Crystallography Flashcards
X-Ray Diffraction (XRD)
- single crystal X-ray diffraction (SXD): analyses 1 crystal
- powder X-ray diffraction (PXRD): analyses powders/bulk solids
~ millions of crystallites or amorphous (or in between i.e. poorly crystalline)
SXD
use just one crystal
- rotate in 3D (data is 3-dimensional)
- measure I(hkl): direct determination of crystal symmetry
determines crystal structure:
- unit cell
- symmetry
- atomic coordinaties
- atomic vibrations
- derived parameters: bond lengths (i.e. bond strength) and angles
PXRD
Bragg’s Law: nλ = 2dsinθ
scan 2θ and measure I
peak positions (2θ): unit cell parameters
peak heights/areas (intensity): structure and quantity
applications:
- qualitative analysis: diffraction pattern for a material is unique
- quantitative analysis: a mixture exhibits a diffraction pattern that is the sum of the diffraction pattern of each component, scaled by the amount present
- unit cell parameter refinement: 2θ values give crystal system and Bravais lattice
sample preparation:
1. fine powder (light grinding)
2. mounted: flat-plate holder/between thin foils/glass capillary
(glass capillary gives better intensity data, but difficult to fill without breaking)
3. measurement: accurate position (2θ) of Bragg peak requires good sample preparation
coordinate system: cartesian system in real space
3 mutually perpendicular axes with same unit length scale along each axes (Ångstroms, Å)
1Å = 10^-10 m (similar to sizes of atoms)
NaCl cartesian system
coordinate system: crystallographic system
3 non-planar vectors defining repeat of the lattice in 3-dimensional space
- vectors define size of the unit cell
- given by symbols 𝐚, 𝐛, 𝐜
- magnitude a, b, c
- inter-vector angles α, β, γ
position of an atom: 𝐫 = x𝐚 + y𝐛 + z𝐜
coordinates are fractions of a, b, c
NaCl crystallographic system
rationale for crystallographic xyz
- to identify common structural types
- to enable generation of coordinates of atoms within the unit cell
- to enable the calculation of the intensity of diffraction peaks
crystallographic system: describing crystal structure based on atoms WITHIN the unit cell
0 ≤ x < 1
0 ≤ y < 1
0 ≤ z < 1
exceptions
- conventional to put centres of mass of the molecule within the unit cell
- since molecules do not have sharp edges, some of the molecule may be outside the unit cell
rationale for cartesian system
best for calculating parameters of chemical interest: bond lengths, bond angles, torsion angles, intramolecular or intra-ionic distances
distance, r between two points S and O given by Pythagoras’ rule
rationale for crystallographic system
best for finding the coordinaes of symmetry related objects
- mirror symmetry
- rotation axes
mirror planes
rotation axis: twofold rotation of points in space by 180 degrees
rotation axis: threefold roation of points in space by 120 degrees
no need to use numbers like √3 as in a Cartesian system
interatomic distances
orthogonal axis systems: cubic, tetragonal, orthorhombic
X = xa
Y = yb
Z = zc
describing crystal structures: BaSO4
forms of description
- words
- visual images
- tables of crystallographic and chemical parameters
WORDS:
discrete Ba2+ cations and SO42- anions
- isolated and not linked
SO42- anions are tetrahedral
10 x O of SO42- anions lie within 3.2Å or Ba2+
TABLES OF CRYSTALLOGRAPHIC AND CHEMICAL PARAMETERS:
CIF files (Crystallographic Information File)
typically contain:
- unit cell parameters
- space group symmetry: symmetry operators
- coordinates of the atoms: derived bond lengths, angles and torsion angles
- atomic displacement parameters
atomic coordinates
symmetry operators
enable one to determine the coordinates of symmetry related atoms within a crystal structure (usually either in the unit cell or in an adjacent one)
a mirror plane perpendicular to c located at z=1/4 has the symmetry operator (x, y, 1/2-z)
so there is another O(3)’ at:
translational symmetry
symmetry operators may generate atom coordinates outside the unit cell
translational symmetry provides additional symmetry operators that allows the generation of atoms within the unit cell
additional translational symmetry may be provided by a centred lattice (so none for a P lattice)
