Powder Diffraction Patterns

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Lecture 9: Powder diffractionpatterns

Contents1 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 12

1 Introduction

The original reason for doing x-ray diffraction is to identify the structureof the unknown material. Single crystals are favorable but for most metals,ceramics, intermetallics, single crystals are not necessarily available. Forsuch materials powder patterns from polycrystals are used for identifyingthe 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 (2)

2. The arrangement of atoms in the unit cell - this is from the relativeintensities of the different lines.

To give an example, for a cubic system the lattice constant a determines thevalues of 2 for the various planes. The arrangement of atoms in the cubicsystem, whether simple cubic, bcc, or fcc, determines the relative intensitiesand the absence and presence of some lines. Thus, given a structure it is

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MM3030: Materials Characterisation

easy to calculate the diffraction pattern, especially for simple metals and in-

termetallics. But doing the reverse (which is what is expected from X-raydiffraction) is not easy. This is especially true for following phase transforma-tions in multi component systems where more than one system, with closelyspaced diffraction lines, is present. Finding the different phases and theirrelative amounts becomes challenging. There are three major steps involvedin phase determination.

1. From the angular position (2) of the lines we get an idea of the shapeand size. We start by assigning a crystal system to the material (outof 7) and based on that calculate Miller indices to the various lines. If

they dot 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 monochro-matic, K lines are also present along with the K line then there willbe extra lines in the diffraction pattern. Usually, these can be min-

imized by using the appropriate filters. Also, the extra lines have aspecific angular relation with the lines from the K radiation whichcan be calculated and then eliminated.

2. Impurities in the unknown material - any presence of crystalline impu-rities in the sample will again cause extra lines. This depends on thespecimen properties and can be eliminated by processing.

2 Indexing pattern for cubic crystals

A cubic crystal gives diffraction lines where the angle () obeys the followingrelation

sin2

h2 +k2 +l2 =

sin2

s2 =

2

4a2 = constant (1)

where a is the lattice constant, is the x-ray wavelength, and (hkl) refers tothe Miller indices of the plane. Equation1is obtained using Braggs law andthe 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. Thevalues that s can take changes for the different cubic systems (sc, bcc, and

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Table 1: 2 values for Al. First 5 diffraction lines

Intensity 2 (in deg)

100 38.3745.4 44.6123.0 64.9022.7 77.976.2 82.14

Table 2: 2 values for Al with corresponding values of sin2s

for the various

cubic systems

sin2s

Intensity 2 (in deg) dA sin2 sc bcc fcc

100 38.37 2.343 0.108 0.108 0.054 0.03645.4 44.61 2.029 0.144 0.072 0.038 0.03623.0 64.90 1.435 0.288 0.03622.7 77.97 1.224 0.396 0.0366.2 82.14 1.172 0.431 0.036

fcc) and this is based on the structure factor rules. The problem is findingthe values ofs for the different 2 values. Thes values for the different cubicsystems 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 ofsusing equation1. This information is summarized graphically in figure1forthe different cubic systems. These can be tried against the different sets forthe 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 ofincreasing 2 are given in table 1. The radiation used is Cu K with wave-length 1.54A. Using equation1 it is possible to calculate the values of sin

2s

taking different values ofsfor simple cubic, fcc, and bcc. These are tabulatedin table 2. From table 2 it is clear that only for the fcc system does sin

2s

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 fromElements of X-ray diffraction - B.D. Cullity.

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structure. The lattice spacing can be calculated using equation 1where thevalue of the constant from table2is 0.036.

sin2

s =

2

4a2 = 0.036 (2)

From equation2the lattice constanta of Al is 4.06A. Given that the densityof Al () is 2.7 gcm3 it is possible to calculate the number of atoms per unitcell, Z. This is given by the relation

= Z. at.wt

a3NA(3)

where NA is Avogadros number. From equation3 the value of Z is 4. Thus,Al has a fcc structure with 4 atoms per unit cell.

3 Indexing for non-cubic systems

Cubic systems are easy to solve, since they have only one lattice constantand the angle are all 90. Things before more difficult if we have non-cubicsystems. Usually, graphical methods are used for solving these systems.Consider a tetragonal system with a = b = c and all 3 angle 90. The

relation between dspacing and the lattice constants for this system is1

d2 =

h2

a2 +

k2

b2 +

l2

c2

1

d2 =

h2 + k2

a2 +

l2

c2

1

d2 =

1

a2[(h2 + k2) +

l2

(c/a)2]

(4)

Taking logarithm on both sides give

2log d = 2log a log[(h2

+ k2

) + l2

(c/a)2 ] (5)

If there are 2 planes with spacing d1 and d2 and Miller indices (h1k1l1) and(h2k2l2) then equation5modifies to

2log d1 2log d2 = log[(h2

1+k2

1) +

l21

(c/a)2] log[(h2

2+k2

2) +

l22

(c/a)2] (6)

Equation6 shows that the logarithm of the difference between d spacing for2 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 secondterm in the right hand side of equation5vs. (c/a) for all possible (hkl). Thiswill give a series of curves. The experimental pattern can be superimposedon these curves and the (c/a) value can be obtained. Such types of curvesare called Hull-Davey curves. A partial Hull-Davey curve for the tetragonalsystem is shown in figure2. A complete one for the body centered tetrag-onal system is shown in figure3. Hull-Davey curves can be constructed fordifferent crystal systems taking into account the relation between the latticeconstants and the lattice angles. In the case of the hexagonal system therelation between the d spacing and the lattice constants is

1d2

= 43

(h2

+ k2

+ hk)a2

+ l2

c2 (7)

After a similar manipulation followed for the tetragonal system, this can berearranged as

2log d = 2log a log[4

3(h2 + k2 + hk) +

l2

(c/a)2] (8)

A Hull-Davey chart can be constructed similar to than for the tetragonalsystem to get the lattice constants.

As the number of independent lattice constants of the of the crystal in-creases (length and angles) it becomes more difficult to use the graphicalmethods. Now, there are computer programs that are used to index patternsby searching and matching with known databases. The powder diffractionpatterns for known materials are stored in the ICDD (International Centerfor Diffraction Data) database. For unknown systems with more than onetype of atom in the unit cell we need the intensities of the lines to find theatom positions. This is done by relating then intensities to the structure fac-tor, 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 experimen-tal pattern and refinement is carried out to the trial structure. This processis 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 determi-nation. 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 exper-imental pattern superimposed. Taken from Elements of X-ray diffraction -

B.D. Cullity. 7


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Figure 3: Complete Hull-Davey curve for the body centered tetragonal sys-tem. Taken from Elements of X-ray diffraction - B.D. Cullity.

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thermal analysis followed by microstructure information. But this does notgive structure information of the phases. For this we need diffraction. Thediffraction pattern for each phase is independent of the other phases. It isalso possible to get quantitative information on the relative amounts of thevarious phases i.e. phase boundaries can also be constructed using XRD. Itis also important how the changing composition of the different phases canaffect the diffraction patterns

1. If there is solid solubility then as the concentration increases the d-spacing changes. This is because the lattice constant changes. Therecan either be an increase or decrease in lattice constant depending on

the relative atomic sizes of the constituent elements. This leads to ashift in the position of the lines.

2. If there is a 2 phase region then as the concentration change therelativeintensity of the different lines changes but there is no change in lineposition.

The following are the principles for collecting x-ray diffraction patterns forphase diagram determination

1. Each alloy must be at equilibrium at the temperature of interest. Forhigh temperature phases not stable at room temperature there are twooptions for studying the crystal structure.

(a) Quench to room temperature and do diffraction.

(b) Use x-ray diffraction with high temperature attachment for di-rect determination. This option is preferred when available for iteliminates the need for preparing large number of samples.

2. The phase sequence: a horizontal line (constant temperature)must passthrough a single phase region and 2 phase region alternatively. Aline cannot pass from one 2 phase region to the next without pass-

ing through a single phase region, can be a line compound.

These principles can be understood by looking at figure 4. If we draw ahorizontal line then the phases go from single phase to a mixture of+ and then a line compound . From we again get a two phase region+ and then finally single phase . Within a single phase region as thecomposition changes then line position changes but in the two phase regionthe relative line intensities change. This information is captured in the seriesof diffraction patterns for the phase diagram shown in figure4and shown infigure5.

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Figure 4: Phase diagram and lattice constant of a hypothetical binary phasediagram. Taken from Elements of X-ray diffraction - B.D. Cullity.

In the single phase region where we have a solid solution, these can be oftwo 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. In-terstitial 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 commonthan defect structures. Depending on the relative sizes of the two atomsthe lattice constants can increase or decrease. In defect structures theincrease in concentration of atom B is accompanied by creating holeswhere A atoms are present. This is prevalent in compounds that havepartial covalent characteristics. They can affect the peak intensities by

affecting the structure factor. An example of a defect structure is inNiAl which has a simple cubic structure with Ni atoms at the cornerand Al at the center. The phase exists over a composition range 45-61%Ni. For off stoichiometry compositions there will be Ni or Al vacanciesin the lattice i.e. defect structures.

