Shape coexistence and superdeformation in 28Si

Master thesis Shape coexistence and superdeformation in 28Si Inter-University Master Degree on Nuclear Physics Author: Dorian Grzegorz Frycz Advisors: Javier Men´endez S´anchez & Antonio M´arquez Romero July 2023 Acknowledgements I would like to express my gratitude towards my advisors Javier Men´endez S´anchez, Antonio M´arquez Romero and Arnau R´ıos Huguet for their invaluable support and guidance throughout the whole research as well as their meticulous review of this Master thesis, which I am truly proud of the outcome. The knowledge and skills I have gained under their supervision are truly priceless. Furthermore, I would like to express my heartfelt gratitude to the new friends I have made during this time. Lastly, I am deeply thankful to my family and long-standing friends for their unwavering support and encouragement in all aspects of my life. Contents 1 Introduction to deformation in nuclei 4 1.1 Collective models ............................. 5 1.2 Quadrupole moment and deformation parameters ................................ 6 2 Theoretical framework 10 2.1 Spherical mean field ........................... 10 2.2 Analytical SU(3) scheme ......................... 12 2.3 Nuclear shell model ............................ 16 2.3.1 Exact diagonalization ....................... 17 2.3.2 Variational method ........................ 19 3 Shape coexistence in 28Si 24 3.1 Ground state and oblate band ...................... 24 3.2 Prolate band and modification of the interaction ............ 29 3.3 Superdeformation ............................. 33 4 Conclusions and Outlook 41 2 Abstract In this Master thesis I analyze the 28Si nucleus within the nuclear shell model using state-of-the-art numerical diagonalizations complemented with the generatorcoordinate method (GCM) with quadrupole-constrained Hartree-Fock-Bogoliubov (HFB) wavefunctions. Experimentally, 28Si presents shape coexistence between the oblate ground state and an excited prolate structure with bandhead at 6.7 MeV. Although the standard USDB interaction, which is very successful in describing the nuclear structure of nuclei in this mass region, reproduces well the oblate ground state, it fails at establishing a prolate band. Therefore, a modification of the USDB interaction must be introduced to reproduce the experimental spectrum. Guided by analytical Elliot’s SU(3) scheme, I show that this is achieved by slightly lowering the gap between the nearly degenerate 1d5/2-2s1/2doublet and the 1d3/2orbital. My calculations suggest that the oblate ground state is mostly 0p-0h configurations in the 1d5/2-2s1/2orbitals, whereas the prolate band consists mainly of 4p-4h excitations into the 1d3/2orbital. Additionally, I study whether 28Si can exhibit a superdeformed structure at higher energies. In order to achieve such deformations, excitations from the sd to the pf shell must be taken into account. I find that most of the deformation contribution comes from the 1f7/2-2p3/2doublet and that the most favorable states are prolate 2p-2h and 4p-4h excitations into the pf shell. In contrast to previous studies, my numerical calculations suggest that this superdeformed structure would mix with normal-deformed configurations, and therefore 28Si would not present a superdeformed band. 1. Introduction to deformation in nuclei The strong interaction among protons and neutrons allows the existence of selfbound systems known as atomic nuclei. As a result of the intricate characteristics of the strong interaction, collective phenomena emerge, wherein nucleons cooperate to exhibit behaviors such as rotations or vibrations centered around a specific intrinsic shape. Apart from the spherical shape, atomic nuclei can exhibit axially symmetric quadrupole deformations, including prolate shapes, which are axially symmetric spheroids elongated along one axis, and oblate shapes, which are flattened. Furthermore, atomic nuclei can manifest triaxial deformations, in which elongation takes place along distinct axes, resulting in an intermediate shape that presents additional complexity. Across the nuclear chart, an abundance of ground states exhibit quadrupole deformations, which originate from the quadrupole-quadrupole interactions of the nucleons. Among these deformations, axially symmetric shapes and prolate configurations stand out as the most common shapes. However, the scenario becomes even more complex as multiple states with distinct intrinsic shapes can coexist within the same nucleus, occupying a narrow energy range of a few MeV. The occurrence of this phenomenon, known as shape coexistence, has been observed across various regions of the nuclear chart [1], indicating its widespread presence. Hitherto, medium-mass nuclei, including 32S [2], 36Ar [3], 40Ar [4], 40Ca [5], 42Ca [6], and 44Ti [7] have been identified as examples where competition between spherical and various types of deformed states occurs. For instance, the 40Ca nucleus is as a doubly magic nucleus with a spherical ground state. However, Fig. 1.1 shows that at an excitation energy of ∼3 MeV, a prolate normal deformed shape emerges, and at ∼5 MeV it presents a prolate superdeformed shape [8], which is of particular interest due to its extreme deformation. This superdeformed states have been studied in numerous nuclei such as 24Mg [9], 28Si [10,11], 32S [12], and 40Ar [13,14]. In this work, I explore the nuclear structure of 28Si, an even-even nucleus composed of 14 protons and 14 neutrons, which exhibits the phenomenon of shape coexistence between oblate and prolate deformations. Previous studies [15,16] have shown that the three lowest-energy 0+states are bandheads of an oblate rotational band, a vibrational band of the oblate ground state and a prolate rotational band, as shown in Fig. 1.1. These rotational bands are manifestations of a permanent intrinsic deformation, shared among the states that belong to the band, each of which is labeled by its total angular momentum number, J. Therefore, 28Si demonstrates shape coexistence between prolate and oblate states, while the spherical state does not persist, in contrast with the prediction of the spherical mean field of Mayer and Jensen [17,18]. In addition, there have been recent attempts at finding a superdeformed band in 28Si [19], predicted by a theoretical calculation [16]. The objective of this study is to gain insight into the nuclear structure of 28Si by employing both simple analytical models such as SU(3) as well as state-of-the-art numerical shell model calculations that shed light on its shape coexistence. 4 CHAPTER 1. INTRODUCTION TO DEFORMATION IN NUCLEI 5 0 1 2 3 4 5 6 7 8 9 10 vib ND GS 2+ 4+ 6+ 0+ 2+ 4+ 0+ 2+ 28Si 0+ Excitation energy (MeV) Oblate Prolate Vibration ND SD Sph 40Ca 2+ 4+ 6+ 0+ 2+ 4+ 0+ 0+ Spherical Prolate ND Prolate SD Figure 1.1: Experimental data of the band structure of 28Si, from [15], and 40Ca, from [5]. The 28Si nucleus exhibits an oblate rotational band and its vibration as well as a prolate rotational band. On the other hand, 40Ca presents a spherical ground state and two prolate rotational bands: one with normal deformation and another one that is superdeformed. The arrows’ width indicate the B(E2) transition strength between states of a particular band. Outband transitions and other lowenergy states are not shown for clarity. 