# Systematic study of bremsstrahlung emission in reactions with light nuclei
in cluster models

S. P. Maydanyuk1,2<EMAIL_ADDRESS>V. S. Vasilevsky3
<EMAIL_ADDRESS>(1)Wigner Research Centre for Physics, Budapest, 1121,
Hungary  (2)Institute for Nuclear Research, National Academy of Sciences of
Ukraine, Kyiv, 03680, Ukraine  (3)Bogolyubov Institute for Theoretical
Physics, Metrolohichna str., 14b, Kyiv, 03143, Ukraine

###### Abstract

A new model of bremsstrahlung emission in the scattering of light nuclei is
constructed with main focus on strict cluster formulation of nuclear
processes. Analysis is performed in frameworks of the folding approximation of
the formalism with participation of $s$-nuclei. Reactions $p+\isotope[4]{He}$,
$\isotope[2]{D}+\isotope[4]{He}$, $\isotope[3]{H}+\isotope[4]{He}$,
$\isotope[3]{He}+\isotope[4]{He}$ are included to analysis. Systematic
analysis of properties of emission of bremsstrahlung photons in the wide
region of kinetic energy of relative motion of two nuclei from 7 to 1000 MeV
is performed. Influence of the oscillator length on the calculated spectra of
bremsstrahlung emission is analyzed. On the example of
$\isotope[3]{H}+\isotope[4]{He}$, dependence of the bremsstrahlung spectra on
parameters of nuclear component of interacting potential is established (at
first time for the light nuclei). Experimental bremsstrahlung data for the
proton-deuteron scattering and proton-$\alpha$-particle scattering are
analyzed on the basis of this model.

bremsstrahlung, photon, nucleon-nucleon potential, folding approximation,
$s$-nuclei, electric form factors of nuclei, oscillator length, coherent
emission, tunneling

###### pacs:

41.60.-m, 03.65.Xp, 23.50.+z, 23.20.Js

##  I Introduction

The bremsstrahlung emission of photons accompanying nuclear reactions is an
important topic of nuclear physics, it has been causing essential interest for
from many researchers long time (see reviews Amusia.1988.PhysRep ;
Pluiko.1987.PEPAN ; Kamanin.1989.PEPAN ; Bertulani.1988.PhysRep ). This is
explained by that the spectra of bremsstrahlung photons are calculated on the
basis of nuclear models with included mechanisms of reactions, interactions
between nuclei, dynamics, many other physical issues. Measurements of these
photons and their analysis provide information about all these aspects,
verifying suitability of the developed models.

Investigations of bremsstrahlung emission in the proton-nucleus scattering
have shown important role of incoherent bremsstrahlung processes. In
particular, in Ref. Maydanyuk.2012.PRC ; Maydanyuk_Zhang.2015.PRC  (see also
Ref. Liu_Maydanyuk_Zhang_Liu.2019.PRC.hypernuclei ) a formalism which takes
into account both coherent and incoherent processes has been formulated and it
was found that the incoherent contribution is essentially larger than the
coherent one in the full bremsstrahlung emission. Moreover, inclusion of the
incoherent processes to formalism improves agreement between calculated cross
section and experimental data Goethem.2002.PRL  for $p+\isotope[197]{Au}$ at
energy of proton beam $E_{\rm p}=190$ MeV. Another useful advance of
incoherent bremsstrahlung is explanation of plateau in the middle part of
experimental cross sectionGoethem.2002.PRL , while coherent contribution gives
only logarithmic behavior of the calculated cross section that is not enough
for good description of the data Goethem.2002.PRL . In such a formalism, full
operator of emission of bremsstrahlung photons can be explicitly separated on
two groups of terms. One group (coherent bremsstrahlung) includes terms with
momentum defined on relative distance between center-of-mass of the nucleus-
target and the scattered proton (for example, for the $\alpha$-nucleus
scattering this is Eq. (B10) in Ref.
Liu_Maydanyuk_Zhang_Liu.2019.PRC.hypernuclei ). The second group (incoherent
bremsstrahlung) includes rest of terms without momentum from relative distance
between the nucleus-target and the scattered proton. But it includes momenta
of relative distances between individual nucleons of the nucleus-target and
the scattering proton (for the $\alpha$-nucleus scattering this is Eq. (B11)
with addition (B12) in Ref. Liu_Maydanyuk_Zhang_Liu.2019.PRC.hypernuclei ). By
simple words, incoherent bremsstrahlung has origin from many nucleon dynamics,
while coherent bremsstrahlung is related with two body (proton-nucleus)
dynamics.

Thus, consideration of nuclear scattering as many-nucleon quantum mechanical
problem allows one to increase essentially accuracy of description of
bremsstrahlung spectra. Another important message from those investigations is
understanding about important role of magnetic moments of nucleons in nuclei
in reactions (i.e. magnetic emission based on magnetic moments of nucleons in
nuclei is more intensive essentially than electric emission based on electric
charges of nucleons in nuclei). Attempts to determine accurately relations
between parameters of individual nucleons in the studied nuclear process and
emission of bremsstrahlung photons give more deep understanding about nuclear
(nucleon-nucleon) interactions. This increases motivation to construct the
fully cluster formalism for the nucleon-nucleus and nucleus-nucleus scattering
in the bremsstrahlung problem.

In these regards, one can remind important investigations of bremsstrahlung
emission in reactions with light nuclei on the basis of cluster models. Such
investigations can reveal new information about properties of wave functions
of continuous cross section states of two colliding nuclei. This process is
complementary to the capture reaction or photodisintegration. However,
contrary to these two processes, the bremsstrahlung is more complicated from
numerical point of view as a calculation of the cross section involves two
wave function of states in continuous energy region. This process has been
investigated within microscopic two-cluster models 1985NuPhA.443..302B ;
1990PhRvC..41.1401L ; 1990PhRvC..42.1895L ; 1991NuPhA.529..467B ;
1991PhRvC..44.1695L ; 1992NuPhA.550..250B ; Liu.1992.FBS ; 1993nuco.conf..423K
; Dohet-Eraly.2013.PRC ; 2013JPhCS.436a2030D ; Dohet-Eraly.2013.PhD ;
2014PhRvC..89b4617D ; 2014PhRvC..90c4611D  [see also Ref. Baye.2012.PRC86 ].
In Tabl. 1 we collect all available information about such investigations.

Nucleus | Clusterization | Papers
---|---|---
[8]Be | $\alpha+\alpha$ |  1985NuPhA.443..302B , 1991NuPhA.529..467B , 1992NuPhA.550..250B , Liu.1992.FBS , 1993nuco.conf..423K , 2013JPhCS.436a2030D , Dohet-Eraly.2013.PhD , 2014PhRvC..90c4611D
[7]Be | $\alpha+\isotope[3]{He}$ |  1990PhRvC..41.1401L , 1991PhRvC..44.1695L
[5]Li | $\alpha+p$ |  1990PhRvC..42.1895L , 1991PhRvC..44.1695L , Dohet-Eraly.2013.PhD , 2014PhRvC..89b4617D
Table 1: Nuclei and clusterizations used in investigations of bremsstrahlung
in the nuclear scattering on the basis of microscopic two-cluster models.

The first investigation has been performed in Ref. 1985NuPhA.443..302B  where
interaction of two $\alpha$ particles were considered. Theoretical data was
obtained for the initial energy of interacting $\alpha$ particles $E_{i}\leq
10$ MeV. Different geometry of the reaction has been discussed in detail and
optimal geometry of experiments were recommended. In Ref. 2013JPhCS.436a2030D
the same model was used to study the bremsstrahlung in $\alpha$-$\alpha$
collisions with a realistic nucleon-nucleon potential. And a wider energy
range of two $\alpha$ particles ($E_{i}\leq 50$ MeV) was analyzed. The
bremsstrahlung emission in the $\alpha+^{3}$He collision has been studied in
Ref. 1990PhRvC..41.1401L  within the resonating group method. Contribution of
narrow resonance states $7/2^{-}$ and 5/2- to the bremsstrahlung cross
sections are thoroughly studied.

The resonating group method has been emplied in Ref. 1990PhRvC..42.1895L  to
investigate the bremsstrahlung process in the interaction of protons with an
$\alpha$ particle. The model correctly reproduced phase shift of the elastic
$p+\alpha$ scattering and the parameters of the 3/2- and $1/2^{-}$ resonance
states in 5Li. A good agreement between available experimental data and
results of calculations was achieved. The bremsstrahlung cross section has
been calculated for the energy of incident proton $0\leq E_{p}\leq 25$ MeV.

Note that there is essentially larger volume of experimental bremsstrahlung
data obtained with higher precision, which has not been studied by the cluster
approaches above (ideology of cluster formalism). These are data
Goethem.2002.PRL  for the proton-nucleus scattering (these are data for
$p+\isotope[197]{Au}$ at energy of proton beam $E_{\rm p}=190$ MeV and inside
photon energy region $E_{\gamma}=40$ – 170 MeV, see also Refs.
Edington.1966.NP ; Koehler.1967.PRL ; Kwato_Njock.1988.PLB ; Pinston.1989.PLB
; Pinston.1990.PLB ), data Boie.2007.PRL ; Boie.2009.PhD  for $\alpha$ decay
of [210]Po (see also Refs. D'Arrigo.1994.PHLTA ; Kasagi.1997.JPHGB ;
Kasagi.1997.PRLTA ; Maydanyuk.2008.EPJA ; Giardina.2008.MPLA  for nuclei
[210]Po, [214]Po, [226]Ra, [244]Cm), data Eremin.2010.IJMPE ;
Maydanyuk.2010.PRC  for the spontaneous fission of heavy nuclei [252]Cf (see
also Refs. Ploeg.1995.PRC ; Kasagi.1989.JPSJ ; Luke.1991.PRC ;
Varlachev.2007.BRASP ; Hofman.1993.PRC ; Pandit.2010.PLB ). There are
experimental investigations of dipole $\gamma$-ray emission with incident
energy in the $\isotope[32]{S}+\isotope[100]{Mo}$ and
$\isotope[36]{S}+\isotope[96]{Mo}$ fusion reactions at $E_{\rm lab}=196$ MeV
and 214.2 MeV, respectively, with aim to probe evolution of fusion with
incident energy Pierroutsakou.2005.PRC . $\gamma$-decays of excited states in
the energy region of the Pygmy dipole states in heavy nuclei have been
observed Kmiecik.Maj.2020.ActaPP  (see also Ref. Lanza.2023.ProgPartNuclPhys
).

One can note that the most of forces were put by researchers on study of
bremsstrahlung in the proton-nucleus scattering and fission. However, strict
cluster model has not been constructed to describe those processes up to the
level of good description of experimental data. Some authors wrote in
published papers about attractive perspective to realize this approach
Kopitin.1997.YF . Note on the microscopic models of $\alpha$ decay
Thomas.1954.PTP ; Delion.1992.PRC ; Xu.2006.PRC ; Delion.2013.PRC ;
Silisteanu.2012.ADNDT ; Ivascu.1990.PEPAN ; Lovas.1998.PRep ;
Hodgson.2003.PRep , however emission of bremsstrahlung photons has not been
studied on the basis of those approaches. Also there are theoretical
investigations on emission of bremsstrahlung during ternary fission of heavy
nuclei [252]Cf Maydanyuk.2011.JPCS . As it was found in that paper, the
bremsstrahlung spectra are essentially dependent on the different scenarios of
dynamics of fission, that can be studied by means of bremsstrahlung analysis.
Properties of hypernuclei in the scattering have been studied in connection
with analysis of bremsstrahlung emission
Liu_Maydanyuk_Zhang_Liu.2019.PRC.hypernuclei . However, those investigations
above were performed without cluster basis.

Summarizing all issues described above, we see attractive perspective to
construct such a formalism and to describe existed experimental information of
bremsstrahlung in nuclear reactions in the unified way. So, the first step in
realization of this program is aim of this paper, where we put focus on
scattering of nuclei with small number of nucleons in frameworks of the
folding approximation.

The paper is organized in the following way. In Sec. II a new cluster model of
emission of bremsstrahlung photons in scattering of light nuclei is
formulated. In Sec. III the folding approximation of cluster model is
described with main formulas for practical calculations for $s$-nuclei. In
Sec. IV properties of emission of bremsstrahlung photons for scattering are
studied on the basis of the model presented above. Here, we analyze parameters
which have essential influence on the accuracy of calculations of the spectra,
we calculate spectra in dependence on kinetic energy of relative motion
between two nuclei in the wide energy region, we estimate spectra for
different nuclei at the same energy of relative motions between nuclei, we
analyze role of oscillator length of nuclei in calculations of the spectra, we
look for sensitivity of the shape of the spectra on the nuclear component of
the interaction potential, we describe experimental bremsstrahlung data for
proton-deuteron scattering on the basis of the model. Conclusions and
perspectives are summarized in Sec. V. Some useful details of the model are
presented in Appendixes. Here, we give formalism and calculations of the
matrix elements in the folding approximation (see Appendix A), multiple
expansion of the matrix elements of bremsstrahlung (see Appendix B), formalism
of polarizations of the emitted photon (see Appendix C), calculation of
angular integrals (see Appendix D).

##  II Cluster formalism

To study the bremsstrahlung emission in interacting of light nuclei, we are
going to employ cluster models. First of all we need to specify the term
_“cluster model”_. By saying cluster model we mean one of numerous
realizations (versions) of the resonating group method (RGM). The resonating
group method suggests that the properties of atomic nuclei and different types
of nuclear reactions can be described by assuming there are stable formations
of nucleons comprising clusters. A nucleon-nucleon interaction of nucleons
belonging to different clusters creates a cluster-cluster interactions. Main
difference of various types of the RGM consists (i) in different shapes of
wave functions describing internal structure of clusters and (ii) in different
algorithms of solving the many-particle Schrödinger equation or equivalent
effective two-body equations of the RGM derived by J. Wheeler
1937PhRv…52.1083W , 1937PhRv…52.1107W .

Note that models which consider clusters as structureless particles we will
call them as _potential models_.

Now, we shall consider interaction between clusters, when antisymmetrization
is taken into account or neglected. More precisely, we employ totally
antisymmetric functions for a two–cluster configuration $A=A_{1}+A_{2}$

$\Psi^{(A)}=\widehat{\mathcal{A}}\left\\{\Phi_{1}\left(A_{1}\right)\
\Phi_{2}\left(A_{2}\right)\ \phi\,(\mathbf{r})\right\\}$ (1)

and also the functions of so-called _folding model_

$\Psi^{(F)}=\Phi_{1}\left(A_{1}\right)\ \Phi_{2}\left(A_{2}\right)\
\phi\,(\mathbf{r}),$ (2)

when antisymmetrization of nucleons of different cluster are neglected. To
describe relative motion of clusters, one can use distance between clusters
$\mathbf{r}$

$\mathbf{r}=\left[\displaystyle\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\mathbf{r}_{i}-\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\mathbf{r}_{j}\right].$ (3)

It is assumed, that the wave functions $\Phi_{i}\left(A_{i}\right)$,
describing internal structure of clusters, are translatoinally invariant and
antisymmetric ones. In Eq. (1), the antisymmetrization operator
$\widehat{\mathcal{A}}$ permutes nucleons between clusters and thus realize
the Pauli principle correctly. It is well-known, that the Pauli principle play
an important role especially for low-energy region of interacting clusters.
This approximation we will call the standard version of the resonating group
method (RGM). The second approximation which is presented by wave function (2)
is called the Folding model (FM) or folding approximations.

In what follows, wave functions $\phi\left(\mathbf{r}\right)$ are to
represented in the spherical coordinates

$\phi\left(\mathbf{r}\right)=R_{El}\left(r\right)Y_{lm}\left(\widehat{\mathbf{r}}\right),$
(4)

where $\widehat{\mathbf{r}}=\mathbf{r}/r$, $|\mathbf{r}|=r$. Wave functions
(1) and (2) suggest approximate solutions for the Schrödinger equation

$\displaystyle\left(\widehat{H}-E\right)\Psi^{(A)}$ $\displaystyle=$
$\displaystyle 0,$ (5) $\displaystyle\left(\widehat{H}-E\right)\Psi^{(F)}$
$\displaystyle=$ $\displaystyle 0$ (6)

with a microscopic Hamiltonian $\widehat{H}$ which consists of the kinetic
energy operator in the center of mass motion and a sum of pairwise nucleon-
nucleon potentials.

By multiplying these equations from the left on the product
$\Phi_{1}\left(A_{1}\right)\ \Phi_{2}\left(A_{2}\right)$ and integrating over
internal spacial, spin and isospin coordinates of nucleons, we obtain integro-
differential equation for $\ \phi\left(\mathbf{q}\right)$ when the Pauli
principle is treated correctly, or differential equation when the folding
approximation is used. The later can be written as

$\left\\{-\frac{\hbar^{2}}{2m_{N}\mu}\Delta_{\mathbf{r}}+\widehat{V}^{(F)}\left(\mathbf{r}\right)-E\right\\}\phi\left(\mathbf{r}\right)=0,$
(7)

where $\mu$ is the reduced mass

$\mu=\frac{A_{1}A_{2}}{A_{1}+A_{2}}$ (8)

and $m_{N}$ is mass of nucleon. It is important to underline that the folding
potential $\widehat{V}^{(F)}\left(\mathbf{r}\right)$ is a key component of a
nonlocal inter-cluster potential appeared in the standard version of the RGM.
The folding potential $\widehat{V}^{(F)}\left(\mathbf{r}\right)$ is totally
determined by the shape of nucleon-nucleon potential and density distributions
of nucleons in each cluster.

###  II.1 Potential in the folding approximation

In the folding approximation, as it was pointed out above, the inter-cluster
potential is local and may be easily calculated, especially when simple shell-
model functions $\Phi_{i}(A_{i})$ are used to describe internal state of
clusters.

