Braz J PhysDOI 10.1007/s13538-015-0372-5

NUCLEAR PHYSICS

Current Status of Nuclear Physics Research

Carlos A. Bertulani1 ·Mahir S. Hussein2,3

Received: 1 September 2015© Sociedade Brasileira de Fı́sica 2015

Abstract In this review, we discuss the current status ofresearch in nuclear physics which is being carried out indifferent centers in the world. For this purpose, we sup-ply a short account of the development in the area whichevolved over the last nine decades, since the discovery ofthe neutron. The evolution of the physics of the atomicnucleus went through many stages as more data becameavailable. We briefly discuss models introduced to discernthe physics behind the experimental discoveries, such as theshell model, the collective model, the statistical model, theinteracting boson model, etc., some of these models may beseemingly in conflict with each other, but this was shownto be only apparent. The richness of the ideas and abun-dance of theoretical models attests to the important fact thatthe nucleus is a really singular system in the sense that itevolves from two-body bound states such as the deuteron,to few-body bound states, such as 4He, 7Li, 9Be, etc. andup the ladder to heavier bound nuclei containing up to morethan 200 nucleons. Clearly, statistical mechanics, usuallyemployed in systems with very large number of particles,would seemingly not work for such finite systems as the

� Mahir S. Husseinhussein@if.usp.br

Carlos A. Bertulanicarlos.bertulani@tamuc.edu

1 Department of Physics and Astronomy, Texas A&MUniversity-Commerce, Commerce, TX 75429, USA

2 Instituto de Estudos Avançados and Instituto de Fı́sica,Universidade de São Paulo, 05314-970 São Paulo, SP, Brazil

3 Departamento de Fı́sica, Instituto Tecnológico deAeronáutica, DCTA, 12.228-900 São José dos Campos, SP,Brazil

nuclei, neither do other theories which are applicable to con-densed matter. The richness of nuclear physics stems fromthese restrictions. New theories and models are presentlybeing developed. Theories of the structure and reactionsof neutron-rich and proton-rich nuclei, called exotic nuclei,halo nuclei, or Borromean nuclei, deal with the wealth ofexperimental data that became available in the last 35 years.Furthermore, nuclear astrophysics and stellar and Big Bangnucleosynthesis have become a more mature subject. Due tolimited space, this review only covers a few selected topics,mainly those with which the authors have worked on. Ouraimed potential readers of this review are nuclear physicistsand physicists in other areas, as well as graduate studentsinterested in pursuing a career in nuclear physics.

Keywords Nuclear physics · Exotic nuclei · Quark-Gluonplasma · Applications

1 Nucleons, Nuclear Structure, and Politics

Nuclear Physics evolved in a rather logical manner; fromsimple ordered shell model tailored in accordance withatomic structure to reproduce the magic numbers [1–3], tocollective models needed to describe vibrations and casesinvolving non-spherical nuclei. The collective model isnow considered an elaborate extension of the shell modelthrough the inclusion of nucleon-nucleon correlations,which are predominantly of the pairing plus quadrupole type[4]. In fact, the shorter-range pairing correlations were intro-duced into nuclear physics at almost the same time that theBCS theory of superconductivity was developed [5]. Thequadrupole-quadrupole interaction is responsible for long-range correlations and gives rise to deformed mean fieldnecessary in the description of deformed nuclei with their

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vibrational and rotational spectra [4]. For general texts onnuclear physics theory, we suggest refs. [4, 6–14]

Correlations involving particles and holes are also used inthe description of excited states in nuclei, the most notableof these is the so-called giant resonances, similar to plas-mon resonances in atom clusters [15–20]. In the 1970s,algebraic models based on group theory were introduced,guided by the pairing plus quadrupole models alluded toabove. The resulting interacting Boson model (IBM) hasas building blocks pairs of correlated particles (neutrons orprotons) coupled to angular momentum 0 and 2 and treatedas bosons [8, 21, 22]. The resulting Hamiltonian can beanalyzed using symmetry concepts and the correspondingspectrum is obtained analytically with the help of grouptheory.

With the development of models of nuclear structure,which pointed out certain degrees of freedom to be studiedto test these models, there was the accompanying develop-ment of a theory of nuclear reactions, aimed to probe thesedifferent degrees of freedoms. The concepts of compoundnucleus [7, 23–27] and direct reactions [10, 11] became acommon language with the bombarding energy of the prob-ing projectile being the deciding factor in applying eitherone of the two concepts: low energy projectiles are capturedby the target to form a highly excited compound nucleuswhich then decays into the different open channels, whilehigher energy projectiles excites the target nucleus or trans-fer a nucleon or more [7]. A unifying concept that dealswith both compound and direct processes in an average wayis the optical model [24], where the scattering Schrödingerequation is solved with a complex potential whose imagi-nary part is meant to simulate processes that remove fluxfrom the elastic channel described by this same equation.It is this rich variety of possibilities which prompted thegreat advances made in the quantum scattering theory thatunderlines nuclear reactions. In this review, we give a briefaccount of these development.

As decades passed by, the main interests in nuclearphysics were broadened and strong overlaps with otherareas of science emerged. Our knowledge about the nucle-ons also changed with time. Now we know that they arebuilt by deceiving confined particles; the quarks and gluons,generically known as partons. New theories and experi-ments are now dedicated to probe the sub-nucleon degreesof freedom and how they change in the nuclear medium,in stars and in the evolution of our universe. We are wit-nessing an increasing enthusiasm in the nuclear physicscommunity with its stronger involvement in problems withinterface with many other areas, such as Chemistry, Biology,Cosmology, Astrophysics, Energy, Medicine, etc.

Needless to mention that Nuclear Physics carries a bigburden of being tightly involved with political issues such asnational security and non-proliferation of nuclear weapons,

a serious threat to the survival of the human race. But,it is also rewarding to witness the involvement of manynuclear physicists in politics, a good example being our col-league E. Moniz, a nuclear theorist and since 2013, the U.S.Secretary of Energy. Moniz was pivotal in worldwide non-proliferation negotiations. He has been regularly in Brazilduring the 1980s and 1990s, e.g., during meetings orga-nized by the present authors. He contributed to numerousimportant theoretical works in Nuclear Physics [28, 29].

The stories we tell in this brief review are divided intotwo main topics: the nuclear physics proper, and nuclearastrophysics. Everyone says that figures are worth a thou-sand words. But due to the limited space, we will onlyuse words (and some equations, as we are physicists, OK?)to tell our stories and point to a few open questions inthis intriguing, beautiful, and extremely difficult field ofscience.

In Section 2, we present an account of the different facetsof nuclei and discuss the major advances in nuclear structureand put special emphasis on the recent research activities inthe area of neutron- and/or proton-rich nuclei, which havebeen produced and studied since the 1980s. In Section 3,we discuss reaction studies and discuss the different the-ories developed for the purpose. In Section 4, we discussthe new nuclei we have discovered in the last few decades.In Section 5, we discuss the importance of nuclei in the Cos-mos, and in the synthesis of elements. Finally, in Section 6,concluding remarks are given.

2 Different Facets of Nuclei

2.1 Quantum Mechanics, Nuclei, and the Origins of Life

Nuclear Physics is the driving science behind stellar evo-lution. It determines energy production, element abundanceand has ultimately led to the knowledge development inother basic sciences such as Astronomy. Imagine a worldwithout Astronomy and the benefits its observations havebrought to such mundane things as the human sense of time,distance and motion, through navigation, and the discover-ies of new worlds on Earth and in the skies. The symbiosisbetween nuclear physics and astronomical observations hasskyrocketed in the last decades with the launch of astronomydedicated satellites and the development of new observationtechnology. In the end, Nuclear Physics is the crucial pieceof science to interpret the observations.

Arguably, the field of nuclear astrophysics started withthe monumental work of Hans Bethe, SubrahmanyanChandrasekhar, and other great pillars of science. A humor-ous event might have happened (nobody really knows) inwhich during the 1930s Hans Bethe told his girlfriend undera bright night sky watch that he was the only person on

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Earth who knew how stars shine. His girlfriend was appar-ently not impressed with his statement because girlfriendsunder a bright sky are more interested in something else.What Bethe knew and possibly nobody else did was thatone gains a lot of energy by nuclear fusion of 4 protonsinto a 4He (α-particle) nucleus. Two protons turn into neu-trons by positron emission (beta-decay process), releasingabout 28 MeV (∼ 4.5 × 10−12 J) of energy per event. Atthe time Bethe proposed a mechanism by means of whicha series of reactions involving carbon and nitrogen nucleiserved as catalyzers for the production of the α-particle(the CN cycle). The catalyzers are recycled, and the pro-cess is analogous to the work carried out by enzymes inbiological processes. The original model proposed by Bethewas later extended to include oxygen nuclei and is nowknown as the CNO cycle [30]. Many decades passed by untilwe fully understood that Bethe’s model is responsible forenergy generation in massive, hot stars, only. For example,in contrast to Bethe’s model, our Sun catalyzes α-particlesby means of proton-induced fusion reactions, also known asthe pp-chain [31].

The tunnel effect is responsible for the long time scaleof stellar burning. Without tunneling, there would be nostars as we know them. It is not a coincidence that allmain physics related to the understanding of how stars shineoccurred in parallel with advances in Nuclear Physics. Themore we learn about Nuclear Physics, the more we learnabout stars and the universe. Sorry, stellar modelers, youhave to wait for us! The first application of the tunnel effectoccurred in Nuclear Physics to understand the process ofα-emission by heavy nuclei. For example, 210Po emits α-particles that tunnel through a Coulomb barrier to get freeoutside the nucleus. This was explained by Gamow, Gurney,and Condon who calculated nuclear half-lives by α-decay(or α-emission) using first principles of quantum mechan-ics available during late 1920s [32, 33]. The inverse processof capture of α-particles or any other charged particle (suchas proton capture) is also explained as an inverse tunnelingprocess through a repulsive Coulomb barrier.

