A systematic study of knockout reactions from exotic nuclei with anomalous ratio N/Z

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UNIVERSIT `

A DEGLI STUDI DI PISA

Facolt`

a di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Magistrale in Scienze Fisiche

TESI DI LAUREA

A systematic study of knockout reactions

from exotic nuclei with anomalous ratio N/Z

Candidato:

Relatore:

Gianluca Salvioni

Dott.ssa Angela Bonaccorso

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Nothing is over

until it is over...

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Abstract

This work is concerned with a systematic theoretical analysis of neutron knockout reac-tions of exotic nuclei that have large differences in valence neutron and proton separation

energies |Sn− Sp|. A large set of published data already exists in the literature. The data

consist of absolute cross sections for one-nucleon knockout as well as inclusive parallel momentum distributions of the residual nucleus. They have been collected by various experimental groups in order to deduce information on the structure of the nuclei, in particular of their valence nucleons because this kind of reaction is so peripheral that only the particles on the surface of the nucleus are involved. The absolute value of the cross section is proportional to the occupancy number (spectroscopic factor) of the initial bound state of the nucleon while the shape of the momentum distribution is related to the angular momentum of the removed particle.

Some anomalies have been found in comparing the experimental results to the theoretical analyses, which used structure model such as the shell model and the eikonal reaction model. The two main anomalies were an apparent reduction in the occupancy of the deeply bound nucleons and deformed momentum distributions as compared as those from the eikonal model. In order to understand and, if possible, clarify the origin of such anomalies, we will discuss in this work an alternative reaction model, and when possible, use also a different structure model.

We shall compare our calculations to data already present in the literature and also apply them to new unpublished data that involve light nuclei.

The plan of this work is as follows. First a brief introduction of the nuclear properties is given to outline the fundamental differences between stable and exotic nuclei.

The radioactive ion beam production mechanisms are described to understand better the difficulties associated to the reactions which involve these short-living nuclides.

We then introduce the formalism of the transfer-to-the-continuum model from which we will obtain our theoretical knockout cross sections. The formalism contains two parts: one deals with the valence neutron breakup from the projectile, the other accounts for the probability that the projectile residual nucleus (called the core) survives the interaction with the target. The fact that the core survives intact is fundamental in the knockout experiments because the residue is the detected product of an inclusive reaction.

The ingredients necessary to describe the neutron breakup are: the initial neutron wave

function and the neutron-target (9_{Be) interaction, which is properly described by the}

op-tical model scattering matrices developed to reproduce the n+9_{Be scattering data. We}

have performed the calculations with two different scattering matrices from two optical potentials in order to quantify the sensitivity of the cross section. On the other hand the core survival probability is described by a S-matrix to which a correction is introduced in order to take into account the fact that the valence proton is not knocked out from the

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2 Abstract projectile even if in some cases it is much less bound than the valence neutron.

The comparison between the experimental differential cross section distribution and the calculated distribution from our model will be presented as well as their integrated values. To deduce further information on the structure of the lightest nuclei, valence nucleon wave functions from Variational MonteCarlo model will be used, in addiction to the standard shell-model wave functions calculated in a Woods-Saxon potential.

Our results seem to indicate that the anomalies can be explained in terms of a more suit-able and accurate reaction theory which takes into account kinematics as well as other single-particle initial states available for the valence nucleons. The strong absorption ra-dius, the most important parameter to describe the core-target scattering, also seems to play a fundamental role in determining the absolute value of the inclusive knockout cross section.

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Chapter 1 Introduction

The atomic nucleus with mass number A can be considered as a bound system formed by Z protons and N neutrons. The nucleons in turn are composed of quarks but here in order to study reactions at medium energy (≈ 100 A.MeV) the quark structure is neglected.

1.1 Nuclear properties

1.1.1 Nuclei and binding energy

One fundamental property of the nucleus is its mass. Its mass measurements [1] can be made indirectly by determining the energy relation between the nuclides involved in a reaction, once the products are known and/or evaluated. Typical reactions for indirect measurement are decays, where measuring the decay products the mass of the parent nuclides can be estimated, neutron or proton captures, and transfer reactions. Detect-ing techniques available for direct mass measurements usually are made with the help of electromagnetic fields that interact with the charged nuclides. Examples of these direct methods are mass spectrometers, time-of-flight spectrometers, spectrometers with a Cy-clotron Frequency such as the Penning Traps.

The mass represents an indirect information about the strength of interactions between nucleus constituents in fact the nuclear mass energy is the sum of neutrons and protons

energy plus the binding energy. Once that the mass of the nucleusA_ZXN is measured, then

the nuclear binding energy is obtained as in Ref.[2] by

BE(N, Z) = Zm(1H)c2 + N mnc2− m(AZXN)c2, (1.1)

with m(1H) mass of the hydrogen atom and mn mass of the neutron.

Sometimes in mass tables the mass energy excess ∆ is given, it is

∆(N, Z) = m(A_ZXN)c2− uAc2, (1.2)

where u is the Atomic Mass Unit defined by

u = m(12C)/12 = 931.49M eV /c2. (1.3)

The behaviour of the binding energy can be approximated with the form

BE(N, Z) = αvA − αsA2/3− αc Z(Z − 1) A1/3 − αsym (N − Z)2 A + δ, (1.4) 3

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4 1. Introduction

which is used to describe a nucleus by the liquid-drop model. The values of α-constants

are adjusted to give the best agreement with the experimental data. αv ≈ 15.5 MeV

describes the volume energy term, αs ≈ 17 MeV for the surface term, αs ≈ 0.7 MeV for

the Coulomb term and αsym ≈ 23 MeV, which together with δ represent the symmetry

energy term.

δ is the pairing energy term approximated as

δ =     

αpA−3/4 for Z and N even

−αpA−3/4 for Z and N odd

0 for A odd

(1.5)

with αp ≈ 34 MeV.

1.2 Stable nuclei vs radioactive nuclei

The nuclei presently known are collected in the nuclear chart, Fig. 1.1, also called the

Segr`e plot. It shows on a grid of proton number Z (y-axis) versus neutron number N

(x-axis) the distribution of the nuclei observed up-to-date. The ’stable’ nuclei, identified by

Fig. 1.1: Nuclear chart with the mean decay mode for each nucleus given by EURISOL project [3].

“Stable” nuclei are in black, radioactive nuclides by β- or β+ decay are respectively in light blue and red, in yellow α-decay and in green nuclei that decay by spontaneous fission. Proton and neutron drip lines are shown.

the black colour, occupy the centre of the distribution. Stable means that the time scale of their possible decay is of the order of the time scale of our universe or larger. Today the number of known stable nuclei is about 200 plus 90 primordial radionuclides, radioactive

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1.2 Stable nuclei vs radioactive nuclei 5

nuclei formed in the built-up of terrestrial matter and having half-lives larger than 1015s.

Thus only 300 of about 3300 nuclei shown in the nuclear chart are present on the Earth. The non-naturally occurring radioactive nuclides are called “exotic” because they lie far off stability and have properties different from those of stable nuclei. Their half-life is

shorter than 1015_{s, down to about 10}−15_{s. They were first discovered as products in}

natural radioactive chains or in cosmic radiation, later they have been created artificially in reactors and/or accelerators.

The neutron to proton ratio N/Z starts with 1 for light nuclei (nuclei with small mass number A) going to 1.44 for the heavier stable nuclides. Heavier nuclei have more neutrons than protons due to the Coulomb repulsion. The neutron presence is favoured as can be deduced from Eq.(1.4).

