Studies of Alpha Clustering in Nuclei

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Studies of Alpha Clustering in Nuclei

Jinesh Kallunkathariyil

Thesis Promoter : Dr hab. Andrzej Wieloch

Co-promoter

: Dr Piotr Pawłowski

Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian

University

This dissertation is submitted for the degree of

Doctor of Philosophy

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iii Wydział Fizyki, Astronomii i Informatyki Stosowanej

Uniwersytet Jagiello´nski

O´swiadczenie

Ja ni˙zej podpisany mgr Jinesh Kallunkathariyil doktorant Wydziału Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiello´nskiego o´swiadczam, ˙ze przedło˙zona przeze mnie rozprawa doktorska pt. "Studies of Alpha Clustering in Nuclei" jest oryginalna i przed-stawia wyniki bada´n wykonanych przeze mnie osobi´scie, pod kierunkiem dr hab. Andrzej Wieloch. Prac˛e napisałem samodzielnie.

O´swiadczam, ˙ze moja rozprawa doktorska została opracowana zgodnie z Ustaw ˛a o prawie autorskim i prawach pokrewnych z dnia 4 lutego 1994 r. (Dziennik Ustaw 1994 nr 24 poz. 83 wraz z pó´zniejszymi zmianami).

Jestem ´swiadom, ˙ze niezgodno´s´c niniejszego o´swiadczenia z prawd ˛a ujawniona w dowolnym czasie, niezale˙znie od skutków prawnych wynikaj ˛acych z ww. ustawy, mo˙ze spowodowa´c uniewa˙znienie stopnia nabytego na podstawie tej rozprawy.

Kraków, dnia ... ... podpis doktorant

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Abstract

In this thesis, I present theoretical studies for the possible existence of alpha clustering in nuclei. During my PhD work, I was doing calculations based on a new Quantum Molecular Dynamics like model originally developed by Dr hab. Zbigniew Sosin in Fortran language. I wrote a C++ version of this model with the inspiration from Fortran code. The interaction between nucleons are described by a new form of Equation of State (EOS) of nuclear matter, which depends on both spins and isospins. For the finite nuclei, the Coulomb interaction between protons and the surface effects are also taken into account. In the preliminary step, we calculated the ground state properties of nucleus by the minimization of its total energy, which will be a starting point for future nuclear dynamics calculations. The calculation results are compared with the experimental binding energy, r.m.s. charge radius and the radial density distribution of nuclei. Also it is presented a systematic study of binding energy curve based on the extended Bethe assumption on number of alpha-alpha bonds.

Streszczenie

W niniejszej rozprawie przedstawiam badania teoretyczne dotycz ˛ace ewentualnego istnienia struktur alfowych w j ˛adrach atomowych. W trakcie pracy doktorskiej, robiłem obliczenia oparte o nowy model typu Kwantowej Dynamiki Molekularnej, pierwotnie opracowany przez dr hab. Zbigniewa Sosina w j˛ezyku Fortran. Napisałem wersj˛e C++ tego modelu bazuj ˛ac na kodzie Fortran. W modelu interakcja mi˛edzy nukleonami opisana jest przez now ˛a form˛e równania stanu (EOS) materii j ˛adrowej, która uwzgl˛ednia zarówno spiny jak i izospiny nuk-leonów. Model uwzgl˛ednia równie˙z, oddziaływanie kulombowskie mi˛edzy protonami i efekty powierzchniowe dla skończonych j ˛ader. We wst˛epnym etapie obliczyli´smy wła´sciwo´sci stanu podstawowego j ˛adra poprzez minimalizacj˛e jego całkowitej energii. Rachunek ten b˛edzie punktem wyj´scia dla przyszłych obliczeń dynamiki j ˛adrowej. Wyniki obliczeń porównano z eksperymentaln ˛a energi ˛a wi ˛azania, ´srednim promieniem kwadratowym rozkładu ładunku i radialnym profilem g˛esto´sci masy j ˛ader. Ponadto w pracy zaprezentowano wyniki badań nad krzyw ˛a energii wi ˛azania j ˛ader w oparciu o rozszerzon ˛a hipotez˛e Bethe na temat liczby wi ˛azań pomi˛edzy cz ˛astkami alfa-alfa.

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Chapter 1 Introduction

Clustering can be considered as a universal mechanism. Galaxy clusters are the largest structures known today and Virgo cluster [1] is such a cluster. Galaxy itself is a cluster of stars. Most of these stars are surrounded by a cluster of planets. If we go to the small scale structures, atoms are clustered inside crystals. Protons and neutrons are clustered inside alpha particle. Quarks are clustered to form protons and neutrons. Thus it would be highly surprising, if we cannot find any cluster structures inside atomic nucleus [2]. Indeed the idea of alpha clustering in nuclei had appeared just after the discovery of atomic nuclei. Today the most accepted idea in the scientific community is that there may be alpha clustering in the low density region and at excited states of nuclei.

In this thesis, we are searching for the possible existence of alpha like clusters in nuclei. These studies are in line with the works of Bethe-Bacher [3] and Hafstad-Teller [4]. Ac-cording to Bethe-Bacher, nuclei can be considered as constituted of alpha particles, where

8_Be_{has two alphas with one bond between them, in}12_{C, there are three alphas with three}

bonds between them, and four alphas with six bonds are forming16O. They also predicted that after12C, for each addition of alpha particle, there is an increment of 3 bonds. All the considered nuclei are even-even nuclei with equal number of protons and neutrons commonly called alpha conjugate nuclei. Hafstad-Teller plotted the total binding energy as a function of number of bonds and found that the binding energy linearly increase with the number of bonds approximately. Unfortunately, such a prediction does not hold beyond56Ni, where the binding energy decreases rapidly as shown in Fig. 1.1. The Coulomb repulsion is understood to be the reason behind the drop in binding energy after this peak. Many theoreticians questioned the existence of alpha clustered nuclei above56Ni, even if they existed below this peak. Our work can be considered as a continuation of the approach of Bethe and Hafstad, in which we try to explain the structure of binding energy curve based on a microscopic model.

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Z 0 10 20 30 40 50 B/A 7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6

Experimental binding energy

56Ni

Fig. 1.1 Experimental binding energy curve: binding energy per nucleon as a function of charge Z. Source: [5].

The binding energy of light to medium nuclei with N=Z=even (alpha conjugate nuclei) is larger compared to the near neighbours. In the beginning of the nuclear studies, this large binding energy of alpha conjugate nuclei was explained by the alpha clustering phenomena, but later on when the shell model was developed, alpha clustering picture faded. A historical theoretical and experimental development of the nuclear models, alpha clustering and some of the recent experimental evidences for alpha clustering are briefly discussed in Chapter 2. This study is mainly based on a microscopic Quantum Molecular Dynamics (QMD) like model developed by Zbigniew Sosin and it treats a nuclear system wavefunction as a product of Gaussian nucleon wave functions. The interactions between nucleons are represented by a new form of parametrised Equation of State (EoS) of nuclear matter expanded around the saturation density ρ0 which also includes spin asymmetry term in addition to the isospin

asymmetry term of the standard version. In the case of a finite nuclei, model also consider the Coulomb effects and the surface effects. The nucleon wave packets has three time dependant variables, mean position, mean momentum and the position variance. The time evolution of these variable can be calculated by the time dependant variational principle. On the other hand, ground state of nucleus can be calculated by the energy minimization. The model and the nuclear interactions are explained in Chapter 3.

In the energy minimized states obtained by the minimization, we could observe the alpha like localization of nucleons. In this minimization process, first the nucleon wave packets are distributed inside the nuclear volume randomly. Next the configuration is changed in such a way that the energy is minimized. It appears that in the minimized state, the mean positions of nucleon wave packets are arranged in a clustered way. This is due to the specific form of interaction which takes into account both spin and isospins of nucleons. On the surface of the nucleus, where the low density region exists, the width of the alpha like clusters are

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3 comparable to the radius of alpha particle, but on the center, the widths are mixed and no longer separable. This minimization calculations could fairly well reproduce the binding energies and the root mean square (r.m.s.) charge radii of the N = Z =even nuclei from8Be till 40Ca, where the EoS parameters are calculated from the ground state binding energy and r.m.s. charge radius of light nuclei2H,3H,3Heand4He. The minimization procedure, results and some predictions are shown in Chapter 4.