The immediate conclusion in this initial story-tellingabout nuclei and stars is that there would be no life in ourUniverse, or even no Universe as we know it (or the lit-tle we know about it!), without quantum mechanics rulingthe physics of the nuclear world. The relevance of nuclearphysics for stellar evolution is deeply rooted in processesgenerating energy and forming heavier elements. But thereis a caveat: nuclear physics is arguably one of the mostdifficult research areas of all sciences. This is because thenucleon-nucleon interaction is not so well known and alsobecause nucleons are composite objects ruled by the physicsof quarks and gluons, effectively manifested through otherparticles which mediate the nuclear force (i.e., the mesons).Moreover, nucleons in nuclei are not so many to allow

for simplifications traditionally available to understand thephysics of bulk objects. In other words, nuclear structureand nuclear reactions are very difficult to handle using theunderlying theory of quantum mechanics. In more mundanewords, nuclear physics research is not for everyone. Onlythe strong and stubborn scientists survive in nuclear physicsresearch. We hope that this statement will only motivategraduate students to choose research in nuclear physics.That is because difficult problems are more challenging andmotivating for real scientists.

It is extremely likely that life would not exist withoutthe presence of the elements we have identified in nature.The light elements were produced in nuclear reactions dur-ing the Big Bang and heavier ones after the formation ofthe first stars. Carbon is a crucial element for life, althoughit is not ruled out that other elements such as silicon mightbe the basis of life in other strange planetary environments[34]. A crucial step in understanding the origins of lifeas we know was the discovery of the triple-alpha captureprocess when three alpha-particles join to form a carbonnucleus. It was early concluded from many experiments thatthere would be no possibility to form carbon as abundantas we observe without resorting to the triple-alpha process,which is part of a path to the formation of heavier elements.If the triple-alpha capture process would be non-resonant,then the reaction rate would also be too small in a stel-lar environment. It was necessary to postulate the existenceof a resonance in the continuum of the beryllium nucleus(unbound) and also in the carbon nucleus close its alphaemission threshold, more precisely at 7.65 MeV [35]. Theresonance in beryllium allows for a short time interval forthe collision with another alpha-particle to occur, thereforethe reason for coining it as the triple-alpha process. The res-onance in carbon would be responsible for the large value ofthe capture cross section necessary to explain the existenceof “abundant” carbon atoms in the Universe.

This purely theoretical hypothesis, due to Sir Fred Hoyle[35], was later confirmed by William Fowler and collabora-tors with an experiment at the Kellogg Radiation Laboratoryat Caltech [36]. The resonance is known as the Hoyle stateor the nuclear state of life. Some think of it as a sup-port to the anthropic principle: we exist, therefore this state[36, 38] in carbon is necessary, or has to be preconceived.Many scientist dislike the use of this principle in scienceand even argue that there is no principle at all, includingthe Nobel laureate Steven Wienberg [37]. Such statementsare not even wrong, as used to say another Nobel laureateWolfgang Pauli. Believing in it is very much like an orna-ment fish’s thinking: there must be a God; otherwise, whowould throw food in my aquarium? It would be embarrass-ing to find out that human brains are only as far able tothink as a fish. Despite such spiritually inspired nonsense,due to its immense relevance, the triple-alpha reaction

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still leads to genuine theoretical and experimental interest[38–40].

2.2 Gluons, Quarks, and Nucleons

In 1949, again Hans Bethe [41] introduced the concept ofscattering length and effective range to describe nucleon-nucleon scattering at low energies. To simplify his descrip-tion, in the absence of the Coulomb interaction, the scatter-ing phase shift expansion for a partial wave l is given by

k2l+1 cot δl(E) = − 1al

+ rl k2

2+ · · · , (1)

where E = (�k)2/2μ, μ is the reduced mass of the system,al is the scattering length and rl the effective range.

Bethe had no idea of the existence of an inner nucleonstructure. Neither did Hideki Yukawa when he introduceda simple model to explain nuclear forces based on pionexchange [42]. His model was later extended to includeother mesons such as the rho and omega mesons. Now thatwe know that nucleons are made of quarks and gluons, wealso know that the meson-exchange theory of nuclear forcesis an effective theory by means of which the quarks and glu-ons degrees of freedom are hidden. Such a meson-exchangeapproach is very much like Bethe’s theory of effective rangeexpansion (ERE), (1), which describes many features of thenucleon-nucleon forces in terms of the scattering length andthe effective range. They hide the nuisances of the nuclearinteraction at very low energies.

Since we know that quarks and gluons flourish withinnucleons, we also discovered that the QCD Lagrangian [43]

LQCD = ψ̄n[i(γ μDμ)mn − mqδmn

]ψm− 1

4GamnG

mna , (2)

is the basic theory describing the inner works of thenucleon-nucleon interaction. In (2), γ μ are Dirac matri-ces, ψn(x) is the quark field, depending on space and time,indexed by m, n, · · · , and mq are quark masses. Moreover,Gaμν = ∂μAaν − ∂νAaμ + gf abcAbμAcν, (3)where g is a coupling constant, f abc are the structure con-stants of SU(3), and Aaμ(x) are non-Abelian gluon fields,also functions of space and time, in the adjoint representa-tion of the SU(3) gauge group, indexed by a, b, · · · . TheGaμν are the QCD analogous to the electromagnetic fieldstrength tensor, Fμν , of quantum electrodynamics [44].

There has been a strong effort to obtain nuclear propertiesfrom first principles by solving (2). The first step, namely,reproducing hadron ground state and resonances has beenrather successfully achieved, especially for light hadrons,by means of, e.g., Lattice QCD (LQCD) (http://www.usqcd.org) [45–48]. The LQCD is formulated to solve (2) on a gridor lattice of points in space and time [48]. There is still a

long list of hadronic problems to be solved before a fullyconsistent set of results is achieved, also for heavy hadronmasses. It is more difficult to describe scattering states fromLQCD [49] than hadronic ground state properties. It willcertainly take decades or many human generations to extendthe LQCD calculations to describe ground and excited stateproperties of many nucleon systems. First attempts in thisdirection are just at their infancy stage [50].

Using LQCD to obtain many bound and unbound nuclearproperties also seems unnecessary and far fetched. It ismuch like killing a fly with a cannon. The world “effec-tive” is in the root of a better solution for the problem. Onedoes not need to use all the complex non-linear features ofQCD for low energy nuclear physics because such featuresarise mostly at short ranges which are rather unaccessible inlow energy processes. Thus, one needs an effective theoryto handle nucleon-nucleon and multi-nucleon interactionsin the same way as Bethe’s ERE theory was developedfor scattering theory with wave mechanics. In 1979, StevenWeinberg [51] proposed the use of effective field the-ory (EFT) for low-energy processes. In nuclear physics, itmeans something like replacing the Lagrangian (2) by

LQCD −→ LEFT = N†(

i∂t + ∇2

2mN

)

N +(μ

2

)D−4

×{−C0(N†N)2 + C2

8

[N†N

(N†

←→∇ 2N)]}

+ · · · , (4)

where N are nucleon fields for the isospin doublet, N =(p, n), which are functions of spacetime, and

←→∇ 2 = ←−∇ 2 +2(←−∇ · −→∇

)+ −→∇ 2. In this expression, μ is a constant to

give the second term a correct dimension for any D. Theconstants Ci are called low energy constants (LECs) andcarry the hidden information on the short-range physics, i.e.,they carry the complex information about quarks and gluonsinside the nucleons. The simpler Lagrangian (4) carries allrequired symmetries from the underlying QCD Lagrangianand can lead to more precise results as those dots on theright-hand side are replaced by more terms in the expansion.Some say that what matters in EFT is how one treats thosedots.

In quantum field theory (QFT), given a Lagrangian onecan solve scattering problems by “reading” the Feynmanrules out of it. Then one follows the usual procedure ofQFT with regularization of integrals, renormalization due toinfinite sums, and all those seemingly boring stuff. It is acompletely different approach than solving the Schrödingerequation with a nucleon-nucleon potential. Thus, the solu-tion of nucleon-nucleon problems using (4) can be donewith increasing levels of approximation, namely, to leading

http://www.usqcd.orghttp://www.usqcd.org

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order (LO), next-to-leading order (NLO), next-to-next-to-leading order (NNLO, or N2LO), etc. Thus, the level ofprecision of the calculation depends on how far does onego with the approximations (the dots). And they are (inprinciple) under control. In contrast, there is no predictivepath to improve the Schrödinger potential method becausenobody knows how to theoretically improve a phenomeno-logical potential. EFT in fact gives a precise prescription ofhow to improve calculations [52–60]. Applications of EFTto solar nuclear reactions and to other stellar evolution ingeneral have been reported in numerous works (see, e.g.,[61–64]. For example, calculations of radiative capture reac-tions using EFT and lattice gauge theory have been reported(see, e.g., refs. [63, 64]). Calculations of the triple-alphareaction and the state of life based on chiral-EFT have alsobeen the recent focus in Nuclear Physics research [65–69].

Nuclear interactions are also strongly modified by themedium and solving the nuclear many-body Schrödingierequation is a task that has taken more than 80 years andis still being developed. The theoretical understanding ofnuclear properties based on the QCD Lagrangian is a slowlygrowing computational effort, despite the advent of fastcomputers [70]. It might require another 80 years of theo-retical efforts. But the use of QFT, EFT, QCD, and renor-malization group techniques for low-energy nuclear physicshas been useful to unify all potential models used in nuclearstructure calculations. For example, a particular interactioncoined as Vlow−k is shown to parameterize a high-order chi-ral effective field theory two-nucleon force. In the theory, acutoff dependence can be used as a tool to assess the errorin the truncation of nuclear forces to two-nucleon inter-actions and introduce low-momentum three-nucleon forces[71–79].

