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The Nature of Samples for X-ray Diffraction Analysis

Bulk Samples

Samples used in crystalline silica analyses are usually of two types. One type of sample is the small block or cut section from a large block, and the other is a crushed sample. A bulk sample is defined as one whose thickness is sufficient so that no further increase adds to the intensity of the measured diffracted beam. This thickness is termed the "infinite" thickness, and is related to the linear absorption coefficient by the relation

t= 3.2 1 ρ sinθ, (9)


μM ρ'

where ρ and ρ' are the particle density and the sample density including void space respectively. For a sample dominated by quartz with ρ = 2.65 gcm-3 and μ = 91.3 cm-1, this thickness is around 0.33 mm. This thickness would require about 0.2 g of sample for a typical diffractometer. This amount is two orders of magnitude larger than the typical filter sample.

Bulk samples are employed where the question is the level of crystalline silica in a product. Usually, there is little difficulty obtaining sufficient sample for the analysis, and the sample can reasonably represent the product if the sampling procedure followed good statistical practices. As will be discussed below, the most common problems encountered quantifying bulk samples are reducing the particle size to the <5 m range and preparing the particles in a truly randomly oriented manner.

The preparation of bulk samples is an art that is not easy to master. Hutchinson (1974), Smith and Barrett (1979), Jenkins et al. (1986), Bish and Reynolds (1989) and Miola and Ramani (1991) have reviewed sample preparation techniques in considerable detail. The biggest single problem is preferred orientation of the crystallites in the sample. Calvert et al. (1982) have tested the many methods proposed to eliminate orientation by comparing the results with a sample of MoO3, a very difficult material to prepare in a random mount. Side drifting and spheroidizing are the most successful. Smith and Snyder (1979a and b) have developed the spray-drying technique specifically for X-ray diffraction. If there is sufficient sample, usually several grams, then the spheroidizing approach may be employed.

Bulk samples require absorption corrections or the use of one of the reference methods. All the theory is derived for the infinitely thick sample, and a thinner sample presents additional correction problems. Calibrations based on an intensity response for quartz compositions in standard samples would have to be applied carefully. If the matrix absorption effects are matched, the standard curve can apply. The measurement of the effective absorption value for each sample has been used to correct such calibration curves with some success, but such corrections usually reduce the accuracy because of the additional measurements involved. Because of the size of bulk samples, it is usually convenient to add an internal standard and use appropriate calibration curves or to use the external standard method.

Thin Samples

The second type of sample is the thin sample which is ideal if the layer is closely packed and one grain thick such that no single grain masks any other grain. This type of sample is prepared by drawing a fluid suspension of particles (either air or liquid) through a porous membrane filter. The filter is usually an organic film so it does not contribute significant to the diffraction, but silver is also used where a diffraction effect is desirable.

Techniques for preparing thin samples on filter substrates have been developed since the early 1950's. They have been used extensively in the analysis of clay minerals where samples are prepared to deliberately orient the clay particles, and that literature should be consulted (Bish and Reynolds, 1989). Kupel et al.(1968), Leroux and Powers (1969), and Bumsted (1973) developed techniques specifically for respirable quartz which has been improved by Altree-Williams (1977), Kolk (1985), Kohyama (1985). Pollack (1975), Davis and Johnson (1982a), Lippman (1983) and Davis (1986) have expanded the technique to prepare filters from small to bulk samples not collected in an personal sampler. Carsey (1987) has devised a larger chamber system, LISA, to evaluate sampling techniques and samplers. Bartley and Doemeny (1986) have discussed sampling procedures.

The thin sample does not require an absorption correction because no particle masks any other particle. Thus, the intensity response is linear with respect to the amount present on the filter within the area of the filter irradiated by the incident X-ray beam. Calibration curves may be prepared by distributing known or measurable weights of silica on the filter and measuring the integrated intensities of the peaks. Even the presence of highly absorbing additional particles will not affect the linearity of the response because there are no masking effects.

