Abstract

In this thesis work, two topics, triaxiality and reflection asymmetry, have been discussed. Band structures in Tm were studied in a “thin” target experiment as well as in a DSAM lifetime measurement. Two new excited bands were shown to be characterized by a deformation larger than that of the yrast sequence. These structures have been interpreted as Triaxial Strongly Deformed bands associated with particle-hole excitations, rather than with wobbling. Moreover, the Tilted-Axis Cranking calculations provide a natural explanation for the presence of wobbling bands in the Lu isotopes and their absence in the neighboring Tm, Hf and Ta nuclei. A series of so-called “unsafe” Coulomb excitation experiments as well as one-neutron transfer measurements was carried out to investigate the role of octupole correlations in the Pu isotopes. Some striking differences exist between the level scheme and deexcitation patterns seen in Pu, and to a lesser extent in Pu, and those observed in Pu and in many other actinide nuclei such as Th and U, for example. The differences can be linked to the strength of octupole correlations, which are strongest in Pu. Further, all the data find a natural explanation within the recently proposed theoretical framework of octupole condensation.

EXOTIC COLLECTIVE EXCITATIONS AT HIGH SPIN:

TRIAXIAL ROTATION AND OCTUPOLE CONDENSATION


A Dissertation


Submitted to the Graduate School

of the University of Notre Dame

in Partial Fulfillment of the Requirements

for the Degree of


Doctor of Philosophy
by
Xiaofeng Wang,  B.S., M.S.


                                                            

Umesh Garg, Director


                                                            

Robert V. F. Janssens, Director


Graduate Program in Physics

Notre Dame, Indiana

December 2007

© Copyright by

Xiaofeng Wang

2007

All Rights Reserved

[10mm]

Dedicated to my wife
in heartful recognition of
her love and encouragement.

Contents

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Figures

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Tables

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Preface

Shell structure is one of the cornerstones of our description of atomic nuclei. Systems with none or a small number of nucleons outside a closed shell are generally spherical, while those away from closed shells are usually deformed because of the long-range forces between valence nucleons. The purpose of this thesis is to explore two exotic modes of collectivity that have only been proposed recently.

Direct evidence for triaxial nuclear shapes has, historically, been difficult to obtain. Nevertheless, early in the 21st century, evidence was found in nuclei with proton number and mass for wobbling, a collective mode uniquely associated with triaxiality. In the present work, the properties of band structures discovered in a nucleus close to those where wobbling was reported are examined. It is shown that these bands are associated with triaxial rotation, but not wobbling.

During the past year, the concept of octupole condensation has been proposed in order to account for band structures observed in some neutron-deficient actinide nuclei. In the present work, the strength of octupole correlations in plutonium isotopes is investigated. It is shown that the rotational sequences observed in Pu find a natural interpretation within the new concept.

For clarity and ease of reading, this thesis is divided into five chapters. In the first, the theoretical concepts relevant to the problems under discussion are described. The second chapter is devoted to the various experimental techniques and data analysis methods. In the next chapter, the results obtained for Triaxial Strongly Deformed bands in Tm are discussed; a general introduction of triaxiality in nuclei is followed by the presentation of the data and a detailed interpretation. The fourth chapter discusses an investigation of octupole correlations in three even-even Pu isotopes (, 240, 242). The first three sections of this chapter contain a general introduction on reflection asymmetry in nuclei, a motivation of the present work and relevant information about the experiments and the data analysis. The data for each nucleus are then presented one by one in the next three sections, and this is followed by the interpretation within the available models. Finally, this thesis ends with a brief summary of the present work and a perpective on possible future measurements.

Acknowledgments

This thesis is a summation of my research efforts over the past four and one half years. In this long and hard period, I worked with as much wisdom and enthusiasm as I am capable of, but, honestly, I do not think that I could have finished this job without the help and support of many people. Here, I would like to deeply thank every contributor to this work from the bottom of my heart. Please forgive me if some of their names are not mentioned within the limited available space.

Dr. Umesh Garg, my advisor at Notre Dame, opened the door for me and introduced me to the world of nuclear physics, which may become my lifetime career. He is a mentor for me not only in work, but also in life.

Dr. Robert Janssens is my advisor at Argonne. I feel very lucky that I was able to work on my thesis in his group, at the side of a wonderful instrument – Gammasphere. His guidance and enthusiasm throughout the course of this work were determinant for its accomplishment. The things that have made a great impact on me in the past years and will stay with me in the future are his profound knowledge of science as well as his positive attitude towards work and life.

My gratitude also goes to Drs. Stefan Frauendorf and Takashi Nakatsukasa; their excellent theoretical work made a nice interpretation of the data possible.

Drs. Frank Moore, Michael Carpenter, Torben Lauritsen, Shaofei Zhu, Constantin Vaman, Daryl Hartley and N. S. Pattabiraman will never be forgotten because all of them have initiated me to the many data analysis techniques used in this work. Dr. Ingo Wiedenhöver, is thanked for the many fruitful discussions on the Pu data. These made me understand better the structure of the heavy elements.

I thank Dr. Donald Peterson for his patience in teaching me how to use Latex, and Dr. Mario Cromaz, for his answers to my questions about the Blue database.