Bethe-Bacher’s picture of alpha clusters in nuclei breaks after56Ni, especially the rule saying on the increase in the number of bonds. With the help of a simple equation, we try to explain the behaviour of binding energy curve after56Ni, where we propose the increase in number of bonds is one for each addition of alpha particle. The results of calculations with this equation is presented in Chapter 5. We hope that these results can be confirmed by the future calculations with our model.

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Chapter 2 An overview of alpha clustering studies

in nuclei

2.1 Early models of nucleus

In 1904, after the discovery of electrons, J.J. Thomson proposed a ’plum pudding model’ for the structure of atom, even before the discovery of protons and neutrons. According to him, since atoms are neutral, negatively charged electrons are randomly distributed within a sphere of equal positive charge. After the discovery of α-particle decay in 1899, and its charge in 1907, Ernest Rutherford devised an experiment to test J.J. Thomson’s model. Hans Geiger and Ernest Marsden performed this experiment in 1911 by directing alpha particles from a radioactive source (radium) to gold foil and then by detecting the scattered alpha particles. Rutherford thought, if atoms obey the model of J.J. Thomson, then alpha particles should not interact much with the neutral nucleus, and they should feel only a weak deflection. The result of the experiment was quite surprising: a part of the particles were deflected at very large angles. Since the alpha particles are much heavier than electrons, it was not possible to observe the alpha particles at such large angles, if atoms were build like Thomson predicted. To explain this observation, Rutherford postulated a new model of atom, where the electrons and positive charges were separated and the positive charges were confined into a very small area. This lead to the idea of an atom with a massive positively-charged nucleus and negatively charged electrons orbiting around it. Next in 1919, Rutherford also found each nucleus is built of hydrogen nuclei and he termed them ’Protons’. Nevertheless it was still not known how it is possible for positively charged protons to held together in nuclei heavier than hydrogen.

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Gamow’s alpha particle nucleus

In 1929 G. Gamow came up with an idea of the nuclei constituted like liquid drop off alpha particles [6, 7]. At that time neutron was not discovered and the α-particle was considered to be constituted of 4 protons + 2 electrons. According to him, nucleus can be considered as a staple configuration of positively charged α-particles, with an attractive force between them which come into play only for a close approach and overbalance at short distances due to the electrostatic repulsion. The experiment which lead him to think so was the scattering of alpha particles from helium. He noticed that at lower energies, alpha scattering obeyed Coulomb repulsion, but at high energies it showed an increase of cross-section of alpha particles scattered at a specific angle. This effect was interpreted as an attractive interaction. At high energies alpha particles can go closer to the nuclei and at these distances, the attractive interaction dominates and change the course of alpha particles. He proposed an equation for the mutual potential energy of two alpha particles at distance r apart from each other as

U(r) = +4e2/r − f (r). (2.1) The positive sign corresponds to the repulsive Coulomb force between α-particles and negative sign corresponds to the attractive nuclear force. The function f (r) which he postulated as Ae−αr decreases very quickly with distance. Then he considered a collection of alpha particles (nucleus) attracting each other with the force that is very rapidly decreasing with distance. The attractive part of the potential energy of α particle is given by

−

Z _∞

d

f(r)4πr2ρ dr, (2.2) where d is the closest approach between the α particles and ρ is the alpha particle density. In this case the forces diminish rapidly with distance, so the integral converge quickly and need to consider only up to a distance r∗, where the attraction is negligible beyond. On the surface of such nucleus, forces try to drag the particles towards inside. Thus he concluded that such a collection of alpha particles behaves like a liquid drop, where the inside pressure and the forces of surface tension are balanced. This pressure is generated by the kinetic energy due to the quantized motion of alpha particles, while the surface tension forces are trying to diminish the drop radius. In such a drop, the density of particles decrease regularly towards the surface. The total energy of each alpha particle inside a nucleus is the sum of its kinetic energy Tα and the potential energy Vα.

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2.1 Early models of nucleus 7 The average momentum p of a single particle in a nucleus of radius r is given by

p∼ h

2r, (2.4)

thus the kinetic energy Tα is

T_α ∼ 1 2mp

2_∼ h2

8mr2. (2.5) From the virial theorem

V_α = −2Tα, (2.6)

he calculated the average potential energy per alpha particle as V_α ∼ − h

2

4mr2. (2.7)

Thus the total energy of α particle inside nucleus, calculated as [7], is T_α+Vα ∼ −

h2

4mr2. (2.8) Then, in a separate calculation, he equated the surface tension S(r, Nα) to the internal

pressure P(r, Nα), both of which are functions of r and Nα (number of alphas), and found

that r = RNα1/3, where R = 2 × 10−13cm. Now the total internal energy Eα of the Nα alpha

particles is E_α = N_α(T_α+V_α) ∼ − h 2 4mR2N 1/3 α . (2.9)

He also calculated the Coulomb effects by approximating nuclei to be a uniformly charged sphere of radius r and charge +2eN_α as

E_C∼ (2eN) 2 r = (2e)2 R N 5/3 α . (2.10)

Finally the total energy of the nucleus is given by E= Eα+ EC= − h2 4mR2N 1/3 α + (2e)2 R N 5/3 α . (2.11)

The results of Eq. 2.11 are shown in the Fig. 2.1. As one can see, the calculation of Gamow fits the experimental data only for light nuclei (Nα < 15). Even though this model could not

explain the properties of heavier nuclei, it was a good starting point for the future nuclear calculations.

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Fig. 2.1 Gamow’s plot of the total nuclear energy E (×102) versus the number of alpha particles N (Nα in Gamow’s notation) in nuclei, filled circles are data. Source: [7].

Extension of Gamow’s Model

In 1932, James Chadwick found that nucleus is not only constituted of positively charged protons but also of neutrons, ie. neutral particles with almost the same mass as protons [8]. After the discovery of neutrons, Heisenberg came up with the idea of proton and neutron bound together inside nucleus by exchanging electrons between them [9–11]. According to this idea, in2Hnuclear energy reaches saturation, but it was known that alpha particles are much more stable than deuteron. So in 1933, Majorana modelled a new interaction between protons and neutrons by exchanging both charge and spin (intrinsic angular momentum) [12]. According to this model, the most bounded nucleus was4He. Heisenberg then adopted Majorana’s exchange force in his calculations [13]. He calculated the total energy of nucleus in terms of the total number of protons and neutrons. In 1935 Heisenberg’s student Carl Friedrich von Weizsacker introduced semi empirical formula for the calculation of the mass of nuclei [14].

The nuclear energy can be derived in a more detailed way as shown by Bethe [3]. If the nuclei was constituted of protons and neutrons, then it is natural to think that its mass is the sum of the masses of the individual protons and neutrons. Surprisingly, the masses of the nuclei were found to be lower. If every nucleon in a nucleus interact with each other, then the nuclear binding energy should be roughly proportional to the total number of interacting pairs, which is A(A − 1) ∼ A2, while the experimental findings suggest that the binding energy increases linearly with the increasing number of particles. From this, one

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2.1 Early models of nucleus 9 can conclude that the nuclear forces are very short range forces and the nucleons interact only with nearest neighbours analogous to a liquid drop. As an example given by Bethe, two hydrogen atoms are bounded together strongly to form H2molecule, but a third hydrogen

atom does not strongly bind with the H2molecule. This means, the forces in H2molecule are

saturated. In an assembly of hydrogen atoms, for example a drop of liquid hydrogen, the total binding energy is approximately equal to the binding energy per hydrogen molecule times the corresponding number of molecules, which is again proportional to the total number of atoms present. In reality, the binding energy is a bit more than this because the forces between the H2molecules are van der Waals type, but it is still proportional to the number of

atoms present in the drop.

