Bremsstrahlung emission from nucleon-nucleus reactions in dense medium of compact stars

Light from stars provides a main information about them. Stars have nuclei from the lightest up to heavy (for example, see book [1] for nuclei in Sun). Convergence in the numeric calculations of the spectra of photons emitted in nuclear processes is obtained, if to take into account space region up to atomic shells. So, it is important to study nucleus as system of evolving nucleons as solution of many body quantum mechanical problem. Such a way is proper basis to study emission of photons produced in nuclear reactions in stars. Similar information about photons could be extracted from nuclear collisions at relativistic velocities. Such photons is subject of research in this paper.

Interest in bremsstrahlung photons emitted in nuclear reactions is explained by that photons are measured. Analysis of the bremsstrahlung spectra provides additional information about nuclear mechanisms, interactions, etc.. In some cases, this information is more deep in comparison with investigations of nuclear processes without bremsstrahlung analysis. One can explain this by the following idea. There are experimental limitations in measurements of cross-sections of nuclear reactions (without measurements of photons), however one can overcome these limitations in the bremsstrahlung measurements and analysis. Some proposals have already been realized on such a way. This is study of internal structure of nucleons in the nucleon-nucleus scattering analyzed from low energies via bremsstrahlung analysis [2]. On the basis of the bremsstrahlung analysis strangeness in hypernuclei is studied [3], deformations of nuclei in α decays are extracted from experimental data [4], information about dynamic of ternary fission of heavy nuclei is extracted [5−7], structure and clusters in nuclei in scattering are studied [8, 9], Δ-resonances in nuclei in proton-nucleus scattering are investigated [10], etc.. Many predictions by such a model have been proposed for further experimental study. Attractive idea is to enlarge study of bremsstrahlung processes to conditions of dense medium, which can be in collisions of nuclei at high energies. Here, possible measurements of bremsstrahlung photons can give new unequal information about physics of such processes. However, for such investigations a new bremsstrahlung model should be constructed on strict quantum mechanical basis (which does not exists now), taking properties and mechanisms of nuclear scattering into account.

Study of nuclear forces [12, 13] in the extreme conditions in compact stars enlarges possibilities to understand those deeper, where star is a good laboratory [14−16]. This idea is one of motivations of research in this paper. Many-nucleon unified theories of nucleus and nuclear reactions (see microscopic cluster models based on resonating group method [17−19] and generator coordinate method [20]), shell models, collective models, relativistic mean-field (RMF) theory [21−33], Ab initio calculations theory [34], QCD approaches for systems of nucleons, quark-meson models [35] have been developed for study of nuclei. In stars a main focus is given to obtain equation of state (EoS) [36]. Many models are applied for such a task (for example, see Refs. [37−40] for RMF theories, see Ref. [36] for Ab initio calculations theory, etc.). APR (Akmal, Pandharipande, Ravenhall) model [41] based on quantum mechanics is one of the most successful (see Refs. [42−45]).

In this paper we study emission of bremsstrahlung photons from scattering of neutrons and protons off nuclei in conditions of experiments and in compact stars [46−48]. We develop a new bremsstrahlung model for nuclear reactions where we generalize the quantum mechanical model of the deformed oscillator shells (DOS) of nucleus [49−52]. We analyze how much the spectra of photons are different in emission from conditions of Earth and in dense medium of stars. As it was shown in Ref. [53, 54], even for the same full stationary wave functions with the same boundary conditions there are different nuclear processes (with the same nuclei and energies) where difference in cross-sections can be up to 3-4 times. RMF theories cannot explain such a quantum phenomenon, which is not small and important for understanding nuclear processes. By such a motivation we use basis of quantum mechanics for analysis. It turns out that the DOS model is enough convenient for such a research. ¹

The paper is organized in the following way. In Sec. II emission of the bremsstrahlung photons in the scattering of protons and neutrons off nuclei in the dense stellar medium is formulated. In Sec. III the matrix elements of emission of photons on the basis of DOS model are derived for even-even nuclei. Emission of photons from the nucleon-nucleus scattering for nuclei from ⁴He to ⁵⁶Fe in stars is analyzed in Sec. IV. Conclusions are summarized in Sect. 5. A and B include useful analytical formulas for some light nuclei, derivation of correction of energy of nucleus due to influence of stellar medium.

We will study, how a dense medium of star influences on emission of photons during nuclear reaction. Such a question has not been studied yet. Some calculations were done for proton-capture reactions in stars [55, 56]. However, nuclei in those calculations were studied as stable, without influence of stellar medium. Now we will take deformation of nuclei into account due to influence of stellar medium. To be closer to that analysis, we will choose scattering of nucleons on nuclei.

Let us write down hamiltonian for the scattering of proton off nucleus in the stellar medium as

Bremsstrahlung photons emitted in this nuclear process can be described via additional new operator $ \hat{H}_{\gamma} $ as

A concrete form of this operator $ \hat{H}_{\gamma} $ should be defined explicitly. $ V_{\rm star} (|{\bf{r}}_{i} - {\bf{r}}_{j}|) $ will be determined in Sec. III B.

Emission of bremsstrahlung photons without influence of stellar medium [i.e. without last term in Eq. (2)] for the scattering of protons off nuclei in conditions of experiments on Earth was studied by many researchers. Here, agreement between theory and existed experimental information has been obtained with the highest precision for this reaction in frameworks of approach [56, 57] (this is data [58] for $ p+^{208}\text{Pb} $ at proton energy beam of $ E_{\rm p}=145 $ MeV (see PhD thesis [59] also), data [60] for $ p+^{12}\text{C} $, $ p+^{58}\text{Ni} $, $ p+^{107}\text{Ag} $, $ p+^{197}\text{Au} $ at proton energy beam of $ E_{\rm p}=190 $ MeV and corresponding calculations in Figs. 5–8 in Refs. [56]). So, we will generalize bremsstrahlung formalism for these reactions from above-zero energies up to intermediate energies in stars, basing on formalism and results of papers [56, 57] (see developments in Refs. [2, 3, 55, 61]).

This bremsstrahlung formalism has been tested on the experimental data of bremsstrahlung in α-decay, proton-nucleus scattering, spontaneous fission [2, 4, 6, 7, 55−57]. It predicts the bremsstrahlung spectra for π-nucleus scattering, radioactivity with emission of proton, ternary fission, scattering of α-particles on hypernuclei [3, 5, 57, 61]. At first, spin of proton was included to such a formalism in Ref. [57] where scattering of protons off nuclei was studied. But, such a bremsstrahlung was of coherent type, and some experimental bremsstrahlung data were explained by such a way. However, more careful measurements of bremsstrahlung emission in the scattering of protons off nuclei were obtained by TAPS collaborations [60] (we use those data for tests of calculations in the current paper). Those data cannot be described on the coherent bremsstrahlung basis only (there is plateau in the data not explained by such a way). By such a motivation, the previous formalism was improved, where spins of nucleons of the nucleus-target were taken into account. For each nucleon of nucleus and the scattering proton spin and magnetic moments are used in calculations of spectra. A new contribution of incoherent emission was appeared in the matrix element. That model [56] describes better experimental data of TAPS for the proton-nucleus scattering. It was estimated than incoherent emission is essentially larger than coherent one. In this paper we predict such ratios between incoherent and coherent bremsstrahlung contributions for different nuclei from ⁴He to ⁵⁶Fe.

We define cross-section of bremsstrahlung emission for reactions in experiments in the laboratory frame in frameworks of Ref. [57], where the matrix element of emission is

matrix elements are

coefficients are defined as

and integrals are defined as

Here, $ {\bf{r}} $ is radius-vector from center-of-mass of the nucleus to the scattered proton, $ \mu = m_{\rm p}\, m_{A} / (m_{\rm p} + m_{A}) $ is reduced mass, A is number of nucleons in nucleus, $ c_{\rm p} = m_{\rm p}/(m_{\rm p}+m_{A}) $, $ {\bf{e}}^{(\alpha)} $ are unit vectors of polarization of the photon emitted [$ {\bf{e}}^{(\alpha), *} = {\bf{e}}^{(\alpha)} $], $ {\bf{k}}_{\rm ph} $ is wave vector of the photon and $ w_{\rm ph} = k_{\rm ph} c = \bigl| {\bf{k}}_{\rm ph}\bigr|c $, $ E_{\rm ph} = \hbar w_{\rm ph} $ is energy of photon. Vectors $ {\bf{e}}^{(\alpha)} $ are perpendicular to $ {\bf{k}}_{\rm ph} $ in Coulomb gauge. We have two independent polarizations $ {\bf{e}}^{(1)} $ and $ {\bf{e}}^{(2)} $ for the photon with impulse $ {\bf{k}}_{\rm ph} $ ($ \alpha=1,2 $). $ \mu_{\rm pn}^{\rm (an)} = \mu_{\rm p}^{\rm (an)} + \mu_{\rm n}^{\rm (an)} $, $ \mu_{\rm p}^{\rm (an)} $ and $ \mu_{\rm n}^{\rm (an)} $ are anomalous magnetic moments of proton and neutron.

The matrix elements $ M_{p}^{(E,\, {\rm dip},0)} $ and $ M_{p}^{(M,\, {\rm dip},0)} $ describe coherent bremsstrahlung emission of photons of electric and magnetic types, the matrix elements $ M_{\Delta E} $ and $ M_{\Delta M} $ describe incoherent bremsstrahlung emission of photons of electric and magnetic types. $ M_{P} $ is related with motion of full nuclear system, which we neglect in this paper. Effective electric charge and magnetic moment of system in the dipole approximation (i.e. at $ {\bf{k_{\rm ph}}} {\bf{r}} \to 0 $) are

$ m_{\rm p} $ and $ z_{\rm p} $ are mass and charge of proton, $ m_{A} $ and $ z_{A} $ are mass and charge of nucleus. We introduce magnetic moment of nucleus $ {\bf{M}}_{A} $

where $ \mu_{j}^{\rm (an)} $ is anomalous magnetic moment of proton or neutron in nucleus, $ m_{Aj} $ is mass of nucleon with number j in nucleus, $ {\boldsymbol{\sigma}} $ is operator of spin.

In this paper we use folding approach in description of scattering of nucleon on nucleus. This approach is approximation of more complicated and fully microscopic (cluster) formalism describing nuclear reactions with structure on nuclei on strict basis of quantum mechanics and using nucleon-nucleon interactions in the basis (see also App. C for some details). Here formalisms of center-of-masses of nucleus and center-of-mass of the full nucleon-nucleus system in the scattering are natural parts of that microscopic formalism. Some developments of that approach for reactions with light nuclei are given by us in Refs. [8, 9] where we included formalism of emission of bremsstrahlung photons. In these papers there are references on other papers with explanations for more detailed cluster formalism without emission of photons but with analysis of different peculiarities of that approach.

