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 E0
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
1.65
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interest in the target, and  is the x-ray take-off angle. The values for the coefficients a1
and 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 =
The term
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.
()Ecsc.
()E
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 Mn K 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.
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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.
E9.3 The 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
Z =  CjZj
(E9.3-1)
j
and that of the standard, an atomic number correction is required. For example in an Al-3
wt% Cu alloy, the value of Z 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);
E0
Ri
(Q/S)dE
Ec
Zi =
E0
R i*
(Q/S*)dE
Ec
(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
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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
S i = (const) Z 1 ln C1 E
AE
J
(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) R Factor
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 E9.31 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:
Rij = R1' - R2' ln(R3'Zj + 25)
(E9.3-4)
where
R 1' = 8.73 x 10-3 U 3 - 0.1669U 2 + 0.9662U + 0.4523
R 2' = 2.703 x 10-3 U 3 - 5.182 x 10-2 U 2 + 0.302U - 0.1836
R 3' = (0.887U 3 - 3.44U 2 + 9.33U - 6.43)/U 3
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
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R i =  Cj R ij
(E9.3-5a)
j
for the standard, and
R i* =  CjR ij
(E9.3-5b)
j
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 (E9.34), the value of Rij for each element in the specimen or standard.
(2) S Factor 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
J = 9.76Z + 58.8Z -0.19
(eV)
(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
S ij = 2(const) Zj ln 583 E0 +Ec
A E +E
Jj
222(const) j 0 c
(E9.3-7)
where i represents the element i which is measured and j represents each of the elements
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present in the standard or specimen including element i. The constant in Equation (E9.37) 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,
S i =  CjS ij
(E9.3-8a)
j
for the standard and
S i* =  CjS ij*
(E9.3-8b)
j
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
Zi = Ri /Ri* S i*/S i
(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*i and 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.
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Table E9.3-1 Atomic Number Correction for an Al-3 wt % Cu Alloy
(a) Input data Al-3 wt % Cu alloy
______________________________________
Al
Cu
______________________________________
Z
13
29
A
26.98
63.55
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 = 314.05
JA1 = 163.0
SCuCu = 0.144
SCuAl = 0.179
R1' = 1.641
R2' = 0.189 R3' = 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:
using an Al standard:
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UAl = 9.62
SAlAl = 0.238
SAlCu=0.189
R1' = 2.073
R2' = 0.332 R3' = 0.623
RAlAl = 0.910
RAlCu = 0.823
Si*= 0.236
Ri* = 0.907
Si = 0.238
ZAl = 0.997
Ri = 0.910
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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.
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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

i (z)d(z)
Zi =
0

*
i (z)d(z)
(E9.4-1)
0
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 depthz,
is then the ratio of I(z) divided by I(z). Thez) 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 .
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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 ofz) 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) vsz 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 thez) curve
as m at z equalsRm, and the ultimate depth of x-ray generation where z) =
0 at z equalsRx (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 xray line has a similar but higher energy to that of Cu K, so that it is not
fluoresced by Cu K. The measured Cu Kz) 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 thez) curve and the variation of the curve with atomic number and initial
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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.
Calculation of z) curves
Numerous researchers have attempted to accurately model z) curves,
for example Wittry (1957), Kyser (1972), and Parobek and Brown (1978). A key
observation was made by Packwood and Brown (1981) with regard to the
functionality of the z) curves. They showed that by plotting ln z) versus
z)2, straight line variations were obtained (Figure E9.4-5), at least beyond the
maxima, m, in the z) curves. This variation implies that the z) curves are
Gaussian in character centered at the surface of the sample. However, there is
also a loss of intensity near the surface. In general the z) curves can be
represented by the following equation as given by Packwood and Brown (1981)

z) =  exp - 2(z)2 { 1- [( - (0)) exp - z ] /  }
(E9.4-2)
The four parameters, , and ), describe the modified Gaussian curve and
are a function of E0, absorption edge energy, and matrix atomic number and
atomic weight. The functional behavior of the terms in the Packwood Brown
z) equation is shown in Figure E9.4-6.
In equation E9.4-2, the Gaussian expression for the z) curve as a
function of mass depthz is given by
exp - (2 (z)2)
(E9.4-3)
in which 1)  can be regarded as a scaling factor or surface intensity for the basic
surface centered Gaussian and 2)  represents the decay rate of the Gaussian and
describes the electron beam penetration to Ec for a given x-ray line. The basic
surface centered Gaussian function is modified in the near surface regions by a
transient function
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-(-(0))exp (-2(z)2 -z)
(E9.4-4)
which reduces the x-ray intensity at the surface. The term (0) or 0 is the
relative x-ray intensity generated at the sample surface and is the same as 0 for
the z) curve. The  term describes the transition process by which the
collimated electron probe which enters the specimen changes to scattering in
random directions in the specimen. Once the four parameters are determined for a
sample and standard for each element i present in the sample, Zi and Ai can be
determined by integration of Equation E9.4-1. One advantage of the Packwood
Brown equation is that it is possible to directly integrate the equation in order to
obtain the corrections for atomic number and absorption.
The performance of the equation E9.4-2 in the z) approach for matrix
correction is, of course, largely dependent upon the successful parameterization of
the , and (0) terms. Packwood and Brown (1981) fit equation E9.4-2 to an
extensive array of experimental z) data using an optimization method.
Improvements were also made by considering, theoretically, electron interactions
with the specimen. Since the initial Packwood Brown method was proposed,
numerous investigators, for example Brown and Packwood (1982), Bastin et al,
(1984), and Bastin and Heijligers (1986), have proposed newer values for the 4
terms in the z) equation. Most of these improvements were due to the
availability of new experimental z) data, and new k ratio measurements on
well characterized specimens, particularly samples containing the light elements.
Another approach to the calculation of z) curves has been developed
by Pouchou and Pichoir (1984). They use a polynomial expression forz), the
coefficients of which are given by four parameters, the surface ionization(0) or
0, the depth of maximum ionization, Rm, the maximum depth of ionization, Rx,
and the integral F of the z) distribution. The 0, Rm and Rx parameters are
shown in Figure 9.11, Chapter 9.
The integral F is equal to

