Theory and Applications of Nuclear DirectReactions

Horst LenskeInstitut fur Theoretische Physik, Justus-Liebig-Universitat Giessen

D-35392 Giessen, Germany∗

March 4, 2018

Contents

1 Preface and Lecture 1: Probing Physical Systems by Scattering Experi-ments 3

2 Lecture 2: Kinematics and Cross Sections 72.1 Kinematics of Two-Body Reactions . . . . . . . . . . . . . . . . . . . . . . 72.2 The Concept of a Cross Section . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Lecture 3: Formal Scattering Theory in a Nutshell 103.1 Basics of Reaction Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Green’s Functions in Scattering Theory . . . . . . . . . . . . . . . . . . . . 103.3 Integral Equation for the Scattering Amplitude . . . . . . . . . . . . . . . 123.4 Partial Wave Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Applications of the T-Matrix Formalism . . . . . . . . . . . . . . . . . . . 15

3.5.1 Example 1: Pion-Nucleon Scattering . . . . . . . . . . . . . . . . . 153.5.2 Example 2: Elastic Scattering of Nucleons . . . . . . . . . . . . . . 16

4 Lecture 4: The Optical Model for Elastic Scattering 194.1 Elimination of Coupled Channels Effects in Feshbach Theory . . . . . . . . 194.2 The Optical Potential for Elastic Scattering . . . . . . . . . . . . . . . . . 20

5 Lecture 5: Perturbative Approach to Nuclear Scattering: DistortedWave Born Approximation (DWBA) and Related Methods 265.1 The Concept of Direct Nuclear Reactions . . . . . . . . . . . . . . . . . . . 26

∗Electronic address: horst.lenske@physik.uni-giessen.de

1

5.2 Treatment of Channel Coupling . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Reaction Theory with Distorted Waves . . . . . . . . . . . . . . . . . . . . 275.4 The DWBA Reaction Amplitude and Cross Section . . . . . . . . . . . . . 28

6 Lecture 6: Theory of Transfer Reactions 306.1 Probing Nuclear Single Particle Dynamics by Transfer Reactions . . . . . . 306.2 The Deuteron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3 The (d,p) Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 Lecture 7: (d,p) Reactions as a Spectroscopic Tool 367.1 Probing the Shell Structure of 40Ca . . . . . . . . . . . . . . . . . . . . . . 367.2 Continuum Spectroscopy at the Dripline with (d,p) Reactions . . . . . . . 39

8 Lecture 8: Exploring Nuclear Excitations by Inelastic Scattering 418.1 General Aspects of Inelastic Nuclear Reactions . . . . . . . . . . . . . . . . 418.2 Interactions in Momentum Representation . . . . . . . . . . . . . . . . . . 428.3 Inelastic Form Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.4 The Collective Model of Inelastic Scattering . . . . . . . . . . . . . . . . . 458.5 Proton Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.6 Choice of Interaction and the Double Folding OMP . . . . . . . . . . . . . 55

9 Lecture 9: Heavy Ion Single Charge Exchange Reactions 57

10 Lecture 10: Charge Exchange Reactions and Double Beta-Decay 59

A Scattering on a Complex Potential 66A.1 Non-Hermiticity and Bi-Orthogonality . . . . . . . . . . . . . . . . . . . . 66A.2 Non-conservation of the Probability Flux . . . . . . . . . . . . . . . . . . . 67A.3 Non-Unitarity of the S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 67A.4 Reaction, Total, and Elastic Cross section . . . . . . . . . . . . . . . . . . 68A.5 Results for s-wave Scattering on a Square Well Potential . . . . . . . . . . 70

B Problem 1: Kinematics of a Two-Body Reaction 72

C Problem 2: Probability Current of a Scattered Particle 73

D Problem 3: Scattering on a 3-D Square Well Potential 74

E Problem 4: Scattering on a 3-D Complex Square Well Potential 74

F Problem 5: Deuteron-induced Stripping Reactions 76

2

1 Preface and Lecture 1: Probing Physical Systems

by Scattering Experiments

In practically all disciplines of physics, scattering experiments are the only way how toobtain information out of microscopic system. Light, i.e. photons, scattered on atomsled to the discovery of the electronic structure of matter. Interactions of high energyphotons, i.e. X-rays, have been essential for understanding the inner (lattice) structure ofsolids. X-rays are also an indispensable probe for medical and biological investigations.Light bent in the gravitation field of stars gave the first prove for Einstein’s theory ofgeneral relativity. Rutherford with Geiger and Marsden discovered the nucleus by alpha-particle scattering on atoms. The famous, Nobel prize honored (1961), electron scatteringexperiments by Hofstadter were a breakthrough for our understanding of shapes and sizesof nuclei. Nuclear power plants (and nuclear weapons) depend on low energy neutronscattering on fissile nuclei, known as compound nuclear reactions. In the late 1950tieswe started to learn how to probe nuclear structure by peripheral reactions of protons,deuterons, alpha particles and other light ions on nuclear targets, thereby changing themass partition by transfer reactions and exciting the target nucleus by inelastic reactions.Pions, kaons, and antiprotons are being used in scattering experiments as well.

A schematic spectrum of a nuclear direct reaction, observed e.g. in inelastic protonscattering off a heavy target nucleus, is shown in Fig.1. An evolution from discrete states atlow energy loss (i.e. low excitation energy in the target) to a continuous energy distributionwith resonance-like structures imposed on a smooth background is seen. One of the meritsof nuclear reaction theory is that those structure can be assigned to nuclear structurephenomena: at low excitation energies the spectrum is dominated by vibrational modesof the nuclear surface, while the giant resonances are e.g. dipole oscillations of protonsagainst neutrons. In recent years, new excitation modes were detected in exotic nuclei,known as pygmy dipole and quadrupole modes.

Direct reactions are fast reactions occurring in grazing projectile-target configurations.They are dominated by single impact events, i.e. one-step reactions. For our purpose ofreactions among nuclei the complexities of nuclear many-body systems inhibit closed formdescriptions. Rather, any nuclear scattering theory relies on resummation techniques,by which certain classes of interactions are described by effective operators, allowing totreat the phenomena of explicit interest by properly chosen perturbative methods. In thelanguage of modern effective field theory, this amounts to separate scales: Typically, ina given reaction one is interested only a certain window of nuclear phenomena, e.g. onelastic scattering defined by a vanishing net energy transfer. Thus, in that case inelasticprocesses may be projected out and subsumed into effective elastic scattering operators.In nuclear physics, this is achieved by the introduction of optical potentials which containin an unresolved manner all reaction contributing as virtual intermediate states to elasticscattering. The next step is to extend the investigations to reactions exciting only a fewintrinsic nuclear degrees of freedom. This leads to the direct reaction scenario: Nucleiare interacting by exchanging amounts of energy and momentum small compared to the

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Figure 1: Schematic picture of the energy loss (or excitation energy) spectrum of directnuclear reaction a+A→ b+B at incident energies of a few 10 MeV to a few 100 MeV pernucleon. The elastic peak, dominating the spectrum, has been omitted. At low excitationenergies discrete states of particle-hole and vibrational or rotational character are seen.Above the particle emission threshold (Esep ∼ 8 MeV) the spectrum develops a continuousenergy distribution because the nuclear states obtain a decay width into particle-plus-corescattering states. The most prominent continuum structures are due to the excitation ofgiant resonances, where the giant dipole resonance (GDR) with Jπ = 1− is the strongestone. For light ions a, at energy losses close to the total incident kinetic energy fusion-likeconfigurations approaching statistical equilibrium may be reached.

nucleon rest mass, i.e. tM2nucleon, thus probing single particle and collective particle-hole

degrees of freedom on the nuclear wave functions. This is done by transfer reactions where anucleon is exchange between projectile and target, and inelastic reactions where vibrationaland rotational nuclear states are excited. A common feature of such direct reactions is thatthey are peripheral reactions where projectile and target interact in grazing configurationsonly. The more violent central collisions, exciting the full spectrum of many-body and evensub-nuclear phenomena, are only taken into account indirectly as they give raise to inducedeffective interactions. on the formal level, this separation of scales is expressed in terms ofprojection techniques, to be discussed later. The spectra typically observed in a nuclearreaction with incident energies of a few 10 MeV to a few 100 MeV is illustrated in Fig. 1.Another defining characteristic of direct nuclear reactions is their forward peaked angulardistribution. This property is illustrated in Fig. 2 where spectra observed in the 54Fe(p, p′)reaction for a series of scattering angles are displayed. At forward angles the spectra areof a shape as sketched in Fig. 1, With increasing scattering angle. however, the directcomponent is increasingly suppressed and in the backward hemisphere only the emissionof protons from the highly excited compound nucleus survives. Hence, from the spectralangular distribution we identify indeed two distinct reaction scenarios: direct reactionswith strongly forward focused angular distributions and the statistical compound nucleuscomponent with an angular distribution which in leading order is isotropic.

A prime example of a direct reaction is elastic scattering probing the nucleus in a

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Figure 2: Spectral angular distributions of inelastically scattered protons observed in thereaction 54Fe(p, p′) at the incident energy TLab = 61.7 MeV. The data were taken byBertrand and Peelle (Phys. Rev. C8 (1973) 1045). The forward peaked direct reactionspectra are clearly distinguished from the compound emission spectra with an almostisotropic angular distribution.

coherent manner. An important case is elastic electron-nucleus scattering, the famousHofstadter experiment: e− + A cross sections are given by form factors determined bythe Fourier transforms of nuclear density distributions which corresponds to the coherentsuperposition of scattering events on all target protons 1. giving insight into the shapesand sizes of the nuclear charge distributions. Theoretically, the description of a Hofstadterexperiment on a heavy nucleus is already challenging because the strong electromagneticfield produced by a high-Z target requires a full treatment of the electron-target interac-tions. In this case, that is achieved by solving the Dirac equation for the motion of theelectron in the nuclear electromagnetic field. Such closed form descriptions are not possiblefor reactions proceeding through strong interaction. The scattering equations involve shortrange non-hermitian optical potentials and Coulomb interactions defined by the nuclearcharge distributions. Their solutions can be achieved only by numerical methods.

All reaction studies have in common that the interpretation of data relies stronglyand essentially on quantum mechanical scattering theory. The program of these series oflectures is to present an introduction into the principles of scattering theory. The focus

1At higher momentum transfer, also the intrinsic charge distribution of neutrons becomes visible

5

will be on the guiding principles. So, for example, we neglect in many cases particle spinsalthough spin degrees of freedom are of paramount importance in nuclear structure andreaction physics. This allows to keep angular momentum coupling on a tractable level. Insection 2 kinematics of a two-body reaction a+A→ b+B are discussed. As a reminder, werefer to classical scattering theory, serving also to recall and introduce the concept of a crosssection. In section 3 an introductory, summarizing overview on formal scattering theory isgiven. The derivation of effective interactions is a central theme of nuclear structure andreaction physics. This is the topic of section 4 where projection techniques are used toderive the induced dispersive self-energies resulting from channel coupling. This leads usto introduce the optical potential, implying the use of non-Hermitian Hamiltonians as theappropriate tool for nuclear reactions. The properties of the resulting wave functions andscattering amplitudes are briefly discussed in section ??. After these preparatory steps, weintroduce in the perturbative Distorted Wave Born Approximation (DWBA) approach tonuclear direct reaction in section 5. The remaining sections are devoted to the discussionof the most frequently used types of direct reactions. The theory of transfer reactionsis considered in section 6, applications are presented in section 7. Inelastic scattering isthe topic of section 8. Charge exchange reactions are introduced in section 9. In section10 recent applications of heavy ion charge exchange reactions as probes for 0ν2β-decayprocesses are discussed. In Appendix A, scattering states in a complex potential andrelated observables are discussed.

Quantum scattering theory is a key tool for understanding interactions of microscopicsystems and elementary particles from the early days of quantum mechanics on. The firstcomprehensive book with a broad influence on the field was published by Mott and Masseyin the 1933 (Theory of Atomic Collisions, Clarendon, Oxford, 1933). Among others, a clas-sical textbook on scattering theory is Collision Theory by Goldberger and Watson (Wiley,1964, Dover reprint, 2004) the Direct reactions are widely discussed in the literature. AfterButler introduced the terminology of Direct Reactions in the early 1950ties, the theoreticalfoundations have been worked out essentially by the end of the 1960ties by various authorsand the results were later published in text books. The probably most influential ones wereHerman Feshbach, Norman Austern, Gerald R. Satchler, Taro Tamura, and David Brink.Especially the books of Austern (Direct nuclear reaction theories, Wiley, 1970), Satch-ler (Direct Nuclear Reactions, Oxford University Press, 1983), and Feshbach (TheoreticalNuclear Physics: Nuclear Reactions, Wiley, 1992) are recommended for further studies.In these books and even more so the original research papers advanced presentations ofnuclear reaction theory, numerical methods, and practical applications are found, goingmuch beyond the level of an introductory lecture. Lately, experiments involving exotic nu-clei have given a new impact on direct reaction theory and applications to data analyses.Direct reactions are a versatile tool for various types of nuclear spectroscopic studies fromsingle particle spectroscopy by transfer reactions and vibrational and rotational excitationin inelastic scattering to the production of hypernuclei and studies of nucleon resonancesin nuclei.

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2 Lecture 2: Kinematics and Cross Sections

2.1 Kinematics of Two-Body Reactions

In an idealized scattering experiment, a sharp beam of particles (a) of definite momentumk are scattered from a localized target (A). A number of different exit channels will beobserved as illustrated in Fig. 3. In a Lorentz-invariant notation we define the total (P )the relative k 4-momentum by the linear transformation

ka = −k + xaP ; kA = k + xAP xa + xA = 1 (1)

The transformation coefficients are given by the Lorentz-invariant forms

xa =s−m2

A +m2a

2s; xA =

s−m2a +m2

A

2s. (2)

In the lab-frame under fixed-target conditions we have ka = (TLab +ma,qa)T and kA =

(mA,0)T , respectively. The total energy available in the a+A rest system is given by theMandelstam energy s = (ka + kA)2, hence

s = (Ea(pLab) +mA)2 − q2a = (TLab +ma +mA)2 − p2Lab (3)

where we have used Ea =√p2Lab +m2

a = TLab + ma. In the barycentric frame, defined bythe condition that the total 3-momentum P vanishes, we find P = (Ea + EA,0) leading to

s = P 2 = (Ea + EA)2 with Ea,A =√q2s +m2

a,A and the relative 3-momentum qs, evaluated

on the energy-shell, i.e. qs = qs(√s) with the result

q2s =1

4s

(s− (ma +mA)2

) (s− (ma −mA)2

)(4)

A quantity of central interest for the interpretation of reaction observables is the momentumtransfer. For a two-body reaction a + A → b + B there are two kinds of momentumtransfers, defining the so-called t-channel and u-channel processes. In Lorentz-invariantnotation, they are defined in terms of 4-momenta

t = (ka − kb)2 =(Ea(q)− Eb(q′))2 − (q− q′

)2(5)

u = (ka − kB)2 =(Ea(q)− EB(q′))2 − (q + q′

)2. (6)

where q and q′ are the relative 3-momenta before and after the reaction. By momentumconservation, we find t = (kA − kB)2 and u = (kA − kb)2. If the energy transfer is largerthan the 3-momentum transfer one speaks of time-like reactions, otherwise the momentumtransfer is called space-like. Elastic scattering is always a space-like process. The non-relativistic limit is obtained for all quantities by a Taylor series expansion in powers ofq2s/m

2a,A.