We apply the nuclear model of deformed oscillatoric shells (DOS) (see Appendix A) to describe structure of nucleus, where Hamiltonian of A nucleons is defined as [49]

Potential energy of two-nucleon nuclear interactions and potential energy of Coulomb forces between protons are defined as [see Eqs. (2.5), (2.6) in Ref. [52], also Eq. (15) in Ref. [49]]

and

where proton density (for nuclei with even number of protons) is

Here, summation is performed over all states of configuration, $ H_{n}(x) $ are Hermitian polynomials [we use definition from Ref. [62], p. 749, (а,6)], a, b, c are oscillator parameters along axes x, y, z. Neutron density $ F_{n} (n; {\bf{r}}_{i}, {\bf{r}}_{j}) $ is obtained after change of proton configuration and numbers of states on the corresponding neutron characteristics.

Kinetic energy of nucleons in center-of-mass frame is defined as [see Eq. (2.4) in Ref. [52]]:

Radiuses of proton and neutron density distributions are defined as [49]

Here, $ a_{\rm p} $, $ b_{\rm p} $, $ c_{\rm p} $ are oscillator parameters along axes x, y, z for protons, $ a_{\rm n} $, $ b_{\rm n} $, $ c_{\rm n} $ --- for neutrons, we use approximation $ a_{\rm p} = a_{\rm n} = a $, $ b_{\rm p} = b_{\rm n} = b $, $ c_{\rm p} = c_{\rm n} = c $.

The potential energies for Coulomb and nuclear interactions, kinetic energy are integrated partially analytically for even-even nuclei (see calculations for some light nuclei in Sec. A.3). However, for general analysis of properties of nuclei in the large region on numbers of protons and neutrons it is more effective to integrate such energies numerically. It turns out that such a way allows to analyze easily influence of dense stellar medium of compact star on nuclei in this wide region in a unified way. In Table 1 we present results of calculations of binding energy (per nucleon), oscillating lengths and radiuses of proton and neutron density distributions for ground states of even-even nuclei by such an approach.

Nucleus	$ E_{full}/A $, MeV	$ E_{nucl} $, MeV	$ E_{Coul} $, MeV	$ E_{kin} $, MeV	a, fm	b, fm	c, fm	$ R_{\rm p} $, fm	$ R_{\rm n} $, fm
4He	−8.462	−125.18	1.11	90.22	1.03	1.03	1.03	1.09	1.09
6He	−2.520	−109.90	0.64	94.14	1.59	1.29	1.29	1.56	1.93
8He	−0.748	−100.92	0.43	94.51	1.50	1.77	1.77	1.93	2.42
10He	−0.091	−98.61	0.43	97.29	1.91	1.91	1.91	2.22	2.77
8Be	−5.549	−207.83	1.29	162.17	1.54	1.15	1.15	1.84	1.84
10Be	−3.496	−204.32	1.29	168.07	1.62	1.46	1.27	2.04	2.11
12Be	−2.475	−203.91	1.23	172.98	1.69	1.55	1.54	2.22	2.33
14Be	−1.785	−213.95	1.07	187.88	1.94	1.59	1.58	2.44	2.97
12C	−5.170	−291.56	1.71	227.80	1.48	1.34	1.34	2.00	2.00
14C	−4.603	−307.63	1.49	241.69	1.52	1.51	1.38	2.13	2.16
16C	−3.672	−314.38	1.41	254.21	1.73	1.55	1.44	2.31	2.73
18C	−2.915	−319.15	1.87	264.79	1.78	1.76	1.49	2.46	3.09
16O	−6.283	−407.27	2.10	304.64	1.41	1.41	1.41	2.08	2.08
18O	−5.255	−417.09	2.20	320.30	1.60	1.44	1.44	2.21	2.57
20Ne	−5.966	−497.98	2.95	375.69	1.66	1.39	1.39	2.59	2.59
22Ne	−5.118	−502.28	3.23	386.44	1.68	1.55	1.42	2.68	2.86
24Mg	−5.587	−575.07	4.82	436.15	1.62	1.62	1.39	2.85	2.85
26Mg	−4.923	−580.27	4.87	447.38	1.64	1.63	1.53	2.93	3.03
28Si	−5.312	−649.71	5.83	495.12	1.60	1.60	1.60	3.03	3.03
30Si	−5.301	−687.11	5.94	522.11	1.64	1.63	1.58	3.06	3.06
32S	−5.898	−771.79	6.21	576.82	1.63	1.62	1.54	3.03	3.03
34S	−5.970	−818.32	6.05	609.28	1.65	1.62	1.56	3.05	3.04
36Ar	−6.545	−899.80	7.03	657.13	1.65	1.59	1.57	3.03	3.03
38Ar	−6.687	−949.80	6.93	688.73	1.64	1.61	1.60	3.05	3.04
40Ar	−6.218	−965.37	6.84	709.82	1.66	1.63	1.65	3.11	3.14
40Ca	−7.285	−1044.28	7.56	745.29	1.60	1.60	1.60	3.01	3.01
42Ca	−6.963	−1072.27	7.82	772.45	1.62	1.62	1.62	3.06	3.09
44Ca	−6.665	−1097.52	7.93	796.32	1.67	1.64	1.62	3.11	3.25
50Cr	−6.788	−1260.28	8.43	912.41	1.73	1.65	1.62	3.30	3.39
52Cr	−6.629	−1293.79	8.53	940.84	1.73	1.69	1.64	3.33	3.49
54Fe	−6.830	−1364.49	8.93	986.71	1.74	1.66	1.64	3.42	3.49
56Fe	−6.788	−1405.45	9.43	1015.87	1.75	1.70	1.64	3.45	3.58
58Fe	−6.662	−1440.90	9.92	1044.59	1.75	1.71	1.67	3.47	3.65

Table 1.  Energies, oscillating parameters and radiuses of proton and neutron density distributions for ground states of even-even nuclei from ⁴He up to ⁵⁶Fe calculated by Eqs. (10), (11), (13), (14). We also include contributions from potential energy of nuclear two-nucleon interactions, Coulomb and kinetic terms. One can see that role of Coulomb interactions in calculations of binding energy is small.

Note that this formulation of hamiltonian of nucleus in frameworks of DOS model includes also term for correction of center-of-mass of nucleus [see last term in Eq. (13)]. But different variants of correction of center of mass were analyzed by A.I. Steshenko and G.F. Filippov in that model, and finally they provided the most suitable formulation of hamiltonian, which we use now. Also we calculated binding energies for different nuclei, starting from light ⁴He. We found that calculated binding energies for light nuclei are enough close with experimental data. We found that model DOS in unified way explains existence of bound states of isotopes of light nuclei, and non-existence of bound states of further isotopes of these nuclei after additional increasing of number of neutrons. The model also explained that only ⁴He is spherical, but other isotopes of Helium and other nuclei have deformations. The model calculates deformations for those nuclei. In Refs. [49, 50] authors compared calculated deformations with available experimental information for light nuclei, and found not bad agreement. Those all results of model DOS (see also Table 1) are based at the chosen form of kinetic term with fixed correction of center-of-mass of nucleus.

Note on nucleon-nucleon potentials based on meson exchange which are used in calculations of binding energies of nuclei: Argonne potential [63], CD-Bonn potential [64], Nijmegen potential [65], Reid93 potential [65], Paris potential [66]. Electromagnetic interactions (and parametrizations) are some different for these potentials. Coulomb interactions is part of electromagnetic interactions which have magnetic moments of protons and neutrons. Note on approach [67] providing accurate study of Coulomb interactions on the basis of Hartree-Fock calculations within Bardeen-Cooper-Schrieffer approximation using seniority force. Our bremsstrahlung formalism includes anomalous magnetic moments of protons and neutrons of nucleus and the scattered proton or neutron. Matrix elements of emission can be calculated on the basis of wave functions of DOS model. So, the bremsstrahlung formalism calculates the spectra of photons, and these spectra are dependent on magnetic moments of protons and neutrons of the nuclear system. Bremsstrahlung approach allows to study magnetic moments of nucleons in nuclear reaction.

A role of magnetic moments of nucleons in producing incoherent bremsstrahlung is more essential than in calculations of binding energies. We estimate the following ratios between incoherent and coherent contributions to the full spectrum: $ \dfrac{\sigma_{\rm incoh} (^{4}\text{He})}{\sigma_{\rm coh} (^{4}\text{He})} = 3.2340 $, $ \dfrac{\sigma_{\rm incoh} (^{8}\text{Be})}{\sigma_{\rm coh} (^{8}\text{Be})} = 58.629 $, $ \dfrac{\sigma_{\rm incoh} (^{12}\text{C})}{\sigma_{\rm coh} (^{12}\text{C})} = 299.20 $, $ \dfrac{\sigma_{\rm incoh} (^{16}\text{O})}{\sigma_{\rm coh} (^{16}\text{O})} = 947.7 $, $ \dfrac{\sigma_{\rm incoh} (^{24}\text{Mg})}{\sigma_{\rm coh} (^{24}\text{Mg})} = 4817.7 $, $ \dfrac{\sigma_{\rm incoh} (^{40}\text{Ca})}{\sigma_{\rm coh} (^{40}\text{Ca})} = 3.720 \cdot 10^{4} $, $ \dfrac{\sigma_{\rm incoh} (^{56}\text{Fe})}{\sigma_{\rm coh} (^{56}\text{Fe})} = 1.7592 \cdot 10^{5} $.

Radius of neutron star is about 10–14 km, radiuses of typical white dwarfs are about 5000 km [46], while radius of nuclei is up to about 10–15 fm. Difference between sizes of nuclei and space regions of compact stars is huge. So, in the proper approximation one can study influence of medium of star on nucleons of nucleus in the homogeneous approximation (near this nucleus), while influence of inhomogeneous effects can be estimated as corrections. Evolution of particle in the homogeneous external field is standard topic of quantum mechanics (for example, see book [62], p. 103–105), where linear force acting on particle is used, that corresponds to such an approximation. Note on folding approach describing structure of nuclei via integration of density distribution of nuclear matter. This approach is successful in description of experimental data of nuclear scattering, fusion, spontaneous and ternary fissions with really good accuracy.