i (z)dz
(E9.4-4)
0
which is equivalent to the generated intensity, Ii gen , divided by Ii (z). The
integral F of z) is calculated theoretically by an evaluation of the atomic
number correction Ri / Si. The atomic number correction of Pouchou and Pichoir
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(1987, 1991) differs in detail from that of Duncumb and Reed (1968) which is
discussed in Section E9.3.
Bastin and Heijligers (1990, 1991) have adopted the Pouchou and Pichoir
method of calculating the integral F of z). They use the calculation of F to
obtain a more correct value of the parameter  in the Packwood - Brown equation
(E9.4-2). Using this procedure, the values of m and Rm in the z) curve,
Figure 9.11, Chapter 9, are more accurately described. This improvement in the
z) procedure has been incorporated in a new calculation scheme called
PROZA. The PROZA method is used to calculate the examples discussed later in
this section and the z) curves shown in Chapter 9. The detailed equations for
F, and the parameters , and (0) in the Packwood - Brown equation are
given by Bastin and Heijligers (1990, 1991a,b).
E9.5 Atomic Number Correction - Zi , z) Approach
Equation E9.4-1 gives the formulation of the atomic number correction
Zi. The Packwood - Brown equation for z), equation E9.4-2, can be integrated
for specimen and standard in closed form as indicated by Brown and Packwood
(1982). According to Bastin et al (1984), the atomic number correction Zi is
{ - [ - (0)] R (} / 
Zi = ------------------------------------
(E9.5-1)
{ - [ - (0)] R (*
where R represents the value of the fifth-order polynomial in the approximation
for the complementary error function for the argument in parentheses. In the PAP
and PROZA approaches to the calculation of z) discussed in the previous
section, E9.4, the integral F of z) is calculated theoretically and
Zi = F/F*
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E 9.6 The Absorption Correction, A – PDH Formulation
Formulation
Since the x-rays produced by the primary beam are created at some non - zero
depth in the specimen, they must pass through the specimen on their way to the detector.
On this journey, some of the x-rays undergo photoelectric absorption due to interactions
with the atoms of the various elements in the sample, as discussed in Chapter 6.
Therefore, the intensity of the x-ray radiation finally reaching the detector is reduced in
magnitude. Following the initial formulation of Castaing (1951), the intensity, dIi, of
characteristic radiation, without absorption, generated from element i in a layer of
thickness dz having density  at some depth z below the specimen surface is
dIi = Ii ( z)(z)d(z)
(E9.6-1)
where iz) is defined as the distribution of characteristic x-ray production of element i
with depth and Ii (zis the x-ray intensity of an isolated thin film (See Chapter 9,
Section 9.6.1 for a description of z)). In the absence of absorption, the total flux
generated, for element i, Ii gen , is

Ii gen = Ii ( z)
i (z)d(z)
0
(E9.6-2)
Considering absorption of the generated x-rays, the total flux emitted, Ii em , is

Ii em = Ii ( z)
i (z)exp - /) i csc(z) d(z)
0
(E9.6-3)
where u/i is the x-ray mass attentuation coefficient of the specimen for i, the
characteristic x-ray line of element i, is the take-off angle, the angle between the
direction of the measured x-ray and the sample surface (See figure 9.16 in Chapter 9).
The path length over which absorption takes place is z csc (See figure 9.18 in Chapter
9). The quantity )i csc  is called .
Philibert (1963) referred to the generated intensity, Ii gen , as F(0), when is
zero. He also referred to the emitted intensity Ii em as F(). Using these terms, the ratio
F()/F(0) is called f(), which is equivalent to
401253795
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06/21/16

i (z)exp - /) i csc(z) d( z)
f(  ) =
0

i (z)d(z)
0
(E9.6-4)
The ratio f() is called the standard absorption term, Philibert (1963).
For the determination of the absorption correction, Ai, for any element i in a
composite specimen, we use the equation
Ai = f()std/f( )spec
(E9.6-5a)
where std and spec refer to standard and specimen, respectively. An alternative
nomenclature, found in the literature, gives
Ai = f()/f( ) *
(E9.6-5b)
where the specimen alloy is noted by an asterisk and the standard is left unmarked.
Expressions for f()
The absorption correction factor, f() of a specific characteristic line of element i,
depends upon the respective mass absorption coefficient , the x-ray emergence angle
, the initial energy of the electron beam, E, the critical excitation energy Ec for K, L,
or M radiation from element i, and the mean atomic number and mean atomic weight of
the specimen. Hence we can write
f( ) = f / csc,E0,Ec,Z,A
(E9.6-6)
There have been a number of experimental measurements of the z) for pure
elements starting with those of Castaing (1951). Direct measurements of z) curves
was discussed in the Enhancements, E9.4. If measured z) curves are available, f()
and f()* can be obtained directly using equations E9.6-2 and E9.6-3.
Another approach to obtaining f() and f()* is by using equations which describe
z) curves for various elements, x-ray lines and initial electron beam energies, E0.
This approach is now called the z) method and is described in Chapter 9. The most
commonly used absorption correction, that of Philibert-Duncumb-Heinrich, was
developed however by using an empirical expression for z). Thus the "ZAF" and the
"z)" methods are in many cases interrelated.
401253795
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Philibert (1963) developed an equation for the functionality of z) using a
simplified equation to fit the form of experimental z) curves available at that time. In
order to simplify the calculations of f(), he chose a z) equation in which 0, the
intersection of the z) curve with the surface,z = 0, was set equal to 0. As discussed
in Chapter 9 and as shown in Figure E9.4-2, 0 is always > 1.0 due to the effect of
backscattered electrons.
The equation for f() developed by Philibert (1963) is;
1 = 1 +
f( )


