X-ray spectroscopy is used widely for laser plasma diagnostics^1–3 and is suitable for investigating inertial thermonuclear plasmas. However, for this diagnostic tool to be applicable, one must know the spectroscopic parameters of various multicharged ions, among which the most important are the wavelengths of radiative transitions. Once measured, these values can be used not only to investigate inertial confinement fusion (ICF) but also across a range of fundamental (atomic physics, quantum electrodynamics, laboratory astrophysics) and applied (e.g., x-ray lithography, medical and biological radiography) fields.

For a given ion, a full set of spectroscopic properties can be obtained only by theoretical calculations. However, only an atomic system with one electron has an exact relativistic equation; the spectroscopic characteristics of ions with two or more electrons are obtained using approximate means such as variational methods or perturbation theory, and in neither case are accuracy estimates available a priori to help choose the most suitable calculation method. The only way to estimate the accuracy of the atomic calculations is to compare (i) the energy-level structure obtained using a particular method for a particular ion with (ii) the wavelengths of spectral lines emitted by this ion, a procedure that requires precise spectroscopic measurements.

The past 100 years have seen investigations aimed at observing, identifying, and measuring the wavelengths of spectral lines emitted by highly charged ions. This activity began with pioneering work on vacuum sparks⁴ and then developed over the 1970s, 1980s, and 1990s with ICF-oriented investigations of hot plasmas in experiments involving, for example, tokamaks,⁵ beam-foil setups⁶ in heavy-ion accelerators, and electron-beam ions traps,⁷ which allowed measurements to be made on a huge variety of highly charged ions including ultra-heavy ones such as H-like uranium.⁸

Since the 1970s, many spectroscopic measurements have also been made using laser facilities that are capable of generating high-intensity laser pulses. Even the moderate (by modern standards) facilities with intensities of no more than 10¹⁴ W/cm² yielded much experimental data on ions with low ionization potential, which exist in plasmas with sub-kiloelectronvolt temperature (e.g., see SPECTR-W3^9,10). However, significantly more powerful kilojoule laser facilities^[11–19] are required to investigate heavy ions, and spectroscopists have gained access to such facilities only recently. Large-scale facilities usually necessitate careful experimental planning because the allocated time (and sometimes even the number of laser pulses) is limited. Therefore, all numerical estimations and calculations should ideally be done in advance so that their results can be compared there and then with the experimental ones and the experimental conditions changed promptly if needed.

To obtain highly accurate wavelength measurements, obviously one must know the dispersion curve of any spectrometer being used, and this can be obtained by observing spectral lines with well-known wavelengths, known as reference lines. However, for a plasma of multicharged ions, the available data on x-ray reference lines are quite limited. Despite there being much experimental data on ions in low-charge states, the accuracy of the measured spectral-line wavelengths is either unavailable or so low that one cannot, for example, choose the suitable atomic-characteristic calculations for the investigated ion.

The best candidates as reference lines are the spectral lines of H-like ions. Calculated analytically considering quantum electrodynamics effects (e.g., see Ref. 20), the wavelengths of these lines are known with a precision better than that achievable experimentally. On one hand, the close ionization potentials of H-like and He-like ions of the same element means that the fraction of H-like ions in the plasma is sufficiently high in only a very narrow temperature range. On the other hand, the significant difference (approximately fourfold) in the He- and Li-like ionization potentials means that the concentration of He-like multicharged ions is high in a wide temperature range. A temperature that is sufficiently high for ionizing an Li-like ion (a process preceding the creation of the He-like state) is too low to ionize a He-like ion. Consequently, the spectral lines of He-like ions are the ideal reference data if their wavelengths have been measured sufficiently accurately based on, for example, the reference line positions of H-like ions.

Herein, we show how to determine with high accuracy the radiative spectrum of He-like K XVIII ions, including dielectronic satellites due to transitions in the Li-like K XVIII ions, by using spectral lines of Al XIII and Si XIV H-like ions as reference lines. Of course, this approach could be implemented for many other He-like ions, and the precisely measured positions of He-like-ion lines could be used as references for investigating ions with more-complicated energy-level structures with ground states corresponding to partially filled L and M shells.

The proposed experiment is aimed at measuring the x-ray spectral-line wavelengths emitted by various K ions. For such measurements, one must create the investigated multicharged ions in excited states and then register their emission via a calibrated spectrometer. The feasibility of generating the required multicharged ions is determined by the parameters of the lasers used, in particular their energy and intensity, and the required high-precision measurements can be made by using a focusing spectrometer with spatial resolution (FSSR)^21–23 with unique properties.

