Microscopic cluster model of elastic scattering and bremsstrahlung of light nuclei

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UNIVERSITÉ LIBREDEBRUXELLES

Microscopic cluster model of elastic scattering and bremsstrahlung of light nuclei

Thèse de doctorat présentée en vue de l’obtention du grade académique de Docteur en Sciences de l’ingénieur

J´ er´ emy Dohet-Eraly

Academic year 2012 - 2013

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Remerciements

Je tiens à exprimer ici ma gratitude à l'ensemble des personnes qui ont contribué à la réalisation de cette thèse.

Je pense en premier lieu à mon co-promoteur, le professeur Daniel Baye, qui, par ses cours et sa pédagogie "à l'ancienne" (comprenez: craie et tableau vert), éveilla mon intérêt pour la mécanique quantique et plus tard pour la physique nucléaire. Il fut pour moi, durant ces quatre années de thèse, la personne ressource : celle qui fournit des idées, des pistes de solutions voire des solutions elles-mêmes ; celle qui relit mes écrits, corrige mon anglais (ce qui nécessite courage et patience) ; celle qui m'encourage à voyager et qui m'en donne l'opportunité ; etc. Je le remercie vivement pour son encadrement.

Je tiens également à remercier mon promoteur, le professeur Jean-Marc Sparenberg, pour avoir soutenu ma candidature au F.R.S.-FNRS il y a quatre ans, me permettant ainsi de com- mencer cette aventure doctorale. Je le remercie pour la conance qu'il m'a accordée dès le début.

Mes remerciements vont aussi aux autres membres du service, en particulier, au professeur Pierre Descouvemont pour l'aide et les codes qu'il a pu me fournir. Je remercie également mes collègues ou anciens collègues, professeurs, docteurs ou doctorants: Edna, Janina, Veerle, Alix, Oscar, Pierre, Simone, Thomas, Wouter,... pour l'ambiance agréable qu'ils ont fait régner dans le service durant ces quatre années.

Je remercie Soa Quaglioni de m'avoir invité un mois au Lawrence Livermore National Lab- oratory dans le cadre d'un programme de coopération académique. Je la remercie elle et les autres membres du Computational Nuclear Physics Group, en particulier Guillaume Hupin, pour l'accueil chaleureux qu'ils m'ont réservé. J'ai bénécié pour ce séjour d'une bourse du F.R.S.-FNRS.

Je remercie ma famille, mes amis et ma conjointe de fait (!) Florence pour leur soutien.

J'ai eectué cette thèse sous un mandat d'aspirant du F.R.S.-FNRS. Je remercie cet organ- isme pour son soutien nancier.

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Publications

The investigations performed during my PhD research resulted in the following publications:

• J. Dohet-Eraly and D. Baye. Phys. Rev. C 84 (2011) 014604.

• J. Dohet-Eraly, J.-M. Sparenberg, and D. Baye. J. Phys.: Conf. Ser. 321 (2011) 012045.

• J. Dohet-Eraly, D. Baye, and P. Descouvemont. J. Phys.: Conf. Ser. 436 (2013) 012030.

• J. Dohet-Eraly and D. Baye. Phys. Rev. C 88 (2013) 024602.

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Introduction

For several decades, the elastic scattering between light nuclei has been studied experimentally.

For some systems such asα+ N,α+³He, orα+α, elastic cross sections have been measured for a large number of scattering angles making possible a phase-shift analysis of the cross sections.

Building nuclear models that are able to reproduce these phase shifts and to predict the phase shifts of other collisions, which are not measured, is one of the most important challenges of nuclear physics. Since they include a small number of nucleons, the elastic collisions between light nuclei are the collisions that are the most suitable to be described from fundamental approaches verifying the rst principles.

For a long time the elastic scattering between light nuclei has been studied by microscopic cluster models, namely the Resonating-Group Method (RGM) [1,2] or the equivalent Generator- Coordinate Method (GCM) [3]. The microscopic models take account of the internal structure of the nuclei and the Pauli principle over the full colliding system, two aspects that are essential in the theoretical explanation of the properties of the collisions between light nuclei [1]. The term cluster means that the microscopic model is based on an assumed cluster structure or, in other words, on the occurrence of correlated subsystems in the fully antisymmetric wave function of the colliding system [1, 2]. The validity of the cluster approximation depends strongly on the nucleon-nucleon (NN) interaction that is considered.

In the RGM-GCM, the system made up of the colliding nuclei is described by a superposition of antisymmetrized states of two clusters separated by a variable distance. Each cluster is described by a Slater determinant of harmonic-oscillator states. After elimination of a Gaussian center-of-mass (c.m.) factor, the wave function of the system becomes invariant under translation (if the oscillator parameters of the two clusters are equal) and can be projected on angular momentum and parity [4]. While it has the required symmetries, the wave function does not have the correct asymptotic behavior of a scattering wave function. This problem is solved, in this work, by the Microscopic R-matrix Method (MRM) [5, 6], which enforces the appropriate asymptotic behavior.

