QUANTITATIVE ANALYSIS BY X-RAY DIFFRACTION

By K. NORRISH and R. M. TAYLOR

Division of Soils, C.S.I.R.O., Adelaide, South Australia.

[Received 18th April, 1962]

ABSTRACT Certain minerals in soil clays may be directly estimated from diffraction line intensity and the mass absorption coefficient of the sample to the radiation used. The main advantages of the method described are that no internal standards or calibration charts are required, and any dif- fraction line of the component to be estimated may be chosen. The results obtained compare favourably with chemical determinations. Orientation of particles during sample preparation makes the estimation of some minerals very difficult. The uncertainty involved in choosing the background level under a diffraction line is the chief source of error in estimating small amounts of very fine-grained minerals.

INTRODUCTION

Because the intensity of an X-ray diffraction pattern is directly proportional to the concentration of the component producing it, when due allowance is made for absorption effects (Klug and Alex- ander, 1954), it has been possible to develop methods of quantitative analysis based on diffracted intensities. The diffracted intensity of any hkl reflection from any crystalline compound can be related to the composition of the compound and its matrix, and to the instru- mental geometry (Klug and Alexander, 1954; yon Engelhardt, 1961). However, for any particular reflection, many of the parameters can be reduced to a constant.

The following derivation of the relationship between diffracted intensity and absorption is reproduced in part from Klug and Alex- ander (1954). Therein, it is demonstrated that if the incident and diffracted X-ray beam enter and emerge symmetrically on the same side of a 'thick' flat powder specimen,

I~ =KV (l)

where Ix is the measured intensity of a diffraction line of a crystMline component of the sample, V is the volume fraction of the component, /~ is the linear absorpt on coefficient of the specimen, and K is a con- stant for any particular line of a particular mineral. This constant depends on the incident X-ray intensity, the diffracting power of the spacing being measured, the geometry of the instrument, etc. The sample has to be sufficiently 'thick' to completely attenuate the in- cident beam.

98

QUANTITATIVE DIFFRACTION ANALYSIS 99

If O is the apparent density of the specimen and p~ the true density of the component being estimated, then

KX /~--pxAx (2) where X=pxV/p-the weight fraction of the component being esti- mated, and A x =tx/p the mass absorption coefficient of the specimen.

If now the same diffraction line is measured on a standard sample for which X is known and equal to S, then

KS /~-- (3)

psAs Since ps=px, combination of equations (2) and (3) gives

x AxI~S A,ls (4)

Generally a pure material can be selected as the standard, in which case S = 1.

These equations were derived assuming that the crystallites of the specimen were small enough not to give micro-absorption effects (Brindley, 1945), an assumption that will be justified for clay separ- ates of < 5 tL equivalent spherical diameter. With larger particles, the incident and diffracted beams may be attenuated to different degrees in the mineral to be estimated and the matrix, and the use of the average measured mass absorption coefficient in equation (4) would not necessarily give the correct estimation. The maximum permissible crystallite size at which micro-absorption effects are still negligible will depend on the difference between the absorptions of the component and the matrix to the radiation being used. Von Engelhardt (1961) has classified the particles of a sample by the pro- duct of their particle size (in cm.) and their linear absorption co- efficient to a particular radiation. When this product is Jess than 0-01, no correction for micro-absorption is necessary.

Most commercial diffractometers have a geometry such that the above equations are satisfied and the measurement of Ix and Is with them presents no problem in principle. Leroux, Lennox and Kay (1953) used equation (4) to determine quartz in various mixtures, using a diffractometer to measure Ix and Is. Ax and As were cal- culated from the measured absorption coefficient of the samples to shorter 'white' radiation. In principle this method cannot be used to obtain absorption coefficients with any high or constant degree of accuracy for two reasons: (a) because the beam is not monochro- matic, the absorption coefficient will change with the sample thick- ness; (b) if the absorption coefficient is measured for any wavelength other than that used in measuring diffracted intensities, the presence of any element with an absorption edge between the two wavelengths would give rise to serious errors.

