Matrix Effects

As primary radiation penetrates into a specimen, it is absorbed. Once a fluoresced x-ray is emitted from an atom, it is also absorbed as it travels out of the sample toward the detection system. Further, fluoresced x-rays may act to enhance the intensity of fluoresced lower-energy x-rays. The absorption of primary radiation and absorption and enhancement of fluoresced characteristic lines is termed matrix effects.

The intensity of the emitted characteristic radiation should be related to the concentration of the associated element. However, to perform quantitative analysis the relation between element concentration and characteristic x-ray intensity needs to be established. The forms of the following equations are given to reflect the flow of this theory section. They do not necessarily follow the forms given by cited authors.

One of the early theoretical basis for using x-ray fluorescence to determine chemical composition of an unknown sample was proposed by von Hamos [7] in the 1940’s. He demonstrated that the emitted intensity of an element characteristic x-ray line in a binary system was proportional to the composition of that element in the specimen;

(5.1)

where Ri is the ratio between the measured x-ray intensity of element i in an unknown and the x-ray intensity measured for a pure specimen of element i. The constant Ki is a function of the composition of the specimen, the mass absorption coefficients of specimen constituents, and the measurement geometry. This equation represents a purely empirical method of determining element concentrations from measured counts or count rates.

Figure 5.1. Incident and takeoff angles of x-rays penetrating and fluoresced from a sample.

Sherman Equation

Use of x-ray fluorescence to determine chemical composition of unknown specimens became more common in the following decade [8]. With this came the need to better understand x-ray absorption and enhancement. Sherman [9] derived a more specific equation for the fluoresced x-ray intensity from a multi-element specimen subjected to a monochromatic non-divergent incident radiation of energy E that only accounted for primary absorption;

(5.2)

where

Ii Intensity of observed characteristic line of element i.

E Energy of incident radiation.

Ei Energy of the characteristic line of element i being measured.

S Irradiated surface area of specimen.

Ci Concentration of element i in the specimen.

gi Proportionality constant for characteristic line of element i.

y1 Angle between the specimen surface and the incident x-rays.

y2 Angle between the specimen surface and the detector.

W Solid angle subtended by the detector.

k(Ei,Ii) Response of instrument at energy Ei of characteristic line energy of element i.

mi(E) Mass absorption coefficient of element i at incident energy E.

m(E) Total absorption coefficient of specimen at incident energy E.

m(Ei) Total absorption coefficient of specimen at characteristic line energy of element i.

Also note that;

(5.3)

Sherman [10] later developed his theory to express the emitted x-ray intensity from a multi-element specimen subjected to a polychromatic radiation source. Sherman’s theory was then further refined by Shiraiwa and Fujino [11];

(5.4)

where;

Ji Jump ratio of the photoelectric mass absorption coefficient at the absorption edge for the line of element i being measured.

wi Fluorescent yield for the line of element i being measured.

Io(E) Intensity of incident radiation at energy E.

ti(E) Mass photoabsorption coefficient of element i at incident energy E.

ti(Ei) Mass photoabsorption coefficient of element i at energy Ei of characteristic line energy of element i.

pi Transition probability of observed line of element i.

Ei edge Energy of the absorption edge of the characteristic line of element i.

Emax Maximum energy of the incident radiation.

In general this equation is referred to as the Sherman equation. The sum over j is the sum over all characteristic lines of all elements strong enough to excite the observed line of element i. The first term in the above equation represents the primary absorption of the incident and characteristic line of element i in the specimen. The second term represents the secondary enhancement of the characteristic line of element i by all other characteristic lines fluoresced by the specimen. Though not included here, Shiraiwa and Fujino also gave an expression for the tertiary enhancement of the observed line of element i by all other characteristic lines in the sample.

The Sherman equation set the stage for modern x-ray fluorescence spectroscopy.