Analogues to the dependence of total binding energy to the number of atoms present in a liquid drop of H2, he proposed that the total binding energy of a nuclear drop is

also proportional to the total number of nucleons, if we assume the nuclear forces exhibit saturation. He considered that the most saturated subatomic ’molecule’ of nucleons is the most bounded α-particle, with its binding energy ∼ 7 MeV /A. So in the case of most bounded nα nucleus56Niwith binding energy ∼ 8.6 MeV /A, around ∼ 7 MeV of 8.6 MeV comes from the ’chemical binding energy of the molecule’4He, while the remaining 1.60 MeV is due to the ’van der Waals’ forces between the α-particles. This "van der Waals" forces or bonding between α-particles can be considered as attractive at larger distances, falling off very rapidly with increasing distances, and repulsive at small distances, giving a mutual impenetrability for two α-particles.

As we already know, the α-particles contain two neutrons and two protons. It means, the forces between neutrons and protons saturate when two neutrons and two protons are brought together and are practically null when a third proton or neutron is in the neighbourhood of the first 4 nucleons. An explanation to this is Pauli principle, which says that no two identical fermions can reside in the same quantum state. Thus alpha particle as subunits are more favourable in nucleus.

From the discussion so far, binding energy is proportional to the total number of nucleons Aand it gets maximal when neutrons and protons are present in equal numbers. This can be represented by a rough formula as

En= αVA+ αA

N − Z N+ Z

2

, (2.12)

where αV is the binding energy per nucleon for a nucleus containing equal number of protons

and neutrons, and αA is a constant which represents the dependence of binding energy on

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We cannot neglect the electrostatic force between protons in the nucleus. At nuclear distances, the electrostatic potential energy between two protons is ∼1₄MeV. This interaction can be neglected for small nuclei (binding energy per nucleon is about 8.5 MeV ), whereas for the larger nuclei, the Coulomb repulsion between protons is important, since there is no saturation for the Coulomb forces. If we suppose nucleus to be a sphere of equally distributed protons, then the nucleus has same charge density everywhere inside the sphere. The Coulomb interaction must be equal to the number of pair of protons in the nucleus, that is 1₂Z(Z − 1) times the potential energy of single proton pair. The energy from Coulomb interaction can be represented as

EC≈ 3 5 1 4πε0 Z2_e2 R . (2.13)

Experimental measurements showed that the nuclear radius is proportional to A1/3 as R≈ r₀A1/3, (2.14) where r0= 1.25 f m, thus EC can be expressed as

E_C≈3 5 1 4πε0 Z2e2 r₀A1/3 = 3e2Z2 20πε0r0A1/3 ≈ 3e 2_{Z(Z − 1)} 20πε0r0A1/3 = a_CZ(Z − 1) A1/3 , (2.15) with aCgiven by a_C= 3e 2 20πε0r0 . (2.16)

From the literature, Wick was first who pointed out that the nuclear binding energy will be reduced for nuclei by the existence of surface [15]. Particles on the surface interact only with half as many as other particles do in the interior of the nuclei, analogous to the surface tension of liquids. The surface effect was quantitatively calculated by Weizacker. It has a classical and quantum mechanical explanation. In classical thinking, the nucleus can be considered to have a sharp boundary at a certain distance R from the center. Those particles near to the boundary will not contribute their full share to the binding energy thus increasing the total energy (mass) of the nucleus. While in quantum theory, the boundary of the nucleus can never be sharp, because this would give an infinite derivative to the particle wave functions and as a result, the kinetic energy of the particles at surface would tend to infinity. Thus the surface layer must spread out over a certain region of the order of magnitude one particle wave length. The narrower the surface region, the larger will be the additional kinetic energy. On the other hand broader the surface region, bigger will be the potential energy. This will cause the nuclear surface to choose an optimum breadth to saturate both effects.

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2.2 Bethe-Bacher’s arguments for the α-particles subunits in heavier nuclei 11 Weizsacker’s semi-empirical mass formula

As we have seen from the previous discussions, Weizsacker proposed a semi-empirical method of calculating the mass of nuclei. According to this formula, the total energy (mass) of the nucleus can be calculated as

M= NMn+ ZMp− αVA+ αSA2/3+ αC

Z2 A + αA

(A − 2Z)2

A , (2.17) where N is the number of neutrons, Z is the number of protons, A = N + Z, Mnis the mass of

neutron and Mpis the mass of proton. The third term in the above equation is proportional

to the volume of nucleus, which is the dominant term in binding energy. Fourth term is the surface energy, fifth term is the energy excess due to the Coulomb interaction, and sixth term is the energy excess due to the asymmetry in the proton-neutron number. This formula could approximately well describe the total mass and the binding energy, but still could not explain the internal structure of nucleus.

Fig. 2.2 Von Weizsacker’s calculation of binding energy from the Eq. 2.17, compared with experimental data (dots). Source: [14].

2.2 Bethe-Bacher’s arguments for the α-particles subunits

in heavier nuclei

In 1936, Bethe and Bacher gave arguments to support and also to oppose the concept of α -particle subunits in nuclei. These arguments and the disagreements are given bellow in the subsections.

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Arguments to support the α-particle subunits in nuclei

1. The binding energy in heavier nuclei is approximately same as if the nuclei was only constituted of α-particles as referenced in section 2.1, where most contribution is coming from the binding energy of individual α-particles (≈ 28 MeV ) and relatively a small additional energy (≈ 7 MeV ) is from each α-α interactions.

2. The assumption of α-particles as subunits therefore seems to offer a straight forward method for the theoretical calculation of binding energy of heavier nuclei. Thus already in zero approximation, the binding energy of the heavy nucleus can be calculated as the sum of binding energies of each α-particles contained in it. If it can be shown that the alpha particles attract each other, then there will be a justified hope to arrive at a theoretical binding energy reasonably close to the observed one. In contrast to this the Hartree approximation, which assumes the nucleons move independently in the nucleus, fails to give satisfactory results for the binding energy.

3. Among the light nuclei, the ones which can be considered as consisting exclusively of α -particles such as8Be,12C,16O,20Ne, etc. have higher binding energy per nucleon than any of their neighbours.

4. Radioactive nuclei emit α-particles.

Arguments to oppose the α-particle subunits in nuclei

Bethe overruled the third argument by following explanations. First, the strong interaction between neutron and proton alone make the nuclei more stable for which the number of neutrons and protons are equal. Secondly, according to the Pauli principle, the nuclei are more stable, if they contain equal number of protons and neutrons.

To oppose the fourth argument, one can show that α-particles are the only favourable particles which can be emitted from radioactive nuclei, by the simple considerations involving energy and probability. Firstly, the binding energy of the α-particle is much higher than that of the preceding nuclei1H,2H,3H and3He. Therefore a given nucleusAZ may have higher total energy than the nucleiA−4(Z − 2) and4Hetogether, but will in general have lower energy than the nuclei for example,A−3(Z − 1) and3Hetogether. Thus the nucleus is energetically more unstable against α-particle decay, but stable against emission of any other lighter particles. Again, this nucleus will in general be energetically unstable against the breaking up in to12Cor16O, if it is unstable against α emission. But it is highly impossible for a heavy nucleus such as 12C to tunnel through the high and broad potential barrier

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2.2 Bethe-Bacher’s arguments for the α-particles subunits in heavier nuclei 13 existing between it and the residual nuclei, while the comparatively light α-particle with its comparatively small charge may tunnel through quite easily.