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Figure 3: Reaction channels populated by the incident system a+ A

2.2 The Concept of a Cross Section

Both classical and quantum mechanical scattering phenomena leading from an incidentchannel α to an exit channel β are characterized by the scattering cross section. Considera collision experiment in which a detector measures the number of particles per unit time,Ndt, scattered into an element of solid angle dΩ in direction (θ, φ). This number is pro-portional to the incident flux of particles, jI , defined as the number of particles per unittime crossing a unit area normal to direction of incidence. Collisions are characterised bythe differential cross section defined as the ratio of the number of particles scattered intodirection (θ, φ) per unit time per unit solid angle, divided by the incident flux,

dσ

dΩ=N

jI(7)

From the differential cross section we obtain the total cross section by integrating overpolar (θ) and azimuthal (ϕ) angles, collecting into the solid angle Ω(θ, ϕ)

σ =

∫dΩ

dσ

dΩ=

∫ 2π

0

dϕ

∫ +1

−1d cos θ

dσ

dΩ(8)

The cross sections are typically depending sensitively on energy and the projectile-targetcombination. They have dimensions of an area, which in nuclear physics is given either infm2 = 10−26cm2 or mb = 10−27cm2 = 10−1fm2. The total cross section can be separatedinto σelastic, σinelastic, σabs, and σtotal. The non-elastic events are giving rise to the totalreaction cross section σreaction = σinelastic + σabs. The conservation of the probability fluxlead to the relation

σtotal = σelastic + σreaction =4π

qsIm (f(0)) (9)

known as the Optical Theorem, relating the total cross section to the quantum mechan-ical scattering amplitude f(Ω) at forward angles.

Before turning to quantum scattering, let us consider briefly classical scattering theory.In Fig. 4 the essential ingredients for the description of a scattering event by classicalmechanics are summarized. The application is illustrated by the first observation of nu-clear Rutherford scattering (alpha particles from a radium source scattered on a gold foil)by Geiger and Marsden, performed between 1908 and 1913 under supervision of ErnestRutherford at the Physical Laboratories of the University of Manchester.

8

Recap of Classical Scattering Theory:

Figure 4: Aspects of classical scattering theory and Rutherford scattering

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3 Lecture 3: Formal Scattering Theory in a Nutshell

3.1 Basics of Reaction Physics

Scattering is a time-dependent process as expressed by the Schrodinger-equation

i~∂

∂tΨ(r, t) = H(r,p)Ψ(r, t) (10)

connecting the time and the spatial evolution of the state vector Ψ(r, t). The naturaldescription is in terms of wave packets, being prepared in the beam source at time t0,chosen often as t0 = −∞, and reaching the detector at time t = +∞. Eq.(10). The formalsolution is Ψ(r, t) = e−iH(r,p)(t−t0)Ψ(r, t0) which, in fact, is more of a symbolic nature thanof practical use. In the realistic case, Ψ will be a state vector involving the intrinsic wavefunctions Φa,A ≡ |aA〉 of projectile and target and the relative motion wave functions ψaA,depending on the available total energy w =

√s and the interactions between projectile

and target in their particular intrinsic states.In general Ψ(r, t0) may be expanded into a superposition of stationary eigenstates of the

system’s Hamiltonian H. Having in mind the time-dependent (Schrodinger) picture, thatis done in terms if the eigenfunctions of the asymptotically separated reaction partnerswhere the relative motion is of plane wave nature. The evolution of the a + A systemunder the mutual interactions is the central topic of quantum mechanical scattering theoryand, moreover, of quantum field theory. The situation simplifies considerably if Ψ(r, t0)coincides with one of the exact stationary eigenstates of H because then for all times Ψ(r, t)remains in an eigenstate. However, typically eigenstates are known only for a reducedmodel problem, described by a Hamiltonian H0 containing only a subset of interactions ina subspace of the full Hilbert-space. Thus, the model configurations Ψ0 will mix once theyare under the action of the full Hamiltonian H = H0 + V . Assuming that the eigenstateproblem for H0 has been solve by H0Ψn = ωnΨn, we may write

Ψ(r, t) =∑n

cne−iHte+iH0te−iωntΨn(r). (11)

Thus, we are left with the problem how to evaluate the remaining products of exponentialoperators which is highly non-trivial because in general [V,H0] 6= 0. The treatment of thatproblem is considered in the interaction picture which we, however, will not discusshere.

3.2 Green’s Functions in Scattering Theory

In scattering theory, a successful approach is obtained by mapping the time-dependenceby a temporal Fourier transform to an energy dependence. Then a system of coupledequations is obtained for Φn|bnBn〉 = cnΨn

(H0 − E) Φn|aA〉+∑k

V Φk|bnBn〉 = 0 (12)

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which has to be solved for the proper boundary conditions

Ψn → eiki·ri |aiAi〉δin + fin(Ωn)eiknrn

rn|bnBn〉 (13)

with an incoming plane wave state only in channel i and outgoing spherical waves in allchannels including i. That point of view is taken in this course: we consider scatteringproblems involving stationary eigenstates only. After integrating out the time coordinateand we have to deal only with spatial degrees of freedom, typically the coordinates r andmomenta p 7→ −i∇.

Green functions are the resolvents of the differential equations underlying H. In fact, asknown from electrodynamics Green functions are a mean to transform differential equationsinto integral equations, incorporating at the same time also physical boundary conditions.In a coordinate-independent notation the Hamilton operator H has the Green function

(H − E)G(+)(E) = −1↔ G(E) = (E ± iη −H)−1 (14)

where the treatment of the singularity is defined by the infinitesimal η → 0+, whichapproaches zero from the positive side. In coordinate representation, this corresponds toa source term of Dirac delta function-type:

(H(r,p)− E)G(±)(r, r′) = −δ(r− r′). (15)

Asymptotically, G(+) leads to outgoing and G(−) to incoming spherical waves, respectively.With the stationary eigenfunctions |n〉 of H,

(H − En) |n〉 = 0 (16)

and their dual elements 〈n| with 〈n|k〉 = δnk we find the spectral representation

G(±)(E) =∑n

|n〉〈n|E ± iη − En

+

∫d3k

(2π)3|k〉ρ(k)〈k|E ± iη − ωk

(17)

where the second term accounts for the continuous part of the spectrum of H with spectraldensity ρ(k).

Green functions are a perfect tool for investigating (formal and numerical) solutionsof wave equations. Consider H = H0 + U where H0 = K is given by the kinetic energyoperator K. The eigenvalues of H0 are known to be Ek ∼ k2 and the eigenfunctions areplane waves ϕ. The full wave function ψ is found by writing

(H0 − E)ψ = −Uψ. (18)

Introducing the propagator G0(E) belonging to H0

(H0 − E)G(±)0 = −1 (19)

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the general solution is

ψ(±) = ϕ+G(±)0 (E)Uψ(±) =

(1−G(±)

0 (E)U)−1

ϕ (20)

where we have indicated the two kinds of boundary conditions relevant for scatteringproblems. If for r → ∞ the interaction U vanishes like 1/r1+α, α > 0, the asymptoticforms of the wave are

ψ(±) → eik·r + f (±)(Ω)e±ikr

r(21)

where f (+)(Ω) = f(Ω) is the scattering amplitude into the solid angle Ω and f (−)(Ω) =f (+)∗(Ω). A more detailed analysis shows the important relation ψ(−)∗(k, r) = ψ(+)(−k, r),valid all over space and indicating a relation typical for time-reversal. Moreover, if thewave equation is coupled to a source s(r)

(H0 − E)ψ = −s (22)

the special solution is given by

ψ(±)s (r) =

∫d3r′G(±)(r, r′)s(r′) (23)

Thus, firstly we have obtained an integral equation for ψ. Secondly, this relation allows aperturbative expansion of the wave function in terms of powers of the interaction U wherethe rate of convergence is determined by G0U ∼ O(U/E). Last but not least, Eq.(20)serves to derive an integral equation for the scattering amplitude, as shown in the nextsection.

3.3 Integral Equation for the Scattering Amplitude

The exact scattering amplitude for the interaction U

f(Ω) = − 2m

4π~2T (k′,k) (24)

is given by T-matrixT (k′,k) = 〈k′|U |ψ(+)〉, (25)

where |k′〉 denotes a plane wave state. We introduce the plane wave matrix elements of U

U(k′,k) = 〈k′|U |k〉. (26)

From Eq.(20) we find the Lippmann-Schwinger integral equation for the T-matrix,i.e. the scattering amplitude,

T (k′,k) = U(k′,k) +

∫d3q

(2π)3U(k′,q)G

(+)0 (E,Eq)T (q,k). (27)

12

The solution of the Lippmann-Schwinger equation can be split into two steps. We noticethat the propagator can be separated into a Cauchy principal value and a pole term

G(+)0 (E,Eq) =

P

E − Eq− iπδ(E − Eq). (28)

We treat the principal value part first leading to the K-matrix equation

K(k′,k) = U(k′,k) + P

∫d3q

(2π)3U(k′,q)

1

E − EqK(q,k) (29)

and obtain the full T-matrix by solving the Heitler-equation

T (k′,k) = K(k′,k)− iπ∫

d3q

(2π)3K(k′,q)δ(E − Eq)T (q,k). (30)

where the integration reduces to an angle integral. Occasionally, the K-matrix approach isexpressed in terms of the somewhat symbolic relation,

T = (1− iρkK)−1K (31)

where ρk is the density of states at (on-shell) momentum k.In practical applications, the tree-level Born amplitudes and the T-matrices are ex-

panded into their partial wave components α = L, S, J.... This leads to a system ofcoupled integral equations of the type

Tαβ(q, q′|qs) = Vαβ(q, q′|qs) +∑γ

∫dkk2Vαγ(q, k|qs)G0(k, qs)Tγβ(k, q′|qs) (32)

where qs is the on-shell relative momentum, fixed by√s and q, q′ denote in principle

arbitrarily chosen off-shell momenta. In practice, one of them is fixed to qs, resultingin the half-off-shell scattering amplitudes. For a numerical solutions of the partial waveLippmann-Schwinger equations, standard integration techniques are used allowing a con-version into a system of linear equations.

The NN scattering problem can be solved accordingly in nuclear matter. The Pauli-principle has to be taken into account properly, which is achieved by the Pauli-projectorQF ,accounting for the blocking of the occupied states within the Fermi-sphere. In addition, thenucleonic self-energies from the interaction with the background medium must be included.Both effects are modifying the NN-propagator and to a lesser extent also the interactionvertices. The resulting K-matrix scattering amplitudes are known as Brueckner G-matrixor, in covariant relativistic formulation, as Dirac-Brueckner G-matrix.

3.4 Partial Wave Expansion

Nuclear reactions are typically described in the partial wave representation. A well knowncase, known already from quantum mechanics courses, is the partial wave expansion of a

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plane wave into spherical Bessel functions j`(x) and Legendre-polynomials P`(cos θ)

eik·r =∑`

(2`+ 1)i`P`(cos θ)j`(kr). (33)

In general, we define the partial waves by

ψ`(k, r) = i−`1

2

∫ +1

−1dtP`(t)Ψ

(+)(k, r) (34)

integrated over t = cos θ where θ is the angle between k and r. With u`(k, r) = e−iσC`

1krψ`(k, r),

where by convention the Coulomb phase shifts σC` are extracted , we obtained the standardform

Ψ(+)(k, r) =4π

kr

∑`m

i`Y ∗`m(k)Y`m(r)eiσC` u`(k, r) (35)

In the asymptotic region, the partial waves approach

ψ`(k, r)→ j`(kr) + C`(k)h(+)(kr), (36)

where h(±)(x) = n`(x)± ij`(x) are a spherical Hankel functions given in terms of sphericalBessel- and von Neumann- functions. The partial wave scattering amplitudes are dimen-sionless complex numbers. For charged particles Bessel and Hankel functions have to bereplaced by the corresponding Coulomb and Whittaker functions. Using a partial aveexpansion also for the scattering amplitude

f(θ) =∑`

(2`+ 1)P`(cos θ)f`(k) (37)

and taking into account the asymptotic form h(+)(x)→ i−`eix/x we find by comparison

f`(k) =1

kC`(k) (38)

A convenient parametrization is

C` =1

2i(S` − 1) (39)

where the partial wave S-matrixS` = η`e

2iδ` (40)

is given by the scattering phase shift δ` and the inelasticity 0 < η` ≤ 1. Considering firstthe case of U is Hermitian (i.e. given by a real potential function) we have η` = 1 and

C` = eiδ` sin δ`. (41)

We define the K-matrix elementsK` = tan δ` (42)

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Figure 5: Theory of meson-nucleon scattering. The types of Born-terms are indicateddiagrammatically. The Lippmann-Schwinger equation is solved in K-matrix representationand the full multi-channel T-matrix is then restored. For details see H. Lenske, M. Dhar,Th. Gaitanos, Xu Cao, Prog.Part.Nucl.Phys. 98:119-206 (2018).

allowing to write

C` =K`

1− iK`

(43)

Thus, outside of the interaction region, in addition to Eq.(36) the following equivalentforms of the partial waves are found

ψ`(k, r) = eiδ` (cos δ`j`(kr) + sin δ`n`(kr)) (44)

=1

1− iK`

(j`(kr) +K`n`(kr)) (45)

= − 1

2i

(h(−)` (kr)− S`h(+)

` (kr)), (46)

finally describing standing waves in different notations.

3.5 Applications of the T-Matrix Formalism

3.5.1 Example 1: Pion-Nucleon Scattering

One of the few cases which are accessible by solving the coupled channels T-matrix prob-lem in meson production on the nucleon via pion- and photon-induced reactions in theregion to

√s ∼ 2 GeV. At least for the single meson channels the dimensionality of the

problem is such that only a finite number of coupled reaction channels must be considered.The ingredients of the so-called Giessen model for pion- and photo-meson production areindicated in Fig.5. On the Born-level, s-, t-, and u-channel interactions, two-pion produc-tion channels, and explicit resonance production tree-level amplitudes are entering into theK-matrix equation. By means of the Heitler equation the full T-matrix is restored.

15

Figure 6: Total pion-nucleon cross section. Excitations of nucleon resonances are indicatedfor selected cases. Further results are found in H. Lenske et al. Prog.Part.Nucl.Phys.98:119-206 (2018).

Typical results for pion-nucleon elastic scattering are shown in Fig.6. The Delta reso-nance and the second and third resonance regions are well resolved.

3.5.2 Example 2: Elastic Scattering of Nucleons

The momentum representation is a standard method in elementary particle physics. Nucleon-nucleon scattering, for example, is described successfully by meson-exchange interactions.The interaction U is taken as a superposition of interactions describing the exchange ofthe isovector pions, rho- and delta-mesons and their isoscalar counterparts (eta, sigma andomega mesons). Together with cut-off form factors, regularizing the momentum integrals,they are defining the Born-terms given by the matrix elements U(k′,k). A decompositioninto good total angular momentum is performed leading to a decoupled block-structure ofthe Lippmann-Schwinger equation. In Fig. 7 results for the singlet-even NN-channel (s-wave scattering in the S=0, I=1 channel) are displayed. The Bonn-A potential of Machleidtet al. was used to construct the Born term. In Fig. 7, the resulting scattering amplitudeobtained in Born approximation is compared to the full K-matrix scattering amplitude. Atlow energies, the summation of the higher order terms by the Lippmann-Schwinger leadsto substantial contributions, increasing close to threshold the strength by large factors.At larger energies, the full amplitude approaches the Born-term, roughly reflecting thesuppression of the correlation integral by ∼ 1/s induced by the intermediate propagator.

The resulting scattering phase shifts for neutron-proton scattering in the singlet-even

16

Figure 7: NN scattering in the SE channel. The S = 0, I = 1 s-wave K-matrix and thecorresponding Born term are shown.

Figure 8: Neutron-proton s-wave scattering phase shifts in the SE and TE channels.

L = 0, S = 0, I = 1 and the triplet-even L = 0, S = 1, I = 0 channels are shown inFig. 8. In the spin-singlet configuration, the neutron-proton interaction is attractive butnot strong enough for the formation of a bound state. That property is reflected in thelow-energy behaviour of the corresponding phase shift, starting with a positive slope upto a maximum. A different situation is found in the spin-triplet configuration: In thatchannel, the deuteron exists as a np-bound state and as a consequence, the phase shiftstarts at threshold with a negative slope. Actually, the triplet channels are involving acoupling of partial waves with |L−L′| = 2 by the rank-2 tensor interaction from pion- andrho-meson exchange, i.e. s- and d-waves, p- and f-waves and so on are coupled. That typeof mixing is also present in the deuteron.

The evolution of the np total cross section with incident energy is illustrated in Fig. 9.Close to threshold the cross section is of the order of one barn. With increasing incidentenergy s sharp decrease is observed, leveling of at about σtot ∼ 30 mb for energies aboveTLab ∼ 300 MeV. Below TLab ∼ 300 MeV the scattering is purely elastic. Beyond that

17

Figure 9: Neutron-proton total cross section. Theoretical results obtained by the meson-exchange model of Love and Franey (see Phys.Rev. C24 (1981) 1073) for the NN T-matrix(red line) are compared to data (symbols)

energy, the pion production channels n + p → n + p + π0, n + p → n + n + π+, andn+p→ p+p+π− are opening, leading to coupled channels effects by which a considerableamount of the elastic flux is redirected into the non-elastic pion-production channels. Theexcitation of nucleon resonances (e.g. ∆(1232) as intermediate states and their subsequentdecay into nucleon-pion configurations is an important mechanism for pion production

As will be discussed below, the coupling among open reaction channels corresponds tothe redistribution of the incident probability flux over the channels accessible at a giventotal center-of-mass energy

√s. The effect is clearly visible already in NN-scattering.

As discussed below, theoretically the flux redistribution is expressed by the partial waveinelasticities 0 < ηLJ ≤ 1 and the reaction factors RLJ = 1 − η2LJ . In Fig. 10, thespin-singlet and spin-triplet s-wave reaction factors R0J for J = 0, 1 are shown for theneutron-proton system. Above the pion production threshold they are seen to increaserapidly.

18

Figure 10: Inelasticity in neutron-proton scattering. The s-wave reaction factors R = 1−η2in the S = 0 and the S = 1 channels are shown.

4 Lecture 4: The Optical Model for Elastic Scattering

4.1 Elimination of Coupled Channels Effects in Feshbach Theory

Nuclear reactions at typical energies of a few ten to a few hundred MeV per particle excitea large number of intrinsic nuclear degrees of freedom. From a reaction theoretical pointof view this amounts to a huge coupled channels problem. If we attempt to single outof the multitude of open channels a few for spectroscopic work, it is still necessary toaccount properly for the effects exerted by the other channels, not included explicitly intothe analysis. The problem is of course known since the very beginning of nuclear reactiontheory. For low-energy, thermal nuclear reactions leading to the formation and decay ofcompound nuclei, statistical nuclear reaction theory was developed, allowing to analysecompound nuclear spectra by quantum-statistical methods. Such investigations, by theway, led in the 1970ties to the discovery of chaotic dynamics in thermal nuclear systems.

In Feshbach theory, the full Hilbert space is split into two orthogonal, disjoint subspaces:the P-space, containing the states to be treated explicitly, and its complement, the Q-space.In terms of projectors, this amounts to use

P +Q = 1; P 2 = P ; Q2 = Q; P ·Q = 0 (47)

Accordingly, the Hamiltonian H is split into

HPP = PHP, HQQ = QHQ; HPQ = PHQ, HQP = QHP = H†PQ (48)

Each state vector Ψ, being eigenstate H|Ψ〉 = E|Ψ〉 is decomposed into the parts in P-and Q-space:

Ψ = (P +Q)Ψ = ΨP + ΨQ (49)

19

where

(HPP − E) ΨP +HPQΨQ = 0 (50)

(HQQ − E) ΨQ +HQPΨP = 0 (51)

For a scattering problem the channel with the incoming plane wave is part of the P-space.The Q-space contains only channels with asymptotically purely outgoing spherical waves,i.e. the Q-space channels are fed by the P-space. That is taken into by the special solution

ΨQ → Ψ(+)Q = G

(+)QQHQPΨP (52)

Inserting this into the P-space wave equation, we find the induced self-energy

ΣPP (E) = VPQG(+)QQ(E)VQP (53)

resulting in the modified, energy (and momentum) dependent Hamiltonian

HeffPP = HPP + ΣPP (E) (54)

such that (HeffPP − E

)ΨP = 0. (55)

By decomposing the Q-space propagator as

G(+)QQ =

1

E + iη −HQQ

=1

E −HQQ

− iπδ(E −HQQ) (56)

we find thatΣPP (E) = Σr(E)− iΣi(E) (57)

is a non-hermitian operators and consequently also Heff†PP 6= Heff

PP has lost hermiticity. Atthe end, we have obtained a reduction by projection of the full Hilbert-space problem toa subspace of lower dimension, but to the expense of dealing with a considerably morecomplicated operator. Thus, further simplifications are necessary before an approach ofpractical use is obtained.

4.2 The Optical Potential for Elastic Scattering

The most obvious – and physically relevant – case is that the P-space contains only a singlestate. The situation is realized by elastic scattering where the P-space contains the statevector describing the reaction a + A→ a + A. By splitting the P-space Hamiltonian intoa kinetic (K), the intrinsic nuclear Hamiltonian HN with HN |aA〉 = (ma + mA)|aA〉 anda non-dispersive projectile-target interaction (U), i.e. HPP = K + U , we find the totaleffective P-space interaction

Ueff (E) = U + V(E)− iW(E) (58)

20

We separate the channel states into a relative motion part, either a plane wave (ϕ) or anoutgoing interacting wave (ψ(+)) and the intrinsic nuclear states |aA〉. The (exact) elasticscattering amplitude is

TPP = 〈ϕkaA|Ueff |aAψ(+)〉 (59)

which seems to require knowledge of the full operator structure of the effective interaction.However, this is an integral quantity which can be evaluated by introduction of an auxiliarypotential Uopt(E). Thus, we write

TPP = 〈ϕkaA|Ueff − Uopt(E) + Uopt(E)|aAψ(+)〉 (60)

and if we determine the auxiliary complex optical potential such that

〈ϕkaA|Ueff − Uopt(E)|aAψ(+)〉 ' 0 (61)

we can replace the exact scattering amplitude by the optical model amplitude

Topt = 〈ϕkaA|Uopt|aAψ(+)〉 ' 〈ϕkaA|Uopt|aAχ(+)〉 (62)

where the distorted wave is given by the optical model wave equation

(K + Uopt − ω)χ(+) = 0, (63)

with the kinetic energy ω = E −ma −mA and leaving us with the remaining error term

∆PP =∣∣〈ϕkaA|Uopt|aAψ(+)〉 − 〈ϕkaA|Uopt|aAχ(+)〉

∣∣2 . (64)

Thus, we have found a scheme justifying on formal mathematical grounds the introductionof an optical potential as the auxiliary operator reproducing as good as possible the exactP-space scattering amplitude. Hence, the full multi-channel problem has been mapped tothe much simpler problem of determining a single channel optical potential. The caveat isthat Uopt and the related distorted waves χ(+) will describe the full P-space solution ψ(+)

strictly spoken only in the asymptotic region. In the interaction region, the optical modelapproach accounts for major effects as contained in the reduction of the elastic flux by theabsorptive effects due to channel coupling.

Phenomenological global optical potentials have been determined especially for nucle-ons and deuterons, and to a lesser extent also for the mass-3 projectiles and alpha-particles.For heavier masses, global parameter fits are missing, although parameter sets for selectedprojectile-target combinations at fixed incident energy are available. In Fig. 11 typicalresults of optical model investigations for proton elastic scattering at energies up to aboutTLab ∼ 50 MeV on Ni- and Cu-targets are shown. The parameter set, determined byBecchetti and Greenlees, has become a widely used standard for proton scattering. How-ever, while at forward angles the data are well described, deviations are showing up inthe backward angle region, indicating the limitations of the chosen potential model. Thereason is that at these low energies elastic scattering can proceed through the compound

21

Figure 11: Angular distributions normalized to the Rutherford cross section for elasticp + A scattering. Data are compared to optical model calculations using the Becchetti-Greenlees potential (see Phys. Rev. 182:1190 (1969)). At backward angle compound-elastic scattering (CE) is seen to become important at these low energies.

22

Figure 12: Parametrization of the d + A optical potential in terms of Wood-Saxon formfactors f(r, R, a) = 1

1+e(r−R)/a .

Figure 13: Global optical model parameters for d+A elastic scattering in the energy rangeTLab = 58.7 MeV to TLab = 85 MeV (see Phys. Rev. 38:1153 (1988)).

nucleus with subsequent (isotropic) emission of nucleons. Those compound elastic (CE)contributions must be included explicitly, as seen in Fig. 11.

Some years ago Hinterberger et al. have been measuring deuteron elastic scatteringcross sections systematically at several incident energies and over a wide range of targetmasses. They fitted optical potentials to their data, thus determining a global set ofparameters for the real and imaginary parts. The potential was chosen as a superpositionof Wood-Saxon form factors of volume and 1st derivative shape, as shown in Fig. 12

In Fig. 14 typical results at an incident deuteron energy TLab = 58.7 MeV are displayed(see Phys. Rev. C38:1153 (1988)). It is seen that the phenomenological approach describesthe angular distributions very satisfactory. The derived set of parameters is found in Tab,13.

In Fig. 15 optical model elastic scattering cross sections for d+14 C at TLab = 20 MeVand p+15C at TLab = 17 MeV are shown. These reactions will be part of the exercises. Atforward angles, Rutherford scattering strongly dominates the angular distributions. Shortrange strong interactions become sizable only at larger scattering angles. Since Coulomband strong interaction amplitudes are contributing coherently to the quantum mechanicalcross section, they are interfering. The interference pattern is seen more clearly by nor-malizing the elastic angular distribution to the Rutherford cross section which behaves like

23

Figure 14: Elastic scattering d + A angular distributions at TLab = 58.7 MeV normalizedto the Rutherford cross section (see Phys. Rev. 38:1153 (1988)).

1/ sin4 θ/2.Finally, it is worthwhile to point out that the same mechanisms are also present in

nuclear structure theory. Induced interactions from outside the model space are playinga central role in shell model approaches, irrespective whether one uses phenomenologicalinteractions or the so-called ab initio chiral effective field theory methods. Induced in-teractions are typically appearing as many-body interactions, modifying considerably theinteractions derived in the limited model space.

24

Figure 15: Elastic scattering angular distributions

25

5 Lecture 5: Perturbative Approach to Nuclear Scat-

tering: Distorted Wave Born Approximation (DWBA)

and Related Methods

5.1 The Concept of Direct Nuclear Reactions

As discussed in the context of Fig. 1 direct nuclear reactions are a special class if reactions,occurring preferentially in grazing projectile-target interactions. They are fast directions,proceeding on time scales of τDR ' 10−22 s or faster. That has to be compared to statisticalreactions leading to a compound nucleus which are characterized by reaction time of τCN ∼10−18 s or slower.

Under more general nuclear physics aspects, direct reactions are populating doorwayconfigurations which decay subsequently into more complex configurations. The majorityof direct reactions is well understood by assuming interactions probing the diagonal (elasticscattering) and non-diagonal pieces (inelastic and transfer reactions) of the projectile andtarget one-body density operators:

• Elastic scattering involves the ground state density operator ρ =∑

i nia+i ai with

ground occupation probabilities, given in the simplest case by filling all levels up tothe Fermi-level by λF , ni ∼ Θ(λF − ei). Elastic scattering is proceeding throughthe two-body projectile-target NN-interaction. It is the by far dominating reactionchannel.

• Transfer reactions include a mixed one-body density operator, e.g. for a strippingreaction transferring a nucleon form the projectile to the target, ρm ∼ a+j akΘ(λFa −ek)Θ(ej−λFA). Transfer reactions are in leading order a mean-field process because asingle particle bound in the mean-field of one nucleus is transferred into a mean-fieldorbit of the other nucleus.

• Inelastic reactions, i.e. reactions leaving the mass partition unchanged, are probingnuclear transition mediated by particle-hole-type transition densities ρph ∼ a+p ah,either in one of the colliding nuclei or simultaneously in both. These reactions arerequiring a two-body projectile-target interactions as given by the NN-interaction.

Typically, direct reactions are one-step reactions. i.e. the projectile-target interaction actonly once. However, occasionally also higher order processes become important, e.g. inreactions populating the transitional region into the statistical regime. Another situationrequiring multi-step processes is given when special nuclear configurations are involvedwhich cannot be excited by one-body operators. A case of large recent attention aredouble charged exchange reactions because of their similarity to rare double beta decayevents.

26

5.2 Treatment of Channel Coupling

The P-Q scheme introduced in section 4 must be considered in more detail. The P-spacewith dim(P ) = NP dim(Q) now contains NP reaction channels which are coupled bythe effective residual P-space projectile-target interaction VPP . Note that VPP includesalso induced interactions from the eliminate Q-space.

For these investigations, we define the channel states by ψ(+)n |anAn〉 where the dis-

torted waves ψ(+)n describe the relative motion degrees of freedom. Integrating out the

intrinsic nuclear states, we obtain a system of coupled equations for the relative motionwave functions ψn, n ∈ P :

(hn − En)ψ(+)n +

∑k∈P

Fnkψ(+)k = 0. (65)

hn = Kn+Uopt,n denotes the optical model Hamiltonian in channel n, including the inducedself-energies from the Q-space. Fnn′ = 〈anAn|VPP |an′An′〉 is the channel coupling formfactor depending on the relative motion coordinates. The system is solved numerically byfirst deriving the set of NP fundamental solutions ΨP defined as NP dimensional columnvectors. The physical channel configuration, given by the boundary condition that allchannels contain outgoing spherical weaves and only one channel has an incoming planewave (see Eq. 13), is constructed by matching a superposition of the fundamental solutionsand their derivatives to the asymptotic state vector and its derivative at a radius r = R∞.Denoting the pane wave state vector (with a single non-zero entry at row i) by ϕP , a vector[H

(+)PP ⊗ fPP

]n

given by the nth column of the diagonal matrix of outgoing spherical waves

multiplied by the matrix of scattering amplitudes fPP (Ω):

NP∑k=1

ckΨk =[H

(+)PP ⊗ fPP

]n

+ ϕP (66)

NP∑k=1

ckΨ′

k =[H

(+)′

PP ⊗ fPP]n

+ ϕ′

P (67)

Thus, a 2Np × 2NP -dimensional linear system for the 2NP unknowns ck, fnk has to besolved. In practice, this is done in partial representation. In the same manner, also solutionwith asymptotically incoming spherical waves is obtained.

A formally equivalent solution is obtained with Green’s functions. Denoting the so-lutions of the homogeneous set of equations by χk, the coupled channels system s solvedby

Ψ(+)P = χ

(+)P +G

(+)PPFPPΨ

(+)P (68)

where obvious vector and matrix notations have been used.