Leroux (1957) in later work used monochromatic Mo Ka radiation (reflected from the 101 plane of quartz) instead of the white radiation

I00 K . N O R R I S H A N D R. M. T A Y L O R

direct from a Cu target, to measure A ~ and As. Using this technique, he estimated quartz with a standard deviation of 2 per cent. from the nominal values. Short wavelengths, such as Mo Ka, are more satisfactory for absorption measurements because much thicker samples can be used, but these wavelengths do not give sufficient dispersion for diffraction studies of clay minerals.

Despite the simplicity of equation (4) and the fact that its applica- tion does not involve the tedium of making calibration curves with internal standards '(Ballard, Oshry and Schrenk, 1940; Klug, 1953), or with artificial samples, it has been used very little. The various other methods of analysis (internal standards, etc.) are essentially indirect methods of allowing for A~/A~. Leroux, Lennox and Kay (1953) used Ax and As but did not measure these quantities directly, possibly because of difficulties associated with making specimens thin enough for direct measurement.

Since not all the advantages of applying equation (4) as a method of analysis are immediately obvious, they are worth brief mention. For most estimations, a method in which As and As are measured eliminates the need of any sample pretreatment other than that necessary to obtain a suitable diffraction trace. Moreover, because Ax and As do not occur as angular functions, as they do in photo- graphic techniques, any diffraction peak in a routine trace may be selected for the measurement of Ix. This is particularly important in the analysis of rocks and soils where, because of the numbers of minerals which may occur and the complexity of their patterns, there is a high probability that any preselected line may suffer interference from adjacent or coincident lines. The technique therefore has the big advantage that the selection of a suitable line for measurement can be made after the diffractometer trace has been obtained.

In the present study it is shown that satisfactory analyses can be made using Cu or Co radiations for both absorption and diffraction measurements.

EXPERIMENTAL

Methods. In preliminary experiments a Norelco high-angle diffractometer using a Geiger-counter detector in conjunction with a ratemeter and recorder was used to estimate I~ and Is. This equip- ment has since been modified by the addition of a pulse-height analyser and the Geiger counter has been replaced by a scintillation detector. Using the scintillation detector, counting losses are less and non-monochromatic radiation can be partially eliminated, thus increasing the line to background ratio.

A monochromatic beam of high intensity for absorption measure- ments, was obtained by putting single-crystal slabs (e.g., quartz, lithium fluoride, or sodium chloride) or oriented powder specimens (e.g., graphite) in the sample position of the diffractometer, the de- tector being set to the 20 position corresponding to the reflecting plane. The absorption of the sample was measured by placing it ia

QUANTITATIVE DIFFRACTION ANALYSIS 10I

front of the receiving slits of the detector and measuring the re- duction in intensity.

It is very desirable to use a reflecting plane which has very weak second and third order reflections, otherwise it is necessary to run the X-ray tube at a low kilo-voltage to avoid contaminating the mono- chromatic beam with harmonics.

Sample Preparation. In general, the samples analyzed were < 5V- fractions. For the various artificial mixtures the individual com- ponents of the required particle sizes were mixed together by grinding. Specimens for diffractometry were prepared by lightly pressing the powder into rectangular metal holders, 2 cm • 1 cm (Klug and Alexander, 1954). Sometimes minerals with a pronounced fibrous or platy habit became preferentially oriented during this treatment, in which case the diffracted intensity of a peak was no longer linearly related to the mineral concentration. It was found that orientation effects could be reduced a little if, during the preparation, the front surface of the specimen was pressed against a surface of ground glass or coarse paper rather than a polished surface, as this preserved some degree of randomness in the surface packing.

Specimens for the measurement of the mass absorption coefficients were prepared by pressing the powdered sample into a �89 in. (1.27 cm) diameter hole in an ~ in. Perspex holder. Pressures varying between 200 and 2000 kg/cm 2 were necessary to ensure self supporting samples, (Preferred orientation does not affect this determination.) Samples of uniform thickness were obtained with the aid of jigs. The sample holders were weighed before and after loading. Samples weighing less than 0.03 g are difficult to make, and 0.07-0.10 g is generally required.

