X-ray crystallography Flashcards
Why use crystallography?
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.
Experimental requirements for crystallography
A single crystal
Monochromatic (single wavelength) X-rays and a diffractometer
What is a single crystal composed of?
Small repeating units
Unit cell
The smallest volume portion of highest symmetry in a crystal
Number of unit cell types
7
What are the 7 unit cell types known as?
The 7 crystal systems
How is the crystal system determined?
By the relationships between the unit cell parameters
Cubic crystal system
a=b=c
alpha=beta=gamma=90
Hexagonal crystal system
a=b=/=c
alpha=beta=90, gamma=120
Trigonal (rhombohedral) crystal system
a=b=c
alpha=beta=gamma=/=90
Tetragonal crystal system
a=b=/=c
alpha=beta=gamma=90
Orthorhombic crystal system
a=/=b=/=c
alpha=beta=gamma=90
Monoclinic crystal system
a=/=b=/=c
alpha=gamma=90, beta=/=90
Triclinic crystal system
a=/=b=/=c
alpha=/=beta=/=gamma=/=90
a=b=c
alpha=beta=gamma=90
Cubic crystal system
a=b=/=c
alpha=beta=90, gamma=120
Hexagonal crystal system
a=b=c
alpha=beta=gamma=/=90
Trigonal (rhombohedral) crystal system
a=b=/c
alpha=beta=gamma=90
Tetragonal crystal system
a=/=b=/=c
alpha=beta=gamma=90
Orthorhombic crystal system
a=/=b=/=c
alpha=gamma=90, beta=/=90
Monoclinic crystal system
a=/=b=/=c
alpha=/=beta=/=gamma=/=90
Triclinic crystal system
How many lattice types are there?
4
Name the 4 lattice types
P (primitive)
I (body-centred)
F (face-centred)
C (centred on 2 opposing faces)
Number of lattice points in a primitive unit cell
1
Number of lattice points in a body-centred unit cell
2
Number of lattice points in a face-centred unit cell
4
Number of lattice points in a C-type unit cell
2
Order of symmetry of the crystal systems
Cubic Hexagonal Trigonal (rhombohedral) Tetragonal Orthorhombic Monoclinic Triclinic
Combining the 4 lattice types with the 7 crystal systems gives…
…the 14 Bravais lattices
All crystal structures belong to 1 of these 14 lattices
How is the Bravais lattice type determined?
By the presence of lattice points in addition to those at the unit cell corners
Cubic lattice types
P, I, F
Hexagonal lattice types
P
Trigonal (rhombohedral) lattice types
P
Tetragonal lattice types
P, I
Orthorhombic lattice types
P, I, F, C
only crystal system with all 4 lattice types
Monoclinic lattice types
P, C
Triclinic lattice types
P
Why does the orthorhombic crystal system have all possible types of lattice?
Because this system has 3 different axial values (a, b, c) but all 3 axial angles (alpha, beta, gamma) equal to 90
Why do some crystal systems have ‘missing’ lattices?
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
Method for determining the crystal system and Bravais lattice
- Identify the smallest motif in the XRD pattern
- Replace each motif with a dot
- Place the first lattice point anywhere, and the others at positions of identical environment
- Join the lattice points to form boxes (this is the unit cell)
- Select the smallest unit cell of highest symmetry - this will be one of the 7 crystal systems
- 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
Notation for Bravais lattices
Crystal system followed by lattice type
e.g. cubic P
Important note about lattice points
They don’t necessarily tell you where atoms/ions/molecules are, just where the environments are the same
Joining of lattice points
Lattice points can be joined in 2D to give lattice lines and in 3D to give lattice places
Methods of indexing lattice planes
- Weiss indices
2. Miller indices
Weiss indices
= the intercepts of the line/plane with the axial system, so are different for every single line/plane
Miller indices
= 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
What is the axial system based on?
The unit cell parameters
i.e. a, b, c
Indexation in 2D
Parallel lines will have the same (h, k) and will have the same d-spacing between adjacent pairs
Indexation in 3D
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
What planes cannot have a Weiss index?
Planes that contain the axis
But these planes can be Miller indexed by looking at the Miller indices of a parallel plane
Bragg’s law
n(lambda) = 2dsin(theta)
n=1
Assumption of Bragg’s law
Treats ‘diffraction’ as ‘reflection’
What does Bragg’s law denote?
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)
Derive Bragg’s law
Draw
What information is needed to determine a crystal structure?
Need to measure the intensity of diffraction for each Miller index (each h, k, l) (Ihkl)
Crystals are 3D, which means…
…they produce a 3D diffraction pattern
What is the diffraction pattern of a crystal?
The Fourier transform of the crystal
and vice versa - the crystal is the Fourier transform of the diffraction pattern
What do we get from the X-ray diffraction pattern of a single crystal?
