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2. physics
3. optics

Lecture 9: Powder diffraction
patterns
Contents
1 Introduction
1
2 Indexing pattern for cubic crystals
2
3 Indexing for non-cubic systems
5
4 Phase diagram determination
6
5 Super lattice structures
1
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Introduction
The original reason for doing x-ray diffraction is to identify the structure
of the unknown material. Single crystals are favorable but for most metals,
ceramics, intermetallics, single crystals are not necessarily available. For
such materials powder patterns from polycrystals are used for identifying
the crystal structure. Powder patterns give two useful information
1. The shape and size of the unit cell - this is from the position of the
diffraction lines (2θ)
2. The arrangement of atoms in the unit cell - this is from the relative
intensities of the different lines.
To give an example, for a cubic system the lattice constant a determines the
values of 2θ for the various planes. The arrangement of atoms in the cubic
system, whether simple cubic, bcc, or fcc, determines the relative intensities
and the absence and presence of some lines. Thus, given a structure it is
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easy to calculate the diffraction pattern, especially for simple metals and intermetallics. But doing the reverse (which is what is expected from X-ray
diffraction) is not easy. This is especially true for following phase transformations in multi component systems where more than one system, with closely
spaced diffraction lines, is present. Finding the different phases and their
relative amounts becomes challenging. There are three major steps involved
in phase determination.
1. From the
and size.
of 7) and
they do’t
angular position (2θ) of the lines we get an idea of the shape
We start by assigning a crystal system to the material (out
based on that calculate Miller indices to the various lines. If
fit go back and reassign a new crystal system and iterate.
2. From the density, known chemical composition, and shape and size of
the unit cell the number of atoms per unit cell are calculated.
3. Finally, from the intensity of the lines the atom positions are calculated.
There are some sources of error in this approach.
1. Lack of truly monochromatic source - if the X-ray is not truly monochromatic, Kβ lines are also present along with the Kα line then there will
be extra lines in the diffraction pattern. Usually, these can be minimized by using the appropriate filters. Also, the extra lines have a
specific angular relation with the lines from the Kα radiation which
can be calculated and then eliminated.
2. Impurities in the unknown material - any presence of crystalline impurities in the sample will again cause extra lines. This depends on the
specimen properties and can be eliminated by processing.
2
Indexing pattern for cubic crystals
A cubic crystal gives diffraction lines where the angle (θ) obeys the following
relation
sin2 θ
λ2
sin2 θ
=
=
= constant
(1)
h2 + k 2 + l2
s2
4a2
where a is the lattice constant, λ is the x-ray wavelength, and (hkl) refers to
the Miller indices of the plane. Equation 1 is obtained using Bragg’s law and
the fraction is a constant for diffraction lines from a given x-ray source. Since,
hkl are integers s is also an integer and can only take certain values. The
values that s can take changes for the different cubic systems (sc, bcc, and
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Table 1: 2θ values for Al. First 5 diffraction lines
Intensity
2θ (in deg)
100
45.4
23.0
22.7
6.2
38.37
44.61
64.90
77.97
82.14
Table 2: 2θ values for Al with corresponding values of
cubic systems
Intensity
2θ (in deg)
d Å
sin2 θ
100
45.4
23.0
22.7
6.2
38.37
44.61
64.90
77.97
82.14
2.343
2.029
1.435
1.224
1.172
0.108
0.144
0.288
0.396
0.431
sin2 θ
s
for the various
sin2 θ
s
sc
bcc
fcc
0.108 0.054 0.036
0.072 0.038 0.036
0.036
0.036
0.036
fcc) and this is based on the structure factor rules. The problem is finding
the values of s for the different 2θ values. The s values for the different cubic
systems are
simple cubic - All (hkl): 1, 2, 3, 4, 5, 6, 8, 9, 10....
bcc - (h + k + l) = even: 2, 4, 6, 8, 10, 12 .....
fcc - (hkl) all even or odd: 3, 4, 8, 11, 12....
From the diffraction lines it is possible to calculate the various values of s
using equation 1. This information is summarized graphically in figure 1 for
the different cubic systems. These can be tried against the different sets for
the cubic systems. If there is no match then the system is not cubic.
