Continuum Shell Model Research Articles

Continuum shell model and nuclear physics at the edge of stability

Studies of nuclei far from the valley of stability are currently in the center of modern nuclear physics. For such loosely bound systems, the continuum effects are vitally important. We develop the continuum shell model based on an effective non-Hermitian Hamiltonian. This rigorous quantum-mechanical method is powerful for description of open quantum systems unifying their structure and reactions. The formalism is explained and examples of its application are given; the results are in a very good agreement with recent experiments on exotic nuclei. We show also how this approach can be successfully applied to a general problem of a signal transmission through an open quantum system.

Continuum Shell Model for Buckling of Single-Walled Carbon Nanotubes with Different Chiral Angles

In this paper, an equivalent thick cylindrical shell model is proposed for the buckling analysis of short single-walled carbon nanotubes (SWCNTs) with allowance for different chiral angles. Extensive, molecular dynamics (MD) simulations are first performed using the adaptive intermolecular reactive bond order potential to determine the critical buckling loads/strains. The MD simulations buckling results are then used as reference solutions to calibrate the properties of the thick cylindrical shell model. Central to this development is the establishment of an empirical expression for the Young's modulus that is a function of both the diameter and the chiral angle of the SWCNT. For the shell model, we have assumed that the Poisson ratio ν = 0.19 and the shell thickness h = 0.066 nm . It will be shown that the proposed shell model furnishes good estimates of the critical buckling loads for SWCNTs with different chiral angles. The critical buckling strains are also evaluated from the critical buckling load with the aid of the stress–strain relation of SWCNTs.

CONTINUUM SHELL MODEL FOR BUCKLING OF ARMCHAIR CARBON NANOTUBES UNDER COMPRESSION OR TORSION

Molecular dynamics (MD) simulations are performed using adaptive intermolecular reactive bond order potential to analyze single-walled and double-walled carbon nanotubes. These carbon nanotubes were analyzed for buckling under compression and under torsion. The MD simulations create a comprehensive database for the critical buckling loads/strains and critical buckling torques/twist angles for armchair SWCNTs and DWCNTs of varying diameters and lengths. Using MD results as a computational benchmark, an equivalent thick shell model of CNT is proposed, which is amenable for analysis using a commercially available software ABAQUS. Based on our MD results, an empirical equation that describes the size-dependent Young's modulus for a single-walled carbon nanotube is established. Buckling analysis of CNT under compression and under torsion are performed with the equivalent shell model using size-dependent Young's modulus, Poisson's ratio = 0.19 and shell thickness h = 0.066 nm. We show that the equivalent shell model gives good estimate of critical buckling load/strain and critical buckling torque with respect to the MD results. Variation of critical twist angle with length of CNT, predicted by the shell model is in good qualitative agreement with MD simulation. However, the equivalent shell model underestimates the critical twist angle by 30% because the continuum shell model overestimates torsional stiffness of CNT compared to an atomistic model of CNT. The equivalent shell model is less computational intensive to implement as compared with MD. Its accuracy for predicting the buckling states for long carbon nanotubes allows it to be used for moderately long CNTs under compression/torsion, in-lieu of MD simulations.

Finite element simulation of single carbon nanotube pull-outs from a cementitious nanocomposite material using an elastic-plastic-damage and cohesive surface models

Carbon nanotubes (CNTs) have recently been integrated in the most widely used material in the world 'concrete' for improving its mechanical properties and fracture resistance. This paper computationally investigates the pull-out behaviour of a single CNT from cement. The effects of: 1) CNT-cement interfacial shear strength, stiffness, and fracture energy; 2) the cement mechanical properties; 3) CNT's mechanical properties, aspect ratio, and surface area to volume ratio on the pull-out strength from a cement matrix are investigated through simulating the single straight CNT pull-out. A coupled elastic-plastic-damage constitutive model is adopted to simulate the behaviour of the cement matrix, whereas the continuum shell model is used to simulate the elastic behaviour of CNT. The surface-based cohesive behaviour is employed for modelling the interface between CNT and cement matrix. It is shown that CNT pull-out is mainly governed by the interfacial fracture energy, and not the interfacial shear strength.

Theoretical Study of Temperature dependent Enthalpy of Absorption, Heat Capacity, and Entropy changes for Protonation of Amines and Amino Acid Solvents.

