Lecture Series-4: X-ray Powder Diffraction:

Analytical Applications

2 sind nθ λ=o Every crystalline substance has a unique X-ray powder pattern because:

• the peak position depends on unit cell size and

• the peak intensity depends on the type of atoms present and on their arrangement in the crystal (elemental or compound form).

o Therefore, a powder diffraction data consists of a record of x-ray photon intensity versus detector angle 2θ and is commonly called diffractogram.

o X-ray diffractogram gives us peak intensity, position and line profile from which a large number of information can be obtained about the sample.

o Based on the peak positions and intensities of powder lines qualitative and quantitative phase analysis.

o There are a number of applications of X-ray powder diffraction which just depend on line position, such as the determination of unit cell parameters and the study of order/disorder in solids.

o Also, there are a variety of applications based on line profile, for example the study of preferred orientation and the determination of particle size.

Qualitative Phase Analysis:

The goal of a qualitative analysis is the identification of elements or compound present in the sample and for this analysis we proceed as follows:

Search/Match Procedures The manual procedure for identifying an unknown single substance is pretty straightforward and involves the following steps:

Measure the powder pattern of the unknown from low angles to high, and derive values of interplanar spacings and relative intensities.

Select the three most intense lines. 1

Scan the search manual to locate the group, which includes the d value of the strongest line.

Allow for the possibility of experimental error, say + 0.02 Å.

If a match is found for the three strongest reflections, then compare the d values of the next five most intense lines.

If these match, then note the PDF number and remove the appropriate Data Card from the drawer file.

Compare the entire d and I/I1 data with those of the unknown. • 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

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2 sind nθ λ=

328.0000 25.7200 380.0000 25.6800 456.0000 25.6400 732.0000 25.6000 1216.0000 25.5600 1720.0000 25.5200 2104.0000 25.4800 1892.0000 25.4400 1488.0000 25.4000 1088.0000 25.3600 752.0000 25.3200 576.0000 25.2800 460.0000 25.2400 372.0000 25.2000

Intensity [cts]

Position [°2θ]

Raw Data

{202}

{113}

{006}

{110}

{104}

{012}

hkl

1.4 1.9680

100.0 2.0903

1.9 2.1701

36.1 2.3852

85.8 2.5583

49.8 3.4935

Relative Intensity (%)

dhkl (Å)

Reduced dI list

2 2 2

adh k l

=+ +

ICDD DATA BASE

International Center Diffraction Data (ICDD) or formerly known as (JCPDS) Joint Committee on Powder Diffraction Standards is the organization that maintains the database of inorganic and organic spectra.

The database is available from the Diffraction equipment manufacturers or from ICDD direct.

MMD search/match and look-up software package will use the PDF II format which more enhanced version of PDF-I.

X-ray spectrum search-match process

d vales of most intense peaks d value range index

Card number for detail

Chemical Formula

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Card file for detail analysis consist of:

o Card number

Chemical formula

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• Name of the compound o Analytical conditions

Sample Characteristics • Sample history

o d values of possible peaks Relative peak intensity Corresponding planes

Applications of Powder Diffractometry • Phase identification • Thin-film analysis • Lattice parameter determination • Purity/quality control of materials • Determination of crystallinity of

polycrystalline materials

• Stress analysis • Orientation of single crystals • Determination of Texture • Particle size determination • Quantitative analysis

(PRACTICAL: Identification of an unknown substance by XRD) EXAMPLE

o Figure 1 shows the diffraction pattern of a single inorganic substance recorded using Cu Kα radiation of wavelength 1.5405 Å.

o Measure the 2θ values as carefully as you can, and then convert them into d values using the Bragg Equation, putting n equal to one.

o Measure peak heights above background in mm, and then scale the values up so that the tallest peak has a value of 100.

o Worksheet for identification of unknown substance

Line No. 2θ (degrees)

d Spacing (Å)

Peak Height (mm) I/I1

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

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Line 2θ d Peak height I/I1 Number (degrees) (Å) (mm) 1 37.3 2.41 33.3 91 2 43.3 2.09 36.5 100 3 63.2 1.47 21.0 58 4 75.6 1.26 5.5 15 5 79.5 1.20 4.5 12 6 95.1 1.04 3.0 8 7 107.3 0.956 2.5 7 8 111.3 0.933 8.0 22 9 12.9.3 0.852 6.5 18 10 146.5 0.804 2.5 7

Fig. 3.1i. Powder diffraction data of unknown single substance

Fig. 3.1j. JCPDS data card for NiO

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Advantages and Limitations of X-Ray Powder Diffraction Analysis

Advantages

(a) Only a small quantity of material is required, particularly if the camera method is used.

(b) The method is non destructive. Therefore the method is ideal for providing forensic evidence and in the study of antiquities and works of art.

(c) The actual chemical compounds are determined. Most methods of analysis only give the elements present, so it is not always possible to state the compounds present.

Example: Portland cement for instance. Its chemical analysis can only be expressed in terms of lime, silica, iron (III) oxide and magnesium oxide. X-ray diffraction analysis shows it to consist of Ca3SiO5, a-Ca2SiO4, Ca (AlO3)2, 4 CaO.Al2O3. Fe2O3 and MgO.

(c) If a material can exist in several polymorphic forms, then the analysis automatically determines the crystal type, eg SiO2 may be quartz, cristobalite or tridymite.

Most methods of analysis would not distinguish them.

(d) Almost anyone can be trained to record diffraction patterns, measure them and use the JCPDS Powder Diffraction File.

Analyses can be carried out without any knowledge of X-ray diffraction.

(f) Inspection of a powder pattern by an experienced analyst can throw light on other aspects of the sample e.g., particle size, existence of preferred orientation, and presence of amorphous material, etc.

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Limitations Like any method of analysis, X-ray diffraction analysis has a number of limitations.

For example it is not possible to identify the following using X-ray diffraction.

(a) Non crystalline (amorphous) substances, since these do not give normal diffraction patterns. (b) Components in a mixture occurring below about 5% by weight since there would be insufficient of the materials present to give measurable diffraction lines.

Difficulty may be experience in identifying the following materials.

(a) Multi-component mixtures using manual search/match proce-dures due to overlapping reflections, and phases occurring in low concentrations since only a few of their lines will show up.

(b) Patterns containing strong very low angle reflections which have not been recorded.

Since very low angles are not scanned to avoid the risk of exposing the detector to the direct beam.

(c) Materials exhibiting preferred orientation

(e) Materials exhibiting order/disorder Summary Every crystalline substance has a unique X-ray powder diffraction pattern from which a characteristic set of d and I/I1 values can be derived.

Materials are identified from these values in conjunction with the JCPDS Powder Diffraction File.

This contains sets of cards containing X-ray data for most known crystalline phases.

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The data include d and I/I1 values, Miller Indices, unit cell dimensions, etc. Similar data are also stored on magnetic tape and disc.

Manual search/match procedures for the identification of single substances involves matching the d and I/I1 values of eight strongest lines of the pattern by systematically scanning a Search Manual.

The appropriate Data Card is retrieved and if the experimental data match the standard data then analysis has been successful.

The relevant Data Cards are found by using an Alphabetical Index of the JCPDS Powder Diffraction File.

X -ray diffraction analysis is non destructive and only requires a small amount of sample.

The method determines the actual compound or compounds present, and can distinguish different crystalline forms of the same compound.

Quantitative Phase Analysis A mixture of two or more substances gives a diffraction pattern made up of the superimposed patterns of the individual components.

αγ

The intensities of the individual patterns are proportional to the con-centrations of the phases present. Therefore, by measuring the intensities of patterns, some idea of the relative amounts of each phase can be determined. Figure: X-ray Diffractometer pattern of steel,

containing ferrite (α) and austenite (γ) phases. Chromium radiation Three methods are

commonly used in quantitative analysis:

One requires an internal standard,

one requires an external standard, and

one does not require a standard at all.

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Analysis without the Use of a Standard

Suppose we have a mixture of two phases called A and B (or γ and α ).

The weight percentages of A and B are wA and wB respectively.

Suppose that the intensities of two selected diffraction lines are IA and IB.

Since intensity is proportional to concentration, then by proportion we can write,

w A I A wB IB. (3.1)

∝

We can remove the proportionality sign by introducing a constant. Therefore Eq. 3.1 becomes,

w A K .I A (3.2) wB IB.

=

Since we are dealing with a two-component mixture,

wA + wB = 100 (3.3) Combining Eqs. 3.2 and 3.3 so that wA is eliminated, we get an expression for the weight percent of B. (3.4) Therefore, simply by measuring the two intensities and knowing the value of K, we can readily calculate wB and thence wA, which is equal to 100 - wB.

To determine K we simply make up a mixture of A and B of known weight percentages, determine the intensities of the two selected lines, and then solve Eq. 3.4 for K.

If only a rough and ready analysis is required, then we need only measure peak heights.

For slightly more reliable results we can consider the peaks as approximating to triangles and determine their area (½ x base x height).

