Structure of K α 1 , 2 - and K β 1 , 3 -emission x-ray spectra for Se, Y, and Zr

The K α and K β x-ray spectra of Se, Y, and Zr were studied experimentally and theoretically in order to obtain information on the K α 1 line asymmetry and the spin doublet in K β 1 , 3 diagram lines. Using a high-resolution antiparallel double-crystal x-ray spectrometer, we obtained the line shapes, that is, asymmetry index and natural linewidths. We found that the corrected full width at half maximum of the K α 1 and K α 2 lines as a function of Z is in good agreement with the data in the literature. Furthermore, satellite lines arising from shake-off appear in the low-energy side of the K α 1 and K α 2 lines in Se but, in Y and Zr, it was very difﬁcult to identify the contribution of the shake process to the overall lines. The K β 1 , 3 natural linewidth of these elements was also corrected using the appropriate instrumental function for this type of x-ray spectrometer, and the spin doublet energies were obtained from the peak positions. The corrected full width at half maximum (FWHM) of the K β 1 x-ray lines increases linearly with Z , but this tendency was found to be, in general, not linear for K β 3 x-ray lines. This behavior may be due to the existence of satellite lines originated from shake processes. Simulated line proﬁles, obtained using the multiconﬁguration Dirac-Fock formalism, accounting for radiative and radiationless transitions and shake-off processes, show a very good agreement with the high-resolution experimental spectra.


I. INTRODUCTION
The Kα and Kβ x-ray emission spectra of the 3d transition metals exhibit several peculiar asymmetric line profiles not observed in other elements [1], whose origin has been under investigation and debate [2][3][4][5][6]. Several mechanisms, such can be fully explained by considering only the diagram and the 3d spectator transitions [20].
On the other hand, the Kβ 1,3 x-ray emission spectrum includes Kβ and Kβ satellites on the low-and high-energy sides of the Kβ 1,3 peak position, respectively, as explained in the case of copper [4,5]. These satellite lines have also been investigated until now both experimentally and theoretically [1,6,18,[21][22][23][24][25][26] for all 3d transition metals. Shake-off from the 3d shell was also shown to account reasonably well for the measured Kβ 1,3 line shape, although a complete quantitative fitting has not been reported, and possible contributions from other shells were not investigated [21,24,25]. More recently, Ito et al. [27] measured systematically the Kβ x-ray spectra of the elements from Ca to Ge, using a high-resolution antiparallel double-crystal x-ray spectrometer. They reported that each Kβ 1,3 natural linewidth has been corrected using the instrumental function of this type of x-ray spectrometer, the spin doublet energies have been obtained from the peak position values in Kβ 1,3 x-ray spectra, and the contributions of satellite lines were considered to be originated from [KM] shake processes.
In order to elucidate the influence of the shake processes on the spectral profile, we investigated in this paper the contribution of [1s3d] shake-off to the asymmetry of Se, Y, and Zr Kα 1,2 emission lines, from both experiment and theory, and the natural width of each line in the Kβ 1,3 emission spectra of the same elements to obtain the energy values of the spin doublet in detail, using a high-resolution double-crystal x-ray spectrometer.

II. EXPERIMENTAL METHODOLOGY
In the present paper we used a RIGAKU (3580E) doublecrystal x-ray spectrometer. The experimental conditions for the measurements are given in Table I. Using Bragg reflections with this spectrometer, the true FWHM of the emission FIG. 1. The observed Kα 1,2 spectra in elements Se, Y, and Zr are shown with the Lorentzian functions used in the fitting processes [20,27]. These spectra were measured using the antiparallel double-crystal x-ray spectrometer described in the text. For Se, on the left side is shown the result of a two-asymmetric Lorentzian fitting analysis, and on the right side the result of a four-symmetric Lorentzian fitting analysis, according to Ito et al. [20]. In this figure, Kα 11 is the Kα 1 diagram line, and Kα 21 is the Kα 2 line, whereas Kα 12 and Kα 22 are the corresponding satellite lines. The ratio of the Kα 12 to Kα 11 line intensities is used in Fig. 9. The spectra of Y and Zr were analyzed by a two-symmetric Lorentzian fitting. A single scan of three repeat measurements is shown in each element. Each χ 2 r is a value in a single scan measurement. See the text for details.
