15 - Lecture notes for Clay Mineralogy

Required reading:

Moore and Reynolds, 298-316
Brindley and Brown, pages 411-436

XRD Quantification of clay minerals

Elemental analysis tells you what elements are in a mixture... Qualitative XRD tells you where the elements are in a mixture... Quantitative XRD and elemental analysis should agree.

The basic premise of quantitative XRD is that the intensity of a diffracted beam is proportional to amount of material present.

Remember! There are intrument, sample, and specimen sensitive factors can affect the intensity of a diffracted X-ray beam.

One of the major drawbacks of directly comparing peak intensities to the amount of material present in the mixture has to do with non-linear responses resulting from differences in mass absorption of each phase in a mixture.

Specifically, when a mixture contains a weak absorber and strong absorber, the lines of the weaker component appears weaker and the strong component appears stronger in a non-linear fashion.

Mass absorption coefficient

Recall from our previous lecture 3 , the mass absorption coefficient of a mixture is equal to the sums of the mass absorption coefficients of the components times their weight fraction:

where:

µ* = mass absorption coefficent of sample
û*= mass absorption coefficent of mixture
I, J, K... = components of a mixture
N = Number of components in mixture
W = weight fraction of a component
i, j, k... lines in the diffaction pattern

Primary and secondary extinction (the need for small particle sizes):

A perfect crystal normally exhibits a small amount of absorption. However, it is possible for additional attenuation of the beam in regions of Bragg reflection. This occurs in several ways.

(1) The beam becomes increasingly weaker as it passes to deeper and deeper planes.

(2) Some interfere destructively as they undergo secondary reflection from the underside of the atomic planes. The diffracted beam is attenuated by back reflected wavelets.

These effects result in a loss in beam intensity and is referred to a primary extinction. This effect varies as a with the angle of incidence, becoming less at higher angles.

Most crystals are composed of a mosaic of tiny imperfect structures. In such a crystal there are a large number of "crystal blocks" (coherent scattering domains) that are at the correct Bragg angle for reflection.

To minimize primary extinction, the particle size should be reduced to 5 - 10 µm.

See Table 5-9 (page 366 ) in Klug and Alexander, which shows the variation in intensity measurements of the quartz peak at 3.34Å using different particle sizes. In essence, the table illustrates that for 10 replicates of quartz intensity measurements, the peak areas for particles ranging from 10 to 50µm and particles ranging from 5 to 15 µm have mean percentage deviations of 18.2% and 2.1%, respectively.

The total intensity of a diffracted beam is that which comes throughout the entire range of Bragg reflection. Slight misorientation of crystallites in a powder results in the additional contribution of diffracted wavelets. The crystallites near the surface, will reduce the intensity of diffracted wavelets from deeper within the crystal. This has an effect of a greater apparent µ* value, than that of a perfect crystal. This effect (known as secondary extinction) is greater for the stronger reflections.

Assumptions for quantitation:

It is assumed that the sample is a uniform mixture of N components and there is no extinction of microabsorption (i.e., the particle size is sufficiently small).

The total intensity of the J^th component of the mixture represented by some plane (hkl) = i is given by:

Where: K_iJ depends upon the nature of component J and instrumental parameters (e.g., geometry).

f_J = the volume fraction of J in the mixture
r = radius of diffractometer
e and m = charge and mass of an electron
M = Multiplicity factor

It can be shown that the total intensity of the J^th component of the mixture represented by some plane (hkl) = i can also be given by:

This equation is the underlying basis for all quantitative analysis by X-ray diffraction.

Putting the intensity equation to use:

Case 1: Mixtures where the µ* of the unknown is the same as the entire mixture. (not common).

In this case, one can simply ratio the intensity of the i^th line in a mixture to same line in the pure component.

Let (I_iJ)_o= intensity of line i of pure component J.

Then:

Example: Mixture of quartz and cristobalite:

Case 2: Mixture of two components with different (i.e., µ*₁ not equal to µ*₂)

For a pure phase:

In a mixture:

The theoretical curve for mixture of the two components is described by the equation:

The lines of the weaker component appears weaker and the strong component appears stronger

Mixtures of quartz (µ* = 34.4 cm²/g) and KCl (µ* = 124 cm²/g) or quartz and BeO (µ* = 7.9 cm²/g)

Think about geologic mixtures with gibbsite and geothite for example....