Indirect Methods in Nuclear Astrophysics with Relativistic Radioactive Beams

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Indirect methods in nuclear astrophysics with relativistic radioactive beams

Thomas Aumanna, Carlos A. Bertulanib,a

aInstitut f¨urKernphysik, Technische Universit¨atDarmstadt, Germany bDepartment of Physics and Astronomy, Texas A & M University-Commerce, Commerce, Texas 75429, USA

Abstract Reactions with radioactive nuclear beams at relativistic energies have opened new doors to clarify the mechanisms of stellar evolution and cataclysmic events involving stars and during the big bang epoch. Numerous nuclear reactions of astrophysical interest cannot be assessed directly in laboratory experiments. Ironically, some of the information needed to describe such reactions, at extremely low energies (e.g., keVs), can only be studied on Earth by using relativistic collisions between heavy ions at GeV energies. In this contribution, we make a short review of experiments with relativistic radioactive beams and of the theoretical methods needed to understand the physics of stars, adding to the knowledge inferred from astronomical observations. We continue by introducing a more detailed description of how the use of relativistic radioactive beams can help to solve astrophysical puzzles and several successful experimental methods. State-of-the-art theories are discussed at some length with the purpose of helping us understand the experimental results reported. The review is not complete and we have focused most of it to traditional methods aiming at the determination of the equation of state of symmetric and asymmetric nuclear matter and the role of the symmetry energy. Whenever possible, under the limitations of our present understanding of experimental data and theory, we try to pinpoint the information still missing to further understand how stars evolve, explode, and how their internal structure might be. We try to convey the idea that in order to improve microscopic theories for many-body calculations, nuclear structure, nuclear reactions, and astrophysics, and in order to constrain and allow for convergence of our understanding of stars, we still need considerable improvements in terms of accuracy of experiments and the development of new and dedicated nuclear facilities to study relativistic reactions with radioactive beams. Keywords: nuclear astrophysics, rare isotopes, nuclear reactions, equation of state of nuclear matter, nuclear experiments. PACS: 24.10.-i, 26., 29.38.-c

Contents 2.5 The equation of state of neutron stars ...... 11 2.6 Microscopic theories of homogeneous nuclear 1 Introduction 2 matter ...... 11 2.6.1 The EoS from Hartree-Fock mean field 2 Nuclear physics in astrophysics 3 theory ...... 11 2.1 Nucleosynthesis ...... 3 2.6.2 The EoS in relativistic mean field models 13 2.2 Some specific reactions ...... 4 2.6.3 Linear response theory and collective 2.2.1 CNO cycle ...... 4 excitations ...... 14 2.2.2 Radiative capture, beta-decay and elec- 2.7 The incompressibility modulus and skewness tron capture ...... 5 parameter of nuclear matter ...... 15 2.2.3 Neutrino induced reactions ...... 6 2.8 The slope parameter of the EoS ...... 16 2.3 Challenges in Measurements of Low Energy As- 2.8.1 Nuclear radii ...... 17 trophysical Reactions ...... 7 arXiv:1910.14094v2 [nucl-th] 23 Dec 2019 2.8.2 Neutron skins ...... 18 2.3.1 Reaction rates in astrophysical environ- 2.8.3 Dipole polarizability and electromagnetic ments ...... 7 response ...... 20 2.3.2 Environment electrons in stars and on earth ...... 8 3 Indirect methods with relativistic radioactive beams 23 2.3.3 Clusterization in light nuclei ...... 9 3.1 Electron scattering ...... 23 2.4 Neutron stars ...... 9 3.1.1 Elastic electron scattering ...... 23 2.4.1 General observations ...... 9 3.1.2 Inelastic electron scattering ...... 23 2.4.2 Structure equations ...... 10 3.1.3 Electron radioactive-ion collider . . . . 24 3.1.4 The electron parity violation scattering 24 3.2 Direct reactions at relativistic energies . . . . . 25 Email addresses: [email protected] (Thomas Aumann), [email protected] (Carlos A. Bertulani)

Preprint submitted to Elsevier December 25, 2019 3.2.1 Nucleus-nucleus scattering at high en- is followed by D + 3H → α + n which liberates about 17.6 ergies ...... 25 MeV of energy. In contrast, 7Li hydrides, in the chemical form 3.2.2 Elastic scattering ...... 27 of 7LiH, is a good moderator in nuclear reactors, because it re- 3.2.3 Dirac phenomenology ...... 28 acts with less probability with neutrons, therefore forming less 3.2.4 Total nuclear reaction cross sections . . 30 tritium, 3H. 3.3 Coulomb excitation ...... 31 After so many decades since the Cockcroft and Walton ex- 3.3.1 The Coulomb dissociation method . . . 34 periment, both 6Li and 7Li isotopes are still of large interest for 3.4 Relativistic coupled-channels method . . . . . 36 nuclear physics and nuclear astrophysics and other areas of sci- 3.4.1 Continuum states and relativistic cor- ence. For example, in cosmology both isotopes are at the center rections ...... 36 of the so-called “lithium puzzle” in the Big Bang Nucleosyn- 3.4.2 Nuclear excitation of collective modes 38 thesis (BBN) theory [3, 4, 5] which is an important part of our 3.4.3 Angular distribution of γ-rays . . . . . 39 understanding of how the universe evolved from a primordial 3.5 Excitation of pygmy resonances ...... 39 soup of fundamental particles to the present day universe with 3.6 Extracting dipole polarizabilities ...... 42 planets, stars and galaxies. 3.7 Extracting neutron skins from nucleus-nucleus The reaction in Eq. (1) was made possible due to the use collisions ...... 44 of electric and magnetic (EM) fields to accelerate charged par- 3.7.1 Removing nucleons from nuclei at high ticles. In the decades following the Cockcroft and Walton ex- energies ...... 44 periment, advances in using EM fields in nuclear accelerators 3.7.2 Isospin dependence ...... 45 developed enormously and almost all stable nuclear isotopes 3.7.3 Ablation model for neutron removal . . 45 are now amenable to accelerate with different charge states, in 3.7.4 Total neutron removal cross sections as some cases even with fully electron-stripped nuclei. In the last a probe of the neutron skin ...... 46 few decades a new era in nuclear physics emerged with a large 3.7.5 Coulomb excitation followed by neu- investment on nuclear accelerators using short-lived unstable tron emission ...... 48 nuclei. This enabled the nuclear science community to study 3.7.6 Nuclear excitation followed by neutron the structure and dynamics of unstable nuclei used as projectiles emission ...... 48 incident on stable nuclear targets. It also provided the access to 3.7.7 Neutron changing and interaction cross key information on astrophysical nuclear reactions by means of sections ...... 49 indirect techniques as we will discuss in this review. 3.8 Polarized protons and the neutron skin . . . . . 50 Assuming that the extraction of any nuclear isotope from 3.9 Charge-exchange reactions ...... 55 ion sources would be possible, a back-of-the-envelope estimate 3.9.1 Fermi and Gamow-Teller transitions . . 55 of the lifetime limit for a nucleus down the beam line can be 3.9.2 Double-charge-exchange and double-beta- done. Taking as an example 100 m along the accelerator and decay ...... 56 projectiles moving close to the speed of light, the lowest life- 3.10 Central collisions ...... 57 time admissible for a nuclear beam, before it decays, would be τ = 100/3 × 108 ∼ 0.3 × 10−6 s, or about one µs1. However, ion 4 Conclusions 59 production and release times from ion sources increases this number, down to a few ms. Accelerated unstable nuclei with 1. Introduction such short lifetimes have only been possible with the developments in the last decades, with better techniques for production About 8 decades have passed by since the first nuclear in- and extraction of nuclei from ion sources and the accomplish- duced reaction was carried out in a terrestrial laboratory. This ment of better accelerator technologies. Target manufacturing happened in 1932 at Cambridge University, in an experiment and new detector construction ideas have also been crucial to lead by J. Cockcroft and E. Walton [1, 2], who used a new study reactions induced with short-lived nuclei. Reactions in- 11 technique at the time to accelerate and collide protons with a volving short-lived nuclear isotopes such as Li (T1/2=1.5 ms) 7 100 Li target, yielding two alpha particles as reaction byproducts, or Sn (T1/2=1.6 s) are common in radioactive nuclear beam namely facilities. By relativistic we mean beam energies in the range p + 7Li −→ α + α. (1) 100 − 2000 MeV/nucleon, typical bombarding energies available in a few radioactive beam facilities in the world such as Till now reactions involving lithium are of great interest for the RIKEN/Japan or the GSI/Germany facility. These labora- mankind for worse or for better. For example, 6Li deuterides, tories have the advantage of probing properties of neutron-rich i.e., a chemical combination in the form 6LiD, can be used as matter, with large beam luminosities, and newly developed de- fuel in thermonuclear weapons. Nuclear fission triggers explo- tectors using inverse kinematic techniques. sion in such weapons to first induce heat and compress the 6LiD 6 Among many subjects, nuclear astrophysics deals with the deuteride, and to bombard the LiD with neutrons. The ensuing synthesis of nuclei in high temperature and pressure conditions reaction 6Li + n −→ α + 3H (2) 1In fact, it would be longer due to Lorentz contraction at very high bombarding energies. 2 existing within stars. Using the big bang theory, astrophysicists In this review, we will focus on a few of the indirect tech- were able to explain how thermonuclear reactions lead to the niques that have emerged in the last decades and that are a main production of 75% of hydrogen and 25% of helium observed in part of our own research agenda. Previous reviews covering the universe. The small traces of other elements have also been other parts of this vast subject have been published, e.g., in explained in terms of stellar processing of nuclear fuel. Popu- Refs. [3, 9]. In section 2 we make a short description of the lation III stars (oldest ones) were formed from light primordial mechanisms involved in nucleosynthesis, the challenges in per- nuclei originated from big bang nucleosynthesis. Population forming measurements at very low energies, the physics of neu- III stars with large masses ejected heavier elements during en- tron stars and their equation of state. Some microscopic theo- ergetic explosions. Population II stars, very poor in metals, are ries used for the purpose are also discussed. In particular, we the next generation of stars and are found in the halo of galax- focus the discussion on constraining the slope parameter of the ies. They can also form carbon, oxygen, calcium and iron which equation of state. In section 3 we discuss some of the indirect can be ejected by stellar winds to the interstellar medium. The techniques used to tackle the astrophysical problems we raised stardust generated by stellar explosions and stellar winds gen- in section 2. These include electron scattering off exotic nuclei, erated the population I stars found in the disk of galaxies con- elastic and inelastic hadronic scattering, total nuclear reaction taining large metallicities. The cores left over during supernova cross sections, Coulomb excitation, pygmy resonances, dipole explosions can either form a neutron star or a black hole. To polarizability and electromagnetic response, charge exchange understand all these physical processes, one needs to develop a reactions and central collisions. In section 4 we present our large number of theoretical models, pursue dedicated astronom- conclusions. ical observations and, if possible, perform nuclear experiments We apologize to the authors and their publications that we on earth. might inadvertently missed to cite properly in this review. The nuclear physics contribution to the synthesis of light elements during the big bang, the formation of medium-heavy 2. Nuclear physics in astrophysics elements in stellar cores and of heavier elements, up to ura- nium, in supernova explosions and in neutron star mergers, in- 2.1. Nucleosynthesis volves a very large number of unsettled puzzles. Some of these Cosmology and stellar evolution involve many aspects of puzzles can be tackled by studying nuclear reactions in nuclear nuclear physics. These areas of science deal with a dynami- physics laboratories with either extremely low energy beams or cal scenario involving matter and radiation densities, thermody- extremely large energies. In particular, reactions involving rare namics, chemical composition, hydrodynamics, and other phys- nuclear species often require the use of fast nuclear beams. It ical quantities and processes. Nuclear physics enters in most of has been realized in the last few decades that relativistic reac- the dynamical parts of the relevant scenarios through the de- tions with radioactive beams can fill the gap of our knowledge termination of rates of nuclear transmutations, such as particle, in many aspects of nuclear astrophysics related to stellar evo- electromagnetic and weak-decay processes, as well as fusion lution, stellar structure and the synthesis of the elements in the and rearrangement reactions. For example, in the hot stellar universe. plasmas the two-body reaction rate, or number of reactions per Neutron stars are interesting objects because they consist unit volume and per unit time involving particles i and k, is of nuclear matter compressed to incredibly high densities. The given by behavior of such dense matter under compression is determined nink hσvi by the so-called neutron star equation of state (EoS) which also Γik = , (3) 1 + δ determines their basic properties, such as their masses and radii ik [6]. Many theoretical predictions exist for the EoS and learn- where ni(k) is the number density of particle i(k), v is the relative ing about it under such extreme conditions is important to ad- velocity between particle i and k, and hσvi is the average of the vance our knowledge of nuclear physics. Recently, gravita- cross section σ for the reaction over the Maxwell-Boltzmann tional waves have been detected by the LIGO-Virgo collabo- relative motion distribution of the particles. The Kronecker- ration [7]. The gravitational waves are thought to be caused delta factor δik prevents double counting for the case i = k. by merging black holes or neutron stars. The event named Thus, the reaction rate is given by GW170817 [8] is likely caused by the orbiting of two neutron  1/2 Z ∞ ! stars during the coalescence phase. The shape of the gravitational- nink  8  E Γ j,k =   σ(E) exp − EdE, (4)  3 3  k T wave signal depends not only on the masses of the neutron stars 1 + δik πmikkBT 0 B but also in their so-called tidal deformabilities which describe how much they are deformed by tidal forces during the merging where mik is the reduced mass of the i-k pair. phase. The effect of tidal deformability could modify the orbital In rare cases, stellar modeling also needs information of re- decay caused by the emission of gravitational waves. Such as- action rates involving three particles, such as in the triple-alpha 12 tronomical observations are important to determine the neutron capture process, α + α + α−→ C + γ. This case can be treated star EoS. Nuclear physics experiments, in particular involving as a sequential process in which two α-particles are radiatively 8 neutron-rich nuclei, are also crucial for a consistent determi- captured to form the unbound Be nucleus, followed by another 12 nation of the EoS of both symmetric and asymmetric nuclear α capture to create C. The reaction rate for this reaction is matter 3 given by stellar and cosmological scenarios. For example, to simulate the BBN one needs information about a chain of nuclear reac- ! !3/2 8π m 7 Γ = 3 ~ αα (5) tions involving light nuclei (Ai < 7), up to the formation of B ααα 2 7 mαα 2πkBT and Li. For solar physics one needs to know features of the re- Z ∞ ! σαα(E) E actions in the pp-chain which convert hydrogen to helium and × exp − hσvi 8 EdE, 8 α Be dominates the energy generation and nucleosynthesis in stars 0 Γα( Be; E) kBT with masses less than or equal to that of the Sun. In heav- where E is the αα energy relative to the αα threshold, σαα is ier stars, with masses larger than the Sun, the energy and nu- the αα elastic cross section described by a narrow Breit-Wigner cleosynthesis process are dominated by the Carbon-Nitrogen- 8 8 function, and Γα( Be; E) is the α-decay width of the Be ground Oxygen (CNO) cycle, fusing hydrogen into helium via a six- 8 state treated as energy dependent. In Be, the width of this reso- stage sequence of reactions. To describe neutron star merg- nance is about Γα = 6 eV. The reaction rate hσviα−8Be in Eq. (6) ers one needs to know cross sections for rapid neutron cap- is obtained in the same way as in Eq. (4), but with the integral ture (r-process) reactions, a rapid sequence of neutron capture 0 0 running over E , where E is now the energy with respect to the followed by beta-decay occurring in neutron-rich environments 8 8 α- Be threshold (which varies with the Be formation energy E) [10, 11, 12, 13, 14, 15, 16, 17, 18]. 0 0 0 and the integrand is replaced by σα−8 Be(E ; E) exp(−E /kBT)E . The two mechanisms involving most neutron-capture ele- The reaction rates for nuclear fusion are crucial for nuclear ments are the r-process and the s-process (slow neutron cap- astrophysics, namely the study of stellar formation and evo- ture). The r-process is thought to generate about half of the lution. To determine the chemical evolution, stellar modelers nuclides with A & 100 [10]. The high abundance patter around need to solve a set of coupled equations for the number densi- Z ∼ 56 − 76, known as the main r-process, is very consis- ties in the form tent with halo star and the solar system r-process abundances ∂n X X X [19]. A second (A ∼ 130) and third (A ∼ 195) r-process peaks i = mi Γ + mi Γ + mi Γ , (6) ∂t j j j,k j,k j,k,l j,k,l are also observed. Abundances around the second and third r- j j,k j,k,l process peaks, as well as for the intermediate nuclei between them, are dependent on nuclear properties of the r-process un- where mi are positive or negative integers specifying how many Y der steady beta-decay flow and fission cycling [20]. Fission particles of species i are created or destroyed in a reaction Y cycling occurs under sufficiently high neutron-rich conditions out of the combination of the particles forming them. Such where the r-process extends to nuclei decaying through fission reactions can be due to (a) decays (Y = i), electron/positron channels. Fission has an impact on the r-process, terminating capture, neutrino-induced reactions, and photodisintegrations, its path near the transuranium region leading to material return- in which case Γi = λini, where λi is the decay-constant, (b) two- ing to the A ∼ 130 peak. The fission products then become { } particle reactions (Y = i, j ), and (c) three-particle reactions new seed nuclei for the r-process, facilitating steady beta-decay { } (Y = i, j, k ), as described above. flow. Using the concept of nuclear abundances defined as Yi = Many stellar nucleosynthesis scenarios, require knowledge ni/(ρNA), where ρ is the density and NA the Avogadro number, of reactions involving short-lived nuclei. These are available for a nucleus with atomic weight Ai, then AiYi is the mass frac- P 2 now in large quantities in radioactive-beam facilities. They are tion of this nucleus in the environment and AiYi = 1 . The usually produced in flight and therefore the extraction of in- reaction network equations (6) can be re-written as formation of interest for astrophysics are done indirectly using dY X X N experimental and theoretical techniques, many of which devel- i = Ni λ Y + i ρN hσvi Y Y dt j j j N !N ! A j,k j k oped in the last few decades. j j,k j k X N + i ρ2N2 hσvi Y Y Y . (7) 2.2. Some specific reactions N !N !N ! A j,l,k j k l j,k,l j k l 2.2.1. CNO cycle An example of a reaction network of interest for nuclear Here N denotes the number of nuclides i produced in the reac- i astrophysics is the CNO cycle shown schematically in Figure 1. tion, being a negative number for destructive processes. It has a cycle I, also known as CN cycle, involving the reactions The energy generation in stars per unit volume per unit time 2 is obtained by adding the mass excess ∆Mic of all nucleus i 12 13 + 13 14 15 + 15 12 C(p, γ) N(e νe) C(p, γ) N(p, γ) O(e ν) N(p, α) C, created during the time step, i.e., (9) where an α-particle is synthesized out of four protons, effec- d X dYi 2 = − NA∆Mic . (8) →4 + dt dt tively as 4p He + 2e + 2νe, and an energy of Q = 26.7 i MeV is released. The 15N(p, γ)16O reaction leads to a breakout Upon including reaction networks in stellar modeling one from the CN cycle, returning to the CN cycle via a number of obtains the chemical evolution and energy generation in diverse reactions within the CNO cycle II, also known as ON cycle, 15 16 17 + 17 14 N(p, γ) O(p, γ) F(e νe) O(p, α) N . (10) 2This only works in cgs units. 4 osynthesis rates. Measuring the radiative capture cross sections b+ 14 15 16 16 17 18 at the relevant energies is very difficult due to the vanishingly O O O F F F small reaction probability. Moreover, when one of the participating nuclei is radioactive, it is hard to achieve the necessary beam intensity to enable an accurate measurement at low en- p, g 13 14 15 15 16 17 N N N O O O ergies. These difficulties has induced the use of indirect experimental techniques often involving the use of relativistic ra- p, a 12 13 14 14 15 16 dioactive beams leading to much higher reaction yields. But, C C C N N N using the measured cross sections and indirectly determining the radiative capture cross sections of astrophysical interest can I (CNO cycles) II only be done reliably with the help of nuclear reaction theory. Radiative capture cross sections are related to photo-decay Figure 1: The CNO cycles I and II. Cycle I, also known as CN cycle [24].Cycle constants through the detailed balance theorem, leading to the II is a breakout of the CN cycle and is sometimes called by ON cycle. The decay rate for the reaction i + j → k + γ, given by stable nuclei are represented in boxes with bottom-right gray shaded areas. !3/2 !3/2 ωiω j AiA j m kT Q λ T u hσ i − , k,γ( ) = 2 v i, j exp (11) ωk Ak 2π~ kT Two low-energy neutrinos are produced in the beta decays of where Q is the energy released, m is the mass unit, T is the 13N(t = 10 min) and 15O(t = 122 s). The ON cycle u 1/2 1/2 temperature, A are the nuclear mass numbers, and ω (T) = is slower by a factor of 1000 than the CN cycle because the i i P (2J + 1) exp(−E /kT) are partition functions. S-factor for 15N(p, α)12C is about 1000 times larger than the m im im An example of actual relevance of radiative capture reac- S-factor for the 15N(p, γ)16O reaction [21]. tions is the pp-chain in the sun which begins with the p(p,e+ν )d The determination of the cross sections for the reactions e reaction, followed by the production of 4He via three possible in the CNO cycle and its breakout cycles are still a matter of reaction pathways involving nuclei with A 8. The radiative intense investigation. For example, asymptotic giant branch 6 capture reactions d(p,γ)3He, 3He(α, γ)7Be, and 7Be(p,γ)8B are (AGB) stars are believed to be one of the major source of in- important parts of the pp chain and for the production of solar terstellar medium (ISM) in the galaxies. The composition of neutrinos from 7Be and 8B decays. The precise determination the AGB ejecta, which constitute the late phase in the evolution of the reaction rates of the last two reactions still remain a goal of stars with masses in the range 1M M 8M , is strongly . . of contemporary nuclear astrophysics [25]. dependent on the mixing processes occurring during the stellar The late evolution stages of massive stars are strongly in- lifetime. A telltale of these processes are the isotopic ratios of fluenced by weak interactions, which determine the core en- carbon (12,13C), nitrogen (14,15N), and oxygen (16,17,18O). Sys- tropy and electron-to-baryon ratio Y in pre-supernovae and in- tematic astrophysical modeling of stars in the red giant branch e fluences its core mass, driving the stellar matter neutron richer. (RGB) show that they undergo the so-called dredge-up mecha- Electron capture reduces pressure support by the remaining elec- nism, a convective mixing process that carries nuclei from inter- trons, while β-decay acts in the opposite direction. Both elec- nal layers to the surface. Some calculations have shown that the tron capture and β-decay generate neutrinos, which escape the observed ratios cannot be explained on the basis of the hydro- star and carry away energy and entropy from the core when den- static H-burning through the CNO cycle. An accurate exper- sities are less than 1011 g/cm3. Electron captures and beta de- imental determination and theoretical description of the reac- cays occur during hydrostatic burning stages in the very dense tions involved in these cycles are still being pursued. For more stellar core where the Fermi energy, or chemical potential, of discussion on this subject, see Refs. [22, 23]. the degenerate electron gas is sufficiently large to overcome the negative Q values for the capture reactions. The capture 2.2.2. Radiative capture, beta-decay and electron capture rates are dominated by Fermi (F) and Gamow-Teller (GT) tran- To model a stellar environment, one needs to input densi- sitions in the nuclei. These rates depend on the temperature and ties and temperatures in the reaction networks, Eq. (7), and to electron-number density, which can be related to the electron acquire knowledge on the decay constants λ, both in direct and capture cross sections by inverse β-decay (electron capture), particle and photon emis- Z ∞ sion, cross sections for the elastic scattering, and the reaction 1 X 2 λec(T) = peσec(e, i, f ) f (e, µe, T)de, (12) rates for disintegration and formation of nuclei. 2 3 0 π ~ e Radiative capture reactions, involving the emission of elec- i, f tromagnetic radiation, are involved in the pp chain and CNO where m is the electron rest mass, 0 = max(Q , m c2), and cycles. They are also prominent in the explosive conditions e e i f e p = (2 − m2c4)1/2/c is the momentum of the captured electron in novae, x-ray bursts, and supernovae. Often, radiative cap- e e e with energy . In a supernova collapsing core the conditions tures are the only proton- or α-induced reactions possible with e are such that the electrons obey the Fermi-Dirac distribution positive Q values. They proceed slowly compared to strong interactions, and frequently control the reaction flow and nucle- −1 f (e, µe, T) = [1 + exp (e − µe)/kBT] , (13)

5 with the electron chemical potential, depending on the electron where HW is the weak interaction Hamiltonian and Ji (J f ) is density, given by µe. The cross section for electron capture the initial (final) angular momentum. Tσ(κ, J f ) depends on the with an energy e, leading to a transition from a proton single- neutrino leptonic traces and on the nuclear matrix elements [29, 3 particle state with energy i to a neutron single-particle state f , 30, 31] : is denoted by σec(e, i, f ). The spectrum of emitted neutrinos 2 " 4πG X D E 2 produced by electron captures on a particular nucleus is given Tσ(κ, J f ) = | J f ||O∅J||Ji | L∅ 2Ji + 1 by (here ~ = c = 1) J X D E 2 1 1 X + | J f ||OMJ||Ji | LM φ p2σ , , f , µ , T , ν( ν) = 2 3 e ec( e i f ) ( e e ) (14) M=0±1 λec π ~ i, f D ED E # − 2< | J f ||O∅J||Ji J f ||O0J||Ji L∅z ,(16) A similar method is used to obtain positron capture rates. with G = (3.04545 ± 0.00006)×10−12 in natural units, and 2.2.3. Neutrino induced reactions 12 3 ! At high densities (ρ > 10 g/cm ) neutrino scattering cross |p| cos θ qz pz L∅ = 1 + , L∅z = + , sections on nuclei and electrons also become important. Such E` Eν E` densities occur in core-collapse supernovae. Densities of or- 2qz pz |p| cos θ der 1011 − 1015 g cm−3 and temperatures ranging from 1 to 50 L0 ≡ Lz = 1 + − , EÈν E` MeV are reached. Neutrinos are produced in large numbers via ! + − qz pz qz pz electron-positron annihilation (e e ↔ νν¯), nucleon-nucleon L±1 = 1 − ± − S 1, (17) bremsstrahlung, etc. Among these, the charged current absorp- EÈν Eν E` tion and emission processes νn ↔ pe− andν ¯p ↔ ne+ dominate. where These and other processes couple the neutrinos to dense nuclear Eν(|p| cos θ − Eν) |p|(|p| − Eν cos θ) matter, influencing the diffusive energy transport in the core to qz = kˆ ·q = , pz = kˆ ·p = , the less dense outer layers where the neutrinos stream freely. κ κ (18) The neutrino heating in the tenuous layers behind the shock is with S = +1 (−1) for neutrino (antineutrino) scattering. For thought ignite the supernova explosion [26, 27, 28]. The neu- 1 simplicity, we omitted the isospin operators τ that are respon- trino wind is also thought to be a site for r-process nucleosyn- ± sible for one unit isospin change in the matrix elements written thesis. Numerous neutrino induced cross sections are needed to above. understand the whole mechanism of supernova explosions. The operators in (16) are We use the notation p` ≡ {p`, E`} and qν ≡ {q, Eν} for the lepton and the neutrino momenta, and k = Pi − P f ≡ {k, k∅}, MV MA MA O∅J = gV J + 2igA J + i(gA + gP1) zJ, for the momentum transfer, where Pi (P f ) is the momentum A V V O = i(δ zg − gA + Mg )M + 2g M − δ zg M , of the initial (final) nucleus, M is the nucleon mass, m is the MJ M P2 W MJ V MJ M V J ` (19) charged lepton mass, and gV , gA, gM and gP are the dimension- less effective vector, axial-vector, weak-magnetism and pseu- with the notation kˆ = k/κ, κ ≡ |k|, the coupling constants de- doscalar coupling constants, respectively. Their numerical val- fined as ues are gV = 1, gA = 1.26, gM = κp − κn = 3.70, and gP = 2 2 κ κ κ gA(2Mm`)/(k + mπ). The cross section within first-order per- gV = gV ; gA = gA ; gW = (gV + gM) ; − 2M 2M 2M turbation theory for the process νe + (Z, A) → (Z + 1, A) + e , κ q κ κ g = g ∅ ; g = g , (20) with momentum k = p` − qν, is given by P1 P P2 P 2M m` 2M m` |p |E Z 1 and σ(E , J ) = ` ` F(Z ± 1, E ) d(cos θ)T (q, J ), (15) ` f 2π ` σ f −1 V A −1 ∇ M = jJ(ρ)YJ(rˆ); MJ = κ jJ(ρ)YJ(rˆ)(σ · ); J X where A J−L−1 MMJ = i FMLJ jL(ρ) [YL(rˆ) ⊗ σ]J ; 2πη Z Z α L F(Z ± 1, E ) = , with η = ± e , X ` MV −1 J−L−1 ⊗ ∇ exp(2πη) − 1 v` MJ = κ i FMLJ jL(ρ)[YL(rˆ) ]J, L is the Fermi function (Z± = Z + 1, for neutrino, and Z − 1, (21) for antineutrino), θ ≡ qˆ · pˆ is the angle between the emerging electron and the incident neutrino, and the transition amplitude where ρ = κr. for initial (final) angular momentum Ji (J f ) is Neutrino scattering cross sections are nearly impossible to measure on earth, especially for all neutrino reactions occurring 1 X X D E 2 Tσ(κ, J f ) = J f M f |HW | Ji Mi , 2Ji + 1 3 s`,sν Mi,M f Indices ∅ and z are used for time- the third-component of four-vectors, respectively. 6 in stellar environments. Therefore, neutrino scattering cross sections are obtained by calculation of Eq. (15) in a theoretical framework such as the Random Phase Approximation (RPA) and its variations [32]. But, the reactions occurring in typical stellar scenarios have small momentum transfer, so that ρ 1 and the angular dependence can be neglected. One obtains for example, for charged-current,

h D A E 2 Tσ(κ, J f ) ∼ C | J f ||Σk=1τ±(k)||Ji | 2 D A E 2i + gA| J f ||Σk=1σ(k)τ±(k)||Ji | , (22) where C depends on the particle energies. The operator τ+ changes a neutron into a proton, τ+ | ni = | pi, and τ− changes a proton into a neutron, τ−| pi = | ni. The matrix elements are known as Fermi and Gamow-Teller matrix elements, respectively. Next-to-leading-order (NLO) terms for the expansion of the matrix elements in powers of ρ probe the nucleus at shorter scales and for small k. They are a consequence of the expansion exp(ik · r) ∼ 1 + ik · r, leading to “first forbidden” terms PA PA i=1 riτ3(i) and i=1[ri ⊗ σ(i)]J=0,1,2τ3(i) within the matrix elements. The NLO terms generate collective radial excitations, such as giant resonances. They to dominate the cross sections for high energy neutrinos in supernova environments. Neutrinos detected on earth originating from stellar events Figure 2: Upper figure: Cross section data for the reaction 3He(α, γ)7Be as a give rise to a number of events in the neutrino detectors given function of the relative energy. Bottom figure: Same data expressed in terms of by the astrophysical S-factor defined in Eq. (23). Data are from Ref. [34]. Z ∞ Nα = Nt Fα(Eν) · σ(Eν) · (Eν)dEν, 0 3 7 where α = νe, ν¯e, νx and (νx = ντ, νµ, ν¯µ, ν¯τ) stands for neutrino He(α, γ) Be reaction relevant for solar physics. To avoid deal- and antineutrino types, Nt is the number of target nuclei used ing with the trivial exponential fall of the capture cross section in the detector, Fα(Eν) is the neutrino flux arriving the detector, due to decreasing tunneling probability as the relative energy σ(Eν) is the neutrino-target nucleus cross section, (Eν) is the decreases, it is convenient to define the astrophysical S factor detector efficiency for the neutrino energy Eν. Dark matter detection experiments are often based on neutrino detectors and S (E) = Eσ(E) exp(2πη), (23) vice-versa. Very sensitive detectors are being built to detect 2 neutrinos from the Sun, the atmosphere, and from supernova. where η = Z jZke /~v is the Sommerfeld parameter, and E is A new window in neutrino physics and astrophysics has been the relative energy, and v is the velocity of the ions j and k. 2 ∝ opened with the development of such new detectors (see, e.g., The “quantum area” is λ 1/E, where λ is the wavelength for − Ref. [33]). the relative motion. Then exp( 2πη) is a rough estimate for the In Ref. [31] it was shown that while microscopic calcu- quantum tunneling probability for s-wave scattering. Therefore, lations using different models for neutrino induced cross sec- the S-factor definition in Eq. (23) removes the steep decrease of tions might agree reasonably well at low neutrino energies, they the cross section with the decrease of the relative energy, as we substantially deviate from each other at larger energies, e.g., see in Fig. 2 (lower panel). In terms of the S-factor, the reaction rate, Eq. (3), for the pair jk becomes Eν & 5 MeV. This has a large impact on the simulation of detector efficiencies of high energy neutrinos streaming off super- !1/2 Z ∞ " # n jnk 8 1 E b nova explosions. S E − − , Γ jk = 3/2 ( ) exp 1/2 (1 + δ jk) πm jk (kT) 0 kT E 2.3. Challenges in Measurements of Low Energy Astrophysical (24) b πηE1/2 m 1/2πe2Z Z / m Reactions where = 2 = (2 jk) j k ~ and jk the reduced mass in units of mu. 2.3.1. Reaction rates in astrophysical environments For reactions induced by neutrons no Coulomb barrier ex- Many fusion reactions involve charged nuclei that need to ists, but quantum mechanics yields a non-zero reflection proba- tunnel through a large Coulomb barrier at the very low rel- bility. The transmission probability of a neutron to an attractive ative energies in stars. This renders cross sections that drop potential region is proportional to its velocity v. This, combined many orders of magnitude as the relative energy decreases. See with the proportionality of the cross section to the “quantum for example, Fig. 2 (upper panel) for the cross section of the area”, 1/E, implies that the cross sections for neutron capture 7 Ion under consideration

