Generation of X-Rays in the SEM Specimen

The electron beam generates x-ray photons in the beam-specimen intearac­tion volume beneath the specimen surface. X-ray photons emerging from the specimen have energies specific to the elements in the specimen; these are the characteristic x-rays that provide the SEM’s analytical capabilities (see Fig. 6.1). Other photons have no relationship to specimen elements and constitute the continuum background of the spectrum. The x-rays we analyze in the SEM usually have energies between 0.1 and 20 keV. Our task in this chapter is to understand the physical basis for the features in an x-ray spectrum like that shown in Fig. 6.1.

6.1. Continuum X-Ray Production (Bremsstrahlung)

Beam electrons can undergo deceleration in the Coulombic field of he specimen atoms, which is the positive field of the nucleus modified by the negative field of the bound electrons, as shown in Fig. 6.2. The loss in electron energy AE that occurs in such a deceleration event is emitted as a photon. The energy of this photon is AE = hv, where h is Planck’s constant and v is the frequency of the electromagnetic radiation. This radiation is referred to as bremsstrahlung, or “braking radiation.” Because the interactions are random, the electron may lose any amount of energy in a single deceleration event. Therefore, the bremsstrahlung can take on any energy value from zero up to the original energy of the incident electron E0, forming a continuous electromagnetic spectrum.

A calculated x-ray spectrum, without absorption and peak broadening effects, is shown in Fig. 6.3. The x-ray spectrum consists of both the continuous and characteristic components that would be generated by an electron beam within the interaction volume of a solid copper specimen. Note that the bremsstrahlung continuum is an electron-generated radiation. (X-ray spectra from a radioactive source or from an x-ray fluorescent source will.

Energy (keV)

Figure 6.1. X-ray spectrum of copper showing K-series and L-series x-ray peaks and the continuous x-ray spectrum (bremsstrahlung or continuum) obtained with a Si(Li) EDS detector with an ultrathin (diamond) x-ray window. Natural widths of peaks are much narrower than measured here. A noise peak is measured at very low energies.

In describing x-rays, we make use of both the energy of the x-ray E (keV) and its associated wavelength X (nm), which are related by the following expression:

where E is x-ray photon energy in keV.

The maximum x-ray energy observed in the spectrum corresponds to beam electrons that have lost all of their incident energy in a single event (Fig. 6.2). Since x-ray wavelength is inversely proportional to en­ergy, the most energetic x-rays will have have a minimum wavelength λ_min or λ_SWL called the short-wavelength limit or Duane-Hunt limit (Duane and Hunt, 1915), which can be related to the incident energy E₀ through Eq. (6.1). Measurement of the Duane-Hunt limit can be accomplished with any calibrated x-ray spectrometer and provides an unambiguous measure of the true electron beam energy as the electrons actually reach the specimen. Figure 6.4 shows an EDS x-ray spectrum of a carbon specimen at electron beam energies of 10,15, and 20 keV. The Duane-Hunt limit is the energy where the extrapolated x-ray background goes to zero.

Figure 6.2. Schematic illustration of the origin of the x-ray continuum which results from the deceleration of beam electrons in the Coulombic field of the atoms.

Figure 6.3. Calculated x-ray spectrum “as generated" in a copper target by a 20-kcV electron beam. The continuum background (heavy line) and the Cu Ka, Cu Kß, and Cu La characteristic x-ray lines are shown. Compare the background intensity with that of Fig. 6.1 and note that most of the low-energy coninuum is absorbed in the specimen or the x-ray window when the spectrum is measured.

The intensity of the x-ray continuum I_cm at any energy or wavelength has been quantified by Kramers (1923) as

where i_p is the electron probe current, Z is the average atomic number based upon mass (weight) fractions of the elemental constituents of the specimen, E₀ is the incident beam energy, and E_v is the continuum photon energy at some point in the spectrum. At low photon energies, the intensity increases rapidly because of the greater probability for slight deviations in

0 5 10 15 20

Energy (keV)

Figure 6.4. Duane-Hunt limit for simulated spectra from a carbon specimen at beam energies of 10, 15, and 20keV. The Duane-Hunt limit is the energy where the continuum background intensity goes to zero.

trajectory caused by the Coulombic field of the atoms. The intensity of the continuum increases with increasing beam current, atomic number of the specimen, and beam energy. The intensity of the continuum increases with the atomic number because of the increased Coulombic field of the nucleus (more charge).

The intensity of the continuum radiation is important in analytical x-ray spectrometry because it forms a background under the characteristic peaks. Once a photon is created with a specific energy, it is impossible to determine whether it is a continuum or a characteristic x-ray. Thus, the background intensity due to the continuum process, occurring at the same energy as a characteristic x-ray, sets a limit to the minimum amount of an element that can be detected. The continuum is therefore usually regarded as a nuisance to the analyst. However, it should be noted from Eq. (6.2) that the continuum carries information about the average atomic number in the specimen and hence the overall composition. Thus regions of different average atomic number in a specimen will emit different amounts of conti­nuum intensity at all x-ray energies. This fact can prove useful in measuring background with wavelength-dispersive x-ray spectrometers (WDS) and forms the basis for an important correction scheme (Marshall-Hall method, peak-to-background method) for quantitative analysis of particles, rough surfaces, and biological specimens. The atomic number dependence of the bremsstrahlung also causes an important artifact in x-ray mapping of minor constituents, which can lead to serious misinterpretation if not recognized and corrected.