Let us write down hamiltonian of nucleus with influence of stellar medium on nucleons as

where $ T_{\rm full} $ is defined in Eq. (13). In the first approximation, we shall assume that influence of stellar medium on nucleons of the studied nucleus is homogeneous. Force $ {\bf{F}} $ of such an influence depends on distance R between center of star and center of mass of the studied nucleus. Potential of such an influence depends on relative distances between nucleons of the studied nucleus. ² Following to quantum mechanics (see Ref. [62], p. 100–102, for details), we define it as ³

Correction $ \Delta E_{\rm star} $ to the full energy of nucleus due to inclusion of influence of star on nucleons of nucleus can be defined as

Such a matrix element is calculated in B. For even-even nuclei (at $ Z=N $) we obtain:

In particular, for ⁴He we have [see Eq. (69) at $ a=b=c $]

where $ F_{P} (R) = |{\bf{F}}_{P} (R)| $. In a general case, we obtain:

So, we have the following picture of influence of stellar medium on the studied nucleus. Forces of stellar medium press on nucleons of nucleus. The deeper this nucleus is located in star, the stronger such forces press on nucleus. However, binding energy (it is negative for nucleus in the external layer of star) is increased at deeper location of this nucleus in star. Starting from some critical distance from nucleus to center of star, the binding energy becomes positive. So, full energy of individual nucleons of the studied nucleus is already larger than mass of this nucleus, i.e. we obtain unbound system of nucleons and nucleus is disintegrated on nucleons. Now it could be interesting to estimate if such a phenomenon is appeared in white dwarfs and neutron stars. The kinetic energy is increased at deeper location of nucleus in star. At decreasing distance from the studied nucleus to center of star, change of kinetic energy is unlimited, while change of nuclear energy is limited. So, ratio between kinetic energy of nucleons of nucleus and nuclear energy of nucleus is changed.

For estimations of bremsstrahlung emission for nuclear reactionbs in stellar medium, we apply perturbation theory. We will take into account influence of stellar medium on emission as

From here we find the matrix element of emission in star as

Following to perturbation theory, the first correction is determined on the basis of two unperturbed wave functions, i.e. we take wave functions as in the matrix element (3):

where

Here $ \varphi_{\rm p-nucl} ({\bf{r}}, k_{f}) $ is wave function of scattering of proton off nucleus (energy has continuous spectrum, as emitted photon takes some energy of proton-nucleus system), $ \varphi_{0}^{2} ({\bf{r}}) $ is wave function of nucleus (energy has only discrete levels, nucleons are only in bound states). We renormalize wave function of scattering of proton off nucleus ⁴. For ⁴He we have

Substituting Eq. (26) to (24) for ⁴He (at $ a=b=c $), we obtain:

Wave function $ \varphi_{\rm p-nucl} $ of relative motion between proton and center-of-mass of nucleus is calculated numerically concerning to the proton-nucleus potential as $ V (r) = v_{c}(r) + v_{N}(r) + v_{\rm so}(r) + v_{l} (r) $, where $ v_{c}(r) $, $ v_{N}(r) $, $ v_{\rm so}(r) $, and $ v_{l} (r) $ are Coulomb, nuclear, spin-orbital, and centrifugal terms with parameters in Eqs. (46)–(47) in Ref. [56]. Note that in such calculations of wave functions parameters of proton-nucleus potential above are varied on the basis of the folding potential (with tests). Such way gives more fast calculations of final cross sections of bremsstrahlung with enough good approximation and stability.

Equations of state (EOSs) have been obtained by many authors. These data are presented in form of tables containing grid of calculated values of matter density, baryon (number) density and pressure. Note on useful analytical representations derived by Haensel and Potekhin for two EOSs for not-rotating and rotating neutron-star matter [68]: FPS and SLy [see Eqs. (14), (17) in that paper]. That result allows to apply those EOSs in study of other problems in fast way. So, we use that result to start own calculations (and for test) on the basis of pressure concerning to the interested density of stellar matter.

However, to obtain corrections to binding energies of nuclei in the stellar medium, we need to reproduce EOSs in compact stars. One of objects, where polytropic model is successfully applied, is white dwarf (see Ref. [15], p. 364–370; Ref. [16], p. 213–233, p. 475–496). According to Ref. [14] (see Fig. 2.2 in that book, p. 33; also Fig. 103 in Ref. [15], p. 365), density in center of such a star is $ 10^{+5} {\rm g}\, {\rm cm}^{-3} $ – $ 1.4 \cdot 10^{+9}\, {\rm g}\, {\rm cm}^{-3} $. Solving Lane-Emden equation at polytropic index $ n=3 $ by the finite-difference method, we obtain radiuses for white dwarfs in region from 3011.28 kilometres (at $ \rho_{\rm cr} = 1.4 \cdot 10^{9}\, {\rm g}\, {\rm cm}^{-3} $) to 72 576.27 kilometres (at $ \rho_{\rm cr} = 10^{5}\, {\rm g}\, {\rm cm}^{-3} $). We start analysis for such a star.

Let us estimate force acting on the studied nucleus under influence of stellar medium. Pressure is force applied perpendicular to the surface of an object per unit area over which that force is distributed. So, pressure acting on selected layer in medium, can be represented as force acting on unit area of this layer. Force acting on full such a layer can be found as the pressure multiplied on the full area of this layer. So, force acting on the nucleus with surface $ S_{\rm nucl} $ in stellar medium can be determined as pressure multiplied on area such a surface of this nucleus as

In good approximation, nucleus can be studied as spherical with $ S_{\rm nucl} = 4\pi\, R_{\rm nucl}^{2} $, $ R_{\rm nucl} = a \cdot A^{1/3} $. Force acting on nucleus ⁴He in star is shown in Fig. 1 (a). ⁵.

Figure 1.  (Color online) Panel (a): Force $ F_{R} $, acting on nucleons of nucleus ⁴He, in dependence on its renormalized distance ξ to the center of star (at polytropic index $ n=3 $) [Force is defined in Eq. (28), oscillator parameter $ a=1.05 $ fm is fixed for estimations, that is close to minimum of full energy of ⁴He in natural conditions in Earth]. Panel (b): Correction $ \Delta E $ to energy of nucleus ⁴He due to influence of stellar medium in dependence on distance to center of star (at $ n=3 $) [Correction of energy $ \Delta E $ is defined in Eq. (20)].

Calculated corrections to the full energy of nucleus ⁴He from such an influence are presented in Fig. 1 (b). One can see that in the white dwarfs and inner and external crusts of neutron stars (densities of $ 10^{5}\, {\rm g}\, {\rm cm}^{-3} $ – $ 1.4 \cdot 10^{9}\, {\rm g}\, {\rm cm}^{-3} $) nuclei are not disintegrated. But, this phenomenon really happens for higher densities, starting from some critical distances from center of stars [see upper brown dashed line (at $ \rho_{\rm cr} = 10^{16}\, {\rm g}\, {\rm cm}^{-3} $) and green dash-double dotted line (at $ \rho_{\rm cr} = 10^{14}\, {\rm g}\, {\rm cm}^{-3} $) in figure]. This case corresponds to densities of core of the neutron stars.

We shall analyze possibility of nucleus to disintegrate on individual nucleons in the neutron star. ⁶ Different types of energy for nucleus ⁴He are shown in Fig. 2. One can see that the full energy of nucleus can be already positive after taking correction to energy into account shown in previos Fig. 1 (b). This means that nucleons do not form bound system, i.e. the nucleus is disintegrated. Oscillator length a corresponding to the minimum of the full energy of system of nucleons is decreased in comparison with its value for nucleus in vacuum. So, relative distances between nucleons of ⁴He are decreased. This is explained by influence of pressure of stellar medius on nucleons. We obtain similar tendencies for other isotopes of He and Be.

Figure 2.  (Color online) Panel (a): Energy of nuclear system ⁴He inside star at distance $ \xi=2.42 $ from center of star with density at center $ \rho_{\rm cr} = 10^{14} {\rm g}\, {\rm cm}^{-3} $ (at $ n=3 $). At minimum of the full energy of nucleus, we obtain: $ a = 0.8374 $ fm, $ E_{\rm full} = 5.038 $ MeV, $ E_{\rm full\, per\, nucl} = 1.327 $ MeV, $ E_{\rm kin} = 138.096 $ MeV, $ E_{\rm Coul} = 1.363 $ MeV, $ E_{\rm nucl} = -164.492 $ MeV, $ E_{\rm star} = 30.313 $ MeV. Panel (b): Full energy of nucleus inside star in comparison with full energy of this nucleus in Earth.

In Fig. 3 we show energies of nuclear systems of ⁸Be, ¹²C, ¹⁶O, ²⁴Mg, ⁴⁰Ca, ⁵⁶Fe. Location of disintegration of nucleus inside star is presented in Fig. 4.

Figure 3.  (Color online) Full energy of "nuclei" ⁸Be (a), ¹²C (b), ¹⁶O (c), ²⁴Mg (d), ⁴⁰Ca (e), ⁵⁶Fe (g) inside dense stellar medium (see blue solid lines) in comparison with full energy of these nuclei in Earth (see red dashed lines). Here, we use the same position as in previous Fig. 3 (distance $ \xi=2.42 $ from center of star, density at center $ \rho_{\rm cr} = 10^{14} {\rm g}\, {\rm cm}^{-3} $). At minimum of the full energy of ⁸Be we obtain: $ a = 0.64 $ fm, $ E_{\rm full} = 623.90 $ MeV, $ E_{\rm full\, per\, nucl} = 77.98 $ MeV, $ E_{\rm kin} = 638.77 $ MeV, $ E_{\rm nucl} = -431.47 $ MeV, $ E_{\rm star} = 415.31 $ MeV. At minimum of the full energy of ¹²C we obtain: $ a = 0.51 $ fm, $ E_{\rm full} = 2004.93 $ MeV, $ E_{\rm full\, per\, nucl} = 167.0 $ MeV, $ E_{\rm kin} = 1694.04 $ MeV, $ E_{\rm nucl} = -757.27 $ MeV, $ E_{\rm star} = 1066.45 $ MeV. At minimum of the full energy of ¹⁶O we obtain: $ a = 0.43 $ fm, $ E_{\rm full} = 4172.6 $ MeV, $ E_{\rm full\, per\, nucl} = 260.7 $ MeV, $ E_{\rm kin} = 3308.0 $ MeV, $ E_{\rm nucl} = -1090.6 $ MeV, $ E_{\rm star} = 1953.2 $ MeV. At minimum of the full energy of ²⁴Mg we obtain: $ a = 0.34 $ fm, $ E_{\rm full} = 11116 $ MeV, $ E_{\rm full\, per\, nucl} = 463.2 $ MeV, $ E_{\rm kin} = 8672 $ MeV, $ E_{\rm nucl} = -1748 $ MeV, $ E_{\rm star} = 4189 $ MeV. At minimum of the full energy of ⁴⁰Ca we obtain: $ a = 0.31 $ fm, $ E_{\rm full} = 41130 $ MeV, $ E_{\rm full\, per\, nucl} = 1028.2 $ MeV, $ E_{\rm kin} = 19536 $ MeV, $ E_{\rm nucl} = -2999 $ MeV, $ E_{\rm star} = 24585 $ MeV, $ R_{\rm p} = R_{\rm n} = 0.58 $ fm. At minimum of the full energy of ⁵⁶Fe we obtain: $ a = 0.28 $ fm, $ E_{\rm full} = 110406 $ MeV, $ E_{\rm full\, per\, nucl} = 1971.5 $ MeV, $ E_{\rm kin} = 363000 $ MeV, $ E_{\rm nucl} = -4068 $ MeV, $ E_{\rm star} = 78164 $ MeV, $ R_{\rm p} = 0.57 $ fm. $ R_{\rm n} = 0.59 $ fm.