1.1 Collective models An atomic nucleus in a deformed state can exhibit collective rotational or vibrational motion, which can be discerned by a characteristic energy spectrum. In order to understand collective motion in nuclei, two frames of reference are required: the laboratory frame and the intrinsic frame. For axially symmetric shapes, the quantum number associated to the projection of the total angular momentum ( J) in the laboratory frame is denoted as Mwhile in the intrinsic frame it is K, as represented in Fig. 1.2. Rotational motion is exclusive to permanently deformed nuclei, as any rotation around a symmetry axis leaves the surface invariant, except for an insignificant global phase in the wavefunction [20]. In consequence, spherical shapes are invariant under any rotation and axially-symmetric nuclei are invariant under rotations around their symmetry axis. A straightforward approach to modeling the rotations of a deformed nucleus is to treat it as an ideal rigid rotor [20] characterized by a constant moment of inertia, denoted as I. The Hamiltonian of an axially-symmetric rotor about a principal axis, denoted as 3 in the intrinsic frame of reference, is Hrot =( J2 1+ J2 2) 2I=( J2− J2 3) 2I,(1.1) where  Jiare the components of the total angular momentum  J. By quantizing the values of the total angular momentum and its projection on the symmetry axis we CHAPTER 1. INTRODUCTION TO DEFORMATION IN NUCLEI 6 Figure 1.2: Projections of the total angular momentum (  J) of a prolate spheroid on the laboratory frame of reference (M), defined by the {X,Y,Z}axes, and on the intrinsic frame of reference (K), where the symmetry axis is labeled as 3. end up with the expression Erot =J(J+ 1) −K2 2I,(1.2) where Jis the total angular momentum number and natural units are employed (ℏ= 1). The energy levels exhibit a proportionality to J(J+ 1), manifesting a distinct pattern that can be readily identified in the spectrum. This succession of levels with energies proportional to J(J+ 1) that share a permanent deformation is called a rotational band. The values of angular momentum Jare dependent on their projection along the symmetry axis Ksuch as K= 0 −→ J=K, K + 1, K + 2...;M=−J, ..., J ; (1.3) K= 0 −→ J= 0,2,4...;M=−J, ..., J , (1.4) where the K= 0 case only admits even values of Jdue to axial symmetry [20]. This sequence of levels that are spaced out by the E∝J(J+ 1) proportionality provides a strong indication of the presence of deformation in a nucleus. 1.2 Quadrupole moment and deformation parameters The interaction between the nucleus and the electromagnetic field yields valuable information regarding the nuclear properties. By employing gamma radiation, it is possible to excite the nucleus to higher energy levels, and analyzing the characteristics of the emitted radiation can offer valuable insight into its structure. The surface of a nucleus can be described in terms of the spherical harmonics CHAPTER 1. INTRODUCTION TO DEFORMATION IN NUCLEI 7 Yλµ(θ, φ) [20]: R(θ, φ) = R0"1 + ∞ X λ=0 λ X µ=−λ αλµYλµ(θ, φ)#,(1.5) where R0is the radius of the spherical nucleus, (r, θ, φ) are the spherical coordinates and αλµ are the expansion coefficients. In the context of nuclear excitations, the monopole term (λ= 0) corresponds to a change of the nuclear volume, which necessitates a significant amount of energy due to the high incompressibility of nuclear matter. On the other hand, the dipole term (λ= 1) describes a shift of the center of mass, which is an insignificant translation. The remaining terms represent different deformations of the nucleus, although modes with λ≥5 are generally negligible. The most common types of deformation across the nuclear chart are the quadrupole deformations (λ= 2). The distribution of the electric charge in a nucleus can be described by the electromagnetic multipole moments Qλµ =rλYλµ(θ, φ).(1.6) We can also define its hermitian average operator as [21] Qλµ =1 2[Qλµ + (−1)µQλ−µ].(1.7) For quadrupole deformations, there are three relevant quadrupole moments Q20, Q21 and Q22. With these operators, we can define a set of dimensionless quadrupole deformation parameters (β, γ) that characterize an arbitrary triaxial shape β=4π 3R2 0Ab2q⟨Q20⟩2+ 2⟨Q22⟩2,(1.8) γ= arctan √2⟨Q22⟩ ⟨Q20⟩!,(1.9) with the nucleus mean radius R0≃1.2A1/3fm and ⟨O⟩≡⟨Ψ|O|Ψ⟩indicates the mean value of an operator for a given nucleus state |Ψ⟩. The harmonic oscillator parameter bcan be derived using the expression [22] b=rℏ mNω=rℏ mNω·√ℏc √ℏ√c2=ℏc p(mNc2)(ℏω)≃197.33 MeVfm √938.9 MeV ·ℏω,(1.10) where ℏrepresents Planck’s reduced constant, cdenotes the speed of light, ωrepresents the angular frequency of the harmonic oscillator, and mNcorresponds to the mass of a nucleon. By utilizing the Blomqvist–Molinari formula [23], ℏω= (45A−1/3−25A−2/3) MeV, and simplifying the constants as 41.4 MeV fm2, we can arrive at the standard parametrization b2≃41.4/(45A−1/3−25A−2/3) fm2(1.11) for the harmonic oscillator parameter. CHAPTER 1. INTRODUCTION TO DEFORMATION IN NUCLEI 8 Figure 1.3: Hill-Wheeler coordinates (β, γ) for the representation of quadrupole deformations. Oblate and prolate shapes are represented with principal axes along the z-axis (green), y-axis (red) and x-axis (blue). Figure extracted from [24]. Revisiting the set of deformation parameters (β, γ), the parameter βquantifies the magnitude of deformation in a nucleus, which can be related to the axis ratio of spheroid shapes. In normal deformed states (ND), typical values fall within the range of 0.2≤β≤0.3, corresponding to an axis ratio of 3:2. On the other hand, superdeformed states (SD) exhibit β≥0.5, resulting in a 2:1 axis ratio. The parameter γcharacterizes the type of deformation. Initially, γcan assume values between 0◦and 360◦. However, considering that Q21 determines only the orientation of the shape in space, it can be set to Q21 = 0, narrowing down the interval to 0◦≤γ≤60◦as illustrated in Fig. 1.3. A value of γ= 0◦corresponds to a prolate shape, while γ= 60◦represents an oblate shape. Any other value of γ within this range indicates a triaxial deformation. In the case of axially symmetric shapes, it is standard to simplify the description by using only the parameter β. The differentiation between prolate and oblate shapes is then determined by the sign of β. Positive values indicate a prolate shape, while negative values indicate an oblate shape. This two shapes are for instance represented in green in Fig. 1.3, taking z as the symmetry axis. Using the set of quadrupole deformation parameters (β, γ), we can establish an intrinsic shape for the nucleus. However, we still need a method to connect this intrinsic description with the experimental data in the laboratory frame of reference. The electric quadrupole moment provides a measure of the nucleus’ deformation, as it characterizes the deviation of the charge distribution from a spherical shape. In the case of spherical shapes, the quadrupole moment vanishes. In the laboratory frame, a microscopic description of the quadrupole moment is [22] Qspec ≡r16π 5sJ(2J−1) (J+ 1)(2J+ 1)(2J+ 3)(J||Q2||J),(1.12) where Q2=PA j=1 ejr2 jY20(θj, φj): ejare the effective nuclear charges of the nucleons CHAPTER 2. THEORETICAL FRAMEWORK 15 2s1/2orbits is merely 0.7 MeV, in contrast to the approximately ∼5 MeV separation observed between the doublet and 1d3/2(single-particle energies extracted from the USDB interaction [38,39]). To solve this issue, alternative SU(3) schemes sacrifice some degree of symmetry in exchange of a more realistic view [40]. One of the variants of the SU(3) model is the quasi-SU(3) scheme, specifically adapted to accommodate orbitals with ∆j= 2 angular momentum differences. This scheme takes advantage of the fact that ∆j= 1 single-particle matrix elements of q(2) 0are weaker compared to those with ∆j= 2 [41]. An example within the sd shell involves the 1d5/2-2s1/2doublet. Figure 2.2 b) illustrates the Q0values corresponding to the quasi-SU(3) model for this doublet. An additional variation of the SU(3) model is known as the single-orbit limit. This situation occurs when an orbit is significantly separated from the others. For example, the 1d3/2orbit is placed between the nearly degenerate 1d5/2-2s1/2doublet and a major shell closure. The quadrupole moment contribution of a single orbit with total angular momentum jand projection mis [40] Q0/e =X m>0P+3 2j(j+ 1) −3m2 2j(j+ 1) b2,(2.19) which verifies the fact that any filled orbit corresponds to a spherical shape Q0= 0, as described in the spherical mean field picture. The contribution of a single 1d3/2 orbit to Q0is represented in Fig. 2.2 c). Given the substantial energy gap between the 1d5/2-2s1/2doublet and the 1d3/2 orbit, a (1d5/2-2s1/2)+1d3/2prescription holds the greatest physical significance, where the 1d5/2-2s1/2orbits are treated as degenerate. The particle-hole notation is introduced as np-nh, where ndenotes the number of particles that are promoted from the lower space, 1d5/2-2s1/2, to the upper one, 1d3/2. For instance, filling the levels from Fig 2.2 and recovering the appropriate units, the oblate 0p-0h configuration has intrinsic quadrupole moment Q0= (4 ·(−2) + 2 ·0.5−3) ·(0.5+1.5)e·3.42 fm2=−58.1efm2,(2.20) and the 4p-4h prolate configuration has Q0= (2 ·3.5+2·0.5+2·1.4 + 3) ·(0.5+1.5)e·3.42 fm2= 84.1efm2.