I am indebted to Dr. Irshad Ahmad and John Greene for the high quality targets used in this work, and to Drs. Filip Kondev, Sean Freeman, Augusto Macchiavelli, Neil Hammond, R.S. Chakrawarthy, S.S. Ghugre and G. Mukherjee for their participation in the thesis experiments.

I thank Drs. Birger Back, Susan Fischer, Kim Lister, Darek Seweryniak, Cheng-lie Jiang, Teng Lek Khoo, Sujit Tandel, Partha Chowdhury and Xiaodong Tang for involving me in their research projects. In this way, I obtained valuable research experience, different from that from my thesis work.

Drs. Philip Collon, Gordon Berry, Ani Aparahamian, Michael Wiescher, Kathy Newman, and James Kaiser are thanked for their continuous support and help throughout the course of my graduate studies.

I owe much to Dr. Walter Johnson, my former advisor. It was his kindness and generousness that made the transfer of my research interests from atomic theory to experimental nuclear physics smooth. The period of my first two years at Notre Dame when I worked with him left me with many good memories.

My friends, Lou Jisona and Nate Hoteling, made my stay at Argonne easier and happier.

My appreciation is also due to the Argonne Physics Division, the Notre Dame Physics Department and the administrative staffs therein (Allan Bernstein, Colleen Tobolic, Janet Bergman, Barbara Weller, Barbara Fletch, Jennifer Maddox, Shelly Goethals, Shari Herman, Sandy Trobaugh, Lesley Krueger ) for all the help I received in the past years.

Last, family is most important for everyone. I would not have accomplished anything without the love and support from my family. My Mom and Dad gave me birth, education and love, I cannot adequately express my gratitude to them in any word. This thesis is dedicated to my wife, Canli, for her love, support and encouragement.

Chapter 1 Theoretical Background

1.1 Fundamental properties of nucleus

Since the discovery of the atomic nucleus following the famous Rutherford scattering experiment [1] in 1911, numerous facts regarding this small (10 to 10 m in diameter, only one ten-thousandth of an atom in size), but heavy (about of an atom in mass) object have been studied and characterized. The well-known properties of the nucleus include the fact that () it consists of protons, positively charged particles, and neutrons, electrically neutral particles; () between nucleons a strong, but short-ranged nuclear force exists, which overcomes the Coulomb repulsion and results in a bound system; () the binding energy per nucleon, which originates from the fact that the mass of a given nucleus is less than the sum of its constituent nucleons, keeps increasing as a function of mass until reaching a maximum of about 8 at mass number A 60, and above this value remains approximately constant [2] (see Figure 1.1); and, () the nuclear force saturates as indicated by the trend of the binding energy per nucleon as a function of mass and by the fact that the nuclear density is almost constant.

Figure 1.1: Average binding energy per nucleon as a function of atomic mass number . Adapted from Ref. [2].

There is strong evidence that nuclei with certain numbers of protons (Z) or neutrons (N) are more stable than others. This is seen, for example, in the neutron and proton separation energy, the energy of first excited states, These specific N or Z numbers, called “magic numbers”, provide an insight into the fact that the nucleons inside nucleus occupy shells, similar to those occupied by the electrons surrounding the nucleus of the atom. On the other hand, the existence of the collective motion of a large number of nucleons in a nucleus, , the collective rotation and vibration of the nucleus, , to be discussed later in this chapter, is also firmly supported by a large number of experimental observations. It is the occupation of shells by two types of nucleons that gives the nucleus its special character. Such occupation is under some conditions responsible for the so-called single-particle aspects of nuclear structure and under some other accounts for its collective behavior. Understanding these two fundamental modes and the interplay between them is one of the most important goals of nuclear structure research.

1.2 Shell model and deformation

1.2.1 The nuclear shell model

In order to account for the shell structure found in the nucleus, the Nuclear Shell Model [3, 4] was developed, and it has proved to be a most successful model. In the shell model, each nucleon is described as moving in an average potential generated by all the other nucleons, the so-called mean field potential. Hence, the ordering and energy of nuclear states can be calculated by choosing an appropriate form of the potential and solving the Schrödinger equation:

(1.1)

One of the applicable potentials can be expressed as,

(1.2)

where is the orbital angular momentum and is the intrinsic spin. The first term, the harmonic oscillator potential, leads to the sequence of levels given in the left column of Figure 1.2, where is the principal quantum number. This first term accounts only for the first three magic numbers. The addition of a term removes some of the degeneracy, as shown in the middle column of Figure 1.2, but it still does not result in the correct magic numbers. Therefore, an additional spin-orbit coupling term  [4] is necessary to obtain the sequence of levels in the right column where the “magic numbers”, , 2, 8, 20, 28, 50, 82, 126, 184, can be understood. The states obtained in this way are occupied by the nucleons in an order of ascending energy starting from the lowest level while obeying the Pauli Exclusion Principle, , a maximum of two nucleons can fill into any single level.

Figure 1.2: Energy levels from calculations using a modified oscillator potential. The three columns of states correspond to the three terms in the potential, and the numbers shown in the right column with red color are the “magic numbers” (see text for details). Adapted from Ref. [2].