Bethe ruled out the rest of arguments by some general considerations. For nuclei heavier than about A = 30, the preference for nuclei composed exclusively of α-particles ceases to exist. The reason for this behaviour is the Coulomb repulsion between α-particles. This repulsion makes it necessary that stable heavy nuclei contain some extra neutrons as ’mortar’ keeping the α-particles together. Thus if the concept of α-particles as nuclear subunits is a good approximation, we would expect that all interactions between α-particles and additional neutrons must be small compared to the alpha particle binding energy. But the experimental evidences show that binding energy of this extra neutron is almost same as the binding energy per nucleon of α-particle, about 7 − 8 MeV . A binding energy of 7 − 8 MeV per additional neutron must correspond to a large perturbation of α-particles so that it become very doubtful to what extend one may speak about their existence as subunits in nuclei at all, at least for nuclei containing large number of extra neutrons.

Geometrical structure of nuclei composed of α-particles

From the special shape of interaction potential as a function of distance, proposed by Heisenberg, Bathe-Bacher tried to gave a purely geometrical arrangement of α-particles inside the nuclei. Two neighbouring α-particles in nucleus would have in general a mutual distance corresponding to the minimum of the interaction potential. Thus the 3α-particles of

12_C_{would be arranged in an equilateral triangle, the 4 αs in}16_O_{form a tetrahedron, etc. as}

shown in the Fig. 2.3. The binding energy would then in first approximation be proportional to the number of pairs of neighbouring α-particles which is 1 for8Be, 3 for12C(triangle), 6 for16O(tetrahedron) and then three more for each additional α-particle. Experimentally the mass of8Beis almost exactly same as that of two α-particles. Thus the mutual attraction of one pair of α’s is not sufficient to overcome the kinetic energy associated with their relative motion and as a result it is unstable against the alpha decay.

The final remarks given by Bethe-Bacher is that, at present, it cannot be decided, how much truth is in the assumption of α-particle subunits in nuclei. Also they pointed out that the above assumption is not to be taken literally, since the α-particles undergo considerable deformations (polarizations) inside the nucleus. On the other hand, the approximation assuming the elementary particle to move independently (Hartree approximation), is certainly not correct either, but must be supplemented by introducing correlations between the particles. Such correlations would lead at least in the direction of α-particle subunits approximation. So in their opinion, the truth will probably lie between the two extremes as Heisenberg pointed out.

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8_Be 12_C 16_O

Fig. 2.3 Bethe-Bacher prediction of geometrical structure of nuclei based on the alpha clustering model.

2.3 Hafstad’s and Teller’s work

In 1938, Hafstad and Teller discussed the α-particle model for light to medium even nuclei [4] consistent with that of Bethe-Bacher. Besides α-conjugate nuclei, authors also explained the structure of nuclei with an additional neutron or proton and also with a proton or neutron hole. Their treatments was in analogy with the molecular bonding in atoms, where alpha particles form the molecules while additional proton or neutron is treated as electrons moving in the field of these molecules. One important weakness of this approach as pointed out by the authors, particularly in heavy nuclei, is that the consideration of energy and space occupation make the stability of α-particle as subunits somewhat doubtful. Nevertheless they could obtain many interesting results by this simple approach. It is very significant fact that, while the Hartree and α-particle approximations starts from quit different assumptions, the results obtained agree in many features.

In order to describe the structure of nucleus constituted of alpha particles, Hafstad also considered the saturation character of nuclear forces. The saturation character of α-α in-teraction corresponds to the fact that, the binding energy of nuclei is proportional to the total number of particles contained in it (as if the particles interacts only with the nearest neighbours). On the other hand if each particle interact with all other particles, then the binding energy will be proportional to the square of the number of particles. Thus they assumed α-particles repels each other at small distances. In order to build a nucleus con-taining more than one α-particle, they introduced an attractive interaction at large distances and also a repulsive Coulomb interaction at even larger distance, as shown in the Fig. 2.4. Here, the repulsive force at small distance comes from the Pauli principle and attraction at larger distance comes from the van Der Waals interactions. The interaction forces are of an additive in nature, that is the interaction of two particle is not changed by the presence of a third α-particle. As discussed in the previous section, this kind of interaction should give Dumb-bell like shape for8Be, triangle for12C, tetrahedral structure for16Oetc. and in such a nuclei the interaction between α-particles will be short range one. The binding

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2.3 Hafstad’s and Teller’s work 15

Fig. 2.4 Interaction potential between two alpha particles as a function of their mutual distance. Source: [4].

energy should be proportional to the number of these short range bonds between adjacent α -particles. The bond can be considered only between the adjacent pairs of α -particles which has a geometrical distance about r0as marked in Fig. 2.4. In such an approximation

the binding energy plotted against the number of bonds can be found in Fig. 2.5. As can be seen from the figure, a very good linear relationship can be found for all nuclei presented till32Sexcept 8Be. For8Be, it is assumed that the kinetic energy of alpha particle is large enough to escape through the potential barrier. However authors doubt about the validity of such a relationship for heavier nuclei.

Nuclei of 4n + 1 type

For the nuclei containing a single proton or neutron in addition to the α-particles, the total binding energy is given by the binding energy of group of α-particles plus an additional energy due to the binding of single nucleon to the aforementioned α-particle group. In the simplest case of5He, the additional energy bN is given directly by the difference between

masses of5Heand4He+ n, thus

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Fig. 2.5 Binding energy of n − α nuclei as a function of number of bonds. Source: [4]. The case of5Liin which a proton is added to the α-particle will differ only slightly from that of5Hedue to the effect of Coulomb forces.

According to Hafstad, the interaction between neutron and the α-particle has a short range nature. This fact makes it plausible that in case of nuclei with more than one alpha particle, the neutron may only interact simultaneously with one α-particle which is at some average distance. For9Be, we have two α-particles, each with an associated wave function ψ for the neutron. These wave functions are designated as ψ1and ψ2. The total wave function

may be any linear combination of ψ1 and ψ2. Since the long wavelengths corresponds to

the lowest energies, they chose the combination ψ1+ ψ2. Thus the average energy of the

neutron will be given by, ¯ E(ψ1+ ψ2) = R (ψ1+ ψ2)H(ψ1+ ψ2) R (ψ1+ ψ2)2 , (2.19) where Hamiltonian H is given as H = V1+V2+ T and V1,V2refer to the potential energy of

neutron due to the interaction with first and second α-particle, respectively whereas T is the kinetic energy of neutron. On the expanding the nominator of the Eq. 2.19, author stated that the terms can be identified as binding, interaction and exchange terms, designated as bN,

Rand Q respectively. The bN term has the formRψ1(V1+ T )ψ1and refers to the potential

and kinetic energy of neutron due to interaction with α1(first α particle), when the neutron

is near to it. The term R in the formR

ψ1(V2)ψ1refers to the additional potential energy of

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2.4 Rotational energy 17 the formR

ψ1Hψ2. If we assign S = R

ψ1ψ2, then Eq. 2.19 can be written as

¯ E(ψ1+ ψ2) = 2(bN+ R + Q) R ψ₁2+Rψ₂2+ 2Rψ1ψ2 = bN+ R + Q 1 + S . (2.20) As we have seen earlier, neutron mainly interact with one α-particle at a time, thus ψ1and

ψ2do not overlap strongly and S ≪ 1, so we have

¯

E(ψ1+ ψ2) ∼= bN+ R + Q. (2.21)

Applying the same principle to the13C, as in the case of8Bewill give the energy of neutron as bN+ 2R + 2Q. The interpretation for this form can be understood as follows. As far as the

term bN is concerned, neutron will interact only with one α-particle at a time. So this term

will be the same as in the case of5Heand9Be. On the other hand for the interaction terms, Rand Q, the neutron located near to one alpha can simultaneously interact with other two neighbouring alpha particles. Therefore these terms are multiplied by a factor of two in case of13C. From the above discussions, one can approximately calculate the additional neutron energy as M(He5) − M(He4+ n) = B, M(Be9) − M(Be8+ n) = B + (R + Q) M(C13) − M(C12+ n) = B + 2(R + Q) M(O17) − M(O16+ n) > B + 3(R + Q), (2.22) where Hafstad set the value of B as 1.2 MeV and R + Q = −3.2 MeV .