(2.21) The quadrupole moments corresponding to remaining prolate and oblate np-nh configurations are indicated in Table 2.1. The significant quadrupole moment observed Table 2.1: Quadrupole moments (efm2) for the experimental GS and ND bands, the possible spherical state and the np-nh configurations in the analytical (1d5/2- 2s1/2)+1d3/2scheme. I also present SU(3) predicted values for prolate and oblate configurations, following Eq (2.17) and Fig. 2.2. Q0,spherical Qexp,GS Qexp,ND 0 -57.3±0.7 70±7 Analytical 0p-0h 2p-2h 4p-4h 6p-6h 8p-8h Pure SU(3) Q0,prolate 37.6 60.9 84.1 71.1 58.1 92.3 Q0,oblate -58.1 -71.1 -84.1 -60.9 -37.6 -92.3 CHAPTER 2. THEORETICAL FRAMEWORK 16 in the oblate 0p-0h configuration is remarkable due to the substantial correlation gain achieved by the simple addition of the 2s1/2orbit, compared to the filled 1d5/2 spherical picture. The proximity of the s1/2orbit to the 1d5/2orbit, coupled with the substantial increase in collectivity, leads the system to prefer an oblate deformed shape over a spherical one. In the case of the ND prolate band, the primary contenders are the 2p-2h and 4p-4h prolate configurations. The prolate 0p-0h configuration exhibits significantly lower deformation compared to the oblate configuration, rendering it energetically unfavorable. Conversely, prolate np-nh configurations with n≥6 display lower deformation than the 4p-4h configuration at the expense of promoting more particles. I anticipate that the 4p-4h prolate configuration will prevail over the 2p-2h configuration. This is because both the 2p-2h and 4p-4h configurations yield the same increase in quadrupole moment compared to the prolate 0p-0h configuration. However, since the correlation energy is quadratically related to the quadrupole moment and the single-particle energy of np-nh configurations increases linearly with the number of excited particles, the 4p-4h configuration may be a more favorable candidate, provided that the quadrupole force is strong enough. 2.3 Nuclear shell model The interacting nuclear shell model, or in short the nuclear shell model (NSM), is a method of solving the nuclear many-body problem formulated as an effective manybody Hamiltonian in a valence space. There are three key features for this method: a relatively small configuration space, an effective interaction and how to solve the Schr¨odinger equation (2.2). The necessity of a valence space arises naturally from the approach to the problem itself when dealing with medium-mass or heavy nuclei. Initially, the starting point involves the spherical mean-field, whose states serve as the single-particle basis. However, as the single-particle Hilbert space is infinitely large, the associated many-body space becomes infeasible to handle. Consequently, a boundary is required to render the problem tractable. The simplest solution involves considering a truncation of the Hilbert space, primarily delimited by the major shells of the mean field potential from Eq. (2.4), although only for light nuclei. The large energy gaps between these shells, due to the magic numbers [17], make mixing between them difficult. Nonetheless, selecting the appropriate valence space is more intricate, as certain operators may depend on properties such as parity changes, which cannot be accommodated solely within a major shell. Moreover, the dimensions of the many-body basis, the Slater determinants, expand combinatorially, rendering some choices of the space numerically unfeasible. This introduces the necessity of an inert core comprising a fraction of the nucleons, which can be represented by a single Slater determinant obtained through the filling of the spherical mean-field levels. Taking into account all these considerations, the analysis leads us to identify three distinct parts of the single-particle Hilbert space: 1. The inert core, which comprises orbits represented by a single Slater determinant that fills up the core particles Acore =Ncore +Zcore to a lower boundary of the single-particle Hilbert space. CHAPTER 2. THEORETICAL FRAMEWORK 17 2. The valence space, which encompasses orbits where the remaining nucleons Av=Nv+Zvcan interact according to the effective interaction. 3. The empty space, which consists of orbits that are always assumed to be unoccupied, defined by an upper boundary of the single-particle Hilbert space. Since we have introduced a change from the original Hamiltonian, defined in the infinite Hilbert space, to a specific valence space, the Hamiltonian must undergo an adaptation to an effective form that suits the problem at hand. Consequently, the effective Hamiltonian Heff incorporates the single-particle energies and the two-body matrix elements associated to the orbits within the valence space. This adjustment ensures the appropriate description of the system within the reduced valence space. The effective Hamiltonian captures the effects, such as correlations, that are disregarded in the simplified mean field approach, Eq. 2.3, and can be understood as a residual interaction Hres: Heff = A X i (ti+vi) + A X i,j Vij − A X i vi!=H0+Hres.(2.22) In summary, the initial many-body problem is simplified to the task of diagonalizing an effective Hamiltonian within the defined valence space: H|Ψ⟩=E|Ψ⟩ −→ Heff|Ψeff⟩=E|Ψeff⟩.(2.23) The bare operators also must undergo a transformation to assume a new effective form that suits the characteristics of the valence space ⟨Ψ|O|Ψ⟩ −→ ⟨Ψeff|Oeff|Ψeff⟩.(2.24) The final step involves seeking a suitable method for determining eigenvectors and eigenenergies of the problem. 2.3.1 Exact diagonalization The first method I have used to solve the many-body Schrodinger equation involves performing an exact diagonalization of the effective Hamiltonian in the valence space [26]. The basis for this method consists of all possible Slater determinants |Φα⟩that can be built within the valence space, denoted as m-scheme. Then, the eigenstates |ΨNSM⟩are linear combinations of the spherical mean field states |ΨNSM⟩=X α Cα|Φα⟩,(2.25) which is configuration mixing of states. However, it results in a basis with maximal dimensions, which may pose numerical challenges due to the large size of the problem. The number of Slater determinants that can be constructed within a valence space with Ωistates, where i=Zrepresents protons and i=Nneutrons, and Zv valence protons and Nvneutrons is dim = ΩZ ZvΩN Nv.(2.26) CHAPTER 2. THEORETICAL FRAMEWORK 18 For instance, a calculation of 28Si in the sd-shell (ΩN= ΩZ= 12) with a core of 16O (Nv=Zv= 6) involves dim = ΩZ ZvΩN Nv=12 612 6≈1·106,(2.27) which is a dimension that is not demanding for modern computer standards. Nevertheless, a calculation involving the sd and pf major shells increases the dimension significantly to around dim = ΩZ ZvΩN Nv=12 + 20 612 + 20 6=32 632 6≈1·1012,(2.28) which exceeds the scope of this project as even record calculations of d≈1·1011 [42] do not reach the required dimensions. It is important to mention that in practice, these dimensions can be slightly reduced by taking advantage of certain symmetries, for instance not necessitating all the total angular momentum projections, m. As demonstrated by this example, direct diagonalization of such large matrices is impracticable due to limitations imposed by computer capabilities. Hence, alternative algorithms are implemented such as the Lanczos method. The Lanczos method entails constructing an orthonormal basis by initiating with a pivot state |Ψ1⟩and subsequently applying the Hamiltonian operator Heff to it. In the first step (k= 1): Heff|Ψ1⟩=E11|Ψ1⟩+E12|Ψ1⟩,(2.29) where E11 =⟨Ψ1|Heff|Ψ1⟩. Then, E12 is obtained by normalization: E12|Ψ2⟩=Heff|Ψ1⟩−E11|Ψ1⟩.