The intrinsic spin of a nucleon is , so for a given there are two values of total angular momentum , , corresponding to different spin orientations with respect to the direction of the orbital angular momentum. In spectroscopic notation, the value is added as a subscript, for example, and , and the multiplicity of the states is . As can be seen in Figure 1.2, for , the energy splitting between and states will be large enough to lower the state from one oscillator shell () to the shell below (). Such levels are known as intruder states and are of opposite parity, , to the shell that they eventually occupy.

It should be noted that the magic numbers mentioned here apply to nuclei close to stablity and recent evidence [5] indicates that these numbers are modified for exotic, neutron-rich nuclei. It is also worth pointing out that the scheme of proton states is slightly different at high energy from that of neutrons because of Coulomb repulsion.

Another often used and more realistic potential is the Woods-Saxon (WS) potential,

(1.3)

where , , and . It can also reproduce the “magic numbers” and the shell structure. In the WS potential, the total angular momentum, , and the parity, , are the only good quantum numbers [6].

Until this point, the nuclear problem is considered as one where each nucleon is treated as an independent particle moving in an average potential representing the effective interaction of all other nucleons with the one being described. This description is often referred to as the mean-field approximation. The assumption above is not accurate and, in fact, the nuclear problem should be treated as a many-body problem, due to the mutual interactions between the nucleons. These types of interactions, called residual interactions, must be taken care of if an accurate description of the nucleus is to be achieved.

1.2.2 Deformation

The Shell Model, using the nuclear potentials with spherical symmetry described above, has been successful in explaining many of nuclear phenomena and in predicting the properties of spherical or near-spherical nuclei, in which the number of nucleons outside a closed shell is small. However, when considering nuclei away from closed shells, the residual interactions between the valence nucleons (nucleons beyond a closed shell) can not be described by the spherical Shell Model. In such nuclei, the long-range effective forces between valence nucleons will lead to collective motion. In some cases, these collective effects can be strong enough to drive to a breaking of the spherical symmetry, and a permanent deformation of the nucleus is then established as the total energy of nuclear system with a deformed shape becomes lower than that associated with a spherical shape. The nuclear shape can be described using a radius vector in terms of a set of shape parameters in the following way:

(1.4)

where is the distance from the center of the nucleus to the surface at angles , is the radius of a sphere having the same volume as the deformed nucleus, the factor is due to nuclear volume conservation, and is a spherical harmonic function of and .

In expression 1.4, the lowest multipole, , corresponds to a shift of the position of the center of mass. It can be easily eliminated by requiring the origin of the coordinate system to coincide with the center of mass. The terms associated with represent the quadrupole deformation. In such a case, the nucleus is either of oblate deformation (with two equal semi-major axes) or of prolate deformation (having two equal semi-minor axes), or of triaxial deformation (having three unequal axes). The latter case is one of the two foci of this thesis work. The terms introduce octupole deformation, which is reflection asymmetric with a pear shape as one of the typical shapes; this is the other emphasis of this work. For the issues addressed in the present thesis, the (hexadecapole) and higher order terms are sufficiently small that they can be ignored.

In the case of pure quadrupole deformation, Eq. 1.4 can be simplified to

(1.5)

The choice of an appropriate coordinate system where the principal axis is lined up with the axis of symmetry of the nuclear shape leads to , . Using the so-called Lund convention (shown in Figure 1.3), the coefficients and can be expressed as

(1.6)
(1.7)

where the parameters and represent the excentricity and non-axiality of the nuclear shape, respectively (see Figure 1.3), and are defined by the Lund convention as:

(1.8)
(1.9)
(1.10)
Figure 1.3: The nuclear deformations described in the Lund convention. Adapted from Ref. [7].

For axially symmetric deformation, , can be derived from the equations above as:

(1.11)

where is the difference between the major () and minor () axis of the ellipsoid. It can be concluded from Eq. 1.11 that for oblate deformation, in which , whereas for a prolate shape, in which . Typical values for found in nuclei are: 0.2 – 0.3 for normal deformation and 0.4 – 0.6 for superdeformation. Another popular deformation parameter is often used in the literature. For small deformation,

(1.12)

The parameter is the one to describe the degree of triaxiality. As can be seen in Figure 1.3, the nuclei are axially deformed only when is equal to multiples of , while intermediate values of describe various degrees of triaxiality with the maximum degree of triaxial deformation being reached when is an odd multiple of .

1.2.3 The deformed shell model

As was mentioned above, the spherical Shell Model has difficulies when dealing with issues regarding deformed nuclei. Therefore, the deformed shell model was introduced.

The modified harmonic oscillator potential, , Nilsson potential [8], allows to take deformation into account. The Hamiltonian in this case can be written as:

(1.13)

where the term represents the spin-orbit force, and the term was introduced by Nilsson to simulate the flattening of the nuclear potential at the bottom of the well (as obtained with a WS potential). The factors and determine the strength of the spin-orbit and term, respectively. The terms are the one-dimensional oscillator frequencies which can be expressed as a function of the deformation. In the axially-symmetric case,

(1.14)

where is the oscillator frequency () in the spherical potential, for which . Using the deformation-dependent Hamiltonian, the single-particle energies can be calculated as a function of the deformation . A plot of single-particle energies versus deformation is known as a Nilsson diagram; two examples of which are given in Figures 1.4 and 1.5 for and , respectively.

