Statistical properties of nuclei by the shell model Monte Carlo method
|
|
- Roderick Banks
- 5 years ago
- Views:
Transcription
1 Statistical properties of nuclei by the shell model Monte Carlo method Introduction Yoram Alhassid (Yale University) Shell model Monte Carlo (SMMC) method Circumventing the odd particle-number sign problem Statistical properties of nuclei in SMMC Applications to mid-mass and heavy nuclei Projection on good quantum numbers (e.g., spin and parity) Projection on an order parameter (associated with a broken symmetry): level density vs. deformation in the spherical shell model Conclusion and outlook Recent review: Y. Alhassid, arxiv: (2016), in a review book, ed. K.D. Launey
2 Introduction Statistical properties of nuclei: level density, partition function, entropy, Statistical properties, and in particular, level densities, are important input in the Hauser-Feshbach theory of compound nuclear reactions, but are not always accessible to direct measurement. The calculation of statistical properties in the presence of correlations is a challenging many-body problem. - Often calculated using empirical modifications of the Fermi gas formula - Mean-field approximations (e.g., Hartree-Fock-Bogoliubov) often miss important correlations and are problematic in the broken symmetry phase. The configuration-interaction (CI) shell model takes into account correlations beyond the mean-field but the combinatorial increase of the dimensionality of its model space has hindered its applications in mid-mass and heavy nuclei.
3 Conventional diagonalization methods for the shell model are limited to spaces of dimensionality ~ The auxiliary-field Monte Carlo (AFMC method) enables microscopic calculations in spaces that are many orders of magnitude larger (~ ) than those that can be treated by conventional methods. Also known in nuclear physics as the shell model Monte Carlo (SMMC) method. G.H. Lang, C.W. Johnson, S.E. Koonin, W.E. Ormand, PRC 48, 1518 (1993); Y. Alhassid, D.J. Dean, S.E. Koonin, G.H. Lang, W.E. Ormand, PRL 72, 613 (1994). SMMC is the state-of-the-art method for the microscopic calculation of statistical properties of nuclei.
4 Hubbard-Stratonovich (HS) transformation Assume an effective Hamiltonian in Fock space with a one-body part and a two-body interaction : i are single-particle energies and densities. ˆ ij = a i a j R Ĥ = P i iˆn i ˆ P v ˆ 2 are linear combinations of one-body The HS transformation describes the Gibbs ensemble at inverse temperature β = 1/T as a path integral over time-dependent auxiliary fields e β H = D σ G σ U σ G = e v 2 ( )d is a Gaussian weight and U σ is a one-body propagator of non-interacting nucleons in time-dependent auxiliary fields Û = T e R 0 ĥ ( )d e β H ( ) with a one-body Hamiltonian ĥ ( ) = P i iˆn i + P s v ( )ˆ s =1 for v < 0, and s = i for v > 0
5 Thermal expectation values of observables hôi = Tr (Ôe Ĥ ) Tr (e Ĥ ) = R D[ ]G h Ôi Tr Û R D[ ]G Tr Û where hôi Tr (ÔÛ )/Tr Û Grand canonical quantities in the integrands can be expressed in terms of the single-particle representation matrix U of the propagator : Tr Û =det(1 + U ) ha i a ji Tr (a i a jû ) Tr Û = h 1 1+U 1 iji Canonical (i.e., fixed N,Z) quantities are calculated by an exact particle-number projection (using a discrete Fourier transform). The integrand reduces to matrix algebra in the single-particle space (of typical dimension ~100).
6 Quantum Monte Carlo methods and sign problem σ The high-dimensional integration is done by Monte Carlo methods, sampling the fields according to a weight W G Tr A Û Tr A U / Tr A U is the Monte Carlo sign function. For a generic interaction, the sign can be negative for some of the field configurations. When the average sign is small, the fluctuations become very large the Monte Carlo sign problem. Good-sign Ĥ = P interactions i iˆn i + 1 P We can rewrite 2 v ( + ) where is the time-reversed density. ) Sign rule: when all v < 0, Tr U > 0 for any configuration and the interaction is known as a good-sign interaction. v < 0 Proof: when all, we have ĥ = P i iˆn i + P (v + v ) and h = h ) the eigenvalues of U appear in complex conjugate pairs and. { i, i } Tr Û = Q i 1+ i 2 > 0
7 A practical method for overcoming the sign problem Alhassid, Dean, Koonin, Lang, Ormand, PRL 72, 613 (1994). The dominant collective components of effective nuclear interactions have a good sign. A family of good-sign interactions is constructed by multiplying the bad-sign components by a negative parameter g H = H G + gh B g =1 Observables are calculated for 1 <g<0 and extrapolated to. In the calculation of statistical and collective properties of nuclei, we have used good-sign interactions.
