Fundamentals of Rietveld Refinement

I. XRD Pattern Calculation and Refinement Parameters

An Introduction to Rietveld Refinement

using
PANalytical X’Pert HighScore Plus v3.0e

Scott A Speakman, Ph.D.

MIT Center for Materials Science and Engineering
speakman@mit.edu
prism.mit.edu/xray
 Outline

• Calculating a Diffraction Pattern from a Crystal Structure

• Using HighScore Plus to Simulate a Diffraction Pattern
• Manually Refining a Model to Match Experimental Data
 A Brief Re-introduction to Miller Indices

• The Miller indices (hkl) define the reciprocal axial

intercepts of a plane of atoms with the unit cell
𝑎 𝑏
– The (hkl) plane of atoms intercepts the unit cell at ℎ, 𝑘,
𝑐
and 𝑙
– The (220) plane drawn to the right intercepts the unit
cell at ½*a, ½*b, and does not intercept the c-axis.
• When a plane is parallel to an axis, it is assumed to
intercept at ∞; therefore its reciprocal is 0
b

• The vector dhkl is drawn from the origin of the unit c

cell to intersect the crystallographic plane (hkl) at a a
90° angle.
– The direction of dhkl is the crystallographic direction.
– The crystallographic direction is expressed using []
brackets, such as [220]
 A given (hkl) refers to a family of atomic planes,
not a single atomic plane
• The Miller indices are determined by The (100) plane includes two
faces of the cubic unit cell.
using the plane of atoms that is These are produced by
closest to the origin without passing drawing the first plane
(shaded orange) at the (1,0,0)
through it. intercept, and then
translating it by [-100].

• The other members of the family of

atoms are determined by translating The (200) plane also includes
the (hkl) plane of atoms by dhkl two faces of the cubic unit
cell. These are produced by
drawing the first plane
• A family of planes will always have (shaded orange) at the (½, 0,
one member that passes through 0) intercept, and then
translating it by ±[½, 0, 0].
the origin
The (400) plane includes
• Some planes of atoms may belong members of the (100) and
to more than one family (as (200) families, as well as the
planes at the (¼, 0, 0) and (¾,
illustrated to the right) 0, 0) intercepts. These are
produced by drawing the first
plane (shaded orange) at the
(¼, 0, 0) intercept, and then
translating it by ±[n*¼, 0, 0].
 The position and intensity of peaks in a diffraction
pattern are determined by the crystal structure
Counts
SiO2 Glass
4000

2000

0
4000 Quartz
3000
2000
1000
0
Cristobalite
4000

2000

0
20 30 40 50
Position [°2Theta] (Copper (Cu))

• These three phases of SiO2 are chemically identical.

• Differences in the atomic structure are reflected in the X-ray
scattering patterns.
 The diffraction peak position is a product of the
atomic bond distances in the crystal

Bragg’s Law: λ = 2dhklsinθ

• Bragg’s law calculates the angle where constructive

interference from X-rays scattered by parallel planes of
atoms will produce a diffraction peak.
– In most diffractometers, the X-ray wavelength l is fixed.
– Consequently, a family of planes produces a diffraction peak only at
a specific angle 2q.
– The lattice parameters of the unit cell are used to calculate dhkl
 dhkl is a function of the size and shape of the unit cell
d *2hkl  h 2 a *2  k 2b *2 l 2c *2 2hka * b * cos   2hla * c * cos   2klb * c * cos 

1
hkl 
2
• dhkl* is the reciprocal of dhkl: d * 2
d hkl
bc sin 
a* 
• dhkl is calculated based on the reciprocal abc
space units of the lattice parameters cos  cos   cos 
cos  * 
sin  sin 
• the diffraction peak position is a product
of the average atomic distances in the
crystal
• anything that changes the average bond
distances (temperature, pressure, etc.)
will change dhkl and therefore change the
diffraction peak positions
 The amplitude of light scattered by a crystal is determined
by the arrangement of atoms in the diffracting planes
2
𝐼ℎ𝑘𝑙 ∝ 𝐹ℎ𝑘𝑙