FIG. 2. The observed Kβ 1,3 spectra in elements Se, Y, and Zr are shown with the fitting Lorentzian functions. These spectra were measured using the antiparallel double-crystal x-ray spectrometer described in detail in Ref. [20,27]. Kβ are satellite lines on the high energy side of the Kβ 1,3 spectra. A single scan of three repeat measurements is shown for each element together with the value of χ 2 r . line can be determined by a simple subtraction of the convolution in the crystal dispersion from the FWHM of the measured emission line [28] (see Refs. [20,27] for details). With a Rh end-window x-ray generator operating at 40 kV and 60 mA, the emitted Kα and Kβ spectra (see Figs. 1 and 2) were recorded under a vacuum with a sealed Xe gas proportional counter in the symmetric Si(220) Bragg reflection of the double-crystal spectrometer at an angular step of 0.0005 • in 2θ for Kα spectra, 0.001 • for Se, 0.0025 • for Y, and 0.0005 • for Zr in 2θ for Kβ spectra. The slits vertical divergence is 0.573 • in this spectrometer. Temperature in the x-ray spectrometer chamber is controlled within 35.0 ± 0.5 • C. Acquisition time was 7-120 s/point (see Table I). Neither smoothing nor correction were applied to the raw data. Each spectrum was repeated three times. The energy values of Bearden [29] were taken as starting points for the diagram line fitting parameters. We used metal powder (99.9%, Nacalai Tesque) for Se, metal plate (99%, Nilaco Corporation) for Y, and metal foil (99.2%, Nilaco Corporation) for Zr. The powder Se was confirmed to be the metallic form using a x-ray diffractometer and the double-crystal x-ray spectrometer. The instrumental function of the double-crystal spectrometer can be very well described from Monte Carlo simulations as has been shown in Refs. [30][31][32] and by simply computing the rocking curve of the Si crystals through dynamical diffraction theory [33]. From these instrumental functions one can obtain the natural linewidths as well as some other broadening mechanisms. In the present case, given the large natural widths of the diagram lines of neutral atoms when compared to the spectrometer instrumental function, we can use the simple broadening method described by Tochio et al. [28] without increasing the final uncertainty.

III. THEORETICAL CALCULATIONS
The level energies, transition amplitudes, and shake probabilities needed to calculate the diagram, and satellite x-ray emission spectra were computed with the relativistic atomic structure code MCDFGME, developed by Desclaux [34] and Indelicato and Desclaux [35]. This code fully implements the multiconfiguration Dirac-Fock (MCDF) method but, in the present calculations, the electronic correlation was only included up to the level of the configuration mixing, and not at the level of a multiconfiguration calculation.

A. Basics of the MCDF method
The N-electron atomic system in the MCDF method is described by the Dirac-Coulomb-Breit (DCB) Hamiltonian, where h D a is the one-electron Dirac Hamiltonian, Here α a and β a are the 4 × 4 Dirac matrices and V N a describes the interaction of one electron with the atomic nucleus. In the length gauge, the two-electron interaction can be writ-052820-3 ten as r ab cos(ω ab r ab ) where r ab is the interelectrons distance and ω ab is the energy of the exchanged photon between the two electrons. The first term 1/r ab describes the instantaneous Coulomb interaction and the remaining terms are known as the Breit interaction. In the limit ω ab r ab 1, the Breit interaction becomes The first term in this relation is known as the magnetic (Gaunt) interaction, and the second term is the lowest-order retardation interaction, which are calculated in the MCDFGME code as part of the self-consistent variational method. The remaining Breit retardation terms in Eq. (3) were also included perturbatively. Furthermore, the code also accounts for radiative corrections, namely, self-energy and vacuum polarization. For details on the theory QED corrections in atomic systems, we refer the reader to Ref. [36]. The atomic wave functions are calculated in the framework of the variational principle using energy eigenfunctions that are written as linear combinations of configuration state functions (CSF). The CSF are expressed as linear combinations of Slater determinants with the same parity and are eigenfunctions of the Hamiltonian, total angular momentum, and projection of the total angular momentum on the quantization axis of the atomic system. All energy levels for one-and two-hole configurations were calculated with complete relaxation, meaning that both the mixing coefficients and the radial orbitals in the CSF were optimized in the variational method.