5.3 Reaction Theory with Distorted Waves

In many cases the channel coupling is rather weak which makes possible to treat thereaction process perturbatively. For nuclear reactions, however, an expansion in terms of

27

plane wave matrix elements, i.e. the Born series of scattering theory, is not a successfulapproach. The initial and final state interactions exerted by the optical potentials on therelative motion of the projectile target system are typically so strong that they must betaken into account to all orders. Thus, a successful scheme is to solve first the homogenousequations with the (auxiliary) optical potentials only:

(Hnn − ωn)χn = (Kn + Uopt,n − ωn)χn = 0 (69)

where the channel energy is ωn = E − man − mAn . The wave equation may be solved

either for the distorted wave χn = χ(+)n or χn = χ

(−)∗n . The corresponding elastic scattering

amplitude is given by the optical model T-matrix

T optnn (kn,k′n) = 〈ϕn|Uopt,n|χ(+)

n 〉 = 〈χ(−)n |Uopt,n|ϕn〉 (70)

where kn,n′ with kn = kn′ are the channel momenta carried by the distorted and planewaves, respectively.

Now, that we have taken into account the strongest component the question arises howto evaluate properly the scattering amplitudes given by the coupling form factors from theresidual interaction V . The proper answer is that the non-elastic T-matrix elements aregiven by

Tnn′ = 〈χ(−)n′ |Fn′n|Ψ

(+)n 〉 (71)

and we note that the distorted waves χ(−)∗(k, r) = χ(+)(−k, r) appear. Without going intothe (formally rather involved) details we mention that at first sight unexpected appearanceof the χ(−) waves is related to a fundamental relation of scattering theory:

Sαβ = 〈Ψ(−)α |Ψ

(+)β 〉 = δαβ(2π)3δ(kα − kβ)− 2iπδ(Eα − Eβ)Tαβ(kβ,kα) (72)

which relates the S-matrix and the on-shell T-matrix, generalizing the relation of Eq.(40)formerly derived for partial wave scattering amplitudes. Applying the above formula tothe two potential case, given here by Uopt and V , one ends up with Eq.(71). A detaileddiscussion is found e.g. in the book of Goldberger and Watson.

5.4 The DWBA Reaction Amplitude and Cross Section

Apart from the fact that the auxiliary optical potential are used in place of the inducedself-energies, the T-matrix defined in Eq.(71) is the exact result for the two potential casebecause of the full appearance of the coupled channels wave functions Ψ(+). Using

Ψ(+)n = χ(+)

n +G(+)opt

∑k

FnkΨ(+)k (73)

we find the distorted wave Born series as a formal power series in the coupling form factors

Ψ(+)n = χ(+) +G

(+)opt

∑k

Fnkχ(+)k +O

((F/E)2

)(74)

28

Inserting this series into the T-matrix we find

Tnn′ = 〈χ(−)n′ |Fn′n|χ

(+)n 〉+O

((F/E)2

)(75)

In most cases, the series is rapidly converging. If so, the full T-matrix is well approximatedby the Distorted Wave Born Approximation given by the first term in the aboveexpansion, i.e. Tnn′ ' T

(DWBA)nn′

T(DWBA)αβ = 〈χ(−)

β |Fn′n|χ(+)α 〉 (76)

where, in practice, n = α is the incident channel, and n′ = β is one of the exit channels.Up to now we did not pay much attention on the quantum numbers of the nuclear

states, except for energy. In fact, the initial and final nuclear states are carrying of courseangular momenta Ja, JA and Jb, JB and the corresponding magnetic quantum numbersMa,MA,Mb,MB. Moreover, for reactions with projectiles having bound excited states,we have to consider that the total excitation energy will be shared by the two reactingnuclei. That effect is taken into account by introducing spectral distributions Sb,B(E). Forenergy-sharp states they are given by Dirac-delta distributions, Sb,B(E) = δ(E − Eb,B).For a reaction measuring angular distributions and simultaneously recording the energystates of projectile and target triple differential cross section are the proper observables

d3σαβdΩdEdE ′

=mαmβ

(2π~2)2kβkα

1

(2Ja + 1)(2JA + 1)∑bB

∑Ma,MA∈α;Mb,MB∈β

∣∣∣T (DWBA)MaMA,MbMB

(kα,kβ)∣∣∣2 SB(E ′)Sb(E − E ′), (77)

where we have assumed unpolarized projectile and target nuclei for which an average overinitial pin orientations and summation over the final spin directions is appropriate. By asuitable integration over energy and/or angle, the various double, single, and total crosssections, respectively, are obtained.

29

6 Lecture 6: Theory of Transfer Reactions

6.1 Probing Nuclear Single Particle Dynamics by Transfer Re-actions

Historically, a (p, d) reaction stood at the beginning of nuclear direct reaction theory. Inearly proton-induced experiment a deuteron yield was observed which was in contradictionto the production rates expected from statistical models. Also, the angular distributionswere unusually forward peaked. Corresponding features were also observed for (d, p) reac-tions. Shortly after, similar characteristics were also found in cross sections of inelasticallyscattered protons, deuterons, and alpha particles. Butler was one the first in the 1950tiesto understand that a new class of nuclear reaction had been discovered, given by a singlestep direct interaction of projectile and target. The new kind nuclear reactions was namedby Butler as Direct Reactions (DR).

Here, we consider in more detail the striping reaction A(d, p)B where B = A + n.The change of the mass partition in such transfer reactions requires a careful treatment ofrecoil effects. As seen in Fig. 40, the rearrangement of the masses leads to a change ofthe channel coordinates such that rβ 6= rα. A more detailed consideration shows that thechannel and the intrinsic coordinates are related by a linear transformation of the type

r1 = s1rα + t1rβ (78)

r2 = s2rα + t2rβ. (79)

where si and ti are given by the masses of the participating nuclei.We also see that we are in fact dealing with at least a 3-body problem, defined by

the target A and d = n + p. Thus, one might attempt to describe the reaction as aA+n+ p problem. Such an approach might be feasible for low incident energies the orderof 1 MeV where only a few partial waves are involved. With increasing energy and targetmass, the number of partial waves increases with wave number and nuclear radius likekRA ∼

√TLabA

1/3. For (d, p) reactions on a light target nucleus like 12C and moderateenergies of a few 10 MeV this amounts to include for the d + A system about 30 partialwaves or more. In order to avoid the conceptually and numerically rather involved 3-bodydescription, reduction schemes have been developed by which the task is reduced to anapproximately equivalent two-body problem.

An obvious first step is to treat the incident d+A channel as a two-body channel, thusnot resolving the internal structure of the deuteron. The Q-space would then contain alsoall breakup contributions where the deuteron disintegrates into n+ p unbound continuumconfigurations. Austern and collaborators were the first investigating the breakup channelsin detail, thus restoring to some extent the 3-body nature of d+ A scattering.

The 2-body approach implies to use in the incident channel the Hamiltonian

Hα = KaA + UaA +HA +Ha + Vα (80)

which is known as the prior representation. The intrinsic nuclear states of the (isolated)nuclei A and a are described by the Hamiltonians HA and Ha, respectively. The a+A elastic

30

Figure 16: Coordinates describing a stripping reaction a+A→ b+B with a = b+ x andB = A+ x.

interactions are subsumed into the optical potential UaA. Vα accounts for all interactionswhich are not covered by the other parts of Hα.

The exit channel β = b+B is described by the Hamiltonian in post representation

Hβ = KbB + UbB +HB +Hb + Vβ (81)

with the b + B optical potential UbB and the intrinsic nuclear Hamiltonians HB and Hb,describing the (bound) n+A system and the (isolated) projectile core b. Interactions notcovered by the other parts of Hβ are supposed to be contained in the effective interactionVβ.

In the literature extended discussions are found whether to use the Hamiltonian eitherin prior (Hα) or the post representation, i.e. Hβ. Finally, this discussion is superfluousbecause both are describing the same total system, consisting of two core nuclei A andb which are sharing a particle x. Hence, solely by energy conservation we have Hα =Hβ. However, within the chosen reduction to a description in terms of effective two-bodyreaction channels, one of the representations may be of advantage when used in practicalnumerical work.

The discussions on the advantages and short comings of the one or the other representa-tion will not be repeated here. Rather, we are satisfied to note that the post representationis the most convenient description for stripping reactions like (d, p). In the post represen-tation, Vβ is identified with the interaction of particle x with the projectile core b, i.e.Vβ ' Ubx. More precisely, Vβ carries an operator structure given by the one-body transferoperator

TaB = ΨB(rAx)Ψ†a(rbx) (82)

with the single particle field operators

ΨB(rAx) =∑

x=Exjxmx

ψx(rAx)a+x (83)

Ψ†a(rbx) =∑

x=Exjxmx

ψ∗x(rbx)b+x (84)

31

(85)

where the single particle wave functions ψx and φx and a+x and b+x denote particle and holewave functions and creation operators, respectively. Then, the transfer matrix elementbecomes

〈bB|Vβ|aA〉 = Ubx(rbx)〈B|ΨB|A〉〈b|Ψ†a|a〉 (86)

The nuclear matrix elements are given by

〈B|ΨB|A〉 =∑x

ψx(rAx)X(p)(BA, x) (87)

〈a|Ψ†a|a〉 =∑x

ψx(rbx)X(h)(ba, x). (88)

where we have introduced the particle and hole spectroscopic amplitudes

X(p)(JBMBJAMA, jm) =

〈JBMB|a+jm|JAMA〉 =(−)JA−MA

j(JBMBJA −MA|jm) X(p)(JBJAj) (89)

X(h)(JbMbJaMa, jm) =

〈JbMb|b+jm|JaMa〉 =(−)Ja−Ma

j(JbMbJa −Ma|jm) X(h)(JbJaj). (90)

where j ≡√

2j + 1 and the Wigner-Eckardt theorem was used for a separation intoClebsch-Gordan coefficients and the reduced matrix elements

X(p)(JBJAj) = 〈JA||a+j ||JA〉 (91)

X(h)(JbJaj) = 〈Jb||b+j ||Ja〉 (92)

It is seen that the reduced amplitudes contain the complete information on the structureof the nuclear states. Their determination is the central task, either by extraction fromexperimental data by a quantitative nuclear reaction calculation or by prediction fromnuclear structure calculations. For heavy nuclei of mass number A 3, one may use as aleading order description Hartree-Fock theory, which in second quantization is

|JBMB〉 = |(jJA)JBMB〉 =∑mnMA

(jnmnJAMA|BMB) a+jnmn|JAMA〉 (93)

and pairing can be included as well. Then, the reduced amplitudes are easily evaluatedwhich is left as an exercise.

In the interacting many-body shell model, the reduced amplitudes are denoted by

X(p)(JBJAj) ' 〈JAj|JB〉 (94)

where the right side is the coefficient of fractional parentage (cfp), which is the probabilityamplitude for finding the specific configuration [JAj] in the many-body state JB.

32

Collecting results, the form factor – i.e. the kernel of the DWBA reaction amplitude –for a stripping reaction in post representation is obtained as

Fβα(rβ, rα) = X(p)(JBMBJAMA, jpmp)X(h)(JbMbJaMa, jhmh)Ubx(rbx)ψjpmp(rBx)ψ

∗jhmh

(rbx)(95)

which in general is a highly non-local object, as seen by recalling that the core-particlecoordinates rbx,Bx must be expressed in terms of the channel coordinates rα,β. Also nu-merically, this is a quite demanding task which we will not discuss here. As a solution, theExact Finite Range (EFR) reaction theory and the corresponding computer codes havebeen developed by several groups, in connection both with DWBA and coupled channelsmethods. For example, the EFR-DWBA matrix element is

TEFRβα (kβ,kα)) = Jαβ∫d3rβ

∫d3rαχ

(−)∗β (kβ, rβ)Fβα(rβ, rα)χ(+)

α (kα, rα) (96)

and in total a 6-dimensional integration needs to be evaluated. The Jacobian for thecoordinate transformation from natural to channel coordinates if denoted by Jαβ.

6.2 The Deuteron

The d = n+ p deuteron bound state is determined by

Hd = Knp + Vnp (97)

(Hd + εd) Φd = 0, (98)

where εd = mp +mn − Ed = +2.224 MeV is the deuteron separation energy.The deuteron is in a spin-triplet (S = 1)/isospin-singlet (I = 0) configuration. The

Pauli principle requires for a two-Fermion state that (−)S+L+I = −1. Thus, only evenrelative partial waves are allowed for a spin-triplet/isospin-singlet state, L = 0, 2..., knownas Triplet-Even (TE) configuration. The deuteron as the energetically lowest np state isgiven as a superposition of S- and D-wave configurations

|Φd〉 = xS|3S1〉+ xD|3D1〉 (99)

with xS = 0.97... and xD = 0.22..., corresponding to S- and D-state probabilities of |xS|2 ∼95 % and |xD|2 ∼ 5%, respectively.

The S-state and D-state radial wave functions uS,D(rpn), including the configurationamplitudes xS,D are shown in Fig.17. Because of its small binding energy, the deuteron isa quite diffuse object with a rms-radius of about 2 fm.

The relevant interaction component is Vpn = VTE(rpn)PTE. VTE(x) denotes the radialform factor of the triplet-even interaction. The spin-isospin degrees of freedoms are takencare of by the projector

PTE =1

16(3 + σp · σn) (1− τp · τn) (100)

with the projector properties P 2TE = PTE and PTE|Φd〉 = |Φd〉.

33

Figure 17: The wave functions of the deuteron S- and D-state, respectively. The full energydependent Bonn interaction was used (Machleidt et al., Phys.Rept. 149 (1987) 1

6.3 The (d,p) Reaction

The target transfer matrix element is evaluated as discussed before. The d→ p transition,however, requires a closer inspection because there is no properly defined core nucleus.Taken that into account, an approach is used oriented on the np bound state properties.In the following, we neglect the small D-state admixture. We split the S-wave part into aradial and a spin-isospin wave function

Φd =1

rw(r)Y00(r)

1√4π|SM, IMI〉 (101)

where r = rpn. The zeroth-order spherical harmonics is Y00 = 1/√

4π; the spin and isospinwave functions are

|JM〉 =∑mpmn

(jpmpjnmn|JM) |jpmp〉|jnmn〉 (102)

for J = S, I and jq, q = p, n, denotes either the spin (sp,n = 12) or the isospin (tp,n = 1

2)

of the proton and the neutron, respectively. Since I = 0, the isospin coupling coefficientreduces trivially to the value 1.

In the post-representation the d → p overlap function is defined by projecting thedeuteron wave function on the spin and isospin degrees of freedom of the outgoing protonand the removed neutron:

DSM(rpn) = VTE(rpn)〈spmp|asnmn|Φa(rpn, ξ) (103)

The result is

DSM(rpn) =1√4π

(spmpsnmn|SM)VTE(rpn)1

rpnw(rpn) (104)

34

which is a spherical symmetric object. The isospin part is trivial because of I = 0. Thus,we may write

DSM(rpn) =1√4π

(spmpsnmn|SM)W0(rpn) (105)

with

W0(rpn) = VTE(rpn)1

rpnw(rpn) (106)

The deuteron wave equation leads to

W0(rpn) = − (εd +Kpn)1

rpnw(rpn), (107)

and for a S-wave, the differential operator reduces to

W0(rpn) =1

rpn

(εd −

~2

2mpn

d2

dr2pn

)w(rpn) (108)

where εd is the (positive) dissociation energy of the deuteron and

1

mpn

=1

mp

+1

mn

∼ 2

mN

(109)

is the reduced mass, frequently approximated in terms of the free space nucleon mass mN .Asymptotically, the radial S-wave function behaves as w(r) ∼ e−κr with the inverse decaylength κ ∼ √εd.