More electrons

More positive ions

Debye Radius

Figure 3: Schematic view of the Debye-Huckel¨ sphere approximation used to describe electron screening in plasmas. Figure 4: Left: Schematic representation of the stopping o low energy ions in nuclear targets. Right: Calculated stopping power in p + 4He collisions at energies of astrophysical relevance. are better rewritten as R(E) σ(E) = , (25) ative electrons than positive ions within the sphere leads to an v enhancement factor, f (E), in the plasma so that the reaction where R(E) has now also a flatter dependence on E. rate in the plasma is reduced compared to the reaction rate in Short-lived nuclei are very common in explosive stellar en- a charge-free environment, hσviplasma = f (E)hσvibare. The vironments. They are the seeds of other reactions leading to screening causes a change in the Coulomb potential between 2 stable nuclear species, which might pass by a large variety of two ions in the form V(r) = ZIZ je exp(−r/RD)/r and one finds unstable and stable nuclei. Cross sections for reaction involv- that f = exp(Ue/kBT), where Ue, with energy dimensions√ is ing short-lived nuclei are not well know in many cases. Dur- know as the screening potential. In the Sun, RD ∼ kT/n ∼ ◦ ing stellar hydrostatic burning stages, reactions with charged- 0.218 A, and f ∼ 1.2, a 20% effect for the reaction 7Be(p,γ)8B, particles at very low energies are very hard to study in direct important for the high-energy neutrino production in the Sun. measurements and frequently impossible to be done. But indi- In Ref. [35] the authors calculate the enhancement f (E) for rect techniques using radioactive beams have been developed to weakly screened thermonuclear reactions, taking into account access partial or full information on decay constants and cross their dependence on the velocity of the colliding ions. They find sections of interest for nuclear astrophysics. These techniques enhancements are appreciably smaller than those given by the also allow the extraction of information for stellar reactions in- standard adiabatic Debye-Huckel¨ approximation if the Gamow volving stable nuclei that were not possible prior to the advent velocity is greater than the ion thermal velocity. The mean of radioactive beam facilities. field approximation following the Debye-Huckel¨ picture is not strictly valid under the conditions prevailing in the core of the 2.3.2. Environment electrons in stars and on earth Sun. A kinetic approach should be implemented. In stellar plasmas the free electrons shield the nuclear charges In Ref. [36] the authors state that the energy exchange be- and reduce the nuclear reaction rates. When the potential en- tween any two scattering ions and the plasma is positive at low ergy is smaller than the kinetic energy of the particles, one can relative kinetic energies and negative at high energies. The account for electron screening effects using the Debye-Huckel¨ turnover in a hydrogen plasma occurs at Ekin−rel ∼ 2kT < approximation, so that the reaction-rate in Eq. (3) is modified EGamow ∼ 6kT for the p-p reaction. The net energy exchange, with i.e., the sum over all pairs of scattering particles, vanishes in hσviscreened = fi j(ρ, T)hσvibare, (26) equilibrium. They claim that fluctuations and non-spherical effects are crucial in affecting the screening. They derive screen- where ing corrections, which for the p-p reaction is found to enhance 1/2 1/2 the transition rates, while for higher Z reactions, like 7Be(p,γ)8B, ZiZ jρ ξ X 2 2 fi j(ρ, T) = 1 + 0.188 , with ξ = (Zk + Zk) Yk. are suppressed relative to the classical Salpeter, or Debye-Huckel,¨ T 3/2 6 k theory. A detailed discussion of the screening in stellar plasmas (27) can be found in Ref. [25], where no conclusion is reached on Here, T is the plasma temperature in units of billions of degree 6 the apparent contradiction among the several models existing Kelvin. The Debye-Huckel¨ approximation, shown schemati- in the literature. In the “strong screening” regime at low den- cally in Figure 3 is valid for electron densities n such that e sity plasmas other models are more appropriate [25]. Screening within a radius R a mean field approximation is valid, n R D e D induced in plasmas with intermediate densities obeying the re- 1. Based on these assumptions, the agglomeration of more neg- lation neRD ≈ 1 require more complicated models, raging from 8 the inclusion of dynamical to quantum field theory contributions [25]. Reaction rates of astrophysical interest measured in the laboratory are also increased by the presence of atomic electrons bound in the nuclei [37, 38, 39], which reduce the Coulomb barrier. Experimental findings on the incremental factors are at odds with some apparently well founded electron screening theories [40, 41, 42, 43, 44]. Due to screening, he fusion cross section measured at laboratory energy E is equal to that at energy E + Ue, with Ue known as the screening potential. That Figure 5: Left: Schematic view of polarization and orientation as nuclei with is, large probabilities for cluster-like structures approach each other, shown here U for 6Li + 6Li. Right: Barrier penetrabilities for d + d and for d + 6Li reactions σ (E + U ) = exp πη(E) e σ(E) , (28) e E as a function of the relative motion energy. since the factor S (E)/E has a much smaller dependence on the energy than the term exp −2πη(E) . Dynamical calculations as well as the consideration of several atomic effects have not (left). It was shown that this is possibly the only way to explain 6 6 been able to explain the fact that Ue as measured experimentally why the reaction Li + Li → 3α yields the experimentally is about a factor of two larger than that obtained theoretically observed cross sections, much higher in value than those pre- [45, 37, 38, 39, 46, 47, 48, 49]). dicted by theory. In fact, if the Coulomb barrier penetrability In Refs. [50, 51] one has questioned if the stopping power used in the 6Li + 6Li were due to structureless 6Li ions, the corrections used in the experimental analysis were properly ac- cross section for 6Li + 6Li → 3α would be nearly zero, or at counted for. As shown in Figure 4 (left) the fusion of a low least one could not measure it, but it is observed experimentally energy ion can occur at any point within the target, and the stop- at low energies. Therefore, one expects that it is highly proba- ping power, S , accounts for the energy loss, S = −dE/dx, of ble that the deuterons within 6Li penetrate a smaller barrier and the ions as they penetrate the target. The proper reaction en- form α particles, thus explaining the puzzle. In Refs. [63, 64] it ergy Ee f f = Eion − hS.dxi, in laboratory experiments of fusion was shown that several reactions of astrophysical interest with reactions, need to account for the average energy loss, hS.dxi. light nuclei can be explained in this way. This indicates that The stopping power at very low energies was further studied more precise experiments need to be carried out to allow for a in Ref. [52, 53] for H+ + H, H+ + He, and He+ + He col- critical review of theory versus experimental values of the elec- lisions. These are the simplest few electron systems that can tronic screening potentials Ue and the role of clusterization in be treated with a relatively accurate theory, and one has ver- astrophysical reactions. ified that the stopping power is in fact smaller than the those predicted by the experimental extrapolations of the Andersen- 2.4. Neutron stars Ziegler tables [54]. Because at very low ion energies the elec- 2.4.1. General observations trons in the atoms respond nearly adiabatically to the time- Neutron-star masses can be deduced from observations of dependent interaction, the main cause of stopping are charge supernova explosions and from binary stellar systems. For ex- exchange, i.e., when an electron jumps from one atom to the ample, Newtonian mechanics (i.e., Kepler’s third law) relates other, or by Rutherford scattering, i.e., straggling, in the tar- the mass of a neutron star MNS and its companion mass MC in get (usually denoted as “nuclear stopping”). Such findings are a binary system though in agreement with previously determined stopping-power values reported in Ref. [55]. This is shown in Figure 4 (right) 3 3 (MC sin θ) TNS v + θ , [53]. The same trend was found for atomic He +He [53]. A 2 = (29) (MNS + MC) 2πG “quenching” of the nuclear recoil contribution to the stopping power was observed experimentally in Ref. [56] and explained where TNS is the period of the orbit, vθ is its orbital velocity in Ref. [53]. Several fusion reactions were further studied in projected along the line of sight, and θ is the angle of inclina- deuterated metals and a large increase of the cross sections were tion of the orbit. But this equation is not enough to determine found [57, 58, 59, 60, 61, 62]. No plausible theoretical expla- MNS as we need to know the mass of the companion, too. Gen- nation seems to exist to explain such discrepancies. eral relativity predicts an advance of the periastron of the orbit, dω/dt, given by 2.3.3. Clusterization in light nuclei !5/3 2/3 Recently, it has been proposed that a possible solution of the dω 2π 2/3 (MNS + MC) = 3 T , (30) “electron screening puzzle”, maybe due to clusterization and dt TNS 1 − e polarization effects in nuclear reactions involving light nuclei 3 −6 at very low energies [63, 64]. Different tunneling distances for where and T = GM/c = 4.9255 × 10 s. The observation of each cluster induce a reduction of the overall tunneling prob- this quantity and comparison to this equation, together with the ability. Such clustering effects can also be induced by polar- range parameter R = T MC associated with the Shapiro time ization as the nuclei approach each other, as shown in Figure 5 9 Outer crust: nuclei + e-- Atmosphere 2.0 Inner crust: nuclei + e-- Outer core: + n + nuclear pasta nuclear matter 1.5 K --

+ n + p + e + µ M /

Inner core: M 1.0 hyperons, quarks? ∼ 106 g/cm3 0.5 ∼ 1 km ∼ 4 x 1011 g/cm3 0.0 ∼ 1014 g/cm3 ∼ 10 km 8 9 10 11 12 13 14 ∼ 5 x 1014 g/cm3 radius (km)

Figure 6: Schematic view of the structure of a neutron star, showing the ap- Figure 7: Mass of a neutron star (in units of solar masses) versus radius in proximate sizes and features of the star crust and the core holding the bulk of kilometers calculated (solid, dashed and dotted curves) with a few realistic EoS. its neutron matter. The horizontal band displays the limits of the observed masses [74].

delay of the pulsar signal as it propagates through the gravi- a “star quake” when there is a rupture in their stiff crust. As tational field of the companion star, and the Keplerian equa- a consequence, the equatorial radius contracts and angular motion 29, allows one to determine the neutron star mass MNS . mentum conservation leads to an increase of rotation. Glitches Other so-called post-Keplerian parameters such as the the or- could also be due to vortices in the superfluid core transiting bital decay due to the emission of quadrupole gravitational ra- from a metastable state to a lower-energy state [73]. diation, or the Shapiro delay shaper parameter, can be used The hitherto compiled knowledge about neutron stars seems together with Eq. 29 to determine the mass of a neutron star to indicate that they contain a dense core surrounded by a much [65]. Typical masses obtained with this procedure range from thinner crust with mass . M /100 and a thickness of . 1 km MNS = 1.44M from observations of binary pulsars systems [6] (see Fig. 6). The crust, divided into outer and inner crust, [66] to MNS = 2.01M from millisecond pulsars in binary sys- consists of neutron-rich nuclei coexisting with strongly degen- 11 −3 tems formed by a neutron star and a white dwarf [67, 68]. erate electrons and for densities ρ & 4 × 10 g cm free neu- The radius of a neutron star is more difficult to determine trons drip from nuclei. The density increases going from crust because they are very small. But measurements of the X-ray to the core and at their interface the density is ≈ ρ0/2, where 14 −3 flux Φ stemming from a neutron star in a binary system at a dis- ρ0 = 2.8 × 10 g cm is the saturation density of nuclear mat- tance d from us can be used, assuming that the radiation stems ter. The outer core is probably composed of neutrons, protons from a blackbody at temperature T, leading to the effective ra- and electrons, whereas the inner core, found in massive and dius r more compact neutron stars might contain pion, kaon or hy- Φd2 peron condensates, or quark matter [75]. The Equation of State R = . (31) e f f σT 4 (EoS), i.e., how pressure depends on the density in the core of This can be used together with a relativistic correction to get a neutron star, is usually separated in an EoS for the outer core the neutron star radius R from (ρ . 2ρ0) and another for the inner core (ρ & 2ρ0). r 2GM 2.4.2. Structure equations R = R 1 − NS . (32) e f f Rc2 The equations of hydrostatic equilibrium for a neutron star are modified to account for special and general relativity cor- The radii of neutron stars found with this method range within rections. The so-called Tolman-Oppenheimer-Volkoff (TOV) 9 − 14 km [69, 70, 71, 72]. equation is the state-of-the-art equation for the structure of a Neutron stars can rotate very fast due to the conservation spherically symmetric neutron star in static equilibrium, given of angular momentum when they were created as leftovers of by supernova explosions. In a binary system the matter absorption " #" #" #−1 from the companion star can increase its rotation. The angu- dp(r) G p(r) p(r) 2Gm(r) = − ρ(r) + m(r) + 4πr3 1 − , lar speed can reach several hundred times per second and turn dr r2 c2 c2 c2r it into an oblate form. It slows down because its rotating mag- (33) netic field radiates energy into free space and its shape becomes where m(r) is the total mass within radius r, more spherical. The slow-down rate is nearly constant and ex- Z r tremely small, of the order of −(dΩ/dt)/Ω = 10−10 − 10−21 m(r) = 4π r02ρ2(r0)dr0, (34) s/rotation. Sometimes a neutron star suddenly rotates faster, a 0 phenomenon know as a “glitch”, thought to be associated with

10 and ρ(r) and p(r) is the local density and pressure, respectively. energy which measures the contribution to the E/A from the The second terms within the square brackets in Eq. (33) stem difference between N and Z, parametrized as δ = (N − Z)/A. from special and general relativity corrections to order 1/c2, For symmetric nuclear matter, the proton and neutron den- and in their absence one has the equations of hydrostatic equi- sities are equal, ρn = ρp, and the total nucleon density is ρ = librium following Newton’s gravitation theory. To obtain how ρp + ρn = 2ρn. The energy per nucleon is related to the en- mass increases with the radial distance, the two equations above ergy density, (ρ), by means of (ρ) = ρE(ρ)/A, which includes need to be supplemented by an EoS linking the pressure to den- the nucleon rest mass, mN . The density dependent function sity, p(ρ). The relativistic corrections factors have all the effect E(ρ)/A − mN has a minimum at ρ = ρ0 with a value E(ρ0)/A = of enhancing the gravity at a distance r from the center. −16 MeV, obtained with The TOV equation is integrated enforcing the boundary con- ! ! ditions M(0) = 0, ρ(R) = 0 and p(R) = 0, where R denotes the d E(ρ) d (ρ) = = 0 at ρ = ρ0 . (36) neutron star radius. It is usually integrated radially outward for dρ A dρ ρ p, ρ and m until it reaches p ' 0 at a star radius R. A maximum The EoS of homogeneous nuclear matter is the relation of the value of the neutron star mass is obtained for a specific value pressure and density, of the central density. Typical results are plotted as in Figure 7 with the star mass M given in terms of the central density ρc, or d[(ρ, δ)/ρ] p(ρ, δ) = ρ2 , (37) in terms of the radius R. An EoS based on non-interacting neu- dρ tron gas yields the maximum mass of a neutron star as ∼ 0.7M , whereas a stiffer equation of state yields ∼ 3M [76]. The hori- with δ = (ρn − ρp)/ρ. zontal band displays the limits of the observed masses [74]. At The value of incompressibility of nuclear matter, K0, or the very high pressures, Eq. (33) is quadratic in the pressure and curvature of the energy per particle, or the derivative of the pres- if a star develops a too high central pressure, it will quickly de- sure at the saturation density ρ = ρ0, is velop an instability and will not be able to support itself against " !# dp(ρ) ∂2(E/A) d2 gravitational implosion. K = 9 = 9ρ2 = 9 ρ2 . (38) 0 dρ ∂ρ2 dρ2 ρ ρ0 ρ0 2.5. The equation of state of neutron stars It has been extracted from the analysis of excitations of isoscalar Neutron stars are almost exclusively made of neutrons with giant monopole resonances in heavy ion collisions. a small fraction, ∼ 1/100, of electrons and protons. The neu- The symmetry energy S (ρ) can also be expanded around tron is transformed into a proton and an electron via the weak x = 0 → e− ν E 2 decay process n p + + ¯e, liberating an energy of ∆ = 1 ∂ E 1 2 S = = J + Lx + Ksym x + ··· (39) mn −mp −me = 0.778 MeV which is carried away by the electron 2 ∂δ2 2 and the neutrino4. In weak decay equilibrium, as many neutrons δ=0 − where J = S (ρ ) is known as the bulk symmetry energy, L decay as electrons are captured in p+e → n+νe, which can be 0 expressed in terms of the chemical potentials for each particle determines the slope of the symmetry energy, and Ksym is its curvature at the saturation density ρ = ρ . The slope parameter as µn = µp + µe. The chemical potential is the energy required 0 is given by to add one particle of a given species to the system. The Pauli dS (ρ) principle impedes decays when low-energy levels for the pro- L = 3ρ0 . (40) dρ ton, electron, or the neutron are already occupied. Since the ρ0 matter is neutral, the Fermi momenta of protons and electrons Experimental values of masses, excitation energies and other are the same, kF,p = kF,e. nuclear properties have been compared to microscopic models A large number of experimental data on stable nuclei has yielding J ≈ 30 MeV [77]. The situation is not so clear for the obtained important results for symmetric nuclear matter (Z = slope parameter. See Ref. [78] for a recent review on the EoS 3 N) such as an equilibrium number density ρ0 = 0.16 nucleons/fm , of neutron stars. and a binding energy per nucleon at saturation of E/A = −16 MeV, where (ρ) is the energy density. The saturation density 2.6. Microscopic theories of homogeneous nuclear matter ρ0 corresponds to a Fermi momentum of kF = 263 MeV/c, 2.6.1. The EoS from Hartree-Fock mean field theory small compared with m = 939 MeV/c2 and thus justifying N The analysis of experimental data is often done by compar- a non-relativistic treatment. A Taylor expansion of the energy ison to microscopic calculations where, e.g., Skyrme interac- per particle for asymmetric nuclear matter (Z N) can be done, , tions are used in traditional Hartree-Fock-Bogoliubiov calcula- E E 1 1 tions. Most of the Skyrme interactions are able to describe suc- (ρ, δ) = (ρ , 0) + K x2 + Q x3 + S (ρ)δ2 + ··· , (35) A A 0 2 0 6 0 cessfully a large number of nuclear properties, such as a global description of nuclear masses and even-odd staggering of ener- here x = (ρ − ρ0)/3ρ0, K0 is known as the incompressibility gies [79]. Assuming that the nuclear interaction between nucle- parameter, Q0 the so-called skewness, and S (ρ) is the symmetry ons i and j has a short range compared with the inter-nucleon spacing, the Skyrme interaction is an expansion which keeps at

4In this and in the next equations we use ~ = c = 1. 11 most terms quadratic in the nucleon momenta, single-particle states in these fields are not orthogonal which should be specially corrected. 1 2 02 vSk = t0 (1 + x0Pσ) δ + t1(1 + x1Pσ)(k δ + δk ) Expressions for the energy density in infinite matter can be 2 obtained by neglecting the Coulomb interaction and assuming 0 1 α + t2 (1 + x2Pσ) k · δk + t3 (1 + x3Pσ) ρ δ plane waves for the orbitals. For uniform and spin-saturated nu- 6 clear matter, the EoS for arbitrary neutron and proton densities · 0 + iW0k(σi + σ j)δ k (41) at zero temperature is given in terms of the Skyrme parameters where k and k0 are the initial and final relative momenta of by [80, 81] a colliding pair of nucleons, P is the spin-exchange opera- 2 σ 3k 1 tor between the two nucleons with spins σ, δ = δ(r − r0 ), (ρ, δ) = F f + t ρ 2(x + 2) − (2x + 1) f i j 10m 5/3 8 0 0 0 2 k = (1/2i)(∇ − ∇ ), acting on the wave function on the right, I j 1 k0 is its adjoint, and ρ(r) is the local density at r = r + r . + t ρα+1 2(x + 2) − (2x + 1) f 1 2 48 3 3 3 2 In this form, the Skyrme interaction has 10 parameters: ti, xi, ( 3 W0 and α. The Energy Density Functional, E[ρ], can be easily 2 + kF ρ [t1(x1 + 2) + t2(x2 + 2)] f5/3 obtained from the Skyrme functional yielding a precious depen- 40 ) dence around the nuclear saturation density which can be used 1 + [t (x + 2) − t (2x + 1)] f , (45) to infer properties of neutron stars [80]. 2 2 2 1 1 8/3 In the Hartree-Fock theory, the nuclear many-body wavefunction, Φ, is described by a Slater-determinant of single-particle where the first term is due to the kinetic energy density, kF = 2 2 m m orbitals and obeys the Schrodinger¨ equation HΨ = EΨ. Each 3π ρ/2 and fm = [(1 + δ) + (1 − δ) ]/2. orbital wavefunction, φk, obeys an equation of the form Pairing is an important part of the total energy, in particular leading to phenomena such as the odd-even staggering effect 2 Z ~ 2 3 of nuclear binding energies, and the pairing energy needs to be − ∇ φk + U(r1)φk(r1) − d r2W(r1, r2)φk(r2) = iφk(r1), 2m added to Eq. (45). It can be included in the HF method by (42) using the Bardeen-Cooper-Schrieffer (BCS) theory [82], or its where k are the single-particle energies. The potential U(r1), Hartree-Fock-Bogoliubov (HFB) extension [83, 84]. The BCS is the direct contribution to the mean-field, theory is better described in the operator formalism. Then the Z ground state of the nucleus is defined as 3 X 2 U(r1) = − d r2vSk(r1, r2) |φk(r2)| , (43) † † k<F |BCS i = Π u + v a a |−i , (46) k>0 k k k k¯ where F is the Fermi energy. The third term in Eq. (42) arises where |−i is the vacuum with no particles, normalized as h0|0i = due to the exchange potential 1, a† is the creation operator of the state |ki, a† is the creation k k¯ X ∗ operator of the state W(r1, r2) = vSk(r1, r2)φk(r2)φk(r1). (44) E k<F ( jk−mk) k¯ = −1 |k; nk jklk, −mki , (47) The set of N coupled equations for the N orbitals self-consistent 2 field is nonlinear. One can solve them in an iterative way start- where (nk jklk, mk) are single-particle quantum numbers, |vk| is ing from some reasonable guess for a mean field where the par- the probability that the state |ki is occupied, and is related to uk 2 2 by the condition |u| + |v | = 1, and u¯ = u and v¯ = −v . ticles are embedded, finding the single-particle eigenstates φ0k k k k k k The BCS state defined in Eq. (46) is not an eigenstate of the and the eigenvalues 0k, filling the lowest states and calculating the resulting fields U and W above. With the new Eq. (42) particle-number operator and the BCS equations are obtained we repeat the whole cycle. However, the nonlinear HF equa- by applying the variational principle to the expectation value of tions have many solutions. Each of them provides a relative the operator H = H − λN , where H is the nuclear hamiltonian, energy minimum as compared to the “nearest neighbors” in the N is the particle number operator and λ is a Lagrange multiplier. space of the Slater determinants. The iteration procedure does The parameters to be changed in the variational procedure now not determine which solution corresponds to an absolute min- are the vk and uk coefficients. The application of the variational imum among Slater determinants. For example, spherical and principle implies to get the solutions of the set of BCS equations deformed mean field are possible as solutions with the same 2 2 original interaction. An additional investigation should show uk + uk ∆k = 2ukvkηk (48) which solution is energetically favorable. One can also look for where the solutions with the distribution of the empty and filled orbitals different from the normal Fermi gas in the ground state. 1 X p D E ∆k = − 2 jm + 1umvm mm0|vpair|kk0 , (49) For example, the solutions exist with some holes inside F . The p 2 jk + 1 m self-consistent field determined by such a distribution of particles is different from the ground-state field. Therefore the and ηk = hk|T|ki − λ, where T is the kinetic energy and a renor-

12 malization term was dropped. The matrix element with the ent types of pairing called volume, sur f ace and mixed pairing, pairing interaction, vpair, indicates that the single-particle wave reflecting pairing fields localized in the volume, surface, or a functions are coupled to angular momentum zero. mix of the two. The volume interaction is not dependent on the The solution of the BCS equations provides the values of density (η = 0), and for this reason it is easier to handle. The vk and uk, which together with φk allows the evaluation of the nuclear compressibility is strongly sensitive to the surface prop- expectation values of various ground state quantities related to erties of the nucleus [87]. Theoretical models using couplings the BCS ground state. The pairing added in the BCS model with collective vibrations require pairing fields peaked at the does not change the nuclear field which means a lack of self- surface of the nucleus [88] and, e.g., Ref. [79] has shown that consistency. The more general approach which is the most ad- surface pairing (η = 1) reproduces nuclear masses with better vanced version of the self-consistent HFB approximation [83, accuracy as compared to other parametrizations. 84], widely used in condensed-matter physics in order to take The density dependence of the pairing interaction gives rise into account simultaneously effects of Coulomb forces or im- to a rearrangement term in the single-particle Hamiltonian. This purities and superconducting pairing. Pairing correlations are is because the energy functional in a Skyrme approximation has very important to describe single-particle motion and collective the form: E = Ekin + ES kyrme + Epair + ECoul with the respec- modes as rotation. Pairing modifies the distribution of particles tive kinetic, Skyrme, pairing, and Coulomb terms. In the HFB over orbitals considerably. As a result, the self-consistent field method the single particle Hamiltonian is obtained from a func- is different than without pairing, influencing nuclear shapes, tional derivative of the energy with respect to the density; and moments of inertia, mass parameters, transition probabilities the contribution from Epair is usually called rearrangement term and reaction cross sections. Since nuclear shape defines single- [89], δE δEskyrme δEpair δE particle orbits and conditions for pairing, a complex interplay h = kin + + + coul . (52) of various residual interactions is not accounted for in the HF δρ δρ δρ δρ approach which does not include pairing correlations on equal Similarly, the residual fields giving rise to the Quasi-particle footing. The pairing added in the BCS model does not change Random Phase Approximation (QRPA) matrix (see below) are the nuclear field which means a lack of self-consistency. There- functions of the second derivatives of the densities and the re- fore, the HFB approximation is required for accuracy in the de- arrangement term is: scription of nuclear properties. The HFB approximation a gen- ! eral canonical (Bogoliubov) transformation and introducing the δh δ δEpair rearr = . (53) concept of quasi-particles which does not conserve the particle δρ δρ δρ number. In the second variant of HFB calculations (HFB+LN), one performs an approximate particle number projection using Without an explicit density dependence of the pairing term, no the Lipkin-Nogami (LN) method. For more details on the HFB rearrangement term appears either in the HFB or in the QRPA approach, we refer to Refs. [83, 84]. matrix, as is the case of the volume pairing. But mixed and A common parametrization of the pairing interaction is given surface pairing parametrizations give rise to a non zero rear- by [85, 86] rangement term. In Fig. 8 we show a Skyrme (SLy4) Hartree-Fock calcula- vpair(1, 2) = v0 gτ[ρ, βτz] δ(r1 − r2), (50) tion of the nuclear matter (SNM) equation of state (EoS) compared to the result pure (ANM) neutron matter. The two curves where ρ = ρn + ρp is the nuclear density and β denotes here the are roughly separated by the bulk symmetry energy factor S (ρ0). asymmetry parameter β = (ρn − ρp)/ρ. One can introduced The EoS for neutron stars is likely to be somewhat different than an isospin-dependence of the pairing interaction through the the ANM curve shown and lie somewhere within the hypothet- density-dependent term, gτ, determined by the pairing gaps in ical region shown in the figure. nuclear matter. A convenient functional form, usually termed isoscalar + isovector pairing, is 2.6.2. The EoS in relativistic mean field models

!αs !αn Relativistic Mean Field (RMF) theories [90, 91, 92, 93] ρ ρ gτ[ρ, βτz] = 1 − fs(βτz)ηs − fn(βτz)ηn , (51) are another way to calculate nuclear-matter properties. RMF ρ0 ρ0 is based on using the Euler-Lagrangian equations using a La- grangian for nucleon and mesons fields and their interactions. where ρ0 is the saturation density of SNM and fs(βτz) = 1 − h i The Lagrangian density with nucleons described as Dirac parti- f (βτ ) with f (βτ ) = βτ = ρ (r) − ρ (r) τ /ρ(r), with τ n z n z z n p z z cles interacting via the exchange of mesons and the photon Aµ. being the isospin operator. This form of the paring interaction The usual mesons considered are the scalar sigma (σ), which introduces 5 additional parameters, v , η and α in the HF 0 s(n) s(n) represents a large attractive field resulting from complex mi- calculation. Most often only two parameters are used, one for croscopic processes, such as uncorrelated and correlated two- strength, and another for the radial dependence of the pairing pion exchange, the vector omega (ω) describing the short-range interaction [79]. repulsion between the nucleons, and the iso-vector vector rho The density dependence of the pairing interaction replicates (~ρ) carrying the isospin. The (~ρ) meson is responsible for the the effect of pairing suppression at high density (momenta). isospin asymmetry. With the nucleon mass denoted by m and The parameters V , η are chosen so that one can describe differ- 0 the corresponding lessons masses (coupling constants) denoted 13 The last term in Σµ is a rearrangement term due to the density dependence of the coupling between the sigma meson and the nucleon, and ρs is the scalar density of nucleons, defined below. The variation of the Lagrangian density with respect to the meson field operators yields the Klein-Gordon equations for mesons, neutron stars 2 3 [−∆ + mσ] σ = −gσρs − g2σ − g3σ (58) µ µ µ ν [−∆ + mω] ω = gω j − c3ω (ω ων) (59) S(r0) = 30 MeV h i µ µ −∆ + mρ ρ = gρ ¯j . (60) The nucleon spinors provide the relevant source terms -16 MeV X µ X µ ρs = ψ¯ iψini, j = ψ¯ iγ ψini, i i µ X µ ¯j = ψ¯ iγ ~τψini, (61) i Figure 8: Skyrme (SLy4) Hartree-Fock calculation of the nuclear matter (SNM) equation of state (EoS) compared to the result pure (ANM) neutron matter. The In infinite matter and at finite temperature, the Fermi-Dirac statis- two curves are roughly separated by the bulk symmetry energy factor S (ρ0). tics imply that the occupation numbers ni of protons and neu- The EoS for neutron stars is likely to be somewhat different than the ANM trons are −1 curve shown and lie somewhere within the hypothetical region shown in the (Ei−µ)/kBT figure. ni = e + 1 , (62) where µ is the chemical potential for a neutron (proton). For the EoS of neutron stars, one is mostly interested in T ∼ 0. For a spherical nucleus, there are no currents in the nucleus and the by mσ (gσ), mω (gω) and mρ (gρ), one has (with ~ = c = 1) spatial vector components of ωµ, ρµ, and Aµ vanish. Thus, only h µ µ µ L = ψ¯ iγ ∂µ − m − gσσ − gωγ ωµ − gργ ~τ · ~ρµ the time-like components, ω0, ρ0, and A0 remain. # The introduction of pairing in RMF is usually done simi- µ 1 − τ3 − eγ Aµ ψ larly as in the HF theory by using RMF + BCS, or an HFB-like 2 procedure, as described previously. The energy density of uni- 1 1 1 1 + ∂µσ∂ σ − m2 σ2 − ωµνω + m2 ωµω form nuclear matter is, for T = 0 2 µ 2 σ 4 µν 2 ω µ X 1 1 1 i 1 h 2 2 2 2 2 0,3i µν 2 µ µν = kin + mσσ + mωω0 + mρρ − ~ρ · ~ρµν + mρ~ρ · ~ρµ − A Aµν, (54) 2 4 2 4 i=n,p µ 1 1 1 where γ are Dirac gamma matrices and + g σ3 + g σ4 + c ω2, (63) 3 2 4 3 3 3 0 ωµν = ∂µων − ∂νωµ where ~ρµν = ∂µ~ρν − ∂ν~ρµ 2 Z Aµν = ∂µAν − ∂νAµ. (55) i d3k E k , kin = 3 ( ) (64) (2π) |k|

14 with the transition amplitudes,

0 0 Akk0 ≡ Aph,p0h0 = kδkk0 + ph |Vext|p h Neutron EoS 0 0 Bkk0 ≡ Bph,p0h0 = pp |Vext|h h , (67)

Where k = p − h is the energy of the bare p-h excitation. This eigenvalue problem does not correspond to the diagonalization of a hermitian operator (because of the matrix σz in the right hand side). Therefore, one cannot guarantee that the eigenvalues ~ω are real. An imaginary part of frequency would signal the instability of the mean ﬁeld placed in the base of the RPA. The backward amplitude Yk incorporates correlations in the ground state and reveal the presence of the holes below F and of the particles above F in the actual ground state which does not coincide with the HF vacuum.

