A Derivation of Eqs. (59) and (61)

Comparison of potential models of nucleus-nucleus bremsstrahlung

Abstract

At low photon energies, the potential models of nucleus-nucleus bremsstrahlung are based on electric transition multipole operators, which are derived either only from the nuclear current or only from the charge density by making the long-wavelength approximation and using the Siegert theorem. In the latter case, the bremsstrahlung matrix elements are divergent and some regularization techniques are used to obtain finite values for the bremsstrahlung cross sections. From an extension of the Siegert theorem, which is not based on the long-wavelength approximation, a new potential model of nucleus-nucleus bremsstrahlung is developed. Only convergent integrals are included in this approach. Formal links between bremsstrahlung cross sections obtained in these different models are made. Furthermore, three different ways to calculate the regularized matrix elements are discussed and criticized. Some prescriptions for a proper implementation of the regularization are deduced. A numerical comparison between the different models is done by applying them to the bremsstrahlung.

pacs:
25.20.Lj,24.10.-i,25.55.-e

I Introduction

Nuclear bremsstrahlung refers to a radiative transition between nuclear states which lie in the continuum. This paper principally focuses on nucleus-nucleus bremsstrahlung, where the photon emission is induced by a collision between two nuclei or a nucleus and a neutron. However, the emission of bremsstrahlung photons can also accompany proton decays, decays, or fissions. The common essential feature of these processes is that both initial and final states are not square-integrable in stationary approaches. This feature leads in some bremsstrahlung models Tanimura and Mosel (1985); Langanke and Rolfs (1986a); Garrido et al. (2012, 2014) to divergent matrix elements, which have to be replaced by some finite values via some regularization prescription. Then, the difficult problem of analyzing the influence of the regularization techniques on the results arises. This problem is avoided in other bremsstrahlung models Philpott and Halderson (1982); Baye and Descouvemont (1985); Langanke (1986); Langanke and Rolfs (1986b); Liu et al. (1990a, b); Baye et al. (1991, 1992); Liu et al. (1992); Maydanyuk (2011, 2012), which are based from the beginning only on convergent matrix elements. To understand the presence or the absence of divergence problems in different bremsstrahlung models, it is required to discuss the fundamental bases of these models. This discussion is also useful to highlight the links between these models.

The description of electromagnetic transitions in nuclear systems relies on the interaction between the nuclear current and the electromagnetic field. When the long-wavelength approximation (LWA) can be applied, the interaction between the nuclear current and the electric field does not explicitly depend on the nuclear current anymore but can be deduced exclusively from the charge density. This property is referred to as the Siegert theorem Siegert (1937). This is particularly useful in nuclear physics where the current density is usually less well known than the charge density. However, in the study of radiative transitions between continuum states, the long-wavelength approximation leads to mathematical divergences and the dependence on the nuclear current cannot thus be fully removed in bremsstrahlung models.

To avoid this divergence problem, most authors decided not to apply the Siegert theorem in bremsstrahlung models Philpott and Halderson (1982); Baye and Descouvemont (1985); Langanke (1986); Langanke and Rolfs (1986b); Liu et al. (1990a, b); Baye et al. (1991, 1992); Liu et al. (1992); Maydanyuk (2011, 2012). For potential models of bremsstrahlung, where the colliding nuclei are treated as point-like particles interacting with an effective nucleus-nucleus interaction, some authors preferred to apply the Siegert theorem and to replace the divergent integrals by convergent expressions by using some regularization techniques Tanimura and Mosel (1985); Langanke and Rolfs (1986a); Garrido et al. (2012, 2014). Even if applying the Siegert theorem seems to simplify the expressions of the matrix elements required to evaluate the bremsstrahlung cross sections, the regularization techniques used in Refs. Tanimura and Mosel (1985); Langanke and Rolfs (1986a); Garrido et al. (2012, 2014) break this apparent simplicity.