Fhkl   N j f j exp 2ihx j  ky j  lz j  
m

j 1

• The structure factor, Fkhl, quantifies the amplitude of light scattered

by a crystal
– The patterns of atoms in the unit cell scatters strongly in some
directions and weakly in others owing to interference of the wavelets
scattered by the atoms
• Fhkl sums the result of scattering from all of the atoms in the unit
cell to form a diffraction peak from the (hkl) planes of atoms
• The amplitude of scattered light is determined by:
– where the atoms are on the atomic planes
• this is expressed by the fractional coordinates xj yj zj
– what atoms are on the atomic planes
• the scattering factor fj quantifies the relative efficiency of scattering at any
angle by the group of electrons in each atom
• Nj is the fraction of every equivalent position that is occupied by atom j
The scattering factor, f, quantifies the efficiency of
scattering from a group of electrons in an atom

Fhkl   N j f j exp 2ihx j  ky j  lz j  
m

j 1

• X-rays are electromagnetic radiation that interact with an electron,

which reradiates the X-ray as a spherical wave
• The number of electrons around an atom defines how strongly it will
scatter the incident X-ray beam
– the initial strength of the atomic scattering factor is equal to the number
of electrons around the atom
• The scattering factor, f, is a product of several terms describing the
interaction of the X-ray with the electrons around an atom

2
   B sin 2 q  
f
2

  f 0 exp    f ' 
  f ' ' 2

  l 2
 
The atomic scattering factor f0 can be found using tables or
equations determined experimentally or from quantum
mechanical approximations
40
• f0 at 0° is equal to the number of
Y
electrons around the atom 35 Y(3+)
– Y and Zr are similar, but slightly Zr
different, at 0° 30 Zr(4+)
– Zr and Zr4+ are slightly different at O(2-)
0° 25

fo
– Y3+ and Zr4+ are identical at 0°
20
• the variation with (sin θ)/λ depends
on size of atom
15
– larger atoms drop off quicker
– at higher angles, the difference 10
between Y3+ and Zr4+ is more
readily discerned 5
– at higher angles, the difference
between different oxidation states 0
(eg Zr and Zr4+) is less prominent 0 0.5 1 1.5
(sin q/l
Thermal motion of the atom changes the scattering
factor 2
 B sin q 
f  f 0 exp  
• the efficiency of scattering by an atom  l 2

is reduced because the atom and its 40
B=0
electrons are not stationary 35 B=1
– the atom is vibrating about its B=10
equilibrium lattice site 30
– the amount of vibration is quantified by
25
the Debye-Waller temperature factor:
B=8π2U2 20

f
• U2 is the mean-square amplitude of the
vibration 15
– this is for isotropic vibration: sometimes
10
B is broken down into six Bij anistropic
terms if the amplitude of vibration is not 5
the same in all directions.
– aka temperature factor, displacement 0
factor, thermal displacement parameter 0 0.5 1 1.5 2
(sin q/l
 Anomalous scattering of X-rays with energies near the
absorption edge of the atom changes the scattering factor
2
   B sin q 
2

f
2
  f 0 exp    f ' 
  f ' ' 2

  l 2
 

• scattering factor has a real and imaginary term

• these terms only matter when the energy of the incident X-ray
approaches the ionization energy of atom
– when the wavelength of the incident X-ray is near the absorption
edge of the atom
• anomalous scattering describes how some photons with
energy approaching the ionization energy will be scattered
out of phase with the majority of photons
With a crystal structure to solve Bragg’s Law and Fhkl, you
can calculate the diffraction pattern from an ideal crystal

• Ideal powder diffraction patterns can be simulated if you

know ...
– space group symmetry
– unit cell dimensions
– atom types
– relative coordinates of atoms in unit cell
– atomic site occupancies
– atomic thermal displacement parameters
• ... for each and every phase in the sample
 Where to Get Crystal Structures

• publications
• Commercial Databases
– ICDD PDF4+ Inorganic
– Inorganic Crystal Structure Database (ICSD)
– Linus Pauling File (LPF)
• this is included in ICDD PDF4+ Inorganic
– NIST Structural Database (metals, alloys, intermetallics)
– CCDC Cambridge Structure Database (CSD) (organic materials)
• Available online http://beta-www.ccdc.cam.ac.uk/pages/Home.aspx
• MIT Site license available for download from IS&T website
• Free Online Databases
– Crystallography Open Database http://www.crystallography.net/
– ICSD- 4% available as demo at http://icsd.ill.eu/icsd/index.html
– Mincryst http://database.iem.ac.ru/mincryst/index.php
– American Mineralogist http://www.minsocam.org/MSA/Crystal_Database.html
– WebMineral http://www.webmineral.com/
– Protein Data Bank http://www.rcsb.org/pdb/home/home.do
– Nucleic Acid Database http://ndbserver.rutgers.edu/
– Database of Zeolite Structures http://www.iza-structure.org/databases/
 A Crystal Structure