8 Circumventing the odd-particle sign problem in SMMC Mukherjee and Alhassid, PRL 109, (2012) Applications of SMMC to odd-even and odd-odd nuclei has been hampered by a sign problem that originates from the projection on odd number of particles. We introduced a method to calculate the ground-state energy of the odd-particle system that circumvents this sign problem. Consider the imaginary-time single-particle Green s functions for even-even nuclei: G Ta a for orbitals =Σ ν = nl j ν( τ) m νm( τ) νm(0) The energy difference between the lowest energy of the odd-particle system for a given spin j and the ground-state energy of the evenparticle system can be extracted from the slope of ln G ν ( τ ). Minimize Ej ( A± 1) to find the groundstate energy and its spin j. G ν (τ )~ e [ E j ( A±1) E gs ( A)] τ
9 Statistical errors of ground-state energy of direct SMMC versus Green s function method Green s function method Direct SMMC Green s function method direct SMMC Direct SMMC Pairing gaps in mid-mass nuclei from odd-even mass differences SMMC in the complete fpg 9/2 shell (in good agreement with experiments)
10 Statistical properties in the SMMC Nakada and Alhassid, PRL 79, 2939 (1997) Partition function Calculate the thermal energy E(β) = < H > versus β and integrate ln Z/ β = E( β) to find the partition function Z( β ). Level density The level density (E) Laplace transform: The average state density is found from approximation: ρ ( E) is related to the partition function by an inverse (E) = 1 2 i 1 2 2πT C e R i1 i1 d e E Z( ) ( ) S E Z( β ) in the saddle-point S(E) = canonical entropy SE ( ) = lnz+β E C = canonical heat capacity 2 C β E/ = β
11 Mid-mass nuclei CI shell model model space: complete fpg9/2-shell. Single-particle Hamiltonian: from Woods-Saxon potential plus spin-orbit Interaction: includes the dominant components of effective interactions pairing: g 0 = MeV determined from odd-even mass differences multipole-multipole interaction terms -- quadrupole, octupole, and hexadecupole: determined from a self-consistent condition and renormalized by k 2 =2, k 3 =1.5, k 4 =1. Level densities in nickel isotopes Excellent agreement with experiments: (i) level counting, (ii) p evaporation spectra (Ohio U., 2012), (iii) neutron resonance data. Bonett-Matiz, Mukherjee, Alhassid, PRC 88, R (2013)
12 Heavy nuclei (lanthanides) CI shell model space: protons: shell plus 1f 7/2 ; neutrons: shell plus 0h 11/2 and 1g 9/2 Single-particle Hamiltonian: from Woods-Saxon potential plus spin-orbit Interaction: pairing (g p,g n ) plus multipole-multipole interaction terms quadrupole, octupole, and hexadecupole. Heavy nuclei exhibit various types of collectivity (vibrational, rotational, ) that are well described by empirical models. However, a microscopic description in a CI shell model has been lacking. Can we describe vibrational and rotational collectivity in heavy nuclei using a spherical CI shell model approach in a truncated space?
13 The various types of collectivity are usually identified by their corresponding spectra, but AFMC does not provide detailed spectroscopy. T The behavior of!" 2 J versus is sensitive to the type of collectivity: 162 Dy 162 Dy J!" 2 = 6 E 2 + T!" 2 e E 2 + /T J =30 is rotational 148 Sm (1 e E 2 + /T ) 2 is vibrational Alhassid, Fang, Nakada, PRL 101 (2008) Ozen, Alhassid, Nakada, PRL 110 (2013)
14 Level densities in samarium and neodymium isotopes Ozen, Alhassid, Nakada, PRL 110 (2013) Level counting n resonance SMMC ρ(e x ) (MeV -1 ) Sm 150 Sm 152 Sm 154 Sm Nd 146 Nd 148 Nd 150 Nd 152 Nd E x (MeV) Good agreement of SMMC densities with various experimental data sets (level counting, neutron resonance data when available).
15 Rotational enhancement in deformed nuclei Alhassid, Bertsch, Gilbreth and Nakada, PRC 93, (2016) A deformed nucleus ( 162 Dy ): Hartree-Fock (HF) vs SMMC Particle-number projection is carried out in the saddle-point approximation Exact particle-number projection: Fanto, Alhassid, Bertsch, arxiv: ρ (MeV -1 ) SMMC HF ρ (MeV -1 ) E x (MeV) E x (MeV) The enhancement of the SMMC density (compared with HF) is due to rotational bands built on top of the intrinsic bandheads. The rotational enhancement gets damped above the shape transition.
16 Projection on good quantum numbers: spin distributions in mid-mass nuclei [Alhassid, Liu, Nakada, PRL 99, (2007)] σ 2 = 7.92 σ 2 = Fe 55 σ 2 60 Fe = 8.11 Co E σ 2 = 11.8 E x = 4.39 MeV x = 5.6 MeV E σ 2 x = 3.39 MeV = 7.96 σ 2 = 10.9 ρ J /ρ 0.02 Low-energy data 0.01 AFMC data scaled to AFMC energy J Spin cutoff model: ρj = ρ 2J πσ e J( J+ 1)/2σ 2 2 σ = spin cutoff parameter Staggering effect (in spin) for even-even nuclei Analysis of experimental data [von Egidy and Bucurescu, PRC 78, R (2008)] confirmed our prediction.
17 Spin distributions in heavy nuclei: 162 Dy Gilbreth, Alhassid, Bonett-Matiz E x = 7.9 MeV E x = 6.2 MeV E x = 3.9 MeV s J / E x = 2.9 MeV E x = 2.5 MeV E x = 2.3 MeV SMMC spin-cutoff staggered s.-c. J / J Good agreement with spin-cutoff (s.-c.) model at higher excitations. Odd-even staggering in spin at low excitation energies. J J ρj 2J + 1 = 3 ρ 2 2πσ e J( J+ 1)/2σ 2 AFMC distributions agree well with an empirical staggered spin cutoff formula based on low-energy counting data (von Egidy and Bucurescu).
