Rietveld refinement

Method of refining powder diffraction data to find the crystal

structure, developed by Hugo M. Rietveld
 Practical reasons to perform a
Rietveld refinement

• solving an unknown crystal structure

• calculating the amount disorder or mixing on a Wyckoff site

• quantitatively determining the percentages of different phases

in your sample

• determining the crystallite sizes in your samples

 Rietveld refinement
Fits the whole pattern at once and refines:
• atomic positions

• disorder or mixing between atomic sites

• lattice parameters

• profile parameters (i.e, the peak shape)

• background parameters
 Factors affecting peak intensities

1. Structure factors (form factors)

2. Multiplicity
3. Lorentz factor } LP factor
4. Polarization factor }
5. Temperature factor or atomic displacement
6. Absorption
7. Preferred orientation
8. Extinction coefficients
 Scattering considerations

incident beam

absorbing substance

fluorescent
x-rays heat
transmitted electrons
beam

scattered x-rays
(coherent & incoherent)
 Scattering considerations

• Thompson/elastic scattering by electrons

I = I0 (K/r2) sin2 α

• Form factor, f: scattering by an atom over

the scattering power of an electron.

• At θ = 0, f = atomic number, but drops off

as sin θ/λ

• Compton/inelastic scattering increases as

atomic # decreases

Figure from Cullity & Stock, 2001.

 Structure Factors
amplitude of the wave scattered
by all the atoms in a unit cell
| Fhkl | =
amplitude of the wave scattered
by one electron

Fhkl = ∑ fn exp [2πi (hu + kv + lw)]

It describes how the atom arrangement (u, v, w) affects the

scattered beam. fn is the atomic scattering factor. The
intensity of a diffracted beam is proportional to |F | 2.

Note: exp(πi) = -1
exp(2πi) = +1
exp(nπi) = exp(-nπi)
 Multiplicity (M)
In a cubic structure (a = b = c)

{100} is (100), (010), (001), (-100), (0-10), (001) so M = 6

{110} is (110), (-110), (1-10), (-1-10) so M = 12

The {110} peak is expected to be TWICE as strong as the {100}

 How distortions affect multiplicity
From cubic (a = b = c) → tetragonal (a = b ≠ c)
(100) M = 6 → (100), (-100), (010), (0-10) M=4
(001), (00-1) M=2

(110) M = 12 → (110) M=4

(101) M=8

(111) M = 8 → (111) M=8

cubic

tetragonal
(100) (110) (111)
 Lorentz factor

The Lorentz factor is a measure of the amount of time that a

point of reciprocal lattice remains on the sphere of reflection
during the measuring process

• small deviations from Bragg’s Law – depends on 1/(sin 2θ)

• orientation of crystals – depends on (cos θ)
• fraction of the diffraction cone that intercepts the detector –
depends on 1/(sin 2θ)

Lorentz factor = 1/(4 sin2θ cosθ)

 Polarization factor

Polarization factor = ½ (1 + cos22θ)

LP factor = (1+cos22θ)/(sin2θ cosθ)

Note 1. If a monochromator is used, then the polarization

factor is ½ (1 + cos22θ cos22θM) where θM is the Bragg
angle for the monochromator.
Note 2. For neutron diffraction, polarization is a constant.
LP vs θ figure from Cullity & Stock, 2001.
 Temperature factor

Thermal vibrations
1. Unit cell expansion causes changes in the 2θ positions.
2. Decrease in the intensities of diffracted lines.
3. Increase in the intensity of background scattering.

Figure from Cullity & Stock, 2001.

 Temperature factor or atomic
displacement
temperature factor = exp (-2M)
f = f0 exp (-M)
M = 8π2ū2 (sinθ / λ)2
M = B (sinθ / λ)2
ū is the mean square displacement of the atom in a direction
normal to the diffractinng plane
Debye-Waller temperature factor
B = 8π2ū2
Isotropic factor B or ū
Anisotropic factor B11, B22, B33, B12, B13, B23
ū11, ū22, ū33, ū12, ū13, ū23
 Absorption

Idiffracted = Iincident exp (-µt)

µ is the linear absorption coefficient, and t is the thickness of

the sample.
 Preferred orientation
• crystals are not randomly oriented
• examples: plate-like crystals or needle-shaped crystals

March-Dollase function Spherical harmonic function

P(α) = (r2cos2α+sin2α/r)-3/2 • measurement of the pole density
distributions of a number of
P(α) pole distribution
diffraction planes
α angle between hkl & PO vector
• more complex and thus more
r adjustable parameter powerful
 Extinction
Destructive interference from re-reflections within the
crystals. The strong peaks appear weaker. The solution
is to grind the powdered sample more.
 Lorentzian vs Gaussian:
peak profile shape functions
Gaussian
2 ln 2  − 4 ln 2 2
I i ,k =
πH k
exp
H 2 (2θi − 2θk ) 
 k 

Lorentzian
−1
2  4 2
I i ,k = 1 +
πH k  H k2
(2θi − 2θk ) 


psuedo-Voigt
Ii,k = η Li,k + (1- η) Gi,k
 Crystallite Size (τ)

Scherrer equation τ = 0.9λ B/cosθB

If Lorentzian Bexp = Bsize + Binst
If Gaussian Bexp2 = Bsize2 + Binst2

Figure from Cullity & Stock, 2001.

 Putting it all together:
the calculated diffraction pattern
yci = s ∑ Lk |Fk |2 φ (2θi – θk) Pk A + ybi
yci calculated intensity at the ith step
s scale factor
k the Miller indices, h, k, l, for a Bragg reflection
Lk multiplicity, Lorentz and polarization factor
Fk structure factor for the kth Bragg reflection
φ reflection profile function
Pk Preferred orientation
A absorption factor
ybi background intensity at the ith step
 Rietveld Method in Practice
Basic requirements
1. Accurate diffraction data
2. A reasonable starting structural model
a. Space group symmetry
b. Approximate atomic positions
c. model may be from: isostructural materials, theoretical
simulations, high-resolution atomic imaging
3. A Rietveld refinement program
a. GSAS (Larson and von Dreele)
b. Fullprof (Rodriguez – Carvajal)
c. Others: BGMN (Bergmann), DBW (Wiles and Young),
LHPM-Rietica (Hunter), MAUD (Lutterotti), Rietan (Izumi),
Simref (Ritter)
 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 Wyckof x y z Biso sof

f 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

• each letter in the Wyckoff notation specifies a site that sits

on a different combination of symmetry elements
• 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)
– 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
 The Site Occupancy Factor, sof, indicates what
fraction of a site is occupied by a specific atom
[ ]
Fhkl = ∑ N j f j exp 2πi (hx j + ky j + lz j )
m

j =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
 Goodness of fit: R’s and χ2

RF =
∑ I K (' obs' ) 1/ 2 − I K (' calc' ) 1/ 2
R-structure factor
∑I
1/ 2
K (' obs')

RB =
∑ I K (' obs') − I K (' calc')
R-Bragg factor
∑I K (' obs' )

Rp =
∑ yi (obs) − yi (calc)
R-pattern
∑ y (obs)
i

 ∑ wi ( yi (obs) − yi (calc)) 2 
1/ 2

Rwp =   R-weighted pattern

 ∑ wi ( yi (obs)) 2 

2
 Rwp  Reduced Chi-squared
χ = 
2

 Rexp 

Which R should you believe?

 Some common problems
• Wrong symmetry

• Wrong impurity peaks

• Missed peak

• High background

• Can’t find O (or some other light element) in

a compound containing heavy elements.
Basics of X-Ray Powder Diffraction
 2012 was the 100th Anniversary of X-Ray Diffraction
• X-rays were discovered by WC Rontgen in 1895
• In 1912, PP Ewald developed a formula to describe the
passage of light waves through an ordered array of scattering
atoms, based on the hypothesis that crystals were composed
of a space-lattice-like construction of particles.
• Maxwell von Laue realized that X-rays might be the correct
wavelength to diffract from the proposed space lattice.
• In June 1912, von Laue published the first diffraction pattern
in Proceedings of the Royal Bavarian Academy of Science.

