**1. Introduction**

The long-wavelength range of the bremsstrahlung (BS) spectra has been studied for a long time because it is the region where the famous "infrared catastrophe" arises [1]. The relative intensity of BS in the long-wavelength range was first measured in [2] on thin metal targets bombarded by low-energy electrons. The experimental data were described well by the nonrelativistic Sommerfeld theory [3] which, in fact, is the theory of BS on a Coulomb center, accounting for the exponential screening of nuclei by the atomic electrons [4]. BS spectra play an important role in engineering and technical applications such as ion beam monitoring [5], plasma diagnostics [6], medical tomography imaging [7], etc.

In addition to the BS photon spectra, the dependence of BS cross sections on the scattering electron energy at a fixed photon energy (isochromatic spectra) is also of interest. These spectra were first measured for solid targets in [8], and the results are cited in a well-known monograph [9]. Korsunsky et al. [8] observed an increase of the BS probability from the zero value (at the minimal possible electron energy equal to the fixed photon energy) and a decrease of the BS probability at higher electron energy. A similar shape of the isochromatic spectra was registered for atomic targets in [10,11] in which the quasi-resonance character of the isochromatic spectra was observed. It should be noted that the detection technique for the isochromatic BS soft X-photons was proposed in [12,13]. A technique for detecting isochromatic BS photons in the UV range was proposed earlier in [14].

An analysis of the Sommerfeld [3] formula performed in [10] showed that an isochromatic spectrum maximum is achieved at the initial electron energy, *Ei*, equal to the following:

$$E\_i^{\text{max}} = 1.53 \,\text{\AA}\omega \,, \tag{1}$$

where *ω* is the photon frequency.

It should be noted that the knowledge of BS cross sections on atomic targets is important for studying BS generated in electron scattering on molecules [15] and clusters [16].

The "atomic" BS data are also used in Monte Carlo simulation of BS on solid targets (see, e.g., [17,18]).

All the above-cited works studied relative BS intensities only. At the same time, there exist some absolute BS cross-section value measurements. For example, high-precision (5.5%) absolute cross sections of BS on thin C, Al, Te, Ta, and Au targets have been recently measured in [19], which also contains references to earlier absolute BS cross-section measurements on solid targets.

To the authors' best knowledge, the first absolute BS intensity measurements on atoms (Ne, Ar, Kr, and Xe) were performed in [20] for incident electron energies ranging from 28 to 50 keV. The experimental results were shown to be in qualitative agreemen<sup>t</sup> with the theory that takes into account not only the traditional BS mechanism (developed by Sommerfeld [3] for the nonrelativistic case and by Bethe and Heitler [21] and Sauter [22] for the relativistic case) but also the polarization bremsstrahlung (PBS) theory [23–25], which takes into account the photons emitted by the electrons of the atomic target. However, the quantitative difference between the experimental and theoretical results was quite significant, and this discrepancy increased with the decrease of BS photon energy. Similarly, discrepancy exists between the experimental cross sections measured in [19] and the data tabulated in [26,27], which also increases with the decrease of the BS photon energy. García-Alvarez et al. [19] noted this fact as one important result of their study. Similar discrepancy between theory and experiment was also observed in recent calculations [28].

However, the authors in [19,20] did not consider BS for photon energies in the ultrasoft X-ray region. Therefore, the absolute BS cross sections measured in [12,13] in the lowenergy photon energy range are of grea<sup>t</sup> interest. These authors recorded the BS photon spectra for 600 eV electrons scattered on Ar, Kr, and Xe atoms, as well as the isochromatic spectra for the electron energies from 0.4 to 2 keV. Their results differ by 3–4 times from the calculations by Pratt et al. [29,30], who used the radial electron wavefunctions obtained in partial-wave series by numerically integrating the radial Dirac equation with a relativistic self-consistent screened potential. In recent calculations [28], the Dirac equation for the continuum wave function was solved by the power-series method with the interaction potential obtained from the Kohn–Sham density functional theory.

In the present work, we develop an interpretation of the experimental results of [12,13] using the soft-photon approximation (SPA); the validity conditions are discussed below. The general SPA formulas are presented in Section 2. SPA was used for interpretation of the ultrasoft X-ray spectra recorded in [31,32]. The BS photon spectra were measured in [12,13] for Ar, Kr, and Xe gaseous targets. Two types of BS spectra were recorded:


The results are discussed in Section 4. The main conclusions are given in Section 5.