When a sample cannot be made sufficiently thin, it may be diluted with a material with low absorption, such as boric acid. The measured absorption coefficient must then be corrected for the amount of diluent added. This type of mixture is particularly prone to errors from micro-absorption effects, so great care must be taken to ensure complete mixing of sufficiently fine particles. The measure- ment of the absorption coefficient is generally made on duplicate samples.

Measurement of Mass Absorption Coefficient. If a sample of thickness I is inserted in a monochromatic X-ray beam of intensity Io, the attenuated beam, intensity I, is related to the incident beam intensity by the relationship

I-- Ioexp( -tzl) so that measurement of I and Io determine txl. The weight of the sample and its area (1.27 cm z) gives its mass/cm 2, pl, and the mass absorption coefficient, A=tz/p--izl/pl.

Ratios of I/Io as low as 3 • 10 -5 (/xl= 10)* could be measured to *These measurements were made using fluorescent radiation. It is doubtful

if radiat ion taken direct f rom an X-ray generator could give such results as it has associated with it much more 'white ' radiation.

102 K. NORRISH AND R. M. TAYLOR

give accurate measurements of A. Table 1 shows some mass ab- sorption measurements compared with those calculated from standard tables.

TABLE 1--Comparison of measured and calculated mass absorption coefficients.

Sample

Aluminium Quartz, SiOz* He natite, FezO3* Goethitet Kaolins Dolomite, CaMg(CO3)2* Calcite, CaCO3*

Cu Ka

Meas- Calc~ ured latec

48.7 48.9 36.1 35-0

- - 230 ! 87 200 32.0 31.6

Co Kct

�9 Meas- ured

75"0 55"1 44 "7 43 '4 48 "0 69

108

Calcu- lated

73.4 54.8 47.7 43.7 47.0 71

106

Fe Ka

Meas- Calcu- ured lated

94-5 93.8 69 '2 67.6 56.4 56.3 53.0 51.4

*Chemical composition assumed to correspond to formula. tAnalysis: Fe203 86"7, H20, 13.2 Si02 0.15. SAssumed: Si02 45, A1203 40.5, H20 14"5.

The deviations between calculated and experimental absorption coefficients are generally less than the uncertainty in published mass absorption coefficients of the elements. The data available on experimentally determined absorption coefficients (Compton and Allison, 1935; Hodgman, Weast and Selby, 1956) are very incom- plete so that calculated coefficients (Henry, Lipson and Wooster, 1953), had to be used and for low atomic number elements the latter often disagree seriously with the former. Mass absorption co- efficients could generally be measured with a reproducibility of about 1 per cent. and counting errors were reduced to 1 per cent. The main source of error in the determination is the deviation of the sample from a uniform parallel sided slab (error in l). Sample preparation and the measurement of mass absorption requires only a few minutes.

Measurement of Line Intensities. The most accurate method of measuring diffraction intensities is to take a large number of counts at the required angle. Then, if sufficient counts are taken, the ac- curacy is limited by output stability of the X-ray set--and this can be reduced to a small fraction of 1 per cent. If no other lines lie near that being measured and if the background in this region is flat, there is no problem in obtaining an accurate measure of the peak height of the line above background.

For complex natural mixtures, however, the integrated area (or a modification of this) was chosen as the most convenient measure o f / , for several reasons. In the analyses of soils and rocks, the back- ground is rarely flat so that averaging the background on either side o f the line does not necessarily give an accurate estimate of the line

QUANTITATIVE DIFFRACTION ANALYSIS 103

background, particularly if other lines lie near that being measured. In these instances it is desirable to see the intensity distribution over an angular region on either side of the line, because the background estimation requires some judgement. If diffraction lines suffer broadening because of small particle size or other causes, the measure- ment of peak height will not give a good estimate of line intensity (line broadening is common in soil minerals). If the crystallite size of specimens is not sufficiently small (< 5t~), relatively large errors can occur in the measurement of the peak height of lines due to the small number of particles contributing to diffraction at a particular angle (Klug and Alexander, 1954). Many more crystallites are considered when the line intensity is integrated over a small angular range and the errors will therefore be reduced.