The diffraction pattern of copper sulfate, published in 1912

 The Laue diffraction pattern

• Von Laue’s diffraction pattern supported

two important hypotheses
– X-rays were wavelike in nature and therefore
were electromagnetic radiation
– The space lattice of crystals

• Bragg consequently used X-ray diffraction

to solve the first crystal structure, which
was the structure of NaCl published in The second diffraction
June 1913. pattern published was of
ZnS. Because this is a
• Single crystals produce “spot” patterns higher symmetry
material, the pattern was
similar to that shown to the right.
less complicated and
• However, powder diffraction patterns look easier to analyze
quite different.
 An X-ray powder diffraction pattern is a plot of the intensity of
X-rays scattered at different angles by a sample

• The detector moves in a circle around the

X-ray sample
tube – The detector position is recorded as
the angle 2theta (2θ)
w
2q – The detector records the number of X-
sample
rays observed at each angle 2θ
– The X-ray intensity is usually recorded
as “counts” or as “counts per second”

• Many powder diffractometers use the

Intensity (Counts)

Bragg-Brentano parafocusing geometry

10000
– To keep the X-ray beam properly
focused, the incident angle omega
5000 changes in conjunction with 2theta
– This can be accomplished by rotating
0 the sample or by rotating the X-ray
35 40 45 50 55
Position [°2Theta] (Cu K-alpha) tube.
 X-rays scatter from atoms in a material and therefore contain
information about the atomic arrangement
Counts
SiO2 Glass
4000

2000

0
4000 Quartz
3000
2000
1000
0
Cristobalite
4000

2000

0
20 30 40 50
Position [°2Theta] (Copper (Cu))
• The three X-ray scattering patterns above were produced by three chemically identical
forms SiO2
• Crystalline materials like quartz and cristobalite produce X-ray diffraction patterns
– Quartz and cristobalite have two different crystal structures
– The Si and O atoms are arranged differently, but both have long-range atomic order
– The difference in their crystal structure is reflected in their different diffraction patterns
• The amorphous glass does not have long-range atomic order and therefore produces
only broad scattering features
Diffraction occurs when light is scattered by a periodic array with
long-range order, producing constructive interference at
specific angles.

• The electrons in each atom coherently scatter light.

– We can regard each atom as a coherent point scatterer
– The strength with which an atom scatters light is proportional to the number of
electrons around the atom.
• The atoms in a crystal are arranged in a periodic array with long-range
order and thus can produce diffraction.
• The wavelength of X rays are similar to the distance between atoms in a
crystal. Therefore, we use X-ray scattering to study atomic structure.
• The scattering of X-rays from atoms produces a diffraction pattern, which
contains information about the atomic arrangement within the crystal

• Amorphous materials like glass do not have a periodic array with long-range order, so
they do not produce a diffraction pattern. Their X-ray scattering pattern features broad,
poorly defined amorphous ‘humps’.
 Crystalline materials are characterized by the long-
range orderly periodic arrangements of atoms.
• The unit cell is the basic repeating unit that defines the crystal structure.
– The unit cell contains the symmetry elements required to uniquely define the
crystal structure.
– The unit cell might contain more than one molecule:
• for example, the quartz unit cell contains 3 complete molecules of SiO2.
– The crystal system describes the shape of the unit cell Crystal System: hexagonal
– The lattice parameters describe the size of the unit cell Lattice Parameters:
4.9134 x 4.9134 x 5.4052 Å
(90 x 90 x 120°)

• The unit cell repeats in all dimensions to fill space and produce the
macroscopic grains or crystals of the material
 The diffraction pattern is a product of the unique
crystal structure of a material
Quartz
8000

6000
Quartz
4000

2000

0
Cristobalite
8000

6000

4000

2000 Cristobalite
0
20 30 40 50 60
Position [°2Theta] (Copper (Cu))

• The crystal structure describes the atomic arrangement of a material.

• The crystal structure determines the position and intensity of the
diffraction peaks in an X-ray scattering pattern.
– Interatomic distances determine the positions of the diffraction peaks.
– The atom types and positions determine the diffraction peak intensities.
• Diffraction peak widths and shapes are mostly a function of instrument
and microstructural parameters.
 Diffraction pattern calculations treat a crystal as a
Counts
collection of planes of atoms

112
Calculated_Profile_00-005-0490

20

110

102

200
111

201
10

003
0
35 40 45 50
Position [°2Theta] (Copper (Cu))

• Each diffraction peak is attributed to the scattering from a specific set of

Peak List

parallel planes of atoms.

• Miller indices (hkl) are used to identify the different planes of atoms

• Observed diffraction peaks can be related to planes of atoms to assist in

analyzing the atomic structure and microstructure of a sample
 A Brief Introduction to Miller Indices
• The Miller indices (hkl) define the reciprocal
axial intercepts of a plane of atoms with the
unit cell
– The (hkl) plane of atoms intercepts the unit cell
𝑎 𝑏 𝑐
at , , and
ℎ 𝑘 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

• The vector dhkl is drawn from the origin of the

unit cell to intersect the crystallographic plane
(hkl) at a 90° angle.
– The direction of dhkl is the crystallographic
direction.
– The crystallographic direction is expressed using
[] brackets, such as [220]
The diffraction peak position is a product of interplanar
spacing, as calculated by Bragg’s law

Bragg’s Law
  2d hkl sin q

• Bragg’s law relates the diffraction angle, 2θ, to dhkl

– In most diffractometers, the X-ray wavelength l is fixed.
– Consequently, a family of planes produces a diffraction peak only at a
specific angle 2θ.
• dhkl is a geometric function of the size and shape of the unit cell
– dhkl is the vector drawn from the origin to the plane (hkl) at a 90° angle.
– dhkl, the vector magnitude, is the distance between parallel planes of
atoms in the family (hkl)
– Therefore, we often consider that the position of the diffraction peaks are
determined by the distance between parallel planes of atoms.
The diffraction peak intensity is determined by the arrangement
of atoms in the entire crystal
2
𝐼ℎ𝑘𝑙 ∝ 𝐹ℎ𝑘𝑙

Fhkl   N j f j exp 2i hx j  ky j  lz j  
m

j 1

• The structure factor 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 efficiency of X-ray scattering at any
angle by the group of electrons in each atom
– The scattering factor is equal to the number of electrons around the atom at 0° θ,
the drops off as θ increases
• Nj is the fraction of every equivalent position that is occupied by atom j
 Bragg’s law provides a simplistic model to understand
what conditions are required for diffraction.
s
[hkl]

  2d hkl sin q q q

dhkl dhkl
• For parallel planes of atoms, with a space dhkl between the planes, constructive
interference only occurs when Bragg’s law is satisfied.
– In our diffractometers, the X-ray wavelength  is fixed.
– A family of planes produces a diffraction peak only at a specific angle 2q.

• Additionally, the plane normal [hkl] must be parallel to the diffraction vector s
– Plane normal [hkl]: the direction perpendicular to a plane of atoms
– Diffraction vector s: the vector that bisects the angle between the incident and
diffracted beam
 Many powder diffractometers use the Bragg-Brentano
parafocusing geometry.
Detector

s
X-ray
tube

w
2q

• The incident angle, w, is defined between the X-ray source and the sample.
• The diffraction angle, 2q, is defined between the incident beam and the detector.
• The incident angle w is always ½ of the detector angle 2q .
– In a q:2q instrument (e.g. Rigaku H3R), the tube is fixed, the sample rotates at q °/min and the
detector rotates at 2q °/min.
– In a q:q instrument (e.g. PANalytical X’Pert Pro), the sample is fixed and the tube rotates at a rate -q
°/min and the detector rotates at a rate of q °/min.
• In the Bragg-Brentano geometry, the diffraction vector (s) is always normal to the
surface of the sample.
– The diffraction vector is the vector that bisects the angle between the incident and scattered beam
A single crystal specimen in a Bragg-Brentano diffractometer would
produce only one family of peaks in the diffraction pattern.

[100] [110] [200]

s s s

2q

At 20.6 °2q, Bragg’s law The (110) planes would diffract at 29.3 The (200) planes are parallel to the (100)
fulfilled for the (100) planes, °2q; however, they are not properly planes. Therefore, they also diffract for this
producing a diffraction peak. aligned to produce a diffraction peak crystal. Since d200 is ½ d100, they appear at
(the perpendicular to those planes does 42 °2q.
not bisect the incident and diffracted
beams). Only background is observed.
 A polycrystalline sample should contain thousands of crystallites.
Therefore, all possible diffraction peaks should be observed.