Figure 1.4: Nilsson diagram for protons, . Adapted from Ref. [9].
Figure 1.5: Nilsson diagram for neutrons, . Adapted from Ref. [9].

The Nilsson orbitals can be characterised by the so-called asymptotic quantum numbers

(1.15)

where is the principal quantum number from the harmonic oscillator, is the projection of the single-particle angular momentum onto the symmetry axis (), is the projection of the orbital angular momentum onto the symmetry axis and is the number of oscillator quanta along the symmetry axis. While and are strictly valid quantum numbers for the Hamiltonian (Eq. 1.13), and become good quantum numbers only for large deformations and are approximate quantum numbers otherwise. The parity of the state, , is determined by . The projection of the intrinsic spin of the nucleon onto the symmetry axis is , thus we can define . The asymptotic quantum numbers for the Nilsson model are shown schematically in Figure 1.6.

Figure 1.6: Asymptotic quantum numbers for the Nilsson model: is the angular momentum of collective rotation; and are the projections of the total angular momentum onto the laboratory axis and the symmetry axis respectively, and is equal to if no collective rotation occurs or the axis of collective rotation is perpendicular to the symmetry axis. Adapted from Ref. [7].

It should also be remembered that if is even, then () must also be even. Similarly if is odd, then () must be odd. It can be seen in Figure 1.4 and Figure 1.5 that at zero deformation, the ()-fold degeneracy of a given state is not lifted. When the deformation is introduced, the states split into two-fold degenerate levels, the number of which for a state is .

Many properties of nuclear excitations based on orbitals in the Nilsson model can be understood with these quantum numbers. For example, in Figure 1.5, it can be seen that the shell, with negative parity, , lies in a region of predominantly positive-parity orbits. As a result, the various trajectories of the orbits of parentage are rather straight and the associated wavefunctions are rather pure, while the orbits in the neighboring shells, , , , are bent and changing slopes much more often. Two levels with the same and quantum numbers can not cross because the deformed potential couples them and causes a repulsion. In contrast, only levels with different or cross, because the axial symmetric, reflection symmetric potential has no matrix elements due to its symmetry. Hence, it is just the high- values and negative parities of orbits, being different from those of the neighboring orbits, that lead to the above observation in the Nilsson diagram.

In order to calculate the total energy of the nucleus, a summation of all populated single-particle energies can be made. The big shell gaps at finite values of , seen in the Nilsson diagram, suggest the existence of stable deformations. Thus, within the framework of the Nilsson model, it is possible to predict the magnitude of the deformation for nuclei away from closed shells. This is only a very rough estimate. An accurate method is described in the following section.

1.2.4 The Strutinsky-shell correction

The method to predict the existence of stable, deformed nuclei by calculating the total energy of the nuclear system with the shell model, described earlier, has proved to be successful in interpreting many microscopic aspects of the nucleus, mostly properties of excited states relative to the ground state. However, it fails to accurately reproduce some of the bulk properties of the nucleus, such as the total binding energy. In contrast, another approach, the liquid drop model [10], where the nucleus is described in analogy to a liquid drop, has difficulty in predicting properties related to shell structure, but is often able to provide an adequate interpretation of the macroscopic properties of the nucleus. A new approach that can incorporate the advantages of both of these models was proposed by Strutinsky [11, 12] to accurately reproduce, for example, the observed nuclear ground-state energies. In the Strutinsky approach, the total energy is split into two terms: the first is a macroscopic term, , derived from the liquid drop model, and the second is the microscopic term , which accounts for the fluctuations in the shell energy,

(1.16)

In Eq. 1.16, the quantity is calculated independently for protons and neutrons. It is defined by the difference between the actual discrete level density and a “smeared” level density. The actual discrete level density consists of a sequence of -functions, and the smeared density uses a Gaussian distribution instead. The respective definitions are:

(1.17)

and

(1.18)

Here, is an energy of the order of the shell spacing , , and is a correction function for keeping unchanged the long-range variation over energies much larger than . The shell energy can thus be calculated using

(1.19)

where the factor 2 arises because of the double degeneracy of the deformed levels. Calculations using this method have predicted well, for example, the existence of stable reflection-asymmetric deformation in nuclear ground states [13, 14].

1.3 Rotation and cranked shell model

1.3.1 Nuclear rotation and rotational band

Because of deformation, discussed earlier, the collective rotation of the nucleus becomes possible. This started attracting people’s attention early in the 1950s [15, 16, 17]. In a quantum mechanical description, a system with a symmetry axis (conveniently named as the z-axis) is given by a wave function which is an eigenfunction of the angular momentum operator , and any rotation about this axis produces only a phase change. The rotating system has, therefore, the same wave function and the same energy as the ground state. This simply means that this system can not rotate about the symmetry axis collectively [18]. The spherical nuclei are symmetric with respect to any axis, therefore, it is not possible to observe collective rotation in them. In the case of an axially deformed nucleus, there is a set of axes of rotation, perpendicular to the symmetry axis. A rotation around such an axis is presented schematically in Figure 1.7, and it gives rise to a distinct rotational pattern.

