Shell Model

Using the REX-ISOLDE facility at CERN the Coulomb excitation cross sections for the 0 + gs → 2 + 1 transition in the β-unstable isotopes 100,102,104 Cd have been measured for the first time. Two different targets were used, which allows for the first extraction of the static electric quadrupole moments Q(2 + 1) in 102,104 Cd. In addition to the B(E2) values in 102,104 Cd, a first experimental limit for the B(E2) value in 100 Cd is presented. The data was analyzed using the maximum likelihood method. The provided probability distributions impose a test for theoretical predictions of the static and dynamic moments. The data are interpreted within the shell-model using realistic matrix elements obtained from a G-matrix renormalized CD-Bonn interaction. In view of recent results for the light Sn isotopes the data are discussed in the context of a renormalization of the neutron effective charge. This study is the first to use the reorientation effect for post-accelerated short-lived radioactive isotopes to simultaneously determine the B(E2) and the Q(2 + 1) values.

This paper investigates the transverse and torsional wave in single-and double-walled carbon nanotubes (SWCNTs and DWCNTs), focusing on the effect of carbon nanotube microstructure on wave dispersion. The SWCNTs and DWCNTs are modeled as nonlocal single and double elastic cylindrical shells. Molecular dynamics (MD) simulations indicate that the wave dispersion predicted by the nonlocal elastic cylindrical shell theory shows good agreement with that of the MD simulations in a wide frequency range up to the terahertz region. The nonlocal elastic shell theory provides a better prediction of the dispersion relationships than the classical shell theory when the wavenumber is large enough for the carbon nanotube microstructure to have a significant influence on the wave dispersion. The nonlocal shell models are required when the wavelengths are approximately less than 2.36 Â 10 À9 and 0.95 Â 10 À9 m for transverse wave in armchair (15,15) SWCNT and torsional wave in armchair (10,10) SWCNT, respectively. Moreover, an MD-based estimation of the scale coefficient e 0 for the nonlocal elastic cylindrical shell model is suggested. Due to the small-scale effects of SWCNTs and the interlayer van der Waals interaction of DWCNTs, the phase difference of the transverse wave in the inner and outer tube can be observed in MD simulations in wave propagation at high frequency. However, the van der Waals interaction has little effect on the phase difference of transverse wave.

In this work neutron cross section measurements for various elements and for energies between 3.0 and 6.7 MeV, are reported. Monoenergetic neutrons were produced by means of the D(d,n)3He reaction and collimated using the associated particle technique. The most relevant characteristics of the neutron facility are described.

Shell model calculations are performed for magnetic dipole excitations in 8 Be and 10 Be in which all valence configurations plus 2hω excitations are allowed (large space). We study both the orbital and spin excitations. The results are compared with the 'valence space only' calculations (small space). The cumulative energy weighted sums are

In this work we study the accu- racy of modern higher-order shell finite element formulations in computation of 3d stress state in elastic shells. In that sense we compare three higher-order shell models: (i) with seven displacement-like kinematic parameters, and (ii, iii) with six displacement-like kinematic parame- ters plus one strain-like kinematic parameter in- troduced by two different versions of enhanced assumed strain (EAS) concept. The finite ele- ment approximations of all shell models are based on 4-node quadrilateral elements. Geometrically nonlinear and consistently linearized forms of considered formulations are given. Several nu- merical examples are presented, where computed stresses are compared with analytical solutions. It was found that through-the-thickness varia- tion of some (non-dominant) stress tensor com- ponents, including through-the-thickness normal stress, may be computed very inaccurately. The reliable representation for those stresses can be in- terp...

In this work we consider the geometrically exact shell model subjected to finite rotations, making use of rotation vector parameters for handling the corresponding constrained rotation for smooth shells. A modification of such a parameterization which is based on the incremental rotation vector and thus capable of avoiding the singularity problem is also discussed. Copyright © 2001 John Wiley & Sons, Ltd.

