$K{\alpha }_{1,2}$ x-ray linewidths, asymmetry indices, and $\left[KM\right]$ shake probabilities in elements Ca to Ge and comparison with theory for Ca, Ti, and Ge

We have investigated systematically the $K\alpha$ x-ray spectra in elements from Ca to Ge within Berger's two-Lorentzian functions model, using a high-resolution antiparallel double-crystal x-ray spectrometer, in order to obtain in detail the physical meaning of the asymmetry in the spectral profiles. The overall tendency of the corrected full width at half maximum of the $K{\alpha }_1$ and $K{\alpha }_2$ lines as a function of $Z$, as well as the linewidths, are in good agreement with the data reported in the literature. It is found, from both the experiments and calculation, that satellite lines arising from shake-off appear between the $K{\alpha }_1$ and $K{\alpha }_2$ lines. The asymmetry index of $K{\alpha }_1$ in $3d$ elements from Sc to Zn is ascribed to the existence of a $3d$ spectator hole. Moreover, the observed Sc $K{\alpha }_1$ line shows a symmetric profile unlike the Ti $K{\alpha }_1$ line profile, this discrepancy being explained by the existence of satellite lines on both sides of the Sc profile. Our experimental results yielded around 25% probability for the probability of shake processes creating a second hole in the $3p$ or $3d$ subshells in Sc and around 18% probability for creating a second hole the in the $3p$ in Ca. For the latter process our calculated value, using multiconfiguration Dirac-Fock wave functions and the sudden approximation yielded a 10% probability.

Figure 1

The double flat crystal x-ray spectrometer used in this work. The first and second crystals are located in the (+,+) configuration with the detector. Upon being bombarded by a primary x-ray beam, secondary x rays are emitted from a sample. The secondary x rays are diagram lines characteristic of the constituent elements of the sample. The angle (marked by an arrow) made from the first and the second crystals moves in tune with the first crystal and the sample. Unnecessary x rays (marked by a broken line) are discarded at the second crystal due to being out of Bragg angle. The most suitable crystals should be selected for the samples.

Figure 2

The observed $K{\alpha }_{1,2}$ spectra in elements Ca to Mn are shown with the Lorentzian functions used in the fitting process [6, 9, 11]. These spectra were measured using the antiparallel double-crystal x-ray spectrometer described in the text. $K{\alpha }_{11}$ is the $K{\alpha }_1$ diagram line, and $K{\alpha }_{21}$ is the $K{\alpha }_2$ line. $K{\alpha }_{12}$ and $K{\alpha }_{22}$ satellite lines are due to $2p\to 1s$ electron transitions in the presence of an extra $3d$ hole resulting from shake processes. The $K{\alpha }^{\prime \prime }$ line is a satellite line ascribed to a $3p$ spectator hole.

Figure 3

The observed $K{\alpha }_{1,2}$ spectra in elements Fe to Ge are shown with the Lorentzian functions used in the fitting process [6, 9, 11]. These spectra were measured using the antiparallel double-crystal x-ray spectrometer described in the text. $K{\alpha }_{11}$ is the $K{\alpha }_1$ diagram line, and $K{\alpha }_{21}$ is the $K{\alpha }_2$ line. $K{\alpha }_{12}$ and $K{\alpha }_{22}$ satellite lines are due to $2p\to 1s$ electron transitions in the presence of an extra $3d$ hole resulting from shake processes.

Figure 4

(a) The corrected FWHM (CF) of the $K{\alpha }_{11}$ line of elements Ca to Ge together with the semiempirical values reported by Krause and Oliver [13], and recommended values of Campbell and Papp [55]. The widths of Krause and Oliver are semiempirical values and those reported by Campbell and Papp are recommended values based on experimental results. (b) The CF of the $K{\alpha }_{21}$ line of elements Ca to Ge is shown together with the recommended values [55]. Open circles and thick crosses are the values subtracting the Coster-Kronig broadening effects reported by Nyholm et al. [56]. The CF values for both the $K{\alpha }_{11}$ and $K{\alpha }_{21}$ diagram lines were obtained from the observed FWHM through Tochio's method [17]. (c) Ratio of satellite to diagram line intensities for elements Ca to Ge, compared with our theoretical values and those of Ref. [57].

Figure 5

Values of the asymmetry index of the $K{\alpha }_1$ and $K{\alpha }_2$ lines in elements Ca to Ge are shown in (a) and (b), respectively (see text for details).

Figure 6

The calculated $K{\alpha }_{1,2}$ spectra of Ti, Ca, and Ge are shown in (a) for Ti, (b) for Ca, and (c) for Ge, respectively, together with the observed ones. The spectra include the contributions of the $K{\alpha }_1$ and $K{\alpha }_2$ diagram lines, and the satellite lines (in different colors, identified in the legend) predicted on the basis of the shake processes probabilities, $\left[1s3d\right]$ and $\left[1s3p\right]$ for Ti, $\left[1s3p\right], \left[1s3s\right]$, and $\left[1s4s\right]$ for Ca, and $\left[1s3d\right], \left[1s3p\right], \left[1s3s\right], \left[1s4p\right]$, and $\left[1s4s\right]$ for Ge, respectively. The Lorentzian functions used in the fitting process are also identified, as in Figs. 2 and 3.