2.4 Rotational energy

In order to better understand the structure of nucleus and symmetry properties of it’s states, it is needed to consider the orientation of nuclear constituents in space and its rotation. Taking into account nuclear rotation will allow us to calculate the excited states of nucleus. Additionally, if one also consider nuclear vibrations, then we get the higher level rotational excited states.

For an alpha conjugate nuclei, the rotational levels may be obtained very simply by comparison with the analogous molecules as presented by Wheeler [16]. For example8Be presents the case of diatomic molecule. Let’s consider a structure, which has a form of the diatomic molecule in the first approximation. It consists of two point masses m1and m2at

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fixed distances l1and l2from their center of mass as shown in the Fig. 2.6. Since the distance ℓ ℓ₂ ℓ₁ m₂ m₁ Center of mass

Fig. 2.6 Rigid rotator

between two masses is fixed, this structure is referred to as rigid rotator. Even though, a diatomic molecule vibrates as it rotates, the vibrational amplitude is small compared with the bond length. So considering the bond length fixed is a good approximation.

Consider the molecule which rotates about its center of mass at a frequency of f . The velocities of two masses are v1= 2πl1f and v2= 2πl2f, which we write as v1= l1ω and

v₂= l2ω , where ω = 2π f is the angular velocity. Thus the kinetic energy of this rigid rotator

is T = 1 2m1v 2 1+ 1 2m2v 2 2= 1 2(m1l 2 1+ m2l22)ω2 = 1 2Iω 2_, (2.23) where I is the moment of inertia. Using the fact that the location of center of mass is given by m1l1= m2l2, the moment of inertia can be written as I = µl2where l = l1+ l2and µ is

the reduced mass. Thus the angular momentum L = Iω and kinetic energy is T = L

2

2I. (2.24)

There is no potential energy in the absence of any external forces, therefore the Hamiltonian operator for rigid rotator is

ˆ

H= ˆT = −¯h

2

2I∇

2_. _(2.25)

The application of above operator will give energy of rotational states as E_J= ¯h

2

2IJ(J + 1) = BJ(J + 1) J= 0, 1, 2, ..., (2.26) where B = ¯h_2I2 is called the rotational constant of molecule.

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2.5 Hoyle state 19 If we come back to the case of 8Be nucleus, which can be compared to a diatomic molecule, then the rotational energy state is given by Eq. 2.26. However, since the alpha particles have spin zero, which obeys Bose statistics, the corresponding antisymmetrical wave functions for J = 1, 3, 5, ... are excluded. This kind of approach is used in recent experimental studies that will be discussed later on.

2.5 Hoyle state

In 1939, Bethe discussed the energy and nuclei production in stars [3]. He showed that no elements heavier than 4He can be built up in ordinary stars. This is due to the instability of 8Be which dramatically reduces the formation of heavier elements. He also checked the possibility of triple α collision leading to the formation of12Cnucleus, but it appeared that the probability is lower than 10−56per α particle. This is a very small probability as compared to the amount of12Cseen in the Universe. Additionally such a reaction needs a high temperature about 109degrees. The production of neutrons in stars is likewise negligible. So he concluded that the heavier elements found in stars must have existed already when the star was formed.

In 1954, Hoyle proposed that the triple-α process [17, 18] may happen in the following way

α + α ⇌8Be

8_Be_{+ α →}12_C_{+ γ.} (2.27)

In the first step, two α particles fuse to form8Be, where the instability of8Beagainst α decay results in an equilibrium concentration of8Be. Next step is the capture of α particle by8Be to form12Cat an excitation energy close to 7.65 MeV followed by γ decay into the ground state of12C. Hoyle’s prediction of such a favourable energy state motivated Dunbar and team to investigate it experimentally. They found that this energy state has a value of 7.68 MeV , but more refined data came out recently and now it is measured to be 7.65 MeV [18]. This is the second excited state of12C, Jπ _{= 0}+_{. The Hoyle state at 7.65 MeV resonantly boosts}

the capture process by a factor close to 107− 108_{. Thus Hoyle state solved the problem of}

connecting the gap between8Beand the12C. The prediction of the existence of 7.65 MeV state by Hoyle was an outstanding example of the anthropic principle. Anthropic principle relies on the fact that intelligent life exists, so there must exist certain properties of the universe which supports life, i.e. we exist so does the 7.65 MeV state in12C. The strong conclusion from historians is that Hoyle’s reliance was on the challenge of understanding

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the natural abundance of12C and 16O(the latter produced by capture of an α-particle by

12_{C) that drove his prediction, rather than a deeper connection to his own existence. It is also}

assumed that at this resonance, the structure of nucleus is more favoured for alpha clustering.

2.6 Ikeda diagram

Ikeda in his famous paper [19] predicted that nuclear states near to α-decay threshold may be alpha clustered. According to this idea, as the threshold for cluster decay is approached, the nucleus can take on a structure which will need minimal reconfiguration to decay by the alpha cluster fragment. This is shown in the so called Ikeda diagram, Fig. 2.7. After his

Fig. 2.7 Ikdea diagram. Source: [18].

paper, it was generally accepted that nucleus may be clustered into alpha particles at nuclear states near to the alpha decay threshold.

2.7 Alpha elastic scattering methods

There are many experimental methods applied to investigate the alpha clustering in nuclei. Here we concentrate on the α elastic resonant scattering methods. This kind of investigation can be found in [20].

In nuclear reactions involving a projectile and target, the low to intermediate energy projectile transfers energy or picks-up/loses nucleons from/to the target nucleus in a single event on a time scale of the order of 10−21s. If an excited compound nucleus is created it lives until one of its particle gets enough energy to overcome coulomb barrier. The time scale

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2.8 Recent experimental results 21 for such a process is typically of the order of 10−19s. In some cases the emitted particle will have the same nature and energy as the projectile particle. This kind of scattering is called elastic resonant scattering. In an elastic scattering process, the kinetic energy of a particle is conserved in the center-of-mass frame, but the direction of propagation is modified.

For example, to investigate the compound nuclear states in32S, and thus its structure, one can perform alpha scattering experiments with28Si. At Coulomb energies, alpha particles are scattering from28Siand it would give a Rutherford cross-section (smooth variation of scattered alpha particle intensity with angle). If the energy of the alpha particle is enough to overcome the Coulomb barrier and the energy matches with the energy states in the compound nuclei32S, then a resonance will take place between alpha particle and the target nucleus, and angular distribution of the alpha particle depends on the spin of the resonant level. In this case, the reaction will be

α +28Si→32S∗→28Si+ α. (2.28) So by analysing the widths, energy and angular distributions of the resonances, one can verify the alpha-clustered excited and bounded states. In order to get these information, mainly three experimental techniques are used. They are: thin target stepping technique, thick target backscattering technique and thick target inverse kinematics (TTIK) technique [20]. Example of results received with the help of all three techniques are shown in Fig. 2.8. From the figure one can see that all of them are able to measure the main resonance structures. It should be mentioned that the TTIK method can be very attractive for the experimentalists, because it consumes very little beam time as compared to the rest of the two techniques and also one can use radioactive targets.

2.8 Recent experimental results

From the previous discussions, one can assume 8Be as a cluster of alpha particles and the Hoyle state in 12C also can be considered as alpha clustered. Some of the recent experiments also can give more deep understanding to the α-particle subunits in nuclei. Recently, Akimune [21] and co-workers performed the4He(56Ni,4He) inelastic experiment at an incident energy of 50 MeV /u at GANIL. The main aim of the experiment was to search for the compression modes in56Ni. This experiment was done using TTIK method. From the measurements authors were able to extract the multiplicity of alpha particles (also protons, 2H and 3He). The maximum multiplicity they observed for alpha particles was seven as shown in Fig. 2.9. This result is striking, because such high multiplicity cannot

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Fig. 2.8 The shaded area shows the thin target data, the thick line represent the thick target data and the thin line represent the TTIK method. Source: [20].