We discuss a theoretical formulation of shell model accounting for through-the-thickness stretching, which allows for large deformations and direct use of 3d constitutive equations. Three different possibilities for implementing this model within the framework of the finite element method are examined: one leading to 7 nodal parameters and the remaining two to 6 nodal parameters. The 7-parameter shell model with no simplification of kinematic terms is compared to the 7-parameter shell model which exploits usual simplifications of the Green-Lagrange strains. Two different ways of implementing the incompatible mode method for reducing the number of parameters to 6 are presented. One implementation uses an additive decomposition of the strains and the other an additive decomposition of the deformation gradient. Several numerical examples are given to illustrate performance of the shell elements developed herein.

In this work we study the time-stepping schemes for shell models which describe the shell director vector motion by the nite rotations. Di erent possibilities for choosing director rotations are examined and their relationships are cast in terms of the commutative diagram. The Newmark time-stepping schemes, making use of di erent rotation parameters, are then developed. The mid-point scheme modi ed to either conserve or dissipate the total energy is further examined. Several numerical simulations are presented to illustrate the performance of each developed scheme.

In this work we present a new modelling paradigm for computing the complete failure of metal frames by combining the stress-resultant beam model and the shell model. The shell model is used to compute the material parameters that are needed by an inelastic stress-resultant beam model; therefore, we consider the shell model as the meso-scale model and the beam model as the macro-scale model. The shell model takes into account elastoplasticity with strain-hardening and strain-softening, as well as geometrical nonlinearity (including local buckling of a part of a beam). By using results of the shell model, the stress-resultant inelastic beam model is derived that takes into account elastoplasticity with hardening, as well as softening effects (of material and geometric type). The beam softening effects are numerically modelled in a localized failure point by using beam finite element with embedded discontinuity. The original feature of the proposed multi-scale (i.e. shell-beam) computational model is its ability to incorporate both material and geometrical instability contributions into the stress-resultant beam model softening response. Several representative numerical simulations are presented to illustrate a very satisfying performance of the proposed approach.

The statistics of quiescent times t L between successive bursts of solar flares activity, performed using 20 years of data, displays a power law distribution with exponent a Ӎ 2.4. This is an indication of an underlying complex dynamics with long correlation times. The observed scaling behavior is in contradiction with the self-organized criticality models of solar flares which predict Poissonlike statistics. Chaotic models, including the destabilization of the laminar phases and subsequent restabilization due to nonlinear dynamics, are able to reproduce the power law for the quiescent times. A shell model of MHD turbulence correctly reproduces all the observed distributions.

We study the generalized seniority properties of the low-lying states of two semimagic pseudonuclei 22 O and 46 Ca in the presence of one-and two-body random interactions. When the single-particle splittings are sufficiently large, S-pair correlation is found to be important in the low-lying states of these systems in the presence of random interactions.

A general microscopic procedure for determining the interacting boson model parameters is presented. A set of correlated fermion coupled pair operators is generated using an approximate seniority shell model. With Marumori mapping the boson images corresponding to s and d are constructed and the interacting boson model parameters determined. The shell effects do show up in our calculation specially at the beginning and at the end of the 50-82 shell. The calculated interacting boson model parameters agree well with the corresponding phenomenologically determined values. NUCLEAR SPECTROSCOPY Interacting boson model, microscopic boson theory, broken pair approximation, shell effects.

A method for estimating the part of the effective shell-model interaction that results from intruder~tate singularities within the unit circle is presented. The method is applied to an exactly solvable test problem which involves several such intruder-state singularities and it is found to be accurate to roughly 5-10~. It id then applied to a realistic description of the l'O Js~0* system, in which a 4p-2h intruder state is known to occur. Using the Nilssoa model to describe the intruder state, we estimate the contributions of the associated branch-cut singularities to a perturbation theory (PT) expansion of the effective interaction. E +,n *~of the errors associated with the use of rth order PT is the presence of these braach~ut sinanlarities are also obtained. The poasibllity of ev~tually oombinirtg this method and dis~~+ er PT into a useful theory of the effective interaction is .discussed .