‘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
What does each ‘spot’ on the frame of data correspond to?
Diffraction from 1 set of Miller planes
i.e. from 1 set of (h, k, l)s
Different spots have different intensities
Low theta
Low (h, k, l) values
Stronger reflection intensities
High theta
High (h, k, l) values
Weaker reflection intensities
Wavelength of radiation used in X-ray diffraction experiments and how this relates to resolution
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
What does the position of a spot in the diffraction pattern tell us?
It is related to the unit cell and associated h, k, l
What does the intensity (brightness) of each spot in the diffraction pattern tell us?
It contains embedded information on every atom type and its location (x, y, z) in the unit cell
What are (x, y, z)?
Fractions along a, b, and c, respectively,
= fractional coordinates
How to relate the positions of the spots in the diffraction pattern to the unit cell/Miller planes
Using reciprocal space/the reciprocal lattice
How to relate the diffraction spots to the contents of the unit cell (i.e. the atom types present in molecules)
Using their intensities (Ihkl) and how they relate to electron density at any location (x, y, z) in the unit cell
The reciprocal lattice
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
Why is a diffraction pattern said to be in reciprocal space?
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
How is the position of a spot in the diffraction pattern determined?
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
What does a set of planes in direct space appear as in reciprocal space?
A spot
These spots in reciprocal space are then referred to as the reciprocal lattice
Data collection process
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
What can intensity (Ihkl) data be used for?
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
Why is X-ray diffraction so successful at structure determination?
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)
What are the necessary conditions for observing diffraction, according to Bragg’s law?
Wavelength of incident radiation
Angle of incidence between this X-ray beam and a Miller set
d-spacing of the Miller index
Reciprocal space
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
What information is contained in the intensity of each diffraction spot?
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
What needs to be produced in order to solve a crystal structure?
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))
What does the electron density map use?
The square root of the I(hkl) values
= F(hkl) = structure factor
I(hkl) =
= [F(hkl)]^2
F(hkl) =
sqrt[I(hkl)]
What is the phase problem?
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
Why do different atoms scatter X-rays differently?
Because different atoms have different electron clouds
Atomic Scattering Factor
Describes the characteristic way that the electron cloud in a particular atom interacts with X-rays
Notation for atomic scattering factor
f(j), where j = particular atom type in question
Effect of Bragg angle on scattering power
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
How does the atomic scattering factor contribute to the structure factor?
Equation on flashcards
There is one structure factor…
…for each Miller index (because each Miller index has only one intensity I(hkl))
What does the structure factor contain information on?
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
Notation for a calculated structure factor
F(hkl) with superscript calc
Notation for an observed structure factor (i.e. from experimental data)
F(hkl) with superscript obs
What is known and still unknown at the end of data collection and integration?
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
What governs the spatial relationship between molecules in the unit cell?
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
Space group
A description of the symmetry of the crystal
Types of symmetry element
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
What are the 4 non-translational symmetry elements?
Rotation
Inversion
Reflection
Rotation-inversion
Rotation
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
Inversion
Non-translational symmetry element
Notation = -
Symbol on space group diagram = o (parallel/perpendicular)
Reflection
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
Rotation inversion
Non-translational symmetry element
Notation = n-bar
Symbol for e.g. 4-bar would be slanted oval within diamond (perpendicular)
Non-translational symmetry elements…
…do not cause absences in crystal data
What are the 2 translational symmetry elements?
Screw axis
Glide plane
Screw axis
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)
Glide plane
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) = —
Translational symmetry elements…
…do cause absences in crystal data
Number of combinations of symmetry elements
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
What is a space group diagram?
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
What is an equivalent position diagram?
It shows how the objects in the unit cell are related by the symmetry elements in the space group
How many space groups does the triclinic crystal system have?
2 of the 230
P1 and P1-bar
Rules for generating an equivalent position diagram
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
P1
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
P1-bar
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
When are special positions possible?
In any space group with non-translational symmetry elements
Where are special positions?
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
Property of an object located at a special position
The object itself must have that symmetry element within it
How to approximate the volume of 1 molecule
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
Centrosymmetric space group
Contains a centre of inversion
Non-centrosymmetric space group
Has no inversion centre
Enantiomorphous space group
Can only contain molecules of one hand
Non-enantiomorphous space group
Must contain pairs of enantiomers
Symmetry elements in monoclinic space groups
All concern the b-axis (beta = 90)
Symmetry elements in orthorhombic space groups
Listed in terms of how they affect the a, b and c axes, respectively
Rotations and screw axes
Always parallel to the associated axes
Mirrors and/or glide planes
Reflection component always perpendicular to the associated axis
Symmetry elements in tetragonal space groups
All concern the c axis because c is the ‘odd one out’
a=b=/=c, alpha=beta=gamma=90