6.2. Characteristic X-Ray Production

1. Origin

A beam electron can interact with the tightly bound inner shell elec­trons of a specimen atom, ejecting an electron from a shell. The atom is left as an ion in an excited, energetic state, as shown in Fig. 6.5. The in­cident beam electron leaves the atom having lost at least Ek, where Ek is the binding energy of the electron to the K shell. The ejected orbital electron leaves the atom with a kinetic energy of a few eV to several keV, depending on the interaction. The atom itself is left in an excited state with a missing inner shell electron. The atom relaxes to its ground state (lowest energy) within approximately 1 ps through a limited set of allowed transi­tions of outer shell electron(s) filling the inner-shell vacancy. The energies of electrons in the shells (atomic energy levels) are sharply defined with values characteristic of a specific element. The energy difference between electron shells is a specific or characteristic value for each element. The excess energy can be released from the atom during relaxation in one of two ways (two branches in Fig. 6.5). In the Auger process, the difference in shell energies can be transmitted to another outer shell electron, eject­ing it from the atom as an electron with a specific kinetic energy. In the characteristic x-ray process, the difference in energy is expressed as a pho­ton of electromagnetic radiation which has a sharply defined energy. For the case of the neon atom shown schematically in Fig. 6.5, creation of a K-series x-ray involves filling the vacant state in the innermost electron shell (K shell) with an electon transition from the next shell out (L shell). Thus, the energy of the Ka x-ray produced is equal to the difference in energy between the K shell and the L shell. The actual situation is more complicated than this because the L shell and the M shell are split into subshells, as we will see shortly. A comprehensive treatment of the properties of characteristic x-rays is beyond the scope of this book, and the interested reader is referred to the literature (e.g., Bertin, 1975). However, certain basic concepts that are fundamental to x-ray microanalysis will be discussed in this chapter.

Figure 6.5. Inner shell electron ionization in an atom and subsequent de-excitation by election transitions. The incident electron is elastically scattered The unscattcred direction of the incident electron is shown by the dotted line. The difference in energy from an electron transition is expressed either as the ejection of an energetic electron with characteristic energy (Auger process) or by the emission of a characteristic x-ray photon.

6.2.2. Fluorescence Yield

The partitioning of the de-excitation process between the x-ray and Auger branches (Fig. 6.5) is described by the fluorescence yield w. For the production of K radiation the fluorescence yield is

The x-ray process is not favored for low atomic numbers; for example, wk ~ 0.005 for the carbon K shell (Fig. 6.6). The characteristic x-ray process dominates for high atomic numbers; for example, wk=0.5 for germanium, increasing to near unity for the heaviest elements. The fluo­rescence yields of the L and M shells are also shown in Fig. 6.6.

Figure 6.6. Fluorescence yield w as a function of atomic number for electron ionization within the K, L, and M electron shells

6.2.3. Electron Shells

Electrons of an atom occupy electron shells around the atom that have specific energies. In order of increasing distance from the atomic nucleus, these shells are designated the K shell, the L shell, the M shell, etc. (Fig. 6.5). These shells are directly related to the quantum numbers of atomic physics. For shells beyond the K shell, the shells are divided into subshells. For example, the L shell is composed of three subshells that are closely spaced in energy, and the M shell has five subshells. Elec­trons populate these subshells in a definite scheme as shown in Table 6.1.

Table 6. 1. Electron Shells and Subshells of Atoms

More detailed information can be found in the Enhancements CD, Chapter 6.

4. Energy-Level Diagram

Figure 6.5 schematically shows electrons in orbits around the nucleus; an alternative way of representing the energies of specific electrons is with an energy-level diagram such as Fig. 6.7. Such diagrams originate from the more accurate quantum mechanical picture of an atom that results from the solution of the Schrodinger wave equation. Here the energy of an atom is represented by the energies of the various vacant states (electrons removed) that can be created by the electron beam. For example, ionization of an electron in the K shell increases the energy of the atom to that of the “K level.” If an L-shell electron moves into the K shell to fill the vacancy, the energy of the atom decreases to the "L level,” but there is still now a vacant state in the L shell, which would be filled by an electron of some lower energy, and so on.

