit cell and where atoms of different types are located within each molecule

78
Q

What needs to be produced in order to solve a crystal structure?

A

An electron density map
This allows you to determine where atoms are located in terms of electrons per cubic Angstrom at all locations (x, y, z) in the unit cell
(i.e. p(xyz))

79
Q

What does the electron density map use?

A

The square root of the I(hkl) values

= F(hkl) = structure factor

80
Q

I(hkl) =

A

= [F(hkl)]^2

81
Q

F(hkl) =

A

sqrt[I(hkl)]

82
Q

What is the phase problem?

A

The X-ray experiment measures I(hkl) values, from which the magnitude of the F(hkl) values can be calculated
However, the sign of the F(hkl) values cannot be calculated from these experimental measurements

83
Q

Why do different atoms scatter X-rays differently?

A

Because different atoms have different electron clouds

84
Q

Atomic Scattering Factor

A

Describes the characteristic way that the electron cloud in a particular atom interacts with X-rays

85
Q

Notation for atomic scattering factor

A

f(j), where j = particular atom type in question

86
Q

Effect of Bragg angle on scattering power

A

At low Bragg angles, the scattering power is proportional to the number of electrons in the cloud
The ability of electron clouds surrounding atomic nuclei to scatter X-rays tails off at high Bragg angles

87
Q

How does the atomic scattering factor contribute to the structure factor?

A

Equation on flashcards

88
Q

There is one structure factor…

A

…for each Miller index (because each Miller index has only one intensity I(hkl))

89
Q

What does the structure factor contain information on?

A 
The Miller set of planes (h, k, l)
The positions (xj, yj, zj) of ALL atoms in the unit cell (j = 1 to j = n)
ALL atom types present in the unit cell (fj)

Therefore, if all atom types/positions in the unit cell are known, F(hkl) values (and their phases) can be calculated and an electron density map constructed

90
Q

Notation for a calculated structure factor

A

F(hkl) with superscript calc

91
Q

Notation for an observed structure factor (i.e. from experimental data)

A

F(hkl) with superscript obs

92
Q

What is known and still unknown at the end of data collection and integration?

A

KNOWN: unit cell parameters, crystal system, I(hkl) for all Miller indices with theta values up to 27.5 degrees
UNKNOWN: symmetry relationships between ‘molecules’ in the unit cells

93
Q

What governs the spatial relationship between molecules in the unit cell?

A

Symmetry
The associated symmetry elements can be determined by looking at the reflections for which I(hkl) = 0 - these absent reflections often form patterns (absences)
Space group determination involves relating these absences to the symmetry elements that govern the spatial relationships between molecules in the unit cell to each other

94
Q

Space group

A

A description of the symmetry of the crystal

95
Q

Types of symmetry element

A

There are a total of 6 types of symmetry element possible

Fall into two categories: 4 non-translational symmetry elements and 2 translational symmetry elements

96
Q

What are the 4 non-translational symmetry elements?

A

Rotation
Inversion
Reflection
Rotation-inversion

97
Q

Rotation

A

Non-translational symmetry element
ALWAYS ANTICLOCKWISE
Notation = n (n= integer)
Symbol on space group diagram if n=2 and parallel to plane of projection =
Symbol on space group diagram if n=2 and perpendicular to plane of projection = oval
Symbol on space group diagram if n=3 and parallel to plane of projection = —
Symbol on space group diagram if n=3 and perpendicular to plane of projection = triangle
Symbol on space group diagram if n=4 and parallel to plane of projection = —
Symbol on space group diagram if n=4 and perpendicular to plane of projection = square
Symbol on space group diagram if n=6 and parallel to plane of projection = —
Symbol on space group diagram if n=6 and perpendicular to plane of projection = hexagon

98
Q

Inversion

A

Non-translational symmetry element
Notation = -
Symbol on space group diagram = o (parallel/perpendicular)

99
Q

Reflection

A

Non-translational symmetry element
Notation = m
Symbol on space group diagram if perpendicular to plane of projection = thicker line
Symbol on space group diagram if parallel to plane of projection = backwards r

100
Q

Rotation inversion

A

Non-translational symmetry element
Notation = n-bar
Symbol for e.g. 4-bar would be slanted oval within diamond (perpendicular)

101
Q

Non-translational symmetry elements…

A

…do not cause absences in crystal data

102
Q

What are the 2 translational symmetry elements?