Fig. 2.9 Multiplicity distributions of4He(A), proton (B), deuteron (C) and3He(D). The histograms and solid lines correspond to the experimental data and the results of fitting, respectively. Source: [21].

be explained by using a simple statistical-decay-model calculations. In order to understand this observation, they performed a simulation based on the ideal alpha-gas model. In this model, the momentum distribution follows the Boltzmann distribution. After taking into account the acceptance of the detector for multi-alpha particles events, authors got a fairly well agreement with the experimental data. This result strongly suggests the existence of an alpha like clustering in56Niat these incident energies. Norrby et al. has shown the results of elastic α scattering experiments done using TTIK method [22]. They examined nuclei

32_S,34_S,36_A_and40_{Ca. The large amount of strong and narrow resonances found in these}

nuclei cannot be explained by theory so far. They analysed the data with R-Matrix method and the spin distributions of the observed resonances have been interpreted as an indication

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2.8 Recent experimental results 23

Fig. 2.10 Comparison of resonances in 32S,34S,36A,40Ca. The diamonds represent aver-age energies of resonances with the same spin, weighted with the reduced widths of the resonances. Source: [22].

of a quantum mechanical rotator with the rotational levels split into many close lying states with the same spin as shown in Fig. 2.10. According to the author, the extracted moments of inertia point to a rotation involving a few alpha particles orbiting an inert core. This type of dynamical structure needs to be further verified experimentally and explained theoretically. In another experiment done by M. Barbui et al. [23], they explored alpha clustering in

24_Mg_{by studying the reaction}20_Ne_{+ α with TTIK technique. The}20_Ne_{beams of energy}

3.74 MeV /A and 11 MeV /A were delivered by the K150 cyclotron at Texas A&M University. The reaction chamber was filled with4Hegas at a pressure sufficient to stop the beam before the detectors. The energy of the light reaction products was measured by three silicon detector telescopes. They studied both single and multiple α-particle decays. These new yet preliminary results obtained on elastic resonant α scattering, as well as on inelastic processes leading to high excitation energy systems decaying by multiple α-particle emission showed the events with α-multiplicity one and two.

So far we discussed some historical development of nuclear structure theories and also the alpha cluster models and the experimental evidences for alpha clustering in nucleus. The alpha clustering model of Bethe did not advance further, because of the lack of experimental evidences during that time, and also due to the difficulties to explain the alpha cluster structures for nuclei beyond56Ni. On the other hand, Hoyle state and the nucleo-synthesis in stars cannot avoid concept of alpha clustering in nuclei. Also the Ikeda’s studies stressed the importance of alpha like clustering in the excited states in nuclei. The experimental

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evidences discussed above also show the possible existence of alpha clustering in nucleus. Thus a commonly accepted picture is that nuclei may be clustered at excited states and at subsaturation densities.

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Chapter 3 Concepts of Microscopic Liquid Drop

Model (MLDM)

As we have seen from the previous chapter, the accurate description of structure of nuclei is still not very well known. Such an accurate prediction of nuclear states needs a reliable approach to the interactions involved. The Coulomb interaction can be well approximated by Coulomb law, but the strong nuclear interaction is not well understood. We are trying to describe the nuclear interaction and the structure with the help of a QMD like model.

3.1 Model description

In this model [24] the nuclear system is treated as a four component fluid composed of protons with spin up, protons with spin down, neutrons with spin up and neutrons with spin down having corresponding densities ρp↑(r,t) , ρp↓(r,t), ρn↑(r,t) and ρn↓(r,t) respectively,

where the densities are position and time dependant. In such a system of A nucleons, the total wave-function is defined as the product of nucleon Gaussian wave-packets.

Φ(r, t) =

A

∏

k=1

φIkSk(r,t). (3.1)

Each nucleon Gaussian wave packet has a form φIkSk(r,t) = 1 2πσ2 k(t) 3/4exp − [r − rk(t)]2 4σ_k2(t) + i ¯hr pk(t) ! , (3.2)

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where parameters σ_k2, r_kand p_kare position variance, position centroids and the associated momentum for kthnucleon. Every nucleon wave function has a label I_k is n or p and S_kis ↑ (spin up) or ↓ (spin down) informing about the isospin and spin associated with that nucleon. The time evolution of such a system (Eq. 3.1) can be calculated by the variational principle, while the probability of finding the kth nucleon at position r and at time t is

P_k(r,t) =φIkSk(r,t)

2

. (3.3)

One can calculate the local densities ρ_p↑, ρp↓, ρn↑and ρn↓by summing up the probabilities

given by Eq. 3.3 for nucleons, with corresponding spin and isospin, as one should remember that P_k(r,t) can be interpreted as a density ρk(r,t) of individual nucleons.

dTot Entries 500 Mean 1.3 RMS 0.8661 radius (fm) 0 1 2 3 4 5 ) -3 density (fm 0 0.02 0.04 0.06 0.08 0.1 dTot Entries 500 Mean 1.3 RMS 0.8661 ↓ (r), p k P ↑ (r), n k P ↓ (r), n k P ↑ (r), p k P (r) ↑ p ρ (r) ↓ p ρ (r) ↓ n ρ (r) ↑ n ρ (r) p ρ (r) n ρ (r) ρ

Fig. 3.1 An example of density distributions inside nucleus. For details see text. In order to get an idea of the densities (see Fig. 3.1) considered in this model, we present an example of symmetric nuclear system composed of 2-protons with spin ↑, 2-protons with spin ↓, 2-neutrons with spin ↑ and 2-neutrons with spin ↓. Such a system can be treated as

8_{Be. The radial density distributions P}

k(r) (here k = 1 or 2) of individual nucleons from the

aforementioned pairs are represented by the 8 dashed curves. For example for protons with spin ↑, we have two black dashed curves, one representing the density of proton located at

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3.1 Model description 27 the center of the nucleus and another representing the density of proton which is not located at the center. Next the solid black line shows the total density of both protons with spin ↑, in the figure denoted as ρp↑. Likewise the solid red line represents the ρp↓, solid blue

line represents ρ_n↓ and solid green line represents ρ_n↑. Once we have these densities, the total proton density is calculated as a sum of ρ_p↑and ρ_p↓(solid thick yellow line) and in the similar way the total neutron density is calculated (solid thick purple line). Finally, the total nuclear density is the sum of individual nucleon densities. All these densities are used to calculate the nuclear interaction and it will be discussed in details in the following sections. The nuclear matter is assumed to be made up of infinite number of nucleons interact-ing only by strong interaction, without surface effects and the density of nucleons dis-tributed uniformly inside volume. For such a matter there exists a scalar energy field ε (r) = ερp↑(r), ρp↓(r), ρn↑(r), ρn↓(r) which is a function of local densities (for the

read-ability of equation, we are omitting the dependence of t). In case of a nucleus (finite piece nuclear matter) this field may have a form as shown in the bottom panel of Fig. 3.2 (black solid line). But one should remember that finite nuclei have surface effects which are not included in this energy field.

dTot Entries 500 Mean 1.449 RMS 0.933 0 1 2 3 4 5 ) -3 density (fm 0 0.01 0.02 0.03 0.04 0.05 0.06 Entries dTot 500 Mean 1.449 RMS 0.933

(r)

1

P

(r)

2

P

(r)

ρ

> eosField Entries 500 Mean 1.514 RMS 0.9715

radius (fm)

1 2 3 4 5 MeV 12 − 10 − 8 − 6 − 4 − 2 − 0 eosField Entries 500 Mean 1.514 RMS 0.9715 (r) ∈ (r)] 1 [e 1 σ (r)] 2 [e 2 σ

Fig. 3.2 The relationship between the single nucleon densities Pk(r), total density ρ(r),

energy field ε(r), mean energy ⟨e(r)⟩, and the individual nucleon variances σ_k2[e_k(r)] are plotted as a function of radius.