18 I calculated from Thermal moment of inertia I vs. excitation energy σ 2 = I T /! 2 Three methods to determine (ii) Fit to spin cutoff model (ii) From σ 2 = < J 2 z >= < J! 2 > /3 (iii) From ratio of level to state density Σ J ρ J (E x )= ρ M =0 (E x ) = 1 2πσ ρ(e x ) σ 2 I E x Fit to J / From <J 2 >/3 From ratio of state to level density Rigid-body Moment of inertia I is suppressed by pairing correlations below E x ~ 5 MeV Parity ratio vs excitation energy Parity ratio is equilibrated at the neutron separation energy ρ / ρ Dy E x (MeV)
19 Projection on an order parameter (associated with a broken symmetry) Example: nuclear deformation in a spherical shell model approach Alhassid, Gilbreth, Bertsch, PRL 113, (2014) Modeling of fission requires level density as a function of deformation. Deformation is a key concept in understanding heavy nuclei but it is based on a mean-field approximation that breaks rotational invariance. The challenge is to study nuclear deformation in a framework that preserves rotational invariance. We calculated the distribution of the axial mass quadrupole in the lab frame using an exact projection on (novel in that [Q 20, H ] 0 ). Q 20 P β (q) = δ (Q 20 q) = 1 Tr e βh dϕ e iϕ q Tr (e iϕ Q 2π 20 e βh )
20 Application to heavy nuclei T = 0.10 MeV T = 1.23 MeV T = 4.00 MeV 154 Sm Sm Rigid rotor SMMC (deformed) P T (q) q (fm 2 ) q (fm 2 ) q (fm 2 ) At low temperatures, the distribution is similar to that of a prolate rigid rotor a model-independent signature of deformation. T = 0.10 MeV T = 0.41 MeV T = 4.00 MeV 148 Sm Sm (spherical) P T (q) q (fm 2 ) q (fm 2 ) q (fm 2 ) The distribution is close to a Gaussian even at low temperatures.
21 Intrinsic shape distributions P T (β,γ ) Alhassid, Mustonen, Gilbreth, Bertsch Information on intrinsic deformation β,γ can be obtained from the expectation values of rotationally invariant combinations of the quadrupole tensor. To 4 th order: q q β 2 ; (q q) q β 3 cos(3γ ) ; (q q) 2 β 4 ( β,γ are the intrinsic deformation parameters) ln P T (β,γ ) at a given temperature T is an invariant and can be expanded in the quadrupole invariants (Landau-like expansion) ln P T = Aβ 2 Bβ 3 cos 3γ + Cβ The expansion coefficients A,B,C can be determined from the expectation values of the invariants, which in turn can be calculated from the low-order moments of q 20 = q in the lab frame. q 2µ < q q >= 5 < q 2 20 >; <(q q) q >= < q 3 >; <(q q) 2 >= < q 4 > 20
22 Expressing the invariants in terms of in the lab frame and integrating over the µ 0 components, we recover P[q 20 ] in the lab frame. q 2µ T = 4.00 MeV T = 1.19 MeV T = 0.10 MeV Sm SMMC expansion P(q) q (fm 2 ) q (fm 2 ) q (fm 2 ) We find excellent agreement with P[q 20 ] calculated in SMMC! T =0.1 3 T = T = lnp(β,γ ) Sm Mimics a shape transition from a deformed to a spherical shape without using a mean-field approximation!
23 β,γ We divide the plane into three regions: spherical, prolate and oblate. oblate 6 prolate spherical Integrate over each deformation region to determine the probability of shapes versus temperature Compare deformed ( 154 Sm), transitional ( 150 Sm) and spherical ( 148 Sm) nuclei
24 Level density versus intrinsic deformation Convert P T ( β,γ ) to level densities vs E x, β,γ Deformed Transitional Spherical E x (MeV)
25 Conclusion SMMC is a powerful method for the microscopic calculation of level densities in very large model spaces; applications in nuclei as heavy as the lanthanides. Microscopic description of rotational enhancement in deformed nuclei. Spin distributions: odd-even staggering in even-even nuclei at low excitation energies; spin cutoff model at higher excitations. Deformation-dependent level density in a rotationally invariant framework (CI shell model). Outlook Other mass regions (actinides, unstable nuclei, ). Gamma strength functions in SMMC by inversion of imaginary-time response functions.
Auxiliary-field quantum Monte Carlo methods for nuclei and cold atoms
Introduction Auxiliary-field quantum Monte Carlo methods for nuclei and cold atoms Yoram Alhassid (Yale University) Auxiliary-field Monte Carlo (AFMC) methods at finite temperature Sign problem and good-sign
More informationThe shape distribution of nuclear level densities in the shell model Monte Carlo method
The shape distribution of nuclear level densities in the shell model Monte Carlo method Introduction Yoram Alhassid (Yale University) Shell model Monte Carlo (SMMC) method and level densities Nuclear deformation
More informationAuxiliary-field quantum Monte Carlo methods in heavy nuclei
Mika Mustonen and Yoram Alhassid (Yale University) Introduction Auxiliary-field quantum Monte Carlo methods in heavy nuclei Auxiliary-field Monte Carlo (AFMC) methods at finite temperature Sign problem
More informationThe shell model Monte Carlo approach to level densities: recent developments and perspectives
The shell model Monte Carlo approach to level densities: recent developments and perspectives Yoram Alhassid (Yale University) Introduction: the shell model Monte Carlo (SMMC) approach Level density in
More informationAuxiliary-field Monte Carlo methods in Fock space: sign problems and methods to circumvent them
Auxiliary-field Monte Carlo methods in Fock space: sign problems and methods to circumvent them Introduction Yoram Alhassid (Yale University) Finite-temperature auxiliary-field Monte Carlo methods in Fock
More informationarxiv: v1 [nucl-th] 18 Jan 2018
Nuclear deformation in the configuration-interaction shell model arxiv:181.6175v1 [nucl-th] 18 Jan 218 Y. Alhassid, 1 G.F. Bertsch 2,3 C.N. Gilbreth, 2 and M.T. Mustonen 1 1 Center for Theoretical Physics,
More informationShell model Monte Carlo level density calculations in the rare-earth region
Shell model Monte Carlo level density calculations in the rare-earth region Kadir Has University Istanbul, Turkey Workshop on Gamma Strength and Level Density in Nuclear Physics and Nuclear Technology
More informationParticle-number projection in finite-temperature mean-field approximations to level densities
Particle-number projection in finite-temperature mean-field approximations to level densities Paul Fanto (Yale University) Motivation Finite-temperature mean-field theory for level densities Particle-number
More informationThe nature of superfluidity in the cold atomic unitary Fermi gas
The nature of superfluidity in the cold atomic unitary Fermi gas Introduction Yoram Alhassid (Yale University) Finite-temperature auxiliary-field Monte Carlo (AFMC) method The trapped unitary Fermi gas
More informationShell Model Monte Carlo Methods and their Applications to Nuclear Level Densities
Shell Model Monte Carlo Methods and their Applications to Nuclear Level Densities H. Nakada (Chiba U., Japan) @ RIKEN (Feb. 14, 2013) Contents : I. Introduction II. Shell model Monte Carlo methods III.
