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Basic Theory and its Significance in Silica Determinations

Fundamental Equations

The fundamental equations of quantitative X-ray powder diffraction analysis were first formulated by Alexander and Klug (1948). For a phase J in a mixture, there is an equation for the intensity of each ith diffraction peak with the form

IiJ= KiJ xJ (1)


where xJ is the weight fraction of phase J, ρJ is the density of phase J, KiJ is the intensity constant for peak i, and *M is the mass absorption coefficient for the mixture. KiJ may be determined from the pure sample by
IiJ= KiJ
= KiJ (2)

ρJμ*J μJ

where *J and J are the mass and linear absorption coefficient respectively for phase J, both of which may be found in tables. If the absorption coefficient can be measured for the mixture, equation 1 may be solved for the weight fraction of J.

Where the sample is not infinitely thick, the expression contains an additional term:

IiJ= xJKiJ
(1- e -2μM McosecθiJ) (3)


M is the weight per unit area of the sample, and θiJ is the Bragg angle of the diffraction peak. Expressions 1 - 3 are known as the direct method or absorption correction method. The diffraction experiment is setup to allow the measurement of both the peak intensity and to determine the effective mass absorption coefficient by experiment. Use of the absorption method is discussed by Leroux et al. (1953), Williams (1959) and Davis and Johnson (1987).

The Internal Standard Method

Because of the complications of measuring the absorption coefficient in most cases, especially in very thin samples, other methods have been devised. If one adds a known, fixed amount of a reference phase to every mixture, the ratio of analyte to standard is a linear function of the weight fraction of the analyte through the relation:

xJ= k'JS IiJ (4)


where IkS is the intensity of a specified peak of the added reference material and k' is a constant that may be determined by calibration with known mixtures. This approach is known as the "internal standard method." It has been further formulated by Chung (1974a) as the "adiabatic" method. Although this method has been used for respirable dust analysis, it is not practical for silica analysis using membrane filters because of the difficulty of adding the standard. The internal standard method has the advantage that individual phases of the mixture may be analyzed independently of all the other phases in the mixture. Examples of the use of an internal standard in silica determinations are given by Gordon et al. (1952), Griffen (1955), Talvatie and Brewer (1962) Kupka (1967), Orberg (1968), Bumsted (1973), Altree-Williams et al. (1977) and others.

The External Standard Method

Another method is called the "external standard method" which is also known as the "reference intensity method" by Davis (1986) or the "matrix flushing method" by Chung (1975). It is most simply expressed as

=K IjJ (5)

xJ IjJ

where xI and xJ are the weight fractions of phases I and J respectively and IiI and IjJ are the intensities of selected peaks in the diffraction pattern. The calibration constant K can be determined in a 1:1 mixture of I and J where xI/xJ=1 as
K= IjJ (6)


K values may be determined against any reference material S (commonly α-alumina, corundum) using 1:1 mixtures then
= KJ (7)


If the weight ratio of every phase in the mixture is determined and the mixture is totally crystalline, than the relation
∑ Xk = 1

allows the individual weight fractions to be determined. Sometimes this method is referred to as "quantitative analysis without standards", but the standards are external. Karlak and Burnett (1966) were the first to formulate this approach which was followed up by Chung (1974b and 1975). Many authors have discussed this method and the measurement of the reference intensity values for many materials. Hubbard et al. (1976) evaluate the use of these RIR values for quantitative analysis. Davis (1984, 1986b and 1988), Hubbard and Snyder (1988) and Snyder (1991) further discuss the RIR method of analysis.

In recent years, the term "standardless" quantitative analysis has appeared in the literature for an extension of the external-standard method. When there are many samples with the same phases, all the diffraction data may be collected first, then the full set of equations may be solved simultaneously. Usually, additional physical or chemical constraint equations are included in the set of equations. The sum of the weight fractions of all the phases being equal to unity is one example of a physical constraint. The term was first introduced by Zevin (1977), and the approach has been further developed by Zevin and Zevin (1983), Rius (1987), Wang (1988) and Wang et al. (1991). This approach is not useful for the determination of silica because it does not establish a procedure to process many samples where only one phase is being quantified.