4. Electron Transitions

Characteristic x-ray lines result from transitions between subshells; however, atomic theory tells us that only transitions between certain sub- shells are allowed (see Table 6.1). This means that not all transitions between

Figure 6.7. Energy level diagram for an atom. The energy of the atom increases upon ioniza- tion of the K, L, M, or N shell (excitation). As the atom’s energy returns to normal, Ka, KB, La, and Ma x-rays are emitted from the atom. Each horizontal line represents the energy of an electron state. Zero energy represents an atom at rest with no electrons missing (normal).

Figure 6.8. Critical excitation energies (absorption edge energies) for the K, L, and M series as a function of atomic number.

shells result in characteristic x-rays. Returning to the example of the copper K series, election transitions (note Table 6.1) to fill a vacant K state from the L shell in the Cu atom are as follows:

6.2.6. Critical Ionization Energy

Because the energy of each shell and subshell is sharply defined, the minimum energy necessary to remove an election from a specific shell has a sharply defined value as well. This energy is called the critical ionization or excitation energy E_c, also known as the excitation potential or x-ray ab­sorption edge energy for example, E_K and E_Lm. Each shell and subshell of an atom requires a different critical ionization energy for electron removal. Figure 6.8 shows the critical excitation energies for the K, L, and M series as a function of atomic number. As an example, consider the wide range in critical ionization energies for the K, L, and M shells and subshells of platinum (Z = 78) listed in Table 6.2. A 20-keV electron beam can ion­ize the L and M shells of Pt, but not the K shell. As the atomic number decreases, as shown in Table 6.2 for Nb (Z = 41) and Si (Z = 14), the ionization energies decrease. The critical ionization energy is an important parameter in calculating characteristic x-ray intensities. As discussed in following chapters, for x-ray microanalysis we typically operate the SEM at energies two to three times the critical ionization energies of the elements of interest. Tabulations of the critical ionization energy and characteristic

x-ray energies for the K, L, and M shells are available in the tables (Bearden, 1967a,b) given in the accompanying CD.

7. Moseley’s Law

X-rays emitted during an electron transition are called characteristic x-rays because their specific energies and wavelengths are characteristic of the particular element which is excited. The energies of the electron shells vary in a discrete fashion with atomic number, so that the x-rays emitted in the process have energies characteristic of that atomic number. The difference in energy between shells changes in a regular step when the atomic number changes by one unit This fact was discovered by Moseley (1913,1914) and can be expressed in the form of an equation:

where E is the energy of the x-ray line, and A and C are constants which differ for each x-ray series (C = 1.13 for the K series and approximately 7 for the L series). Thus, Eq. (6.4) may be used to find the energy of any element K or L line. Moseley's law forms the basis for qualitative x-ray analysis, that is, the identification of elemental constituents.

7. Families of Characteristic Lines

For elements with atomic numbers > 11 (sodium), the shell structure is sufficiently complex that when an ionization occurs in the K shell, the transition to fill that vacant state can occur from more than one outer shell. As shown in Fig. 6.7, following ionization of a K -shell electron, a transition to fill the vacant state can occur from either the L shell or the M shell. X-rays resulting from transitions of electrons from the M shell to the K shell are designated KB x-rays. Because the energy difference between the K and M shells is larger than between the K and L shells, the KB x-ray energy is larger than that of the Ka. For example, for copper, the Ka x-ray has an energy of 8.04 keV and the KB x-ray has an energy of 8.90 keV.

Primary ionization involves removing an electron from a bound state in a shell to an effective infinity outside the atom. However, characteristic x-rays are formed by transitions between bound states in shells. Therefore, the energy of the characteristic x-ray is always less than the critical ionization energy for the shell from which the original electron was removed. Thus, E_ka = E_k-E_L and E_KB = E_K- E_M; therefore, E_Ka < E_KB < E_K.

As noted in Table 6.1 and Fig. 6.9, the L, M, and N shells may be further divided into subshells of slightly different energy, giving rise to

figure 6.9. Electron transitions in the atom that give rise to M-series, L-series, and K-series x-rays. By reference to a table atomic energy levels, the x-ray energy of each photon shown can be calculated as the difference between the associated energy 'els (Woldseth, 1973).

additional x-ray lines. For copper, various transitions into the K shell produce Ka1 (8.048 keV, a transition from the L_III subshell) and Ka2 (8.028 keV a transition from the L_II subshell) and several KB lines from the M and N shells: KB_I (8.905 keV, from M_III), KB₂ (8.976 keV, from N_II, _III), and KB₅ (8.977 keV, from M_IV._V) (see Fig. 6.9). Not all observable x-ray lines have been assigned Greek letter designations. A number of x- ray lines with low intensity relative to the principal lines are designated in an alternative style which is used in atomic physics. In this scheme, the first letter denotes the shell in which the ionization initially takes place, with the subscript indicating the specific subshell, and the second letter and number indicates the shell and subshell from which the inner shell vacant state is filled. Thus, the weak M_IIN_IV line arises from an ionization in the M_II subshell followed by an electron transition from the N_IV subshell.