[110] [200]
[100]
s s
s
2q 2q 2q

• For every set of planes, there will be a small percentage of crystallites that are properly
oriented to diffract (the plane perpendicular bisects the incident and diffracted beams).
• Basic assumptions of powder diffraction are that for every set of planes there is an equal
number of crystallites that will diffract and that there is a statistically relevant number of
crystallites, not just one or two.
 Powder diffraction is more aptly named polycrystalline
diffraction
• Samples can be powder, sintered pellets, coatings on substrates, engine blocks...
• The ideal “powder” sample contains tens of thousands of randomly oriented
crystallites
– Every diffraction peak is the product of X-rays scattering from an equal
number of crystallites
– Only a small fraction of the crystallites in the specimen actually contribute to
the measured diffraction pattern
• XRPD is a somewhat inefficient measurement technique
• Irradiating a larger volume of material can help ensure that a statistically relevant
number of grains contribute to the diffraction pattern
– Small sample quantities pose a problem because the sample size limits the
number of crystallites that can contribute to the measurement
 X-rays are scattered in a sphere around the sample
• Each diffraction peak is actually a Debye diffraction cone produced by the tens of
thousands of randomly oriented crystallites in an ideal sample.
– A cone along the sphere corresponds to a single Bragg angle 2theta
• The linear diffraction pattern is formed as the detector scans along an arc that
intersects each Debye cone at a single point
• Only a small fraction of scattered X-rays are observed by the detector.
 X-Ray Powder Diffraction (XRPD) is a somewhat
inefficient measurement technique

• Only a small fraction of crystallites in the sample actually

contribute to the observed diffraction pattern
– Other crystallites are not oriented properly to produce diffraction from
any planes of atoms
– You can increase the number of crystallites that contribute to the
measured pattern by spinning the sample
• Only a small fraction of the scattered X-rays are observed by
the detector
– A point detector scanning in an arc around the sample only observes
one point on each Debye diffraction cone
– You can increase the amount of scattered X-rays observed by using a
large area (2D) detector
Diffraction patterns are collected as absolute intensity vs 2q, but
are best reported as relative intensity vs dhkl.

• The peak position as 2q depends on instrumental characteristics such as

wavelength.
– The peak position as dhkl is an intrinsic, instrument-independent, material
property.
• Bragg’s Law is used to convert observed 2q positions to dhkl.

• The absolute intensity, i.e. the number of X rays observed in a given peak,
can vary due to instrumental and experimental parameters.
– The relative intensities of the diffraction peaks should be instrument
independent.
• To calculate relative intensity, divide the absolute intensity of every peak by the
absolute intensity of the most intense peak, and then convert to a percentage. The
most intense peak of a phase is therefore always called the “100% peak”.
– Peak areas are much more reliable than peak heights as a measure of
intensity.
 Powder diffraction data consists of a record of photon
intensity versus detector angle 2q.
• Diffraction data can be reduced to a list of peak positions and intensities
– Each dhkl corresponds to a family of atomic planes {hkl}
– individual planes cannot be resolved- this is a limitation of powder diffraction versus
single crystal diffraction
Raw Data Reduced dI list
Counts
Position Intensity DEMO08
hkl dhkl (Å) Relative
3600
[°2q] [cts] Intensity
25.2000 372.0000 (%)
25.2400 460.0000
25.2800 576.0000 1600
{012} 3.4935 49.8
25.3200 752.0000 {104} 2.5583 85.8
25.3600 1088.0000
25.4000 1488.0000
{110} 2.3852 36.1
400
25.4400 1892.0000 {006} 2.1701 1.9
25.4800 2104.0000
25.5200 1720.0000 {113} 2.0903 100.0
25.5600 1216.0000 0
25 30 35 40 45 {202} 1.9680 1.4
25.6000 732.0000 Position [°2Theta] (Copper (Cu))

25.6400 456.0000
25.6800 380.0000
25.7200 328.0000
Applications of XRPD
 You can use XRD to determine
• Phase Composition of a Sample
– Quantitative Phase Analysis: determine the relative amounts of phases in a
mixture by referencing the relative peak intensities
• Unit cell lattice parameters and Bravais lattice symmetry
– Index peak positions
– Lattice parameters can vary as a function of, and therefore give you
information about, alloying, doping, solid solutions, strains, etc.
• Residual Strain (macrostrain)
• Crystal Structure
– By Rietveld refinement of the entire diffraction pattern
• Epitaxy/Texture/Orientation
• Crystallite Size and Microstrain
– Indicated by peak broadening
– Other defects (stacking faults, etc.) can be measured by analysis of peak
shapes and peak width
• We have in-situ capabilities, too (evaluate all properties above as a
function of time, temperature, and gas environment)
 Phase Identification
• The diffraction pattern for every phase is as unique as your fingerprint
– Phases with the same chemical composition can have drastically different
diffraction patterns.
– Use the position and relative intensity of a series of peaks to match
experimental data to the reference patterns in the database
The diffraction pattern of a mixture is a simple sum of
the scattering from each component phase
 Databases such as the Powder Diffraction File (PDF) contain dI
lists for thousands of crystalline phases.
• The PDF contains over 300,000 diffraction patterns.
• Modern computer programs can help you determine what phases are
present in your sample by quickly comparing your diffraction data to all of
the patterns in the database.
• The PDF card for an entry contains a lot of useful information, including
literature references.
 Quantitative Phase Analysis

• With high quality data, you can determine how

much of each phase is present 60

..
– must meet the constant volume assumption (see

I(phase a)/I(phase b)
later slides)
50
• The ratio of peak intensities varies linearly as a 40
function of weight fractions for any two phases
in a mixture 30
𝐼α 𝑋
– = K * 𝑋α 20
𝐼β β

– need to know the constant of proportionality 10

• RIR method is fast and gives semi-quantitative
results 0
𝑅𝐼𝑅 0 0.2 0.4 0.6 0.8 1
– 𝐾 = 𝑅𝐼𝑅α
β

• Whole pattern fitting/Rietveld refinement is a X(phase a)/X(phase b)

more accurate but more complicated analysis
 You cannot guess the relative amounts of phases based
only on the relative intensities of the diffraction peaks

• The pattern shown above contains equal amounts of TiO2 and Al2O3
• The TiO2 pattern is more intense because TiO2 diffracts X-rays more efficiently

With proper calibration, you can calculate the amount of each phase present in the sample
 Unit Cell Lattice Parameter Refinement

• By accurately measuring peak positions over a long range of

2theta, you can determine the unit cell lattice parameters of
the phases in your sample
– alloying, substitutional doping, temperature and pressure, etc can
create changes in lattice parameters that you may want to quantify
– use many peaks over a long range of 2theta so that you can identify
and correct for systematic errors such as specimen displacement and
zero shift
– measure peak positions with a peak search algorithm or profile fitting
• profile fitting is more accurate but more time consuming
– then numerically refine the lattice parameters
 Crystallite Size and Microstrain
• Crystallites smaller than ~120nm create broadening of diffraction peaks
– this peak broadening can be used to quantify the average crystallite size of nanoparticles
using the Scherrer equation
– must know the contribution of peak width from the instrument by using a calibration curve
• microstrain may also create peak broadening
– analyzing the peak widths over a long range of 2theta using a Williamson-Hull plot can let
you separate microstrain and crystallite size
• Careful calibration is required to calculate accurate crystallite sizes!
00-043-1002> Cerianite- - CeO2

K
B2q  
Intensity (a.u.)