(2.30) The next step (k= 2) involves Heff|Ψ2⟩=E21|Ψ1⟩+E22|Ψ2⟩+E23|Ψ3⟩.(2.31) Since the Hamiltonian is hermitian E12 =E21. Then E22 =⟨Ψ2|Heff|Ψ2⟩and E23|Ψ3⟩= (Heff −E22)|Ψ2⟩−E21|Ψ1⟩.(2.32) The following steps follow the relation Heff|Ψk⟩=Ekk−1|Ψk−1⟩+Ekk|Ψk⟩+Ekk+1|Ψk+1,⟩(2.33) and the normalization is applied in each step. Through this procedure, a tridiagonal matrix is constructed, wherein the mean values of the Hamiltonian appear sequentially. These mean values are regarded as eigenvalues when the difference between each iteration falls below a certain tolerance. It is important to note that the Lanczos iteration process exhibits characteristics similar to those of variational methods, as the eigenvalues converge progressively rather than being immediately exact. However, the convergence of the eigenvalues is highly efficient and exhibits a well-behaved behavior [26]. Therefore, despite not being strictly an exact method, I refer to this approach as an “exact solution” due to its reliable convergence properties. In practice, all states obtained in this work by exact diagonalization are converged in energy to better than 0.5 keV, which exceeds the precision required for nuclear-scale phenomena. In this work I use the nuclear shell model ANTOINE code [43–45], which performs the exact diagonalization of the effective Hamiltonian by the Lanczos method. CHAPTER 2. THEORETICAL FRAMEWORK 19 2.3.2 Variational method The exact diagonalization method provides the most accurate solution to Eq. (2.2) within the valence space. However, its applicability is limited due to the combinatorial scaling of the many-body basis, as shown in Eq. (2.26), which quickly restricts the dimensions of the space. Hence, approximate methods are necessary to solve the eigenvalue problem [46]. The dimensionality issue arises from using all possible Slater determinants in the valence space as the many-body basis. To overcome this challenge, a viable approach is to employ a smaller yet more complex set of wavefunctions obtained through variational techniques. Subsequently, configuration mixing of these variational wavefunctions allows for the determination of approximate solutions instead of exact diagonalization [47]. According to the Ritz variational principle [48], the energy obtained from a trial wavefunction |ϕ0⟩for the ground state is always higher than or equal to the lowest energy obtained from the exact diagonalization of the Hamiltonian (E0): E[|ϕ0⟩] = ⟨ϕ0|Heff|ϕ0⟩ ⟨ϕ0|ϕ0⟩≥E0,(2.34) where both energies agree when the trial wavefunction coincides with the exact ground state. Variational methods involve finding the wavefunction that exhibits the closest resemblance to the exact wavefunction within a variational space. This is accomplished by minimizing a set of parameters that determine the characteristics of the possible wavefunctions. When employing product-like wavefunctions, the implementation of this variational method leads to the formulation of Hartree-Fock equations. However, this assumption is overly restrictive since correlations between particles play a crucial role in describing atomic nuclei. Therefore, in order to expand the variational space, certain symmetries such as particle number, rotational invariance or parity are intentionally broken. For instance, to incorporate pairing correlations between fermions, the independent particle description is modified to include quasiparticles, which results in a violation of the conservation of particle number. This approach is the Hartree-Fock-Boguliubov (HFB) method [49], where the HFB wavefunction |ϕHFB⟩ is the vacuum of a set of fermionic operators βk: βk|ϕHFB⟩= 0 → |ϕHFB⟩=Y k βk|−⟩;∀k= 1, .., M,(2.35) where Mis the number of single-particle states of the valence space, and |−⟩ is the true vacuum. The HFB transformation is defined as β† k=X l (Ulkc† l+Vlkcl),(2.36) βk=X l (U∗ lkcl+V∗ lkc† l) (2.37) with Ulk and Vlk matrices that define the state and act as the variational parameters and c† l(cl) the creation (annihilation) operators of a single particle basis. In this work, I am using the TAURUS code [21], which employs the spherical mean field levels as the single particle basis and considers real Uand Vmatrices. CHAPTER 2. THEORETICAL FRAMEWORK 20 As the transformation itself violates the conservation of particle number, it becomes essential to implement constraints to conserve quantities at the mean-field level. This is achieved by modifying the energy functional to minimize H′ eff =Heff −λZOZ−λNON−X i λiOi,(2.38) where OZand ONare particle number operators for the protons and neutrons spaces, respectively, and λiare the Lagrange multipliers. Additional constraints such as quadrupole moments or parity correlations can be added to respect the expectation value of any given operator O ⟨O⟩ ≡ ⟨ϕHFB|O|ϕHFB⟩.(2.39) The minimization process is carried out using a gradient method, with the following schematic procedure. It begins with an initial trial wavefunction |ϕ0⟩defined by the HFB matrices (U0, V0). These matrices can be defined to preserve any desired symmetry, such as spherical symmetry, axial symmetry, or parity. Afterwards, the gradient of the energy is computed, and a direction for minimization is determined. A new wavefunction is proposed, and if it minimizes the Hamiltonian, it is retained. This iterative process continues until convergence is achieved. For a more detailed understanding, refer to [21]. In this study, the focus is on investigating the shapes displayed by 28Si. To explore the energy surfaces associated with possible deformations, constraints on the quadrupole moments are utilized, as defined in section 1.2. Rather than imposing constraints directly on the quadrupole moments ⟨Q20⟩and ⟨Q22⟩, the restrictions are applied to the deformation parameters (β, γ). Additionally, the constraint ⟨Q21⟩= 0 is included, so the definitions made in Section 1.2 hold. To restore the symmetries violated by the previously obtained HFB functions |ϕHFB⟩, quantum number projection is required. This involves restoring proton number, neutron number, and total angular momentum. The HFB wavefunctions can be expressed as linear combinations of states with good quantum numbers: |ϕHFB⟩=X N,Z,J aN,Z,J |ϕN,Z,J ⟩.(2.40) To obtain symmetry-conserving wavefunctions, the corresponding projectors (PO) are applied to the HFB states |ϕN,Z,J ⟩=PNPZPJ MK|ϕHFB⟩.(2.41) The particle number projectors for neutrons and protons are PN=1 2πZ2π 0 eiφN(N−N)dφN,(2.42) PZ=1 2πZ2π 0 eiφZ(Z−Z)dφZ,(2.43) with φNand φZbeing the gauge angle, associated to this U(1) symmetry. Finally, the restoration of good total angular momentum involves PJ MK =2J+ 1 16π2ZΩ DJ∗ MKR(Ω)dΩ,(2.44) CHAPTER 2. THEORETICAL FRAMEWORK 21 where DJ∗ MK are the Wigner Dmatrices [25] with Mand Kthe third components of Jin the laboratory and intrinsic frames of reference, as explained in section 1.1. Then, the SO(3) rotation operator is R(Ω) = e−iαJze−iβJye−iγJz, with Ω ≡(α, β, γ) the Euler angles [49]. The expected value of an operator Oof the projected states can be calculated as ⟨ONZJ ⟩=⟨ϕ|OPNPZPJ MK|ϕ⟩ ⟨ϕ|PNPZPJ MK|ϕ⟩,(2.45) where I have used that (Pi)2=Pi, (Pi)†=Piand the fact that the involved operators commute with each other. The restoration of symmetries in the HFB wavefunctions is computationally the most demanding step in this variational process, as it requires the evaluation of multiple numerically intensive integrals. Having obtained a set of wavefunctions with well-defined quantum numbers and different deformations, they can be employed to approximate an exact solution by considering linear combinations of states characterized by different deformation parameters, akin to configuration mixing in exact diagonalization. This approach is known as the generator-coordinate method (GCM), where the generator coordinates (q) are represented by the deformation parameters (β,γ). The GCM wavefunction can be expressed as |ΨNZJM σ,GCM ⟩=X qK fJMNZ σ;qK PNPZPJ MK|ϕHFB(q)⟩,(2.46) where |ϕHFB(q)⟩are the unprojected HFB wavefunctions, fJMNZ σ;qK is the coefficient of each projected wavefunction and σorders the states with the same quantum numbers Γ≡NZJM. However, the set of states considered might be linearly dependent, which requires a previous diagonalization of the norm overlap matrix NΓ qKq′K′=⟨ϕHFB(q, K)|PNPZPJ KK′|ϕHFB(q′, K′)⟩(2.47) to find an independent basis, denoted as the natural basis: X q′K′NΓ qKq′K′uΓ λ;q′K′=nΓ λuΓ λ;qK.