Figure 4.  (Color online) Critical distance ξ from center of star in dependence on its density at center, where disintegration of nucleus ⁴He on nucleons takes place.

A polytropic model based on solution of the Lane-Emden equation is not so accurate in description of neutron stars, where effects of gravity described by general relativity are important. So, for analysis of bremsstrahlung processes inside neutron stars we solve equation of Oppenheimer and Volkoff (see Ref. [14], p. 34–36), where not polytropic dependence of pressure on density for relativistic neutrons is used (see Ref. [46], p. 23–29). Difference between these two approaches for densities in center of star in region of $ \rho_{\rm cr} = 10^{14}\, {\rm g}\, {\rm cm}^{-3} $ – $ 10^{16}\, {\rm g}\, {\rm cm}^{-3} $ is presented in Fig. 5.

Figure 5.  (Color online) Dimensionless density θ inside star in dependence on distance r from center of star, calculated by solution of Oppenheimer–Volkoff equation with Chandrasekar (not polytropic) EOS (OV+C) and by solution of Lane-Emdan equation with polytropic EOS at $ n=3 $ (LE+P) [parameters of calculations: Chandrasekar EOS is defined by Eqs. (2.3.5)–(2.3.6) in Ref. [46] for relativistic neutrons at $ \mu_{e}=1 $ (see p. 25–26 in that book), dimensionless density is defined from condition $ \rho (r) = \rho_{\rm c} \cdot \theta^{n=3} $ for simplicity of comparison between two approaches]. In approach with solution of Oppenheimer–Volkoff equation with not polytropic EOS we obtain: mass $ M=1.575\; M_{\rm sun} $ and radius $ R=36.00 $ km for star with density at center $ \rho_{\rm cr} = 10^{14}\, {\rm g}\, {\rm cm}^{-3} $, mass $ M=2.276\; M_{\rm sun} $ and radius $ R=19.95 $ km for star with density at center $ \rho_{\rm cr} = 10^{15}\, {\rm g}\, {\rm cm}^{-3} $, mass $ M=1.655\; M_{\rm sun} $ and radius $ R=11.07 $ km for star with density at center $ \rho_{\rm cr} = 10^{16}\, {\rm g}\, {\rm cm}^{-3} $ ($ M_{\rm sun} $ is mass of Sun).

Inverse beta decay is important in study of structure of neutron stars, the simplest EOS taking into account these processes is provided by Harrison and Wheeler. Such an EOS also provides information about the most probable nuclei in dependence on stellar density. Such estimations are given in Fig. 6.

Figure 6.  (Color online) Distribution of nuclei with mass numbers A (a) and charge numbers Z (b) along radius for star with density at center of $ 10^{14}\, {\rm g}\, {\rm cm}^{-3} $ [here Harrison–Wheeler Equation of State is used in calculations (we take EOS parameters from Ref. [46], see p. 42–48). This allows to take into account processes of inverse beta decay for higher densities, and provide information about chemical composition inside star. The finite-difference method is used for solution of Oppenheimer-Volkoff equation for this star). We obtain radius $ R=52.859 $ km and mass of star $ M=0.91072\; M_{sun} $ for such a type of model.

One can conclude that nuclei can co-exist with neutrons inside enough long layer in dense star with density at center of $ 10^{14}\, {\rm g}\, {\rm cm}^{-3} $. But this gives restrictions for consideraction of possibility of emission of bremsstrahlung photons from different nuclei. However, we prefare to provide full picture that is useful for general understanding about emission of photons from nuclear scattering.

We start calculations of spectra from a case when experimental data are known and agreement between theory and experimental data was obtained with highest accuracy. This is a case of bremsstrahlung in scattering of protons of nuclei $ p + $¹⁹⁷Au at proton beam energy of 190 MeV, where experimental bremsstrahlung data [60] were obtained. In framework of new formalism above, we reconstruct the spectra with inclusion of incoherent bremsstrahlung emission (for conditions of measurements).

Important (not-small) role of incoherent bremsstrahlung emission in the proton-nucleus scattering can be seen from the simple calculations. One can use only the largest coherent and incoherent terms at $ l_{i}=0 $, $ l_{f}=1 $, $ l_{\rm ph}=1 $. I.e., we define the coherent term $ M_{\rm coh} $ on the basis of term $ M_{p}^{(E,\, {\rm dip})} $ and incoherent term $ M_{\rm incoh} $ on the basis of term $ M_{\Delta M} $ in calculation of full matrix element as (see Ref. [10, 56] for details, reference therein)

where $ M_{p}^{(E,\, {\rm dip},0)} $ and $ M_{\Delta M} $ are calculated in Eqs. (4). Such calculations in comparison with experimental data are presented in Fig. 7 (a). ⁷ One can see that the full spectrum with included both coherent and incoherent contributions (see blue solid line in this figure) is essentially better describes experimental data, than just the coherent bremsstrahlung (see purple dash-dotted line in this figure). Note essential difference between these two spectra, that confirms large role of incoherent emission in the full bremsstrahlung. A main conclusion from this analysis is that incoherent processes are essentially more intensive than coherent processes, this difference is increased at increasing of energy of photon emitted. From this analysis it is easy to extract normalization factor, which can be used for other calculations of bremsstrahlung in cases where there is no any experimental information about bremsstrahlung.

Figure 7.  (Color online) Panel (a): The calculated bremsstrahlung spectra in the scattering of protons off the ¹⁹⁷Au nuclei at energy of proton beam of $ E_{\rm p}=190 $ MeV in comparison with experimental data [60] [matrix elements are defined in Eqs. (1)– (3), $ Z_{A} (k_{\rm ph})\simeq Z_{A} $ is electric charge of nucleus, we normalize each calculation of the second point of experimental data]. Here, open circles are experimental data obtained by TAPs collaboration with high precision [60], blue solid line is full spectrum (with coherent and incoherent terms), purple dash-dotted line is coherent contribution, red dashed line is incoherent contribution [all spectra are renormalized on 1 point of experimental data]. One can see that full spectrum with inclusion of coherent and incoherent contributions is in essentially better agreement with experimental data, than only coherent bremsstrahlung. This result confirms important and not small role of incoherent emission in bremsstrahlung during scattering of protons off heavy nuclei (inside such an energy region). One can see also that role of incoherent processes is increased at increasing of energy of photon. Panel (b): The calculated bremsstrahlung spectra for the scattering of protons off the ⁴He at energy of proton beam of $ E_{\rm p}=100 $ MeV in comparison with the calculated spectra and experimental data [60] for the scattering of protons off ¹⁹⁷Au at $ E_{\rm p}=190 $ MeV.

It is also useful to understand how much difference between bremsstrahlung emission for heavy nucleus and light nucleus in the similar conditions. Such calculations (on conditions of experimental laboratory on Earth) are presented in Fig. 7 (b) with comparison between spectrum for ¹⁹⁷Au at $ E_{\rm p}=190 $ MeV [given in Fig. 7 (a)] and spectrum for ⁴He at $ E_{\rm p}=100 $ MeV. One can see that difference between spectra is really large. In particular, more heavy nuclei and larger energies of proton beam produce essentially more intensive emission of bremsstrahlung photons. Similar conclusion we obtained before in study of bremsstrahlung emission in fission of nuclei [6, 7] (when we studied separation of parent nucleus ²⁵²Cf on two heavy fragments of similar masses with emission of photons) comparing that with bremsstrahlung in α decay of nuclei (see Refs. [4, 55], reference therein).

As a next step, we will estimate bremsstrahlung in the stellar medium. Here there are many new questions which have never been studied yet. As a demonstration, we will be interesting in how spectra are changed in result of influence of stellar medium. The simplest analysis is obtained from calculations of spectra for ⁴He. We use normalization factor extracted from previous study for $ p + $¹⁹⁷Au at 190 MeV with existed experimental data [see Fig. 7 (a)]. New calculations for $ p + $⁴He in dense stellar medium are shown in Fig. 8 (b). We conclude the following.

Figure 8.  (Color online) Bremsstrahlung emission of photons in scattering of protons off nuclei ⁴He inside star at energy of protons of $ E_{\rm p}=100 $ MeV at polytropic index $ n=3 $ [we calculate spectrum on the basis of the leading matrix element $ M_{p}^{(E,\, {\rm dip},0)} $, which gives the largest contribution to full spectrum, according to analysis in Refs. [56, 57]]. Contribution on the basis of matrix element $ \langle \Psi_{f} |\, \Delta \hat{H}_{\gamma} |\, \Psi_{i} \rangle $ in Eq. (23) (a), and full spectrum on the basis of matrix element $ \langle \Psi_{f} |\, \hat{H}_{\gamma} |\, \Psi_{i} \rangle_{\rm star} $ in Eq. (23) (b) are shown in these figures.

• For the white dwarfs, according to Fig. 2, influence of stellar medium on emission is not larger than $ 0.1\; MeV^{2} $. This means that influence of stellar medium imperceptibly affects on emission of bremsstrahlung photons. In particular, such a conclusion can be formulated for nuclear reactions in Sun. I.e., we have obtained accurate description of emission of bremsstrahlung photons during nuclear reactions in Sun, white dwarfs and stars at similar densities.

• For the neutron stars, influence of the stellar medium is more intensive, it crucially changes shape of the bremsstrahlung spectrum (see Fig. 8). In the simplest approximation, maximum of probability of the emitted photons is for their energy which is half of energy of the scattered protons: $ E_{\rm ph} \simeq E_{\rm p} / 2 $. The most intensive emission is formed in the bowel of stars, while the weakest emission is from the periphery (for the same energy of the scattered proton).

Calculations of the spectra of bremsstrahlung photons in the scattering of protons off ⁸Be, ¹²C, ¹⁶O, ²⁴Mg, ⁴⁰Ca, ⁵⁶Fe in dense stellar medium are shown in Fig. 9.

