**2. Methods**

#### *2.1. General Formalism of Soft-Photon Approximation (SPA)*

In the nonrelativistic SPA framework, the double-differential cross section, d *σ*BS, of BS in the electron scattering on a target can be factorized. In particular, d *σ*BS is the product of the differential cross section, d *σs*, of the electron scattering on the same target without *γ*-radiation, by the probability, *dwγ*, of the photon radiation [1]:

$$\mathbf{d}\sigma\_{\rm BS} = \left. \mathbf{d}\sigma\_{\rm s} \, \mathbf{d}w\_{\gamma\prime} \right. \tag{2}$$

$$\mathbf{d}w\_{\gamma\prime} = \left. \frac{\alpha}{4\pi^2} \left[ \frac{\mathbf{k}}{\omega} (\mathbf{v}\_i - \mathbf{v}\_f) \right]^2 \frac{\mathbf{d}\omega}{\omega} \, \mathbf{d}\Omega\_{\gamma\prime}$$

where *α* 1/137 is the fine structure constant, **<sup>v</sup>***i*, **<sup>v</sup>***f* are the electron velocity in the initial and final state, respectively; **k** is the photon wave vector, and dΩ*γ* is the element of solid angle to which the photon is radiated. The cross section in (2) is averaged over the electron spin in the initial state and summed over the photon polarizations and the electron spin in the final state.

If the ejection direction of the electron is not fixed, we should integrate Equation (2) over this direction. To do this, we choose the *z* axis directed along the **<sup>v</sup>***i* vector and write d*<sup>σ</sup>s* in the following form:

$$\operatorname{d}\sigma\_{\mathfrak{s}}(E\_{i},\theta) = \sigma\_{\mathfrak{s}}(E\_{i},\theta)\operatorname{d}\Omega\_{\mathfrak{v}\_{f}\prime} \tag{3}$$

where *θ* is the polar angle of the **<sup>v</sup>***f* vector, *Ei* is the energy of the impact electron. Then, Equation (2) can be written in the following form:

$$\begin{split} \mathbf{d}\sigma\_{\mathrm{BS}} &= \quad \frac{\boldsymbol{\alpha}}{4\pi^{2}\omega} \,\boldsymbol{v}\_{\mathrm{S}}(\boldsymbol{E}\_{i},\boldsymbol{\theta}) \Big[ \frac{\boldsymbol{v}\_{i}^{2}}{c^{2}} \cos^{2}\boldsymbol{\Theta} - \frac{2\boldsymbol{v}\_{i}}{c\omega} (\mathbf{k}\mathbf{v}\_{f}) \cos\boldsymbol{\Theta} \\ &+ \frac{(\mathbf{k}\mathbf{v}\_{f})^{2}}{\omega^{2}} \Big] \,\mathrm{d}\Omega\_{\mathbf{v}\_{f}} \,\mathrm{d}\omega \,\mathrm{d}\Omega\_{\mathbf{v}} \\ &= \quad \frac{\boldsymbol{\alpha}}{4\pi^{2}c^{2}\omega} \,\boldsymbol{v}\_{\mathrm{S}}(\boldsymbol{E}\_{i},\boldsymbol{\theta}) \Big[ \boldsymbol{v}\_{i}^{2} \cos^{2}\boldsymbol{\Theta} \\ &- 2\boldsymbol{v}\_{i} \boldsymbol{v}\_{f} \cos\boldsymbol{\Theta} (\sin\boldsymbol{\Theta}\sin\boldsymbol{\theta}\cos\boldsymbol{\varphi} + \cos\boldsymbol{\Theta}\cos\boldsymbol{\theta}) + \\ &+ \boldsymbol{v}\_{f}^{2} (\sin\boldsymbol{\Theta}\sin\boldsymbol{\theta}\cos\boldsymbol{\varphi} + \cos\boldsymbol{\Theta}\cos\boldsymbol{\theta})^{2} \Big] \mathrm{d}\Omega\_{\mathbf{v}\_{f}} \,\mathrm{d}\omega \,\mathrm{d}\Omega\_{\mathbf{v}}. \end{split}$$

where *ϕ* is the azimuthal angle of the **<sup>v</sup>***f* vector, Θ is the polar angle of the **k** vector (i.e., the angle between **<sup>v</sup>***i* and **k** vectors), and the azimuthal angle of the **k** vector is assumed to be zero. After integration over *ϕ*, we obtain the following:

$$\mathrm{d}\sigma\_{\mathrm{BS}} = \frac{\mathfrak{a}}{2\pi c^2 \omega} \,\sigma\_s(E\_i, \theta) Q(\Theta, \theta) \sin\theta \,\mathrm{d}\theta \,\mathrm{d}\omega \,\mathrm{d}\Omega\_{\gamma\prime} \tag{4}$$

where

$$Q(\Theta,\theta) = v\_i^2 \cos^2\Theta - 2v\_i v\_f \cos^2\Theta \cos\theta + v\_f^2 \left(\frac{1}{2}\sin^2\Theta \sin^2\theta + \cos^2\Theta \cos^2\theta\right). \tag{5}$$

## *2.2. SPA: Validity Conditions*

SPA is valid when the irradiated photon energy is low as compared to the scattering electron energy, *Ei*:

$$
\omega \llcorner \to\_i / \hbar. \tag{6}
$$

Apart the condition (6), it is also required for SPA validity that the momentum, **q** = **p***i* − **p***f* , transferred to the atom from the scattering electron, is much higher than the electron momentum change due to irradiation of the photon [1]. In the nonrelativistic approximation, this condition takes the following form [1]:

$$
\hbar\omega/(v\_i\eta) \ll 1.\tag{7}
$$

It is seen that the condition (7) is violated at low transferred momentum values, i.e., at small scattering angles, *θ*. However, the contribution of the small angles into BS cross section is small due to the angular factor, *Q*(<sup>Θ</sup>, *θ*) in Equation (4). This factor is plotted in Figure 1 as function of the scattering angle, *θ*, for the value of Θ = 83◦ used experimentally in [12,13]. It is clearly seen that *Q*(<sup>Θ</sup>, *θ*) is small for *θ* → 0.