PRACTICAL: Analysis of a Two Phase Mixture

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Quantitative Analysis Using an Internal Standard For most mixtures of phases, there is not a linear relationship between the intensity of a particular diffraction peak and concentration. This may be overcome by the use of an internal standard. The method is based on the comparison of a diffraction line of the phase to be determined, with a line from a standard substance mixed with the sample in known proportions. Suppose we have a mixture of A and B to which has been added an internal standard S. If the intensities of particular diffraction lines of A and S are 1A and 1s respectively, then for a constant weight percent of S, the weight percent of A in the original sample is given by Eq. 3.6.

(3.6)

where K' is a constant. Equation 3.6 shows that the intensities 1A /1s varies linearly with wA.

Therefore to carry out a quantitative analysis we require to prepare known mixtures of A and B. We take the same amount of each mixture (x grams) and to each we add the same amount of internal standard (y grams). We also add y grams of standard to x grams of unknown mixture. We then measure the intensities of particular diffraction lines of A and S in the unknown mixture and prepare a calibration graph of 1A/1s versus wA. This should yield a straight line passing through the origin. The ratio 1A/1s is determined for the unknown sample and the calibration graph used to determine the amount of A in the unknown sample.

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The internal standard method has been widely used in industry.

Examples include the following.

(a) The

measurement of respirable quartz in airborne mine dust.

Fig. 4.3h. Extrapolation of a standard additions graph

(b) The analysis of

cement.

(c) The analysis of clay minerals.

(d) The determination of asbestos in airborne dust.

Since the internal standard method is applicable to a wide variety of mixtures, the JCPDS has generalised the method by choosing α-corundum (A12O3) as the reference material.

This was chosen on account of its chemical stability, purity, availability in very small particle sizes, and freedom from preferred orientation.

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Quantitative Analysis Using an External Standard o In this method of analysis, instead of adding a standard to a mixture, we

use a pure sample of the material we wish to determine in the mixture, as the standard.

o Suppose we have a binary mixture of A and B.

o Let the weight fractions of A and B be wA and wB, and their mass

absorption coefficients be (μm)A and (μm)B respectively. o If IA is the intensity of a particular line in the mixture, and IPA is the

intensity of the same line using a pure sample of A, then providing the intensities were recorded under identical conditions, the ratio of intensities is given by Eq. 3.7.

I A = w A(μm) A (3.7) IPA wA [(μm)A - (μm)A] + (μm)B

o If mass absorption coefficients are not known, then a calibration curve of

(IA/IPA) versus weight percent of A can be prepared from mixtures of known composition.

o This may then be used to determine the composition of an unknown sample

of A and B. o It is essential that the intensities are recorded under identical conditions.

o The calibration graphs are usually curved since the intensity of a diffraction

line from one phase depends on the absorption coefficient of the other phase.

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Usefulness and Limitations of Quantitative X-ray Diffraction Analysis

o Quantitative phase analysis by X-ray diffraction is accurate to about +1

%. o This is not particularly spectacular, but there are no other instrumental

methods which can perform such an analysis so simply. o The X -ray method is however, restricted to the analysis of crystalline

solids, and it is not usually possible to detect less than about 5% of a constituent in a mixture.

o There are a number of phenomena which can affect observed intensities,

such as preferred orientation, X-ray absorption, and extinction. o Their effects can be removed or significantly by making making sure

that particle sizes of the samples are small. o Uunless the sample is a polycrystalline aggregate, and must be analysed

in that form, it is usual to grind samples to a fine powder, in a ball mill, prior to the analysis.

Standardless Method

Of the three techniques described, the method without a standard is probably the most simple, and it is applicable to powders and poly-crystalline aggregates. However, the method can only be used when diffracted intensity is independent of the absorption coefficients of the mixtures. The method is usually restricted to the analysis of mixtures of polymorphs, such as anatase and rutile, the common forms of TiO2.

Internal Standard Methos

An obvious disadvantage of the internal standard method is that it is only applicable to powdered specimens.

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Also it can be somewhat time consuming to make up mixtures of known composition. However, a rough and ready analysis can rapidly be achieved if there is an I/Ic entry for the compound of interest in the JCPDS File.

External Standard Method

The external standard method relies on the availability of a pure sample of the material to be determined. The method is thus applicable to powders and polycrystalline aggregates. Providing mass absorption coefficients are available, the result can be calculated from these in conjunction with the intensity data. Alternatively the more time consuming task of preparing mixtures for the production of a calibration graph may be undertaken. This does not require knowledge of mass absorption coefficients.

