Line shapes of atomic lines and soft x-ray emission bands measured with a wavelength dispersive spectrometer (WDS) with the Electron Probe Micro Analyzer (EPMA) are reviewed. Least square fitting to pseudo-Voigt profiles of the digitally measured spectra are used to account for the presence of non-diagram features (high and low energy satellites) and instrumental induced distortions. The effect of line width and relative intensities on the quality of fits is illustrated. Spectral distortions resulting from the presence of absorption edges within the analyzed wavelength region are illustrated for the case of FeL[alpha].[beta] emission bands for pure Fe and iron oxides. For quantitative analysis, an analytical approach is presented where the measured soft x-ray emission bands are corrected for self absorption before extracting the intensities from the experimental data.
Key words: atomic lines; distortions induced by absorption edges; pseudo-Voigt profiles; satellites; soft x-ray bands; WDS instrumental distortions.
1. Introduction
It is well recognized that the peak height of an x-ray emission line measured with a wavelength dispersive spectrometer (WDS) is a sufficient approximation for quantitative microanalysis on a routine basis using an Electron Probe Micro Analyzer (EPMA). This approach assumes that the observed x-ray line is symmetrical around the peak maximum occurring at a Bragg angle characteristic of the analyzed emission. A symmetrical peak never exists even for the case of atomic lines resulting from radiative transitions involving only core levels because of the presence of high and low energy satellites and instrumental distortions induced during measurement. While the approximation of the peak height for the x-ray intensity measurement remains valid in most analytical problems with the EPMA, this simplified approach is no longer sufficient in the presence of severe peak overlaps as is the case for L emission spectra of rare-earth elements (1) or when soft x-ray emissions are used in the analytical procedure (2-4).
We define soft x-ray emission as x-ray emission with energies lower than 1 keV such as the K emission series characteristic of low atomic number elements including carbon, oxygen, nitrogen, etc., and the L emission series chosen for intermediate atomic number elements. The soft x-ray emissions result from radiative transitions involving valence electrons. Consequently, the shape and the position of the maximum of soft emission bands are complex and depend on the electronic structure of the element within the matrix.
In the soft x-ray region, the peak height is no longer proportional to the peak area and several approaches have been proposed to determine the intensity of a soft x-ray emission band. For example, the use of predetermined "peak-to-area factor" as discussed by Bastin and Heijligers (5,6) or by measuring the peak area by summing the number of counts in each channel analyzed by stepping the monochromator across the wavelength domain containing the analyzed emission bands. The merits and limitations of these procedures have been discussed by Fialin et al. (7) accounting for the dependence of the peak shape on the self-absorption effect, the peak overlaps, and the resolution and the detection efficiency of the monochromator. However, it is still unclear whether the total area or only some spectroscopic features present in the measured spectra must be used to determine the intensity of the analyzed emission.
It is the aim of this paper to review the different features that lead to the complex shape of an x-ray line particularly within the soft x-ray emission domain. Practical considerations for WDS spectra processing using least-squares fitting techniques will be discussed. Applications to the interpretation of WDS spectra to the study of the chemical environment and quantitative microanalysis using soft x-ray emission bands will be illustrated using the Fe L[alpha],[beta] emission bands measured from pure iron and iron oxides.
2. Contributions of X-Ray Generation Mechanisms and of Instrumental Factors to the Shape of a WDS X-Ray Line
2.1 The Diagram Lines
The energy loss due to inelastic scattering events produces a hole in the inner-shell of an ionized atom. The de-excitation processes leads to the emission of a mono-energetic photon which is characteristic of the atom. The energy of the emitted photon is equal to the energy difference [DELTA]E of the energy levels involved in the radiative transition (in a non-radiative transition, the excess of energy [DELTA]E contributes to the emission of an Auger electron).
The emitted photon is characterized by a Lorentzian energy distribution with a width at half maximum [GAMMA] (natural or physical width) related to the life time, [tau], of the hole on the initial state according to:
[GAMMA][tau] = h/2[pi]. (1)
The natural profile of a radiative transition is a convolution of energy distributions of each of the levels involved in the transition, which is a Lorentzian curve whose FWHM is equal to the sum of the FWHM of the two levels. Broadening may occur with low energy levels if non-radiative transitions are possible such as Coster-Kronig transitions. The natural profile for transitions involving valence electrons are also broader than those resulting only from core holes.
The probability [P.sub.if] for a radiative transition between two levels i and f can be expressed according to:
[P.sub.if] = [[omega].sub.i] [Z.sub.if] [N.sub.i] (2)
where [[omega].sub.i] is the fluorescence yield, [z.sub.if] is the weight of the line and [N.sub.i] is the number of atoms in the initial level per unit volume.
The fluorescence yield [omega] expresses the probability that the atom de-excites according to a radiative transition with the production of an x-ray photon. The probability to have a non-radiative transition with the emission of an Auger electron is (1-[omega]).
The fluorescence yield [[omega].sub.i] of the level j is
[[omega].sub.j] = [N.sub.R]/([N.sub.R]+[N.sub.NR]) (3)
where [N.sub.R] and [N.sub.NR] are the radiative and non-radiative transition rates, respectively.