Within the SEM beam energy range, each element can emit x-rays of the K series only (light elements), the K series plus the L series (in­termediate elements), or the L series plus the M series (heavy elements). Figure 6.9 shows the electron transitions responsible for essentially all the possible x-ray lines. Because of the relatively low resolution of the energy- dispersive spectrometer (EDS), only some of the possible x-ray lines are observed (see chapter 8, Fig. 8.1). Many more of the x-ray lines shown in Fig. 6.9 are observed with the wavelength-dispersive spectrometer (WDS). As the atomic number of the atom increases above 10, the K-shell x-rays split into Ka and KB pairs. Above atomic number 21, an additional family, the L shell, begins to become measurable at 0.2 keV When atomic number 50 is reached, the M-shell family begins to appear above 0.2 keV The com­plexity of the shells is extended further by the splitting of lines due to the energy divisions of subshells. Figure 6.10 shows elemental x-ray spectra at the energy resolution typical of an EDS system for several elements of increasing atomic number. These spectra show that K-, and M-series x-rays increase in energy with increasing atomic number from Zn (Z = 30) to Nb (Z = 41) to La (Z = 57) to Pt (Z = 78). Note that many x-ray lines for the lighter elements are not resolved (e.g.. La, LB, Ly) because of the close proximity in energy of the peaks and the relatively poor resolution of the energy-dispersive x-ray spectrometer.

When a beam electron has sufficient energy to ionize a particular electron shell in an atom to produce characteristic x-ray lines, all other characteristic x-rays of lower energy for the atom will also be excited, provided there are electrons available for such transitions. The production of lower energy x-rays occurs because of (1) direct ionization of those lower energy shells by beam electrons and (2) propagation of the vacant state outward (K to L to M shells) as the atom returns to the ground state.

6.2.9. Natural Width of Characteristic X-Ray Lines

Energy levels are so sharply defined that the energy width of a charac­teristic x-ray is a small fraction of the line energy. For example, elements such as calcium (Z = 20) have an x-ray linewidth at half-height intensity of only 2 eV (Salem and Lee, 1976). This narrow, linelike character of peaks superimposed on the broad, slowly changing x-ray background has led to the frequent use of the term x-ray “lines" in the literature when referring to characteristic x-ray peaks. Note that the width of measured peaks in Fig. 6.1 is governed by the resolution of the EDS x-ray spectrometer used to acquire the spectrum. Typically the peak is about 70 times wider than the natural linewidth.

Figure 6.10. EDS spectra for families of x-ray lines (a) Zinc (Z = 30), (b) niobium (Z =41), (c) lanthanum (Z = 57), and (d) platinum (Z = 78). Note that for zinc the L series is an unresolved single peak, but as the atomic number increases the individual L-series peaks spread out into separate lines.

10. Weights of Lines

Although many possible electron transitions can occur in high-atomic-number elements to fill a vacancy in an inner shell, which in turn gives rise to the families of x-rays (Fig. 6.9), the probability for each type of transition varies considerably. The relative transition probabilities for the lines arising

from an ionization of a specific shell are termed the “weights of lines.” The weights of lines are dependent upon atomic number and vary in a complex fashion, but in general, the greater the energy difference in the electron transition, the less probable and less intense is the x-ray line. Thus, the KB lines arc less intense than the Ka lines. The intrafamily weights of K lines are the best known; for atomic numbers above Z = 18 the Ka to Kfi ratio is about 10:1. General values are presented in Table 6.3 for the lines of significant intensity that can be readily observed in energy-dispersive x-ray spectra. Although these values may not be exact for a specific ele­ment, these weights are a useful guide in interpreting spectra and assigning peak identifications in energy-dispersive x-ray spectrometry. Note that the values in Table 6.3 are weights of lines for one element. It is difficult to compare x-ray line intensities of different elements even for the same spectral series because the intensity depends on a variety of factors includ­ing the fluorescent yield, absorption (see Section 6.4), and the excitation energy.

10. Cross Section for Inner Shell Ionization

Numerous cross sections expressing the probability for inner shell ionization can be found in the literature; these have been reviewed by Powell (1976a, b; 1990). The basic form of the cross section is that derived by Bethe (1930):

where n, is the number of electrons in a shell or subshell (e.g., n_s = 2 for a K shell), bs and c_s are constants for a particular shell, E_c is the critical ionization energy (keV) of the shell, and U is the overvoltage:

where E is the instantaneous beam energy. The cross section for K-shell ionization for silicon is plotted as a function of overvoltage in Fig. 6.11. The cross section increases rapidly from an overvoltage of l to a maximum at about U = 3. As beam electrons lose their energy because of inelastic scattering, they can interact with atomic inner shells down to overvoltages as low as U = 1.