Figure 9.  (Color online) Bremsstrahlung emission of photons in scattering of protons off nuclei ⁸Be (a), ¹²C (b), ¹⁶O (c), ²⁴Mg (d), ⁴⁰Ca (e), ⁵⁶Fe (g) inside dense stellar medium at energy of protons of $ E_{\rm p}=100 $ MeV [we calculate spectra on the basis of the coherent matrix element $ M_{p}^{(E,\, {\rm dip},0)} $ defined in Eqs. (4)–(6) and the matrix element $ M_{\rm star} $ defined in Eq. (24) at $ N = A(A-1) $]. We obtain the following ratios between the coherent and incoherent contributions to the full bremsstrahlung spectrum at photon energy $ E_{\rm ph}=33 $ MeV without influence of stellar medium on nuclear process and emission: $ \frac{\sigma_{\rm incoh} (^{4}\text{He})}{\sigma_{\rm coh} (^{4}\text{He})} = 3.2340 $, $ \frac{\sigma_{\rm incoh} (^{8}\text{Be})}{\sigma_{\rm coh} (^{8}\text{Be})} = 58.629 $, $ \frac{\sigma_{\rm incoh} (^{12}\text{C})}{\sigma_{\rm coh} (^{12}\text{C})} = 299.20 $, $ \frac{\sigma_{\rm incoh} (^{16}\text{O})}{\sigma_{\rm coh} (^{16}\text{O})} = 947.7 $, $ \frac{\sigma_{\rm incoh} (^{24}\text{Mg})}{\sigma_{\rm coh} (^{24}\text{Mg})} = 4817.7 $, $ \frac{\sigma_{\rm incoh} (^{40}\text{Ca})}{\sigma_{\rm coh} (^{40}\text{Ca})} = 3.720 \cdot 10^{4} $, $ \frac{\sigma_{\rm incoh} (^{56}\text{Fe})}{\sigma_{\rm coh} (^{56}\text{Fe})} = 1.7592 \cdot 10^{5} $. One can see that role of incoherent emission is highly increased for heavier nuclear nuclei.

We also calculate bremsstrahlung emission during scattering of neutrons of nuclei in conditions of the dense stellar medium. From Eqs. (3)–(7) one can see that such a magnetic emission exists. If to take into account that incoherent emission is larger than coherent contribution [see Fig. 7], and incoherent emission comes from anomalous magnetic moments of protons and neutrons of nucleus and magnetic moment of the scattered neutron, one can conclude that this emission is comparable with bremsstrahlung for the proton–nucleus scattering. From Eqs. (3)–(7) we rewrite the matrix element of emission for neutron-nucleus scattering as

But, in this paper we will restrict ourselves only by main contribution, i.e. we will estimate cross-sections on the basis of matrix elements $ M_{\Delta M} $ and $ M_{\rm star} $. Such calculations of the bremsstrahlung spectra for the ⁴He, ¹²C, ¹⁶O, ²⁴Mg, ⁴⁰Ca, ⁵⁶Fe "nuclei" in dense stellar medium are shown in Fig. 10.

Figure 10.  (Color online) Bremsstrahlung emission of photons in scattering of neutrons off nuclei ⁴He (a), ¹²C (b), ¹⁶O (c), ²⁴Mg (d), ⁴⁰Ca (e), ⁵⁶Fe (g) inside dense stellar medium at energy of neutrons of $ E_{\rm n}=100 $ MeV [we calculate spectra on the basis of the incoherent matrix element $ M_{\Delta M} $ defined in Eqs. (4)–(6) and the matrix element $ M_{\rm star} $ defined in Eq. (24 at $ N = A(A-1) $, in calculations of normalization of full cross-sections we take into account ratios between the coherent and incoherent bremsstrahlung contributions given in caption to Fig. 9)].

After general analysis with unified picture of emission of bremsstrahlung photons during scattering of protons and neutrons off nuclei given in previous Sect. 2, we will try to analyze influence of structure of matter in neutrons stars on such emission. In this paper we will restrict ourselves by obtaining only the first preliminary estimations, while more accurate data should be obtained after careful analysis, as structure of matter inside dense stellar medium is deeply studied (for example, see Ref. [69], reference therein).

According to Ref. [46] (see p. 57 in that book), at density of $ 4 \cdot 10^{11}\, {\rm g} / {\rm cm}^{3} $ ratio between neutrons and protons reaches a critical value, and free neutrons, electrons and nuclei co-exist inside (i.e., neutron drips are formed). When density is higher than $ 4 \cdot 10^{12}\, {\rm g} / {\rm cm}^{3} $, neutrons give larger pressure than electrons, and neutrons play more important role than electrons. So, we will choose this condition as starting point for analysis of bremsstrahlung during scattering of neutrons off nuclei, where we will use nucleus ⁵⁶Fe for calculations. Such a density in star with density at center of $ 10^{16}\, {\rm g} / {\rm cm}^{3} $ is in external layer, close to external surface of star [see Fig. (a), this is point at distance of 10.63 km from center of star, radius of this star is 11.47 km].

However, nuclei in this space region of star are located at very close distances between each other. In approximation, one can estimate the distance between two closest nuclei. Let's write density as

where $ n_{^{56}\text{Fe}} $ is number of nuclei ⁵⁶Fe per unit volume, $ N_{^{56}\text{Fe}} $ is number of nuclei ⁵⁶Fe per volume V, $ m_{^{56}\text{Fe}} $ is mass of nucleus ⁵⁶Fe. From this formula we find averaged volume per one nucleus as

If to take into account that the most probably nuclei inside such a dense medium are located in the most compact way, then a distance between two closest nuclei can be estimated as

On the basis of this formula, for density of $ 4 \cdot 10^{12}\, {\rm g} / {\rm cm}^{3} $ we calculate distance and obtain $ d=28 $ fm. Here, consideration of geometry and surface of Wigner-Seitz cell will be more accurate in such estimations and study [69]. But, in this paper we restrict ourselves by simple geometric schemes of location of nuclei.

Let us consider neutron located between two closest nuclei. At some non-zero kinetic energy $ E_{\rm n} $, this neutron is incident (scattered) on one nucleus, i.e. we have the neutron-nucleus scattering process. According to grounds of QED, for non-zero kinetic energy of relative motion (of neutron and nucleus) there is non-zero probability of emission of bremsstrahlung photons during such a type of scattering.

Usually, for conditions of experiments on the Earth, the bremsstrahlung cross-sections are calculated on the basis of square of full matrix element of emission [see Eqs. (3)–(6)]. Multipole expansion of wave function of photons provides enough accurate calculation of matrix elements of emission for different nuclear processes. So, in frameworks of our formalism, we applied this expansion in current task. In result, each matrix elements is represented as multiplications of radial integrals and coefficients which describe angular distribution of emission of photons in the studied reaction. A main difficulty in this problem is in numeric calculations of radial integrals. These integrals can be written in general form as

where

and r is distance between neutron and center-of-mass of nucleus. For conditions of experiments in Earth, in order to reach convergence in calculations of the bremsstrahlung spectra we need to use $ R_{\rm max} $ at distances of atomic shells (more 10000 fm) or higher from nuclear center. So, it is natural to suppose that other nuclei located at distances close 28 fm should give strong changes in the full spectrum of bremsstrahlung emission. However, such a model above is constructed on space representation (in contrast to models, constructed on the basis of Feynman diagrams in momentum representation), and it provides accurate way how to estimate such an influence from other nuclei.

In order to perform such estimations, we calculate the spectra on the basis of integral $ I_{1} $ in Eqs. (35), where we chose $ R_{\rm max,1}=d/2=14 $ fm. In such a case, corrections should be given after inclusion of another integral $ I_{2} $ in Eqs. (35), where we chose $ R_{\rm min}=R_{\rm max,1}=14 $ fm and $ R_{\rm max}=10000 $ fm. One can compare the spectrum calculated on the basis of the first radial integral $ I_{1} $ only with the full spectrum with inclusion of this first integral $ I_{1} $ and addition of the second integral $ I_{2} $ also.

Results of such calculations are presented in Fig. 11 (b). In this figure one can see that spectrum calculated on the basis of the first radial integrals $ I_{1} $ at $ R_{\rm max,1}=14 $ fm (see red solid line in this figure) is very close with another spectrum calculated on the basis of full radial integral I at $ R_{\rm max}=10000 $ fm (see brown dash-dotted line in this figure). We also add calculated correction from integral with boundaries $ R_{\rm min}=14 $ fm and $ R_{\rm max}=10000 $ fm to this figure multiplied on factor $ 10^{10} $ (see blue dash-double dotted line in this figure). This correction is extremely small that confirms that main part of bremsstrahlung emission is formed inside space region up to 14 fm from nuclear center. By such a reason, we conclude that influence of another nucleus (located more far than 28 fm from the studied nucleus) on emission of these bremsstrahlung photons is extremely small, and the obtained estimations are correct. This result indicates that the bremsstrahlung emission of photons from such nucleon-nucleus nuclear processes can be studied for conditions even inside neutron stars. It is supposed that such results can be applied for conditions of inner crust of neutron star (at further more accurate corrections on the basis of details of structure there).

Figure 11.  (Color online) [Pannel a]: Density of matter inside star with density at center of $ 10^{16}\, {\rm g} / {\rm cm}^{3} $ [calculations are obtained by solution of Oppenheimer–Volkoff equation with not polytropic EOS, EOS is defined by Eqs. (2.3.5)–(2.3.6) in Ref. [46] for relativistic neutrons at $ \mu_{e}=1 $ (see p. 25–26 in that book)]. [Pannnel b]: Bremsstrahlung emission of photons in scattering of nucleons off nucleus ⁵⁶Fe at energy of neutrons of $ E_{\rm n}=100 $ MeV located at density $ 4 \cdot 10^{12}\, {\rm g} / {\rm cm}^{3} $. Here, red solid line is the spectrum calculated on the basis of the first radial integrals $ I_{1} $ at $ R_{max,1}=14 $ fm in Eqs. (35), brown dash-dotted line is the spectrum calculated on the basis of the full radial integrals I at $ R_{max}=10000 $ fm in Eq. (34), blue dash-double dotted line is the correction calculated on the basis of the second radial integrals $ I_{2} $ at $ R_{max,1}=14 $ fm and $ R_{max}=10000 $ fm in Eqs. (35) and multiplied on factor of $ 10^{10} $, green dotted line is the full spectrum calculated on the basis of the full radial integrals at $ F=100\,MeV^{2} $.

