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INTRODUCTION TO XRD
The parameters that define a unit cell are:
a, b, c = unit cell dimensions along x, y, z respectively
, , = angles between b,c (); a,c (); a,b ()
z
y
x
Unit cell: the building block of crystalline solids
90;cba
Shapes of unit cells
All the possible shapes of a unit cell are defined by 7 crystal systems, which are based on the relationship among a,b,c and , , ,
Cubic system
90;cba Tetragonal system
90;cba Orthorhombic system
90;cba Triclinic system
12090 ;;cba Hexagonal system
90;cba Rhombohedral system
90;cba Monoclinic system
Primitive lattice: the unit cell has a lattice point at each corner only (P)
Body centred lattice: the unit cell has a lattice point at each corner and one in the centre (I)
Types of Unit Cells
Face-centred lattice: the unit cell has a lattice point at each corner and one in the centre of one pair of opposite faces (A), (B), (C)
All-face-centred lattice: the unit cell has a lattice point at each corner and one in the centre of each face (F)
CHARACTERIZATION OF THE STRUCTURE OF SOLIDS
Three main techniques:
X-ray diffraction Electron diffraction Neutron diffraction
Principles of x-ray diffraction
Single crystalPowder
X-rays are passed through a crystalline material and the patterns produced give information of size and shape of the unit cell
X-rays passing through a crystal will be bent at various angles: this process is calleddiffraction
X-rays interact with electrons in matter, i.e. are scattered by the electron clouds of
atoms
WHAT IS DIFFRACTION?
Diffraction – the spreading out of waves as they encounter a barrier.
What is a Diffraction pattern?
- an interference pattern that results from the superposition of waves.
- Mathematically, this process can be described by Fourier transform, if the diffraction is kinematic (electron or X-ray has been scatted only once inside the object).
Laser diffraction pattern of a thin grating films, where the size of holes is closed to the wavelength of the laser (Ruby red light 594 um) .
Fourier transform of regular lattices:
Reciprocal spaceReal space
INTERACTION BETWEEN X-RAY AND MATTER
d
wavelength Pr
intensity Io
incoherent scattering
Co (Compton-Scattering)
coherent scattering
Pr(Bragg´s-scattering)
absorbtionBeer´s law I = I0*e-µd
fluorescense
> Pr
photoelectrons
The angles at which x-rays are diffracted depends on the distance between adjacent layers of atoms or ions. X-rays that hit adjacent layers can add their energies constructively when they are “in phase”. This produces dark dots on a detector plate
Scattering of x-rays by crystallographic planes
We need to consider how x-rays are diffracted by parallel crystallographic planes
diffracted x-rays
lattice planes
atoms on lattice planes
d
incident x-rays
X-rays diffracted in phase will give a signal. “In phase” means that the peak of one wave matches the peak of the following wave
d
D
E
F
C
The two x-ray beams travel at different distances. This difference is related to the distance between parallel planes
We connect the two beams with perpendicular lines (CD and CF) and obtain twoequivalent right triangles. CE = d (interplanar distance)
d
DE sin DEd sin EFDE EFd sin
length path in difference sin2 DEEFd
The angle of incidence of the x-rays is is
The angle of diffraction is the sum of these two angles, 2
The angle at which the x-rays are diffracted is equal to the angle of incidence,
These conditions are met when the difference in path length equals an integral numberof wavelengths, n. The final equation is the BRAGG’S LAW
sin2dn Data are collected by using x-rays of a known wavelength. The position of the sampleis varied so that the angle of diffraction changes
When the angle is correct for diffraction a signal is recorded
With modern x-ray diffractometers the signals are converted into peaks
Inte
nsity
(a.
u.)
2 degrees
(200) (110) (400)
(310)(301)
(600) (411) (002)(611)(321)
Reflection (signal) only occurs when conditions for constructive interference between the beams are met
TEST
NaCl is used to test diffractometers. The distance between a set of planes inNaCl is 564.02 pm. Using an x-ray source of 75 pm, at what diffraction angle (2) should peaks be recorded for the first order of diffraction (n = 1) ?
