**4. Discussion**

In the geometry of the experiment [13] (Θ = 83◦), according to Equation (5), the main contribution into the BS cross section (8) is made by the following term:

$$\left(\frac{\mathrm{d}\sigma\_{\mathrm{BS}}}{\mathrm{d}\Omega\_{\gamma}}\right)\_{0} = \frac{\alpha}{4\pi c^{2}} \frac{\Delta\lambda}{\lambda} \, v\_{f}^{2} \sin^{2}\Theta \, \mathcal{J}\_{02}(E\_{i}),\tag{10}$$

where the factor J02(*Ei*) is defined by Equation (9). The *Ei*-dependence in Equation (10) is due not only to the *v*2*f*factor but also to the J02(*Ei*) dependence.

For the total (scattering + impact ionization) cross section *<sup>σ</sup>*tot(*Ei*), this dependence on *Ei* in the range 0.5–2.0 keV, according to [36], can be approximated by the following equation:

$$
\sigma\_{\rm tot}(E\_i) \sim E\_i^{-B},\tag{11}
$$

where parameter *B* varies from 0.90 for He to 0.59 for Xe. These values make the slow dependence (11) unable to compensate the linear increase of *v*2*f* . However, the dependence of J02 on *Ei* can differ from (11). Of course, the discrepancy between theory and experiment can be also due to inaccuracy of SPA in this case.

In the case of argon (see Figure 3), the maximum in the isochromatic spectrum measured in [13] appears at the electron energy, *E*exp<sup>t</sup> *i* = 0.7 keV, which is greater compared to that given by Equation (1). For the energies less than 0.7 keV, our theoretical results are in satisfactory accord with the experiment. However, theoretical dependence of BS cross section on *Ei*, obtained using the differential cross section of elastic scattering from [34] has a less sharp maximum at *E*theor *i* = 1.11 keV. When the differential cross-section data from [35] are used, the maximum appears at *E*theor *i* = 1.31 keV. In both cases, the calculated cross section decreases with *Ei* slower compared to the experiment. Unfortunately, we have no explanation of such a behavior of the cross section. We note that using the elastic cross sections from [35] results in lesser BS cross section compared to using the data from [34] that make better agreemen<sup>t</sup> with the experiment. This can be caused by the fact that the results in [35] give (i) a more sharp maximum at the forward differential cross section and (ii) less magnitude of the cross section in the *θ* > 15◦ domain, as compared to [34] (see also Figure 2). It is this angular domain that makes the main contribution into the value of J02(*Ei*) in (9) due to sin<sup>3</sup> *θ* in the integral. Thus, in the case of Ar, we have quantitative agreemen<sup>t</sup> with the experiment at the electron energy *Ei* < *E*exp<sup>t</sup> *i* = 0.7 keV, and a qualitative agreemen<sup>t</sup> for higher energies.

In the case of krypton (see Figure 4), the experimental isochromatic spectrum [13] has the maximum at *E*exp<sup>t</sup> *i* = 1.0 keV which is higher than the value given by (1) as well. Our BS cross section calculated using the differential cross section of elastic scattering from [34], slowly increases in the studied energy range, *Ei* = 0.4–2.0 keV. Unfortunately, the authors in [35] do not provide differential cross sections of elastic *e*–Kr scattering, so our calculations for Kr can achieve only qualitative agreemen<sup>t</sup> with the experiment yet.

An even worse situation takes place for xenon (see Figure 5). Both experimental and theoretical isochromatic spectra increase with the electron energy, *Ei*, the former [13] increasing much faster than the latter. Extrapolating the Ar and Kr data, one can suppose that *E*exp<sup>t</sup> *i* increases with the nuclear charge of the atomic target. Probably, one has *E*exp<sup>t</sup> *i* > 2 keV for Xe. A similar discrepancy between theoretical and experimental BS spectra in *e*–Xe scattering with fixed electron energy, *Ei* = 0.6 keV, was noted in our earlier work [33]. A possible explanation of this discrepancy could be due to greater role of so-called polarization BS [23,31,37] for Xe, as compared to the lighter rare gases. Indeed, the main atomic

feature responsible for the polarization bremsstrahlung is the dipole polarizability [23]. The contribution of polarization BS into the experimental BS spectra of rare gases should increase with the nuclear charge since the polarizabilities of these gases are 11, 17 and 27 bohr<sup>3</sup> for Ar, Kr and Xe, respectively [38]. The discrepancy between theory and experiment for Kr isochromatic spectra can be also due to polarization BS, as well as to some inelastic processes, such as impact ionization (in Equations (4) and (8), we take into account conventional BS and elastic cross section only).

The main disadvantage of the presented theory is that it does not reproduce the BS intensity decrease for sufficient high electron energy, *Ei* > *E*exp<sup>t</sup> *i*, as seen from Figure 4.
