• neutron stars
• two-neutron transfer reactions
• pairing interaction
• Sn isotopes
• superfluidity
• GANIL
• direct reactions

The aim of this Master thesis is the study of the superfluid properties in nuclear matter in general, especially in what concerns very neutron rich systems such as ``exotic" nuclei, infinite nuclear matter and neutron stars which, in a certain sense, can be thought as ``giant nuclei". Superfluidity in nuclear matter is due to the pairing of nucleons which is similar to what happens in metals when superconductivity occurs thanks to the formation of the Cooper pairs of electrons. An experimental evidence of the pairing in nuclei is, for example, the odd-even
staggering effect, namely the nuclei having an even number of nucleons are more stable than those having an odd one. In neutron stars, the pairing interaction influences the cooling time of the inner crust and the ``glitches", that is the observed sudden change in the rotation period of these stars. So, our aim is to investigate the properties of the pairing interaction which accounts for superfluidity and to find an interaction which is as realistic as possible and better fits to the already existing experimental data. The thesis is structured in four Chapters, each one being dedicated to a particular subject, as we will better specify in what follows. In Chapter 1, we give a brief description
of neutron stars which represent the final stage evolution of massive stars and, as the name says, they are mainly made of neutrons. The masses of these stars are of the order of the solar
one, but their radii are of the order of 10 km and thus their densities are comparable and greater than the nuclei saturation density. They are compact objects and we need to use the theory of general relativity to describe their structure. This was done by Tolman, Oppenheimer and Volkoff (TOV) who solved the Einstein equation (for non rotating spherical symmetric mass distribution) and obtained the relativistic hydrostatic equilibrium equations. Once we have the equation of state (EoS), namely the behavior of the pressure with density, we can solve the structure of the star and so obtain the profile of the mass (M), density (d) and pressure (P) all along the star. We use two different equations of state: the Friedman-Panharipande-Skyrme (FPS) and Skyrme-Lyon (SLy) EoS. These EoS describe the nucleon-nucleon interaction and
belong to the so called ``effective" interactions, whose parameters are chosen in order to reproduce some properties of the
infinite nuclear matter and finite nuclei. We solve these equations using the Runge-Kutta method and thus obtain the M, P and d profiles and see the differences between the two EoS. We also calculate the maximum mass that the neutron star can have for each EoS and we find that the one given by the SLy4 EoS is grater than the one given by the FPS EoS. The knowledge of the
maximum allowed mass Mmax of neutron stars is very important, since it represents the boundary between two different classes of compact stars: neutron stars (M<Mmax) and black holes (M>Mmax). Anyway, in this work, we especially focus on the study of the superfluid properties in the inner crust of neutron stars due to the pairing of the unbound neutrons whose distribution in the crust is given by the Negele and Vautherin model, but first we see (in the second Chapter) how is the pairing interaction studied in nuclei and infinite nuclear matter. In Chapter 2, we study some nuclei and the infinite nuclear matter in the mean field approximation of the non relativistic quantum many-body problem. In these approximations, nucleons in nuclei are supposed to move in a mean field they themselves create. We use the Hartree-Fock (HF) and the Hartree-Fock-Bogoliubov (HFB) approximations to describe finite nuclei and the Bardeen-Cooper-Schrieffer (BCS) approximation, which is a particular case of the HFB theory, to describe infinite nuclear matter and the inner crust of the neutron stars where there is also a gas of unbound superfluid neutrons which can be approximated with infinite nuclear matter made of neutrons. The HF, HFB and BCS work well in that concerns the reproduction of
ground state or static properties of the mean and heavy mass nuclei, but they fail to reproduce the dynamic properties concerning the excited states. The description of the dynamic properties of nuclei is done using, for example, the Quasi-particle-Random-Phase-Approximation (QRPA) theory. The HF approximation describes well the closed-shell nuclei, such as the 16O, but it fails to describe nuclei where the pairing
correlations among nucleons become important and these correlations are taken into account in the HFB approximation. We use the Skyrme-Lyon 4 (SLy4) ``phenomenological" and ``effective"
zero-range interaction to describe the nucleon-nucleon interaction inside nuclei. The term ``effective" refers to the difficulties that are encountered while describing the nucleon-nucleon