The microscopic foundation of the configuration-mixing interacting boson model, as proposed by Duval and Barrett, is explored. In this model, the normal shell-model configurations are supplemented by intruder configurations that arise by exciting a proton pair from one major shell to the next. A computationally-efficient generalized-seniority formalism for treating the mixing of these two configurations is described. Boson mixing parameters are obtained using the Otsuka-Arima-lachello mapping procedure. Specific application to configuration mixing in the Cd isotopes is reported. The successes and failures of the calculations are assessed and suggestions for future work are made.

Low-lying states in nuclei are investigated using an ensemble of random interactions. Both in the nuclear shell model and in the interacting boson model we find a dominance of J P = 0 + ground states. It is shown that this feature is not due to time reversal symmetry. In the shell model, evidence is found for the occurrence of pairing properties, and in the interacting boson model for both vibrational and rotational band structures. Our results suggest that these features represent general and robust properties of the model space, and do not depend on details of the interactions.

This paper presents a continuum finite element mechanics approach to model the vibration behaviours of singlewalled carbon nanotubes (SWCNTs) of varying lengths, aspect ratios, chiralities, boundary conditions, axial loads and with initial strain applied. The results are in good agreement with the open literature and show that resonancebased carbon nanotubes sensors have the potential to meet the high level performance requirements inherent of many sensor based applications such as mass detectors, biomedical sensors, monitoring for metal deposition and chemical reactions amongst others. Currently, the sensitivity of many electromechanical transducers used for these applications have reached their respective theoretical limit. The merit of carbon nanotubes is that, due to their miniature dimensional structures, the sensitivity of these sensor based applications is vastly improved.

In this work we present interrelations between different finite rotation parametrizations for geometrically exact classical shell models (i.e. models without drilling rotation). In these kind of models the finite rotations are unrestricted in size but constrained in the 3‐d space. In the finite element approximation we use interpolation that restricts the treatment of rotations to the finite element nodes. Mutual relationships between different parametrizations are very clearly established and presented by informative commutative diagrams. The pluses and minuses of different parametrizations are discussed and the finite rotation terms arising in the linearization are given in their explicit forms.

In this work we study the accu- racy of modern higher-order shell finite element formulations in computation of 3d stress state in elastic shells. In that sense we compare three higher-order shell models: (i) with seven displacement-like kinematic parameters, and (ii, iii) with six displacement-like kinematic parame- ters plus one strain-like kinematic parameter in- troduced by two different versions of enhanced assumed strain (EAS) concept. The finite ele- ment approximations of all shell models are based on 4-node quadrilateral elements. Geometrically nonlinear and consistently linearized forms of considered formulations are given. Several nu- merical examples are presented, where computed stresses are compared with analytical solutions. It was found that through-the-thickness varia- tion of some (non-dominant) stress tensor com- ponents, including through-the-thickness normal stress, may be computed very inaccurately. The reliable representation for those stresses can be in- terp...

In this work we consider the geometrically exact shell model subjected to finite rotations, making use of rotation vector parameters for handling the corresponding constrained rotation for smooth shells. A modification of such a parameterization which is based on the incremental rotation vector and thus capable of avoiding the singularity problem is also discussed. Copyright © 2001 John Wiley & Sons, Ltd.

OF THESIS submitted by Jonathan Patrick Entwisle, September 2016, to the University of Manchester for the degree of Doctor of Philosophy and entitled: Changes in proton occupancies pertaining to putative neutrinoless double beta decay in 130Te and 136Xe A systematic study of the change in proton single-particle occupancies in two neutrinoless double beta decay candidates, 130Te→130Xe and 136Xe→136Ba, has been performed. Final states in 129Sb, 129I, 135I and 135Cs have been populated using the (d,3He) single proton removal reaction. The deuterons were accelerated to 101 MeV using the coupled azimuthally varying field and Ring cyclotrons at the Research Center for Nuclear Physics, University of Osaka. The outgoing ejectiles were momentum analysed using the Grand Raiden magnetic spectrometer. Absolute cross sections were measured for states up to 7 MeV in excitation. Transferred angular momenta were identified through a comparison of angular distributions with those calculated using th...