Abstract The equilibrium constants for protonation of amine and alkanolamine solvents for post combustion CO 2 capture are studied in this work using density functional theory coupled with the continuum and explicit solvation shell model introduced by da Silva et al. [da Silva, E. F.; Svendsen, H. F.; Merz, K. M. J. Phys. Chem. A. 2009, 113, 6404]. The temperature dependency of the protonation equilibrium constants is very crucial in understanding the thermodynamics of any solvent for post combustion carbon capture. Theoretically calculated temperature dependent protonation equilibrium constants for CO 2 capture solvent are given as input to the Gibbs Helmholtz equations to calculate thermodynamic data of a particular solvent. From the comparison of calculated results from molecular modeling with experimental results, it has been seen that the errors are within experimental error bars.

Recoil-corrected continuum shell model calculations for11B(p,n)11C,11B(p,p′)11B*,12C(e,e′x)

Background: The recoil-corrected continuum shell model provides coupled-channels solutions for bound and unbound wave functions from realistic effective interactions. The wave functions are antisymmetric and contain no spurious components because the calculations are performed in the center-of-mass system.Purpose: This model has now been extended to include 1$\ensuremath{\hbar}\ensuremath{\omega}$ excitations in the structure of $p$-shell target (residual) nuclei, hence allowing 0$s$-shell knockout processes. Several reactions involving the ${}^{12}$C compound system are investigated to demonstrate the utility of the model.Methods: The states of ${}^{11}$B and ${}^{11}$C are constructed in the nonspurious 0$\ensuremath{\hbar}\ensuremath{\omega}$ plus 1$\ensuremath{\hbar}\ensuremath{\omega}$ model space. An interaction, fitted to Cohen and Kurath (8-16) plus Reid soft core $g$-matrix elements, is employed. One nucleon is coupled to these states to create a basis for the bound and scattering states for ${}^{12}$C.Results: Calculated elastic and inelastic cross sections agree well with available data. The calculated transverse response at high momentum transfer is lower than that extracted from data. Although significant, meson exchange currents are not sufficient to give agreement with data. Likewise inclusion of 0$s$-shell knockout is not sufficient to provide agreement. The high-energy octupole resonance appears at low momentum transfer and an energy of 106/${A}^{1/3}$.Conclusions: The model should provide meaningful predictions for states near the proton drip line via the ($p$,$n$) reaction. Coupled-channels solutions are necessary for describing ${}^{12}$C($e$,${e}^{\ensuremath{'}}x$) at low momentum transfer. Lack of strength at low energy and momentum transfer in optical model calculations of ${}^{12}$C($e$,${e}^{\ensuremath{'}}x$) is at least partly attributable to the omission of giant resonances. Data for (${\ensuremath{\pi}}^{+}$,${\ensuremath{\pi}}^{+\ensuremath{'}}p$)/(${\ensuremath{\pi}}^{\ensuremath{-}}$,${\ensuremath{\pi}}^{\ensuremath{-}\ensuremath{'}}p$) could verify this conclusion. Lack of strength in the transverse response may be attributable to recoil terms which are omitted in most calculations.

Structure of8B from elastic and inelastic7Be+pscattering

Background: Detailed experimental knowledge of the level structure of light weakly bound nuclei is necessary to guide the development of new theoretical approaches that combine nuclear structure with reaction dynamics.Purpose: The resonant structure of ${}^{8}$B is studied in this work.Method: Excitation functions for elastic and inelastic ${}^{7}$Be+$p$ scattering were measured using a ${}^{7}$Be rare isotope beam. Excitation energies ranging between 1.6 and 3.4 MeV were investigated. An $R$-matrix analysis of the excitation functions was performed.Results: New low-lying resonances at 1.9, 2.54, and 3.3 MeV in ${}^{8}$B are reported with spin-parity assignment 0${}^{+}$, 2${}^{+}$, and 1${}^{+}$, respectively. Comparison to the time-dependent continuum shell (TDCSM) model and ab initio no-core shell model/resonating-group method (NCSM/RGM) calculations is performed. This work is a more detailed analysis of the data first published as a Rapid Communication. J. P. Mitchell, G. V. Rogachev, E. D. Johnson, L. T. Baby, K. W. Kemper et al., [Phys. Rev. C 82, 011601(R) (2010)].Conclusions: Identification of the 0${}^{+}$, 2${}^{+}$, 1${}^{+}$ states that were predicted by some models at relatively low energy but never observed experimentally is an important step toward understanding the structure of ${}^{8}$B. Their identification was aided by having both elastic and inelastic scattering data. Direct comparison of the cross sections and phase shifts predicted by the TDCSM and ab initio no-core shell model coupled with the resonating group method is of particular interest and provides a good test for these theoretical approaches.