B. Calculation of the line intensities
Radiative transition amplitudes were calculated between levels of the K-shell one-hole configurations with full relaxation, that is, initial and final bound-state wave functions were optimized independently. This so-called optimal levels scheme does not ensure the orthogonality of the initial-and final-state spin orbitals. To deal with the nonorthogonality of the wave functions, the code uses the formalism prescribed by Löwdin [37]. Radiationless transition amplitudes were calculated between initial levels of K-shell one-hole configurations and final levels of bound two-hole configurations and an electron in the continuum. The initial levels wave functions were also obtained from the previous energy calculations, but to ensure orthogonality, no orbital relaxation was allowed between the initial and the final bound-state wave functions. Nonetheless, the radiationless rates were calculated using the more accurate transition energies obtained in the first step.
From the results of these calculations, we computed all diagram line intensities from initial levels i to all possible final levels f where the indices i and f stand for the electronic configurations C i and C f , the total angular momenta J i and J f , respectively, and all other quantum numbers required to completely specify these levels, Here, N i is the population of level i, E if = E i − E f is the transition energy, B if is the x-ray emission branching ratio, and ω i is the initial level fluorescence yield, where A R if and A NR if are the radiative and radiationless transition amplitudes, respectively, between two levels. In Eqs. (6) and (7), f and f stand for all possible levels that can be reached from level i by radiative and radiationless transitions, respectively. The populations of levels i are taken to be statistical, which means that all states of the initial levels of a given configuration have the same probability of being populated where the summation runs over all levels belonging to configuration C i . The radiative transition amplitudes in the length gauge can be written as the Einstein A coefficients for the photon emission, where Q J are the many-electron multipole transition operators of rank J as defined in Ref. [38]. All electric and magnetic multipole transitions with J 3 were included in the present calculations. The radiationless transition amplitudes were calculated assuming the sudden approximation using perturbation theory. The calculation was performed in the frozen-core approximation, including both direct and exchange terms [39]. Satellite lines correspond to transitions where a second "spectator" hole is present and occur when initial double-hole levels exist. This double ionization may result from shake-off, that is, the ejection of an outer electron due to the sudden change in nuclear potential when an inner hole is created. This process was considered by Bloch [40] and used for the first time by and Demekhin and Sachenko [41] to calculate for the first time shake satellite intensities in the x-ray spectra. Åberg [42] also used the sudden approximation for the calculation of Kα 1 α 2 satellite line intensities in Ne-like ions. Frequently, the number of satellite lines is so large that they are seen in the spectra as bands as in the present spectra.
To compute satellite branching ratios, we used radiative transition amplitudes between levels of two-hole configurations, calculated in the same way as for the one-hole configurations but limited to J 1. An exact calculation of satellite intensities also requires the calculation of radiationless transition amplitudes between initial two-hole and final three-hole configuration levels to obtain the satellite level fluorescence yield ω sat i . This, however, would be computationally  735  8597  Total  786  210  4013  Total  10040  4662  71204  Total  32906  16579  215335 very demanding and, instead, we approximated the fluorescence yield of the satellite-line initial level by the K-shell fluorescence yield, where i labels the satellite level belonging to a [KX ] (X = K, L, M, . . .) two-hole configuration. The calculation of the satellite intensities was performed using the expression equivalent to Eq. (5) but with each line intensity weighted by the shake-off probability of the initial level. The shake-off probabilities were calculated in the sudden approximation, according to the method of Carlson and Nestor [43], using the overlap integrals between orbitals in the neutral and the one-hole K-shell configurations.
To produce the theoretical spectra, the transition intensities distribution was convoluted with a line-shape function that accounted for the natural and experimental broadening. The line-shape function consisted of two Lorentzian with different widths: the natural width, calculated from the partial sum of radiative and radiationless amplitudes, and the experimental width. The choice of two Lorentzian is based on the fact that the rocking curve of the Si crystals in double-crystal spectrometers resembles much better a Lorentzian than a Gaussian profile.
The number of transitions calculated in this paper is shown in Table II. The simulations of the Kα 1,2 and Kβ 1,3 lines of Se, Y, and Zr, as well as the corresponding resulting satellite bands can be seen in Fig. 3 together with the experimental plots. In Fig. 4 are presented, as an example, the individual satellite bands due to shake-off from the different orbitals for Y , and the resulting overall satellite band. In Table V we list the probabilities (in percentages) of shake processes as the result of a sudden 1s vacancy production for all shells.