1+ h
1+h
(E9.6-7)
where
(E9.6-8)
h = 1.2A/Z 2
and A and Z are, respectively, the atomic weight and number of element i. The absorption
parameter  equals  csc , where is the mass absorption coefficient of element i
in itself. The parameter  (the Lenard coefficent) is a factor which accounts for the
voltage dependence of the absorption or loss of the primary electrons.
Duncumb and Shields (1966) proposed that the dependence of Ec should be taken
into account in the formulation of  in equation (E9.6-7). Later Heinrich (1969), after
critical examination of existing experimental f() data, suggested a formula for ,
namely,
5
 = 4.5 x 1 0
E01.65 - Ec1.65
(E9.6-9)
This development of the absorption effect [Equations (E9.6-7 to E9.6-9)] is known as the
Philibert-Duncumb-Heinrich (PDH) equation and is currently the most popular
expression for f(). Accordingly, the PDH equation is commonly used in microprobe
correction schemes to calculate f().
In using the PDH equation to calculate f() or f()* of element i in
multicomponent samples, the effect of other elements in the sample or standard must be
401253795
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considered. These elements have an effect on the values of h, and . After h is
evaluated for each element in the specimen, an average h value is obtained using
h =  Cjhj
(E9.6-10)
j
where j represents the various elements present in the sample including element i and Cj
is the mass fraction of each element j. To obtain the mass absorption coefficient for
element i in the sample, ()ispec, must be calculated. The mass absorption coefficient
is given by
i
(/)spec
=  (/)ji Cj
j
(E9.6-11)
where ()ij is the mass absorption coefficient for radiation from element i in element j
and Cj is the weight fraction of each element in the specimen including i. The  value is
obtained from Equation (9.20) using the Ec value of element i.
Practical Considerations
The effects of the errors in input parameters (, , E0) have been considered in
detail by Yakowitz and Heinrich (1968). The major conclusions of their study are:
1. Serious analytical errors can result from input parameter uncertainties. Mass
absorption coefficients for the light elements are a particular problem.
2. In order to reduce the effects of these input parameter uncertainties, the value
of the absorption function f() should be 0.7 or greater.
3. To maximize f(), the path length for absorption in the sample should be
minimized. Samples should be run at low overvoltage ratios and instruments should have
high x-ray emergence angles.
The PDH absorption correction is particularly sensitive to errors when the amount
of absorption is high and the majority of the emitted signal comes from regions close to
the sample surface. Typical examples of this situation include measurements of the light
element x-rays (C, N, O) in metal matrices (Ti, Fe, Cu). In this analytical case, the
functionality of the z) curve assumed by Philibert (1963), close to the sample surface,
is particularly in error. In such cases, the calculation of f()* may have rather substantial
errors.
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Calculations of the Absorption Factor, A
The three major variables which influence the absorption factor Ai are the
operating voltage E0, the take-off angle , and the mass absorption coefficient for the
element of interest i in the specimen, ()ispec. Since Ai is determined by the ratio
f(x)/f(x)*, both terms should be similar if the absorption factor Ai is to approach l. As Ai
approaches l, the measured intensity ratio becomes a better approximation for the
concentration ratio of element i from sample to standard.
The significance of the absorption factor can be illustrated by considering two
binary systems, Fe-Ni and Al-Mg. In both binaries the atomic numbers of the two
elements are so close that no atomic number Zi correction need be made. We will
consider Ni K absorption in Fe and Al K absorption in Mg. In both cases secondary
fluorescence (Ni K by Fe or Al K by Mg) does not occur so that a Fi correction need
not be made. Calculations of Ai = f(x)/f(x)* were made for both systems using the PDH
correction, Equations (E9.6-5), (E9.6-7)—(E9.6-10).
Tables E9.6-1 and E9.6-2 contain the input data for the absorption calculation and
also list the various terms X, h which are evaluated in the calculation. In the case of
Fe-Ni, a 10 wt% Ni alloy was considered (Table E9.6-1). Calculations were performed
for two operating voltages, 30 and 15 keV, and two take-off angles, 15.5° and 52.5°. The
case of E0 = 30 keV,  = 52.50 is illustrated in Figure 9.9, Chapter 9 with experimental
data. The smallest f() and f() * factors are calculated at E0 = 15 keV and  = 52.5°.
The amount of absorption is minimized because x-rays are generated close to the surface
and the absorption path length is smaller at high take-off angles. In this case the ANi
factor is 1.05, requiring only a 5% correction. On the other hand at 30 keV, and a low
take-off angle of 15.5°, the absorption factor is 1.60, requiring almost a 60% correction.
It is clear that Ai will be minimized at low overvoltage and high take-off angles.
Table E9.6-2 illustrates the absorption corrections necessary for Al K in a Mgl0wt% Al alloy. At E0=l5 and 30 keV,the f(x) * values are so low that corrections of
over 200% are necessary. The very high (/) value of Al K in Mg (4376.5 cm2/g) is
responsible for the large absorption correction. Since Ec for Al is only 1.56 keV, it is
possible to perform an x-ray microanalysis at a lower operating voltage than 15 keV. The
excitation region will be closer to the surface and the absorption path length will be
decreased. At an operating voltage of 7.5 keV, and a take-off angle of 52.5° (note Table
E9.6-2) the absorption factor is only 1.306. In this case a reasonably small correction can
be applied.
It is clear from these calculations that the analyst should be wary of large (/)
values and operation at high overvoltages and low take-off angles.. Clearly a reasoned
choice of SEM operating conditions and x-ray lines with small mass absorption
coefficients can help minimize the corrections needed in the ZAF procedure.
Alternatively, the analyst could turn to the (z) method which incorporates a more
401253795
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advanced absoprtion correction specifically devised to consider the high absorption
situations, which often cannot be avoided in practical analysis situations.
Table E9.6-1 Absorption of Ni K in a Fe-Ni Binary
(a) Input data Fe-Ni binary
___________________________________________________
Ni
Fe
___________________________________________________
/ Ni K (cm2/g) absorber
58.9
379.6
Z
28
26
Ec(keV)
8.332
7.111
A
58.71
55.85
(b) Output data Ni-Fe binary
________________________________________________________________________