(2.48) In principle, the diagonalization of the overlap matrix would yield a spectrum of non-zero eigenvalues, corresponding to linearly independent natural basis states, as well as null eigenvalues that represent linear combinations of the independent basis states. However, due to the use of numerical methods in the diagonalization process, it becomes challenging to distinguish between small eigenvalues and true zero eigenvalues due to numerical errors. In practice, only eigenstates uΓ λ;qK with eigenvalues nΓ λabove a certain threshold, typically in the range of nΓ λ≈10−10, are taken into account. However, the specific threshold value is arbitrary, and including more states than necessary can result in numerical difficulties that lead to divergence of operator mean values [50]. Figure 2.3 displays the convergence behavior of the first three 0+states of 20Ne. It is evident that as more states are included in the natural basis, the difference between the exact diagonalization and the variational approach decreases. Specifically, when CHAPTER 2. THEORETICAL FRAMEWORK 22 -45 -40 -35 -30 -25 0 5 10 15 20 25 30 Energy (MeV) Number of natural states GCM 0+ 1 GCM 0+ 2 GCM 0+ 3 Diag. 0+ 1 Diag. 0+ 2 Diag. 0+ 3 Figure 2.3: Energy evolution of the first three 0+states in 20Ne with the GCM method as the number of included natural states increases using the USDB interaction. The exact energies are shown in thick dashed lines. 20 states of the natural basis are considered, out of the initial 47 projected HFB wavefunctions, the discrepancy reaches nearly 10 keV. However, upon including additional states, there is a sharp decline in the energy of the ground state (85 MeV), which falls below the value obtained from exact diagonalization. This sudden drop in energy is also observed for the other excited states. Such behavior serves as an indication that the employed basis exhibits issues such as linear dependence. With the choice of a threshold, the natural basis states are |ΛΓ λ⟩=X q′K′ uΓ λ;q′K′ (nΓ λ)1/2PNPZPJ MK|ϕHFB(q)⟩.(2.49) Then, the GCM wavefunction in the natural space is |ΨΓ σ,GCM⟩=X λ GΓ σ;λ|ΛΓ σ⟩(2.50) and the coeficients GΓ σ;λare determined solving the Hill-Wheeler-Griffin (HWG) equation X λ′⟨ΛΓ λ|Heff|ΛΓ λ′⟩GΓ σ;λ′=EΓ σGΓ σ;λ,(2.51) which is also an eigenvalue problem. In addition to the energies of the GCM states, the mean value of any given operator can be obtained as ⟨ΨΓ σ|O|ΨΓ′ σ′⟩=X λλ′X qKq′K′ GΓ∗ σ;λ uΓ∗ λ;qK (nΓ λ)1/2⟨ϕΓ(q)|O|ϕΓ′(q′)⟩GΓ′ σ′;λ′ uΓ′ λ′;q′K′ (nΓ′ λ′)1/2.(2.52) CHAPTER 2. THEORETICAL FRAMEWORK 23 Among these operators, the electromagnetic transitions and quadrupole moments can be evaluated to explore the deformations of different GCM states for a given nucleus. Finally, the probability of finding the GCM state in one of its HFB states can be calculated using the collective wave functions |FΓ σ(q)|2=|X Kλ GΓ σ;λuΓ λ;qK|2.(2.53) Taurus [47] is a numerical suite that incorporates the complete variational process outlined in this section. Initially, the quadrupole-constrained HFB wavefunctions are generated. Then, the projection onto good quantum numbers is carried out. Finally, the GCM is used to combine the projected HFB functions and generate the complete 28Si states. 3. Shape coexistence in 28Si In this section I describe the properties of 28Si employing the nuclear theory techniques discussed in Section 2. The section is organized based on the different collective structures either observed or proposed in previous works among the low-lying states of 28Si. These structures include the observed oblate rotational band associated with the ground state (Section 3.1), the measured prolate rotational band (Section 3.2), and the potential existence of a superdeformed structure (Section 3.3). 3.1 Ground state and oblate band The ground state of 28Si cannot be adequately explained by a simplistic spherical mean-field model, wherein the nucleons occupy the 1d5/2orbit and the nucleus adopts a purely spherical shape. By incorporating particles into the 2s1/2orbit using the analytical SU(3) model, which takes into account the nearly degenerate 1d5/2- 2s1/2doublet, significant quadrupole correlations are gained (Q0=−58.1efm2). These correlations establish the oblate deformation (β≃ −0.21) as the ground state in 28Si, which is in agreement with the experimental data [15] as indicated in Table 2.1. However, to gain further insight in the structure of 28Si I employ state-of-the-art shell model calculations with an exact diagonalization performed using the ANTOINE code [43], and a variational method using the TAURUS suite [21,47]. Both methods are implemented within a valence space that encompasses the sd shell (Figure 2.2) and utilize the USDB interaction [38], which is the established effective interaction for this space. For electromagnetic properties, I assume the standard effective nuclear charges for the sd shell: eZ= 1.5eand eN= 0.5e. The inert core consists of a 16O nucleus, while the remaining 12 particles interact with each other explicitly within the valence space. First, I discuss the variational approach to solving the many-body problem. As I have shown in section 2.3.2, the initial step is to generate a a set of quadrupoleconstrained HFB wavefunctions. The constraints are imposed on the particle numbers ⟨Nv⟩=⟨Zv⟩= 6 and the quadrupole parameters (0 ≤β≤0.27,0◦≤γ≤60◦) with spacing of δβ= 0.02 and δγ= 15◦respectively. In addition, I constrain ⟨Q21⟩= 0, to be consistent with the definitions of Section 1.2. Figure 3.1 (left panel) presents the unprojected energy surface. The plot shows that the absolute minimum energy corresponds to an oblate shape (γ= 60◦). Meanwhile, prolate shapes (γ= 0◦) are energetically disfavoured, even when compared to spherical shapes (β≃0). Based on these results, the ground state acquires an oblate deformation with β≃ −0.24, which aligns with the predictions of the SU(3) model as well as experimental observations. However, these HFB wavefunctions mix states with good quantum numbers, thus requiring projection to obtain well-defined quantum number states. In our basis, 24 CHAPTER 3. SHAPE COEXISTENCE IN 28SI 31 0 2 4 6 8 10 vib ND GS GS ND vib GS ND vib 0+ 1 2+ 1 4+ 1 111 149 0+ 3 2+ 2 4+ 3 103 124 0+ 2 2+ 4 36 128 149 111 123 94 32 146 28 50 70 67 USDB-MOD GCM Theory Experiment USDB-MOD DIAG. Theory 0+ 1 2+ 1 4+ 1 6+ 1 0+ 3 2+ 2 4+ 3 0+ 2 2+ 4 0+ 2+ 4+ 6+ 0+ 2+ 4+ 0+ 2+ Oblate Prolate Excitation energy (MeV) Figure 3.7: Band structure of the lowest-lying positive parity states of 28Si: left, experiment [15]; center, calculation using the USDB-MOD interaction with the TAURUS suite and right, using USDB-MOD. It has no impact on the quadrupole moment of the np-nh configurations presented in Table 3.1. Instead, it enables the structure to withstand configuration mixing within the complete sd space. It is worth noting that this change is relatively small, affecting only ϵ(d3/2) among the three single-particle energies and the 63 two-body matrix elements that constitute the USDB interaction [38]. By employing the USDB-MOD interaction and conducting an exact diagonalization, I have obtained the states shown in Figure 3.7 (right). Notably, the transition B(E2,4+ 3−→ 2+ 2) now corresponds to 123 e2fm4, in very good agreement with the experimental