Introduction

It has been said that a theory should be as simple as possible, but no simpler.  We'll try.  To begin with, we will limit the scope of this discussion to what happens only within the irradiated specimen, and ignore what happens in the x-ray tube, filters, slits, collimators, analyzing crystals and/or detectors, pulse processors, etc.   Here, we will try to describe only those physical processes that seem to be most important in accounting for the major interelement x-ray absorption and indirect fluorescence effects observed in the quantitative x-ray fluorescence analysis of thick, flat, homogeneous specimens.   Further, we will make certain assumptions of a mathematical nature, so that the expressions needed to model the XRF experimental situation are both easier to understand and easier to calculate than might otherwise be the case.   Whenever an assumption is stated here, the reader should keep in mind that these assumptions have been traditional since the time of rather primitive computers.   Therefore, much more detailed mathematical-physical modeling might now be appropriate for certain kinds of specimens, measurement conditions, and standard reference materials.   The constraints on our models are no longer so much a matter of computer speed and storage capacity, but rather the need for more detailed specifications of the experimental conditions and more detailed values of the relevant physical properties of the elements.   The scope of this discussion of modeling also stops short of treating the many different ways that a mathematical-physical model can be used for the purpose of computing specimen composition from measured x-ray intensities. Implementation is a separate topic that can be treated independently of the particular mathematical-physical model chosen.

Direct and Indirect Fluorescence

An x-ray tube (or isotope or other source) irradiates the specimen with x-ray photons having a range of energies (or wavelengths).   Any incident photon whose energy is greater than the binding energy of any electron shell (or subshell) of an atom in the specimen is capable of ionizing that shell of the atom.   The vacancy thus created can be filled by an electron from another, more outer shell of the atom, often accompanied by the emission from the atom of a fluorescence x-ray photon whose energy is the difference in the binding energies of the two shells. That fluorescence photon energy is characteristic of the particular element.   A single ionization can even lead to two or more different characteristic lines from the same atom.   The processes just described will be called "direct" fluorescence by the incident radiation.   "Indirect" fluorescence also can occur.   Sufficiently energetic direct fluorescence photons within the specimen also can ionize atoms, leading to secondary fluorescence.   Secondary fluorescence photons can produce tertiary fluorescence, etc.   In each case, there are physical properties of the elements that are combined to predict the likelihood of fluorescence of a particular characteristic line at each point -- for example, x-ray mass absorption coefficients, subshell absorption coefficients (via "jump factors"), radiationless transition probabilities, subshell fluorescence yields, relative yields of different lines, etc.   Since these considerations are simply a matter of applying appropriate probabilities to what happens within a single atom at a single point, they can be treated as completely as desired without affecting the general mathematical complexity of the model, which depends mostly on the geometry of the experimental arrangement, the shape of the specimen, and the processes affecting x-ray penetration within the specimen, as described next.

X-Ray Absorption Within the Specimen

As a simplest case, imagine that the specimen is extremely thin, a single layer of atoms.   Then, every atom would be fully exposed to the incident radiation, every atom would be visible to the detector (or detection system, such as collimator, crystal, and detector), and thus the number of detected characteristic photons of a particular element would be directly proportional to the number of that element's atoms that were in the specimen.   Quantitative analysis would be simple and direct.

In reality, XRF is an indirect method of analysis.   We do not measure the elements' concentrations, or anything else that is simply proportional to concentrations.   We measure x-ray intensities, which can be very indirect measures, as described next.   Real specimens are not atomic monolayers and so atoms partially shield each other, from both the incident photons and the fluorescence photons.   The extent of that shielding -- by way of x-ray absorption and scattering -- is different for different elements and different photon energies.   Thus, how well radiation penetrates through the specimen depends on every element present, no matter whether it is measured or not, and the measured intensity of a particular characteristic line is not simply proportional to the concentration of the corresponding element.   A measured fluorescence intensity from any single element depends on the concentrations of all elements present.