Figure 6.11. Plot of the cross section for inner shell ionization of the silicon K shell as a function of overvoltage U = E/E_t

6.2.12. X-Ray Production in Thin Foils

X-ray production from a thin foil can be estimated using the cross section for inner shell ionization. A “thin” foil has a thickness which is small compared to the elastic mean free path of a beam electron. In a “thin” foil the average beam electron passes through the foil along its incident trajectory without significant deflection. The energy loss due to inelastic scattering is also negligible when the foil is thin. To convert the ionization cross section given in Eq. (6.5), which has dimensions of ionizations/e-/(atom/cm²), to x-ray production per electron n„, in units of photons/e", the following dimensional argument is used:

where w is the fluorescence yield. No is Avogadro's number, A is the atomic weight, p is the density, and I is the thickness.

13. X-Ray Production In Thick Targets

A thick specimen target is one for which elastic scattering, inelastic scattering, and energy loss are significant. A thick specimen has a thickness several times the elastic mean free path, or nominally about 100 nm or more. Specimen targets greater than 10 um in thickness are effectively infinite in thickness to beam electrons at typical energies for the SEM (<30 keV). To accurately calculate x-ray production for thick targets, one must consider that the x-ray cross section varies over the entire range of electron energy from E₀ down to E_e. The calculation of the x-ray intensity under these conditions will be presented in Chapter 9 as part of the physical basis for quantitative x-ray analysis.

A number of workers have reported experimental measurements of the generated characteristic x-ray intensity I_c. This is the intensity produced in the specimen prior to its absorption as the radiation propagates through the specimen (Green, 1963; Lifshin et al.. 1980). The experimental expressions have the general form

where i_p is the electron probe current and a and n are constants for a particular element and shell. The value of n is normally in the range of 1.5-2.0. Thus, for a particular characteristic x-ray line, the line intensity increases as the beam energy increases.

13. X-Ray Peak-to-Background Ratio

The most important factor in determining the limits of detection in x-ray spectrometric analysis is the presence of the continuum background, that is, noncharacteristic radiation at the same energy as the characteristic radiation of interest. By dividing l_c by l_cm [Eq. (6.8) by Eq. (6.2)], we can calculate the peak-to-background ratio and observe the variables that influence the limits of detection:

From Eq. (6.9), the peak-to-background ratio increases as the difference Eq — E_c increases. An increase in P/B allows the analyst to measure a smaller elemental mass fraction in a specimen. It would therefore seem to be advantageous to make Eo as large as possible for a specific excitation energy E_c of an element in the specimen. However, Eq. (6.9) only accounts for generated x-rays. As noted in the discussion of interaction volume in Chapter 3, beam electrons penetrate deeper into the specimen as the beam energy increases. Correspondingly, x-rays are produced deeper into the specimen as the beam energy is increased. Besides degrading the spatial resolution of analysis, another consequence of increasing the beam energy is an increased absorption of x-rays (see Section 6.4) within the specimen before they are measured in the x-ray detector. This absorption reduces the measured x-ray intensity, degrades the limit of detection, and increases the uncertainty of the correction for absorption that must be applied in the quan­titative analysis procedure (Chapter 9). Thus, for a thick, bulk specimen there is an optimum beam energy beyond which further increases in beam energy actually degrade analytical performance. This limit depends on the energy of the characteristic x-rays and the composition of the specimen. An overvoltage U of 2-3 is usually optimum for a given element.

6.3. Depth of X-Ray Production (X-Ray Range)

Depending on the critical excitation energy E_cy characteristic x-rays may be generated over a substantial fraction of the electron interaction volume as shown in Fig. 3.5. To predict the depth of x-ray production (x-rav range) and the lateral x-ray source size (x-ray spatial resolution), the starting point is the electron range, such as given by the Kanaya-Okayama range expression Eq. (3.4). Electron range expressions have the following general form:

where Eo is the incident electron beam energy, p is the density, K depends on material parameters, and n is a constant between 1.2 and 1.7. This formulation of the range considers electrons that lose all their energy while scattering in the specimen. Characteristic x-rays can only be produced within that portion of the electron trajectories for which the energy exceeds £_c for a particular x-ray line, whereas bremsstrahlung x-rays continue to be produced until the electron energy equals zero. The range of direct primary x-rav generation is therefore always smaller than the electron range. To account for the energy limit of x-ray production, the range equation is modified to the form

1. Anderson-Hasler X-Ray Range

By fitting Eq. (6.11) to experimental data, Anderson and Hasler (1966) evaluated the constants K and n, obtaining an analytical expression for the x-ray range useful for most elements:

where R_x has units of /*m when £ is in keV and p is in g/cm³. Figure 6.12 shows A1 Ka and Cu Ka ranges in an A1 specimen and Cu La and Cu Ka in a Cu specimen as a function of beam energy. Note that the x-ray ranges in Al are all less than the electron range.