Free vibration analysis of nanocones embedded in an elastic medium using a nonlocal continuum shell model

The effect of elastic foundation on the free vibration characteristics of embedded nanocones is investigated in this paper. The nanocone is modeled as a thin shell and the nonlocal elasticity theory is used to derive the governing equations of motion. Also the elastic medium is simulated using Winkler and Pasternak foundation models. Based on the modal analysis technique and applying the Galerkin method the governing equations are solved to obtain the natural frequencies. The drawn results emphasis the effects of geometry and small-scale parameter on the natural frequencies of nanocone. Also the effect of elastic foundation modulus on the resonance frequencies of the nanocones are studied and some conclusions are outlined.

Toward understanding the microscopic origin of nuclear clustering

AbstractOpen Quantum System (OQS) description of a many‐body system involves interaction of Shell Model (SM) states through the particle continuum. In realistic nuclear applications, this interaction may lead to collective phenomena in the ensemble of SM states. We claim that the nuclear clustering is an emergent, near‐threshold phenomenon, which cannot be elucidated within the Closed Quantum System (CQS) framework. We approach this problem by investigating the near‐threshold behavior of Exceptional Points (EPs) in the realistic Continuum Shell Model (CSM). The consequences for the alpha‐clustering phenomenon are discussed.

Asymptotic normalization coefficients and continuum coupling in mirror nuclei

Background: An asymptotic normalization coefficient (ANC) characterizes the asymptotic form of a one-nucleon overlap integral required for description of nucleon-removal reactions. Purpose: We investigate the impact of the particle continuum on proton and neutron ANCs for mirror systems from $p$- and $sd$-shell regions. Method: We use the real-energy and complex-energy continuum shell model approaches. Results: We studied the general structure of the single-particle ANCs as a function of the binding energy and orbital angular momentum. We computed ANCs in mirror nuclei for different physical situations, including capture reactions to weakly-bound and unbound states. Conclusions: We demonstrated that the single-particle ANCs exhibit generic behavior that is different for charged and neutral particles. We verified the previously proposed relation [N.K. Timofeyuk, R.C. Johnson, and A.M. Mukhamedzhanov, Phys. Rev. Lett. 91, 232501 (2003); Phys. Rev. Lett. 97, 069904(E) (2006); N. K. Timofeyuk and P. Descouvemont, Phys. Rev. C 72, 064324 (2005)] between proton and neutron mirror ANCs. We find minor modifications if the spectroscopic strength is either localized in a single state or broadly distributed. For cases when several states couple strongly to the decay channel, these modifications may reach 30%.

Stability analysis of nanocones under external pressure and axial compression using a nonlocal shell model

A nonlocal continuum shell model is developed to study the stability of nanocones under combined loading: external pressure and compression force. The nonlinear governing equations of motion of nanocone are obtained using Hamilton's principle and the external loads are considered as prestress. Based on Eringen's nonlocal elasticity theory the small-scale effect is accounted in the governing equations of motion. To obtain the critical loads, the equations are solved using Galerkin technique and the effect of small-scale parameter and geometry on the stability of nanocone is studied.

Nonlocal Continuum Modeling and Molecular Dynamics Simulation of Torsional Vibration of Carbon Nanotubes

This paper investigates the size effects in the dynamic torsional response of single-walled carbon nanotubes (SWCNTs) by developing a modified nonlocal continuum shell model. The purpose is to facilitate the design of devices based on CNT torsion by providing a simple, accurate, and efficient continuum model that can predict the frequency of torsional vibrations and the propagation speed of torsional waves. To this end, dispersion relations of torsional waves are obtained from the proposed nonlocal model and compared to classical models. It is seen that the classical and nonlocal models predict nondispersive and dispersive behavior, respectively. Molecular dynamics simulations of torsional vibrations of (6, 6) and (10, 10) SWCNTs are also performed, the results of which are compared with the classical and nonlocal models and used to extract consistent values of the nonlocal elasticity constant. The superiority and accuracy of the nonlocal elasticity model in predicting the size-dependent dynamic torsional response of SWCNTs are demonstrated.

Free vibration analysis of nanocones using a nonlocal continuum model

This Letter aims at the investigation of free-vibration properties of nanocones based on a nonlocal continuum shell model. A novel approach is used for derivation of the governing equations of motion and the Galerkin technique is used to obtain the natural frequencies of vibrations. The effects of small-scale and geometrical parameters of the nanocone on the natural frequencies are studied and some conclusions are drawn.

Porter-Thomas distribution in unstable many-body systems

We use the continuum shell model approach to explore the resonance width distribution in unstable many-body systems. The single-particle nature of a decay, the few-body character of the interaction Hamiltonian, and the collectivity that emerges in nonstationary systems due to the coupling to the continuum of reaction states are discussed. Correlations between the structures of the parent and daughter nuclear systems in the common Fock space are found to result in deviations of decay width statistics from the Porter-Thomas distribution.