In a recent paper Dohet-Eraly and Baye (2013), an extension of the Siegert theorem Schmitt et al. (1990), which does not rely to the long-wavelength approximation and which does not lead to divergent matrix elements, was proposed to greatly reduce the dependence of the electric transition multipole operators on the nuclear current. This method was applied to a microscopic description of nucleus-nucleus bremsstrahlung, namely the  Dohet-Eraly and Baye (2013) and systems Dohet-Eraly (2014). In this paper, the method developed in LABEL:DEB13 is applied to a potential model of bremsstrahlung. With this method, the expressions of bremsstrahlung cross sections obtained after regularization in the Siegert approach based on the long-wavelength approximation can be justified without introducing divergent integrals.

In Sec. II, the potential models of bremsstrahlung are outlined. In Sec. III, the different forms of the electric transition multipole operators are derived in a common framework. The interest of a Siegert approach in the potential models of bremsstrahlung is discussed. In Sec. IV, the calculation of the matrix elements of the electric transition multipole operators is explained and the basic idea of the regularization techniques is presented. In Sec. V, three implementations of regularization techniques are presented and compared: the fixed method proposed by Garrido, Fedorov, and Jensen in LABEL:GJF12, the integration by parts (IP) method inspired by Tanimura and Mosel’s work Tanimura and Mosel (1985), and the contour integration (CI) method, more adapted for numerical calculations, based on the contour integration proposed by Vincent and Fortune Vincent and Fortune (1970). In Sec. VI, the different versions of the potential model of nucleus-nucleus bremsstrahlung are applied to the system and the bremsstrahlung cross sections are compared. Concluding remarks are presented in Sec. VII.

Ii Bremsstrahlung cross sections

In the center-of-mass (c.m.) frame, two spinless nuclei with charges and , masses and , respectively, and reduced mass collide with initial relative wave vector in the direction and relative energy . After emission of a photon in direction with energy , the nuclear system has final relative vector in direction and relative energy given by

 Ef=Ei−Eγ, (1)

up to small recoil corrections.

The bremsstrahlung cross sections are evaluated from the multipole matrix elements, which are proportional to the matrix elements of the electromagnetic transition multipole operators between the incoming initial state in the direction with energy and the outgoing final state in direction with energy ,

 uσλμ(Ωf)=ασλ⟨Ψ−f(Ωf)|Mσλμ|Ψ+i⟩, (2)

where is the order of the multipole, is its component, or corresponds to an electric multipole and or corresponds to a magnetic multipole, and is given by

 ασλ=−√2π(λ+1)iλ+σkλγ√λ(2λ+1)(2λ−1)!!. (3)

The differential bremsstrahlung cross section is given by Baye et al. (1991)

 dσdEγ=Eγπ2ℏ5cp2f1+δ12∑σλμ∫π0|uσλμ(θf,0)|22λ+1sinθfdθf, (4)

where is equal to unity if nuclei and are identical and to zero otherwise. The division by is added to take the possible identity of both nuclei into account. Other differential bremsstrahlung cross sections are also obtained from the multipole matrix elements . Explicit formulas can be found in LABEL:BSD91.

In the potential model, nuclei are treated as point-like particles interacting with an effective nucleus-nucleus interaction. The initial and final states and are solutions of the Schrödinger equation

 HΨ=EΨ (5)

with energy and , respectively. The internal Hamiltonian reads

 H=−ℏ22μMΔρ+V(ρ), (6)

where is the relative coordinate between the nuclei, is the norm of , and is a local potential describing the interaction between both nuclei. The potential is assumed to be real and central. Some comments about more general potentials are given in Sec. III. The potential can be defined by subtracting the bare Coulomb potential from the potential ,

 U(ρ)=V(ρ)−Z1Z2e2ρ. (7)

It is assumed to have a finite range.