Quantum mechanics provides natural basis for study of such nuclear processes in the dense matter. Effective way to describe such types of nuclear processes in condition of dense matter can be obtained after unification of (1) many nucleon formalism on structure of nucleus and nuclear reactions on the basis of quantum mechanics (new developments in line of Refs. [17−20], see also Refs. [49−52]), (2) methods of quantum mechanics with high precision and tests (see method of Multiple Internal Reflections (MIR), also there is method of phase functions in nuclear scattering), which describe processes of relative motion of two nuclei (or nuclear fragments) in nuclear scattering with possibility of formation of compound nuclear system and fusion (or opposite processes of nuclear decays), and (3) additional formalism describing influence of dense medium on nuclear processes in compact star.

In particular, the method MIR was initially constructed for α decays of nuclei, then applied for decays of nuclei with emission of protons, then generalized for opposite process of capture of α particles by nuclei (see Refs. [53, 54], reference therein). This method provides own independent parameters of nuclear potentials for the studied fusion reactions with high (the highest) accuracy of agreement with available experimental information (for example, see Ref. [53] for α-nucleus reactions, with details, comparison with alternative approaches, etc.). The method was investigated to study different stages and mechanisms of nuclear scattering (for example, resonant and potential scattering are the simplest demonstrations). Then, that method was extended for fusion of nuclei in case of close distance between those (like located in lattice cites of compact star, that is not existed in Earth). Accuracy and effectiveness of this method were demonstrated in study of possibility of synthesis of new more heavy nuclei from close nuclei in neutron star at low energies (pycnonuclear reactions). This was presented on example with isotopes of Carbon in Refs. [77, 78].

Note that this method has accuracy of the calculated cross sections of fusion about $ 10^{-14} $, in comparison with existed alternative approaches of about $ 10^{-3} $ in best cases. Tests were developed in order to check and estimate calculations obtained by method MIR and other methods (see Refs. [53, 54], reference therein, for details). In Refs. [77, 78] there are demonstrations related with study of details of fusion for isotopes of Carbon. Some new physical effects (which are not small and have not been studied by other approaches) were studied in Ref. [53] (for conditions of labs in Earth). In particular, in that publication probabilities of fusion from experimental data were extracted, and new probabilities of fusion were predicted for new reactions with close isotopes for new experimental tests in future. It is important to note that formalism of method MIR is in direct connection with methods of inverse scattering problem (for example, see Refs. [79−82]), that provides save basis for further study of nuclear forces in connection with experimental data. So, we estimate for conditions of dense matter this can be naturally extended, if to use results of those publications and this paper. We estimate this research as a good perspective in study nuclear processes in dense matter, and to extend that for collisions of nuclei with higher energies.

Emission of bremsstrahlung photons from nucleon-nucleon scattering is important topic of research. It helps to understand many aspects of nucleon-nucleon interactions, development of different relativistic models, etc. But in the nucleon-nucleus reactions and nucleus-nucleus reactions, many nucleon problem is already important.

Role of many-nucleon processes (effects of many body problem) in calculation of cross sections of emission of photons is larger, when number of nucleons in nuclei is larger. The most easy way to check this phenomenon can be obtained in the study of proton-nucleus scattering where experimental data for bremsstrahlung are also obtained with high precision (we found that phenomenon in Refs. [55−57], see also Ref. [10] with reference therein).

Different aspects of many-nucleon effects can be explicitly described by operator of emission of photons (and corresponding matrix elements) in the scattering of protons off nuclei composed of nucleons in unified formalism in frameworks of quantum mechanics. Here, both electric charges and (full) magnetic moments of protons and neutrons in nucleus-target and protons in beam have important role. In study of many nucleon effects in general, magnetic moments of nucleons are essentially more important, than electric charges of these nucleons. In particular, magnetic moments of nucleons provide basis for incoherent emission of photons. It turns out that for nuclei of middle and heavy masses, role of incoherent processes becomes significantly larger than role of coherent processes (i.e., case when emission of photons is produced due to relative motion (evolution) between nucleus-target as whole object and proton in beam in scattering). After inclusion of incoherent emission to the bremsstrahlung model, understanding on full emission of photons in nuclear processes is changed significantly (shapes of cross sections of full bremsstrahlung are changed even after possible re-normalizations of these spectra on the same experimental data). At the same time, for light nuclei (such as deuterium, isotopes of Helium, Beryllium, see Ref. [8]) the coherent emission is similar with incoherent emission of photons.

One can study even influence of clusters in nuclei on full bremsstrahlung emission (that characterizes structure of these nuclei and is pure phenomenon of many body quantum problem) if to use such many-nucleon formalism of quantum mechanics [see Refs. [8, 9]].

So, one can suppose that such many-nucleon effects should be important in study of different processes in evolution of neutron stars. In particular, it will be interesting to know role of many-nucleon dynamics produced emission of photons in cooling of neutron stars. Such mechanisms as incoherent processes in different nuclear processes with different isotopes of nuclei of different masses can be interesting for study. It would be good at first to obtain simple estimations providing clear picture of this question. We would like to indicate good perspective of this research direction in further study.

There is interesting question in study of emission of neutrino in cooling process of neutron star, where many-nucleon dynamics in nuclear reactions can play important role. To construct a model for such a study, we should solve some problems.

One problem is unification of theory of structure of nuclei and nuclear reactions on strict basis of quantum mechanics (in space representation; for example, see Ref. [17−20]), and formalism describing emission of neutrino in reactions (described in QFT in good way, in momentum representation). We take idea for construction of this model from study of production of lepton-antilepton pairs (dileptons) in proton-nucleus scattering (heavy ion collisions) on the basis of quantum mechanics (see Ref. [83]). The simplest process of production of dileptons can be described via intermediate process of exchange of virtual photon (between scattered nucleon and produced dilepton) on the basis of connection of hadronic and leptonic fluxes. Here, one flux describes emission of virtual photon from one nucleon of nucleus-target or proton in beam, another flux describes production of lepton pairs from virtual photon. In case of emission of neutrino, one can generalize this formalism with new implementation of weak fluxes describing emission of neutrino (in frameworks of theory of weak interactions).

Another question not clear for us in this study is related with observation of neutrinos emitted from nuclear reactions in such stars. We see good perspective in further development of quantum mechanical model for study emission of neutrino in cooling process in such stars.

One can think that bremsstrahlung emission from nuclear processes in stars is small. But, from our experience in bremsstrahlung study, such photons in nuclear reactions bring more rich and deep information about such nuclear processes (structure of nuclei, clusters, nuclear deformations, mechanisms of reactions, dynamical properties in scatttering, new properties of hypernuclei, even nuclear interactions, study of quarks in nuclear reactions, etc.; for example, see Ref. [8−10] with references) than direct study of these reactions itself, without inclusion of photons. Usually experimental cross sections or probabilities of emitted bremsstrahlung photons are really small, but such investigations exist for long time. For example, this even gives possibility to construct new type of microscopy to explore physics of micro-world on the basis of analysis of bremsstrahlung emission. Such attempt was done by us and now this idea has been already accepted by physical community (see Ref. [2]). By simple words, this is transition from study of some elementary processes with emission of photons known from textbooks on QED by many people to principally new type of microscopy (constructed on the basis of bremsstrahlung analysis) in explore more tiny microscopic structure of matter (without additional construction of new experimental facility, just to use already existed facilities and detectors for photons). Now we would like to find similar possibility for more effective way in study of internal structure of stars, using emission of such photons and our previous investigations in the basis.

Before our study, many questions have not been known. For example, it was not clear how emission of bremsstrahlung photons is changed in dependence on density of matter in star (now we give some information and estimations, with help of Referee).

We have hypothesis that this model will give possibility to measure densities of matter inside neutron stars, if it is possible to measure spectra of photons emitted from this object. Star can emit photons. But mainly photons emitted from reactions with higher intensities are not bremsstrahlung. Those have essentially different shape of spectrum with peaks. Peaks indicate processes of synthesis or other nuclear processes at some fixed energies of photons. Another part of spectrum of photons is of bremsstrahlung type. Such a spectrum looks like spectra in Fig. 7 (spread, without peaks). Now, from the measured spectrum of photons one can find the most probable value of kinetic energy of the scattered nucleon in nucleon-nucleus processes (see Figs. 8–10). Here, one can propose a simple formula for such estimations as $ E_{\rm p,n}^{\rm (kin)} = E_{\rm ph, max}/2 $ [here, $ E_{\rm p,n}^{\rm (kin)} $ is the estimated kinetic energy of the scattered nucleon, $ E_{\rm ph, max} $ is energy of photons in spectrum at bremsstrahlung cross section with maximum]. According to Eqs. (25), bremsstrahlung cross-section at maximum indicates on value of F that is related with density of stellar medium with the studied reaction and chosen EoS [see Figs. 8–10]. By such a way, one can try to extraction of information about density of matter in compact object from possible measurements of photons. In this paper we construct some basis for that investigation.

Another perspective is to apply that investigation for collisions of heavy nuclei at close relativistic velocities (where high density of matter can be supposed, comparing to saturation density of nuclear matter). Of course, to study that we need in working model, which should be tested on different tasks before. One can suppose that bremsstrahlung photons emitted in such collisions can bring new information about dense nuclear matter of colliding nuclei or compound nuclear system during collision. It will be interesting to check that.

In this paper the bremsstrahlung formalism [56, 57] (see Refs. [2, 3, 55, 61, 70−75]) for scattering of nucleons off nuclei is generalized for conditions of dense medium of compact stars. For that, the nuclear model of deformed oscillator shells [49, 50, 52] is applied via new inclusion of influence of stellar medium on nucleons of nucleus. We obtain unified picture of bremsstrahlung emission in the nucleon-nucleus scattering including conditions of experiments in the Earth and dense medium of compact stars.

In frameworks of that formalism stellar medium does not affect on emission of photons in the white dwarfs. Concluding, we obtain enough accurate description of emission of bremsstrahlung photons during nuclear reactions in Sun, white dwarfs and stars with similar densities. For neutron stars influence of stellar medium is essentially more intensive. It changes essentially shape of the spectrum of the bremsstrahlung photons. The most intensive emission is formed in the bowel of the star, while the smallest emission is from the periphery. One can obtain the following approximated formula: Maximum of probability of the emitted photons is at energy which is half of energy of the scattered nucleon ($ E_{\rm ph} \simeq E_{\rm p} / 2 $ for protons). We estimate emission of photons due interactions of protons and neutrons with ⁸Be, ¹²C, ¹⁶O, ²⁴Mg, ⁴⁰Ca, ⁵⁶Fe inside dense stellar medium (see Figs. 9, 10). We find that emission of bremsstrahlung photons in the scattering of neutrons of nuclei is not small. This is explained by large contribution of incoherent processes due to important role of magnetic moments of protons and neutrons in nuclei. All these results are obtained at first time.