L cos q

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
2q (deg.)
 Preferred Orientation (texture)

• Preferred orientation of crystallites can create a systematic

variation in diffraction peak intensities
– can qualitatively analyze using a 1D diffraction pattern by looking at
how observed peak intensities deviate systematically from the ideal
– a pole figure maps the intensity of a single peak as a function of tilt
and rotation of the sample
• this can be used to quantify the texture
10.0 (111) 00-004-0784> Gold - Au

8.0

(311)
Intensity(Counts)

6.0 (200)
(220)
4.0

2.0
(222)
(400)
x10 3
40 50 60 70 80 90 100
Two-Theta (deg)
 Non-ideal samples: Texture (i.e. preferred
crystallographic orientation)

• The samples consists of tens of thousands of grains, but the

grains are not randomly oriented
– Some phenomenon during crystallization and growth, processing, or
sample preparation have caused the grains to have preferred
crystallographic direction normal to the surface of the sample
350

300

250

Intensity(Counts)
200

150

100

50

0 (111)
(221) JCS#98> CaCO3 - Aragonite
(021) (012)
(102) (112) (220) (041) (132) (113)
(002) (121) (211) (040) (212) (222) (042)
25 30 35 40 45 50 55
Two-Theta (deg)

The preferred orientation creates a

systematic error in the observed
diffraction peak intensities.
Overview of the Diffractometer
 Essential Parts of the Diffractometer

• X-ray Tube: the source of X Rays

• Incident-beam optics: condition the X-ray beam before it hits
the sample
• The goniometer: the platform that holds and moves the
sample, optics, detector, and/or tube
• The sample & sample holder
• Receiving-side optics: condition the X-ray beam after it has
encountered the sample
• Detector: count the number of X Rays scattered by the sample
 X-radiation for diffraction measurements is produced
by a sealed tube or rotating anode.
H2O In
•
H2O Out
Sealed X-ray tubes tend to operate at 1.8
to 3 kW.
• Rotating anode X-ray tubes produce
much more flux because they operate at
9 to 18 kW. Be
Cu ANODE
Be
window
– A rotating anode spins the anode at 6000 window

rpm, helping to distribute heat over a e-

larger area and therefore allowing the XRAYS XRAYS
FILAMENT
tube to be run at higher power without (cathode)
melting the target. metal

• Both sources generate X rays by striking

the anode target with an electron beam (vacuum) (vacuum)
glass

from a tungsten filament.

– The target must be water cooled.
– The target and filament must be
contained in a vacuum.

AC CURRENT
The wavelength of X rays is determined by the anode of
the X-ray source.
• Electrons from the filament strike the target anode, producing characteristic
radiation via the photoelectric effect.
• The anode material determines the wavelengths of characteristic radiation.
• While we would prefer a monochromatic source, the X-ray beam actually
consists of several characteristic wavelengths of X rays.

K
L
M
 Spectral Contamination in Diffraction Patterns
Ka1 Ka1

Ka2

Ka2 Ka1

Ka2

W La1
Kb
• The Ka1 & Ka2 doublet will almost always be present
– Very expensive optics can remove the Ka2 line
– Ka1 & Ka2 overlap heavily at low angles and are more separated
at high angles
• W lines form as the tube ages: the W filament contaminates
the target anode and becomes a new X-ray source
• W and Kb lines can be removed with optics
 Monochromators remove unwanted wavelengths of radiation
from the incident or diffracted X-ray beam.

• Diffraction from a monochromator crystal can be used to select one

wavelength of radiation and provide energy discrimination.
• Most powder diffractometer monochromators only remove K-beta,
W-contamination, and Brehmstralung radiation
– Only HRXRD monochromators or specialized powder monochromators
remove K-alpha2 radiation as well.
• A monochromator can be mounted between the tube and sample
(incident-beam) or between the sample and detector (diffracted-
beam)
– An incident-beam monochromator only filters out unwanted wavelengths
of radiation from the X-ray source
– A diffracted-beam monochromator will also remove fluoresced photons.
– A monochromator may eliminate 99% of K-beta and similar unwanted
wavelengths of radiation.
– A diffracted-beam monochromator will provide the best signal-to-noise
ratio, but data collection will take a longer time
 Beta filters can also be used to reduce the intensity of
K-beta and W wavelength radiation
• A material with an absorption
edge between the K-alpha and
K-beta wavelengths can be
used as a beta filter Ni filter
• This is often the element just

Suppression
below the target material on
the periodic table
– For example, when using Cu
radiation
• Cu K-alpha = 1.541 Å
• Cu K-beta= 1.387 Å
• The Ni absorption edge= 1.488 Å
– The Ni absorption of Cu radiation
is:

Cu Ka
Cu Kb
W La
• 50% of Cu K-alpha
• 99% of Cu K-beta Wavelength
 H He
Li Be
Fluorescence B C N O F Ne
Na Mg Al Si P S Cl Ar
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
Cs Ba L Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn
Fr Ra A
• Some atoms absorb incident X-rays and fluoresce them as X-rays of a different
wavelength
– The absorption of X-rays decreases the diffracted signal
– The fluoresced X-rays increase the background noise
• The increased background noise from fluoresced X-rays can be removed by using:
– a diffracted-beam monochromator
– an energy sensitive detector
• The diffracted beam signal can only be increased by using a different wavelength of
radiation
• The most problematic materials are those two and three below the target material:
– For Cu, the elements that fluoresce the most are Fe and Co
 The X-ray Shutter is the most important safety device
on a diffractometer
H2O In H2O Out

• X-rays exit the tube through X-ray

transparent Be windows. XRAYS
Be
window

Cu ANODE
Be
window

• X-Ray safety shutters contain the Primary

Shutter
e-

beam so that you may work in the Secondary

Shutter FILAMENT
(cathode)
XRAYS

diffractometer without being exposed Solenoid metal

to the X-rays. glass

(vacuum) (vacuum)

• Being aware of the status of the

shutters is the most important factor
in working safely with X rays. AC CURRENT

SAFETY SHUTTERS
 The X-ray beam produced by the X-ray tube is divergent.
Incident-beam optics are used to limit this divergence

  2d hkl sin q
• X Rays from an X-ray tube are:
– divergent
– contain multiple characteristic wavelengths as well as Bremmsstrahlung radiation
• neither of these conditions suit our ability to use X rays for analysis
– the divergence means that instead of a single incident angle q, the sample is actually
illuminated by photons with a range of incident angles.
– the spectral contamination means that the smaple does not diffract a single wavelength
of radiation, but rather several wavelengths of radiation.
• Consequently, a single set of crystallographic planes will produce several diffraction peaks
instead of one diffraction peak.
• Optics are used to:
– limit divergence of the X-ray beam
– refocus X rays into parallel paths
– remove unwanted wavelengths
 Most of our powder diffractometers use the Bragg-
Brentano parafocusing geometry.
• A point detector and sample are
moved so that the detector is always
at 2q and the sample surface is
always at q to the incident X-ray
beam.
• In the parafocusing arrangement, the
incident- and diffracted-beam slits
move on a circle that is centered on
the sample. Divergent X rays from the
source hit the sample at different
points on its surface. During the
diffraction process the X rays are
refocused at the detector slit. F: the X-ray source
• This arrangement provides the best DS: the incident-beam divergence-limiting slit
SS: the Soller slit assembly
combination of intensity, peak shape, S: the sample
and angular resolution for the widest RS: the diffracted-beam receiving slit
number of samples. C: the monochromator crystal
AS: the anti-scatter slit
 Divergence slits are used to limit the divergence of the
incident X-ray beam.
• The slits block X-rays that have too great a
divergence.
• The size of the divergence slit influences
peak intensity and peak shapes.
• Narrow divergence slits:
– reduce the intensity of the X-ray beam
– reduce the length of the X-ray beam hitting
the sample
– produce sharper peaks
• the instrumental resolution is improved so
that closely spaced peaks can be resolved.
One by-product of the beam divergence is that the length of the beam
illuminating the sample becomes smaller as the incident angle
becomes larger.

• The length of the incident beam

is determined by the divergence
slit, goniometer radius, and
incident angle.
• This should be considered when 185mm Radius Goniometer, XRPD
choosing a divergence slits size: 40.00
– if the divergence slit is too large,
the beam may be significantly 35.00
L
longer than your sample at low I
e 30.00
angles r
n
– if the slit is too small, you may r
g 25.00
not get enough intensity from a 2°DS
t
your sample at higher angles d
h 20.00
i
– Appendix A in the SOP contains a a
guide to help you choose a slit 15.00
1°DS
(

t
size. e
m
m 10.00
• The width of the beam is d 0.5°DS
)

constant: 12mm for the Rigaku 5.00

0.15°DS
RU300.
0.00
0 20 40 60 80 100
2Theta (deg)
 Some systems use parallel-beam optics for a parallel
beam geometry

s Detector
X-ray
tube

w 2q

• Parallel beam optics do NOT require that the incident angle w is always ½
of the detector angle 2q .
• A coupled scan with parallel-beam optics will maintain the diffraction
vector in a constant relationship to the sample.
– If w is always ½ of 2q then the diffraction vector (s) is always normal to the surface of
the sample.
– If w = ½ 2q + τ, then s will be always tilted by τ away from the vertical position.
• That direction will not change as long as both omega and 2theta change in a coupled
relationship so that w is always equal to ½ 2q + τ
 Parallel beam optics allow for the possibility of grazing
incidence X-ray diffraction (GIXD)
Detector