• ZrO2
• Space Group Fm-3m (225) site occupancy factor,
• Lattice Parameter a=5.11 sof, is equivalent to N
in the Fhkl equation

Atom Wyckoff x y z Biso sof

Site

Zr 4a 0 0 0 1.14 1

O 8c 0.25 0.25 0.25 2.4 1

Wyckoff Site notation is a shortcut to indicate if the
type of site that the atoms occupies in the crystal
Atom Wyckoff Site x y z Biso sof
Zr 4a 0 0 0 1.14 1
O 8c 0.25 0.25 0.25 2.4 1

• each letter in the Wyckoff notation specifies a site that sits on a different
combination of symmetry elements
– The Wyckoff site indicates if x, y or z are can change or if they must be fixed to preserve the
symmetry of the crystal
– An element on the 4a site in the Fm-3m space group can only occupy 0,0,0
• Moving the element to a different position would change the surroundings of the site and
would represent a change in the interaction between that atom and its neighbors
– An element on the 32f site in the Fm-3m space group will occupy x,x,x (indicating that the x,
y, and z positions must be the same number)

• the number in the Wyckoff notation indicates the number of atoms put into the
unit cell if the atom is on that site
– 4a- an atom on the a site will populate the unit cell with 4 atoms
– 8c- an atom on the c site will populate the unit cell with 8 atoms
– Wyckoff notation gives you a quick way to check if stoichiometry is correctly preserved
when you create a crystal structure
– you can deduce z, the number of molecules per unit cell, from the Wyckoff notation (z=4 for
ZrO2)
 Retrieve information about space groups using the
Symmetry Explorer in HighScore Plus

• To use the Symmetry Explorer

– select the menu Tools >
Symmetry Explorer
– In the left pane, select the Space
Group you would like to examine
• The entries in the Symmetry
Explorer contain information
similar to what you would find
in the International Tables of
Cyrstallography
• The Special Positions tab will
show the Wyckoff sites for that
space group
 The Site Occupancy Factor, sof, indicates what
fraction of a site is occupied by a specific atom
 
Fhkl   N j f j exp 2ihx j  ky j  lz j 
m

j 1

Atom Wyckoff Site x y z Biso sof

Zr 4a 0 0 0 1.14 1
O 8c 0.25 0.25 0.25 2.4 1

• The sof is N in the structure factor equation

• sof = 1, every equivalent position for xyz is occupied by that atom
• sof < 1, some of the sites are vacant
• two atoms occupying the same site will each have a fractional sof
• remember that the observed XRD pattern is an average of
hundreds or thousands of unit cells that make a crystallite
 How to manually enter a crystal structure
and calculate an ideal X-ray powder diffraction pattern
using HighScore Plus
For pattern calculations and Rietveld refinement in
HSP, the “Structures” desktop is recommended

• Desktops can be
changed in the menu:
– View > Desktops

• All slides in this tutorial

were created with HSP in
the “Structures” desktop
 With this crystal structure information, we can
build a crystal in HSP

• Start X’Pert HighScore Plus

• Create a New Document File>New
• Analysis>Rietveld>Enter New Structure
 Step 1: Enter Title

• Follow the entry form from top to bottom, via the

numbered steps
• The Title can be any name that you want
• In this example, we are creating an entry for Zirconia
 Step 2: Enter Space Group
• To enter the space group:
– you can just enter number into the space group field
• Each space group has a unique number (1 thru 230)
– You can use the drop-down menu to select a space group
– You can open the symmetry explorer by clicking on the ... button
• HSP will assume the standard origin and setting unless you use the
Symmetry Explorer to select otherwise
• In this example, type 225 for the Fm-3m space group
The symmetry explorer allows you to choose the space
group and a specific origin, and to see information about
the symmetry elements and special positions
 Step 3: Enter Unit Cell Lattice Parameters