As in the HBF theory, the Quasi-particle Random Phase Ap- Figure 9: Equation of state of pure (EoS) neutron matter as predicted by numer- proximation (RPA) is based on Bogoliubov transformed opera- † ous Skyrme interactions. At the neutron saturation density they tend to agree, tors b, b that are introduced to annihilate and create a “quasi- but diverge substantially as they depart from it. Such dispersive behavior of the particle” with a well-defined energy, momentum and spin but as numerous Skyrme parametrizations used in numerical calculations of the EoS a quantum superposition of particle and hole state. They carry was observed quite early [94, 95]. The relativistic mean field parameterizations also show a similar behavior for the several parameterizations used to describe coefficients u and v given by the eigenvectors of a Bogoliubov nuclear properties. matrix. The framework of the time dependent superfluid local density approximation (TDSLDA) [96, 97, 98, 99, 100, 101] is more powerful because it allows one to study large amplitude collective motion. This is an extension of the Density Func- body ground state is not a stationary state any longer. Instead it tional Theory (DFT) to superfluid nuclei that can assess the re- is a wave packet changing in time with different Fourier com- sponse to external time-dependent fields. The time evolution of ponents of the packet. If the perturbation V is weak, the ext the nucleus is determined by the time-dependent mean field main component still corresponds to the ground state and excited states are admixed to it by the perturbation (linear approx-   ∂  U(r, t)  imation). Changing the frequency ω of the perturbation yields   i~   resonances in the response. They appear when real transitions ∂t  V(r, t)  with energy conservation, yielding the excitation spectrum of      h(r, t) − µ ∆(r, t)   U(r, t)  the system. In the next order, transitions between the excited =     , (68) states appear which in general show no resonance with the ex-  ∆∗(r, t) −h∗(r, t) + µ   V(r, t)  ternal field. This is the (neglected in the RPA) “noise” superim- posed onto the coherent response, giving the origin of the term where h(r, t) = ∂E(r, t)/∂ρ is the single-particle Hamiltonian, “random phases”. ∆(r, t) = ∂E(r, t)/∂ρpair is the pairing field, and µ is the chemi- The standard form of the RPA equations can be reached if cal potential. Both are calculated self-consistently using an en- one considers the ground state filled as in the Fermi gas when ergy functional such as a Skyrme interaction. A Fourier transform of the time evolution of the nuclear density yields the en- the basis can be subdivided into the particle (p) states, np = 0 ergy spectrum. if p > 0, and hole (h) states, nh = 1 if h < 0. If one considers the set (ph) = k as a unified label of the p-h excitation and introduces the notations for the matrix elements, 2.7. The incompressibility modulus and skewness parameter of nuclear matter ω ω (ρω)ph = Xk , (ρω)hp = Yk , (65) Nuclei display collective excitation modes which have been studied for more than 6 decades. Baldwin and Klaiber [102, where ρω is a time independent matrix (Fourier transform of ρ, 103] reported the existence of highly collective modes in pho- the density matrix) with matrix elements labeled by the pairs toabsorption experiments. A decade before that, Bothe and (ph) of particle-hole states. In terms of the HF ground state, Gentner [104] had already obtained very large cross sections |0i, the density matrix expressed in an arbitrary basis is ρph = for the photoproduction of radioactive elements on numerous D † E 0|ahap|0 . targets, by two orders of magnitude larger than predicted by The canonical RPA equations are theory, what was a first indication that such resonances might         exist. They used high-energy (15 MeV) photons at the time. We ω ω  AB  X  1 0  X  now understand that the origin of these absorption resonances     = ~ω     (66) B∗ A∗ Yω 0 −1 Yω are due to the collective response of the nuclei to the photon electric dipole (E1). The dipole collective states were also earlier predicted by Migdal [105]. Later, giant resonances were 15 2 where mN is the nucleon mass, and hr i is the mean square radius of the nucleus. Extracting the value of the incompressibility of nuclear matter K0 from KA is strongly model dependent [87, 115]. There- fore, the most reliable way to obtain K0 is from a microscopic calculation which reproduces accurately the experimental data for a large number of nuclei. To extract the centroid√ of the giant √resonance (EISGMR) one uses the ratios m1/m0, m1/m−1 and m3/m1, where the moments are defined as Z ∞ k mk = E S (E)dE, (70) 0 where the strength is

X 2 S (E) = |h0|F0| ji| δ(E − E j), (71) j Figure 10: Schematic view of giant resonance vibrations in nuclei. The isoscalar giant monopole resonance (ISGMR) in a spherical nucleus consists where |0i is the ground state and | ji is an eigenstate with energy of a breathing mode-like vibration, the Isovector Giant Dipole Resonance (IVGDR), here shown for a deformed nucleus, consists of vibrations of pro- E j of the QRPA. The monopole operator is of the form: tons against neutrons, and the Isoscalar Giant Quadrupole Resonance is like a prolate-oblate vibration mode in the nucleus. A X 2 F0 = ri . (72) i=1 further studied with photoabsorption processes with increased The integral of Eq. (70) should run from zero to infinity, but, accuracy Refs. [106]. The centroid of giant resonances are in practice, the QRPA calculations yield discrete values for the located above the particle emission threshold, mostly decay- eigenstates and the integral of Eq. (70) reduces to a finite sum. ing by neutron emission. The large Coulomb barrier in nuclei The results of HFB calculations reported on Refs. [89, 116] usually prevents charged particle decay. Therefore, photoab- are shown in Table 1. 20% of the nuclei investigated in Refs. sorption cross sections are usually observed from neutron yields [89, 116] are well explained with the SLy5 interaction, and 10% [107, 108]. with the SkM* interaction. For the majority of the nuclei under The incompressibility of nuclear matter, Eq.(38), has been investigation the centroid energy of the ISGMR is better repro- extracted from several experiments. The giant resonances, very duced using the soft interaction Skxs20 (K0 ≈ 202 MeV) in collective nuclear vibrations, of several multi polarities (see contrast to the generally accepted value for KNM ≈ 230 MeV. Figure 10) has been very valuable for such studies. The isoscalar Therefore, there is still some uncertainty in the generally ac- giant monopole resonance (ISGMR) in a spherical nucleus con- cepted value of the incompressibility of nuclear matter. sists of a breathing mode-like vibration, while the Isovector Gi- The skewness parameter Q0 is more difficult to ascertain ant Dipole Resonance (IVGDR), here shown for a deformed nu- and estimated values in the range −500 ≤ Q0 ≤ 100 MeV cleus, consists of vibrations of protons against neutrons, and the [117]. The lack of accurate knowledge of the high powers in Isoscalar Giant Quadrupole Resonance is like a prolate-oblate the Taylor expansion means that densities extrapolated to val- vibration mode in the nucleus. In particular, the study of the ues well below and well above the saturation density tend to isoscalar giant monopole resonance (ISGMR) has been one of yield conflicting results [118]. Therefore, there is a strong in- the best tools [109] to extract the magnitude of K0 [110]. Re- terest in the literature to pinpoint those Skyrme interactions that sults in the range K0 = 210−240 MeV have emerged from such better describe neutron matter properties. analysis [111, 112, 87, 89, 113]. Not only Skyrme-type calculations, but also those based on a relativistic mean field approach, 2.8. The slope parameter of the EoS e.g., in Ref. [114] show good estimates of the ISGMR centroids Both the HFB as well as the RMF method are the abun- with the same value of the incompressibility modulus. dantly used in the literature to describe nuclear properties and The energy centroid of the ISGMR is related with the nu- to study, by extrapolation, the properties of neutron stars. As an example, we show in Table 1 the predictions of a few Skyrme clear incompressibility modulus KA which is the equivalent for a nucleus of the elastic constant of a spring. Nuclei with a low models (there are hundreds of them) [119, 120, 121, 122, 123, 124, 125, 81] for some basic quantities needed for the EoS of value of KA are called “soft nuclei” and those with high values neutron stars. Whereas K and J tend to consistently agree of KA as called “stiff nuclei”. The relation between KA and the 0 energy centroid of the ISGMR is [87] among the models, the slope parameter L can differ widely. Because the theoretical models from Refs. [119, 120, 121, r K 122, 123, 124, 125, 81], as well as from numerous other Skyrme E A , ISGMR = ~ 2 (69) and RMF models, have been fitted to reproduce nuclei at labo- mN hr i ratory conditions, it is far from clear which value of L should be 16 K0 J L K0 J L 3.4 SIII 355. 28.2 9.91 SLY5 230. 32.0 48.2 B 3.2 Be SKP 201. 30.0 19.7 SKXS20 202. 35.5 67.1 Li SKX 271. 31.1 33.2 SKO 223. 31.9 79.1 3.0 ] m f 2.8 [

HFB9 231. 30.0 39.9 SKI5 255. 36.6 129.

I R 2.6 Table 1: Predictions by some Skyrme models for the properties of nuclear matter at the saturation density in MeV units. The parameters for the Skyrme forces 2.4 He C were taken from [119, 120, 121, 122, 123, 124, 125, 81] 1/3 . 2.2 ...... 1.18 A fm

2.0 6 8 10 12 14 16 18 adopted in the EOS of neutron stars. Some constraints based on A observations and new laboratory experimental data have been used in a few references, see, e.g., Ref. [81] to eliminate some Figure 11: Left: An artistic drawing of a 11Li nucleus, showing the two neutrons 4 “outdated” Skyrme or RMF functionals. But there is still a long building a halo around a He nucleus core. Right: Interaction radii of light nuclei (errors bars not shown). The large deviation from the expected radii of a way to go to reach a consensus on what are the best models to nucleus (dotted line) is evident for the so-called “halo” nuclei. be extrapolated to neutron stars. In fact, it is easy to understand the strong dependence of the neutron EoS on the symmetry energy S . For pure neutron natural explanation of the narrow component of the momentum matter, δ = 1, and at the saturation density ρ ∼ ρ0 one has distribution. In fact, a narrow momentum distribution is related p = Lρ0/3. Hence, the neutron matter and its response to gravitational pressure is directly connected to the slope parameter to a large spatial extent of the neutrons, which is a due to their L. It is also worthwhile to mention that the explosion of a core- small separation energy. These na¨ıve conclusions were later collapse supernova is strongly depends on the symmetry energy confirmed with Coulomb breakup experiments [147] and with of the nuclear EOS. For more discussion of these subjects see, theory [17,18]. The influence of nuclear haloes in astrophysics e.g., Refs. [126, 127, 128, 129, 75, 76, 130, 131, 79, 132, 133, is observed in numerous reactions. Perhaps, the most celebrated 7 8 89, 134, 135, 136, 137, 138, 139, 140, 141, 142]. one is the Be(p,γ) B reaction, of relevance for the production of energetic solar neutrinos via the 8B decay. Theoretically, it is 2.8.1. Nuclear radii found that an accurate value of the S-factor (at Ecm ∼ 20 keV) Mean field theories such as HFB and RMF can obtain phys- for continuum to bound state radiative capture is only possible ical observables of nuclear interest such as the root mean square if one integrates the nuclear matrix elements up to distances of radius, the charge radius, and the neutron skin thickness. The r = 200 fm [148]. nuclear charge radius can be measured experimentally by the Recently, isotope-shift measurements have been performed scattering of electrons or muons by the nucleus. As for the neu- for He, Li and Be isotopes to determine their charge radii [149, tron distribution, hadronic probes are often used. It was in this 150]. The isotope shift, δνIS , has contributions from the mass way that abnormally large radii of light nuclei were found. In shift , δνMS , and the field shift , δνFS , i.e., , δνIS = δνMS +δνFS . 11 The latter contribution is proportional to the charge radius of Figure 11 (left) an artistic drawing of a Li nucleus is shown, 2 with the two neutrons building a halo around a 9Li nucleus core. the nucleus, δνFS = C hric. To obtain the radius with accuracy a variational calculation is done for the electronic wave- The interaction radius, RI is defined as the radius in the interac- 2 function, and QED corrections are included. The measure- tion cross section, σ = πRI , for reactions leading to any number of removed nucleons. Interaction radii of light nuclei are plotted ments show good agreement with ab-initio nuclear structure (errors bars not shown) in Figure 11 (right). The large deviation calculations [150]. Such experiments open the window to ex- from the expected radii of a nucleus (dotted line) is evident for plore a larger number of radii of light nuclei providing valu- the so-called “halo” nuclei [143, 144, 145]. able constraints to refine current nuclear models. The tech- The unexpected large size of the 11Li nucleus was also made nique has been extended to heavy nuclei and in particular, the clear by measuring the momentum distributions of 9Li frag- mean-square nuclear charge radii has been measured along the 108−134 ments in nucleon-removal reactions [146]. A superposition of even-A tin isotopic chain S n by means of collinear laser two approximate gaussian-shaped distributions was used to re- spectroscopy at ISOLDE/CERN using several atomic transi- produce the data. The wider peak was connected to the knock- tions [151]. out of neutrons from a tightly bound 9Li core, while the nar- For medium-heavy and heavy nuclei, matter radii are not rower peak was thought to arise from the knockout of the loosely so well known. In particular, for rare nuclear isotopes not only bound valence neutrons in 11Li. The two-neutron separation en- matter radii but also information on charge radii is scarce, de- ergy in 11Li is small, of the order of 300 keV, explaining why spite the new data obtained with isotope-shift experiments men- their removal only slightly perturbs the 9Li core, leading to a tioned above. Most of our knowledge on these nuclear proper- 17 6 rexp. - rth. 0.2 (a) SLy5 + surface 5 SkM* + surface 0.15 Skxs20 + surface O 4 Ca 0.1 (fm)

n Ni r

Zr fm Sn 0.05 3 Pb 1/3 1.139N 0 2 0 20 40 60 80 100 120 140 N -0.05 80 100 120 140 160 180 200 220 A

Figure 13: Diﬀerence between the experimental [152] and the theoretical charge radii of Zr, Mo, Cd, Sn, Sm and Pb isotopes(isotopes of the same element are connected by a curve) [89].

theories for the nucleus. An electron-rare-isotope ion collider would be the best possibility, as we discuss later.

2.8.2. Neutron skins A neutron excess in a nucleus leads to a larger neutron pressure than that due to the protons. Such a neutron pressure con- Figure 12: Neutron (a) and proton (b) radii of various isotopes and isotones tributes to the energy per nucleon which then becomes a func- calculated by means of the HF + BCS model with isoscalar + isovector pairing tion of the nuclear density and its nucleon asymmetry. Neutron interaction [86]. The solid lines are the empirical fits used in Ref. [138]. skins in nuclei are therefore expected to be directly correlated to the symmetry energy and the slope parameter of the nuclear matter EoS. This is expected even for a large stable nucleus 208 ties come from microscopic mean field theories, such as those such as Pb. The neutron skin in nuclei is defined as we described previously. For example in Figure 12, from Ref. D E1/2 D E1/2 ∆r ≡ r − r = r2 − r2 . (73) [86], we show the neutron (a) and proton (b) radii of various iso- np n p n p topes and isotones calculated by means of the HF + BCS mo- Very little is known about neutron skin in nuclei. There del with isoscalar + isovector pairing interaction. Solid lines are few experimental data using, e.g., antiprotonic atoms [153]. are empirical fits used in Ref. [138]. One notices in Figure The antiproton annihilation method consists of the study of the 12(a) that the HF + BCS model yields larger neutron radii for residual nuclei with one unit mass number smaller than that nuclei with N < 40 than the simple phenomenological formula of the target A . The assumption is that such residues origi- 1/3 T Rn = 1.139N fm but yields a smaller radii for nuclei with nate from events in which all produced pions miss the target, N > 120. In Figure 12(b) the proton radii for N = 20, 28, and the target remains with a very low excitation energy so 40, 50, 82, and 126 isotones are shown as a function of proton that compound-nucleus evaporation or fission does not occur. 1/3 number Z. The simple Rp = 1.263Z fm dependence is also When both the AT − 1 products are γ-emission unstable, their shown. The simple formula reproduces rather well the HF + yields are determined by standard nuclear-spectroscopy tech- BCS calculations and could be used as a good starting point to niques. These yields are directly related to the proton and neu- describe the isospin dependence of nuclear charge radii in neutron densities at the annihilation region in the nucleus. The ra- tron rich nuclei. However, as expected from phenomenological dial distance of this position in the nucleus, is assumed to be models, one sees a deviation with the HF + BCS model, espe- almost independent of the target atomic number Z and is com- cially for heavy N = 50 and N = 82 isotones. pared to calculations [154] based on antiproton-nucleus optical The differences between known cases of charge radii of Zr, potentials, generated by the so-called t-ρ approximation, and Mo, Cd, Sn, Sm and Pb isotopes and calculations based on the antiproton atomic orbits together with the proton orbitals in the three Skyrme interactions with surface pairing are shown the nucleus involved in the annihilation process. in Figure 13 [89]. The determination of nuclear charge radii The antiprotonic atoms annihilation data is not very accu- in very neutron-rich nuclei is one of the most challenging ex- rate, as shown in Figure 14 by the filled circles for the neutron perimental goals. It would be an excellent test of microscopic

18 0.3 232 130 Th 106 116 Cd 96 Cd Te exp 1 0.2 48Zr 124 238 Ca Sn U exp 2 57 116 0.2 56 90 Sn 208 IS+IV 54 Fe Fe Zr Pb Fe 209 0.1 Bi 128 Te 126 (fm)

(fm) 112 Te 0.1 122 120 p p Sn

Te124 Sn -r -r n

n Te 0 60 r r Ni 40 64 Ca Ni exp 0 58 Sn -0.1 Ni IS+IV 59 Co -0.2 -0.1 0 0.05 0.1 0.15 0.2 100 110 120 130 (N-Z)/A A Figure 15: Combined experimental data for the neutron skin for Sn isotopes. are Figure 14: Neutron skin as a function of isospin parameter δ = (N − Z)/A taken from Refs. [155, 153]. The triangles represent numerical results obtained calculated by means of the HF + BCS model with (isoscalar + Isovector) IS + with the HF + BCS model with IS + IV interaction [86]. IV interaction [86]. Experimental data are taken from Ref. [153] based on the annihilation of antiprotons at the surface of the nuclei.

tion above is due to the Coulomb repulsion of protons. One usually neglects the term contains the surface diffuseness (i.e., skin in several nuclei as a function of isospin parameter setting an = ap). The first term in Eq. (75) depends in leading N − Z order on the asymmetry parameter δ = (N − Z)/A, δ = (74) A 3r J δ − δ t = 0 c , (76) and with empirical values obtained by antiprotonic atom exper- 2 Q 1 + (9J/4Q)A−1/3 iments in a wide range of nuclei from 40Ca to 238U [153]. It 1/3 is clear that the statistics are not very good. Also shown in the where r0 pertains to the relation R = r0A fm, figure (triangles) are results of calculations based on the HF + e2Z BCS model with Isoscalar + Isovector (IS + IV) pairing inter- δ , c = 1/3 (77) action [86]. It is evident that the data can accommodate a wide 20Jr0A range of Skyrme parameters and is not very constraining. and Q is the so-called surface stiffness coefficient, being a pa- Another tool to investigate the neutron-skin thickness is the rameter of the droplet model involving the dependence of the excitation of the spin-dipole resonance (SDR) [155]. It is based surface tension energy on the isospin and neutron skin thick- on the idea that the total L = 1 strength of the SDR is sen- ness. Therefore, it is not obvious from the transcendental rela- sitive to the neutron-skin thickness [156, 157]. The SDR is tion, Eq. (76) that the neutron skin thickness is proportional to be strongly excited in the (p,n) reaction in inverse kinematics the slope parameter. using radioactive nuclear beams. A few radioactive isotopes A bit of phenomenology can help to illuminate the ∆r vs. have been studied using this technique. In Figure 15 we show np L correlation, as shown in Ref. [161]. The key to the proof is the combined experimental data of neutron skin for Sn isotopes to identify [160] [155]. The triangles represent numerical results obtained with ! the HF + BCS model with IS + IV interaction [86]. It is again J 1 S (A) ∼ − 1 A1/3, evident that the data are not good enough to distinguish the most Q (1 + (9J/4Q)A−1/3) J adequate Skyrme interactions. The liquid drop model [158, 159] is perhaps the best way where J to understand the correlation between the neutron skin and the S (A) = slope parameter of the symmetry energy [160]. In the droplet 1 + (9J/4Q)A−1/3 model the neutron skin is given by is the droplet model symmetry energy for a finite nucleus. For heavy nuclei S (A) ∼ S (ρ) [161]. Therefore, one gets, r " 2 # 3 e Z 5 2 2 ∆rnp = t − + (a − a ) , (75) " # n p 2r Ksym 5 70J 2R t = 0 L 1 − x xA1/3(δ − δ ) (78) 3J 2L c where J = S (ρ0) is the volume term of the symmetry energy, Eq. (39), R is the mean nuclear radius and an(p) is the surface with the parameters defined as in Eqs. (35) and (39). Therefore, diffuseness for neutrons (protons). The second term in the equa- to lowest order, t = (2r0/3J)Lδ, showing a linear correlation 19 0.35 0.4 208Pb 0.30 0.3 Sn ] m 0.25 ] f [ m

0.2 f p [ n

r 0.20 p n ∆ r 0.1

0.15 ∆

0.10 0.0 0 20 40 60 80 100 120 140 L [MeV] 0.1 50 60 70 80 90 Figure 16: Neutron skin thickness of 208Pb as a function of the slope param- N eter of the nuclear symmetry energy at saturation as predicted by numerous nuclear mean-field interactions. The different colored circles represent calculations with a particular interaction, either Skyrme or Gogny-type, or RMF inter- Figure 17: The points represent neutrons skin, ∆rnp calculated for tin isotopes actions. The shaded area reveals the uncertainty range. with the 23 Skyrme interactions [116]. Each one of the lines correspond to one of the interactions and are also guide to the eyes. between the neutron skin and the slope parameter, assuming the validity of the assumptions taken above. of the predicted slope parameter L. The lines are guides to the In Fig. 16 we show the neutron skin thickness of 208Pb as a eyes. Each of the connected curves corresponds to a different function of the slope parameter of the nuclear symmetry energy lead isotope with neutron number N [116]. One observes that at saturation, as predicted by numerous nuclear mean-field in- the nearly linear correlation shown in Fig. 16 also varies as the teractions. The mean field calculations shown in the plot range number of neutrons is changed. As expected, greater variations from Skyrme zero-range forces, as in Eq. (41) and Gogny finite are observed for nuclei with larger neutron excess. Based on the range forces [162, 163, 164, 165, 166, 167, 168, 169, 170]. In results, it is apparent that we are faced with the quest to extrap- the Gogny models, some of the delta functions in Eq. (41) are olate the physics of neutron skins due to a few extra neutrons 57 replaced by gaussians with widths representing the range of nu- to that of a neutron stars with 10 nucleons, as schematically clear forces. Also shown in the figure are results for relativistic shown in Figure 19. It is a quest that requires accurate exper- forces based on effective field theory Lagrangians with meson iments in symbiosis with well developed microscopic nuclear self-interactions, density-dependent meson-nucleon couplings, theories. and point couplings, as in Eq. (54) [171, 172, 173, 174, 175, 176, 177, 178, 179, 180]. 2.8.3. Dipole polarizability and electromagnetic response In Figure 17 we show results for the neutron skin, ∆rnp cal- The electric polarizability of a nucleus is a measure of the culated with more than 20 Skyrme interactions [116]. It is visi- tendency of the nuclear charge distribution to be distorted by an ble that neutron skins calculated with different Skyrme interac- external electric field, e.g. tions tend to spread out as the neutron number increases. The electric dipole moment same is observed for Ni and Pb isotopes [116]. Calculations for αD ∼ . (79) the stable tin isotopes with masses A = 116, 118, 120, yield external electric field neutron skin in the range 0.1−0.3 fm, depending on the Skyrme The action of an external field Feiωt + F†e−Iωt, of dipolar form, force adopted. As observed in Fig. 16, the predictions for the slope param- A X J eter are all over the place within the range L ∼ 5-140 MeV. It FJM = r YJM(r)τz(k), (L = 1, for dipole), (80) is also evident that a near linear correlation exists between the k neutron skin and the slope parameter within a 5% uncertainty yields a response proportional to the static dipole polarizability in ∆rnp for a given value of L and a 15% uncertainty in L for a given value of ∆rnp. The question is: what are the best mi- 8πe2 X 1 8πe2 α = |hi|F |0i|2 = m , (81) croscopic models? As it stands the Figure 16 implies an overall D 9 E 1 9 −1 theoretical uncertainty of the neutron skin of 208Pb within the i range ∆rnp = 0.1-35 fm and for the slope parameter the uncer- where m−1 is the inverse-energy-weighted moment (see Eq. (70)) tainty is about L = 5 − 140 MeV. of the strength function defined as in Eq. (71). Isovector energy- In Fig. 18 we show the neutrons skins, ∆rnp, calculated weighted sum rules (EWSR) for dipole and quadrupole excita- with numerous Skyrme interactions [81, 119, 120, 121, 187, 123, 124, 125, 181, 182, 183, 184, 185, 186, 188] as function

20 0.4 where E is the photon energy, N Z dB(E1; E) X Pb 138 dE ≡ B(E1; E → E ) (85) 0.3 dE 0 j 126 j< jmax ]

m 0.2 is known as the response function to the external ﬁeld, and f 116 [ σabs(E) is the photo-absorption cross section. B(E1; E0 → E j) p 108 n

r 0.1 is the reduced transition strength from an initial state 0 to a ﬁ- 102

∆ nal state j in the nucleus induced by the photo-absorption. The 0.0 96 notation dB/dE for the response function is often used when many excited states within an energy interval dE are accounted for, such as in transitions to the continuum (E > E ). 0.1 j threshold 0 20 40 60 80 100 120 140 The dipole polarizability αD can be extracted from isovec- L [MeV] tor excitations of the nuclei, such as Coulomb excitation experiments. For stable nuclei the dipole response is mostly concentrated in the isovector giant dipole resonance (IVGDR), ex- Figure 18: Neutrons skin, ∆rnp, calculated with numerous Skyrme interactions [81, 119, 120, 121, 122, 123, 124, 125, 81, 181, 182, 183, 184, 185, 186, 187, hausting almost 100% of the energy-weighted sum rule. One 188] as function of the predicted slope parameter L. The lines are guides to the understands this excitation mode as an oscillation of neutrons eyes. Each of the connected curves corresponds to a different lead isotope with against protons where the symmetry energy, S in Eq. (35) acts neutron number N [116]. as a restoring force. Studies of symmetry energy using dipole polarizability and its relation to the symmetry energy were already known in the literature [189, 190]. A renewed interest in isovector excitations arose due to the possibility that in neutron- rich nuclei a softer symmetry energy, changing slowly with density, predicts larger values for the dipole polarizability at the lower densities. This means that the quantity m−1, proportional to αD is highly sensitive to the density dependence of the symmetry energy and suggests a correlation so that the larger skin ∆rnp the larger αD [191]. It is worthwhile noting that Migdal [105] obtained the dipole polarizability assuming a sharp-edged liquid drop energy density and found that D E A r2 m = . (86) Figure 19: The neutrons form a neutron skin in a heavy nucleus, e.g., Pb, within −1 48J a have a range of 0.1−0.2 fm. Studying the experimental value of neutron skins and their predictions from microscopic models, one hopes to clarify nuclear This approximation implies a change in the proton density which interactions in many-nucleon systems, the EoS of asymmetric nuclear matter, varies linearly with the position in the dipole (z) direction. It in- and possibly the properties of neutron stars, with ∼ 1057 neutrons. duces then a “tilting” collective mode rather different in nature from the “sliding” Goldhaber-Teller mode [192]. Using the liq- tions are, respectively, uid drop model with surface corrections, Ref. [192] was able to show that the dipole polarizability can be related to the param- 2 NZ eters of the droplet model as m(D) = ~ (1 + κ ), (82) 1 2m A D D E N 2 " ! # A r L L2 δ2 15 J α ≡ m− = 1 − x − 3M − 2 + , and D 1 48J J K J A1/3 Q 2 D E 0 4 (Q) ~ 50 2 (87) m1 = A r (1 + κQ), (83) 2mN 4π where M is a coefficient specifying the deviation from a quadratic where mN is the nucleon mass, A, N, and Z are the charge, dependence on δ. Using the same arguments leading from Eq. neutron and charge number, and κ j is a correction due to the (76) to Eq. (78) one is able to show that [193] momentum dependence of the nucleon-nucleon interaction.  q  2 D 2E 3 e2Z sur f Using a different notation, the electric polarizability is pro- πe A r  ∆rnp + − ∆rnp   5 5 70J  portional to the inverse energy weighted sum rule of the electric αD ' 1 +  , (88) 54J  2 2 1/2  dipole response in the nucleus, i.e.,  r (δ − δc) 

2 Z Z 8πe dE dB(E1; E) ~c dE sur f 208 α = = σ (E), (84) where ∆rnp ∼ 0.09 fm for Pb, is almost constant for the D 9 E dE π2 E2 abs 2 EDFs. This shows that, in the LDM, ∆rnp (and L) are better correlated with αD J than directly with αD for a heavy nucleus

21 10

24 ] 3 m

f 9

23 V e 8 M [

0 7 ] 22 1 3 / m J f D

[ 6

21 D

α 5 0.10 0.15 0.20 0.25 0.30

20 ∆rnp [fm]

Figure 21: Same as in Fig. 21, but with the dipole polarizability multiplied by 19 the volume term of the symmetry energy, J.