E0
Ni-Fe Sample (keV) (deg)

h
f( )
f( )* ANi
________________________________________________________________________
Fe- 10% Ni Sample
438
1870 0.0982
—
0.794
30
52.5
1.21
Ni Standard
74.2 1870 0.0899
0.959
—
Fe-10% Ni Sample
Ni Standard
1300 1870
220.4 1870
0.0982
0.0899
—
0.886
0.555
—
Fe-10% Ni Sample
Ni Standard
438
74.2
8310
8310
0.0982
0.0899
—
0.990
0.945
__
Fe-10% Ni Sample
Ni Standard
1300 8310
220.4 8310
0.0982
0.0899
—
0.972
0.853
—
Fe-50% Ni Sample
820.4 8310 0.0945
—
0.902
Ni Standard
220.4 8310 0.0899
0.972
—
_______________________________________________________________________
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Table E9.6-2. Absorption of Al K in a Al-Mg Binary
(a) Input data Al-Mg binary
___________________________________________________
Al
Mg
___________________________________________________
/ Al K (cm2/g) absorbera
385.7
4376.5
A
26.98
24.305
Z
13
12
Ec(keV)
1.56
1.303
(b) Output data Al-Mg binary
________________________________________________________________________

E0
Al-Mg Sample (kV) (deg)

h
f( )
f( )*
AAl
________________________________________________________________________
Sample Mg-10% Al
Al Standard
5,013
486
1,657
1.657
0.201
0.192
0.165
Sample Mg-10% Al
Al Standard
14,884
1,443
1,657
1,657
0.201
0.192
Sample Mg-10% Al
5,013
5,286
0.201
Sample Mg-10% Al
Al Standard
14,884
1,443
5,286
5,286
0.201
0.192
0.753
Sample Mg-50% Al
Al Standard
8,910
1,443
5,286
5,286
0.197
0.192
0.753
Sample Mg-l0% Al
5,013
17,506
0.201
0.738
0.04
0.469
0.443
0. 178
0.291
0.742
7.5
52.5
1.306
Al Standard
486
17,500 0.192 0.969
________________________________________________________________________
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E 9.7 Absorption correction, Ai –
The absorption correction, Ai, can be expressed in terms of z) using Equation
E9.6-4 for both f() and f()*. In this case