Consider the case of Al. The first five diffraction lines for Al, in order of
increasing 2θ are given in table 1. The radiation used is Cu Kα with wave2
length 1.54 Å. Using equation 1 it is possible to calculate the values of sins θ
taking different values of s for simple cubic, fcc, and bcc. These are tabulated
2
in table 2. From table 2 it is clear that only for the fcc system does sins θ
become a constant as indicated in equation 1. Thus, Al crystallizes in a fcc
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Figure 1: Calculated diffraction patterns for the various lattices. Taken from
Elements of X-ray diffraction - B.D. Cullity.
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structure. The lattice spacing can be calculated using equation 1 where the
value of the constant from table 2 is 0.036.
λ2
sin2 θ
=
= 0.036
s
4a2
(2)
From equation 2 the lattice constant a of Al is 4.06 Å. Given that the density
of Al (ρ) is 2.7 gcm−3 it is possible to calculate the number of atoms per unit
cell, Z. This is given by the relation
ρ =
Z. at.wt
a3 NA
(3)
where NA is Avogadro’s number. From equation 3 the value of Z is 4. Thus,
Al has a fcc structure with 4 atoms per unit cell.
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Indexing for non-cubic systems
Cubic systems are easy to solve, since they have only one lattice constant
and the angle are all 90◦ . Things before more difficult if we have non-cubic
systems. Usually, graphical methods are used for solving these systems.
Consider a tetragonal system with a = b 6= c and all 3 angle 90◦ . The
relation between d spacing and the lattice constants for this system is
h2
k2
l2
1
=
+
+
d2
a2
b2
c2
2
2
h +k
l2
1
=
+
d2
a2
c2
2
1
1
l
= 2 [(h2 + k 2 ) +
]
2
d
a
(c/a)2
(4)
Taking logarithm on both sides give
2 log d = 2 log a − log[(h2 + k 2 ) +
l2
]
(c/a)2
(5)
If there are 2 planes with spacing d1 and d2 and Miller indices (h1 k1 l1 ) and
(h2 k2 l2 ) then equation 5 modifies to
2 log d1 − 2 log d2 = log[(h21 + k12 ) +
l12
l22
2
2
]
−
log[(h
+
k
)
+
] (6)
2
2
(c/a)2
(c/a)2
Equation 6 shows that the logarithm of the difference between d spacing for
2 planes in the tetragonal system depends only on the logarithm of the (c/a)
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ratio and the Miller indices. It is possible to make log plots of the second
term in the right hand side of equation 5 vs. (c/a) for all possible (hkl). This
will give a series of curves. The experimental pattern can be superimposed
on these curves and the (c/a) value can be obtained. Such types of curves
are called Hull-Davey curves. A partial Hull-Davey curve for the tetragonal
system is shown in figure 2. A complete one for the body centered tetragonal system is shown in figure 3. Hull-Davey curves can be constructed for
different crystal systems taking into account the relation between the lattice
constants and the lattice angles. In the case of the hexagonal system the
relation between the d spacing and the lattice constants is
4 (h2 + k 2 + hk)
l2
1
=
+
d2
3
a2
c2
(7)
After a similar manipulation followed for the tetragonal system, this can be
rearranged as
l2
4
]
2 log d = 2 log a − log[ (h2 + k 2 + hk) +
3
(c/a)2
(8)
A Hull-Davey chart can be constructed similar to than for the tetragonal
system to get the lattice constants.
As the number of independent lattice constants of the of the crystal increases (length and angles) it becomes more difficult to use the graphical
methods. Now, there are computer programs that are used to index patterns
by searching and matching with known databases. The powder diffraction
patterns for known materials are stored in the ICDD (International Center
for Diffraction Data) database. For unknown systems with more than one
type of atom in the unit cell we need the intensities of the lines to find the
atom positions. This is done by relating then intensities to the structure factor, F , which is related to the atomic scattering factors, and atom positions.
This is usually a trial and error process, where an initial structure is assumed
and the diffraction pattern calculated. This is matched with the experimental pattern and refinement is carried out to the trial structure. This process
is repeated until there is a match. Sometimes, for complex molecules (e.g.
organic), single crystals are needed for structure determination.