1. X-Ray Spatial Resolution

For an electron beam normal to a surface, the maximum width of the electron interaction volume or x-ray generation volume projected up to the specimen surface is approximately equal to the spatial resolution. Figure 6.13 shows how the x-ray range and the x-ray spatial resolution L x are defined in cross section. As the atomic number and density of the tziget increase, the depth of production for the principal line decreases. The depth of production is also a function of the critical ionization energy of the line. These effects are also shown in Fig. 6.13. For the same beam energy the figure shows how the x-ray range and the x-ray spatial resolution vary for Al Ka and Cu Ka in a matrix of density ~3 g/cm³ (e.g., aluminum) and in a matrix of density ~10 g/cm³ (e.g., copper). In the low-density sample,

Figure 6.12. X-ray generation range (Anderson-Haslet) for Al Ka and Cu Ka generated in an Al specimen and Cu A'cr and Cu La in a Cu spccimcn as a function of beam energy. Note that the electron range for aluminum Rm is larger than the x-ray range of the x-ray lines generated by the electron beam.

a trace amount of Cu is present in an Al sample, whereas in the high-density sample, a trace element amount of Al is present in a Cu sample. Both Al Ka and Cu Ka are produced at greater depths in the low-density matrix than in the high-density matrix. The shapes of the interaction and x-ray generation volumes differ considerably in the two targets, with the low-density matrix having a pear shape and the high-density matrix giving a less distinct neck.

Figure 0.13. Comparison of x-ray production regions from specimens with densities of 3 g/cm* (left) and 10 g/cm³ (right) at a beam energy of 20keV. The x-ray spatial resolution L» is found by projecting the maximum diameter of the x-ray distribution to the surface of the specimen.

6.3.3. Sampling Volume and Specimen Homogeneity

The range of primary x-ray generation is the critical parameter in es­timating the “sampling volume” for x-ray microanalysis. From Figs. 6.12 and 6.13, it is clear that the sampling volume for x-ray microanalysis de­pends on the beam energy and the energy of the x-ray line measured. When two different analytical lines are available for a given element with widely differing energies (Cu Ka and Cu La in Cu, Fig. 6.12), the depth of the sampling volume may differ by a factor of two or more. If the specimen is heterogeneous in composition over a depth equivalent to the electron range, the effect of differing sampling volumes will often lead to incorrect results. Therefore, a fundamental requirement for accurate results by conventional quantitative x-ray microanalysis is that the specimen must be homogeneous over the electron range.

6.3.4. Depth Distribution of X-Ray Production, fi,(pz)

Although the single numerical value given by the x-ray range defines the effective spatial limit of x-ray production, even a cursory examination of the Monte Carlo electron trajectory/x-ray plots in Fig. 3.5 suggests that the depth distribution of x-ray production is not uniform within the x-ray range. In fact, the distribution of x-ray generation beneath the surface is nonuniform both laterally and in depth. The lateral distribution, as projected up to the surface plane, is important in defining the spatial resolution of x-ray microanalysis. The x-ray distribution in depth is important because the x-rays produced below the surface must propagate through the specimen to escape, and some fraction of these x-rays is lost because of photoelectric absorption (described below). Figure 6.14 shows a Monte Carlo simulation

Figure 6.14. (Right) Monte Carlo simulation of the generation of Al Ka x-rays in pure Al at 30 keV. The electron beam enters the specimen at the surface, 0, and progresses into the specimen as a function of distance Z. (Left) A histogram of the x-ray generation function with depth, fi(pz)- X-ray generation increases to a maximum just below the surface and then decreases as fewer x-rays are generated deeper within the specimen

of the generation of Al Ka x-rays in pure Al at 30 keV Each dot represents a generated x-ray. The histogram on the left of the figure shows the number of x-rays generated at various depths z into the specimen. The x-ray depth distribution function, called fi(pz), is an important tool in developing quan­titative x-ray microanalysis correction procedures (Castaing, 1951) and is discussed in Chapter 9.

6.4. X-Ray Absorption

X-rays, as photons of electromagnetic radiation, can undergo the phenomenon of photoelectric absorption upon interacting with an atom. That is, the photon is absorbed and the energy is completely transferred to an orbital electron, which is ejected with a kinetic energy equal to the pho­ton energy minus the binding energy (critical ionization energy) by which the electron is held to the atom. For x-rays of an incident intensity I₀ in photons per second propagating through a slab of thickness t and density p, the intensity on the exit surface is attenuated according to the expression

where (u/p) is the mass absorption coefficient (cm²/g) of the sample atoms (absorber) for the specific x-ray energy of interest. Fortunately, there is no change in the energy of the x-ray photon that passes through the absorber. For the micrometer dimensions of interest in electron beam x-ray micro- analysis, inelastic scattering of x-rays is insignificant. Typical values of mass absorption coefficients for Cu Ka radiation traveling through vari- ous pure elements are listed in Table 6.4. Note that for Cu Ka radiation the mass absorption coefficient is high for Co as an absorber. The energy of Cu Ka is slightly higher than E_c for Co, but is much lower than E_c for Cu as an absorber. Because the energy of an element's characteristic radiation is always less than E_c, the absorption coefficient of an element for its own ra­diation is low. Therefore, an element passes its own characteristic line with little absorption. Extensive compilations of mass absorption coefficients are available in the literature (Heinrich 1966, 1986). A selection of mass absorption coefficients for Ka and La characteristic lines is tabulated in the accompanying CD.