The integrated intensity can itself be measured by two methods, either by measuring the area enclosed by the peak and the estimated background, or by taking the product of the maximum peak height and the peak width at half maximum height.

There are, however, further problems associated with the accurate measurement of peak areas. A diffraction line has the profile of a tailed triangle (Fig. 1) for which the level of the base (background) is

z

40 39 38 37

2 e DEGREES

FIG. 1--Profile of a typical X-ray diffraction line, the 130 line of goethite.

the most uncertain feature, whilst the area changes markedly with small changes of background because of line width in this region. To overcome the large errors which might thus occur, and to take cognisance of line width, the product of the peak height and the width at half peak height was chosen as the most satisfactory compromise

104 K. NORRISH AND R. M. TAYLOR

:for the measurement of L Table 2 shows the errors introduced into the various measures of line intensity by purposeful errors in back- ground.

TABLE 2--Variations in line intensity due to background uncertainty (assumed • 5% of peak height).

Mineral and line width

Goethite 130 line Width at ~- peak height 0-3 ~ Quartz 101 line Width at �89 peak height 0.175 ~

Method of Measuring line intensity

Peak height

+5% --5% +5% -5%

Area

+22 .4~ -15.8%

--16-1%

Peak height • width at �89 peak height

+12"2~ - - 8 " 8 % + 3"7% --13 '2%

In obta in ing diff ractometer traces for measurement the scanning speed o f the de tec tor and the t ime cons tant o f the ra temeter were ad jus ted so that l ine profiles were no t al tered appreciably .

RESULTS AND DISCUSSION Fig. 2 shows exper imenta l values o f Ix~Is for Co K a and Cu K~

radia t ions , for artificial mixtures o f goethi te and kaol ini te , and it is

1.0

O 8

~C

0 6

�9 ~ 0 4 .=.

0 2

~ / //�84

" , / >~

0.2 0 4 0 6 0.8 0

t x / i s

Fzo. 2---Variatiort of Ix~Is with goethite content for Co and Cu radiations. Continuous lines show the predicted relationship; • experimental points.

QUANTITATIVE DIFFRACTION ANALYSIS 105

seen that these values lie close to the curves predicted by equation (4). The two curves are quite different because the mass absorption coefficients of kaolinite and goethite for Cu Ka differ widely (32 and 187 respectively) whereas for Co Ka radiation they are almost the same. Hence there is an almost linear relation between I~/I~ and goethite concentration for Co Kct.*

In obtaining the theoretical curves for Fig. 2, A x was calculated from the relationship,

Ax=xAg + (1--x)A.,

Ag and AK being the mass absorption coefficients of goethite and kaolinite, and x the proportion of goethite present.

TABLE 3--Calculated and nominal compositions for various synthetic mixtures.

Mixtures in Kaolin

Sample 1

Sample 2 Sample 3

Sample 4

Sample 5

Sample 6

Nominal composition

(%)

50 Goethite

5 Haematite 50 Quartz

20 Quartz

50 Goethite

25 Haematite 25 Goethite

37.5 Haematite 25 Quartz

Calculated comp. (~)

Using Using areas products

47 49.2 50.5

4'25 <{% s o.,

249.4 $20-2 $21.5 %22 ~20.4

48.2 .5 23.2

f24.6" ( ~ ' 5 \22"2

37 "5 21 20"7

Line

130 111 102 101 112 10l 112 130 111 102 130 lll 102 101

* Calculated from the intensity of the compound peak at 2"69 ~ by subtracting the intensity duo to the 102 line of the hematite component

Table 3 shows the results obtained by the application of equation (4) to some artificial mixtures, the quantities As, Is, Ax and Ix being determined experimentally. The agreement between the actual and determined amounts is satisfactory, in that the errors are of the same order as errors in A and L Analyses made using different lines o f a diffraction pattern gave good agreement, even when the lines have a large angular separation (Table 3).

Table 4a presents the analyses of some soils and clays for goethite and haematite, these particular results being given for comparison

*This is a very good reason, apart from the high background associated with Ctt radiation, for using Co radiation on soil samples high in iron.