Hint: To calculate the angle from sin , the sin-1 function on the calculator must be used
7.62 2 ; 3.81
0.066 pm 564.02 2
pm 75 sin
sin pm 564.022pm 751
dn
sin2
Lattice Planes and Miller Indices
Atoms or ions in lattices can be thought of as being connected by lattice planes.Each plane is a representative member of a parallel set of equally spaced planes.
A family of crystallographic planes is always uniquely defined by three indices, h, k, l, (Miller indices) usually written (h, k, l)
The Miller indices are defined byZ
l,Y
k,X
h111
Note - plane // to axis,intercept = ∞ and 1/∞ = 0
X, Y, Z are the intersections of one plane with on a, b, c respectively
)hk(
)lh(
)kl(
0
0
0family of lattice planes parallel to
z
y
x
How to Determine Miller Indices
EXAMPLES OF CRYSTALLOGRAPHIC PLANES
(111)
(212)
a
b
c
(100)
a
b
c
0.5
a
b
c
Inter-Planar Spacing, dhkl, and Miller Indices
The inter-planar spacing (dhkl) between crystallographic planes belonging to the same family (h,k,l) is denoted (dhkl)
Distances between planes defined by the same set of Miller indices are unique for each material
2D
d'h’k’l’
dhkl
Inter-planar spacings can be measured by x-ray diffraction (Bragg’s Law)
The lattice parameters a, b, c of a unit cell can then be calculated
The relationship between d and the lattice parameters can be determined geometrically and depends on the crystal system
Crystal system dhkl, lattice parameters and Miller indices
Cubic
Tetragonal
Orthorhombic
2
22
a
l k h
d
2
2
1
2
2
2
1
c
l
a
k h
d
2
2
2
22
2
2
1
c
l
b
k
a
h
d
22
2
The expressions for the remaining crystal systems are more complex
THE POWDER TECHNIQUE
An x-ray beam diffracted from a lattice plane can be detected when the x-ray source, the sample and the detector are correctly oriented to give Bragg diffraction
A powder or polycrystalline sample contains an enormous number of small crystallites, which will adopt all possible orientations randomly
Thus for each possible diffraction angle there are crystals oriented correctly for Bragg diffraction
Each set of planes in a crystalwill give rise to a cone of diffraction
Each cone consists of a set of closely spaced dots each one of which represents a diffraction from a single crystallite
FORMATION OF A POWDER PATTERN
Single set of planes
Powder sample
Experimental Methods
To obtain x-ray diffraction data, the diffraction angles of the various cones, 2, must be determined
The main techniques are: Debye-Scherrer camera (photographic film) or powder diffractometer
Debye Scherrer Camera
Powder Diffractometer
The detector records the angles at which the families of lattice planes scatter (diffract) the x-ray beams and the intensities of the diffracted x-ray beams
The detector is scanned around the sample along a circle, in order to collect all the diffracted x-ray beams
The angular positions (2) and intensities of the diffracted peaks of radiation (reflections or peaks) produce a two dimensional pattern
This pattern is characteristic of the material analysed (fingerprint)
Each reflection represents the x-ray beam diffracted by a family of lattice planes (hkl)
Inte
nsity
2 degrees
(200) (110) (400)
(310)(301)
(600) (411) (002)(611) (321)
APPLICATIONS AND INTERPRETATION OF X-RAY POWDER DIFFRACTION DATA
Number and positions (2) of peaks
crystal class
lattice type
cell parameters
Intensity of peakstypes of atoms
position of atoms
Information is gained from:
Identification of unknown phases
Determination of phase purity
Determination and refinement of lattice parameters
Investigation of phase changes
Structure refinement
Determination of crystallite size
Powder diffraction data from known compounds have been compiled into a database (PDF) by the Joint Committee on Powder Diffraction Standard, (JCPDS)
This technique can be used in a variety of ways
The powder diffractogram of a compound is its ‘fingerprint’ and can be used to identify the compound
‘Search-match’ programs are used to compare experimental diffractograms with patterns of known compounds included in the database
Identification of compounds
PDF - Powder Diffraction File
A collection of patterns of inorganic and organic compounds
Data are added annually (2008 database contains 211,107 entries)
Example of Search-Match Routine
Outcomes of solid state reactions
Product: SrCuO2?Pattern for SrCuO2from database
Product: Sr2CuO3?
Pattern for Sr2CuO3from database
CuO2SrCO3 2SrCuO
3CuOSr2?
When a sample consists of a mixture of different compounds, the resultant diffractogram shows reflections from all compounds (multiphase pattern)
Phase purity
Sr2CuO2F2+
Sr2CuO2F2+ + impurity
*
http://www.talmaterials.com/technew.htm
ZrO2 (monoclinic)
3 mol % Y2O3 in ZrO2 (tetragonal)
8 mol % Y2O3 in ZrO2 (cubic)
Effect of defects
Determination of crystal class and lattice parameters
X-ray powder diffraction provides information on the crystal class of the unit cell (cubic, tetragonal, etc) and its parameters (a, b, c) for unknown compounds
Indexing Assigning Miller indices to peaks
1
Determination of lattice parameters
Bragg equation and lattice parameters
2
2222
22
4lkh
asin
Cubic system
Crystal class comparison of the diffractogram of the unknowncompound with diffractograms of known compounds(PDF database, calculated patterns)
3
PROBLEM
NaCl shows a cubic structure. Determine a (Å) and the missing Miller indices( = 1.54056 Å).
2 () h,k,l
27.47 111
31.82 ?
45.62 ?
56.47 222
Selected data from the NaCl diffractogram
? ?
2222
22
4lkh
asin
638.5
2473.56
sin4
12541.1
sin4 2
2
2
2222
lkh
a
Use at least two reflections and then average the results
(222)Å
a (Å)
Miller Indices
2222
22
4lkh
asin
A
2222 lkhAsin
018670
63854
540561
4 2
2
2
2
..
.
aA
2222
lkhA
sin
82312 .40264
0186702
82312
..
.sin 2004222 lkh
62452 . 80528018670
262452
..
.sin 2208222 lkh
Systematic Absences
Conditions for reflection
number)(even 2,, 222222 nlklhkh Fi.e indices are all odd or all even
I nlkh 2222
P No conditions
For body centred (I) and all-face centred (F) lattices restriction on reflections fromcertain families of planes, (h,k,l) occur. This means that certain reflections do not appear in diffractograms due to ‘out-of-phase” diffraction
This phenomenon is known as systematic absences and it is used to identifythe type of unit cell of the analysed solid. There are no systematic absences forprimitive lattices (P)
Considering systematic absences, assign the following sets of Miller indices to either the correct lattice(s).
Lattice Type
Miller Indices P I F
1 0 0 Y N N
1 1 0 Y Y N
1 1 1 Y N Y
2 0 0 Y Y Y
2 1 0 Y N N
2 1 1 Y Y N
2 2 0 Y Y Y
3 1 0 Y Y N
3 1 1 y N Y
Autoindexing
Generally indexing is achieved using a computer program.This process is called ‘autoindexing’
Input: •Peak positions (ideally 20-30 peaks)•Wavelength (usually =1.54056 Å)•The uncertainty in the peak positions•Maximum allowable unit cell volume
Problems: •Impurities•Sample displacement•Peak overlap
Derivation of 2222
22
4lkh
asin
sin2d2
22
a
l k h
d
2
2
1
222
2
22
22
2222
2
l k h4a
l k h2a
l k h
1a
l k h
1ad
l k h
ad
2
2
2
222
2
;
2sin
sin
sin
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