2.8 48 0.10 0.15 0.20 0.25 0.30 Ca 2.6

∆rnp [fm]

] 2.4 3

Figure 20: Predictions for the dipole polorizabitly and for the neutron skin of m 2.2 f

238 [

Pb [194]. The ﬁlled circles are Skyrme predictions with the interactions SIII 2.0

[119], SkI3 and SkI4 [121], SkM [184], SkO [124], SkP [120], SkX [122], D

SLy4 and SLy6 [186, 123], Sk255 [188], BSk17 [195], LNS [196], and UN- α EDF0 and UNEDF1 [182].The filled triangles are results from the NL3/FSU 1.8 interaction [197, 198], the open circles are DD-ME interaction [199, 174]. The 1.6 filled squares are predictions from the Skyrme-SV interaction [200]. The experimental constraint for the dipole polarizability from a Coulomb excitation experiment [201] is shown as a horizontal band. Figure 22: Electric dipole polarizability in 48Ca as predicted by chiral-EFT calculations (circles) and mean-field models (squares). More details on the χ- [161, 193]. EFT calculations, see Ref. [202]. The experimental data [203] is represented In Fig. 20 we plot the predictions for the dipole poloriz- by the solid triangle and the experimental error by the horizontal band. abitly and for the neutron skin of 238Pb [194]. The filled circles are Skyrme predictions with the interactions SIII [119], SkI3 one multiplies the polarizability, α by the volume term of the and SkI4 [121], SkM [184], SkO [124], SkP [120], SkX [122], D symmetry energy, J. This is clearly seen in Figure 21 where SLy4 and SLy6 [186, 123], Sk255 [188], BSk17 [195], LNS we plot all data displayed Fig. 20 but now in the form of the [196], and UNEDF0 and UNEDF1 [182].The filled triangles product α J. The evident correlation between α J and ∆r are results from the NL3/FSU interaction [197, 198], the open D D np has indeed been confirmed by microscopic calculations [160]. circles are DD-ME interaction [199, 174]. The filled squares are In Fig. 22 we show the electric dipole polarizability in predictions from the Skyrme-SV interaction [200]. The experi- 48Ca as predicted by chiral-EFT [202] calculations (circles) and mental constraint for the dipole polarizability from a Coulomb mean-field models (squares). The experimental data [203] is excitation experiment [201] is shown as a horizontal band. The represented by the solid triangle and the experimental error by first clear observation is that the linear dependence of the dipole the horizontal band. Based only on these interactions and func- polarizability on the basis of the interactions used is not as clear tionals, the authors of Ref. [203] extract a neutron skin in 48Ca as expected. The second is that the experiment of Ref. [201] ex- of 0.14-0.20 fm, which, together with the ab-initio χ-EFT re- tracted a neutron skin of ∆r = 1.56+0.025 fm, whereas the av- np 0.021 sults imply that the volume symmetry energy ranges within erage ∆r = (0.168 ± 0.022) fm, is extracted from mean field np J = 28.5-33.3 MeV and a slope parameter of L = 43.8-48.6 calculations [194] based on all interactions mentioned previ- MeV, which sets a very accurate constraint for the slope pa- ously. This value that is close to the one of the experiment of rameter. This surprisingly tight constraint still needs to be con- Ref. [201]. firmed by other experimental and theoretical developments. Figure 20 shows a significant amount of scatter between the results for different functionals. However, as described in details in Ref. [160], a much better correlation can be obtained if

22 3. Indirect methods with relativistic radioactive beams mean ﬁeld 3.1. Electron scattering 26 Mainz 3.1.1. Elastic electron scattering Saclay Since at the energies considered here, the electron can be 28 treated as a point-like particle, its interaction with the nucleus is extremely well described by quantum electrodynamics (QED),

/sr)] 4

2 exp(-2pqa)/q and its momentum transfer can be used as a variable quantity for 30 a given energy transfer, they are ideal probes of nuclear charge

(cm

distributions, transition densities, and nuclear response func- ⌦ tions. /d

32 Under the conditions of validity of the plane-wave Born ap- [d proximation (PWBA), the elastic electron scattering cross sec- log tion in the laboratory is 34 ! dσ σ M |F (q) |2, = 2 ch (89) q ~ p/kR dΩ PWBA 1 + (2E/MA) sin (θ/2) 36 4 2 2 −4 where σM = (e /4E ) cos (θ/2) sin (θ/2) is known as the 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Mott cross section. The denominator accounts for a recoil cor- 1 rection, E is the electron total energy, MA is the nucleus mass q(fm ) and θ is the electron scattering angle. q = 2k sin (θ/2) is the momentum transfer, k is the electron momentum, and E = Figure 23: Data on elastic electron scattering off Pb from the Saclay and p ~ 2k2c2 + m2c4. The Mott cross section has a dependence on Mainz experimental groups. The dashed lines are form factors calculated with ~ e Skyrme-Hartree-Fock theory [204]. The dotted lines show that the exponential the momentum transfer as decay of the cross sections by many orders of magnitude are due to the nuclear E diffuseness while the dips are reflect the nuclear radius. σ ∼ . (90) M q4 where Jn are Bessel functions of order n. Therefore, in the dif- The charge form factor Fch (q) is, for a spherical charge dis- ferential cross sections for elastic electron scattering the distribution, Z ∞ tance between minima is a direct measure of the nuclear size, 2 Fch (q) = dr r j0 (qr) ρch (r) . (91) whereas their exponential decay reflects the surface diffuseness 0 of the charge distribution. This is shown in Figure 23 for the Elastic electron scattering essentially measures the Fourier electron scattering off lead. The dashed lines are form factors transform of the charge distribution through the form factor Fch. calculated with Skyrme-Hartree-Fock theory [204]. The dotted The physics displayed in the cross section is better understood lines show that the exponential decay of the cross sections by if we take for simplicity the one-dimensional case. For light many orders of magnitude are due to the nuclear diffuseness nuclei, well described by a Gaussian distribution, we have while the dips are reflect the nuclear radius. One can also deduce that, at low-momentum transfers, Eq. Z Z eiqx π F (q) ∼ eiqxρ(x)dx ∼ dx = e−qa, (92) (91) yields to leading order ch a2 + x2 a q2 D E q4 D E While, for a heavy nucleus, the density ρ is better described by F (q) /Z = 1 − r2 + r4 + O(q6), (95) ch 3! ch 5! ch a Fermi function or Woods-Saxon charge distribution, yielding where hrni is the n-th moment of the charge density distribu- Z eiqx F (q) ∼ dx ∼ (4π) sin(qR) e−πqa, (93) tions, Z ch 1 + e(x−R)/a hrni = rnρ(r)d3r. (96) valid for R a, and qa 1. Upgrading the plane waves to eikonal wavefunction (see text ahead), one has At the lowest order, elastic electron scattering at low-momentum transfers obtains the root mean squared radius of the charge dis- Z D E1/2 iχ(b) 2 Fch (q) ∼ dbbJ0(qb)[1 − e ] tribution, rch . With increasing data at higher momentum transfers, more details of the charge distribution are probed. Z J (qb) ∼ 0 dbb b−R 1 + exp[( a )] 3.1.2. Inelastic electron scattering R Inelastic electron scattering is another powerful tool for in- ∼ J (qR) exp(−πqa), (94) q 1 fer the properties of nuclear excited states, such as their spins, parities, and transition strengths between ground and excited

23 states (e.g. Refs. [205, 206]). In the plane-wave Born approxi- view, see [215]). mation (PWBA), one obtains for the cross section for inelastic To investigate the asymmetry properties of bulk nuclear mat- electron scattering [205, 206] ter, a systematic study of the charge-density distributions, of ! ( nuclei with large proton-neutron asymmetry is necessary. To- dσ 8πe2 p0 X EE0 + c2p · p0 + m2c4 = |M (q; CL)|2 gether with information about the hadronic matter distribution dΩ 4 p q4 (~c) L in nuclei, one can get information on the nuclear matter incom- EE0 − c2 (p · q)(p0 · q) − m2c4 pressibility [216, 217], the bulk of the nuclear symmetry en- + 2 ergy [218], and its slope parameter to extrapolate to the pure 2 2 2 c q − q0 neutron matter equation of state and its dependence on density h io [95]. Despite heroic efforts, this era of nuclear physics is still × |M (q; ML)|2 + |M (q; EL)|2 (97) in its infancy as an electron-radioactive-beam collider with the properties needed to investigate unstable nuclei is still unavail- where J J is the initial (final) angular momentum, (E, p) and i f able [215]. (E0, p0) are the initial and final four-momenta of the electron, Since the electron scattering theory is very well understood, and (q , q) = (∆E/ c, ∆p/ ) is the four-momentum transfer in 0 ~ ~ electron scattering is potentially a great tool for inferring the the reaction. M (q; ΠL) are matrix elements for Coulomb (C), i j charge distribution in unstable nuclei in a rare isotope ion-electron electric (E) and magnetic (M) multipolarities, Π = C, E, M, collider. In conjunction with hadronic probes, such as elas- respectively. For small momentum transfers and scattering at tic proton scattering, a better study of the evolution of neu- forward angles, electron scattering involves the same matrix el- tron skins in nuclei can be achieved. The implications of elas- ements as scattering by real photons and by Coulomb excita- tic and inelastic electron scattering in determining the prop- tion [207]. Both in Coulomb excitation as in the scattering by erties of light, neutron-rich nuclei have been studied in Ref. real photons the energy and momentum transfers are related by [219, 210, 220]. But while the electron-ion collider is not avail- |∆p| = ∆E/c, whereas in electron scattering the momentum and able, other sources of information on the difference between energy transfer are varied independently, allowing for an addi- neutron and proton distribution in neutron-rich nuclei need to tional information on the nuclear response. be used. At very forward angles and small energy transfers, one can use the Siegert theorem [208, 209] to prove that the electric and 3.1.4. The electron parity violation scattering Coulomb form factors appearing in Eq. (97) are proportional In order to include weak interaction scattering in electron to each other. Then, for electric multipole excitations, one can scattering, the electromagnetic potential Ze2/r has to be gener- write [210] alized to exp (−Mr) (EL) V(r) = ke2g g . (100) dσ X dNe E, Eγ, θ e T = σ(EL) E , (98) 4πr dΩdE dΩdE γ γ γ L γ where here we use the notation ~ = c = 1, ge is the electron charge and gT is the charge of the target particle in units of valid for excitation energies Eγ ~c/R, with the “equivalent the electron charge e. The constant k denotes the strength of photon number” given by [210] the coupling, with M being the mass of the exchanged particle, (EL) " #2L−1 where for M = 0 one has the usual electromagnetic interaction, dNe E, Eγ, θ 4L α 2E θ = sin while, for M = MZ (mass of Z0-boson) the interaction is due to dΩdEγ L + 1 E Eγ 2 the neutral weak current. One also has k = 1 for electromag- 2 −3 2 cos (θ/2) sin (θ/2) netism and k = (sin θW cos θW ) for weak interaction scatter- × 2 2 ing. The exchanged particle in the neutral weak interaction, i.e. 1 + 2E/MAc sin (θ/2)  !2  the Z0 boson, has both vector and axial vector couplings. For 1 2E L θ θ  ×  2 2  spinless nuclei, the net axial coupling to the nucleus is absent.  + sin + tan  . (99) 2 Eγ L + 1 2 2 Moreover, the Z0 has a much larger coupling to the neutron than the proton and it also contains a large axial coupling to the Therefore, under these conditions (very forward angles and small electron. That is the reason for a parity-violating amplitude in energy transfers) the cross section for electron scattering are electron scattering. proportional to the cross sections for real photons inducing elec- (EL) In Table 3.1.4 we list the electromagnetic and weak charges, tric multipolarity EL excitations, σγ Eγ . Because it is nearly or coupling constants, of the electron and light quarks. impossible to study photo-nuclear cross sections with real pho- In longitudinally polarized electron scattering, the weak charge tons and radioactive nuclei as targets, this might be a way to of a relativistic electron depends on its helicity, so that gR , gL, access this information in electron-ion facilities, if they are ever where gR and gL now denote the the charge of an electron with realized in practice [211, 212, 213, 214, 215]. right-handed and left-handed helicity, respectively. Therefore, it is more convenient to write them in terms of the vector and 3.1.3. Electron radioactive-ion collider axial-vector weak charges, gR = gV + gA and gL = gV − gA. The Electron radioactive-beam colliders have been proposed and weak and electromagnetic charges of the electrons and relevant already exist in some nuclear physics facilities (for a recent re-

24 where Particle gem gV gA − − 1 2 1 GF h 2 i e -1 4 + sin θW 4 A r − θ ρ r − ρ r ( ) = 3/2 (1 4 sin W ) p( ) n( ) (106) 2 1 2 2 1 2 u 3 4 − 3 sin θW − 4 The electron-nucleus interaction via the axial potential is of or- d, s − 1 − 1 + 1 sin2 θ 1 3 4 3 W 4 der one eV while the electromagnetic interaction with the nucleus is of order MeV. Therefore, it only becomes important Table 2: The electromagnetic and weak charges, or coupling constants, of the 2 ∼ electron and light quarks. in parity violating scattering. Since sin θW 0.23, the factor 2 . (1 − 4 sin θW ) is small and A(r) depends mainly on the neutron distribution ρn(r). As shown previously, the presence of the parity-violating asymmetry quantity in Eq. (103) will involve the interference quarks are given in Table 1. The electroweak mixing angle is between VC(r) and A(r) and one gets for it a simple expression known to high precision, as θW = 0.23116 ± 0.00013 [221]. 2 " # With the potential of Eq. (100), the scattering cross section GF Q 2 Fn(Q) of Eq. (89) becomes (neglecting recoil corrections) APV (r) = √ 4 sin θW − 1 + , (107) 4πe2 2 Fp(Q) ! !2 dσ 2kg g θ 2 µ e T |F (q) |2 2 , where Q is the four-momentum transfer, Q = qµq > 0, and = 2 2 ch cos (101) dΩ PWBA q + M 2 Fn(Fn) is the neutron(proton) form factor. Therefore, the parity- 2 2 violating asymmetry is proportional to Q /MZ because GF ∝ When the momentum transfer is small so that q MZ, the 2 2 1/MZ. Besides, since 1−4 sin θW 1 and Fp(q) is known in a weak interaction is negligible compared to the electromagnetic nucleus such as 208Pb the asymmetry yields a direct measure of charge, so that the charge form factor is an sum of form factors Fn(q) and is a good way to study the neutron skin of the nucleus for light-quarks distributions in the nucleus: [222]. Based on these ideas, an experiment named the Lead Ra- (e.m.) 2 1 F /e = Fu − (Fd + Fs) . (102) dius Experiment (PREX) for electron scattering experiments ch 3 3 on 208Pb was carried out at the Jefferson Lab (JLab) for the In experiments aiming at studying the effects of parity vio- doubly-magic nucleus whose first excited state was discrimi- lating electron scattering one measures the asymmetry nated by high resolution spectrometers. The neutron skin in 208Pb was obtained by an analysis of the parity violating asym- dσR − dσL 208 +0.16 APV = . (103) metry, yielding ∆rnp( Pb) = 0.33−0.18 fm [223, 224]. If we dσR − dσL compare this experimental result to the plot in Fig. 16, we see for polarized electrons with right and left helicity. If this quan- that it practically agrees with any of the mean-field model, and tity is different than zero, it measures the strength of parity vio- a wide range of the slope parameter L. The PREX experiment was a benchmark run and a proposal exists for the Calcium Ra- lation. In this case, it is dominated by the interference between 48 the weak and electromagnetic amplitudes, so that dius Experiment (CREX) electron scattering on Ca. For this nucleus, microscopic nuclear theory calculations have been de- q2 veloped and shown to be sensitive to poorly constrained three- A − PV = 2 2(MZ cos θW sin θW ) nucleon forces [202]. The realization of this experiment can set 1 2 2 1 1 2 a tighter constraint on the slope parameter of the ANM EoS. ( − sin θW )Fu + (− + sin θW )(Fd + Fs) × 4 3 4 3 2 1 3 Fu − 3 (Fd + Fs) 3.2. Direct reactions at relativistic energies (104) 3.2.1. Nucleus-nucleus scattering at high energies Direct reactions in nucleus-nucleus collisions at relativis- This provides a different combination of quark distributions in tic energies have a fundamental problem: No-one really knows the nucleon than Eq. (102) and is one of the reasons why parity- how to treat accurately relativistic many-body collisions, when violating electron scattering is a useful tool in nuclear physics. retardation, simultaneity, and the cumulative dynamics of local A formulation following the same path as described above (strong) and long-range (Coulomb) pair-wise interactions are can be used to directly connect the scattering observables to the treated consistently. Evidently, non-relativistic DWBA meth- neutron and proton densities in the nucleus [222]. To a good ap- ods are inappropriate to study nuclear excitation and other in- proximation, the total interaction of the electron and a nucleus, elastic process at laboratory energies of 100 MeV/nucleon and including the electromagnetic, V , and the axial potential, A, C above, where Lorentz contraction and other relativistic effects arising from the parity violating term, can be written as at the level of 10% or more factor in the kinematics and in the dynamics of the system. In particular, it is highly doubtful if the V(r) = VC(r) + γ5A(r), (105) concept of an optical potential, a non-relativistic concept, can be adapted to a covariant theory involving four-dimensional potentials. 25 The relativistic treatment of direct reactions at high-energy not play well with an exact relativistic treatment of the colli- collisions is amenable to a reasonable theoretical treatment only sion, as simultaneity is invoked, and retardation is neglected. if a sudden approximation is justified. In this case, the theoret- But the advantage of using only the transverse directions to the ical description of the relativistic scattering waves can be sepa- beam is that the expression is Lorentz invariant. rated from the internal structure of the nuclei which then can be The above formulation is difficult to implement as one needs treated in a non-relativistic fashion. to account for the transverse distances b j between all nucleons The best way to treat “soft” collisions between composite participating in the collision. An often used approximation is particles at high energies is the formalism of the Glauber theory called the Optical Limit (OL) of the Glauber theory amounting [225]. The wavefunction of a high energy projectile after it to an average over all the collisions what means that a continu- interacts with a target is given by ous “frozen” nucleon density can be used to calculate S (b). In other words iQ xµ |Ψ(t)i = e µ S (b) ψ(r , r , ··· , r (108) 1 2 AP D P E iχOL(b) i j χ j(b j) S OL(b) = e = e . (114) where S (b) is the eikonal “survival” amplitude, also known as “eikonal scattering matrix”, and |ψi is the internal wave func- As explained in Ref. [226], one can obtain the OL eikonal phase tion of the projectile. In this way, its wavefunction is factorized by relating it to the observables in nucleon-nucleon collisions so in a scattering times an internal part. Anti-symmetrization of that projectile and target nucleons is neglected. Moreover, Qµ = Z 0 0 00 00 3 0 3 00 (E f − EI, p f − pI) is the four-momentum transfer and xµ = (t, r) χOL(b) = ρP(r )Γ(s − r + r )ρT (r )d r d r , (115) are the corresponding coordinates. Assuming that the total energy of the projectile is much larger than the energy transfer in where ρP(ρT ) is the projectile(target) density and the “profile the collision, i.e., EI = E f , one has the stationary wavefunction function” Γ(b). For most cases, involving spherically symmetric densities, one can use the Fourier transform of the nuclear |Ψi = eIq·rS (b) ψ(r , r , ··· , r , (109) 1 2 AP densities, yielding the simple expression where q is the momentum transfer. S (b) can also be cast in the Z 0 ˜ form S (b) = exp(iχ(b)), where χ(b) is known as the eikonal χOL(b) = ρ˜P(q ) Γ(q)ρ ˜T (q) J0(qb) q dq, (116) phase χR + iχi, with a real imaginary part χi accounting for inelasticity. In the eikonal theory (for more details, see Ref. where J0 is the ordinary Bessel function of zeroth-order, and [226], the projectile survival probability is given by the profile function in momentum space

2 P b |S b | . i + αNN −β q2 survival( ) = ( ) (110) Γ˜(q) = σ e NN . (117) 4π NN The eikonal scattering wavefunction for a potential U(r) at high energies is given by [225, 226] In the equation above, σNN is the total nucleon-nucleon cross section, αNN is the ratio of the real to the imaginary part of h i ψ(+)(r) = exp ik · r + χ(b, z) , (111) the NN-scattering amplitude, and βNN is the momentum depen- eik dence parameter. where The parameters of the nucleon-nucleon cross scattering am- Z z 1 0 0 plitudes for E ≥ 100 MeV/nucleon are shown in Table 3.2.1, χ(b, z) = − U(b, z ) dz . (112) lab ~v −∞ extracted from Refs. [227, 228]. An isospin average of these But here, again, we run into the problem we mentioned pre- quantities can be done in most practical cases, while in other viously that if U is taken as an ordinary optical potential, the situations they can be used to separate cross sections sensitive- formalism is not in compliance with the laws of transformation ness to the proton and neutron densities separately. in special relativity. The Coulomb interaction also adds to the eikonal phase. In The good news is that the scattering part of the wavefunc- the OL, the total eikonal phase is χ = χOL + χC, where the tion in Eq. (108) is Lorentz invariant because S (b) only de- Coulomb eikonal phase, χC is to ﬁrst-order pends on the transverse coordinate b, usually associated with an χ (b) = 2η ln(kb) , (118) impact parameter. But further simpliﬁcations need to be done C when dealing with a many-body system. First, the survival am- 2 where η = Z1Z2e /~v, Z1 and Z2 are the charges of projec- plitude, S (b), is cast as an incoherent sum of scattering matrices tile and target, respectively, v is their relative velocity, k their for each individual nucleon nucleon collision, i.e., wavenumber in the center of mass system. Eq. 118 reproduces Y P the exact Coulomb scattering amplitude when used in the cal- S (b) = eiχ j(b j) = ei j χ j(b j), (113) culation of the elastic scattering with the eikonal approximation j where χ j(b j) is now the eikonal phase acquired in a binary collision j. Notice that the assumption of incoherent collisions do

26 where Θ is the step function. Assuming a Gaussian distribution Elab σpp αpp βpp σpn αpn βpn of charge radius R for light nuclei, the Coulomb phase becomes [MeV] [fm2] [fm2] [fm2] [fm2] n 1 o 40 7.0 1.328 0.385 21.8 0.493 0.539 χ (b) = 2η ln(kb) + E (b2/R2) , (122) C 2 1 60 4.7 1.626 0.341 13.6 0.719 0.410 where the error function E1 is deﬁned as 80 3.69 1.783 0.307 9.89 0.864 0.344 Z ∞ e−t 100 3.16 1.808 0.268 7.87 0.933 0.293 E1(x) = dt , (123) x t 120 2.85 1.754 0.231 6.63 0.94 0.248 which also also converges, as b → 0. 140 2.65 1.644 0.195 5.82 0.902 0.210 At intermediate energy collisions (Elab ' 50 MeV/nucleon), 160 2.52 1.509 0.164 5.26 0.856 0.181 a correction due to the Coulomb deﬂection can be done by re- placing the impact parameter b within Eq. (116) by the distance 180 2.43 1.365 0.138 4.85 0.77 0.154 of closest approach in Coulomb scattering, 200 2.36 1.221 0.117 4.54 0.701 0.135 s !2 a a 240 2.28 0.944 0.086 4.13 0.541 0.106 b0 = 0 + 0 + b2 , (124) γ γ 300 2.42 0.626 0.067 3.7 0.326 0.081 2 2 425 2.7 0.47 0.078 3.32 0.25 0.0702 where a0 = Z1Z2e /mv is half the distance of closest approach 550 3.44 0.32 0.11 3.5 -0.24 0.0859 in a head-on collision of point charged particles. This correction leads to a considerable improvement of the eikonal ampli- 650 4.13 0.16 0.148 3.74 -0.35 0.112 tudes for the scattering of heavy systems in collisions at inter- 2 2 −1/2 700 4.43 0.1 0.16 3.77 -0.38 0.12 mediate energies. The Lorentz factor γ = (1 − v /c ) is remnant of the dynamic correction of the elastic Coulomb scat- 800 4.59 0.06 0.185 3.88 -0.2 0.12 tering, as shown in Ref. ([230]). 1000 4.63 -0.09 0.193 3.88 -0.46 0.151 It is also useful for consistency purposes to extract optical potentials by using an inversion method of the eikonal phases. 2000 4.67 0. 0.12 3.88 -0.50 0.151 In this approach, one uses the Abel transform [225]

Table 3: Parameters [227, 228, 229] for the nucleon-nucleon amplitude, as Z ∞ ~v d χ(b) given by Eq. (117). The compilation of the interpolated values to several ener- U r r dr . opt( ) = 2 2 1/2 (125) gies and adapted to collisions at lower energies were taken from Ref. [229]. iπr dr r (b − r ) This procedure has been tested in Ref. [231] leading to eﬀec- tive potentials with tails very close to those obtained with phe- [226]: nomenological potentials of Ref. [232]. Under certain approximations, and for Gaussian density distributions, the potential 2 Z1Z2e n h 2 i o obtained through Eq. (125) coincides with that obtained with fC(θ) = exp − iη ln sin (θ/2) + iπ + 2iφ0 2µv2 sin2(θ/2) the double folding procedure [231]. (119) where 3.2.2. Elastic scattering ∞ ! X η η Using the scattering waves in the eikonal approximation, φ = arg Γ(1 + iη/2) = −ηC + − arctan , one can show that the elastic scattering amplitudes are given by 0 j + 1 j + 1 j=0 [225, 226] (120) Z ∞ The Coulomb phase in Eq. (118) diverges at b = 0, but n h io fel(θ) = ik db b J0(qb) 1 − exp iχ(b) , (126) since the strong absorption suppresses scattering at small im- 0 pact parameters this fact does not matter numerically. One can also assume a uniform charge distribution with radius R and the where q = 2k sin(θ/2), θ is the scattering angle and χ includes Coulomb phase is ﬁnite for b = 0: contribution of both nuclear and Coulomb scattering. The elastic scattering cross section is ( χC(b) = 2η Θ(b − R) ln(kb) + Θ(R − b) dσ 2 el = f (θ) . (127) dΩ el " p × ln(kR) + ln(1 + 1 − b2/R2) #) p 1 − 1 − b2/R2 − (1 − b2/R2)3/2 , (121) 3 27 eikonal phase χ is given by [225, 226] 107 S Z ∞ 6 1 10 χS (b) = − USO(b, z) dz , (132) ~v −∞ 5 12C(12C,12C)12C 10 where the spin-orbit potential is given by (s·L)USO(r), with r =

] 135 MeV/nucl (b, z). However, as we discussed previously, this approach is r 4

s 10 not compatible with the relativistic transformation laws, which /

b require four-potentials to describe interactions. 103

m In Fig. 24 we show the experimental data on the diﬀeren-

[ 12 12 tial cross section for the elastic scattering of C from C at 102

Ω 135 MeV/nucleon. The solid curve shows the result of the non-

d relativistic optical model calculation, whereas the dashed curve / 1

σ 10 uses the Glauber formalism described in the text [233]. De- d 0 spite the good agreement of the experimental data with a non- 10 relativistic DWBA model, the Glauber calculation is superior in -1 physics ingredients, as no parameters were introduced to fit the 10 data, except for the observables in nucleon-nucleon scattering entering the profile function, Eq. (117). One sees that at large 10-2 0 2 4 6 8 10 12 14 scattering angles the reproduction of the experimental data is not so good. Modifications of the theory to do a better job at θc. m. [deg] higher energies has been introduced by several authors and, in general, the Glauber theory and its improvements have done a very good job in reproducing elastic scattering in high-energy Figure 24: Experimental data on the differential cross section for the elastic scattering of 12C from 12C at 135 MeV/nucleon. The solid curve shows the re- heavy ion collisions (see, e.g., Refs. [234, 235]). sult of the non-relativistic optical model calculation, whereas the dashed curve Fig. 25 shows the differential cross sections, dσ/dt ver- uses the Glauber formalism described in the text [233]. sus the four-momentum transfer squared (−t = Q2), for p4He, 6 8 p He and p He elastic scattering at energies Ep = 628 and 721 MeV, in inverse kinematics, respectively. The data are the ex- Adding and subtracting the Coulomb amplitude, fC(θ) in eq. 126, one gets perimental values from Refs. [236, 237]. The calculated cross sections are based on the Glauber theory for elastic scattering Z ∞ h in h 0 io using nucleon-nucleon scattering data as input [238]. The anal- fel(θ) = fC(θ)+ik dbbJ0(qb) exp iχC(b) 1−exp iχ(b ) , ysis of these data were useful to determine the nuclear matter 0 (128) distribution in 6He and 8He. 0 where we replaced b in χOL(b) by b as given by Eq. (124) to account for the nuclear recoil. In contrast to Eq. (126), Eq. 3.2.3. Dirac phenomenology (128) converges quickly because the argument in the integral A very successful phenomenological approach was devel- drops to zero at large distances. oped in Ref. [239] based on an optical potential consisting of For proton-nucleus elastic scattering, the cross sections ac- two parts: U0(r), transforming like the time-like component of a quire two components for spin-up and spin-down protons. The Lorentz four-vector; and a second potential, US (r), is a Lorentz eikonal elastic scattering cross section becomes [225, 226] scalar. U0 and US are regarded as effective interactions arising from nucleons interacting via the meson exchange, folded with 2 2 dσel proton and neutron densities. They depend on the masses and = F(θ) + G(θ) , (129) dΩ coupling constants can be of the neutral vector ω and scalar σ where the spin-up scattering amplitude is bosons of the one boson exchange (OBE) model of the two- nucleon interaction. The Dirac equation for the scattering wave Z ∞ h i for the proton-nucleus system is F(θ) = fC(θ) + ik db b J0(qb) exp iχC(b) 0 n h i h io h i h i α · p + β m + US (r) + U0(r) + VC(r) Ψ(r) = EΨ(r), (133) × 1 − exp iχ(b) cos kb χS (b) (130) where, m is the proton mass, E the nucleon total energy in the and the spin-down scattering amplitude is c.m. frame, and αk (k = 1, 2, 3) and β are Dirac matrices. One Z ∞ h i h i problem with this method is the difficulty to disentangle the G(θ) = ik db b J1(qb) exp iχC(b) + iχ(b) sin kb χS (b) . contributions of ρ0(r) from ρS (r), including medium effects on 0 (131) the meson couplings and masses. In the equation above q = 2k sin(θ/2), where θ is the scatter- An effective Schrodinger¨ equation can be obtained from this procedure, which still carries the essence of relativistic cor- ing angle, J0 (J1) is the zero (first) order Bessel function. The

28 104

6 3 10 10 p (181 MeV) + 40Ca

102

] 8

2 p He, Ep = 678 MeV ] ) r c s

/ 1

5 / 10 V

10 b e m G 0 [ (

10 /

b Ω d m 6 -1 / [ p He, Ep = 721 MeV 10

σ 4 t

10 d d / 10-2 σ d 10-3 103 10-4 0 10 20 30 40 50 60 70 80 90 0.00 0.01 0.02 0.03 0.04 0.05 θc. m. [deg] -t [GeV/c]2 Figure 26: Elastic p-40Ca cross sections at 181 MeV. The curves are the results Figure 25: Differential cross sections, dσ/dt versus the four-momentum trans- of the relativistic Dirac phenomenology optical model analysis [239]. fer squared (−t = Q2), for p4He, p6He and p8He elastic scattering at energies Ep = 628 and 721 MeV,respectively. The data are the experimental values from Refs. [236, 237]. The calculated cross sections are based on the Glauber theory in terms of a sum of real and imaginary Woods-Saxon poten- for elastic scattering using nucleon-nucleon scattering data as input [238]. tials. However, as an inheritance of their relativistic nature, their depths are quite different than those in low-energy scattering. Typically, e.g., for p+40Ca at E = 200 MeV one has rections and has been shown to be very successful to describe p ReU ∼ −350 MeV, ImU ∼ −100 MeV, ReU ∼ −550 MeV, proton-nucleus scattering at high energies [239, 240, 241, 242]. 0 0 S and ImU ∼ −100 MeV [239]. There are therefore big cance- The Dirac equation (133) can be rewritten as two coupled equa- S lations in Eq. (135), leading a central depth of U compatible tions for the upper (Ψ ) and lower (Ψ ) components of the Dirac cent u l with the non-relativistic models. But relativity also introduces wave function, Ψ(r). Keeping only the two upper components √ modifications in U rendering them different than a simple of the wavefunction, using the definition Ψ = Bφ, where cent u Woods-Saxon form. Moreover, the approach is more consistent m + U + E − U − V than in the non-relativistic case, as there is a prescription on B(r) = S 0 C , m + E how to get the spin-orbit potential out of U0 and US . Hence in the Dirac approach, the spin-orbit potential appears natu- one obtains the coupled equations rally when one reduce the Dirac equation to a Schrodinger-like¨ h i h i second-order differential equation, while in the non-relativistic 2 · − 2 − 2 p + 2E (Ucent + USOσ L) φ(r) = (E VC) m φ(r) Schrodinger¨ approach, one has to insert the spin-orbit potential (134) by hand. with Fig. 26 shows the elastic p-40Ca cross sections at 181 MeV. 1 h i The curves are the results of the relativistic optical model anal- U = 2EU + 2mU − U2 + U2 − 2V U + U (135) cent 2E 0 S 0 S c 0 D ysis [239]. The agreement with the data is excellent. Many other examples can be found in the literature. For relativistic where the Darwin potential is given by unstable isotopes, data can be inferred from inverse kinematics ! !2 using proton targets. So far, most data has been obtained at low 1 ∂ 2 ∂B 3 ∂B UD(r) = − r + , (136) energies. At high energies a few reactions have been carried out 2r2B ∂r ∂r 4B2 ∂r with light projectiles, such as 8Be+p at 700 MeV [243]. In this case, the data displays an exponential smooth decrease with an- and the spin-orbit potential by gle due to the rather transparent charge distribution in the light 1 ∂B nucleus. U (r) = − . SO 2EBr ∂r Elastic scattering data is a simple way to access sizes, density profiles, and other geometric features of nuclei. For exam- The potentials U0 and US are treated exactly in the same ple, the beautiful exponential decrease of the cross sections with way as the non-relativistic optical potentials, usually parametrized 29 3.2.4. Total nuclear reaction cross sections Since the survival probability is defined as in Eq. (110), it 0.30 is evident that the reactions cross sections is given by 0.25 Z h 2i ] 0.20 σR = 2π dbb 1 − S (b) m f

[ 0.15  2 Z * + p  Y  n   r 0.10 = 2π dbb 1 − Ψ0 exp iχ j(b j Ψ0  , ∆   0.05 j (137) 0.00

0.05 where Ψ0 is the projectile+target intrinsic (translation-invariant) ground state wave functions accounting for the position of the 112 114 116 118 120 122 124 nucleons, and b is the projection onto the collisional transverse Mass Number A i plane of the relative coordinates between the nucleons in a bi- Figure 27: Neutron skin thickness of tin isotopes obtained by various exper- nary collision j. The product accounts for all possible binary imental methods, namely, elastic proton scattering at 295 MeV [244] (black collisions. triangles), elastic proton elastic scattering at 800 MeV [245] (green stars), giant Only in a few cases the equation (137) is solved exactly, as, dipole resonance [246], spin dipole resonance [155] (red circles), and antipro- e.g., in Ref. [250]. Usually the reaction cross section is calcu- tonic x-ray atoms [153]. The dashed blue line represents the RMF predictions of Ref. [247], whereas the solid red line are HF calculations using the SkIII lated with help of the optical limit of the Glauber model [225]. parametrization [248] and the dotted black line use the SkM* parametrization It is worth noticing that the reaction cross section includes any [249]. channel out of the elastic one and therefore includes the excitation of the colliding nuclei. We will discuss these cases later, the nuclear diffuseness is clearly seen in elastic scattering data but first we concentrate on the the case where Eq. (137) applies, at large energies. In contrast, inelastic scattering requires many which some authors prefer to call interaction cross section. As other pieces of information about the intrinsic nuclear proper- we discussed previously in connection with the data presented ties and are sensitive to the models used to describe nuclear in Fig. 11. Often, experimental results are compared to the phenomenological formula σ = πr2(A1/3 + A1/3)2. And this excitation. Often, the coupling to many excitation channels has R 0 P T to be considered. This contrasts to the nice features of elastic was used to interpret the data shown in Fig. 11, leading to the scattering as a probe of the nuclear geometry and density pro- discovery of halo nuclei. files. Experimental data on reaction cross sections for nucleus- In Ref. [244], the neutron density distributions of tin iso- nucleus collisions at high energies are not very abundant, spe- topes have been studied by measuring the cross sections and cially for radioactive nuclear species. In Fig. 28 we plot the nucleon-nucleon (top panel) and the total reaction cross sec- analyzing powers for proton elastic scattering at 295 MeV. The 12 12 relativistic Love-Franey interaction was tuned to explain the tions for C on C (bottom panel) as a function of projec- proton elastic scattering off 58Ni whose density distribution is tile energy. The blue points are data from Refs. [251] (100 to well known. The compiled results for this experiments added 400 MeV/nucleon), 790 MeV/nucleon) [252], and [253] (950 to other data is shown in Figure 27. The RMS radii of the MeV/nucleon). Black triangles represent the result of a parameter- point proton and neutron density distributions were extracted free eikonal calculation using the Glauber optical limit. The red and compared with theoretical mean-field calculations. Skyrme- diamonds include the effects of Pauli blocking [254]. Hartree-Fock calculations using the SkM* parametrization were Fig. 28 shows the free total nucleon-nucleon cross sections in good agreement with the RMS radii of both point proton [255]. A very useful chi-square fit of the experimental data, and neutron density distribution, as shown in Fig. 27. Neutron skin thickness of tin isotopes obtained by various experimental methods, namely, elastic proton scattering at 295 MeV [244] (black triangles), elastic proton elastic scattering at 800 MeV [245] (green stars), giant dipole resonance [246], spin dipole resonance [155] (red circles), and antiprotonic x-ray atoms [153]. The dashed blue line represents the RMF predictions of Ref. [247], whereas the solid red line are HF calculations using the SkIII parametrization [248] and the dotted black line use the SkM* parametrization [249]. The results show a clear increase in neutron skin thickness with mass number, although the values obtained were not as large as what some RMF models predict.