106 K. NORRISH AND R. M. TAYLOR

with independent chemical estimates of the total Fe203. Table 4b compares the percentages of gorceixite and crandalli te measured by diffraction methods with those calculated from the P content. These calculat ions were made assuming that the minerals had an invariant composi t ion. The two methods of est imation agree well bu t several factors prevent an absolute comparison. The chemical variabili ty o f gorceixite prevents accurate estimations from the phosphorus

TABLE 4a--X-ray diffraction estimations of iron oxide minerals in soils.

Sample

Cookls. whole soil ... West. Aust.

< 2/* Lateritic . . . . . . West. Aust.

< 2 1. Lateritic . . . . . . West. Aust.

< 2 1. Lateritic . . . . . .

Victoria < 5 t * . . . . . . . . .

Tasmania < 0.51. . . . . . . . . .

Barbados W. I. < 2 1. Terra Rossa ...

Diffraction estimation

(%)

Goethite 42

Goethite 30

Goethite 39

Haematite 1.2 Goethite 2

Haematite 6 Goethite 3

Goethite 3.5

Goethite 6.2

Free FezO3 (%)

49 "5*

29"6t

37 "8"["

4'6

9.5

4

7.5

Fe203 accounted for

(%)

76

91

93

65

92

79

74

*A small amount of ilmenite or haematite was identified from diffraction photographs. i 'A very small amount of the iron of these clays is probably present as ilmenite.

TABLE 4b---Estimation of gorceixite in three fractions of a soil.

Sample

Whole Soil . . . . . . 0.5--5 ~ .. . . . . . . . 0.5--5 1. after HF treatment

Gorceixite (%)

�9 ' Calcuiated By diffraction from total

P content

3.2 2.6 12.5 11.5 70 56

Ratio of estimations

1-23 1.10 1.25

content , whilst this same variability means that the s tandard mineral is no t identical with that of the soil, so that absolute estimations cannot be made accurately by diffraction methods. Most complex minera ls - -par t icu lar ly those forming mul t i -component solid solutions - - s h o w such chemical variability and are therefore difficult to analyze absolutely. Most clay minerals fall into this category. The failure to account for much of the measured free Fe203 in some samples, (Table 4a), can also be at t r ibuted to a similar cause. Norrish and

QUANTITATIVE DIFFRACTION ANALYSIS 107

Taylor (1961) have shown that aluminium isomorphously replaces iron in soil goethites. Replacements up to 25 mole per cent. have been measured and there are corresponding changes in the diffracted intensities. The diffraction line used to measure Ix for goethite in these soils was the 111 reflection whose intensity varies linearly with the degree of substitution. When correction is made for this factor the diffraction estimates agree better with the chemical determination.

Preferred orientation of particles during the preparation of the flat sFecimen used with focusing diffractometers may also give rise to serious errors. Materials which have a good cleavage or a platy or fibrous habit are very prone to orientation effects. In analyses for

150

o 12o

9o

o

60 J i (,',1 (13o)

(,ooo ,,~,) L a " ' . . a ' " L . ~ - . - J

i

]

(11,)

NON- ORIENTED

4 3 42 3q 3~ <.3

~ O DEGREES 2 0 DEGREES 3 0 3 b

3 9 3 8

FIG. 3 - -The effect o f orientat ion on the relative intensities o f geothite 130 and 111 diffraction peaks.

goethite orientation effects were observed, the 111 line decreasing and the 130 line increasing in intensity with increased orientation. Fig. 3a demonstrates the increase in intensity of the 130 and an associated decrease in the 111 peaks after pressing the sample on a fiat surface to a pressure of 4000 p.s.i. Fig. 3b shows the height of two goethite peaks when the sample was prepared under a slight pressure on a filter paper surface, which tends to eliminate orienta- tion. Satisfactory results were obtained only from those specimens for which the intensity ratio of the 111 and 130 lines was the same as that of the standard. Here, the ratio used was 1.43 and probably corresponds to random or almost random packing. This value agrees

108 K. NORRISH AND R. M. TAYLOR

well with the ratio of 1.38 calculated from the structure of goethite. With minerals of more pronounced habit it has not been possible to make specimens in which the particles are even approximately randomly orientated. The degree of orientation of a mineral in samples prepared by identical methods will vary with its particle shape and size. Table 5 gives results for various kaolin minerals,

TABLE 5--Orientation in kaolins as measured by the intensity ratios of the 002 and 060 lines.