It is interesting that the nuclear physics communityalmost forgot about Bethe’s ERE which was quicklyadopted in other fields. Nuclear physicists in the 1950–1990were mostly interested either in nuclear spectroscopy inlow-energy nuclear physics, or in the study of a quark-gluonphase transition which might be discovered in relativisticheavy ion central collisions [80]. This phase transition mighthave occurred during the Big Bang and a colder versionof it might also occur in the core of neutron stars. Theresearch efforts and funding feeding the relativistic heavyion community almost extinguished the scientific interestin low energy nuclear physics. Concepts such as ERE werebeing long forgotten in nuclear physics, although thrivingin other areas. However, in the last three decades or so, theERE method resurrected in the nuclear physics literatureand spilled out to other fields. It was in fact another nuclearphysicist, Herman Feshbach from MIT, that developed theconcept of closed and open channels in reaction theory andwho also introduced the concept Feshbach resonances. Bothof these concepts became cornerstone tools to study atomic

physics phenomena and in particular were well suited forthe physics studied with ion and atom traps [81]. ERE isnow a standard tool used to understand many features inoptical lattices [82], Bose-Einstein condensates [83, 84], orthe unitary Fermi gases [85–88].

2.3 The Nuclear Chart, Varieties of Elements

Assuming that we fully understand the nuances of thenucleon-nucleon interactions and how it emerges fromQCD, we still would like to know how the nearly 250 stablenuclei emerge from a consistent theory. Most of these nucleiare composed of even number of protons and even num-ber of neutrons, namely, even-even nuclei (see, e.g., [13]).But even-odd and odd-even nuclei are also abundant. Onthe other hand, the number of odd-odd stable nuclei can becounted in our hands, nominally: they are rare and amountto only five. Among these, four are very light, 2H, 6Li, 10B,and 14N. The fifth stable odd-odd nucleus is very heavy andexists in an excited isomeric state, 180mTa. It has a high spin,9−, which makes its decay to the ground state, 1+, by γ -emission or through β-decay extremely unlikely since thesedecay processes only change the angular momentum by oneunit. Thus to decay to the ground state, 180Ta, with spin andparity 1+, and whose half-life is about 8.15 h, there must bean exceedingly weak and thus very slow multi-photon emis-sion or a similarly very slow β-decay. As such, this isotopeof Tantalum is stable in this isomeric state (E = 77.1keV ),and it is the only stable nucleus to exist in such a state.Another feature of 180mTa is that it is the rarest isotope innature amounting to only 0.012 % abundance in the nat-ural ore composed mostly of the odd-even nucleus 181Ta(99.988 %) [89].

Isotopes of neutron-rich nuclei extend till the neutrondrip line, where the neutron separation energy is zero, whileisotopes of proton-rich nuclei are fewer and extend to theproton drip-line which lies closer to the valley of the stabil-ity than the neutron drip line. The drip lines act as naturalboundaries of bound nuclei. One important fact to mentionis the nonexistence of stable nuclei with A = 5 and 8. Thisfact has an important consequence on Big Bang nucleosyn-thesis and especially on the carbon production. Recently,it has been possible to produce neutron-rich and proton-rich radioactive nuclei as secondary beams, which allowedresearchers to extend the study of nuclear structure to theseexotic species of nuclei [90, 91]. This area of research hasbecome a priority in many countries and major projectsinvolving the construction of large facilities of acceleratorsare in the process of being developed in Germany, the USA,China, Japan, and elsewhere. In Brazil, the activity in thisarea is centered in RIBRAS, installed in the Nuclear PhysicsLaboratory of the Institute of Physics of the University ofSão Paulo [92, 93].

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Another important extension of nuclear research has beenin the production of the so-called superheavy elements,with proton numbers exceeding 100. These elements areproduced through the fusion of a medium mass nucleuswith a heavy one resulting in an excited compound nucleusthat, after emitting several γ -rays and cooling down, startsemitting α-particles till reaching the superheavy elementthat survives enough time to be studied both physically andchemically (see, e.g., [94]). Some of these elements areZ = 101 Mendelivium Md; Z = 102, Nobelium, No; Z = 103Lawrencium, Lr; Z = 104, Rutherfordium, Rf; Z = 105, Dub-nium Db; Z = 106, Seaborgium, Sg; Z = 107, Bohrium, Bh;Z = 108, Hassium, Hs, 109, Meitnerium, Mt; Z = 110, Z =112; etc. [95–103].

There is a huge effort in Nuclear Physics to understandhow nuclear masses emerge from a consistent theory ofthe nuclear forces. For medium and heavy mass nuclei, themean field theories are the tool of choice. Starting from anucleon-nucleon interaction including two and three-bodyterms, such as

v =∑

i

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usual studies of the nuclear response, and to actually inflictchanges in the properties of nuclei which are hitherto notpossible and could reach the realm of freedom of manip-ulation practiced in cold atomic gases and Bose-Einsteincondensation research.

3 Nuclear Reactions

3.1 The Optical Model and Beyond

Research in nuclear reactions accompanied closely thedevelopment of models of nuclear structure [135]. Twoaspects of nuclear structure were in apparent conflict. Theshell model which assumes a mean field felt by the nucle-ons and a corresponding long mean free path, and theBohr model of the compound nucleus which asserts thatthe nucleons, once captured suffer many collisions with theother nucleons and with the mean field, which implies ashort mean free path. These conflicting models were latershown to be different manifestation of the same underlyingnucleon-nucleon force and the Pauli exclusion principle. Amodel for the nuclear reaction, which is an extension of theshell model to positive energies, was then developed, wherethe nucleus is taken to exhibit refraction due to the real meanfield and diffraction due to the absorption caused by the for-mation of the compound nucleus, which was developed byFeshbach, Porter, and Weisskopf [136], and is based on theuse of a complex average potential felt by the impingingnucleon.

The optical model equation is

(K + U)∣∣∣�(+) 〉 = E

∣∣∣�(+) 〉 . (7)

where K is the kinetic energy operator and U(r) = V (r) −iW(r) is the complex potential, whose real part, V (r), isrelated to the mean field used in the shell model (at nega-tive energies), while the imaginary part, −W(r), accountsfor the flux lost into the formation of the compound nucleuswhich is treated separately using the statistical model. Infact, a simple manipulation of the above Schrödingier equa-tion can be used to derive a continuity equation throughwhich an absorption cross section can be derived,

σabs(E) = kE

〈�(+) |W(r)| �(+)〉 . (8)

The above cross section accounts for the compoundnucleus, treated within the optical model, as a sink of flux.This cross section can be expanded into partial waves,yielding,

σabs(E) = πk2

∞∑

l=0(2l + 1)Tl(E) , (9)

where the lth transmission coefficient, Tl(E) is

Tl(E) = 4kE

∫ ∞

0dr|ψ(+)l (r)|2W(r) , (10)

and ψ(+)l (r) is the radial wave function of the optical modelwith scattering boundary conditions.

In the case of fusion cross sections, (9) can be used with= 1/k√�2/2mE = �/mv being the reduced wavelength

(remember the reduced Compton wave length C = �/mc).Equation (9) can be interpreted as if the cross section is pro-portional to π 2, the area of the quantum wave. Classically,different parts of the wave have different impact parametersand different fusion probabilities, Tl . Large impact param-eters correspond to large angular momenta, leading to theweight (2l + 1). For fusion, it is very useful to use theconcept of astrophysical S-factor, such that

σF (E) = 1E

S(E) exp [−2πη(E)] , (11)

where η(E) = Z1Z2e2/�v, with v equal to the relativevelocity. The exponential approximately accounts for thebarrier transmission. While σ(E) decreases rapidly as theenergy decreases, the astrophysical S-factor remains ratherflat [137].

Empirical optical potentials used in the analysis of elasticscattering of nucleons and heavy ions have been developed.Pion-nucleus scattering was also extensively studied usingsemi-microscopic optical potential based on salient featuresof the pion-nucleon scattering t-matrix including the exci-tation of the �-resonance. In the case of heavy-ions, wemention the recent development of the São Paulo Potential,based on the double folding of the densities of the two nucleiin conjunction with an effective N-N interaction includ-ing non-locality owing to Pauli exchange [138, 139]. Thispotential has been quite successful in accounting for theelastic scattering of both stable and radioactive nuclei, afterthe addition appropriate polarization potentials that simu-late the effect of the couplings to strongly coupled channels[140], such as the breakup one in the latter case [141]. Itsuse in coupled channels calculations has been proven to bequite successful in dealing with the scattering of nuclei fromdeformed targets, and in fusion reactions [142].

3.2 Compound, Pre-equilibrium, and Direct Reactions

Low-energy reactions are dominated by the mechanismof the formation and subsequent decay of the compoundnucleus (CN), which is the nucleus formed from the cap-ture of the projectile by the target nucleus. This compoundsystem is formed at a relatively high excitation energy andangular momentum. According to Bohr’s hypothesis, theformation and decay of the CN are independent as the whole

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process is statistical and it takes time for the system to equi-librate after the capture process. Accordingly, it is assumedthat the cross section to go from channel α (the entrancechannel) to a final decay channel β, is for a given value ofthe angular momentum,

σα,β = πk2

(2l + 1)ηαηβ . (12)

Summing over β gives the total absorption cross section forangular momentum l,

∑β σα,β = (π/k2)ηα

∑β ηβ which

can be further simplified by recognizing that the transmis-sion from channel α, Tα = k2/π(2l + 1))σα , can be writtenas Tα = ηα ∑β ηβ . Thus

∑α Tα = (

∑β ηβ)

2. These

manipulations lead to ηα = Tα/√∑

β Tβ , and the cross

section becomes,

σα,β = πk2

(2l + 1) TαTβ∑γ Tγ

. (13)

The above expression is the Hauser-Feshbach (HF) crosssection [143]. The importance of the work of [143] whichis an extension of the Ewing-Weisskopf theory [144], is theobservation that the transmission coefficients Tδ , where δdesignates any decay channel of the compound nucleus,including the entrance channel, is directly related to theabsorption cross section of the optical model [144]. The caseof compound elastic, α = β, is distinct as there are obviouscorrelations between the entrance and exit channels, and theHF cross section above has to be multiplied by a factor of 2[145]. The angular distribution of compound nucleus crosssection is symmetric around θ = π , and in fact is isotropicas expected of a statistical process.

