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
(*I _{i}*)
of

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 This changes the µ* of the mixture and the cancellation of the µ* is no longer valid.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 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-Al_{2}O_{3}
(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/I_{c}).
In this case, the strongest line intensity of the phase of
interest is ratioed to the corundum (113) intensity. The I/I_{c}
value assumes CuK_{α1}
radiation. I/I_{c} 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, W_{J}/W_{K}
= 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/I_{c}
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 W_{J} 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-*W*_{c}) and *W*_{c} become
constant.

The ratio then simple becomes:

Example:

Snyder and Bish (

The Rietveld method* is a popular minimization technique that employs the kinematic XRD intensity equation I(2θ) = Lp |G