Figure 1.7: Schematic coupling of angular momenta for collective rotation of an axially deformed nucleus: is the collective angular momentum, is the intrinsic angular momentum, and is the projection of the total angular momentum, , onto the symmetry axis . Adapted from Ref. [19].

Here, the rotational angular momentum is generated by the collective motion of many nucleons about the axis , which is perpendicular to the symmetry axis . The intrinsic angular momentum, , is the sum of the angular momenta of the nucleons, , . The total angular momentum is then , and its projection onto the symmetry axis , , is equal to the sum of the projection of the angular momentum of the individual nucleons onto the symmetry axis, , , in this case.

The classical kinetic energy of the rotating rigid body is, , where is the angular momentum and is the moment of inertia. In analogy, for a quantum system, the rotational energy is the expectation value of the Hamiltonian of rotation. For the rotating nuclear system schematically described in Figure 1.7, the Hamiltonian of rotation is given by:

(1.20)

where , are the projection of the total angular momentum onto the axis of rotation and onto the symmetry axis , respectively. The approximation assumes that the and components of can be neglected (strong coupling), which is often the case.

The state of the rotating system can be described in terms of three quantum numbers, the total angular momentum (), its projection onto the axis of rotation (), and its projection onto the symmetry axis (). Hence, the energy of the rotating system can be obtained:

(1.21)

It can be seen in Figure 1.7 that the quantum number is associated with the intrinsic degrees of freedom of the valence nucleons, thus, in the energy of Eq. 1.21, one term depends on intrinsic degrees of freedom, and the other depends on the total angular momentum of the system. The latter, generally called the rotational energy, can be written as:

(1.22)

The total wavefunction of the rotating system is the combination of the rotational wavefunction () and the single-particle wavefunction (), and can be expessed as:

(1.23)

In this expression, the second term reflects the property that a rotation by around the axis of rotation leaves the system unchanged. This rotational invariance results in two degenerate states, and , which form a single series of rotational states with spins give by:

(1.24)

The phase factor is called the signature. If the and components are taken into account,

(1.25)

The new terms compared with Eq. 1.20, and , are called Coriolis interactions. They represent the influence of rotation on the motion of the individual nucleons. Among other things, they disturb the regular sequence (Eq. 1.24). According to the signature quantum number, , the states of Eq. 1.24 can be divided into two distinct sets with an opposite value of the signature:

(1.26)

and

(1.27)

and, each set of states is just a so-called rotational band. This means that the rotational bands are restricted to favored bands and unfavored partners with opposite signature. In an odd-A nucleus, for example, the levels in the favored bands possess spins, , while the unfavored partner bands are characterized by spins, , and opposite signature.

In the excitation mode of rotation, a nucleus deexcites mostly in the form of emitting rays, therefore, it is necessary to briefly introduce the fundamental properties of the rays here.

As shown in Figure 1.8, the energy of a ray that decays from an initial level with energy to a final level with energy is:

(1.28)

Since each nuclear state has a definite angular momentum , and parity , a photon must take out angular momentum (its eigenvalue is ) and parity in accordance with the conservation laws:

(1.29)
(1.30)

The angular momentum of the photon, , is called its multipolarity. For each multipolarity, two types of transitions are possible: the electric transition (EL) or the magnetic transition (ML). Electric transitions have angular momentum and parity , while the magnetic ones are characterized by angular momentum and parity . Therefore, the selection rules for any ray are:

(1.31)

Since the photon has an intrinsic spin of 1, a transition from state to state can not occur. Often, the so-called stretched transitions, , decays from levels with an angular momentum to levels with an angular momentum () and the same parity, dominate in a rotational band, while the transitions, , decays from levels with the angular momentum to levels with the angular momentum () and the opposite parity, dominate inter-band deexcitations, especially in the case of octupole bands discussed in detail in this thesis work.

Figure 1.8: Scheme of a ray. Adapted from Ref. [2].

It is also worth to note that the real nucleus is intermediate between two extremes, a rigid body and a superfluid, as the measured moments of inertia are less than the rigid body values at low spin and larger than those calculated for the rotation of a superfluid. Furthermore, experimentally the moment of inertia of nucleus is found to change as a function of spin. For the rotating nucleus, the important angular rotational frequency, , can be written as

(1.32)

where is called the aligned angular momentum and is the projection of the total angular momentum onto the rotation axis. In the simplest case, . For a rotational band, where states are linked by transitions, the angular rotational frequency can be approximated as

(1.33)

where is the energy of ray between two consecutive levels in the rotational band. For a rotational band, two spin-dependent moments of inertia, which are related to two different aspects of nuclear dynamics, have been introduced in terms of the derivatives of the excitation energy with respect to the aligned angular momentum. The kinematic moment of inertia is the first order derivative

(1.34)

and can be used to express the transition energy, , in a rotational band with Eq. 1.22 as:

(1.35)

through Eq. 1.21; while the dynamical moment of inertia is the second order derivative:

(1.36)

and can be related to the energy spacing of consecutive rays in a rotational band

(1.37)

Moreover, the two moments of inertia have the following relation,

(1.38)

and if is constant in a band.