• lattice parameters are automatically constrained by the

space group that you chose
– For example, in a cubic space group: a=b=c, 90
• In this example, a=5.11
 Step 4: Enter Atom Information

• You can type the element symbol or select from a drop-down menu
– The element can be an atom or ion (select oxidation state)
• the use of oxidation states in modifying the scattering factor for an atom is set in
Customize>Program Settings, in the Rietveld tab
• by default it is set to “Ignore Oxidation States”

• The name is the element by default, but it can be changed to anything

you like (useful if you have the same element on multiple sites)

• Enter the atom positions, sof, and Biso

– If you enter the Wyckoff site, constrained x, y, and z positions are automatically
entered
– if you enter xyz, the Wyckoff site is automatically assigned

• When finished, click OK to create that crystal structure!

 We can now simulate the ideal X-Ray powder
diffraction pattern in HSP

• We have entered in all of the information necessary to

simulate an X-ray diffraction pattern
• Click on the ‘Start Pattern Simulation’ button to produce
the theoretical diffraction pattern
The pattern simulation parameters can be
changed in Program Settings
• To change settings go to
Customize> Program
Settings
• In Simulation tab, you
can change the
parameters for the
simulated scan
– Default range for
simulation is 5 to 70
°2theta
– change start and end
angles to 25 and 90
°2theta
– click OK to enter change
and close window
• click ‘Start Pattern
Simulation’ again
The wavelength of radiation used in the simulation
can be changed in Document Settings
• To change settings go to
Customize> Document
Settings
• In Document Wavelength
tab, you can change the X-
ray source used
– there is an “Other” option if
you used a non-
characteristic wavelength,
such as at a synchrotron
• click ‘Start Pattern
Simulation’ again
Parameters for the crystal structure and the simulation can
be accessed in the Refinement Control list

• the parameters for the pattern

simulation are shown in the
Refinement Control List
• The values for global parameters
and each individual crystal
structure entered into the
simulation are listed in a nested
tree format
• You can trim the number of tabs in
the Lists Pane in the View menu
 The Object Inspector shows additional details for
any object you select in other windows
• If you click on the phase
name (eg Zirconia) in
Refinement Control list, you
will be able to see
additional information about
that phase in the Object
Inspector
– This includes information
such as calculated density,
chemical formula, and
formula mass
– You can use these to
check that you created the
crystal structure correctly
– You can change details
such as the phase name
and color of simulated
diffraction pattern
We want to keep this pattern as a reference during
the next exercises

• Go to Scan List in the Lists Pane

• Right Click on the current scan
that is shown in the list (which is
the simulated pattern)
• Select “Convert calc. Profile to
obs. Profile”
• Now the original simulation will
remain in the Main Graphics
View and we can compare it to
new simulations
 We can change parameters for the crystal structure and
observe how they change the diffraction pattern

• The unit cell lattice parameters determine where the

diffraction peaks are observed
• the peak intensity is determined by
– what atoms are in the crystal and their positions xyz
– the site occupancy of the atom
– the relative scattering strength of the atom, scattering factor,
which is a product of the number of electrons around the atom
– the thermal parameter of the atom

• This is not just useful for these tutorial purposes, but also
for anticipating what your real data might look like and
for evaluating limits of detection
 To change values for the simulation

• In the Refinement Control,

– expand the Zirconia entry
– Expand the Unit Cell entry
– Change the Unit Cell value
• After typing, you must press ‘Enter’ key
to register the change
• if you try recalculating the simulation
without pressing enter, you may lose the
changed value
• recalculate the diffraction pattern by
clicking on “Start Pattern Simulation”
button again
• change the lattice parameter and
recalculate the pattern several times
to get a feel for the relationship
To change atom parameters

