Enhancements, Chapter 9
E9.1 Continuum Background Modeling
Kramers Equation
According to the Kramers equations, the number of photons NE in the energy interval from E to E + E per incident electron is given by
(E9.1-1)
where E0 is the incident electron energy in keV, E is the x-ray photon energy in keV, Z is the specimen average atomic number, and kE is a constant, often called Kramers' constant, which is supposedly independent of Z, E0, and (E0 - E).
Lifshin (1974) empirically tested the Kramers equation based on EDS spectra from known standards. He found a better fit of the experimental data could be achieved through the use of a more general relation for N(E):
(E9.1-2)
in which a and b are fitting factors, where fE and PE refer to absorption in the sample and the efficiency of the detector while a and b are constants. Fiori et al., (1976) found that it is possible to determine a and b for any target by measuring N(E) at two or more separate photon energies and solving the resulting equations in n unknowns. This procedure presumes that both fE and PE in Equation (E9.1-2) are known. Since the determination of N(E) is made from measurements on the target of interest only, the inaccuracies of Kramers' law with respect to atomic number do not affect the method. The points chosen for measurement of N(E) must be free of peak interference and detector artifacts (incomplete charge collection, pulse pileup, double-energy peaks, and escape peaks). Since a and b are determined for each spectrum being fitted the quadratic term adds great "robustness" to the quality of the fit. Even equation E9.1-2 can break down at low energy and over the years further refinements had to be added to commercial analyzer systems.
For characteristic primary x-ray photons, the absorption within the target is taken into account by a factor called f(). (See Section E9.6) The factor f() can be calculated from Equation (E9.6-7) (PDH equation), or a simpler form due to Yakowitz et al., (1973) can be used:
(E9.1-3)
where is E01.65 - EC1.65. EC the excitation potential in keV for the shell of interest, and = ()icsc. The term ()i is the mass absorption coefficient of the element of interest in the target, and is the x-ray take-off angle. The values for the coefficients a1and a2 are a1 = 2.4 • 10-6 gcm-2keV-1.65, a2 = 1.44 • 10-12 g2cm-4keV-3.3.
If it is assumed that the generation of the continuum has the same depth distribution as that of characteristic radiation and if E ~ EC, then Equation (E9.1-3) can be modified to
(E9.1-4)
where C = ()Ecsc. The term ()E is the mass absorption coefficient for continuum photons of energy E in the target and fE is the absorption factor for these photons. Similar assumptions and expressions have been used by other authors.
As pointed out by Ware and Reed (1973), the absorption term fE is dependent upon the composition of the target. Consequently, when the composition of the target is unknown, it is necessary to include Equation (E9.1-2) in an iteration loop when the ZAF method is being applied.
E 9.2 Background Filtering
The “averaging” used in the background filtering method is done in the following manner. The filter, Figure E9.2-1, is divided into three sections: a central section, or positive lobe, and two side sections, or negative lobes. The central lobe is a group of adjacent channels in the original spectrum from which the contents are summed together and the sum divided by the number of channels in the central lobe. The side lobes, similarly, are two groups of adjacent channels from which the contents are summed together and the sum divided by the total number of channels in both lobes. The “average” of the side lobes is then subtracted from the “average” of the upper lobe. This quantity is then placed in a new spectrum into a channel which corresponds to the center channel of the filter.
The effect of this particular averaging procedure is as follows. If the original spectrum is straight, across the width of the filter, then the “average” will be zero. If the original spectrum is curved concave upward, across the width of the filter, the “average” will be negative. Similarly, if the spectrum is curved convex upward the “average” will be positive. The greater the curvature, the greater will be the “average.” The above effects can be observed, for a Gaussian superposed on a linear background, in Figure E9.2-1. In order for the filter to respond with the greatest measure to the curvature found in spectral peaks, and with the least measure to the curvature found in the spectral background, the width of the filter must be carefully chosen. For a detailed treatment of the subject see Schamber (1978) and Statham (1977). In general, the width of the filter for any given spectrometer system is chosen to be twice the full width at half the maximum amplitude (FWHM) of the MnK peak, with the number of channels in the upper lobe equal to or slightly more than the combined number of channels in the side lobes.