The folding potential is the integral:

$\widehat{V}^{(F)}(\mathbf{r})=\displaystyle\sum_{i\in
A_{1}}\displaystyle\sum_{j\in A_{2}}\displaystyle\int
dV_{1}\,dV_{2}\;|\Phi(A_{1})|^{2}\,\widehat{V}(\mathbf{r}_{i}-\mathbf{r}_{j})\>|\Phi(A_{2})|^{2},$
(9)

where integration is performed over all coordinates

$\begin{array}[]{llllll}dV_{1}=\displaystyle\prod\limits_{i\in
A_{1}}d\mathbf{r}_{i},&dV_{2}=\displaystyle\prod\limits_{i\in
A_{2}}d\mathbf{r}_{i}.\end{array}$ (10)

As wave functions $\Phi_{1}\left(A_{1}\right)$ and$\
\Phi_{2}\left(A_{2}\right)$ are translationally invariant, they actually
depends on coordinates

$\displaystyle\mathbf{r}_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\mathbf{r}_{i}-\mathbf{R}_{1},\quad\mathbf{R}_{1}=\displaystyle\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\mathbf{r}_{i},\quad i\in A_{1},$ (11)
$\displaystyle\mathbf{r}_{j}^{\prime}$ $\displaystyle=$
$\displaystyle\mathbf{r}_{j}-\mathbf{R}_{2},\quad\mathbf{R}_{2}=\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\mathbf{r}_{j},\quad j\in A_{2},$

respectively. Thus we have to switch to these coordinates

$\begin{array}[]{llllll}\widehat{V}^{(F)}(\mathbf{r})=\displaystyle\sum_{i\in
A_{1}}\displaystyle\sum_{j\in
A_{2}}\displaystyle\int\,dV_{1}^{{}^{\prime}}\;dV_{2}^{\prime}\;|\Phi(A_{1})|^{2}\,\widehat{V}(\mathbf{r}_{i}^{\prime}-\mathbf{r}_{j}^{\prime}+\mathbf{r})\,|\Phi(A_{2})|^{2},\end{array}$
(12)

where

$\displaystyle\mathbf{r}$ $\displaystyle=$
$\displaystyle\mathbf{R}_{1}-\mathbf{R}_{2}=\displaystyle\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\mathbf{r}_{i}-\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\mathbf{r}_{j}$ (13) $\displaystyle dV_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\displaystyle\prod\limits_{i\in
A_{1}}d\mathbf{r}_{i}^{\prime},\quad
dV_{2}^{\prime}=\displaystyle\prod\limits_{j\in
A_{2}}\,d\mathbf{r}_{j}^{\prime}$

and $\Phi(A_{\alpha})$ is a many-particle shell model function, describing
internal motion of $A_{\alpha}$ nucleons. As we deal with two body potential,
then we can perform integration over all single-particle coordinates
$\mathbf{r}_{i}^{\prime}$ ($\mathbf{r}_{j}^{\prime}$) but one. As a results
integration over all but one coordinates leads us to the density distribution

$\rho_{\alpha}\left(\mathbf{r}\right)=\displaystyle\int
dV_{\alpha}^{\prime}\Phi_{\alpha}\left(A_{\alpha}\right)\displaystyle\sum_{i}\delta\left(\mathbf{r-r}_{i}^{\prime}\right)\Phi_{\alpha}\left(A_{\alpha}\right).$
(14)

And thus

$\widehat{V}^{(F)}(\mathbf{r})=\displaystyle\int
d\mathbf{r}_{1}\,d\mathbf{r}_{2}\,\rho_{1}\left(\mathbf{r}_{1}\right)\widehat{V}(\mathbf{r}_{1}-\mathbf{r}_{2}+\mathbf{r})\,\rho_{2}\left(\mathbf{r}_{2}\right).$
(15)

For $s$-nuclei density distribution equals

$\rho_{\alpha}\left(\mathbf{r}_{\alpha}\right)=N_{\alpha}\exp\left\\{-\displaystyle\frac{r^{2}}{b^{2}}\,\displaystyle\frac{A_{\alpha}}{A_{\alpha}-1}\right\\},$
(16)

where $b$ is the oscillator length.

By using Fourier transformation we can reduce (15) to the form

$\widehat{V}^{(F)}(\mathbf{r})=\int
d\mathbf{k}\;\mathcal{V}(\mathbf{k})\>\exp\\{i\,\mathbf{k}\,\mathbf{r}\\}\,\Phi_{1}(\mathbf{k})\,\Phi_{2}(\mathbf{k}).$
(17)

Here, $\mathcal{V}$ denotes the Fourier transform of a nucleon-nucleon
interaction

$\widehat{V}(\mathbf{r})=(2\pi)^{-3/2}\displaystyle\int
d\mathbf{k}\,\exp\\{-i\mathbf{k}\mathbf{r}\\}\,\mathcal{V}(\mathbf{k})$ (18)

and $\Phi_{\alpha}(\mathbf{k})$ denotes form-factor of subsystem with
$A_{\alpha}$ nucleons. For the $s$-shell nuclei, it easy to find that

$\Phi_{\alpha}(\mathbf{k})=\exp\left\\{-\displaystyle\frac{k^{2}b^{2}}{4}\,\displaystyle\frac{A_{\alpha}-1}{A_{\alpha}}\right\\}.$
(19)

For a nucleon-nucleon interaction, having Gaussian form

$\widehat{V}(ij)=V_{0}\,\exp\left\\{-\displaystyle\frac{(\mathbf{r}_{i}-\mathbf{r}_{j})^{2}}{a^{2}}\right\\}$
(20)

all calculation can be done analytically in closed form and results reads:

$\widehat{V}^{(F)}(\mathbf{r})=V_{0}\,z^{3/2}\exp\left\\{-\displaystyle\frac{r^{2}}{a^{2}}z\right\\},$
(21)

where

$\begin{array}[]{llllll}z=\left(1+\displaystyle\frac{b^{2}}{a^{2}}[2-\mu^{-1}]\right)^{-1}.\end{array}$
(22)

It is worth while to notice, that when ratio $b^{2}/a^{2}$ is rather small (it
takes place for a very wide potential well), then $z\approx 1$ and the folding
potential almost coincides with the nucleon-nucleon potential. In other limit
case, when $b^{2}/a^{2}\gg 1$ (it may realize for potential with small radius
of a core), then intensity, as well as radius of potential are significantly
redetermined.

###  II.2 Case of two-cluster systems

General formulas, obtained above, may be adopted to the case of interest,
namely, to two-cluster systems. Here we consider two-cluster systems where one
of the clusters is an alpha-particle ($A_{1}$=4) and the second cluster
consists of $A_{2}$ nucleons with 1$\leq A_{2}\leq$4\. In this case we can
write :

$\begin{array}[]{llllll}V_{NN}^{(F)}&=&\displaystyle\frac{A_{2}}{4}(9V_{33}+3V_{31}+3V_{13}+V_{11})\left(1+\displaystyle\frac{b^{2}}{a^{2}}\displaystyle\frac{3}{4}\right)^{-3/2}\exp\left\\{-\,\displaystyle\frac{R^{2}}{a^{2}}\Bigl{(}1+\displaystyle\frac{b^{2}}{a^{2}}\displaystyle\frac{3}{4}\Bigr{)}^{-1}\right\\},\end{array}$
(23)
$V_{NN}^{(F)}(\mathbf{r})=\frac{A_{2}}{4}(9V_{33}+3V_{31}+3V_{13}+V_{11})z^{3/2}\exp\left\\{-\frac{R^{2}}{a^{2}}z\right\\},$
(24)

where

$\displaystyle z$ $\displaystyle=$
$\displaystyle\left(1+\frac{b^{2}}{a^{2}}\left[2-\mu^{-1}\right]\right)^{-1},$
$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{A_{1}A_{2}}{A_{1}+A_{2}}$

and where $V_{33}$, $V_{31}$, $V_{13}$, $V_{11}$ are intensities of the
central nucleon-nucleon interaction (denoted as $V_{2S+1,2T+1}$) with the
fixed value of the spin $S$ and isospin $T$ of interaction nucleons. Each
component of the potential is presented by sum of two or three Gaussians

$\begin{array}[]{llllll}V_{2S+1,2T+1}\left(r\right)=\displaystyle\sum_{i=1}^{N_{G}}V_{2S+1,2T+1}^{\left(i\right)}\exp\left\\{-\displaystyle\frac{r^{2}}{a_{i}^{2}}\right\\}.\end{array}$
(25)

Expressions, obtained for $NN$-interaction with Gauss spatial form, can be
easily transformed to the case of the Coulomb forces. For this aim we shall
use the well-known relation:

$\displaystyle\frac{1}{r}=\displaystyle\frac{2}{\sqrt{\pi}}\displaystyle\int_{0}^{\infty}\,dx\,\exp\\{-r^{2}x^{2}\\}.$
(26)

Then Coulomb interaction between cluster with the number of protons $Z_{1}$
and $Z_{2}$

$\widehat{V}_{C}^{\left(F\right)}(\mathbf{r})=\displaystyle\frac{Z_{1}\,Z_{2}\,e^{2}}{b}\displaystyle\frac{2}{\sqrt{\pi}}\,\displaystyle\int
d\gamma\,z^{3/2}\exp\left\\{-\,\displaystyle\frac{r^{2}}{b^{2}}\gamma^{2}z\right\\}.$
(27)

By introducing new variable for integration:

$a=\displaystyle\frac{\sigma\gamma^{2}}{1+\sigma\gamma^{2}},\,\sigma=2-\mu^{-1},$
(28)

we obtain the integral

$\widehat{V}_{C}^{\left(F\right)}(\mathbf{r})=\displaystyle\frac{Z_{1}Z_{2}e^{2}}{b}\frac{2}{\sqrt{\pi}}\displaystyle\frac{1}{2\sqrt{\sigma}}\,\displaystyle\int_{0}^{1}da\,a^{-1/2}\,\exp\left\\{-\,\displaystyle\frac{r^{2}}{\sigma\,b^{2}}a\right\\},$
(29)

which leads to the error function:

$\widehat{V}_{C}^{\left(F\right)}(\mathbf{r})=\displaystyle\frac{Z_{1}\,Z_{2}\,e^{2}}{R}\;{\rm
erf}\left(\displaystyle\frac{r^{2}}{\sigma\,b^{2}}\right).$ (30)

For large values of $R\gg 1$, we have got

$\widehat{V}_{C}^{\left(F\right)}(\mathbf{r})\approx\displaystyle\frac{Z_{1}\,Z_{2}\,e^{2}}{r}.$
(31)

###  II.3 Operator of emission of bremsstrahlung photons

The translation invariant operator of interaction of photon with atomic nuclei
is

$\widehat{H}_{e}\left(\mathbf{k}_{\gamma},\mathbf{\varepsilon}_{\mu}\right)=\frac{1}{2}\frac{e\hbar}{m_{N}c}\sum_{i=1}^{A}\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\left[{\hat{\mbox{\boldmath$\pi$}}}_{i}^{\ast}\mathbf{A}^{\ast}\left(i\right)+\mathbf{A}^{\ast}\left(i\right)\widehat{\mbox{\boldmath$\pi$}}_{i}^{\ast}\right]$
(32)

where

$\begin{array}[]{lllllllll}\vspace{1.5mm}\mathbf{A}^{\ast}\left(i\right)=\mathbf{\varepsilon}_{\mu}\exp\left\\{-i\left(\mathbf{k}_{\gamma}\mbox{\boldmath$\rho$}_{i}\right)\right\\},&\widehat{\mbox{\boldmath$\pi$}}_{i}^{\ast}=i\nabla_{\mbox{\boldmath$\rho$}_{i}},&\mbox{\boldmath$\rho$}_{i}=\mathbf{r}_{i}-\mathbf{R}_{cm},\\\
\mathbf{R}_{cm}=\displaystyle\frac{1}{A}\displaystyle\sum_{i=1}^{A}\mathbf{r}_{i},&\widehat{\mbox{\boldmath$\pi$}}_{i}=\widehat{\mathbf{p}}_{i}-\widehat{\mathbf{P}}_{cm},&\widehat{\mathbf{P}}_{cm}=\displaystyle\frac{1}{A}\displaystyle\sum_{i=1}^{A}\widehat{\mathbf{p}}_{i},\end{array}$
(33)

and $\mathbf{k}_{\gamma}$ is wave vector of photon and
$\mathbf{\varepsilon}_{\mu}$ is its circular polarization.

It is worthwhile noticing that operator
$\widehat{H}_{e}\left(\mathbf{k}_{\gamma},\mathbf{\varepsilon}_{\mu}\right)$
has to be projected momentum $\lambda$ which determine the multipolity of the
emitted photon. The projected operator we denote as
$\widehat{H}_{e}^{\left(\lambda\right)}\left(\mathbf{k}_{\gamma},\mathbf{\varepsilon}_{\mu}\right)$.
The projected operator
$\widehat{H}_{e}^{\left(\lambda\right)}\left(\mathbf{k}_{\gamma},\mathbf{\varepsilon}_{\mu}\right)$
will be proportional to spherical functions
$Y_{\lambda\mu}\left(\widehat{\mathbf{r}}_{i}\right)$. We do not dwell on
projection of the operator
$\widehat{H}_{e}\left(\mathbf{k}_{\gamma},\mathbf{\varepsilon}_{\mu}\right)$,
we will project matrix elements of this operator calculated between Slater
determinants. Later we will also consider the operator

$\begin{array}[]{lllllllll}\widehat{H}_{0}=\displaystyle\sum_{i=1}^{A}\displaystyle\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\exp\left\\{i\left(\mathbf{k}_{\gamma}\mathbf{r}_{i}\right)\right\\},\end{array}$
(34)

which determines emission or absorption of photon in the capture reaction or
photodisintegration reaction, respectively. This operator can be easily
projected on quantum number $\lambda$

$\begin{array}[]{lllllllll}\widehat{H}_{0}^{\left(\lambda\right)}=\displaystyle\sum_{i=1}^{A}\displaystyle\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)j_{\lambda}\left(k_{\gamma}r_{i}\right)Y_{\lambda\mu}\left(\widehat{\mathbf{r}}_{i}\right)\end{array}$
(35)

To calculate matrix elements of the operator between Slater determinant wave
functions, it is more expedient to use the single-particle operator

$\widehat{\overline{H}}_{e}\left(\mathbf{k}_{\gamma},\mathbf{\varepsilon}_{\mu}\right)=-\frac{1}{2}\frac{e\hbar}{m_{N}c}\sum_{i=1}^{A}\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\left[\mathbf{\varepsilon}_{\mu}\exp\left\\{-i\left(\mathbf{k}_{\gamma}\mathbf{r}_{i}\right)\right\\}\widehat{\mathbf{p}}_{i}^{\ast}\right].$
(36)

After calculations we obtain

$\begin{array}[]{lllllll}\vspace{0.5mm}\widehat{\overline{H}}_{e}\left(\mathbf{k}_{\gamma},\mathbf{\varepsilon}_{\mu}\right)\par&=&-\displaystyle\frac{1}{2}\frac{e\hbar}{m_{N}c}\displaystyle\sum_{i=1}^{A}\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\left[\mathbf{\varepsilon}_{\mu}\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{r}_{i}-\mathbf{R}_{cm}\right)\right\\}\left(\widehat{\mathbf{p}}_{i}^{\ast}-\widehat{\mathbf{P}}_{cm}\right)\right]\par\times\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{R}_{cm}\right)\right\\}\\\
&-&\displaystyle\frac{1}{2}\frac{e\hbar}{m_{N}c}\displaystyle\sum_{i=1}^{A}\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{r}_{i}-\mathbf{R}_{cm}\right)\right\\}\times\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{R}_{cm}\right)\right\\}\left(\mathbf{\varepsilon}_{\mu}\widehat{\mathbf{P}}_{cm}\right).\end{array}$
(37)

###  II.4 Cross section of bremsstrahlung emission

We follow papers 1990PhRvC..41.1401L ; 1990PhRvC..42.1895L  of Liu, Tang and
Kanada to consider the bremsstrahlung emission in light nuclei. The
differential cross section of the bremsstrahlung emission in the complanar
laboratory framework is

$\begin{array}[]{lllll}\vspace{1.0mm}\displaystyle\frac{d\,\sigma^{(1)}}{d\Omega_{1}\,d\Omega_{2}\,d\Omega_{\gamma}}&=&\displaystyle\frac{E_{\gamma}}{(2\pi\hbar)^{4}}\,\Bigl{(}\displaystyle\frac{p_{f}}{\hbar
c}\Bigr{)}\,\displaystyle\frac{\sin^{2}\theta_{1}\,\sin^{2}\theta_{2}}{\sin^{5}(\theta_{1}+\theta_{2})}\;\par\displaystyle\frac{1}{2J+1}\displaystyle\sum\limits_{\mu
m_{i}}\Bigl{|}\Bigl{\langle}\Psi_{E_{f}l_{f}}\Bigl{|}\hat{H}_{\gamma}(\mathbf{k}_{\gamma},\varepsilon_{\mu})\Bigr{|}\Psi_{E_{i}l_{i}}\Bigr{\rangle}\Bigr{|}^{2}.\end{array}$
(38)

New version

$\frac{d^{3}\sigma^{(2)}}{d\Omega_{1}d\Omega_{2}dE_{\gamma}}=\frac{p_{1}^{4}v_{f}}{\left(2\pi\hbar\right)^{4}\hbar}\frac{\sin^{2}\theta_{1}\sin^{2}\theta_{2}}{\sin^{5}\left(\theta_{1}+\theta_{2}\right)}\sum_{\mu}\left|\left\langle\Psi_{E_{f}l_{f}}\left|\widehat{H}_{e}\left(\mathbf{k}_{\gamma},\mathbf{\varepsilon}_{\mu}\right)\right|\Psi_{E_{i}l_{i}}\right\rangle\right|^{2},$
(39)

where $p_{1}$ is the momentum of the incident nucleus (cluster) with $A_{1}$
nucleons.