4
Phase diagram determination
Another area where x-ray diffraction is useful is in phase diagram determination. If we want to construct a phase diagram the classical way to do it is
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Figure 2: Partial Hull-Davey curve for the tetragonal system with the experimental pattern superimposed. Taken from Elements of X-ray diffraction B.D. Cullity.
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Figure 3: Complete Hull-Davey curve for the body centered tetragonal system. Taken from Elements of X-ray diffraction - B.D. Cullity.
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thermal analysis followed by microstructure information. But this does not
give structure information of the phases. For this we need diffraction. The
diffraction pattern for each phase is independent of the other phases. It is
also possible to get quantitative information on the relative amounts of the
various phases i.e. phase boundaries can also be constructed using XRD. It
is also important how the changing composition of the different phases can
affect the diffraction patterns
1. If there is solid solubility then as the concentration increases the dspacing changes. This is because the lattice constant changes. There
can either be an increase or decrease in lattice constant depending on
the relative atomic sizes of the constituent elements. This leads to a
shift in the position of the lines.
2. If there is a 2 phase region then as the concentration change the relative
intensity of the different lines changes but there is no change in line
position.
The following are the principles for collecting x-ray diffraction patterns for
phase diagram determination
1. Each alloy must be at equilibrium at the temperature of interest. For
high temperature phases not stable at room temperature there are two
options for studying the crystal structure.
(a) Quench to room temperature and do diffraction.
(b) Use x-ray diffraction with high temperature attachment for direct determination. This option is preferred when available for it
eliminates the need for preparing large number of samples.
2. The phase sequence: a horizontal line (constant temperature)must pass
through a single phase region and 2 phase region alternatively. A
line cannot pass from one 2 phase region to the next without passing through a single phase region, can be a line compound.
These principles can be understood by looking at figure 4. If we draw a
horizontal line then the phases go from single phase α to a mixture of α + γ
and then a line compound γ. From γ we again get a two phase region
γ + β and then finally single phase β. Within a single phase region as the
composition changes then line position changes but in the two phase region
the relative line intensities change. This information is captured in the series
of diffraction patterns for the phase diagram shown in figure 4 and shown in
figure 5.
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Figure 4: Phase diagram and lattice constant of a hypothetical binary phase
diagram. Taken from Elements of X-ray diffraction - B.D. Cullity.
In the single phase region where we have a solid solution, these can be of
two types
1. Interstitial - when solute atom is much smaller than the solvent atom
e.g. C, N, H, B atoms then we can have interstitial solid solutions. Interstitial solid solutions always lead to an increase in lattice parameters.
For non cubic structures not all lattice constant change equally.
2. Substitutional - These are of 3 types - random, ordered, or defect.
Random and ordered substitutional solid solutions are more common
than defect structures. Depending on the relative sizes of the two atoms
the lattice constants can increase or decrease. In defect structures the
increase in concentration of atom B is accompanied by creating holes
where A atoms are present. This is prevalent in compounds that have
partial covalent characteristics. They can affect the peak intensities by
affecting the structure factor. An example of a defect structure is in
NiAl which has a simple cubic structure with Ni atoms at the corner
and Al at the center. The phase exists over a composition range 45-61%
Ni. For off stoichiometry compositions there will be Ni or Al vacancies
in the lattice i.e. defect structures.
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Figure 5: XRD patterns for different alloys from the hypothetical binary
phase diagram in figure 4. Taken from Elements of X-ray diffraction - B.D.
Cullity.
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Figure 6: Ordered and disordered configuration in AuCu3 . Taken from Elements of X-ray diffraction - B.D. Cullity.
5
Super lattice structures
These are also called order-disorder transformations. In this a substitutional
solid solution that has atoms located at random positions at high temperature transforms into an ordered structure where the different kinds of atoms
are located at specific positions. In x-ray diffraction an order-disorder transformation will not affect the positions of the peaks but relative intensities will
change. Sometime new peaks are also formed. Ordered structures are also
called super lattice structures. The new lines seen in the diffraction pattern
are called super lattice lines. The original lines are called fundamental
lines.
