1. IntroductionThe assessment study of solid matter can be done

in various ways, one of which is X-ray diffraction. X-ray diffraction techniques give information about thestructure of solids, that is, the arrangement of theatoms that compose the solid.

Among various assessment methods the X-raydiffraction method is commonly used because itpermits nondestructive structure analysis, although itis relatively low in sensitivity. The types ofinformation this method can provide are:(1) The kinds of materials that compose a solid

(qualitative analysis).(2) The quantities of materials that compose the

solid (quantitative analysis).(3) The quantities of materials that are

crystallized (crystallinity).(4) The amount of stress present in the solid

(residual stress).(5) The size of crystallites that compose the

solid (crystallite size).(6) Average orientation of crystallites that

compose the solid (texture).As tools for analysis, X-ray diffraction equipment

may be classified by sample forms into those for thepolycrystal method (powder method), those for thesingle crystal method, and those for the amorphousmethod. Likewise, according to the way of detectingX-rays, they may be classified into those for thephotographic method using film and those for thecounter method using a photon counter. X-ray cam-eras play a major role for the photographic methodand so do X-ray diffractometers for the countermethod. The latter devices are being widely utilizedjointly with various attachments.

Fig. 1 shows-X-ray diffraction equipment asclassified by sample forms.

In the past, because of its rather low sensitivity, the X-ray diffraction technique was not actively used, inparticular, for surface analysis and micro area anal-ysis. In recent years, however, in order to upgrade thesensitivity, efforts have been made to improve themeasuring optical systems, detectors, and X-raysources. This has resulted in its increasing use for theassessment study of solids.

In micro area X-ray diffraction techniques, thePSPC (position sensitive proportional counter) is used to simultaneously detect diffracted rays generated atdifferent angular positions in the shortest possibletime. X-ray information in efficiently obtained frommicro areas down to 0.01 mm in diameter. Also, inthin film X-ray diffrac­tion techniques, X-ray inci-dence is made at a mini­mum angle to the samplesurface so as to get as much information as possibleon the surface. Further, in the thin film study, amonochromator is used to improve the peak-to-background ratio [1].

Described below are the aforementioned X-raydiffraction techniques (1 ) through (6).

2. Processing of X-ray Diffraction DataIt is necessary to correctly read the intensity and

position of diffracted rays from X-ray diffraction dataobtained by measurement with a diffractometer.While various methods are available for this purpose,computer processing is now prevalent.2.1 Background Subtraction

The background must be removed when attem-pting to correctly read the intensities of diffracted rays from X-ray diffraction data. The background iscaused by fluorescent X-rays emitted from the sam-ple, scattered X-rays from amorphous substances inthe sample, and so on.

There are different ways of removing the back-ground. Sonneveld et al. [2] used measured values atevery 20 points (e.g. point n) as the first approxi-

34 The Rigaku Journal

The Rigaku Journal

Vol. 6/ No. 2/ 1989

Technical Note

MICRO AREA X-RAY DIFFRACTION TECHNIQUES

H. ARAKI

Rigaku Corporation, Tokyo, Japan

Vol. 6 No. 2 1989 35

mation of the background. But in view of a likelihoodthat some of the selected points fall on diffractionlines, they conceived sample Pi (i= 2, 3, . . ., n - 1 ) atpoint n - 2 by removing both ends in order to achieve abetter approximation. In this case, calculation of (mi-1

+Pi+1)/2 is to be made, and if Pi > mi, then Pi should bereplaced with mi.

A smooth curve can be obtained by repeating theabove procedure several times and sequentiallyconnecting each point. This curve is regarded as thebackground and is to be subtracted from themeasurement data. If the background changesforming a curve as shown in Fig. 2(b) instead ofchanging almost linearly in terms of 2θ, then Pi>mi+Cmay be used to replace the formula Pi > mi.

There is R. P. Goehner's [3] method as anothermethod along with other pertinent ones which havebeen devised according to the memory size and speedof computers to be used.2.2 Detection of Peak Position and Its Measurement

Various methods are available for peak positiondetermination as well. What is frequently used is oneby means of a quadratic differential curve of themeasurement data [4], [5], [6]. (See Fig. 3.) Thequadratic differential method is advantageous in thatit allows peak detection even in the case of raw data as well as in the case of a peak overlapped with anotherpeak at its shoulder, thus making its detection difficult by the linear differential method.

36 The Rigaku Journal

Fig. 4 Example of qualitative analysis result.

Fig. 2 Background determination. (a) When backgroundis roughly a straight line. (b) When background has somecurvature.

Fig. 3 Peak detection and positional determination withlinear and quadratic curves.