The region occupied by the stable nuclei is called the valley of β-stability because it is formed by nuclei stable to β-decay. For a given A the mass excess value, Eq.(1.2), is minimum for stable nuclei and it increases when adding proton or neutron to the stable

configuration. The time scale for the β-decay ranges from 10−3s to years (1010_{s). By}

the Heisenberg’s indetermination principle we can understand that the bigger the mass excess the faster the decay. Increasing N from the valley of stability, the decay mode is

Fig. 1.2: Valence neutron separation energy Sn and valence proton separation energy Sp respectively

for the known nuclei [4]. Correspondence between the energy values in KeV and the colour gradients are given in the legends. In dark green are represented loosely bound nuclei, in blue and violet the unbound nuclei, which lie beyond the drip-lines.

β−. When the available decay energy becomes so high that the strong interaction can

dominate the decay process, the neutron emission becomes the decay mode. In a family

of isotopes this happens where the neutron separation energy Sn becomes zero, i.e. the

ground state for the nucleus system is above the neutron threshold. Sn is defined as

Sn= BE(N, Z) − BE(N − 1, Z), (1.6)

the difference between the binding energy of the nucleus and the binding energy of the nucleus without the valence neutron. The borderline over which for increasing N no bound nuclei are found is called neutron drip line. Some rare cases of unbound nuclei

lying beyond the drip line have been studied. As example in Ref.[5] the 9_{He nucleus}

is studied and it is shown that its ground state is 1.1 MeV above the neutron-emission stability.

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6 1. Introduction

Increasing Z from the valley of stability we reach the region in which instability is due

to electronic capture and β+ _{decay. On the borderline of this region proton emission is a}

possible decay mode, where the proton separation energy Sp becomes zero, i.e.,

Sp = BE(N, Z) − BE(N, Z − 1). (1.7)

At present the driplines are not well defined because a small difference in the binding energy or different mass models can lead to different available decay modes.

For heavy unstable nuclei and for a few light nuclei such as 5Li, 8Be, 9B, the α-particle

emission can be the decay mode if the available decay energy allows the alpha particle to overcome the Coulomb barrier. Superheavy nuclides can also decay by spontaneous fission, but this decay mode is very sensitive to the potential shape and to the shell effects such that the borderline of the nuclear existence is not defined. It is estimated that more than 6000 nuclides can exist and most of the nuclei to be discovered are located in the heavy corner of the nuclear chart, in the region called “terra incognita”.

1.2.1 Nuclear radius and halo nuclei

Another important characteristic of a nucleus is its radius. Inside the nucleus the density is stated to be nearly constant, implying that the mass is proportional to the volume, thus the nuclear radius R varies with mass number following the behaviour

R = r0A1/3. (1.8)

For stable nuclei, this relation can be a good approximation with the standard parameter

r0=1.25 fm. Some exotic nuclei, in which the valence nucleon or nucleons are very weakly

bound, have a radius larger than expected by Eq.(1.8) with the standard parameter. For

example Tanihata et al. [6] demonstrated first for 6_{He and} 11_{Li, that the value of the}

r0 parameter has to be modified. These nuclei are called “halo nuclei” because the

va-lence nucleons behave like a wide halo that surrounds the other nucleons packed in the

core. In such cases it has been found that r0 has to be as large as r0=1.5 fm. This

phe-nomenon can be explained with some help from quantum mechanics: the valence nucleon are so weakly bound that their wave function can largely extend out of the classically-allowed nuclear region due to the tunnelling effect. If a system is well clustered and can be described in terms of a short range potential a halo state will appear once the bind-ing energy is sufficiently small. The typical bindbind-ing energy scale for halo nucleons is less than about 1 MeV in light nuclei, decreasing to the order of 0.1 MeV for heavier nuclei [7].

1.2.2 Nuclear shell model and magic numbers

In the liquid drop model, the nuclei are studied by their number A, N and Z. On the other hand the nuclear structure can be also analysed from the point of view of the in-teractions among its constituents such as neutrons and protons. This model is called the nuclear shell model and can be seen as a way to organize the nucleons in a sequence of levels inside the nucleus similarly to the electrons in the atomic levels. The first approx-imation to the shell configuration is the Independent Particle Model. Here each nucleon is treated as a particle inside a mean field due to the presence of the other nucleons.

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1.3 Radioactive nuclides production 7

Solving the Schr¨odinger equation with the Harmonic Oscillator potentials, square well or

Woods-Saxons form, gives energy eigenvalues and eigenstates for various quantum num-bers. When a correction to the potential is introduced by the spin-orbit term l·s that split the degenerated level energies, one-particle states are obtained. Each state can contain a number of neutron or proton up to 2j + 1, where j = l + s is the total angular momentum. Energy eigenvalues are called levels and are filled from the lowest to the highest with N neutrons and Z protons for a nucleus with A = N + Z. If a level is fully occupied the shell is said to be closed otherwise the shell is open. In the case that the outermost shell is open, all the closed shell below form the inner core while the nucleons in the outer uncomplete level are the valence nucleons. The levels tend to arrange themselves in bands well separated from each other such that the complete filling of a band can give a better stability.The number of nucleons in each band with all closed shells corresponds to a magic number. These numbers, shown in Fig. 1.1, can be obtained for example in a Woods-Saxon plus spin-orbit potential where the spin-orbit term is responsible for the energy splitting and relative position of the nuclear shells.

A more accurate shell model is calculated in the Hartree-Fock formalism where more

re-alistic two-body and three-body interactions are used to solve the Schr¨odinger equation.

Usually the inner core is neglected and the calculations are developed only for valence nucleons. The form of the interaction can vary from model to model including angular momentum, spin, momentum, isospin dependencies in order to reproduce better the nu-clear levels experimentally measured. Applying the two-body potential to two valence nucleons (two neutrons or two protons) it can be seen that they tend to form a state

Jπ₌₀+ _{because its energy is the lowest for the two-body potential. This effect is called}

pairing and it makes reasonable to use the shell model also for heavy nuclei, in which essentially all nucleons are paired-off to J=0 and only the valence nucleons will determine the low energy nuclear structure properties.

Nuclei with a magic proton or neutron number have the role of strongholds for the de-velopment of the nuclear chart because the neighbouring nuclei can be studied by adding a particle or subtracting a nucleon to their stable configuration. If a nucleus presents both proton and neutron magic numbers it is called doubly-magic. Only few of them are

stable like 4He,16O,40Ca, 48Ca and 208Pb, while the majority are exotic nuclei like48Ni,

56_Ni,78_Ni,100_{Sn and}132_{Sn. Magic numbers are derived from the spherical shell model, but}

the nuclei can be deformed by their internal interactions and correlations. Deformations and high speed rotation are observed for heavy nuclei. They can be explained by two-body and three-body correlations from the core and the valence nucleons. Deformation leads to different shell model structure and in fact new magic numbers have been discovered as

in the case of 54Ca where N =34 is the new magic number [8].

1.3 Radioactive nuclides production

The experimental way to enlarge the knowledge of the nuclear chart and of the nuclear properties is to make nuclear reactions.

After the discovery of radioactivity made by H. Becquerel in 1896, radioactive nuclides have been systematically studied and employed to obtain different nuclei. In 1919 E. Rutherford was the first to deliberately transmute one element into another. He used

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8 1. Introduction

α-particles in a nitrogen gas, which was converted in oxygen and hydrogen (protons)

through the nuclear reaction 14_{N+ α →}17_{O + p. In 1934 I. Curie and J.F. Joliot used a}

nuclear reaction between two stable nuclei to obtain a new radioactive nuclide, for

exam-ple 27Al+α →30P + n. Taking advantage by the controlled fission reactor that E. Fermi

built in 1942, a large number of new neutron-rich radioactive isotopes was discovered. The production of exotic nuclei grew up with the advent of powerful accelerators, which permitted various production process like fission, spallation, fragmentation or fusion re-actions.