Because the top hat filter “averages” a number of adjacent channels, the effects of counting statistics in any one channel are strongly suppressed. Consequently, in addition to suppressing the background under spectral peaks, the digital filter also “smooths” a spectrum. Note that in Figure E9.2-1, the top hat filter converts the sloped background into a flat background.
E9.3The Atomic Number Correction, Z – R and S Factors
The atomic number effect arises from two phenomena, namely, electron backscattering, R, and electron retardation, S, both of which depend upon the average atomic number of the target. Therefore, if there is a difference between the average atomic number of the specimen given by
(E9.3-1)
and that of the standard, an atomic number correction is required. For example in an Al-3 wt% Cu alloy, the value of is 13.3, so a somewhat larger atomic number effect for a Cu (Z = 29) analysis would be expected than for an Al (Z = 13) analysis if pure element standards are used. In general, unless this effect is corrected for, analyses of heavy elements in a light element matrix generally yield values which are too low, while analyses of light elements in a heavy matrix usually yield values which are too high.
The most accurate formulation of the atomic number correction, Zi, for element i appears to be that given by Duncumb and Reed (1968);
(E9.3-2)
where Ri and Ri* are the backscattering correction factors of element i for standard and specimen respectively:
Ri = total number of photons actually generated in the sample / total number of photons generated if there were no backscatter
Q is the ionization cross section and defined as the probability per unit path length of an electron with a given energy causing ionization of a particular inner electron shell of an atom in the target (see Chapter 3), and S is the electron stopping power —(1/)(dE/dX) (see Chapter 3) in the region l E 50 kV. The stopping power Si given by Bethe (1930) is
(E9.3-3)
where C1 is a constant and J is the so-called mean ionization potential. To calculate the term (C1E/J), the units commonly used are C1 equal to 1.166 when E and J are given in kilovolts.
(1) RFactor The R factor represents the fraction of ionization remaining in a target after the loss due to the backscattering of beam electrons. Values of R lie in the range 0.5-1.0 and approach l at low atomic numbers. The backscattering correction factor varies not only with atomic number but also with the overvoltage U = E0/Ec. Figure E9.3-1 shows this variation. As the overvoltage decreases towards l, fewer electrons are backscattered from the specimen with voltages greater than Ec, and consequently less of a loss of ionization results from such backscattered electrons.
There are several tabulations of R values as a function of the pure element atomic number and the overvoltage U (Duncumb and Reed, 1968; Green, 1963; Springer, 1966). The tabulation given by Duncumb and Reed very nearly agrees with experimental determinations where comparisons can be made. The electron backscattering correction factors R from Duncumb and Reed (1968) were fitted by Yakowitz et al. (1973) with respect to overvoltage U and atomic number Z as follows:
(E9.3-4)
where
The term i represents the element i which is measured and the term j represents each of the elements present in the standard or specimen including element i. The term Rij then gives the backscattering correction for element i as influenced by element j in the specimen.
Duncumb and Reed (1968) have postulated that
(E9.3-5a)
for the standard, and
(E9.3-5b)
for the sample. To evaluate Ri, one must obtain from the Duncumb-Reed tabulations, as given in the data base of the Enhancements, Fitting Parameter or by Equation (E9.3-4), the value of Rij for each element in the specimen or standard.
(2) SFactor There are several relations for Q as discussed in Chapter 3. Despite the fact that Q values differ by several percent depending on the value of the constants, Heinrich and Yakowitz (1970) have shown that discrepancies in Q values have only a negligible effect on the final value of the concentration. A simplifying assumption is often made that Q is constant and therefore cancels in the expression for Zi, equation (E9.3-2). With the advent of fast laboratory computers, the integration can be done directly.
The value of J to be used in Equation (E9.3-3) for the value of Si is a matter of controversy since J is not measured directly but is a derived value from experiments done in the MeV range. The original Bethe expression was developed for hydrogen only. Present J values allow the expression to be used for other elements. The most complete discussion of the value of J is given by Berger and Seltzer (1964). These authors postulate, after weighing all available evidence, that a “best” J vs. Z curve is given by
(E9.3-6)
The Berger and Seltzer J values as a function of Z are listed in the data base of the Enhancements.