Higher energy data on particle spectra have indicateddeviations from the pure compound nucleus evaporationform, indicating the operation of another mechanism ofdecay. This particle emission process is associated with theexcitation of 2p-1h (2 particle—1 hole), 3p-2h, etc, configu-rations in the compound nucleus before a fully equilibratedsystem is reached. Such pre-equilibrium emission has beenthe subject of investigation both experimentally and theo-retically. The angular distribution of the emitted particlesis, however, not isotropic, indicating the contribution of yetanother mechanism besides that of the CN. This other mech-anism involves the excitation of the target nucleus throughthe excitation of 1p-1h, 2p-2h configurations with the pro-jectile particle remaining in the continuum. The combinedeffect of multistep compound and multistep direct processesconstitutes the model of pre-equilibrium reactions which isused in ref. [146]. For a recent review of compound nuclearpre-equilibrium reactions, see ref. [147].

3.3 Feshbach’s Theory

Nuclear reaction data indicate clearly the existence of twotypes of processes. The ones dominated by the compoundnucleus which are slow processes and treatable with the sta-tistical model of Bohr and formalized through the Hauser-Feshbach theory, and others, fast processes, describable bythe optical model equation as generalized to many channels.Processes, such as inelastic scattering, transfer reactions,and breakup reactions are commonly called direct reactionsand characterized by forward peaked angular distributionswith oscillations indicative of their coherent, non-statisticalnature. To accommodate both types of reactions, the slow,compound ones and the fast, direct ones, Feshbach [148,149], developed a formal theory based on projection opera-tors.

Call the open channels space projection operator, P , andthe closed channels space Q, with P + Q = 1 and PQ =QP = 0, PP = P , QQ = Q, then the Schrödingierequation of the colliding nuclear system can be written as,

(E − PHP)P |� 〉 = PHQ |� 〉 (14)(E − QHQ)Q |� 〉 = QHP |� 〉. (15)

These equations summarize the whole theory of nuclearreactions. To describe the direct reactions, one eliminatesthe compound nucleus Q-space and averages out the cor-responding resonances, to obtain the following effectiveequation,(

E − PHP − PHQ〈

1

E − QHQ〉QHP

)P |� 〉 = 0 .

(16)

The Hamiltonian H in the above equations is the sumH = K + h1 + h2 + V1,2, where K is the kinetic energyoperator, hi is the intrinsic hamiltonian of nucleus i, andV1,2 is the real interaction operator between the two nuclei.The effective polarization operator

PHQ

〈1

(E − QHQ)〉QHP = PHQ

[1

(E − QHQ + iI )]

QHP ,

where the energy averaging interval, I , is much larger thanthe average width of a resonance. This equation is complexand by adding it to PV1,2P , defines the complex opticalpotential operator,

Veff = PV1,2P + PHQ 1E − QHQ + iI QHP, (17)

and the optical model equation becomes,

(E − K − h1 − h2 − Veff )P |� > = 0. (18)This equation is an equation for the optical wave function

P |� > which contains many components (channels). Assuch, it is a set of coupled channels equations that describe

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the direct reactions involving the channels projected byP . The compound nucleus is now completely hidden inthe complexity of Veff . To extract the contribution of thecompound nucleus, one needs to resort to statistical con-siderations involving the fluctuation component of P |� >.The treatment of this issue relies on the use of assumedrandom properties of the P − Q couplings and the wholeapparatus of quantum chaotic scattering theory is employed.The Hauser-Feshbach cross section and the Ericson correla-tion function [150, 151] are then obtained, making the abovetheory a complete one.

The set of equations (18) constitutes the coupled chan-nels theory (CCT) of nuclear reactions. Most reactions atlow and intermediate energies require treatment with theCCT [152]. At higher energies or weak couplings, a pertur-bative treatment is adequate, through the use of the distortedwave born approximation (DWBA). This theory has beenvery useful in the study of the nuclear structure as the ampli-tude is linear in the coupling that induces the transition andaccordingly spectroscopic information are unambiguouslyextracted [10]. Variance of this theory, which is commonlyused in the case of transfer of nucleons in cases involvingdeformed nuclei, is the coupled channel born approxima-tion (CCBA), where the distorted waves are generated froma coupled channels calculation, whereas the transfer cou-pling is treated perturbatively. Such theory is still in use inthe case of the scattering of exotic neutron- and proton-richnuclei from deformed target nuclei [153].

3.4 Coupled Channels

The advent of radioactive beams has brought into focus sev-eral important features of reaction dynamics. The CCT is anexample. The wave function P |�(+) > is taken to be com-posed of a sum of several terms representing the importantchannels operating in the collision of a stable, or neutron-or proton-rich projectile with a target nucleus. Writing P =Pel + Pbup + Ptrans + P ′ renders the optical model equa-tion a set of coupled equations for the elastic channel, Pel ,the breakup channel, Pbup, the transfer channel, Ptrans , andthe other channels that are treated in an average way, P ′.For exotic halo nuclei, such as the one-neutron halo 11Be,the two-neutron halo 11Li, 20C, or one-proton halo, 8B, thebreakup coupling is quite important owing to the close tothreshold breakup channel. As such, it becomes importantto consider Pel and Pbup in the projected P-space opticalequation, when treating elastic scattering.

The adiabatic model, i.e., the neglect of the breakup Q-value, is generally used at higher energies, as done in [154],and [155]. In [154], the Glauber approximation while in[155] the usual adiabatic Schrd̈inger model are used for theradial wave function. The result of the analysis of the totalreaction cross section in [154] of 11Li, 11Be, and 8B and a

12C target, demonstrated that the breakup channel couplingnecessarily increases the reaction cross section and accord-ingly requires a larger rms radius of the halo nucleus, asseemed to be required by the data on the interaction crosssection (the total reaction cross section without the projec-tile inelastic or breakup contributions) at the time, whichwere analyzed using the optical, eikonal, approximation[156–158]. This result is easily understood using the generalstructure of the transmission coefficient Tl(x) ≡ T (b, x) =1 − exp [−2χ(b, x)], where χ(b, x) is the imaginary part ofthe eikonal phase. b refers to the impact parameter and theparameter x refers to the separation distance between thecore and the center of mass of the excess nucleons. The orig-inal, Tanihata analysis [158], considered the optical averageexp [−2 〈χ(b, x)〉x], whereas the adiabatic model deals with〈exp [−2χ(b, x)]〉x . Jensen’s inequality [159, 160] comesinto play to dictate that 〈exp [−2χ]〉x ≥ exp [−2 〈χ〉x],and thus the transmission coefficient in the adiabatic modelTad(b) = 1−〈exp [−2χ]〉x is smaller than the optical trans-mission coefficient, Topt (b) = 1 − exp [−2 〈χ〉x] renderingthe reaction cross section σR,ad = 2π

∫∞0 bdbTad(b),

smaller than the optical one, σR,opt = 2π∫∞

0 bdbTopt (b).To compensate for the reduction in σR,ad compared toσR,opt , a larger rms radius, e.g., for 11Li, < r211 >= 3.55fm, than the one extracted by Tanihata [158], 3.1 fm, isrequired to fit the data. The same considerations wereapplied in the treatment of the elastic scattering of 11Be +12C [155]. Improvement of the adiabatic model was thendeveloped by the Pisa [161–169], Commerce [170, 171],Brussels [172–174], Osaka [175–178], and Seville [179–181] groups, through the relativistic continuum discretizedcoupled channels (CDCC) model and the dynamical eikonalmodel.

At low energies, the fusion process becomes important.A lot of attention has been dedicated to the influence ofthe excess nucleons on the tunneling probability which dic-tates the value of the fusion cross section at near Coulombbarrier energies. The discussion of this process and otherreaction processes at low energies requires the solution ofcoupled channels equations involving the projections Pel ,Pbup, and Ptransf exactly. The breakup channel involvesthree or four clusters in the continuum, requiring for itstreatment a formidable three- or four-body scattering cal-culation. In practice, however, the continuum is discretizedinto a finite number of pseudo-states treated as inelasticchannels. The resulting CDCC equations, [182–185], aresolved for a truncated number of pseudo states using knownnumerical methods. Recent improvement and extensions ofthe CDCC have been made [186, 187]. Computationally,the CDCC relies on an ad hoc truncation of the discretizedcontinuum by keeping only a few pseudo-states. The effectof the neglected pseudo-states on the convergence of theresults is seldom analyzed. Recently, the idea of using

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statistical methods to treat these neglected pseudo-stateswas advanced [188], and a model for a statistical CDCC(sCDCC), based on quantum chaotic scattering theory (seebelow), is currently being developed [189].

The results of CDCC calculations for fusion reactionshave taught us several things. The excess nucleons in thehalo nuclei results in two important effects. The excita-tion of low-lying dipole mode, the pigmy dipole resonance(PDR) (also known as the Ikeda resonance) [190–195],and the close-to-threshold breakup. These two effects werefound to influence the fusion in two distinct ways. Thebreakup coupling results in a suppression of fusion at ener-gies above the top of the Coulomb barrier, whereas thedipole excitation results in an enhancement of fusion atsub-barrier energies [204, 205]. This latter effect is inti-mately related to the longer extension of the matter densityof the halo nucleus. An extensive effort, both experimentaland theoretical, has been dedicated to the low-energy reac-tions of exotic nuclei and several reviews were published,[206–209].