value. Furthermore, the states 0+ 3, 2+ 2, and 4+ 3exhibit a clear proportionality to J(J+ 1). Due to the modification that reduces the energy cost of promoting particles to the d3/2orbit, the prolate band head lies at a lower energy level compared to the USDB interaction. While the oblate band and its vibration experience some additional collectivity, their overall properties remain largely similar to the results obtained with the USDB interaction. Consequently, by lowering the d3/2orbit, I have successfully established a well-defined prolate band while introducing only minor alterations to the oblate band and its vibration. In order to gain a better understanding of the structure of the newly found prolate band, I conduct a variational calculation using the USDB-MOD interaction. The resulting energy surfaces, both unprojected and projected to J= 0, are presented in Figure 3.8. In comparison to the USDB results, the unprojected surface exhibits a more pronounced oblate minimum, while the prolate region appears to be relatively flatter. When considering the projected surface, the preference for a spherical shape diminishes further, and the prolate minimum at β= 0.23 is no longer discernible. Using the projected HFB states as a basis, I use the GCM to determine the configuration-mixed eigenstates of 28Si with the USDB-MOD interaction. The con- CHAPTER 3. SHAPE COEXISTENCE IN 28SI 32 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 15 30 45 60 USDB-MOD Unprojected 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 15 30 45 60 USDB-MOD J = 0 132 130 128 126 124 Energy (MeV) 135.0 133.5 132.0 130.5 129.0 127.5 Energy (MeV) Figure 3.8: Energy surfaces of 28Si using the USDB-MOD interaction for the unprojected (left) and projected to J= 0 (right) quadrupole-constrained HFB wavefunctions. Each black circle represents a HFB wavefunction of the mesh. vergence of the first three 0+states is illustrated in Figure 3.9, where the number of natural states included in the basis is varied. I find that the convergence of these states is achieved with the inclusion of up to 47 states from the initial 62 projected HFB states, before encountering any linear dependencies. The convergence pattern observed in USDB-MOD is similar to that of the USDB interaction, demonstrating a favorable agreement with exact diagonalization, as shown in Fig. 3.7. By computing the GCM states, I am able to determine the collective wavefunctions, which provide insight into the observed enhancement of collectivity indicated by the increased B(E2) transition strengths. In Figure 3.10, the collective wavefunctions of the 0+ 3, 2+ 2, and 4+ 3states belonging to the normal deformed prolate band are presented. The 0+ 3bandhead primarily resides in the β≈0.3 region, indicating a larger deformation compared to the USDB result with β≈0.23. Subsequently, the 2+ 2state mixes with both the β≈0.25 region and incorporates some contribution from the β= 0.17 minimum. On the other hand, the 4+ 3state exhibits reduced deformation as it primarily resides in the β= 0.17 minimum, although it retains a significant component from the β≈0.25 region. Overall, this band demonstrates a progressive decrease in deformation as Jincreases. However, it is noteworthy that the deformations obtained with the USDB-MOD interaction are larger than those obtained with USDB, and the enhanced transition from 4+ 3to 2+ 2is justified within these results. These findings are consistent with the complementary information obtained from the analysis of the USDB-MOD interaction with exact diagonalization. The bottom panel of Fig. 3.6 shows that the 0+ 3state of the full sd shell space acquires nearly double the component of the |0+ 4p-4h⟩sd fixed configuration. This additional probability is significant and supports the formation of a normal-deformed prolate band, which was not observed with the non-modified USDB interaction. CHAPTER 3. SHAPE COEXISTENCE IN 28SI 33 -138 -136 -134 -132 -130 -128 Energy (MeV) GCM 0+ 1 GCM 0+ 2 GCM 0+ 3 Diag. 0+ 1 Diag. 0+ 2 Diag. 0+ 3 0.1 1 10 0 5 10 15 20 25 30 35 40 45 50 USDB-MOD ETau-EAnt (MeV) Number of natural states 0+ 1 0+ 2 0+ 3 Figure 3.9: (Top panel) Energy evolution of the first three 0+states in 28Si with the GCM method as the number of included natural states increases using the USDBMOD interaction. (Bottom panel) Energy difference between the GCM and exact diagonalization as a function of the number of natural states. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 15 30 45 60 28 Si , 0+ 3 USDB-MOD 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 15 30 45 60 28 Si , 2+ 2 USDB-MOD 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 15 30 45 60 28 Si , 4+ 3 USDB-MOD 0.00 0.05 0.10 0.15 0.20 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Figure 3.10: Collective wavefunctions of the first three states of the oblate rotational band in 28Si, calculated using the USDB-MOD interaction. 3.3 Superdeformation In sections 3.1 and 3.2 I have discussed how the oblate rotational band, vibrational band, and prolate rotational band, which exhibit normal deformations, can be effectively described within the sd shell. However, the use of this valence space limits on the available deformations among the states of 28Si. Fig. 3.11 shows the deformation parameter value of different np-nh prolate configurations. The analytical (d5/2-s1/2) + d3/2prescription reaches a maximum deformation of approximately β≈0.3 at 4p-4h (orange line), which falls short of producing a superdeformed state. CHAPTER 3. SHAPE COEXISTENCE IN 28SI 34 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2 4 6 8 10 12 0 25 50 75 100 125 150 175 200 β Q0 (efm2) np-nh sd+pf sd+(f7/2-p3/2) (d5/2-s1/2)+(f7/2-p3/2) Diag. sd+(f7/2-p3/2) (d5/2-s1/2)+d3/2 Figure 3.11: Deformation parameters (β) and the intrinsic quadrupole moment (Q0) for prolate states achieved using different SU(3) variations and numerical calculations. Therefore, to reach more extreme deformations, the inclusion of orbitals from the pf shell becomes necessary. The formation of superdeformed bands by promotion of particles from the sd shell to the pf shell is a phenomenon that has been found in 40Ca [8], 36Ar [3], 40Ar [13,14], 44Ti [7] among others, and suggested for 28Si [16] even though it has not been found experimentally despite some recent efforts [19]. We can begin our exploration by considering the SU(3) limit, which encompasses the sd and pf shells. In this limit, all orbits within the same major shell are assumed to have degenerate energies. The deformations resulting from promoting nparticles from the sd shell to the pf shell are as follows: β= 0.32,0.45,0.57,0.61,0.67 for n= 0,2,4,6,8, as shown by the red crosses in Fig. 3.11. Within this framework, promoting 2 particles is nearly sufficient to reach a superdeformed state. However, it is important to note that the SU(3) limit is not realistic due to the assumption of energy degeneracy among the orbits. As the previous deformations are overestimated, I seek for a more realistic model that can accurately predict the relevant configurations of np-nh configurations that may form a superdeformed band. Based on my previous analysis of the sd shell, the nearly degenerate doublet of (d5/2-s1/2) orbitals contributes significantly to the deformation in the sd shell. Considering that promoting particles to the d3/2orbit does not yield sufficient deformation, it is likely that these particles occupy states within the pf shell instead. As a result, the d3/2orbital is expected to contain only a small fraction of particles. The primary source of deformation within the pf shell arises from the quasi-SU(3) orbits, specifically the (f7/2-p3/2) orbitals. These orbits are closer in energy to the sd shell, as illustrated in Figure 2.1. This observation is further supported by Figure 3.11, where the difference between the blue squares representing the sd +pf shells and the red crosses representing only the sd shell CHAPTER 3. SHAPE COEXISTENCE IN 28SI 35 with