In order to avoid the integration in Equation (E9.3-2), Thomas (1964) proposed that an average energy E may be taken as 0.5(E0 + Ec), where Ec is the critical excitation energy for element i. The average energy is substituted for E in equation (E9.3-3) for Si with little loss in accuracy. Hence
(E9.3-7)
where i represents the element i which is measured and j represents each of the elements present in the standard or specimen including element i. The constant in Equation (E9.3-7) need not be evaluated since the constant will cancel out when the stopping power for the sample and standard are compared, equation (E9.3-9). To evaluate Si and Si*, there is experimental evidence that a weighted average of Sij and Sij*, respectively, can be used, namely,
(E9.3-8a)
for the standard and
(E9.3-8b)
for the specimen. The final values of Si and Si* are obtained using equation (E9.3-8).
If the integration in Equation (E9.3-2) is avoided and Q is constant, we have the final form of Zi, namely
(E9.3-9)
The major variable which influences the atomic number factor Zi is the difference between the average atomic number of the specimen and the standard.
A sample calculation of the atomic number correction for this alloy is given in Table E9.3-1. The atomic number correction ZCu and ZAl is obtained for an operating energy, E0, of 15 keV. The values of JCu and JAl are obtained from equation (E9.3-6). The values of SCu and SA1 for either Cu K radiation (Ec = 8.98 keV) or Al K radiation (Ec = 1.56 keV) are calculated using equation (E9.3-7). The constant in equation (E9.3-7) is set equal to l since it eventually drops out in later calculations. The backscattering corrections RCu and RAl for either Cu K or Al K are obtained using equation (E9.3-4). The Si , S*iand Ri , R*i terms are calculated using equations (E9.3-5) and (E9.3-8). The final corrections ZCu and ZAl are obtained from Equation (E9.3-9) and are 1.16 and 0.998. For 30 keV, ZCu and ZAl are 1.11 and 0.999. It is interesting to note that the ZCu correction is decreased by ~50% from 1.16 to 1.08 at 15 keV when a lower atomic number standard of phase, CuAl2 (Z = 21.6) is used in favor of pure Cu (Z = 29). The ZCu correction is also decreased from 1.11 to 1.05 when operating at a higher E0 of 30 keV.
Table E9.3-1 Atomic Number Correction for an Al-3 wt % Cu Alloy
(a) Input data Al-3 wt % Cu alloy
______
AlCu
______
Z1329
A26.9863.55
Ec (keV) 1.56 8.98
______
(b) Output data Al-3 wt % Cu alloy
______
Operating conditions:E0 = 15 keV
For Cu K:UCu = 1.67
JCu = 314.05JA1 = 163.0
SCuCu = 0.144SCuAl = 0.179
R1' = 1.641R2' = 0.189R3' = 0.792 RCuCu = 0.910 RCuAl = 0.968
Si* = 0.178
Ri* = 0.966
using a Cu standard:Si = 0.144
ZCu = 1.16
Ri = 0.910
For Al K:UAl = 9.62
SAlAl = 0.238SAlCu=0.189
R1' = 2.073R2' = 0.332R3' = 0.623 RAlAl = 0.910 RAlCu = 0.823 Si*= 0.236
Ri* = 0.907
using an Al standard:Si = 0.238
ZAl = 0.997
Ri = 0.910
Heinrich and Yakowitz (1970) have investigated error propagation in the Zi term. In general, the magnitude of Zi decreases as the overvoltage U = E0/Ec increases, but very slowly (5% for a tenfold increase in U). The uncertainty in Zi remains remarkably constant as a function of U since the uncertainties in R and S tend to counterbalance one another. Thus, no increase in the error of Zi is to be expected at low U (U > 1.5) values and hence the choice of operating with low U to minimize errors in the absorption correction is still valid for obtaining the highest accuracy.
E9.4 z) Technique
Introduction
The z) method uses calculated z) curves for the determination of the atomic number, Z, and absorption, A, in the microchemical analysis of specimens in the SEM - EPMA using Equation 9.4 in Chapter 9. The concept of z) curves was introduced in Chapter 9 along with a description of how the curves can be used to calculate the atomic number and absorption corrections. The atomic number correction for a given element i is obtained by calculating the area under the z) curve, which represents the total number of x-rays of element i generated in the specimen, divided by the area under the z) curve for element i in a standard for the same operating conditions (Figure 9.1 in Chapter 9). Zi can be expressed as
(E9.4-1)
where the specimen is noted by an asterisk and the standard is left unmarked.