An often used approximation is to approximate W0 by a Dirac delta-function

1√4πW0(rpn) 7→ D0δ(rp − rn) (110)

Expressing rpn in terms of the channel coordinates, we find

δ(rpn) = J −1βα δ(rβ −mA

mB

rα) (111)

where the pre-factor is the Jacobian of the coordinate transformation. The zero-rangeconstant is found to be D2

0 ' 1.58MeV 2fm3. Evaluating the DWBA integral in thisapproximation, the Jacobian is cancelled and we are left with (using r = rα)

T dpβα ∼ D0

∫d3rχ(−)∗(kβ,

mA

mB

r)ψjm(r)χ(+)(kα, r) (112)

where we have left out factors from angular momentum coupling. The angular differentialcross section for a reaction with unpolarized beam and target is

dσ

dΩ=

mαmβ

(2π~2)2kβkα

1

(2Sd + 1)(2JA + 1)

∑md,mp,MAMB

∣∣∣∣∣∑`jm

T(`jm)MAmd,MBmp

(kβ,kα)

∣∣∣∣∣2

(113)

where the full dependence on magnetic quantum numbers is shown.

35

7 Lecture 7: (d,p) Reactions as a Spectroscopic Tool

7.1 Probing the Shell Structure of 40Ca

In textbooks, 40Ca is usually presented as a perfect double-magic nucleus with Z = N = 20major shell closures. In the single particle shell model protons and neutrons fully occupythe 1s, the 1p, and the 2s and 1d sub-shells adding up to 40 nucleons in total. If that pictureis true, then transfer reaction should populate only negative parity states belonging to thefollow-up (2p, 1f) proton and neutron major shells. In particular, the population of low-lying neutron or proton states of positive parity should be blocked by the Pauli-principle.However, in nuclear matter nucleons undergo residual interactions by which nuclear wavefunctions are polarized. As a result. single particle states will be coupled to core excitedconfigurations. Thus, a single particle above the Fermi-level, e.g. a neutron in 41Ca, willscatter into a 2p1h configuration. Correspondingly, a nucleon from the Fermi-sea will bepartly in a 2h1p configuration. In order to understand nuclear many-body dynamics, suchpolarization effects must be studies experimentally.

This was the purpose of an experiment exploring the shell closure by 40Ca(d, p)41Careactions, performed some time ago at the tandem accelerator laboratory at TU Munchenby the group of G. Graw from LMU Munchen. A beam of polarized deuterons was shotwith an incident energy TLab = 20 MeV at a 40Ca target. Using the Munich Q3D spectro-graph with high acceptance and energy resolution and appropriate detectors a very preciseand complete measurement of the 41Ca spectrum up to the neutron emission thresholdcould be performed. The incident deuteron energy of TLab = 20 MeV is high enough tosuppress compound nuclear effects which affected considerably the results of former (d, p)experiments at lower energies. 2

In the investigated energy range, states with jπ = 12

±, 32

±, 52

±, 12

−and 9

2

+were observed

in 41Ca. Thus, clear signatures for only a partial closure of the neutron valence shells weredetected. The date were analyzed by large scale DWBA and – as a cross check for higherorder interactions – also by coupled channels calculations. The Fig. 18, transfer angulardistributions of cross sections and vector analyzing powers leading to the population of72

−and 5

2

−states in 41Ca are shown. Overall, both observables are rather well described

by the DWBA calculations. The quantity of central interest is the spectroscopic factorC2Sj = |X(JAJB, j)|2. Remarkably, for the 41Ca(7

2

−, g.s.) a single particle strength of only

C2S = 0.682 is found , thus indicating that about 1/3 of the shell model strength is shifted

to higher lying 72

−configurations because of core polarization.

As a further example, underlining even stronger the importance of core polarization, isthe observed population of positive parity states. As an example, In Fig. 19, results for52

+and 3

2

+are shown. In these cases, coupled channel effects are actually important.

2Details are found in the original paper, Nucl. Phys. A506 (1990) 156, by F. Eckle, H. Lenske et al.

36

Figure 18: Angular distributions of cross sections (left) and vector analysing powers (right)for the reaction 40Ca(d, p)41Ca at TLab = 20 MeV are shown, populating states in the f7/2and the f5/2 sub-shells. The 7

2

−ground state carries a single particle spectroscopic factor

of only C2S = 0.682, indicating that part of the spectroscopic strength is shifted to other72

−configurations (see F.-J. Eckle, H. Lenske et al. Nucl. Phys. A506 (1990) 156).

37

Figure 19: Angular distributions of cross sections (left) and vector analysing powers (right)for the reaction 40Ca(d, p)41Ca at TLab = 20 MeV are shown, populating states in the d5/2and the d3/2 sub-shells. The single particle spectroscopic factor are rather small, ranging

from C2S = 0.167 for the first 52

+state down to only a few percent or even promille (see

F.-J. Eckle, H. Lenske et al. Nucl. Phys. A506 (1990) 156).

38

Figure 20: 10Li energy spectrum for the d(9Li, p)10Li reaction at 100 MeV incident energyand angular region θCM = [5.5; 16.5]. The partial wave contributions obtained by theo-retical calculations are shown together with the sum which is indicated by the solid blackline. The best-fitting sum of three core-excited resonances, convoluted with the experi-mental energy resolution, is shown in the inset (see M. Cavallaro et al., Phys. Rev. Lett.18:012701 (2017)).

7.2 Continuum Spectroscopy at the Dripline with (d,p) Reac-tions

An intriguing application of (d, p) reactions is to explore dripline nuclei by adding a furtherunit of mass to an already neutron-rich nucleus. An outstanding example is the reactiond(9Li, p)10Li reaction which is a (d, p) reaction in inverse kinematics: A 9Li beam is col-liding with a (liquid or frozen or deuterated-solid) deuteron target. Instead of recordingthe projectile-like outgoing heavy nucleus, the recoiling proton is detected. 10Li is alreadybeyond the dripline, thus the reaction populates in fact 9Li + n continuum states. Thereaction is unique because it allows to study the staggering of binding properties at theneutron dripline because 11Li is bound again.

Since 9Li by itself is a short-lived nucleus (t1/2 ' 178 ms), it must be produced onsite. Thus, the experiment can only be done at a RIB facilities. A first experiment wasdone by Jeppesen et al. at REX-ISOLDE (Jeppesen et al., Phys.Lett. B642 (2006) 449)at TLab = 3.14 AMeV. A few years later, we have analysed the data in a microscopicapproach (see S. Orrigo and H. Lenske, Phys.Lett. B677 (2009) 214), explaining the

39

observed structures as pairing resonances in the continuum. Similar measurements werepreformed also at the MSU-NSCL fragmentation facility.

Very recently, the experiment was repeated at the TRIUMF ISAC-II facility with a9Li beam of 100 MeV incident energy impinging on a CD2 target (see Phys. Rev. Lett.18:012701 (2017)). In this case the recoiling proton and the outgoing 9Li were detected incoincidence. The conditions were such that a much higher statistics and a considerably im-proved energy and angle resolution could be achieved. Results of a microscopic description,including this time also core-excited continuum resonances (so-called Fano resonances) areshown in Fig.20. The measured spectrum is seen to be described rather well, allowing alsoto assign quantum numbers with clearly visible 9Li+ n p-wave and d-wave resonances.

40

8 Lecture 8: Exploring Nuclear Excitations by Inelas-

tic Scattering

8.1 General Aspects of Inelastic Nuclear Reactions

Inelastic direct reactions of the type A(a, a′)A∗ where the target A is left in an excited stateare another class of reactions widely used for spectroscopic studies. We assume that theprojectile a remains in its ground state and only the target undergoes a transition into anexcited state. The gross structure of inelastic reaction form factors is given by the folding ofthe NN interactions between projectile and target nucleons and nuclear transition densities,describing the rearrangement of nucleons within the given nucleus in the excitation processA → A∗. Inelastic reactions are driven by the interactions between target and projectilenucleons. Different to transfer reactions, the channels coordinates do not changes duringthe reaction and we have rβ = rα. The NN-interactions are given by spin-isospin exchangecomponents VST with spin and isospin operators of tensorial ranks S = 0, 1 and T = 0, 1.In non-relativistic reduction, the interaction is given by central (C), spin-orbit (LS), andtensor (Tn) components:

VNN = V(C)NN + V

(LS)NN + V

(Tn)NN (114)

which are in the given order of tensorial rank 0, 1 and 2. In the following, we considermainly the central, rank-0, interactions, thus using VNN = V

(C)NN . In coordinate space, the

NN interaction is given by

V(C)NN (x) =

∑S,T=0,1

VST (x) [σa · σA]S [τa · τA]T + VLS(x) + VT (x) (115)

VST (x) are radial form factors where x = ra− rA denotes the relative distance of a nucleonin the projectile a and in the target A. As before, x may be expressed in terms of theintrinsic coordinates ri and the channel coordinate rα:

x = r1 − r2 + rα (116)

where r1 denotes a nucleonic coordinate in the projectile. The scalar products of spinand isospin operators – corresponding to rank-0 tensors – are conveniently expressed inspherical representation, e.g.

σa · σA =∑

MS=−1,0,+1

(−)MSσa,MSσA,−MS

. (117)

where

σ± = ∓ 1√2

(σx ± iσy) ; σ0 = σz. (118)

A widely used approach is to define interactions which already include anti-symmetrization.In spin and isospin, this amounts to act on VNN with the exchange operator

Px(1, 2) = −Pσ(σ1, σ2)Pτ (τ1, τ2) (119)

Pσ(σ1, σ2) =1

2(1 + σ1 · σ2) ; Pτ (τ1, τ2) =

1

2(1 + τ1 · τ2) (120)

41

The remaining anti-symmetrization of the spatial coordinates is taken care of by intro-ducing an effective momentum-dependent contact potential VST = HST (k2α)δ(x) which isjustified by the fact that the exchange channel (often called u-channel in analogy to theMandelstam notation) involves high momentum components ∼ 2kF ∼ 500 MeV/c. Theclear advantage of this method is that the reaction form factors become local functions inthe channel coordinates. Using this convention, we agree on

VST (x) 7→ VST (x) +HST (k2α)δ(x) (121)

The choice of VNN will be discussed later.

8.2 Interactions in Momentum Representation

An elegant method is to presenting VNN in momentum space:

VNN(p) =

∫d3p

(2π)3eip·xVNN(x). (122)

which, in fact, is the more general representation because it gives access to the originalderivation of the static NN-potentials from scattering amplitudes.

The Fourier transform of the form factors are

VST (p2) = 4π

∫ +∞

0

dxx2j0(px)VST (x) +HST (k2α) (123)

where j0(x) = sinxx

is the 0th order spherical Bessel function.We introduce the nuclear transition operators

OST,IN(p, r) = eip·r [σ]S [τ ]T (124)

These tensor operators are one-body operators acting either in projectile or target. Theinteraction is then obtained as

VNN(x) =∑ST

∫d3p

(2π)3eip·rαVST (p2)OST (p, ra) ·O†ST (p, rA) (125)

The dot-product indicates that the transition operators have to be contracted to totalscalars in spin and isospin.

8.3 Inelastic Form Factors

An important advantage of the momentum representation is the separation of projectileand target degrees of freedom. The nuclear matrix element is now given by

Fβα(rα) =∑ST

∫d3p

(2π)3eip·rαVST (p2)F

(a)ST (p) · F (A)†

ST (p) (126)

42

with the nuclear transition form factors

F(a)ST (p) = 〈a∗|OST |a〉 (127)

F(A)ST (p) = 〈A∗|OST |A〉 (128)

which can be further reduced into irreducible tensor components according to the orbital,spin, and total angular momentum transfer allowed by the transitions. The nuclear con-figuration spaces include, of course, also the ground states of projectile and target. If,for example, the projectile remains in its ground state, i.e. a∗ = a, F (a)(p) is essentiallygiven by the Fourier transform of the projectile isoscalar or isovector ground state densitydistributions ρ0,1 = ρn ± ρp. If a is an even-even nucleus with ground state spin Jπa = 0+,then only the S = 0 components of VNN are selected. If, in addition, a is an N = Znucleus, also the isovector (T = 1) parts will be strongly suppressed, up to minor isospinsymmetry breaking induced by electromagnetic interactions. Hence, by an appropriatechoice of projectile, we have at hand to enhance selected interaction channels and suppressothers, for example:

• If a is a nucleon all spin and isospin channels will contribute.

• If a is a deuteron, only T = 0 isoscalar interactions will be active.

• If a is an alpha-particle only S = 0 = T spin and isospin-scalar interactions areprobed.

For a nucleon projectile, the form factors F(a)ST (p) reduce to constants, corresponding to

a delta-function distribution in coordinate space. Thus, a nucleon is treated as a pointparticle. That is an acceptable approximation as long as the energy and momentum transferis small compared to the rest mass and the size of the nucleon. Otherwise, internal degreesof freedom of the nucleon can be excited, i.e. nucleon resonances N∗ will become accessible,as e.g. the ∆33(1232) state, which are decaying strongly into pions and the nucleon.

The transition operators include in fact a rich multipole structure as is evident whenexpanding the plane waves into partial waves. That leads to the multipole transitionoperators

TLST ;IN(p, r) = jL(pr)[iLYL(r)⊗ σS

]INτ T (129)

coupled to total angular momentum transfer I,N . These operators have good parityπL = (−)L. For a given nuclear transition, say JπA → Jπ

∗A∗ the selection rules are |JA−JA∗| ≤

I ≤ JA + JA∗ and ππ∗πL = +1. In order to evaluate the nuclear transition densitiesexplicitly, second quantization is the appropriate method. We postulate a complete setof single particle functions ϕq and the related set of Fermion operators (a+q , aq), whereq = (nlsjm) denotes a complete set of quantum numbers. In this way and using theWigner-Eckardt theorem, we obtain in second quantization

TLST ;IN(p, a+a) =∑qq′

〈ϕq||jL[iLYL ⊗ σS

]Iτ T ||ϕq′〉

[a+q aq′

]IN

(130)

43

thus representing the multipole operator in terms of a set of reduced matrix elements –which are C-numbers – and particle-hole type transition operators of good total angularmomentum and parity. Above, ajm = (−)j−maj−m denotes the time-reversed operators,in the present context to be identified with the hole creation operator b+jm. We define thecoupled particle-hole state operators

A+IN(ph) =

[a+p ah

]IN

=∑mpmh

(jpmpjh −mh|IN) a+jpmp ajhmh (131)

and the related time-reversed operator is AIN = (−)I+NAI−N . Excited nuclear states arerepresented in that basis by the correlated state operator as a coherent superposition ofparticle-hole components

Ω+JCMC

=∑qq′

[xJC (qq′)A+

JCMC(qq′)− y∗JC (qq′)AJcMC

(qq′)]

+O(A+2, A2) (132)

creating the state |JCMC〉 by acting on the ground state, |JCMC〉 = Ω+JCMC

|A〉. The

part linear in the state operators A+, A is known as the RPA state vector. However, ingeneral higher order terms will contribute also. The configuration amplitudes are obtainedfrom nuclear structure calculations, diagonalizing HA with the above ansatz, 〈C|HA|D〉 =ECδCD.