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One can calculate the average energy associated with the given nucleon k as ⟨ε⟩_k=

Z

P_k(r)ε

ρp↑(r), ρp↓(r), ρn↑(r), ρn↓(r) d3r, (3.4)

and also the variance of that energy can be calculated by σ_k2(ε) =

Z

P_k(r)ε ρp↑(r), ρp↓(r), ρn↑(r), ρn↓(r) − ⟨ε⟩k

2

d3r. (3.5) To take into account the surface effects in nuclei, it is assumed that the total energy ek

associated with each nucleon can be expressed by the mean value ⟨ε⟩_k and the variance σ_k2(ε) of the field ε(ρp↑, ρp↓, ρn↑, ρn↓) as

e_k= ⟨ε⟩_k+ λ σk(ε), (3.6)

where λ is a parameter related to the surface energy as can be understood later. To give an impression about the meaning of the variance term of the Eq. 3.6, we present two situations, one for nucleon density distributed near to the center of nucleus (blue dashed line) and other density for the nucleon situated near to the surface (red dashed line). As one can notice from Fig. 3.2, the variance (blue solid line) of nucleon situated near to the center of nucleus, where there is a constant density region (solid aqua color line) is practically zero. While for the nucleon situated near to the surface of nucleus, where the density changes has a non zero variance (red solid line). Such a surface effects in the nucleus is taken in to account by the non-zero variance term of Eq. 3.6.

Since the density distribution in nuclear matter is uniform, the variance of energy asso-ciated with the field ε(ρp↑, ρp↓, ρn↑, ρn↓) must be zero. Thus in such an infinite system, the

mean value of this field only determines the average energy per nucleon, this implies that the field ε(ρp↑, ρp↓, ρn↑, ρn↓) can be treated as an EoS of nuclear matter and it will provide the

information about its energy. In the next few sections, we derive the EoS.

3.2 Equation of state of nuclear matter

The nuclear matter (NM) is an ideal system of infinite number of nucleons (neutrons and protons) interacting only by nuclear forces (without any Coulomb forces) and filling an infinite volume with constant density. Infinite volume of NM implies that there are no surface effects and they obey translational invariance. To develop EoS, usually an ideal system of symmetric nuclear matter (infinite with equal number of protons and neutrons without electrons) is used. The EoS of a nuclear matter is a thermodynamic equation which

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3.2 Equation of state of nuclear matter 29 gives the relationship between two or more state functions such as temperature, pressure, volume, internal energy etc. under a given set of physical conditions. Neutron star is usually considered to be an example of such a system, but since it has more neutrons than protons it does not obey translational invariance. Also the pressure of the neutron star varies from zero on the surface to an unknown large value in the core. The EOS of nuclear matter can be also applied to the finite systems such as stars and atomic nuclei, but necessary constrains should be added. For example in the case of nuclei, surface and Coulomb effects should be included. Liquid drop model is an example of such an approach.

First of all, let’s consider temperature. When we cool and compress plasma, we get gas. If we further compress it, we get liquid followed by solid. In such matter phases in which thermal effects dominate, most of the particle energy levels are unfilled and particles are free to move in to these unfilled states. As we compress the matter, we are increasing the density, particles progressively fill the lower energy states and additional particles are forced to occupy states of higher energy, thus the particles is starting to be well defined in space and becomes incompressible. This incompressibility is due to the Pauli’s exclusion principle, which states that no two fermions can be in the same quantum state. In such a matter, all lower energy states are occupied by particles and no thermal energy can be extracted. This matter is called degenerated matter. The particle positions are well localised and according to Heisenberg’s uncertainty principle, ∆p∆x ≥ h₂, particle’s momenta are highly uncertain and this fact generates a pressure known as degeneracy pressure.

Now let’s consider the case of a system of non interacting fermions in the degenerated state. The Fermi energy is the maximum kinetic energy of an individual fermion in this ground state. Fermi gas model of atomic nucleus is a statistical model where nucleus is considered as a fermionic, degenerated gas of protons and neutrons obeying the Fermi-Dirac statistics. At 0 K the particles fill all the states up to an energy Ef called Fermi energy, and

each level is occupied by two particles with opposite spins. The total kinetic energy of the nucleus in Fermi gas model can be written as

E_kin(N, Z) = 3 10M ¯h2 R2₀ 9π 4 2/3 N5/3+ Z5/3 A2/3 . (3.7)

This energy has a minimum at N = Z for fixed mass number, hence from Fig. 3.3, the binding energy B gets maximum at N = Z. In order to introduce the asymmetry term δ = (N−Z)_A one can use the following substitutions Z = A₂−N−Z₂ , N = A₂+N−Z₂ and expand the Eq. 3.7 around δ as E_kin(N, Z) = 3 10M ¯h2 R2₀ 9π 8 2/3 A 1 +5 9δ 2_{+ ...} . (3.8)

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Fig. 3.3 Energy levels of protons and neutrons in the potential well of depth V0. EFpand EFn

are the Fermi energy for protons and neutrons respectively while B is the binding energy. Application of this kinetic energy in EoS can be seen in the next section.

3.2.1 EOS definitions and conventions

We can derive the nuclear matter equation of state as shown in [25]. Suppose that nuclear matter consists of only nucleons of density ρ(⃗r,t) and at temperature T (⃗r,t) extended over all space. The total energy contained in a volume V cut out from that nuclear matter can be expressed as

E_V =

Z

V

ρ e(ρ , T )dV, (3.9) where the term e(ρ, T ) is the per nucleon energy density, which contains all the important global properties of nuclear matter.

In this case pressure p is given by

p= −∂ e(ρ , T )

∂V |T=0. (3.10) But we know that ρ = _VA where A is the number of nucleons. So according to the chain rule,

∂ ∂V = ∂ ∂ ρ ∂ ρ ∂V = ∂ ∂ ρ ∂ AV−1

∂V one can write

∂ ∂V = − A V2 ∂ ∂ ρ, (3.11)

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3.2 Equation of state of nuclear matter 31 hence p= A V2 ∂ e ∂ ρ |T=const= A2 V2 ∂ ∂ ρ e A |T=const= ρ2 ∂ ∂ ρ e(ρ, T ) |T=const. (3.12)

The second derivative (curvature) of e(ρ) near to ρ = ρ0characterises the compressibility of

equilibrated nuclear matter. A larger value of compressibility means a greater energy needed to compress the nuclear matter. The curvature determined from density oscillations (giant monopole resonances) in spherical nuclei is

K₀= 9ρ2 ∂

2

∂ ρ2e(ρ, T ) |ρ =ρ0= 210 ± 30MeV. (3.13) In the case of asymmetric nuclear matter, isospin asymmetry term δ should be included, where

δ = ρn− ρp ρn+ ρp

= ρn− ρp

ρ . (3.14)

Thus the energy per nucleon can be written as E

A = e(ρ, T, δ ). (3.15) On the other hand energy per nucleon can be also expressed as the following sum [26]

E

A = e(ρ, T, δ ) = eT(ρ, T, δ ) + eC(ρ, T = 0, δ ) + e0(δ ), (3.16) where eT(ρ, T, δ ) is the energy associated with the thermal excitation, eC(ρ, T = 0, δ ) is the

energy associated with the compression and e0(δ ) is the saturation energy. Thermal energy

is defined as the additional kinetic energy above Fermi gas ground state energy. When the thermal excitation is relatively small, one can neglect this thermal energy term, then one have

E

A = e(ρ, T, δ ) = eC(ρ, T = 0, δ ) + e0(δ ) = e(ρ, δ ). (3.17) Many theoretical models have shown that the properties of infinite asymmetric nuclear matter at relatively low temperature can be described by an approximate form of EoS as

e(ρ, δ ) = TF(ρ, δ ) +V0(ρ) + δ2V2(ρ), (3.18) where TF(ρ, δ ) = 3¯h2 10M 3π2_ρ 2 2/3 h δ5/3+ (1 − δ )5/3 i , (3.19)