The z) curves
Definition
Figure E9.4-1 shows the scheme for defining the depth distribution of the generated x-rays, z). The x-ray generation volume in the specimen is divided into 10 to 20 or more thin layers of equal mass thickness, (z). We can calculate or measure the number of x-rays generated for a given x-ray line of interest in each layer, (z), z in mass depth from the specimen surface and z) in mass thickness, for a given number of beam electrons. For normalization purposes we can also calculate or measure the number of x-rays generated for the same x-ray line in a thin film of the specimen (same composition) with the same mass thickness, z), isolated in space (See Figure E9.4-1b), again for the same number of beam electrons. We define the x-ray intensity in the isolated thin film as I(z). The depth distribution of generated x-rays, z), at a mass depthz, is then the ratio of I(z) divided by I(z). Thez) term varies with mass depth, z, and with depth, z, from the specimen surface z = z = 0, to the position where x-rays are no longer generated, z = Rx, z = Rx .
Figure E9.4-2 shows, schematically, the variation of I(z) with mass depth, z, using 20 layers of equal mass thickness, z). The depth distribution of generated x-rays, z) generated, obtained by dividing the intensities I(z) in Figure E9.4-1 by I(z). The total generated intensity , Igen , can be obtained by adding together the contributions ofz) from each layer z) in mass thickness, that is, taking the area under the z) curve and multiplying that area by the x-ray intensity of the isolated thin film, I(z). The total generated intensity, Igen , can be determined for a specific x-ray line, specimen, and initial electron beam energy and can also be used to calculate the atomic number effect.
The general shape of the depth distribution of the generated x-rays, the z) vsz curve (Figure 9.11 in Chapter 9), provides information on the effect of Z, and also on the effect of E0. For convenience, we define the intersection of the z) curve with the surface,z = 0, as 0, the maximum in thez) curve as m at z equalsRm, and the ultimate depth of x-ray generation where z) = 0 at z equalsRx (See Figure 9.11). The major reason that 0 is larger than 1.0 in solid samples is the effect of the backscattered electrons. The backscattered electrons excite x-rays in solid samples as they leave the sample. In a thin film specimen of z) in mass thickness, beam electrons are not able to backscatter as in the solid specimen. Therefore the ratio of the intensities from the top layer in the solid sample to the isolated thin film, 0 is > 1.0. The value increases as backscattering increases, that is, it increases with the atomic number of the specimen.
Measurement of z) curves
Numerous z) curves have been measured experimentally. The experimental set up first proposed and used by Castaing (1951), called the tracer technique, to obtain z) curves is shown in Figure E9.4-3. We illustrate the experiment by the measurement of z) for Cu K A thin film, z) in thickness, of Zn, the tracer, is deposited on a substrate and is coated by a number of successive layers of Cu (matrix), each z) in mass thickness. The emitted intensity of Zn K radiation from the tracer is measured by placing the beam on each successive deposit above the tracer (Figure E9.4-3a) . The intensity from the thin film Zn tracer itself serves as the isolated film in space (Figure E9.4-3b). The generated z) curve can be calculated after correction for the absorption of ZnK from the tracer in the overlying matrix layers. Zn was selected as the tracer in this case because it is of similar atomic number to Cu and the Zn K x-ray line has a similar but higher energy to that of Cu K, so that it is not fluoresced by Cu K. The measured Cu Kz) curve, using Zn K as a tracer, at 25 keV (Brown and Parobek, 1972) is shown in Figure E9.4-4 and illustrates the type of z) vs z curve that can be measured. The general shape of thez) curve and the variation of the curve with atomic number and initial electron beam energy were discussed in some detail in Chapter 9, Section 9.6.1.
During the last 30 years considerable effort has been made to increase the number of measured z) curves so that generalized expressions can be obtained for z). Once these generalized expressions became available, the quantitative analysis scheme using the z) method was developed. The next section considers the development of generalized expressions for the x-ray generation function.