The interest in fusion of neutron-rich nuclei stems fromthe desire to produce a compound system that could sur-vive long enough to be studied and analyzed. Heavier nucleiwith large amount of excess neutrons have been produced assecondary beams, and the existence of the pigmy resonancewas clearly established in, e.g., 134Sn [191, 193–195]. Thesenuclei with a thick neutron skin were produced at high ener-gies. It would be very interesting to produce the neutron skinnuclei at lower energies and study their fusion with a heavytarget to test the idea of an enhanced fusion and a potentialproduction of a superheavy nucleus with a Z and A beyondthe known ones. Such investigation will become possible inthe near future with the new nuclear facilities being builtaround the world.

It is worthwhile mentioning that recent experiments havealso investigated the nuclear neutron skin by means of com-pletely different physics methods. For example, we cite the208Pb radius experiment (PREX) based on parity violationin electron scattering [196, 197]. Such experiments seemto be less dependent on the data interpretation base on amodel description for scattering by pions [198], protons[199–201], or antiprotons [202, 203].

3.5 Incomplete Fusion

Besides complete fusion [210], which involves the captureof the whole projectile, the breakup may lead to the cap-ture of one of the fragments, resulting in what is known asincomplete fusion. Other names of this process are used inthe literature, such as massive transfer, inclusive non-elasticbreakup, surrogate reaction, etc. This process is importantat it supplies a mean to study the fusion of a nucleus whichis otherwise difficult to produce as a beam, such as neutrons

(in a deuteron induced reaction). Several methods and the-ories have been proposed to calculate the incomplete fusionreactions. We mention the ones which are currently in use[211–216].

The standard treatment rely on the spectator model,which says that the two-cluster projectile a = x + b inter-acts with the target only through the participant nucleus x,leaving the spectator b unaffected and only suffer elasticscattering. The cross section for observing b is invariablyderived to be,

d2σ

d�bdEb= 2

�vaρb(Eb)

〈ρ̂x |Wx | ρ̂x

〉, (19)

where Eb is the energy of the outgoing fragment b, andρb(Eb) = mbkb/(8π3�2), is the density of states of b. Thesource function, ρ̂x(rx), is the wave function of the partic-ipant fragment x in the projectile as it reaches the targetnucleus. The imaginary part of the x-A optical potential isdesignated by Wx(rx).

In refs. [212, 213, 216], the source function is the overlap

ρ̂x(rx) =(χ

(−)b (rb)

∣∣∣χ(+)a (rx, rb)φa(rb − rx)

〉, (20)

where χ is the optical model wave function (distortedwave), and φa is the intrinsic wave function of the projectile.The coupling interaction in refs. [211–213] was taken to beVx,b (post), while in [214, 215], the interaction is the dif-ference in the optical potentials, Ux(rx) + Ub(rb) − Ua(ra)(prior). Furthermore, in refs. [211, 214, 215], the incidentwave function of a is taken to be beyond the distortedwave. A debate has been going on concerning which ofthe above approaches is more appropriate for the calcula-tion of incomplete fusion [212, 213, 217, 218]. This debatecontinues.

The important point to mention here is that the structureof the cross section above is similar to the one suggestedby Serber [219, 220], namely the spectrum of the spectatorfragment is proportional to the squared Fourier transform ofthe intrinsic wave function of the projectile, times the totalreaction cross section of the participant fragment. Furtheranalysis was performed in refs. [212, 213] using the eikonalapproximation for the distorted waves gave for the incom-plete fusion cross section the simple form of an integral overimpact parameter of terms of the type, 〈(1 − Tb)Tx〉, whereT is the optical transmission coefficient, and the average isover the internal motion of b inside the projectile. If the Tbis set to zero and the average is ignored, one recovers theSerber expression [219],

d2σ

d�bdEb∼ ρb(Eb)|φa(kb)|2σx(Ex) . (21)

The precise determination of the incomplete fusion isimportant not only for the purpose of capture reactions ofonly a part of the projectile, such as neutrons in deuteron

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induced reactions, but also in fusion studies as in manyinstances the data give the total fusion which is the sumof the complete fusion plus the incomplete fusion. There-fore to get the complete fusion, one needs to subtract fromthe data a believable incomplete fusion cross section. Theresearch in the area of incomplete fusion is currently pur-sued by several groups. A semiclassical model has recentlybeen developed [223]. In fact only in this year, three papershave been written and will appear soon published in thejournals [224–226].

3.6 Quantum Chaotic Scattering

A subject of continuous interest in nuclear physics researchis the understanding of fluctuations in the cross sections.Back in the early 50s, Wigner [227, 228] recognized thatit is meaningless to try to analyze the many resonancesseen in the neutron capture cross sections in the regionof overlapping resonances. He suggested that a statisticaltreatment is more appropriate and introduced the conceptof random matrices for the purpose. The idea is to assumethat the nuclear Hamiltonian be considered a member ofan ensemble of random matrices and averages of differentquantities be perfumed through ensemble averages. The ran-dom matrix theory (RMT) of nuclear reactions was born.The RMT was then greatly developed by Dyson [229] andapplication to fluctuation phenomena in other fields besidesnuclear physics ensued.

At the same time, Ericson [150, 151, 234] introducedthe cross-section correlation function as a measure of thedegree of coherence in the otherwise chaotic nuclear sys-tem. Through an analysis of the correlation function or theaverage density of maxima [234], one is able to extract thecorrelation width, which had been previously estimated byWeisskopf to be

�corr = D(2π)

∑

β

Tβ, (22)

where D is the average spacing between the overlappingresonances. These developments were quite important asthey are universal in nature and can be employed in manysystems exhibiting fluctuations in the observables. This syn-ergy, supplied by research in nuclear physics, attests to therichness of the ideas and concepts introduced in the field[235]. As a matter fact, recent application of RMT has beenmostly in mesoscopic systems, such as electronic conduc-tance in quantum dots and graphene [236]. The test of RMThas been made possible using, among others, microwaveresonators [237, 238].

The quantum chaotic scattering theory relies basically onan expression of the S-matrix which exhibits its relation

to the random Hamiltonian of the system. It is normallywritten as,

S(ε) = 1 − 2πiW †(ε − H + iπWW †)−1W . (23)

In the equation above, H is the random Hamiltonianwhich pertains to one of the universal classes of ran-dom matrix ensembles, the Gaussian orthogonal ensemble(GOE), the Gaussian unitary ensemble (GUE), or the Gaus-sian simpletic ensemble (GSU), considered by Dyson andexcellently reviewed by Bohigas and Giannoni [229, 230].The coupling matrix W couples the internal degrees of free-dom to the open channels, and is taken to be fixed (notrandom). The observables such as the average cross sectionand the correlation function are then calculated by perform-ing an average over the ensemble to which H pertains ofproducts of S-matrices of the type given above. For moredetails, we refer the reader to [237, 238].

Experimentally, the emergence of chaos in quantum sys-tems can be verified by looking at the statistical propertiesand eigenvalues of the wave functions, and in in the fluctu-ation properties of the scattering matrix elements of somesystems, such as quantum dots, quantum wires and Diracquantum dots (quantum dots on a graphene flake, where theelectrons are massless and obey the Dirac equation ratherthan the Schrd̈inger equation), etc. Microwave billiards pro-vide very useful systems to study quantum chaos, becausethey already contain a degree of chaoticity in their classi-cal dynamics. In fact, the eigenvalues and wave functions ofquantum microwave billiards have been studied by AchimRichter’s group [231–233] with a high precision insight intoquantum chaos phenomena.

4 New Nuclei

4.1 Halo Nuclei, Efimov States, and Borromean Nuclei

A rather simple but ingenious experiment reported in 1985by Isao Tanihata and his group on interaction cross sec-tions of light nuclei close to the drip line was the seed ofa new era in nuclear physics [156, 157]. This experimentand others following it have shown that some nuclei such as11Li possess a long tail neutron distribution. The long tailis due to the low binding energy of the valence nucleons.It is simple feature, but one that nobody saw before, andit took Tanihata’s genius to realize the apparently obviousnuclear property. His experiments were put within a nicecontext by Hansen and Jonson [239] who relied on the anal-ysis of Coulomb breakup of 11Li, shown to be enormous dueto the loosely bound character of the nucleus [240]. AfterHansen and Jonson’s paper was published, the commu-nity suddenly realized Tanihata’s discovery and the number

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of works and citations to the emerging field of exotic, halo,and neutron-rich nuclei skyrocketed.

These developments lead to the then existing nuclearfacilities in Europe (GSI/Germany and GANIL/France), inthe USA (NSCL/MSU), and in Japan (RIKEN) to almostentirely turn their beams and detectors to the productionof radioactive beams. Experiments with nuclei far from thestability became a routine in nuclear physics. The open-ing of this new research field in nuclear physics was oneof the reasons why the U.S. National Science Foundationdecided not to close the NSCL facility in Michigan StateUniversity (MSU) in 1992. The Indiana Cyclotron facilitywas not so lucky at the time, perhaps because they missedthose historical developments. Since then, new facilitieshave evolved and planned in all continents, in particu-lar we mention the new RIKEN radioactive beam facilityand the FRIB/MSU and FAIR/Darmstadt facilities presentlyunder construction (http://www.fair-center.eu; http://www.frib.msu.edu). In particular, it took the relentless and dili-gent work of Konrad Gelbke and collaborators to elaborateand get funds for the FRIB/MSU facility in the USA. Alsoin China and Korea, other facilities are being planned orunder construction. For a review of the theoretical effortsconducted at the period of 1990s, we suggest the reader toread refs. [9, 90, 91, 241].