As mentioned above, precise wavelength measurements require the spectrometer dispersion curve to be known exactly. Generally speaking, it can be obtained analytically by using geometrical optics, but such calculations involve a cumbersome procedure to obtain various parameters associated with the relative positions of the spectral source (the plasma), dispersive crystal, and detector. Even if the dispersion curve is derived exactly for the ideal experimental geometry, unavoidable uncertainties associated with the spectrometer alignment introduce substantial errors. Based on much research experience, the absolute accuracy of the wavelength measurements cannot better several tens of milliångstroms unless additional spectral references (distinct lines with well-known wavelengths) are used.

The error can be reduced significantly (down to several milliångstroms) by combining the calculated dispersion curve with a reference line, thereby avoiding one of the main sources of measurement error, namely determining inaccurately the absolute spectral-line positions on the detector. A further improvement in measurement accuracy (reaching several tenths of a milliångstrom) results from using at least three reference lines, this being because, for the chosen spectrometer observation range, the dispersion curve can be fitted very precisely (with error less than 0.1 mÅ) with a parabolic function whose coefficients are determined easily from three reference lines. In that case, the accuracy of the wavelength measurement is independent of FSSR parameters such as the dispersive-crystal interplanar spacing and geometry (degree of sphericity) and is affected only by the signal-to-noise ratio of the detected signal and the natural width of the crystal rocking curve,²³ which results in additional line broadening. For example, such an approach has been used for high-precision measurements of the spectral-line wavelengths of Kr multicharged ions.²⁴

Herein, we propose using transitions in the H-like ions Al XIII and Si XIV as reference lines. Being known accurately, they should allow us to obtain the radiation spectra of the He-like K XVIII ion, including dielectronic satellites that arise because of transitions in the Li-like K XVII ion (hereinafter referred to as Li-like satellites), with accuracy no worse than 0.6 mÅ. For that purpose, targets with a complex chemical composition were irradiated by laser pulses. The plasma kinetic calculations presented herein show that all essential lines, both investigated and reference ones, should be excited effectively in the plasma at the same electron temperature, thereby making the proposed experiment feasible. Note also that there is no blending of the measured lines and those emitted by the reference material.

A highly abundant feldspar, namely KAlSi₃O₈ (microcline or orthoclase²⁵), can be used as the laser target. The radiation spectrum of the plasma, which comprises the aforementioned elements, is shown in Fig. 1. The calculations were performed using the PrismSPECT radiative-collision code²⁶ using the steady-state approximation for the electron temperature T_e = 1000 eV and the electron density N_e = 10²¹ cm⁻³, i.e., the critical electron density for the laser wavelength of 1 µm. The temperature was chosen to provide comparable intensities of the measured and reference spectral lines.

The dispersive elements (spherically bent crystals) of the x-ray spectrometer that is used may reflect radiation effectively in different diffraction orders, so Fig. 1 shows how the spectral-line intensities depend on nλ (λ - wavelength) for n = 1–5. The spectrometer was aligned to observe wavelengths between λ₁ and λ₂ in first-order reflection, and also spectra emitted between λ₁×n and λ₂×n. This is very important because it extends significantly the range of applicability of the reference lines. If the measured and reference lines are registered in different orders of reflection, then one must consider the difference between refraction indices for significantly different wavelengths. This is usually done by including an effective interplanar distance (2d)_n that depends on the reflection order (e.g., see Ref. 27). Figure 1 shows that the lines of K and the reference ions can be registered simultaneously in different spectral ranges (as mentioned already). Two cases are considered below: the first anticipates an application involving an α-quartz crystal, and the second a mica crystal.

Consider a spherically bent α-quartz crystal with Miller indices (100) (2d = 8.512 Å), radius of curvature R = 150 mm, located 626 mm from the plasma source, and aligned according to the one-dimensional²⁸ scheme with a central wavelength of 6.55 Å. In that case, the wavelength range of 5.8 Å–7.17 Å [see Fig. 2(a)] would be observed, and the corresponding length of the spectrum on a detector would be 29 mm. The precision of measuring the spectral-line positions is limited by detector pixel size. Widely used image plates (such as TR by Fujifilm) can be scanned with a spatial resolution of up to 25 µm, giving the accuracy of wavelength measurements as 1 mÅ. The accuracy improves to 0.5 mÅ when using a CCD camera with a pixel size of 13.5 µm, and to 0.26 mÅ for a fine-grain x-ray film. For K spectral lines, the accuracy would be two times better given that in the proposed scheme they undergo second-order reflection. Figure 2(a) shows that high-precision data can be obtained for the 1s2p → 1s² and 1s3p → 1s² resonance lines of K XVIII ions and their dielectronic satellites.

Consider a spherically bent mica crystal (2d = 19.9149 Å) with radius of curvature R = 150 mm, located 440 mm from the plasma source, and aligned according to the two-dimensional²⁸ scheme with a central wavelength of 13.5 Å. In that case, the wavelength range of 11.65 Å–15.11 Å [see Fig. 2(b)] would be observed.