s
X-ray
tube

w 2q

• The incident angle, w, is set to a very shallow angle (between 0.2 and 5 deg).
– This causes the X-rays to be focused in the surface of the sample, limiting the penetration depth of the X-
rays
• Only the detector moves during data collection
– The value τ is changing during the scan (where τ = ½*2q - w)
– As a consequence, the diffraction vector (s) is changing its direction during the scan
• Remember the diffraction only comes from crystallites in which dhkl is parallel to s
– Therefore, the direction being probed in the sample changes
– This is perfectly ok for ideal samples with randomly oriented grains; however, for samples with preferred
orientation this will cause a problem.
• Regular GIXD will constrain the X-ray beam in the top few microns of the surface
• IP-GIXD can be configued to constrain diffraction to the top 10-20 nm of the surface.
 Other optics:
• limit divergence of the X-ray beam
– Divergence limiting slits Parallel Plate Collimator & Soller
– Parallel plate collimators Slits block divergent X-rays, but
do not restrict beam size like a
– Soller slits divergent slit
• refocus X rays into parallel paths
– “parallel-beam optics”
– parabolic mirrors and capillary lenses
– focusing mirrors and lenses
• remove unwanted wavelengths
– monochromators
– Kb filters

Göbel Mirrors and capillary lenses collect

a large portion of the divergent beam and
refocus it into a nearly parallel beam
 Detectors

• point detectors
– observe one point of space at a time
• slow, but compatible with most/all optics
– scintillation and gas proportional detectors count all photons, within an
energy window, that hit them
– Si(Li) detectors can electronically analyze or filter wavelengths
• position sensitive detectors
– linear PSDs observe all photons scattered along a line from 2 to 10° long
– 2D area detectors observe all photons scattered along a conic section
– gas proportional (gas on wire; microgap anodes)
• limited resolution, issues with deadtime and saturation
– CCD
• limited in size, expensive
– solid state real-time multiple semiconductor strips
• high speed with high resolution, robust
 Area (2D) Diffraction allows us to image complete or
incomplete (spotty) Debye diffraction rings

the area observed by a linear detector the area observed by a linear detector

Polycrystalline thin film on a Mixture of fine and coarse grains

single crystal substrate in a metallic alloy
Conventional linear diffraction patterns would miss
information about single crystal or coarse grained materials
 PANalytical X’Pert Pro Multipurpose Diffractometer
• Prefix optics allow the configuration to be quickly changed to accommodate a
wide variety of data collection strategies.
• This diffractometer can be used to collect XRPD, GIXD, XRR, residual stress, and
texture data.
• A vertical-circle theta-theta goniometer is used so that the sample always lies flat
and does not move.
– Sample sizes may be as large as 60mm diameter by 3-12mm thick, though a more typical
sample size is 10-20mm diameter.
• Data collection modes can be changed between:
– high-speed high-resolution divergent beam diffraction
• Programmable divergence slits can maintain a constant irradiated area on sample surface
– parallel beam diffraction using incident Gobel mirror and receiving-side parallel plate
collimator
• eliminates errors due to irregular sample surfaces, sample displacement, and defocusing during
glancing angle measurements
• A variety of sample stages include:
– 15 specimen automatic sample changer
– open Eulerian cradle with automated z-translation as well as phi and psi rotation for
texture, reflectivity, and residual stress measurements
– furnace for heating a sample to 1200°C in air, vacuum, or controlled atmosphere
– a cryostat for cooling a sample to 11 K in vacuum
 Back Reflection Laue Diffractometer

• The sample is irradiated with white radiation for Laue

diffraction
• Use either Polaroid film or a two-dimensional multiwire
detector to collect back-reflection Laue patterns
– The 2D multiwire detector is not currently working
• Determine the orientation of large single crystals and thin film
single crystal substrates
Sample Preparation
 Important characteristics of samples for XRPD

• a flat plate sample for XRPD should have a smooth flat surface
– if the surface is not smooth and flat, X-ray absorption may reduce the
intensity of low angle peaks
– parallel-beam optics can be used to analyze samples with odd shapes
or rough surfaces
• Densely packed
• Randomly oriented grains/crystallites
• Grain size less than 10 microns
– So that there are tens of thousands of grains irradiated by the X-ray
beam
• ‘Infinitely’ thick
• homogeneous
 Preparing a powder specimen

• An ideal powder sample should have many crystallites in random

orientations
– the distribution of orientations should be smooth and equally distributed
amongst all orientations
• Large crystallite sizes and non-random crystallite orientations both lead to
peak intensity variation
– the measured diffraction pattern will not agree with that expected from an
ideal powder
– the measured diffraction pattern will not agree with reference patterns in the
Powder Diffraction File (PDF) database
• If the crystallites in a sample are very large, there will not be a smooth
distribution of crystal orientations. You will not get a powder average
diffraction pattern.
– crystallites should be <10mm in size to get good powder statistics
 Preferred orientation

• If the crystallites in a powder sample have plate or needle like

shapes it can be very difficult to get them to adopt random
orientations
– top-loading, where you press the powder into a holder, can cause
problems with preferred orientation
• in samples such as metal sheets or wires there is almost
always preferred orientation due to the manufacturing
process
• for samples with systematic orientation, XRD can be used to
quantify the texture in the specimen
 Non-Ideal Samples: a “spotty” diffraction pattern

• The sample does not contain tens of thousands of grains

– The Debye diffraction cone is incomplete because there are not a
statistically relevant number of grains being irradiated
Counts
Mount3_07

3600

1600

400

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

The poor particle statistics cause random

error in the observed diffraction peak
intensities.
 Non-ideal samples: Texture (i.e. preferred
crystallographic orientation)

• The samples consists of tens of thousands of grains, but the

grains are not randomly oriented
– Some phenomenon during crystallization and growth, processing, or
sample preparation have caused the grains to have preferred
crystallographic direction normal to the surface of the sample
350

300

250

Intensity(Counts)
200

150

100

50

0 (111)
(221) JCS#98> CaCO3 - Aragonite
(021) (012)
(102) (112) (220) (041) (132) (113)
(002) (121) (211) (040) (212) (222) (042)
25 30 35 40 45 50 55
Two-Theta (deg)

The preferred orientation creates a

systematic error in the observed
diffraction peak intensities.
 Ways to prepare a powder sample

• Top-loading a bulk powder into a well

– deposit powder in a shallow well of a sample holder. Use a slightly
rough flat surface to press down on the powder, packing it into the
well.
• using a slightly rough surface to pack the powder can help minimize
preferred orientation
• mixing the sample with a filler such as flour or glass powder may also help
minimize preferred orientation
• powder may need to be mixed with a binder to prevent it from falling out
of the sample holder
– alternatively, the well of the sample holder can be coated with a thin layer of
vaseline
• Dispersing a thin powder layer on a smooth surface
– a smooth surface such as a glass slide or a zero background holder (ZBH) may
be used to hold a thin layer of powder
• glass will contribute an amorphous hump to the diffraction pattern
• the ZBH avoids this problem by using an off-axis cut single crystal
– dispersing the powder with alcohol onto the sample holder and then allowing
the alcohol to evaporate, often provides a nice, even coating of powder that
will adhere to the sample holder
– powder may be gently sprinkled onto a piece of double-sided tape or a thin
layer of vaseline to adhere it to the sample holder
• the double-sided tape will contribute to the diffraction pattern
– these methods are necessary for mounting small amounts of powder
– these methods help alleviate problems with preferred orientation
– the constant volume assumption is not valid for this type of sample, and so
quantitative and Rietveld analysis will require extra work and may not be
possible
Experimental Considerations
 Varying Irradiated area of the sample

• the area of your sample that is illuminated by the X-ray beam varies
as a function of:
– incident angle of X rays
– divergence angle of the X rays
• at low angles, the beam might be wider than your sample
– “beam spill-off”
• This will cause problems if you sample is not homogeneous
185mm Radius Goniometer, XRPD
40.00

35.00
L
I
e 30.00
r
n
r
g 25.00
a 2°DS
t
d
h 20.00
i
a 15.00
1°DS
(

t
m
e
m 10.00
d 0.5°DS
)

5.00
0.15°DS
0.00
0 20 40 60 80 100
2Theta (deg)
 Penetration Depth of X-Rays

• The depth of penetration of x-rays into a material depends on:

– The mass absorption coefficient, μ/ρ, for the composition
– The density and packing factor of the sample
– The incident angle omega
– The wavelength of radiation used
3.45 sin ω
• Depth of penetration, t, is 𝑡 = μ
ρ∗ρ𝑏𝑢𝑙𝑘
• Depth of penetration at 20 degrees omega
– W
• With 100% packing: 2.4 microns
• With 60% packing (typical for powder): 4 microns
– SiO2 (quartz)
• With 100% packing: 85 microns
• With 60% packing (typical for powder): 142 microns
 The constant volume assumption

• In a polycrystalline sample of ‘infinite’ thickness, the change in the

irradiated area as the incident angle varies is compensated for by
the change in the penetration depth
• These two factors result in a constant irradiated volume
– (as area decreases, depth increases; and vice versa)
• This assumption is important for any XRPD analysis which relies on
quantifying peak intensities:
– Matching intensities to those in the PDF reference database
– Crystal structure refinements
– Quantitative phase analysis
• This assumption is not necessarily valid for thin films or small
quantities of sample on a zero background holder (ZBH)
 Many sources of error are associated with the focusing
circle of the Bragg-Brentano parafocusing geometry

• The Bragg-Brentano parafocusing

geometry is used so that the
detector divergent X-ray beam reconverges
tube
Receiving at the focal point of the detector.
Slits • This produces a sharp, well-
defined diffraction peak in the
data.
• If the source, detector, and sample
are not all on the focusing circle,
sample errors will appear in the data.
• The use of parallel-beam optics
eliminates all sources of error
associated with the focusing circle.
 Sample Transparency Error
• X Rays penetrate into your sample
detector – depth of penetration depends on:
tube
Receiving • the mass absorption coefficient of your
sample
Slits
• the incident angle of the X-ray beam
• This produces errors because not all
X rays are diffracting from the same
location in your sample
– Produces peak position errors and
peak asymmetry
sample – Greatest for organic and low
absorbing (low atomic number)
samples
• Can be eliminated by using parallel-
beam optics
• Can be reduced by using a thin
sample
 Other sources of error
• Flat specimen error
– The entire surface of a flat specimen
detector cannot lie on the focusing circle
tube
Receiving – Creates asymmetric broadening toward
Slits low 2theta angles
– Reduced by using small divergence slits,
which produce a shorter beam
• For this reason, if you need to increase
intensity it is better to make the beam
wider rather than longer.
– eliminated by parallel-beam optics

sample
• Poor counting statistics
– The sample is not made up of thousands
of randomly oriented crystallites, as
assumed by most analysis techniques
– The sample might have large grain sizes
• Produces ‘random’ peak intensities and/or
spotty diffraction peaks

A good reference for sources of error in diffraction data is available at http://www.gly.uga.edu/schroeder/geol6550/XRD.html

 Axial divergence

• Axial divergence
– Due to divergence of the X-ray beam in plane with the sample
– creates asymmetric broadening of the peak toward low 2theta angles
– Creates peak shift: negative below 90° 2theta and positive above 90°
– Reduced by Soller slits and/or capillary lenses

Counts
0.04rad Soller Slits
0.04rad incident Soller slit and 0.02rad detector Soller Slit
0.02rad Soller Slits
60000

40000

20000

3 4 5 6
Position [°2Theta] (Copper (Cu))
 Grazing Incident Angle Diffraction (GIXD)

• also called Glancing Angle X-Ray Diffaction

• The incident angle is fixed at a very small angle (<5°) so that X-rays are
focused in only the top-most surface of the sample.
• GIXD can perform many of analyses possible with XRPD with the added
ability to resolve information as a function of depth (depth-profiling) by
collecting successive diffraction patterns with varying incident angles
– orientation of thin film with respect to substrate
– lattice mismatch between film and substrate
– epitaxy/texture
– macro- and microstrains
– reciprocal space map
 X-Ray Reflectivity (XRR)

• A glancing, but varying, incident

angle, combined with a matching
detector angle collects the X rays
reflected from the samples
surface
• Interference fringes in the
reflected signal can be used to
determine:
– thickness of thin film layers
– density and composition of thin
film layers
– roughness of films and interfaces
 Back Reflection Laue

• Used to determine crystal orientation

• The beam is illuminated with ‘white’ radiation
– Use filters to remove the characteristic radiation wavelengths from the
X-ray source
– The Bremmsstrahlung radiation is left
• Weak radiation spread over a range of wavelengths
• The single crystal sample diffracts according to Bragg’s Law
– Instead of scanning the angle theta to make multiple crystallographic
planes diffract, we are effectively ‘scanning’ the wavelength
– Different planes diffract different wavelengths in the X-ray beam,
producing a series of diffraction spots
 Small Angle X-ray Scattering (SAXS)

• Highly collimated beam, combined with a long distance between the

sample and the detector, allow sensitive measurements of the X-rays that
are just barely scattered by the sample (scattering angle <6°)
• The length scale of d (Å) is inversely proportional to the scattering angle:
therefore, small angles represented larger features in the samples
• Can resolve features of a size as large as 200 nm
– Resolve microstructural features, as well as crystallographic
• Used to determine:
– crystallinity of polymers, organic molecules (proteins, etc.) in solution,
– structural information on the nanometer to submicrometer length scale
– ordering on the meso- and nano- length scales of self-assembled molecules
and/or pores
– dispersion of crystallites in a matrix
 Single Crystal Diffraction (SCD)

• Used to determine:
– crystal structure
– orientation
– degree of crystalline perfection/imperfections (twinning, mosaicity,
etc.)
• Sample is illuminated with monochromatic radiation
– The sample axis, phi, and the goniometer axes omega and 2theta are
rotated to capture diffraction spots from at least one hemisphere
– Easier to index and solve the crystal structure because it diffraction
peak is uniquely resolved
 Available Free Software

• GSAS- Rietveld refinement of crystal structures

• FullProf- Rietveld refinement of crystal structures
• Rietan- Rietveld refinement of crystal structures

• PowderCell- crystal visualization and simulated diffraction

patterns
• JCryst- stereograms
X-Ray Powder Diffraction Data Analysis
 An X-ray diffraction pattern is a plot of the intensity
of X-rays scattered at different angles by a sample

• The detector moves in a circle around

X-ray the sample
tube – The detector position is recorded
ω as the angle 2theta (2θ)
2θ – The detector records the number of
sample
X-rays observed at each angle 2θ
– The X-ray intensity is usually
recorded as “counts” or as “counts
per second”
Intensity (Counts)

• To keep the X-ray beam properly

10000
focused, the sample will also rotate.
– On some instruments, the X-ray
5000 tube may rotate instead of the
sample.
0
35 40 45 50 55
Position [°2Theta] (Cu K-alpha)
 Each “phase” produces a unique diffraction pattern
Quartz • A phase is a specific chemistry and
atomic arrangement.
• Quartz, cristobalite, and glass are all
different phases of SiO2
Cristobalite
– They are chemically identical,
but the atoms are arranged
differently.
– As shown, the X-ray diffraction
Glass
pattern is distinct for each
different phase.
– Amorphous materials, like glass,
do not produce sharp diffraction
15 20 25 30 35 40 peaks.
Position [°2Theta] (Cu K-alpha)

The X-ray diffraction pattern is a fingerprint that lets you figure out what is in your sample.
 The diffraction pattern of a mixture is a simple sum of
the diffraction patterns of each individual phase.
Quartz Mixture

Cristobalite

Glass

0
15 20 25 30 35 40
15 20 25 30 35 40
Position [°2Theta] (Cu K-alpha) Position [°2Theta] (Copper (Cu))

• From the XRD pattern you can determine:

– What crystalline phases are in a mixture
– How much of each crystalline phase is in the mixture (quantitative
phase analysis, QPA, is covered in another tutorial)
– If any amorphous material is present in the mixture
Qualitative Analysis of XRD Data
 Experimental XRD data are compared to reference
patterns to determine what phases are present

• The reference patterns are represented by sticks

• The position and intensity of the reference sticks should match the data
– A small amount of mismatch in peak position and intensity is
acceptable experimental error
 Specimen Displacement Error will cause a small
amount of error in peak positions

Peaks that are close The peak shift follows a cosθ

together should be behavior, so peak shift might
shifted the same change direction over a large
direction and by the angular range
same amount