The nuclear structure of the 17O nucleus has been investigated using shell model and self-consistent
Hartree-Fock calculations. In particular, elastic and inelastic electron scattering form factors, energy levels, and
transition probabilities are calculated for positive and negative low-lying states. Two different shell model spaces
have been used for this purpose. The first one is the psdpn model space for positive parity states and the second
one is p1/2sd model space for negative parity states. For all selected excited states, Skyrme interactions are
adopted to generate from them a one-body potential in Hartrre-Fock theory to calculate the single-particle matrix
elements. The deduced results are discussed for the longitudinal and transverse form factors and compared with
the available experimental data. It has been confirmed that combining the shell model plus Hartree-Fock mean
field method with the Skyrme interaction can accommodate very well the nuclear excitation properties, and work
better for low lying states than for higher excitations. Furthermore, the combination can be used to reproduce the
positive and negative parity states after choosing the suitable model space, effective two-body interaction, and
parameterization to reach highly descriptive and predictive results.

The lifetimes of the 2+1 , the 2 + 2 and the 3 − 1 states of Po have been measured in the Pb(C, Be)Po transfer reaction by the Doppler-shift attenuation method. The result for the lifetime of the 2+1 state is about three times shorter than the adopted value. However, the new value still does not allow for a consistent description of the properties of the yrast 2+1 , 4 + 1 , 6 + 1 , and 8 + 1 states of Po in the framework of nuclear shell models. Quasi-particle Phonon Model (QPM) calculations also cannot overcome this problem thus indicating the existence of a peculiarity which is neglected in both theoretical

In this paper we report measurements of the form factor and the structure factor of a sterically stabilized colloidal dispersion consisting of silica spheres coated with octadecane in toluene by small angle neutron scattering (SANS). The phase diagram of this system shows the liquid-liquid coexistence line and also a jamming transition at higher concentrations, where the jamming line intersects the coexistence line roughly at the critical point. We have performed SANS experiments at a temperature well above the transition temperature and at various volume fractions ϕ, spanning from the very dilute regime (ϕ=0.2%) to the critical concentration (ϕ=16%) and the highly viscous regime (ϕ=39.2%). Except for the very dilute regime, we observe a structure factor S(q) in all other cases. We fitted our data over the whole concentration regime using a global fitting routine with a core-shell model for the form factor P(q), taking into account the structure factor, which we describe with the Ro...

β-decay spectroscopy of the proton-rich nucleus 24 Si was performed. The decay scheme was reconstructed from results of delayed γ-ray and proton measurements. We observed two β branches to bound states in 24 Al for the first time. The branching ratios were determined to be 31(4)% and 23.9(15)% for the 1 + 1 state at 0.426 MeV and the state at 1.090 MeV, respectively. The observation of an allowed transition to the 1.090-MeV state enabled us to firmly determine its spin-parity as 1 +. In the proton measurements performed with the E-E method, we observed a new unbound level at 6.735 MeV. The branching ratios to three unbound states, including the new level, were also determined for the first time. Based on the decay scheme, the B(GT) values of 24 Si were deduced. The B(GT) values were smaller than those of the mirror nucleus 24 Ne by 22% and 10% for the 1 + 1 and 1 + 2 states, respectively. The mirror asymmetries of B(GT), observed in both the 1 + 1 and the 1 + 2 states, indicate changes in configuration in the wave function associated with the Thomas-Ehrman shift. To clarify the mechanism of this asymmetry, a comparison with shell-model calculations is also discussed. The calculations attribute the changes in configuration to the lowering of the 1s 1/2 orbital.

We employ a generalized-seniority-based truncation scheme to calculate double beta decay rates in 76Ge, 82Se, ~28Te, and 13°Te, avoiding problems associated with other shell-model calculations and the quasiparticle random phase approximation (QRPA). Our two-neutrino matrix elements are small, reflecting a suppression mechanism first observed in the QRPA. The new results should be most accurate for neutrinoless decay because the closure approximation is reliable and the matrix elements less suppressed. We set limits on the Majorana mass of the neutrino that differ only by factors of two or three from those of previous methods, suggesting that major difficulties in predicting double beta decay are restricted to the two-neutrino process.

In this work, we examine critically the relation between orbital magnetic dipole (scissors mode) strength and quadrupole deformation properties. Assuming a simple K = 0 ground state band in an even-even nucleus, the quantities Q(2 + 1) (i.e., the static quadrupole moment) and B(E2) 0 1 →2 1 both are described by a single parameter-the intrinsic quadrupole moment Q 0. In the shell model, we can operationally define Q 0 (Static) and Q 0 (BE2) and see if they are the same. Following a brief excursion to the sd shell, we perform calculations in the f p shell. The nuclei we consider (44,46,48 Ti and 48,50 Cr) are far from being perfect rotors, but we find that the calculated ratio Q 0 (Static)/Q 0 (BE2) is in many cases surprisingly close to one. We also discuss the quadrupole collectivity of orbital magnetic dipole transitions. We find that the large orbital B(M 1) strength in 44 Ti relative to 46 Ti and 48 Ti cannot be explained by simple deformation arguments.

We perform shell model calculations using a quadrupole-quadrupole interaction (Q.Q).We show results in single j shell spaces and the full S-D shell. We show that one gets useful results with Q.Q in both spaces. We emphasize the importance of the choice of single particle energies in order to obtain the results of Elliott using a Q.Q interaction without the momentum terms. We show a J(J+1) spectrum for a ground state band but with B(E2)'s different from the rotational model. We also show results not found in textbooks such as J (J+1), J(J-1) and J(J+3) excited bands. We find spectra starting with J=0 which have both even J and odd J members.

We perform shell model calculations using a quadrupole-quadrupole interaction (Q.Q) in a single j shell space. We show that this one-parameter interaction is a good predictor of where nuclear isomerism occurs and where it does not occur. The limitations of this interaction are also discussed. We then include other interactions, e.g. in the f 7 / 2 shell those obtained from the (local) spectrum of a two particle system and then from a two hole system. In the g 9 / 2 where there is insufficient empirical data from the two hole system and so various empirical interactions are used.

The g factor of the 2 + 1 state in radioactive 44 Ti has been measured for the first time with the technique of α transfer to 40 Ca beams in inverse kinematics in combination with transient magnetic fields, yielding the value, g(2 + 1) = +0.52(15). In addition, the lifetimes of the 2 + 1 , τ = 3.97(28) ps, and the 4 + 1 states, τ = 0.65(6) ps, were redetermined with higher precision using the Doppler shift attenuation method. The deduced B(E2)'s and the g factor were well explained by a full fp shell model calculation using the FPD6 effective NN interaction. The g factor can also be accounted for by a simple rotational model (g = Z/A). However, if one also considers the B(E2)'s and the E(4 + 1)/E(2 + 1) ratios, then an imperfect vibrator picture gives better agreement with the data.

A new experimental approach to the famous problem of the anomalously slow Gamow-Teller (GT) transitions in the decay of the A 14 multiplet is presented. The GT strength distributions to excited states in 14 C and 14 O were studied in high-resolution d; 2 He and 3 He; t charge-exchange reactions on 14 N. No-core shell-model calculations capable of reproducing the suppression of the decays predict a selective excitation of J 2 states. The experimental confirmation represents a validation of the assumptions about the underlying structure of the 14 N ground state wave function. However, the fragmentation of the GT strength over three 2 final states remains a fundamental issue not explained by the present no-core shell model using a 6@! model space, suggesting possibly the need to include cluster structure in these light nuclei in a consistent way.