50 for proton-proton collisions, and pp 45  √  89.4 − 2025/ E + 19108/E − 43535/E2  40   (for E < 300 MeV) np  (mb) 35  σ   30  −5 2 −9 3  14.2 + 5436/E + 3.72 × 10 E − 7.55 × 10 E  25 σ =  ≤ np  (for 300 MeV E < 700 MeV)   8800 200 400 600 800 1000 1200   E (MeV)  × −3 − × −6 2 860  33.9 + 6.1 10 E 1.55 10 E   · −10 3 840  +1.3 10 E  820  (for 700 MeV ≤ E ≤ 5 GeV) (mb) R (139) σ 800 Glauber Glauber with for proton-neutron collisions, where E is the laboratory energy. 780 Pauli correction The fits are represented by dotted and solid curves in the top 760 Experiment panel of Fig. (28). The experimental data for the total reaction cross section 0 200 400 600 800 1000 1200 for 12C on 12C as a function of beam energy is shown as filled E (MeV/nucleon) circles in the bottom panel of Fig. 28. Its is worth noticing that Figure 28: Nucleon-nucleon (top panel) and the total reaction cross sections few data exist in the energy region of most relevance for the 12 12 for C on C (bottom panel) as a function of projectile energy. The blue study of radioactive nuclei in the present and planned rare iso- points are data from Refs. [251] (100 to 400 MeV/nucleon), 790 MeV/nucleon) [252], and [253] (950 MeV/nucleon). Black triangles represent the result of tope facilities. The optical limit the Glauber theory was used to a parameter-free eikonal calculation using the Glauber optical limit. The red obtain the black triangles using the above parameterization for diamonds include the effects of Pauli blocking [254]. the NN cross sections. It is interesting to notice that medium corrections such as the inclusion of the effects of the Pauli prin- yields the expressions obtained in Ref. [256], ciple [256] (shown in the figure as filled diamonds) are relevant even at high energies because one of the nucleons in the binary  √ collision can end up in a low energy state, blocked by other  19.6 + 4253/E − 375/ E + 3.86 × 10−2E  nucleons occupying that state. The incorporation of in medium   (for E < 280 MeV)  effects in nucleon-nucleon collisions is thus a relevant (see also,  Ref. [257]).   In Fig. 29 we show measured interaction cross sections for  −2 −7 3  32.7 − 5.52 × 10 E + 3.53 × 10 E 20−32Na and 20−33Mg (open circles) projectiles at 950 MeV/nucleon   −10 4 on a C target [258]. The crosses are the Glauber optical limit  −2.97 × 10 E σpp =  (138) calculations using HFB densities as inputs in Eq. (115). The  (for 280 MeV ≤ E < 840 MeV)  agreement with the experimental data is quite good. In fact,   one can turn this argument around and use the experimental   −3 −7 2 data and adjust the matter radius in the densities to extract den-  50.9 − 3.8 × 10 E + 2.78 × 10 E  sities which can be used as a constraint to theoretical models  −15 4  +1.92 × 10 E [258].   (for 840 MeV ≤ E ≤ 5 GeV) 3.3. Coulomb excitation As with the case of electron scattering, Coulomb excitation (Fig. 30) is one of the most useful probes of nuclear structure because the interaction is well-known. However, unless the collision occurs ate very low energies, the excitation or breakup caused by the strong interaction can lead to large contributions, and also to nuclear-Coulomb interference. Experi- mentalists investigate the best cases for the bombarding energies, angular distributions, and excitation energies that can be safely described as Coulomb excitation. Coulomb excitation cross sections are directly related to the

31 1500

1400

] 12 b 1300 Na + C m [

1200

σ 1100

1000 20 22 24 26 28 30 32 1400

] 12 b 1300 Mg + C

m Figure 30: Schematic description of the Coulomb excitation of a projectile a

[ ∗ 1200 leading to an excited nucleus, a , or a direct breakup of the projectile.

σ 1100 and 1000 20 22 24 26 28 30 32 34 2 Z Ex 2 Exb gm(Ex) = 2π dbbKm exp −2 χI(b) , (144) mass number γ~v γ~v

where χI(b) is the imaginary part of the eikonal phase and the Figure 29: Top: The measured interaction cross sections for 20−32Na (open circles) projectiles at 950 MeV/nucleon on a C target [258]. The crosses are the functions Gπλm(c/v) were obtained in Ref. [260] and, for the Glauber calculations using HFB densities. Bottom: Same as top panel, but for lowest multipolarities are given by 20−33Mg projectiles at the same bombarding energy. 1 √ G (x) = 8πx = −G (x) E11 3 E1,−1 photonuclear cross sections by means of [207] 4 √ 1/2 G (x) = −i π x2 − 1 , (145) E10 3 dσ (E ) X n (E ) X n (E ) C x Eλ x σγ (E ) Mλ x σγ (E ) , r = Eλ x + Mλ x 1/2 dEx Ex Ex 2 π 2 Eλ Mλ GE22 (x) = − x x − 1 = GE2,−2 (x) , (146) (140) 5 6 γ r where σπλ (Ex) are the photonuclear cross sections for the mul- 2 π 2 tipolarity πλ and E is the excitation energy. GE21 (x) = i 2x − 1 = iGE2,−1 (x) , (147) x 5 6 The photonuclear cross sections are related to the reduced √ 1/2 matrix elements, for the excitation energy Ex, through the rela- 2 2 GE20 (x) = πx x − 1 , (148) tion [207] 5 for the electric E1 and E2 excitations, and (2π)3(λ + 1) E 2λ−1 dB (πλ, E ) σπλ(E ) = x x (141) γ x 2 1 √ λ [(2λ + 1)!!] ~c dEx G (x) = −i 8π = G (x) , G (x) = 0, (149) M11 3 M1,−1 M10 where dB/dEx is deﬁned in Eq. (85). For diﬀerential cross sections one obtains for the M1 excitations. In Figure 31 we show a calculation (with Eγ ≡ Ex) of the dσ (E ) 1 X dn C x = πλ (E , θ)σπλ(E ), (142) virtual photons for the E1 multipolarity, “as seen” by a pro- dΩ E dΩ x γ x x πλ jectile passing by a lead target at impact parameters b > 12.3 fm, at three projectile energies. As the bombarding energy in- where Ω denotes to the solid scattering angle. creases, virtual photons with larger energies become available At high energies, above Elab = 100 MeV/nucleon, the eikonal for the reaction. The number of states accessed in the excitation formalism developed in Ref. [259] yields for the virtual photon process is concomitantly increased. numbers in Eq. (140) The cross sections for Coulomb excitation are usually larger for electric dipole (E1) excitations and for the isovector giant λ(2λ + 1)!!2 X n (E ) = Z2α |G |2 g (E ) , (143) dipole resonance (GDR). It leads overwhelmingly to neutron πλ x 1 π 3 λ πλm m x (2 ) ( + 1) m decay and can be calculated as [207, 108] Z dE σ−n = n (E)σGDR(E), (150) C E E1 γ

32 200 1000 28 MeV/nucl. 70 MeV/nucl. 136Xe 280 MeV/nucl. ]

V 150 e M ) ) / ! b (E

m 100 [

E1 100

n DGDR E d

Pb target / σ 50 d bmin = 12.3 fm

10 0 246810 0 10 20 30 40 50

E! [MeV] ! Excitation energy

Figure 31: Number of virtual photons for the E1 multipolarity, as “seen” by Figure 32: Experimental cross sections for the excitation of 136Xe (700 a projectile flying by a lead target at impact parameters b > 12.3 fm, at three MeV/nucleon) projectiles incident on lead (solid circles) and carbon targets projectile energies. (open circles). The dashed curve is a calculation including the excitation of isoscalar and isovector giant quadrupole resonances and the isovector giant dipole resonance (IVGDR). Altogether, these resonances compose the large where the equivalent photon number is obtained from bump in the spectrum. The double giant dipole resonance (IVGDR) is identified as the bump at double the energy of the IVGDR. Data are from Ref. 2 !2 2Z α ωc Z [264]. n (E) = T db b E1 π γv2 " # 1 as the random phase approximation (RPA) [263]. One can also × K2(x) + K2(x) Λ(b), (151) 1 γ2 0 use a simple phenomenological parameterization of the GDR −2 1.91 width in the form, ΓGDR = 2.51 × 10 EGDR MeV, with EGDR with v equals to the projectile velocity, the Lorentz contraction in units of MeV. The centroid of the ISGQR can be taken as factor given by γ = (1 − v2/c2)−1/2, α being the fine-structure 1/3 EIS GQR = 62/AP MeV. constant and Kn the modified Bessel function of nth-kind as One of the most dramatic findings in the application of rel- a function of x = ωb/γv. We denote the excitation energy ativistic Coulomb excitation was the discovery of the Double GDR by E = ~ω. The photo-nuclear cross sections σγ can be Giant Dipole Resonance (DGDR). It was in fact first found in parametrized by a Lorentzian shape pion scattering at the Los Alamos Pion Facility [265]. The excitation of the DGDR can be described as a two-step mechanism E2Γ2 σGDR(E) = σ , (152) induced by the pion-nucleus interaction. Using the Axel-Brink γ 0 2 2 2 2 2 (E − EGDR) + E Γ hypotheses, the cross sections for the excitation of the DGDR with pions tend to agree within the experimental possibilities. −1/3 −1/6 where EGDR = 31.2AP + 20.6AP is a fit to the mass depen- Five years after the DGDR excitation experiments with pions dence of the centroid of the experimentally observed GDR. It were revealed, the first DGDR Coulomb excitation experiments is a mixture of the mass dependence predicted by the hydrody- were carried out at the GSI facility in Darmstadt/Germany [264, namical Goldhaber-Teller and Steinwedel-Jensen models [261, 266]. The excitation of multiple giant resonances had been pre- 262]. The parameter σ0 can be chosen to yield the Thomas- dicted a few years before in Ref. [267, 268]. In Fig. 32 we Reiche-Kuhn (TRK) sum rule show the result of one of these experiments [264], which de- Z N Z tected neutron decay channels of giant resonances excited with dEσGDR(E) = 60 P P MeV mb, (153) relativistic projectiles. The excitation spectrum of relativistic γ A P 136Xe projectiles incident on Pb are compared with the spec- which is a nearly model independent results for the nuclear re- trum using C targets. The comparison proves that nuclear con- sponse to a dipole operator [206]. NP(ZP) are the neutron(charge) tribution to the excitation is small for large-Z targets. An ad- number of the excited projectile. ditional experiment [266] used the photon decay of the DGDR The width Γ of the GDR is more complicated to explain. as a probe. A bump in the spectra of coincident photon pairs It has a strong dependence on the nuclear shell structure. Ex- was observed around an energy twice as large as the GDR cen- perimental systematics provides widths ranging within 4 − 5 troid energy in 208Pb targets excited with relativistic 209Bi pro- MeV for a closed shell nucleus and can grow to 8 MeV for a jectiles. The advantages of relativistic Coulomb excitation of nucleus between closed shells. Rare nuclear isotopes are not heavy ions over other probes (pions, nuclear excitation, etc) was easy to investigate experimentally using photo nuclear reac- clearly demonstrated in several GSI experiments, and reviewed tions. One often adopts a microscopic theoretical model such in Refs. [269, 263].

33 6000 EM dissociation 238U 5000 total

] 4000 b m

[ 1n

3000 σ

2000 fission

Figure 34: Left panels: Diﬀerential cross sections, with respect to excitation 1000 energy E∗, obtained with electromagnetic dissociation of 130Sn and 132Sn. The 2n arrows indicate the neutron-separation thresholds. Corresponding right panels: 3n Deduced photo-neutron cross sections. The curves represent ﬁtted Gaussian 0 (blue dashed line) and Lorentzian (green dash-dotted line) distributions, as- 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 signed to the Pygmy Dipole Resonance (PDR) (centroid indicated by an arrow) and Giant Dipole Resonance (GDR), respectively, and their sum (red solid line), E [GeV/nucleon] after folding with the detector response. Data are from Ref. [275].

Figure 33: Cross section for electromagnetic dissociation of 238U. The Coulomb fission cross sections (diamonds), measured with 208Pb beams on olds. In the right panels we show the deduced photo-neutron 238U-targets are from Ref. [270], whereas the fission cross sections, as well cross sections. The curves represent fitted Gaussian (blue dashed as the xn cross sections are obtained in inverse kinematics with 238U beams line) and Lorentzian (green dash dotted line) distributions, as- [271]. The data obtained in Ref. [272] with Au-targets were scaled to the data signed to the Pygmy Dipole Resonance (PDR) (centroid indi- obtained with Pb targets. The curves show theoretical calculations using two sets of experimental GDR parameters as an input. cated by an arrow) and Giant Dipole Resonance (GDR), respectively, and their sum (red solid line), after folding with the detector response. It was earlier on recognized [273] that the electromagnetic excitation of nuclei in relativistic heavy ion collisions lead to 3.3.1. The Coulomb dissociation method large cross sections because of the excitation of giant resonances. The idea behind the Coulomb dissociation method is rela- The decay of these resonances, including the DGDR, lead to tively simple [276]. The (differential, or angle-integrated) Coulomb several decay channels, such as xn evaporation and fission. This breakup cross section for a + A → b + c + A follows from Eq. was demonstrated by comparison off experiment to theory in (140). It can be rewritten as Ref. [108, 274]. Therefore, EM dissociation leads to large in- πλ πλ teraction cross sections, which can easily measure in inverse dσ (Eγ) dn (Eγ; θ; φ) C = σπλ (E ), (154) kinematics. In Fig. 33 we show the cross section for elec- dΩ dEdΩ γ+a → b+c γ tromagnetic dissociation of 238U [274]. The Coulomb fission where E is the energy transferred from the relative motion to cross sections (diamonds), measured with 208Pb beams on 238U- γ the breakup, σπλ (E ) is the photo-dissociation cross sec- targets are from Ref. [270], whereas the fission cross sections, γ+a → b+c γ tion for the multipolarity πλ and photon energy E , and j are as well as the xn cross sections are obtained in inverse kine- γ k the spins of the particles involved. Using time reversal invari- matics with 238U beams [271]. The data obtained in Ref. [272] ance, the radiative capture cross section b + c → a + γ from with Au-targets were scaled to the data obtained with Pb tar- σπλ (E ), becomes gets. The curves show theoretical calculations using two sets γ+a → b+c γ of experimental GDR parameters as an input. Hence, it is clear 2(2 j + 1) k2 that, except for light targets, one should expect that EM dissoci- σπλ E a σπλ E , b+c→a+γ( γ) = 2 γ+a → b+c( γ) (155) ation will always be relevant in any nucleus-nucleus collision at (2 jb + 1)(2 jc + 1) kγ relativistic energies and might be either used as a spectroscopic 2 tool, or become a background for processes where the effects of where k = 2mbc(Eγ − S ) with S equal to the separation energy, the strong interaction are studied. and kγ = Eγ/~c. In Fig. 34 in the left panels, we show the differential cross The Coulomb Dissociation (CD) method was originally pro- sections, with respect to excitation energy E∗, obtained with posed in Ref. [276]. It has been tested successfully in a number electromagnetic dissociation of 130Sn and 132Sn. Data are from of reactions of interest to astrophysics. The best known case is 7 8 Ref. [275]. The arrows indicate the neutron-separation thresh- the reaction Be(p, γ) B, first studied in Ref. [280], followed by numerous other similar experiments. As an example of the 34 30

14 15 25 C(n,γ) C

] 20 b µ [

15 γ n

σ 10

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

ECM [MeV]

6 Figure 35: Electric dipole response function for He. The shaded area repre- Figure 36: Neutron capture cross section of 14C leading to the 15C ground sents the experimental results from the Coulomb dissociation experiment re- state. Solid red circles are the results of the experiment of Ref. [282], and ported in Ref. [277]. Dashed and dotted lines correspond to calculations using the blue squares represent those from a direct capture measurement [283]. The three-body models Refs. [278, 279]. dot-dashed curve is a calculation based on the direct radiative capture model, whereas the solid curve is the same calculation but includes experimental resolution. application of the CD method, we quote the two-neutron capture on 4He that may play a role in the post-collapse phase in type-II supernovae. A nucleosynthesis bottleneck prevents for- there is only one excited state at 0.74MeV, and the neutron cap- mation of nuclei with A ≥ 9 from nucleons and α-particles and ture to this state is estimated to be very small [284, 285]. Such the reaction 4He(2n, γ)6He could be a solution to bridge the in- fortunate situations may be less probable for heavier neutron- stability gap at A = 5. It is worth mentioning that one believes rich nuclei, but we may expect lower level densities near closed that the most probable candidate is the (αn, γ) process in a type- shells, such as N = 50 and 82, where Coulomb breakup can be II supernova. A Coulomb dissociation experiment to study this used. Nuclei in this region of the chart may be relevant to the reaction is shown in Figure 35 with the electric dipole response r-process, and thus of importance for studies using Coulomb function for 6He. The shaded areas represent the experimental breakup. 17 18 results from the Coulomb dissociation experiment reported in Another example concerns the C(n, γ) C reaction at as- Ref. [277]. The dashed and dotted lines correspond to calcu- trophysical energies. Elements heavier than iron are thought lations with three-body models of Refs. [278, 279]. The ex- to be created in reactions in the slow (s-) and rapid (r-)neutron perimental analysis of this experiment concluded that 10% of capture processes [286, 19] since they are not suppressed by the CD cross section proceeds via the formation of 5He, with the Coulomb barrier at low energies [287]. An evidence is an estimate of 1.6 mb MeV for the photoabsorption cross sec- that the abundance pattern observed in ultra metal-poor stars tion of 6He(γ, n)5He, which agrees with theoretical calculations [288, 289, 290, 291] that are attributed to the r-process is re- [281]. One has concluded that the cross sections for formation markably close to solar abundance within 56 ≤ Z ≤ 76, sug- of 5He and 6He via one (two) neutron capture by 4He are not gesting the existence of a generic production mechanism. Sev- large enough to compete with the (αn, γ) capture process [107]. eral possible scenarios with nucleosynthesis flows are sensi- This attests the usefulness of the CD method to help us under- tive to reaction rates of light neutron-rich nuclei existent in standing basic questions of relevance for nuclear astrophysics. core-collapse Type II supernova (SN) explosions or neutron In Fig. 36 we show the neutron capture cross section of 14C star mergers [292]. The final heavy element abundances were leading to the 15C ground state. The solid red circles are the found to change up to an order of magnitude as compared to 17 results of the experiment of Ref. [282], and the blue squares calculations without light nuclei and the neutron capture on C represent those from a direct capture measurement [283]. The was considered critical as the rate was solely based on Hauser- dot-dashed curve is a calculation based on the direct radiative Feshbach calculation. Up to recently, no experimental informa- 17 capture model, whereas the solid curve is the same calculation tion on the neutron capture cross sections of C was available. including the experimental resolution. The experimental result The reaction was studied in Ref. [293] using the CD method is consistent with a final state ground state capture to 15C being and the experimental analysis yielded the results shown in Fig. a halo state with a dominant s-wave component. The agreement 37 for the reaction rates in a r-process site. The figure plots 17 of the Coulomb breakup result with the direct capture measure- the reaction rate for neutron capture on C with respect to the ment suggests that Coulomb breakup is a good method to ob- stellar temperature in T9 units (billion K). The data from Ref. tain neutron capture cross sections involving radioactive nuclei. [293] (grey band) are compared to rate calculations using the A word of caution is that in the inverse reaction of Coulomb Hauser-Feshbach theory [292] (dashed blue line) and a direct breakup the reaction is restricted to the ground state. For 15C, capture model [294] (dotted red line). The lower panel shows

35 continuum states

threshold

bound states

Figure 37: Reaction rate for neutron capture on 17C with respect to the stellar temperature in T9 units (billion K). The data from Ref. [293] (grey band) are compared to rate calculations using the Hauser-Feshbach theory [292] (dashed blue line) and a direct capture model [294] (dotted red line). The lower panel Figure 38: Schematic description of a coupled-channels calculation includ- shows the percentage contribution of experimental data for transitions to the ing the discretized continuum states. The arrows represent transitions between ground state in 18C. states. the percentage contribution of experimental data for transitions numbers E` m. The function Γk(E) is assumed to peak at an to the ground state in 18C. energy E in the continuum. This leads to nearly discrete states |φk`mi allowing for an easy implementation of the coupled-states 3.4. Relativistic coupled-channels method calculations. At relativistic energies, the projectile moves at forward angles with nearly constant velocity. Thus one can 3.4.1. Continuum states and relativistic corrections replace z = vt in the above equations to obtain a series of Eq. (154) is obtained with first-order perturbation theory. coupled-equations in coordinate space. As shown below, the One also assumes that the nuclear contribution is small, or that amplitudes a are become the scattering matrices S . The it can be extracted in other types of experiments. For exam- j`m j`m coupled-channels approach with the inclusion of the contin- ple, 8B has a small proton separation energy (≈ 140 keV). uum is commonly know as Continuum Discretized Coupled- Multiple-step, or higher-order effects, are important for such Channels (CDCC) method and is schematically shown in Fig. loosely-bound systems [295, 296]. A calculation of reorienta- (38) where the arrows represent transitions between the states tion effects in the break up of halo nuclei was first presented in due to the perturbative field V . Ref. [295]. In the (non-relativistic) projectile frame of refer- ex The relativistic Coulomb potential for the multipolarity ΠL ence, the time-dependent Schrodinger¨ equation due to a (rela- (Π = E or M; L = 1, 2, ··· ) as seen by the projectile P passing tivistic) electromagnetic interaction for its internal wave func- by a target T are given in Ref. [297]: tion expanded in terms of bound states and continuum states, P  Ψ(t) = a j`m φ j`m, with energies E j, yields to the coupled- r j`m 2π γZ e  ∓b (if µ = ±1) channel equations V = ξY ξˆ T  √ E1µ 1µ 3/2  3 b2 + γ2z2  2z (if µ = 0) da j`m X D E 0 −i(E −E j)t/ (158) i = φ |V | φ 0 0 0 a 0 0 0 e j ~. (156) ~ dt j`m ex j ` m j ` m j0`0m0 for the E1 (electric dipole) field, and r where we reserve the index j = 0 for the ground state and Vex 3π 2 γZTe VE2µ = ξ Y2µ ξˆ is a combination of time dependent scalar φ(r, t) and vector po- 10 b2 + γ2z25/2 tentials, A(r, t), so that h|Vext|i ≡ δρφ + δj · A, with δρ and δj  being transition densities and currents.  b2 (if µ = ±2)  The continuum discretization can be done in several ways.  2 ×  ∓(γ + 1)bz (if µ = ±1) (159) An obvious way is to use pseudo-states, e.g., selected states  √  2 2 − 2 generated by the same potential as the bound states, or by the  2/3 2γ z b (if µ = 0) use basis of time-dependent discrete states are defined as for the E2 (electric quadrupole) field. The intrinsic coordinate −iEk`mt/~ |φk`mi = e |0i , if Ek`m < 0 (157) between the nucleons are denoted by ξ and b is the impact pa- Z rameter (or transverse coordinate) in the collision of P and T, −iEkt/~ p = e Γk(E) |E`mi , if Ek`m > 0 which is defined by b = x2 + y2 with R = (x, y, z), the relative coordinate of P from T in the Cartesian representation. The where |E`mi being continuum wavefunctions of the projectile −1/2 Lorentz contraction factor is denoted by γ = 1 − v2/c2 , fragments with good energy and angular momentum quantum 36 where v is the velocity of P. These relations are obtained with where so-called far-field approximation [298] with R assumed to be (b) nucl(b) Coul(b) always larger than ξ. The coupling potentials in E-CDCC are Fcc0 (z) = hΦc|UCT + UvT|Φc0 iξ = Fcc0 (z) + Fcc0 (z), obtained with Eqs. (158)–(159) as shown below. The derivation (164) of these equations is described in Ref. [299]. nucl(b) D nucl nucl E F 0 (z) = Φc|U + U |Φc0 , (165) In addition, for magnetic dipole excitations, cc CT vT ξ Coul(b) D Coul Coul E F 0 (z) = Φc|U + U |Φc0 , (166) r  cc CT vT ξ 2π v γZ e  ±b (if µ = ±1) V (b, z, ξ) = i M (ξ) T  M1µ 1µ 3/2  where v and C denote two clusters composing the projectile. 3 c b2 + γ2z2  0(if µ = 0) Φ (ξ) are the internal wave functions of P, ξ are the coordi- (160) c nate of v relative to C, and U (U ) is the potential between C where M (ξ) is the intrinsic M1 operator. CT vT 1µ (v) and T including nuclear and Coulomb parts. The multipole A second method used to handle higher-order effects in high expansion for each term on the right-hand-side of Eq. (164) is energy collisions is to solve the time-dependent Schrodinger¨ X equation in the frame of reference of the excited nucleus by a nucl(b) nucl(b) Fcc0 (z) = Fcc0,λ (z), (167) lattice discretization of the four dimensional space. This method λ was developed in Ref. [296] and amounts to expand the wave- Coul(b) X Coul(b) F 0 (z) = F 0 (z). (168) function is a spherical basis, cc cc ,λ λ 1 X Ψ(r , t) = u (r , t)Y (rˆ ), The explicit form of the nuclear multipoles is given in Ref. [305]. k r `m k `m k k `m In order to include the dynamical relativistic effects the conjecture similar to that in Ref. [306], one can make the replace- use the orthogonal properties of the spherical harmonics and ments [302, 303] obtain the time-dependent equation in the lattice λ(b) λ(b)   Fcc0 (z) → γ fλ,m−m0 Fcc0 (γz). (169)  `(` + 1) 2µcx  ∆k − − Vex(rk) u`m(rk, t)  2  f rk ~ The factor λ,µ is equal to unity for nuclear couplings, while for Coulomb couplings one sets X (`m) 2µcx ∂u`m(rk, t) + S 0 0 ΠL; r , t u 0 0 (r , t) = − , ` m k ` m k ∂t `0m0 ~   1/γ (λ = 1, µ = 0) (161)   2 fλ,µ =  (γ + 1)/(2γ)(λ = 2, µ = ±1) (170) (`m)  where S 0 0 ΠL; r , t is a source function of multipolarity ΠL  ` m k  1 (otherwise) which couples the different partial waves and depends on the perturbative potential Vex. For more details, see Refs. [296, following Eqs. (158) and (159). Correspondingly, 300, 301]. 2 2 In Refs. [297, 302, 303] a proposal has been made to in- ZPZTe ZPZTe clude relativistic dynamical effects also using non-relativistic δcc0 → γ p δcc0 (171) R b2 + (γz)2 nuclear potentials and extending them to include relativistic dynamics in an approximate way. The starting point are the non- in (163). With these changes we note that Eq. (162) is Lorentz relativistic eikonal-CDCC (E-CDCC) equations [304, 305] for covariant, as desired. a three-body reaction between the and the target Solving Eq. (162) under the boundary condition

i 2 d X (b) ~ (b) (b) (b) i(kc0 −kc)z lim ψ (z) = δc0, (172) k ψ (z) = F 0 (z) ψ 0 (z) e , (162) c E dz c cc c z→−∞ c0 where 0 denotes the incident channel, one gets the following where c are the channel indices {i, `, m}; i > 0 (i = 0) denotes the excitation amplitude of the channel state c by solving the the ith discretized-continuum (ground) state, and ` and m are, coupled-channels equations (156) with respectively, the orbital angular momentum and its projection (b) on the z-axis taken to be parallel to the incident beam. The in- Ac0(b) ≡ limψc (z) (173) ternal spin indices are suppressed for simplicity. Note that in z→∞ Eq. (162) b is relegated to a superscript since it is not a dynam- Incorporating nuclear absorption at small impact parameters (b) ical variable. The reduced coupling potential Fcc0 (z) is given implies that the eikonal S-matrix in the coupled-channels cal- by culations yield 2 (b) (b) ZPZTe Fcc0 (z) = Fcc0 (z) − δcc0 , (163) n o R S c(b) = Ac0(b) exp − Im χOL(b) (174)

with χOL(b) given by Eq. (116).