Sample Origin Ratio

2.99 Kaolinite

Kaolinite < 2 t* Kaolinite < 2 t~ Kaolinite

Kaolinite Dickite Hallyosite*

Halloysite* Halloysite*

Mt. Crawford Sth. Aust.

West Aust. West. Aust. McNamee, Sth.

Carolina Germapy N.S. Wales Black Spur,

N.S. Wales Tasmania Bedford,

Indiana

Area of Area of 002 line 060 line

55 18'5

174 10.6 103 "5 1 l 56.1 24 "9

202 6 '5 138 15.2 69 "9 28.3

78 32.5 62.1 24-7

16"4 9"4 2"25

31 .I 9'1 2 "46

2-4 2.56

*All haUoysites were measured after being oven dried for 4 hours at 105~

orientation being measured by the intensity ratio of the 002 to the 060 line. The halloysites, which are tubular in habit, give the lowest values for this ratio and are probably nearly randomly oriented.

Even within the one specimen it is rash to assume that various clay minerals will have the same degree of orientation, an assumption which is implicit in many recent clay and soil analyses made using diffractometers. A powder diffraction photograph of an oriented flake of a soil clay clearly shows how the various clay components may be very differently oriented.

Though there does not yet appear to be any reliable method of overcoming orientation problems, the authors are experimenting with various techniques for either overcoming orientation or allowing for it. A satisfactory technique, however, will probably require more elaborate methods of sample preparation and subsequent analysis than those described here.

CONCLUSIONS

From the results presented it is clear that this method gives very good results on artificial samples or when estimating large amounts of a mineral under ideal conditions. The accuracy will normally be less when analyzing an unknown sample, the limit being set by nature

QUANTITATIVE DIFFRACTION ANALYSIS 109

of the mineral, its orientation, micro-absorption variability, con- centration, and the measurement of the diffracted intensity above the background. However, these problems are present in any method of quantitative diffraction analysis.

Even considering these limitations this method gives results as good as those obtained by other methods and avoids the necessity of calibration curves (which imply a knowledge of the sample) or additions of an internal standard which itself may introduce into an otherwise satisfactory sample micro-absorption or orientation effects.

Acknowledgements.--Dr E. W. Radoslovich of this Division is thanked for his critical reading of the manuscript.

REFERENCES

BALLARD, J. W., OSHRY, H. i., and SCHRENK, H. H., 1940. Rep. Invest. U.S. Bur. of Min., No. 3520.

B~NOLEY, G. W., 1945. Phil. MAR., 36, 347. COMPTON, A. H., and ALLISON, S. K., 1935. X-ray in Theory and Experiment.

Van Nostrand, New York. HE~Y, N. F. M., LXPSON, H., and WOOSTER, W. A., 1953. The Interpretation

of X-ray Diffraction Photographs. Macmillan, London. HODGMAr~, C. D., WEAST, R. C., and SELBY, S. M., 1956. Handbook of Chemistry

and Physics. Chemical Rubber Publishing Co., Ohio. KLUG, H. P., 1953. Analyt. Chem. 25, 704. KLUG, H. P., and ALEXANDER, L. E., 1954. X-ray Diffraction Procedures.

Wiley, New York. LEROLrX, J., LENNOX, D. H., and KAY, K., 1953. Analyt. Chem., 25, 740. LEROtrX, J., 1957. Norelco Reporter, 4, 107. "CON ENG~LHARDT, W., 1961. Science and Industry (Philips Electrical Industries

Ltd.), 8, 2. NORRISH K., and TAYLOR, R. M., 1961. J. Soil Sci., 12, 294.