these operators appy to every atoms within the cell
in NaCl, Cl- are at:
0,0,0
1/2,1/2,0
1,2, 0, 1/2
0, 1/2, 1/2
general position
substitution of O(3) coordinates into the symmetry operators will generate 8 symmetry-related atoms of type O(3) per unit cell
- 8 symmetry operators and 8 sets of coordinates: O(3) is said to be in a GENERAL POSITION within the unit cell
special position
when the number of atomic positions is less than the number of symmetry operators per unit cell
- atom is said to be in a SPECIAL POSITION
- i.e. its position is on a symmetry element
ionic solids: NaCl (halite) + NiAs
NaCl
- cubic close packed layers of Cl- with Na+ in all Oh holes
- 6:6 coordination
NiAs
- hexagonal close packed layers of As- with Na+ in all Oh holes
- 6:6 coordination
ionic solids: ZnS (zinc blende/sphalerite) + ZnO (wurtzite/zincite)
ZnS
- cubic close packed layers of S2- with Zn2+ in half of the Td holes
- 4:4 coodination
ZnO
- hexagonal close packed layers of O2- with Zn2+ in half of the Td holes
ionic solids: Li2O (anti-fluorite strucure) + CaF2 (fluorite structure)
Li2O
- cubic close-packed layers of O2- with Li+ in all Td holes
- 4:8 coordination
CaF2
- anion and cation sites are swapped with each other to give an 8:4 coordination
- primitive cubic array of F-
- cubic close-packed layers of Ca2+ with F- in all td holes
ionic solids: CdI2 + CdCl2
CdI2
- hexagonal close-packed (ABABAB) layers of I- with Cd2+ in Oh holes - but in alternate layers
i.e. 100%, 0%, 100%, 0%…
CdCl2
- cubic close-packed (ABCABC) layers of Cl- with Cd2+ in Oh holes - but in alternate layers
spinel AB2O4 structure
structure
O2- anions with two types of cation
A is normally 2+
B is normally 3+
face-centred cubic - close-packed layers of oxide anions can be seen when model is viewed perpendicular to body diagonal of cube (a 2D view)
structure 1:
- close-packed O2- anions
- all Oh holes filled with cations
structure 2:
- close-packed O2- anions
- half of Oh holes filled with cations
structure 3:
- close-packed O2- anions
- all Td holes filled with cations
structure 4:
- close-packed O2- anions
- 1/8 (12.5%) Td holes filled with cations
structure 5:
- close-packed O2- anions
- half of Oh holes filled with cations
- 1/8 (12.5%) Td holes filled with cations
inverse spinel B(AB)O4 structure
close-packed O2- anions (shown as 2x2x2 fcc cube)
BUT
3+ cations on Td site
2+ and 3+ cations on Oh site (disordered)
eamples of AB2O4 and B(AB)O4 structure
A = 2+ cation
B = 3+ cation
NORMAL: spinel structure - MgAl2O4
- larger Mg2+ ion prefers tetrahedral geometry
- smaller Al3+ ion prefers octahedral geometry
INVERSE: spinel structure - CoFe2O4
- larger Co2+ ion prefers octahedral geometry
- smaller Fe3+ ion swaps to tetrahedral geometry (whilst other stays octahedral)
close packing for more highly charged cations
close sphere packing works well for low charged cations (1+ and 2+)
- not good for highly charged cations (Ti4+ or Si4+ - ionic with some covalency)
perovskite oxides (ABO3)
parent compound: CaTiO3
sum charges
ABO3 oxides: A + B = 6+
ABX3 (less common) halides A + B = 3+
examples:
SrCoO3
NaMgF3
cubic, tetragonal, orthorhombic or monoclinic
importance of perovskites
ceramic high Tc superconductors
YBa2Cu3O7-x
perovskites of recent interest: MAPI
Methyl Ammonium Lead (Pb) Iodide
perovskite ABO3 structure (ideal)
corner shared octahedrons of [BO6] units
- form a primitive unit cell with a = 3.7 - 4.5 Å
large cation A at the centre of the cube (1/2, 1/2, 1/2)
smaller cation B is at (0,0,0)
O2- anions are at (1/2, 0, 0), (0, 1/2, 0), and (0,0,1/2)
can be viewed as cubic close packed layers of AO3 with B in 25% Oh holes
perovskite ABO3 structure (actual)
if A cation too small:
cooperative rotations lower the symmetry
- tilting of [BO6] octahedra
- reduces distance between A site cation and O2-
- O2- can get closer to A site
- many tilt combinations possible
other distortions caused by the B cation moving slightly off the centre of the O6 octahedron
- results in interesting electrical and magentic properties
volume of unit cell
density
linking MO6 octahedra
simple corner-sharing octahedra:
perovskite structure
mixing corner-sharing with edge-sharing:
several permutations possible
-Anatase
-Rutile
-Brookite
Anatase
Tetragonal Bravais Lattice
- TiO6 octahedra with both corner and edge sharing
slice through structure parallel to c
- within the slice, [TiO6] units are corner linked
double slide through the structure parallel to c
- slices are linked together along c via edge-shared [TiO6] units
consequences of vertex and 2D edge sharing in Anatase
- distorted internal angles within octahedron
- coordination around O2- deviates from 120°
Ti-O-Ti: 101.9° (×2), 156.2°
Rutile
Tetragonal Bravais Lattice
- TiO6 octahedra with both corner and edge sharing
slice through structure parallel to c
- within the slice [TiO6] units are corner linked (but differently to Anatase)
edge sharing occurs in 1D columns
consequences of vertex and 2D edge sharing in Rutile
- less distorted angles within octahedron
- coordination around O2- has smaller deviation from 120° (compared to anatase)
Ti-O-Ti: 130.6° (×2), 98.8.2°
density comparison
Rutile: 4.25 g cm-3
Anatase 3.89 g cm-3
Rutle is the thermodynamically stable form
Anatise is the kinetic form
Brookite
Orthorhombic lattice
- TiO6 octahedra with both corner and edge sharing
silica (SiO2) and silicates
tetrahedral [SiO4] group
- either as a discrete anion, (SiO4)2- or as oxygen-linked tetrahedral neutral/anionic species
-O-Si(-O-)2-O-Si(-O-)2-O-
crystalline quartz SiO2
common crystalline form of SiO2
its crystalline nature can be seen in PXRD pattern
quartz glass (silica) SiO2
tough clear solid with very high melting point (softens at > 950°C)
no sharp diffraction peaks in PXRD pattern - indicates a purely amorphous (glassy) substance
SiO2 is polymorphic
multiple crystalline forms + an amorphous (glassy) form
local structure is the same for all forms
difference in the way the [SiO4] tetrahedra are linked
random [SiO4] linking occurs when cooling from the liquid state - leads to a glass as evidenced by the lack of Bragg peak in PXRD pattern
[SiO4] tetrahedra may be linked so as to lead to solids varying from high to low density
zeolites: low density SiO4 networks
[SiO4] or [AlO4]- tetrahedra (often simplified to [TO4])
T = Si or Al
primary building unit:
[TO4] unit linked via O bridging
secondary building unit:
T-O-T rings
Lowensteins’ rule
Al-O-Al linkages in zeolitic frameworks are forbidden
- all [AlO4]- must be linked to 4 [SiO4] units
Sodalite (SOD)
simplest zeolite framework structure
Na8(Al6Si6O24)Cl2
[Si]/[Al] ratio is 1:1
cubic with a = 8.9 Å
Faujasite (FAU)
cubic, naturally occuring zeolite with a = 24.7 Å
(Na2,Ca,Mg)3.5[Al7Si17O48].32(H2O)
synthetic versions are known as zeolites X/Y
X is where [Si]/[Al] = 2 to 3
Y is where [Si]/[Al] > 3
diffraction: scattering by a single atom
x-rays are scattered by electrons in an atom
incoming x-ray photon acts as a plane wavelet as source is a long way from the sample
scattered x-ray photon acts as a spherical wavelet
diffraction in 1-dimension
multiple slits: a 1-D optical diffraction grating
optical grating: nλ = dsinθ
n - order
d - slit spacing
if atoms are in a regular array, diffraction will occur if an appropriate wavelength source is used
diffraction in 2-dimension
2-D optical diffraction grating
useful to demonstrate how symmetry information can be derived from a single image
- inversion symmetry added during the process of diffraction
- diffraction space unit cell vectors are in a different direction to real space ones when the unit cell angle is not 90
- diffraction space has natural origin (0,0,0)
- small dimensions become big and vice versa (property of diffraction space: e.g. volume, V* = 1/V)
- translational symmetry in 2-dimensional space due to a centred lattice leads to missing intensity spots in a systematic matter in 2-D diffractional space
inversion symmetry is added during the process of diffraction
(not exact when object itself has no inversion symmetry but very close to inversion symmetry)
diffraction pattern contains a point of inversion in the absence of anomalous dispersion, even when this symmetry element is not present in the crystal structure
diffraction space unit cell vectors are in a different direction to real space ones when the unit cell angle is not 90°