• Specimen displacement is a systematic peak position error due to

misalignment of the sample.
ିଶ௦ ୡ୭ୱ ఏ
• The direction and amount of peak shift will vary as
ோ
 Most diffraction data contain K-alpha1 and K-alpha2
peak doublets rather than just single peaks
K-alpha1
K-alpha1 K-alpha2

K-alpha2
K-alpha1
K-alpha2

• The k-alpha1 and k-alpha2 peak doublets are further apart at higher
angles 2theta
• The k-alpha1 peaks always as twice the intensity of the k-alpha2
• At low angles 2theta, you might not observe a distinct second peak
 The experimental data should contain all major peaks
listed in the reference pattern

Minor reference peaks could

If a major reference peak is
be lost in the background
not observed in the data, then
noise, so it may be acceptable
that is not a good match
if they are not observed

This is an example of a bad match between the data and the reference pattern
 The X-ray diffraction pattern is a sum of the diffraction
patterns produced by each phase in a mixture
Counts

Rutile, syn;

Hematite, syn;

Rutile, syn;
Hematite, syn;
3600

Anatase, syn;
Hematite, syn;

Rutile, syn;
Hematite, syn;
1600

Anatase, syn;

syn;

Rutile, syn;
Anatase, syn;

Anatase, syn;
syn;

Hematite, syn;
Rutile,
Hematite,
400

0
25 30 35 40 45
Position [°2Theta] (Copper (Cu))

Each different phase produces a different combination of peaks.

 You cannot guess the relative amounts of phases based
upon the relative intensities of the diffraction peaks

• The pattern shown above contains equal amounts of TiO2 and Al2O3
• The TiO2 pattern is more intense because TiO2 diffracts X-rays more efficiently

With proper calibration, you can calculate the amount of each phase present in the sample
 Diffraction peak broadening may contain information
about the sample microstructure
• Peak broadening may indicate:
– Smaller crystallite size in nanocrystalline materials
– More stacking faults, microstrain, and other defects in the crystal structure
– An inhomogeneous composition in a solid solution or alloy
• However, different instrument configurations can change the peak width, too

These patterns show the difference between bulk These patterns show the difference between the
ceria (blue) and nanocrystalline ceria (red) exact same sample run on two different instruments.
When evaluating peak broadening, the instrument profile must be considered.
Quantitative Analysis of XRD Data
 Diffraction peak positions can be used to calculated
unit cell dimensions
Counts

25.321 deg
d= 3.5145 Å
24.179 deg
1000 d= 3.6779 Å

500

0
23 24 25 26
Position [°2Theta] (Copper (Cu))

• The unit cell dimensions can be correlated to interatomic distances

• Anything the changes interatomic distances- temperature,
subsitutional doping, stress- will be reflected by a change in peak
positions
 To calculate unit cell lattice parameters from the
diffraction peak positions
• Convert the observed peak positions, °2theta, into dhkl values
using Bragg’s Law: λ
݀௛௞௟ =
2 sin θ
• Determine the Miller indices (hkl) of the diffraction peaks
from the published reference pattern
– If you do not have access to a reference pattern that identifies (hkl)
then you will need to index the pattern to determine the (hkl)
• Use the d*2 equation to calculate the lattice parameters
– Most analysis programs contain an unit cell refinement algorithm for
numerically solving the lattice parameters
– These programs can also calculate and correct for peak position error
due to specimen displacement
2
d *hkl = h 2 a *2 + k 2b *2 +l 2c *2 +2hka * b * cos γ * +2hla * c * cos β * +2klb * c * cos α *
 The diffraction peak width may contain microstructural
Counts
information
Width=0.007 rad
XS ~ 19 nm
Width=0.002 rad
1000 ௄஛
XS> 90 nm Size =
ௐ௜ௗ௧௛∗ୡ୭ୱ ఏ

500

0
23 24 25 26
Position [°2Theta] (Copper (Cu))
• Nanocrystallite size will produce peak broadening that can be quantified
– Once the crystallite size is larger than a maximum limit, the peak broadening cannot be quantified.
This creates an upper limit to the crystallite size that can be calculated.
– The upper limit depends on the resolution of the diffractometer.
• Non-uniform lattice strain and defects will also cause peak broadening
• Careful evaluation is required to separate all of the different potential causes of peak broadening
 The weight fraction of each phase can be calculated if
the calibration constant is known
Counts

TiO2, Rutile 49.4 %

Fe2O3, Hematite 28.7 %
3600 TiO2, Anatase 21.9 %

1600

400

0
25 30 35 40 45
Position [°2Theta] (Copper (Cu))

• The calibration constants can be determined:

– By empirical measurements from known standards
– By calculating them from published reference intensity ratio (RIR) values
– By calculating them with Rietveld refinement
 All calculations are more accurate if you use more
peaks over a longer angular range

• If you use one or two peaks, you must assume:

– That there is no specimen displacement error when calculating lattice parameters
– That there is no microstrain broadening when calculating crystallite size
• If you use many peaks over a long angular range (for example, 7+ peaks over a
60° 2theta range), you can:
– Calculate and correct for specimen displacement when solving lattice parameters
– Calculate and account for microstrain broadening when calculating crystallite size
– Improve precision by one or two orders of magnitude
 There are different ways to extract peak information
for quantitative analysis

• Numerical methods reduce the diffraction data to a list

of discrete diffraction peaks
– The peak list records the position, intensity, width and shape of
each diffraction peak
– Calculations must be executed based on the peak list to produce
information about the sample
• Full pattern fitting methods refine a model of the sample
– A diffraction pattern is calculated from a model
– The calculated and experimental diffraction patterns are
compared
– The model is refined until the differences between the observed
and calculated patterns are minimized.
– The Rietveld, LeBail, and Pawley fitting methods use different
models to produce the calculated pattern
 A peak list for empirical analysis can be generated in
different ways

• The diffraction data are reduced to a list of diffraction peaks

• Peak search
– Analysis of the second derivative of diffraction data is used to identify
likely diffraction peaks
– Peak information is extracted by fitting a parabola around a minimum
in the second derivative
– This method is fast but the peak information lacks precision

• Profile fitting
– Each diffraction peak is fit independently with an equation
– The sum of the profile fits recreates the experimental data
– Peak information is extracted from the profile fit equation
– This method provides the most precise peak information
 Profile Fitting produces precise peak positions, widths,
heights, and areas with statistically valid estimates

• Empirically fit experimental data

with a series of equations
– fit the diffraction peak using the
profile function

Intensity (a.u.)
• The profile function models the
mixture of Gaussian and
Lorentzian shapes that are typical
of diffraction data
– fit background, usually as a
polynomial function
• this helps to separate intensity in
28.5 29.0 29.5 30.0
peak tails from background 2θ (deg.)

• To extract information, operate

explicitly on the equation rather
than numerically on the raw data
 Diffraction peak lists are best reported using dhkl and
relative intensity rather than 2θ and absolute intensity.

• The peak position as 2θ depends on instrumental characteristics

such as wavelength.
– The peak position as dhkl is an intrinsic, instrument-independent,
material property.
• Bragg’s Law is used to convert observed 2θ positions to dhkl.
• The absolute intensity, i.e. the number of X rays observed in a given
peak, can vary due to instrumental and experimental parameters.
– The relative intensities of the diffraction peaks should be instrument
independent.
• To calculate relative intensity, divide the absolute intensity of every peak by
the absolute intensity of the most intense peak, and then convert to a
percentage. The most intense peak of a phase is therefore always called the
“100% peak”.
– Peak areas are much more reliable than peak heights as a measure of
intensity.
Calculations must be executed on the peak list to yield
any information about the sample

• This peak list itself does not tell you anything about the
sample
– Additional analysis must be done on the peak list to extract
information
• From the peak list you can determine:
– Phase composition: by comparison to a database of reference
patterns
– Semi-quantitative phase composition: calculated from peak
intensities for different phases
– Unit cell lattice parameters: calculated from peak positions
– Crystal system: determined by indexing observed peaks and
systematic absences
– Crystallite size and microstrain: calculated from peak widths and/or
shapes
– A number of engineering indexes are also calculated from peak list
information
 Full pattern fitting methods use different models to
produce a calculated pattern

• The Rietveld method uses fundamental calculations from

crystal structure models to produce the calculated diffraction
pattern
– Analysis produces a refined crystal structure model for all phases in
the sample
• Peak positions and intensities are constrained by the crystal
structure model
– Crystallite size, microstrain, and preferred orientation can be
extracted from empirical models included in the refinement