i (z)exp - /)i csc(z) dz
0

i (z)dz
0
Ai =

*
i (z)exp - /)i csc(z) dz
0

*
i (z)dz
0
(E9.7-1)
We can calculate the emitted intensity, Iem, for element i, which results when
the generated intensity, Igen, for element i is absorbed within the specimen, in a
several step process. In the first step, the value of z) for each layer, z, in mass
depth from the surface of the specimen and z in mass thickness, is multiplied by
exp - [u )i csc (z)]. Figure 9.16 in Chapter 9 shows the geometrical
relationships within the specimen that are considered for the calculation of the x-ray
intensity for one x-ray line and for one layerz) in mass thickness, z in mass
depth from the surface. These calculations yield a z) vsz curve called z)
emitted (See Figure 9.9). In the second step, the area under the z) vsz emitted
curve is obtained. We call this areaz)em Area. Finally the area under the z)
vs z emitted curve (z)em Area) for element i is multiplied by the x-ray intensity
from element i in the isolated thin film, I(z). The value of Iem which is obtained
contains the combined effects of atomic number, through the z) vsz curve, and
absorption.
Figure E9.7-1 shows the z) vsz (emitted) curves for Cu K in pure
Cu and Al K in pure Al calculated at an initial electron beam energy of 15 keV. A
400 take off angle was assumed and the calculation was made using the PROZA
program (Bastin and Heijligers, 1990, 1991). The Cu Kz) emitted curve falls
essentially on the z) generated curve since the mass absorption coefficient is so
small, 52 cm2/g. The Al K z) emitted curve for pure Al falls below that of
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z) generated curve and the position of the maximum x-ray production, Rm , is
somewhat closer to the surface. Both of these changes to thez) generated curves
are due to the much higher Al K mass absorption coeffecient, and the increasing
amount of absorption with mass depth. The values of 0 are not changed because
absorption at the surface is limited.
Figure E9.7-1 also shows the Cu Kand Al K z) generated and emitted
curves in an Al - 3 wt% Cu alloy. As previously shown, (Figure 9.13 in Chapter 9),
the generated z) curve for Cu K in the Al - 3 wt% Cu alloy has a lower0 value
and a lowerRm value due to the smaller amount of electron backscattering in the
lower atomic number specimen. As in the pure element case, the Cu K emitted
curve (Figure E9.7-1) shows little effect of absorption and parallels the generated
curve. The Al K z) generated and emitted curves for the alloy parallel those of
pure Al since the atomic number of the Al - 3 wt% Cu specimen is almost the same as
pure Al. The Al K emitted curve in the alloy is slightly lower than the Al
Kemitted curve in pure Al because the presence of Cu increases the Al K mass
absorption coefficient in the alloy.
E 9.8 The Characteristic Fluorescence Correction, F
If the energy of a characteristic x-ray peak E from element j in a specimen is
greater than Ec of element i, then parasitic fluorescence must be accounted for in the
correction procedure for element i. Such a fluorescence correction is necessitated
because the energy of the x-ray peak from element j is sufficient to excite x-rays
secondarily from element i. Thus, more x-rays from element i are generated than would
have been produced by electron excitation alone. The correction becomes negligible,
however, if (E — Ec ) is greater than 5 keV.
Electrons are attenuated more strongly than x-rays of comparable energy. Thus,
fluorescent radiation can originate at greater distances from the point of impact of the
electron beam than primary radiation. (Note Figure 6.18, Chapter 6.) Hence, the mean
depth of production of fluorescent radiation is greater than that of primary radiation.
Therefore, the intensity of fluorescent emission that can be measured by the x-ray
detector relative to that of primary emission increases with increasing x-ray emergence
angle.
Since x-ray fluorescence always adds intensity for element i, an equation of the
following form can be used:
*
f
f
I
I
Fi = 1 +  ij
1 +  ij
Ii
Ii
j
j
(E9.8-1)
/
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f
The correction factor Iij /Ij relates the intensity of radiation of element i produced by
f
fluorescence by element j, Iij , to the electron-generated intensity of radiation from
element i, Ii. The total correction factor is the summation of the fluorescence of element i
f
by all the elements j in the sample. The most popular version of the correction factor Iij /Ij
was derived by Castaing (1951) and modified by Reed (1965). For element i fluoresced
by element j in a specimen containing these or additional elements, we have
Iijf /Ij = Cj Y 0Y 1Y 2Y 3Pij
(E9.8-2)
Cj is the concentration of the element causing the parasitic fluorescence, i.e., the
fluorescer; Y0 is given by
Y 0 = 0.5 ri r- 1 j A i
i
Aj
where ri is the absorption edge jump ratio for element i--for a K line (ri - 1)/ri is 0.88
and for an L line (ri - 1)/ri is 0.75 although each term has a small atomic number
dependence (Armstrong, 1988); j is the fluorescent yield for element j (see
Enhancements, Data Base); Ai is the atomic weight of the element of interest and Aj is
the atomic weight of the element causing the parasitic fluorescence; Y1 = [(Uj- 1)/(Ui1)]1.67 where U = E0/Ec . The term Y2 = (/ji)/(/)jspec where (/)ji is the mass
absorption coefficient of element i for radiation from element j, and (/)jspec is the
mass absorption coefficient of the specimen for radiation from element j. The values of
, A, and (/) are given in the Enhancements data base. The other term Y3 accounts for
absorption:
Y3 =
ln(1+ u) ln(1+ v)
+
u
v
with
i /(/) j
u = (/)spec
spec csc
where (/)ispec is the mass absorption coefficient of the specimen for radiation from
element i, Equation (E9.6-11) and
v=
401253795
E01.65
3.3 x 10 5
j
- Ec1.65 (/)spec
23
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where Ec is evaluated for element i. Finally Pij is a factor for the type of fluorescence
occurring. If KK (a K line fluoresces a K line) or LL fluorescence occurs, Pij = l. If LK
or KL fluorescence occurs, Pij = 4.76 for LK and 0.24 for KL. In the Y3 factor, an
exponential function was used to represent the z) curve, assuming a point source at
the surface.
The Reed relation is used in most computer-based schemes for correction.
Heinrich and Yakowitz (1968) tested the Reed model for its response to input parameter
uncertainties. They found that j produces the worst uncertainties; the other variables
f
produce negligible errors. The extension of Iij /Ij ,to fluorescence of element i by more
f
than one element j is given by Equation (E9.8-1). The term Iij /Ij , is calculated by
Equation (E9.8-2) for each element j which fluoresces element i. The effects of all these
elements are summed as shown in Equation (E9.8-1). If the standard is a pure element or
element i is not fluoresced by other elements present in a multicomponent standard,
Equation (E9.8-1) for Fi can be written in the more standard form:
*
f
I
Fi = 1 / 1 +  ij
Ii
j
(E9.8-3)
The fluorescence factor Fi is usually the least important factor in the ZAF
correction, since secondary fluorescence may not occur or the concentration Cj in
Equation (E9.8-2) may be small.
The significance of the fluorescence correction Fi can be illustrated by
considering the binary system Fe—Ni. In this system, the Ni K characteristic energy,
7.478 keV, is greater than the energy for excitation of Fe K radiation, Ec = 7.11 keV.
Therefore an additional amount of Fe K radiation is produced. In this system the atomic
number correction Zi is < 1% and can be ignored. Calculations of FFe in a 10 wt% Fe90 wt% Ni alloy were made using the expression given in Equation (E9.8-3).
Table E9.8-1 contains the input data used for the Fi calculation. Calculations of
FFe were performed for two operating voltages, 30 and 15 keV, and two take-off angles,
15.5° and 52.5°.
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Table E9.8-1 Fluorescence of Fe K in a l0 wt% Fe-90 wt%Ni Alloy
______________________________________________________
(a) Input data for Fe-Ni binary
____________________________________________________
Fe
Ni
____________________________________________________
/ Fe K (cm2/g) absorber
/ Ni K (cm2/g) absorber