Where the deposits are made on silver filters, it has been shown that even in thin samples, some of the particles are drawn into the pores of the filters. These particles are partially masked from the X-ray beam by the surrounding silver and cause the intensity response to depart significantly from linearity. If the loading of the sample is too heavy, greater than 2 mg, then particles will deposit on top of each other thus masking the hidden grains. For such a sample, a matrix correction is necessary as indicated in equations 1-3 which can be obtained by measuring the attenuation of the intensity of a peak from the substrate if it is crystalline such as silver. It is preferable in the filter method to avoid the need for the absorption correction if possible because the correction is not very accurate for a 2 to 3 mg layered sample.

Because of the small number of particles in the filter sample, it is difficult to add an internal standard for sample calibration purposes. CaF2 has been used by Orberg (1968) and Bumsted (1973) and others. MgO, NiO and other compounds have also been used. The use of reference intensity methods is also difficult because the values reported in the literature apply to bulk samples not to thin samples, and appropriate values would have to be measured. Obtaining appropriate RIR values for thin samples of quartz would require depositing thin, single-grain layer samples of mixtures of quartz and a standard. As is discussed in the section on statistics, there are insufficient particles to yield much accuracy to the RIR values measured in this manner.

Also, because of the small size of the thin-layer sample, there is always concern as to how representative the particles are of the sample that was tested. In the case of dust analyses, the sample is not just the particles collected, but the volume of air sampled. Thus, the accuracy of the measurement is limited by the number of particles as discussed in the section on crystallite statistics.

Crystallite Statistics

This section is taken from Smith (1992). Regardless of the technique employed, the conditions for accuracy, which are very sample dependent, are the same--total randomness of the crystallite orientations, sufficient crystallites in the experimental sample to meet statistical requirements and sufficient intensity measured to meet counting statistics. Randomness may be described by selecting an equivalent general direction in all crystallites and examining the distribution of this direction vector in space for all the crystallites in the sample. "Particle statistics" requires determining how many of these crystallites will diffract in the experiment and whether they are sufficient to allow intensity measurements to the desired accuracy. These conditions will be examined in more detail.

The conditions of randomness may be described by circumscribing a sphere of unit radius about the sample and plotting the intersection of the poles (direction vector or diffraction vector if the pole is perpendicular to the Bragg plane) on the surface of the sphere. Because the direction selected was general, every direction in the crystallites should behave similarly. Randomness requires the pole density (number per unit area) to be uniform over the surface of the sphere. Randomness may be achieved with any number of poles (crystallites in the sample) as shown in Figure 4. When the number of poles is small, the angle between adjacent poles is large, so that only a few grains in the sample can meet the conditions for diffraction. The average angle between poles is

αp= sin-1(4/√πη) (10)

where η is the number of crystallites in the sample.

Figure 4. Stereogram of random pole distribution: Three distributions of 6 pts, 12 pts and n pts.

The number of crystallites in a sample depends on the volume irradiated by the X-ray beam and the crystallite size. Assume the irradiated area is 1 cm2. The volume of the sample depends on the depth of penetration of the beam. A good estimate of the effective depth is twice the half-depth of penetration

t= 3.2 1 ρ sinθ, (9)


μM ρ'

or SiO2 and CuKα, = 97.6 cm-1 which is approximately 100 cm-1. Thus, the effective volume is essentially 20 mm3. The crystallite size in the sample depends on how the sample was prepared. If the sample was crushed and sieved, the maximum particle size will be determined by the screen size as shown in Table 5. Because the 400 mesh screen is a commonly used size, 40 m crystallites are generally thought to be sufficiently small for accurate quantification. The effect of this size will be compared to 10 m and 1 m sizes in the discussion which follows. Crystallite size will also be considered equal to particle size.
Table 5. Screen mesh and particle sizes


Screen Mesh
Effective Particle Size (μm)
200 74
325 47
400 38
600 25
1000 10

Table 6 compares the particle populations in a sample. There are less than a million crystallites in the 40 m sample compared to over thirty billion in the 1 m sample. The low population of the 40 m crystallites yields an average angle between the poles of 10 minutes of angle compared to 2.5 seconds in the 1 m sample. This difference has a significant effect under the diffraction conditions of a sample.
Table 6. Particle Size Comparisons