1.3.2 Pairing interaction

The pairing interaction is a force responsible for binding together two identical nucleons with opposite intrinsic spins in the same orbit, and this interaction is such that the energy of the configuration of opposite spins for the two nucleons is much lower than the one of any other configuration. The existence of pairing forces in the nucleus is firmly supported by many experimental results, for example: (1) the ground state of even-even nuclei always has spin and parity, (2) the ground-state spin of odd-mass nuclei is always determined by the spin of the last nucleon, which is the only unpaired one, and (3) the binding energy of an odd-mass nucleus is found to be always smaller than the average values for two neighboring even-even nuclei. The strength of the pairing interaction, , which favors the maximum spatial overlap between the wave functions of nucleons, is lower for protons () than for neutrons (). The Hamiltonian describing pairing is usually written in the form:

(1.39)

where and are pair creation and annihilation operators, respectively, is the chemical potential, and is the number operator.

Near to the Fermi surface, , near the last filled level, some unoccupied orbits are present. The pairing interaction scatters pairs of nucleons with from occupied states into empty states and this will result in a “smearing” of the Fermi surface. In the absence of pairing, the Fermi surface would be a sharp rectangle (see Figure 1.9).

Figure 1.9: Schematic representation of the smearing of the Fermi surface due to the pairing interaction. Adapted from Ref. [20].

The smearing of the Fermi surface leads to the concept of quasi-particles [21, 22], where particle and hole wave functions are combined. The probability that a state is occupied by a hole is given by the expression:

(1.40)

while the corresponding expression for the occupation by a particle is given as:

(1.41)

where is the single particle energy, and is the average Fermi energy associated with a certain particle number (see Ref. [23] for a detailed discussion of these quantities). The probabilities are normalised such that . It can be seen that, far below the Fermi surface () , and far above the Fermi surface () . Close to the Fermi surface, the occupation probabilities are mixed. Following the treatment described in Ref. [21], the quasi-particle energy can be expressed by:

(1.42)

As the nucleus rotates, the induced Coriolis force, in analogy to the one of the classical rotations, competes with the pairing interaction and attempts to break the pair and align the individual angular momenta of the two nucleons with the rotation axis. More generally, the rotational motion weakens the pairing interaction in the nucleus, , while some pairs of nucleons are broken and align at specific rotational frequencies, pairing is affected for all pairs. This is known as the Coriolis anti-pairing effect (CAP) [24].

1.3.3 The cranked shell model

In order to understand the interplay between the collective and intrinsic degrees of freedom of the nucleons, the cranked shell model (CSM) was developed by Bengtsson and Frauendorf [25], built on the original cranking concepts introduced by Inglis in 1954 [26]. In this model, the nucleons can be viewed as particles independently moving in an average potential, which rotates around the principal axis (), which is perpendicular to the symmetry axis of the nucleus (an example is shown in Figure 1.7).

The cranking model is formulated in the body-fixed frame. The transformation from the laboratory frame to the body-fixed frame can be made easily using the the rotation operator, , where is the projection of the total angular momentum onto the rotational axis . The time-dependent Schrödinger equation of a single particle in the laboratory system can be written as:

(1.43)

Using the rotation operator , the wavefuction in the laboratory frame can be expressed in terms of the intrinsic wavefunction ,

(1.44)

and the Hamiltonian in the laboratory frame can be expressed in terms of the intrinsic Hamiltonian , , the non-rotating Hamiltonian expressed in the body-fixed frame,

(1.45)

Hence, the time-dependent Schrödinger equation of a single particle in the intrinsic (body-fixed) frame can be obtained,

(1.46)

by replacing and with the expressions 1.44 and 1.45, respectively, and computing the time derivation in Eq. 1.43. The single-particle cranking Hamiltonian becomes:

(1.47)

where the term represents the Coriolis and centrifugal forces resulting from the rotating frame. The eigenvalue of the single-particle cranking Hamiltonian, , derived from the Schrödinger equation,

(1.48)

is the single-particle Routhian, where is the single-particle eigenfunction in the rotating frame. Taking into account the pairing interaction, the single-particle (quasi-particle) cranking Hamiltonian becomes:

(1.49)

where is the pair gap. For a given configuration, the total Routhian can be deducted by diagonalizing as:

(1.50)

The single-particle (quasi-particle) aligned angular momentum (alignment), which is the projection of angular momentum onto the axis of rotation, can be obtained from the slope of the single-particle (quasi-particle) Routhian versus the rotational frequency, , . Similar to the Routhian, the total alignment, , is given as:

(1.51)

Therefore, after being appropriately transformed into the rotating frame (to be discussed in Sec. 1.3.4), the measured Routhian and alignment values as a function of the rotational frequency can be compared with the results of calculations from Eqs. 1.50 and 1.51.

