Absolute measurement of the relativistic magnetic dipole transition in He-like sulfur

We have made an absolute, reference-free measurement of the $1s2s{}^{3}S_1\to 1s^2{}^{1}S_0$ relativistic magnetic dipole transition in He-like sulfur. The highly charged sulfur ions were provided by an electron-cyclotron resonance ion source, and the x rays were analyzed with a high-precision double-crystal spectrometer. A transition energy of 2430.3685(97) eV was obtained and is compared to most advanced bound-state quantum electrodynamics calculations, providing an important test of two-electron QED effects and precision atomic structure methods in medium-$Z$ species. Thanks to the extremely narrow natural linewidth of this transition and to the large dispersion of the spectrometer at this energy, a complementary study is also performed to evaluate the impact of different silicon crystal atomic form-factor models in the transition-energy analysis. We find no significant dependence on the model used to determine the transition energy.

Figure 1

Experimental pair of dispersive and nondispersive (parallel) spectra for the $1s2s{}^{3}S_1\to 1s^2{}^{1}S_0$ transition in He-like sulfur. (a) Top: Experimental dispersive spectrum for the $1s2s{}^{3}S_1\to 1s^2{}^{1}S_0$ transition in He-like sulfur (black points) with 68 993 integrated counts, fit with a simulated response function and linear background (red curve) with a corresponding reduced ${\chi }^2$ of 1.38. Bottom: The fit residuals. (b) Top: Experimental nondispersive (parallel) spectrum for the $1s2s{}^{3}S_1\to 1s^2{}^{1}S_0$ transition in He-like sulfur (black points) with 15 205 integrated counts, fit with a simulated response function and linear background (red curve) with a corresponding reduced ${\chi }^2$ of 1.36. Bottom: The fit residuals.

Figure 2

Experimental dispersive spectra fit with simulated response functions with different Gaussian Doppler widths, from 0 to 500 meV. The colors from black to red indicate minimal and maximal simulated Doppler widths, respectively.

Figure 3

${\chi }^2$ of the fit to the dispersive spectrum as a function of Gaussian width (black points). The trend has been fitted with an eighth-degree polynomial (red line).

Figure 4

Doppler width as extracted from each dispersive spectrum with the statistical error bars. The weighted average and its uncertainty are shown as the solid line and shaded bar, respectively.

Figure 5

Fitted two-dimensional angular offset function from Eq. (2) and experimental results (white spheres) for an example pair of experimental spectra for the sulfur $M1$ transitions. The fit is performed while taking into account the statistical error bars for each point.

Figure 6

Results of the transition-energy analysis for the different pairs of dispersive and nondispersive spectra recorded during the experiment. The error bars on the points correspond to the statistical, temperature, and angle-encoder measure uncertainties added in quadrature. The black solid line represents the weighted average considering only the statistical uncertainty of each point. The gray shaded region corresponds to the weighted standard deviation. The blue dashed lines indicate the standard uncertainty on the weighted average including both statistical and systematic uncertainties.

Figure 7

Comparison between experiment and theory for the $1s2s{}^{3}S_1\to 1s^2{}^{1}S_0$. $Z=16$: this work, $Z=18$: [17], $Z=20$: [43], $Z=21$: [44], $Z=22$: [45], $Z=23$: [46], $Z=29$: [20]. Theory (2005): [40], Theory (2022): [41].

Figure 8

Si(111) rocking curves for unpolarized radiation $\frac{1}{2}(\pi \mathrm{polarized}+\sigma \mathrm{polarized})$ calculated with the different models used to produce the DCS response functions. The curves have been evaluated for the theoretical energy of the $1s2s{}^{3}S_1\to 1s^2{}^{1}S_0$ transition of 2430.3511 eV taken from Ref. [40].

Figure 9

Bayesian evidence (BE) curves for different Doppler widths for a single experimental dispersive spectrum. Fits are shown with response functions generated with different atomic form-factor models. The BE uncertainties are smaller than the size of the points. The BE trends as a function of width are shown fit with eighth-degree polynomials, from which the maximum evidence and standard BE uncertainty are obtained, shown by the solid lines and colored bars, respectively.

Figure 10

Comparison of the transition-energy results from analyses using different models for the reflective-profile calculation. The error bars on the points correspond to the statistical, temperature, and angle-encoder measure uncertainties added in quadrature. The black dots correspond to the energy values extracted in the analysis using simulations performed with xop [31, 32, 33] calculated profiles, the red triangles are from those using NIST profiles [53], and the blue squares correspond to those using JENA [55] profiles. The black, red, and blue solid lines represent the weighted averages of the different energy values using the respective points of same color. In the weighted-average calculation of the energy for each model, only the statistical uncertainty of each point is considered. The shaded areas correspond to the statistical uncertainty.