diffraction space indicated with *
b* perpendicular to a
a* perpendicular to b
c and c* are colinear and perpendicular to the plane of the screen
unit cell angles that are 90° in the crystal will exhibit 90° angles in reciprocal space
diffraction and crystals
crystalline objects produce diffraction spots
symmetry relationship between crystal (real) space and diffraction (reciprocal) space
crystal space: 3-dimensional symmetry across all space
diffraction space: 3-dimensional symmetry around a point (h,k,l = 0,0,0)
amorphous objects (glasses) do not produce diffraction spots
single-crystal experiment
modern single-crystal x-ray diffractometer has 3 rotation axes (ω, κ (or χ), φ)
rotate crystal in 3-dimensional space, plus a rotation axis 2θ for the detector
orient the crystal with respect to the incident x-ray beam so as to satisfy Bragg’s law by rotations about the different instrument axes
- as the crystal is slowly rotates, the intensity of the diffracted x-rays is measured using an area detector
2θ value for every spot provides d spacing via Braggs law
λ = 2d_hkl sinθ_hkl
d-spacing relates to the 6 unit cell parameters: a, b, c α, β, γ
d-spacing for orthogonal symmetry
d-spacing for cubic symmetry
coordinate system for diffraction spots is defined in terms of 3 vectros
a*
b*
c*
a* is perpendicular to b and c in the crystal
b* is perpendicular to c and a in the crystal
c* is perpendicular to a and b in the crystal
the vector dot products are unity
a.a* = 1
b.b* = 1
c.c* = 1
the diffraction coordinate space has a ‘reciprocal’ relationship to crystal coordinate space (hence, the name reciprocal space)
diffraction space coordinates: position of a point, d* on the lattice formed by the 3 vectors, a, b, c*
d* = ha* + kb* lc*
hkl correspond to the same Miller indices hkl of the lattice planes from which the diffraction intensity I(hkl) arises
diffraction space coordinates: if crystal axes (a,b,c) are orthogonal
diffraction space axes (a* ,b* ,c*) are orthogonal and in the same direction
lengths of the axes are given by:
a* = 1/a
b* = 1/b
c* = 1/c
diffraction space coordinates: length of d*
if cell angles are 90,
consider a unit cell with orthogonal axes
Fourier Transform links diffraction data to crystal structure
real space (crystal) → reciprocal space (diffraction)
atoms → scattering factor
unit cell and lattice → interference term
atomic displacements → Debye-Waller factors
scattering of x-rays by atoms
x-rays interact strongly with the electrons in an atom (or ion)
elastic interaction → diffraction
absorption & re-emittance → fluorescence (loss of coherent scattering)
elastic interaction is proportional to number of electrons (atomic number, Z)
- scattering factor given by symbol, f
x-ray scattering factors, f (2θ)
electron density of atom has gaussian distribution
fourier transform of the electron cloud corresponds to the X-ray form factor
intensity of diffraction I(hkl)
position of the atoms is determined by I(hkl)
- peak maximum is the position of the peak at 2θ
- position of the peaks, 2θ, determined by Bragg’s Law
peak area is the intensity of the peak I(hkl)
peak has height H and width W (for powder diffracion)
- I(hkl) ∝ H ∝ W
intensity of the peak, I(hkl) is given by the ‘Structure Factor’ equation (symbol F(hkl))
I(hkl) ∝ IF(hkl)I^2
the proportionality terms are:
c - the scale factor
~ counting time
~ amount of diffracting material
j_hkl - multiplicity (this only applies to PXRD)
~ number of symmetry equivalent reflections
Lp(2θ) - geometric factors
- Lorentz, L
- polarisation, p
the structure factor equation
summation is taken over all atoms, n within a single unit cell
summation could be taken over all atoms in the crystal, but since all unit cells are the same:
multiply F_hkl by the number of unit cells
the Interference Term
amount of constructive/destructive interference of X-ray wave depends on the scattering power of each atom, and its relative position (r_n) within the unit cell
also depends on the diffracting planes hkl whose normal can be defined in terms of d* as d* = ha* + kb* + lc*
definition of structure factor F_hkl
the Fourier transform of the electron density ρ_xyz
- x-rays ‘see’ electrons NOT atoms