With so much of science at stake, it was imperative toobtain funding for new nuclear physics laboratories with thesole goal of studying nuclei far from the stability line. Itis worthwhile mentioning that before this era, most physicsstudied in those facilities had to do with central collisionswith the purpose to study the equation of state (EOS) ofnuclear matter at high densities and temperatures [244].This EOS (pressure versus density), specially at low temper-atures, is crucial for understanding the physics of supernovaexplosions and neutron stars. In fact, masses and radii ofneutron stars are constrained by the EOS of nuclear matter.In contrast to the study of EOS, the physics of radioac-tive secondary beams was mostly driven by peripheral,direct reactions. Stripping, Coulomb and nuclear excitation,two- and three-body breakup, and other direct reactionswere and remain the principal tools to access spectroscopicinformation of interest to model and develop theories forshort-lived nuclei, far from the stability line. Their relevancefor astrophysics purposes is discussed below.

Typically, neutron binding energies are of the order of8 MeV for most stable nuclei. But neutron separation ener-gies as low as few keV have been observed in short-livednuclei. The long (visible only in a logarithmic scale!) tail ofthe weakly bound neutrons and their corresponding densi-ties is known as “halo” and the nuclei with this property areknown as halo nuclei. Not only the small separation ener-gies but also nucleon-nucleon correlations are vital for thehalo properties. As a typical example, neither 10Li nor two

neutrons form a bound system, but together in 11Li theybind by about 300 keV [89]. The nucleus 11Li became a sortof nuclear superstar in the 1990s. There was not a singlemonth or weeks without a nuclear physics preprint report-ing either an experiment or a theoretical calculation on 11Li.Nuclear physicists also love fashion, fame, and applauses,not differently than anybody else.

Three-body systems theorists were very happy to see that11Li fell in their category of delicate structures arising fromloosely bound three-body systems. In fact, 11Li is a proto-type of a structure known as the Efimov effect [243], i.e.,the appearance of (many) bound states in a system of two-body subsystems with large scattering lengths [242, 243].Jan Vaagen named such systems as Borromean systems,because they reminded him of the heraldic symbol of thearistocratic Borromeo family from the fifteen century [241].In fact, the Borromean rings shown in the heraldic symbolsare three inter-connected rings, such that if one is cut loose,the remaining two also become free. A large activity in thearea of three and few-body physics have profited from thisearly work on nuclear physics in the 1980s and 1990s. Efi-mov states in Borromean systems have become a commonfeature in atomic [248] and nuclear [249] physics.

4.2 New Magic Numbers, Clusters, Majorana Particles

In 1949, Maria Goeppert-Mayer and Hans Jensen proposedthat nucleon forces should include a spin-orbit interactionable to reproduce the nuclear magic numbers 2, 8, 20,28, 50, 82, 126, clearly visible from the systematic studyof nuclear masses [1–3]. Now, we know that as nucleimove away from the line of stability (Z ∼ N), where Z(N) is the proton (neutron) number) those magic numbersmight change due to correlations and details of the nucleon-nucleon interaction, e.g., the tensor-force [250–256]. Theseand other properties of nuclei far from the stability and inparticular those close to the drip-line (i.e., when addingone more nucleon leads to an unbound nucleus) increasedenormously the interest for low-energy nuclear physics,in particular for the prospects of its application to otherareas of science [257]. The applications in astrophysics areevident. For example, the rapid neutron capture process (rp-process) involves nuclei far from the stability valley, manyof which are poorly, or completely, unknown.

It is not easy to predict what one can do with short-lived nuclei on Earth. Sometimes, they have lifetimes ofmilliseconds, or less. It is tricky to explain to the laymenthe importance of such discoveries in the recent history ofnuclear physics. For example, in 1994, the unstable doubly-magic nucleus 100Sn was discovered at the Gesellschaftfür Schwerionenforschung (GSI), Darmstadt, in Germany[258]. At the same time, the laboratory developed a newcancer therapy facility based on the stopping of high-energy

http://www.fair-center.euhttp://www.frib.msu.eduhttp://www.frib.msu.edu

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protons in cancer cells. To celebrate the new facility, thelaboratory invited high-ranked authorities from the Ger-man government. They surely brought along the main newsmedia in the country. Wisely, the laboratory managementincluded the discovery of 100Sn alongside the proton ther-apy facility in the celebration agenda. At one moment, areporter asked a nuclear physicist what is 100Sn good forpractical purposes, what was followed by a long silence anda curious answer: “I think one can develop better, lighter,cans.” Maybe so, but such a can would last fractions of sec-ond! The only sure prediction for applications of short-livednuclei is that it is impossible to understand stellar evolutionwithout a dedicated study of their nuclear properties.

Clustering phenomena in nuclei is a difficult problem,which requires physics beyond the shell model (SM). TheSM describes protons and neutrons moving in shells. Othermodels start from a molecular viewpoint that employs com-posite particles, such as the α-particles as the buildingblocks for nuclear structure. A major task is underway toconnect the two seemingly disjoint points of view and hownuclear excitations can emerge from a unified picture of thenucleus [259–268, 270–275].

Majorana had already shown in 1937 that if the neu-trino would be its own anti-particle, then the beta-decaytheory would not change. If the neutrinos are their own anti-particle (Majorana neutrinos), then neutrinoless beta-decayin double beta-decay is also possible, with two neutronsbeing converted into two protons in the decay. But this isa very rare process and extremely suppressed, what makesit difficult to access experimentally. There has been someearly hints that the process might occur in nature but upto the present date there has been no definitive proof of itsexistence [276, 277]. The shell-model and nuclear structurecalculations are crucial to compare theory to experiments.The plethora of nuclear structure techniques involve, amongothers, the interacting shell model [278, 279], the quasipar-ticle random-phase approximation [280, 281], the interact-ing boson model [282], and the energy density functionalmethod [283, 284].

5 Nuclear Astrophysics

5.1 The Repulsive Coulomb

Although stars are very hot, the kinetic energy of particlesin their plasma are very small compared to the Coulombrepulsion energies, or the Coulomb barriers, in nuclei. Forexample, in the Sun’s core, the temperature translates toabout 10 keV of relative energy of the nuclei. It is very dif-ficult to measure reactions at such energies because of the

smallness of the cross sections due to the Coulomb repul-sion. In general, i.e., either for the Sun or for other stellarscenarios, many of the reactions of interest are not knownat the level of accuracy required by stellar modelers, or theyare simply not measurable directly at all. Many reactionsalso involve short-lived nuclei which are difficult, if notimpossible, to manipulate.

One option for experimental nuclear astrophysicists is touse, e.g., stable or radioactive beams and study reactionswith inverse kinematics techniques. Reactions with unsta-ble nuclei use a stable target (e.g., a proton gas target) andby inference one can access information on cross sectionsof astrophysical interest. This information relies heavily onreaction theory [135]. We cite a few phenomena (only a few,indeed) of current interest using these techniques.

5.1.1 Collective Resonances, Neutron Skins, and NeutronStars

At low energies, of a few MeV/nuclei, heavy nuclei arenot able to surpass the Coulomb barrier and nuclei basi-cally interact only via the Coulomb force. At high energies,above the Coulomb barrier, experimental techniques havebeen developed to extract Coulomb excitation events fromother sort of concurring reactions. A plethora of processesare studied using this technique which has the advantageof using a well-known interaction and therefore the reac-tion mechanism is very much under control. A review ofthe applications of Coulomb excitation to several reactionsof astrophysical interest is found in refs. [285–287]. One ofthe processes of interest for astrophysics is the excitation ofcollective resonances above the particle emission threshold.The most notable of these are the so-called giant resonances.They exist in all sorts of modes. There are monopole, dipole,quadrupole, and octupole vibrations, related to the charac-ter of nuclear shape vibration. They can also be of isoscalaror isovector character, due to their isospin vibration type(T = 0 ot T = 1). There are also electric and magneticcollective excitations.

Collective vibrations in nuclei are intimately related tothe compressibility of nuclear matter, or equation of state ofnuclear matter. This equation give the density dependenceof the energy density in nuclear matter as

E(ρ, α)=E(ρ, 0)+S(α)α2+O(α2), where α = ρn−ρpρ

,

(24)

with ρ = ρp + ρn equal to the nuclear density. The energy�(ρ) = E(ρ, 0) is the energy for symmetric nuclear matter.From it, one can calculate quantities of interest for astro-physics, such as the pressure, P(ρ) = ρ2∂�(ρ)/∂ρ, and the

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matter incompressibility, K = 9∂P (ρ)/∂ρ. These quanti-ties, together with the entropy density, are key to determinethe relation between mass and radii of neutrons stars.

The EOS of nuclear matter can only be inferred by a inti-mate combination of theory and experiment because nothingon Earth can resemble the matter at extreme densities insidea neutron star. The main part of the EOS, i.e., the depen-dence of nuclear pressure on nuclear density, can only beinferred in a small range of interest for neutron star mod-elers. These are often strongly guided by theoretical meanfield calculations. Traditionally, theorists are able to calcu-late the nuclear matter energy incompressibility modulus,K , using energy density functional theories, which startfrom the nucleon-nucleon interaction in the nuclear mediumand calculates E(ρ, α) using a microscopic theory suchas the Hartree-Fock, relativistic mean field theories, ran-dom phase approximation, or variations of these theories[107–112, 123, 288].

It is well-known that although some effective meanfield interactions can reproduce many nuclear propertiesquite well, their prediction strongly deviate from eachother as one departs from the nuclear density saturation(ρ = ρ0 ≈ 0.17 nucleons/fm3 )[289]. As an example ofsuch microscopic calculations, by comparing Hartree-Fock-Bogoliubov results for the excitation of giant monopoleresonances in nuclei, one can infer what sort of densityfunctionals are most appropriate to reproduce the experi-mental data. Giant monopole resonances are the most directcompressible mode of a nucleus and and clearly related tothe compressibility modulus. Inspecting the most acceptedforms ot the microscopic interactions has shown that acompressibility modulus, K 200 MeV [123], can beobtained, although values of K 230 MeV are also accept-able by studies of additional mean-field calculations for theexcitation and decay of other giant resonance modes.