3. General X-ray Diffraction Techniques

3.1 Qualitative AnalysisThis analytical procedure is the so-called search-

match of X-ray diffraction data, such that the diffrac-tion pattern of an unknown sample is measured and iscompared with already known standard patterns(JCPDS cards) to obtain an identification.

When the unknown sample consists of a singlematerial, its qualitative analysis is rather simple.When it is a mixture, on the other hand, the analysisrequires high-level skills because of the many diffrac-tion present. A variety of combinations of standardpatterns should be taken into account. To cope withthe situation, an attempt to carry out search-matchwith a computer was initiated in the 1960s, and impro-vements have been made year after year. The present-

day search-match has gone so far as to include asearch of the complete JCPDS files, comprising over48,000 patterns.3.2 Quantitative Analysis

This procedure estimates the quantity of theanalyte material by taking advantage of the fact thatthe peak heights in an X-ray diffraction pattern areproportional to the quantities of materials that com-pose a solid.

In quantitative analysis by X-ray diffraction, themean mass absorption coefficient depends on the dif-ference in the quantity ratio of material, resulting in adifference in the diffracted ray intensity. It is impor-tant, therefore, to correct for absorption due to thematerial. Quantitative analysis techniques are classi-fied according to differences in the absorption correc-tion.

Quantitative analysis may be made in two ways;one is the internal standard method designed to mix aknown quantity of material into an unknown sample,and the other is the external standard method desi-gned for separate measurement without mixing. Al-though the external standard method is preferable forquantitative analysis, errors are liable to occur in thismethod. For this reason, the internal standard methodis more often used.3.3 Crystallinity

In X-ray diffraction data a pattern due to a crystal-line material and a pattern due to an amorphousmaterial may overlap with each other. They differ asshown in Fig. 6.

While various methods are available for thedetermination of crystallinity, they are basically thesame. That is, they employ a way of examining thedegree of crystallinity from a ratio between the

Vol. 6 No. 2 1989 37

Fig. 6 When crystalline material and amorphous material area mixed together.

Fig. 5 Difference in calibration curve due to absorptioncoefficient.

pattern area of the crystalline material and that of theamorphous material. There is such a case, however, as with graphite, for instance, where the position of itsdiffraction line will vary depending on the degree ofcrystalline properties. This is also referred to ascrystallinity.3.4 Residual Stress

When force is applied solid matter within itselastic limit, it will be deformed in proportion to themagnitude of the force. In other words, the crystallattice interplanar spacing (d-value) of the materialwill change. This distortion is uniform and it shouldbe distinguished from nonuniform distortion referredto in Fig. 7.

When changes in the diffraction angle (2θ) areexamined by varying the angle ψ formed by thenormal to the sample plane and that to the latticeplane, the stress value can be obtained by using thefollowing equation.

( )( )

( )σ

νθ π ∂ θ

∂ φ= −

+⋅ ⋅ ⋅E

2 1 180

20 2

cotsin

( )( )

= ⋅K∂ θ

∂ φ

22sin

whereσ: Stress (kg/mm2)E: Young's module (kg/mm2)ν: Poisson's ratioθ0: Standard Bragg angleK: Constant determined by material and measure-ment wavelength (called a stress constant)The optical system of the stress measuring system

should be selected according to the shape of the object for measurement and the stress measuring direction.3.5 Crystallite Size

The peak width in an X-ray diffraction pattern isrelated to the size of crystallites that compose thematerial.

Besides minuteness of the crystallite, nonuniformdistortion of the crystallite (Fig. 7) is another factorthat causes broadening of the peak width. According-ly the size of the average crystallite can be determined by measuring the peak width. Scherrer's equation andHall's equation are often used for calculations of thecrystallite size.(Method by Scherrer)

DK

hkl = ⋅λβ θcos

where

λ: X-ray wavelength for measurement (Å)β: Breadth of diffracted rays due to the crystal-

lite size (rad)θ: Bragg angle of diffracted raysK: Constant (which differs depending on β and

D constants)(Method by Hall)

β β β λε θ

η θ= + = +1 2 2cos

tan

∴ = +β θλ

η θλ ε

cos sin2

1

whereβ1: Integral width of the breadth of diffracted rays

due to the crystallite size (rad)β2: Integral width of nonuniform distortion η, the

relation with the nonuniform distortion is: β1 = 2tan θ

ε: Crystallite size (rad)

38 The Rigaku Journal

Fig. 7 Uniform distortion and nonuniform distortion.

Fig. 8 Stress and lattice interplanar spacing.