The initial and final states and can be expanded in partial-wave series Baye et al. (1991)

 Ψ+i = ∑liψili0, (8) Ψ−f(Ωf) = 2√π∑lfmf(2lf+1)−1/2Ymf∗lf(Ωf)e−2i(σlf+δlf)ψflfmf, (9)

where and are the Coulomb and quasinuclear phase shifts. Their dependence on energy is dropped to simplify the notation. The normalized spherical harmonics are defined by following the Condon and Shortley convention. If colliding nuclei are identical bosons (resp. fermions), partial-wave expansions (8) and (9) are restricted to even (resp. odd) values of and to satisfy the Pauli principle.

The partial waves can be written, by splitting the radial and angular dependences, as

 ψclcmc(ρ)=Clcuclc(ρ)ρYmclc(Ωρ), (10)

where is the angular part of the spherical coordinates of , or designates the initial or final channel, and is a complex coefficient defined by

 Clc=2√π(2lc+1)1/2ilcei(σlc+δlc). (11)

The radial function is a real solution of the radial Schrödinger equation at energy ,

 −βuc′′lc(ρ)+[Vlc(ρ)+V(ρ)]uclc(ρ)=Ecuclc(ρ), (12)

where the prime designates the derivative with respect to , , and is the centrifugal potential

 Vlc(ρ)=βlc(lc+1)ρ2. (13)

The normalization of is fixed by its asymptotic behavior,

 uclc(ρ)⟶ρ→∞uc,aslc(ρ) = (1+δ12)1/2√vckc[Flc(ηc,kcρ)cosδlc+Glc(ηc,kcρ)sinδlc] (14) = i(1+δ12)1/22√vckc[e−iδlcIlc(ηc,kcρ)−eiδlcOlc(ηc,kcρ)], (15)

where is the Sommerfeld parameter, and are the regular and irregular Coulomb functions, and and are the incoming and outgoing Coulomb wave functions. From Eqs. (8) and (9), the expansion of in partial-wave series can be written as Baye et al. (1991)

 uσλμ=2√πασλ∑lilf(2lf+1)−1/2Yμlf(Ωf)(liλ0μ|lfμ)e2i(σlf+δlf)⟨ψflf||Mσλ||ψili⟩, (16)

where the reduced matrix elements are defined following the convention

 ⟨ψflfmf|Mσλμ|ψilimi⟩=(liλmiμ|lfmf)⟨ψflf||Mσλ||ψili⟩. (17)

Since only the electric transitions () are concerned by the Siegert approach and since they dominate for light-ion bremsstrahlung at low photon energy Philpott and Halderson (1982), the magnetic transitions are not considered hereafter. The potential models of nucleus-nucleus bremsstrahlung used in Refs. Baye et al. (1991); Tanimura and Mosel (1985); Langanke and Rolfs (1986b); Langanke (1986); Langanke and Rolfs (1986a); Garrido et al. (2012, 2014) differ by their definitions of the electric transition multipole operators, which are given in the next section.

Iii Electric transition multipole operators

The electric transition multipole operators can be defined from the nuclear current by Bohr and Mottelson (1969)

 MEλμ=√λλ+1(2λ+1)!!kλγc∫J(r)⋅AEλμ(r)dr, (18)

where is the nuclear current density and is the electric multipole defined, in the Coulomb gauge, as Messiah (1962)

 AEλμ(r)=ikγ√λ(λ+1)χλμ(kγ,r) (19)

with

 χλμ(k,r)=(k2r+∇∂∂rr)jλ(kr)Yμλ(Ω) (20)

and . The usual notation is used to designate the spherical Bessel functions (first kind) of order  Abramowitz and Stegun (1965).

The suppression of the current dependence of the electric transitions at low photon energies relies on the fact that is reduced to a gradient term at the long-wavelength approximation, i.e., by keeping only the lowest order term in in the expression of the electric multipole,

 AEλμ(r)⟶kγ→0i√λ+1kλ−1γ√λ(2λ+1)!!∇rλYμλ(Ω). (21)

To reduce the current dependence without applying the long-wavelength approximation, the idea is to introduce an approximate electric transition multipole operator, denoted by , in which is approximated only by a gradient term

 ˜MEλμ=√λλ+1(2λ+1)!!kλγc∫J(r)⋅∇Φλμ(r)dr, (22)

where is chosen such that and have the same behavior at low photon energies,

 ||∇Φλμ||||AEλμ||⟶kγ→01. (23)

Practical choices of are specified in Eqs. (30) and (31).

After integrating by parts and by using the continuity equation

 ∇⋅J(r)+iℏ[H,ρ(r)]=0, (24)

where is the charge density, the operator can be written as

 ˜MEλμ=i√λλ+1(2λ+1)!!kλγℏc∫[H,ρ(r)]Φλμ(r)dr, (25)

when is assumed to lead to a vanishing surface term at infinity. If the partial waves and are assumed to be exact eigenstates of the Hamiltonian defined by Eq. (6), the matrix elements of the approximate electric transition multipole operators between initial and final states are given by

 ⟨ψflfmf|˜MEλμ|ψilimi⟩=−i√λλ+1(2λ+1)!!kλ−1γ∫⟨ψflfmf|ρ(r)Φλμ(r)|ψilimi⟩dr, (26)

where Eq. (1) is used. The r.h.s. of Eq. (26) defines the Siegert form of the approximate electric transition multipole operator, denoted as , which depends on the charge density and not on the current density,

 ˜ME(S)λμ=−i√λλ+1(2λ+1)!!kλ−1γ∫ρ(r)Φλμ(r)dr. (27)

The electric transition multipole operator can be written from by adding a correcting term,

 MEλμ=˜MEλμ+(MEλμ−˜MEλμ). (28)

At low photon energies, the contribution of the correcting term should be weak compared to the contribution of . By analogy with Eq. (28), the Siegert form of the electric transition multipole operator, denoted by , can be defined by Schmitt et al. (1990); Dohet-Eraly and Baye (2013)

 ME(S)λμ=˜ME(S)λμ+(MEλμ−˜MEλμ). (29)

Since at low photon energies the contribution of , which is current-independent, dominates, the current dependence is well reduced in the Siegert operator in comparison with the non-Siegert operator . The non-Siegert and Siegert operators, defined by Eqs. (18) and (29), exactly lead to the same results if consistent current and charge densities are considered and the exact eigenstates of Hamiltonian (6) are used. Consequently, the non-uniqueness of or equivalently the arbitrary nature of the choice of is not problematic since it has theoretically no influence.

Possible choices of , which avoid divergent integrals in bremsstrahlung calculations, are given by

 Φλμ(r)=i√λ+1kγ√λφλ(kγr,ϵ)Yμλ(Ω), (30)

where

 φλ(x,ϵ)=jλ(x) or xλ(2λ+1)!!e−ϵx or xλ(2λ+1)!!e−ϵ2x2 (31)

with . These choices are named respectively the Bessel, exponential, and Gaussian choices. The parameter has no meaning for the Bessel choice but is denoted for having a common notation. The Bessel choice is used in Refs. Dohet-Eraly and Baye (2013); Dohet-Eraly (2014). The exponential and Gaussian choices are used in the next section to make some formal link between the results based on the extended Siegert theorem and the ones based on the regularization techniques.

At the long-wavelength approximation, the Siegert operator is reduced to the operator defined by

 ME(S,LWA)λμ=∫ρ(r)rλYμλ(Ω)dr, (32)

where the current-dependence is fully dropped. However, in the time-independent approaches, since the continuum states have an infinite extension, applying the long-wavelength approximation is not rigorously justified in the study of bremsstrahlung.

Let me particularize the electric transition multipole operators to the potential model. To limit the complexity of the calculations, the charge and current densities for free nucleons are considered. For spinless nuclei, the charge and current densities are given by

 ρ(r)=eZ1δ(m2mρ−r)+eZ2δ(m1mρ+r) (33)

and

 J(r)=e2{Z1m1[pρ,δ(m2mρ−r)]+−Z2m2[pρ,δ(m1mρ+r)]+}. (34)

The shorthand notation is used for where and can be scalar or vector operators. For a real central potential, these current and charge densities exactly verify the continuity equation. Consequently, differences between Siegert and non-Siegert approaches can only come from numerical inaccuracies in the resolution of the radial Schrödinger equation or in the computation of the integrals. Therefore, choosing the Siegert or non-Siegert approach is a matter of convenience and should have no significant impact on the results.

Let me note that the continuity equation (24) is generally not verified if the nucleus-nucleus potential is not purely central. For instance, if the potential contains some parity-dependent terms, an extra current should be considered to verify Eq. (24). Similarly, if the spins of the colliding nuclei are considered and if the potential contains a spin-orbit term, a spin-orbit contribution should be added to the nuclear current for verifying Eq. (24). In both cases, neglecting these extra currents should have a smaller importance in the Siegert approach than in the non-Siegert approach, especially at low-photon energy. If only the convection current, defined by Eq. (34) is considered, the Siegert approach should thus be preferred. In the optical models, the interaction between nuclei is described by a so-called optical potential, i.e., a potential containing an imaginary part which simulates the effects of the open channels not explicitly described. For complex potentials, the Hamiltonian is not Hermitian and Eq. (26) is not valid. In these models, the non-Siegert electric transition multipole operators have thus to be considered.

Let me restrict again to real central potentials. Inserting the charge and current densities defined by Eqs. (33) and (34) in Eq. (18) leads to the explicit definition (35) of the non-Siegert electric transition multipole operators,

 MEλμ=ie(2λ+1)!!μMckλ+1γ(λ+1)[Z1χλμ(m2mkγ,ρ)+(−1)λZ2χλμ(m1mkγ,ρ)]⋅pρ (35)

with . These non-Siegert electric transition multipole operators are used in several models of bremsstrahlung Philpott and Halderson (1982); Langanke (1986); Langanke and Rolfs (1986b); Baye et al. (1991).

The approximate non-Siegert and Siegert operators are written in the potential model as

 ˜MEλμ=ie(2λ+1)!!2μMckλ+1γ[Z1∇ρφλ(m2mkγρ,ϵ)Yμλ(Ωρ)+(−1)λZ2∇ρφλ(m1mkγρ,ϵ)Yμλ(Ωρ),pρ]+ (36)

and

 ˜ME(S)λμ=e(2λ+1)!!kλγ[Z1φλ(m2mkγρ,ϵ)+(−1)λZ2φλ(m1mkγρ,ϵ)]Yμλ(Ωρ). (37)

The explicit expression of the Siegert electric transition multipole operator in the potential model is obtained from Eqs. (29), (35), (36), and (37). Inserting the charge density defined by Eqs. (33) in Eq. (32) or applying the long-wavelength approximation to Eq. (37) leads to the explicit definition of the long-wavelength-approximated Siegert electric transition multipole operators,

 ME(S,LWA)λμ=eZ(λ)effρλYμλ(Ωρ), (38)

where the effective charge is defined by

 Missing or unrecognized delimiter for \left (39)

The LWA Siegert multipole operators are used in Refs. Tanimura and Mosel (1985); Langanke and Rolfs (1986a); Garrido et al. (2012, 2014). Intrinsically, the operator includes an extra approximation in comparison to the operator . However, it has the advantage of having a simpler form which does not include any derivative of the radial wave function. Nevertheless, this apparent advantage can be lost with some regularization techniques, as in the IP method. This fact is highlighted in Sec. V.

The multipole matrix elements converge by using the electric transition multipole operators , , and whereas they diverge by using the operators . This property is made apparent in the next section but can already be understood. Since the wave functions are not square-integrable, the matrix elements converge only if the electric transition multipole operators tend asymptotically to zero, rapidly enough. Thus, since the operators are increasing functions of , they lead to divergent values of the multipole matrix elements . For discussing the asymptotic behavior of , the scalar product is advantageously written as

 χλμ(k,ρ)⋅pρ=−iℏλ(λ+1)ρjλ(kρ)Yμλ(Ωρ)∂∂ρ−iℏρ2[∂∂ρρjλ(kρ)][∇ΩρYμλ(Ωρ)]⋅∇Ωρ, (40)

which can be deduced from the properties of the spherical Bessel functions Abramowitz and Stegun (1965). The angular operator is implicitly defined by Varshalovich et al. (1988)

 ∇ρ=ρρ∂∂ρ+1ρ∇Ωρ. (41)

Since the spherical Bessel functions behave asymptotically as oscillating functions divided by  Abramowitz and Stegun (1965), Eq. (40) shows that the electric transition multipole operators behave asymptotically as oscillating functions divided by . The radial wave functions and thus the partial waves behave asymptotically as oscillating functions, as it can be seen from Eq. (14) or (15). By combining both these properties, the matrix elements and thus the matrix elements are proved to be convergent. It can be shown by a similar reasoning that and also lead to convergent matrix elements .

Iv Matrix elements of the electric transition multipole operators between partial waves

The reduced matrix elements of the non-Siegert multipole operators between partial waves are given by Baye and Descouvemont (1985)

 ⟨ψflf||MEλ||ψili⟩=(2λ+1)!!kλ+1γeℏμMcYlfλliC∗lfCli[Z1Iλ(m2mkγ)+(−1)λZ2Iλ(m1mkγ)], (42)

where is a shorthand notation for the following reduced matrix element Edmonds (1957)

 Ylfλli=⟨Ylf||Yλ||Yli⟩=(−1)λ(4π)−1/2(2λ+1)1/2(lfλ00|li0) (43)

and where is given by

 Iλ(k)=λ(λ+1)+li(li+1)−lf(lf+1)2(λ+1)∫∞0uflfuiliρ2[ρjλ(kρ)]′dρ+λ∫∞0uflfjλ(kρ)(uili/ρ)′dρ. (44)

The dependence on of the radial functions and is dropped to simplify the notations. For continuum to continuum transitions, the integrands in Eqs. (44) behave asymptotically as oscillating functions divided by , as anticipated in Sec. II. The integrals thus converge but slowly. The convergence rate can be improved by using the contour integration method proposed in LABEL:VF70 and largely used in bremsstrahlung models Philpott and Halderson (1982); Baye and Descouvemont (1985); Langanke (1986); Langanke and Rolfs (1986b); Liu et al. (1990a); Dohet-Eraly and Baye (2013); Dohet-Eraly (2014). The principle of this method is explained in Sec. V.3.

The reduced matrix elements of the approximate multipole operators between partial waves are given in the Siegert approach by

 ⟨ψflf||˜ME(S)λ||ψili⟩=(2λ+1)!!ekλγYlfλliC∗lfCli[Z1˜I(S)λ(m2mkγ)+(−1)λZ2˜I(S)λ(m1mkγ)], (45)

where

 ˜I(S)λ(k)=∫∞0uflfuiliφλ(kρ,ϵ)dρ (46)

and in the non-Siegert approach by

 ⟨ψflf||˜MEλμ||ψili⟩=(2λ+1)!!kλ+1γeℏ2μMcYlfλliC∗lfCli[Z1˜Iλ(m2mkγ)+(−1)λZ2˜Iλ(m1mkγ)], (47)

where

 ˜Iλ(k)=∫∞0∂φλ(kρ,ϵ)∂ρWfidρ+[li(li+1)−lf(lf+1)]∫∞0uflfuiliρ2φλ(kρ,ϵ)dρ (48)

and designates the Wronskian of and ,

 Wfi=uflfui′li−uf′lfuili. (49)

For continuum to continuum transitions, the integrals in Eqs. (46) and (48) converge slowly. Again, the contour integration method can be used for accelerating the convergence.

The reduced matrix elements of the Siegert multipole operators between partial waves are given by

 ⟨ψflf||ME(S)λ||ψili⟩=⟨ψflf||˜ME(S)λ||ψili⟩+⟨ψflf||MEλ||ψili⟩−⟨ψflf||˜MEλμ||ψili⟩. (50)

For real potentials, if the exact radial wave functions are considered, Eqs. (45) and (47) are equivalent and consequently, Eqs. (42) and (50) are equivalent, too.

The reduced matrix elements of the LWA Siegert multipole operator between partial waves are given by

 ⟨ψflf||ME(S,LWA)λ||ψili⟩=eZ(λ)effYlfλliC∗lfCli∫∞0ρλuflfuilidρ. (51)

For continuum to continuum transitions, the integrands behave asymptotically as an oscillating function times with and the integrals diverge, as anticipated in Sec. II. To obtain a finite value, the technique used in Refs. Tanimura and Mosel (1985); Langanke and Rolfs (1986a); Garrido et al. (2012) is to replace the divergent integral by a limit of convergent integrals

 ⟨ψflf||ME(S,LWA)λ||ψili⟩reg=eZ(λ)effYlfλliC∗lfClilimϵ→0Jλ(ϵ), (52)

where

 Jλ(ϵ)=∫∞0ρλuflfuilif(ϵ,ρ)dρ. (53)

The index reg is added to denote the regularized reduced matrix elements. The regularization factor is defined such that is finite for any strictly positive value of , the limit of for is finite, and

 limϵ→0f(ϵ,ρ)=1. (54)

More explicitly, the regularization factor is chosen to be an exponential in Refs. Tanimura and Mosel (1985); Langanke and Rolfs (1986a),

 f(ϵ,ρ)=e−ϵρ (55)

and a Gaussian in LABEL:GJF12,

 f(ϵ,ρ)=e−ϵ2ρ2. (56)

In the next section, it is proved that both choices of defined by Eqs. (55) and (56) are equivalent. The ways used in Refs. Tanimura and Mosel (1985); Langanke and Rolfs (1986a); Garrido et al. (2012) to evaluate the limit introduced in Eq. (52) are also explained. A new way to evaluate this limit, based on the contour integration method is also presented.

To conclude this section, let me note that the regularized reduced matrix elements defined by Eq. (52) can also be deduced from Eqs. (45) and (46) without introducing divergent integrals. Let me consider only the exponential and Gaussian choices of , for which has a meaning. At low photon energies, the reduced matrix elements of should be good approximations of the reduced matrix elements of for values of small enough. Since any value of which is strictly positive leads to convergent integrals , an arbitrary small value of can be considered, which is equivalent to take the limit for of . For , the reduced matrix element tends to , which justifies Eq. (52) without using the divergent expression (51).

V Implementation of the regularization techniques

v.1 Fixed ϵ0 method

The idea that Garrido, Jensen, and Fedorov have proposed in LABEL:GJF12 is simply to approximate the limit for by considering a small but finite value of , denoted here by ,

 limϵ→0Jλ(ϵ)≈Jλ(ϵ0). (57)