The kinematic relations for initial ($E_{i}$), final ($E_{f}$) energies of a
two-cluster system and the photon energy ($E_{\gamma}$) are

$E_{i}=E_{f}+E_{\gamma}$ (40)

and

$E_{\gamma}=E_{1,i}\left[1-\frac{1}{A_{2}}\frac{A_{1}\sin^{2}\theta_{1}+A_{2}\sin^{2}\theta_{2}}{\sin^{2}\left(\theta_{1}+\theta_{2}\right)}\right],$
(41)

where $E_{1,i}$ is the energy of the incident cluster $A_{1}$. Energies
($E_{i}$) and ($E_{f}$) are determined in the center of mass of coordinate
system.

##  III Matrix elements in folding approximation

Matrix element of bremsstrahlung emission of photons for two $s$-clusters
(i.e., for clusters with $1\leq A_{\alpha}\leq 4$ or for $n$, $p$, $d$,
${}^{3}{\rm H}$, ${}^{3}{\rm He}$, ${}^{4}{\rm He}$) is [see App. A for
details]

$\begin{array}[]{lllll}\vspace{1.0mm}&\Bigl{\langle}\Psi_{E_{f}l_{f}}\Bigl{|}\hat{H}_{\gamma}(\mathbf{k}_{\gamma},\varepsilon_{\mu})\Bigr{|}\Psi_{E_{i}l_{i}}\Bigr{\rangle}=\\\
\vspace{1.0mm}=&\sqrt{\displaystyle\frac{A_{2}}{A_{1}\,A}}\biggl{\langle}R_{E_{f}l_{f}}(r)\,Y_{l_{f}m_{f}}(\widehat{\mathbf{r}}_{i})\biggl{|}\exp{-i\,\sqrt{\displaystyle\frac{A_{2}}{A_{1}\,A}}\,(\mathbf{k}_{\gamma},\mathbf{r})}\,(\mbox{\boldmath$\varepsilon$}_{\mu},\hat{\mbox{\boldmath$\pi$}})\biggr{|}R_{E_{i}l_{i}}(r)\,Y_{l_{i}m_{i}}(\widehat{\mathbf{r}}_{i})\biggr{\rangle}\,F_{1}-\\\
-&\sqrt{\displaystyle\frac{A_{1}}{A_{2}\,A}}\biggl{\langle}R_{E_{f}l_{f}}(r)\,Y_{l_{f}m_{f}}(\widehat{\mathbf{r}}_{i})\biggl{|}\exp{i\,\sqrt{\displaystyle\frac{A_{1}}{A_{2}\,A}}\,(\mathbf{k}_{\gamma},\mathbf{r})}\,(\mbox{\boldmath$\varepsilon$}_{\mu},\hat{\mbox{\boldmath$\pi$}})\biggr{|}R_{E_{i}l_{i}}(r)\,Y_{l_{i}m_{i}}(\widehat{\mathbf{r}}_{i})\biggr{\rangle}\,F_{2}.\end{array}$
(42)

In the standard approximation of resonanting group method, form factor
$F_{\alpha}$ equals ($\alpha=1,2$)

$\begin{array}[]{lllll}F_{\alpha}=&\Bigl{\langle}\Phi_{\alpha}(A_{\alpha})\Bigl{|}F_{0}^{(\alpha)}\Bigr{|}\Phi_{\alpha}(A_{\alpha})\Bigr{\rangle}=Z_{\alpha}\,\exp{-\displaystyle\frac{1}{4}\,\displaystyle\frac{A_{\alpha}-1}{A_{\alpha}}\,(k,b)^{2}},\end{array}$
(43)

with $b$ is oscillator length. Thus, to determine cross section of the
bremsstrahlung emission, we need to calculate matrix element ($i=1,2$)

$\begin{array}[]{lllll}\vspace{2.0mm}\mathbf{I}\,(\alpha_{i})=\left\langle
R_{E_{f}l_{f}}(r)\,Y_{l_{f}m_{f}}(\widehat{\mathbf{r}})\Bigl{|}\exp\left\\{-i\alpha_{i}\left(\mathbf{k}_{\gamma},\mathbf{r}\right)\right\\}\widehat{\mbox{\boldmath$\pi$}}\Bigr{|}R_{E_{i}l_{i}}(r)\,Y_{l_{i}m_{i}}(\widehat{\mathbf{r}})\right\rangle,\\\
I_{\mu}\,(\alpha_{i})=\mathbf{\mbox{\boldmath$\varepsilon$}}_{\mu}\,\mathbf{I}\,(\alpha_{i})=\left\langle
R_{E_{f}l_{f}}\left(r\right)Y_{l_{f}m_{f}}\left(\widehat{\mathbf{r}}\right)\Bigl{|}\exp\left\\{-i\alpha_{i}\left(\mathbf{k}_{\gamma},\mathbf{r}\right)\right\\}\left(\mathbf{\mbox{\boldmath$\varepsilon$}}_{\mu}\widehat{\mbox{\boldmath$\pi$}}\right)\Bigr{|}R_{E_{i}l_{i}}(r)\,Y_{l_{i}m_{i}}(\widehat{\mathbf{r}})\right\rangle\end{array}$
(44)

for two values of the parameter

$\alpha_{1}=\sqrt{\frac{A_{2}}{A_{1}A}},\quad\alpha_{2}=-\sqrt{\frac{A_{1}}{A_{2}A}}.$
(45)

###  III.1 Multipole expansion

Applying the multipolar expansion, the integral is [see App. B, Eq. (89)]

$\begin{array}[]{ll}\mathbf{I}_{1}(\alpha_{i})=\biggl{<}\phi_{f}\biggl{|}\,e^{-i\alpha_{i}\mathbf{k_{\gamma}r}}\displaystyle\frac{\partial}{\partial\mathbf{r}}\,\biggr{|}\,\phi_{i}\biggr{>}_{\mathbf{r}}=\sqrt{\displaystyle\frac{\pi}{2}}\>\displaystyle\sum\limits_{l_{\gamma}=1}\,(-i)^{l_{\gamma}}\,\sqrt{2l_{\gamma}+1}\;\displaystyle\sum\limits_{\mu=\pm
1}\mbox{\boldmath$\xi$}_{\mu}\,\mu\,\Bigl{[}p_{l_{\gamma}\mu}^{M}-i\mu\>p_{l_{\gamma}\mu}^{E}\Bigr{]},\end{array}$
(46)

where (at $l_{i}=0$)

$\begin{array}[]{lcl}\vspace{3mm}p_{l_{\gamma}\mu}^{M0m_{f}}&=&-I_{M}(0,l_{f},l_{\gamma},1,\mu)\cdot
J_{1}(0,l_{f},l_{\gamma},\alpha_{i}),\\\
\vspace{1mm}p_{l_{\gamma}\mu}^{E0m_{f}}&=&\sqrt{\displaystyle\frac{l_{\gamma}+1}{2l_{\gamma}+1}}\cdot
I_{E}(0,l_{f},l_{\gamma},1,l_{\gamma}-1,\mu)\cdot
J_{1}(0,l_{f},l_{\gamma}-1,\alpha_{i})\;-\\\
&-&\sqrt{\displaystyle\frac{l_{\gamma}}{2l_{\gamma}+1}}\cdot
I_{E}(0,l_{f},l_{\gamma},1,l_{\gamma}+1,\mu)\cdot
J_{1}(0,l_{f},l_{\gamma}+1,\alpha_{i})\end{array}$ (47)

and

$\begin{array}[]{ccl}J_{1}(l_{i},l_{f},n,\alpha_{i})&=&\displaystyle\int\limits^{+\infty}_{0}\displaystyle\frac{dR_{i}(l_{i},r)}{dr}\>R^{*}_{f}(l_{f},r)\,j_{n}(\alpha_{i}kr)\;r^{2}dr.\end{array}$
(48)

Here, $\mbox{\boldmath$\varepsilon$}^{(\alpha)}$ are unit vectors of _linear_
polarization of the photon emitted
($\mbox{\boldmath$\varepsilon$}^{(\alpha),*}=\mbox{\boldmath$\varepsilon$}^{(\alpha)}$),
$\mathbf{k}_{\gamma}$ is wave vector of the photon and
$w_{\gamma}=k_{\gamma}c=\bigl{|}\mathbf{k}_{\gamma}\bigr{|}c$. Vectors
$\mbox{\boldmath$\varepsilon$}^{(\alpha)}$ are perpendicular to
$\mathbf{k}_{\gamma}$ in Coulomb gauge. We have two independent polarizations
$\mbox{\boldmath$\varepsilon$}^{(1)}$ and
$\mbox{\boldmath$\varepsilon$}^{(2)}$ for the photon with impulse
$\mathbf{k}_{\gamma}$ ($\alpha=1,2$). $\mbox{\boldmath$\xi$}_{\mu}$ are
_vectors of circular polarization_ with opposite directions of rotation (see
Ref. Eisenberg.1973 , (2.39), p. 42; see App. C). Also we have properties:

$\begin{array}[]{lclc}\Bigl{[}\mathbf{k}_{\gamma}\times\mbox{\boldmath$\varepsilon$}^{(1)}\Bigr{]}=k_{\gamma}\,\mbox{\boldmath$\varepsilon$}^{(2)},&\Bigl{[}\mathbf{k}_{\gamma}\times\mbox{\boldmath$\varepsilon$}^{(2)}\Bigr{]}=-\,k_{\gamma}\,\mbox{\boldmath$\varepsilon$}^{(1)},&\Bigl{[}\mathbf{k}_{\gamma}\times\mbox{\boldmath$\varepsilon$}^{(3)}\Bigr{]}=0,&\displaystyle\sum\limits_{\alpha=1,2,3}\Bigl{[}\mathbf{k}_{\gamma}\times\mbox{\boldmath$\varepsilon$}^{(\alpha)}\Bigr{]}=k_{\gamma}\,(\mbox{\boldmath$\varepsilon$}^{(2)}-\mbox{\boldmath$\varepsilon$}^{(1)}).\end{array}$
(49)

Also we have property:

$\begin{array}[]{llll}\displaystyle\sum\limits_{\alpha=1,2}\mbox{\boldmath$\varepsilon$}^{(\alpha)}\cdot\mathbf{I}_{1}(\alpha_{i})=\sqrt{\displaystyle\frac{\pi}{2}}\>\displaystyle\sum\limits_{l_{\gamma}=1}\,(-i)^{l_{\gamma}}\,\sqrt{2l_{\gamma}+1}\;\displaystyle\sum\limits_{\mu=\pm
1}\mu\,h_{\mu}\,\bigl{(}p_{l_{\gamma},\mu}^{M}+p_{l_{\gamma},-\mu}^{E}\bigr{)},\\\
\end{array}$ (50)

where [see Eqs. (101), App. C]

$\begin{array}[]{llllll}h_{\pm}=\mp\displaystyle\frac{1\pm i}{\sqrt{2}},\hskip
28.45274pth_{-}+h_{+}=-i\,\sqrt{2},&\hskip
28.45274pt\displaystyle\sum\limits_{\mu=\pm
1}\mu\,h_{\mu}=-h_{-}+h_{+}=-\sqrt{2}.\end{array}$ (51)

###  III.2 Case of $l_{i}=0$, $l_{f}=1$, $l_{\gamma}=1$

In a case of $l_{i}=0$, $l_{f}=1$, $l_{\gamma}=1$ integral (46) is simplified
to

$\begin{array}[]{llllll}\mathbf{I}_{1}(\alpha_{i})=-i\,\sqrt{\displaystyle\frac{3\pi}{2}}\>\displaystyle\sum\limits_{\mu=\pm
1}\mbox{\boldmath$\xi$}_{\mu}\,\mu\;\Bigl{[}p_{l_{\gamma}\mu}^{M}-i\mu\>p_{l_{\gamma}\mu}^{E}\Bigr{]},\end{array}$
(52)

where [see Eqs. (47)]

$\begin{array}[]{lll}\vspace{3mm}p_{l_{\gamma}\mu}^{M0m_{f}}=-I_{M}(0,1,1,1,\mu)\cdot
J_{1}(0,1,1,\alpha_{i}),\\\
p_{l_{\gamma}\mu}^{E0m_{f}}=\sqrt{\displaystyle\frac{2}{3}}\,I_{E}(0,1,1,1,0,\mu)\cdot
J_{1}(0,1,0,\alpha_{i})-\sqrt{\displaystyle\frac{1}{3}}\,I_{E}(0,1,1,1,2,\mu)\cdot
J_{1}(0,1,2,\alpha_{i}).\end{array}$ (53)

Results of calculation of angular integrals are [we omit details of
calculations in this paper]:

$\begin{array}[]{llllll}\vspace{0.8mm}I_{E}\,(0,1,1,1,0,\mu)=\sqrt{\displaystyle\frac{1}{24\pi}},&I_{M}\,(0,1,1,1,\mu)=0,&I_{E}\,(0,1,1,1,2,\mu)=\displaystyle\frac{47}{240}\sqrt{\displaystyle\frac{3}{2\pi}},\end{array}$
(54)

and matrix elements (53) are simplified to

$\begin{array}[]{lllllllll}\vspace{2mm}p_{l_{\gamma}\mu}^{M0m_{f}}=0,&p_{l_{\gamma}\mu}^{E0m_{f}}=\displaystyle\frac{1}{6}\sqrt{\displaystyle\frac{1}{\pi}}\cdot
J_{1}(0,1,0,\alpha_{i})-\displaystyle\frac{47}{240}\sqrt{\displaystyle\frac{1}{2\pi}}\cdot
J_{1}(0,1,2,\alpha_{i}).\end{array}$ (55)

We substitute these solutions to Eq. (52) and obtain:

$\begin{array}[]{llllll}\mathbf{I}_{1}(\alpha_{i})&=&-\displaystyle\frac{1}{6}\;\sqrt{\displaystyle\frac{3}{2}}\>\Bigl{(}J_{1}(0,1,0,\alpha_{i})-\displaystyle\frac{47}{40}\sqrt{\displaystyle\frac{1}{2}}\cdot
J_{1}(0,1,2,\alpha_{i})\Bigr{)}\;\bigl{(}\mbox{\boldmath$\xi$}_{\mu=+1}+\mbox{\boldmath$\xi$}_{\mu=-1}\bigr{)}.\end{array}$
(56)

###  III.3 Action on vectors of polarization

Now we calculate summation over vectors of polarization. We use definition of
vectors of polarization [see Eq. (97)], and find:

$\begin{array}[]{cc}\mbox{\boldmath$\varepsilon$}^{(1)}=\displaystyle\frac{1}{\sqrt{2}}\,\bigl{(}\mbox{\boldmath$\xi$}_{-1}-\mbox{\boldmath$\xi$}_{+1}\bigr{)},&\mbox{\boldmath$\varepsilon$}^{(2)}=\displaystyle\frac{i}{\sqrt{2}}\,\bigl{(}\mbox{\boldmath$\xi$}_{-1}+\mbox{\boldmath$\xi$}_{+1}\bigr{)}.\end{array}$
(57)

Using (56), we obtain:

$\begin{array}[]{llll}\mbox{\boldmath$\varepsilon$}^{(1)}\cdot\bigl{(}\mbox{\boldmath$\xi$}_{\mu=+1}+\mbox{\boldmath$\xi$}_{\mu=-1}\bigr{)}=0,&\mbox{\boldmath$\varepsilon$}^{(2)}\cdot\bigl{(}\mbox{\boldmath$\xi$}_{\mu=+1}+\mbox{\boldmath$\xi$}_{\mu=-1}\bigr{)}=i\,\sqrt{2}.\end{array}$
(58)

On such a basis, from Eq. (56) we find:

$\begin{array}[]{llllll}\vspace{1.5mm}\mbox{\boldmath$\varepsilon$}^{(1)}\cdot\mathbf{I}_{1}(\alpha_{i})&=&0,&\vspace{1.0mm}\mbox{\boldmath$\varepsilon$}^{(2)}\cdot\mathbf{I}_{1}(\alpha_{i})&=&-\,i\;\displaystyle\frac{\sqrt{3}}{6}\,\Bigl{(}J_{1}(0,1,0,\alpha_{i})-\displaystyle\frac{47}{40}\sqrt{\displaystyle\frac{1}{2}}\,J_{1}(0,1,2,\alpha_{i})\Bigr{)}.\end{array}$
(59)

Using definition (46) for $\mathbf{I_{1}}\,(\pm\alpha)$, the matrix element
(42) is written down as

$\begin{array}[]{lllll}\Bigl{\langle}\Psi_{E_{f}l_{f}}\Bigl{|}\hat{H}_{\gamma}(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu})\Bigr{|}\Psi_{E_{i}l_{i}}\Bigr{\rangle}=-\,i\hbar\,\Bigl{\\{}\sqrt{\displaystyle\frac{A_{2}}{A_{1}\,A}}\,\mbox{\boldmath$\varepsilon$}_{\mu}\,\mathbf{I}_{1}\Bigl{(}+\,\sqrt{\displaystyle\frac{A_{2}}{A_{1}\,A}}\,\Bigr{)}\,F_{1}-\par\sqrt{\displaystyle\frac{A_{1}}{A_{2}\,A}}\,\mbox{\boldmath$\varepsilon$}_{\mu}\,\mathbf{I}_{1}\Bigl{(}-\,\sqrt{\displaystyle\frac{A_{1}}{A_{2}\,A}}\,\Bigr{)}\,F_{2}\Bigr{\\}}.\end{array}$
(60)

##  IV Analysis

Cross sections of bremsstrahlung emission are determined on the basis of
integrals defined in Eqs. (48) [see Eqs. (39) or (40), Eqs. (60), (59)]. These
integrals involve radial wave functions $R_{El}(r)$. These radial wave
functions are calculated numerically by solving the Schrödinger equation with
the corresponding potential of interaction between two studied clusters
(nuclei). Some details of normalization, asymptotic behavior of such wave
functions and their numeric calculations are given in Appendix E. In Fig. 1
such cluster-cluster potentials constructed within our formalism above are
presented. These potentials include nuclear and Coulomb components and are
determined by using the Minnesota nucleon-nucleon potential kn:Minn_pot1 ;
1970NuPhA.158..529R  and formulas (24) and (30).