The corresponding size of the spectrum on a detector is 57 mm, and the accuracy of measuring the spectral-line positions is again limited by the detector pixel size. With an image plate, the wavelength measurement accuracy is 1.5 mÅ for first-order reflection, improving to 0.8 mÅ with a CCD camera with a pixel size of 13.5 µm, and to 0.36 mÅ with a fine-grain x-ray film. The accuracy for K spectral lines would be four times better (and approximately that with the quartz crystal) given that in the proposed scheme they undergo fourth-order reflection. Figure 2(b) shows that high-precision data can be obtained for the 1s2p → 1s² and 1s3p → 1s² resonance lines of K XVIII ions and their dielectronic satellites.

As mentioned above, the measured and reference spectra have close intensity values under steady-state conditions with temperature T_e = 1000 eV, thereby facilitating high-precision measurements. Results of time-dependent plasma kinetics calculations using PrismSPECT show that the plasma ionization becomes steady after approximately 200 ps, which means that the optimal way to create the plasma is to use subnanosecond laser pulses with an on-target intensity of approximately 10¹⁶ W/cm². Filters that block radiation in the ultraviolet and visible parts of the electromagnetic spectrum are usually used during the experiments. These filters are also not fully transparent for soft x rays, and their opacity is higher for the softer reference spectra. Therefore, a slightly lower plasma temperature (600–800 eV) may be required to adjust the intensities of the registered lines, thereby facilitating more-moderate laser intensities of between 10¹⁵ W/cm² and 10¹⁶ W/cm². Note that for a 100-µm laser focal spot, the laser pulse energy should be approximately 200 J.

Regarding the possibility of using lasers with shorter pulse duration for plasma generation, note that the plasma ionization would then not become steady. However, if the temporal intensity dependencies of the investigated and reference lines are quite similar, then even a non-steady-state plasma could be used for x-ray spectral measurements.

Figure 3 shows the results of calculations performed with a time-dependent kinetic model for a plasma with electron temperature T_e = 1000 eV, electron density N_e = 10²¹ cm⁻³, and containing only singly ionized ions at time t = 0. As shown, steady ionization is reached after 0.2 ns, but before then the intensities of all relevant lines have comparable values after only 30 ps–40 ps. This means that not only subnanosecond but also picosecond lasers can be used for the described experiment.

The feasibility of the discussed experiment was demonstrated at the PALS laser facility.²⁹ In the experiment, a laser pulse with a duration of 480 ps and a full energy of 600 J was focused on the surface of a flat and approximately 1-mm-thick orthoclase plate to a spot with a diameter of 100 µm. The indicated laser parameters correspond to intensity I = 1.5 × 10¹⁶ W/cm². The spectra were obtained using an α-quartz crystal (100) spherically bent to a radius of 150 mm. The FSSR was installed close to the target normal to ensure symmetry of the observed line broadening due to the macroscopic Doppler shift induced by accelerated ions. Fuji IX80 x-ray film³⁰ covered by a 1.5-μm-thick Al foil to avoid exposure from visible light was used as an x-ray detector. The recorded spectra were digitized using an Epson V700 scanner with a resolution of 4800 dpi (pixel size 5.29 µm). The recorded spectrum is shown in Fig. 4.

The set of observed lines agrees exactly with Fig. 2(a) and contains three H-like ion spectral lines, namely Ly_α (λ = 7.170 908 Å), Ly_β (λ = 6.052 53 Å) of Al XIII, and Ly_α (λ = 6.180 428 Å) of Si XIV, thereby allowing us to obtain the exact dispersion curve and measure the positions of the He-like K-ion spectral resonance lines. The wavelengths indicated in parentheses are theoretical values calculated for isolated ions. In a plasma, their observed positions can be shifted by the Doppler effect, asymmetrical Stark broadening, shape distortion by plasma, or laser satellites. For the last of these, higher laser intensity is required. Stark broadening is symmetrical for the given experimental values of electron density and does not cause any shifts. The Doppler-effect influence can be significant, but the shifts should be the same for all observed lines because the investigated and reference ions move similarly. Therefore, none of these effects can affect the measurement results.

The Li-like satellites comprising three overlapping peaks correspond to different groups of transitions in Li-like K ions, which are labeled according to the notation introduced in Ref. 31. All the measured spectral-line positions are listed in Table I. The values were obtained under the assumption that the observed peaks have Gaussian shape. The measurement accuracy is considered as being twice the pixel width at the center of a particular peak. The Li-like satellites comprise three overlapping peaks corresponding to different groups of transitions in Li-like K ions, thus the measured wavelengths correspond to the group centers.

The wide abundance of different mineral targets of various chemical compositions opens broad possibilities for extensive application of the approach described herein. Our view is that the first targets to be investigated should be those that facilitate high-precision data for multicharged He- and Ne-like ions with fully filled inner shells. Such ions exist in a wide temperature range and have a rather simple spectral structure. The measured wavelengths of those spectral lines could then be used as references in studies of the spectral emission from other ions that have more-complicated emission features.