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Figure 5: XRD patterns for different alloys from the hypothetical binaryphase diagram in figure4. Taken from Elements of X-ray diffraction - B.D.Cullity

.

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Figure 6: Ordered and disordered configuration in AuCu3. Taken from Ele-ments of X-ray diffraction - B.D. Cullity.

5 Super lattice structures

These are also called order-disorder transformations. In this a substitutionalsolid solution that has atoms located at random positions at high tempera-ture transforms into an ordered structure where the different kinds of atomsare located at specific positions. In x-ray diffraction an order-disorder trans-formation will not affect the positions of the peaks but relative intensities willchange. Sometime new peaks are also formed. Ordered structures are also

called super lattice structures. The new lines seen in the diffraction patternare called super lattice lines. The original lines are called fundamentallines.To understand the formation of super lattice lines in XRD consider the ex-ample ofAuCu3. The disordered and ordered structure for this is shown infigure6. AuCu3 has an fcc structure with 4 atoms per unit cell. From theformula, 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 struc-ture, Au atoms are located at the corners and the Cu atoms are located atthe face center positions. The order-disorder transition temperature for thissystem is 390C.

Consider the completely disordered structure. The probability of a site beingoccupied by Au atom is 1

4while the probability of occupation by Cu is 3

4.

Hence it is possible to define an average atomic factor term, fav that is givenby

fav = 1

4fAu +

3

4fCu (9)

where fAu and fCu are the atomic scattering factors for Au and Cu. Thedisordered 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(ih+k) + exp(ik +l) + exp(il +h)] (10)

The structure factor rules for the disordered structure are also similar to aregular fcc lattice i.e. the structure factor Fhkl vanishes when hkl are mixedand is non-zero when they are all even or all odd. The difference arises whenwe have the ordered structure. Now the Cu and Au atoms are located atspecific positions and hence the structure factor is calculated by using thesespecific positions. This gives the structure factor for the ordered structureas

Fhkl = fAu + fCu [exp(ih+k) + exp(ik +l) + exp(il +h)] (11)

Using equation11we can see that the structure factor does not vanish forcertain (hkl).

F = (fAu + 3fCu)when (hkl) are all evenor odd

F = (fAu fCu)when (hkl) aremixed

Thus, the ordered structure has extra diffraction lines, which are called superlattice lines. This can be seen in the case of powder patterns ofCuAu3 in

figure7where extra lines are visible.Complete order and complete disorder represent the two extremes. In mostcases, it is possible to get a mixture of both. In such cases, it is possible todefine 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 refersto the fraction of A atoms in the material. In the case of complete order rA= 1 and henceS= 1. In complete disorderrA= FAandS= 0. It is possibleto calculate the long range order parameter by comparing the intensity of thesuper 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 intensityand are not visible in the powder pattern. Consider the case of CuZn. Thedisordered structure is a bcc unit cell with Cu and Zn atoms randomly locatedeither at the corner locations or the body center. In the ordered structurethe Cu atoms are located at the corner and the Zn atoms at the center. Thisis shown in figure8. InCuZn the order-disorder transformation takes placeat 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 orderedAuCu3. Taken from Elements of X-ray diffraction - B.D.Cullity.

Figure 8: Ordered and disordered structures inCuZn. Taken from Elementsof X-ray diffraction - B.D. Cullity.

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average atomic scattering factor defined by the average offCu and fZn. Inthe ordered structure the scattering factor is given by

Fhkl = fCu + fZn [exp(ih+k+l)] (13)

Using equation 13 it is possible to calculate the structure factors for thevarious values of (hkl)

F = (fCu + fZn)when (h+k+l) iseven Fundamental line

F = (fCu fZn)when (h+k+l) isodd Super lattice line

Since the super lattice line is given by the difference of the atomic scatteringfactors its intensity is very weak compared to the fundamental line. Sinceintensity is directly proportional to the square of the structure factor

IsuperIfundamental

= (fZn fCu)

2

(fZn + fCu)2 (14)

For = 0 the atomic scattering factors are equal to the atomic numbers sothat fCu = 29 and fZn = 30. Substituting in equation14 this gives the ratioto be 3 104. It is thus possible for the super lattice lines to be too weak tobe 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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