• click on Atomic Coordinates in the

Refinement Control List
• The atomic information appears in the
Object Inspector
• Expand the Object Inspector window
so that you can see more clearly
• You know that you are working with the
atoms because the Object Inspector
pane says “Selected Object: Atomic
Coordinates”
 Change the Site Occupancies (sof) for the atoms
• Changing the occupancy of Zr has
40
a big effect Y
– change the SOF of Zr to 0.5 and 35 Y(3+)
recalculate the pattern Zr
30 Zr(4+)
• Changing the occupancy of O has a O(2-)
small effect 25
– reset the SOF of Zr to 1.0 and

fo
change the SOF of O to 0.5, then 20
recalculate the pattern
15
• Remember, the atomic scattering
factor, f0, is equal to the number of 10
electrons around the atom
5
– Zr atoms scatter much more
strongly than O, so the XRD 0
diffraction pattern is more sensitive 0 0.5 1 1.5
to the Zr atoms (sin q/l
 We can predict the change in peak intensities
when we dope the sample

• replace 20% Zr with Y

– right click on the Object Inspector and select “add new atom”
– To create the new atom, type “Y” for the Element
– tab over to Wyckoff Site and type “4a” and enter
• this automatically places the atom at 0,0,0
– make the sof 0.2
– make the Biso 1.14
– make the sof for the Zr atom 0.8
– This means that the 4a site is occupied by 80% Zr and 20%Y
– recalculate the simulation
• little change in intensity
• replace 20% Zr with Ce
– change the atom type of Y to Ce and recalculate the simulation
• how could we quantify amount of Y doping in YSZ?
 In substitutional doping, peak intensities and/or
peak positions may change
• In a substitutionally doped material:
– peak intensities will change depending on the difference between
the number of electrons (and therefore scattering factor f) between
the original and dopant ion
– Peak positions will change depending on the difference between the
size of the original and dopant ion
• A difference in ionic radii will produce a change in the average
interatomic distances and therefore a change in the unit cell lattice
parameter
• This change cannot be predicted by Rietveld modeling; it requires
empirical calibration or advanced ab-initio computer simulation.

• Y is significantly larger than Zr

– The ionic radius for Y is 1.019
– The ionic radius for Zr is 0.84
– The relationship between Y content and lattice parameter in YSZ
has been empirically determined from known standards
– The amount of Y in YSZ can be determined by calculating the lattice
parameter of the unknown specimen
 The thermal parameter, Biso, changes how scattering
efficiency changes with as a function of (sin q/l

40
• change thermal parameter for Zr B=0
– reset the SOF of Zr to 1 and the SOF 35 B=1
of the other 4a atom (Ce or Y) to 0 B=10
30
• the diffraction pattern will be
representative of ZrO2 with no doping 25

– change Biso and recalculate the 20

f
diffraction pattern
15
• change from 1.14 to 2.3
• change to 5 10

• change to 0.1 5

0
0 0.5 1 1.5 2
(sin q/l

 B sin 2 q 
f  f 0 exp  
 l 2

These simulations do not necessarily represent the
data that your diffractometer will produce
• There are more factors that contribute to the diffraction pattern that
you collect with an instrument

• Some factors are calculated from fundamental equations

– The simulations automatically account for:
• multiplicity
• K2 radiation
• Lorentz polarization factor

• Some factors are approximated using empirical formulas

– The diffraction peak width and shape is a product of the diffractometer
optics, sample preparation, and the specimen microstructure
– Diffraction peak widths and shapes in these simulations have been
calculated using default parameters in an empirical formula
• Some software attempts to model the instrument contribution to peak width and
shape using “Fundamental Parameters”. This is not yet implemented in
HighScore Plus.
 Other Things that effect intensity
IK=SMKLK|FK|2PKAKEK
• S is an arbitrary scale factor
– used to adjust the relative contribution of individual phases to the overall
diffraction pattern
• M is the multiplicity of the reflection
– accounts for the fact that some observed diffraction peaks are acutally
the product of multiple equivalent planes diffracting at the same position
2theta (for example, (001) (100) (010) etc in cubic)
– automatically calculated based on the crystal structure
• L is the Lorentz polarization factor
• P is the modification of intensity due to preferred orientation
• A is absorption correction
• E is extinction correction
• F is structure factor, which is the amplitude of scattered light due to
the crystal structure (already discussed)
The Lorentz polarization factor corrects for several
effects
• The Lorentz factor
– Different planes remain in the diffracting condition 1  POL cos 2 2q
over a different angular range Lp 
– high angle planes remain in the diffracting position sin 2 q cos q
longer, increasing their apparent intensity
– 1/sin2qcosq relationship for Bragg-Brentano
Monochromator POL
– varies with geometry of the diffractometer
• Polarization K-beta Filter 1
– Scattering of light by the atom polarizes the X-rays Cu diffracted beam 0.8
– this causes the scattered intensity to vary as graphite (point
1+cos2(2q)/2
detector)
• Monochromator Polarization
– The monochromator uses diffraction from a crystal Cu X’Celerator 1.25
to remove unwanted wavelengths of radiation.
Cr diffracted beam 0.588
– This diffraction will also polarize the X-ray beam,
causing intensity to vary as cos2(2q) (point detector)
• POL is the correction factor for the monochromator