At the time when radioactive nuclei were “commonly” obtained as reaction products, the necessity of new production methods emerged to study better their properties. For this reason from the second half of the XX century the development of the Radioactive Ion Beams (RIBs) production methods started at dedicated facilities, where radioactive nuclides can be collected in beams that can then be used as projectiles for collision with stable targets. It means that a larger bulk of nuclear reactions can be obtain and more in-formation on the exotic nuclei structure can be deduced. The two fundamental methods of RIB production are the Isotope Separation On Line (ISOL) and the In-Flight separation.

1.3.1 ISOL method

The ISOL method was first used in 1951 at the Niels Bohr Institute in Copenhagen (Den-mark), where neutrons produced by deuterons break-up were directed on to an Uranium target to obtain Krypton isotopes. However it was only in 1989 that with this technique secondary beams of radioactive nuclei were produced at Louvain-la-Neuve (Belgium).

In the Isotope Separator On Line (ISOL) method, the radioactive nuclei can be

pro-Fig. 1.3: Schematic representation of an ISOL facility [10]. The various phases of the process are here represented. The primary beam that collides on the target, the ionization of the products, the separation of the obtained isotopes and the post-acceleration that finally gives a radioactive ion beam of a specific nuclide.

duced by spallation, fission or fragmentation reactions of a light projectile against a thick target (primary target). The reaction products are thermalized in a catcher and then they diffuse out of the target through the transfer line towards the ion source, where they are ionized. Subsequently they are extracted, mass-separated on-line and (in some cases) re-accelerated. The whole process from production to detection was performed in

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1.3 Radioactive nuclides production 9

a continuous way, from here the term “On Line”.

The secondary beams obtained with this technique are of excellent quality. The energy resolution is well defined while the thermalization process can be slow and inefficient to produce radioactive nuclides with short-life.

Around the world we can find several facilities that work with this technology: REX-ISOLDE at CERN in Geneva (Switzerland), which since the birth of the REX-ISOLDE facility in 1964 has become a benchmark for all other ISOL projects, SPIRAL at Caen (France) and TRIUMF at Vancouver (Canada).

1.3.2 In-Flight method

After the 1960s, a class of kinematic separators started to be employed to select reac-tion products coming from high-energy heavy ion collisions. This technique was called In-Flight separation because the production of RIBs occurs quickly after the reactions. With this method, exotic nuclides are produced by the fragmentation or fusion of a

high-Fig. 1.4: Schematic representation of an In-Flight facility [10]. The fundamental process in this technique is the kinematic separation that can be made by magnetic fields, electrical fields and/or solid degraders.

energetic and heavy projectile on a thin target. The reaction products, emerging with beam-like velocities, are then separated in-flight by magnetic and electric devices. The pioneering work was performed in 1985 at the Bevalac in Berkley (USA): through

frag-mentation of 20_{Ne beams, I.Tanihata [6] obtained the first} 11_{Be and} 11_{Li beams.}

This method is very fast, allowing for the production of very short-living isotopes (half-life on the order of µs) while the optical beam quality is limited. The In-Flight separators are highly successful in exploring the limits of stability, because only the flight time from the production target to the measuring station can induce decay losses. Other advantages are their high transmission and their good particle identification.

At the present in the world we have a number of RIBs facilities working with In-Flight method: GANIL at Caen (France), GSI at Darmstadt (Germany), NSCL/MSU in Michi-gan (USA), RIKEN at Tokyo (Japan).

A high-quality beam of short-living radioactive ions can be produced mixing the best of both techniques, using a gaseous catcher in the slowing down process (which can happen before ionization in the ISOL method or after separation in the In-Flight method). The use of RIBs has obtained such a success in nuclear physics and its applications that a new generation of future facilities are presently under construction in Europe, North

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10 1. Introduction

America and Asia. Improving the quality of RIBs for a better use of this front-line physics needs a strong international collaboration, like the EURISOL project in Europe. Italy will take a driver-role in the field with the SPES project at LNL-INFN Legnaro laboratories (Padova).

1.3.3 RIBs in the future and their applications

The possibility to extend the knowledge of the nuclear chart depends on the technical ad-vances in radioactive beams physics such as employing additional target material or new ion source at ISOL-facilities, increasing primary beam intensities or enhancing selectivity and detection efficiency at In-Flight facilities.

The aim of the second generation RIBs is to obtain more intense and pure beams, with energies ranging from MeV/nucleon to GeV/nucleon. It should be remembered that

stan-dard radioactive beam intensity is around 106 nuclides/s against 1013 particles/s in stable

beams. The beam-target combination and the beam energy have to be optimized in order to achieve the highest production rate of a certain nucleus. Any manipulation of the reaction products, like ionization, purification, acceleration, transport to the detectors, has to be efficient. The time between production and arrival at the experimental set-up should be reduced to minimum to avoid losses by radioactive decay. The whole equipment has to select the isotopes of interest from unwanted species. Production techniques have to offer a wide variety of RIBs, from the lightest elements to the heavier ones.

Radioactive beams, as well as enlighten the understanding of the nuclear physics, can be useful in other physics fields.

The nucleus offers a good test site for strong and electro-weak interactions. Working on radioactive nuclei far off stability can give new opportunity to probe the Standard Model in the same way as the β-decay was fundamental to the study of exotic nuclei.

A deeper understanding of the reaction mechanisms of radioactive nuclei can help nuclear astrophysics providing more accurate information to model nucleosynthesis processes that occur in stars.

In solid-state physics, radioactive beams can have a diagnostic function offering informa-tion on the host material in which the ions are implanted and also they can be a useful tool for deeper implantation inside the samples.

Today nuclear radiation is used for diagnostic and therapy in medicine. For example

the 14O nucleus can be employed as positron emitter in Positron Emission Tomography

(PET) cameras. The Bragg peak, the sharp maximum in the energy distribution for a projectile nucleus entering into a material, can emit a ionizing dose to malicious tissues in a specific body region.

With the advent of the new generation of RIBs facilities and their improved qualities new opportunities to employ exotic radionuclides in other technological fields can be offered.

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Chapter 2 Knockout reaction model

The purpose of this thesis is a systematic analysis of one-nucleon knockout reactions with exotic nuclei. This kind of reaction consists in the removal of a valence nucleon from a projectile nucleus due to the projectile nuclear interaction with a target. The fact that only one nucleon breaks-up from the projectile means that the reaction is so peripheral that only the particles on the surface of the nucleus are directly involved.

The comprehension of the knockout mechanism is useful to better understand the strength of the nuclear interaction in a many-body problem in such extreme conditions as in the exotic nuclei: the valence nucleon-core, the nucleon-target and the core nucleon-nucleon interactions are the ingredients that regulate the reaction dynamics. On the other hand, these reactions give further information on the structure of the exotic nuclei as the occu-pancy of the valence nucleon in the outermost levels.

A large set of published data is present in the literature. The data consist of absolute cross sections for one-nucleon knockout as well as momentum distributions of the resi-dual nucleus, called the “core”. This reaction is inclusive because the residue parallel momentum and mass are measured after the collision, while the target and the broken-up nucleon are not measured.

The theoretical analyses of the knockout reactions are commonly made by the eikonal model [11], an approximate scattering theory that is valid when the incident energy is higher than the depth of the scattering potential. In comparing the experimental results to the theoretical analyses, some anomalies have been found such as an apparent reduction in the occupancy of the deeply bound nucleons and deformed momentum distributions as compared as those from the eikonal model.