• Le-Bail and Pawley fitting methods use unit cell models

combined with empirical fitting of peak intensities
– Analysis produces a refined unit cell model but does not immediate
yield information about parameters related to peak intensities
 Other analytical methods

• Total scattering methods (whole pattern fitting) attempts

to model the entire diffraction pattern from first
principal calculations
– Calculations include
• Bragg diffraction peaks,
• diffuse scatter contributions to background,
• peak shapes based on diffractometer optics,
• peak shapes based on crystallite size, shape, defects, and
microstrain
• Pair distribution functional analysis uses Fourier
analysis to produce an atomic pair density map
– Can yield atomic structure information about non-crystalline,
semi-crystalline, and highly disordered materials
 Properties of neutrons

Particle-like properties:

•Mass = 1.68×10-27 kg (photon mass = zero)

•Charge = zero (photon charge = zero)
•Spin = ½ (photon spin = 1)
•Magnetic dipole moment = -9.66×10-27 JT-1 (photon moment zero)

Wave-like properties:

1 h
E = mn v 2 = k BT λ=
2 mn v
Neutron type Energy (meV) Temperature (K) Wavelength (Å)
“Cold” 0.1 – 10 1 – 120 4 - 30
“Thermal” 5 – 100 60 – 1000 1–4
“Hot” 100 – 500 1000 – 6000 0.4 – 1

•For diffraction experiments thermal neutrons are used. Velocity is of the order of ~2000
ms-1 for “room temperature” neutrons (photons 3×108 ms-1).
Interactions of neutrons and X-rays with matter

www.ncnr.nist.gov
 Elastic scattering of X-rays from electrons

“Elements of Modern Xray Physics”

(J. Als-Nielsen & D. McMorrow)

Ratio of radiated electric field magnitude to incident electric field magnitude is:

Erad ( R, t ) ⎛ e2 ⎞ e i kR
= −⎜⎜ ⎟
2 ⎟
cosψ
E in ⎝ 4πε 0 mc ⎠ R Thomson scattering length r0 = 2.82 × 10-5 Å
(the “ability” of an electron to scatter X-rays)
Minus sign indicates that incident and radiated fields are 180°out of phase
 Elastic scattering of neutrons from nuclei

www.ncnr.nist.gov

•Neutron-nucleus interaction involves very short-range forces (on the order of 10-15 m). A
metastable nucleus + neutron state is formed which then decays, re-emitting the neutron as a
spherical wave with a phase change of 180°

•Radius of nucleus is ~10-17 m – much smaller than wavelength of thermal neutrons (10-10 m),
thus can be considered “point-like”
 Basics of diffraction

k k'

Bragg’s Law: nλ = 2d hkl sin θ

 Basics of diffraction

2θ rj

unit cell

“Elements of Modern Xray Physics”

(J. Als-Nielsen & D. McMorrow) motif

atomic form factor = number of electrons (X-rays)

Scattering amplitude for crystal: nuclear scattering length (neutrons)

F crystal
(Q ) = ∑ f j e iQ⋅r j
∑ e iQ ⋅ R n integer multiple of
2π when Bragg’s
rj Rn law is satisfied
lattice vectors -Rn
unit cell structure factor lattice sum
 Neutron v X-ray diffraction: atomic number

•For neutrons there is no systematic trend in scattering length with atomic number- it depends
on the nucleus (isotope, nuclear spin). Scattering length is negative for some nuclei!

•Adjacent atoms in the periodic table often have very different neutron scattering lengths,
allowing them to be distinguished easily.
Absorption for thermal neutrons and 8keV X-rays

• Neutrons are
absorbed by nuclear
processes that destroy
the neutrons, emitting
secondary radiation
(α, β, or γ) as a result.

• For most atoms,

neutrons penetrate
much further into the
sample than X-rays.

www.ncnr.nist.gov
 Neutron diffraction – magnetic structure

F 2
hkl =F 2
Nuc ( hkl ) +q F 2 2
Mag ( hkl )

q = 1 − (ε ⋅ κ )
2 2

unit vector in direction of Magnetic structure factor

reciprocal lattice vector for
plane hkl

unit vector in direction of spin

•Nuclear and magnetic scattering intensities are additive.

 Neutron diffraction – magnetic structure
N
FMag ( hkl ) = ∑ pn e 2πi ( hxn + ky n + lz n ) − M n
e
1
Atomic
coordinates Thermal factor
(x,y,z)

electron charge neutron magnetic moment

magnetic
⎛ eγ ⎞ 2
form factor
Magnetic scattering amplitude: pn = ⎜⎜ ⎟ gSf m
2 ⎟
(tabulated)

⎝ 2mc ⎠
electronic spin
electron mass
Landé g factor, usually ~2
 Neutron diffraction – magnetic structure

Neutron diffraction can give:

•The positions of magnetic atoms within the unit cell

•The directions of their ordered magnetic moments

•The magnitudes of their ordered magnetic moments

Neutrons only ever see the components of the magnetization that are perpendicular to the scattering vector!
 Introduction to magnetic symmetry
Paramagnet (T > TC)
n

intensity
n n
n

2θ
Ferromagnet (T < TC)
n n+m
intensity

n+m Unit cell same size

n
n+m Additional intensity
appears on top of
existing peaks

2θ
 Introduction to magnetic symmetry
Paramagnet (T > TN)
n

intensity
n n
n

2θ
Antiferromagnet (T < TN)
n n+m
intensity

m
n+m n
m Unit cell is larger
m
n New peaks appear

2θ
Example: multiferroic TbMn2O5
Neutron scattering facilities

www.veqter.co.uk
 Neutron sources – nuclear reactor

•A steady supply of neutrons is produced by the 235U fission chain reaction (2.5 neutrons
per fission event, 1.5 are reabsorbed by the fuel).

•Neutrons are extracted from the core by neutron guide tubes and slowed down by a
moderator. A particular wavelength can then be selected using a crystal monochromator.

Broad spectrum

www.euronuclear.org
Wavelength selected by monochromator
 Neutron sources – spallation (pulsed) source
•Protons are accelerated in a synchrotron ring and then collide with a heavy metal target,
which emits many subatomic particles including neutrons (more than 10 per proton).

•A white beam of thermal neutrons is produced.

http://www.isis.stfc.ac.uk
Spallation sources – time-of-flight diffraction technique
arrival time of neutron
at detector

λ ⎛ h ⎞⎛ 1 ⎞ ht hkl
d hkl = =⎜ ⎟⎜ ⎟=
2 sin θ ⎝ mv ⎠⎝ 2 sin θ ⎠ 2mL sin θ

neutron mass, velocity distance from source

(momentum = mv = h/λ) to detector (v = L/thkl)

•Detector is kept at fixed position (analogous to X-ray Laue technique).

•Arrival time of diffracted neutrons at detector is determined (“time-of-flight”).

•Often a large bank of many detectors covering a range of angles is used.

•Resolution in dhkl can be increased by increasing distance L from the source.

 Neutron v X-ray diffraction
+ Neutrons are highly penetrating towards matter (neutral particles)- absorption is low for most
elements. Allows use of heavy sample environment (cryostats, pressure cells, magnets etc.) and
probes the whole sample.

+ There is often strong contrast in scattering between neighbouring elements (eg. can
distinguish Mn from Fe).

+ Light elements can give strong scattering eg. 2D, 12C, 14N, 16O.

+ Strong interaction with magnetic moments- can determine magnetic structures routinely.

+ No radiation damage to samples- important for organics / biological samples.

- Neutron sources are much weaker than X-ray sources – in general large samples are needed.

- Some nuclei strongly absorb neutrons and cannot be probed eg. 10B, 113Cd, 157Gd.

- Some nuclei are almost transparent to neutrons and cannot easily be probed eg. 51V. Some
nuclei have strong incoherent scattering giving high background eg. 1H.
 Neutron diffraction: summary

• Sensitive to light elements

• Bulk samples and big sample environment
• Distinguish neighbouring elements
• Sensitive to magnetic moments

• Neutron sources are relatively weak

• Some elements are strongly absorbing or give incoherent scattering

• Complementary to X-ray diffraction

• Larmor diffraction will be a more sensitive probe of structural phase

transitions and sample inhomogeneity / strain