Determination of the appropriate reference-intensity ratios, RIR, is best done by experiment on the same equipment and with the same sample preparation technique as is being used for the analysis, but sometimes appropriate samples are impossible to acquire or synthesize. The alternative approach is to use calculated diffraction data to generate a RIR value based on known crystal structures. The most common program for this purpose is POWD12 (Smith et al., 1982). Discussions on the use of calculated references data are given by Hubbard et al. (1976), Hubbard and Smith (1977), Altree-Williams (1977), Goehner (1982) and Smith et al. (1988). These calculated RIR values are for infinetly thick samples. Appropriate changes must be made when they are to be used for thin samples. This problem is discussed by Davis (1984).

Intensity Scales

The magnitude of the measured intensities depend on several factors: the scattering power, KiJ, of the compound in the sample and the amount present, xJ; the strength of the incident X-ray beam; the count time selected for the measurement; and the efficiency of the detection system. Consequently, the intensities must be placed on some common scale for interpretation. There are three scales which are used in diffraction studies: the relative scale, the relative-absolute scale and the absolute scale. The relative scale normalizes the strongest peak to 100 and lists all the other peaks proportionately lower. This scale is useful for identification. The relative-absolute scale may be achieved by referencing measurements to some standard, the reference intensity method. This method is the most common method used in bulk quantitative analysis, and its basis was first described by Visser and deWolff, (1964). The absolute scale would involve all the factors in the expanded intensity equation to place the intensity on a scale in terms of intensity units. This scale is only used in very specialized diffraction experiments and requires very elaborate instrumental calibrations.

Advantages of QXRPD for crystalline silica

Sensitivity of XRD to Specific Phases

The basic principle of quantification with X-ray diffraction is that the measured intensity is proportional to the amount of the phase in the mixture modified by the effect of absorption as seen in equation 1. The diffraction effect is dependent on the nature of the crystalline structure of the phase resulting in diffraction patterns which are usually distinctive as is the case for the silica polymorphs. It is apparent in Figure 3 that the particular polymorph may be easily identified, which is impossible from chemical information alone and difficult from other spectroscopic measurements.

The principal source for diffraction pattern information that distinguishes the different phases is the Powder Diffraction File using the data in Table 3 which have been reported in the PDF (1991). If the form of silica is not known, the procedure would be to collect a fast scan diffraction pattern and then to match the d's and I's from the pattern with the data for these phases or other phases in the PDF. Although the probability that a different form of silica will be encountered other than quartz or cristobalite is small, the possibility exists, and XRD is the only method to provide positive identification. Also, the other forms of silica may be toxic, so it is necessary to have the ability to confirm the presence of the other forms. If any of the other forms are encountered in any significant quantities, then it will be necessary to establish a quantification procedure for its determination.

Interferences in XRD Both quartz and cristobalite have relatively few diffraction peaks in their patterns as shown in Figure 3, and quantification of these phases is usually little affected by minor amounts of other phases. However, atmospheric samples and most bulk product samples usually do have interfering phases present. Atmospheric samples, especially those from mines, may contain other "predictable" minerals. Some of the minerals do not cause problems, such as calcite and dolomite (Emig and Smith, 1989), but micas and clay minerals which often accompany quartz do interfere. Knight and Zawadski (1989) discuss some interferences in mining environments. If the interference only affects the strongest of the analyte peaks, the alternate peaks may be used with a concommitment decrease in detectability and accuracy due to the weaker intensities employed.

Table 4, modified from Pickard et al. (1985) and HSE/MDHS 5 1/2 (1988), lists some of the interferences from common "impurity" minerals. This list pertains specifically to atmospheric samples but could apply to other situations. Most of the minerals overlap with the strongest quartz peak, (101), requiring an alternate choice of quartz peaks or a rather unsatisfactory correction of the (101) intensity by estimating the amount of the interfering phase and subtracting its contribution to the quantification peak. For phases such as aragonite and kaolinite, there is no option because all the quartz quantification peaks are overlapped. The alternative approach when overlap is severe is the whole-pattern fitting procedures discussed elsewhere in this review.

Table 4. Potential mineral interferences with quartz. Modified after MDHS 5 (1988) and Pickard et al.