F_hkl is a vector quantity
- can be thought of a wave expression with magnitude IF_hklI and a relative phase angle Φ
calculating Fhkl for CsCl
consider the hypothetical case where the atoms(/or ions) are stationary in space
Debye-Waller factor exp(-Wn)=1 since the atomic displacements are zero (e^0=1)
100 planes
Cs atoms lie in the planes with the Cl atoms half-way between them - therefore destructive interference occurs
200 planes
both the Cs atoms and Cl atoms lie in the same planes - therefore constuctive interference occurs
calculating F_hkl for Fe metal
reflection intensities from planes such as 110, 220, 211, 220, etc. (even) of Fe will be strong
reflection intensities from planes such as 100, 111, 210, 300, etc. (odd) of Fe have zero intensity
such reflections are said to be missing/extinct or SYSTEMATICALLY ABSENT
for the cases of non-zero intensities, they are usually referred to as a REFLECTION CONDITION
Debye-Waller factor exp(-W_n)
effect of atomic motion
atoms vibrate
- usually as harmonic oscillators
- giving rise to a normal (Gaussian) distribution about a point
- Fourier transform of this motion is a normal distribution
simplest case is for isotropic vibration
Lorentz-polarisatiion term - effect on PXRD
structure factor has a marked effect on the RELATIVE intensity of peaks in a PXRD
- e.g. CsCl: weak 100 but strong 110
Lorentz-polarisation term also has a big effect
ICDD (International Centre for Diffraction Data) PDF (Powder Diffraction File) database
or JCPDS (Joint Committee on Powder Diffraction Standards) database
PDF database consists of a set of cards, one for each material with its powder diffraction pattern parameters
contains:
- file number for the database entry
- chemical formula, chemical name, and mineral/common name (if one exists)
- quality factor of the data entry: star indicates data of the highest quality and data is indexed with hkl values
- experimental details on how the data was obtained (radiation used - Cu Kα1 and wavelength - 1.5405Å)
- crystallographic information: crystal system, lattice parameters, no. formula units per unit cell, calculated density
- physical properties: mp, optical refraction indices and colour
- the PXRD data
the PXRD data of PDF
PXRD pattern represented as a set of d-spacings (from high to low values)
relative peak intensities (I/I1) and hkl values for each observed peak (i.e. the fingerprint region)
d spacings of the 3 most intense peaks in the powder pattern and their relative intensities are used to constuct HANAWALT search/match tables
Hanawalt
most intense peaks from each phase are the easiest to identify in a mixture of phases
sorted into d-spacing ranges (e.g. 3.74-3.60 Å)
faster for manual search-match
lists d-spacings for 4th-8th strongest peaks in addition
qualitative PXRD analysis ~ search-match
for a match, d-spacing values should match within experimental error
reasons for mis-match
- wrong material made
- measurement error
- instrumental error
- not comparing like with like (e.g. if measurement and databse entry refers to different temperatures OR comparing doped vs pure phase materials)
practical qualitative PXRD
material at lower temperatures is almost amorphous
- peaks sharpen with increasing T
narrower peaks indicate that crystallite size is getting bigger with higher T
difficult to make definitive assignment with broad peaks using database
- use the highest temperature dataset
matching the strong peaks against a database entry can show the presence of a particular major componenet in a PXRD sample
HOWEVER
purity is demonstrated by accounting for ALL of the peaks in the measured PXRD pattern
quantitative PXRD analysis
concept of quantitative analysis by PXRD methods is based on the fact that scattering is proportional to the amount of material present
indexing PXRD data - simple cubic material
the diffraction technique
single crystal x-ray diffraciton (SXD)
- IF(hkl)I^2 values provide detail of the material/compound/complex
- atomic postions, average atomic motion, unit cell and crystal symmetry
powder x-ray diffraction (PXRD)
- used for qualitative and quantitative analysis
- can determine: is this the right material? it is single phase? purity?
