2.1. The Intensity Ratio in the Elements from Mg to Cu
In contrast to the
intensity ratio, almost all published measurements of the
intensity ratio were performed using semiconductor detectors (SD), since the energy difference between the
and
complexes, 500–900 eV for the transition elements, is considerably larger than the 200 eV energy resolution of the SD. We were able to find in the literature only a single experiment using a high-resolution crystal spectrometer (CS) [
9] with a SD.
The double-crystal spectrometer (DCS) with the proportional counter is much more suitable than CS for excitation by photons, due to its much higher resolution, and the almost absence of background due to bremstrahlung, allowing for a clear separation of X-ray satellite lines, as seen in
Figure 1. As can be seen from that figure, the background due to the photon excitation is almost negligible.
The intensity ratios and asymmetries of the
lines for each element, obtained in this work, are shown in
Figure 2a,b, respectively, together with previously reported data. The asymmetry index is defined as the ratio of the width of the low-energy part to the width of the high-energy part of the half-width [
20,
21]. Generally, the asymmetry index is larger when using higher order Bragg reflections. If we use Si(220) and Si(440) crystals in the X-ray spectrometer, the asymmetry index should be larger in the latter, as the influence of the profile base is smaller. The DCS measured
(the
lines resulting from
double ionization) and
spectra were recorded in only a single run for each element, with the exception of Mg, Al, and Si
(the
lines resulting from
triple ionization) and
spectra. As can be seen from
Figure 2a, in the case of
elements the
intensity ratio fluctuates around the purely statistical 0.5 value due to the contribution of the [
] or [
] shake processes. Although Hölzer et al. [
9] obtained the
and
intensity ratio for each element, they are not referenced in
Figure 2a since their intensity ratios were obtained by a different fitting method. Chantler et al. [
22] also increased the number of peaks in order to obtain zero residuals in the multiple peaks fitting of the
lines and did not obtain an intensity ratio taking into account the contribution of the shake process.
Using an asymmetry fitting analysis, the area intensities of
,
,
(satellites resulting from the [
] shake processes),
, and
were determined to obtain the
or
intensity ratios, as shown in
Figure 1. Hölzer et al. [
9] obtained
intensity ratios, from Cr to Cu, that may be compared with our
intensity ratios. From Fe to Cu the difference between the theoretical ratios and the corrected measured intensity ratios increases. This may result from the fact that the experimental data include satellite intensities unlike the theoretical calculations. We found that the inclusion of all satellites in the
intensity ratios calculations increases the results by 69.2% for Al, 3.79% for Ca, 3.01% for Ti, and 3.65% for Cu, for instance.
Figure 2a shows a large spread in the
intensity ratios, about
for Sc to V, and about
for Mn and Fe. The values of this ratio for elements Sc to V can be attributed to the contribution of the [
] shake process, which leads to the
satellite, that can be seen, for example, in the Ti
line of
Figure 1, because this shake process contributes to the higher energy side intensity of each diagram line. In other words, the asymmetric indices of the
spectra is less than
because there is no contribution from the [
] shake process in Ca and very little [
] contribution in Cr. For elements above Cr, the
spectral lines are influenced by the contribution of the [1s3d] shake process, so that the
intensity ratio for Mn and Fe is about
. The contribution of the [
] shake process to the lines in the figure is on the low energy side. Compared to the variation of the
intensity ratio of the
elements, those of Mg, Al, and Si are relatively small, and the hidden satellites due to the shake process do not affect this ratio so much. In contrast, those of
elements are affected by the hidden satellites.
The formula used for the corrected ratio (corr) as a function of the measured ratio (meas) is
where
S,
D, and
R are the self-absorption, the detector efficiency, and the crystal integrated reflectivity, respectively. The corrected values for the
intensity ratios were calculated according to Equation (
1), and are given in
Table 1. For example, in the case of Cu the measured intensity ratio is
, the detection efficiency for the
line is
, and that for the
lines is
. The detection efficiency ratio
is therefore
. The self-absorption ratio
for each of those lines, and the reflection intensity ratio
for two reflections. The correction value is therefore 0.1487. The correction values for the other elements were obtained using this procedure.
The
and
lines were measured simultaneously. Furthermore, the Cu
line intensities were repeatedly measured for each
line measurement, in order to check the intensity variation in the spectrometer. This allowed for a better assessment of the intensity. The uncertainty in the peak positions of Cu
and
emission lines was estimated as less than 0.1%. The energy values of the
lines determined by the multiple peak analysis are in good agreement with those of Bearden [
27], and Deslattes et al. [
28] (
Table 2).
The ability to accurately measure X-ray profiles is one of the features of this wide-area scanning double-crystal X-ray spectrometer. For Cu measurements, the uncertainty of the intensity ratio is about , while the uncertainty of the intensity ratio is about . The normalized intensity of the and lines were calculated by fitting asymmetric Lorentz functions to each element, except for Mg, Al, and Si, where symmetric Lorentz functions were used due to the lack of asymmetry in the diagram lines of these elements.
Table 1 shows the corrected
intensity ratios for each element in this study, together with the values of Salem et al. [
25], Ertugrul et al. [
26], Hölzer et al. [
9], the recommended values of Hamidani et al. [
1], obtained from the average over a large number of previous experimental values, and the theoretical values obtained in this work, and by Scofield [
3]. Our systematic MCDF (GRASP) calculations were performed for various valence electronic configurations (
and
, where
n is the number of valence electrons) for
elements. On the other hand, the MCDFGME calculations were performed including only the diagram line intensities and, in selected cases, the diagram and satellite intensities, as shown, in the 11th and 12th columns.
lines of all 3rd period elements result from electron transitions from an occupied state in a valence orbital (valence band) to an inner shell ( orbital). In the free neutral Mg atom, only orbitals (valence electrons) are occupied. However, for metallic Mg, and due to the low energy difference between the and levels, the latter are also contained in the valence band (occupied band). The hybrid nature of these orbitals allows that electron transitions between and levels are possible in metallic Mg, leading to the presence of lines in the measured spectra. This possibility was not taken in account in our theoretical calculations.