Figure 1: (Color online) Potentials for different nuclear systems [see text in
Sect. IV for details].

Also in this paper we will restrict ourselves by the bremsstrahlung cross
sections integrated over angles.

###  IV.1 Which parameters have essential influence on the accuracy of
calculations of the spectra

Achieving the needed accuracy of calculation of the spectra is important in
this problem. It turns out that just direct derivation of the bremsstrahlung
spectra at some chosen parameters of calculations does not give satisfactory
convergence in calculations for some nuclei. It causes necessity to understand
what determines the accuracy of the calculations and what causes errors in the
calculation on the computer. Note that it would seem that it would be possible
to increase the number of intervals in the selected region of integration in
order to obtain higher convergent calculations of the spectra. By such a way,
we chose minimum number of intervals equal to 4 per one oscillation of the
radial wave function. However, it turned out that just increasing the number
of intervals did not allow us to increase the accuracy of determining the
cross section. From formalism one can find that it is more important to
analyze the full integrant function of the radial integral, rather than the
wave functions themselves.

The integrant function of the bremsstrahlung matrix element for
$\isotope[3]{H}+\isotope[4]{He}$ at energy of relative motion of 15 MeV is
shown in Fig. 2.

Figure 2: (Color online) Shape of the full integrant function of the
bremsstrahlung matrix element for $\isotope[3]{H}+\isotope[4]{He}$ at energy
of relative motion of 15 MeV in the center of mass system [Parameters of
calculations: energy of photons is chosen as 4 MeV for demonstration ].

From figures one can clearly see that the radial coordinate region from zero
up to 700 fm is the minimum region that includes complete shape of the all
harmonics. However, for calculations of the spectra with minimal satisfactory
accuracy it is better to take into account at least a few of these complete
shapes. This specifies the minimum value for $R_{\rm max}$. Hence, it is clear
that this parameter cannot be small. For calculations in this paper we chose
value $R_{\rm max}=20000~{}{\rm fm}$ and 2500000 intervals for radial region
of integration.

###  IV.2 Spectra in dependence on kinetic energy of relative motion between
two nuclei

At first, let us amalyze how the spectra are chenged in dependence on kinetic
energy of relative motion between two nuclei. In Fig. 3 we present the
bremsstrahlung cross sections for ${}^{3}{H}+^{4}{He}$ in such a dependence
inside energy range from 7 MeV to 1000 MeV calculated by our approach.

Figure 3: (Color online) Cross sections of bremsstrahlung emission for
${}^{3}{H}+^{4}{He}$ at different kinetic energies $E_{kin}$ of relative
motion. [Parameters of calculations: $R_{max}=20000$ fm and 2500000 intervals
are chosen in the numerical integration. Time of computer calculation is 53
min for each cross section in figure (a) (50 points for each cross section), 1
hour 12 min for each cross section in figure (b) (70 points for each cross
section).] One can see tendency in each figure that the spectra are increased
monotonously at increasing of energy of relative motion $E_{kin}$. There is
essential decreasing of the cross section at increasing of energy of photons
for the smallest energy $E_{kin}$ = 7 MeV in figure (a), that is explained by
tending to the kinematic limit at higher energies of photons. Rate of
increasing of the spectra from energy $E_{kin}$ is not monotonous, where one
can find maximums at some energies.

From these figures one can see that our approach gives unified picture of
emission of bremsstrahlung photons inside this wide energy region.

###  IV.3 Cross sections for different nuclei at the same energy of relative
motions between nuclei

Cross sections calculated for $p+\isotope[4]{He}$,
$\isotope[2]{D}+\isotope[4]{He}$, $\isotope[3]{H}+\isotope[4]{He}$,
$\isotope[3]{He}+\isotope[4]{He}$ at energy 15 MeV are presented in Fig. 4.

Figure 4: (Color online) Cross sections of bremsstrahlung emission for
$p+\isotope[4]{He}$, $\isotope[2]{D}+\isotope[4]{He}$,
$\isotope[3]{H}+\isotope[4]{He}$, $\isotope[3]{He}+\isotope[4]{He}$ at kinetic
energy of relative motion of $E_{\rm kin}=15$ MeV in center-of-mass frame.
[Parameters of calculations: cross sections are defined in Eq. (38), $R_{\rm
max}=20000\,{\rm fm}$ and 2500000 intervals are chosen in the numerical
integration, time of computer calculation for each cross section is 53 min].

From this figure one can see that (at the same energy $E_{\rm kin}=15$ MeV)
(1) the most intensive emission of photons is for $p+\isotope[4]{He}$, (2)
emissions of photons for $\isotope[3]{H}+\isotope[4]{He}$ and
$\isotope[3]{H}+\isotope[4]{He}$ are very similar, (3)
$\isotope[2]{D}+\isotope[4]{He}$ emits photons with the smallest intensity.
One of explanations is in different ratios between form-factors (43) for these
systems (more precisely, one can calculate factor $f=\sqrt{A_{2}/(A_{1})}\cdot
F_{1}-\sqrt{A_{1}/A_{2}}\cdot F_{2}$ for such estimations).

###  IV.4 Role of oscillator length $b$ of nuclei in calculations of the
spectra

Let’s clarify how oscillator length influences of the spectra of
bremsstrahlung emission. According to Eqs. (LABEL:eq.simplestcase.1.8),
oscillator length $b$ is included to the matrix element as

$\begin{array}[]{llll}\vspace{2.0mm}\Bigl{\langle}\Psi_{E_{f}l_{f}}\Bigl{|}\hat{H}_{\gamma}(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu})\Bigr{|}\Psi_{E_{i}l_{i}}\Bigr{\rangle}=-\,i\hbar\,\Bigl{\\{}\sqrt{\displaystyle\frac{A_{2}}{A_{1}\,A}}\,\mbox{\boldmath$\varepsilon$}_{\mu}\,\mathbf{I}_{1}\Bigl{(}+\,\sqrt{\displaystyle\frac{A_{2}}{A_{1}\,A}}\,\Bigr{)}\,F_{1}-\sqrt{\displaystyle\frac{A_{1}}{A_{2}\,A}}\,\mbox{\boldmath$\varepsilon$}_{\mu}\,\mathbf{I}_{1}\Bigl{(}-\,\sqrt{\displaystyle\frac{A_{1}}{A_{2}\,A}}\,\Bigr{)}\,F_{2}\Bigr{\\}},\\\
F_{\alpha}\;=\;Z_{\alpha}\,\exp{-\displaystyle\frac{1}{4}\,\displaystyle\frac{A_{\alpha}-1}{A_{\alpha}}\,(k,b)^{2}}.\end{array}$
(61)

One can see that in order to study influence of the oscillator length on the
spectra, the exponent in the second formula in Eqs. (61) should be taken into
account in calculations. From this formula one can see that inclusion of the
oscillator length suppresses the full bremsstrahlung cross section
additionally. Here a natural question is appeared: how much strong is such a
suppression or can it be practically invisible? A next question is in where is
such an influence the most strong? The simplest way to obtain answer comes
from numerical estimations of exponent in dependence on different energies.
More precise information is obtained from calculations of the spectra. The
bremsstrahlung spectra with influence of the oscillator length for different
energies are shown in Fig. 5.

Figure 5: (Color online) Cross section of bremsstrahlung emission for
$\isotope[3]{H}+\isotope[4]{He}$ at different oscillator lengths $b$
[Parameters of calculations: cross section is defined in Eq. (38), $R_{\rm
max}=10000\,{\rm fm}$ and 5000000 intervals are chosen in the numerical
integration].

Analyzing such spectra, we conclude the following.

  * •

Changes of the bremsstrahlung spectra due to change of the oscillator length
are practically not visible for energies below 50 MeV.

  * •

The oscillator length begins to play visible role for higher energies of the
emitted photons (i.e., hundreds of MeV). This requires to use more large
energies of scattering of nuclei. Such highest sensitivity in cross sections
at higher energies can be explained by the following. Each matrix element of
bremsstrahlung emission includes two wave functions of relative motion, i.e.
these are wave functions for states before emission of photon and after such
an emission. Wave function before emission of photon is defined concerning to
higher energy of relative motion $E_{\rm kin}$, so its wave length is shorter
that helps to distinguish more details (more tiny microstructure) in the shape
of potential with barrier. Wave function after emission of higher energetic
photon is defined concerning to smaller energy $E_{\rm kin}$. That allows to
analyze underbarrier tunneling effects and shape of barrier. In the matrix
element properties of these two wave functions are combined. Indeed, this is
visible Fig. 5 that confirms such a logic.

  * •

There is no sense to analyze spectra at low energies (energies of photons and
energies of relative motion between two incident nuclei). Instead of this, it
needs to choose window in high energy region for such analysis. In this place
cross sections are smaller essentially, that makes to be more difficult
experimental measurements and theoretical calculations. This aspect gives own
restriction on maximally large energy, higher that study of this question will
be not realistic (impossible).

###  IV.5 Is it possible to see in the spectra the influence of the nuclear
component of the interaction potential and where?

Let us analyze whether it is possible to see in the spectra of photon emission
the influence of parameters of the nuclear part of the interaction potential.
Calculations of the spectra for $\isotope[3]{H}+\isotope[4]{He}$ in dependence
on the depth of the nuclear component of the potential are shown in Fig. 6.

Figure 6: (Color online) Cross sections of bremsstrahlung emission for
$\isotope[3]{H}+\isotope[4]{He}$ in dependence on depth of nuclear potential
$V_{0}$ at energy of relative motion of 15 MeV (a) and 200 MeV (b) in center-
of-mass frame [Parameters of calculations: cross section is defined in Eq.
(38), $R_{\rm max}=2000\,{\rm fm}$ and 2500000 intervals are chosen in the
numerical integration].

In the figure (a), calculations are given at $E_{\rm kin}=15$ MeV, where one
can chose a region in the range of higher photon energies (10–14 MeV), where
there is small visible difference between the spectra (this area in the figure
is highlighted with a rectangle). One can see change in the spectra in
dependence on the depth of the nuclear part $V_{0}$ of the interaction
potential. It would seem that this picture should be good indication of the
place where it is worth looking for the influence of the parameters of the
nucleus part of the potential on the bremsstrahlung cross section: _This is a
region with higher photon energies (at any energy of relative motion between
nuclei)_.

However, let us analyze whether such a dependence will persist with an
increase in the energy of the relative motion of the nuclei. The spectra at
energy of relative motion of nuclei of 200 MeV are presented in Fig. 6 (b).
Here there is another region at range of 25–130 MeV, where difference between
the spectra is clearly visible. Note that this type of dependence has never
been found yet in study on bremsstrahlung emission in nuclear reactions. Such
a dependence is likely to have opposite character than the previous dependence
revealed in Fig. 6 (a). It seems, this dependence is more suitable for
experimental investigations. Because it allows use of essentially lower photon
energies for measurements. This dependence is much more reliable (i.e., it is
shown in a wider region of photon energies). The spectra have more intense
emission, that is easier to measure and gives higher convergence in computer
calculations. If now, after such a detection of dependence at $E_{\rm
kin}=200$ MeV, to go back to Fig. 6 (a) at $E_{\rm kin}=15$ MeV, then one can
find a suitable difference between the spectra of a similar nature for lower
photon energies (at 2–6 MeV). This reinforces confidence in the analysis
obtained.

###  IV.6 Analysis of experimental bremsstrahlung data

####  IV.6.1 Bremsstrahlung in the proton-deuteron scattering

We will analyze bremsstrahlung from the proton-deuteron scattering. There are
different aspects of use of angles in calculations of the bremsstrahlung cross
sections. Moreover, in the different papers authors sometimes used re-
normalized experimental data for that reaction. But, one can obtain
understanding about general tendency of bremsstrahlung cross section from
normalized calculations. By such a reason, we will provide normalized
calculations of spectra in this paper. Calculations of bremsstrahlung spectra
on the basis of our model in comparison with experimental data of Clayton et
al. Clayton.1992.PRC.p1810  are presented in Fig. 7.

Figure 7: (Color online) Normalized cross sections of bremsstrahlung emission
for $p+\isotope[2]{D}$ at energy of proton beam of 145 MeV in comparison with
experimental data Clayton.1992.PRC.p1810  [Parameters of calculations: cross
section is defined in Eq. (38), $R_{\rm max}=20000\,{\rm fm}$ and 2500000
intervals are used in the numerical integration; kinetic energy $E_{\rm kin}$
of relative motion of proton and deuteron is used in calculations of
bremsstrahlung matrix elements, which is $E_{\rm kin}=2/3\cdot E_{\rm p}=98$
MeV]. Bremsstrahlung calculation of Herrmann, Speth and Nakayama for $p+n$ at
150 MeV Herrmann.1991.PRC  is added for comparative analysis (see the red
dashed line in figure).

From these figures one can see that our approach describes these data in Ref.
Clayton.1992.PRC.p1810  with enough good agreement. One can suppose that one
can improve agreement between our calculations and experimental data
Clayton.1992.PRC.p1810  in the middle part of the spectra, if to add influence
of magnetic moments of nucleons (related with spins of these nucleons) on
emission of photons.

For comparative analysis we add also calculation Herrmann.1991.PRC  of
Herrmann, Speth and Nakayama, which was used in analysis in Ref.
Clayton.1992.PRC.p1810 . After renormalization, general tendency of the
spectra are not changed actually (differences are very small). By such a
reason, we think that our calculation is in better agreement with experimental
data Clayton.1992.PRC.p1810  than calculation Herrmann.1991.PRC  (see red
dashed line in figure).

####  IV.6.2 Bremsstrahlung in the scattering of the $\alpha$-particles on
protons

In this section we will analyze bremsstrahlung emission in the scattering of
$\alpha$ particles on protons. Photons of bremsstrahlung were measured over
large range for the reaction of $\alpha$ particles with protons, using photon
spectrometer TAPS at AGOR facility of the Kernfysisch Verneller Institut
Hoefman.2000.PRL . In this experiment the beam of 200 MeV $\alpha$ particles
was incident on a liquid hydrogen target. Experimental bremsstrahlung data for
this study were presented in papers Hoefman.2000.PRL ; Hoefman.1999.NPA  and
PhD thesis Hoefman.1999.PhD .

From these presented data we choose data in Fig. 3 in Ref. Hoefman.2000.PRL
for analysis. Note that inclusive data given in Fig. 1 and exclusive data
given in Fig. 3 in that paper are different at $E_{\gamma}<20$ MeV (with
smaller difference at $E_{\gamma}>20$ MeV). We explain our choice by the
following. (1) Data in Fig. 3 in Ref. Hoefman.2000.PRL  were obtained on the
basis of double and triple coincidences. (2) On the basis of these data
authors of that paper obtained information about resonances (radiative capture
populating the unbound ground and first excited states) of shortly lived
nucleus [5]Li, authors extracted parameters presented in Tabl. 1 in Ref.
Hoefman.2000.PRL , and calculated the bremsstrahlung contributions for these
resonances shown in Figs. 1 in that paper.

As it was shown in Ref. Hoefman.2000.PRL , for explanation of experimental
data it needs to take into account presence of two unbound states of [5]Li for
capture in addition to the main bremsstrahlung emission during the $p+\alpha$
scattering (it is clearly seen in Figs. 1 and 3 in that paper). In particular,
in Fig. 1 in Ref. Hoefman.2000.PRL  such states were obtained as two Gaussian
peaks. Parameters of these Gaussian peaks were deduced using fitting procedure
(see Tabl. 1 in that paper). In our paper we do not take into account
incoherent part and magnetic part of bremsstrahlung. So, our approach
reproduces main contribution during scattering, but to take into account two
resonant states of [5]Li we follow to results of research Hoefman.2000.PRL .
Calculations of bremsstrahlung spectra on the basis of our model in comparison
with experimental data of Hoefman et al. Hoefman.2000.PRL  are presented in
Fig. 8.

Figure 8: (Color online) Calculated cross sections of bremsstrahlung emission
for $p+\isotope[4]{He}$ at energy of beam of the $\alpha$ particles of 200 MeV
in comparison with experimental data Hoefman.2000.PRL  [Parameters of
calculations: cross section is defined in Eq. (39), $R_{\rm max}=20000\,{\rm
fm}$ and 2500000 intervals are used in the numerical integration]. [Pannel
(a)]: Main bremsstrahlung contribution of photons emitted in the $p+\alpha$
scattering calculated by our model (see blue dashed line) in comparison with
experimental data. [Pannel (b)]: Full bremsstrahlung cross section (see red
solid line) obtained as summation of main contribution in the $p+\alpha$
scattering (see blue dashed line) and additional two contributions from
capture at unbound ground and first excited states of [5]Li (see green dash-
dotted line and brown dash-double dotted line) in comparison with experimental
data.