interaction inside nuclei, since it becomes repulsive at short distance, and the term ``phenomenological" means that the parameters of the interaction are adjusted to reproduce some
properties of both the infinite nuclear matter and finite nuclei. At first, we perform some HF calculation using the SLy4 interaction for the 16O and the 68Ni. We reproduce well
some known properties, such as the radii and the binding energies. Then, in order to better understand the pairing interaction, we perform HFB calculations concerning the tin isotopes 124Sn and 136Sn. We focus on these nuclei since the 124Sn is stable and has already been experimentally studied, while the 136Sn is very neutron rich and thus pairing correlations are supposed to have a great importance (this nucleus has not yet been experimentally studied). So, apart from the SLy4 interaction, we put in the hamiltonian describing these nuclei a paring interaction. We use different paring interactions which belong to the so called ``density-dependent-delta interactions" (DDDI), which, as the name says, depend on the nucleonic density, are local interactions and, according to their dependence on the density, they can be ``isoscalar" or ``isovector". The parameters
of the DDDI are fitted once again in order to reproduce some properties of infinite nuclear matter and nuclei. For example, we use two DDDI pairing interactions, the ``surface" (which is peaked at the surface of the nucleus) and the ``mixed" (which is located in a broad region inside the nucleus) ones, whose parameters are
chosen so that we can reproduce the two-neutron separation energies in the tin isotopes. In particular, we focus on the pairing gap among neutrons and thus, we calculate it for each kind
of interaction and for each nucleus to see the differences among the different pairing interactions. In the case of the stable 124Sn, all the DDDI give a mean neutron pairing gap which reproduces well the experimental one, while in the case of the exotic 136Sn, there are important differences among the mean neutron pairing gap values given by the different DDDI. In particular, the ``surface" interaction gives a quite high pairing gap compared to the other DDDI. We also calculate the pairing gap in infinite symmetric (same fraction of protons and neutrons), asymmetric and pure neutron infinite nuclear matter in the BCS theory and for each of the considered DDDI. In this case too, we see that the ``surface" interaction gives a very high mean pairing gap compared to the other interactions. As we have already said, we use the information obtained in the case of the pure neutron infinite matter to calculate the pairing gap also in the inner crust of neutron stars where the gas of unbound neutrons in the crust is described by the Negele and Vautherin model and we see, once again, how does the pairing gap given by the different DDDI vary along the crust. As expected, the pairing gap due to the ``surface" interaction is peaked in a more narrow region compared to the other interactions, while the ``mixed" one is located in a more broad region inside the crust of the star. After having seen the behavior of the pairing gap in nuclei and infinite matter, we perform Local Density Approximation (LDA) calculations over the
isotopic chain of the tin nuclei where the number of neutrons is varied from 44 to 110. The LDA, as the name says, consists in locally approximating the nuclei with infinite nuclear matter and
thus it tells us how accurate are we when we use the information obtained from finite nuclei to describe infinite nuclear matter and vice versa. The better or worse agreement between LDA and HFB
calculations depends, as expected, both on the considered nucleus and pairing interaction. In particular, we see that the LDA works better for the mid-shell nuclei (far from nuclei having a
magic number of neutrons) and that in general the worse agreement between HFB and LDA is obtained in the case of the ``surface"
interaction. Until now, we have seen, from a theoretical point of view, how does the pairing gap varies in different nuclear systems in connection with which kind of pairing interaction we consider, but, as we know, the final tests that physical theories have to pass are observations and experiments: to this is dedicated the third Chapter. In Chapter 3, we give a brief description of the experimental methodologies and setups which are used to
investigate the pairing properties and in particular we refer to an experiment which took place at the Grand Accélérateur National
d'Ions Lourds (GANIL) located in Caen (France). In this experiment the 69Ni is studied and we give just a very brief summary of it, since it goes beyond the aim of this work. Anyway, the
methodology used in this experiment is the same as the one used to explore the pairing interaction with exotic beams. In particular,
we give a brief description of the heavy and light particle detectors, such as the MUST2 detector. The pairing interaction properties can be experimentally explored performing direct