Since the non-rotating single-particle wavefunctions are not eigenfunctions of , the rotation leads to a mixing of the single-particle states and breaks the time-reversal symmetry. Thus, for the single-particle states the only remaining good quantum numbers are the parity, , which is a conserved quantum number as long as the shape of the potential can be expanded in even multipoles, and the signature, , which is related to the properties of a nucleonic state under a rotation by around an axis () perpendicular to the symmetry axis. The signature is defined by:

(1.52)

where denotes a wavefunction with signature . While the parity is or , the signature of a single particle state can be written as or conventionally. In a non-rotating potential (if , ), the time-reversed states with the quantum number and , , the projection of spin onto the symmetry axis (), are energetically degenerate. Although they do not have a good signature with respect to a rotation perpendicular to the symmetry axis, it is always possible to form linear combinations of and . These linear combinations can then be used as basis states when solving the cranking equation 1.48. which is then split into four independent sets of equations, each one corresponding to a particular combination of the parity, , and the signature, . The solutions, , Routhians of quasi-particles, can therefore be classified by the quantum numbers (), which have four available values: , , , and . The Routhians are calculated as a function of the rotational frequency, , for a given deformation and pairing gap using the cranking equation. They are usually summarized through quasi-particle diagrams. An example is given in Figure 1.10 which presents the quasi-proton diagram calculated for Er [27]. In the figure, the trajectories (orbitals) are labeled by (), and it is especially noticeable that orbitals with the same () do not cross; they rather come within some energy and then repel each other. The interaction regions can be interpreted as virtual crossings between different quasi-particle configurations, resulting in changes in alignment and energy. The experimental observation associated with a virtual crossing between the occupied and unoccupied quasi-particle orbitals is characterized by a sudden, large increase of the angular momentum along with a decrease in rotational frequency; , the curve bends back and up. The same happens in a plot of the moment of inertia vs. the rotational frequency. This phenomenon has been called “backbending”. It was first observed in the ground state rotational bands of Er and Dy [28]. The underlying physical explanation is the decoupling of a pair of high- quasi-particles from time reversed orbitals, where they have opposite intrinsic spins, and the alignment of their spins with the rotational axis (x) due to the increase of the Coriolis force with rotation [29]. Hence, the rearrangement of the quasi-particle configuration of the nucleus represents the rotational alignment of a pair of quasi-particles.

Figure 1.10: Quasi-proton Routhians of Er calculated with the following parameters: the quadrupole deformation, ; the hexadecapole deformation (a high-order term in Eq. 1.4 in Sec. 1.2.2), ; the triaxiality, ; the pairing gap, ; and the Fermi energy, . Further details about these parameters may be found in Refs. [30, 25], and the energy scale is defined in Ref. [30] (for a spherical shape, commonly, ). The figure is taken from Ref. [27].

1.3.4 Transfering the experimental data to the intrinsic frame of nucleus

As described earlier in this section, the Cranking shell model provides an opportunity to make predictions about the properties of a nuclear system, particularly the alignment and the quasi-particle energy (Routhian) , in the rotating frame of reference. On the other hand, the measured values of the alignment and Routhian can be extracted from experimental data. Hence, after transferring the data from the laboratory frame to the rotating frame, a comparison between experiment and theory can be made to give the data an appropriate theoretical interpretation as well as to test how well the model predicts the experimental observations.

For a rotational band, where states are linked by transitions, the angular rotational frequency can be derived by using the transformed expression of Eq. 1.33 in Sec. 1.3.1,

(1.53)

from the measured -ray energy (for the transition ), the spin , and the known value which represents the projection of onto the axis of symmetry. The two spin-dependent moments of inertia can then be deduced from the measured and spin values for three consecutive levels in the band, , , , and, , through applying the following formula which are deduced from Eqs. 1.34 and 1.36:

(1.54)
(1.55)

where , and are in the same form with only substituted by and , respectively, and, is deduced by replacing with in Eq. 1.53. For the simplest case, in which , , and, , where () is the energy spacing of consecutive rays in the band.

The experimental Routhian is given in terms of the excitation energy of level , the angular frequency , and the angular momentum on the symmetry axis as:

(1.56)

To evaluate the quasi-particle contribution alone, a reference energy must be subtracted in order to eliminate the contribution of the rotating core. This is commonly done by parameterizing the moments of inertia , as functions of the rotational frequency , as described in Ref. [31]:

(1.57)
(1.58)

and are called Harris parameters. Hence, the fits of measured and values, obtained from Eqs. 1.54 and 1.55, in the form of Eqs. 1.57 and 1.58, where values are extracted from Eq. 1.53, will give the value of the Harris parameters. The reference aligned angular momentum can be deduced by using the expression as:

(1.59)

and the reference energy is given by:

(1.60)

omitting the last term, if . Finally, the experimental Routhian in the rotating frame, refering to the quasi-particle contribution alone, is written as:

(1.61)

and, in the same manner, the experimental alignment in the rotating frame, which reflects the quasi-particle contribution only, can be given as:

(1.62)

where is just defined above.