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Summary For the quantitative analysis of two polymorphs A and B, the intensities of two diffraction lines, IA and IB, are measured. Substitution into Eq. 3.4 allows the calculation of the weight percent of B.

wB = 100 1 + K (IA/IB) (3.4)

K is calculated from Eq. 3.4 using intensity data collected from a mixture of known composition. The internal standard method is applicable to multi-phase samples. The same amount of standard is added to mixtures of known composition and to the unknown. The intensity of a diffraction line of A (IA) and of the standard (IS) are measured, and a calibration graph prepared of (IA/IS) versus weight percent of A. (IA/IS) is determined for the unknown and the calibration graph used to determine the percentage of A. Similar procedures are carried out for the other phases present in the mixture. In the external standard method, a calibration curve is prepared of (IA/IPA) versus weight percent of A, where J A is the intensity of a particular diffraction line of A in mixtures of known composition, and J PA is the intensity of the same line from a sample of pure A. J AI I PA is determined for the unknown and the calibration curve used to determine the percentage of A. Alternatively, if mass absorption coefficients (t.1m) are known, then the weight percent of A (w A) in the unknown mixture can be calculated. For a two component mixture of A and B apply Eq. 3.7.

I A = w A(μm) A IPA wA [(μm)A - (μm)B] + (μm)B

(3.7) The methods are accurate to about + 1 %, but are restricted to crystalline solids whose constituents have concentrations greater than about 5 %. For good accuracy the samples should contain crystallites of small size.

MISCELLANEOUS APPLICATIONS o The qualitative phase analysis is based on the positions and intensities of

powder lines, and quantitative phase analysis depends on the accurate determination of line intensity.

o There are a number of applications of X-ray powder diffraction which just depend on line position, such as the determination of unit cell parameters and the study of order/disorder in solids.

o Also, there are a variety of applications based on line profile, for example the study of preferred orientation and the determination of particle size.

Accurate Unit Cell Parameters and Their Uses

Many applications of x-ray diffraction require precise knowledge of the lattice parameter (or parameters) of the material under study, e.g., solid solutions.

Since the lattice parameter of a solid solution varies with the concentration of the solute, the composition of a given solution can be determined from a measurement of its lattice parameter.

Thermal expansion coefficients can also be determined, without a dilatometer, by measurements of lattice parameter as a function of temperature in a high-temperature camera. Since, in general, a change in solute concentration or temperature produces only a small change in lattice parameter, rather precise parameter measurements must be made in order to measure these quantities with any accuracy.

The process of measuring a lattice parameter is a very indirect one, and is fortunately of such a nature that high precision is fairly easily obtainable.

The parameter a of a cubic substance is directly proportional to the spacing d of any particular set of lattice planes.

If we measure the Bragg angle θ for this set of planes, we can use the Bragg law to determine d and, knowing d, we can calculate a.

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( )2222

22

4lkh

asin ++=

λθ

By measuring Bragg’s angle on a powder diffraction pattern and knowing λ, we can calculate lattice parameter.

Now the determination of unit cell size from a powder pattern is subject to a number of systematic errors.

These arise from: o inaccurate measurement of line position, o the specimen not being exactly in the centre of the instrument used, o X-ray absorption, o film shrinkage, and so on.

It has been shown by Nelson and Riley, that collectively, the errors vary linearly according to the expression given in Eq. 3.11.

Errors α cos2θ + cos2θ

sin2θ 2θ (3.11)

Thus if the lattice parameter, a, of a cubic substance is calculated from each line and plotted against the Nelson Riley expression, a straight line graph should be obtained, as indicated in Fig. 3.3h.

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There are a number of uses of accurate lattice parameters. Examples include the following.

o Accurate unit cell parameters can be used

to calculate accurate values of ionic, atomic or covalent radii. Fig.3.3h. Extrapolation graph for

accurate determination of cubic lattice parameter

o Solid solution analysis.

o Determination of linear thermal expansion coefficients.

Example: Solid solution analysis.

• Suppose cubic materials A and B have unit cell parameters aA and aB respectively.

• If the materials form a solid solution of composition AxBy then its unit cell parameter axy will be intermediate between those of aA and aB ·

• In fact here is usually a linear relationship between lattice parameter and composition. This is known as Vėgard's Law.

• Thus a calibration graph can be constructed as shown in Fig. 3.3k which can be used to determine the composition of AxBy. Alterna-tively, the composition can be calculated using similar triangles.

AB =CD BE DE (3.12)

i.e. a B - a A = a xy - a A 100 %B (3.13) %B = (a xy - a A ) x 100 aB - aA

• The method can be extended to the determination of the composition of interstitial solids, since the presence of the interstitial material expands the lattice without distortion.

Fig. 3.3k. Calibration graph for solid solution analysis

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