Relativistic heavy ion collisions also induce a mattercompressibility during a short-time interval, with conse-quences such as side-flow of nuclear matter which canbe measured and compared to microscopic calculationsusually done with transport theories such as the Boltzmann-Uehling-Uhlenbeck (BUU) equation [244, 290],

∂f

∂t+(

pmN

+∇pU)

· ∇rf −∇rU · ∇rf =∫

d3p2

∫d� σNN (�) |v1−v2|

× {f ′1f ′2 [1−f1] [1−f2]−f1f2

[1−f ′1

] [1 − f ′2

]}.

(25)

Here, the number of particles in the volume element d3rd3pat time t , is given by f (r,p, t) d3rd3p, in terms of thedistribution function f (r,p, t). The mean-field, U (r,p, t)accounts for the effect of each particle interacting with all

others, with forces on each particle given by −∇rU (r,p, t)and −∇pU (r,p, t) . The distribution function changes dueto nucleons leaving (or entering) the volume d3rd3p,accounted for by the collision term on the right side, whereσNN is the nucleon-nucleon scattering cross section, v1 andv2 are the velocities of two colliding nucleons, and � theirscattering angle. The factors (1 − f ) account for Pauliblocking of final occupied states.

In further constrain, the EOS of nuclear matter, the func-tion S in (24), called by the symmetry energy is needed. Itis usually expanded around x = 0 where x = (ρ −ρ0)/3ρ0,yielding

S = 12

∂2E

∂α2

∣∣∣∣α=0

= J + Lx + 12Ksymx

2 + · · · (26)

where J is the bulk symmetry energy, L determines theslope, and Ksym is the curvature of the symmetry energy atsaturation, ρ = ρ0 (≈ 0.17 nucleons/fm3). The use of heavyion central collisions and the interpretation of experimentaldata using transport equations has been proven to be a goodmethod to study the effects of symmetry energy in nucleiand in nuclear matter (see, e.g., refs. [245–247]).

Collective vibrations of nuclei are also a good tool toinvestigate the symmetry energy. Some features of neutronrich nuclei are also though to be very sensitive to the thesymmetry energy, such as their neutron skin �r = rp − rn,where rp(n) is the proton (neutron) matter distribution [291,292]. Collective vibrations in neutron-rich nuclei at lowenergies are usually so-called pigmy resonances or Ikedaresonances [190, 193, 293–295]. According to equation (14)of ref. [296], the energy of pigmy resonances, EPR , areelated to the neutron skin, �r , and the neutron excess, Nexc,by means of

EPR =(

3Sn�2

2�rRmNNexc

)1/2, (27)

where R is the nuclear radius and Sn the neutron separationenergy. Inserting typical values for Sn, R, Nexc, and �r formedium mass neutron-rich nuclei, one obtains EPR = 5−8MeV.

Study of pigmy resonances helps constrain the symmetryenergy dependence of the EOS of nuclear matter in neutronstars [244, 297]. They are also related to the neutron skins innuclei, as shown in (27) [291, 298, 299]. Pigmy resonancesare a possible energy sink during supernovae explosions.Their existence can lead to noticeable changes in the abun-dance of heavy elements by means of rapid neutron capturereactions, or r-processes for short [300]. In the process, aneutron capture occurs on a very short time-scale, ∼ 0.01–10 s. This is often much faster than accompanying β-decayprocesses occurring between the neutron captures. The r-process path therefore approaches the neutron drip-line asnuclei get more neutron-rich. A large number of isotopes

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in the range 70 < A < 209 are produced in this way.The stellar site of r-processes are not well known. It mightoccur during supernovae explosions, or during neutron starmergers processes.

The nuclear dipole polarizability αD has recently beensuggested [301–303] as an alternative observable constrain-ing both neutron skin and symmetry energy. The polar-izability is related to the photoabsorption cross section

αD = �c2π2e2

∫σabs(ω)

ω2dω, (28)

where ω is the photon energy. Because of the inverse energyweighting, the low-energy response sensitive to the symme-try energy is probed. Recently, experiments carried out inJapan and led by Peter von Neumann-Cosel allowed to inferthe dipole polarizability of several nuclei [304–306]. Earlywork by Takashi Nakamura and Thomas Aumann [191, 193,194, 293, 294] have also unravelled the electromagneticresponse of pigmy resonances with gain of insight into thesymmetry energy.

5.2 Nuclear Reactions in Stars

5.2.1 Reaction Rates and Radiative Capture

In stars, most nuclear reactions occur in the form of binarycollisions. One of the exceptions is the triple-alpha reac-tion mentioned in the introduction. For nuclei j and k in anastrophysical plasma, the reaction rate in binary collisionsis given by rj,k = 〈σv〉 njnk , where the reaction rate 〈σv〉is the average of the product of the cross section and the rel-ative velocities, σv, over the temperature distribution. Morespecifically,

rj,k = 〈σv〉j,knjnk

1 + δjk , (29)

where

〈σv〉j,k=(

8

mjkπ

)1/2(kT )−3/2

∫ ∞

0Eσ(E) exp

(− E

kT

)dE.

(30)

Here, mjk denotes the reduced mass of the target-projectilesystem. The factor 1 + δjk in (29) accounts for the case ofidentical particles, i = k. The rate should be proportionalto the number of pairs of interacting particles in the vol-ume. If the particles are distinct, that is just ∝ njnk . Butif the particles are identical, the sum over distinct pairs is∝ njnk/2.

In (30), the velocity, or kinetic energy E, distribution ofthe nuclei in the plasma at temperature T is assumed tobe Maxwellian. It has been shown in many instances thateven a small depart from the Maxwell-Boltzmann veloc-ity distribution would modify dramatically the outcome of

the stellar evolution. For example, recently, it has beenshown that alternative statistics would ultimately have to beextremely close to Boltzmann-Gibbs statistics to reproducethe observed relative abundance of elements evolved duringthe big bang nucleosynthesis [307].

The direct measurement of the reaction cross sectionsσ for nuclear astrophysics is difficult because it eitherinvolves neutron-induced reactions such as in the r-process,or charged particle reactions which are strongly inhibitedby the Coulomb repulsion, or Coulomb barrier. Many ofsuch reactions also involve nuclei far from the stabilityor neutron-rich nuclei. These nuclei are not usable as tar-gets in laboratory experiments. Thus, most experiments withnuclei far from the stability requires the use of radioactivebeam facilities. They are also done at much higher ener-gies than the typical nuclear relative motion energy in thestars. Therefore, many indirect methods have been devisedto extract indirectly the cross section for nuclear reactionsin stars.

One of such indirect methods is known as the Coulombdissociation method. This method is useful when a projec-tile is loosely-bound and can be dissociated (a → b + c)in the Coulomb field of a target with a large charge Z. TheCoulomb breakup cross section for a + A −→ b + c + A isgiven by [308, 309],

dσC

dEd�=∑

�L

dn�L

dEd�σ�Lγ+a →b+c(E), (31)

where � = E(electric) or M(magnetic), L = 1, 2, 3, · · ·is the multipolarity, and σπλγ+a→b+c(E) is the the photonu-clear cross section for photon energy E. dN�L/dEd� arethe equivalent photon numbers, depending on E, and thescattering angle � = (θ, φ) for a + A −→ b + c + A.The equivalent photon numbers are obtained theoretically[308] for each multipolarity �L. Time reversal allows oneto relate the radiative capture cross section b + c −→ a + γto the photo-breakup cross section σπλγ+a → b+c(ω). Thismethod was proposed in ref. [309] and has been appliedto several reactions of interest for astrophysics. For exam-ple, the reaction 7Be(p, γ )8B relevant for the productionof high-energy solar neutrinos has been studied with thismethod [310–312].

For many systems studied via the Coulomb dissociationmethod, the contribution of the nuclear breakup is rele-vant and cannot be ignored. Such contributions has beenexamined by several authors (see, e.g. [315]). In the caseof loosely-bound systems such as 8B, it has been shownthat multiple-step, or higher-order effects, are important[316–318] due to continuum-continuum transitions. Thefragments can make excursions in the continuum and iter-ate with the Coulomb field of the target many times beforethey become asymptotically free. Multistep processes in the

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continuum led to the development of new techniques, suchas the continuum-discretized coupled-channels (CDCC)method, e.g., as applied to a loosely-bound nucleus in ref.[316]. The effects of high multipolarities (such as E2 con-tributions) have also been studied [321, 323, 324] for thereaction 7Be(p, γ )8B). It has also been shown that the influ-ence of giant resonance states can be neglected [327]. Thechallenge is to develop reliable nuclear models to obtain thescattering states including resonances.

5.2.2 Single-Particle, Ab Initio Methods

Taking radiative capture reactions as an example, the crosssections for direct capture are given by [324]

σ d.c.EL,Jb ∝ |〈lcjc ‖O�L‖ lbjb〉|2 , (32)where a proportionality factor involving phase-space quan-tities is omitted, O�L is the electromagnetic operator, and〈lcjc ‖O�L‖ lbjb〉 is a multipole matrix element involvingbound (b) and continuum (c) wavefunctons. For electricmultipole transitions and in the long-wavelength approxi-mation, O�L = rLYLM ,

〈lcjc ‖OEL‖ lbjb〉 ∝∫ ∞

0dr rLub(r)uc(r), (33)

where ui are radial single-particle wavefunctions. If thewavefunctions are obtained in a many-body model, includ-ing anti-symmetrization and correlations, the equations canget more involved. If a relation with the single-particlemodel can be made, then the total direct capture crosssection is obtained by adding all multipolarities and finalspins of the bound state (E ≡ Enx),σ d.c.(E) =

∑

L,Jb

(SF )Jb σd.c.EL,Jb

(E) , (34)

where (SF )Jb are spectroscopic factors. The spectroscopicfactors are adjusted to a shell model calculation for thesingle-particle occupancy amplitudes. This kind of calcu-lations have been routinely used in the literature for the7Be(p, γ )8B) and other radiative capture reactions (see,e.g., refs [324, 328–333]).