• the correct POL value can be found in HSP Help

– go to the Help Index, type in POL, and the first
option describes the Lorentz polarization factor
 Setting the Polarization Correction Coefficient,
POL, for your simulation
• in the Refinement Control list,
click on Global Parameters
• The Global Parameters will
show up in the Object
Inspector
• In the “General Properties”
expansion of the Object
Inspector, you can set the
Polarization Correction
Coefficient (POL)
 Modeling Preferred Orientation

• The default calculation assumes that there are an equal number of

crystallites contributing to every diffraction peak
– the orientation of the crystallites is random and there are a statistically
relevant number of grains
• preferred crystallographic orientation produces a non-random
distribution in the orientation of the crystallites
– the observed diffraction peak intensities will vary systematically
The March distribution function empirically models
the preferred orientation effect

• The March distribution function is a model for fitting

preferred orientation of cylinder or needle shaped
crystals.

 
 32
Pk  r 2 cos2  k  r 1 sin 2  k
k is the angle between the preferred orientation vector and the
normal to the planes generating the diffracted peak
r is a refinable parameter in the Rietveld method

• There are other models for Preferred Orientation, such as

spherical harmonics
– The spherical harmonics model will be added in HighScore Plus v4
 Modeling Preferred Orientation in HSP

• In the Refinement Control list,

click on the phase that has
preferred orientation
• In the Object Inspector,
expand the Preferred
Orientation settings
– set the h,k, and l directions to
represent the [hkl] of the
preferred orientation

– set the Preferred Orientation parameter

• this parameter is found in the Object Inspector as “Parameter
(March/Dollase)”
• this parameter is also found in the Refinement Control List as “Preferred
Orientation”
• 1 means the orientation of [hkl] is completely random
• <1 means that the [hkl] direction is the preferred orientation of the crystallites
• >1 means that the [hkl] direction is preferentially avoided
• recalculate the simulation to observe the change in peak intensities
 Modeling Preferred Orientation in HSP

• For high symmetry phases,

you should enable the
“Summed March/Dollase
function”
• In the Refinement Control list,
click on Global Parameters
• In the Object Inspector, scroll
to the Refinement settings
• Check the option “Use
summed March/Dollase
function”
 Additional Intensity Factors

• Absorption Correction
– accounts for the transmission and absorption of the X-rays through the irradiated
volume of the sample.
– depends on sample geometry: flat plate or cylindrical
– depends on the mass absorption coefficient of the sample
• significant for cylindrical or transmission samples with a high sample absorption
• significant for a flat plate sample with a low sample absorption
• Extinction Correction
– Secondary diffraction of scattered X-rays in a large perfect crystal can decrease
the observed intensity of the most intense peaks
– scattering by uppermost grains limits the penetration depth of a majority of X-
rays, causing a smaller irradiated volume
– want “ideally imperfect” crystals– grains with enough mosaicity to limit extinction
– better to reduce extinction by grinding the powder to a finer size
• These two corrections are rarely used. Caution is advised. Always check to
make sure that the correction values are physically meaningful.
 We now know how to calculate the diffraction peak
intensity, but there a couple of more factors involved in
simulating real diffraction data
• The intensity, Yic, of each individual data point i is calculated using the
equation: k2
Yic  Yib   Gik Ik
k  k1
• We already know how to calculate IK, the intensity of the Bragg diffraction
peak k: IK=SMKLK|FK|2PKAKEK
• Yib is the intensity of the background at point i in the pattern
• k1 - k2 are the reflections contributing to data point i in the pattern
– sometimes multiple Bragg diffraction peaks overlap, resulting in multiple
contributions to the observed intensity at a single data point
• Gik is the peak profile function
– this describes how the intensity of the diffraction peak is distributed over a range
of 2theta rather than at a single point
– this profile is due to instrument broadening, sample broadening, etc
 Diffraction peaks have profiles that must be
empirically modeled
• The intensity equation Ik predicts that diffraction peak intensity occurs at a
single point 2theta– but in reality the diffraction peak intensity is spread out
over a range of 2theta
– The total area of the diffraction peak profile is the predicted intensity Ik
• Diffractometers contribute a characteristic broadening and shape to the
diffraction peak based on their optics and geometry
– each separate combination of divergence slit apertures, Soller slits, beta filter
or monochromators, and detectors will have their own characteristic
instrument profile