A full f p calculation is performed for states which were degenerate in a single-j-shell calculation in which isospin-zero two-body matrix elements were set to zero energy. Most of the splitting in a complete shell calculation (but not all) comes from the T = 0 part of the interaction.

The spin-orbit splittings in the spectra of nuclei with mass numbers 5, 15 and 17 are studied within the framework of shell-model configuration mixing calculations including 2hω excitations. The contributions of the two-body spinorbit and tensor components of the nucleon-nucleon interaction are studied in various model spaces. It is found that the effects of the two-body spin-orbit interaction are dominant and quite sensitive to the size of the model-space considered. The effects of the tensor interaction are weaker. The correlations effects which are included in the larger (0+2)hω shell-model space reduce the spin-orbit splitting in the case of A = 5 by 20%, and enhance it for A = 15 by about the same 20%. However, it is found that the correlations have a very small effect on the d 3/2 − d 5/2 splitting in A = 17.

g factors and lifetimes have been measured for the first 2 ϩ and 4 ϩ states of 46,48 Ti and 50,52 Cr employing the combined technique of projectile Coulomb excitation and transient magnetic fields. The same target was used in all measurements. The individual isotopes were provided by the ion source of the accelerator. These conditions guarantee high reliability and precision as demonstrated in earlier experiments. While the global behavior of the data is well explained by full f p shell model calculations, distinct deviations from theoretical predictions for 46,48 Ti might be attributed to excitations of the 40 Ca core.

The g factor of the 2 + 1 state in 40 Ar has been measured by use of Coulomb excitation in inverse kinematics and the transient magnetic-field technique. The resulting g factor, g(2 + 1) = −0.015(42), is in reasonable agreement with shell-model calculations within the (full sd) π (full fp) ν space, without including core excitations. Although highly deformed admixtures in the wave function cannot be completely ruled out, they are small, in contrast to the case of 42 Ca.

The g factors and mean lifetimes of several short-lived low-lying states in 70 30 Zn 40 have been measured using the techniques of projectile Coulomb excitation in inverse kinematics combined with transient magnetic fields and the Doppler-shift attenuation method. The present results have been interpreted within the framework of large-scale shell-model calculations that include the g 9/2 orbital.

The g factors of the first 2 + and 4 + states of the radioactive 100 Pd nucleus have been investigated for the first time, using an alpha-particle transfer reaction from 12 C to 96 Ru. The transient magnetic field technique in inverse kinematics was used. The 100 46 Pd54 nucleus is a suitable candidate for studying single-particle proton and neutron effects in the nuclear wave functions near the N = Z = 50 shell closures. The results are discussed within the frameworks of both large-scale shell-model calculations and collective-model predictions.

Using the interaction $Q \cdot Q ~+~ xV_{S.O.}$ where $V_{S.O.}$ is a two-body spin-orbit interaction, we study the effects of varying x on the static electric quadrupole and magnetic dipole moments of the $2^+_1$ and $2^+_2$ states of $^{10}Be$. This is done both in the valence space $s^4p^6$ and in a larger space in which 2 $\hbar \omega$ excitations are allowed. In the former case, for x=0, we have the Wigner Supermultiplet limit in which the $2^+_1$ and $2^+_2$ states are degenerate and correspond to K=0 and K=2 rotational states, with equal and opposite static quadrupole moments. Turning on the spin-orbit interaction with sufficient strength in the valence space gives an energy splitting to the two $2^+$ states in accord with experiment. When higher shell admixtures are allowed, we get quite a different behaviour as a function of x than in the valence space. Of particular interest is a value of x for which $Q(2^+_1)$ and $Q(2^+_2)$ both nearly vanish, and so does (somewhat coin...