Shell Model for Open Quantum Systems

The postulate of Fano to develop the configuration interaction description of weakly bound or unbound energy levels which would be fully symmetric treatment of bound, resonance, and scattering single-particle states has been realized recently with the development of a modern continuum shell model. We discuss salient non-perturbative effects of the configuration mixing induced by the configuration interaction in nuclear many-body states near the dissociation limit (the particle-emission threshold), which find a natural formulation in the non-hermitian framework of the continuum shell model.

Vibration Properties of Single Walled Carbon Nanotubes—A Comparison Between the Atomistic Simulations and Continuum Shell Modelling

Free vibration analysis of single-walled carbon nanotubes (SWCNTs) is carried out to study the natural frequencies and mode shapes. Appropriate continuum parameters are proposed for equivalent continuum modelling of SWCNTs. Atomistic simulations and continuum shell modelling are employed for the purpose. The formulation of the stiffness matrix computation of the atomistic model is brought out using a finite difference method to bypass the usage of Young’s modulus and wall thickness of SWCNTs. The importance of using equilibrium configuration is highlighted. The large scattered values of Young’s modulus and wall thickness of SWCNTs are reported based on the prior atomistic models. The appropriate set of continuum parameters are proposed based on the free vibration analysis of continuum shell model of SWCNTs. The computed natural frequencies and mode shapes given by the continuum shell model are close to that of atomistic computations, provided the wall thickness values (between 0.06 nm to 0.09 nm) and corresponding Young’s modulus of SWCNTs are chosen appropriately.

Low-lying states inB8

Excitation functions of elastic and inelastic $^{7}\mathrm{Be}+p$ scattering were measured in the energy range between 1.6 and 2.8 MeV in the c.m. An $R$-matrix analysis of the excitation functions provides strong evidence for new positive parity states in $^{8}\mathrm{B}$. A new ${2}^{+}$ state at an excitation energy of 2.55 MeV was observed, and a new ${0}^{+}$ state at 1.9 MeV is tentatively suggested. The $R$-matrix and time-dependent continuum shell model were used in the analysis of the excitation functions. The new results are compared to the calculations of contemporary theoretical models.

Open problems in the theory of nuclear open quantum systems

Is there a connection between the branch point singularity at the particle emission threshold and the appearance of cluster states which reveal the structure of a corresponding reaction channel? Which nuclear states are most impacted by the coupling to the scattering continuum? What should be the most important steps in developing the theory that will truly unify nuclear structure and nuclear reactions? The common denominator of these questions is the continuum shell-model approach to bound and unbound nuclear states, nuclear decays and reactions.

Torsional buckling of carbon nanotubes based on nonlocal elasticity shell models

Abstract This paper investigates size-effects in the torsional response of single walled carbon nanotubes (SWCNTs) by developing a modified nonlocal continuum shell model. The purpose is to facilitate the design of devices based on SWCNT torsion by providing a simple, accurate and efficient continuum model that can predict the corresponding buckling loads. To this end, Eringen’s equations of nonlocal elasticity are incorporated into the classical models for torsion of cylindrical shells given by Timoshenko and Donnell. In contrast to the classical models, the nonlocal model developed here predicts non-dimensional buckling torques that depend on the values of certain geometric parameters of the CNT, allowing for the inclusion of size-effects. Molecular dynamics simulations of torsional buckling are also performed and the results of which are compared with the classical and nonlocal models and used to extract consistent values of shell thickness and the nonlocal elasticity constant (e0). A thickness of 0.85 A and nonlocal constant values of approximately 0.8 and 0.6 for armchair and zigzag nanotubes respectively are recommended for torsional analysis of SWCNTs using nonlocal shell models. The size-dependent nonlocal models together with molecular dynamics simulations show that classical shell models overestimate the critical buckling torque of SWCNTs and are not suitable for modeling of SWCNTs with diameters smaller than 1.5 nm.

Formulation of 2D Graphene Deformation Based on Chiral‐Tube Base Vectors

The intrinsic feature of graphene honeycomb lattice is defined by its chiral index (n, m), which can be taken into account when using molecular dynamics. However, how to introduce the index into the continuum model of graphene is still an open problem. The present manuscript adopts the continuum shell model with single director to describe the mechanical behaviors of graphene. In order to consider the intrinsic features of the graphene honeycomb lattice—chiral index (n, m), the chiral‐tube vectors of graphene in real space have been used for construction of reference unit base vectors of the shell model; therefore, the formulations will contain the chiral index automatically, or in an explicit form in physical components. The results are quite useful for future studies of graphene mechanics.