More informationStabilizing Canonical- Ensemble Calculations in the Auxiliary-Field Monte Carlo Method
Stabilizing Canonical- Ensemble Calculations in the Auxiliary-Field Monte Carlo Method C. N. Gilbreth In collaboration with Y. Alhassid Yale University Advances in quantum Monte Carlo techniques for non-relativistic
More informationBenchmarking the Hartree-Fock and Hartree-Fock-Bogoliubov approximations to level densities. G.F. Bertsch, Y. Alhassid, C.N. Gilbreth, and H.
Benchmarking the Hartree-Fock and Hartree-Fock-Bogoliubov approximations to level densities G.F. Bertsch, Y. Alhassid, C.N. Gilbreth, and H. Nakada 5th Workshop on Nuclear Level Density and Gamma Strength,
More informationarxiv: v2 [nucl-th] 10 Oct 2017
INT-PUB-7-4 Nuclear deformation in the laboratory frame C.N. Gilbreth, Y. Alhassid, 2 and G.F. Bertsch,3 Institute of Nuclear Theory, Box 3555, University of Washington, Seattle, WA 9895 2 Center for Theoretical
More informationMoment (and Fermi gas) methods for modeling nuclear state densities
Moment (and Fermi gas) methods for modeling nuclear state densities Calvin W. Johnson (PI) Edgar Teran (former postdoc) San Diego State University supported by grants US DOE-NNSA SNP 2008 1 We all know
More informationMean field studies of odd mass nuclei and quasiparticle excitations. Luis M. Robledo Universidad Autónoma de Madrid Spain
Mean field studies of odd mass nuclei and quasiparticle excitations Luis M. Robledo Universidad Autónoma de Madrid Spain Odd nuclei and multiquasiparticle excitations(motivation) Nuclei with odd number
More informationNuclear models: Collective Nuclear Models (part 2)
Lecture 4 Nuclear models: Collective Nuclear Models (part 2) WS2012/13: Introduction to Nuclear and Particle Physics,, Part I 1 Reminder : cf. Lecture 3 Collective excitations of nuclei The single-particle
More informationJoint ICTP-IAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation August Introduction to Nuclear Physics - 1
2358-19 Joint ICTP-IAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation 6-17 August 2012 Introduction to Nuclear Physics - 1 P. Van Isacker GANIL, Grand Accelerateur National d'ions Lourds
More informationNuclear vibrations and rotations
Nuclear vibrations and rotations Introduction to Nuclear Science Simon Fraser University Spring 2011 NUCS 342 February 2, 2011 NUCS 342 (Lecture 9) February 2, 2011 1 / 29 Outline 1 Significance of collective
More informationThe 2010 US National Nuclear Physics Summer School and the TRIUMF Summer Institute, NNPSS-TSI June 21 July 02, 2010, Vancouver, BC, Canada
TU DARMSTADT The 2010 US National Nuclear Physics Summer School and the TRIUMF Summer Institute, NNPSS-TSI June 21 July 02, 2010, Vancouver, BC, Canada Achim Richter ECT* Trento/Italy and TU Darmstadt/Germany
More informationQuantum Theory of Many-Particle Systems, Phys. 540
Quantum Theory of Many-Particle Systems, Phys. 540 Questions about organization Second quantization Questions about last class? Comments? Similar strategy N-particles Consider Two-body operators in Fock
More informationEffective shell model Hamiltonians from density functional theory: Quadrupolar and pairing correlations
PHYSICAL REVIEW C 77, 6438 (8) Effective shell model Hamiltonians from density functional theory: Quadrupolar and pairing correlations R. Rodríguez-Guzmán and Y. Alhassid * Center for Theoretical Physics,
More informationNuclear Structure (II) Collective models
Nuclear Structure (II) Collective models P. Van Isacker, GANIL, France NSDD Workshop, Trieste, March 2014 TALENT school TALENT (Training in Advanced Low-Energy Nuclear Theory, see http://www.nucleartalent.org).
