Single-Peak Methods II. Multi-Component Systems

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Multi-Component Systems

Over the years much ingenuity has been expended on devising methods for determining the weight fractions of each component in a mixture, and in particular for eliminating the absorption problem (discussed below). There are distinct advantages in methods which utilise some form of component-addition to the original mixture coupled with measurements of intensity of a diagnostic peak for each component. By measuring the intensity of different peaks from one sample and from one pattern we know that many factors, such as absorption, will be the same for each and therefore can be made to cancel out (on division). The following are typical examples of this approach:

1. Use of Internal Standard: In this method a known amount of additional independent phase, β, is added to an original mixture consisting of components α (α = A, B ... etc.). The internal standard, β, should not interfere chemically with the mixture and its diagnostic (hkl)_β peak should not overlap with those of the mixture; it is also desirable that it should be abundant and cheap; corundum and rutile are popular candidates. The weight fraction of each component, w_α', in the modified mixture is then given by:

   wα' wβ

   = kα

   I(hkl) α I(hkl) β

   which must obviously then be corrected for the addition of the internal standard to finally give the desired original fractions, w_α. The normalisation constants, k_α, for each component are determined from separate diffraction measurements in which binary mixtures of each component are mixed in turn with the internal standard (e.g. for 50:50 binary mixtures the normalising constants will simply be (I(hkl)_β / I(hkl)_α) for the binary mixture. In principle this method can also evaluate the amount, if any, of amorphous content in the mixture from the unaccounted remainder 1− Σw_α.

2. Use of External Standard: This is similar to the internal standard approach given above except that no standard is added; rather each component is compared to corundum (as though it were hypothetically present in infinitesimal quantities) by using the I/I_cor values in the ICDD database. Obviously the method only works if all components are listed in the database, and then these values have to be accepted by the user with no control over their accuracy or reliability. There have been many variations on this theme, one particular variant in 1974 being known as the Chung matrix flushing method after its inventor F.H.Chung, (see footnote ¹ ).
3. Spiking (or Method of Standard Additions): This is similar to the internal standard method except that a series of multiple fractional additions (Y_α) of pure phase α is added to the original mixture, and each time the intensity of the diagnostic peak, I_α, is compared to that, I_β, of another (constant) phase, β, within the mixture which may or may not be of interest. It is easily shown algebraically that a plot of I_α /I_β versus Y_α yields a straight line plot whose negative intercept on the Y_α axis is the required weight fraction of phase α.
4. Calibration methods: This is included as a general category to denote any method (above or otherwise) whereby a series of known additions are used to "calibrate out" undesirable effects such as X-ray absorption. If one has a mixture from which the weight fraction of a component is required then a series of artificial mixtures, of known compositions, is first made up using pure forms of the components and measurements/determinations of the weight fraction are carried out by the required method. A calibration graph is made of the determined result versus true result (since the artificial mixtures are known) and the corresponding result from any unknown mixture is read off by interpolation between points on the calibration graph. An example of this kind of approach is given in the next part (The Absorption Problem). The main disadvantage of this method is that pure forms of the components are required and obviously the method can become very time-consuming. The big advantage is that the method does not rely on any special assumptions, and can provide compensation for effects such as micro-absorption which are otherwise difficult to correct for.

¹ F.H.Chung, J. of Applied Crystallography 7, 526-531 (1974)

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