In the early days of modern x-ray fluorescence spectroscopy, the computing power required to determine the integrals of equation 5.4 was not readily available to spectroscopy laboratories. Thus efforts turned to empirical approximations to the above equations. Using Sherman’s first equation for an incident beam of monochromatic radiation (equation 5.2 above), Beattie and Brissey [12] showed that by taking the ratio between counts measured for element i in an unknown to the counts measured from a pure specimen of element i, a system of simultaneous equations was created;

(5.5)

(5.6)

By measuring the intensity ratios Ri for a set of standards of known composition, this system of equations could be solved for each element in the substance to determine the constants Aij.

The appearance of Ci on both sides of equation 5.5 meant that the system of equations had to be solved numerically. Observing that concentration was roughly proportional to measured x-ray intensity, Lucas-Tooth and Pyne [13] rearranged the above equation to yield;

(5.7)

Once the parameters ai, bi, and kij had been determined using a set of standards, the concentration could be determined directly from this equation for each element in the specimen. Though limited in accuracy away from concentrations of the standards used to determine the coefficients, this method was highly attractive in the days before inexpensive laboratory computers.

Empirical Alpha Models

A problem with the formulation of Beatie and Brissey was that the system of equations had no constant terms and so was over-determined. In the mid sixties, LaChance and Traill [14] made the rather obvious observation that if equation 6 is substituted into equation 5, the over-determination of the system of equations was removed. This equation became the basis for the empirical alphas equations that followed;

(5.8)

Later attempts were made to find an empirical equation that more accurately accounted for the real relationship between measured x-ray intensity and specimen concentration. Claisse and Quintin [15] took the original Sherman equation (equation 5.2) and modeled for polychromatic incident radiation by taking the superposition of mass absorption coefficients at multiple energies. They obtained an equation of the form;

(5.9)

Though there is no direct theoretical support of this, it was generally found that the LaChance and Traill equation accounted for minor enhancement of x-ray intensities with negative alpha coefficients [17]. Rasberry and Heinrich [16] observed that strongly enhancing elements in binary mixtures yielded a concentration/intensity plot that did not follow the hyperbolic dependence of the LaChance and Traill equation. This led them to propose a modified form of the equation where a new term was to be used in place of the LaChance and Traill alpha coefficient for analytes causing significant secondary enhancement;

(5.10).

By the middle of the seventies, other forms of the alpha correction models had been proposed. Most notable are the equations of Tertian [18], [19] who, observing that alpha coefficients are more properly not constant with specimen composition, proposed forms of the LaChance and Traill, and Rasberry and Heinrich equations utilising alpha coefficients that were linear functions of element concentration Ci. Later, Tertian also showed [20] that for a binary system, his modified form of the Rasberry and Heinrich equation reduced to the Claisse and Quintin equation.

Fundamental Parameters Method

Sherman’s equation (equation 5.4) expresses the intensity of a characteristic x-ray fluoresced from an element contained in a specimen of known composition. By determining the concentrations of elements required to produce the measured set of intensities the composition of a specimen can be determined. The direct use of Sherman’s equation is termed the fundamental parameters method. Instrument and measurement geometry effects are removed by measuring characteristic line intensities emanating from standards of known composition. Since this equation accounts for all absorption and enhancement, in theory only one standard is required for each element. It should be noted that the standard should also account for reflection from the surface of the specimen. As such, the surface texture of the standard should be similar to that of the unknown.

Equation 5.4 requires a knowledge of all elements contained in the specimen, the values of the total mass absorption and mass photoabsorption coefficients of each of these elements, and the step ratios of the mass photoabsorption coefficients at the absorption edges of the measured characteristic lines. A knowledge of the incident x-ray tube intensity distribution is also required. To account for secondary enhancement in the specimen, a knowledge of shell fluorescent yields and line transition probabilities are required.

Criss and Birks [21] were among the first to utilise the full fundamental parameters method. They were able to obtain uncertainties in concentrations for nickel and iron-base alloys between 0.1% and 1.7%. Aside from the requirement for significant computing power to evaluate the above integrals, the method is limited by the accuracy of the fundamental parameters themselves, and how well the tube spectrum is known. Determining a tube spectral distribution is no trivial matter. Due to the intensity of the primary radiation, direct measurement is not feasible. A common approach was to measure the reflected distribution from sugar, but then this involved properties of reflection. In the original Criss and Birks paper, the measured spectra of Gilfrich and Birks [22] were utilised. Later developments either continued to use the spectra of Gilfrich and Birks [23], allowed user-entered spectra which usually implied the use of the spectra of Gilfrich and Birks [24], [25], [26], or utilised Kramer’s Law to generate the spectrum [27].