For most experimental situations and most specimens, by far the dominant physical process affecting x-ray travel within the specimen is x-ray photoelectric absorption.   X-ray scattering usually has been either ignored or lumped into the absorption treatment.   Specifically, our physical model assumes that a photon travels in a straight line, without any loss of photon energy, until it is totally absorbed (photoelectrically).   The mathematical model for this process is the exponential absorption law

, (Eqn 1)

where T(x) is the number of photons traveling a distance x and q is the linear x-ray absorption coefficient of the specimen for photons of a particular energy.   Most texts use instead of the q above, but we will use below to represent the mass absorption coefficient, which is q divided by , the density of the specimen. The way that specimen composition affects x-ray absorption is expressed by

. (Eqn 2)

That is, the mass absorption coefficient of the specimen as a whole (for photons with a particular wavelength ) is just the mass-fraction average of the mass absorption coefficients of the various elements (for that wavelength).   Some authors take each element's to be the total absorption coefficient, which includes scattering components.   However, it is might be better to take to be just the photoelectric absorption coefficient.   The reasoning here is that the total coefficient applies strictly to losses in narrow-beam attenuation experiments, whereas the transport in a bulk specimen is a broad-beam situation, and many of the photons scattered out of each path's original direction are compensated by photons scattered into that direction from other, nearby paths.   In many cases both values (total absorption and photoelectric only) are practically the same.   No matter what value is used for , the form of the equations is unaffected in this model.

Experimental and Specimen Geometries

The preceding took care of the physical model.   The physics was easy to state and easy to express mathematically.   The only remaining problems with the physical aspect per se is to know the spectral distribution of the radiation striking the specimen and to know the values of the required physical properties of the elements -- absorption coefficients, transition probabilities, yields, etc. The rest of the problem of modeling interelement effects within the specimen is mathematical, rather than physical.

In order to apply the preceding two equations to an XRF experiment, it has been common to assume that all the incident radiation makes the same angle with the specimen surface.   We know that the incident beam is really divergent, with a wide range of incidence angles, but this geometrical assumption helps make the equations simpler, and has proven reasonable for a wide variety of applications.   Also, we assume that any particular characteristic fluorescence radiation that we measure is emitted from the specimen at a single angle.   The angle might be different for different lines, as for some multiple-spectrometer systems.   We know that for energy-dispersive detection systems, the detectors generally accept radiation that is emitted over a wide range of angles but, again, the assumption is very valuable mathematically and has worked well in practice, especially if one chooses the best nominal angle for the kind of specimen being analyzed.   Also, as stated at the beginning, we assume that the specimen is homogeneous (on a very fine scale), has a flat surface, and is effectively infinitely thick.   We see problems when any of these specimen assumptions is invalid.   We also ignore edge effects.

To summarize the geometrical model, we assume a homogeneous, flat, semi-infinite specimen, with a parallel incident beam and a parallel measured beam.

Equations for Predicted Intensities

In addition to the geometrical assumptions just stated, it is common to limit how many orders of indirect fluorescence are to be considered.   With the model as given here, if we consider only the direct fluorescence, the resulting equation for predicted fluorescence intensity is very simple. The intensity of the primary fluorescence radiation for a particular characteristic line l from element i, fluoresced directly by the incident radiation, is given by

, (Eqn 3)

wheredescribes the incident spectrum, as the intensity of the radiation per unit wavelength, and and are the angles made by the incident and fluorescence rays with the specimen surface. The range of integration (done numerically in practice) is limited to those wavelengths that are shorter than the absorption edge wavelength for the particular element-i subshell whose ionization leads to the emission of the line l of interest. The coefficient is related to the intensity emitted from a specimen of purely element i, and cancels out when one deals with ratios of fluorescence intensities from specimens of different compositions, the usual practice. The constant includes a number of constants relating to the fluorescence, at a point, of the line l from element i caused by ionization by photons with wavelength -- those constants might include relative subshell absorption coefficients (via "jump factors"), constants that describe possible rearrangements of subshell vacancies, subshell yields, and the relative yield of the particular line l of interest. Of special note for this discussion is that the fluorescence intensity expressed by this equation is not just proportional to, the mass fraction of the emitting element i, but also depends on the mass fractions of all the elements present, by the way that the 's in the denominator depend on composition via Eqn 2, above. The denominator includes the specimen's x-ray absorption effects for both the incident radiation (wavelengths ) and the escaping fluorescence radiation (line l).