In order to clarify the origin of such anomalies, we apply an alternative reaction model, the transfer-to-the-continuum, in the analysis of these reactions.

In a knockout reaction, the typical experimental set-up is formed by a radioactive ion beam directed on a target of stable nuclear material. In the reactions, we are interested

in, the target is 9_{Be, a nucleus commonly used in such experiments because it does not}

have bound excited states and, above all, because it has the largest number of interactions per energy loss for any simple target.

We schematically represent this situation with a two-body collision model, then we make appropriate corrections to take into account experimental uncertainties such as the beam spread as well as target thickness.

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12 2. Knockout reaction model

The projectile is formed by a radioactive nucleus, that is represented by a core plus one or two valence nucleons. This picture is used to take into account the fact that the nucleon will be stripped from the projectile during its interaction with the target nucleus. Its

va-lidity is based on the hypothesis that the core density ρc and the valence nucleon density

ρn span different regions of the projectile-target distance. In some nuclei the nucleons

are so weakly bound that the projectile can be considered an halo nucleus. We fix the reference frame in the laboratory, with the origin centered at the target centre (see Fig. 2.1). The motion of the projectile nucleus is described by R(t), which is the position of the centre of mass of the incident nucleus relative to the target, while r represents the position of the neutron relative to the target. We make the approximation that the core projectile moves on a classical path (straight line) with velocity v along the z-axis. The

core-target and neutron-target impact parameters are respectively bc and bv. The

pro-jectile trajectory is simply R(t) = bc+ vt. This approximation is reasonable because the

incident energy is well above the Coulomb barrier and the projectile residual is detected intact. Furthermore we have made the no-recoil approximation which means neglecting the difference between the center-of-mass of the projectile and the center-of-mass of the core. For light targets, this is a good approximation as shown in Ref.[9].

Fig. 2.1: Coordinate system used to describe the neutron knockout reaction.

The inclusive cross section σexp _{has been measured for various nuclei and its value can be}

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2.1 Transfer to the continuum formalism 13

2.1 Transfer to the continuum formalism

Following the core-valence neutron hypothesis, the total theoretical cross section for neu-tron break-up in the transfer to the continuum model can be written [12] as

σtheor =X i (C2Si) Z +∞ 0 dbcP−n(bc)|Sct(bc)|2 (2.1)

where the integration is done over the core-target impact parameter bc.

P−n(bc) is the neutron break-up probability, obtained from the neutron transition

ampli-tude from the initial to the final state. |Sct(bc)|2 represents the core-target S-matrix and

contains the information that the core survives after the interaction with the target. We start by calculating the breakup probability for a valence neutron. The transition probability from the initial to the final state is

P (jf, ji) = 1 2ji+ 1 X ni,nf |Af i(jf, ji)|2 (2.2)

where ji and jf are the initial and final state angular momentum, ni and nf are their

z-components and Af i is the transition amplitude, which depends on the initial and final

neutron wave functions.

Following the calculation shown in App. A, we obtain

dP−n dεf =X jf |1 −Sjf | 2 + 1 − |Sjf | 2_B(j f, ji). (2.3)

The inclusive breakup cross section is the sum over all possible final angular momenta

jf. Sjf means that the S-matrix is averaged on energy. From the square of the the

transfer amplitude and the energy average one obtains |1 −Sjf |

2_{+ 1 − |}_S

jf |

2_{. The}

splitting of the scattering effects in two terms shows the two components of the reaction

mechanisms: |1 −Sjf |

2 _{represents the neutron elastic breakup , 1 − |}_S

jf |

2 _{gives the}

absorption due to inelastic scattering of the breakup neutron by the target nucleus and due to compound nucleus formation.

The function B(jf, ji) in Eq.(2.3) is defined B(jf, ji) =

2jf + 1 2(2lf + 1)

(1 + R)B(lf, li). (2.4)

In Eq.(2.4), B(lf, li) is an elementary transfer probability depending on the initial and

the final state, on the core-target distance of closest approach bc, on the energy of relative

motion: B(lf, li) = 1 4 ~ mv 2 m ~2kf |Ci|2(2lf + 1) e−2ηbc ηbc Mlfli. (2.5)

In order to obtain P−n(bc) an integration over all final energies εf is performed

P−n(bc) = Z +∞ 0 dεf dP−n dεf . (2.6)

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14 2. Knockout reaction model

P−nis the probability that the valence neutron is stripped from the projectile and it ends

up in the continuum of the target.

We pay attention on the factor |Sct(bc)|2 present in Eq.(2.1). Sct(bc) represents the

projectile core-target S-matrix and its square can be seen as the probability of elastic

scattering Pel between core and target.

A useful parametrization is given in Ref.[12] |Sct(bc)|2 = exp

−ln2 e(Rs−bca )

, (2.7)

where Rs is the strong absorption radius and a is the diffuseness like parameter. This

expression introduces a smooth cut-off around bc = Rs. It means that for bc < Rs the

total cross section is small because the core interacts closely with the target such that it cannot survive intact and in the elastic channel.

2.2 Knocking out a strongly bound neutron while

skipping a weakly bound proton

In some reactions discussed in this thesis, a strongly bound valence neutron is knocked out while a weakly bound valence proton remains in the projectile core. For this situation, we suggest a correction to Eq.(2.7) to show explicitly the proton survival effect:

Pel(bc) = |Sct(bc)|2e−P−p(bc). (2.8)

Pel(bc) is the probability of elastic scattering between the core and the target and can

replace |Sct(bc)|2 in Eq.(2.1). e−P−p(bc) ≈ 1 − P−p(bc) is the probability that the weakly bound proton is not knocked out from the projectile.

We justify Eq.(2.8) following Ref.[13]. There it was shown that the elastic scattering

probability Pel can be given in terms of the imaginary part of an optical potential phase

shift δI(bc) Pel= e−4δI(bc), (2.9) with δI(bc) = − 1 2~ Z +∞ −∞ dt (WV(R(t)) + WS(R(t))) = δIV(bc) + δIS(bc). (2.10)

where we have assumed that the optical potential is given by the sum of a volume potential

WV plus a surface potential WS. The “volume” part of phase shift δVI can be related to the

core-target scattering while the “surface” part of phase shift δS

I accounts for the peripheral

nature of the scattering, which involves the proton and the target. The surface optical

potential WS can be linked to the proton transfer probability P−p by

Z +∞

−∞

dtWS(R(t)) = −~

2P−p, (2.11)

where WS is identified as the one channel proton breakup imaginary potential. Thus the

core survival probability becomes Pel(bc) = e−4δ V I(bc)e−4δ S I(bc)= |S ct(bc)|2e−P−p(bc) (2.12) as assumed in Eq.(2.8).

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2.3 Nucleon-target S-matrix 15

2.3 Nucleon-target S-matrix

The S-matrix entering in Eq.(2.3) is fundamental in the cross section calculation. It repre-sents the valence neutron-target interaction in the final state and contains the dynamical

information on the scattering. In our work we use two different n-9Be potentials given in

Ref.[14]. Both are phenomenological optical-model potentials, built in order to reproduce the available experimental data [16] for the total, elastic and inelastic cross sections of the

system n-9Be. The first potential, that we indicate with AB, is highly phenomenological

because it is constrained only to fit the total, elastic and inelastic cross section. The S-matrix is obtained by a potential of the form:

U (r) = −[VW S(r) + δV (r) + iW (r)]. (2.13)

The real part of the potential is given by VW S, which contains a Woods-Saxon plus

spin-orbit part VW S(r) = VR 1 1 + expr−RR aR − ~ mπc 2 Vso r l · σ d dr   1 1 + exp r−Rso aso  , (2.14)

and a correction δV , that takes care of the surface-deformation effects. The imaginary part of the optical-model potential has a volume and a surface contribution.