Thus, our spectra calculations are fully ab initio without resorting to any kind of fit other than the energy offset of all transitions due to the lack in the calculations of physical effects, such as Auger shifts and small corrections due to higher-order multiconfiguration electronic correlation. The values of the energy offset that were obtained by fitting the final theoretical spectral shape (satellites included) to the experimental data are (in eV) 1 The use of the multi-Lorentzian fitting for diagram and satellite lines, although not strictly valid, serves the purpose of allowing the comparison with a common framework used extensively by experimentalists in the x-ray emission spectrometry field. As can be seen in Fig. 4, the shape of the satellite bands cannot be truly fitted by a single Lorentzian, however, it is remarkable that for the Kα lines of Se, the inclusion of only one Lorentzian for each diagram and satellite line results in such a low χ 2 r (see Fig. 1).

IV. RESULTS AND DISCUSSION
As described in Sec. I, asymmetric line shapes in x-ray emission spectra were attributed as early as 1927 to the existence of one-electron transitions in the presence of a spectator hole, resulting from shake processes after ionization by photons, electrons, and other particles [1,6,7]. After Deutsch et al. [4] demonstrated that the line profiles of Cu Kα and Kβ emission lines can be fully accounted for by contributions to the diagram lines from 3d-spectator transitions only, Hölzer et al. [6] concentrated on the investigation of Kα and Kβ diagram lines based on the results of Deutsch et al. [4] in order to elucidate the origin of the asymmetry using a high-resolution x-ray crystal spectrometer. According to Hölzer et al. [6], only few of the previous data were corrected for instrumental broadening, although it was realized early that broadening from this origin has a significant effect [44]. A two-flat-crystal x-ray spectrometer, using a crystal instead of the slit in a single-flat-crystal x-ray spectrometer, is used for our studies. This spectrometer was developed by Gohshi et al. [45] with the two crystals linked. Tochio et al. [28] evaluated the instrumental broadening in this type spectrometer. Ito et al. [20] investigated systematically the asymmetry index, FWHM, and intensity ratio of Kα 1,2 diagram lines in 3d transition elements using the double-crystal x-ray spectrometer in order to elucidate the origin of the asymmetry in the line profile. They showed that the asymmetry index of Kα 1 in the elements from Sc to Ge is ascribed to the existence of a 3d spectator hole. In which element the asymmetry of Kα lines appears was discussed.
In the present paper, the Kα 1,2 and Kβ 1,3 spectra of elements Se, Y, and Zr, have been measured three times each, using a high-resolution double-flat-crystal x-ray spectrometer. The values of the obtained averaged line energies and averaged relative intensity ratios for each line in each Lorentzian model for these elements are shown in Tables III and IV [46,47]. The corrected FWHM values were taken from the observed FWHM through the method of Tochio et al. [28].

A. The observed Kα 1,2 emission spectra
The two-asymmetric and the four-symmetric Lorentzian fittings are performed only for Se. The two-symmetric Lorentzian fittings are performed also for Y and Zr. This is because the contribution of the [1s3d] shake process could not be clearly confirmed from the observed spectra through the fitting method as seen in Fig. 1. The asymmetry index is defined as the ratio of the half width at half maximum TABLE III. The observed Kα 1,2 spectra in elements Se, Y, and Zr are shown with the Lorentzian functions used in the fitting processes [4,20]. Kα 11 spectra are the Kα 1 diagram line, and Kα 21 spectra are the Kα 2 diagram line. Kα 12 and Kα 22 satellite lines are due to the shake processes. The corrected FWHM was obtained by the method of Tochio et al. [28]. All energy values are in eV. The values of the instrumental broadening are around 0.17, 0.21, and 0.22 eV for Se, Y, and Zr Kα 1 photon energy, respectively. The asymmetry index is defined as the ratio of the half width at half maximum on the low-and high-energy sides of each peak, respectively.   [27]. Kβ spectra are satellite lines (see Ref. [4,5,27] and references therein). The fitting analyses in Se, Y, and Zr were executed with symmetric Lorentzians. The corrected FWHM was obtained by the method of Tochio et al. [28]. All energy values are in eV. on the low-and high-energy sides of each peak, respectively, providing a measure of the peak asymmetry [49].  [20] were used for only Si(220) analyzing crystals in a vacuum. The corrected FWHM values for the Kα 1 diagram lines in the elements Se, Y, and Zr were obtained from the observed FWHM through the method of Tochio et al. [28] relative intensity ratios for each line in the asymmetric model are given in Table III, and averaged line energies, averaged observed FWHM, averaged corrected FWHM, and averaged relative intensity ratio in four symmetric Lorentzian model are also shown in Table III. The instrumental broadening was found to be 0.17, 0.21, and 0.22 eV of Kα 1 in Se, Y, and Zr, respectively, using the method of Tochio et al. mentioned above, and the CFWHMs were obtained by subtraction from the FWHMs. Moreover, for Y and Zr, the values of the obtained averaged line energies, averaged FWHM, averaged corrected FWHM, and averaged relative intensity ratio in Kα 1 and Kα 2 lines, in each symmetric Lorentzian model are shown in Table III as well as the other reported data, that is, Krause  [20]. Crosses in (a) are from the observed data of Kα 2 in Ref. [20]. The corrected FWHM values for the Kα 2 diagram line in the elements Se, Y, and Zr were obtained from the observed FWHM through the method of Tochio et al. [28]. and Oliver semiempirical natural linewidths [47], and the recommended linewidth values of Campbell and Papp [46].
We checked the Cu Kα 1,2 diagram lines as standard lines in order to evaluate the reproducibility in the double-crystal x-ray spectrometer, whereas measuring the Kα 1,2 and Kβ 1,3 diagram lines for each element [20,50]. The fitting process with the Lorentzian functions shown in Fig. 1 for Kα 1,2 and  [46].The linear function W was obtained by the least-squares method using Kβ 1 linewidths in the elements Zn and Ge [27]. (a) Ito et al. [27]. (b) Hölzer et al. [6]. (c) Anagnostopoulos et al. [18]. (d) Pham et al. [26].  [20,48] marked by crosses (a). According to Ito et al. [20], this satellite is mainly considered to be a contribution of the [1s3d] shake process to the asymmetry of the Kα 1 emission line. Open rhombohedra (b) from Ref. [20] and open triangles from statistical errors resulting from the fitting processes and the limited reproducibility of the experimental setup. To obtain realistic uncertainties, the errors originating from the energy calibration have to be considered. Absolute Kα 1,2 and Kβ 1,3 photon energies for all 3d elements between Cr and Cu can be found in Ref. [6].
In Figs. 5 and 6 we can see that the overall variation behavior with Z of the Kα 11 (or Kα 1 ) and Kα 21 (or Kα 2 ) lines is well reproduced by the data reported by Campbell and Papp [46] and Krause and Oliver [47]. We obtained the corrected FWHM values for both Kα 11 (or Kα 1 ) and Kα 21 (or Kα 2 ) diagram lines from the observed FWHM through the method of Tochio et al. [28]. Ito et al. [20] attributed the difference in Kα 11 and Kα 21 diagram lines corrected FWHM values observed in 3d elements to the broadening effect of the Coster-Kronig transitions.
In what concerns the asymmetry of the Kα 1 diagram line, Ito et al. [20] concluded that it is due to the [1s3d] shake processes as seen in Fig. 9. Through precise experiments and theoretical considerations, we investigated up to which elements the influence of the [1s3d] shake processes contributes to the Kα 1 emission spectrum. In Table V and Fig. 9 (shake probability ratio), the [KX ] (X = M, N, O) shake probabilities are shown for Se, Y, and Zr. The [1s3d] shake probability is relatively large, at least, until element Zr. However, an asymmetric Kα 1 profile can be observed for Se, whereas in Y and Zr it is very difficult to distinguish the satellite lines from the diagram lines. One should consider that the shake probabilities of [KN] or [KO] hidden satellites are large in these elements as seen in Table V. Therefore, it is hard to discuss the asymmetry in the Kα 1 spectra of elements with atomic numbers larger than Y based on both the experimental results and the theoretical calculations of the shake process as seen in Figs. 1 and 3 because, according to Ito et al. [20], the asymmetry depends on the [KM] shake probability and  1.1906 for elements higher than Y, the influence of the shake process due to [KN] and [KO] becomes larger than the influence of [KM] shake process. It is noteworthy that for these elements the observed spectra are almost symmetrical. From the theoretical calculations and the simulated line shapes, we conclude that the [1s3d] satellites present themselves as small peaks aligned with the diagram centroid with shoulders on the low-energy side of the diagram line as expected from Fig. 1. Nevertheless, the [1s3p] satellites also contribute to a small asymmetry on the high-energy side of the simulated diagram peaks, although these could not be confirmed experimentally. It may be due to the detection limit of the measurement. This cannot be seen in the Kα spectra of all the measured elements but is more pronounced in Zr and, to some extent, Se. Other than that very subtle effect, the ab initio simulated Kα spectra present a very good agreement with the experimental data (see Fig. 3) both on the overall shape and the intensity ratios.

