Absolute measurements and simulations of x-ray line energies of highly charged ions with a double-crystal spectrometer

A double-crystal spectrometer (DCS) has been recently constructed and used with the electron cyclotron resonance ion source (ECRIS) of the Source d'Ions Multichargés de Paris (SIMPA). This spectrometer provides absolute and accurate measurements of x-ray energies from inner-shell transitions of highly charged ions (HCI) inside the ECRIS plasma. These measurements can reach accuracies of a few ppm's and are sensitive to quantum electro dynamic (QED) effects in transition energies of HCI, such as two loop corrections ([1, 2] and references therein).

The SIMPA-ECRIS is jointly operated by the Laboratoire Kastler Brossel and by the Institut des NanoSciences de Paris and a detailed description of this source of HCI can be found elsewhere [3]. Unlike other DCSs (e.g. [4]), the SIMPA-ECRIS has a feature of both crystals and the detector being mounted on a single precision-machined optical table that rotates around the first crystal axis. This setup is necessary since our x-ray source (ECRIS) is fixed. A detailed description of the SIMPA-DCS can be found in [5].

In a DCS, the first crystal being kept at a fixed angle acts as a collimator and defines the direction and the energy of the incoming x-ray beam. The outgoing x-ray beam is then analyzed by the second crystal. A first peak is obtained by scanning the second crystal angle when the two crystals are parallel (non-dispersive mode). In this setting, the peak is not dispersive in energy. A second peak is obtained when both crystals deflect the beam in the same direction, which is dispersive in energy (dispersive mode, see figure 1). The difference in angle settings of the second crystal between the non-dispersive and the dispersive modes is directly connected to the crystal Bragg angle. The rotation of the crystals can be set and measured to sub-arcsecond precision with angle encoders. Figures 2(a) and (b) show spectra in non-dispersive and dispersive modes obtained with the DCS. The spectra presented in these figures clearly show that even a relatively small, permanent magnet ECRIS provides high enough intensities for precision measurements of transitions in HCI with a DCS.

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The peak in figure 2(b) corresponds to the 1s2s ³S₁ → 1s^{2 1}S₀ relativistic M1 transition in He-like argon and was measured with a 2.5 ppm accuracy without reference to any theoretical or experimental energies [2]. The final measured value is 3104.1605(77) eV, which is 1.6σ above the most recent ab initio QED theoretical value [6]. A comparison of our measured value with other theoretical values can be found in [2].

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We have developed a ray-tracing program to obtain line profiles for the DCS, in the dispersive and non-dispersive modes. The results of this simulation program are used to analyze the experimental data. The program is based on the Monte-Carlo method and includes all relevant geometrical components of the experiment, along with the crystal reflectivity curve calculated by dynamical diffraction theory using x-ray oriented programs (XOP) code [7]. This makes the simulation code capable of taking into account multiple reflections in the crystal, and considers various corrections to the Bragg law, such as index of refraction corrections and energy-dependent absorption. With assigned distribution functions (Gaussian and Lorentzian functions) to the selected energy of the randomly created photon bunches, the simulation is capable of providing a line width analysis for our measurement results. The experimental data presented in figures 2(a) and (b) were fitted with several simulations containing different Gaussian width. Only the dispersive mode is sensitive to the Gaussian widths. The simulation with the optimum width is represented in figure 2(b) by the red line. The Gaussian distribution takes into account the Doppler effect in the energy of the photons emitted by the moving thermal ions. This motion is isotropic since the ions are formed and trapped at the B-minimum potential of the ECRIS (see [3] and references therein). In this case the average velocity is null, which is not the case for experiments performed with ion beams at storage rings. During the measurements no extraction of the ions was performed. Therefore, the Doppler effect makes a broadening of the energy lines and not a shift. The Lorentzian distribution takes into account the natural broadening of an energy transition. By performing a line width analysis on measured results, it is possible to obtain plasma temperatures and natural line widths [8, 9].

The ray tracing program consists of a succession of three xyz (orthogonal) coordinate systems that follow the central line (see figure 3), which is the line connecting the geometrical centers of the different components of an ideally aligned spectrometer. Each randomly generated ray will be represented in these coordinate systems within the three different parts of the experiment.

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The uncertainties associated with the DCS are estimated with the help of the simulation program. The energy deduced from simulated spectra, with various parameters varied, is evaluated and compared with the simulation input energy. The contributions from geometrical and diffraction profile uncertainties are listed in table 1.

The alignment procedure provides an uncertainty of 0.01°, which is due to crystal tilts or vertical misalignments and corresponds to 0.2 meV uncertainty in energy. To estimate possible uncertainties due to fluctuations of the plasma size, we performed two simulations for an x-ray plasma diameter of 12 mm (diameter of the collimator) and another for a 6 mm plasma diameter. We found a difference of 1.3 meV, which is used as a largely over-estimated uncertainty. Besides XOP, another code, X0h [10], was used in order to obtain the reflectivity curve. Moreover, we use the capacity of XOP to allow choosing different form factor values of the crystals. By comparing simulations performed with the diffraction curves from the two different programs and with the different form factors, we obtained an uncertainty of 2 meV for the diffraction profile. The uncertainty due to unknown polarization of the x-rays was also estimated using the simulation code. We performed two simulations: one with a diffraction profile containing only the σ polarization and another with σ + π polarization (unpolarized). From their difference, a maximum uncertainty of 1.4 meV was estimated due to the presence of any polarized light. The integrated reflectivity in π polarization is 6% of the σ polarization. This would lead to roughly 230 times fewer counts. The width of the π polarization profile is roughly 30% less than the σ polarization. The agreement between experimental profiles widths and simulations widths performed for unpolarized x-rays was found to be excellent. This confirms, within the statistical uncertainty in the experimental spectra, that the x-ray beam from the ECRIS is not polarized and justifies the uncertainty listed in table 1. Other sources of negligible uncertainties can be found in [5].

The third category is due to uncertainties in the knowledge of fundamental constants and crystal properties such as the lattice spacing as well as due to instrumental uncertainties, such as the encoder and the crystal temperature uncertainties.

We experimentally demonstrate that a combination of an ECRIS and a DCS provides absolute ppm measurements of x-ray line energies of transitions in He-like Ar. Several uncertainties were evaluated with the use of an ab initio simulation developed by our group. This simulation describes very accurately the experimental line shapes and thus it is possible to obtain plasma temperatures and natural line widths.

Laboratoire Kastler Brossel is 'UMR no. 8552' of the ENS, CNRS and UPMC. The SIMPA ECRIS was financed by grants from CNRS, MESR and UPMC. The experiment is supported by grants from BNM 01 3 0002, the ANR ANR-06-BLAN-0233 and the Helmholtz Alliance HA216/EMMI, the FCT (through projects PEstOE/FIS/UI0303/2011 and PTDC/FIS/117606/2010), the PESSOA Program no. 441.00, the Acções Integradas Luso- Francesas (no. F-11/09) and the Programme Hubert Curien PESSOA 20022VB. MG acknowledges the support of FCT, under contract no. SFRH/BD/38691/2007. PA acknowledges the support of FCT, under contract no. SFRH/BD/37404/2007 and the German Research Foundation (DFG) within the Emmy Noether program under contract no. TA 740 1-1.