The second potential, indicated as DOM, is a dispersive optical model potential that follows the DOM method given in Ref.[15]. In that theory, the optical-model potential

is represented by a non-local self-energy Σ(r, r0; E) with a real and an imaginary part.

Through approximations Σ can be replaced by an effective energy-dependent local poten-tial

U (r, E) = VHF(r, E) + ∆V (r, E) − iW (r, E). (2.15)

VHF, given by volume, surface and spin-orbit components describes the effects of the mean

field similarly to the Hartree-Fock potential. W is the imaginary potential given by a sum of the volume plus surface parts. ∆V is the dispersion correction defined as

∆V (r, E) = 1 π Z W (r, E0) 1 E0_{− E} − 1 E0_{− E} F dE0. (2.16)

with the Fermi energy EF.

Through these potentials we obtain S-matrices that depend on the energy Elab of the

neutron relative to the target. The energy interval is chosen from 1 to 180 MeV, with 1

MeV step. For each Elab the scattering matrix depends also on the neutron final angular

momentum jf = |lf ±1₂|, the orbital angular momentum range is taken from lf = 0 up to

lf = 19, in the sum over partial waves giving the total cross section. With these

condi-tions two different S-matrices have been extracted by the phenomenological optical model (AB) and by the dispersive optical model (DOM). Both these S-matrices, by construction,

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16 2. Knockout reaction model d σ /d E [ b ar n /M e V ] 0.1 1 Elab [MeV] 1 10 100 tot data el data abs data abs (p) data tot AB tot DOM el AB el DOM abs AB abs DOM

Fig. 2.2: Experimental (point) and calculated (line) cross sections for the reaction n+9Be. In the legend colors and lines are specified. The proton absorptive data are included to increase the available number of data.

In Fig. 2.2 the total, elastic and inelastic (absorptive) cross section are shown as well as the corresponding experimental data. The proton absorptive data, obtained from the

p+9_{Be reaction, are included in the inelastic data to increase the available number of}

data. They lie mainly in the range between 20 and 50 MeV, where the Coulomb effect can be considered negligible and their behaviour can be related to nuclear interaction. In our calculation we use an energy step of 1 MeV while in Ref.[14] they use 0.1 MeV step,

thus for low Elab the behaviour of our calculated cross sections is less close to the data.

Starting from 1 MeV we lose information on the peak around 0.7 MeV, which is considered

a resonance state p1/2 and we have a reduced contribution from the peak at 2.9 MeV, a

resonance state d5/2. With 1 MeV precision the differences discussed in Ref.[14] between

the two model at low energy are not so relevant. The dispersive optical potential, which is very accurate in the resonance regions, is here not fully exploited because in the reactions that we will analyse the cross section peaks are localized at energy higher than 3 MeV. At higher energy both potentials reproduce the total scattering cross section. Looking at the elastic cross section (dashed lines) and at the absorptive cross section (dotted lines)

the differences between the two model emerge. In the low energy region (Elab <10 MeV)

the elastic AB cross section (red dashed line) has an higher peak and a deeper decrease than elastic DOM curve (blue dashed line). After 20 MeV the elastic AB curve falls down quickly. Overall the elastic DOM better reproduces the elastic experimental data (dark cyan points). In the inelastic case the cross sections become different from zero after 2 MeV. The absorptive DOM curve (red dotted line) presents a peak at 3 MeV and then it decreases with a long tail while the absorptive AB curve (blue dotted line) grows up until 5 MeV and after it decreases smoothly with a long and large tail. The AB cross section fits better the neutron and proton inelastic data.

After these considerations we expect that the different behaviour of the cross sections

calculated for the reaction n+9_{Be will be reflected in the calculation of the neutron}

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Chapter 3 Kinematic parameters

We discuss now how to use the formalism of the transfer-to-the-continuum reactions and in particular some useful parameters that it contains to understand better the reaction mechanism.

3.1 Parallel momentum distribution

In inclusive reactions, where only the residual projectile core is detected, the core parallel momentum is the observable experimentally measured.

Before the interaction with the target, the beam shape usually has Gaussian or square form, with the peak centred at

P_kproj = 1 c

q

(TP + MPc2)2− MP2c4, (3.1)

where TP = AP Eincis the kinetic energy of the projectile (Einc is the incident energy per

nucleon) and MP is the projectile mass. This is because the incident secondary beam has

some production difficulties (see Sec. 1.3) that does not make it completely monochro-matic.

After the scattering process, the residual core parallel momentum distribution is mea-sured. In Fig. 3.1 an example of the residual core distribution as function of the parallel momentum is given and for comparison the initial beam shape is also shown.

By energy and momentum conservation and under the hypothesis that the core moves on a straight trajectory with respect to the target, the core parallel momentum distribution in the laboratory reference frame is given by:

P_kcore = 1 c q (TR+ MRc2)2− MR2c4 = 1 c q (TP + i− f)2+ 2MRc2(TP + i− f) (3.2)

where TR is the residual nucleus kinetic energy, MR is the mass of the residual nucleus

with (AP-1) nucleons, TP is the projectile kinetic energy, −i is the valence neutron

sep-aration energy in the projectile while f is the neutron final energy with respect to the

target.

In a transfer-to-continuum process, f in principle can takes the values from 0 → ∞.

Thus there is a maximum allowed value for Pcore

k corresponding to f = 0. The peak of

the core parallel momentum distribution corresponds instead to the best matching con-dition between the nucleon initial separation energy, its final state energy and its kinetic

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18 3. Kinematic parameters

Fig. 3.1: Projectile (RHS squares) and core (LHS dots) parallel momentum distributions for46_Ar+9_Be

reaction, taken from Ref.[17]. The Gaussian-like shape for the projectile Pk distribution is centered at

16.3 GeV/c, at which corresponds by Eq.(3.1) to the incident energy of 65 A.MeV. The curves used to reproduce the core Pk data in an eikonal model are centered at 15.9 GeV/c, which will be the center

of the distribution in the best transfer condition for the neutron stripped according to Eq.(3.2). The distribution of the core Pk can give information on the stripping process between the projectile and the

target.

energy at the distance of closest approach between projectile and target. This matching

corresponds to the condition |i − f| = 1₂mv2 (see Sec. 3.2). It means that all the

ini-tial available neutron energy (neutron binding energy and incident energy per nucleon) is converted into the energy of neutron final state.

In order to compare experimental differential cross section spectra to theoretical calcu-lations some changes of variables are necessary. The differential cross section formula

dσ

dεf, obtained integrating Eq.(2.1) over bc and using Eq.(2.3), can be converted into the

differential form _dPdσcore k

by the Jacobian of Eq.(3.2) with respect to εf

_dPcore k dεf

.

Other corrections have to do with the experimental set-up. The target thickness is not negligible, thus we take care of this fact correcting the incident energy and accordingly the core parallel momentum distribution. The thickness effect can be understood by the following model. Before colliding with the target each nucleon of the beam has an in-cident energy more or less close to the peak value of the distribution curve. Entering into the target the beam particles lose some energy due to the projectile interaction with the target matter. When the projectile arrives in the middle of the target, we assumes that the reaction takes place. Thus we use as input for our computation the nucleon incident energy at the middle of the target. After the collision the residual nucleus passes through the target matter before being detected. From collision to detection it further

loses kinetic energy. We treat this effect by shifting the calculated Pcore

k distribution by

the momentum value corresponding to the energy loss after collision. These corrections are calculated with LISE++ software for each reaction on the bases of the experimental apparatus specifications such as the target thickness [18].