37 The angular distribution of the inelastically scattered parti- for isoscalar monopole, cles can be obtained from the S-matrix above, yielding 2 ! IV d 1 d Z ∞ U (r) = δ + R U(r) , (183) 1 1 dr 3 dr2 fc(θ) = ik dbbJm(qb)S c(b). (175) 0 for isovector dipole, and The inelastic scattering cross section is obtained by an average dU(r) over the initial spin and a sum over the final spin: U (r) = δ , (184) l l dr dσ 1 X inel | |2 = fc(θ) . (176) for l ≥ 2. In these equations, δi = βiR, are known as deforma- dΩ 2`0 + 1 m0,m f tion lengths. The matrix elements can be calculated using well-known 3.4.2. Nuclear excitation of collective modes sum-rules leading to relations between the deformation length, Using the collective particle-vibrator coupling model the and the nuclear sizes and the excitation energies. For isoscalar matrix element for the transition j −→ k becomes [307] excitations one has the sum rules [307] δ < ` m |U |` m >= − √ λ < ` m |Y |` m > 2 1 2π 2 1 k k N(λµ) j j k k λµ j j α2 = 2π ~ , δ2 = ~ λ (2λ + 1) 2λ + 1 0 m < r2 > AE λ≥2 3 m AE × Y (rˆ) U (r) (177) N x N x λµ λ D E (185) where A is the atomic number, r2 is the r.m.s. radius of the where δ is the vibrational amplitude (deformation parameter), λ nucleus, and E is the excitation energy. U (r) is the transition potential, and x λ For dipole isovector excitations [307] " #1/2 (2I + 1)(2λ + 1) `k−mk k 2 < `kmk|Yλµ|` jk j > = (−1) 2 π ~ A 1 4π(2` j + 1) δ1 = , (186) ! ! 2 mN NZ Ex ` λ ` ` λ ` × k j k j . (178) −mk µ m j 0 0 0 where Z (N) the charge (neutron) number. The transition potential in this case is modified from Eq. (183) to account for the The transition potentials can be related to the real part of isospin dependence [307]. It becomes the optical potentials, U = Re[Uopt], originated from low energy collisions. Notice that one does not include the imaginary N − Z dU 1 d2U UIV (r) = −Λ + R , (187) part of the optical potentials because the inelastic cross section 1 A dr 3 dr2 already includes the absorption part in the factor exp[iχOL(b)] in Eq. (174). The transition densities are given by [307] where the factor Λ depends on the difference between the proton and the neutron matter radii as ! d δρ (r) = −α 3U + r ρ (r), (179) 2(N − Z) Rn − Rp ∆rnp 0 0 dr 0 Λ = = . (188) 3A 1 R 2 (Rn + Rp) for isoscalar monopole excitations, The strength of isovector excitations increases with the neutron ! d 1 d2 skin ∆rnp which is larger for neutron-rich nuclei. δρIV (r) = βIV + R ρ (r) (180) 1 1 dr 3 dr2 0 As shown in Ref. [308], for the isoscalar giant dipole resonance, the transition density in the deformed model is expressed for isovector dipole excitations, and as (IS ) " d (IS ) β1 2 d 5 D 2E d δρl(r) = βl ρ0(r) (181) δρ (r) = − √ 3r + 10r − r dr R 3 dr 3 dr 2 !# for isoscalar giant quadrupole and higher multipoles. α0 and βi d d − r + 4 ρ0(r), (189) are the vibrational amplitudes. R is the nuclear radius at 1/2 dr2 dr the central nuclear density and ρ (r) are the ground state den- 0 where sities.The same equation above can be used for isovector giant ! 2 4 5 ~ resonances of l ≥ 2 multi polarities. For a thorough discussion = + , (190) E E 3mA on the subject, see Ref. [307]. 2 0 The corresponding transition potentials are [307] with A being the nuclear mass, E2 and E0 are the excitation energies of the giant isoscalar quadrupole and monopole reso- ! d nances, respectively. In Eq. (189) hi means average value. U (r) = −α 3 + r U(r) , (182) 0 0 dr The collective coupling parameter for the isoscalar dipole

38 resonance is coordinates θ and φ of the photon momentum vector k by

2 2 ! X 2 (IS ) 6π R D E 25 D E2 D E M f 0 ~ 4 2 2 W (θ) = (−1) ai−→ f Fk l, l , Ig, I f β1 = 11 r − r − 10 r . (191) mN AEx 3 k=even 0 Mi,M f ,l,l Analogously, the transition potential can be written, for isoscalar   dipole excitations  I I k  √  f f  ∗ ×   2k + 1Pk (cos θ) ∆l∆l0 , (198) (IS )   "  M f −M f 0  (IS ) δ1 2 d 5 D 2E d U1 (r) = − √ 3r + 10r − r R 3 dr 3 dr where Pk are Legendre polynomials, and !# d2 d q − r + 4 U(r), (192) 0 I f −Ig−1 0 dr2 dr Fk l, l , Ig, I f = (−1) (2l + 1)(2l + 1) 2I f + 1 (2k + 1)     The reduced transition probability for electromagnetic tran-  l l0 k   l l0 k  ×     . (199) sitions is defined by [206]      1 −1 0   I f I f Ig  1 2 B (πλ; i → j) = < I j||M(πλ)||Ii > , (193) The angular distribution of γ-rays described above is in the 2I + 1 i reference frame of the excited nucleus. To obtain the distribu- and neglecting the spin of the initial state i, these matrix ele- tion in the laboratory one has to perform the transformation ments can be related to the deformation parameters α0 and δλ ( ) sin θ by means of the sum-rules [307] θ = arctan , (200) L γ cos θ + β " 2 #2 2 3ZeR 2 9 NZe 2 B(E0) = α0 , B(E1) = δ1 , (194) and 10π 4π A 2 2 W(θL) = γ (1 + β cos θ) W(θ) , (201) and p 2 " #2 where γ is the Lorentz contraction factor, and β = 1 − 1/γ . 3 λ−1 2 ph ph B(Eλ) = ZeR δ , (195) The photon energy in the laboratory is E = γEcm (1 + β cos θ). λ≥2 4π λ L 3.5. Excitation of pygmy resonances 3.4.3. Angular distribution of γ-rays E Giant resonances have been observed all along the nuclear After the excitation with energy ~ω, the state I f can decay chart. Pygmy resonances, on the other way, have been observed E by gamma emission to Ig . There are complications due to mainly in neutron-rich nuclei. Its first observation was reported the fact that the nuclear levels are not only populated through in 1961 with the observation of a large number of bunched un- Coulomb excitation, but also by conversion and γ-transitions bound states in γ-rays emitted after neutron capture [310]. The from higher states. To compute the angular distributions one name pygmy resonance, or pygmy dipole resonance (PDR), must know the parameters ∆l (i −→ j) and l (i −→ j) for l ≥ 1 was first used in 1969 in connection with calculations of neu- [309], tron capture cross sections [311]5. The theoretical treatment of 2 −→ −→ 2 −→ l (i j) = αl (i j) ∆l (i j) , (196) the PDR as a collective excitation mode in the nucleus was first where αl is the total l-pole conversion coefficient, and published in Ref. [312] using a three fluid model with a proton and another neutron fluid in the same orbitals, and an additional " #1/2 8π (l + 1) 1 ω2l+1 −1/2 fluid of neutron excess interacting weakly with the rest. The ∆πl = 2I j + 1 l [(2l + 1)!!]2 ~ c model predicted that the neutron excess fluid oscillates against D E a core with N = Z. Finally, the first experimental proposal to × I is(l)M(πl) I , (197) j i measure the effect of Coulomb excitation of pygmy resonances with s(l) = l for electric (π = E) and s(l) = l + 1 for magnetic was submitted in 1987 in Japan at the JPARC facility [313]. (π = M) transitions. The square of ∆πl is the l-pole γ-transition The relevance for the existence of pygmy resonances for nucle- rate (in s−1). M(πl) is the electromagnetic operator. osynthesis processes was reported in Refs. [314, 315, 316]. In the non-relativistic case [309], the gamma ray angular The interest on excitation of pygmy resonances in exotic distributions after the excitation depend on the frame of refer- nuclei surge in the 1990s when a narrow peak associated with ence used. Here the z-axis corresponds to the beam axis and the small separation energies was reported in Coulomb dis- considering a simple situation in which the γ-ray is emitted di- sociation experiments [317, 318]. A few years earlier Refs. rectly from the final excited state f to a lower state g observed [319, 320, 321], a simple expression for the response func- experimentally it was demonstrated in Ref. [299] that the angu- tion emerges for electric multipoles was obtained using simple lar dependence of the γ-rays is given explicitly by the spherical

5We are grateful to Riccardo Raabe (KU Leuven) for sharing this information with us.

39 Yukawa or Hulthen functions for bound states and plane waves 40 for the continuum,

L−1 2 !L dBEL(E) 2 2 ~ ]

= C (2L + 1)(L!) 5 dE L π2 µ 30 √ S (E − S )L+1/2 × Z2e2 , (202) L E2L+2 where CL is a constant depending on how the wave functions 20 are normalized, eL is the eﬀective charge for the multipolarity [MeV fm 2 L, µ is the reduced mass and S is the separation energy of a | two-cluster nucleus. In the case of the electric dipole (E1) ex- sp

I 10 citation, | √ 2 2 2 3/2 dBE1(E) 3Z e ~ S (E − S ) C L . = 1 2 4 (203) dE µπ E 0 Eqs. (202) and (202) predict a maximum of the response func- 1 2 3 tion at Ex = 8S/5 and a width Γ ∼ Ex. They have been used in numerous experimental analyses [322, 323] and compared with Er [MeV] predictions of other theoretical models [324, 321, 325, 326]. Adding final state interactions complicates the theoretical 2 predictions based on Eq. (202). Considering the transition from Figure 39: Is→p calculated using eq. 204, assuming different values of the scattering length and effective range for the n-10Be scattering in the final state a bound single-particle s-state to a continuum p-wave, as in the [210]. Coulomb breakup of 11Be, and using the notation dσ/dΩ ∼ 2 |Is→pI , Ref. [210] finds between the calculation of the response function (in arbitrary " √ # (E − S )3/4 µ 3/2 S (3E − 2S ) units) in 11Li using zero phase-shifts, δ = 0 and δ = 0, I ' n n r n , nn nc s→p 2 1 + 2 2 E 2~ −1/a1 + µr1Er/~ (dashed line), or including the effects of final state interactions (204) (continuous line) [210]. The experimental data are from ref. [323]. where S n is the neutron separation energy, and the effective Following the first calculations on pygmy resonances based 2l+1 range expansion of the phase shift, δ, k cot δ ' −1/al + on two- and three-body models, it was soon realized that the 2 rlk /2, was used to obtain the correction included by the second the role of the continuum and of the nuclear contribution was term within brackets. For l = 1, a1 is the “scattering volume” crucial to explain the experimental data on the EM response 3 (units of length ) and r1 is the “effective momentum” (units of obtained in breakup experiments [295, 300, 330, 331, 332, 333, 1/length). Their interpretation is not as simple as the l = 0 334, 297, 335, 303, 304, 336, 337, 338, 339, 340]. Calculations effective range parameters. based on effective field theories started to appear recently [341]. For a three-cluster nucleus, such as 11Li or 6He, three-body Because the dipole response at low energies were also ob- models are evidently necessary, yielding for the E1 multipolar- served for heavier nuclei, the old idea that pygmy resonances ity [327, 328, 278, 210] could have a collective character emerged again and revived old theoretical predictions [312, 342, 343]. Extending the Goldhaber- dB (E) (E − S )3 E1 ∝ 2n (1 + FSI), (205) Teller (GT) [344] and Steinwedel-Jensen (SJ) [262] hydrody- dE E11/2 namical models to pygmy resonances in “soft” neutron-rich nu- where the final state interaction (FSI) term is given by an in- clei, and using the admixture of the two pictures as in Ref. tegral over hyper angles [210]. Without the FSI term, the E1 [345], the radial transition density of pygmy resonances can be response in the three-body model would imply a peak at around described by [210] E ' 1.8S . r " # 2n 4π d K In both two-body and three-body halos, the FSI model shifts δρ(r) = R Z α + Z α j (kr) ρ (r), (206) 3 GT GT dr SJ SJ R 1 0 appreciably the peak in the EM response.Therefore, the separation energy still roughly determines the peak location of the E1 where Zi are the GT and SJ effective charges [210], αi are amount response but at a different energy and with a different width. of GT and SJ admixture, with αGT + αSJ = 1, K = 9.93 and R is It was also shown that final state interactions can substantially the (mean (sharp density) nuclear radius. j1(kr) is the spherical change the location of the low energy peak [210, 329]. This Bessel function of order 1, with k = 2.081/R. The transition is clear from Fig. 39, where the function Is→p is plotted for density in Eq. 206 na¨ıvely describes a pygmy resonance as a different bu reasonable adopted values for the scattering length collective dipole vibration of protons and neutrons (soft dipole and effective range [210]. In Fig. 40 we show a comparison mode), as shown schematically in Figure 41.

40 For the Goldhaber and Teller model [344] one has αSJ = 0, and the centroid energy of the IVGDR is obtained as,

!1/2 3S~2 EIVGDR = , (207) 2aRmN 12 where a is the nuclear distribution diﬀuseness, mN is the nucleon mass, and S is not the nucleon separation energy but the 11Li energy needed to extract one neutron from the proton environ- 8 ment. This is roughly is the part of the potential energy due to the neutron-proton interaction within the nuclear environment, also roughly proportional to the symmetry energy ∝ (N − Z)/A. Goldhaber and Teller [344] used S = 40 MeV, a = 1 − 2 fm and 4 to obtain E ' 10 − 20 MeV for a medium heavy and heavy nuclei, in rough agreement with the experimentally found centroid energies of the IVGDR.

dB(E1)/dE [arb. units] It is not straightforward to extend the arguments of Gold- 0 123haber and Teller discussed in the preceding paragraph to the case of neutron rich nuclei. But the quantity S should somehow Er [MeV] relate to the symmetry energy. This could be obtained within a microscopic model because the pygmy resonances display fine- structures that seem to be much more dependent on the coupling Figure 40: Comparison between the calculation of the response function (in of phonon states with complex configurations than in giant res- 11 arbitrary units) in Li using zero phase-shifts, δnn = 0 and δnc = 0, (dashed onances. In fact, the collectivity of the pygmy resonance is not line), or including the effects of final state interactions (continuous line) [210]. well established. Hydrodynamical models are surely unfit for The experimental data are from Ref. [323]. halo nuclei. We could approximate S to the separation energy, S = S and expect that the product aR is proportional to S −1. 11 Then, naively, we would get EPDR ∼ βS , with β ∼ 1. For Li, we take a = 1 − 2 fm, R = 3 fm, and S = 0.3 MeV, one gets EPDR = 1 − 2 MeV, which is also compatible with experimental data. But, evidently, hydrodynamical models lack accuracy and can only be used to give physics insight of the problem. Micro- scopic models, often relying on the linear response theory, are a more accurate method to describe giant resonances and pygmy

resonances on the same foot. The electromagnetic response in weakly-bound nuclei was

ﬁrst studied using the RPA equations done in Ref. [346], based

GDR on the continuum RPA model developed in Ref. [347]. Later

giant resonance strength it was also used in Refs. [321, 324] to study the eﬀects of

soft mode clustering and higher multipole EM response. The RPA model results for halo nuclei again showed a pronounced peak at E small excitation energies, again interpreted as a pygmy reso-

single particle excitations excess neutrons nance, but in fact it was just the eﬀect of a small separation energy in the nuclei. For medium heavy and heavy nuclei dif- bound states ferent RPA models have shown that basically all neutron-rich nuclei display a visible bunching in the response at low ener- “core” with p and n gies which has again been attributed to the pygmy resonance. Relativistic mean ﬁeld models also yield similar results, e.g. in Refs. [348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, Figure 41: Schematic view of the excitation of giant resonances (left) as a broad and large peak in the excitation spectrum of nuclei around 10-20 MeV. The 359, 360, 361, 362, 363]. More recently the powerful TDSLA pygmy resonances are viewed as a collective excitation of a neutron “mantle” method, described previously has been used to study pygmy vibrating against a tightly bound nuclear core (left). Besides the giant reso- resonances and also nuclear large amplitude collective motion nance, here a Giant Dipole Resonance (GDR) at higher energies (right), one [364, 96, 97, 98, 99, 100]. First TDSLDA calculations for rel- should see a concentrated bump at lower energies due to the pygmy states. ativistic Coulomb excitation in a collision of 238U + 238U were reported in Ref. [99]. One has reported considerable amount of electromagnetic strength occurs at low energies, around Ex ∼ 7 MeV. This additional structure was attributed to the excitation

41 80

(a) 300 (a) total EM spectrum 10-1 m1(!) FSU -2 200 10 60 68 100

Ni ] ω ]

-3 2 2

! 10 0 [fm

[fm 10 20 30

-4 ) 40

10 !

dE/d (b) -4 -1 x10 10 R( 0.18 fm ! 6 EF(E) 0.21 fm -2 4 0.24 fm 10 2 20 0.26 fm 0 0.29 fm 10-3 4 6 8

-4 0 10 0 5 10 15 20 25 0 10 20 30 40 E! (MeV) [MeV] ω [MeV] ! 0.3 (b) 1 m-1(!) FSU Figure 42: Top: Total energy spectrum (solid line) of emitted EM radiation,

238 238 ] ] dE/d ω, for U + U collision at the impact parameter b = 12.2 fm. The to- 2 ~ 2 0.5 tal quadrupole contribution is shown by a double-dotted line. The other curves 0.2 are contributions from the three target-projectile orientations. Bottom: The /MeV /MeV 2

2 0 electric dipole radiation only emitted from the target nucleus. The inset shows 0 10 20 30

the pygmy resonance contribution to the emitted spectrum, only visible in the [fm [fm main ﬁgure as a slope change at low energies [99]. )

! 0.18 fm 0.21 fm R(

1 0.1

" 0.24 fm

! 0.26 fm of the pygmy dipole resonance (PDR). This is shown in Fig. F(E)/E 0.29 fm 42 where in the upper panel one sees the total energy spectrum (solid line) of emitted EM radiation, dE/d~ω, for 238U + 238 0 U collision at the impact parameter b = 12.2 fm. The total 0 5 10 15 20 25 quadrupole contribution is shown by a double-dotted line. The E! (MeV)[MeV] other curves are contributions from the three target-projectile orientations. In the bottom panel one sees the electric dipole radiation only emitted from the target nucleus. The inset shows the pygmy resonance contribution to the emitted spectrum, only Figure 43: Top: The energy weighted dipole response, as defined in Eq. (81) visible in the main figure as a slope change at low energies [99]. 68Ni computed with the RMF and a family of FSU interactions [197]. Bottom: As we have discussed previously the excitation of exotic The inverse energy weighted dipole response using the same interactions. nuclei by electrons is not possible within the next years. Theo- retically, the electric pygmy dipole resonance in electron scat- nuclei. For example, if we assume S = 100 keV, Ee = 10 tering has been studied at large angles at existing facilities, 3 MeV, ee f f = e, and µ = mN ∼ 10 MeV, we get σe = 25 mb. e.g. at the S-DALINAC in Darmstadt, Germany [365]. They The cross section also increases, despite very slowly, with the demonstrate that the excitation of pygmy resonance states in electron energy which brings no real advantage for high energy (e,e’) reactions is predominantly of transversal character for electrons. large scattering angles. They were also able to extract the fine structure of the pygmy states at low excitation energies. Elec- 3.6. Extracting dipole polarizabilities tron scattering with light and loosely bound nuclei could be As we discussed previously, the nuclear dipole polarizabil- the first sort of inelastic scattering studied experimentally if an ity α defined as in Eq. (84) is a useful quantity to constrain electron-radioactive-ion collider might become available [211, D the symmetry energy [191, 201]. The dipole polarizability are 215, 366]. In Ref. [367] that, for an electron energy E, the to- usually extracted from Coulomb excitation experiments. The tal cross section for the dissociation of a two-body cluster halo advantage of Coulomb excitation is that, for the E1 multipo- nuclei is √ 2 larity, the virtual photon numbers entering Eq. (140) have a ee f f Ee σ E π , nE ∼ ln(1/E) dependence with the excitation energy. To- e( e) = 64 2 2 ln (208) 1 µc S S gether with the 1/E factor this favors the low energy part of the where µ is the reduced mass and ee f f is the effective charge. spectrum thus enhancing the features of the pygmy resonances. This equation predicts an inverse separation energy dependence Therefore, Coulomb dissociation is nearly proportional to the which helps inelastic electron scattering for very loosely bound nuclear dipole polarizability at low energies. This is becomes

42 evident from Fig. (43), where on the left we show the energy 120 weighted dipole response, as deﬁned in Eq. (81) 68Ni computed with the RMF and a family of FSU interactions [197]. On the 110 right we see that the inverse energy weighted dipole response using the same interactions, entering in the calculation of the 100 c.c. dipole polarizability, Eq. (43), is largely enhanced at low energies. 90

The extract of the dipole polarizability and the exploration σ [mb] 80 1st order of their connection to the slope parameter is reported in, e.g., Refs. [201, 275, 368, 369, 370, 371, 372, 373, 374, 375, 376, 70 377, 378, 379, 380, 202, 203, 381]. As mentioned in relation to Fig. (22), a large range of values, of the order of 20-30%, 60 102 103 has been found in experiment studying αD. This uncertainty is not small enough to large to constrain most of the energy func- TLAB [MeV/nucleon] tionals stemming from Skyrme and relativistic models. Devel- opments on nuclear reaction theory are also necessary to obtain Figure 44: Coulomb excitation cross sections of the PDR in 68Ni projectiles the desired accuracy in the experimental analyses [382]. incident on 197Au targets as a function of the bombarding energy. The filled cir- In the Coulomb excitation of pygmy resonances there is a cles are calculations using first-order perturbation theory and the filled squares large excitation probability at small impact parameters, with a represent coupled-channel calculations. strong coupling between the pygmy and giant resonances. This leads to dynamical effects changing the transition probabilities and cross sections for the excitation of the PDR. Such effects In Figure 44 one sees the Coulomb excitation cross sec- have been observed in the case of the excitation of double gi- tions of the PDR in 68Ni projectiles incident on 197Au targets ant dipole resonances (DGDR) [207, 269, 263]. The DGDR is as a function of the bombarding energy. The filled circles are a consequence of higher-order effects in relativistic Coulomb calculations using first-order perturbation theory and the filled excitation due to the large excitation probabilities of giant reso- squares represent coupled-channel calculations. The deviation nances in heavy ion collisions at small impact parameters and a is clearly more pronounced at lower energies. Around 600 strong dynamical coupling between the usual giant resonances MeV/nucleon the PDR excitation cross section changes from and the DGDR arises [383]. A study of this effect in the excita- 80.9 mb obtained with the virtual photon method to 92.2 mb tion of the PDR using the relativistic coupled channels (RCC) obtained with the coupled-channels method. This implies that equations introduced in Eq. (156) was done in Ref. [382]. The the extracted PDR strength from the experimental data would resonances were described by Lorentzian functions centered have an appreciable change of 14% with a a reduction by nearly at the energies EPDR for pygmy dipole resonances and EGDR the same amount in the excitation strength. (EGQR) for the isovector (isoscalar) giant dipole (quadrupole) Coupled-channels calculations were also performed for the resonances. reaction 68Ni+208Pb at 503 MeV/nucleon, corresponding to the 68 197 208 The excitation of Ni on Au and Pb targets at 600 experiment of Ref. [381]. The Coulomb excitation cross sec- and 513 MeV/nucleon were investigated. These reactions have tion of the PDR in 58Ni were found to be 57.1 mb to first or- been experimentally investigated in Refs. [368, 381]. In the der, but including couplings to the giant resonances increase the 68 first experiment a pygmy dipole resonance in Ni was identi- cross section to 61.4 mb, a small but still relevant 7.5% correc- fied at EPDR ' 11 MeV with a width of ΓPDR ' 1 MeV, ex- tion. The value of the dipole polarizability αD extracted from hausting about 5% of the Thomas-Reiche-Kuhn (TRK) energy- the experiment in Ref. [381] is 3.40 fm3 while to reproduce the weighted sum rule. The identification was done by observing experimental cross section with our dynamical calculations we the excitation and decay via gamma emission. Another ex- 3 have αD = 3.27 fm , a small but non-negligible correction. If periment found that the PDR centroid energy was located at a linear relationship between the dipole polarizability and the 9.55 MeV, with a 2.8% fraction of the TRK sum rule, and a neutron skin is assumed [197], a reduction of the neutron skin width of 0.5 MeV. The PDR identification was done by observ- from 0.17 fm, as reported in Ref. [381], to 0.16 fm is expected. ing the neutron decay channel of the PDR. Ref. [382] studied This correction still lies within the experimental error [381]. the effects of the coupling between giant resonances and the Therefore, we conclude that due to the large Coulomb exci- PDR, on the excitation function dσ/dE. The centroid energy tation probabilities of giant resonances in heavy ion collisions EPDR = 11 MeV consistent with Refs. [368, 381] was used with at energies around and above 100 MeV/nucleon, the excitation a width at half maximum of 2 MeV, adopted from theoretical of the PDR is also appreciably modified due to the coupling be- calculations [350, 210, 220, 354, 384, 385, 386] and not from − + − tween the 1 and 2 states. In the future it might be possible to the experimental data [368, 381]. For the (isovector) 1 giant carry out nearly “ab-initio” calculations based on a microscopic dipole resonance (GDR) it was assumed EGDR = 17.2 MeV and theory, coupled with a proper reaction mechanism. A known al- + ΓGDR = 4.5 MeV and for the (isoscalar) 2 giant quadrupole ternative, already used in previous studies of multiphonon res- resonance (GQR) the values EGQR = 15.2 MeV and ΓGQR = 4.5 onances [207], is to use individual states calculated with the MeV were used. 43

number of nucleons in nucleus A(B), this probability becomes

proton target    AB  nucleon   n AB−n P (n, b) =   [T (b) σNN ] [1 − T (b) σNN ] . (211)   s n

• z b The ﬁrst term is the number of combinations for the occurrence • z’ projectile target of n collisions out of AB possible nucleon-nucleon encounters.

The second term is the probability of n collisions, while the last term is the probability of AB − n misses.

For nucleus-nucleus collisions, the thickness function T (b), can be related to the corresponding thickness function for nucleon- Figure 45: Collision geometry for a projectile proton hitting a target nucleon. nucleon collisions by means of Z T (b) = ρ (bB, zB) dbBdzB ρ (bA, zA) dbAdzA t (b − bA − bB) , (212) RPA or other microscopic together with higher order perturba- R tion theory. Advanced mean-ﬁeld time-dependent method such with T (b) db = 1. as that developed in Ref. [99] will also help clarify the reaction The total probability, or diﬀerential cross section, is given dynamics. by

dσ XAB 3.7. Extracting neutron skins from nucleus-nucleus collisions = P (n, b) = 1 − [1 − T (b) σ ]AB , (213) d2b NN 3.7.1. Removing nucleons from nuclei at high energies n=1 Previously, we have discussed several ways to extract the with the total nucleus-nucleus cross section given by neutron skin of nuclei and correlate it to the EoS in neutron stars. Here we will describe a method to extract it from mea- Z 2 n ABo surements of fragmentation reactions. The method is based on σ = d b 1 − [1 − T (b) σNN ] . (214) the theory described in Ref. [235, 387]. We will treat all nucleons in the same way irrespective if they are protons or neutrons. The quantity Latter we show how this can be easily extended to discern pro- 2 AB tons from neutrons. |S (b)| = [1 − T (b) σNN ] (215) Let us assume, as shown in Fig. 45, that a nucleus-nucleus collision occurs at an impact parameter b. The probability of is the square of the scattering matrix, for reasons that become having a nucleon-nucleon collision within the transverse ele- clear when one derives this equation using the eikonal approx- ment area db is deﬁned as t (b) db, where t (b) is known as the imation. In the optical limit of the eikonal approximation, de- thickness function. It is normalized so that scribed previously, where a nucleon of projectile undergoes only one collision in the target nucleus, Z t (b) db = 1. (209) 2 |S (b)| ' exp [−ABT (b) σNN ] (216)

For unpolarized projectiles t (b) = t (b). Frequently, one uses Note that if the densities are normalized to the number of nucle- t (b) ' δ (b), which simplifies the calculations considerably. ons, instead of Eq. (210), then the equation above is exactly the Since the total transverse area for nucleon-nucleon colli- same used to obtain the “survival probability” in the “optical sions is given by the nucleon-nucleon cross section σNN , the limit of the Glauber theory”. Evidently, Eq. (214) is superior probability of an inelastic nucleon-nucleon collision occurring than using the approximation (216). Eq. (216) is derived in within this area is t (b) σNN . The probability of finding a nu- terms of pure statistical theory, while Eq. (216) is an approxi- cleon in dbBdzB is given by ρ (bB, zB) dbBdzB, where the nu- mation when T (b) σNN 1. clear density is normalized to unity: The individual thickness functions for the nucleus A can be Z defined as Z ρ (bB, zB) dbBdzB = 1 . (210) TA (bA) = ρ (bA, zA) dzA , (217)

and similarly for the nucleus B. Then we can rewrite Using these deﬁnitions, it is easy to derive [235] the probability of occurrence of n nucleon-nucleon collisions in a nucleus- Z nucleus collision at impact parameter b. Denoting by A(B) the T (b) = dbAdbB TA (bA) TB (bB) t (b − bA − bB) . (218)

The probability of n nucleons in A colliding with m nucle-

44 ons in B, using t (b) ' δ (b) becomes [235] that differences in proton and neutron scattering are smallest at     such energies. Such differences will be more evident at lower  A   B  energies where the proton-neutron cross section is about three     n B−n P (n, m, bA, bB) =     [TB (bB) σNN ] [1 − TB (bB) σNN ] times the proton-proton and neutron-neutron cross sections.  m   n  As shown in Refs. [273, 391, 392], the Pauli blocking pro- m A−m × [TA (|b − bB|) σNN ] [1 − TA (|b − bB|) σNN ] . jection yields an average nucleon-nucleon cross section for two (219) Fermi gases with relative momenta k0 given by Z d3k d3k 2q Ω The “abrasion-ablation model” for nuclear fragmentation σ (k, ρ , ρ ) = 1 2 σ f ree(q) Pauli , NN 1 2 3 3 k NN 4π [388] is based on this equation. This model can also be ex- (4πkF1/3)(4πkF2/3) 0 tended to account for the isospin dependence of the nucleon- (223) nucleon collisions, as we show below. One can interpret m as the number of holes created in the nucleon orbitals in the target. where 2q = k1 − k2 − k0. The momentum of a single nucleon is Ablation refers to the decay phase of the nuclei after the “holes” denoted by ki. Pauli-blocking enters through the restriction that 0 0 are created in the nucleon orbitals. The equations above can the magnitude of the final nucleon momenta, |k 1| and |k 2|, lie also be deduced quantum-mechanically using the eikonal ap- outside the Fermi spheres, with radii, kF1 and kF2. This leads to proximation [235]. The derivation presented here is much sim- a limited fraction of the solid angle into which the nucleons can pler and only uses classical probability concepts. This model scatter, ΩPauli. has been shown to describe rather well the fragment yields in The numerical calculations can be simplified if we assume relativistic heavy ion collisions (see, for example, [389, 390]). that the free nucleon-nucleon cross section entering Eq. (223) is isotropic [273, 391, 392]. This is a rough approximation be- 3.7.2. Isospin dependence cause the anisotropy of the free NN cross section is markedly P T manifest at large energies. In the isotropic case, a formula We assume again that ρp and ρn are the projectile and target proton and neutron densities, normalized so that which fits the numerical integration in Eq. (223) is [273, 391, 392] Z Z d3rρT (r) = 1, and d3rρT (r) = 1, (220) 1 p n σ (E, ρ , ρ ) = σ f ree(E) NN 1 2 NN !2.75 |ρ1 − ρ2| σ σ 1 + 1.892 while pp and pn are the total (minus Coulomb) proton-proton ρρ˜ 0 and proton-neutron scattering cross sections, respectively.  2/3  37.02˜ρ 2/3 The cross section for the production of a fragment (Z, N)  1 − , if E > 46.27˜ρ  E from a projectile (ZP, NP) due to nucleon-nucleon collisions is  ×  (224) given by   E 2/3      , if E ≤ 46.27˜ρ  231.38˜ρ2/3  ZP   NP  R h iZP−Z     2 − σ(NP, ZP; N, Z) =     d b 1 Pp(b)  Z   N  where E is the laboratory energy in MeV,ρ ˜ = (ρ1 + ρ2)/ρ0, −3 with ρ0 = 0.17 fm , and ρi(r) is the local density at position r ×PZ b − P b NP−N PN b , p( ) [1 n( )] n ( ) (221) within nucleus i. where b is the collision impact parameter. The binomial coefficients in front of the integral account for all combinations 3.7.3. Ablation model for neutron removal selecting Z protons out of the ZP projectile protons, and simi- The differential yield of fragments in which N neutrons are larly for the neutrons [235, 387]. The probabilities for single removed from the projectile is, nucleon survival are denoted by Pp for protons and Pn for neu- X 0 ∗ 0 trons. The probability that a projectile proton does not collide σ−N = ω(NP, ZP, N , E ; N)σ(NP, ZP; N , ZP) (225) 0 with target nucleons is given by [235, 387] N >N 0 ∗ R h R where ω(NP, ZP, N , E ; N) is the probability that the nucleus P b dzd2 sρP s, z −σ Z d2 sρT b − s, z 0 ∗ p( ) = p ( ) exp pp T p ( ) (NP, ZP) with N neutrons removed and excitation energy E , R 2 T i ends up with N neutrons after evaporation. − σpnNT d sρ (b − s, z) , (222) n The removal of N0 neutrons in the first stage leads to “pri- where σpp and σnp are the proton-proton (Coulomb removed) mary yields”. After evaporation the nucleus is left with Nx neu- and proton-neutron total cross sections. A fit of experimental trons, called by “secondary yields”. It is the secondary yields data in the energy range of 10 to 5000 MeV is often used, as in which are usually measured in experiments. Eqs. (138) and (139) (see top panel of Fig. 25). If evaporation is neglected (primary yields), the total cross The primary yields in this model depend on the incident section for neutron removal with all protons remaining is given energy only through the nucleon-nucleon cross sections in the absorption factors. At high energies E/A > 100 MeV/nucleon), these cross sections have nearly the same value. One expects 45