1.3.5 Shape vibrations

In addition to rotation, vibration is also one of the collective excitation modes of the nucleus. One of the foci of this work is the nature of octupole vibrations in the Pu isotopes. It corresponds to a type of oscillation of the shape of the nucleus. When a spherical nucleus absorbs small amounts of energy, its density distribution can start to vibrate around the spherical shape. The magnitude of this vibration can be described by the coefficients , defined in Eq. 1.4 in Sec. 1.2.2. For small amplitude vibrations, the Hamiltonian for a vibration of multipole order , which is actually the difference between the energy of the deformed shape corresponding to the vibration and the energy of the nucleus at rest, can be written as:

(1.63)

With the assumption that the different modes of vibrational excitation are independent from one another, the classical equation of motion can be obtained from the above Hamiltonian,

(1.64)

Therefore, a small vibration can be considered as an harmonic oscillation with the amplitude, , and the angular frequency, . The vibrations are quantized. The quanta are called phonons, and is quantity of vibrational energy for the multipole . Each phonon is a boson carrying angular momentum and a parity . The different modes of low order vibrational excitation () are illustrated in Figure 1.11.

Figure 1.11: A schematic illustration of the different modes of the nuclear vibration.

1.4 Electromagnetic properties of deformed nuclei

1.4.1 Electric quadrupole moment

The nuclear quadrupole moment is one of the most important properties of a deformed nucleus, and the observation of large quadrupole moments in nuclei away from closed shells is one of the direct evidences for the existence of stable nuclear deformation. The intrinsic quadrupole momemt, , in the body fixed frame of a deformed nucleus rotating about its z-axis can be defined in terms of the charge distribution in the nucleus, , and, hence, of the nuclear shape, as:

(1.65)

where and are the lengths of the major and minor axes of nucleus, respectively, and . Therefore, the nuclear quadrupole moment is a direct measure of the nuclear deformation, , for a spherical shape, ; for a prolate shape, ; and for an oblate nucleus, . The moment can also be related to the deformation parameter, . In axially symmetric nuclei with quadrupole deformation only, the first order expression can be given as:

(1.66)

Generally, the experimental quadrupole moments measured in the laboratory frame are the spectroscopic quadrupole moments, , which will be discussed in Chapter 3. As shown in Ref. [32], the intrinsic quadrupole moment, can be obtained by projecting the spectroscopic quadrupole moment onto the frame of reference fixed on the nucleus through the following relation:

(1.67)

where is the projection of onto the symmetry axis, as described in Sec. 1.3.1. For a band, such as the ground state band in even-even nuclei, this relation has the simpler form:

(1.68)

Moreover, as shown in Ref. [33], the experimental transition quadrupole moment, , which can be derived from the measurement of the lifetime of a state (see below), is related to the moment by the relation:

(1.69)

1.4.2 Magnetic moment

In contrast to the nuclear electric moment, the nuclear magnetic moment reflects the contribution of the individual nucleons inside the nucleus. It is convenient to separate the orbital and spin contributions of the neutrons and protons. The magnetic moment operator can be expressed as:

(1.70)

where is the nuclear magneton, and are the orbital and the spin gyromagnetic ratios (the gyromagnetic ratio is the ratio of the magnetic dipole moment to the angular momentum of a nucleus), respectively. Besides this contribution, the rotation of the core as a whole, , the collective rotation, contributes to the nuclear magnetic moments. In units of the nuclear magneton, the latter contribution is proportional to the angular momentum of rotation, . Combining all of the contributions together, the magnetic moment operator can be written after some mathematical treatment as:

(1.71)

The observed nuclear magnetic moment is the expectation value of the magnetic moment operator on a nuclear state :

(1.72)

where is the total angular momentum, is the projection of onto the symmetry axis, and the z-axis is the axis of rotation.

1.4.3 Gamma-ray transition probability and branching ratio

As described in Sec. 1.3.1, in the process of nuclear deexcitation between two levels of energy and (), a ray or a conversion electron is emitted, which carries the energy and angular momentum difference between initial and final states. The angular momentum of the ray (photon) always has an integer value of at least 1, and the photon can be of electric or magnetic character.

As described in Ref. [34, 35], the transition probability for a ray of multipolarity from an excited state to a final state is given by,

(1.73)

where the reduced transition probability for is

(1.74)

and is the electromagnetic operator. The reduced transition probability, , represents a sum of squared matrix elements over the substates of the final state and an average over the substates of the initial state [36]. An approximation to single-particle matrix elements is often used to calculate an approximate unit of strength, which is called Weisskopf unit [34],

(1.75)
(1.76)

where is the mass number, is femtometer ( m), and the units and are and .

In this work, the observed rays are of E1, E2 or M1 character, and the reduced transition probabilities for these three basic cases can be written according to Eq. 1.73,

(1.77)
(1.78)
(1.79)

where is the -ray energy in , and (X is E or M) can be measured experimentally using the relation with the -ray intensity, (the summation goes over all of the emitted rays from the initial state ),

(1.80)

Furthermore, within a rotational band, it is possible to express the reduced transition probabilities of E1, E2 and M1 transitions in terms of the dipole moment (unit: ), , the intrinsic quadrupole moment (unit: ), , and the gyromagnetic ratios, and , respectively, as: