. Fig. 23.34  Relative errors in normalized analyses for Mg and Fe in K411 glass with point beam centered on the image center of mass and with overscanning: a spherical particles, b fractured shards

a

b

K411 Spheres E0 = 20 keV

Point beam

Overscan

K411 Shards E0 = 20 keV

Point beam

Overscan

Mg relative error (%)

23.6.6\ Peak-to-Background (P/B) Method

Specimen Geometry Severely Affects the k-ratio, but Not the P/B

Another approach to establishing quantitative X-ray microanalysis for objects of an irregular shape, such as particles and rough surfaces, is the peak-to-background (P/B) method, which is an extension of the Marshall–Hall method for the

23 correction of mass loss in beam-sensitive materials (see 7  Chap. 20) (Marshall and Hall, 1966; Hall, 1968). The P/B method (Small et al., 1978, 1979a,b; Statham and Pawley,

1978; Statham, 1979; Wendt and Schmidt, 1978; August and

Wernisch, 1991a, b, c) is based on the observation that although the characteristic X-ray intensity emitted from an irregularly shaped object is highly dependent on local geometric effects, the P/B ratio measured between the characteristic X-rays and the continuum X-rays of the same energy is much less sensitive to specimen geometry. .  Table 23.6 contains measurements of the k-ratio (measured relative to bulk K411) and the P/B from the spectra of the SRM-470 (K-411 glass) shards in Fig. 23.13. The shard spectra show significant deviations from the spectrum of the polished bulk K411, especially at low photon energies below 4 keV. Although the k-ratio for Mg measured for these shards varies by a factor of

23.6 · Particle Analysis

P/B peak-to-background

2.95 for the most extreme case, the corresponding P/B ratio for Mg only differs from that of the bulk K411 by a factor of 1.03. For the particular combination of elements in K-411 at this beam energy (20 keV), the most extreme variation in the P/B observed for these shards is 1.13 for Ca.

Although the characteristic and continuum X-rays are produced by different physical processes (inner shell ionization versus deceleration in a Coulombic field) that have different behaviors as a function of the exciting electron energy; especially near the ionization threshold for an element, both characteristic and continuum X-rays are generated in nearly the same volume. Both forms of radiation thus scale similar to the geometric mass effect, because the loss of beam electrons due to backscattering and penetration also robs both characteristic

and continuum generation processes, at least to a first order for photons of the same energy. Both types of radiation have a similar, although not identical, depth distribution; thus, the absorption paths to the detector are alike. As the same photon energy is chosen for characteristic and continuum X-rays, the geometric absorption effect is thus comparable for both. When making corrections for an irregularly shaped object, the exact absorption path is very difficult to determine. Because the continuum radiation of the same photon energy is following the same path to the detector that the characteristic radiation follows, regardless of local object shape, this continuum intensity IB can be used as an automatic internal normalization factor to compensate for the major geometric effects. Furthermore, the P/B ratio is independent of probe current (and thus need not be measured); yet, the quantification results need not be normalized. Because of this, both standards-based and standardless P/B algorithms have been implemented that provide an estimate of the analytical total. More sophisticated models have been developed that account for the second-order (i.e., subtler) differences between the distributions of characteristic and continuum radiation generation (August and Wernisch, 1991a, b, c).

Using the P/B Correspondence

Consider the k-ratio for an object measured relative to a flat,

bulk standard of the same composition, kobject = Iobject/Ibulk. I is the characteristic peak intensity corrected for continuum

background at the same energy, I = P – B. The measured kobject is a strong function of the object’s size and shape, but the ratio

(Iobject/IB,object)/(Ibulk/IB, bulk) involving the background at the same photon energy is nearly independent of object size,

except for very small particles where the anisotropy of the continuum emission becomes significant (Small et al., 1980). This experimental observation, which has been confirmed by theoretical calculations and Monte Carlo simulations, can be employed in several ways (Small et al., 1978; Statham, and Pawley, 1978). One useful approach is to incorporate the following correction scheme into a conventional ZAF method (Small et al., 1978, 1979b). Given that:

a modified particle intensity that compensates for the geo-

metric effects, I*object, can be calculated that is equivalent to the intensity that would be measured from a flat bulk target

of the same composition as the particle:

To apply Eq. 23.5 for the analysis of an irregularly shaped object of unknown composition, the quantities Iobject and

IB,object are determined from the measured X-ray spectrum. Because the composition of the object is unknown, the term

IB,bulk in Eq. 23.5 is not known, as a bulk multi-element