More informationSymmetry breaking and symmetry restoration in mean-field based approaches
Symmetry breaking and symmetry restoration in mean-field based approaches Héloise Goutte GANIL Caen, France goutte@ganil.fr Cliquez pour modifier le style des sous-titres du masque With the kind help of
More informationH.O. [202] 3 2 (2) (2) H.O. 4.0 [200] 1 2 [202] 5 2 (2) (4) (2) 3.5 [211] 1 2 (2) (6) [211] 3 2 (2) 3.0 (2) [220] ε
E/ħω H r 0 r Y0 0 l s l l N + l + l s [0] 3 H.O. ε = 0.75 4.0 H.O. ε = 0 + l s + l [00] n z = 0 d 3/ 4 [0] 5 3.5 N = s / N n z d 5/ 6 [] n z = N lj [] 3 3.0.5 0.0 0.5 ε 0.5 0.75 [0] n z = interaction of
More informationCollective excitations of Λ hypernuclei
Collective excitations of Λ hypernuclei Kouichi Hagino (Tohoku Univ.) J.M. Yao (Southwest Univ.) Z.P. Li (Southwest Univ.) F. Minato (JAEA) 1. Introduction 2. Deformation of Lambda hypernuclei 3. Collective
More informationJoint ICTP-IAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation August Introduction to Nuclear Physics - 2
2358-20 Joint ICTP-IAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation 6-17 August 2012 Introduction to Nuclear Physics - 2 P. Van Isacker GANIL, Grand Accelerateur National d'ions Lourds
More informationThe Nuclear Many-Body Problem
The Nuclear Many-Body Problem relativistic heavy ions vacuum electron scattering quarks gluons radioactive beams heavy few nuclei body quark-gluon soup QCD nucleon QCD few body systems many body systems
More informationEffective quadrupole-quadrupole interaction from density functional theory
PHYSICAL REVIEW C 74, 3431 (26) Effective quadrupole-quadrupole interaction from density functional theory Y. Alhassid, 1 G. F. Bertsch, 2,* L. Fang, 1 and B. Sabbey 2 1 Center for Theoretical Physics,
More informationCentral density. Consider nuclear charge density. Frois & Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 133 (1987) QMPT 540
Central density Consider nuclear charge density Frois & Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 133 (1987) Central density (A/Z* charge density) about the same for nuclei heavier than 16 O, corresponding
More informationAccurate Description of the Spinand Parity-Dependent Nuclear Level Densities
Accurate Description of the Spinand Parity-Dependent Nuclear Level Densities Mihai Horoi Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA Nuclear Level Densities
More informationComputational Challenges in Nuclear and Many-Body Physics. Summary
Computational Challenges in Nuclear and Many-Body Physics Summary Yoram Alhassid (Yale University) Physical systems of interest Challenges in many-body physics Computational methods and their formulation
More informationNuclear structure at high excitation energies
PRAMANA cfl Indian Academy of Sciences Vol. 57, Nos 2 & 3 journal of Aug. & Sept. 2001 physics pp. 459 467 Nuclear structure at high excitation energies A ANSARI Institute of Physics, Bhubaneswar 751 005,
More informationNovel High Performance Computational Aspects of the Shell Model Approach for Medium Nuclei
Novel High Performance Computational Aspects of the Shell Model Approach for Medium Nuclei Mihai Horoi Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA Collaborators:
More informationObservables predicted by HF theory
Observables predicted by HF theory Total binding energy of the nucleus in its ground state separation energies for p / n (= BE differences) Ground state density distribution of protons and neutrons mean
More informationNilsson Model. Anisotropic Harmonic Oscillator. Spherical Shell Model Deformed Shell Model. Nilsson Model. o Matrix Elements and Diagonalization
Nilsson Model Spherical Shell Model Deformed Shell Model Anisotropic Harmonic Oscillator Nilsson Model o Nilsson Hamiltonian o Choice of Basis o Matrix Elements and Diagonaliation o Examples. Nilsson diagrams
More informationMean-field concept. (Ref: Isotope Science Facility at Michigan State University, MSUCL-1345, p. 41, Nov. 2006) 1/5/16 Volker Oberacker, Vanderbilt 1
Mean-field concept (Ref: Isotope Science Facility at Michigan State University, MSUCL-1345, p. 41, Nov. 2006) 1/5/16 Volker Oberacker, Vanderbilt 1 Static Hartree-Fock (HF) theory Fundamental puzzle: The
More informationShell model calculation ー from basics to the latest methods ー
Shell model calculation ー from basics to the latest methods ー Takahiro Mizusaki Senshu university / CNS, the univ. of Tokyo Basics What we will learn? What is Shell Model? Shell model basis states M-scheme
More informationGiant Resonances Wavelets, Scales and Level Densities
Giant resonances Damping mechanisms, time and energy scales Fine structure Wavelets and characteristic scales Application: GQR TU DARMSTADT Many-body nuclear models and damping mechanisms Relevance of
More informationNeutron Halo in Deformed Nuclei
Advances in Nuclear Many-Body Theory June 7-1, 211, Primosten, Croatia Neutron Halo in Deformed Nuclei Ó Li, Lulu Ò School of Physics, Peking University June 8, 211 Collaborators: Jie Meng (PKU) Peter
More informationCoexistence phenomena in neutron-rich A~100 nuclei within beyond-mean-field approach
Coexistence phenomena in neutron-rich A~100 nuclei within beyond-mean-field approach A. PETROVICI Horia Hulubei National Institute for Physics and Nuclear Engineering, Bucharest, Romania Outline complex
More informationMicroscopic analysis of nuclear quantum phase transitions in the N 90 region
PHYSICAL REVIEW C 79, 054301 (2009) Microscopic analysis of nuclear quantum phase transitions in the N 90 region Z. P. Li, * T. Nikšić, and D. Vretenar Physics Department, Faculty of Science, University
More informationMass and energy dependence of nuclear spin distributions
Mass and energy dependence of nuclear spin distributions Till von Egidy Physik-Department, Technische Universität München, Germany Dorel Bucurescu National Institute of Physics and Nuclear Engineering,
More informationTheoretical Nuclear Physics
Theoretical Nuclear Physics (SH2011, Second cycle, 6.0cr) Comments and corrections are welcome! Chong Qi, chongq@kth.se The course contains 12 sections 1-4 Introduction Basic Quantum Mechanics concepts
More information14. Structure of Nuclei
14. Structure of Nuclei Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 14. Structure of Nuclei 1 In this section... Magic Numbers The Nuclear Shell Model Excited States Dr. Tina Potter 14.