For [n.sub.j] ionisations created on level j, the number of photons in the j series is ([[omega].sub.j] * [n.sub.j]).
The intensity [I.sub.if] in the radiative transition is proportional to [P.sub.if] [Eq. 6)] and depends on the convolution of the initial [D.sub.i] and final [D.sub.j] energy level distributions:
[I.sub.if] = [P.sub.if] ([D.sub.i] * [D.sub.j]) (4)
For atomic lines involving only core levels, the convolution product [D.sub.i] * [D.sub.j] is assumed to be a constant. However, this approximation is no longer valid with soft (low energy) x-ray emissions since the final states are valence hole states so that the emission spectra will change with the electronic structure (density of occupied states, DOS) of the material. The position of the maximum energy and the intensity of the emission band will vary as a function of the chemical environment.
In wide band gap materials such as aluminum oxide, the major peak of a soft x-ray emission band (DOS) is usually accompanied by a low energy peak ("bonding peak") resulting from transitions to the initial hole of electrons from mixing [Al.sub.3sp] and [O.sub.2p] energy states in the valence band, as illustrated in Fig. 1. The bonding peak resulting from the mixing of states (referred as K[alpha]' in the literature) is located at approximately -5 eV from the maximum of the OK[alpha] parent peak. The feature labeled k[alpha]" in Fig. 1 occurring on the short wavelength (high energy) side of the diagram peak may result from satellite emissions (see Sec. 2.2) or from instrumental effects, as discussed below.
In wide band gap crystals, some high energy features may also result from transitions involving levels located in the band gap of the energy diagram of the crystal. These levels are associated with intrinsic point defects which are induced either during the crystal growth conditions or induced during the specimen preparation (polishing with abrasives) or by radiolysis mechanisms during the electron irradiation. In oxides, the most frequent defects are [F.sup.+] and F centers, i.e., oxygen vacancies with one or two trapped electrons, respectively. As an example, Jonnard et al. [8] showed that the AIK[beta] emission (3p-1s transition) from alumina crystals is accompanied by a small high energy weak emission peak located 0.6 eV above the top of the valence band.
2.2 The Non-Diagram Lines
An x-ray emission line (or diagram line) resulting from a transition between two levels in the energy-level diagram is frequently accompanied by satellites (or non-diagram lines), i.e., x-ray lines whose energies do not correspond to the difference of two energy levels.
2.2.1 High Energy Satellites
The high energy satellite lines have been intensively studied since the 1930s to 1940s beginning with the detailed works of Parratt (9,10) and Randall and Parratt (11). Satellite lines result from electronic rearrangement concomitant with the ionization process during the de-excitation mechanisms of the ionized atoms.
K Lines: When 1s and 2p vacancies are created simultaneously, the 2p vacancy has a relatively long life-time compared to that of the is vacancy. Thus, the inner vacancy de-excites in presence of a spectator hole which produces a change in the electrostatic potential leading to shifts in the energy levels (Fig. 2). The energy shifts for the K[alpha] lines are given by:
[DELTA]E = [([DELTA]E).sub.1s] - [([DELTA]E).sub.2p]. (5)
The satellite lines resulting from the presence of outer vacancies consist of a number of closely spaced features. For the case of the K[alpha] emission line, the high energy satellites are usually labeled as K[[alpha].sub.3,4.] The high energy satellite resulting from the de-excitation in presence of two outer vacancies is referred as K[[alpha].sub.5,6] and exhibits a very weak amplitude. The energy separation distance between the satellite band and the K[alpha] line ranges from about 10 eV up to about 40 eV for atomic number 12 < Z < 30. Aberg [12] presents an extensive set of values for the relative intensity of the satellite which decreases from about 30% for Z = 10 to 0.5 % at Z = 30.
L and M Lines: The de-excitation of an L or M level in presence of outer holes may also lead to the presence of high energy satellites associated with L or M x-ray peaks. The additional outer vacancies may result from Coster-Kronig transitions or shake-off mechanisms. The Coster-Kronig transitions result from an Auger process between sub-shells of the same shell.
The hole created on the [L.sub.1] sub-shell may be filled by an electron originating from the [L.sub.2] or [L.sub.3] subshell. According to the selection rules, these transitions are not radiative and the excess of energy [L.sub.1]-[L.sub.2], [L.sub.2]-[L.sub.3] or [L.sub.1]-[L.sub.3] is dissipated by the emission of an Auger electron from the M or N levels. The transition rate of non-radiative Coster-Kronig transitions [f.sub.ij], where i and j are two subshells within the same energy level, is not permitted for all elements.
Indirect ionizations resulting from the non-radiative Coster-Kronig process have the following effects on the emission profile:
1) To create additional vacancies so that the total number of ionizations is the sum of the direct ionizations produced by the incident electrons and those created by the non-radiative Coster-Kronig transition. For example the L[alpha] emission line involving ionization on the [L.sub.3] subshell, the number of L[alpha] photons will be:
[n.sub.[L.sub.3]] < [[omega].sub.[L.sub.3]] > = [[omega].sub.[L.sub.3]] [[n.sub.[L.sub.3]] + [f.sub.13] [n.sub.[L.sub.1]] + [f.sub.23] ([n.sub.[L.sub.2]] + [f.sub.12] [n.sub.[L.sub.1]])] (6)
where [f.sub.13], [f.sub.23], and [f.sub.12] are the Coster-Kronig transition probabilities.
