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Introduction
Quantitative Analysis versus Structure Analysis
In an earlier section, Diffraction II (as well as Assignment 3), a
simplified formula was given, relating the intensity of a powder diffraction
peak to a number of instrumental and crystallographic parameters:
On that occasion the intention was to obtain for each peak a value for
the structure factor Fhkl, the ultimate goal being
the determination of the atomic structure of the diffracting sample.
It was a working assumption that only one diffracting
phase was present. In quantitative analysis there is a marked change of
emphasis, since in general the sample will contain more than one
diffracting phase, but we are no longer interested in crystallographic
structure determination.
The relative concentrations of the phases present are
usually denoted as wi (termed the weight fraction for phase
i ), and the diffracted intensity equation must now be written as:
| Ihkl = (wi / ρi )
c jPLA
Fhkl2 |
where ρi is the density of phase i. The aim is now the
elimination of all terms except wi. This is the essence of
quantitative analysis - to be presented with an unknown powder sample, to
collect its powder diffraction pattern, to identify the phases present, and to
determine the weight fractions of each phase (i.e. wi,
i = 1 to i = N) present. Historically
this has been one of the most important applications of powder diffraction. For
example, in the industrial context it can give information on the efficiency of
a process or the purity of the product; in exploration industry it can give
valuable information on mineral composition; in investigative science it
provide an effective "fingerprint" enabling the source of material to be
identified. However, it is true to say that in its fullness the literature on
the subject has become complicated, even messy. This situation has arisen
because the technique has been developed independently down many routes. The
result is that the there are only a few enlightening reviews covering the whole
field ("X-ray Powder Diffractometry" by Jenkins & Snyder comes closer than
most). It would be beyond the scope of this section to rectify this defect; an
introductory look at some typical problems and solutions is all that is
promised!
It has been a common practice to use only one peak (or perhaps just a few
peaks) of a powder pattern to represent the quantity of any one phase present.
The main problem entailed by this approach is that with a heavy reliance on
just one peak, effects such as preferred orientation can strongly distort the
result. The method of whole pattern fitting is therefore preferable,
though computationally more demanding. These considerations are the subject of
the next three sub-sections.
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