the inclusion of (f7/2-p3/2) orbits is relatively small. Thus, the key orbits that play a significant role in determining the deformation are the quasi-SU(3) orbits within the sd and pf shells, forming the dsfp ≡(d5/2- s1/2)+(f7/2-p3/2) valence space. The deformations of the np-nh configurations within this space are indicated in green times symbols in Fig. 3.11. Notably, promoting 2 particles to the (f7/2-p3/2) orbits results in a deformation similar to promoting 4 particles to the d3/2orbit, both yielding a deformation of approximately β≈0.3. The most significant enhancement in deformation, compared to their respective previous configurations, is observed in the 2p-2h and 4p-4h configurations within the (f7/2-p3/2) orbits, yielding an increase in deformation of approximately ∆β≈0.2. Considering that the promotion of particles across a major shell incurs a high singleparticle energy cost, it is anticipated that the np-nh configurations with n≥6 will be less relevant, as the marginal increase in deformation, ∆β≈0.05, is outweighed by the associated energy cost. Therefore, I expect that the superdeformed (SD) state should be composed of the 4p-4h configuration in the dsfp space. The 2p-2h configuration cannot be considered a SD shape since its β < 0.5. Another advantage of the dsfp valence space is that, besides capturing the physics of deformation, exact diagonalization is still manageable (dim ≈1·1010). In contrast, the complete sd+pf space is not feasible due to its enormous dimension (dim ≈1·1012) resulting from the combinatorial growth of the many-body basis, as described by Eq. (2.26). Given the change in the valence space from the sd shell in Sections 3.1 and 3.2 (using the USDB interaction) to the sd+pf shells, a different effective interaction is required. However, finding an interaction specifically tailored for 28Si in the sd +pf space proves challenging, as most interactions are either designed for nuclei with a filled d5/2orbit or assume a neutron-rich composition. Consequently, I have opted to employ the SDPF-NR interaction [52], which was developed for neutron-rich silicon isotopes such as 34,35Si. In the sd +pf space, the single-particle energies of the orbits are given in Table 3.2. Notably, there is a substantial energy gap between the d3/2and f7/2orbits, indicative of a major shell closure. Additionally, the f7/2,p3/2, and f5/2orbits are nearly degenerate in energy. To perform the calculations, I reduce the space from sd+pf to sd+(f7/2-p3/2) for the exact diagonalization. Excluding the p1/2orbit is justified due to its energy being approximately 4 MeV higher than the rest of the pf shell orbits. Moreover, the f5/2orbit can be neglected since, for quadrupole properties, the dominant contribution arises from the quasi-SU(3) f7/2-p3/2orbits. The deformations associated with the np-nh configurations in the sd+(f7/2-p3/2) valence space are shown in Fig. 3.11 in cyan asterisks. The agreement with the dsfp scheme, in green times symbols, is noticeable, even though the exact diagonalization includes the d3/2orbit. This supports that the 4p-4h configuration provides the best candidate for the formation of a superdeformed state. Table 3.2: Single-particle energies (SPE) of the orbits in the sd +pf space for the SDPF and SDPF-4MeV interactions. The SPE of the sd-shell orbits correspond to those of the USDB interaction. SPE d5/2s1/2d3/2f7/2p3/2f5/2p1/2 SDPF-NR (MeV) -3.699 -2.915 1.895 6.220 6.314 6.479 10.950 SDPF-4MeV (MeV) -3.699 -2.915 1.895 2.220 2.314 2.479 6.950 CHAPTER 3. SHAPE COEXISTENCE IN 28SI 36 0 10 20 30 40 50 60 70 80 0 5 10 15 20 25 30 35 SDPF-NR Probability (%) Excitation energy (MeV) 2p-2h convolution 4p-4h convolution 2p-2h pivot 4p-4h pivot Figure 3.12: Probability (%) of finding the lowest energy state with fixed 2p-2h and 4p-4h configuration in the full sd+(f7/2-p3/2) states |0+ σ⟩sdfp with the SDPF-NR interaction. Each eigenstate is convoluted with Gaussians of 200 keV width. Following the same strategy as in Section 3.2, we can use the np-nh configurations as a pivot state to perform an exact diagonalization in the full sd + (f7/2-p3/2) space |0+ np-nh⟩sdfp =1 NX σ S(σ)|0+ σ⟩sdfp.(3.2) Figure 3.12 illustrates the Lanczos strength function of the 2p-2h and 4p-4h configurations using the SDPF-NR interaction. The former primarily resides at an excitation energy of approximately 19 MeV, while the latter exhibits some fragmentation in the region around 30 MeV, with a width of 1 MeV. The approximate single-particle cost of promoting these particles can be estimated by considering the gap between the d3/2and f7/2(or p3/2) orbits, which is almost 10 MeV. For the 2p-2h configuration, the excitation energy is nearly 2 times this value, approximately 19 MeV. Meanwhile, for the 4p-4h configuration, the excitation energy lies around 10 MeV lower than expected, which can be attributed to the increased quadrupole correlations resulting from its high deformation. However, it is important to remark that due to its significantly higher energy, the Lanczos strength function for the 4p-4h configuration is not fully converged in energy. Then, if the SD band is dominated solely from a pure np-nh configuration, my calculations predict that the 2p-2h configuration would appear at 19 MeV, and the 4p-4h configuration at 30 MeV in my calculations. However, this result is in contrast with the findings of the study on superdeformation in 28Si [19], where a structure with 4 particles promoted to the pf-shell yields a superdeformed bandhead at 13 MeV (β≈0.6). The 4p-4h structure I find is less deformed (β≈0.5) and lies much higher in energy than the one found in the aforementioned study. In order to identify a superdeformed state within the full sd+(f7/2-p3/2) space, it would be necessary to compute the |0+ σ⟩sdfp states as well as the |2+ σ⟩sdfp states, CHAPTER 3. SHAPE COEXISTENCE IN 28SI 37 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 SDPF-4MeV Probability (%) Excitation energy (MeV) 2p-2h convolution 4p-4h convolution 2p-2h pivot 4p-4h pivot Figure 3.13: Probability (%) of finding the lowest energy state with fixed 2p-2h and 4p-4h configuration in the full sd+(f7/2-p3/2) states |0+ σ⟩sdfp with the SDPF-4MeV interaction. Each eigenstate is convoluted with Gaussians of 200 keV width. since they are required for calculating the B(E2) transitions and the spectroscopic quadrupole moments Qspec. However, considering an energy of Eexc ≤15 MeV, this would involve calculating fully converged states with σ≃10 for both J+ σ. This calculation is extremely computationally demanding within such space. Therefore, an alternative approach is needed. The variational approach proves to be more convenient in this case. Once the projection onto good quantum numbers of the HFB states is performed, which is computationally the most demanding step, the solution of the HWG equation (2.51) enables the computation of any |0+ σ⟩state. Computing these states is enough to locate the SD states, as we can identify the SD states with the collective wavefunctions. However, as σincreases, the variational approach provides a poorer approximation to the exact diagonalization. To address this limitation, it is helpful to artificially modify the interaction such that higher deformation states appear at lower energies in the spectrum. This can be achieved by reducing the single-particle energies of the pf-shell orbits. The modification I employ, referred to as SDPF-4MeV, is presented in Table 3.2. The chosen adjustment in single-particle energies corresponds to a reduction of 4 MeV, which is the maximum energy that maintains the pf shell above the sd-shell orbits without altering them. However, since the modification strongly favors states with pf-shell excitations, these configurations may mix with lower energy states. Figure 3.13 displays the strength functions, Eq. (3.2), of the same 2p-2h and 4p-4h configurations using the SDPF-4MeV interaction. With this modification, the gap between the d5/2and f7/2decreases from