DD2+++ DD2+++ 0.35 DD2- - DD2- - 550 0.3 (fm) (mb) np

0.25 N r Figure 46: Schematic view of fragment production in nucleus-nucleus colli- ∆ 500 ∆ sions at high energies. σ 0.2

450 by 0.15 Z n o ZP survives 2 ZP − NP σ∆N = σall n decay channels = d b[Pp(b)] 1 [Pn(b)] . 125 130 135 125 130 135 A A (226) This means that the probability that ZP protons survive while all Figure 47: Neutron-skin thickness ∆rnp (left) and the related neutron-removal possible neutron removal occurs is equal to the probability that cross sections σ∆N (right) for Sn isotopes predicted by RMF models based all protons survive (irrespective to what happens to any neutron) on the use of the DD2 interaction [395, 396]. The slope parameter L varies −− minus the probability that all protons and neutrons survive, si- between 25 MeV (for the DD2 in interaction ) and 100 MeV (for DD2+++) multaneously. [396]. Following the same procedure as above, we can obtain a general formula for the the total reaction cross section [235] of 2.478(9) fm [394]. We proton and neutron densities was as- Z sumed to be the same. h iZP 2 NP σR = d b 1 − Pp(b) [Pn(b)] . (227) To estimate the sensitivity of σ∆N with variations of ∆rnp and L, cross sections were calculated using theoretical density The reaction cross section is then due to the probability that distributions obtained with RMF models, as the DD2 interac- anything happens, i.e., the unity, minus the probability that all tion developed in Ref. [395] The slope parameter L was sys- protons and neutrons survive. tematically varied to optimize the isovector parameters reproducing nuclear properties such as masses and radii [396]. The For spherical nuclei, the probabilities Pn(b) and Pp(b) do not depend on the direction of the impact parameter. For de- same procedure was used for the DD interaction [397]. The formed nuclei, the calculation is much more complicated, as left panel in Fig. 47 displays the predicted neutron-skin thick- the orientation of the nuclei have to be taken into account and nesses for tin isotopes. In these calculations, different inter- −− properly averaged. actions were used ranging from L values of 25 MeV (DD2 ) +++ It is clear from Eq. (225) that the calculation is not as to 100 MeV (DD2 ) which accordingly also predict different 132 simple as shown in the previous sub-section. Also note that values of ∆rnp ranging from 0.15 to 0.34 fm in Sn. This PN 0 ∗ results in a corresponding variation of σR from about 2550 to N0=1 ω(NP, ZP, N , E ; N) , 1, because the nucleus can end up with N0 > N and also proton evaporation can occur although 2610 mb, i.e., a 2.5% change. The most sensitive quantity to with a much smaller probability due to the Coulomb barrier. ∆rnp is σ∆N is shown in the right frame of Fig.47. A variation 132 However, if after one, or multiple, neutron-knockout all de- within 460 and 540 mb is clearly visible for Sn, i.e., a cross cay fragments of the same element are detected, then evapora- section change of almost 20%. One concludes that σ∆N has a tion is not relevant in Eq. (225), except for a small uncertainty larger potential to constrain L and is less sensitive to reaction due to proton, γ, α, emission probabilities. Then one can use theory uncertainties. Eq. (226), which is purely geometric and only depends on the Fig. 48 shows the correlation of the value of L obtained 124 132 densities. using the DD2 interaction and ∆rnp for Sn and Sn. A variation of L by ±5 MeV changes the calculated neutron skin in 3.7.4. Total neutron removal cross sections as a probe of the 124Sn by about ±0.01 fm. The same variation in L yields a mod- neutron skin ification of σ∆N by about ±5 mb, i.e., of only ±1%. Therefore, a In Ref. [393] the total neutron removal cross sections has determination of σ∆N within a 1% accuracy in a combination of been proposed as a probe of the neutron skin in nuclei, as shown experiment and theory can be reached to constrain L via a com- schematically in Fig. 46. The exploratory study chose the parison with DFT. The scatter of results for various relativistic neutron-rich part of the tin isotopic chain and specifically the re- and non-relativistic models predicting a given L on σ∆N is ex- actions Sn+12C. The density of 12C was obtained from a model- pected to be similar to the case of the scatter of ∆rnp analyzed independent elastic scattering analysis up to large momentum in Ref. [398], namely, about 10 MeV in the determination of transfers, q2, using the Fourier Bessel expansion [394] and ex- L. The dependence of the cross section on the slope parameter 132 trapolated by using a Whittaker function at very large radii. L is steeper in the case of the more neutron-rich nucleus Sn, The rms radius of 12C was taken as the published best value thus providing an even higher sensitivity. 46

deﬁned as the sum of the two processes, σI = σR + σinel. As 540 124 Sn we discuss later σ contains a nuclear and an electromagnetic 132 inel 520 Sn contribution as well as their interference. The contribution of 12 12 500 σinel will be shown to b e about 1% for C + C at 500-1000 132 12

(mb) MeV/nucleon, but it can attain 100 mb for Sn+ C. This cor- N ∆ 480 responds to 4% or 20% of σ or σ , respectively. Since it σ I ∆N 460 is the neutron-removal cross section which provides most sensitivity to the neutron skin, this additional channel has to be 440 known within less than 5% to achieve the required constraint 0.350 20 40 60 80 100 120 on L. This seems impossible to achieve with reaction theory. 124Sn L (MeV) But it is possible using state-of-the-art kinematically complete 0.3 132Sn experimental measurements to separate the nuclear excitation contribution and to determine its cross section. In fact, the 0.25 angular distributions of the neutrons are very diﬀerent for the (fm)

np two processes: evaporated neutrons (typically with energies of r ∆ 0.2 2 MeV in the projectile rest frame) are kinematically boosted to the forward direction at high bombarding energies and can be 0.15 well detected with nearly the beam velocity and a typical angular distribution covering the angular range of 0 to 5◦, while 0 20 40 60 80 100 120 neutrons stemming from binary nucleon-nucleon collisions dis- L (MeV) play a broader angular distribution scattered within 0 and 90◦ and a maximum around 45◦. The expected overlap region be- Figure 48: Relation between σ∆N (top) and ∆rnp (bottom) with the slope parameter L based on RMF calculations for 124Sn and 132Sn. The lines indicate tween the two processes is thus negligible. the sensitivity of the observables with L for a variation of L within 10 MeV. Base on the discussion above the primary process of binary nucleon-nucleon collisions remains the only significant hurdle to relate σ∆N with ∆rnp or L. This step was investi- As we stressed before, nuclear fragmentation in high-energy gated in Ref. [393] for the symmetric system 12C + 12C, com- collisions is often modeled via two completely disconnected as- pared to the available experimental information on σR, using sumptions: (a) the production of primary fragments via multi- the eikonal scattering theory as described previously. In this nucleon knockout in binary nucleon-nucleon collisions (as we model, the known free nucleon-nucleon cross sections and the described previously), followed by (b) the production of sec- nuclear densities are only input. The total reaction cross secondary fragments due to nucleon evaporation from the energy tion as a function of the laboratory energy are displayed by deposited in the primary fragments. The second step, using the black triangles in the lower frame of Fig. 28. The calculated Hauser-Feshbach theory of compound-nucleus decay, is highly cross sections are larger than the experimental data for ener- model dependent. The method discussed in Ref. [393] does not gies above 200 MeV/nucleon. In-medium effects are expected need to consider the nuclear evaporation step, because the to- to be responsible for the deviations at high beam energies. A tal neutron and charge removal cross sections include the com- large fraction of this effect can be attributed to Pauli block- pleteness of the sum over all possible decay channels. Also, ing, calculated as in Ref. [256], which when included yield the proton or charged-particle evaporation is negligible in the cases red diamonds. However, the high-energy data point at about 124 of Sn and heavier tin isotopes considered in Ref. [393]. For 950 MeV/nucleon is still overestimated, although by only 2%. a typical case, one obtains σ∆N = 485.6 mb for the produc- Below 400 MeV/nucleon, the data start to deviate strongly from tion of primary fragments in the reaction of 580 MeV/nucleon the calculation, probably due to the failure of the method be- 124 12 Sn incident on C. The same cross section calculated after low these energies. In Ref. [251], the effect of Fermi motion the CN evaporation stage, with standard parameters used in the was shown to be important at low energies and to increase the Hauser-Feshbach formalism, yields σ∆N = 483.4 mb. That is, cross sections. In the most relevant energy region (400 − 1200 less than 0.5% of the neutron-removal cross section is moved MeV/nucleon) there are only three data points, and no other to the charge-changing cross section after the primary produc- data in the important region of 400 to 800 MeV/nucleon, where tion stage. For more neutron-rich tin isotopes, this contribution the cross section shows an increase with bombarding energy. will be even smaller. Modifications of the input parameters in Deviations from the eikonal approximation, such as in-medium Hauser-Feshbach calculations are by no means enough to in- and higher-order effects certainly depend on the beam energy, crease this effect appreciably. and high-precision data covering this energy region with less In addition to fragmentation processes induced by nucleon- than 1% accuracy are thus of utmost urgency for a more strin- nucleon collisions, the projectile can emit a nucleon after an gent test of reaction cross sections and further developments in inelastic excitation to collective states in the continuum of the reaction theory. They will also be used for a better the quantifi- projectile, such as giant resonances. For heavy neutron-rich nu- cation of the uncertainties in neutron-changing ceros sections. clei, this process adds almost exclusively to the neutron-removal Further sensitivity on the neutron skin can be obtained by channel and to the total interaction cross section. The later is changing the target. The np and pp cross sections have a very 47

0.9 45 ∆ 0.8 N DD2- - 40 0.7 35 Pb C) +++

12 DD2 ] ( 30 σ

0.6 b

(p)/ 25 m σ 0.5 [

0.4 C Sn ∆Z σ 15 0.3 10 Ni 400 500 600 700 800 5 E (MeV/nucleon) 0 0.2 0.1 0.0 0.1 0.2 0.3 Figure 49: Bombarding energy dependence of the ratios of σR, σ∆Z and σ∆N cross sections for 134Sn projectiles incident on proton and 12C targets. (N-Z)/A

Figure 50: Cross sections in milibarns for the Coulomb excitation of the different energy dependence as seen in Fig. 28. One thus ex- IVGDR in nickel, tin and lead projectiles incident on carbon targets at 1 pects that a corresponding change in the ratio of neutron-removal GeV/nucleon, as a function of the asymmetry coefficient δ = (N − Z)/A of to charge-changing cross sections exists as a function of bom- the projectile. barding energy. This should be most pronounced in the case of a proton target as the proton probes the neutron skin ex- efficient δ = (N −Z)/A of the projectile is shown. The Coulomb clusively via pn collisions, whereas charge-changing reactions cross section shows little dependence on the neutron skin. The are exclusively related to pp collisions. Other additional subtle neutron skin only enters at small impact parameters entering effects such as the proton passing across the nucleus without the imaginary phase χ (b) in Eq. (144). The cross section de- knocking out another nucleon adds a difference to the use of OL pends on the asymmetry coefficient δ, mainly through the iso- 12C targets that probes only the nuclear surface. This becomes topic dependence of the photo-nuclear cross section as shown evident in Fig. 49 where we show the ratios of σ , σ and R ∆Z in Eq. (153). The mass dependence of the resonance centroid σ for 134Sn bombarding proton targets compared with those ∆N energy also has an influence on the virtual photon numbers n on 12C targets. While we do not see an energy dependence E1 because of their fast decrease with the excitation energy. for σ (p)/σ (12C) ratio, the charge-changing σ (p)/σ (12C) R R ∆Z ∆Z As displayed in Fig. 50, the Coulomb cross sections in- and σ (proton)/σ (12C) ratios clearly show an energy de- ∆N ∆N crease linearly with (N − Z)/A. They are very small for nickel pendence. This energy dependence and the fact that the tar- and tin isotopes and negligible compared to the neutron chang- get ratio is significantly larger for σ is due to the strong en- ∆N ing and interaction cross sections. But the cross sections for ergy dependence of the pp cross section (as shown in Fig. 28) lead projectiles are not negligible for carbon targets. The Coulomb leading to a substantial proton survival probability when proton cross sections are proportional to the square of the target charge targets are used around 400 MeV/nucleon and therefore yield- yielding for proton targets cross sections that are smaller than ing a larger σ . This effect becomes visibly smaller at 800 ∆N those for carbon targets by a factor 30−40. By means of a com- MeV/nucleon and above. Therefore, the energy dependence of parison of experimental results obtained with carbon and pro- the σ target ratio provides an additional sensitivity test of the ∆N ton targets one can easily separate the contributions of Coulomb reaction theory, if experimentally obtained with accuracy. Both cross sections to the fragmentation [108]. There is a large vari- ratios for σ and σ display negligible dependence on the neu- R ∆Z ation of the Coulomb cross sections with bombarding energy tron skin whereas the ratio for σ displays a much stronger ∆N [207] which can also help disentangling their contribution from dependence on ∆r as evidenced by using the DD2+++ and np nuclear excitation. DD2−− RMF interactions. Since the rms radii of the charge distribution in the nuclei are known, the charge-changing cross 3.7.6. Nuclear excitation followed by neutron emission sections obtained with proton and carbon targets by varying the bombarding energy will serve as a crucial test on the accuracy Using the deformed potential model, described in section of the calculated cross sections. 3.4.2 to obtain the cross sections for nuclear excitations, we show in Figure 51 the cross sections for nuclear excitation of IS- 3.7.5. Coulomb excitation followed by neutron emission GQR (GQR) and IVGDR (GDR) resonances in nickel, tin and lead projectiles incident on carbon targets at 1 GeV/nucleon. We now extend the discussion of “small” corrections to the The dependence on the asymmetry coefficient δ = (N − Z)/A cross sections σ and σ . We first discuss Coulomb excitation ∆N I of the projectile is displayed. The upper curves in each frame of giant resonances followed by neutron emission. In Figure are calculations for the ISGQR and the lower curves are for the 50 we plot the results for the excitation cross sections of the excitation of IVGDR multiplied by a factor of 20. For 208Pb IVGDR in nickel, tin and lead projectiles incident on carbon tar- projectiles the cross sections are of the order of 43 mb (1.11) gets at 1 GeV/nucleon. The dependence on the asymmetry co- 48 mb, for the ISGQR (IVGDR). 150 The IVGDR cross sections are negligible for N = Z with ] a negligible neutron skin. Light nickel isotopes exhibit a non- b GQR

m 100 [ zero proton skin and a reversing trend of the IVGDR excitation Ni

cross section around δ = 0 is observed. The cross sections for N 50 GDR 20

σ × IVGDR resonances are bemire than a factor 20 smaller than for ISGQR resonances. Hence, they are unimportant for the 0 0.2 0.1 0.0 0.1 0.2 0.3 purposes of extracting neutron skins at such bombarding energies. Nonetheless, the method has been used previously at 80 lower energies, below 100 MeV/nucleon, by measuring diﬀer- ] ential cross sections that can display marked diﬀerences be- b 60 m [ tween angular distributions for L = 1 and L = 2. The energy 40 Sn

dependence of the cross sections at these energies has also been N

σ 20 used as a tool [246]. 0 The ISGQR cross sections decrease along an isotopic chain 0.00 0.05 0.10 0.15 0.20 0.25 as the neutron numbers increase. This can be understood as due to the decrease of the deformation parameter δ2 with the 50 increase of the ISGQR centroid energy with mass number, as ] inferred from Eq. (185). b 40 m

[ 30

Larger theoretical uncertainties in treating nuclear excita- Pb 20 tions in high energy collisions exist as compared to Coulomb N excitation. Whereas the Coulomb interaction is well known, σ 10 optical potentials used in the deformed potential model, Eq. 0 0.10 0.15 0.20 0.25 (185), are not so constrained. Little can be done to improve these models with the state of the art knowledge of high energy (N-Z)/A nuclear reactions. The deformed potential and the Tassie model [399] are should be taken as rough approximations for nuclear Figure 51: Nuclear excitation cross sections of ISGQR (GQR) and IVGDR reactions at high energies. More theoretical efforts are certainly (GDR) resonances in nickel, tin and lead projectiles bombarding carbon targets needed. at 1 GeV/nucleon. The dependence on the asymmetry coefficient (N − Z)/A of the projectile is shown. The upper curves in each frame display the excitation The deformed potential model used in collisions with a proof ISGQR and the lower ones are calculations for the excitation of IVGDR ton target yields cross sections that similar to those shown in multiplied by 20. Fig. 51. Use different targets does not display appreciable changes in the nuclear excitation of giant resonances. A notice- able variation of the Coulomb excitation is possible. By varying on the neutron number with a linear dependence with the neu- the bombarding energies in the range 100-1000 MeV/nucleon tron number for a given Skyrme interaction. This seems to be is will not help either because the cross sections for nuclear ex- a robust result that can be employed in the experimental anal- 208 citation remain practically unchanged. Thus, the 50 mb to 100 ysis. For Pb, the heaviest stable lead isotope, the cross sec- mb of nuclear excitation cross sections mainly contributing to tions range from 537 to 576 mb, an approximate 7% variation the one-neutron decay channel will be hard to control systemat- with the choice of the Skyrme interaction. Therefore, it seems ically without adding other observables to the angle integrated that neutron-changing cross sections can constrain the several cross sections. But, as discussed previously [393], simulations Skyrme models by comparison with calculations. The linear show that nuclear excitation events can be separated using the relation between σ∆N and the neutron number is also a feature angular distribution of neutrons. worthwhile to explore. The same calculations are displayed in Figure 53 now as a 3.7.7. Neutron changing and interaction cross sections function of the neutron skin in the various isotopes. A row of Figure 52 displays neutron-changing cross sections, follow- vertical points correspond to different isotopes and each curve ing the model described in section 3.4.2, for nickel (upper frame) along an isotopic chain is obtained with a single Skyrme inter- and lead (lower frame) isotopes and for several Skyrme interaction. Because ∆rnp and the neutron number show a strong actions, as a function of the neutron number. Nickel isotopes correlation (see Figure 18), no additional information is gained display a very small dependence on the neutron number with a as compared to Figure 52. But such dependencies can be used given Skyrme interaction. The nickel radius is not much larger to deduce the accuracy needed to obtain a given value of ∆rnp. that for carbon and the geometric variation along the isotopic For example, a neutron skin of 0.15 fm in Ni and Pb isotopes, chain with a given Skyrme interaction do not lead to a sizable yield cross sections within the range of 0.32 to 0.42 b and 0.47 cross section variation. For 64Ni, the heaviest stable nickel iso- to 0.52 b, respectively. They correspond to sensitivities of the tope, the cross sections range from 337 to 350 mb, yielding a neutron skin with the choice of Skyrme interactions of 20% for 4% sensitivity on the choice of the Skyrme interaction. How- nickel and 10% for lead isotopes. Plots like Figure 53 are a ever, for lead the cross sections show a very strong dependence combination of the theoretical predictions shown in Figures 18 49 0.5 0.5

] 0.4 0.4 ] b b [

Ni Ni 0.3 [ 0.3

N N ∆ 0.2 ∆ σ 0.2 0.1 σ 0.1 0.0 20 25 30 35 40 45 0.2 0.1 0.0 0.1 0.2 0.65 0.60 0.60 ]

b 0.55 Pb ] [

0.55 b N

0.50 [

∆ Pb σ 0.45 N 0.50 ∆

0.40 σ 0.45 100 110 120 130 140 0.40 N 0.00 0.05 0.10 0.15 0.20 ∆r [fm] Figure 52: Neutron-changing cross sections in barns following the model of np section 3.4.2, for nickel (upper frame) and lead (lower frame) isotopes and the 23 Skyrme interactions [116]. The dependence on the neutron number is Figure 53: Same as in Figure (52) but plotted as a function of the neutron skin shown. The lines are drawn to guide the eyes and each one of them represents ∆r for different isotopes and Skyrme interactions. The lines are drawn to a prediction with one of the Skyrme interactions along an isotopic chain. np simply guide the eyes and are the prediction for each Skyrme interaction along an isotopic chain. and 52. They are useful if a large number of projectile isotopes can be tested with experiment. ter constraint on the proper Skyrme interaction that reproduces Fig. 54 shows the total interaction cross sections for tin experimental data. Both neutron changing and total interaction isotopes bombarding carbon targets at 1 GeV/nucleon. Calcu- cross sections will help constraining these interactions and the lations follow Eq. (221), and use the same Skyrme interactions EoS of symmetric and asymmetric nuclear matter. as above. The upper frame displays the calculations as a function of the neutron number N, while the lower frame shows the 3.8. Polarized protons and the neutron skin same data in terms of the neutron skin ∆rnp. The cross sections The spin-orbit interaction is one of the most celebrated find- change negligibly by varying the Skyrme interaction used for ings in our quest to understand the nature of nuclei. In 1936, a given isotope. The reason is that for a given isotope all in- Inglis [400] investigated the microscopic origin of the nuclear teractions yield essentially the same total matter density. Also, spin-orbit interaction by adding to nuclear forces a copycat of similar values for neutron skins are obtained for different iso- the atomic spin-orbit coupling, also known as the relativistic topes using two or more Skyrme interactions. This is clear from Thomas effect [401]. Years later, the phenomenological inclu- the lower frame of Figure 54 where a much larger change of σI sion of the spin-orbit interaction in the nuclear shell model, to- with ∆rnp is observed. Hence, a systematic study of measure- gether with the Pauli principle, allowed an astonishingly simple ments of neutron-changing and interaction cross sections will explanation of the magic numbers appearing in nuclear energy be useful to test theoretical predictions of nuclear densities. spectra. The interaction was assumed to be much larger than As a final remark, we show in Fig. 55 the ratio of neutron the relativistic effect proposed by Inglis. This achievement had changing cross sections, σ∆N , for 1 GeV/nucleon tin isotopes a large impact in our understanding of nuclear systems and lead obtained with carbon and proton targets. The dependence on to a Nobel prize for Mayer and Jensen in 1950 [402, 403]. The the neutron skin, ∆rnp, is shown. The set of points along each microscopic origins of the nucleon-nucleus spin-orbit force are curve correspond to a single Skyrme interaction. In the lower now explained in terms of a quantum field description of σ and frame we show the same ratio, but this time for the total in- ω meson exchange. teraction cross sections, σI. It is visible in the figure that the The spin-orbit interaction is of fundamental importance to cross sections with proton targets have a steeper variation with explain basic phenomena observed in atomic and nuclear colli- the neutron skin than those obtained with carbon targets. This sions. In nuclear physics, the simplest of all collisional cases, feature is better seen in the ratio of interaction cross sections. namely, elastic collision differential cross sections, display in- Thus, by using both carbon and proton targets allows for a bet- terference of polarized protons scattering through the near side 50 3.2 1.60 ) p

3.1 (

N Sn ] 3.0 1.55 ∆ b [ σ

2.9 / I ) 1.50 σ 2.8 Sn C 2

2.7 1 ( 1.45 N

50 60 70 80 90 ∆ σ N 1.40 3.2

3.1 )

p 2.25 ] 3.0 ( I b Sn [ σ

2.9 / I

) 2.20

σ 2.8 Sn C 2

2.7 1 (

I 2.15 0.1 0.0 0.1 0.2 0.3 0.4 σ ∆r [fm] 2.10 np 0.1 0.0 0.1 0.2 0.3 0.4 ∆r [fm] Figure 54: Dependence on several Skyrme interactions of the total interaction np cross sections for tin isotopes bombarding carbon targets at 1 GeV/nucleon, using Eq. (221). The results in the upper frame are displayed as a function of Figure 55: Top frame: Ratio of neutron changing cross sections, σ∆N , for 1 the neutron number N, while the lower frame displays the same data in terms GeV/nucleon tin isotopes obtained with carbon and proton targets. The depen- of the neutron skin ∆rnp. dence on the neutron skin, ∆rnp, is shown. The set of points along each curve correspond to a single Skyrme interaction. Bottom frame: Same ratio as in defined in the upper frame, but this time for the total interaction cross sections, and the far side of the nucleus. This interference pattern can be σI . explained in terms of the opposite signs of the s · L spin-orbit term due to the angular momentum flip (see, e.g., Ref. [226]) experiments have been concentrated on using quasi-free scat- in changing from the near to the far side. Evidently, other types tering (QFS) in inverse kinematics as a tool to assess the shell- of direct collisions using polarized protons are also influenced evolution in neutron-rich nuclei. Problems such as quenching by the strength of the spin-orbit force and serve as a probe of spectroscopy factors and single-particle properties of neutron- of its modification in the nuclear medium. For example, one rich nuclei have been explored. Recent theoretical work on has speculated modifications of nucleon and meson masses and (p,2p) reactions have also been reported [414, 416, 417, 418]. sizes, and also of meson-nucleon coupling constants in nuclear The induced polarization due to a combination of absorp- medium, motivated by strong relativistic nuclear fields in the tion and spin is known as the “Maris effect” [404, 405, 406]. medium, deconfinement of quarks, and also partial chiral sym- The idea is rather simple and invokes a combination of absorp- metry restoration [407, 408, 409, 410, 411]. A density depen- tion and spin-orbit interaction. Suppose that the primary polar- dence of the nucleon-nucleon interaction is obviously expected, ized proton is detected on the large-angle side of the momentum modifying the expectations for nucleon induced reactions based transfer q. Proton initial momenta directed toward the large- on bare interactions. angle side of q, correspond to spin-up protons with j = l − 1/2 High-energy radioactive beams, and in particular quasifree on the near side and to j = l + 1/2 on the far side. Initially po- (p,2p) and (p,pn) reactions have resurfaced as standard experi- larized nucleons knocked out from the near side will undergo mental tools to investigate nuclear spectroscopy. New and more less attenuation on their way out than those from the far side. efficient detectors have allowed accurate experiments using in- Therefore they are less polarized, P < 0, for j − 1/2. The verse kinematics with hydrogen targets and have also opened N reverse is true, and P > 0, for j = l + 1/2. The resulting net new possibilities for studies of the single-particle structure and N polarization of the knocked out nucleons when summed over nucleon-nucleon correlations in nuclei as the neutron-to-proton their subshells would vanish for a closed-shell nucleus if the ratio in the projectile increases. The detection of all outgoing subshell momentum distributions were identical and if the NN particles has provided kinematically complete measurements of interaction were spin-independent. But they are not and will reactions studied at the GSI/Germany, RIKEN/Japan, and other cause differences in P(near) and P( f ar). Therefore, one expect nuclear physics laboratories. First experiments using (p,2p) and N N that due to absorption and the spin-orbit part of the optical po- (p,pn) with newly developed experimental techniques have al- tential, Maris polarization is approximately twice as large for ready been reported with success [412, 413, 414, 415]. These 1p1/2 as for 1p3/2 and also opposite in sign. The net polariza- 51 40% when the matter density is about 50% of the saturation near side p2 density. These expectations have been verified experimentally p1 L . S < 0 [421]. p 0 θ Before we assess the importance of Maris polarization to probe asymmetric nuclear matter using neutron-rich projectiles, p2 we discuss how well existing experimental data can be under- (a) p stood with our calculations based on a standard theory of quasi- 1 free reactions. The triple differential cross sections for QFS in p0 θ L . S > 0 the Distorted Wave Impulse Approximation (DWIA) is given by [422] far side 3 d σ 2 = C S · KF dΩ1dΩ2dT1 2 (−) (−) (+) × χσ k χσ k τpN χσ k ψ jlm , (228) 2 p2 1 1 0 0

where KF is a kinematic factor, p0 (p1) denotes the incoming (outgoing) proton, p2 the knocked-out nucleon, T2 its en- (b) ergy, C2S is the spectroscopic factor associated with the single- particle properties of p2 in the nucleus and ψ jlm is the nucleon wavefunction, which in the na¨ıve single-particle model is la- belled by the jlm quantum numbers. χ denote distorted σkp scattering waves for the reaction channel with spin σ and momentum p). The DWIA matrix element includes the scatter- Figure 56: (a): a proton with spin up knocks out a nucleon (proton or neutron) ing waves for the incoming and outgoing nucleons, and the in- with spin up. The near and the far side scattering have opposite signs in the spin-orbit part to the optical potential. Near and far side scattering also yields a formation on their spins and momenta, (σk), as well and the shorter or a longer scattering path within the nucleus, changing the absorption t-matrix for the nucleon-nucleon scattering. In first-order per- of the scattered wave and its interference. (b): averaging the collisions of the turbation theory this t-matrix is directly proportional to the NN incoming proton with nucleons within closed subshell tends to keep the initial interaction. For unpolarized protons, Eq. (228) has to be aver- proton polarization. But a net depolarization of the incoming proton will occur due to the spin-dependent part of the NN-interaction. This depolarization effect aged over initial spin orientations besides a sum over final spin will increase as the number of nucleons in the closed subshell also increases. orientations. This formalism has been used previously in sev- The final polarization of the scattered proton will be sensitive to a combined eral calculations and a good description of experimental data effect of the interference between the near and far side paths due to their different absorption attenuation, the spin-orbit parts of the optical potential, and the has been achieved with proper choices of optical potential and number of nucleons in the subshell [404, 405, 406]. the nucleon-nucleon interaction (see, e.g., Refs. [423, 424]). Ref. [414] reported that momentum distributions of the residual nuclei obtained in quasi-free scattering are well described tion of the knocked-out nucleon can be observed using polar- using the eikonal wave functions χki in Eq. (228). The Maris ized proton targets and exploiting the difference between the effect has mostly been studied using the partial wave method (spin-up)-(spin-up) and (spin-down)-(spin-up) cross sections in [404, 405, 406, 419, 420, 421, 422, 423, 424]. triplet and singlet scattering, respectively [404, 405, 406]. Figure 57, taken from Ref. [426] shows the cross sections (p,2p) reactions are thought to be simpler than the elastic 40 39 for Ca(p,2p) K and Ep = 148 MeV,as a function of the recoil nuclear scattering mentioned above. This statement is based momentum, pA−1 of the the residual nucleus. The proton knock- on the argument that in elastic scattering one deals with the 40 out is assumed to be from the 1d3/2 and 2s1/2 orbitals in Ca. scattering amplitudes of all nucleons in the nucleus, whereas The cross sections are integrated over the energy of knocked- (p,2p) reactions involves the scattering amplitude of a single out proton and given in units of µb sr−2 MeV−1. The experimen- nucleon in the nucleus. Absorption in this case is used as a tal data is from Ref. [425]. In the big panel the dashed (solid) benefit to enhance the effective polarization (see Fig. marisf). lines include (do not) the spin-orbit interaction. The optical Polarized protons can also be used to probe the density de- potential of Ref. [427] was used together with NN-interaction pendence of the nucleon-nucleon (NN) interaction. A reduction from Ref. [428]. In agreement with Refs. [425, 429], the spin- of the analyzing power, Ay, due to density dependence has in- orbit effect is found to be rather small for unpolarized protons. deed been discussed in Refs. [90, 419]. An approximate 40% The inset shows the comparison with the experimental data for reduction of A has been predicted in the case of the 12C(p,2p) y the 1s1/2 state as the NN-interaction is changed. The shaded 12 reaction, where the averaged density within C is about 50% of area includes results for seven NN-interactions taken from Refs. the saturation density. In Ref. [420] it was also suggested that [430, 431, 432, 433, 434, 428, 435]. It is clear that the proper the reduction of meson masses and coupling constants in dense choice of the interaction has a greater impact on the results for nuclear matter will cause modifications of spin observables in unpolarized protons than the spin-orbit interaction. The same quasifree reactions, explaining why the A are reduced by about y conclusion applies for the proton removal from the 1d3/2 or-