The combined effect of the maximum value of P_kcore given by Eq.(3.2) and the shift of

the Pcore

k distribution due to energy loss can be observed in several experimental spectra.

The overall effect gives a cut-off to the spectra on the high momentum values.

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3.2 The nucleon transverse momentum η 19

shape, a convolution integral is used. The convoluted differential cross section is dσconv dPcore k = Z +∞ 0 dσth d ¯Pcore k g( ¯P_kcore− Pcore k )d ¯Pkcore (3.3)

where g(¯ζ − ζ) is a function that suitably approximates the initial beam shape.

g(¯ζ − ζ) can be chosen for example as a Gaussian distribution

g(¯ζ − ζ) = √1

2πξe

−( ¯ζ−ζ)2

2ξ2 (3.4)

with the standard deviation ξ calculated from the initial beam shape.

3.2 The nucleon transverse momentum η

The quantity η, appearing in Eq.(2.5) and defined as

η2 = k2₁+ γ_i2 = k2₂− k2_f, (3.5)

is the magnitude of the transverse component ~k⊥= i~η of the neutron momentum. ~k⊥

is conserved during the breakup reaction and it is purely imaginary because the neutron

which in the initial state has negative energy εi is emitted through a potential barrier.

η depends on εi and εf trough the definitions of γi and k1 or εf and k2 given in App. A.

η [ fm -1] 0 0.5 1 1.5 2 2.5 3 3.5 4 k1 [fm-1] −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 ________Sn____Einc_______ 23.18 53.0 (14O) 21.28 116.2 (10C) 17.01 28.5 (13O) 15.04 118.4 (32S) 14.25 63.8 (9C) 10.25 70.2 (57Ni) 8.02 70.0 (46Ar) 6.81 77.8 (10Be) 5.66 36.6 (6Li) 1.22 103.0 (15C)

Fig. 3.2: The η parameter as a function of k1 for various nuclei. In the legend, the neutron separation

energies Sn (MeV) and incident energies Einc (A.MeV) are specified.

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20 3. Kinematic parameters η [ fm -1] 0 0.5 1 1.5 2 2.5 3 3.5 4 εf [MeV] −20 0 20 40 60 80 100 120 140 160 180 200 _________Sn____Einc_______ 23.18 53.0 (14O) 21.28 116.2 (10C) 17.01 28.5 (13O) 15.04 118.4 (32S) 14.25 63.8 (9C) 10.25 70.2 (57Ni) 8.02 70.0 (46Ar) 6.81 77.8 (10Be) 5.66 36.6 (6Li) 1.22 103.0 (15C)

Fig. 3.3: The η parameter as a function of εf for various nuclei. The neutron separation energies Sn

(MeV) and incident energies Einc (A.MeV) are specified in the legend

and target. v is defined by Eq.(A.5) and it is related to the center-of-mass energy ECM

and consequently to the incident energy Einc in the laboratory reference frame, which is

Einc=

AP + AT

AT

ECM. (3.6)

For a given incident nucleus, γi is given by Eq.(A.10) and the minimum of η corresponds

to k1 = 0. When k1 = 0, η2 = γi2 = 2m_~2 Sn, where Sn = |εi|, thus the minimum value of η contains the information on the neutron initial binding energy. The smaller the η, the more weakly bound the neutron is. In Fig. 3.2 we show the behaviour of η for various nuclei, as a function of k1.

η enters the transfer probability Eq.(2.3) via the form factor in Eq.(2.5)

I = e

−2ηbc 2ηbc

, (3.7)

which is due to the combined effects of the initial and final wave function Fourier trans-forms. The maximum of the form factor corresponds to the minimum of η, thus we expect that this behaviour will be reflected in the differential cross section, with the peak at the

value of εf which realizes the condition k1 = 0. k1 = 0 is satisfied for

|Q| = |εi− εf| = 1

2mv

2 _(3.8)

where Q is the optimum Q-value of the transfer reaction and εf = εi+1₂mv2 is the most

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3.2 The nucleon transverse momentum η 21

In Fig. 3.3, η is plotted as a function of the neutron-target final relative energy εf for

different projectiles on a9_{Be target. The minima occur at the most favoured final energy}

for each nucleus. Nuclei with small difference between Sn and Einc have their minimum

at low εf, on the contrary nuclei with higher Einc have minima at higher εf.

The shape of the η-parabola is asymmetric for most of the nuclei. In fact, although it

has the peak in k1 = 0, the permitted values of k1 are between k1min = −

εi+1₂mv2 ~v and kmax 1 = εmax f −εi+1₂mv2

~v . The shorter the left branch of the parabola is, the smaller the

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Chapter 4 Structure inputs for the reaction

model

4.1 Initial state Wood-Saxons wave functions

The neutron and proton single-particle wave functions are needed to describe the initial state in the transfer-to-the continuum formalism.

4.1.1 Neutron wave function

The asymptotic function for the neutron initial state (see Eq.(A.14)) contains the

in-formation on the exact bound-state solution of the Schr¨odinger equation through the

asymptotic normalization constant of the initial state Ci. It is defined as

Ci = lim

r→∞−

χjili(r) γiilih+li(iγir)

. (4.1)

χjili(r), the radial part of the exact bound-state wave function, depends on the initial

momenta ji and li. This wave function is calculated using a one-body potential of

Woods-Saxon form in order to reproduce the experimental value of the neutron separation energy

Sn. To represent a bound-state, we choose the Woods-Saxon potential form as

V (r) = V0(r) + VSO(r)l · σ + Vc(r), (4.2)

where V0(r) is the spin-independent central potential

V0(r) = V0

1 1 + expr−R0

a0

, (4.3)

Vso(r) is the spin-orbit potential

Vso(r) = ~ mπc 2 Vso 1 r d dr   1 1 + expr−Rso aso  , (4.4) 22

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4.2 Spectroscopic factor and center-of-mass corrections 23

with the Pauli-matrices σ, and Vc(r) is the Coulomb potential. It is zero for neutrons

while for protons it is defined as

Vc(r) = (_Z_c_e2 r for r ≥ Rc Ze2 Rc h 3 2 − r2 2R2 c i for r < Rc , (4.5)

with Rc = rc(Acore)1/3 and rc is chosen as 1.3 fm. It has to be remembered that in the

Schr¨odinger equation also the centrifugal potential Vl(r) enters,

Vl(r) = ~

2_{l(l + 1)}

2mr2 . (4.6)

We calculate numerically the neutron wave function using R0 = r0(Acore)1/3 and Rso =

rso(Acore)1/3 with the prescription r0=rso=1.25 fm commonly adopted to describe the

shell-model wave functions. For the diffuseness parameter we choose a0=aso=0.7 fm. We

use Acore and Zcore because the valence nucleon in the nucleus AZXN feels the effect of

the other Acore nucleons as well as Zcore protons. We fix Vso = 5.5 MeV and give li and

ji as input. Tuning V0 to reproduce the experimental Sn, we obtain the corresponding

asymptotic normalization constant Ci and the exact single-particle wave function.

4.1.2 Proton wave function

With the same computer code, including the Coulomb potential, we can calculate the

initial valence proton wave function. Reproducing the proton separation energy Sp, we

obtain the asymptotic normalization constant and the exact single-particle wave function for the valence proton in its initial state.