Diameter 40 μm 10 μm 1 μm
Volume per crystallite 3.35x10-5 5.24x10-7 5.24x10-10
Crystallites per 20 mm3 5.97x105 3.82x107 3.82x1010

Figure 5a shows a sample irradiated with a beam of radial divergence angle γ. The divergence angle is much wider than any of the crystallites in the sample, so it is not this divergence angle which defines the diffraction conditions of any individual crystallite. As shown in Figure 5b, it is the size of the X-ray source which limits the angular range over which a single grain may diffract. Although different crystallites within the sample (see Figure 3a) may diffract within the divergence angle γ, each crystallite is limited to the range s where
αS= (12)
sin-1 ( F + ds )

+ βs
R

Where F is the apparent width of the X-ray source, ds is the diameter of the crystallite, R is the radius of the diffractometer, and βs is the rocking angle of a crystallite. The rocking angle of a crystallite such as quartz is of the order of 15 seconds (0.0042). Thus, as for a typical fine-focus diffraction source and 40 æm crystallites is around 0.044, about 1/4 of the average angle between the direction vectors of the crystallites. It is quite apparent that a 40 m crystallite size will have too few crystallites to meet the diffraction conditions necessary for statistical significance of the intensity measurements.
Figure 5. A. Diffractometer optics 
for incident rays B. Effective ray paths for a single crystalline

To analyze fully the number of crystallites in diffraction in a particular sample, axial divergence effects also need to be considered. The length of the X-ray source also limits the angular range in the axial direction. This length depends on the sollar slit. A medium resolution sollar slit (5) will expose a length, L, of about 0.5 mm to a given crystallite. Thus, the number of crystallites in diffraction is


η = area on unit sphere corresponding to diffraction range

area on unit sphere per particle
(13)
        = Ad

  Ap
      FL  
Ad =
= 2.5 x 10-4 steradins
    R      

Ratioing Ad to Ap in Table 7 gives the number of crystallites in a given sample. The number for a 40 m sample is a surprising 12; hardly enough for good statistics.

To achieve a specific statistical accuracy in the measured intensity, counting statistics of discrete events is a good guide to the number of counts which must be accumulated. The same description may be applied to the number of crystallites in the sample.


    

(14)
 σ =  √n/n
 
Table 7. Particle Distribution Comparisons

Diameter 40 m 10 m 1 m
Area per pole
(Ap, steradians)
2.11x10-5 3.29x10-7 6.58x10-9
Angle between poles
p, degrees)
0.167 0.0209 0.0007
Crystallites in
diffraction
12
760 38000


The standard error, 2.3σ, should be less than 1%, thus, n should be greater than 52900 crystallites. Using this number as a guide, even the 1 m sample fails to meet the desired condition.

This analysis is for a fixed sample with a single diffraction vector per crystallite. Actually, many aspects of the experiment modify the number of effective crystallites in the sample. One of the factors is the diffraction multiplicity due to the crystal symmetry. Because dhkl always equals d-h-k-l for all crystals, there are always two equivalent directions per crystallite. For crystals other than triclinic, the multiplicity may be considerably higher--up to 48 for some cubic reflections. If a variable divergence slit is used, the irradiated area is more than 2 cm2. For low absorbing materials, the depth of penetration is increased. A broader range of crystallite divergence, αs, may be obtained by using the coarser sollar slits and a broad-focus diffraction source.

Spinning the sample in the sample plane considerably improves the particle statistics. In Figure 5a, crystallites in position 1 may not be positioned to diffract; but as the sample is spun to position 2, the crystallite finds a position where it does diffract. Bragg planes whose tilt is within the divergence angle may find some position in which diffraction will occur. The axial divergence will also affect the range of crystallite tilt which will allow diffraction. Analyzing the effect of spinning is not simply counting all the grains within a specific orientational range as shown in Figure 6. Crystallite B in position 1 may also diffract in position 2 whereas crystallite A will not diffract in either position. A crystallite on the spin axis still requires the Bragg planes to be parallel to the sample surface. Spinning usually does bring more crystallites into the irradiated area because the rectangular shape of the irradiated area covers a circular area with spinning. It is evident from this analysis that sample spinning has less effect than is usually assumed. A device for rocking the sample about the diffractometer axis even a few degrees during spinning would bring many more crystallites into diffraction orientation.