The gradient of a straight line obtained by plottingβcosθ/λ and sinθ/λ on the Y-axis and X-axisrespectively, is 2η. The point of intersection with theY-axis is 1/ε. From this, calculation can be made byseparating the ununiform distortion η and the crystal-lite size from each other.3.6 Texture

When the crystal orientation of matter thatcomposes a solid is random, the resultant Debye ringswill be uniform, as shown in Fig. 11 (a), (b), (c). If, onthe other hand, the crystal orientation is in a particulardirection, an arc shape will result instead of a ring, asshown in Fig. 11 (d). Such a state is referred to aspreferred orientation.

Sometimes there are cases in which preferredorientation is specifically given to utilize texture inorder to improve the characteristics of solid materials.The device to measure the state of this texture is a polefigure diffractometer attachment.

4. Assessment of Materials by X-Ray DiffractionTechniques

The aforementioned items 3.1 through 3.6 serve asbasic assessment methods to examine the state andquality of solid matter. By way of example, one of themethods used for analysis of the mechanical behaviors of materials is shown in Fig. 13.

5. Other X-ray Diffraction Techniques for Material Assessment

The above descriptions have been made centeredon bulk analysis. In general X-ray diffractiontechniques there are certain systems in which the X-

Vol. 6 No. 2 1989 39

Fig. 11 Observation of Debye rings in polycrystals: α-Fe (211).

Fig. 10 Particle size and crystallite size.

Fig. 9 Example of residual stress calculation result.

RESULTS OF STRESS ANALYSIS

PEAK POSITION : CENTER OF FWHMSTRESS = -0.36 kg/mm/mm

SIGMA RELIABILITY = +-0.04 kg/mm/mm

ray optics, detector and X-ray source are specificallydeisgned for particular applications, such as:

• Thin film X-ray diffraction method [1]• X-ray surface diffraction method [7]• Cone scanning type microdiffractometer• Curved PSPC type microdiffractometer (PSPC/

MDG)• Small angle scattering measurement method

etc. These methods are aimed at surface analysis,

micro area analysis and the like. In any case theexistence of preferred orientation must be taken intoaccount.

Because the number of crystallites that contributeto diffraction in lessened in the case of micro area X-ray diffraction techniques, this may cause discontin-uous Debye rings. For the purpose of getting ameasurement result with high reproducibility byeliminating this drawback, rotation or oscillation isapplied to the sample during measurement. Threeaxes ω, χ and θ are available as the axis of thesemovements, and they can be run independently orsimultaneously to perform the desired rotation oroscillation.

The small angle scattering measurement methoddeals with diffuse scattering caused in the vicinity of

40 The Rigaku Journal

Fig. 12 Measurement example with a pole figure attachment.

Fig. 13 Relations of material information and X-ray diffraction techniques.

the incident X-ray direction as well as Braggreflections in case of exceedingly large latticeinterplanar spacing. It deals also with a phenomenonthat diffraction due to long periodicity is observedwhen the crystalline properties and the amorphousproperties are arranged periodically in fiber samples.These are utilized for particle size measurement, long-periodicity measurement, and so on.

6. ConclusionIn recent years, higher-intensity X-ray sources

have become available, such as synchrotron radiationX-rays from an electron storage ring and X-rays froma high-power rotating anode X-ray generator.Moreover, the improvement and progress inmeasurement methods now make it possible toconduct structure analysis of micro area objects formeasurement under various conditions by X-raydiffraction techniques.

Vol. 6 No. 2 1989 41

Fig. 14 Measurement example with a thin film attach-ment.

Fig. 15 Measurement example of a rock flake. Fig 17. Sample moving axes in curved PSPC/MDG.

Fig 16. .Measurement example with PSPC/MDG.

References[ 1 ] Kobayashi and Yoshimatsu: Progress of X-Ray Anal-

ysis VII, Kagaku Gijutsu-sha (1975), 49.

[ 2 ] E. J. Sonnevied and J. W. Vieser: L. Appl. Cryst., 8(1975), 1 .

[ 3 ] R. P. Goehner: Anal. Chem., 50 (1978), 1223.

[ 4 ] A. Savitzky and M. J. E. Goley: Anal. Chem., 36 (1964),1927.

[ 5 ] M. A. Mariscotti: Nuclear Instruments and Methods, 50(1967), 309.

[ 6 ] W. N. Schreiner and R. Jenkins: Adv. X-ray Analysis, 23 (1980), 287,

[ 7 ] Kikuta: J. of The Crystallographic Society of Japan, 29(1987), 44.

42 The Rigaku Journal

Fig. 18 Effect of 3-axis oscillation