We need to calculate the no-breakup proton probability P−pentering Eq.(2.8). We

approx-imate the proton wave function as a neutron wave function of larger separation energy according to Ref.[19] and [20]. From the calculations done for various nuclei, we have found that the effective neutron separation energy is larger than the proton separation energy. This effect is due to the Coulomb interaction: the proton wave function is larger in the region inside the Coulomb barrier than the corresponding neutron wave function not constrained by the Coulomb barrier, thus the neutron has to be more bound to reproduce the proton wave function. For the light nuclei studied in this thesis up to A = 16 the

difference Sn(ef f )− Sp changes with A from 0.5 to 1 MeV, for heavier nuclei the difference

varies from 1.5 to 4 MeV. This is in agreement with the fact that the Coulomb barrier

and Rc increase with A.

4.2 Spectroscopic factor and center-of-mass

correc-tions

As mentioned in Sec. 1.2.2 the nuclear structure reflects the interplay between single-particle degrees and collective degrees of freedom. The single-single-particle states are mixed by residual nucleon-nucleon interactions, which can produce nuclear deformations. This mixing modifies the occupancy numbers, called spectroscopic factors and here indicated

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24 4. Structure inputs for the reaction model ra d ia l w av e f u n ct io n [ fm -3 /2] 0.1 0.2 0.3 0.4 r [fm] 0 2 4 6 8 10 Sp=21.28 MeV Sn=21.22 MeV Sn=21.88 MeV Sn=22.54 MeV ra d ia l w av e f u n ct io n [ fm -3 /2] 0.0001 0.001 0.01 0.1 r [fm] 0 2 4 6 8 10

Fig. 4.1: Example of the proton wave function (empty black dots) and the neutron effective wave

functions, changing the effective neutron separation energy Sn for nucleus15C. In the small graphic the

y-scale is logarithmic.

Such structure changes can be found in the new magic numbers of nuclei with a large neutron excess or near the drip lines.

The spectroscopic factor is obtained from reactions in which a nucleon is removed from a specific initial state i in a nucleus A to a final state f in a nucleus A − 1. Its experimental value is the ratio between the measured total cross section and the calculated single particle cross section

(C2S)exp = σ exp σsp

. (4.7)

Following Ref.[21], the theoretical spectroscopic factor for one-nucleon removal is defined as

(C2S)th_lj =

Z ∞

0

dr0r02Ilj(r0). (4.8)

Ilj(r) is the so called radial overlap function Ilj(r0) = 1 √ AhYl(ˆr 0_{) ⊗ χ}τ 1/2 j ⊗ ψ A−1 f |ψ A i i, (4.9)

where Yl is the spherical Bessel function and χτ1/2 is the spin-isospin function of the

removed nucleon with total angular momentum j. The overlap integral between the

projectile nucleus wave function ψA

i and the residue nucleus wave function ψ

A−1

f is carried

out over the coordinates that describe the internal structure of the nucleus A − 1. Thus

Ilj(r0) is a function of the distance r0 between the center of mass of A − 1 and the valence

nucleon.

Spectroscopic factors for nuclei with even number of valence neutrons or protons generally exceed those of the neighbouring nuclei with odd number because pairs of neutrons or protons are coupled to spin zero by the pairing interaction. An empirical approximation for spectroscopic factor can be obtained following Ref.[22]:

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4.3 VMC wave functions 25

(C2S) = 1 − n − 1

2j + 1 (n odd).

where n is the number of neutrons or protons in the outermost shell.

A correction to the spectroscopic factor is the center-of-mass correction which enhances the calculated total cross section. As discussed in Ref.[23] this is a correction due to the effect of the center of mass asymmetry between the A-1 and A nuclei. The independent particle model does not satisfy translational invariance, thus imposing this symmetry in our model produces effects on the nuclear wave function. These changes can be taken as a correction, the amount of which is

Ccm = A A − 1 2Nho+l , (4.10)

where Nho is the principal quantum number of neutron initial wave function in the

spec-troscopic notation and l is the angular momentum of the neutron initial bound state.

4.3 VMC wave functions

In the previous discussion, the valence nucleons wave functions are described as

solu-tions of the Hamiltonian in the Schr¨odinger equation containing a mean field potential of

Woods-Saxon form. With this formalism the shell-model structure of the nucleus as well as structure quantities like the spectroscopic factors and the root mean square radii of core and valence nucleons can be derived.

Recently new theoretical models have been developed in order to obtain structure infor-mation taken from ab initio approaches which do not rely on shell-model approxiinfor-mations. The interactions among the nucleons inside the nucleus can be studied with more realistic Hamiltonians which contain the nucleon-nucleon NN effective potentials (two body in-teraction) and the NNN potentials (three body inin-teraction). The three body interaction is fundamental to describe nuclei with A≥3 and their properties such as their binding energy.

The two body potentials have been adjusted with accurate fits to n-p and p-p

scatter-ing data. The most famous example of these potential is the Argonne v18, presented in

Ref.[24]. Examples of three body potentials, set to reproduce the binding energies of light nuclei, are the Urbana IX, described in Ref.[25], and Illinois, introduced in Ref.[26], based on meson-exchange potentials.

These potentials are employed in the variational MonteCarlo VMC (or in the Green’s function MonteCarlo GFMC) method to calculate the overlap function Eq.(4.9) between the projectile state with A nucleons and the residue state with A-1 nucleons. The VMC method uses a variational principle with parameterized wave functions solutions of

many-body Schr¨odinger equation (see Ref.[27]). Minimizing the expectation value of the

Hamil-tonian with respect to the wave function parameters gives an upper bound to the exact binding energy. This method allows for the calculation of the binding energy and the overlap functions for nuclei up to A=12 (p-shell). This method even if it works with ap-proximate solutions is supposed to give the most accurate wave functions because it takes

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26 4. Structure inputs for the reaction model

into account nucleon-nucleon correlations, which are instead neglected in the shell-model method.

As shown in Ref.[28] (particularly in Fig. 1) the VMC overlap function has however the drawback of not providing the correct behaviour of the wave function at large distances. In fact for distances larger than 5 fm the calculated wave function starts to oscillate and the binding energy is no longer well reproduced. These difficulties are due to the neces-sity of constraining the wave function in the short-range part, in order to get the correct normalization (the spectroscopic factor), while the long-range part should give a consis-tent asymptotic normalization constant. The radial overlap functions, calculated with the VMC method, are not normalized to unity as the Woods-Saxon radial wave functions,

but to the spectroscopic factor SF. Because of this, their asymptotic normalization

con-stant AN C, defined as the ratio between the radial overlap function and the Whittaker

function, is AN C = Ci

√

SF compared to the Woods-Saxon wave function Ci.

The VMC overlap function behaviour can be reproduced by Woods-Saxon wave function with appropriate parameters which fit the exact separation energy. Some examples can be found in Ref.[29]. In the following, we will calculate Woods-Saxon wave functions that fit the VMC wave function available at Ref.[30]. With the structure inputs obtained we will calculate the theoretical knockout cross sections.

It has to be outlined that in the VMC case the spectroscopic factor does not need the center of mass correction because this effect is directly included in the overlap function calculation. ra d ia l w av e f u n ct io n [f m -3 /2] 0.0001 0.01 1 r [fm] 0 5 10 15 ra d ia l w av e f u n ct io n [ fm -3 /2] 0 0.1 0.2 0.3 r [fm] 0 5 10 15 <7Li|6Li> p3/2 overlap

7Be p3/2 VMC eﬀ. neutron

Fig. 4.2: Example of VMC overlap function h7Li|6Lii for the p3/2 state (empty black dots) and the

Woods-Saxon wave function (green line) that fits the VMC function behaviour for r . 5 fm to reproduce the p3/2effective neutron state in 7Be.