With the GCM, a large number of elastic collisions are described with good precision [2,7] by using two-body phenomenological interactions. In these forces, the binding eect of tensor terms is partially simulated in the central terms. Phase shifts in excellent agreement with experimental data can be obtained by tting one or two parameters for each collision or even for each partial wave. The predictive power of the model, which is a strong point of microscopic approaches, is thus limited by the fact that these parameters may dier in each case.

To avoid a specic t of the NN interaction, many eorts have been devoted during the last decade to describe the nuclear collisions in ab initio approaches [816]. In the present framework, the expression ab initio means that the approach is based on an interaction between nucleons that is not adjusted to the studied system and that the resolution of the Schrödinger

1

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equation is as precise as possible. Since the cluster assumption is not made in the ab initio approaches, theses approaches require much more computational eorts. The ab initio models are based on so-called realistic NN potentials such as the Argonne [17, 18], CD-Bonn [19], or N^1,2,3LO [20, 21] interactions. These two-body potentials are tted precisely to the properties of the two-nucleon systems. The agreement with the experimental data is thus very good for the two-nucleon systems but much less good for heavier systems. There are dierences of one or a few MeV between experimental and calculated binding energies for nuclei made up of three or more nucleons [22]. For the α+neutron elastic scattering, the agreement with experiment is qualitatively good [8, 23] but less good than with eective potentials since no parameter can be tted.

To improve the experimental agreement, three-body interactions are added [22,24,25]. They make calculations heavier and thus limit present scattering studies to relatively simple systems such as α+ n[8,26]. These three-body interactions signicantly reduce the gap between experimental and calculated binding energies for a large set of nuclei [22]. However, the situation is less favorable for elastic scattering [8,26]. Indeed, it is shown in Ref. [8] that two dierent three-body interactions, which give the same binding energies in good agreement with experimental data, lead to signicantly dierent elastic phase shifts for the α+ ncollision. This result shows that nuclear interactions cannot be determined only from binding energies but also require scattering data of many-body systems.

Because of their important repulsive core and their tensor component, the realistic interactions are unsuitable for the RGM-GCM cluster approach, where the clusters are described in a independent-particle model based on a central NN potential. A goal of this work is to improve the cluster model by using better-founded two-body interactions where the tensor force is included rather than partly simulated by the central term. To some extent, it is possible to compensate the eects of three-body interactions by a two-body interaction. This property has been studied in a general way in Ref. [27].

To keep a cluster approach simple and thus to maintain acceptable computation times, one can either adapt the cluster wave function or adapt the interaction. The methods of the Jastrow kind [28] follow the rst way while the methods based on a Lee-Suzuki transform [29] follow the second one. Villars proposed [30] an intermediate method: the correlation is inserted in the wave function by a unitary operator or in an equivalent way in the Hamiltonian by applying this unitary transformation. The two points of view can be considered simultaneously. This idea was continued and developed by the GSI-Darmstadt group and led to the Unitary Correlation Operator Method (UCOM) [3133]. This method was used with success in nuclear structure calculations based on a realistic interaction [34] and more recently for collisions [16].

The UCOM is able to create an innity of phase-equivalent two-body interactions by applying a unitary transformation to a realistic two-body interaction. Since the correlated potentials are equivalent to each other for the NN phase shifts and the deuteron binding energy, the unitary transform can be adjusted to some heavier system. In particular, the transform can be adjusted to a collision. The binding energies of the colliding nuclei may not be well reproduced but their values are not essential to obtain accurate phase shifts in the cluster model [7].

By taking account of these ideas, a UCOM NN interaction from the Argonne potentials AV18 [18] is built by adjusting the correlators to the α+α phase shifts studied microscopically with the GCM coupled with the MRM [35, 36]. The validity of the correlated interaction is veried by applying the same model with the same interaction to theα+ n,α+ p, andα+³He collisions and by comparing the obtained phase shifts with experimental data [36].

Besides the study of light-nuclei elastic scattering, this work focuses on another type of nuclear collisions, a channel which is always open, too: radiative transitions within the continuum, usually referred to as nucleus-nucleus bremsstrahlung. This nuclear process can be seen as a

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3 perturbation of the elastic scattering. At the rst order of perturbation, the bremsstrahlung cross sections can be evaluated from the scattering wave functions.

Initially, the NN bremsstrahlung was studied to discriminate among various NN potentials leading to the same elastic phase shifts but having dierent o-shell behavior [37,38]. However, these hopes have come to nothing because dierent NN potentials led to similar bremsstrahlung cross sections in agreement with the experimental data [39]. Nevertheless, later, the theoretical study of nuclear bremsstrahlung including heavier nuclei showed that the bremsstrahlung cross sections could become very sensitive to the model for some congurations and some energies [40], which renewed the interest for bremsstrahlung. Besides probing the nuclear models, the nucleus-nucleus bremsstrahlung can be interesting for itself with, for instance, the recent per- spective of using the t(d, nγ)α bremsstrahlung to diagnose plasmas in fusion experiments [41].