Our model calculates main bremsstrahlung contribution on the microscopic
basis, while in Ref. Hoefman.2000.PRL  this contribution was obtained on the
basis of phenomenological formula (1) from classical electrodynamics with
energy and momentum conservations. The summarized full cross section from our
calculations is shown in Fig. 8 (b) by red solid line. One can see that
agreement between such our calculated full cross section and experimental data
Hoefman.2000.PRL  is comparable with result in Ref. Hoefman.2000.PRL . Also we
add the full bremsstrahlung cross section at low energy region of photons (in
contrast to Ref. Hoefman.2000.PRL ).111Presence of smoothly visible hump in
the calculated cross section at photon energy close to 16 MeV can be explained
by restriction of numerical calculation of Coulomb functions in the close
asymptotic region. Note that Coulomb functions at some parameters are
calculated on the basis of asymptotic series. These series are not convergent,
in principle. But the best calculated values have not bad approximation and
they are generally accepted in physical community for use. However, this forms
origin of small changes in the full bremsstrahlung spectra at some parameters.
We suppose accuracy of calculation of this part of the cross section can be
improved on the basis of technique developed in Appendix in Ref.
Maydanyuk.2010.PRC . However, this technically solid volume of development is
omitted in this paper, and we restrict ourselves by current existed accuracy
of these functions.

Note that for analysis authors of research Hoefman.2000.PRL  also used the
potential model of Baye _et al._ 1992NuPhA.550..250B , the covariant
generalization of the Feshbach-Yennie approach based on Refs. Ding.1989.PRC ;
Liou.1995.PLB , and model of direct capture into unbound states based on Ref.
Weller.1982.PRC  (see Fig. 3 in Ref. Hoefman.2000.PRL ). However, at low
energy region of the bremsstrahlung cross section these approaches are less
successful than our microscopic cluster model.

##  V Conclusions and perspectives

In this paper a new model of bremsstrahlung emission in the scattering of
light nuclei is constructed with main focus on strict cluster formulation of
nuclear processes. But, analysis is performed in frameworks of the folding
approximation of the formalism with participation of $s$-nuclei. On the basis
of such a model we investigate emission of the bremsstrahlung photons in the
scattering of the light nuclei. Conclusions from such a study are the
following.

  * •

On the example of $\isotope[3]{H}+\isotope[4]{He}$, we obtain unified picture
of emission of bremsstrahlung photonsn in dependence on the different kinetic
energies $E_{\rm kin}$ of relative motion of two nuclei inside wide energy
range from 7 MeV to 1000 MeV calculated by our approach (see Fig. 3). Cross
sections are increased at increasing of energy of relative motion $E_{\rm
kin}$. But, rate of increasing of the spectra from energy $E_{\rm kin}$ is not
monotonous, where one can find maximums at some energies.

  * •

We estimate comparable picture of the bremsstrahlung emission for
$p+\isotope[4]{He}$, $\isotope[2]{D}+\isotope[4]{He}$,
$\isotope[3]{H}+\isotope[4]{He}$, $\isotope[3]{He}+\isotope[4]{He}$ at the
same kinetic energy of relative motion of $E_{\rm kin}=15$ MeV (see Fig. 4).
General tendency of the spectra is similar for all systems. We find that (1)
the most intensive emission of photons is for $p+\isotope[4]{He}$, (2)
emissions of photons for $\isotope[3]{H}+\isotope[4]{He}$ and
$\isotope[3]{He}+\isotope[4]{He}$ are very similar, (3)
$\isotope[2]{D}+\isotope[4]{He}$ emits photons with the smallest intensity.

  * •

We have analyzed influence of the oscillator length $b$ on the spectra of
bremsstrahlung emission [see Eqs. (61), Fig. 5] and conclude the following.

    * –

Inclusion of the oscillator length suppresses the full bremsstrahlung cross
section additionally.

    * –

Changes of the bremsstrahlung spectra due to change of the oscillator length
are practically not visible for energies below 50 MeV.

    * –

The oscillator length begins to play visible role for higher energies of the
emitted photons (i.e., hundreds of MeV). This requires to use more large
energies of the incident nuclei.

    * –

There is no sense to analyze spectra at low energies. Instead of this, it
needs to choose window in high energy region for such analysis. In this place
cross sections are smaller essentially, that makes to be more difficult
experimental measurements and theoretical calculations. This aspect gives
restriction on maximally large energy, above that study of this question will
be not realistic (impossible).

  * •

We have found dependence of the bremsstrahlung spectra on parameters of the
nuclear part of the interaction potential. On the example of
$\isotope[3]{H}+\isotope[4]{He}$, we find two regions in the spectra for
better observation of such a dependence: the middle energy region and high
energy region (see Fig. 6). In the middle energy region calculations of the
spectra are more stable, intensity of emission is higher that is convenient
for possible measurements. In the high energy region dependence is more
sensitive to variation of nuclear parameters. But intensity of emission is
smaller that is problematic for possible measurements and computer
calculations.

  * •

The bremsstrahlung cross section calculated on the basis of our model for the
proton-deuteron scattering at $E_{p}=145$ MeV is in good agreement with
experimental data Ref. Clayton.1992.PRC.p1810  (see Fig. 7).

  * •

The bremsstrahlung emission for the scattering of $\alpha$ particles on
protons at energy of beam of $\alpha$ particles of $E_{\alpha}=200$ MeV is
analyzed. The bremsstrahlung cross section obtained as summation of main
contribution in the $p+\alpha$ scattering (calculated on the basis of our
microscopic model) and additional two contributions from captures at unbound
ground and first excited states of [5]Li is in good agreement with
experimental data Hoefman.2000.PRL  (see Fig. 8).

As a perspective, we estimate useful new contribution from inclusion of
magnetic moments of nucleons to the model. This should give additional new
incoherent bremsstrahlung contribution. According to previous study of
bremsstrahlung in the scattering of protons of heavy nuclei
Maydanyuk_Zhang.2015.PRC , this incoherent emission is important and not
small. However, its role can be essentially smaller for light nuclei,
according to our preliminary estimations. We also estimate that this term can
improve agreement between the full calculated cross section and experimental
data.

## Acknowledgements

M.S.P. thanks the Wigner Research Centre for Physics in Budapest for warm
hospitality. This work was supported in part by the Program of Fundamental
Research of the Physics and Astronomy Department of the National Academy of
Sciences of Ukraine (Project No. 0117U000239).

##  Appendix A Matrix element in the folding approximation

Let us consider the operator

$\widehat{H}_{e}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)=\frac{e\hbar}{m_{N}c}\sum_{i=1}^{A}\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\mathbf{A}^{\ast}\left(i\right)\widehat{\mbox{\boldmath$\pi$}}_{i}^{\ast}$
(62)

for two-cluster system with the partition $A=A_{1}+A_{2}$. In this case the
operator
$\widehat{H}_{e}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)$
can be presented as

$\begin{array}[]{lll}\vspace{1.5mm}\widehat{H}_{e}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)&=&\widehat{H}_{e}^{\left(1\right)}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)+\widehat{H}_{e}^{\left(2\right)}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)\\\
&=&\displaystyle\frac{e\hbar}{m_{N}c}\sum_{i\in
A_{1}}\displaystyle\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\mathbf{A}^{\ast}\left(i\right)\widehat{\mbox{\boldmath$\pi$}}_{i}^{\ast}+\displaystyle\frac{e\hbar}{m_{N}c}\sum_{j\in
A_{2}}\frac{1}{2}\left(1+\widehat{\tau}_{jz}\right)\mathbf{A}^{\ast}\left(j\right)\widehat{\mbox{\boldmath$\pi$}}_{j}^{\ast}.\end{array}$
(63)

It is necessary to recall that

$\begin{array}[]{lll}\mathbf{A}^{\ast}\left(i\right)\widehat{\mbox{\boldmath$\pi$}}_{i}^{\ast}&=&\exp\left\\{-i\left(\mathbf{k}_{\gamma}\mbox{\boldmath$\rho$}_{i}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\widehat{\mbox{\boldmath$\pi$}}_{i}^{\ast}\right)\\\
&=&\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{r}_{i}-\mathbf{R}_{cm}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\left[\widehat{\mathbf{p}}_{i}^{\ast}-\widehat{\mathbf{P}}_{cm}^{\ast}\right]\right).\end{array}$
(64)

The first operator
$\widehat{H}_{e}^{\left(1\right)}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)$
we represent in the following form

$\begin{array}[]{lll}\widehat{H}_{e}^{\left(1\right)}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)&=&\displaystyle\frac{e\hbar}{m_{N}c}\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{R}_{1}-\mathbf{R}_{cm}\right)\right\\}\par\displaystyle\sum_{i\in
A_{1}}\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{r}_{i}-\mathbf{R}_{1}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\left[\widehat{\mathbf{p}}_{i}^{\ast}-\widehat{\mathbf{P}}_{1}^{\ast}\right]\right)\\\
&+&\displaystyle\frac{e\hbar}{m_{N}c}\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{R}_{1}-\mathbf{R}_{cm}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\left[\widehat{\mathbf{P}}_{1}^{\ast}-\widehat{\mathbf{P}}_{cm}^{\ast}\right]\right)\par\displaystyle\sum_{i\in
A_{1}}\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{r}_{i}-\mathbf{R}_{1}\right)\right\\},\end{array}$
(65)

where $\mathbf{R}_{1}$ and $\widehat{\mathbf{P}}_{1}^{\ast}$ are coordinate
and momentum of center of mass motion of the first cluster:

$\begin{array}[]{lll}\mathbf{R}_{1}=\displaystyle\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\mathbf{r}_{i},&\widehat{\mathbf{P}}_{1}=\displaystyle\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\widehat{\mathbf{p}}_{i}.\end{array}$ (66)

In similar way we can present the second operator

$\begin{array}[]{lll}\widehat{H}_{e}^{\left(2\right)}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)=\par&=&\displaystyle\frac{e\hbar}{m_{N}c}\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{R}_{2}-\mathbf{R}_{cm}\right)\right\\}\par\displaystyle\sum_{j\in
A_{2}}\frac{1}{2}\left(1+\widehat{\tau}_{jz}\right)\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{r}_{j}-\mathbf{R}_{2}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\left[\widehat{\mathbf{p}}_{j}^{\ast}-\widehat{\mathbf{P}}_{2}^{\ast}\right]\right)\\\
&+&\displaystyle\frac{e\hbar}{m_{N}c}\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{R}_{2}-\mathbf{R}_{cm}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\left[\widehat{\mathbf{P}}_{2}^{\ast}-\widehat{\mathbf{P}}_{cm}\right]\right)\par\displaystyle\sum_{j\in
A_{2}}\displaystyle\frac{1}{2}\left(1+\widehat{\tau}_{jz}\right)\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{r}_{j}-\mathbf{R}_{2}\right)\right\\},\end{array}$
(67)

where

$\begin{array}[]{lll}\mathbf{R}_{2}=\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\mathbf{r}_{j},&\widehat{\mathbf{P}}_{2}=\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\widehat{\mathbf{p}}_{j}.\end{array}$ (68)

Consider in detail the differences

$\begin{array}[]{lll}\mathbf{R}_{1}-\mathbf{R}_{cm},&\mathbf{R}_{2}-\mathbf{R}_{cm}.\end{array}$
(69)

Taking into account their definitions, we obtain

$\begin{array}[]{lll}\mathbf{R}_{1}-\mathbf{R}_{cm}=\displaystyle\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\mathbf{r}_{i}-\displaystyle\frac{1}{A}\displaystyle\sum_{i=1}^{A}\mathbf{r}_{i}=\par\par\par\par\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\sqrt{\displaystyle\frac{A_{1}A_{2}}{A}}\left[\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\mathbf{r}_{i}-\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\mathbf{r}_{j}\right],\\\
\mathbf{R}_{2}-\mathbf{R}_{cm}=\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\mathbf{r}_{j}-\displaystyle\frac{1}{A}\displaystyle\sum_{i=1}^{A}\mathbf{r}_{i}=\par\par\par-\sqrt{\displaystyle\frac{A_{1}}{AA_{2}}}\sqrt{\displaystyle\frac{A_{1}A_{2}}{A}}\left[\displaystyle\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\mathbf{r}_{i}-\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\mathbf{r}_{j}\right].\end{array}$ (70)

Thus

$\begin{array}[]{lll}\mathbf{R}_{1}-\mathbf{R}_{cm}=\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\mathbf{q},&\mathbf{R}_{2}-\mathbf{R}_{cm}=-\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\mathbf{q},\end{array}$
(71)

where

$\begin{array}[]{lll}\mathbf{q}=\sqrt{\displaystyle\frac{A_{1}A_{2}}{A}}\left[\displaystyle\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\mathbf{r}_{i}-\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\mathbf{r}_{j}\right]\end{array}$ (72)

is the Jacobi vector determining distance between clusters. Similarly, we can
transform momenta

$\begin{array}[]{lll}\widehat{\mathbf{P}}_{1}-\widehat{\mathbf{P}}_{cm}=\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\widehat{\mbox{\boldmath$\pi$}}\mathbf{,}&\widehat{\mathbf{P}}_{2}-\widehat{\mathbf{P}}_{cm}=-\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\widehat{\mbox{\boldmath$\pi$}}\mathbf{,}\end{array}$
(73)

where

$\begin{array}[]{lll}\widehat{\mbox{\boldmath$\pi$}}=\sqrt{\displaystyle\frac{A_{1}A_{2}}{A}}\left[\displaystyle\frac{1}{A_{1}}\displaystyle\sum_{i\in
A_{1}}\widehat{\mathbf{p}}_{i}-\displaystyle\frac{1}{A_{2}}\displaystyle\sum_{j\in
A_{2}}\widehat{\mathbf{p}}_{j}\right].\end{array}$ (74)

By concluding we can write down

$\begin{array}[]{lll}\widehat{H}_{e}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)\par\par\par&=&\exp\left\\{-i\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\cdot
F_{1}^{\left(1\right)}+\displaystyle\frac{e\hbar}{m_{N}c}\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\exp\left\\{-i\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\widehat{\mbox{\boldmath$\pi$}}\cdot
F_{0}^{\left(1\right)}\\\
&+&\exp\left\\{i\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\cdot
F_{1}^{\left(2\right)}-\displaystyle\frac{e\hbar}{m_{N}c}\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\exp\left\\{i\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\widehat{\mbox{\boldmath$\pi$}}\cdot
F_{0}^{\left(2\right)},\end{array}$ (75)

where

$\begin{array}[]{lllllll}\vspace{1.5mm}F_{0}^{\left(\alpha\right)}&=&\displaystyle\sum_{i\in
A_{\alpha}}\displaystyle\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{r}_{i}-\mathbf{R}_{\alpha}\right)\right\\},\\\
F_{1}^{\left(\alpha\right)}&=&\displaystyle\frac{e\hbar}{m_{N}c}\displaystyle\sum_{i\in
A_{\alpha}}\displaystyle\frac{1}{2}\left(1+\widehat{\tau}_{iz}\right)\exp\left\\{-i\left(\mathbf{k}_{\gamma},\mathbf{r}_{i}-\mathbf{R}_{\alpha}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\left[\widehat{\mathbf{p}}_{i}^{\ast}-\widehat{\mathbf{P}}_{\alpha}^{\ast}\right]\right)\end{array}$
(76)

with $\alpha=$ 1, 2.

If we consider a two-cluster system in the folding approximation, then a wave
function of the system is

$\Psi_{El}=\Phi_{1}\left(A_{1}\right)\Phi_{2}\left(A_{2}\right)R_{El}\left(q\right)Y_{lm}\left(\widehat{\mathbf{q}}\right),$
(77)

where functions $\Phi_{1}\left(A_{1}\right)$ and $\Phi_{2}\left(A_{2}\right)$
describe internal motion of nucleons of the first and second clusters,
respectively, $\widehat{\mathbf{q}}$ is a unit vector
($\widehat{\mathbf{q}}=\mathbf{q}/q$). Matrix element of the operator (75)
between functions $\Psi_{E_{f}l_{f}}$ and $\Psi_{E_{i}l_{i}}$ is then equal

$\begin{array}[]{lll}\vspace{1.5mm}&&\left\langle\Psi_{E_{f}l_{f}}\left|\widehat{H}_{e}^{\left(1\right)}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)\right|\Psi_{E_{i}l_{i}}\right\rangle\\\
\vspace{1.5mm}&=&\left\langle
R_{E_{f}l_{f}}\left(q\right)Y_{l_{f}m_{f}}\left(\widehat{\mathbf{q}}\right)\left|\exp\left\\{-i\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\right|R_{E_{i}l_{i}}\left(q\right)Y_{l_{i}m_{i}}\left(\widehat{\mathbf{q}}\right)\right\rangle\par\left\langle\Phi_{1}\left(A_{1}\right)\left|F_{1}^{\left(1\right)}\right|\Phi_{1}\left(A_{1}\right)\right\rangle\\\
\vspace{1.5mm}&+&\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\left\langle
R_{E_{f}l_{f}}\left(q\right)Y_{l_{f}m_{f}}\left(\widehat{\mathbf{q}}\right)\left|\exp\left\\{-i\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\widehat{\mbox{\boldmath$\pi$}}\right)\right|R_{E_{i}l_{i}}\left(q\right)Y_{l_{i}m_{i}}\left(\widehat{\mathbf{q}}\right)\right\rangle\par\left\langle\Phi_{1}\left(A_{1}\right)\left|F_{0}^{\left(1\right)}\right|\Phi_{1}\left(A_{1}\right)\right\rangle\\\
\vspace{1.5mm}&+&\left\langle
R_{E_{f}l_{f}}\left(q\right)Y_{l_{f}m_{f}}\left(\widehat{\mathbf{q}}\right)\left|\exp\left\\{i\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\right|R_{E_{i}l_{i}}\left(q\right)Y_{l_{i}m_{i}}\left(\widehat{\mathbf{q}}\right)\right\rangle\par\left\langle\Phi_{2}\left(A_{2}\right)\left|F_{1}^{\left(2\right)}\right|\Phi_{2}\left(A_{2}\right)\right\rangle\\\
\vspace{1.5mm}&-&\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\left\langle
R_{E_{f}l_{f}}\left(q\right)Y_{l_{f}m_{f}}\left(\widehat{\mathbf{q}}\right)\left|\exp\left\\{i\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\widehat{\mbox{\boldmath$\pi$}}\right)\right|R_{E_{i}l_{i}}\left(q\right)Y_{l_{i}m_{i}}\left(\widehat{\mathbf{q}}\right)\right\rangle\par\left\langle\Phi_{2}\left(A_{2}\right)\left|F_{0}^{\left(2\right)}\right|\Phi_{2}\left(A_{2}\right)\right\rangle.\end{array}$
(78)