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Chapter 5 Choice of the strong absorption

radius R

_s

The strong absorption radius Rs is a fundamental parameter in the knockout model

pre-viously discussed, entering in the function that describes the core-target interaction |Sct|2

(see Eq.(2.7)).

It is defined as the distance of closest approach between the core and the target bc when

|Sct(bc)|2=1/2, distance at which both nuclei survive, i.e. the overlap of core wave

func-tions and target wave funcfunc-tions is negligible.

In literature this parameter is usually quantified as Rs = rs(A 1/3 C + A 1/3 T ) (5.1) with rs= 1.4 fm.

When VMC wave functions are used to describe the valence nucleon states, a root-mean-square radius usually larger than the one predicted by the shell model is obtained. In cases like that the wave function are more extended, then it seems reasonable to modify also the strong absorption radius in order to preserve the peripheral nature of the knock-out mechanism. Furthermore we expect that the core itself is large and consequently it should have a larger distance of minimum approach to survive at the interaction with the target.

In the following analysis we will discuss case by case the choice of Rs that give a better

description of the reaction, according to the experimental results.

In literature various prescriptions are available for Rs.

A parametrization can be extracted from the total reaction cross section model used by Kox [31] to describe nucleus-nucleus scattering at intermediate incident energies. This model takes care of a microscopic model for nucleon-nucleon interaction, of the Coulomb barrier and of the distance of closest approach between the projectile and the target.

σ = πR2_s 1 − BC ECM , (5.2) with Rs = Rvol+ Rsurf. (5.3) 27

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28 5. Choice of the strong absorption radius Rs

The microscopic model corrections enter in the volume contribution

Rvol= r0(A1/3_P + A1/3_T ), (5.4)

with r0 set to 1.1 fm, and in the surface contribution

Rsurf = r0 k

A1/3_P A1/3_T A1/3_P + A1/3_T − c

!

. (5.5)

where k is a constant and c is an energy dependent parameter. BC is the height of the

Coulomb barrier BC = e2_Z PZT rC(R 1/3 P + R 1/3 T ) , (5.6)

with rC=1.3 fm, and ECM is the energy in the center-of-mass reference.

The parametrization of Rs in Eq.(5.3) is interesting because Rsurf contains c, an energy

dependent parameter which accounts for the increase of surface transparency as the inci-dent projectile energy increases.

Thus Rs should be energy dependent and decrease at high energy.

In Ref.[32], Rs is distinguished by the kind of scattering it is used to represent. In

rs [ fm ] 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 Einc [MeV] 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 12C 40Ca

Fig. 5.1: Dependence of rs on the incident energy obtained from Sct for12C and40Ca projectiles. Sct

is calculated from a folding potential with realistic core density.

optical model analyses, it is called RCoul and it represents the classical distance of closest

approach for a given l

RCoul =

ξ + (ξ2+ l1/2(l1/2+ 1))1/2

k (5.7)

where l1/2 is the angular momentum corresponding to 1 − |Sl|2=1/2, ξ is the Sommerfeld

parameter (for a semi-classical trajectory ξ = Z1Z2e2 ~ v∞ , k =

µ v∞

~ . With this method

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29

in fm unit.

In the cases where the nuclear interaction dominates the reaction mechanism, it is called

Rint and it is defined in fm units as

Rint = Rcore+ Rtarget+ 3.2. (5.9)

Experimentally Rint− (Rcore+ Rtarget) tends to be smaller than RCoul− (Rcore+ Rtarget).

An example of the Rs dependence on the incident energy is shown in Fig. 5. Here

Rs= rs(A1/3_C +A1/3_T ) is considered and the values of the parameter rsare plotted. They are obtained from the value of bcfor which |Sct(bc)|2=1/2 at various incident energy. For these

calculation |Sct|2 is given by the imaginary phase shift derived from the folding potential

between the neutron-target potential [14] and the core density taken from Ref.[33] for12C

and 40_Ca.

The behaviour of rs for 12C and 40Ca gives us an idea of the range spanned increasing

Einc and increasing A. While the 12C and 40Ca nuclei have N=Z, the exotic nuclei that

we will analyse have different ratio N/Z and then the behaviour of rs can change a bit

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Chapter 6 Systematic analysis of experimental

data

In the following we apply the model presented in the previous chapters to analyse one-neutron transfer-to-the-continuum reactions which involve various projectile nuclei. We

focus our attention on nuclei with large difference in separation energy |Sn− Sp| for

va-lence neutron and proton that are already studied in the literature. For each nucleus information on its structure, particularly on its valence nucleons, is provided as well as the experimental inclusive knockout cross sections. The reference articles are specified for each case. The aim of our work is to calculate the theoretical inclusive cross section with our transfer-to-the-continuum model in order to compare with the existing data in the literature.

When available, it is also interesting to compare our results with the theoretical calcu-lations given in each reference, usually obtained with the eikonal model. It can help understanding the validity of the approximations done in the two models and the differ-ences can give further information on the reaction mechanism.

The inputs and the results of the calculations are presented in tables.

For each reaction, the projectile valence nucleon properties are given in the structure table. The initial single-particle bound state is specified together with the model and parameters used to reproduce its wave function.

As standard parametrization (that in the following we will call standard Woods-Saxon)

we chose a Woods-Saxon plus spin-orbit potential with the parameters r0(= rso)= 1.25

fm, a0(= aso)= 0.7 fm, Vso= 5.5 MeV and rc= 1.3 fm in the proton case, because these

are the values commonly used to describe shell-model states. If in the references different parametrizations are given, we will report them in the table to compare the wave function inputs and the wave functions information we get from it. For example, as discussed in Sec. 4.3, for some light nuclei VMC wave functions are available. In these cases we will give the parameters of an equivalent Woods-Saxon which provides a wave function which fits the VMC wave functions and also the experimental binding energy. We use equiva-lent wave function because our reaction model provides an analytical formula in which the initial state enters only through its asymptotic form (cf. Eq.(A.14)).

For each set of wave function inputs we will show the corresponding information about the single-particle state: separation energy, asymptotic normalization constant and

A systematic study of knockout reactions from exotic nuclei with anomalous ratio N/Z

UNIVERSIT `

A DEGLI STUDI DI PISA

Facolt`

a di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Magistrale in Scienze Fisiche

TESI DI LAUREA

A systematic study of knockout reactions

from exotic nuclei with anomalous ratio N/Z

Candidato:

Relatore:

Gianluca Salvioni

Dott.ssa Angela Bonaccorso

Nothing is over

until it is over...

Contents

Abstract

Chapter 1

Introduction

1.1

Nuclear properties

1.1.1

Nuclei and binding energy

1.2

Stable nuclei vs radioactive nuclei

1.2.1

Nuclear radius and halo nuclei

1.2.2

Nuclear shell model and magic numbers

1.3

Radioactive nuclides production

1.3.1

ISOL method

1.3.2

In-Flight method

1.3.3

RIBs in the future and their applications

Chapter 2

Knockout reaction model

2.1

Transfer to the continuum formalism

2.2

Knocking out a strongly bound neutron while

skipping a weakly bound proton

2.3

Nucleon-target S-matrix

Chapter 3

Kinematic parameters

3.1

Parallel momentum distribution

3.2

The nucleon transverse momentum η

Chapter 4

Structure inputs for the reaction

model

4.1

Initial state Wood-Saxons wave functions

4.1.1

Neutron wave function

4.1.2

Proton wave function

4.2

Spectroscopic factor and center-of-mass

correc-tions

4.3

VMC wave functions

Chapter 5

Choice of the strong absorption

radius R

s

Chapter 6

Systematic analysis of experimental

data

_s