Bremsstrahlung photons also provide useful information in other nuclear processes such as proton decay,αdecay, ssion, ... (see references in Refs. [42,43]). These processes can still be considered as continuum to continuum transitions since the initial decaying state is not square-integrable.

In this work, a new microscopic cluster bremsstrahlung model is proposed. While in previous models of nucleus-nucleus bremsstrahlung [40, 4450], the contributions of the meson-exchange currents were fully neglected, they are partially included in this new model by deriving the photon-emission operator from the nuclear density rather than from the nuclear current. The eects of using such a photon-emission operator in a microscopic cluster approach are studied for the α+α [51] andα+ Nbremsstrahlung.

Chap. 2 presents the theoretical aspects of the microscopic study of the nuclear collisions in a quite general way. Chap. 3 particularizes for a microscopic cluster model the expressions given in Chap. 2. In Chap. 4, the dierent types of potentials describing the interaction between nucleons are discussed. The NN potentials that are considered in this work are specied. The Unitary Correlation Operator Method is presented in Chap. 5. In Chap. 6, the microscopic cluster model is applied to the α+ n,α+ p,α+³He, andα+α elastic scattering for an eective NN potential and for a realistic NN potential modied by UCOM. Theoretical elastic phase shifts are compared with each other and with the experimental one. In Chap. 7, the microscopic cluster approach is extended to the study of nucleus-nucleus bremsstrahlung before being applied, in Chap. 8, to theα+α andα+ Nbremsstrahlung. Finally, Chap. 9 provides some concluding remarks.

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Chapter 2

Microscopic collision theory

In this chapter, the microscopic collision theory is presented with some generality and the microscopic cluster models based on this theory, the Resonating-Group Method and the Generator- Coordinate Method, are introduced. More details about the microscopic cluster model that is used in this work are given in the next chapter.

2.1 Microscopic Hamiltonian

In nuclear physics, the microscopic models are models that take each nucleon of the studied system into account. The protons and neutrons are described in the isospin formalism as dierent projection states of the same particle called nucleon. The dierence between their masses is neglected. Their common mass is denoted bymN. The antisymmetrization between the nucleons is treated exactly to satisfy the Pauli principle.

At non-relativistic energies, the nuclear model has to be Galilean-invariant. To achieve this invariance, the internal operators, i.e. the operators in the center-of-mass frame, are systemati- cally used.

The microscopic description of anA-nucleon system is based, at non-relativistic energies, on the Schrödinger equation

HΨ =ETΨ, (2.1)

where H is the internal Hamiltonian, Ψ the internal wave function, and E_T is the total energy of the system in the c.m. frame. The internal Hamiltonian H can be written as

H=T−T_c.m.+V, (2.2)

where T and Tc.m. are the total and c.m. kinetic energies and V is a potential describing the interaction between nucleons. The total kinetic energy is a one-body operator dened by

T =

A

X

i=1

t_i =

A

X

i=1

p²_i

2m_N, (2.3)

where ti is the kinetic energy of nucleon i and pi is its momentum. The c.m. kinetic energy is given by

T_c.m.= P_c.m.²

2Am_N, (2.4)

where Pc.m. is the c.m. momentum. In a general way, the potential V describing the nuclear interaction can be written as

V =

A

X

i>j=1

v_ij+

A

X

i>j>k=1

v_ijk+· · · , (2.5)

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wherevij is a two-body interaction between nucleonsiandjandvijk is a three-body interaction between nucleonsi,j, andk. The ellipsis in Eq. (2.5) indicates that the potential is not limited to three-body terms but can possibly include four- and more-body terms. In a general way, the presence of N-body forces (3 ≤N ≤ A) in the nuclear interaction means that the interaction between N −1 given nucleons is modied by the presence of an extra nucleon. Assuming that the nuclear forces are carried out by the mesons, this modication can be explained by the multiple exchanges of mesons in which N nucleons take part [52]. Since the three- and more- body forces come from the exchange of several mesons, their range is shorter than the range of the two-body forces. Consequently, the eects of the three- or more-body forces are expected to be much less important than the eects of the two-body forces. For this reason, the three- or more-body forces are often neglected or simulated by eective two-body forces in the many-body calculations. This point is discussed with more details in Chap. 4, where the nuclear interactions that are considered in this work are specied. For now, it is only assumed that the potentialV is invariant under translation, reection, rotation, and time reversal. The internal Hamiltonian dened by Eq. (2.2) is thus invariant under the same transformations.

2.2 Collision channels

To describe a collision wave function, it is useful to dene the notion of collision channel. A collision channel is an asymptotic situation for which the A nucleons are clustered in several distinct nuclei. Each such clustering is called a partition. For a given partition α, a given nucleus k is characterized by its number of nucleons A^α_k and its charge Z_k^αe. The number of nucleons of each nucleus in partition αis linked with the total number of nucleons of the system by

A=

Nα

X

k=1

A^α_k, (2.6)

where Nα is the number of clusters in partition α. In a similar way, the charge of each nucleus of partition α is linked with the total charge of the systemZe by

Z =

Nα

X

k=1

Z_k^α. (2.7)

To dene a channelc, the internal states of the nuclei have to be specied in addition to the partition. By convention, it is assumed that a channel denoted by c corresponds to a partition denoted byα, a channel c⁰ to a partition α⁰, ...

For a given channelc, the internal state of a given nucleus k is characterized by its spinI_k^c, its spin projectionν_k^c, its parityπ^c_k, and its energyE_k^c. Its internal wave function is a solution of the Schrödinger equation

H_k^αΦ^c_I^c

kν_k^c =E_k^cΦ^c_I^c

kν_k^c, (2.8)

where the internal cluster Hamiltonian H_k^α has the same form as the Hamiltonian (2.2) with A replaced by A^α_k. The superscript c in Φ^c_Ic

kν_k^c represents the parity π^c_k and the energy E_k^c of the cluster. The cluster wave function Φ^c_Ic

kν_k^c is antisymmetrized with respect to exchange of two nucleons to satisfy the Pauli principle. From the cluster energies, the threshold energy of a channel ccan be dened by

E_t^c=

Nα

X

k=1

E_k^c. (2.9)

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2.3. SYSTEM OF COORDINATES 7 The energy from the threshold is denoted by E^c and is given by

Ec=ET −E_t^c. (2.10)

The sign of Ec determines if the channelc is open (Ec>0) or closed (Ec<0).

To limit the complexity of the calculations, our approach is restricted to binary channels (Nα = 2). In Sec. 2.10, a brief discussion about the approximate treatment of ternary-or-more- particle channels with the same formalism is done.

2.3 System of coordinates

AnA-nucleon system can be described with the set of theAindividual coordinates of the nucleons {r1, . . . ,r_A}, (2.11) wherer_iis the coordinate of nucleoniwithi= 1, . . . , A. However, since the Hamiltonian dened by Eq. (2.2) is invariant by translation, it is useful to dene a system of coordinates from A−1 internal coordinates, which are invariant by translation, and the total center-of-mass coordinate.

The total c.m. coordinate is dened by

Rc.m.= 1 A

A

X

i=1

ri. (2.12)

The choice of the internal coordinates depends on the considered partition. The center of mass coordinates of the clusters for a binary partition αare dened by

R^α(1)_c.m. = 1 A^α₁

A^α₁

X

i=1

r_i, (2.13)

R^α(2)_c.m. = 1 A^α₂

A

X

j=A^α₁+1

rj. (2.14)

They are linked with the total c.m. coordinate by R_c.m.= A^α₁

A R^α(1)_c.m.+A^α₂

A R^α(2)_c.m.. (2.15)

Although R^α(1)_c.m. and R^α(2)_c.m. depend on the considered partition, Eq. (2.12) clearly shows that Rc.m. is independent onα.

A set ofA−2linearly independent internal coordinates of the nuclei can be dened from the cluster c.m. coordinates by

ξ^α(1)_i = ri−R_c.m.^α(1) fori= 1, . . . , A^α₁ −1, (2.16) ξ^α(2)_j = rj−R^α(2)_c.m. forj =A^α₁ + 1, . . . , A−1. (2.17) Other choices of internal coordinates, such as the Jacobi coordinates, are obviously possible but they are less convenient in the approach presented here.

The relative coordinate between the cluster centers of mass, which is dened by

ρα=R^α(1)_c.m.−R^c(2)_c.m., (2.18) is an internal coordinate linearly independent ofξ^α(1) andξ^α(2). A convenient coordinate system to describe a binary partition α is thus given by

n

Rc.m.,ξ^α(1)₁ , . . . ,ξ^α(1)_Aα 1−1,ξ_A^α(2)α

1+1, . . . ,ξ_A−1^α(2),ρα

o

. (2.19)

The total c.m. coordinate is omitted in this set if one is only concerned by the internal coordinates.

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2.4 Antisymmetrization

To take the Pauli principle into account, the wave function has to be antisymmetrized with respect to exchange of two nucleons. To this end, one denes the antisymmetrization operator by

A= 1 A!

A!

X

p=1

(−1)ⁱ^pP_p, (2.20)

where the operator P_p achieves the permutation p which has the signature i_p and the parity (−1)ⁱ^p. The sum is over theA!possible permutations of the A nucleons. The operator Ais a Hermitian projector since

A² =A=A^†, (2.21)

where the dagger symbol is used to designate the Hermitian conjugate of A.

When the antisymmetrization between all nucleons is performed, there are

n_α= A!

A^α₁!A^α₂!(1 +δA^α₁A^α₂) (2.22) equivalent denitions of ρ_α. To take the possible symmetry of the α partition into account, the projectorPα is introduced [6]

Pα= 1 + (−1)^A^α¹δ_A^α₁_A^α₂P_α 1 +δ_A^α₁_A^α₂ =

(1 if A^α₁ 6=A^α₂,

1+(−1)^Aα¹Pα

2 if A^α₁ =A^α₂, (2.23) where P_α exchanges the A^α₁ nucleons of the rst nucleus with the A^α₂ nucleons of the second nucleus if A^α₁ =A^α₂. The operatorsAand Pα verify the following property [6]

PαA=APα =A. (2.24)

Asymptotically, for the two-cluster wave functions, the antisymmetrization operator can be sim- plied as

AΦ^c_I^c

1ν^c₁Φ^c_I^c

2ν₂^cg(ρα)_ρ−→

α→∞n⁻¹_α PαΦ^c_I^c

1ν^c₁Φ^c_I^c

2ν₂^cg(ρα), (2.25) whereg represents the relative wave function.

2.5 Channel wave function

The channel spin function is obtained by coupling the spins of the clusters Φ^c_Iν(ξ^α(1),ξ^α(2)) =X

ν1ν2

(I1I2ν1ν2|Iν)Φ^c_I₁_ν₁(ξ^α(1))Φ^c_I₂_ν₂(ξ^α(2)), (2.26) whereν =ν1+ν2,(I1I2ν1ν2|Iν)is a Clebsch-Gordan coecient, andI is called the channel spin.

To simplify the notations, the superscript c is omitted for the spins and for their projections.

Let me note that the superscript c has a variable meaning depending on the context. It can represent either the considered channel or a set of quantum numbers characterizing this channel.

For instance, in the l.h.s. of Eq. (2.26), cis written for E₁^cI₁π₁^cE₂^cI₂π^c₂ while, in the r.h.s. of this equation,cis written forE₁^cπ₁^c orE₂^cπ^c₂. When an ambiguity could appear in the meaning of the superscript c, the quantum numbers represented by care explicitly given.

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2.6. ASYMPTOTIC BEHAVIOR OF THE HAMILTONIAN 9 A channel wave function is obtained by coupling the channel spinI with the relative orbital momentum land applyingPα [6]

ϕ^{J M π}_β (ξ^α(1),ξ^α(2),Ωρα) =

1 +δ_A^α₁_A^α₂ 1 +δ^c₁₂

1/2

X

mν

(lImν|JM)PαY^ml (Ωρα)Φ^c_Iν(ξ^α(1),ξ^α(2)), (2.27) where β stands for clI, ρα = (ρα,Ωρα), Y^m_l =i^lY_l^m, and δ₁₂^c is equal to unity if the channel is symmetric, i.e. if both nuclei are identical and in the same state (Φ^c_I

1 = Φ^c_I

2), and equal to zero otherwise. The convention chosen to dene the spherical harmonicsY_l^m is given in Appendix A.

Spherical harmonicsY^m_l are used because they have a more convenient transformation under time reversal than spherical harmonicsY_l^m [53], as discussed in Sec. 2.8. Unless otherwise stated, the spherical coordinates of any vector aare denoted by(a,Ω_a)whereais the radial coordinate and Ωarepresents the colatitudeθaand the azimuthϕa. Coherently with the previous simplications, the indexcis omitted for the relative orbital momentuml. The total parity of the channel wave function ϕ^{J M π}_β is linked with the individual parities of nuclei 1 and2 by

π=π^c₁π₂^c(−1)^l. (2.28)

Thanks to the projectorPα, the channel wave function is symmetric (resp. antisymmetric) with respect to exchange of nuclei 1 and 2 if these are bosons (resp. fermions) with the same number of nucleons. When the channel is symmetric, it can be veried that the channel wave function is identically zero if

(−1)^I 6= (−1)^l. (2.29)

The channel wave functions are orthonormal [6]

hϕ^{J M π}_β |ϕ_β^J⁰0^M⁰^π⁰i=δ_{J J}⁰δ_{M M}⁰δ_ππ⁰δ_ββ⁰ if α=α⁰. (2.30) If the partitions α andα⁰ are dierent, Eq. (2.30) is only valid asymptotically.

2.6 Asymptotic behavior of the Hamiltonian

The microscopic Hamiltonian dened by Eq. (2.2) can be written as

H =H₁^α+H₂^α+Tρα +V₁₂^α, (2.31) where Tρα is the relative kinetic energy between the two clusters of partition α and V₁₂^α is the intercluster potential. These operators are given explicitly by

T_ρ_α = p²_ρ_α

2µ_α, (2.32)

V₁₂^α =

A^α₁

X

i=1 A

X

j=A^α₁+1

vij +

A^α₁

X

i=1 A

X

j=A^α₁+1 A

X

k=j+1

v_ijk+

A^α₁

X

i=1 A^α₁

X

j=i+1 A

X

k=A^α₁+1

v_ijk+. . . , (2.33) wherepρα is the relative momentum andµα is the reduced mass

µ_α=A^α₁A^α₂m_N/A. (2.34)

At large distances between clusters, the nuclear forces between nucleons included in dierent clusters are weak compared to the Coulomb force and can thus be neglected. Moreover, when the distance between clusters is large, the Coulomb force between two nucleons from dierent

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clusters can be assumed in a good approximation to interact between the cluster centers of mass.

For largeρα values, the microscopic Hamiltonian is thus given by H_ρ−→

α→∞H₁^α+H₂^α+H_ρ_α (2.35)

with

Hρα =Tρα +Z₁^αZ₂^αe²

4π₀ρ_α . (2.36)

Asymptotically, H is hence divided into three terms: one depending on the internal coordinates of the rst nucleus, one depending on the internal coordinates of the second nucleus and one depending on the relative coordinateρ_α between nuclei. Asymptotically, the wave function can thus be written for a given channel c from the eigenfunctions of H₁^α with energy E₁^c, H₂^α with energyE₂^c, andHρα with energyEc.

2.7 Coulomb wave functions

Let me focus on the solutions of the Schrödinger equation that is associated with the Hamiltonian H_ρ_α at energyE_c,

HραψC(ρα) =EcψC(ρα). (2.37) The capital letterCrecalls that the potential inH_ρ_αis purely Coulombic. Assuming that channel c is open (Ec>0), the wavenumber kcand the Sommerfeld parameter ηcare dened by

kc =

√2µαEc

~ , (2.38)

ηc = Z₁^αZ₂^αe²µα

4π0 ~²kc

. (2.39)

Since the Coulomb potential is central, the wave function can be factorized in spherical coordinates as

ψC(ρα) =Y_l^m(Ωρα)u^C_l (ρα)ρ⁻¹_α , (2.40) whereu^C_l (ρ_α) is a solution of the radial Schrödinger equation

d²

dρ²_α − l(l+ 1)

ρ²_α − 2kcηc

ρα

+k_c²

u^C_l (ρα) = 0. (2.41) The solutions of Eq. (2.41) are linear combinations of the regular and irregular Coulomb functions F_l(η_c, k_cρ_α)andG_l(η_c, k_cρ_α)[54]. At the origin, the regular function vanishes while the irregular function is unbound. The solutions of Eq. (2.41) can also be written as linear combinations of the incoming and outgoing Coulomb functions Il and Ol dened by

I_l(ηc, kcρα) = G_l(ηc, kcρα)−iF_l(ηc, kcρα), (2.42) Ol(ηc, kcρα) = Gl(ηc, kcρα) +iFl(ηc, kcρα). (2.43) For real arguments, I_l and O_l are complex conjugates. The incoming and outgoing Coulomb functions are called like this because they behave asymptotically like incoming and outgoing waves [54]

I_l(ηc, kcρα) _ρ−→

α→∞ e^−i(k^c^ρ^α^{−lπ/2−η}^c^{ln 2k}^c^ρ^α^+σ^l⁾, (2.44) O_l(η_c, k_cρ_α) _ρ−→

α→∞ e^i(k^c^ρ^α^{−lπ/2−η}^c^{ln 2k}^c^ρ^α^+σ^l⁾, (2.45)

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2.8. COLLISION WAVE FUNCTIONS 11 whereσl is the Coulomb phase shift.

Let us note that for closed channels (Ec < 0), the physical solutions of Eq. (2.41) tend exponentially to zero.

To describe collisions, it is also useful to introduce the scattering Coulomb functionsψ_C⁺and ψ_C⁻, which are dened by [55]

ψ_C^±(k_c,ρ_α, η_c) =e^−πη^c^/2Γ(1±iη_c)e^ik^c^·ρ^α₁F₁(∓iη_c,1,±ik_cρ_α−ik_c·ρ_α), (2.46) where kc is the wave vector, Γ is the Euler function, and 1F1 is the conuent hypergeometric function. The meaning of indices + and − is explained in the next section. The functions ψ_C⁺ and ψ⁻_C are linked with each other by

ψ⁻_C(kc,ρα, ηc) =K0ψ⁺_C(−kc,ρα, ηc), (2.47) whereK0 is the antiunitary complex conjugation operator.

The functions ψ⁺_C and ψ⁻_C are bounded solutions of Eq. (2.37), which have the asymptotic behavior of an outgoing scattering state [56]

ψ_C⁺(k_c,ρ_α, η_c) _ρ−→

α→∞

e^ik^c^·ρ^α^+iη^c^ln(k^c^ρ^α^−k^c^·ρ^α⁾+f_c^C(Ω_ρ_α)e^ik^c^ρ^α^−iη^c^{ln 2k}^c^ρ^α ρα

, (2.48) ψ_C⁻(k_c,ρ_α, η_c) _ρ−→

α→∞

e^ik^c^·ρ^α^−iη^c^ln(k^c^ρ^α^+k^c^·ρ^α⁾+f_c^C∗(Ω_ρ_α)e^−ik^c^ρ^α^+iη^c^{ln 2k}^c^ρ^α ρ_α

, (2.49) where the coecients f_c^C are the Coulomb scattering amplitudes.

Sinceψ_C⁺and ψ_C⁻are bounded solutions of Eq. (2.37), they can be expanded in partial waves using the regular Coulomb functions as [56]

ψ_C^±(kc,ρα, ηc) = 4π k_cρ_α

X

lm

i^le^±iσ^lY_l^m∗(Ω_k_c)Y_l^m(Ωρα)F_l(ηc, kcρα). (2.50)

2.8 Collision wave functions

Let us consider a collision between two nuclei with initial spins I1 and I2 and spin projections ν1 and ν2 at energy E_T in the c.m. frame. The entrance channel of the collision is denoted by c and corresponds to the α partition, in agreement with the adopted convention. The colliding system is described by a collision wave function Ψ^ν_c¹^ν²⁺ that is a solution of the Schrödinger equation (2.1),

HΨ^ν_c¹^ν²⁺=E_TΨ^ν_c¹^ν²⁺. (2.51) Asymptotically, the collision wave function can be divided into an incident plane wave dened by the properties of the nuclear system before the collision and outgoing spherical waves including the whole information about the collision process. In agreement with Eq. (2.35), the asymptotic behavior of the collision wave function is given by

Ψ^ν_c¹^ν²⁺_ρ→∞−→

p1 +δA^α₁A^α₂

√n_αv_c Pαψ_C⁺(kc,ρα, ηc)Φ^c_I₁_ν₁(ξ^α(1))Φ^c_I₂_ν₂(ξ^α(2))

+ X

c⁰ν₁⁰ν₂⁰

q1 +δ_Aα0 1 A^α₂⁰

√n_α⁰v_c⁰ Pα⁰

e^ik^c⁰^ρ^α⁰^−iη^c⁰^{ln 2k}^c⁰^ρ^α⁰

ρ_α⁰ f_c^cν0ν¹₁⁰^νν²₂⁰(Ωρ_α0)Φ^c_I⁰⁰

1ν₁⁰(ξ^α⁰⁽¹⁾)Φ^c_I⁰⁰

2ν₂⁰(ξ^α⁰⁽²⁾).

(2.52)

wherevc is the relative velocity associated with channelc

vc=~kc/µα. (2.53)

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The velocity vc⁰ is dened in a similar way by replacing cby c⁰ and α by α⁰ in Eq. (2.53). The sum in Eq. (2.52) is over the open channels (E_c⁰ >0) since the closed channels do not contribute asymptotically to the collision wave function. The collision wave function is normalized to a unit ux, which leads to more compact expressions for the bremsstrahlung cross sections. The asymptotic expansion (2.52) is dened for ρ → ∞. The absence of an index to ρ, designating the partition, means that dierent asymptotic regions associated with dierent partitions are considered. Indeed, the rst term of the r.h.s. of Eq. (2.52) corresponds to a part of the asymptotic behavior of the channel wave function for ρα → ∞and the other terms to ρ_α⁰ → ∞.

The coecients f_c^cν0ν¹₁⁰^νν²₂⁰ in Eq. (2.52) are called the scattering amplitudes. They enable one to determine the elastic, inelastic, and reaction cross sections [57].

The collision wave function Ψ^ν_c¹^ν²⁺ represents a state that was specied by the wave vector kc and the quantum numbers cν1ν2 in the remote past [55]. Its evolution is determined by the microscopic Hamiltonian H. In the study of bremsstrahlung, a state that will be specied by the wave vector k_c and the quantum numbers cν₁ν₂ in the distant future has to be introduced, too. It is denoted byΨ^ν_c¹^ν²⁻. It can be dened fromΨ^ν_c¹^ν²⁺ by the time-reversal transformation.

Following the convention chosen in [55], let us dene the time-reversal operator by

K =K0e^iπS^y^/~, (2.54)

where Sy is the y-component of the total spin operator S. With this convention, the spherical harmonics Y^m_l are transformed under time reversal as [53]

KY^ml =K₀Yl^m = (−1)^l+mY^−m_l . (2.55) Let|SM_Si be a spin eigenstate ofS² and S_z,

S²|SMSi = ~²S(S+ 1)|SMSi, (2.56)

S_z|SM_Si = ~M_S|SM_Si. (2.57)

Applying the time-reversal operator, |SMSi is modied following

K|SMSi = e^iπS^y^/^~|SMSi (2.58)

= X

M_S⁰

d^J_M⁰

SMS(π)|SM_S⁰i (2.59)

= (−1)^S+M^S|S−MSi, (2.60) whered^J_M0

SMS is the reduced rotation matrix (see Appendix A). Thanks to the phase convention of Y^m_l , the orbital and spin angular momenta are transformed in the same way under time reversal [53]. Moreover, the couplings of the spherical harmonics Y^m_l and spin states are also transformed in the same way under time reversal. The same phase convention is assumed in the denition of the internal cluster states so that they verify

KΦ^c_I_k_ν_k = (−1)^I^k^+ν^kΦ^c_I_k₋_ν_k (2.61) for k= 1,2.

