16 - Lecture notes for Clay Mineralogy

Required reading: Moore and Reynolds, 316-328
Suggested reading:
Brindley and Brown, pages 411-436

XRD Quantification of clay minerals - Internal Standard methods


Recall from the previous notes:

The method of standard additions or the spiking method (adopted from XRF work) is a good/reliable technique if you are interested in a particular component in a mixture (i.e., interested only in the weight fraction of that one phase).

This method relies on the addition of known amounts of the component of interest to the sample.

Any reflection from the other component in the mixture can be selected to be use in the analysis, so long as it provides reasonable intensity (signal to noise) and there is minimal peak overlap from other phases that may be present.

Let J = component of interest and K = any other component in the unknown sample.

Using the above equation and looking at the ratio of intensities (Ii) of J to K we obtain:

Noting that the matrix absorption effect cancels, this equations can be simplified to:

If a known amount (X) of the pure component J is added to the mixture, then the concentration of the component in the mixture becomes:

and the equation becomes:

The intensity ratio can be rewritten:

A plot of the intensity ratio versus the grams of analyte added per gram of sample produces a linear relationship.

Important to remember that not too much of the spike can be added. This changes the µ* of the mixture and the cancellation of the µ* is no longer valid.

Rule of thumb: Do not add more than 20% by weight of the spike. Increments of 5% work well.

Reference Intensity Ratio (RIR) Method.

The RIR method is based on the theory described in a series of articles published by Frank Chung (1, 2, 3). The idea of removing the effects of non-linear and varible matrix absorption on diffracted X-ray intensity is also employed in other methods, namely the "matrix flushing" method of Chung and the "Mineral Intensity Factor" or MIF method of Moore and Reynolds.

The RIR is defined as the intensity of the strongest line of the sample to that of the strongest line for a reference phase in a 1:1 mixture. The reference phase of choice is α-Al2O3 (corundum), but other phases can work just as well, such as ZnO. In the special case of 1:1 mixtures between the sample and corundum, the RIR value is referred to as "I over I-corundum" value (i.e., I/Ic). In this case, the strongest line intensity of the phase of interest is ratioed to the corundum (113) intensity. The I/Ic value assumes CuKα1 radiation.  I/Ic values are published in the ICDD-PDF data base.  Notice that the matrix effect gets "flushed" from the equations. In the equation below and a 1:1 mixture,  WJ/WK = 1 and all the other factors become a new constant, which collectively is the RIR.  If the ICDD-PDF values are used, then you as best should consider the analysis semi-quantitative. In other words your precision may be good, but your accuracy may be in error up to +/- 20%.

Other internal standards can be used (and in fact, may be preferable because of conflict between overlapping lines).  Your best practice is to develop your own  I/Ic values for your own experimental set-up. Just remember that when you buy a jar of reagent or prepare your own internal standard, you need to establish the RIR for that batch. If you renew your internal standard supply, the distribution of coherent scattering domains in that batch may be different from the your previous supply.  So you need to reestablish a new RIR for every batch and for the specifc experimental conditions you are using (i.e., radiation, tube type and current, slit sizes, take-off angles, goniometer radius, etc...).

A 1:1 mixture dilutes the original sample intensity, therefore a lesser amount of internal standard will allow for better quantification of minor phases.  In this case it is best to develop a calibration curve by mixing known amounts of analyte. By adding an internal standard you change the relative weight fraction of the phases of interest. If WJ is the weight without standard added, then W'J is the weight fraction with standard added. In the equations below, the subscript c is for corundum, as it is a common internal standard.

A calibration curve is plotted:

Note that the weight fraction of W'J is the weight after the corundum has been added. Therefore, the weight of J in the original sample is:

If the XRD procedure calls for a standard routine, for example adding 0.2 g of corundum to 0.8 g of sample, then (1-Wc) and Wc become constant.

The ratio then simple becomes:


Full Pattern quantification

Snyder and Bish (Reviews in Mineralogy and Geochemistry; January 1989; v. 20;1; p. 101-144) provide a comprehensive overview of XRD quantitative analysis, which includes a discusion of both single line reflection and full pattern quantification.  The above single line concepts can readily be extended to using all intensities from a full or complete XRD pattern. To accomplish this however, complex computer matrix operations must be performed. Using a wide range of diffraction data minimizes biases from such factors as overlapping lines, preferred orientation, and primary extinction. This can result in more accurate quantification.

The Rietveld method* is a popular minimization technique that employs the kinematic XRD intensity equation I(2θ) = Lp |G2| Φ ) . Recall that symmetry group, unit cell  parameters, atomic coordinates, and scattering domain distributions are included in the intensity equation. The Rietveld method was originally developed to refine the unit cell and atomic coordinates of a single phase (assuming a correct symmetry group). Minimization is the process of using numerical matrix methods to fit the calculated intensities to the observed intensities using the weighted sum of differences between corresponding calculated and observed intensity values.  Although the Rietveld methods was originally designed to refine crystal structures, it has been modified to include the WJ,K,L... terms for each phase present in a mixture (I often refer to it as using a sledge hammer to drive a tack... but it works quite well). In fact, addition of a known amount of internal standard further improves the result.  The challenge for improving the method is accurately matching the initial "guesses" for the input parameters. They need to closely match those of your unknowns. Minimization methods are inherently unstable  if the initial guesses are not close.  By saying "unstable", it is meant that the computer programs may reach false minimums or generate results that crash the program. Additionally, order and disorder as defined by the distribution of coherent scattering domains in the interference function and the strain in the structure function are integrated into the refinement process. Using peak shape functions that emulate the observed peak shapes is a key to successful modeling.