B. The observed Kβ 1,3 emission spectra
The values of the obtained averaged line energies and averaged relative intensity ratios for each line of Kβ 1,3 or Kβ 2 in each Lorentzian model for Se, Y, and Zr, are shown in Table IV. The corrected FWHM for the Kβ 1 and Kβ 3 diagram lines are presented in Table IV, and Figs. 7 and 8, respectively, together with values reported by other authors [6,18,26,27]. The corrected FWHM values were taken from the observed FWHM through the method of Tochio et al. [28]. The Kβ 1,3 spin doublet values observed and calculated in this paper are presented in Table VI and Fig. 10, respectively, together with other observed values [6,18,26,27]. The Lorentzian model was used for an analytic representation of Kβ x-ray lines [27], and the results of the fitting analysis are shown in Fig. 2 for Se, Y, and Zr. The errors quoted in Tables IV and VI are, thus, only statistical errors resulting from the fitting processes and the limited reproducibility of the experimental setup. In Fig. 7, the linear function W shown by a solid line was found by the least-squares method with the corrected FWHM of Sc Kβ 1 as an initial value in order to compare with the recommended FWHM of Campbell and Papp [46]. We can see that the overall variation behavior with Z of the corrected FWHM of the Kβ 1 line may be linearly fitted with the function W = 0.257 × Z − 4.364 and these values are consistent with those of Ref. [46] marked in dots and dashed line between the elements Fe and Zr. What happens to the change in the corrected FWHM, when the Z number is over 40, has to wait for future research. Furthermore, when the value of Z is less than 26, the corrected FWHM largely deviates from the recommended value. This value is a little different from the function W = 0.300 × Z − 5.445 previously reported [27]. This difference is due to the fact that the Kβ 1 linewidths [27] data were evaluated by adding the widths of the Se, Y, and Zr Kβ 1 lines. The obtained W values were corrected using the corrected FWHM of Kβ 1 lines in the same elements. We consider the obtained linear function W to be more reliable. We will discuss this later.
In Fig. 8, the linear function W , represented by a solid line, was obtained using the corrected FWHMs of Zn and Ge Kβ 3 in order to compare with the data of Campbell and Papp [46] because these elements have comparatively small effects of the shake processes [KM] on the diagram lines in 3d elements as seen in Fig. 9. The corrected FWHM values of the Kβ 3 lines between Cu and Zr, with the exception of Se, are consistent with those of the Kβ 3 from Ref. [46], although the corrected FWHM values of the Kβ 3 lines in 3d elements are very different from those reported therein. The cause of the deviation of the Se Kβ 3 line from the linear function W is unknown. The explanation of the tendency of the corrected FWHM of the Kβ 3 line is much more difficult than that in the Kβ 1 line case. It is necessary to collect experimental data of Kβ 3 lines with atomic Z of 40 or more and 20 or less, using this type of x-ray spectrometer in order to elucidate the complexity of the tendency of the corrected FWHM in Kβ 3 as a function of Z.