Figure 6. Effect of sample spinning 
on crystalline diffraction

Several factors reduce the number of available crystallites or alter the distribution of the diffraction vector. Preferred orientation seriously affects the distribution to the situation where the number of crystallites is not representative of the amount of the phase in any direction and the quantification fails. A wider range of diffraction tilt will improve the measurements; but if the orientation is severe, even a full rotation about the diffractometer axis may not achieve randomness unless the sample is also spinning. Defocusing of the diffracted beam results from sample tilting. This defocusing may be accommodated by using a wider focal slit.

The amount of phase in the mixture reduces the number of crystallites proportional to the volume fraction. Thus, even for a 1 m sample, phases in concentrations below 10% may have insufficient crystallites for the analytical accuracy desired. Usually, accuracies are quoted in percent absolute rather than percent relative because of the effect of reduced concentration. Determining quartz in a sample at the 0.1% level to an accuracy of 1% is probably impossible. Comparative studies on the potential accuracy usually suggest 2% absolute as the limit of modern instrumentation.

Factors such as microabsorption and surface roughness affect the quality of the data measured but do not affect the particle statistics. The absorption, however, significantly affects the depth of penetration of the X-ray beam and hence the effective volume of the sample. Low-absorbing compounds such as organic materials increase the number of available crystallites. High absorbing materials usually present serious problems by reducing the number of available crystallites orders of magnitude.

Particle Size effects and Amorphous Surface Layers

It has been recognized by many researchers following Nagelschmidt et al. (1952) and Dempster and Ritchie (1952) that there is a strong effect of particle size on the intensity response of quartz. The general interpretation is that quartz particles develop an X-ray amorphous layer on the surface, and as the particles become smaller, the volume of the amorphous fraction becomes a larger fraction of the total particle volume. Only the crystalline volume contributes to the diffracted peaks, so the intensity response versus weight of sample becomes proportionately smaller.

Early studies by Clelland et al. (1952) and Clelland and Ritchie (1952) considered this surface layer highly soluble which would react with various reagents not normally known to affect crystalline quartz. The layer affects the interpretation of silica quantified by chemical methods. Jephcott and Wall (1955), Gordon and Harris (1956), Brindley and Udagawa (1959), Leroux et al. (1973) and Altree-Williams et al. (1981b) showed how the surface layer also affected X-ray intensity measurements. The layer is estimated to be 0.03 m thick, and for particles 2 m or less in diameter, the diffracted intensity is appreciably diminished. This loss in intensity is not to be confused with extinction effects which are stronger in the larger particles. Gordon and Harris showed that the amorphous layer could be removed by acid treatment. Brindley and Udagawa showed that crushed quartz which may develop considerable mechanical damage was susceptible to the formation of the amorphous surface layer suggesting the presence of induced defects as well as ageing initiated vitrification. Edmonds et al. (1977) showed how critical was the effect of matching the particle size distribution of the calibration material and the analyte. There is a significant loss of diffraction response per unit weight of quartz when the particle size gets smaller than 2 m. In a more recent evaluation of the NIOSH Method 7500, Palassis and Jones (19__) have validated the need to modify NIST SRM-1878 by sieving to eliminate the particles > 10 m. All the problems of particle effects are reviewed by Cline and Snyder (1983, 1985).

Several recent studies have been concerned with this amorphous layer. Nakamura et al. (1989) used careful calibration with mixtures of quartz and amorphous silica for calibration and direct analysis and standard addition to quantify the amorphous content of natural quartz samples. The intensity of the amorphous band attributed to the amorphous form was used to quantify the amorphous component. The paper also shows that there is a significant difference in the diffraction pattern of silica glass and silica gel. This result is not surprising because silica gel contains considerable water, and glass is anhydrous. No comparison was made to the opal diffraction patterns. O'Connor and Chang (1986) and Jordan et al. (1990) used the Rietveld method to examine many quartz samples also. Their results were similar. All quartz samples have some amorphous component.