Radiative capture reactions using EFT techniques, suchas those mentioned in an earlier section, have been reportedin, e.g., refs. [334–336]. The advantage of this approach isthe predictive power of EFT which allows for increasingthe precision of calculations as more terms in the effec-tive theory expansion is included (the dots). One of themain problems of this method is the inclusion of many-bodyintrinsic features such as anti-symmetrization and mediummodification of the interactions.

The resonating group method (RGM) or the genera-tor coordinate method (GCM) are able to include anti-symmetrization of bound and continuum states on equal

foot. These are based on a set of coupled integro-differentialequations of the form [337]∑

α′

∫d3r ′

[HABαα′ (r, r

′) − ENABαα′ (r, r′)]gα′(r

′) = 0, (35)

where H(N)(r, r′) = 〈�A(α, r)|H(1)|�B(α′, r′)〉. Here,H is the Hamiltonian for the system of two nuclei (A andB) with the energy E, �A,B is the wavefunction of nucleusA (and B), and gα(r) is a function to be found by numer-ical solution of (35). This function describes the relativemotion of A and B in channel α with full antisymmetrizationbetween nucleons of A and B. Such calculations have beencarried out successfully for many reactions of astrophysicalinterest (see, e.g., ref. [338]).

To the present date, the RGM is the most practical toolto describe nuclear scattering and fusion of light nucleiat low energies with the inclusion of anti-symmetrization.It was introduced in ref. [339] and by solving it one cancalculate cross sections for reactions involving cluster-likenuclei such as 12C(α, γ )16O [340]. This reaction is rele-vant for cosmology. It is thought that if the cross section forthis reaction would be twice the adopted value, a 25 solarmasses star will not produce 20Ne since carbon burningwould cease. An oxygen-rich star is more likely to collapseinto a black hole while carbon-rich progenitor stars is morelikely to leave behind a neutron star [341]. Can you imaginehow important this nuclear reaction is for Cosmology? Thisreaction determines the density of black holes and neutronstars scattered through the Universe.

Other techniques, beyond the single-particle and RGMmodels, have been pursued. The ultimate goal is to calculatethe cross sections from an ab initio-type of calculations, i.e.,a calculation that starts from the bare nucleon-nucleon inter-action and evolves to solve the many-body nuclear problem,including the continuum. First attempts have been done inrefs. [342, 343] with ab initio-bound state wavefunctionscalculated with the quantum Monte-Carlo shell model orthe no-core shell model. In both cases, the continuum wave-function was derived by assuming a single-particle clustermodel for the scattering states. Later, the RGM has beencombined with NSCM to calculate radiative capture reac-tions [344, 345]. Another example is a calculation of thetransfer reaction 3H(d,n)4He cross section to explain theexperimental data obtained at the National Ignition Facility[346].

5.2.3 Transfer, Trojan Horses, ANCs, and Surrogates

Although some direct measurements are possible, calcu-lating transfer reactions such as 6Li(d,α)4He is relativelydifficult. In general, the measurements also suffer from thesame kind of difficulties we mentioned earlier. Therefore,some indirect methods have been developed. Among these,

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we cite the trojan horse method (THM). Because of off-shelleffects, a nucleus x can be carried inside a trojan nucleusand brought to react at low energies with a heavy target. Theheavier and faster trojan nucleus allows x to overcome theCoulomb barrier [347, 348]. The basic idea is to extract thecross section in the low-energy region of a two-body reac-tion with significant astrophysical impact, a+x −→ c+C,from a suitable three-body reaction, a + b −→ s + c + C,at high energies. An example is the study of the S-factorfor the 2H(d,p)3H reaction by using 3He breakup in the2H(3He,pt)H reaction. The analysis of THM data is doneby applying the well-known theoretical formalism of thequasi-free (QF) process. One of the clusters in the incom-ing projectile is the participant, x (the deuteron in the givenexample), while the spectator s, (4He or p) are only weaklyinvolved in the process. These assumptions allows one torelate the cross section to the momentum distribution of theinter-cluster (x − s) motion inside b

Trojan horse (TH) reactions allow for the use of higherbombarding energies, Ea , high enough to overcome theCoulomb barrier of a + x in the entrance channel of thereaction. Then, the effect of the Coulomb barrier and elec-tron screening effects become small. The triple differentialcross section in the center of mass of the TH reaction can bewritten as

d3σ

dEcd�cd�C= KF |�(psx)|2

∑

ll

∣∣Lll

∣∣2(

dσll

d�c.m.

)HOES,

(36)

where ll is the orbital angular momentum of particles sand x and Lll is a function of relative momentum andkinetic energy in the entrance channel [349]. In this equa-tion, (dσ/d�)cm)HOES is the half-off-energy-shell (HOES)differential cross section for the two-body reaction at thecenter-of-mass energy given by Ecm = Ec−C −Q2b, whereQ2b is the two-body Q-value of the binary process andEc−C is the relative energy between the outgoing parti-cles, KF is a kinematical factor, and �(psx) is the Fouriertransform of the radial wave function χ(r) for the x − sintercluster motion. The success of the THM relies on theQF kinematics (equivalent to �(psx) ∼ 0 [350–354].

One has shown that the THM obtains is the same crosssection energy dependence as those obtained with directmethods [355]. The THM data are renormalized to thoseobtained with direct methods at high energies and the dataat low energies are obtained as a bonus. The THM has beenapplied to a large number of reactions of interest for theproduction of light elements in numerous stellar environ-ments to reactions of interest for the BBN [356]. One of themany advantages of using transfer reaction techniques overdirect measurements is to avoid the treatment of the electron

screening problem [357–359]. This is just because the reac-tion is peripheral, probing the tails of the bound-state wavefunctions. In fact, most astrophysical nuclear fusion pro-cesses read only this information for an accurate theoreticalcalculation.

The asymptotic normalization coefficients (ANC)method, introduced by Akram Mukhamedzhanov and col-laborators [358, 360, 361], assumes that the amplitude forthe radiative capture cross section b + x −→ a + γ iscalculated according to

M =〈I abx(rbx) |O(rbx)| ψ(+)i (rbx)

〉, (37)

where

I abx = 〈φa(ξb, ξx, rbx) |φx(ξx)φb(ξb) 〉 (38)is the overlap function, i.e., an integration over the internalcoordinates ξb, and ξx , of b and x. At low energies, I abx isdominated by contributions of large rbx . Thus, the matrixelement M is dominated by the asymptotic value of

I abx ∼ Cabx1

rbxW−ηa,1/2(2κbxrbx), (39)

where Wα,n is the Whittaker function and Cabx is the ANC.The ANC can be related to single-particle properties bywriting it as a product of the single particle (s.p.) spectro-scopic factor and a normalization constant which dependson the details of the s.p. wave function in the interior partof the nucleus. If the reaction occurs at very low energies,then the radiative capture cross section only probes the tailsof the bound-state wave functions. Therefore, Cabx is theonly unknown part of the bound state wavefunction neededto calculate the direct capture cross section. The ANCscan be obtained by experimental analysis of elastic scatter-ing between nuclei by extrapolation of the scattering phaseshifts data to the bound state pole in the energy plane. Theycan also be accessed in peripheral transfer reactions whoseamplitudes contain the same overlap function as the ampli-tude of the corresponding astrophysical radiative capturecross section.

As an example, consider the proton transfer reactionA(d, a)B, where d = a + p, B = A + p. Using the asymp-totic form of the overlap integral, the cross section is givenby

dσ

d�=∑

JBjd

[(CdAp)

2

β2Ap

][(Cdap)

2

β2ap

]

σ̃ (40)

where σ̃ is a reduced cross section, βap (βAp) is the asymp-totic normalization of the shell-model bound-state protonwave functions in nucleus d(B), which are related to thecorresponding ANC’s of the overlap function by meansof (Cdap)

2 = Sdapβ2ap, where Sdap is the spectroscopic fac-tor. If the reaction A(d, a)B is peripheral, the bound-state

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wave functions used to calculate σ̃ are also approximated bytheir asymptotic form, leading to σ̃ ∝ β2Apβ2ap. Hence

dσ

d�=∑

ji

(CdAp)2(Cdap)

2RBd, where RBd = σ̃β2Apβ

2ap

(41)

is independent of β2Ap and β2ap. Thus for surface-dominated

reactions, the cross section is obtained in terms of the prod-uct of the square of the ANC’s of the initial and the finalnuclei (CdAp)

2(Cdap)2 rather than in terms of spectroscopic

factors.Most of the (n,γ ) reactions of astrophysical interest will

never be measured directly. Transfer reaction methods asthe ones described above form a kind of surrogate reac-tions in which the neutron is carried along by a nucleusto react with a target [362, 363]. An example of surrogatereactions is (d,p) reactions, which exploits the simplic-ity of the deuteron structure. Evidently, the neutron in anuclear environment carries angular momentum and otherquantum numbers which are different than those of freeneutrons. Unless the angular momentum plays a minimalrole in the reaction, this brings difficulties in extracting thedesired neutron-induced reaction from the surrogate equiv-alent. Theoretically, this is the same as the implication thatthe Hauser-Feshbach formalism for (n,γ ) reactions agreeswith the Ewing-Weisskopf formalism [364]. To elaborateon top of our discussion of the Hauser-Feshbach formalismleading to (13), assume that σN(c) is the cross section for theformation of a compound nucleus in the entrance channel c.Using the time-reversal theorem, the cross-section σcc′ canbe related to the cross-section for the time-reversed processc′ → c, and one gets

σcc′ (Ec′ ) dEc′=σCN (c) (2Ic′ +1) μc′Ec′σCN

(c′)ω(Uc′ )dUc′

∑c

∫ Emaxc0 (2Ic+1) μcEc σCN (c) ω(Uc)dUc

,

(42)

where Ic is the angular momentum and μc is t