A
Ec (keV)
C(wt fraction)
71.4
379.6
---55.847
7.11
0.1
90
58.9
0.37
58.71
8.332
0.9
(b) Output data for l0 wt %Fe-90 wt %Ni binary
_____________________________________________________
 E0
f
IFe-Ni
IFe
(deg) (keV)
FFe
AFe
ANi
_____________________________________________________
52.5 15
0.263
0.792
1.002
15.5 15
0.168
0.856
1.008
52.5 30
0.346
0.743
1.011
15.5 30
0.271
0.787
1.030
______________________________________________
1.005
1.015
1.023
1.065
f
The amount of Fe fluoresence given by IFe-Ni /IFe , is listed in Table E9.8-1 and ranges
from 16.8% to 34.6%. It increases with increasing take-off angle and operating voltage.
To minimize the amount of the fluorescence correction, low kilovoltage operation is
suggested. Low-kilovoltage operation will also minimize the absorption correction AFe
and ANi (Table E9.8-1). Although the fluorescence correction is minimized at low 
angles, the error associated with the term Fi does not increase with the take-off angle
(Heinrich and Yakowitz, 1968). Therefore low E0 , high  operation, which is
recommended to minimize Ai , is satisfactory even in cases requiring large fluorescence
corrections.
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If the concentration of Cj , equation (E9.8-2), decreases the amount of
f
fluorescence, Iij /Ii , also decreases. For example if the Ni content changes in a binary FeNi alloy from 90 wt % to 50 wt %, a 50 wt % Fe-50 wt % Ni alloy, the amount of
fluorescence at E0 = 15 keV,  = 15.5° decreases by over a factor of 2 from 16.8% to
6.5%. Clearly the relative effect of fluorescence increases markedly as Ci decreases and
Cj increases. Figure 9.9 in Chapter 9 gives experimental data on the effect of
fluorescence at E0 = 30 keV and = 52.50.
Reed (1990) reviewed the approximations used in the derivation of the
characteristic fluorescence correction as well as the constants used in the calculations.
More recent data for fluorescence yields and absorption edge ratios are preferred. In
addition, it is suggested to replace the (U - 1)1.67 term in the equation for Y1 by (UlnU U + 1). The effect of these changes is generally small when the correction is large for
high atomic numbers (Zi > 20). For low atomic numbers (Zi < 20) where the absolute
size of the correction is small, the relative difference is greater and is caused mainly by
using more recent values of the fluorescence yield.
E 9.9 Calculation of ZAF Corrections
For the traditional ZAF formulation, calculations of Z, A, and F are obtained from
fundamental equations describing the physical phenomena occurring in the sample
(Sections E9.3, E9.6, and E9.8 for Z, A, and F respectively). The three correction factors
are multiplied by each other as given in Equation 9.4, Chapter 9.
For the ZAF formulation based on the z) approach, the combination of Zi and
Ai is given by the equation

i (z)exp - /)i csc(z) dz
Zi Ai =
0

*
i (z)exp - /)i csc(z) dz
0
(E9.9-1)
and these terms are often considered together in the correction procedure.
Combining the characteristic fluorescence correction, Fi, as given in Section E9.8,
we have the complete correction, ZAFi, which can be used along with the measured
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intensity ratio, ki, to calculate the concentration of element i, Ci (See Equation 9.4,
Chapter 9).
Equation E9.9-1 gives the formulation of the atomic number and
absorption correction ZiAi. The absorption correction, equation E9.7-1, is obtained
by dividing the ZiAi correction by the atomic number correction, Zi, as discussed in
the previous section. The Packwood - Brown equation for z), equation E9.4-2, can
be integrated for specimen and standard in closed form as indicated by Brown and
Packwood (1982). According to Bastin et al (1984), the combined ZiAi correction is
[  R() - ( (0)) R(( ) / 2
ZiAi = -------------------------------------------------[  R() - ( (0)) R(( ) / 2*
(E9.9-2)
where R represents the value of the fifth-order polynomial in the approximation for the
error function which comes in when the integrals are solved in closed form. The term 
is u )i csc  as defined in Section E9.6 for the absorption correction.
E9.10 Detector Efficiency
As discussed in Chapter 7, the radiation emitted from the target toward the
detector penetrates several layers of "window" material before it arrives in the "active"
part of the detector. The nominal purpose of the first "window" is to protect the cooled
detector chip from the relatively poor vacuum in the specimen chamber.
The first window is made from a variety of materials. Historically, a beryllium
window, typically about 7.6 m thick (0.3 mils) has been used. During the last several
years window materials with considerably less mass-thickness have been gaining wide
popularity. These window materials are either boron or silicon nitride, diamond, or are
organic.
The second "window" material is a surface-barrier contact (about 20 nm thick and
usually made of gold). The purpose of this window is to provide an electrical contact to
the diode. The gold is not uniform in thickness and tends to form islands.
The third "window" is an inactive layer of silicon extending 200 nm or less into
the detector. This layer is also not uniform and is often called the "dead layer" or "silicon
dead zone". The radiation then enters the active (intrinsic) region of the detector which
has a thickness typically between 2 and 5 mm.
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As an example of how we would "model" the transmission through these various
"windows" we will provide an expression for the "traditional" beryllium window
detector. The absorption losses in the beryllium window, the gold, the silicon dead layer,
and the transmission through the active silicon zone can be calculated from the linear
combination of Beer's Law applied to each material separately:
E
E
  E
 