**Figure 1.** The angular factor, *Q*(<sup>Θ</sup>, *<sup>θ</sup>*)/*v*<sup>2</sup> *i* , from Equation (5) at Θ = 83◦ and *vf vi*.

#### **3. Results on Isochromatic Spectra**

Wavelength dependence of the BS intensity was obtained in experiment [13] and analyzed theoretically in [33].

Now, we proceed to the analysis of the isochromatic spectra. These spectra show the dependence of BS intensity on the electron energy, *Ei*, at a fixed photon energy, *h*¯ *ω*. The authors in [13] list the experimental values for a modified (as compared to that in Equation (4)) differential BS cross section. This modification, as compared to Equation (4), implies calculating the angle-differential cross section of BS photon radiation into a finite small wavelength interval, Δ *λ*, integrated over the scattering angle, *θ*,

$$\frac{d\sigma\_{\rm BS}}{d\Omega\_{\gamma}} = \frac{a}{2\pi c^2} \frac{\Delta\lambda}{\lambda} \int\_0^\pi \sigma\_s(E\_{i\prime}\theta) Q(\Theta\_{\prime}\theta) \sin\theta \,\mathrm{d}\theta,\tag{8}$$

where *λ* = 2*πh*¯ /*k* is the photon wavelength, the factor *Q*(<sup>Θ</sup>, *θ*) is defined in Equation (5). In other words, a change d *ω*/*ω* → Δ *λ*/*λ* should be done in Equation (4). In the experiment presented in [13], photons were registered in the wavelength interval of Δ *λ* = 0.1 nm. Equation (8) requires knowledge of the following integrals:

$$\mathcal{J}\_{ab}(E\_i) = \int\_0^\pi \sigma\_s(E\_i, \theta) \cos^a \theta \sin^{1+b} \theta \,\mathrm{d}\theta,\quad (ab) = (00), (10), (20), (02). \tag{9}$$

In particular, the cross section of elastic electron–atom scattering is *<sup>σ</sup>*el(*Ei*) = 2*π* J00(*Ei*).

Equation (8) implies using differential cross sections of elastic electron–atom scattering, *<sup>σ</sup>s*(*Ei*, *<sup>θ</sup>*), tabulated as functions of the scattering angle, *θ*. Such data can be obtained by numerical simulation using various models for interaction of the incident electrons with atoms. For instance, the cross-section database [34] contains data for the majority of atoms of the Periodic system. These data were used in our earlier work [33] for analysis of experimental BS photon spectra at fixed energy of an electron scattering on rare gas atoms [13].

In the case of electron scattering on argon atom, the data calculated in [35] can be used as an alternative to the NIST database [34]. The results of the cross-section calculation at particular electron energies, *Ei*, using both datasets are presented in Table 1.


**Table 1.** Simulated cross section in elastic electron–atom scattering (Å2).

It can be seen that the cross sections of elastic electron scattering on Ar calculated in [34,35] coincide within 4%. However, the angle-differential cross sections differ significantly. As it is seen from Figure 2, the differential cross section has a sharp maximum for the forward scattering. This maximum calculated in [35] is two times higher as compared to the results of [34], and the differential cross section drops down for the scattering angles *θ* > 15◦. Cross sections of electron scattering on Ar, Kr, and Xe atoms were also calculated in [36]. They are presented in Table 1 as well and their values are less as compared to those of [34]. This difference increases with the electron energy, *Ei*, up to 40% at *Ei* = 2 keV. Unfortunately, the angle-differential cross sections are not listed in [36].

**Figure 2.** The differential cross section of elastic *e*–Ar scattering at *Ei* = 0.4 keV from [34] (solid line) and from Figure 7a of [35] (dashed line).

The results of our calculations of isochromatic BS spectra by Equation (8) are given in Figures 3–5 in comparison with the experimental data shown in Figures 5–7 of [13], respectively. We note a misprint in [13]: the cross section shown in these figures should be measured in mbarn/sr, not barn. As seen from these figures, the calculated BS cross sections increase with the initial electron energy, *Ei*, for Kr and Xe, and slightly decrease for Ar. We recall that calculations of the isochromatic spectra according to Pratt et al. [29,30] (as given in [13]) predict a monotonic decrease of the BS intensity as a function of *Ei*.

**Figure 3.** Isochromatic spectrum for Ar: double-differential BS cross section (8) as a function of the initial electron energy at the fixed photon energy of 177 eV. Solid line: elastic-scattering differential cross sections are taken from [34]; dashed line: from [35]. Black points with error bars: experimental values adopted from Figure 5 in [13] by manual digitization.

**Figure 4.** Same as in Figure 3 for Kr, *h*¯ *ω* = 165 eV. Experimental values adopted from Figure 6 in [13].

**Figure 5.** Same as in Figure 4 for Xe, *h*¯ *ω* = 177 eV. Experimental values adopted from Figure 7 in [13].

Note the maximum value of BS cross sections is reached at *Ei* = *E*max *i* (see Equation (1)) and predicted by Sommerfeld [3] in the pure Coulomb potential. Under the conditions of the experiment of [13], this maximum should be observed at *Ei* ≈ 0.26 keV.