If we consider secondary fluorescence, the equation for is rather more complicated, but it is still a combination of elementary functions, and straightforward to calculate. The formula for involves a summation of the effects of all sufficiently energetic characteristic lines fluoresced directly within the specimen, no matter whether the lines are measured or not. The equation is found in many texts, and does not need to be repeated here.  The equations for primary plus secondary fluorescence is what is usually referred to as the "fundamental-parameters" equation.   That name dates from 1968, when the term was used to distinguish the equations from other equations that expressed such relationships in terms of experimentally-determined "empirical coefficients".   Fundamental-parameters relationships actually had been used long before 1968.   Several workers had dealt with equations expressing just the direct fluorescence, often assuming that the incident radiation either was monochromatic, or could be handled as though it had a single effective wavelength.   The more general equations, which included broad-spectrum excitation and secondary fluorescence, were published by Gillam and Heal in 1952.   Nevertheless, those equations usually are called the "Sherman equation", with reference to equations published by Sherman in 1955 and 1959, which included expressions for tertiary fluorescence (i.e., ) in terms of exponential-integral functions.

Summary of this Fundamental-Parameters Model

The total fluorescence intensity for a particular line l is the sum of the intensities of the direct fluorescence, secondary fluorescence, etc., for as many orders of indirect fluorescence that can occur in the specimen, and that we wish to treat:

(Eqn 4)

Often, intensities calculated from such a model are expressed relative to the intensity predicted from a pure-element specimen. That does not imply that pure elements are ever measured as real specimens; it just provides a unit of measure that eliminates the need to know certain parameters that act simply to scale the intensity level, such as current on the x-ray tube, sizes of various slits and collimators, efficiencies of crystals and/or detectors, etc. The predicted relative intensity is

, (Eqn 5)

where the predicted pure-element intensity typically is just the expression for for a pure element:

. (Eqn 6)

We say "typically" because it is also possible to have secondary fluorescence in a pure-element specimen. For example, if some of the incident radiation can fluoresce an element's K lines, and we are interested in using an L line for analysis, there could be some K to L secondary fluorescence throughout the specimen. Also, we would need to use fluorescence yields in a somewhat more detailed way to include the L fluorescence emission from an individual atom initially ionized in its K shell. Such fine points are important in the design of computer programs, but can be neglected in general discussions of the main effects.

Since the constant appears in the equations for , , etc., it cancels out in Equation 5, and thus does not need to be known. Likewise, the equation for contains in both numerator and denominator, so that the incident spectrum does not need to be known on an absolute scale. Actually, it is not even necessary to calculate a predicted intensity for a pure element, unless such a specimen is actually measured; a relative intensity for a particular line can be calculated as a ratio to any specimen measured under the same conditions, such as multielement standard reference materials. Equations 5, written in terms of a pure element, is useful mainly to scale the units of predicted intensities to a common basis.

The preceding equations referred to the intensity of line l, whereas most discussions, for convenience, just use the subscript i instead, referring to the emitting element. Strictly, the list of emitted lines, whether measured or not, is very different from the list of elements: some elements might not be fluoresced at all, while other elements might emit several different characteristic lines that can be useful in measurement and/or important in calculating secondary fluorescence. So, even though we usually speak of the "intensity from element i", those who write the computer programs and their user interfaces need to make the proper distinctions between lines and elements.

Other, More Detailed Models

A number of simplifying assumptions, both physical and geometrical, were noted above. More detailed models of fluorescence could be developed by imposing fewer, or different, assumptions. Typically, considering more detail can multiply the calculation time by roughly ten to perhaps a hundred times, just to consider one additional variable, such as a range of x-ray incidence or emission angles. Now that laboratory computers are thousands of times faster than those used in the late 1970's, it is reasonable to consider much more detailed models. In fact, there have been a number of more detailed models used over the last twenty years or so, especially for geometrical configurations that could not be handled adequately by the typical fundamental-parameters model given here. For example, several workers have developed equations to predict intensities from thin specimens, multilayer specimens, and various kinds of particulate specimens. Another approach has been the use of Monte Carlo simulations, which can handle very complicated geometries, multiple orders of indirect fluorescence, and even x-ray scattering effects when necessary.

Use of a Model

No matter how detailed the physical-mathematical model might be, one important feature is to be expected: the model will be a way to calculate predicted fluorescence intensity, in terms of specimen composition, fundamental parameters of the elements, and experimental conditions. The model itself is not a calculation of specimen composition, which of course is the goal of XRF. Analysis depends on an implementation of an intensity model, to calculate indirectly a best estimate of composition consistent with the measured intensities from the unknown specimen and standard reference materials. Using an intensity model will be discussed as a completely separate topic.