2) To leave outer vacancies during the re-arrangement of the ionized states between the sub-shells prior to the radiative transition with the emission of an x-ray photon. This process is responsible for the production of high energy satellites. When the atomic number of the emitter decreases, the energy separation distance between the shake-off satellites and the diagram also decreases and may be observed as a shoulder to the major peak.
According to Fabian [13], several line shapes of L emission spectra of elements in the first transition series can be distinguished depending on the incident electron energy region: 1) The Threshold Excitation Region when the incident energy lies between the [L.sub.3] and [L.sub.2] energy thresholds, multiple vacancy satellites are largely reduced, 2) The Satellite Region in which the L[alpha] diagram line becomes distorted by the progressive development of high energy satellites when the incident electron energy increases from the [L.sub.3] sub-shell threshold up to about three times that value, and 3) The Self Absorption Region for incident energies greater than about 3 or 4 times the [L.sub.3] threshold energy, the fine structure vanishes and the effect is attributed to self-absorption. Increasing incident electron energy also increases the absorption path of the generated x-ray photons within the specimen and self-absorption removes the fine structure when a high incident energy is used.
Peak shape changes as a function of the incident energy is illustrated in Fig. 3 for the case of the CuL[alpha] emission from pure copper measured with a TAP monochromator. The variation of the intensity of the high energy satellite relative to that of the diagram peak as a function of the beam energy results from a differential self absorption effect because the [L.sub.3] absorption edge occurs between the two spectral features.
Similarly, high energy satellites associated with M[alpha] lines of elements with high atomic number result from the [M.sub.5] hole de-excitation in presence of simultaneous vacancies in the [M.sub.5] and N sub-shells. Only the envelope of satellites resulting from additional vacancies in the [M.sub.5] sub-shell can be distinguished from the diagram line.
As shown in Fig. 4, peak shape changes as a function of the beam energy are also observed for the case of the AuM[alpha] emission band measured from a pure Au specimen with a PET monochromator at 3 keV and 15 keV successively. The excitation energy thresholds for the [M.sub.1] to [M.sub.5] sub-shells are 3.425 keV ([M.sub.1]), 3.150 keV ([M.sub.2]), 2.743 keV ([M.sub.3]), 2.291 keV ([M.sub.4]) and 20.206 keV ([M.sub.5]), respectively. Thus, a 3 keV incident energy is sufficient to provoke the Au M[alpha] emission involving the initial hole in the [M.sub.5] sub-shell. Additional vacancies may be created in the [M.sub.3] and [M.sub.4] sub-shells with possible outer N vacancies resulting from Coster-Kronig transfer of the type [M.sub.3,4]-[M.sub.5][N.sub.x], producing weak high energy satellites. No additional vacancies are created in the [M.sub.1] and [M.sub.2] levels and transfer of the type [M.sub.1,2]-[M.sub.5][N.sub.x] does not exist for a 3 keV incident energy. Reciprocally, the [M.sub.1] and [M.sub.2] su b-shells are excited with a 15 keV incident energy and the resulting inner vacancies can move to the [M.sub.5] sub-shell with production of outer holes by Coster-Kronig mechanisms, thus the pronounced asymmetry on the high energy side of the Au M[alpha] may reasonably be assigned to the development of satellites resulting from the de-excitation of the [M.sub.5] level in presence of outer N vacancies.
2.2.2 Low Energy Satellites
Several theories are available to describe the K[beta]' low energy feature associated with the K[[beta].sub.1,3] emission resulting from transitions involving the partially filled 3d shells of transition elements and their oxides.
The Radiative Auger Effect (RAE) produces a broad structure at a lower energy than the characteristic diagram line. The RAE process results from a deexcitation of a K vacancy, similar to an Auger process with simultaneous emission of a bound electron and an x-ray photon (Fig. 5). For atomic number 15 According to Salem et al. (15), the interaction between the electrons in the incomplete 3d shell and the hole in the incomplete 3p shell splits both 3p and 3d levels causing a demultiplication of transitions.
The K[beta]' satellite has also been explained in terms of the plasmon oscillation theory (16). During the x-ray emission process, the transition valence electron excites a plasmon in the valence band. The transition energy of the K[[beta].sub.1,3] line will thus be shared between the plasmon and the emitting photon which will be deprived of an energy equal to the plasmon energy. For the transition elements the energy separation distance between the K[beta]' satellite and K[[beta].sub.1,3] diagram line is in the order of magnitude of 10 eV, depending upon the chemical environment (16).
The theories concerning the production of K[beta]' satellite associated with the [K[beta].sub.1,3] line of transition elements were extended to the case of the L x-ray spectra of the lanthanide elements [17]. The [L[beta].sub.2] [L[beta].sub.4] [L[gamma].sub.1] and L[gamma].sub.2] emissions exhibiting low energy effects are associated with transition