52 0.4 2 6 10 Li (1s ) 1/2 0.3 2 10 0.2 ) -1 ) 1

-2 0.1 3 10 10 1 MeV 10 0 MeV -2 -2 0.4 12

b sr 2 2s C (1s )

b sr 1/2 µ 10 1/2 0 50 100 150 0.3 µ ( 1

( 1

2 10 y 0.2 dT A 2 dT 1 Ω 1 0.1

d 10 dT 1 2

Ω 0 Ω 0 d /d

1 10 σ 0.4 3 Ω 40

d 0 10 /d Ca (2s1/2) 1d3/2 (x 0.1) σ 3 d 10 0.2

-1 10 0 50 100 150 p (MeV/c) 0 A-1 2 10 20 25 30 35 40 45 20 25 30 35 40 45

θ1 (deg.) θ1 (deg.) 40 39 Figure 57: Cross sections for Ca(p,2p) K and Ep = 148 MeV, in terms of the recoil momentum, pA−1 of the the residual nucleus. The proton knockout 40 are assumed to be from the 1d3/2 and 2s1/2 orbitals in Ca. The cross sec- Figure 58: Cross sections and analyzing powers for (p,2p) reactions on 6Li, 12C tions are integrated over the energy of removed proton and given in units of µb and 40Ca at 392 MeV. The solid lines contain the spin-orbit part of the optical −2 −1 sr MeV . The data are from Ref. [425]. The big panel displays a dashed potential, the dashed lines are results without the spin-orbit part, and the dotted- (solid) line which include (do not) the spin-orbit interaction. The inset shows lines contain spin-orbit but neglect the density dependence in the NN t-matrix. the modiﬁcation of the calculation for the 1s1/2 state with the inclusion of sev- The data are from Ref. [421]. eral NN-interactions. The shaded lines include a broad range of results obtained with the diﬀerent NN-interactions.

scattering waves χi. The spinor parts of these waves are then bital. Not shown for simplicity are sources of uncertainty in the incorporated into the NN t-matrix, with the nucleon mass re- ∗ numerical results arising with the adoption of different global placed by mN within the nucleon spinors [419]. This effect is optical potentials. They also yield a broad range of results, as somewhat reminiscent of the modification of meson masses and with the case of the NN interactions. coupling constant in the Rho-Brown scaling conjecture [436] so We now turn to the effects of the density dependence on the that the masses of the mesons giving rise to the interaction are ∗ ∗ ∗ cross sections and analyzing power, modified according to mσ/mσ = mρ/mρ = mω/mω = ξ and ∗ ∗ gσN /gσN = gωN /gωN = χ. It has been applied previously in dσ(↑) − dσ(↓) Ref. [420] to study nuclear medium effects in quasi-free scat- A = , (229) y dσ(↑) + dσ(↓) tering, with the parameters ξ and χ varying within the range 0.6 − 0.9. which requires the detection of knocked out nucleons by incom- In Figure 58 we show the results of Ref. [426] for (p,2p) ing polarized protons with opposite polarizations. To describe reactions on 6Li, 12C and 40Ca at 392 MeV, based on Eq. (230) the analyzing power one needs to properly account for the spin in the model proposed by Horowitz and Iqbal [419]. The data variables in the transition matrix of Eq. (228). This procedure are from Ref. [421]. The NN interaction from Ref. [437] and has been described in details in Refs. [404, 405, 406, 419, 420, the Dirac phenomenological optical potential from Ref. [438] 421, 422, 423, 424]. The density dependence of the interaction was used, where the inclusion of the modification in Eq. (230) has been assumed to be of the form proposed in Ref. [419], is straightforward. The solid lines contain the spin-orbit part namely, one assumes that the NN t-matrix is modified because of the optical potential and the calculations are normalized to ∗ the nucleon mass in the nuclear medium, m (r) changes locally 3 the data for d σ/dΩ1dΩ2dT1. Due to the nature of the data according to " # analysis [421], no attempt was made to identify them as spec- ρ(r) m∗ (r) = 1 − 0.44 m , (230) troscopic factors which are also irrelevant for the calculation N ρ N 0 of Ay. The dashed lines display the results without the spin- where ρ(r) is the density at radius r, ρ0 is the nuclear satura- orbit part, and the dotted-lines contain spin-orbit but neglect tion density of 0.17 fm−3, and the factor −0.44 stems from the the density dependence in the NN t-matrix. The usage of s-shell relativistic mean field theory [90]. This effect is obtained from protons is chosen because the interpretation is rather simplified a Schrodinger¨ equivalent form of the Dirac description of the since Maris polarization (discussed below) should be small, al-

53 1.0 1.4

0.8

0.6 1.3 0.4

0.2 ΔA y

y A y 1.2 0 ∆ A

-0.2 With ρ Without ρ -0.4 1.1

-0.6

-0.8 1 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -1.0 40 60 80 100 120 140 ∆R (fm)

E1 (MeV)

Figure 60: Difference between the polarization maxima and minima for the 16 Figure 59: Analyzing powers for the 1p3/2 and 1p1/2 states in O(p,2p) re- 2p1/2 and 2p3/2 subshells in tin isotopes for (p,2p) reactions at 200 MeV as a action at 200 MeV as a function of the kinetic energy of the ejected proton. function of the neutron skin. The solid (open) circles include (do not include) Thin (thicker) lines include (do not include) the medium modification of the density dependence of the NN interaction. We assume that protons are detected ◦ ◦ NN interaction. One proton is measured at 30◦ and the other at −30◦. The open at θ = 35 and θ = −35 , respectively. circles are for 1p1/2 and the solid circles for 1p3/2. The thiner (thicker) lines include (do not include) the medium modification of the NN interaction. ing the tin isotopic chain which is well described with standard mean field theories. The density dependence of the in- though the knocked out nucleon can still acquire a non-zero teraction was included following the prescription in Eq. (230) angular momentum with respect to the (A-1) residue after the with nuclear densities obtained from HFB calculations and the collision. In fact, the target protons are unpolarized and the BSk2 Skyrme interaction, described in Ref. [440]. One needs scattering asymmetry should be nearly equal to the asymme- single-particle energies as well as wavefunctions of the ejected try for the scattering of free protons. But we observe that the nucleon. It is possible, but complicated and not necessarily spin-orbit effect still plays a role even for nucleons knocked reliable, to extract these quantities from the HFB mean field from s-waves because of the non-negligible angular momentum method. A simpler approach was adopted to determine these transfer in the collision. quantities from a global Woods-Saxon potential model. Pro- Maris polarization is a combined action of absorption and tons are ejected from the 2p1/2 and 2p3/2 states were assumed spin-orbit force for a nucleon knocked out a non-zero angular [426]. momentum orbital, such as p1/2 and p3/2 nucleons. The ejected The calculated analyzing powers for the tin isotopes have a nucleon is polarized before the collision and an average over similar feature as that displayed in Fig. 59. The magnitude of the spin tends to wash out this polarization, except that the in- the Maris polarization will be quantified in terms of the differ- teraction with the incoming polarized proton has a strong spin ence between the first maximum of the 2p1/2 state and the first dependence. This leads to a net effective polarization which de- minimum of the 2p3/2 state, denoted by pends on the sub-shell where the nucleon is ejected from. But, p1/2 p3/2 as shown in Ref. [406], the effective polarization is not far from ∆Ay = (Ay )max − (Ay )min. (231) being proportional to Ay. Therefore, Maris polarization is also directly visible in analyzing power data. This is best seen if In Figure 60 we plot ∆Ay in tin isotopes for (p,2p) reac- Ay is displayed for fixed angles of the outgoing nucleons and tions at 200 MeV as a function of the neutron skin obtained scanning the energy of the ejected nucleon, as seen in Figure from the calculated rms radii for the HFB neutron and proton 59. The data are from Ref. [439]. Both nucleons are mea- densities. The protons are assumed to be detected at θ = 35◦ ◦ ◦ sured at 30 . The open circles are for 1p1/2 and the solid ones and θ = −35 , respectively. The solid (open) circles include for 1p3/2. Thiner (thicker) lines include (do not include) the (do not include) the density dependence of the NN interaction. medium modification of the NN interaction. It is evident that adding more neutrons to the system increases Maris polarization in neutron-rich nuclei and its dependence the magnitude of the Maris polarization. The polarization in- on the neutron number was explored in Ref. [426] by study- creases becomes larger than 30% along this isotopic chain. The

54 dependence with the neutron skin comes out nearly linear, although deviations from the linear behavior appears at large neutron numbers. The inclusion of the density dependence of the NN interaction decreases ∆Ay for small neutron numbers and small skins. This difference is stronger for isospin symmetric nuclei than for asymmetric ones. As the nuclear size increases, the asymmetric behavior in analyzing power measurements due to the combination of spin- orbit and absorption effects increase accordingly. We have shown that the determination of neutron skins with such measurements can be done if a separate information on the nuclear charge density is known. An unequivocal determination of the neutron skin requires that the Ay measurements also explore the choices of nuclear interactions and account of medium effects. The choice of NN interactions, some of them also including medium modifications due to Pauli blocking and many-body effects, using e.g., a G-matrix approach, can be tested for a large number of experimental data already available. The Maris polarization effect is an useful tool to investigate Figure 61: Nucleon-nucleon potential (in momentum space) at forward angles. single-particle properties in nuclei and their evolution in neu- The picture shows the separate contributions from the spin-isospin, στ, and the tron rich isotopes. Its sensitiveness to the strength of the spin- isospin, τ, part of the interaction as a function of the laboratory energy. orbit interaction, medium modification of nucleon masses, and nuclear absorption allows for new applications in the studies ena become important when nuclei with masses A ∼ 45 − 120 carried out with secondary radioactive beams. Because experi- become more abundant in the supernova core. Weak interac- ments can now be carried out with a much larger accuracy than tions change the value of Y and electron capture dominates, in the past, new techniques are increasingly being developed to e the Y value is successively reduced from its initial value ∼ extend our knowledge of the nuclear physics of neutron-rich nu- e 0.5. Electron capture yields more neutron-rich and the abun- clei. We have shown that the effective polarization of knocked dance of heavier nuclei, because nuclei with decreasing Z/A out protons in (p,2p) reactions can be added to the new tech- ratios are more bound with increasing nuclear mass. For den- niques to study the nuclear size measurements. indeed, the de- sities ρ ≤ 1011 g/cm3, the weak-interaction processes are dom- termination of neutron skins in nuclei is one of the major re- inated by Gamow-Teller and sometimes by Fermi transitions. search efforts due to its relation to neutron stars and their equa- Ref. [447, 447, 448] reported systematic estimates of elec- tion of state [76]. tron capture rates in stellar environments. However, their cal- 3.9. Charge-exchange reactions culations are only based on the centroid of the Gamow-Teller response. B(GT)-distributions have also been obtained using The rapid neutron capture process (r-process) is responsible modern shell-model calculations [449, 450, 451, 452]. Some for about half of all elements heavier than iron. Despite that, notable deviations from the previous rates reported in [447, 447, the nuclear physics properties of the nuclei involved in the r- 448] have emerged, e.g., Y increases to about 0.445 instead of process are not well known and its astrophysical site has not e the value of 0.43 found previously [447, 447, 448]. been clearly identified. Supernova explosions and neutron star One cannot access these reaction rates directly in labora- mergers are possible stellar sites candidates for the r-process. It tory experiments. The medium nuclear mass range, accurate is uncertain what of these mechanisms are more effective [441, shell-model calculations are difficult and theoretical methods 442]. The neutrino-driven wind model within core-collapse employing mean-field techniques have been introduced which supernovae is a promising candidate for the r-process and it include large uncertainties. Theoretical calculations must be could explain the observation that the abundances of r-nuclei tested against experiment. Another even more difficult prob- in old halo-stars are similar to those in our solar system [443]. lem arises because laboratory-based experiments do not repro- Electron-capture on neutron-rich nuclei with mass number A ∼ duce the conditions (density and temperature) present in stellar 45 − 120 is also thought to become important as the density environments [444, 32]. Thus the numerous electron capture increases during core-collapse supernovae [444]. Electron cap- reactions occurring in stars need coordinated efforts involving ture can also occur on excited states which are energetically not theory and experiments. allowed with atomic electrons on earth [445]. Electron capture by nuclei in p f -shell plays a pivotal role in the deleptoniza- 3.9.1. Fermi and Gamow-Teller transitions tion of a massive star before the core-collapse [446]. During the silicon burning, supernova collapse proceeds via to a com- Charge exchange induced reactions is often used to obtain petition of gravity and the weak interaction, with electron cap- values of Gamow-Teller, B(GT), and Fermi, B(F), matrix el- tures on nuclei and on protons and the β-decay processes play- ements which cannot be extracted from β-decay experiments ing crucial roles. In this scenario, weak-interaction phenom- [453]. This method relies on the similarity in spin-isospin reac- 55 tion operators in charge-exchange reactions and β-decay oper- change, which contains spin and isospin operators. For forward ators. In fact, it can be shown from first principles that, in the scattering, low-momentum transfers, q ∼ 0, and small reaction DWBA approximation, the cross section for charge-exchange q-values, the matrix element (234) becomes at small momentum transfer q is closely proportional to B(GT) (0) and B(F) [454], Mexch(q ∼ 0) ∼ Vexch(q ∼ 0) Ma(F, GT) Mb(F, GT) , (235)

2 (0) dσ ◦ µ k f 2 where v is the spinless part of the interaction, and (θ = 0 ) = ND|Jστ| [B(GT) + CB(F)] , (232) exch dΩ 2π~ ki D ( f ) (i) E Mexch(F, GT) = Ψa,b||(1 or σ)τ||Ψa,b where µ is the reduced mass, ki(k f ) is the reactants relative momentum, ND is a distortion factor (which accounts for initial and are Fermi or Gamow-Teller (GT) matrix elements for the nu- final state interactions), Jστ is the Fourier transform of the GT clear transition. The result above emerges by using eikonal 2 part of the effective nucleon-nucleon interaction, C = |Jτ/Jστ| , scattering waves for the nuclei. In conclusion, extracting Fermi and B(α = F, GT) is the reduced transition probability for non- or Gamow-Teller transition strength from experimental mea- spin-flip (τk is the isospin operator), surements of charge-exchange reactions depends on the validity 1 X of the low-momentum transfer assumption in collisions at high B(F) = |h f || τ(±)||ii|2, energies. 2J + 1 k i k 3.9.2. Double-charge-exchange and double-beta-decay and spin-flip (σ is the spin operator), k The validity of one-step processes in Eq. (232) was proven

1 X (±) 2 to be a rather good assumption for (p, n) reactions with a few B(GT) = |h f || σkτk ||ii| , 2Ji + 1 exceptions. In heavy-ion charge-exchange reactions this as- k sumption might not be so good as shown in Refs. [456, 454]. transitions. The condition that the momentum transfer is small, In Ref. [456] multi-step processes involving the physical ex- q ∼ 0, is assumed to be valid for very small scattering angles, change of a proton and a neutron were shown to still play an so that θ 1/kR, with R being the nuclear radius and k is the important role up to bombarding energies of 100 MeV/nucleon. projectile wavenumber. Ref. [455] explored the isospin terms of the effective interac- At high energies, the charge-exchange reactions proceed via tion to show that deviations from Eq. (232) are common under the exchange of charged pions and rho mesons which carry spin many circumstances. For those important GT transitions whose and isospin. Fig. 61 shows the nucleon-nucleon potential (in strengths are only a small fraction of the sum rule, a direct re- momentum space) at forward angles and the separate contribu- lation between σ(p, n) and B(GT) values may cease to exist. tions from the spin-isospin, στ, and the isospin, τ, part of the Discrepancies have also been observed [457] for reactions in- interaction. One sees that, at E ∼ 100 − 300 MeV, the στ con- volving some odd-A nuclei including 13C, 15N, 35Cl, and 39K tribution is larger than the τ one. This hints to a favored energy and for charge exchange using heavy ions [458]. region for studies of the Gamow-Teller matrix elements needed Double-charge exchange reactions, as shown schematically for astrophysics. At such energies, one expects in Figure 62, could in principle be used to extract matrix elements for double beta decay in nuclei for a number of nu- dσ 2 clei where such decays are energetically allowed. The reaction (q = 0) ∼ KND|Jστ| B(GT), (233) dq mechanism using DWBA would involve the calculation of the amplitude where K is a kinematical constant. When used to obtain B(GT) * + values from experiments, Eq. 233 is purely empirical. It has X 1 M 0 (−) + been used indiscriminately in the analysis of charge exchange (k, k ) = Cγ χk0 Vexch Vexch χk , Ek − γ,k00 − T − Vexch reactions, although it has been also shown that it fails in few γ,k00 (236) cases. It lacks a solid theoretical basis and should be used with where χ is the the distorted scattering wave due to an optical caution to reach the accuracy needed for the electron capture, k potential U in the initial and final channels, k, k0 are the initial beta-decay, or neutrino scattering response functions [455]. and final momenta of the scattering nuclei, k00 is the momen- Eq. (232) can be derived easily in the plane-wave Born- tum of the intermediate state γ with energy 00 and T is the approximation when the charge-exchange matrix element be- γ,k kinetic energy. C are the spectroscopic amplitudes of the inter- comes [454] γ mediate states. By using the Glauber scattering theory one can D E ( f ) ( f ) −iq·ra iq·rb (i) (i) include the interaction U in all orders. Using the assumptions Mexch(q) = Ψa (ra)Ψb (rb) e Vexch(q)e Ψa (ra)Ψb (rb) , (234) of forward scattering, and following the same approximations where q is the momentum transfer, Ψ(i, f ) are the intrinsic wave- as used in Eq. (235) one can show again that a proportionality a,b arises between double charge-exchange reactions and double functions of nuclei a and b for the initial and final states, ra,b beta-decay processes. The typical value of the cross section of are the nucleon coordinates within a and b, and vexch is the part of the nucleon-nucleon interaction responsible for charge ex- a single step charge exchange relation is a few millibarns, while the a double charge exchange cross section is expected to be of 56 0.40 p r U

U p r ]

1 0.35 − m f [

g T β 0.30 β G ν D 0

Figure 62: Schematic view of a double-charge exchange reaction, involving a M 0.25 two-step process induced by the nucleon-nucleon interaction. The potential U is responsible for the elastic scattering of the incoming and outgoing nuclei.

0.20 the order of microbarns or less [454]. 0.20 0.25 0.30 0.35 DGT 1 Usually, double beta decay are ground state to ground state DCE [fm− ] transitions. It can be accompanied by two neutrino emission, or M by no emission of neutrinos. The latter process puts constraints on particle physics models beyond the standard one, such as Figure 63: Correlation between calculated double charge-exchange (DCE) nu- the breaking of the lepton number conservation symmetry. In clear matrix elements (NME) for Gamow-Teller (GT) transitions and neutrino such case, the neutrino is a Majorana particle, e.g., its own anti- less double beta-decay (0νββ) [466]. The calculations have been done for 116Cd → 116Sn, 128Te → 128Xe, 82Se → 82Kr, and 76Ge → 76Se, respectively. particle. To study neutrinoless double beta decay one needs to know the mass of the neutrino and the nuclear transition matrix element. Double beta decays emitting two neutrinos have been been done for 116Cd → 116Sn, 128Te → 128Xe, 82Se → 82Kr, observed [459] but the observation of neutrinoless double beta and 76Ge → 76Se, respectively. The linear correlation is ex- decay remains elusive. plained in terms of a simple reaction theory and if it holds for Usually, the Fermi type operator does not contribute appre- cases of interest, it opens the possibility of constraining neutri- ciably to double beta-decay with neutrinos emitted, since the noless double beta-decay NMEs in terms of the experimental ground state of the final nucleus is not the double isobaric ana- data on DCE at forward angles. For a recent review on the use log of the initial state. Therefore, the important transitions are of charge-exchange reactions as a probe of nuclear β-decay, see those of double Gamow-Teller type. In the case of neutrinoless Ref. [467]. beta-decay one still expects that Gamow-Teller are larger than Fermi transitions [460]. 3.10. Central collisions The problem still remains if one can control the contribu- Finally, we would like to briefly discuss the role of cen- tions of the matrix elements for intermediate states entering Eq. tral collisions at energies of few hundreds of MeV/nucleon as a (236). Perhaps, by measuring transitions to a large number of means to access information on the nuclear EoS (see Fig. 62). intermediate states in one step charge-exchange reactions, such For a gas of particles, with number density n, if their interaction as in (p,n) reactions, one in principle can determine the inco- distance is small compared to their mean separation distance a, herent sum for the double charge-exchange transition. An ob- then na2 1. In this situation, the particles interact only when vious problem is that one is not sure if the same intermediate they collide with the average distance travelled by a particle states excited in (p,n) and (n, p) experiments are involved in between two collisions being known known as its mean free double-charge exchange. These intermediate states might also path λ. Since the interaction cross section between particles is contribute very weakly in one-step reactions and very strongly σ ∼ a2, one has for the mean free path λ = 1/nσ, and for a in double charge-exchange, and vice-versa. Therefore, it seems dilute gas λ a. If the gas is dilute, the probability of three- that the best way to access information on the matrix elements body collisions is much lower than that for two-body collisions needed for double beta-decay is to measure double charge ex- and they can be neglected. change reactions directly. This idea is now being adopted by a Assuming that these conditions are valid, several practical few experimental groups (see, e.g., [461, 462, 463]) not only fo- theoretical methods have been used to describe nucleus-nucleus cused on double charge-exchange reactions related to neutrino- collisions from the microscopic point of view, i.e., using the less double beta-decay but also to populate exotic nuclear struc- collisions between the elementary particles composing the sys- tures. Recent theoretical studies on the relation between nuclear tem. Most models are similar to the cascade model, where the reactions and the neutrino induced matrix elements have also elementary particles (nucleons, mesons, or quarks and gluons) emerged (see, e.g., [464, 465]). move within a mean-field U between collisions. Such mod- In Fig. 63 we show a correlation between calculated dou- els are often called transport models, for example solving the ble charge-exchange (DCE) nuclear matrix elements (NME) for Gamow-Teller (GT) transitions and neutrino less double beta- decay (0νββ) obtained in Ref. [466]. The calculations have 57 Soft EoS

Stiff EoS

Figure 64: Schematic view of a near central collision between two nuclei at high energies. The incoherent and coherent nucleon-nucleon collisions produce numerous particles, and at extreme relativistic energies available at RHIC/Brookhaven and CERN/Switzerland, it can de-confine the quarks and gluons (colored objects) within the nucleons as predicted by Quantum- Chromodynamics. Identifying the properties of the final particles can lead to Figure 65: Pressure associated with neutron matter as a function of nucleon information about the nuclear EoS in the hadronic and de-confined phase. density. The various curves are different theoretical models. The shadow bands show the numerous possibilities for the EoS based on the experiment analyses using a soft (lower shadow region) or stiff (upper shadow region) EoS. (Adapted Boltzmann-Uehling-Uhlenbeck (BUU) equation, from [469]). ∂ f p + + ∇ U · ∇ f − ∇ U · ∇ f = ∂t m p r r r in the overlap region rises above the saturation density, driv- Z Z ing the compressed matter to expand into transverse directions, 3 d p2 dΩ σ (Ω) |v1 − v2| a phenomenon known as collective flow. The detection of the matter distribution in arising form this compression phase, with × f 0 f 0 1 − f 1 − f − f f 1 − f 0 1 − f 0 , 1 2 1 2 1 2 1 2 a comparison with transport theories, allows one to obtain the (237) incompressibility of nuclear matter. Studies of the effect of where σ is the elementary cross sections scattering the particles the symmetry energy can be inferred from collisions involving to within a solid angle Ω and the number of particles with a neutron-rich nuclei. Other observables, such as particle pro- phase space volume is defined in terms of the distribution func- duction and their kinematic properties, also help in the experi- tion f as f (r, p, t) d3rd3 p where p and r are the coordinates mental analysis to deduce K and S . As an example, Ref. [469] of a particle and the labels 1 and 2 refer to the two colliding determined that maximum pressures deduced from experiments aimed at studying central nuclear collisions. Pressures in the particles. The gain and loss (first and second) collisional terms 3 on the right hand side account for binary collisions between range of P = 80 to 130 MeV/fm were deduced in collisions at 2 GeV/nucleon. In Pascal units, this corresponds to 1.3 × 1034 the particles and incorporates the Pauli principle through the 34 − f to 2.1 × 10 Pa). At 6 GeV/nucleon, the deduced pressures are (1 )-terms to avoid scattering into occupied states [235]. 3 34 34 Eq. (237) can be generalized to a covariant equations taking even higher: P = 210 to 350 MeV/fm (or 3.4×10 to 5.6×10 into account elements of relativity, although retardation and si- Pa), about 19 orders of magnitude larger than pressures within multaneity are very difficult to handle. In such transport models the core of the Sun and only comparable to pressures within the mean field U and the elementary cross sections σ are are neutron stars. The experimental analyses are consistent with K correlated and one needs to use a self-consistent microscopic of Eq. (38) within the range K = 170 − 380 MeV [469]. In Fig- approach. In practice, the simulations are often done with a ure (65) (adapted from Ref. [469]), we show the pressure for phenomenological mean field and free nuclear cross sections. neutron matter as a function of the density. The shadow bands Skyrme-type interactions are often adopted with a momentum represent the range of possible theoretical EoS based on soft dependent part [468]. As in the case of mean field calculations and stiff mean-field potentials. of nuclear densities, mentioned in previous sections of this re- According to numerical simulations, in neutron star merg- view, this procedure allows one to deduce the compressibility ers, high temperatures T . 100 MeV can be reached [470]. In K of nuclear matter, which refers to the second derivative of the Fig. 66 we show the largest values of baryon densities (solid compressional energy E with as well as the symmetry energy S lines) and temperatures (dashed lines) reached in relativistic related to the nuclear EoS. heavy ion collisions (RHIC) and in√ neutron stars as a function After an initial compression of the projectile and target den- of the center of mass√ beam energy sNN = 2γc.m.mN . Beam en- sities, the elementary particle collisions start to thermalize mat- ergies in the range sNN = 2.5 − 3 GeV have been considered. ter in the collisional overlap region. The momentum distribu- This is the energy region of the current SIS18 accelerator at tions in this region are centered at zero momentum in the c.m. the GSI/Darmstadt laboratory. The densities and temperatures system, as shown schematically in Figure (65). The density were calculated using a quark-hadron chiral parity doublet mo-

58 6.0 140 0 Temperature RHIC 1.7 120 MeV/nucleon / SLy4 5.5 Temperature Star 120 (124Sn+124Sn) 112 112 y 1.6 ( Sn+ Sn) t

i 5.0 100 s

n 1.5

e 4.5 80 d 1.4 n 4.0 60 o y

DR(n/p) 1.3

r SkM*

a 3.5 40 b

1.2

t Density RHIC 3.0 20 Temperature [MeV] e Density Star N 1.1 2.5 0 2.00 2.25 2.50 2.75 3.00 20 40 60 80 sNN [GeV] ECM [MeV]

Figure 66: Largest values of baryon densities (solid lines) and temperatures (dashed lines) reached in relativistic heavy ion collisions (RHIC) and in neutron Figure 67: Double ratio (DR) of neutron over proton yields in collisions of √ 124Sn+124Sn over 112Sn+112Sn at 120 AMeV as a function of the center-of- star mergers as a function of the center of mass beam energy sNN = 2γc.m.mN . The densities and temperatures were calculated using a quark-hadron chiral mass energy of the emitted particles. The calculations with error bands use parity doublet model for the EoS which depend on the beam energy. ] (Adapted the Boltzmann-Uehling-Uhlenbeck transport equation with either the Skyrme from Ref. [470]). functional SKM* or SLy4. (Adapted from Ref. [472]). del for the EoS which depend on the c.m. energy [471]. This actions at 120 MeV/nucleon. The double-ratio yields between EoS has different properties for different isospin content, and two reactions of different neutron content can allow the extrac- therefore the figure shows that the temperatures in RHIC are tion of nucleonic chemical potentials and the symmetry energy larger and densities slightly smaller at the same relative veloc- via isoscaling [473, 474]. Ratios of n/p yields in reactions with ities. In nucleus-nucleus collisions at relativistic energies the isospin partners should be sensitive to the symmetry depen- isospin per baryon is of the order of -0.1, whereas in NS it is dence of the mean-field potential with density and also test their about -0.38. NS also have a significantly different composition momentum dependence. of strange particles compared to RHIC. The density compres- Ultra-relativistic central collisions, as depicted in Fig. 64, sion in symmetric nuclear matter and in NS matter is very simi- are crucial to study on earth what happens with nuclear mat- lar, but the temperature is quite different. This indicates that the ter within the neutron star core for densities ρ ρ0 where a additional degrees of freedom, such as leptons in beta equilib- description of matter in terms of leptons and nucleons is not ad- rium and non-conserved strangeness, decrease the temperature equate. At such high densities, hyperons and δ isobars may at a given compression. Therefore, the study of neutron star be produced and meson condensations may also occur. Ul- mergers requires the use of a consistent and realistic temper- timately, at extremely high densities, a transition to a quark- ature dependent EoS, probably incorporating quark degrees of gluon phase should occur [475, 476, 477]. But in contrast to rel- freedom. Experiments carried out in the future FAIR/Darmstadt ativistic nuclear collisions on earth, the cold quark-gluon phase facility will be of crucial importance to study de isospin de- within a neutron star poses additional experimental problems, pendence of the EoS, with a connection to quarks and gluons difficult to assess with central collisions. If a cold quark-gluon degrees of freedom. For more details on this subject, see Ref. phase exists inside a neutron star, its maximum mass can be [470]. substantially modified [478]. Such studies have already been The use of radioactive beams has been crucial to understand carried out for decades at several worldwide facilities, such as the role of symmetry energy in central nuclear collisions. Sev- RHIC/Brookhaven/USA or the CERN/ Switzerland laborato- eral indicators have been used experimentally. For example, ries. In a near future, the CBM (Compressed Baryonic Matter) the “coalescence-invariant”, DR(n/p) is one of such quantities experiment will be held at the FAIR (Facility for Antiproton [472], in which the ratio R(n/p) neutrons and protons of all ex- and Ion Research) in Darmstadt/Germany to find out how mat- perimentally measured clusters with A ≤ 4 are measured for ter changes at such high densities, extending previous experi- reactions with different combinations of projectile and target mental efforts on the study of compact stars. isotopes, e.g., 4. Conclusions R(n/p;124 Sn +124 Sn) DR(n/p;124 Sn;112 Sn) = . (238) R(n/p;112 Sn +112 Sn) In this review we have concentrated on the use of relativistic radioactive beams to study nuclear astrophysics with reactions This ratio is plotted in Fig. 67, the double-ratio yields for n/p in inverse kinematics. We have not covered se viral subjects emission being compared to data from Ref. [472] for Sn+Sn re- 59 associated with this topic, as they have been widely discussed lying heavily on theory. As nuclear physics is certainly one of in previous reviews, although we mentioned some of the chal- the hardest problems to tackle in all science, we will need many lenges involved with them. more decades of dedicated scientific work. Instead, we have focused the review on the needs to determined the equation of state of neutron stars and supernova Acknowledgement explosions. We have shown that a great amount of progress has been made in experimental nuclear physics over the past This project was supported by the BMBF via Project No. few decades. Most importantly, the application to solve long 05P15RDFN1, through the GSI-TU Darmstadt co- operation standing problems in astrophysics are remarkable. We have also agreement, the DFG via SFB1245, and in part by U.S. DOE discussed many aspects of theoretical nuclear physics, both for Grant No. DE- FG02-08ER41533 and U.S. NSF Grant No. nuclear structure and nuclear reactions, which need improve- 1415656. We thank HIC for FAIR for supporting visits (C. A. ment. This is particularly true in view of the construction of B.) to the TU Darmstadt. the next-generation rare-isotope facilities like GSI/FAIR in Ger- many and FRIB in the United States, RAON in Korea and ma- References jor upgrades on the way at GANIL in France and TRIUMF in Canada. References The RIBF facility in Japan, will continue to provide a large [1] J. D. Cockcroft, E. T. S. Walton, Experiments with high velocity posi- number of exciting opportunities to advance the experimental tive ions. (i) further developments in the method of obtaining high ve- research in nuclear astrophysics at all fronts. 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