In
Figure 3, our
intensity ratio values are compared with previous theoretical results and experiments employing electron or photon excitation only, since the pronounced multiple ionization that occurs in the case of heavy-ion excitation strongly modifies this ratio [
1,
4]. In the same figure, the recommended values of Hamidani et al. [
1] are also plotted.
As seen in this figure, the intensity ratio increases comparatively abruptly from Mg to Si and then to Ca, and the trend becomes slower for
elements. The reason for this may be related to the exchange interaction between
and
electrons, and between
and
electrons. The calculated
intensity ratios are in good agreement with the recommended experimental and theoretical values. The intensity ratios, measured in this study, of the
elements, except for Ca, are exponentially fitted to the solid lines shown in
Figure 3. For atomic numbers above Fe the experimental values of the present study agree well with those of Hölzer et al. [
9] obtained with a high-resolution single-crystal spectrometer, although they also show differences from previous experimental and calculated values, as well as from the calculated values obtained in the present study. Moreover, the two solid curves in
Figure 3 are linear fittings of the measured
intensity ratios from Mg to Si, and from Sc until Cu.
The slope of the
intensity ratio increase changes around the onset of
elements. It is noteworthy that for these elements, the ratio agrees better with the value of the
coordination interaction marked by open circles in
Figure 3, calculated in this study, than with the recommended average value. This value is slightly different from the one obtained by Scofield, and is the first calculation that shows the need to take into account the configuration interaction between
and
electrons.
Our method is presented as a useful method for safety and security: trivalent Cr is relatively safe, but hexavalent Cr can cause dermatitis and tumors if left adhering to the skin and mucous membranes. Drinking contaminated well water causes vomiting. In addition, we are surrounded by Cr-plated metals, such as Cr-plating. Thus, in order to identify the amount of hexavalent Cr present, a method to identify trivalent and hexavalent Cr is an urgent issue.
The aforementioned asymmetry index, chemical shifts, and FWHMs are used to evaluate the chemical effects of Cr compounds, as follows.
2.2. Cr Compounds
The measurement conditions are shown in
Table 3. The X-ray spectra were measured with a high-resolution double-crystal spectrometer. In this spectrometer, unlike the case of a single crystal spectrometer, the horizontal broadening is suppressed by the second crystal, so it is sufficient to use only the slit to suppress the vertical broadening. The spectrometer used in this study is equipped with a Soller slit with a slit length of
and a spacing of
between each layer. The detector was a gas flow proportional counter, and PR10 gas Ar
(CH
)
was used. The measurement time per point for
and
spectra, without
satellites was
for the metal and Oh-symmetric compounds, and
for the Td-symmetric compounds, due to their rather weak intensity (as seen in
Figure 4). The
step angle of this spectrometer is 0.0005°. The measurement time per point for
spectra are
for Cr metal and Cr
O
,
for K
Cr
O
, FeCr
O
, and CoCr
O
, and
for K
CrO
. The
step angle of this spectrometer is 0.002°. Since the
spectral lines of the Cr compounds are extremely weak, the
lines could be measured separately.
The results of fitting in each measured spectrum with two asymmetric Lorentz functions are shown in
Table 4. As mentioned above, spectra measured with a double-crystal X-ray spectrometer are less affected by the instrument function. The uncorrected values for the broadening effect of the instrumental function are given in
Table 4. The reason for not correcting is that, for example, the
peak of an Oh-symmetric compound may be likely to contain some peaks, each of which is affected as shown in
Figure 4, and it is difficult to distinguish the peaks in the spectral profile. Therefore, we used the values obtained by analyzing the observed data with asymmetric fitting, without any corrections for the instrument function. The present measurement results are in close agreement with previous ones [
29,
30,
31,
32], such as the asymmetry and large half-width of the
peak in Oh-symmetric compounds [
29,
32], and the asymmetry and large half-width of the
peak in Td-symmetric compounds [
29,
31]. In comparison with the results of [
29] (Cr metal, K
CrO
, Cr
O
only) (
Table 5) on the energy shifts of
and
peaks with respect to the metal, they are qualitatively consistent in many aspects, such as the energy shifts of
and
, and the energy difference between
and
peaks.
In addition, the compound effect for Cr is clearly seen in the
intensity ratio (see
Table 4). In other words, the compound with Td symmetry and the compound with Oh symmetry can be distinguished. Moreover, the changes in the values of the
intensity ratio can be associated not only with symmetry but also with the valence of Cr in these compounds. In the K
CrO
and K
Cr
O
compounds, Cr has a valence equal to +6, but for the remaining compounds (in
Table 4), Cr has a valence of +3. It turns out that the
intensity ratio calculated by us (using GRASP package) for Cr with the valence of +6 (3d
4s
configuration) is 0.1610, and for Cr with the valence of +3 (3d
4s
configuration) is 0.1395. As can be seen, the obtained theoretical values are in good agreement with the experimental results presented in
Table 4. Similarly, for metallic Cr, our calculated
intensity ratios are 0.1333 and 0.132, as seen in
Table 1 in agreement with the experimental value of 0.132.