  E  
 
 
D
PE = exp-   tBe +   tAu +   tSi  1-exp-   tSi  
 Be
 Si 
Au
Si 


(E9.10-1)
In equation (E9.10-1), tSiD and tSi are the thicknesses (g/cm2) of the silicon dead
layer and of the active detector region, respectively. The mass attenuation coefficients of
beryllium and silicon at the energy E, ()BeE and ()AuE and ()SiE are calculated
as described by Myklebust et al. (1979). Since sufficiently accurate values are not usually
available from the manufacturer, estimates of the thicknesses can be adjusted to optimize
the fit between the calculated and the experimental.
E 9.11 Sample Homogeneity
A more exacting determination of the range (wt%) and level (%) of
homogeneity involves the use of (1) the standard deviation Sc of the measured values
and (2) the degree of statistical confidence in the determination of N . The standard
deviation includes effects arising from the variability of the experiment, e.g.,
instrument drift, x-ray focusing errors, and x-ray production. The degree of
confidence used in the measurement states that we wish to avoid a risk, , of rejecting
a good result a large percentage (say 95% or 99%) of the time. The degree of
confidence is given as l- and is usually chosen as 0.95 or 0.99, that is 95% or 99%.
The use of a degree of confidence means that we can define a range of homogeneity,
in wt%, for which we expect, on the average, only 5% or 1% of the repeated random
points to be outside this range.The range of homogeneity in wt% for a degree of
confidence l- is
W1- =  C
t1-
n-1 Sc
n1/2 N
(E9.11-1)
where C is the true weight fraction of the element of interest, n is the number
of measurements, N is the average number of counts accumulated at each
1-
measurement and tn-1 is the Student t value for a l- confidence level and for n-l
95
99
degrees of freedom. Student's t values for tn-1 and tn-1 for various degrees of freedom,
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n-1 are given in Table E9.11-1 (Bauer, 1971). At least four individual measurements,
n = 4, should be made to establish the range of homogeneity. If fewer than four
measurements are made, the value of W1-will be too large.
The level of homogeneity, or homogeneity level, for a given confidence level,
l-, in percent is given by
 W1- = 
C
t1-
n-1 Sc (100) (%)
n1/2 N
(E9.11-2)
It is more difficult to measure the same level of homogeneity as the
concentration, present in the sample, decreases. Although W1- is directly
proportional to C, the value of Sc/N will increase as C and the number of x-ray counts
per point decreases. To obtain the same number of x-ray counts per point, the time of
the analysis must be increased.
More sophisticated studies of homogeneity can be performed by making
several measurements on each of the n analysis points on the sample. This type of
analysis includes the difference between analysis points and the error due to counting
statistics. In addition, differences in homogeneity between individual samples can be
considered using a statistical analysis. The appropriate analysis procedures for these
studies are given by Marinenko et al. (1979, 1981).
Table E9.11-1 Values of Student t Distribution for 95% and 99% Degrees of
Confidencea
____________________________________________
n
n-1
t95
n-1
t99
n-1
3
2
4.304
9.92
4
3
3.182
5.841
8
7
2.365
3.499
12 11
2.201
3.106
16 15
2.131
2.947
30 29
2.042
2.750
∞ ∞
1.960
2.576
____________________________________________
aBauer (1971).
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E 9.12 Analytical Sensitivity
Analytical sensitivity is the ability to distinguish, for a given element, between
two concentrations C and C' that are nearly equal. X-ray counts N and N' for both
concentrations therefore have a similar statistical variation. If one determines two
concentrations C and C' by n repetitions of each measurement, taken for the same
fixed time interval, then these two values are significantly different at a certain
degree of confidence, l-, if
1/2
N - N  21/2 (t1-
n-1 )Sc/n
(E9.12-1)
and
C = C-C 
21/2 C(t1-
n-1 )Sc
n1/2 (N - NB)
(E9.12-2)
in which C is the concentration of one element in the sample, N and NB are the
average number of x-ray counts of the element of interest for the sample and the
1-
continuum background on the sample, respectively, tn-1 is the “Student factor”
dependent on the confidence level l- (Table E9.11-1), and n is the number of
measurements. Ziebold (1967) has shown that the analytical sensitivity for a 95%
degree of confidence can be approximated by
C = C-C  2.33 Cc
n1/2 (N - NB)
(E9.11-3)
The above equation represents an estimate of the maximum sensitivity that
can be achieved when signals from both concentrations have their own errors but
instrumental errors are disregarded. Since the actual standard deviation SC is
usually about two times larger than C, C is in practice approximately twice that
given in Equation (E9.11-3).
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E 9.13 Analytical Sensitivity, Trace Element Analysis
By analogy with Equation (E9.12-1) we can also define the detectability limit
DL as (N - N B )DL for trace analysis as
1/2
(N - NB)DL  21/2 (t1-
n-1 )Sc/n
(E9.13-1)
where Sc is essentially the same for both the sample and background
measurement. In this case we can define the detectability limit at a confidence
level l- (Table E9.11-1) that the analyst chooses. The 95% or 99% confidence
level is usually chosen in practice.
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Chapter 9, Enhancements - References
Armstrong, J. T. (1988). In Microbeam Analysis – 1988 (D. E. Newbury, ed.) (San
FranciscoPress, San Francisco), p. 469.
Bastin, G. F. and H. J. M. Heijligers (1990). In Proc. 12th Intl. Cong. Electron. Micros.