More information13. Basic Nuclear Properties
13. Basic Nuclear Properties Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 13. Basic Nuclear Properties 1 In this section... Motivation for study The strong nuclear force Stable nuclei Binding
More informationShape coexistence and beta decay in proton-rich A~70 nuclei within beyond-mean-field approach
Shape coexistence and beta decay in proton-rich A~ nuclei within beyond-mean-field approach A. PETROVICI Horia Hulubei National Institute for Physics and Nuclear Engineering, Bucharest, Romania Outline
More informationCollective excitations of Lambda hypernuclei
Collective excitations of Lambda hypernuclei Kouichi Hagino (Tohoku Univ.) Myaing Thi Win (Lashio Univ.) J.M. Yao (Southwest Univ.) Z.P. Li (Southwest Univ.) 1. Introduction 2. Deformation of Lambda hypernuclei
More informationarxiv:nucl-th/ v2 14 Jan 1999
Shell Model Monte Carlo Investigation of Rare Earth Nuclei J. A. White and S. E. Koonin W. K. Kellogg Radiation Laboratory, 106-38, California Institute of Technology Pasadena, California 91125 USA arxiv:nucl-th/9812044v2
More informationNuclear Matter Incompressibility and Giant Monopole Resonances
Nuclear Matter Incompressibility and Giant Monopole Resonances C.A. Bertulani Department of Physics and Astronomy Texas A&M University-Commerce Collaborator: Paolo Avogadro 27th Texas Symposium on Relativistic
More informationAn Introduction to. Nuclear Physics. Yatramohan Jana. Alpha Science International Ltd. Oxford, U.K.
An Introduction to Nuclear Physics Yatramohan Jana Alpha Science International Ltd. Oxford, U.K. Contents Preface Acknowledgement Part-1 Introduction vii ix Chapter-1 General Survey of Nuclear Properties
More informationEvolution Of Shell Structure, Shapes & Collective Modes. Dario Vretenar
Evolution Of Shell Structure, Shapes & Collective Modes Dario Vretenar vretenar@phy.hr 1. Evolution of shell structure with N and Z A. Modification of the effective single-nucleon potential Relativistic
More informationStructure of the Low-Lying States in the Odd-Mass Nuclei with Z 100
Structure of the Low-Lying States in the Odd-Mass Nuclei with Z 100 R.V.Jolos, L.A.Malov, N.Yu.Shirikova and A.V.Sushkov JINR, Dubna, June 15-19 Introduction In recent years an experimental program has
More informationNuclear Shell Model. 1d 3/2 2s 1/2 1d 5/2. 1p 1/2 1p 3/2. 1s 1/2. configuration 1 configuration 2
Nuclear Shell Model 1d 3/2 2s 1/2 1d 5/2 1d 3/2 2s 1/2 1d 5/2 1p 1/2 1p 3/2 1p 1/2 1p 3/2 1s 1/2 1s 1/2 configuration 1 configuration 2 Nuclear Shell Model MeV 5.02 3/2-2 + 1p 1/2 1p 3/2 4.44 5/2-1s 1/2
More informationNuclear physics: a laboratory for many-particle quantum mechanics or From model to theory in nuclear structure physics
Nuclear physics: a laboratory for many-particle quantum mechanics or From model to theory in nuclear structure physics G.F. Bertsch University of Washington Stockholm University and the Royal Institute
More informationLisheng Geng. Ground state properties of finite nuclei in the relativistic mean field model
Ground state properties of finite nuclei in the relativistic mean field model Lisheng Geng Research Center for Nuclear Physics, Osaka University School of Physics, Beijing University Long-time collaborators
More informationThermodynamics of the nucleus
Thermodynamics of the nucleus Hilde-Therese Nyhus 1. October, 8 Hilde-Therese Nyhus Thermodynamics of the nucleus Owerview 1 Link between level density and thermodynamics Definition of level density Level
More informationChapter 6. Summary and Conclusions
Chapter 6 Summary and Conclusions The basic aim of the present thesis was to understand the interplay between single particle and collective degrees of freedom and underlying nuclear phenomenon in mass
More informationThe Nuclear Shape Phase Transitions Studied within the Geometric Collective Model
April, 203 PROGRESS IN PHYSICS Volume 2 The Nuclear Shape Phase Transitions Studied within the Geometric Collective Model Khalaf A.M. and Ismail A.M. Physics Department, Faculty of Science, Al-Azhar University,
More informationRFSS: Lecture 8 Nuclear Force, Structure and Models Part 1 Readings: Nuclear Force Nuclear and Radiochemistry:
RFSS: Lecture 8 Nuclear Force, Structure and Models Part 1 Readings: Nuclear and Radiochemistry: Chapter 10 (Nuclear Models) Modern Nuclear Chemistry: Chapter 5 (Nuclear Forces) and Chapter 6 (Nuclear
More informationc E If photon Mass particle 8-1
Nuclear Force, Structure and Models Readings: Nuclear and Radiochemistry: Chapter 10 (Nuclear Models) Modern Nuclear Chemistry: Chapter 5 (Nuclear Forces) and Chapter 6 (Nuclear Structure) Characterization
More informationApplication of quantum number projection method to tetrahedral shape and high-spin states in nuclei
Application of quantum number projection method to tetrahedral shape and high-spin states in nuclei Contents S.Tagami, Y.Fujioka, J.Dudek*, Y.R.Shimizu Dept. Phys., Kyushu Univ. 田上真伍 *Universit'e de Strasbourg
More informationB. PHENOMENOLOGICAL NUCLEAR MODELS
B. PHENOMENOLOGICAL NUCLEAR MODELS B.0. Basic concepts of nuclear physics B.0. Binding energy B.03. Liquid drop model B.04. Spherical operators B.05. Bohr-Mottelson model B.06. Intrinsic system of coordinates
More informationarxiv:nucl-ex/ v1 29 Apr 1999
Observation of Thermodynamical Properties in the 162 Dy, 166 Er and 172 Yb Nuclei E. Melby, L. Bergholt, M. Guttormsen, M. Hjorth-Jensen, F. Ingebretsen, S. Messelt, J. Rekstad, A. Schiller, S. Siem, and
More informationProbing neutron-rich isotopes around doubly closed-shell 132 Sn and doubly mid-shell 170 Dy by combined β-γ and isomer spectroscopy.