transfer reactions using exotic beams in inverse kinematics. The term ``direct reaction" means that the passage from the initial to the final state of the reaction is done in one or few steps and proceeds in a short time, while the term ``inverse kinematics" means that the exotic beam (e.g. tin isotopes) is used as projectile and stable nuclei (e.g. protons) as targets. Therefore, we can study the pairing interaction performing, for example, two-neutron transfer reactions, where a couple of neutrons is
transferred from the projectile to the target (or vice versa), but, before performing nuclear transfer reaction experiments, we need theoretical arguments concerning the reactions of interest and this is the subject of Chapter 4. In Chapter 4, we study from a theoretical point of view the following two neutron transfer reactions: 124Sn(p,t)122Sn and
136Sn(p,t)134Sn. We choose these reactions since the first one has already been experimentally studied and thus it allows us to compare our theoretical results with the already
existing data; while, the second one is studied since the pairing correlations are supposed to have a very important role in the very neutron rich 136Sn nucleus. We notice that the
136Sn(p,t)134Sn reaction has not yet been experimentally investigated. Cross section measurements of transfer reactions allow us to compare theoretical and experimental results
concerning the pairing interaction. We therefore calculate theoretical cross sections for each of the considered pairing interactions. The theoretical cross-section of the nuclear reaction is described in the one-step zero-range Distored-Wave-Born-Approximation (DWBA) theory and we use global optical phenomenological potentials both in the entrance and in the exit channels of the reaction. The form factors of the reactions are
provided by the QRPA theory and we use them us inputs for our calculations. Note that the information concerning the pairing interaction is contained in the form factors, since there is a
different form factor for each of the different pairing interactions. In particular, we focus only on the ``surface" and the ``mixed" pairing interactions and consider, for each kind of
the (p,t) reactions, both the transition from the ground state (gs) of the initial nucleus (for example, 124Sn) to the gs of the final nucleus (for example, 122Sn) and the transition from the gs to the first excited (1st0+) state (gs-1st0+). We vary the energy of the incident proton (Ep) from 10 MeV to 35 MeV, to see which should be the
energy of the eventual beam to use in the experiments to constrain the pairing interaction. So, varying Ep from 10 MeV to 35 MeV, we calculate the differential cross sections for the two reactions of interest, for each pairing interaction and for the two transitions (gs-gs and gs-1st0+). The code we use (DWUCK4, that is
Distorted Waves University of Colorado Kunz 4) does not provide normalized cross sections (CS) and thus we can not provide absolute cross sections. For this reason we calculate the ratios
of the absolute CS of the considered transitions, namely CS(gs-gs)/CS(gs-1st0+). Experimental data for the reaction 124Sn(p,t)122Sn already exist for the transition gs-gs (at Ep=20 MeV) and for the transition gs-1st0+ (at Ep=34.9 MeV) and we compare our theoretical results to them. Our theoretical cross sections well reproduce the shapes and the diffraction minima of the differential cross sections in both cases. On the contrary, our cross section ratios are not in good agreement with the experimental ones. This is due to the fact that, at Ep=20 MeV, our calculations are very dependent on the Q of the reaction, since it is very negative and, in this case, the energy of the beam is not high enough so that we can reliably compare our theoretical cross section with the experimental ones. In what
concerns the 136Sn(p,t)134Sn, we do not have experimental data to compare our theoretical results with, but we find that we can disentangle among the different pairing interactions just considering the shape of our cross sections, as
it happens, for example, in the case of the transition gs-1st0+ and at Ep=15 MeV. In this case, at Ep=10 MeV, analogously to what happens for the 124Sn(p,t)122Sn at Ep=20 MeV, our theoretical cross sections are not reliable once again because of the Q of the reaction which is not high enough. Anyway, at higher energies we
do not have this problem and we show that we can experimentally disentangle between the ``surface" and the ``mixed" interactions only for Ep values at around 15 MeV. This result is very
important, since nowadays facilities produce exotic beams whose energies are not higher than 15 MeV and the very next future facilities, such as the SPIRAL2, will study transfer reactions
with the tin isotopes and thus will help us to shed more light on the pairing interaction and all the properties which depend on it,
such as the cooling time of neutron stars, nuclear structure...