Quartz qualification peak (100) (101) (112) (211)
Relative intensity 22 100 14 9
Diffraction angle 20.86 26.66 50.18 59.98

Albite - + + +
Anorthite + + - -
Aragonite + x + +
Barite - x x -
Biotite - x - +
Critobalite - - - +
Graphite - x - +
Kaolinite + - + +
Maghemite + + + -
Microcline + + - -
Mullite - x - +
Muscovite - x - -
Sillimanite - x + +
Tridymite x + + +
Wollastonite - + + +
Wustite - - - x
Zircon - x - +

- no interference
+ minor interference
x major interference

Automation of the Analysis Procedure

There are two primary reasons to automate sample processing, but the dominant one is because of the large number of samples which usually must be processed. The other reason is the intensive but routine calculations which are involved in the reduction of the data. A third reason is to maintain the sanity of the operator. With the increased use of computer-controlled APD's, automation is realistic and relatively easily achieved. Sample automation requires a sample changer on the diffractometer which is usually available for all APD's in use. The control of the sample changer is in the software of the APD. If every sample is processed in a preset manner, the APD software usually is adequate for data collection. However, if the diffractionist wants to interrogate the data, such as testing the main peak to set the count time for subsequent integrations, software patches may be required. Once the intensity data are obtained, the data may be transferred to a data reduction package for producing the quantification values. This software may be part of the APD package, but more often it will have to be supplied by the user. Examples of automation for silica analyses is presented by Bumsted (19__), Abell et al. (1978), Malik and Viswanathan (19__), Snyder et al. (1981, 1982 and 1984), Hubbard et al. (1983) and Wong et al. (1983).

Non-Destructive Nature of X-ray Diffraction

One important and often-quoted advantage of XRD methods is the non-destructive aspect of the treatment of the samples used in the analysis. Other than the need to crush some samples, no other change is induced in the samples by the X-ray experiment. This aspect is particularly useful in silica analyses where alternative procedures may be run in parallel with the X-ray analysis. Also, the sample may be retained for later measurement which is especially valuable in situations involving legal disputes.

One problem, which will be discussed in more detail later, is the controversy whether the collection membrane filter may be used directly in the diffractometer or whether the dust must be reprocessed to achieve the desired accuracy. Because the sample may be saved, the direct method may be used first, and then the more time-consuming method may be employed if there is any reason to doubt the results. Fortunately, membrane filters do not require much space for storage, and the silica phases are stable over time, but the handling must be done carefully so as not to lose any of the dust particles at any step.

Crystallite versus Particle Size

Throughout the discussions of particulate behavior in X-ray diffraction samples and experiments, there is a need to carefully distinguish between particle size and crystallite size. Particle size is the size of the discrete particle which is the entity that is important in interpreting absorption effects. The crystallite size is the size of the effective crystal domain which contributes coherently to the diffraction experiment. This size contributes to peak broadening and is the entity to be considered in the discussions of crystallite orientation and intensity response for a sample. In many cases the particle size and crystallite size are synonymous. Where a quartz crystal is crushed to prepare a reference sample, the fragments are individual crystallites. Crushing of polycrystalline products may lead to single-crystal particles also, but not necessarily. The diffractionist must remember these distinctions. Although particles will be considered individual crystallites in the discussions that follow, the distinction will be kept clear where the distinction could be important.

Problems of QXRPD

Orientation and Crystallite Statistics

X-ray diffraction techniques are not without their problems. There are two very severe sample problems which must always be considered (crystallite orientation and crystallite statistics) and several less severe problems (extinction, detector response, and microabsorption) which must be considered. The orientation problem is whether the crystal domains that make up the sample are randomly oriented in space which is the basic assumption on which the powder method is based. Accurate quantification is impossible if this condition is not met. The statistics question is concerned with whether the effective number of crystallites in diffracting position is sufficiently large to satisfy the statistics of sample representation. The statistics problem has been considered by Alexander et al. (1948), deWolff (1958) and deWolff et al. (1959). Because of the critical nature of these problems, a detailed discussion of the analysis of orientation and crystallite statistics is included in later section. The result of this analysis is to show how critical the crystallite size and the number of crystallites in a sample may be to the potential accuracy of the diffraction experiment.