Probing neutron-rich isotopes around doubly closed-shell 132 Sn and doubly mid-shell 170 Dy by combined β-γ and isomer spectroscopy Hiroshi Watanabe Outline Prospects for decay spectroscopy of neutron-rich
More informationDeexcitation mechanisms in compound nucleus reactions
Deexcitation mechanisms in compound nucleus reactions Curso de Reacciones Nucleares Programa Inter-universitario de Física Nuclear Universidade de Santiago de Compostela March 2008 Contents Elements of
More information1 Introduction. 2 The hadronic many body problem
Models Lecture 18 1 Introduction In the next series of lectures we discuss various models, in particluar models that are used to describe strong interaction problems. We introduce this by discussing the
More informationLesson 5 The Shell Model
Lesson 5 The Shell Model Why models? Nuclear force not known! What do we know about the nuclear force? (chapter 5) It is an exchange force, mediated by the virtual exchange of gluons or mesons. Electromagnetic
More informationShape Coexistence and Band Termination in Doubly Magic Nucleus 40 Ca
Commun. Theor. Phys. (Beijing, China) 43 (2005) pp. 509 514 c International Academic Publishers Vol. 43, No. 3, March 15, 2005 Shape Coexistence and Band Termination in Doubly Magic Nucleus 40 Ca DONG
More informationUniversity of Tokyo (Hongo) Nov., Ikuko Hamamoto. Division of Mathematical Physics, LTH, University of Lund, Sweden
University of Tokyo (Hongo) Nov., 006 One-particle motion in nuclear many-body problem - from spherical to deformed nuclei - from stable to drip-line - from particle to quasiparticle picture kuko Hamamoto
More informationPavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Praha
and Nuclear Structure Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Praha ha,, CZ cejnar @ ipnp.troja.mff.cuni.cz Program: > Shape phase transitions in nuclear structure data
More informationThe collective model from a Cartan-Weyl perspective
The collective model from a Cartan-Weyl perspective Stijn De Baerdemacker Veerle Hellemans Kris Heyde Subatomic and radiation physics Universiteit Gent, Belgium http://www.nustruc.ugent.be INT workshop
More informationNuclear structure aspects of Schiff Moments. N.Auerbach Tel Aviv University and MSU
Nuclear structure aspects of Schiff Moments N.Auerbach Tel Aviv University and MSU T-P-odd electromagnetic moments In the absence of parity (P) and time (T) reversal violation the T P-odd moments for a
More informationTheoretical study of fission barriers in odd-a nuclei using the Gogny force
Theoretical study of fission barriers in odd-a nuclei using the Gogny force S. Pérez and L.M. Robledo Departamento de Física Teórica Universidad Autónoma de Madrid Saclay, 9th May 2006 Workshop on the
More informationSystematics of the K π = 2 + gamma vibrational bands and odd even staggering
PRAMANA cfl Indian Academy of Sciences Vol. 61, No. 1 journal of July 2003 physics pp. 167 176 Systematics of the K π = 2 + gamma vibrational bands and odd even staggering J B GUPTA 1;2 and A K KAVATHEKAR
More informationnuclear level densities from exactly solvable pairing models
nuclear level densities from exactly solvable pairing models Stefan Rombouts In collaboration with: Lode Pollet, Kris Van Houcke, Dimitri Van Neck, Kris Heyde, Jorge Dukelsky, Gerardo Ortiz Ghent University
More informationInstitut für Kernphysik, Universität zu Köln, Germany 2. Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania 3
High-Resolution Study of Low-Spin States in Mark Spieker 1,*, Dorel Bucurescu 2, Janis Endres 1, Thomas Faestermann 3, Ralf Hertenberger 4, Sorin Pascu 1,2, Hans-Friedrich Wirth 4, Nicolae-Victor Zamfir
More informationAsymmetry dependence of Gogny-based optical potential
Asymmetry dependence of Gogny-based optical potential G. Blanchon, R. Bernard, M. Dupuis, H. F. Arellano CEA,DAM,DIF F-9297 Arpajon, France March 3-6 27, INT, Seattle, USA / 32 Microscopic ingredients
More informationXX Nuclear Physics Symposium Oaxtepec, Mexico January 6-9,1997
* - -9 W S w SHELL-MODEL MONTE CARLO STUDES OF NUCLE D J Dean Oak Ridge National Laboratory* Oak Ridge, TN 3783-6373 nvited talk to be published in Proceedings XX Nuclear Physics Symposium Oaxtepec, Mexico
More informationPairing Interaction in N=Z Nuclei with Half-filled High-j Shell
Pairing Interaction in N=Z Nuclei with Half-filled High-j Shell arxiv:nucl-th/45v1 21 Apr 2 A.Juodagalvis Mathematical Physics Division, Lund Institute of Technology, S-221 Lund, Sweden Abstract The role
More informationINVESTIGATION OF THE EVEN-EVEN N=106 ISOTONIC CHAIN NUCLEI IN THE GEOMETRIC COLLECTIVE MODEL
U.P.B. Sci. Bull., Series A, Vol. 79, Iss. 1, 2017 ISSN 1223-7027 INVESTIGATION OF THE EVEN-EVEN N=106 ISOTONIC CHAIN NUCLEI IN THE GEOMETRIC COLLECTIVE MODEL Stelian St. CORIIU 1 Geometric-Collective-Model
More informationTamara Nikšić University of Zagreb
CONFIGURATION MIXING WITH RELATIVISTIC SCMF MODELS Tamara Nikšić University of Zagreb Supported by the Croatian Foundation for Science Tamara Nikšić (UniZg) Primošten 11 9.6.11. 1 / 3 Contents Outline
More informationTackling the Sign Problem of Ultracold Fermi Gases with Mass-Imbalance
Tackling the Sign Problem of Ultracold Fermi Gases with Mass-Imbalance Dietrich Roscher [D. Roscher, J. Braun, J.-W. Chen, J.E. Drut arxiv:1306.0798] Advances in quantum Monte Carlo techniques for non-relativistic
More informationShell evolution and pairing in calcium isotopes with two- and three-body forces
Shell evolution and pairing in calcium isotopes with two- and three-body forces Javier Menéndez Institut für Kernphysik, TU Darmstadt ExtreMe Matter Institute (EMMI) with Jason D. Holt, Achim Schwenk and
More informationCoupled-channels Neutron Reactions on Nuclei
Coupled-channels Neutron Reactions on Nuclei Ian Thompson with: Gustavo Nobre, Frank Dietrich, Jutta Escher (LLNL) and: Toshiko Kawano (LANL), Goran Arbanas (ORNL), P. O. Box, Livermore, CA! This work
More informationOblate nuclear shapes and shape coexistence in neutron-deficient rare earth isotopes
Oblate nuclear shapes and shape coexistence in neutron-deficient rare earth isotopes Andreas Görgen Service de Physique Nucléaire CEA Saclay Sunniva Siem Department of Physics University of Oslo 1 Context
More informationCoupled-cluster theory for medium-mass nuclei
Coupled-cluster theory for medium-mass nuclei Thomas Papenbrock and G. Hagen (ORNL) D. J. Dean (ORNL) M. Hjorth-Jensen (Oslo) A. Nogga (Juelich) A. Schwenk (TRIUMF) P. Piecuch (MSU) M. Wloch (MSU) Seattle,
More informationBand Structure of nuclei in Deformed HartreeFock and Angular Momentum Projection theory. C. R. Praharaj Institute of Physics Bhubaneswar.