For two $s$-clusters (i.e. for cluster with 1$\leq A_{\alpha}\leq$4 or for
$n$, $p$, $d$, $t$, 3He and 4He)

$\begin{array}[]{lll}\left\langle\Phi_{1}\left(A_{1}\right)\left|F_{1}^{\left(1\right)}\right|\Phi_{1}\left(A_{1}\right)\right\rangle=\left\langle\Phi_{2}\left(A_{2}\right)\left|F_{1}^{\left(2\right)}\right|\Phi_{2}\left(A_{2}\right)\right\rangle=0,\end{array}$
(79)

consequently

$\begin{array}[]{lll}\vspace{1.5mm}&&\left\langle\Psi_{E_{f}l_{f}}\left|\widehat{H}_{e}^{\left(1\right)}\left(\mathbf{k}_{\gamma},\mbox{\boldmath$\varepsilon$}_{\mu}\right)\right|\Psi_{E_{i}l_{i}}\right\rangle\\\
\vspace{1.5mm}&=&\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\left\langle
R_{E_{f}l_{f}}\left(q\right)Y_{l_{f}m_{f}}\left(\widehat{\mathbf{q}}\right)\left|\exp\left\\{-i\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\widehat{\mbox{\boldmath$\pi$}}\right)\right|R_{E_{i}l_{i}}\left(q\right)Y_{l_{i}m_{i}}\left(\widehat{\mathbf{q}}\right)\right\rangle\par\left\langle\Phi_{1}\left(A_{1}\right)\left|F_{0}^{\left(1\right)}\right|\Phi_{1}\,\left(A_{1}\right)\right\rangle\\\
&-&\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\left\langle
R_{E_{f}l_{f}}\left(q\right)Y_{l_{f}m_{f}}\left(\widehat{\mathbf{q}}\right)\left|\exp\left\\{i\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\widehat{\mbox{\boldmath$\pi$}}\right)\right|R_{E_{i}l_{i}}\left(q\right)Y_{l_{i}m_{i}}\left(\widehat{\mathbf{q}}\right)\right\rangle\par\left\langle\Phi_{2}\left(A_{2}\right)\left|F_{0}^{\left(2\right)}\right|\Phi_{2}\left(A_{2}\right)\right\rangle.\end{array}$
(80)

In the standard approximation of the resonating group method (or cluster
model), form factor equals

$\begin{array}[]{lll}\left\langle\Phi_{\alpha}\left(A_{\alpha}\right)\left|F_{0}^{\left(1\right)}\right|\Phi_{\alpha}\left(A_{\alpha}\right)\right\rangle=Z_{\alpha}\exp\left\\{-\displaystyle\frac{1}{4}\displaystyle\frac{A_{\alpha}-1}{A_{\alpha}}\left(k_{\gamma}b\right)^{2}\right\\},\end{array}$
(81)

where $b$ is the oscillator length. Thus, to determin cross section of the
bremsstrahlung emission, we need to calculate matrix element

$\begin{array}[]{lll}I_{\mu}\left(\alpha\right)=\left\langle
R_{E_{f}l_{f}}(q)\,Y_{l_{f}m_{f}}(\widehat{\mathbf{q}})\left|\exp\left\\{-i\alpha\left(\mathbf{k}_{\gamma},\mathbf{q}\right)\right\\}\left(\mbox{\boldmath$\varepsilon$}_{\mu}\widehat{\mbox{\boldmath$\pi$}}\right)\right|R_{E_{i}l_{i}}\left(q\right)Y_{l_{i}m_{i}}\left(\widehat{\mathbf{q}}\right)\right\rangle\end{array}$
(82)

for two values of the parameter

$\begin{array}[]{lll}\alpha_{1}=\sqrt{\displaystyle\frac{A_{2}}{A_{1}A}},\quad\alpha_{2}=-\sqrt{\displaystyle\frac{A_{1}}{A_{2}A}}.\end{array}$
(83)

##  Appendix B Multipole expansion of matrix elements

###  B.1 Matric elements integrated over space coordinates

We shall calculate the following matrix elements [see Ref. Maydanyuk.2012.PRC
, Eqs. (24)–(41)]:

$\begin{array}[]{ll}\Bigl{<}\phi_{f}\Bigl{|}\,e^{-i\alpha_{i}\mathbf{k_{\gamma}r}}\,\Bigr{|}\,\phi_{i}\Bigr{>}_{\mathbf{r}}=\displaystyle\int\phi_{f}^{*}(\mathbf{r})\>e^{-i\alpha_{i}\mathbf{k_{\gamma}r}}\>\phi_{i}(\mathbf{r})\;\mathbf{dr},&\hskip
14.22636pt\biggl{<}\phi_{f}\biggl{|}\,e^{-i\alpha_{i}\mathbf{k_{\gamma}r}}\displaystyle\frac{\partial}{\partial\mathbf{r}}\,\biggr{|}\,\phi_{i}\biggr{>}_{\mathbf{r}}=\displaystyle\int\phi_{f}^{*}(\mathbf{r})\>e^{-i\alpha_{i}\mathbf{k_{\gamma}r}}\displaystyle\frac{\partial}{\partial\mathbf{r}}\>\phi_{i}(\mathbf{r})\;\mathbf{dr}.\end{array}$
(84)

####  B.1.1 Expansion of the vector potential $\mathbf{A}$ by multipoles

Let us expand the vectorial potential $\mathbf{A}$ of electromagnetic field by
multipole. According to Ref. Eisenberg.1973  [see Eqs. (2.106), p. 58], in the
spherical symmetric approximation we have:

$\mbox{\boldmath$\xi$}_{\mu}\,e^{i\alpha_{i}\mathbf{k_{\gamma}r}}=\mu\,\sqrt{2\pi}\,\sum_{l_{\gamma}=1}\,(2l_{\gamma}+1)^{1/2}\,i^{l_{\gamma}}\,\cdot\Bigl{[}\mathbf{A}_{l_{\gamma}\mu}(\mathbf{r},M)+i\mu\,\mathbf{A}_{l_{\gamma}\mu}(\mathbf{r},E)\Bigr{]},$
(85)

where [see Eisenberg.1973 , (2.73) in p. 49, (2.80) in p. 51]

$\begin{array}[]{lcl}\vspace{2mm}\mathbf{A}_{l_{\gamma}\mu}(\mathbf{r},M)&=&j_{l_{\gamma}}(\alpha_{i}k_{\gamma}r)\>\mathbf{T}_{l_{\gamma}l_{\gamma},\mu}(\widehat{\mathbf{r}}),\\\
\vspace{2mm}\mathbf{A}_{l_{\gamma}\mu}(\mathbf{r},E)&=&\sqrt{\displaystyle\frac{l_{\gamma}+1}{2l_{\gamma}+1}}\,j_{l_{\gamma}-1}(\alpha_{i}k_{\gamma}r)\>\mathbf{T}_{l_{\gamma}l_{\gamma}-1,\mu}(\widehat{\mathbf{r}})\;-\sqrt{\displaystyle\frac{l_{\gamma}}{2l_{\gamma}+1}}\,j_{l_{\gamma}+1}(\alpha_{i}k_{\gamma}r)\>\mathbf{T}_{l_{\gamma}l_{\gamma}+1,\mu}(\widehat{\mathbf{r}}).\end{array}$
(86)

Here, $\mathbf{A}_{l_{\gamma}\mu}(\textbf{r},M)$ and
$\mathbf{A}_{l_{\gamma}\mu}(\textbf{r},E)$ are _magnetic_ and _electric
multipoles_ , $j_{l_{\gamma}}(\alpha_{i}k_{\gamma}r)$ is _spherical Bessel
function of order $l_{\gamma}$_,
$\mathbf{T}_{l_{\gamma}l_{\gamma}^{\prime},\mu}(\widehat{\mathbf{r}})$ are
_vector spherical harmonics_ , $\mbox{\boldmath$\xi$}_{\mu}$ are _vectors of
circular polarization_ of emitted photon. Eq. (85) is solution of the wave
equation of electromagnetic field in form of plane wave, which is presented as
summation of the electrical and magnetic multipoles (for example, see p. 83–92
in Ahiezer.1981 ). Therefore, separate multipolar terms in eq. (85) are
solutions of this wave equation for chosen numbers $j_{\gamma}$ and
$l_{\gamma}$ ($j_{\gamma}$ is quantum number characterizing eigenvalue of the
full momentum operator, while
$l_{\gamma}=j_{\gamma}-1,j_{\gamma},j_{\gamma}+1$ is connected with orbital
momentum operator, but it defines eigenvalues of photon parity and, so, it is
quantum number also).

We orient the frame so that axis $z$ be directed along the vector
$\mathbf{k}_{\gamma}$ (see Eisenberg.1973 , (2.105) in p. 57). According to
Eisenberg.1973  (see p. 45), the functions
$\mathbf{T}_{l_{\gamma}l_{\gamma}^{\prime},\mu}(\widehat{\mathbf{r}})$ have
the following form ($\mbox{\boldmath$\xi$}_{0}=0$):

$\mathbf{T}_{j_{\gamma}l_{\gamma},m}(\widehat{\mathbf{r}})=\sum\limits_{\mu=\pm
1}(l_{\gamma},1,j_{\gamma}\,\big{|}\,m-\mu,\mu,m)\;Y_{l_{\gamma},m-\mu}(\widehat{\mathbf{r}})\;\mbox{\boldmath$\xi$}_{\mu},$
(87)

where $(l,1,j\,\bigl{|}\,m-\mu,\mu,m)$ are _Clebsh-Gordon coefficients_ ,
$Y_{lm}(\theta,\varphi)$ are _spherical functions_ defined, according to
Landau.v3.1989  (see p. 119, (28,7)–(28,8)). From eq. (85) one can obtain such
a formula (at $\mbox{\boldmath$\varepsilon$}^{(3)}=0$):

$e^{-i\alpha_{i}\mathbf{k_{\gamma}r}}=\displaystyle\frac{1}{2}\,\displaystyle\sum\limits_{\mu=\pm
1}\mbox{\boldmath$\xi$}_{\mu}\,\mu\,\sqrt{2\pi}\,\sum_{l_{\gamma}=1}\,(2l_{\gamma}+1)^{1/2}\,(-i)^{l_{\gamma}}\,\cdot\Bigl{[}\mathbf{A}_{l_{\gamma}\mu}^{*}(\mathbf{r},M)-i\mu\,\mathbf{A}_{l_{\gamma}\mu}^{*}(\mathbf{r},E)\Bigr{]}.$
(88)

####  B.1.2 Central interactions

Using (88), for (84) we find:

$\begin{array}[]{ll}\vspace{1mm}\Bigl{<}\phi_{f}\Bigl{|}\,e^{-i\alpha_{i}\mathbf{k_{\gamma}r}}\,\Bigr{|}\,\phi_{i}\Bigr{>}_{\mathbf{r}}=\sqrt{\displaystyle\frac{\pi}{2}}\>\displaystyle\sum\limits_{l_{\gamma}=1}\,(-i)^{l_{\gamma}}\,\sqrt{2l_{\gamma}+1}\;\displaystyle\sum\limits_{\mu=\pm
1}\Bigl{[}\mu\,\tilde{p}_{l_{\gamma}\mu}^{M}-i\,\tilde{p}_{l_{\gamma}\mu}^{E}\Bigr{]},\\\
\biggl{<}\phi_{f}\biggl{|}\,e^{-i\alpha_{i}\mathbf{k_{\gamma}r}}\displaystyle\frac{\partial}{\partial\mathbf{r}}\,\biggr{|}\,\phi_{i}\biggr{>}_{\mathbf{r}}=\sqrt{\displaystyle\frac{\pi}{2}}\>\displaystyle\sum\limits_{l_{\gamma}=1}\,(-i)^{l_{\gamma}}\,\sqrt{2l_{\gamma}+1}\;\displaystyle\sum\limits_{\mu=\pm
1}\mbox{\boldmath$\xi$}_{\mu}\,\mu\,\times\Bigl{[}p_{l_{\gamma}\mu}^{M}-i\mu\>p_{l_{\gamma}\mu}^{E}\Bigr{]},\end{array}$
(89)

where

$\begin{array}[]{lcllcl}p_{l_{\gamma}\mu}^{M}&=&\displaystyle\int\phi^{*}_{f}(\mathbf{r})\,\biggl{(}\displaystyle\frac{\partial}{\partial\mathbf{r}}\,\phi_{i}(\mathbf{r})\biggr{)}\,\mathbf{A}_{l_{\gamma}\mu}^{*}(\mathbf{r},M)\;\mathbf{dr},&\hskip
19.91692ptp_{l_{\gamma}\mu}^{E}&=&\displaystyle\int\phi^{*}_{f}(\mathbf{r})\,\biggl{(}\displaystyle\frac{\partial}{\partial\mathbf{r}}\,\phi_{i}(\mathbf{r})\biggr{)}\,\mathbf{A}_{l_{\gamma}\mu}^{*}(\mathbf{r},E)\;\mathbf{dr},\end{array}$
(90)

and

$\begin{array}[]{lcllcl}\tilde{p}_{l_{\gamma}\mu}^{M}&=&\mbox{\boldmath$\xi$}_{\mu}\,\displaystyle\int\phi^{*}_{f}(\mathbf{r})\,\phi_{i}(\mathbf{r})\;\mathbf{A}_{l_{\gamma}\mu}^{*}(\mathbf{r},M)\;\mathbf{dr},&\hskip
19.91692pt\tilde{p}_{l_{\gamma}\mu}^{E}&=&\mbox{\boldmath$\xi$}_{\mu}\,\displaystyle\int\phi^{*}_{f}(\mathbf{r})\,\phi_{i}(\mathbf{r})\;\mathbf{A}_{l_{\gamma}\mu}^{*}(\mathbf{r},E)\;\mathbf{dr}.\end{array}$
(91)

####  B.1.3 Calculations of the components $p_{l_{\gamma}\mu}^{M}$,
$p_{l_{\gamma}\mu}^{E}$ and $\tilde{p}_{l_{\gamma}\mu}^{M}$,
$\tilde{p}_{l_{\gamma}\mu}^{E}$

For calculation of these components we shall use _gradient formula_ (see
Eisenberg.1973 , (2.56) in p. 46):

$\begin{array}[]{lcl}\displaystyle\frac{\partial}{\partial\mathbf{r}}\>\phi_{i}(\mathbf{r})&=&\displaystyle\frac{\partial}{\partial\mathbf{r}}\>\Bigl{\\{}R_{i}(r)\>Y_{l_{i}m_{i}}(\widehat{\mathbf{r}})\Bigr{\\}}=\par\sqrt{\displaystyle\frac{l_{i}}{2l_{i}+1}}\>\biggl{(}\displaystyle\frac{dR_{i}(r)}{dr}+\displaystyle\frac{l_{i}+1}{r}\,R_{i}(r)\biggr{)}\,\mathbf{T}_{l_{i}l_{i}-1,m_{i}}(\widehat{\mathbf{r}})-\\\
&-&\sqrt{\displaystyle\frac{l_{i}+1}{2l_{i}+1}}\>\biggl{(}\displaystyle\frac{dR_{i}(r)}{dr}-\displaystyle\frac{l_{i}}{r}\,R_{i}(r)\biggr{)}\,\mathbf{T}_{l_{i}l_{i}+1,m_{i}}(\widehat{\mathbf{r}}),\end{array}$
(92)

and obtain:

$\begin{array}[]{lcl}\vspace{1mm}p_{l_{\rm
ph,\mu}}^{M}&=&\sqrt{\displaystyle\frac{l_{i}}{2l_{i}+1}}\>I_{M}(l_{i},l_{f},l_{\gamma},l_{i}-1,\mu)\cdot\Bigl{\\{}J_{1}(l_{i},l_{f},l_{\gamma},\alpha_{i})+(l_{i}+1)\cdot
J_{2}(l_{i},l_{f},l_{\gamma},\alpha_{i})\Bigr{\\}}\;-\\\
\vspace{3mm}&-&\sqrt{\displaystyle\frac{l_{i}+1}{2l_{i}+1}}\>I_{M}(l_{i},l_{f},l_{\gamma},l_{i}+1,\mu)\cdot\Bigl{\\{}J_{1}(l_{i},l_{f},l_{\gamma},\alpha_{i})-l_{i}\cdot
J_{2}(l_{i},l_{f},l_{\gamma},\alpha_{i})\Bigr{\\}},\\\ \vspace{1mm}p_{l_{\rm
ph,\mu}}^{E}&=&\sqrt{\displaystyle\frac{l_{i}\,(l_{\gamma}+1)}{(2l_{i}+1)(2l_{\gamma}+1)}}\cdot
I_{E}(l_{i},l_{f},l_{\gamma},l_{i}-1,l_{\gamma}-1,\mu)\cdot\Bigl{\\{}J_{1}(l_{i},l_{f},l_{\gamma}-1,\alpha_{i})\;+(l_{i}+1)\cdot
J_{2}(l_{i},l_{f},l_{\gamma}-1,\alpha_{i})\Bigr{\\}}\;-\\\
\vspace{1mm}&-&\sqrt{\displaystyle\frac{l_{i}\,l_{\gamma}}{(2l_{i}+1)(2l_{\gamma}+1)}}\cdot
I_{E}(l_{i},l_{f},l_{\gamma},l_{i}-1,l_{\gamma}+1,\mu)\cdot\Bigl{\\{}J_{1}(l_{i},l_{f},l_{\gamma}+1,\alpha_{i})\;+(l_{i}+1)\cdot
J_{2}(l_{i},l_{f},l_{\gamma}+1,\alpha_{i})\Bigr{\\}}\;+\\\
\vspace{1mm}&+&\sqrt{\displaystyle\frac{(l_{i}+1)(l_{\gamma}+1)}{(2l_{i}+1)(2l_{\gamma}+1)}}\cdot
I_{E}(l_{i},l_{f},l_{\gamma},l_{i}+1,l_{\gamma}-1,\mu)\cdot\Bigl{\\{}J_{1}(l_{i},l_{f},l_{\gamma}-1,\alpha_{i})\;-l_{i}\cdot
J_{2}(l_{i},l_{f},l_{\gamma}-1,\alpha_{i})\Bigr{\\}}\;-\\\
&-&\sqrt{\displaystyle\frac{(l_{i}+1)\,l_{\gamma}}{(2l_{i}+1)(2l_{\gamma}+1)}}\cdot
I_{E}(l_{i},l_{f},l_{\gamma},l_{i}+1,l_{\gamma}+1,\mu)\cdot\Bigl{\\{}J_{1}(l_{i},l_{f},l_{\gamma}+1,\alpha_{i})\;-l_{i}\cdot
J_{2}(l_{i},l_{f},l_{\gamma}+1,\alpha_{i})\Bigr{\\}},\end{array}$ (93)

where

$\begin{array}[]{ccl}J_{1}(l_{i},l_{f},n,\alpha_{i})&=&\displaystyle\int\limits^{+\infty}_{0}\displaystyle\frac{dR_{i}(r,l_{i})}{dr}\>R^{*}_{f}(l_{f},r)\,j_{n}(\alpha_{i}\,k_{\gamma}r)\;r^{2}dr,\\\
J_{2}(l_{i},l_{f},n,\alpha_{i})&=&\displaystyle\int\limits^{+\infty}_{0}R_{i}(r,l_{i})\,R^{*}_{f}(l_{f},r)\>j_{n}(\alpha_{i}\,k_{\gamma}r)\;r\,dr,\\\
I_{M}\,(l_{i},l_{f},l_{\gamma},l_{1},\mu)&=&\displaystyle\int
Y_{l_{f}m_{f}}^{*}(\widehat{\mathbf{r}})\,\mathbf{T}_{l_{i}\,l_{1},\,m_{i}}(\widehat{\mathbf{r}})\,\mathbf{T}_{l_{\gamma}\,l_{\gamma},\,\mu}^{*}(\widehat{\mathbf{r}})\;d\Omega,\\\
I_{E}\,(l_{i},l_{f},l_{\gamma},l_{1},l_{2},\mu)&=&\displaystyle\int
Y_{l_{f}m_{f}}^{*}(\widehat{\mathbf{r}})\,\mathbf{T}_{l_{i}l_{1},\,m_{i}}(\widehat{\mathbf{r}})\,\mathbf{T}_{l_{\gamma}l_{2},\,\mu}^{*}(\widehat{\mathbf{r}})\;d\Omega.\end{array}$
(94)

By the same way for $\tilde{p}_{l_{\gamma}\mu}^{M}$ and
$\tilde{p}_{l_{\gamma}\mu}^{E}$ we find:

$\begin{array}[]{lcl}\tilde{p}_{l_{\gamma}\mu}^{M}&=&\tilde{I}\,(l_{i},l_{f},l_{\gamma},l_{\gamma},\mu)\cdot\tilde{J}\,(l_{i},l_{f},l_{\gamma},\alpha_{i}),\\\
\tilde{p}_{l_{\gamma}\mu}^{E}&=&\sqrt{\displaystyle\frac{l_{\gamma}+1}{2l_{\gamma}+1}}\tilde{I}\,(l_{i},l_{f},l_{\gamma},l_{\gamma}-1,\mu)\cdot\tilde{J}\,(l_{i},l_{f},l_{\gamma}-1,\alpha_{i})-\sqrt{\displaystyle\frac{l_{\gamma}}{2l_{\gamma}+1}}\tilde{I}\,(l_{i},l_{f},l_{\gamma},l_{\gamma}+1,\mu)\cdot\tilde{J}\,(l_{i},l_{f},l_{\gamma}+1,\alpha_{i}),\end{array}$
(95)

where

$\begin{array}[]{lcl}\tilde{J}\,(l_{i},l_{f},n,\alpha_{i})&=&\displaystyle\int\limits^{+\infty}_{0}R_{i}(r)\,R^{*}_{f}(l,r)\,j_{n}(\alpha_{i}\,k_{\gamma}r)\;r^{2}dr,\\\
\tilde{I}\,(l_{i},l_{f},l_{\gamma},n,\mu)&=&\mbox{\boldmath$\xi$}_{\mu}\displaystyle\int
Y_{l_{i}m_{i}}(\widehat{\mathbf{r}})\>Y_{l_{f}m_{f}}^{*}(\widehat{\mathbf{r}})\>\mathbf{T}_{l_{\gamma}n,\mu}^{*}(\widehat{\mathbf{r}})\>d\Omega.\end{array}$
(96)

##  Appendix C Linear and circular polarizations of the photon emitted

We define _vectors of linear polarization_ of emitted photon as (in Coulomb
gauge at $\mbox{\boldmath$\varepsilon$}^{(3)}=0$):

$\begin{array}[]{lll}\mbox{\boldmath$\varepsilon$}^{(1)}=\displaystyle\frac{1}{\sqrt{2}}\>\bigl{(}\mbox{\boldmath$\xi$}_{-1}-\mbox{\boldmath$\xi$}_{+1}\bigr{)},\hskip
8.53581pt\mbox{\boldmath$\varepsilon$}^{(2)}=\displaystyle\frac{i}{\sqrt{2}}\>\bigl{(}\mbox{\boldmath$\xi$}_{-1}+\mbox{\boldmath$\xi$}_{+1}\bigr{)},\end{array}$
(97)

where $\mbox{\boldmath$\xi$}_{\mu}$ are _vectors of circular polarization_
with opposite directions of rotation for the emitted photon used in Eqs. (85).
Rewrite vectors of linear polarization
$\mbox{\boldmath$\varepsilon$}^{(\alpha)}$ through vectors of circular
polarization $\mbox{\boldmath$\xi$}_{\mu}$ (see Ref. Eisenberg.1973 , (2.39),
p. 42; Appendix A in Ref. Maydanyuk.2012.PRC ,
$\mbox{\boldmath$\varepsilon$}^{(\alpha),*}=\mbox{\boldmath$\varepsilon$}^{(\alpha)}$):

$\begin{array}[]{lll}\mbox{\boldmath$\xi$}_{-1}=\displaystyle\frac{1}{\sqrt{2}}(\mbox{\boldmath$\varepsilon$}^{(1)}-i\mbox{\boldmath$\varepsilon$}^{(2)}),&\mbox{\boldmath$\xi$}_{+1}=-\displaystyle\frac{1}{\sqrt{2}}(\mbox{\boldmath$\varepsilon$}^{(1)}+i\mbox{\boldmath$\varepsilon$}^{(2)}),&\mbox{\boldmath$\xi$}_{0}=\mbox{\boldmath$\varepsilon$}^{(3)}.\end{array}$
(98)

We obtain properties:

$\begin{array}[]{ccc}\displaystyle\sum\limits_{\alpha=1,2}\mbox{\boldmath$\varepsilon$}^{(\alpha),*}=h_{-1}\mbox{\boldmath$\xi$}_{-1}^{*}+h_{+1}\mbox{\boldmath$\xi$}_{+1}^{*},\end{array}$
(99) $\begin{array}[]{ccc}\displaystyle\sum\limits_{\mu=\pm
1}\mbox{\boldmath$\xi$}_{\mu}^{*}\cdot\mbox{\boldmath$\xi$}_{\mu}=\displaystyle\frac{1}{2}\,\bigl{(}\mbox{\boldmath$\varepsilon$}^{(1)}-i\mbox{\boldmath$\varepsilon$}^{(2)}\bigr{)}\,\bigl{(}\mbox{\boldmath$\varepsilon$}^{(1)}-i\mbox{\boldmath$\varepsilon$}^{(2)}\bigr{)}^{*}+\displaystyle\frac{1}{2}\,\bigl{(}\mbox{\boldmath$\varepsilon$}^{(1)}+i\mbox{\boldmath$\varepsilon$}^{(2)}\bigr{)}\,\bigl{(}\mbox{\boldmath$\varepsilon$}^{(1)}+i\mbox{\boldmath$\varepsilon$}^{(2)}\bigr{)}^{*}=2,\end{array}$
(100)

where

$\begin{array}[]{lcr}h_{-1}=\displaystyle\frac{1}{\sqrt{2}}(1-i),&h_{1}=-\displaystyle\frac{1}{\sqrt{2}}(1+i),&h_{-1}+h_{1}=-i\sqrt{2}.\end{array}$
(101)

Also there is property (see Eqs. (3), (5) in Ref. Maydanyuk.2006.EPJA ):

$\begin{array}[]{lcr}\displaystyle\sum_{\alpha=1,2}\mbox{\boldmath$\varepsilon$}^{\rm(\alpha),\,*}=\displaystyle\sum\limits_{\mu=\pm
1}h_{m}\,\mbox{\boldmath$\xi$}_{m}^{*}.\end{array}$ (102)

We shall find multiplications of vectors $\mbox{\boldmath$\xi$}_{\pm 1}$. From
Eq. (98) we obtain:

$\begin{array}[]{ll}\mbox{\boldmath$\xi$}_{-1}^{*}=\Bigl{(}\displaystyle\frac{1}{\sqrt{2}}(\mbox{\boldmath$\varepsilon$}^{(1)}-i\mbox{\boldmath$\varepsilon$}^{(2)})\Bigr{)}^{*}=\displaystyle\frac{1}{\sqrt{2}}(\mbox{\boldmath$\varepsilon$}^{(1)}+i\mbox{\boldmath$\varepsilon$}^{(2)})=-\,\mbox{\boldmath$\xi$}_{+1},&\mbox{\boldmath$\xi$}_{+1}^{*}=-\,\mbox{\boldmath$\xi$}_{-1}\end{array}$
(103)

and

$\begin{array}[]{cc}\mbox{\boldmath$\xi$}_{-1}^{*}=-\mbox{\boldmath$\xi$}_{+1},&\mbox{\boldmath$\xi$}_{+1}^{*}=-\mbox{\boldmath$\xi$}_{-1}.\end{array}$
(104)

We check orthogonality conditions as

$\begin{array}[]{llllll}\vspace{1.5mm}\mbox{\boldmath$\xi$}_{-1}\cdot\mbox{\boldmath$\xi$}_{-1}=\mbox{\boldmath$\xi$}_{+1}\cdot\mbox{\boldmath$\xi$}_{+1}=0,&\hskip
14.22636pt\mbox{\boldmath$\xi$}_{-1}\cdot\mbox{\boldmath$\xi$}_{-1}^{*}=\mbox{\boldmath$\xi$}_{+1}\cdot\mbox{\boldmath$\xi$}_{+1}^{*}=-1,\\\
\mbox{\boldmath$\xi$}_{-1}\cdot\mbox{\boldmath$\xi$}_{+1}=1,&\hskip
14.22636pt\mbox{\boldmath$\xi$}_{-1}\cdot\mbox{\boldmath$\xi$}_{+1}^{*}=\mbox{\boldmath$\xi$}_{+1}\cdot\mbox{\boldmath$\xi$}_{-1}^{*}=0.\end{array}$
(105)

We calculate multiplications of vectors as

$\begin{array}[]{lcllcl}\vspace{2mm}\Bigl{[}\mbox{\boldmath$\xi$}_{-1}^{*}\times\mbox{\boldmath$\xi$}_{+1}\Bigr{]}&=&-\,\Bigl{[}\mbox{\boldmath$\xi$}_{+1}\times\mbox{\boldmath$\xi$}_{+1}\Bigr{]}=0,&\hskip
28.45274pt\Bigl{[}\mbox{\boldmath$\xi$}_{-1}^{*}\times\mbox{\boldmath$\xi$}_{-1}\Bigr{]}&=&-\,\Bigl{[}\mbox{\boldmath$\xi$}_{+1}\times\mbox{\boldmath$\xi$}_{-1}\Bigr{]},\\\
\Bigl{[}\mbox{\boldmath$\xi$}_{+1}^{*}\times\mbox{\boldmath$\xi$}_{-1}\Bigr{]}&=&-\,\Bigl{[}\mbox{\boldmath$\xi$}_{-1}\times\mbox{\boldmath$\xi$}_{-1}\Bigr{]}=0,&\hskip
28.45274pt\Bigl{[}\mbox{\boldmath$\xi$}_{+1}^{*}\times\mbox{\boldmath$\xi$}_{+1}\Bigr{]}&=&-\,\Bigl{[}\mbox{\boldmath$\xi$}_{-1}\times\mbox{\boldmath$\xi$}_{+1}\Bigr{]}\end{array}$
(106)

and

$\begin{array}[]{lll}\Bigl{[}\mbox{\boldmath$\xi$}_{-1}\times\mbox{\boldmath$\xi$}_{-1}^{*}\Bigr{]}&=&-\,\Bigl{[}\mbox{\boldmath$\xi$}_{+1}\times\mbox{\boldmath$\xi$}_{+1}^{*}\Bigr{]}=i\,\bigl{[}\mbox{\boldmath$\varepsilon$}^{(1)}\times\mbox{\boldmath$\varepsilon$}^{(2)}\bigr{]}=-\,\bigl{[}\mbox{\boldmath$\xi$}_{-1}\times\mbox{\boldmath$\xi$}_{+1}\bigr{]}.\end{array}$
(107)

Now we take into account that two vectors
$\mbox{\boldmath$\varepsilon$}^{(1)}$ and
$\mbox{\boldmath$\varepsilon$}^{(2)}$ are vectors of polarization of photon
emitted, which are perpendicular to direction of emission of this photon
defined by vector $\mathbf{k}$. Modulus of vectorial multiplication
$\bigl{[}\mbox{\boldmath$\varepsilon$}^{(1)}\times\mbox{\boldmath$\varepsilon$}^{(2)}\bigr{]}$
equals to unity. So, we have:

$\begin{array}[]{lll}\bigl{[}\mbox{\boldmath$\varepsilon$}^{(1)}\times\mbox{\boldmath$\varepsilon$}^{(2)}\bigr{]}=\displaystyle\frac{\mathbf{k}_{\gamma}}{\bigl{|}\mathbf{k}_{\gamma}\bigr{|}}\equiv\widehat{\gamma}.\end{array}$
(108)

By such a basis, we rewrite properties (107) as

$\begin{array}[]{lll}\Bigl{[}\mbox{\boldmath$\xi$}_{-1}\times\mbox{\boldmath$\xi$}_{-1}^{*}\Bigr{]}&=&-\,\Bigl{[}\mbox{\boldmath$\xi$}_{+1}\times\mbox{\boldmath$\xi$}_{+1}^{*}\Bigr{]}=-\,\bigl{[}\mbox{\boldmath$\xi$}_{-1}\times\mbox{\boldmath$\xi$}_{+1}\bigr{]}=i\,\bigl{[}\mbox{\boldmath$\varepsilon$}^{1}\times\mbox{\boldmath$\varepsilon$}^{2}\bigr{]}=i\,\widehat{\mathbf{\gamma}}.\end{array}$
(109)

##  Appendix D Angular integrals $I_{E}$, $I_{M}$ and $\tilde{I}$

We shall calculate the integrals in eqs. (94) and (96) [see Appendix B in Ref.
Maydanyuk.2012.PRC ]:

$\begin{array}[]{lcl}I_{M}\,(l_{i},l_{f},l_{\gamma},l_{1},\mu)&=&\displaystyle\int
Y_{l_{f}m_{f}}^{*}(\widehat{\mathbf{r}})\,\mathbf{T}_{l_{i}\,l_{1},\,m_{i}}(\widehat{\mathbf{r}})\,\mathbf{T}_{l_{\gamma}\,l_{\gamma},\,\mu}^{*}(\widehat{\mathbf{r}})\;d\Omega,\\\
I_{E}\,(l_{i},l_{f},l_{\gamma},l_{1},l_{2},\mu)&=&\displaystyle\int
Y_{l_{f}m_{f}}^{*}(\widehat{\mathbf{r}})\,\mathbf{T}_{l_{i}l_{1},\,m_{i}}(\widehat{\mathbf{r}})\,\mathbf{T}_{l_{\gamma}l_{2},\,\mu}^{*}(\widehat{\mathbf{r}})\;d\Omega,\\\
\tilde{I}\,(l_{i},l_{f},l_{\gamma},n,\mu)&=&\mbox{\boldmath$\xi$}_{\mu}\displaystyle\int
Y_{l_{f}m_{f}}^{*}(\widehat{\mathbf{r}})\,Y_{l_{i}m_{i}}(\widehat{\mathbf{r}})\,\mathbf{T}_{l_{\gamma}n,\,\mu}^{*}(\widehat{\mathbf{r}})\;d\Omega.\end{array}$
(110)

Substituting the function $\mathbf{T}_{jl,m}(\widehat{\mathbf{r}})$ defined by
eq. (87), we obtain (at ${\mathbf{\xi}}_{0}=0$):