The crystallite analysis is directly attributable to the bulk sample which has infinite thickness, but the principles also apply to the thin-layer sample. If the sample is assumed to be one crystallite thick, all the crystallites in the sample may contribute to the diffracted intensity. The effective number of crystallites in the proper orientation to diffract in a thin-layer sample may be estimated from the total number of particles as is done for the bulk sample. If the average particle size of the respirable dust sample is 2 m, and if the total weight of the sample is 2 mg, there are 1.8x108 particles in the sample and 3860 in a position to diffract assuming true randomness. Assuming all the particles are quartz, the predicted absolute accuracy is of the order of 4 %. Smaller quantities of quartz will be proportionately less accurately determined. The orientation and crystallite statistics are the most important property of the sample in defining the accuracy of the QXRPD method.


Extinction effects in crystals may be divided into two categories: primary extinction where the crystal is perfect and causes multiple diffraction within the crystallite and secondary extinction where the crystal is "ideally imperfect" and aligned domains cause the multiple diffraction. Both effects diminish the strength of the diffracted beam compared to what it should have been without the effect and affect the stronger intensities proportionately more than the weaker intensities. Cline and Snyder (1987) have discussed the seriousness of this effect. The order of magnitude of the problem may be illustrated by considering the PDF data for quartz. PDF-33-1161 shows that the intensity for the (100) quartz peak was measured as 22 on the scale of (101) normalized to 100. The POWD12 (Smith et al. 1982) calculated diffraction pattern for quartz shows the theoretical intensity for (100) should be 17. The experimental ratio indicates that around 25% of the intensity of the strong peak was lost to extinction. The usual perfection of quartz crystals used for preparing such standards suggests that primary extinction is the cause. The particle sizes for the PDF pattern were not reported, but it is probable that the average was not less than 10 m. Smaller crystallites show less extinction, but even submicron particles will not be free of the problem. If the perfection of the crystallites can be destroyed by some treatment (such as radiation damage), the effect can be minimized. As long as the reference material used for preparing calibration curves has the same perfection as the samples to be analyzed and the particle size distribution is equivalent, the results will be acceptable.

Detector Response

Modern scintillation and solid-state detectors can handle relatively high counting rates in the diffracted beams, but for quartz, which is a very strong diffractor, departure from linear response might occur on the (101) peak when the amount of quartz is high. Usually, the departure is negligible if the counting rates are below 1x104 counts/sec. At higher count rates the diffractionist should consider this potential error. An example of the effect of deadtime in counting is the comparison of the now deleted PDF-5-490 to the active PDF-33-1161. In PDF-5-490 the intensity for (100) is 35. The more recent measurements show (100) with an intensity of 22. The old diffraction pattern was taken with a Geiger-Mueller counter with a long deadtime. The effect was to diminish the intensity of the (101) peak by over 35%. This intensity difference is not trivial.


Large particles with different absorption magnitudes cause particle masking known as microabsorption. This effect has been analyzed by Brindley (1945) who derived correction formulae when the particle sizes were in excess of 1 m. A detailed discussion of the effects of microabsorption on quantitative analysis was presented by Cline and Snyder (1987).

The Integration of Diffraction Peaks by Profile Fitting

When using the individual peak approach for quantifying one or more phases, all is not lost if there is overlap with peaks of other phases. If the raw data is a digitized trace including the peaks of interest and the peaks which interfere, modern profile fitting may be used to decompose the clusters into individual profiles thus determining the individual peak areas.

Profile fitting was first proposed by Rietveld (1969) where it was coupled with a refinement of the lattice parameters and the crystal structures. Many authors have extended Rietveld analysis to be more versatile and to be usable for quantitative analysis as mentioned in another section. Alternatively, there have been several programs developed which decompose peak clusters into individual component profiles without any constraints or with only constraints imposed by fitting peak positions to be compatible with a specific unit cell. Howard and Snyder (1982) have discussed procedures to be followed in profile fitting. Schreiner and Jenkins (1982) discuss profile parameters such as width and shape to be considered when fitting. The basic concept is to use a single resolved peak of a phase to fix the profile shape parameters before fitting other peaks of the same phase. An example of profile fitting for quantitative applications is given by Werner et al. (1979).

Major problems occur when the sample is a multiphase mixture with some phases having broad profiles while others have sharp profiles. In this situation, each peak must be fitted without restraining the parameters which can lead to incorrect results. The Appendix contains a list of programs for profile fitting. Proper use of an appropriate program can yield good integrated intensities for quantitative analysis.