The nuclear transition form factors are found as

F(CA)LST ;IN(p) = 〈JCMC |TLST ;IN |JAMA〉

= (−)JA−MA (JCMCJA −MA|IN)1√

2I + 1〈JC ||TLST ;I ||JA〉. (133)

The reduced matrix element is found to be given by an integration over the transitiondensity, obtained by inserting a delta-function kernel

ρ(JCJA)LST ;I (r) =

1

I〈JC ||

1

r2δ(r − r′)

[iLYL ⊗ σS

]Iτ T ||JA〉 (134)

With the linear part of the state operator, Eq.(132), we find

ρ(JCJA)LST ;I (r) =

∑qq′

1

I〈ϕq||

1

r2δ(r − r′)

[iLYL ⊗ σS

]Iτ T ||ϕq′〉(

xJC (qq′) + (−)Sy∗JC (qq′))

(135)

The transition densities are of universal character because every one-body operator canbe evaluated correspondingly, provided the specific radial form factor is inserted into theabove integral. The form factor is obtained as

F(CA)LST ;IN(p) =

∫ ∞0

drr2jL(pr)ρ(JCJA)LST ;I (r). (136)

44

Electromagnetic transitions are another important case. The (reduced) transition matrixelements for an electric transition of multipolarity L

MJCJA(EL) = e

∫ ∞0

drrL+21

2

(ρ(JCJA)L00;L (r)− ρ(JCJA)L01;L (r)

)(137)

where e is the electric charge and the sum of T = 0 and T = 1 transition densities projectsout the proton component.

For L ≥ 1, the nuclear transition densities are typically dominated by functional struc-tures peaked at the nuclear surface. This reflects the softness of the nuclear surface againstexternal perturbations. In spherical nucleus the lowest excitations at one or a few MeV ofexcitation energy are always surface vibrations. They are of collective character, i.e. givenby a phase-coherent superposition of a number of particle-hole states. To excite the nuclearinterior requires a much stronger impact. Compressional states like the Giant MonopoleResonance (GMR) are found at much higher energies, typically close to Ex ∼ 20 MeV.

8.4 The Collective Model of Inelastic Scattering

In many cases, the microscopic structure of nuclear excited state is not known or thequantum numbers have to be determined from data before a comparison to theory becomesmeaningful. For those purposes, a schematic and flexible, but realistic model is needed.That problem is addressed by the so-called collective models of inelastic scattering. Thefocus is on the description of matrix elements which are less sensitive to details of theradial shape. That is the purpose of the collective model for vibrational and rotationalexcitations. While the microscopic approaches emphasize the nucleonic, i.e. granularaspects of nuclear matter, the collective models are based on the cooperative propertiesof interacting many-body systems. This a concept well known form classical physics: it isthe basic concept underlying hydrodynamics. Classical hydrodynamics postulates the useof continuum dynamics by the law of large numbers. That is not the case in nuclei whichat the best consist of 200 to 300 particles. However, the genuine quantum mechanicalcharacter of nuclei supports a continuum view. The nuclear wave functions are acting likea smoothing factor. This is clearly visible in the smoothed-out shapes of radial densitydistributions, quite different from those of a classical systems of the same particle number.

The collective model of nuclear excitations incorporates elements of the liquid dropmodel. The description was invented by Bohr and Mottelson and used in studies of vi-brational and rotational excitations. The founding (and defining) idea is to consider thenucleus as incompressible matter but with a deformable surface. That is accomplishedby treating the nuclear radius as a dynamical quantity, oscillating around the equilibriumvalue RA

R(Ω) = RA

(1 +

∑λ≥2,µ

αλµY∗λµ(Ω)− α00

)= RA + δR(Ω) (138)

where αλµ are complex-valued expansion coefficients. The monopole counterterm α00 isfixed by the constraint that the volume integral of the nuclear density distribution ρA(r, R)

45

is conserved: ∫d3rρA(r, R(Ω)) = A (139)

Assuming for the moment a step function distribution, ρA(r, R) ' ρcΘ(R − r) we find upto second order in δR

A.= ρc

1

3

∫dΩR3(Ω) = ρ0

1

34πR3

A

(1 +

3

4πR2A

∫dΩδR2(Ω)

)(140)

finding to the monopole amplitude as

α00 =1

4π

∑λµ

|αλµ|2 (141)

The λ = 1 term vanishes because it would describe a translation of the center-of-mass.Thus, we impose the further constraint∫

d3rrρ(r, R).= 0 (142)

The deformed density is expanded into multipoles

ρ(r, R(Ω)) =∑`m

ρ`m(r)i`Y`m(Ω) (143)

and by inversion

ρ`m(r) =

∫dΩi−`Y ∗`m(Ω)ρ(r, R(Ω)) (144)

We introduce also the mass multipole moments

q`m =

∫d3rρ(r)r`Y`m = i−`

∫ +∞

0

ρ`m(r)r`+2 (145)

Corresponding moments, evaluated for the charge distribution, determine the electric mul-tipole moments M(EL) and transition probabilities B(EL) ∼ |M(EL)|2.

As it stands, the above relation is used in classical physics to describe the surfacevibrations of a droplet or the rotational motion of a deformed droplet. These are also theareas of applications in nuclear physics. For rotational motion of a deformed nucleus, thedeformation amplitudes αλµ are simple numbers and the moments, Eq.(144), are easilyevaluated numerically. In that way one accounts for the effect of a given deformation λ toall orders so it contributes to ` = λ, 2λ, 3λ....

In order to comply with the quantum mechanical properties of vibrational modes, quan-tal aspects are incorporated by replacing the expansion coefficients by quantum mechanicaloperators of bosonic character. Thus,

R(Ω) = RA

(1 +

∑λµ

αλµiλY ∗λµ(Ω) + α00

)(146)

46

The quantal treatment requires an expansion into a Taylor series

ρ(r, R) = ρ(r, RA) +∑n

1

n!

(δR

RA

)nRnA

∂nρ(r, R)

∂Rn |R=RA

(147)

results in a series of single (n = 1 and multiple excitations (n ≥ 2) of the surface modes.For a truncation at n = 1 we find

ρ(1)`m(r) = i`f

(1)` (r) ; f

(1)` (r) = RA

∂ρ(r, RA)

∂RA

(148)

thus the multipole densities are in fact independent of the multipolarity and this orderthe multipoles are given by ` = λ. Inserting this ansatz into the form factor, we find the(dimensionless) transition coefficient

NCAλ ==1√

2λ+ 1〈C||αλ||A〉. (149)

and by convention the multiplicity factor 2λ+ 1 is extracted by defining

NCAλ =βCAλ√2λ+ 1

(150)

NCAλ describes the fractional parentage by which the mode αλ is contained in the transitionA→ C.

In second order, however, we have

ρ(2)`m(r) = i`f

(2)` (r)

1√16π

∑λ1λ2

λ1λ2ˆ

(λ10λ20|`0) [αλ1 ⊗ αλ2 ]`m (151)

and several modes are contributing to an excitation of a given multipolarity `.Other collective model formulations are also in use. A popular version is the Tassie

model, derived from the velocity fields studied in hydrodynamics. The deformation isassigned to the equi-density surfaces by using instead of δR

δr(Ω) = r

[1−

∑λ≥2,µ

(r

RA

)p−2αλµY

∗λµ(Ω) + α00

]) (152)

thus using r → r + δr(Ω). Volume conservation requires

α00 =1

4πR2p+4

∑λµ

|αλµ|2pr2p−3 (153)

The 1st order multipole densities are found as

f(1)` (r) = −

(r

RA

)p−2rdρ(r)

dr(154)

47

Figure 21: Comparison of the standard and Tassie model transition densities (left) andTassie model vs. a microscopic RPA transition density for 208Pb(3−) (right).

In the Tassie model p = ` is chosen. For ` = 2, the multipole functions are those of thestandard model but multiplied by r. For ` ≥ 3 the multipole densities gain an even strongerweighting in the surface region and the maximum is pushed to large radii. In Fig. 21 a fewTassie and standard multipole functions are compared, corresponding to a nucleus withA ∼ 40. Also shown is a comparison for a case of physical interest, namely a comparisonof the Tassie model density to a microscopic transition density for 208Pb(0+, g.s.) →208

Pb(3−1 , 2.6). The surface region is well reproduced but in the nuclear interior deviationsare observed. Translated to momentum transfer, this means that the dominating lowmomentum components (qL ∼ O (L/RA)) of the transition will be well described butdeviations have to be expected at large momentum transfer.

For practical applications we recall that nuclear density distributions are characterizedby the radius and the surface thickness. For A > 20, as in the examples of Fig.21, nucleardensity distributions are well described by a Fermi function

ρ(r, R) = ρ01

1 + e(r−R)/cA, (155)

with the half-density radius R ∼ RA the diffusivity cA. For nuclei not too far away fromthe valley of stability, one finds the nearly mass independent value cA ' c = 0.5 fm andRA ' r0A

1/3, r0 ∼ 1.145 fm. The central density ρ0 is fixed by the constraint∫d3rρ(r, RA) = A (156)

which results in

ρc =3A

4πR3A

1

1 + (πcA/RA)2. (157)

In a heavy nucleus one finds ρc ' 0.16fm−3. Defining x = (r −R)/cA and

f(x) =1

1 + ex(158)

48

Figure 22: A radial Fermi-distribution. The half-density radius R and the surface thicknessδ are indicated.

we find for 1st derivative of a Fermi-function

f ′(r −RcA

) =∂

∂Rρ(r, R) = − ∂

∂rρ(r, R) = − 1

cA

df(x)

dx |x=(r−R)/cA

(159)

which is a function with a maximum at r = R and a full width at half maximum ∆ =cA ln 3+

√8

3−√8∼ 3.525cA which gives ∆ ∼ 1.763 fm for cA = c. Another widely used definition

of the nuclear surface thickness is the distance δ between the point where ρ(r) = 0.9ρ0and ρ(r) = 0.1ρ0. For a Fermi-shaped density one finds δ = cA4 ln 3 ' 4.394c. Hence,δ ' 2.197... fm is the nearly mass-independent value of the surface thickness in well boundnuclei. In Fig.22 a Fermi distributions is shown and the significant features are marked.

An interesting application is to consider the derivative form factors as global transitionform factor, describing in the average the radial dependence of the microscopic transitiondensities sufficiently well. Let the transition JA → JC be described by the microscopictransition density ρTLSI . We use the scaling ansatz

ρTLSI(r) ∼ NTLSIδρ(r) (160)

Fixing the normalization e.g. by

MJCJA(EL) = NJCJAL

∫ ∞0

drrL+2δρ(r). (161)

allows to establish a relation on the level of matrix elements. The amplitude NJCJA;L isfound to be a measure of the transition strength in units of the multipole moments of the

49

derivative form factor. An often used parametrization is

NJCJA(LST ; I) =1√

2I + 1βJCJA(LST ; I) (162)

and to introduce the deformation length δJCJA(LST ; I) = βJCJA(LST ; I)RA where RA isthe nuclear radius. While NJCJA(LST ; I) and βJCJA(LST ; I) will change with the choice ofδρ, it is assumed that the deformation length is a conserved quantity. In many applications,this approach is applied to the optical potential by defining δU according to Eq.(??) andintroduce corresponding deformation lengths which are determined in fits to cross sectiondata.

8.5 Proton Inelastic Scattering

In the literature a large amount of work on inelastic scattering with various projectiles isfound, ranging from electrons and pions to protons, neutrons, deuterons and complex lightnuclei starting at the A=3 systems and the alpha-particle to Li and Be isotopes. Sincethe late 1970ties also heavy ions have been used as probes in peripheral direct reactions.Proton inelastic scattering has been an important tool for spectroscopic studies of nucleifrom low-lying vibrational and rotational excitations up to the region of giant resonances.Initially, many experiments were done at incident energies below 100 MeV. The energyrange was extended close to the 1 GeV region when the Los Alamos facility came intooperation in the 1970ties.

One the first systematic spectroscopic studies with (p, p′) reactions was performed byLewis et al. on a 208Pb target at TLab = 54 MeV. A few examples of the measured inelasticangular distributions are shown in Fig.23. The DWBA calculations were performed withthe collective model approach, in this case applying the procedure directly to the opticalpotential. The deformation amplitudes were determined by fits to the data. Selectedresults are shown in Fig.24

An interesting extension of proton inelastic scattering is to study ejectile spectra overa wide energy loss range becoming comparable to the incident energy. Such investigationsgive access to pre-equilibrium phenomena where the p + A system gradually approachesstatistical equilibrium.

For that purpose, the Multi-Step Direct Reaction (MSDR) theory was developed in the1980ties, known as Tamura-Udagawa-Lenske (TUL) theory. Pre-equilibrium spectra aredescribed by a sequence of DWBA-like one-step and second-DWBA two-step processes.MSDR theory was the first quantum mechanical description of pre-equilibrium spectra. InFig. 25 angular distributions for (p, p′) pe-equilibrium scattering on 27Al and 209Bi at theincident energy TLab = 62 MeV are shown. Nuclear response were calculated microscopi-cally as discussed in the previous paragraph. While at low energy losses the cross sectionsare dominated by one-step scattering, two-step processes are taken over at larger energylosses.

50

Figure 23: Angular distributions of the reactions 208Pb(0+, g.s.)(p, p′)208Pb(Jπ, Ex) atTLab = 54 MeV. Excitation energies and multipolarities are shown in the figure (Lewiset al., Phys.Rev. C7 (1973) 1966.

51

Figure 24: Vibrational amplitudes βL extracted by fits to the208Pb(0+, g.s.)(p, p′)208Pb(Jπ, Ex) data shown in Fig.23 (Lewis et al., Phys.Rev. C7(1973) 1966.

52

Figure 25: Inelastic pre-equilibrium scattering of protons. MSDR angular distributionsare compared to data. The sum of one- and two step cross sections (full lines) and thetwo-step cross sections (dashed lines) are displayed. At large energy losses, second orderDWBA two-step process are dominating (Phys. Rev. C26 (1982) 379).

53

Figure 26: Angular distributions for 12C +12 C elastic scattering at TLab = 240 MeV andTLab = 300 MeV. The cross sections are normalized to the Rutherford cross section (seeNucl.Phys. A668 (2000) 3).

54

8.6 Choice of Interaction and the Double Folding OMP

A question of its own interest is how to choose the NN interaction. In the so-calledtρ- or impulse approximation the free space NN T-matrix is used, VNN = TNN . Thus,the interaction contains complex-valued form factors VST , leading to complex reactionform factors Fβα. The impulse approximation is open to critic because it neglects thein-medium modifications of NN interactions. In particular, contributions from 3-bodyinteractions and other induced dispersive many-body interactions is neglected. In thepast, much work has been spent to overcome these shortcomings. The work of Love andSatchler (see Physics Reports 55, (1979) 183) and Khoa and Satchler (see Nucl.Phys. A668(2000) 3). In the latter work, a generalized double-folding model for elastic and inelasticnucleus-nucleus scattering was presented. It was designed to accommodate effective NNinteractions incorporating 3-body effects that depend on the density of nuclear matterin which the two nucleons are immersed. The density dependence was implemented inparameterized form by adjusting the so-called M3Y-representation of the Brueckner G-matrix obtained from the Paris NN potential such that the saturation properties of infinitenuclear matter are correctly reproduced. The knock-on exchange effects were included ina local momentum approximation, as discussed above.