(L. Peachey and D. B. Williams, eds.) (San Francisco Press, San Francisco),11, 216.
Bastin, G. F. and H. J. M. Heijligers (1991a). In Electron Probe Quantitation (K. F. J.
Heinrich and D. E. Newbury, eds.) (Plenum Press, New York), 163
Bastin, G. F. and H. J. M. Heijligers (1991b). In Electron Probe Quantitation (K. F. J.
Heinrich and D. E. Newbury, eds.) (Plenum Press, New York), 145
Bastin, G. f., and H. J. M. Heijligers (1986). X-ray Spectrom. 15, 143.
Bastin, G. F., F. J. J. van Loo and H. J. M. Heijligers (1984). X-ray Spectrom. 13, 91.
Bauer, E. L. (1971). A Statistical Manual for Chemists, 2nd ed. (Academic Press, New
York), p. 189.
Berger, M. J., and S. M. seltzer (1964). Nat. Acad. Sci./Nat. Res. Council Publ. 1133,
Washington, 205
.
Bethe, H. (1930), ann. Phys. (Leipzig), 5, 325.
Brown, J. D., and L. Parobek (1972) In Proc. 6th Int. Canf. X-ray Optics and
Microanalysis (G. Shinoda, K. Kohra and T. Ichinokawa, eds.) (Univ. of Tokyo Press,
Tokyo), p. 163.
Brown, J. D., and R. H. Packwood (1982). X-Ray Spectrom. 11, 187.
Castaing, R. (1951) Ph.D. thesis, University of Paris
Duncumb, P., and P. K. Shields (1966) In The Electron Microprobe (T. D. McKinley, K.
F. J. Heinrich, and D. B. Wittry, eds.) (Wiley, New York), p. 284.
Duncumb, P., and S. J. B. Reed (1968). In Quantitative Electron Probe Microanalysis (K.
F. J. Heinrich, ed.) National Bureau of Standards Special Publication 298, p. 133.
Fiori, C. E., R. L. Myklebust, K. F. J. Heinrich, and H. Yakowitz (1976). Anal. Chem.
48, 172.
Green, M. (1963) In Proc. 3rd Intl. Conf. On X-ray Optics and Microanalysis (H. A.
Patee, V. E. Cosslett, and a. Engstrom, eds. ) (Academic Press, New York), p. 361.
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Heinrich, K. F. J. (1969) National Bureau of Standards Technical Note 521.
Heinrich, K. f. J., and H. Yakowitz (1968). Mikrochim. Acta 5, 905.
Heinrich, K. F. J., and H. Yakowitz (1970). Mikrochim. Acta 7, 123.
Kyser, D. F. (1972). In Proc. 6th Int. Canf. X-ray Optics and Microanalysis (G. Shinoda,
K. Kohra and T. Ichinokawa, eds.) (Univ. of Tokyo Press, Tokyo), p. 147
Lifshin, E. (1974). In Proc. 9th Ann. Conf. Microbeam Analysis Soc., Ottawa, Canada, p.
53.
Marienenko, R. B., K. F. J. Heinrich, and F. C. Ruegg (1979). “Microhomogeneity
Studies of NBS Standard Reference Materials, NBS Research Materials and Other
Related Samples.” NBS Special Publication 260-265.
Marienenko, R. B., F. Biancaniello, L. DeRoberts, P. A. Boyer, and A. W. Ruff (1981).
“Preparation and Characterization of an Iron-Chromium-Nickel Alloy for Microanalysis:
SRM 479a.” NBS Special Publication 260-270.
Myklebust, R. L., C. E. Fiori, and K. F. J. Heinrich (1979). National Bureau of Standards
Tech. Note 1106.
Packwood, R. H., and J. D. Brown (1981). X-ray Spectrom. 10, 138.
Parobek, L., and J. D. Brown (1978). X-Ray spectrom. 7, 26..
Philibert, J. (1963). In Proc. 34th Intl. Symp. X-ray Optics and X-Ray Microanalysis,
Stanford University (H. H. Pattee, V. E. Cosslett, and A. Engstrom, eds.) (Academic
Press, New York), p. 379.
Pouchou, J. L., and R. Pichoir (1984). Rech. Aerosp. 3, 13.
Pouchou, J. L., and R. Pichoir (1987). In Proc. 11th Int. cong. On X-ray Optics and
Microanalysis (J. D. Brown and R. H. Packwood, eds.) (Univ. of Western Ontario Press),
p. 249.
Pouchou, J. L., and R. Pichoir (1991). In Electron Probe Quantitation (K. F. J. Heinrich
and D. E. Newbury, eds.) (Plenum Press, New York), p. 31.
Reed, S. J. B. (1965), Br. J. Appl. Phys. 16, 913.
Reed, S. J. B. (1990). Microbeam Analysis –1988 (San Francisco Press, San Francisco),
p. 109.
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Schamber, F. H. (1977). In X-Ray Fluorescence analysis of Environmental Samples (T.
G. Dzubay, ed.) (Ann Arbor Science Publishers, Ann Arbor, Michigan), p. 241.
Schamber, F. H. (1978). In Proc. 13th Nat. Conf. Microbeam Analysis Society, Ann
Arbor, Michigan, p. 50.
Springer, G. (1966), Mikrochim. Acta 3, 587.
Thomas, P. M. (1964). U. K. Atomic Energy Auth. Rept. AERE-R 4593.
Ware, N. G., and S. J. B. Reed (1973). J. Phys. E. 6, 286.
Wittry, D. B. (1957). Ph. D. Dissertation, California Institute of Technology
Yakowitz, H., and K. F. J. Heinrich (1968). Mikrochim. Acta 5, 183.
Yakowitz, H., R. L. Myklebust, and K. F. J. Heinrich (1973). National Bureau of
Standards Tech. Note 796.
Ziebold, T. O. (1967). Anal. Chem. 39, 858.
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Figure Titles
Figure E9.2-1 Effect of “top hat” digital filter on a spectrum comprised of a Gaussian
peak plus a sloped linerar background. The filtered spectrum is plotted immediately
below the actual spectrum. The channel correspondence for one calculation of the top hat
filter is shown.
Figure E9.3-1 Fraction of ionization R remaining in a specimen of atomic number Z after
loss of beam electrons due to backscatter (Duncumb and Reed, 1968).
Figure E9.4-1 Schematic definition of z) curve. (a) X-ray generation in a solid
sample. (b) X-ray generation in a thin foil.
Figure E9.4-2 Initial measurement of intensity versus mass-depth to derive a z)
curve for a solid.
Figure E9.4-3 Experimental setup for the measurement of z) curves by the tracer
technique. (a) Tracer. (b) Isolated thin film.
Figure E9.4-4 Measured Cu K z) curve using a Zn K tracer for 25 keV (Brown
and Parobek, 1972).
Figure E9.4-5 Plot of ln z) vs (z)2 showing that the z) curves are Gaussian in
character (Packwood and Brown, 1981).
Figure E9.4-6 Example of a z) curve showing the functional behavior of the four
Gaussian parameters, , and (0) in the Packwood-Brown z) equation.
Figure E9.7-1 Calculated zgenerated and emitted, versus z curves for Al K and
Cu K radiation in Al, Cu, and Al-3 wt% Cu at 15 keV. The take-off angle is 400.
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