The need for computing power sufficient to evaluate the above integral and the lack of good knowledge of the tube spectrum led a number of authors to the use of an effective incident wavelength in place of the actual tube spectral distribution [28], [29], [30], [31]. Comparisons between fundamental parameters software packages utilising effective wavelength and tube spectral distributions have demonstrated the shortcomings of this approach [32].

The strength of the fundamental parameters method is that only one standard is required. Since the method predicts the degree of correction for a given composition, a single standard should be sufficient for all ranges of composition of an unknown specimen. Empirical alpha models of correction require significantly more standards, and these standards need to be of similar composition to the unknown being analysed. Early developments of the fundamental parameters method noted that most of the fundamental parameters drop out for a pure substance. Taking the ratio between x-ray intensities measured from the unknown specimen to those measured from pure substances allowed the most direct use of the fundamental parameters method. As noted by Sherman himself, and later by Criss, Birks and Gilfrich [33], this tends to increase the reliance on the fundamental parameters that are known to be in error by as much as 10% [34]. The degree of correction (and so the error in correction) is reduced by using standards similar in composition to the unknown.

Fundamental Alphas

The strength of the fundamental parameters method is that it is theoretically exact, and requires relatively few standards. Aside from the need for accurate fundamental parameters and a knowledge of the x-ray tube spectrum, the fundamental parameters method is numerically intensive, and so could take a significant amount of time to compute the composition of a specimen on early laboratory mini-computers. To take advantage of the few number of standards required by the fundamental parameters method and the relatively small computing resource needed for the empirical alphas methods, the hybrid fundamental alphas method came into being [35], [36]. These methods use the fundamental parameters method on larger computing facilities to compute empirical coefficients that are later used in traditional empirical alphas equations on a smaller laboratory computer [37], [38].

There are different methods used to compute theoretical alpha coefficients. One approach involves computing synthetic standards using the fundamental parameters method, and then computing the empirical alphas using standard regression techniques [33], [39]. Another approach is to compute the empirical alpha coefficients directly from Sherman’s equation for binary systems [40], [41]. Rouseau [42] proposed a new empirical alphas equation that can be more directly related to Sherman’s equation;

 (5.11)

where rij are another set of alpha coefficients.

As noted by LaChance [43], the fundamental parameters method and theoretical alphas of the fundamental alphas method rely on inherently different concepts. This means that the flexibility gained by the fundamental alphas approach implies a loss of the ability to define those coefficients explicitly from theory.

It is our opinion that with the increase in laboratory computing power available by the mid 1990’s, the need for compromise with the fundamental parameters method has vanished.

Other Approaches

Other methods of reducing measured counts to element concentrations arise under special circumstances. Particularly for geological samples containing heavy trace elements, the integral of the Sherman equation will sometimes reduce to a constant value for the heavy trace elements. This constant is a function of absorption in the sample, and through the observation that the ratio between mass absorption coefficients at a given energy is roughly independent of energy [44], the so-called Ratio method has evolved. To improve counting statistics, the Compton-scattered primary radiation peak, also dependant on total sample mass-absorption is often used.

Semi-Quantitative Analysis

A throw-back to the early days of XRF spectroscopy is semi-quantitative analysis. This method of matrix correction involved simply computing the concentration of an element from the product of the unknown to standard intensity ratio with the concentration of the element in the standard. More sophisticated so-called quantitative analysis methods utilized polynomials and peak counts corrected for background. These approaches are both mathematically and theoretically simplistic, and with well-designed modern XRF software, are completely unnecessary. XRFWIN includes the Polynomial Fit to Standards matrix correction method for legacy support, but most analysis such as quantitative analysis of a qualitative scan should utilize the fundamental parameters method.

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