As an example, one can use Eq. (6.43) to calculate the amount of x-ray absorption in the Be window of an EDS x-ray detector. Suppose fluorine Ka radiation is passing through an 8-ftm thick Be window. The mass absorption coefficient for F Ka radiation in Be is 2039 cm²/g:

Only 5% of the F Ka radiation incident on the detector window passes through. Therefore fluorine is not considered an element that can be prac­tically measured with an EDS detector employing a Be window.

1. Mass Absorption Coefficient for an Element

Photoelectric absorption by electrons in a specific shell requires that the photon energy exceed the electron binding energy for that shell. When the photon energy is slightly greater than the binding energy of the electron in the absorber (£_c), the probability for absorption is highest For a spe­cific absorber, mass absorption coefficients generally decrease in a smooth fashion with increasing x-ray energy given approximately by

However, there is a sharp jump in absorption coefficient in the energy region just greater than the critical excitation energy for each shell of the absorber. Figure 6.15 shows a plot of (n/p) for the element lanthanum (Z = 57) as an absorber of various x-ray energies. Sharp jumps in (ji/p) occur at the energy of the K edge at 38.9 keV the L edges at ~5.9 keV, and the M edges at ~ 1.1 keV These jumps are referred to as “x-ray absorption edges.” X-rays with an energy slightly greater than the x-ray absorption edge (critical ionization energy) can eject a bound electron and therefore are strongly absorbed themselves.

Figure 6.16. Energy-dispersive x-ray spectrum of nickel generated with a primary electron beam energy of 40 koV. The spectrum shows a sharp step in the x-ray continuum background due to increased absorption just above the Ni K absorption edge (arrow).

1. Effect of Absorption Edge on Spectrum

Absorption edges can be directly observed when the x-ray spectrum energy range spans the critical excitation energy for the absorber element in the target. The electron-excited x-ray bremsstrahlung provides a con­tinuous distribution in energy. At the absorption edge, the bremsstrahlung intensity abruptly decreases for x-ray energies slightly above the edge be­cause the mass absorption coefficient increases abruptly at the absorption edge. An example for nickel excited with a 40-keV electron beam is shown in Fig. 6.16. Just above the Ni Ka and Ni K0 lines, at the position of the critical excitation energy for the Ni K shell (E_: = Ek = 8.33 keV), the x-ray bremsstrahlung is much lower than what would be expected if the bremsstrahlung below the peaks were extrapolated to higher energy. This sudden decrease in the continuum corresponds to a sudden increase in the mass absorption coefficient for Ni as shown in Fig. 6.17. The presence of absorption edges in spectra may be difficult to observe if they are hidden under an x-ray peak that has been broadened by the detection process as is often the case in EDS analysis.

1. Absorption Coefficient for Mixed-Element Absorbers

The mass absorption coefficient for a sample containing a mixture of elements is found by taking the summation of the mass absorption coeffi­cient for each element multiplied by its weight (mass) fraction:

Figure 6.17. Mass absorption coefficient of a Ni absorber as a function of x-ray energy. A step increase in u/p occurs at an energy above the Ni K edge. The position of the Zn Ka x-ray line energy is noted on the drawing. The Zn Ka line is just slightly more energetic than the Ni K edge (the E_c for the Ni K series) and thus causes fluorescence of Ni K radiation. At the same time the Zn Ka line will be heavily absorbed in Ni.

where (u/p)', is the mass absorption coefficient for radiation from element i passing through element j and Cj is the weight(mass) fraction of element j. Thus, for Cu Ka radiation propagating through a sheet of Si02 the mass absorption coefficient may be calculated as

It is important to note that x-ray photoelectric absorption is an “all- or-nothing” process. Either the photon is completely absorbed in a single absorption event or else it continues to propagate without modification of its energy. In the absorption process, characteristic x-rays that escape the specimen retain their specific energy.

6.5. X-Ray Fluorescence

For some element pairs in the specimen the primary x-ray of element A created by the electron beam can generate a secondary x-ray of element B by fluorescence. This process is an inevitable consequence of photoelectric absorption of primary A x-rays by a specimen B atom with the ejection of a bound inner shell electron from the absorbing B atom. The primary A pho­ton is absorbed by the specimen B atom and its energy is transferred to the kinetic energy of the ejected electron, the photoelectron. The B atom is left in the same excited state as that produced by inner shell ionization directly by the electron beam. Subsequent deexcitation as the atom returns to the ground state by electron transitions is the same for both cases: The excited atom will follow the same routes to deexcitation, producing either characteristic x-rays or characteristic electrons (Auger electrons). X-ray-induced emission of x-rays is referred to as “x-ray fluorescence.” To distinguish the effects, “primary radiation" will refer to the x-rays produced by electron- beam ionization of atoms and “secondary radiation” will refer to x-rays produced through ionization of atoms by higher energy x-rays. Secondary radiation can be created from both characteristic x-rays and continuum (bremsstrahlung) x-rays. Because the x-ray photon causing fluorescence must have at least as much energy as the critical excitation energy of the atom in question, the energy of the secondary radiation will always be less than the energy of the photon that is responsible for ionizing the inner shell.