The Kβ 1,3 spin doublet values observed and calculated in this paper are presented in Table VI together with other reported values [29,46,51] and in Fig. 10 together with other observed values [6,18,26,27]. From the fitting analysis of the Kβ 1,3 spectra in these elements, the photon energies of the Kβ 1,3 and Kβ or Kβ 2 lines are presented in Tables IV and  VI together with the other reports [29,51] for comparison. The spin doublet energy dependency on the atomic number Z is obtained using the data from Ito et al. [27] and the present paper, and the logarithmic function is almost the same as that reported by Ito et al. [27], that is, log 10 S = 0.065(Z − 24.879) + 0.149. However, the physical meaning of this function is not clarified yet. Now we considered the validity of the logarithmic function of the spin doublet (Fig. 10) in what concerns the Kβ 1 and Kβ 3 lines (Figs. 7 and 8). The FWHM by fitting analysis of the measured data of Ca Kβ 1,3 emission lines was obtained in the following way: The double-crystal x-ray spectrometer was used to measure Ca Kβ 1,3,5 spectra in a CaCO 3 compound, using Si(220) analytical crystals. The primary target was tungsten, and the tube voltage and tube current were 40 kV and 70 mA, respectively. The step angle in 2θ was 0.005 • and the measuring time was 500 s/point. Ca Kβ 1,3 spectral lines cannot be separated even by a high-resolution measurement. The 12655.11 (33) 17015.30 (49) observed FWHM is 1.63 eV, and, when this value is corrected for the instrumental function in this x-ray spectrometer [28], it becomes 1.58 eV. In Fig. 10 [52]. This accuracy confirms the very good agreement between our theoretical and our experimental results. The photon energies of Kα 1,2 and Kβ 1,3 emission lines in the present paper are consistent with those of Bearden [29] and Deslattes et al. [51] and the energy value of our theoretical Y Kβ 2 spectral line, although consistent with our experimental value, disagrees with values of those authors [51].
The simulated shapes of the Kβ lines of Se, Y, and Zr are presented in Fig. 3 and as can be seen, the overall agreement with the experimental data is very good for Se with a slight overestimation of the energy of the [1s3d] satellite of around 4 eV. However, the calculated shake probabilities seem to agree very well for Se. For Zr, the intensity ratio of the multiplet agrees very well with the experimental data, but the calculated [1s3d] shake probability is probably overestimated, leading to a shoulder on the high-energy side of the Kβ 1 peak, FIG. 10. The spin doublet energies of Kβ 1 and Kβ 3 lines for elements Se, Y, and Zr. The least-squares fitting was executed using both data in the present paper and Ito et al. [27]. (a) Ito et al. [27]. (b) Hölzer et al. [6]. (c) Anagnostopolos et al. [18]. (d) Pham et al. [26]. The value of the logarithmic function is well consistent with that in Ito et al. [27]. not visible in the measured spectrum. Furthermore, such as in Se, the satellite centroid is shifted towards the high-energy side of the x-ray diagram line. Still, the overall agreement of the line shape is very good. Regarding Y, the agreement is not as good as for Se and Zr as the Kβ 3 /Kβ 1 ratio is around 10% lower than the experimental one, and for the shake structure, we find a similar behavior as in the Zr spectra with a consistent shake probability and a slight shift of the [1s3d] satellite peak toward the high-energy side.

V. SUMMARY AND CONCLUSIONS
Se, Y, and Zr Kα 1,2 and Kβ 1,3 spectra were measured using a high-resolution two-crystal x-ray spectrometer. The values of the energies, FWHM, and ratio are compared with the literature and will contribute to future theoretical research in this field. The correction values for the FWHM of Kα 1 and Kα 2 agree well with the recommended full width at half maximum values of Campbell and Papp [46].
The values of the spin doublet in the Kβ 1,3 of Se, Y, and Zr are found to agree with the extrapolation of what has been reported by Ito et al. [27]. This is very important for the evaluation of the spin doublet observed by the x-ray spectrometer.
In what concerns the asymmetric index in the elements under study in this paper, the contribution of the [1s3d] shake process is observed only for Se. However, on Y and Zr, the effect of the [KN] and [KO] shake processes increases, whereas the effect of [KM] decreases. Therefore, it is difficult to confirm the existence of asymmetry.
The computed spectra in this paper required a huge computational effort due to the very large number of radiative and radiationless transitions. This high number of transitions is needed to calculate all of the parameters for a simulation of the multiplet line shapes in an ab initio way. The simulated spectra are in good agreement with the high-resolution experimental spectra from the double-crystal spectrometer although, for Y, the simulated line shape of the Kβ 1,3 multiplet presents a discrepancy on the intensity ratio of around 10%. From the theoretical results, we see that the [1s3p] satellite lines might also contribute to the peak asymmetry of these elements, calling for more research on the simulation of other transition metals with Z higher than 40.