Before going on, we note that the formalism discussed above includes as a specialcase also the scenario when a and A remain in their respective ground state, i.e. elasticscattering. Thus, also the elastic double folding potential produced by the interaction VNNis recovered. Assuming 0+ ground states, the isoscalar elastic potential is

U(aA)0 (rα) =

2

π

∫ +∞

0

dpp2V00(p2)j0(prα)ρ0A(p2)ρ0a(p

2) (163)

and the isovector potential U(aA)1 is given by a corresponding expression, involving V01 and

the isovector densities. Using the NN T-matrix, a complex potential is generated whichmay serve as a leading order estimate for an optical potential. The double folding potentialsare of particular interest for reactions were empirical optical potentials are not available.Typical cases are heavy ion reactions and reactions involving exotic nuclei. In both cases,elastic scattering are rare or completely absent. In Fig. 26 cross sections for 12C +12 Celastic scattering at two incident energies are have been analyzed in the Khoa-Satchlerapproach. The theoretical results describe the data rather satisfactorily.

The same interaction was then used in DWBA calculation for inelastic scattering tothe first excited state in 12C, a 2+ state at Ex = 4.44 MeV. The reaction form factorwas obtained by folding collective model transition densities with the density dependentNN-interaction. The deformation length δ2 = β2RA have been used as free parameters.The DWBA angular distributions shown in Fig. 27 agree quite well with the data over awide range of scattering angles. The values of the deformation lengths are shown in thefigure.

55

Figure 27: Inelastic 12C +12 C scattering data at TLab = 240 MeV and TLab = 300 MeV,exciting the 12C(2+

1 , 4.44) state. The DWBA results were obtained by using the inelasticfolded potential with the density dependent treatment of Khoa and Satchler of the mediumcorrection (solid curves). Also shown are cross section results obtained by the phenomeno-logical deformed optical potential approach (dotted curves) - see Nucl.Phys. A668 (2000)3.

56

9 Lecture 9: Heavy Ion Single Charge Exchange Re-

actions

Charge exchange reactions, converting a proton into neutron or vice versa, are a classof inelastic reactions of particular high interest. They are probing the same type of nu-clear configurations as in beta-decay, thus establishing a link between strong and weakinteractions. The similarities of strong and weak interactions are elucidated on in Fig. 28.

In the 1970ties measurements of (p, n) cross sections at the Indiana University CyclotronFacility (IUCF) led to the discovery of the Giant Gamow-Teller-Resonance (GTR), a (L =0, S = 1) 1+ mode, excited in the target nucleus by the rather simple στ− operator. InFig, 29 recent (p, n) and (n, p) spectra, measured at the RCNP facility at Osaka are shown.

Figure 28: Operator structures of strong and weak interaction vertices.

The theory of single charge exchange reactions follows closely the approach sketched insection 8. The excitations are proceeding through operators including the τ± isospin stepoperators. Thus, charge exchange are probing specifically the isovector response of nucleias is done also by beta-decay. For composite projectiles, the direct NN-collisional processmay be contaminated by two-step transfer processes. Their importance, however, dependson Q-values and other matching conditions. At energies above a few 10 MeV, two-steptransfer process are rapidly dying out.

In a recent paper, Fukui and Minato have analyzed the data of Fig. 29 in a microscopicapproach, including 1p1h and 2p2h configurations. In Fig. 30 results of their calculationsare shown.

Alternatively, (3He, t) reactions have become an important tool for studying the nuclearisovector response in single charge exchange reactions. Heavy ion single charge exchangereactions have been found to provide complementary spectroscopic information, not ac-cessible at the same level by light ion reactions. Results of an early study of the reaction12C(12C,12N)12B at TLab = 70 AMeV are shown in Fig.31 .using double folding optical po-tentials and reaction form factors as discussed in section 8 were used, including also rank-2tensor interactions. The interference of central and tensor form factors is seen found to beessential for the reproduction of the measured angular distribution.

57

Figure 29: (p, n) and (n, p) charge exchange cross sections measured the RCNP facility atOsaka. The beam energy was 300 MeV (Phys.Rev.Lett. 103, 012503 (2009)).

58

Figure 30: The differential cross section of the 48Ca(p, n)48Sc(GT ) reaction at Tlab =295 MeV for the low-lying 1+ GTR resonance. The calculated result with (without) the2p2h configuration shown by the solid (dashed) line is compared with the experimentaldata (open circle) taken from the measurements shown in Fig. 29. For further discussionsof the theory see Phys.Rev. C96 (2017) 054608.

10 Lecture 10: Charge Exchange Reactions and Dou-

ble Beta-Decay

Second order quantal processes like heavy ion double charge exchange reactions are ofgenuine reaction theoretical interest. First of all, until now heavy ion DCE reactions havenot been studied, neither experimentally nor theoretically. Some attempts were made on(π+, π−) reactions but their notoriously bad energy definition of the incoming beams isunfavorable for spectroscopic work. Thus, double charge exchange reactions with heavyions heavy ion are much better suited for explorations of weakly populated transitions.Here, we consider collisional charge exchange processes given by elementary interactionsbetween target and projectile nucleons. In accordance with explicit calculations, the mean-field driven transfer contributions are neglected because they are at least of 4th order forDCE reactions considered here. Thus, only processes with changes of the charge partitionsbut leaving the projectile-target mass partition unaltered will be discussed.

A central question is whether we can identify on the elementary level a correspondencesbetween strong and weak interaction processes. The answer is yes, as illustrated in Fig. 32.Under nuclear structure aspects, the 0ν2β decay of a nucleus is nothing but special classof two-body correlation, sustained by the exchange of a (pair of) Majorana neutrino(s)between two nucleons where the interaction vertices are given by the emission of virtualW± gauge bosons. The strong interaction counterpart is a two-nucleon correlation built up

59

Figure 31: Direct charge-exchange differential cross section for 12C(12C,12N)12B at TLab =70 AMeV. The central (long-dashed) and tensor (short-dashed) contributions and the co-herent sum of both (full line) are shown (see H. Lenske et al., Phys.Rev.Lett. 62 (1989)1457).

60

Figure 32: The weak 0ν2β decay in comparison to the analogous strong interaction process.

by the exchange of a virtual charge-neutral quark-antiquark (qq) pair accompanied by theemission of a charge qq component, thus changing at the same time the nucleonic charges.Similar to the weak process, the vertices are originating from gauge bosons, here given bythe initial emission of gluons which materialize into two qq pairs. At the end, the highlyoff-shell qq compounds will decay into mesons, preferentially into pions but also multi-pionconfigurations like the scalar and vector mesons.

Such interaction process may occur frequently in nuclei, both on the weak as well ason the strong interaction scale. A generic type of diagram converting a pair of neutronsinto a pair oh protons is shown in Fig. ??. Processes of that type remain unobserved ifthe emitted charged mesons (or electrons) are reabsorbed by the same nucleon. This willlead to vertex correction and contributions to the nucleon in-medium mass operator of a,however, negligibly small strength. Nevertheless, it is worthwhile to keep in mind thatwe are dealing here with phenomena belonging to the large class of nuclear ground statecorrelations beyond the commonly studied mean-field sector.

Both processes become of interest if they reveal their existence and nature in observablesignals. In this respect, we encounter a fundamental difference between 0ν2β decay and thehadronic process: Only the former may occur in an isolated nucleus while the latter one isforbidden by energy conservation. Thus, in order to observe the double-meson emission bya nucleon pair a partner nucleus is required which takes care of the virtuality of the processby absorbing the two charged virtual mesons. For that purpose, heavy ion double chargeexchange reactions are the ideal tool. Diagrammatically, the microscopic dynamics of sucha reaction are indicated in Fig. 33. The target undergoes a correlated double-meson pairdecay nn→ pp+π−π− and the projectile absorbs the pions by the simultaneous excitationof two np−1 -type configurations. Other meson configurations will contribute as well.

The whole reaction will proceed as a one-step reaction via a special kind of two-bodyinteraction generated by the correlation diagram. Denoting the (in-medium) pion-nucleon

61

Figure 33: Generic diagram illustrating the hadronic surrogate process for 0ν2β decay. Avirtual nn → ppπ−π− scattering process, causing the ∆Z = +2 target transition A → B,is accompanied by nnp−1p−1 double-CC excitation in the projectile. As indicated, otherisovector mesons as e.g. the rho-meson iso-triplet will contribute, too. Contributions ofcrossed diagrams are not displayed.

62

T-matrix by TπN,π′N ′ , the target-part of the interaction is in a somewhat symbolic notation

V (MDCE)(13,24) ∼ Tπ−p,π0n(1,3)Dπ0(1− 2)Tπ0n,π−p(2,4) (164)

where the n→ p target transitions are denoted by 1 and 2, respectively. The coordinates3 and 4 indicate the outgoing charged pions, inducing the complementary transitions inthe projectile. The correlation built up by the neutral pion is described by the propagatorDπ0 . A decomposition into irreducible tensors gives rise in particular to an effective rank-2iso-tensor projectile-target interaction of operator structure [τ1A ⊗ τ2A]2 · [τ3a ⊗ τ4a]2. Werecognize immediately the similarity to the nuclear matrix element of 0ν2β decay, justi-fying the name Majorana-DCE. It we consider, on the other hand, the effective operatorunderlying the conventional double-SCE two-step reaction mechanism, we find

V (DSCE)(13,24) ∼∑cC

TNN(3,4)GcC(2− 4,1− 3)TNN(2,1) (165)

where TNN is the isovector nucleon-nucleon T-matrix and GbB denotes the (full many-body)propagator of the intermediate nuclei reached in the first SCE reaction step. The differ-ences are obvious, visible in particular in the different operator structures. At present,the strengths of the nucleon-nucleon and the pion-nucleon T-matrices are taken from data.More refined description will be scrutinized in future work by referring to the date baseavailable for free space nn → ppπ−π− reaction. In fact, the charge-conjugated reactionpp → nnπ+π+ reaction and other double-pion production channels were investigated atCELSIUS@Uppsala and COSY@Juelich and more recent also at HADES@GSI, supple-mented by several theoretical studies combining meson exchange and resonance excitationhave been performed by the Valencia group and in somewhat extended form by Xu Caoet al. at Beijing and Giessen.

The full DCE reaction amplitude is given by the coherent sum of the MDCE and theDSCE amplitudes:

Mαβ ∼ 〈χ(−)†β , bB|V (MDCE) + V (DSCE)|aA, χ(+)

α 〉 = M(MDCE)αβ +M

(DSCE)αβ (166)

A caveat of heavy ion scattering is the strong role played by initial (ISI) and final (FSI) stateion-ion interactions. The are typically well described by optical potentials. Above, thoseeffects are taken care off by the distorted waves χ

±()α,β . As discussed elsewhere a momentum

space representation allows to separate the ISI/FSI contributions and the nuclear transitionform factors,

Mαβ ∼∫d3pNαβ(p)Mαβ(kα − kβ − p) (167)

where we have introduced the distortion coefficient

Nαβ(p) =1

(2π)3〈χ(−)†

β |eip·r|χ(+)α 〉 (168)

which in the strong absorption limit acts mainly as a scaling factor, typically reducing theforward cross section by several orders of magnitudes compared to the plane wave limit. In

63

Figure 34: Angular distribution of the DCE reaction 18O + 40Ca→ 18Ne+ 40Ar at Tlab =270 MeV. Theoretical results are compared to the data, measured at the LNS Catania byCappuzzello et al., Eur. Phys. J. A51 (2015) 145. The theoretical cross section includesMDCE and DSCE contributions, added up coherently (see H. Lenske, presented at CNNP2017 and Jour. Phys. Conf. Ser. 2018).

the calculations discussed below, ion-ion interactions are treated in the strong absorptionblack disk limit.

First results of a DCE calculation along the line discussed above are shown on Fig.34 and compared to recent NUMEN data of Cappuzzello et al. for the reaction 18O +40Ca → 18Ne + 40Ar at Tlab = 15 AMeV. The transition strengths are taken from QRPAcalculations. The transition potentials were approximated by Gaussians. Only the pioniccontributions were included. The forward peak of the angular distribution is dominated,in fact, by the MDCE component. Overall, the data are described decently well in view ofthe exploratory character of the calculations.

The MDCE and, to a slightly lesser degree also the DSCE process, support large mo-mentum transfers, as it is expected also for the 0ν2β -decay. This is illustrated in Fig. 35where the partial MDCE and DSCE cross sections are shown separately. By comparisonto Fig. 34 it is seen that the coherent sum of both amplitudes is leading to a complexinterference pattern, especially at large momentum transfers. From Fig. 35 is evident thatfor the 18O+ 40Ca system the MDCE process dominates at small scattering angels, at leastfor the considered case of purely pionic Majorana-correlations.

As concluding remark, we emphasize the large research potential of newly introduced re-action scenario for heavy ion double charge exchange reactions. At the diagrammatic level,structures similar to 0ν2β matrix elements have been identified. The hadronic Majorana-DCE process is accessible only by reactions of composite nuclei. The lightest possible

64

Figure 35: Angular distribution of the DCE reaction 18O + 40Ca → 18Ne + 40Ar atTlab = 270 MeV. The MDCE and DSCE partial cross sections are shown separately incomparison to the LNS data of Cappuzzello et al. (Eur. Phys. J. A51 (2015) 145). Bothprocesses support large momentum transfers which is even more pronounced for the MDCEcomponent.

system is the reaction 3H+ 3He→ 3p+ 3n. Here, we have discussed explicitly the case of aDCE reaction with medium mass ions at relatively low incident energy. ISI and FSI ion-ioninteractions were taken into account and the quantum mechanical coherence of the MDCEand the DSCE reaction mechanism was treated properly. The strongly forward peakedmeasured angular distributions indicate a direct mechanism which indeed is confirmed bythe calculations. These first results are very promising by indicating a new way of access-ing second order nuclear matrix elements of charge changing interactions. Together withthe much better studied SCE reactions and their established usefulness for spectroscopicwork, heavy ion DCE reactions are opening a new window to high-precision spectroscopy.Although it will not be possible to insert the extracted matrix elements directly into a0ν2β analysis, DCE reactions provide a unique way to validate nuclear structure modelsunder controllable laboratory conditions by comparison to data on processes of comparablephysical content. New impact on theoretical investigations in both reaction and nuclearstructure theory is demanded for a quantitative understanding of these special reactions.Although the present calculations do not yet include the full spectrum of contributions,they are establishing the hadronic Majorana-DCE reaction mechanism. The refinementsmay lead to changes in detail but will not alter the overall picture. An exciting and en-couraging result is that the MDCE process is clearly visible, even dominating the crosssection at extreme forward angles.

65

A Scattering on a Complex Potential

From the previous section we have to conclude that non-hermitian Hamiltonians are ofcentral importance for the description of nuclear scattering data. In this section, we brieflyreview the most important aspect of non-hermitian quantum mechanics. For that purposewe use a potential of the structure U(r) = V (r)− iW (r), assuming spherical symmetry:

V (r) = V0fV (r) ; W (r) = W0fW (r) (169)

and W0 > 0. The radial form factors are chosen of Wood-Saxon shape

f(r) =1

1 + e(r−R)/a(170)

sometimes extended by derivative terms which we neglect here. If the diffusivity parametera approaches zero, the Heaviside step function is found as the limiting case,

f(r)→ Θ(R− r). (171)

The wave equation is(K + V (r)− ω)χ = 0 (172)

By writing χ(r)0 = u(r) + iv(r) the Schrodinger equation is transformed into the real2-by-2 system

(K + V (r)− ω)u(r) +W (r)v(r) = 0 (173)

(K + V (r)− ω) v(r)−W (r)u(r) = 0 (174)

showing that χ has a non-trivial imaginary part. Asymptotically, i.e. at distances largecompared to the potential range, the partial waves behave as

χ`(r)→ F`(kr) + C`H(+)(kr) (175)

where F`(x) = xj`(x) is a spherical Bessel function and H(+)(x) = xh(+)` (x) is a spherical

Hankel function. For charged systems, the corresponding Coulomb waves are enteringinstead.