6.5.1. Characteristic Fluorescence

If a mixed sample consists of atom species A and B and the energy of the characteristic radiation from element A exceeds the critical excitation energy for element B, then characteristic fluorescence of B by the radiation of A will occur. The fluorescence effect depends on how close the photon energy of A is above the critical excitation energy for B, with the maximum effect occurring when Ea just exceeds Ec for B. The absorption process during characteristic fluorescence is illustrated in Fig. 6.17, where element A is Zn and element B is Ni. Nickel Ka fluorescent radiation is produced, as Zn Ka is strongly absorbed by the sample.

To examine this situation for a complex alloy, consider a sample con­taining a sequence of transition elements, for example, manganese, iron, cobalt, and nickel (Table 6.S). The critical excitation energy for manganese is lower than the Ka energies for cobalt and nickel, and therefore charac­teristic manganese fluorescence will occur from Co and Ni x-ray radiation. The KB energies of iron, cobalt, and nickel exceed the critical excita­tion energy for manganese, and so these radiations also contribute to the

secondary fluorescence of manganese. These arguments can be repealed for each element in the complex alloy. As shown in Table 6.5, the situation for the generation of secondary fluorescence is different for each of the constituents.

For characteristic fluorescence to be a significant effect, the primary x-ray radiation must be strongly absorbed, that is, the absorber must have a high mass absorption coefficient for the primary radiation. Examina­tion of Tables 6.4 reveals how the absorption/fluorescence phenomena depend on proximity to the critical excitation energy. Cu Ka radiation (8.04 keV) is strongly absorbed in cobalt (Z = 27, E_c = 7.709keV, fi/p = 326 cm²/g) with consequent emission of fluorescent Co Ka and Kfi; in nickel, which is only one atomic number higher, the absorption is reduced by almost a factor of seven (Z = 28, £_c = 8.331 keV, ji/p = 49 cm²/g). The Cu Ka line (8.04 keV) is not energetic enough to ionize the K shell of Ni. The magnitude of the fluorescence effect from different x-ray energies on a given element can be estimated by comparing the mass absorption coefficients.

2. Continuum Fluorescence

Characteristic-induced fluorescence can only occur if the primary radi­ation in question is more energetic than the critical excitation energy of the element of interest. In practice, the fluorescence effect is only significant if the characteristic energy is within approximately 3 keV of the critical excitation energy. On the other hand, the continuum (bremsstrahlung) ra­diation provides a source of x-rays at all energies up to the incident beam energy. Continuum-induced fluorescence will therefore always be a com­ponent of the measured characteristic radiation from a specimen. The cal­culation (Henoc, 1968) of the intensity of continuum fluorescence involves integrating the contributions of all photon energies above the critical exci­tation energy for the edge of interest. In practice, the extra intensity ranges from 1 to 7% for Z = 20 to 30 at £q = 20 keV.

Figure 6. IS. Range of the secondary fluores­cence of Fe Ka by Ni Ka in an alloy of composi­tion Ni-10% Fe, Eo=20keV. The innermost semi- carck shows the extent of direct electron-excited production of Ni Ka and Fe Ka by beam elec­trons. The 99% semicircle denotes locations deep in the specimen where some fluorescence of Fe by Ni Ka u still possible.

3. Range of Fluorescence Radiation

Direct, electron-induced characteristic x-ray production is constrained to lie within the interaction volume of the electrons in the target. A good estimate of range of production is given by the x-ray range, Eq. (6.12). However, x-rays penetrate into matter much farther than electrons, so the range of x-ray-induced fluorescence is correspondingly much greater. Con­sider the case of a Ni-10% Fe alloy, where the Ni Ka radiation can induce fluorescence of iron K radiation. The range of Ni Ka in the alloy can be calculated with Eqs. (6.13) and (6.15) using the data in Table 6.4. Consider that the Ni Ka radiation propagates uniformly in all directions from the source, which is the hemisphere containing the direct, electron-induced Ni Ka production. Based on these calculations, the electron and x-ray ranges are compared in Fig. 6.18. The radius of the hemisphere that contains 90% of the Fe Ka induced by Ni Ka is approximately 10 times greater, and the volume approximately 1000 times greater, than the hemisphere that contains all of the direct electron-induced Fe Ka production. The dimen­sions of the 50%, 75%, and 99% hemispheres are also shown in Fig. 6.18. Clearly, the x-ray-induced fluorescence from both characteristic and con­tinuum contributions originates in a much larger volume than the electron- induced characteristic radiation. This has the effect of degrading the spatial resolution of x-ray microanalysis.