• Sample attributes also contribute to the

diffraction peak shape and width
– mass absorption coefficient (sample
transparency)
– sample thickness
– crystallite size
– microstrain
– defect concentration

50
 Profile Parameters

• Rietveld refinement assumes

that peak profiles vary
systematically as a function of
2theta
• Peak profile may also vary
according to crystallographic
direction [hkl]
Peak Width

Peak Shape

51
 The Cagliotti equation describes how peak width
varies with 2theta

H k  U tan q  V tan q  W
2

1/ 2

• Hk is the Cagliotti function where U, V and W are

refinable parameters
• HSP treats the Cagliotti equation as a convolution
between instrument and sample broadening functions
• The change in U and W between an instrument profile
standard (eg NIST 640c) and your sample can indicate
the amount of nanocrystallite size and microstrain
broadening
The systematic variation of peak width with 2theta

• For a standard Bragg-

Brentano
diffractometer
– U and W will be positive
– V will be negative
• This trend may vary for
data collected with
parallel beam, GIXD,
and capillary
geometries

53
Diffraction Peaks from laboratory diffractometers
are a mixture of Gaussian and Lorentzian profiles

Counts
Counts

4000

Gaussian profile shape

2000 Lorentzian profile shape
3000

2000

1000

1000

0
0
42 44 46 48 50
45.50 46
Position [°2Theta]
Position [°2Theta]

54
 Several different functions can be used to describe the
mixing of Gaussian and Lorentzian contributions

• The profile function is set in the Object Inspector for

Global Parameters (found in Refinement Control list)
– the same profile function is used for all phases in a single pattern
– Pseudo Voigt and Pearson VII are the most commonly used
– Pseudo Voigt
– Pseudo Voigt with FJC Asymmetry correction
– Pearson VII
– Voigt
 The Pseudo Voigt equation uses up to 3 parameters to
describe Gaussian and Lorentzian components

G jk  
2
H k

1  4 X 2jk 
1
 1   
C11/ 2
H k  1/ 2

exp  C1 X 2
jk 
   1   2 2q   3 2q 2
• The parameters 1, 2, and 3 are the refinable shape parameters,
listed as Peak Shape 1, Peak Shape 2, and Peak Shape 3 in the
Refinement Control list
– Usually Peak Shape 1 is the only one used
• Hk is the Cagliotti peak width function
• C1 is 4 ln 2


2q i  2q k 
• Xkj is X jk
Hk
Background can be empirically fit with an equation,
manually fit, or evaluated with a blank sample
• refinement of a polynomial function
• refinement using a type I or II Chebyshev polynomial function
• By amorphous sinc function
• linear interpolation between base points
• independent data collection run without the sample in place
– It is difficult to use this technique effectively– it requires careful evaluation of the validity of the background fit

57
 Systematic error in diffraction peak positions can
also be calculated

• The systematic error functions are found in Global

Parameters
• They apply to all phases in the sample
– zero shift
– specimen displacement
 The Parameters of a Pattern Simulation
• Global parameters affect the entire sample rather than individual phases
– Background function
– Zero Shift
– Specimen Displacement
– POL (Lorentz polarization factor)
– Sample Geometry
– selecting the Profile Function
• Other parameters are set individually for each phase
– scale factor
– preferred orientation (March-Dollase function)
– unit cell
– atomic parameters are for each atom in the phase
• position, SOF, thermal parameter
– Peak Profile parameters
• Cagliotti peak width function parameters U,V, and W
• Profile shape parameters
• peak asymmetry profile
• anisotropic broadening function

59
 What is Physically Meaningful?

Wel (1975)

60