Electron-nucleus processes in the crusts of neutron star produce bremsstrahlung emission. Nucleons of nuclei have magnetic moments and electrical charges. Investigations on bremsstrahlung in proton-nucleus scattering indicate on more intensive incoherent bremsstrahlung emission than coherent one. This effect is small for light nuclei, but it is increased essentially for heavy nuclei (for example, ratio in the proton-nucleus scattering for ⁵⁶Fe is $ \dfrac{\sigma_{\rm incoh} (^{56}\text{Fe})}{\sigma_{\rm coh} (^{56}\text{Fe})} = 1.7592 \cdot 10^{5} $). Bremsstrahlung emission in the scattering of protons off nuclei were measured with high accuracy by TAPS collaboration [60]. Plateau in these data cannot be described on the basis of the coherent bremsstrahlung only. Note that previously less accurate data indicating on such a shape of spectra were obtained in Refs. [58, 59]. To explain measured data of bremsstrahlung, at first ratios between incoherent and coherent contributions were extracted for those reactions [56]. Analysis of such data indicated on essentially larger incoherent bremsstrahlung. Later, explicit formulas were obtained allowing to predict such an incoherent bremsstrahlung. A similar situation exists also with electron nucleus scattering in surface layer of stars (electron also has magnetic momentum and non-zero spin). One can suppose that inclusion of incoherent processes to the models of stars can change essentially a picture of ratios (between rates of emission and absorbtion) in such layers in stars. So we see a perspective to check carefully this idea in further research.

The constructed model describes disintegration of nucleus on individual nucleons, starting from some critical distance between this nucleus and center of star. Stellar medium acts on individual nucleons of nucleus. Binding energy (it is negative for nucleus in the external layer of star) is increased at deeper location of this nucleus in star. Starting from some critical distance from nucleus to center of star, the binding energy becomes positive (see Fig. 2). This means that full energy of individual nucleons of the studied nucleus is already larger than mass of this nucleus, i.e. we obtain unbound system of nucleons and nucleus is disintegrated on nucleons. Such a phenomenon is observed in the neutron stars (see Fig. 3), and it is not observed in the white dwarfs.

Authors are highly appreciated to Profs. V. S. Vasilevsky, M. I. Gorenstein, A. V. Nesterov for useful recommendations and interesting discussions concerning to modern many-nucleons nuclear models and physics of nuclear processes inside dense stellar medium.

In this Appendix we give a short review of the model of deformed oscillatoric shells (DOS) (for example, see Refs. [49−52]). Hamiltonian of nucleus as a system on A nucleons (in the bound states) is defined in Eq. (9) as

and the Schrödinger equation is written as

Here, Ψ and E are wave function of nucleus and full energy of nucleus which are unknown and should be found.

Hamiltonian in Eq. (A1) has potential of two-nucleon nuclear and Coulomb interactions between all nucleons (i.e. this is many body problem of quantum mechanics). It is known that, if to ignore dependence on velocities of nucleons, the most general form of potential of nuclear interactions between two nucleons dependent on three vectors: distance r between nucleons and two spins $ {\bf{s_{i}}} $, $ {\bf{s_{j}}} $ of nucleons with numbers i and j. This can be expressed via potential (operator), which has 6 different terms [for example, see book [62], p. 555–559].

where

Here, $ {\bf{n}}_{ij} = ({\bf{r}}_{i} - {\bf{r}}_{j}) /r_{ij} $, $ r_{ij} = |{\bf{r}}_{i} - {\bf{r}}_{j}| $. Operator of exchange of two nucleons $ \hat{P} $ can be written via operators of isospin $ \hat{ {\boldsymbol{\tau}}_{i}} $, $ \hat{ {\boldsymbol{\tau}}_{j}} $ of two nucleons [for example, see book [62], p. 550–551].

The potential $ \hat{V}(ij) $ can be rewritten in another form via four potentials $ \hat{U}_{2S+1, 2T+1} $ where $ T=0,1 $ and $ S=0,1 $ are eigenvalues of operators of full isospin and spin of system of two nucleons [for example, see book [62], p. 559]. In DOS model, this potential is transformed to the following form:

where spin and isospin operators of exchange are [for example, see Ref. [76], p. 24, 55; Ref. [62], p. 289]

Along to the DOS model, the full energy of nucleus and unknown wave function are calculated as solution of many body problem in the first order of perturbation theory in quantum mechanics. One can find next corrections (it is known that second correction improves agreement between calculated binding energies ant their experimental values for wide region of nuclei, we do not focus on this research in this paper). In first order of perturbation theory we calculate energy as

where $ \Psi_{0} $ is unperturbative wave function. In the DOS model this wave function is defined as

Here, $ \beta_{A} $ is the set of numbers $ 1 \ldots A $ of nucleons of the nucleus, $ \psi_{\rm nucl} (\beta_{A}) $ is the many-nucleon function dependent on nucleons of the nucleus. Summation in Eqs. (A8) is performed over all $ A! $ permutations of coordinates or states of nucleons. One-nucleon functions $ \psi_{\lambda_{s}}(s) $ represent the multiplication of space and spin-isospin functions as $ \psi_{\lambda_{s}} (s) = \varphi_{n_{s}} ({\bf{r}}_{s})\, \bigl|\, \sigma^{(s)} \tau^{(s)} \bigr\rangle $, where $ \varphi_{n_{s}} $ is the space function of the nucleon with number s, $ n_{s} $ is the number of state of the space function of the nucleon with number s, $ \bigl|\, \sigma^{(s)} \tau^{(s)} \bigr\rangle $ is the spin-isospin function of the nucleon with number s. One-nucleon space function is eigenfunction of 3D Harmonic oscillator as

where $ H_{n_{x}} $, $ H_{n_{y}} $ and $ H_{n_{z}} $ are the Hermitian polynomials, a, b and c are oscillator lengths. It is supposed that all oscillator lengths are unknown (for each nucleus), these needed parameters are found from condition of minimum of full energy of nucleus at variation of these parameters. By such a way, minimum of full energy (related to binding energy) and all needed parameters for wave function are found for each nucleus.

So, in the DOS model, interacting potential is dependent on spin and isospin operators, nuclear energy is calculated even analytically for even-even nuclei (where both cases $ N \ne Z $ and $ N=Z $ are included). Matrix element in Eq. (10) in the manuscript for nuclear energy is obtained due to calculation with use of wave function in form (A8)–(A9) and potential in form (A5)–(A6) and summation over all spin and isospin states. This formalism can be extended (developed) for nuclei with odd number of protons or/and odd number of neutrons (this case is not analyzed in this manuscript).

A saturation property of nuclear matter is also taken into account, that gives additional relations between components of potential (see logics in book [62], p. 560). Taking this properties into account, and also from analysis of meson theories for one-pion exchange one can obtain (see Eq. (1) in Ref. [49]) also):

For calculations, it is used:

where $ V_{t}=72.5 $ MeV, $ \mu_{t}=1.47 $ fm, $ V_{s}=39.15 $ MeV, $ \mu_{s}=1.62 $ fm. Potentials $ U_{31} $ and $ U_{13} $ in form (A11) with such parameters are obtained from good description of experimental data for scattering lengths and effective radiuses in nucleon-nucleon scattering.

Full energy of each nucleus is calculated at variation of oscillator lengths of one-nucleon space functions. Varying such parameters, one can obtain minimum of full energy. This point indicates on the most stable state of nucleus. This state is chosen as the ground state for the analyzed nucleus, oscillator lengths and full energy of nucleus (and binding energy) are found. Such a procedure is stable, as minima of energies are clear for each nucleus. As a demonstration, we show Fig. 2 and Fig. 3 for nuclei from ⁴He up to ⁵⁶Fe. This procedure is working also if to add additional influence from stellar medium to the model (this is new aspect for DOS model, not studied previously).

From Eq. (12) we calculate the proton and neutron densities for isotopes of ^4,6,8He and ^8,10Be:

where

One can see that the proton and neutron densities are the same for nuclei ⁴He, ⁸Be, they are different for nuclei ⁶He, ⁸He, ¹⁰Be. Also we have properties:

We shall find potential energy of two-nucleon nuclear interactions for nucleus ⁶He on the basis of matrix element in Eq. (10) and found densities (A12). After calculations, we obtain:

where

and we use notations (with change of indexes t and s):

From solutions above one can see that (1) only ⁴He is spherical in the ground state, while other nuclei are deformed; (2) nuclei ⁶He, ⁸Be are axially symmetric in the ground state, while ⁸He, ¹⁰Be are fully deformed. Further calculations of minima of full energy of these nuclei confirm such a logic.

In spherically symmetric approximation ($ a = b = c $), we obtain

where

In this Section we shall find correction of energy of nucleus due to influence of stellar medium (18):

Substituting Eq. (17) to this formula and taking into account the same action of force $ {\bf{F}}_{P}(R) $ for each nucleon, we obtain:

We use property:

Here, summation is performed over all states for the given configuration of nucleus (they are denoted by indexes m and k). All nuclerons are numeberd by indexes i and j. We use representation for one-nucleon wave function:

where $ \varphi_{n_{s}} $ is space function of the nucleon with number s, $ n_{s} $ is number of state of the space function of the nucleon with number s, $ \bigl|\, \sigma^{(s)} \tau^{(s)} \bigr\rangle $ is spin-isospin function of the nucleon with number s. For operator $ \hat{V}\, ({\bf{r}}_{i}, {\bf{r}}_{j}) $ acting on space functions for two nucleons only, we calculate matrix element:

where orthogonalization properties of spin and isospin functions are used:

In particular, for ⁴He Eqs. (B6) are simplified:

and we obtain

where Eq. (A13) for $ F_{0} ({\bf{r}}_{i}, {\bf{r}}_{j}) $ is used. For correction of energy, from (B1) we obtain:

In the spherically symmetric case ($ a=b=c $), we calculate integral:

and obtain solution:

In this Section we describe some aspects of folding approach which we use, and its connection to microscopic approach for nucleon-nucleus scattering. Definitions of the folding model and cluster model in nucleon-nucleus and nucleus-nucleus scatterings are given in Ref. [8] [see Eqs.(1) and (2) in that paper], where determination of folding potential is described in Eqs. (9)–(31) in Ref. [8].