A.1 Non-Hermiticity and Bi-Orthogonality

Since the underlying Hamiltonian is non-hermitian, the dual space elements are not givenby simple complex conjugation. This becomes clear by considering the conjugate Schrodinger-equation (

K + U † − ω)χ = 0 (176)

showing that the conjugate solution is determined by a potential with positive imaginarypart, i.e. a potential creating additional probability flux.

66

The orthogonality condition is changed to the bi-orthogonality condition∫d3rχ(±)†(k, r)χ(±)(q, r) = (2π)3δ(k− q) (177)

and the completeness relation reads∫d3k

(2π)3χ(±)(k, r)χ(±)†(k, r′) = δ(r− r′) (178)

Thus, optical model Green functions are of the structure

G(±)(r, r′) =

∫d3k

(2π)3χ(±)(k, r)χ(±)†(k, r′)

ω + iη − ωk(179)

A.2 Non-conservation of the Probability Flux

Subtracting from the wave equation multiplied by χ† the complex conjugated wave equationmultiplied by χ one finds the divergence of the continuity equation

∇ ·(χ†(r)∇χ(r)− χ(r)∇χ†(r)

) ~2

2m− 2iW (r) |χ(r)|2 = 0 (180)

showing that the probability flux is not conserved because W (r) acts as a sink. The reducedmass has been denoted by m. We define the probability current density

j(r) =(χ†(r)∇χ(r)− χ(r)∇χ†(r)

) ~2im

(181)

where m is the reduced mass. Integrating both side of the above equation over a sphericalvolume VR of radius R, Gauss’ law leads to

R2

∮dSRnR · j(r) =

2m

~2

∫VR

d3rW (r) |χ(r)|2 (182)

A.3 Non-Unitarity of the S-Matrix

The non-conservation of the probability flux is reflected also in the partial wave properties.This is seen immediately when the flux equation, Eq.(182), is decomposed into a sum ofpartial wave contributions. Then, one finds that the flux suppression is related to thepartial wave absorption or reaction cross section

σreac,` =π

k2(2`+ 1)

(1− |S`|2

)(183)

where we recognize the previously introduced inelasticities η` = |S`|. The S-matrix in thefull Hilbert-space is of course unitary, SS† = 1. Here, however, we are dealing with asubset of S-matrix elements, namely those related to the P-space, S = SPP , which is onlya (tiny) part of the full scattering operator as one finds immediately from

S = SPP + SQQ + SSQ + SQP (184)

from which it is obvious that SS† < 1 in the P-space.

67

A.4 Reaction, Total, and Elastic Cross section

The differential elastic cross section for the scattering of (spin-less) particles is defined bythe angular distribution obtained by the coherent sum of partial wave scattering amplitudes

dσelasdΩ

=1

k2

∣∣∣∣∣∑`

(2`+ 1)P`(cos θ)C`

∣∣∣∣∣2

(185)

and integration over the angles leads to the total elastic cross section

σelas =4π

k2

∑`≥0

(2`+ 1)|C`|2 (186)

where we have used the orthogonality of the Legendre polynomials P`.In partial wave representation the reaction cross section is given by summation of the

partial contribution

σreac =∑`≥0

σreac,` =π

k2

∑`≥0

(2`+ 1)(1− |η`|2

), (187)

which we may also write as

σreac =∑`≥0

σreac,` =π

k2

∑`≥0

(2`+ 1)(1− |2iC` + 1|2

). (188)

By using

1− |2iC` + 1|2 = 1− (−2Im(C`) + 1)2 − 4Re(C`)2 = 4Im(C`)− 4 |C`|2 (189)

and we find that the second term leads to total elastic cross section, Eq.(186). The secondterm is known as the optical theorem, which usually is cited in the form

σtot =1

kIm(f(0)) (190)

relating the forward scattering amplitude to the total cross section. Thus, we recover thefundamental relation

σtot = σelas + σreac. (191)

In Fig.39 the three cross sections are shown and the optical theorem is verified for s-wavescattering on a potential roughly corresponding to nucleon scattering on A = 90 (see nextsection).

68

Figure 36: Real (left) and imaginary (right) parts of ` = 0 radial wave functions in complex3-D square well potential at E = 0.1 MeV. Results for −Im(Uopt) = W = 0, 10, 40 MeVare shown. Note the suppression of wave functions in the interior region for W 6= 0.

Figure 37: Real (left) and imaginary (right) parts of ` = 0 radial wave functions in complex3-D square well potential at E = 20 MeV. Results for −Im(Uopt) = W = 0, 10, 40 MeVare shown. Note the suppression of wave functions in the interior region for W 6= 0.

69

Figure 38: Energy dependence of the inelasticities η (left) and the scattering phase shiftsδ for s-wave scattering on the three model potentials.

A.5 Results for s-wave Scattering on a Square Well Potential

As illustrating examples, we consider s-wave scattering of a nucleon on three 3D-square wellpotentials Un = (Vn−iWn)Θ(R−r), keeping the real part Vn = −50 MeV fixed but varyingthe imaginary parts Wn = 0, 10, 40 MeV. The potential radius is given by R(A) = r0A

1/3.Choosing A = 90, we find R = 5.37 fm. Basics of the theoretical-mathematical treatmentare introduced in Problem 3 and Problem 4, appended below.

Real and imaginary parts of s-wave wave functions in a complex 3-D square well po-tential are shown in Fig.36 for an energy of E = 100 keV and in Fig.37 for an energy ofE = 20 MeV, respectively.

Remarkable dependencies on the the energy and the strength of the imaginary partsare observed. At low energy the interior parts of the wave functions (r ≤ R) are small:The outer part of the wave functions (r ≥ R) increase almost linearly in the shown radialregion. That behaviour is fixed by the the scattering length (as) and the effective range(rs) of the potentials, describing the low-energy properties of a scattering system and beingdefined by the power series expansion of the s-wave scattering phase shift:

tan δs ' −1

as+

1

2k2rs +O(k4). (192)

At E = 20 MeV, Fig.37, both v(r) and u(r) are of clear oscillatory structure. Still, in theinterior region the wave function amplitudes are smaller in magnitude than those of theoutgoing waves.

At both energies, a common feature is the additional suppression of the inner wavefunctions by imaginary part of the potential. With increasing W the amplitudes arereduced strongly, reflecting the absorption of probability flux into the other channels whichare generating the non-hermitian part Im(U) of the self-energy as discussed in section 4.

The s-wave inelasticities η and the scattering phase shifts δ for the three potentials areshown in Fig.38 as functions of the energy. As expected, η = 1 at all energies for potentialU1 with W = 0. For U1,2 the corresponding inelasticities are decreasing steadily with

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Figure 39: Elastic (σe), reaction (σr), and total (σtot) s-wave cross sections for a complexsquare well potential, Im(Uopt) = −40 MeV, RA = 5.38 fm, are shown as functions ofenergy. The sum of elastic and reaction cross section, indicated by symbols, reproducesthe independently computed total cross section perfectly well .

energy. That energy dependence is a result of the deeper penetration in the interactionregion with increasing scattering energy. Accordingly, also the scattering phase shifts varyin a manner characteristic for potentials which have bound states. In the present case, thereal part of the potential is strong enough to support two bound s− states. The phase shiftsare only weakly affected by the presence of the imaginary parts, leading to the conclusionthat they are mainly determined by the diffractive real potential components.

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Problems on Scattering Theory

B Problem 1: Kinematics of a Two-Body Reaction

Consider a reaction a + A→ b + B where a projectile a of mass ma impinges on a targetnucleus A of mass mA, converting the two nuclei into the projectile- and target-like residuesb, and B with masses mb and mB, respectively. In the laboratory system, A is initially atrest while a carries the kinetic energy TLab.

1. What are the 4-momenta ka,A of projectile a and target A in the laboratory system?Use the metric in the Bjorken-convention, implying for 4-vectors q = (q0,q)T the(Lorentz-invariant) measure q2 = q02 − q2.

2. Determine the Mandelstam total energy s in terms of the rest masses ma,A and thekinetic energy TLab.

3. We define the total (P ) the relative (k) 4-momentum by the linear transformation

ka = −k + xaP ; kA = k + xAP xa + xA = 1 (193)

Verify that the transformation coefficients are given by the Lorentz-invariant forms

xa =s−m2

A +m2a

2s; xA =

s−m2a +m2

A

2s. (194)

4. Show that k is a space-like, P a time-like 4-vector.

5. What is found for P and xa,A in the barycentric frame?

6. Derive xa,A in the low-energy limit up to second order on k

7. Which is the functional form of s in the barycentric system? Derive k in terms of sand the rest masses.

8. Verify the relation s+ t+u = m2a+m2

A+m2b +m2

B where t and u are the Mandelstam4-momentum transfers.

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.

C Problem 2: Probability Current of a Scattered Par-

ticle

In the asymptotic region, the scattering wave function obtains the form

ψ(+)k (r)→ eik·r + f(Ωk)

eikr

r(195)

where Ωk(θk, ϕk) denotes the solid angle of the scattered particle. Derive the asymptoticparticle flux by evaluating the probability current operator

j(r) =~

2im

(ψ(+)∗∇ψ(+) − ψ(+)∇ψ(+)∗) . (196)

where m is the reduced mass. Show that the current is given by the incoming current fromthe plane wave part and a radially directed outgoing current.Hint: Strongly oscillating terms may be neglected.

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D Problem 3: Scattering on a 3-D Square Well Po-

tential

A (spinless) neutron is scattered on an attractive square well potential in three dimensions,U(r) = −U0Θ(R − r) with U0 > 0. The scattering wave function Φ(k, r) obeys theSchroedinger-equation (

− ~2

2M∇2 + U(r)− E

)Φ(k, r) = 0 (197)

1. Write down the expansion of Φ(k, r) into partial waves Φ`(r) = 1kru`(r) and the wave

equation for u`(r).

2. In the following, we consider s-wave (` = 0) scattering only with radial wave functionu(r) ≡ u0(r). Generalization to arbitrary partial waves ` ≥ 1 is obvious. Express thegeneral s-wave solution in terms of trigonometric functions. What are the physicallymeaningful boundary conditions at the origin and in the asymptotic region?

3. Write down the matching conditions at r = R and derive an explicit expression forK = tan δ where δ is the scattering phase shift. At which energies approaches δmultiples of pi/2?

4. Plot and discuss the s-wave cross section as a function of the wave number k.

E Problem 4: Scattering on a 3-D Complex Square

Well Potential

Solve Problem 3 for a complex optical potential U(r) == −(U0 + iW0)Θ(R − r) withU0,W0 > 0.

1. Use the asymptotic form u(r) = sin(kr) + Ceikr. The scattering amplitude C =12i

(S − 1) is expressed in terms of the S-matrix element S = ηe2iδ. The inelasticity0 < η ≤ 1 accounts for the loss of flux due to the absorptive part W0 of the potential.Determine η and δ. How do these quantities behave as a functions of W0 and k?

2. Compute numerically and plot the (properly normalized) wave function u(r) in theinterior region r ≤ R for a selection of values of W0 ≥ 0.

3. Which form is found for C for W0 = 0? Express C in terms of K = tan δ.

4. Derive the s-wave total cross section σtot, the total reaction cross section σreac, andthe total elastic cross section σelas.

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5. Show that the s-wave probability flux through a spherical surface at asymptoticdistance is given by σreac. Moreover, show that σreac is given by an integral involvingIm(U(r)).Hint: Use the Schroedinger-equations for u(r) and u∗(r).

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Figure 40: Coordinates describing a stripping reaction a+A→ b+B with a = b+ x andB = A+ x.

F Problem 5: Deuteron-induced Stripping Reactions

Here, we study a (d, p) reaction in two extreme limiting cases: the plane wave limit (PWBA)and the strong absorption limit, corresponding to treat distortion effects by the Black Diskassumption. We consider the stripping reaction d+A→ p+B where B = A+n is formedby capturing the neutron into a bound state with wave function ϕn(r). The intrinsicnuclear coordinates (r1,2) and the channel coordinates (rα,β) are shown in Fig. 40.

1. Derive a 2-by-2 linear system relating the set of intrinsic coordinates:

r1 = s1rα + t1rβ (198)

r2 = s2rα + t2rβ. (199)

Determine the coefficients si, ti for the coordinates shown in Fig. 40. In tho followingwe use the so-called zero-range approximation rβ = A

Brα where A and B are the mass

numbers of the target-like nuclei.

2. In the zero-range limit (and the post-representation) the (d, p) reaction amplitude is

Mαβ = JD0〈χ(−)β |Φnδ(rβ −

A

Brα)|χ(+)

α 〉 (200)

with the interaction constant D20 ' 1.5 · 104[MeV fm3] and the Jacobian J of the

coordinate transformation. The nuclear overlap function is Φn(r) given by

Φn(r) = 〈JBMB|Ψn|JAMA〉 =∑jm

1√2j + 1

(JBMBJA −M − A|jm)ϕjm(r)Sj(JAJB)

(201)including the single particle wave functions ϕjm and the spectroscopic amplitudesSj(JAJB) = 〈JB||a+j ||JA〉, defined as the reduced matrix element of the creationoperator a+jm. Evaluate the spectroscopic amplitude, assuming that B is given by asingle BCS quasiparticle configuration with respect to the parent nucleus A.

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3. Write down the reaction amplitude explicitly as an integral. What is obtained whenreplacing the distorted waves by planes waves?

4. As an example, we consider the reaction 14C(0+)(d, p)15C( 12

+

. The neutron is placedinto the 2s 1

2orbit, described reasonably well by the harmonic oscillator wave function

ϕ20jm(r) =

√6√πb3

(1− 2

3x2)e−

x2

2 Y0jm(r) (202)

where x = r/b and Y0jm denotes a spin-spherical harmonics. A suitable choice forthe oscillator length is b ∼ 2.00 fm. Evaluate the scattering amplitude in PWBA andplot the result as a function of the momentum transfer q.

5. In the strong absorption limit, the absorption of the probability flux is simulated byintroducing a lower cutoff RBD and evaluating of the PWBA integral in [RBD,∞],

i.e. χ(−)∗β χ

(+)α ∼ Θ(r − RBD)eq·r where q is the momentum transfer. Contributions

of Re(Uopt) are neglected. Compute the reaction amplitude with RBD = 4.2 fm andfor oscillator length b = 2.22 fm and compare the result to the full DWBA and thePWBA cases. Results are shown in Fig. 41.

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Figure 41: Momentum distribution of the cross section for the stripping reaction14C(0+, g.s.)(d, p)15C(1

2

+, g.s.) at incident energy TLab = 20 MeV. Exact DWBA results

(symbols), using partial waves up to ` = 20, are shown for the incident deuteron en-ergy Tlab = 20 MeV, obtained with realistic deuteron and proton optical potentials. Theform factor is described by a 2s1/2 neutron wave function of the correct separation energy(Sn = 1.218 MeV). For comparison, in the upper panel results of schematic calculationusing the Black Disk approximation with a hard radial cutoff form factor (step functionΘ(r−RBD), RBD = 4.2 fm) are shown. In the lower panel, a soft radial cutoff form factorof Fermi function shape (1 − f((r − RS)/aS), RS = 3.42 fm, aS = 0.4 fm)was used. Inboth plots, also the plane wave PWBA results are shown. The PWBA and the BD crosssections were adjusted by χ2-fits to the DWBA results.

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