approximately 10 MeV to 6 MeV. Consequently, the 2p-2h configuration shifts from 19 MeV to 12 MeV, and the 4p-4h configuration moves from 30 MeV to 15 MeV, albeit fragmenting among several states. Additionally, the 2p-2h configuration also contributes to the ground CHAPTER 3. SHAPE COEXISTENCE IN 28SI 38 0.0 0.1 0.2 0.3 0.4 0.5 0 15 30 45 60 28Si,SDPF-NR Unprojected 0.0 0.1 0.2 0.3 0.4 0.5 0 15 30 45 60 28Si,SDPF-4MeV Unprojected 130 120 110 100 90 80 70 60 Energy (MeV) 130 120 110 100 90 Energy (MeV) Figure 3.14: Energy surfaces of 28Si using the SDPF-NR (left) and the SDPF-4MeV (right) interaction for the unprojected quadrupole-constrained HFB wavefunctions. Each black circle represents a HFB wavefunction of the mesh. state due to the artificial energy modification I have introduced. Nevertheless, the results are consistent with the ones from the non-modified SDPF-NR interaction. These results suggest that the SD structure does not survive configuration mixing. In particular, the analysis strongly disfavors the appearance of the SD band below 15 MeV. We can also examine the impact of the SDPF-NR and SDPF-4MeV interactions on the unprojected HFB surfaces in the sd+(f7/2-p3/2) space, as shown in Figure 3.14. Both panels show that the absolute minimum remains in the oblate region with a deformation of approximately β≈ −0.25, which is consistent with the results obtained using the USDB interaction shown in Figure 3.1. This indicates that the inclusion of pf-shell orbits does not significantly alter the physics of the previously described states within the sd shell. However, the addition of pf-shell orbits enables higher deformations in the prolate region, from approximately β≈0.3 to β≈0.5. These results support the capability of the SDPF-NR interaction to accommodate highly deformed states using the SDPF-NR and SDPF-4MeV interactions. The only difference is that the modified SDPF-4MeV interaction shows a smaller energy range between the minimum and maximum values compared to the original SDPFNR interaction. This reduction in range is a result of the decreased single-particle energy required to promote particles to the pf shell. By performing projection onto good quantum numbers and employing the GCM method to mix the states, I can examine the collective wavefunctions of the SD states in the sd+(f7/2-p3/2) space using the SDPF-4MeV interaction. Figure 3.15 highlights the notable prolate structures in the spectrum. Starting with the 0+ 3 state at an excitation energy of Eex = 7.22 MeV, this state exhibits a moderate deformation with β≈0.3, similar to the prolate bandhead obtained with the USDB- CHAPTER 3. SHAPE COEXISTENCE IN 28SI 39 0.0 0.1 0.2 0.3 0.4 0.5 0 15 30 45 60 0+ 3,SDPF-4MeV E = 7.22 MeV 0.0 0.1 0.2 0.3 0.4 0.5 0 15 30 45 60 0+ 5,SDPF-4MeV E = 12.5 MeV 0.0 0.1 0.2 0.3 0.4 0.5 0 15 30 45 60 0+ 7,SDPF-4MeV E = 14.4 MeV 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.0 0.1 0.2 0.3 0.4 0.5 Figure 3.15: Collective wavefunctions of selected 0+states with prolate deformation in 28Si in the sd+(f7/2-p3/2) space, calculated using the SDPF-4MeV interaction. MOD interaction in Figure 3.10. Furthermore, this state is predominantly contained within the sd shell, with only a fraction of 0.5 out of 12 particles promoted to the pf-shell orbits. This suggests that it corresponds to the equivalent of the ND prolate bandhead from USDB using the SDPF-4MeV or SDPF-NR interaction, as the pfshell orbits play a negligible role in this state. Although not shown, the 0+ 1and 0+ 2 states can also be associated with their respective USDB counterparts for the oblate and vibrational bandheads based on their deformations and occupation patterns. This provides further evidence that the SDPF-NR or SDPF-4MeV interaction is well-suited to describe 28Si. The lowest energy state exhibiting significant deformations is the 0+ 5state at 12.5 MeV, predominantly composed of the β= 0.4 and γ= 0.15◦projected HFB state. The occupation numbers of pf-shell orbits in this particular HFB state are approximately nf7/2≈2 and np3/2≈1, totaling 3 particles in the pf shell. Likewise, the 0+ 7state at 14.4 MeV displays a deformation of β= 0.45 and γ= 0.15◦, with 4 particles occupying the pf-shell. Although these states may exhibit properties resembling SD states in 28Si, it is important to consider the impact of the modified interaction used. The states obtained with the SDPF-4MeV interaction already appear at 12.5 and 14.4 MeV, falling within the energy range expected for SD states in 28Si according to previous studies [16]. However, if we naively add the single-particle energy cost of 4 MeV/particle due to the reduced d5/2-f7/2gap in the SFPF-4MeV interaction, compared to the original SDPF-NR one, the 0+ 5state would be placed at 24 MeV and the 0+ 7state at 28.5 MeV. This supposition is merely qualitative, as the states obtained through configuration mixing would differ, since the energy of the projected HFB states in the high deformation region would be higher and thus less favorable on average. Therefore, according to my calculations, it appears unlikely that a low-lying SD state exists in 28Si within the energy range of Eex ≲15 MeV. Instead, a more realistic view is to suppose that the configurations with particles in the pf shell mix with less deformed states of the sd shell, as suggested by the fragmentation in Fig. 3.13, and the mixed sd-pf nature of the 0+ 5 state in Fig. 3.15. Finally, I have performed a GCM calculation in the full sdpf space, considering all orbits of the sd and pf shells. This calculation is only possible within the CHAPTER 3. SHAPE COEXISTENCE IN 28SI 40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 15 30 45 60 28 Si ; 0+ 2 SDPF-NR-4MeV E = 2.81 MeV 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 15 30 45 60 28 Si ; 0+ 8 SDPF-NR-4MeV E = 14.7 MeV 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.00 0.04 0.08 0.12 0.16 0.20 0.24 Figure 3.16: Collective wavefunctions of selected prolate states in the full sdpf space using the SDPF-4MeV interaction. variational approach, as the exact diagonalization is unfeasible. With the inclusion of the additional pf-shell orbits, the deformations that can be reached increase from β≈0.5 to β≈0.6. This observation aligns with my previous analysis of Fig. 3.11, as the predominant source of deformation in the pf shell arises from its quasi-SU(3) orbits. Figure 3.16 presents the collective wavefunctions of the 0+ 2state at 2.8 MeV and the 0+ 8state at 15 MeV. The 0+ 2state exhibits a deformation of approximately β≈0.35 at an unusually low energy of only 2.8 MeV. The corresponding projected HFB state with β≈0.35 contains 3 particles distributed among the pf-shell orbits. This state is likely an artifact arising from the modification of the interaction. Because the single-particle energy cost of pf shell excitations is greatly reduced, this state captures additional correlations that brings it too low in energy, compared to any experimental state. When searching for a genuine superdeformed (SD) state, the 0+ 8state at 15 MeV emerges as the only candidate. It exhibits a deformation characterized by β= 0.45 and γ= 0.15◦. However, a similar issue arises as observed in the analysis of the sd+(f7/2-p3/2) space. The energy at which the 0+ 8state appears is too high to be considered as a candidate for the formation of low-lying SD states in 28Si. In sum, the results obtained from the fixed np-nh calculation indicate that the formation of superdeformed (SD) structures solely from pure np-nh configurations extending to the pf shell is unlikely in the 10 −15 MeV energy region. This conclusion is further supported by the GCM calculations, which also show no evidence of such SD structures at low energy. Due to the significant energy cost associated with promoting particles to the pf-shell, it is more likely that the mixing of these states with normal-deformed configurations would give rise to moderately deformed states within the the 10-15 MeV energy range rather than SD states.