$\begin{array}[]{lcl}I_{M}\,(l_{i},l_{f},l_{\gamma},l_{1},\mu)&=&\displaystyle\sum\limits_{\mu^{\prime}=\pm
1}(l_{1},1,l_{i}\,\big{|}\,m_{i}-\mu^{\prime},\mu^{\prime},m_{i})\;(l_{\gamma},1,l_{\gamma}\,\big{|}\,\mu-\mu^{\prime},\mu^{\prime},\mu)\;\times\\\
&\times&\displaystyle\int Y_{l_{f}m}^{*}(\widehat{\mathbf{r}})\cdot
Y_{l_{1},\,m_{i}-\mu^{\prime}}(\widehat{\mathbf{r}})\cdot
Y_{l_{\gamma},\,\mu-\mu^{\prime}}^{*}(\widehat{\mathbf{r}})\;d\Omega,\\\
I_{E}\,(l_{i},l_{f},l_{\gamma},l_{1},l_{2},\mu)&=&\displaystyle\sum\limits_{\mu^{\prime}=\pm
1}(l_{1},1,l_{i}\,\big{|}\,m_{i}-\mu^{\prime},\mu^{\prime},m_{i})\;(l_{2},1,l_{\gamma}\,\big{|}\,\mu-\mu^{\prime},\mu^{\prime},\mu)\;\times\\\
&\times&\displaystyle\int Y_{l_{f}m}^{*}(\widehat{\mathbf{r}})\cdot
Y_{l_{1},\,m_{i}-\mu^{\prime}}(\widehat{\mathbf{r}})\cdot
Y_{l_{2},\,\mu-\mu^{\prime}}^{*}(\widehat{\mathbf{r}})\;d\Omega.\end{array}$
(111)
$\begin{array}[]{ccl}\tilde{I}\,(l_{i},l_{f},l_{\gamma},n,\mu)&=&(n,1,l_{\gamma}\,\big{|}\,0,\mu,\mu)\times\displaystyle\int
Y_{l_{f}m_{f}}^{*}(\widehat{\mathbf{r}})\,Y_{l_{i}m_{i}}(\widehat{\mathbf{r}})\,Y_{n0}^{*}(\widehat{\mathbf{r}})\;d\Omega.\end{array}$
(112)

Here, we have taken orthogonality of vectors $\xi_{\pm 1}$ into account. In
these formulas we shall find angular integral:

$\begin{array}[]{l}\vspace{1mm}\displaystyle\int\>Y_{l_{f}m_{f}}^{*}(\widehat{\mathbf{r}})\,Y_{l_{1},\,m_{i}-\mu^{\prime}}(\widehat{\mathbf{r}})\,Y_{n,\,\mu-\mu^{\prime}}^{*}(\widehat{\mathbf{r}})\;d\Omega=\par(-1)^{l_{f}+n+m_{i}-\mu^{\prime}}\;i^{l_{f}+l_{1}+n+|m_{f}|+|m_{i}-\mu^{\prime}|+|m_{i}-m_{f}-\mu^{\prime}|}\;\times\\\
\;\times\sqrt{\displaystyle\frac{(2l_{f}+1)\,(2l_{1}+1)\,(2n+1)}{16\pi}\displaystyle\frac{(l_{f}-|m_{f}|)!}{(l_{f}+|m_{f}|)!}\;\displaystyle\frac{(l_{1}-|m_{i}-\mu^{\prime}|)!}{(l_{1}+|m_{i}-\mu^{\prime}|)!}\;\displaystyle\frac{(n-|m_{i}-m_{f}-\mu^{\prime}|)!}{(n+|m_{i}-m_{f}-\mu^{\prime}|)!}}\;\times\\\
\;\times\displaystyle\int\limits_{0}^{\pi}\>P_{l_{f}}^{|m_{f}|}(\cos{\theta})\;P_{l_{1}}^{|m_{i}-\mu^{\prime}|}(\cos{\theta})\;P_{n}^{|m_{i}-m_{f}-\mu^{\prime}|}(\cos{\theta})\cdot\sin{\theta}\,d\theta,\end{array}$
(113)

where $P_{l}^{m}(\cos{\theta})$ are associated Legandre’s polynomials, and we
obtain conditions:

$\begin{array}[]{llllll}\mbox{for integrals
}I_{M},I_{E}:&\mu=m_{i}-m_{f},&n\geq|\mu-\mu^{\prime}|=|m_{i}-m_{f}+\mu^{\prime}|,&\mu=\pm
1,\\\ \mbox{for integral }\tilde{I}:&m_{i}=m_{f}.&&\end{array}$ (114)

Using formula (113), we calculate integrals (111) and (112):

$\begin{array}[]{lcl}I_{M}\,(l_{i},l_{f},l_{\gamma},l_{1},\mu)&=&\delta_{\mu,m_{i}-m_{f}}\;\displaystyle\sum\limits_{\mu^{\prime}=\pm
1}C_{l_{i}l_{f}l_{ph}l_{1}l_{ph}}^{m_{i}m_{f}\mu^{\prime}}\displaystyle\int\limits_{0}^{\pi}\>f_{l_{1}l_{f}l_{\gamma}}^{m_{i}m_{f}\mu^{\prime}}(\theta)\;\sin{\theta}\,d\theta,\\\
I_{E}\,(l_{i},l_{f},l_{\gamma},l_{1},l_{2},\mu)&=&\delta_{\mu,m_{i}-m_{f}}\;\displaystyle\sum\limits_{\mu^{\prime}=\pm
1}C_{l_{i}l_{f}l_{ph}l_{1}l_{2}}^{m_{i}m_{f}\mu^{\prime}}\displaystyle\int\limits_{0}^{\pi}\>f_{l_{1}l_{f}l_{2}}^{m_{i}m_{f}\mu^{\prime}}(\theta)\;\sin{\theta}\,d\theta,\\\
\tilde{I}\,(l_{i},l_{f},l_{\gamma},n,\mu)&=&C_{l_{i}l_{f}l_{\gamma}n}^{m_{i}\mu}\displaystyle\int\limits_{0}^{\pi}\>f_{l_{i}l_{f}n}^{m_{i}m_{i}0}(\theta)\;\sin{\theta}\,d\theta,\end{array}$
(115)

where

$\begin{array}[]{lcl}\vspace{1mm}C_{l_{i}l_{f}l_{ph}l_{1}l_{2}}^{m_{i}m_{f}\mu^{\prime}}&=&(-1)^{l_{f}+l_{2}+m_{i}-\mu^{\prime}}\;i^{l_{f}+l_{1}+l_{2}+|m_{f}|+|m_{i}-\mu^{\prime}|+|m_{i}-m_{f}-\mu^{\prime}|}\;\times\\\
\vspace{2mm}&\times&(l_{1},1,l_{i}\,\big{|}\,m_{i}-\mu^{\prime},\mu^{\prime},m_{i})\;(l_{2},1,l_{\gamma}\,\big{|}\,m_{i}-m_{f}-\mu^{\prime},\mu^{\prime},m_{i}-m_{f})\;\times\\\
&\times&\sqrt{\displaystyle\frac{(2l_{f}+1)\,(2l_{1}+1)\,(2l_{2}+1)}{16\pi}\displaystyle\frac{(l_{f}-|m_{f}|)!}{(l_{f}+|m_{f}|)!}\;\displaystyle\frac{(l_{1}-|m_{i}-\mu^{\prime}|)!}{(l_{1}+|m_{i}-\mu^{\prime}|)!}\;\displaystyle\frac{(l_{2}-|m_{i}-m_{f}-\mu^{\prime}|)!}{(l_{2}+|m_{i}-m_{f}-\mu^{\prime}|)!}},\end{array}$
(116)

$\begin{array}[]{lcl}\vspace{1mm}C_{l_{i}l_{f}l_{\gamma}n}^{m_{i}\mu}&=&(-1)^{l_{f}+n+m_{i}+|m_{i}|}\;i^{l_{f}+l_{i}+n}\cdot(n,1,l_{\gamma}\,\big{|}\,0,\mu,\mu)\cdot\sqrt{\displaystyle\frac{(2l_{f}+1)\,(2l_{i}+1)\,(2n+1)}{16\pi}\displaystyle\frac{(l_{f}-|m_{i}|)!}{(l_{f}+|m_{i}|)!}\;\displaystyle\frac{(l_{i}-|m_{i}|)!}{(l_{i}+|m_{i}|)!}},\end{array}$
(117)

$f_{l_{1}l_{f}l_{2}}^{m_{i}m_{f}\mu^{\prime}}(\theta)=P_{l_{1}}^{|m_{i}-\mu^{\prime}|}(\cos{\theta})\;P_{l_{f}}^{|m_{f}|}(\cos{\theta})\;P_{l_{2}}^{|m_{i}-m_{f}-\mu^{\prime}|}(\cos{\theta}).$
(118)

We define differential functions on the integrals (115) with angular
dependence as

$\begin{array}[]{lcl}\displaystyle\frac{d\,I_{M}\,(l_{i},l_{f},l_{\gamma},l_{1},\mu)}{\sin{\theta}\,d\theta}&=&\delta_{\mu,m_{i}-m_{f}}\;\displaystyle\sum\limits_{\mu^{\prime}=\pm
1}C_{l_{i}l_{f}l_{ph}l_{1}l_{ph}}^{m_{i}m_{f}\mu^{\prime}}\cdot
f_{l_{1}l_{f}l_{\gamma}}^{m_{i}m_{f}\mu^{\prime}}(\theta),\\\
\displaystyle\frac{d\,I_{E}\,(l_{i},l_{f},l_{\gamma},l_{1},l_{2},\mu)}{\sin{\theta}\,d\theta}&=&\delta_{\mu,m_{i}-m_{f}}\;\displaystyle\sum\limits_{\mu^{\prime}=\pm
1}C_{l_{i}l_{f}l_{ph}l_{1}l_{2}}^{m_{i}m_{f}\mu^{\prime}}\cdot
f_{l_{1}l_{f}l_{2}}^{m_{i}m_{f}\mu^{\prime}}(\theta),\\\
\displaystyle\frac{d\,\tilde{I}\,(l_{i},l_{f},l_{\gamma},n,\mu)}{\sin{\theta}\,d\theta}&=&\delta_{m_{i}m_{f}}\,C_{l_{i}l_{f}l_{\gamma}n}^{m_{i}\mu}f_{l_{i}l_{f}n}^{m_{i}m_{i}0}(\theta).\end{array}$
(119)

##  Appendix E Calculations of the wave functions of relative motion between
two nuclei

###  E.1 Boundary conditions and normalization of the wave functions

In this paper for the wave function of relative motion $\phi(\mathbf{r})$ in
the initial $i$-state and final $f$-state we chose states of the elastic
scattering of one nucleus on another nucleus, for which we have used the
normalization condition for the radial wave function of relative motion
$R_{s}(k,r)$ as (see Ref. Landau.v3.1989 , p. 138)

$\begin{array}[]{lll}\int\limits_{0}^{+\infty}R_{s}^{*}(k^{\prime},r)\,R_{s}(k,r)\>r^{2}dr=\int\limits_{0}^{+\infty}\chi_{s}^{*}(k^{\prime},r)\,\chi_{s}(k,r)\>dr=2\pi\,\delta(k^{\prime}-k),&R_{s}(k,r)=\displaystyle\frac{\chi_{s}(k,r)}{r}.\end{array}$
(120)

Here, $r$ is relative distance between nuclei, $k$, $k^{\prime}$ are wave
numbers. The radial wave function in the asymptotic region can be written as
($s=i,f$)

$\begin{array}[]{ccl}\chi_{s}(k,r)=N_{s}\;\bigl{(}A_{s}G_{s}(k,r)+B_{s}F_{s}(k,r)\bigr{)},\end{array}$
(121)

where $F_{s}$ and $G_{s}$ are the Coulomb functions, $A_{s}$ and $B_{s}$ are
real constants determined concerning the found solution for $\chi_{s}(k,r)$ at
small $r$, $N_{s}$ is unknown normalization factor. At far distances we have

$\begin{array}[]{ccl}F_{s}(k,r)=\sin{\theta_{s}},&G_{s}(k,r)=\cos{\theta_{s}},&\theta_{s}=k_{s}r-\displaystyle\frac{l\pi}{2}+\sigma_{l}(\eta_{s}),\\\
\end{array}$ (122)

where $\sigma_{l}(\eta_{s})=\arg{\;\Gamma(i\eta_{s}+l+1)}$,
$\eta_{s}=\displaystyle\frac{\mu\,m_{N}\,\nu}{k_{s}\,\hbar}$,
$\nu=Z_{1}\,Z_{2}\,e^{2}$ is Coulomb parameter,
$\mu\,m_{N}=m_{1}m_{2}/(m_{1}+m_{2})$ is reduced mass of two nuclei with mass
$m_{1}$ and $m_{2}$, $e$ is electric charge of proton. At such a
representation of the asymptotic Coulomb functions we find

$\begin{array}[]{c}\vspace{2mm}\Bigl{(}A_{s}G_{s}(k^{\prime},r)+B_{s}F_{s}(k^{\prime},r)\Bigr{)}^{*}\Bigl{(}A_{s}G_{s}(k,r)+B_{s}F_{s}(k,r)\Bigr{)}=\\\
=\displaystyle\frac{A_{s}^{2}+B_{s}^{2}}{2}\cos{(\theta^{\prime}+\theta)}+A_{s}B_{s}\sin{(\theta^{\prime}+\theta)}+\displaystyle\frac{A_{s}^{2}+B_{s}^{2}}{2}\cos{(\theta^{\prime}-\theta)}.\end{array}$
(123)

On such a basis, the integral (120) is transformed to the following form:

$\begin{array}[]{c}|N_{s}|^{2}\displaystyle\frac{A_{s}^{2}+B_{s}^{2}}{2}\int\limits_{0}^{+\infty}\cos{(\theta^{\prime}-\theta)}\;dr=2\pi\;\delta(k^{\prime}-k).\end{array}$
(124)

Taking definition of the $\delta$-function into account, we obtain

$N_{s}=\displaystyle\frac{2}{\sqrt{A_{s}^{2}+B_{s}^{2}}}.$ (125)

In bremsstrahlung problems another normalization condition is useful for decay
of nuclear system, so we add this formalism. The wave function should
correspond to emission of cluster from nucleus during unit of time (see Ref.
Landau.v3.1989 , p. 140):

$\begin{array}[]{ll}\displaystyle\oint j_{s}\,(\mathbf{\rm
r})\;r^{2}\>d\Omega=1,&\chi_{s}(r\to+\infty)=N_{s}\cdot(G_{s}(r)+iF_{s}(r)),\end{array}$
(126)

where $d\Omega=\sin{\theta}\>d\theta\>d\phi$ is solid angle element,
$j_{s}({\bf r})$ is probability flux density and the integration is performed
over spherical surface of enough large radius, $R_{\rm max}$. Defining the
flux as $j_{s}\,(\mathbf{\rm r})=i/2m\>(\phi_{s}({\bf
r})\>\nabla\phi_{s}^{*}({\bf r})-{\rm c.c.})$ and choosing the radial
component, $\chi_{s}(r)$, at far distances as the outgoing Coulomb wave (see
second formula in Eqs. (126)), where $N_{s}$ is unknown normalization factor.
We integrate Eq. (126) over the angular variables and find $N_{s}$ as

$\begin{array}[]{ccc}\biggl{(}\displaystyle\frac{k_{s}}{m}\cdot\Bigl{|}\chi_{s}(R_{\rm
max})\Bigr{|}^{2}=\displaystyle\frac{k_{s}}{m}\>|N_{s}|^{2}\Bigl{(}|G_{s}(R_{\rm
max})|^{2}+|F_{s}(R_{\rm
max})|^{2}\Bigr{)}=\displaystyle\frac{k_{s}}{m}\cdot|N_{s}|^{2}=1\biggr{)}&\to&\biggl{(}N_{s}=\sqrt{\displaystyle\frac{m}{k_{s}}}\;\biggr{)}.\end{array}$
(127)

###  E.2 Aspects of numerical calculations of the wave functions

To calculate integrals (94) and (96), we used large but finite range for the
variable $r$: $0\leq r\leq R_{\rm max}$. We separate the full radial region on
the internal and asymptotic parts at point $R_{\rm at}$: The internal region
with strong effects of nuclear and Coulomb interactions between clusters
($0\leq r\leq R_{at}$), and the asymptotic region with the Coulomb interaction
only ($R_{\rm at}\leq r\leq R_{\rm max}$). Calculations of the wave functions
are performed in each region independently by different methods.

In the internal region we use the following way. For the state of elastic
scattering the wave function, $\chi_{s}$, is real. We determine each partial
solution of wave function and its derivative at a selected starting point
$r_{0}$, and then we calculate those in the region close enough to this point
using the _method of beginning of the solution_ (based on expansion on Teylor
series, see Ref. Kunz.1964.book ). For the solution increasing in the barrier
region we choose $r_{0}=0$ as the starting point. For the solution decreasing
in the barrier region we choose $r_{0}=R_{\rm at}$ as the starting point. Then
we calculate both partial solutions and their derivatives independently in the
full nuclear region using the _method of continuation of the solution_ shortly
presented in App. B.3 in Ref. Maydanyuk.2011.JPG , which is improvement of the
Numerov method with constant step Kunz.1964.book . Then, we find the unknown
complex coefficients from the corresponding boundary conditions.

The Coulomb wave functions and their derivatives in the asymptotic region are
calculated by using library programs. Then, they are matched at point $R_{\rm
at}$ with the solutions in the internal region, using continuity conditions
for the wave functions and their derivatives. The $R_{\rm max}$ boundary is
chosen from requirement to achieve convenient stability and convergence in the
calculations of the cross sections (this can be boundary about 200000 fm).

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