Band Structure of nuclei in Deformed HartreeFock and Angular Momentum Projection theory C. R. Praharaj Institute of Physics. India INT Workshop Nov 2007 1 Outline of talk Motivation Formalism HF calculation
More informationIntroduction to shell-model Monte-Carlo methods
Introduction to shell-model Monte-Carlo methods Gesellschaft für Schwerionenforschung Darmstadt Helmholtz International Summer School Dubna, August 7-17, 2007 Introduction Nuclear Shell Model A Concise
More informationChiral effective field theory on the lattice: Ab initio calculations of nuclei
Chiral effective field theory on the lattice: Ab initio calculations of nuclei Nuclear Lattice EFT Collaboration Evgeny Epelbaum (Bochum) Hermann Krebs (Bochum) Timo Lähde (Jülich) Dean Lee (NC State)
More informationNuclear moment of inertia and spin distribution of nuclear levels
PHYSICAL REVIEW C 7, 6436 Nuclear moment of inertia and spin distribution of nuclear levels Y. Alhassid, G. F. Bertsch, L. Fang, and S. Liu Center for Theoretical Physics, Sloane Physics Laboratory, Yale
More informationCHAPTER-2 ONE-PARTICLE PLUS ROTOR MODEL FORMULATION
CHAPTE- ONE-PATCLE PLUS OTO MODEL FOMULATON. NTODUCTON The extension of collective models to odd-a nuclear systems assumes that an odd number of pons (and/or neutrons) is coupled to an even-even core.
More informationBeyond-mean-field approach to low-lying spectra of Λ hypernuclei
Beyond-mean-field approach to low-lying spectra of Λ hypernuclei Kouichi Hagino (Tohoku Univ.) Hua Mei (Tohoku Univ.) J.M. Yao (Tohoku U. / Southwest U.) T. Motoba (Osaka Electro-Commun. U. ) 1. Introduction
More informationWhat did you learn in the last lecture?
What did you learn in the last lecture? Charge density distribution of a nucleus from electron scattering SLAC: 21 GeV e s ; λ ~ 0.1 fm (to first order assume that this is also the matter distribution
More informationFine structure of nuclear spin-dipole excitations in covariant density functional theory
1 o3iø(œ April 12 16, 2012, Huzhou, China Fine structure of nuclear spin-dipole excitations in covariant density functional theory ùíî (Haozhao Liang) ŒÆÔnÆ 2012 c 4 13 F ÜŠöµ Š # Ç!Nguyen Van Giai Ç!ë+
More informationFitting Function for Experimental Energy Ordered Spectra in Nuclear Continuum Studies
Fitting Function for Experimental Energy Ordered Spectra in Nuclear Continuum Studies J.R. Pinzón, F. Cristancho January 17, 2012 Abstract We review the main features of the Hk-EOS method for the experimental
More informationBand crossing and signature splitting in odd mass fp shell nuclei
Nuclear Physics A 686 (001) 19 140 www.elsevier.nl/locate/npe Band crossing and signature splitting in odd mass fp shell nuclei Victor Velázquez a, Jorge G. Hirsch b,,yangsun c,d a Institute de Recherches
More informationRIKEN Sept., Ikuko Hamamoto. Division of Mathematical Physics, LTH, University of Lund, Sweden
RIKEN Sept., 007 (expecting experimentalists as an audience) One-particle motion in nuclear many-body problem - from spherical to deformed nuclei - from stable to drip-line - from static to rotating field
More informationNuclear Structure Theory II
uclear Structure Theory II The uclear Many-body Problem Alexander Volya Florida State University Physics of light nuclei 1 H 4 Li 3 He 2 H 8 7 6 Be 5 Li 4 He 3 B H 10 9 8 7 Be 6 Li 5 He 4 B H 12 11 10
More informationQuantum Chaos as a Practical Tool in Many-Body Physics
Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Statistical Nuclear Physics SNP2008 Athens, Ohio July 8, 2008 THANKS B. Alex
More informationLecture 4: Nuclear Energy Generation
Lecture 4: Nuclear Energy Generation Literature: Prialnik chapter 4.1 & 4.2!" 1 a) Some properties of atomic nuclei Let: Z = atomic number = # of protons in nucleus A = atomic mass number = # of nucleons
More information