In the folding approach, key element is representation of the full wave function for nucleon-nucleus scattering in more approximated form than in the fully microscopic formalism. Following to this formalism, the full wave function of the proton-nucleus scattering is written down as $ \Psi = \Phi ({\bf{R}}) \cdot \Phi_{\rm p - nucl} ({\bf{r}}) \cdot \psi_{\rm nucl} (\beta_{A}) $ [also following to the formalism in Ref. [56] for the proton-nucleus scattering, see Sect. II.B, Eqs. (10)–(13)], many-nucleon structure of the nucleus is described as in Ref. [55]]. Here, $ \Phi ({\bf{R}}) $ is the function describing motion of center-of-mass of the full nuclear system in laboratory frame, $ \Phi_{\rm p - nucl} ({\bf{r}}) $ is the function describing relative motion of the scattered proton concerning to nucleus (without description of internal relative motions of nucleons in the nucleus), $ \psi_{\rm nucl} (\beta_{A}) $ is the many-nucleon function of the nucleus, defined in Eq. (12) Ref. [56] on the basis of one-nucleon functions $ \psi_{\lambda_{s}}(s) $, $ \beta_{A} $ is the set of numbers $ 1 \cdots A $ of nucleons of the nucleus. One-nucleon functions $ \psi_{\lambda_{s}}(s) $ represent the multiplication of space and spin-isospin functions as $ \psi_{\lambda_{s}} (s) = \varphi_{n_{s}} ({\bf{r}}_{s})\, \bigl|\, \sigma^{(s)} \tau^{(s)} \bigr\rangle $, where $ \varphi_{n_{s}} $ is the space function of the nucleon with number s, $ n_{s} $ is the number of state of the space function of the nucleon with number s, $ \bigl|\, \sigma^{(s)} \tau^{(s)} \bigr\rangle $ is the spin-isospin function of the nucleon with number s. Definitions are also in Ref. [11].

In our paper we assume approximated form of the function $ \Phi_{\bar{s}} ({\bf{R}}) $ before and after emission of photon as ($ \bar{s}=i $ is for state before emission of photon, and $ \bar{s}=f $ is for state after emission of photon)

where $ {\bf{K}}_{i} $ and $ {\bf{K}}_{f} $ are moments of full nucleon-nucleus system in states before emission of photon and after such an emission in laboratory frame.

In description of structure of nucleus in this problem we use model of deformed oscilatoric shells (DOS) instead of part of formalism in Refs. [8, 9] describing structure of nucleus. This model DOS is more clear for analysis of structure of nuclei, provides calculations of wave function $ \psi_{\rm nucl} (\beta_{A}) $ of nucleus, binding energy, deformations of different isotopes of light nuclei (that was previously compared with existed experimental data), allows to include influence of stellar medium on nucleus, etc. It also provides enough close calculations for properties of light nuclei, but without clusters inside these nuclei. For many calculations of ⁴He and other light nuclei is it convenient, and we use it. It makes many calculations more fast (some even analytical). In bremsstrahlung formalism DOS model provides wave functions which are important for determination of form factors in calculation of cross sections of bremsstrahlung.

Note that all wave functions should be found from numerical solution of Schrodinger equation for nucleon-nucleus scattering with the given nucleon-nucleon potentials. But in the problem of emission of bremsstrahlung photons in this reaction, these wave functions are already used, we develop formalism for operator of emission of photon (or different contributions to it, like coherent and coherent contribution, etc.), and we calculate numerically the matrix elements of bremsstrahlung emission on the basis of the found before wave functions and this operator of emission. Actually, for practical calculations of bremsstrahlung spectra we need to know wave functions of relative motion between the scattered nucleon and nucleus $ \Phi_{\rm p - nucl} ({\bf{r}}) $ which are calculated numerically concerning to kinetic energy $ E_{T} $ of relative motion between these two objects. Exact formulas are $ E_{T} = \dfrac{A}{A+1}\, E_{\rm p} $ and $ E_{\rm cm} = \dfrac{1}{A+1}\, E_{\rm p} $ where $ E_{\rm p} $ is kinetic energy of the scattered proton in laboratory frame. But, for problems where one object in scattering is nucleon, approximation is enough good where we have $ E_{T} \simeq E_{\rm p} $ (we use that in calculations in this paper).

Formalism (in compact form) of calculations of matrix elements of bremsstrahlung for proton-nucleus scattering in the folding approximation is presented in Ref. [10] with Supplementary Material [11] (here there is reference list for more details with explanations, needed definitions are given). In particular, in the Supplementary Material [11] formalism of inclusion of each function $ \Phi ({\bf{R}}) $, $ \Phi_{\rm p - nucl} ({\bf{r}}) $ and $ \psi_{\rm nucl} (\beta_{A}) $, to calculation of bremsstrahlung cross sections in the nucleon-nucleus scattering is presented.

Equation of equilibrium in star on Newtonian gravity has form (see Eq. (1.2), p. 19, [14]):

It can be rewritten as

Also we should take into account equation of continuity of mass [for example, see Eq. (1.3) in Ref. [14], p. 19]:

Substituting it to Eq. (D2), we obtain [see Eq. (1.4) in Ref. [14], p. 21]

Let’s write down polytropic dependence of pressure P on density ρ as

where K is constant, γ is polytropic index. Stars with such EoS are called as polytropic (see Ref. [14], p. 19). Using it, Eq. (D4) is transformed as

Now we introduce unitless density θ and dimension ξ as

where $ \rho_{\rm c} $ is density at center of star, n is polytropic index, $ R_{0} $ is parameter. Substituting those to Eq. (D6), after derivation we obtain

At choice

we obtain Lane-Emden equation of equilibrium of polytropic star

We have Tolmann-Oppenheimer-Volkoff (TOV) equation

and equation of conservation of mass

Also we use polytropic EoS as

Here, K is constant. We obtain

Substituting this formula to (D11), we find

Now we introduce unitless density and distance as

and Eq. (D15) is transformed as

So, we obtained system of equation on the basis of TOV formalism:

This system of equations can be rewritten in compact form as [see App.??, (??), p.??]:

Here, new coefficients are introduced [see Eqs. (??), p.??]:

This system is solved numerically at initial conditions of $ \theta(0) = 1 $, $ \dfrac{d\theta(0)}{d\xi} = 0 $. Also $ \xi^{*} $ is unitless radius of star, which is found from condition of zero density at boundary $ \theta(\xi^{*}) = 0 $, and radius of star is $ r_{star} = \xi^{*}\, R_{0} $. For solution of system of equations it needs to define parameters $ R_{0} $ and K. Simple formalism of calculations of these parameters is presented in Appendix D.3 [see Eqs. (D34), (D35)].

The system of equations (D19) can be rewritten in form of one differential equation, if to express function θ via m. At first, from the second equation of Eqs. (?? we find function θ an its derivative as

Substituting these solutions to the first formula in Eqs. (D19), we obtain

Mass m is calculated in units of MeV in numerical calculations.

We follow the formalism in Ref. [14] (see p. 29–34 in that book). Let's consider the equation of state of completely degenerate electrons. Since electrons are fermions with a half-integer spin of $ 1/2 $, only one electron with a given spin in a phase space cell can be there. The distribution function at complete degeneracy $ f_{e} (p) $ is given by a shelf in the region $ 0 < p < p_{fe} $, where the maximal momentum of the electrons $ p_{fe} $ is called as Fermi momentum. The volume of one phase space cell is $ (2 \pi, \hbar)^{3} $. Electrons occupy a spherical volume of $ V_{\rm ph} $ with a radius of $ p_{fe} $ in the phase space and

Note that in one cell of the phase space there are two electrons with spins of $ 1/2 $ and $ - 1/2 $. Therefore, concentration of completely degenerate electrons is written down as

This formula defines dependence of $ p_{fe} $ on the concentration of $ n_{rm e} $:

The isotropic pressure of electrons $ P_{\rm e} $ is determined by the momentum flux as

A relativistic relation between velocity and momentum is used here.

The energy of a unit of volume $ \varepsilon_{rm e} $ is written in the same way (D26). The total energy of one electron is equal to $ \sqrt{p^{2} + m_{\rm e}^{2} c^{2}} $. The maximal kinetic energy of one electron in degenerate matter (Fermi energy) is

The integral in (D26) is derived quite not simply. But within non-relativistic and ultra-relativistic limits relations are quite simple

The concentration of electrons due to electroneutrality is expressed in terms of the density of matter

where $ mu_{\rm e} $ is number of nucleons per electron. For hydrogen, $ \mu_{\rm e} = $1. In a substance consisting of helium, carbon, and oxygen, we have $ \mu_{\rm e} = 2 $. From Eq. (D30) one can see that in non-relativistic white dwarfs the matter has a polytropic index of $ \gamma = 5/3 $ ($ n = 3/2 $). For such stars, the mass increases with the increase of the central density as $ M_{\rm wd} \sim \sqrt{\rho_{c}} $ (see Ref. [14], p. 32).

At high densities in the ultra-relativistic limit, the polytropic index is $ \gamma = 4/3 $ ($ n = 3 $) and mass of star in equilibrium with increasing density tends to the limit which is called as the Chandrasekhar limit (see Eq. (2.4), p. 32, Ref. [14]).

Let’s write down the found EqS in polytropic form as

To do that, let's convert formulas (D29) taking concentration (D29) into account.

Non-relativistic approximation ($ p_{fe} \ll m_{\rm e} c^{2} $, $ \gamma = 5/3 $, $ n = 3/2 $):

Ultrarelativistic approximation ($ p_{fe} \gg m_{\rm e} c^{2} $, $ \gamma = 4/3 $, $ n = 3 $):

Calculations for $ ^{4}\text{He} $ give ($ \mu_{e} (^{4}\text{He}) = 2 $)

Also we have

We will be interesting in how emission of photons are changed after change of parameter n in politropic EoS. Such calculations for $ p + $⁴He in dense stellar medium at $ n=3/2 $ are shown in Fig. D1.

Figure D1.  (Color online) Bremsstrahlung emission of photons in scattering of protons off nuclei ⁴He inside star at energy of protons of $ E_{\rm p}=100 $ MeV at polytropic index $ n=3/2 $ [we calculate spectrum on the basis of the leading matrix element $ M_{p}^{(E,\, {\rm dip},0)} $, which gives the largest contribution to full spectrum, according to analysis in Refs. [56, 57]]. Contribution on the basis of matrix element $ \langle \Psi_{f} |\, \Delta \hat{H}_{\gamma} |\, \Psi_{i} \rangle $ in Eq. (23) (a), and full spectrum on the basis of matrix element $ \langle \Psi_{f} |\, \hat{H}_{\gamma} |\, \Psi_{i} \rangle_{\rm star} $ in Eq. (23) (b) are shown in these figures. Comparing these calculations with results in Fig. 8 obtained at $ n=3 $ for the same density, we see that at $ n=3/2 $ emission of photons in stellar medium is larger on about 100 times that emission of photons in similar conditions at $ n=3 $. Concluding we see that EoS has essential influence of calculations of cross sections of emission of photons in medium.