To understand the formation of super lattice lines in XRD consider the example of AuCu3 . The disordered and ordered structure for this is shown in
figure 6. AuCu3 has an fcc structure with 4 atoms per unit cell. From the
formula, there are 3 Cu atoms for 1 Au atom. In the disordered structure,
the 4 atoms are randomly located in the unit cell while in the ordered structure, Au atoms are located at the corners and the Cu atoms are located at
the face center positions. The order-disorder transition temperature for this
system is 390◦ C.
Consider the completely disordered structure. The probability of a site being
occupied by Au atom is 41 while the probability of occupation by Cu is 34 .
Hence it is possible to define an average atomic factor term, fav that is given
by
1
3
fav = fAu + fCu
(9)
4
4
where fAu and fCu are the atomic scattering factors for Au and Cu. The
disordered structure can be considered as a regular fcc structure so that the
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structure factor Fhkl is given by
Fhkl = fav [1 + exp(iπh + k) + exp(iπk + l) + exp(iπl + h)]
(10)
The structure factor rules for the disordered structure are also similar to a
regular fcc lattice i.e. the structure factor Fhkl vanishes when hkl are mixed
and is non-zero when they are all even or all odd. The difference arises when
we have the ordered structure. Now the Cu and Au atoms are located at
specific positions and hence the structure factor is calculated by using these
specific positions. This gives the structure factor for the ordered structure
as
Fhkl = fAu + fCu [exp(iπh + k) + exp(iπk + l) + exp(iπl + h)]
(11)
Using equation 11 we can see that the structure factor does not vanish for
certain (hkl).
F = (fAu + 3fCu ) when (hkl) are all even or odd
F = (fAu − fCu ) when (hkl) are mixed
Thus, the ordered structure has extra diffraction lines, which are called super
lattice lines. This can be seen in the case of powder patterns of CuAu3 in
figure 7 where extra lines are visible.
Complete order and complete disorder represent the two extremes. In most
cases, it is possible to get a mixture of both. In such cases, it is possible to
define a long range order parameter, S, given by
S =
rA − FA
1 − FA
(12)
where rA refers to the fraction of A sites occupied by A atoms and FA refers
to the fraction of A atoms in the material. In the case of complete order rA
= 1 and hence S = 1. In complete disorder rA = FA and S = 0. It is possible
to calculate the long range order parameter by comparing the intensity of the
super lattice lines with the expected intensity when there is complete order
(S = 1).
There are certain cases when the super lattice lines have a low intensity
and are not visible in the powder pattern. Consider the case of CuZn. The
disordered structure is a bcc unit cell with Cu and Zn atoms randomly located
either at the corner locations or the body center. In the ordered structure
the Cu atoms are located at the corner and the Zn atoms at the center. This
is shown in figure 8. In CuZn the order-disorder transformation takes place
at 460 ◦ C. The disordered structure behaves like a bcc structure with an
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Figure 7: Powder patterns of (a) disordered (b) partially ordered and (c)
completely ordered AuCu3 . Taken from Elements of X-ray diffraction - B.D.
Cullity.
Figure 8: Ordered and disordered structures in CuZn. Taken from Elements
of X-ray diffraction - B.D. Cullity.
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average atomic scattering factor defined by the average of fCu and fZn . In
the ordered structure the scattering factor is given by
Fhkl = fCu + fZn [exp(iπh + k + l)]
(13)
Using equation 13 it is possible to calculate the structure factors for the
various values of (hkl)
F = (fCu + fZn ) when (h + k + l) is even − F undamental line
F = (fCu − fZn ) when (h + k + l) is odd − Super lattice line
Since the super lattice line is given by the difference of the atomic scattering
factors its intensity is very weak compared to the fundamental line. Since
intensity is directly proportional to the square of the structure factor
Isuper
If undamental
=
(fZn − fCu )2
(fZn + fCu )2
(14)
For θ = 0 the atomic scattering factors are equal to the atomic numbers so
that fCu = 29 and fZn = 30. Substituting in equation 14 this gives the ratio
to be 3 × 104 . It is thus possible for the super lattice lines to be too weak to
be detected. Order-disorder transition is an example of long range ordering.
It is harder to detect short range order or clustering using x-ray diffraction.
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