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is the Fermi-gas kinetic energy as shown by Eq. 3.8, V0(ρ) and δ2V2(ρ) are the isospin

independent and isospin dependent potential energies. Higher order terms of δ are negligible. The Eq. 3.18 can be further approximated by

e(ρ, δ ) = e(ρ, δ = 0) + δ2e_sym(ρ). (3.20) This equation is known as the empirical parabolic law for the EoS of asymmetric nuclear matter and is considered to be valid only at small isospin asymmetries. However, many non-relativistic and relativistic calculations have shown that it is actually valid up to δ = 1. The first term in Eq. 3.20 represents the energy density per baryon associated to symmetric nuclear matter (δ = 0), which can be expanded around normal nuclear matter density ρ0as

e(ρ, δ = 0) = e(ρ) = e0+ K₀ 18 ρ − ρ0 ρ0 2 + ..., (3.21) where e0is the volume energy term of LDM. This equation as a function of density is plotted

Fig. 3.4 Empirical parabolic energy per nucleon for symmetric nuclear matter. in Fig. 3.4. At ρ = 0, above equation becomes

0 = −16MeV +K0 18 0 − ρ0 ρ0 2 . (3.22) From this case, one gets K0≈ 300MeV . The second term in Eq. 3.20 written as

e_sym(ρ) = 1 2 ∂2e(ρ, δ ) ∂ δ2 |δ =0= 5 9TF(ρ, 0) +V2(ρ), (3.23)

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3.3 New form of EoS 33 is the bulk symmetry energy. It can be evaluated from the two extreme cases of both pure neutron matter and symmetric nuclear matter via

e_sym(ρ) = e(ρ, 1) − e(ρ, 0). (3.24) The symmetry energy which appears in Eq. 3.20 also can be expanded around ρ0as

esym(ρ) = es0+ L_s 3 ρ − ρ0 ρ0 +Ks 18 ρ − ρ0 ρ0 2 + ..., (3.25) where the variables Ls (slope) and Ks (curvature) characterize the density dependence of the

nuclear symmetry energy and are given by L_s= 3ρ0 ∂ es(ρ) ∂ ρ |ρ = ρ0, (3.26) K_s= 9ρ₀2∂ 2_e s(ρ) ∂ ρ2 |ρ = ρ0. (3.27) The coefficients ρ0, e(ρ0, 0), es(ρ0), K0, Lsand Ksallow us to determine the EoS around

the normal nuclear density. Unfortunately, neither of the values of the slope Ls and the

curvature Ks are well determined experimentally nor accurately estimated theoretically. Data

from experiments on giant monopole resonances suggest that Ksvaries from −566 ± 1350

MeV to 34 ± 159 MeV [27]. Theoretical estimates for Ksare model dependent and vary from

−700 to +466 MeV [28]. So far, experimental predictions for the Ls values are rather scarce.

In theoretical models, Lsvaries from −50 up to 200 MeV [29].

As in the case of LDM, we are also using the EoS for the volume energy calculations. The detailed procedure will be presented in the following sections.

3.3 New form of EoS

There are some drawbacks in the EoS presented in the last sections. One is that the EoS is valid only, if the density ρ is sufficiently close to ρ0. Second drawback is that E_A is

not approaching zero as ρ → 0 as can be seen from Eq. 3.21. The total nuclear matter density is often expressed by the sum of partial densities given in Gaussian form (Eq. 3.2). Unfortunately, in such a case, the integral Eq. 3.9 can not be solved analytically. A new form of EoS which is free of such shortcomings is presented below as shown in [30].

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Let’s consider e(ρ, 0) as the following polynomial

e(ρ, 0) = α0ρ + β0ρ2+ γ0ρ3, (3.28)

at ρ = ρ0,

e(0) = α0ρ0+ β0ρ₀2+ γ0ρ₀3. (3.29) The coefficients α0, β0 and γ0 are determined from Eq. 3.21 by knowing the slope and

curvature of EoS ∂ e(ρ = ρ0, 0) ∂ ρ = 0 = α0+ 2β0ρ0+ 3γ0ρ 2 0, (3.30) ∂2e(ρ = ρ0, 0) ∂ ρ2 = K₀ 9ρ₀2 = 2β0+ 6γ0ρ0, (3.31) next from Eq. 3.30, one gets

α0= −2β0ρ0− 3γ0ρ₀2, (3.32) after inserting α0into Eq. 3.29 and doing some rearrangements, e0now is

e₀= −β0ρ2− 2γ0ρ3. (3.33)

From equations 3.29, 3.31 and 3.33 we can find the value of α0, β0and γ0, respectively

by solving the system of 3 equations with three unknowns. So their values are α0= 3e0+ K₀ 18 /ρ0, (3.34) β0= − 3e0+ K₀ 9 /ρ₀2, (3.35) γ0= e0+ K0 18 /ρ₀3. (3.36) In a similar way, the symmetry energy Eq. 3.25 also can be represented in the form of a polynomial as

e_s(ρ) = αsρ + βsρ2+ γsρ3, (3.37)

and at densities around ρ0

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3.3 New form of EoS 35

Fig. 3.5 Upper panel: the equation of state of nuclear matter with new form of EoS. Lower panel: the left part presents the Skyrme-Hartree-Fock approach, and the right one the relativistic mean-field model calculation. See text for more details. Source: [30].

Similarly as was shown for the symmetric EoS, we have two equations for slope and curvature, ∂ es(ρ = ρ0) ∂ ρ = Ls 3 = αs+ 2βsρ0+ 3γsρ 2 0, (3.39) ∂2es(ρ = ρ0) ∂ ρ2 = Ks 9ρ₀2 = 2βs+ 6γsρ0, (3.40) and again we have three equations 3.38, 3.39 and 3.40 with three unknowns αs, βsand γs

which can be solved to find their values αs= 3es0− 2 3Ls+ Ks 18 /ρ0, (3.41) βs= − 3es0− Ls+ Ks 9 /ρ₀2, (3.42)

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γs= es0− L_s 3 + K_s 18 /ρ₀3. (3.43) This new form of EoS is identical to the previous EoS (Eq. 3.20) up to the second order term. The most important advantage of the new EoS is that, if the density ρ is given as sum of individual nucleon Gaussian densities, then the integral can be solved analytically. This will allow such an EoS to be used in QMD-like model calculations. The comparisons of this EoS with the Skyrme-Hartree-Fock (SHF) and Relative Mean Field (RMF) calculations are shown in Fig. 3.5. This comparison was made with the following parameter values: ρ0= 0.15nucleon_{f m}3 , e0= −15.95 MeV and K0= 300 MeV . For SHF fit it was used e0s= 29

MeV, Ls= 11 MeV , Ks= −300 MeV , while for the RMF fit, e0s= 39 MeV , Ls= 150 MeV ,

K_s = 300 MeV were considered. From the figure, one can see that the new calculations could reproduce the RMF and SHF calculations, despite the fact that the search for those parameters was very preliminary. In both cases the solid curves correspond to different values of ρp/ρnwhich are 0, 0.2, 0.4, 0.6, 0.8 and 1 (from top to bottom).

3.4 Extension of EoS

The new form of EoS derived in the previous section depends on the total density and isospin. This form can be extended by including densities which depend on the spin [31]. As shown in section 3.1, total nuclear matter density can be represented as the sum of protons with spin-up density ρ_p↑, protons with spin-down density ρ_p↓, neutrons with spin-up density ρ_n↑ and neutrons with spin-down density ρ_n↓. Formally EoS of such a system can be written as e= e(ρp↑, ρp↓, ρn↑, ρn↓). (3.44)

With addition of spin interaction term, it is convenient to introduce the new coordinates which describe the nuclear interaction as

ξ = ρ − ρ0 ρ0 , (3.45) δ = ρn− ρp ρ , (3.46) ηn= ρ_n↑− ρ_n↓ ρ , (3.47) ηp= ρp↑− ρp↓ ρ . (3.48)

Studies of Alpha Clustering in Nuclei