problem with PXRD data
the Rietveld Method
powder diffraction pattern yi(calc) can be calculated from:
- peak positions (determined by unit cell and any instrument zero offsets)
- peak intensities (determined by structure factor, scale factor, geometric factors etc)
applications of the Reitveld Method
analysis of PND data
- mixed metal oxides (e.g. perovskites)
used to fit all powder diffraction data
- used for organic crystal structures
- often done badly and data over interpreted
zeolite synthesis
used in petroleum industry
- heterogeneous catalyst for hydrocarbon isomerisation reactions
low density crystal structure is not normally favoured
- synthetic methods need to exploit kinetic (and other) effects
- hydrothermal methods
- silica and alumina gels (F- to act as a mineraliser)
- templating organic molecules (amine based e.g. diethylamine to make ZMS22) - removed after synthesis by calcination
- [PO4] to replace [SiO4] to give ALPOs
Metal Organic Framework (MOF)
large, open pore network structures
- created by (semi-) rigid organic linkers and metal cations
syntheis is similar to that of zeolites
- hydrothermal/solvothermal techniques used (crystals grow slowly from a hot solution)
constructed from bridging organic ligands that remain intact throughout the synthesis and act as the templates
- contrast with zeolites where amines are used and have to be removed later
functionality built into MOF by an appropriate choice of ligand
typical ligands for MOFs
oxalic acid
malonic acid
isophthalic acid
terephthalic acid
trimesic acid
metal cation in MOFs
cation’s coordination preference influences size and shape of pores
- dictates how many ligands can bind to metal
structures descibed in terms of primary building blocks (ligands) and secondary building blocks
scientific interest of MOFs
driven by materials for H2 storage/CO2 capture and storage
large pore size - can store lots of gas
pore size can change depending on whether absorbed gas is present or not
example of MOF: MOF-210
prepared from a solvothermal reaction of H3BTE, H2BPDC and zinc(II) nitrate hexahydrate
- synthesis involved 2 topologically different organic linking groups (both carboxylic acids)
has ultrahigh surface area
example of MOF: MOF-5
organic linker: 1,4-benzenedicarboxylate (BDC)
framework formula:
Zn4(O)(BDC)3
metal cluster:
4 tetrahedra of [ZnO4]
corner shared by an O to give a tetrahedral arrangement of Zn around the central O
prepared by diffusion of NEt3 into solution of Zn2+ nitrate and BDC in DMF/C6H5Cl + H2O2
cavity contains DMF and C6H5Cl solvents
diffraction vs spectroscopy
diffraction
- the Fourier transform of the whole unit cell and its contents
- d-spacings do not relate to bond lengths
- crystal structure determination gives structure of the WHOLE molecule - not just the metal and atoms bound to in an organometallic complex
spectroscopy (NMR, IR)
- measures local information
- can be tuned to specific parts of a molecule (i.e. energy of a metal carbonyl bond)