This approach is undeniably the simplest one. It is applied easily for each multipole and does not require the calculation of the derivative of the radial wave functions. Nevertheless, it appears to be unsatisfactory because, as noted in LABEL:GJF12, the value of is very sensitive to the value of . To be acceptable, the choice of has to be such that any smaller value of leads to the same results, within the desired limits of accuracy. For low photon energies, this criterion leads to very small values of . However, the more is small, the more the integral converges slowly, which makes tedious its numerical integration. In practice, to avoid a too slow convergence, the authors of LABEL:GJF12 choose a rather big value of , for which approximation (57) can be very poor, as shown in LABEL:GJF12 and in Sec. VI. Then, the bremsstrahlung cross sections are corrected by some more or less arbitrary cut and integrated to obtain the total bremsstrahlung cross section. The total bremsstrahlung cross section seems stable with respect to small variations of  Garrido et al. (2012) although the differential bremsstrahlung cross section cannot be considered as reliable. The main drawback of this method is not to allow to obtain any reliable accurate differential bremsstrahlung cross sections, due to the fact that the values of considered in practice are chosen too big. Both alternative methods presented in the next subsections do not have this inconvenience because they enable one to consider explicitly the case .

v.2 Integration by parts (IP) method

This section presents a variant of the regularization technique proposed by Tanimura and Mosel and applied by them to the operator in LABEL:TM85. This variant has the advantage to be more easily generalizable to electric multipoles of any order. Moreover, it can be applied for any potential singular at the origin, contrary to the version of Tanimura and Mosel.

The principle of the method is to derive, by some integration by parts and by using the properties of the radial wave functions, an expression of the function which is valid and continuous at . Then, the limit for is simply calculated by putting at zero in this expression.

Let me start by dividing the integral into two integrals: from zero to () and from to infinity to avoid a particular treatment of potential singular at the origin,

 Jλ(ϵ)=∫R0ρλuflfuilif(ϵ,ρ)dρ+∫∞Rρλuflfuilif(ϵ,ρ)dρ. (58)

The regularization method is based on the following relation

 Eγlimϵ→0∫∞Rguflfuilif(ϵ,ρ)dρ=limϵ→0[β∫∞Rg′Wfif(ϵ,ρ)dρ+∫∞RLguiliuflff(ϵ,ρ)dρ]+βg(R)Wfi(R), (59)

where

 L(ρ)=Vli(ρ)−Vlf(ρ) (60)

and is a function of class over with the asymptotic behavior . Roughly speaking, Eq. (59) reduces the power of the divergence of the integral at . Indeed, the integrand in the l.h.s. of Eq. (59) behaves asymptotically as an oscillating function times whereas the integrands in the r.h.s. of Eq. (59) behave asymptotically as oscillating functions times (for ) and , respectively. The case is even more favorable.

The power of the divergence of the second integral of the r.h.s of Eq. (59) can be reduced by applying recursively Eq. (59). For reducing the power of the divergence of the first integral of the r.h.s of Eq. (59), the following relation can be used

 Eγlimϵ→0∫∞RgWfif(ϵ,ρ)dρ=limϵ→0{∫∞R[gh′+2g′h−βg′′′]uflfuilif(ϵ,ρ)dρ+∫∞RLgWfif(ϵ,ρ)dρ}+∫∞R(2gU′+4g′U)uflfuilidρ+Xg(R), (61)

where

 h(ρ) = Vli(ρ)+Vlf(ρ)+2Z1Z2e2ρ−Ei−Ef, (62) Xg(ρ) = [g(h+2U)−βg′′]uiliuflf−2βgui′liuf′lf+βg′(uflfuili)′. (63)

Like Eq. (59), Eq. (61) reduces the power of the divergence of the integral at . The integrand in the l.h.s. of Eq. (59) behaves asymptotically as an oscillating function times whereas the integrands of the first two integrals in the r.h.s. of Eq. (59) behave asymptotically as oscillating functions times (for ) and , respectively. The case is more favorable, again. The last integral converges without the regularization factor since the potential has a finite range. Eqs. (59) and (61) are inspired from equations (A.5) and (A.6) in LABEL:TM85. They are proved in the Appendix.

By applying recursively Eqs. (59) and (61) to , one obtains an expression which converges at having the following form

 limϵ→