A

Screw axis

Glide plane

103
Q

Screw axis

A

Translate and rotate
Translate by small/big and rotate by 360/big
Symbol on space group diagram for 2(1) screw axis parallel to plane of projection = double-headed half arrow
Symbol on space group diagram for 2(1) screw axis perpendicular to plane of projection = oval with 2 curved lines out of top and bottom
Symbol on space group diagram for 4(1) screw axis parallel to plane of projection = —
Symbol on space group diagram for 4(1) screw axis perpendicular to plane of projection = square with lines coming out of each corner (anticlockwise)
Symbol on space group diagram for 4(3) screw axis parallel to plane of projection = —
Symbol on space group diagram for 4(3) screw axis perpendicular to plane of projection = square with lines coming out of each corner (clockwise)

104
Q

Glide plane

A

Translate and reflect
Translate by 1/2 then reflect in mirror either parallel or perpendicular to the translation axis
Notation = a, b, c (denotes direction of translation)
Symbol (perpendicular mirror) = ……. or -.-.-.-
Symbol (parallel mirror, a) = across-down arrow
Symbol (parallel mirror, b) = backwards r arrow
Symbol (parallel mirror, c) = —

105
Q

Translational symmetry elements…

A

…do cause absences in crystal data

106
Q

Number of combinations of symmetry elements

A

Finite - 230
Each of the 230 combinations of symmetry elements is a space group
The 230 space groups are distributed unevenly across the 14 Bravais lattices

107
Q

What is a space group diagram?

A

A 2D projection of a 3D unit cell that shows all the symmetry of the unit cell - i.e. every possible symmetry relationship between all objects in the unit cell
View direction is along c-axis, with b-axis across the page and the a-axis down the page

108
Q

What is an equivalent position diagram?

A

It shows how the objects in the unit cell are related by the symmetry elements in the space group

109
Q

How many space groups does the triclinic crystal system have?

A

2 of the 230

P1 and P1-bar

110
Q

Rules for generating an equivalent position diagram

A 
Only operate using the symmetry elements in the space group title
Only operate (ONCE) on all objects present with each symmetry element in the space group title
111
Q

P1

A

Simplest space group
Only one asymmetric object in the unit cell
No absence conditions
P-type = 1 lattice point per unit cell, therefore 1 lattice point = 1 object
An optically pure compound could crystallise in this space group

112
Q

P1-bar

A

Two asymmetric objects in the unit cell
No absence conditions
P-type = 1 lattice point per unit cell, therefore 1 lattice point = 2 objects related by a centre of inversion (unless special positions are involved)
An optically pure compound could not crystallise in this space group

113
Q

When are special positions possible?

A

In any space group with non-translational symmetry elements

114
Q

Where are special positions?

A

At the locations of non-translational symmetry elements
i.e. special positions are occupied by objects placed exactly at a symmetry element
If an object is located at a special position, there will always be fewer objects in the unit cell than the number of general positions

115
Q

Property of an object located at a special position

A

The object itself must have that symmetry element within it

116
Q

How to approximate the volume of 1 molecule

A

Add up number of each atom type present multiples by 20 A^3

Can divide the volume of the unit cell by the volume of the molecule to estimate how many molecules per unit cell

117
Q

Centrosymmetric space group

A

Contains a centre of inversion

118
Q

Non-centrosymmetric space group

A

Has no inversion centre

119
Q

Enantiomorphous space group

A

Can only contain molecules of one hand

120
Q

Non-enantiomorphous space group

A

Must contain pairs of enantiomers

121
Q

Symmetry elements in monoclinic space groups

A

All concern the b-axis (beta = 90)

122
Q

Symmetry elements in orthorhombic space groups

A

Listed in terms of how they affect the a, b and c axes, respectively

123
Q

Rotations and screw axes

A

Always parallel to the associated axes

124
Q

Mirrors and/or glide planes

A

Reflection component always perpendicular to the associated axis

125
Q

Symmetry elements in tetragonal space groups

A

All concern the c axis because c is the ‘odd one out’

a=b=/=c, alpha=beta=gamma=90

x (7 decks)

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