The wave functionΨ^ν_c¹^ν²⁻ is dened by [55]

Ψ^ν_c¹^ν²⁻(kc) = (−1)^I¹^+I²^−ν¹^−ν²KΨ_c⁻^ν¹⁻^ν²⁺(−kc), (2.62)

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2.9. PARTIAL WAVES 13 where the kc dependence is explicit. Its asymptotic behavior is given from Eqs. (2.62), (2.52), (2.54), (2.47), and (2.61) by

Ψ^ν_c¹^ν²⁻_ρ→∞−→

p1 +δA^α₁A^α₂

√n_αv_c Pαψ⁻_C(kc,ρα, ηc)Φ^c_I₁_ν₁(ξ^α(1))Φ^c_I₂_ν₂(ξ^α(2))

+ X

c⁰ν₁⁰ν⁰₂

(−1)^I¹^+I²^+I¹⁰^+I²⁰^−ν−ν⁰

q1 +δ_Aα0 1 A^α₂⁰

√n_α⁰v_c⁰ Pα⁰

e^−ik^c⁰^ρ^α⁰^+iη^c⁰^{ln 2k}^c⁰^ρ^α⁰ ρ_α⁰

×f_c^c−ν0−ν¹₁⁰^−ν−ν²₂⁰^∗(Ωρ_α0)Φ^c_I⁰⁰

1ν₁⁰(ξ^α⁰⁽¹⁾)Φ^c_I⁰⁰

2ν₂⁰(ξ^α⁰⁽²⁾),

(2.63)

whereν⁰ =ν₁⁰ +ν₂⁰.

2.9 Partial waves

Since Hamiltonian H is invariant under rotation and reection, the total angular momentumJ, its projectionM, and the total parityπ are good quantum numbers. The collision wave function can thus be expanded as a linear combination of the partial waves of the total wave function.

These partial waves are dened by

Ψ^{J M π}_β =X

β⁰

√n_α⁰Aϕ^{J M π}_β0 g_β^{J π}0β(ρ_α⁰). (2.64)

The subscript β recalls that the partial waves Ψ^{J M π}_β are designed for the entrance channel c associated with the momenta l and I. The partial waves are approximate eigenstates of the microscopic Hamiltonian

HΨ^{J M π}_β =ETΨ^{J M π}_β . (2.65)

To be exact solutions of the Schrödinger equation (2.65), the partial waves should include the ternary- and more-particle channels. In a variational point of view, the Schrödinger equation (2.65) can be written as

hδΨ^{J M π}_β |H−E_T|Ψ^{J M π}_β i= 0, (2.66) whereδΨ^{J M π}_β is an arbitrary variation ofΨ^{J M π}_β . Assuming that the variation concerns only the relative wave function, Eq. (2.66) can be written as [12]

X

β⁰

Z ∞ 0

r⁰²[H^{J π}_β⁰⁰_β⁰(r⁰⁰, r⁰)−E_TN_β^{J π}⁰⁰_β⁰(r⁰⁰, r⁰)]g^{J π}_β0β(r⁰)dr⁰ = 0, (2.67)

where the energy kernel H^{J π}_β⁰⁰_β⁰(r⁰⁰, r⁰) and the norm kernelN^{J π}_β⁰⁰_β⁰(r⁰⁰, r⁰)are dened by H^{J π}_β⁰⁰_β⁰(r⁰⁰, r⁰) = h√

nα⁰⁰ϕ^{J M π}_β⁰⁰ δ(ρα⁰⁰−r⁰⁰)

ρ_α⁰⁰r⁰⁰ |H|A√

nα⁰ϕ^{J M π}_β⁰ δ(ρα⁰−r⁰)

ρ_α⁰r⁰ i, (2.68) N^{J π}β⁰⁰β⁰(r⁰⁰, r⁰) = h√

n_α⁰⁰ϕ^{J M π}_β⁰⁰ δ(ρα⁰⁰−r⁰⁰) ρ_α⁰⁰r⁰⁰ |A√

n_α⁰ϕ^{J M π}_β⁰ δ(ρα⁰−r⁰)

ρ_α⁰r⁰ i. (2.69) To dene entirely the functionsg_β^{J π}0β(ρ_α⁰), their normalization has to be xed. This can be done by specifying the asymptotic behavior of g^{J π}_β0β(ρ_α⁰). For open channels, one has

g_β^{J π}⁰_β(ρα⁰)_ρ−→

α0→∞

x^{J π}_β

√v_c⁰kcρ_α⁰[δββ⁰Il(ηc, kcρα)−U_β^{J π}⁰_βOl⁰(ηc⁰, kc⁰ρα⁰)], (2.70)

Microscopic cluster model of elastic scattering and bremsstrahlung of light nuclei