Triaxial Projected Configuration Mixing 1.Collective wave functions? 2.Old results on Zr 3.Few results on 24 Mg 4.Many questions.

Slides:

Advertisements

Similar presentations

Spin order in correlated electron systems

Advertisements

A brief introduction D S Judson. Kinetic Energy Interactions between of nucleons i th and j th nucleons The wavefunction of a nucleus composed of A nucleons.

Collective properties of even- even nuclei Vibrators and rotors With three Appendices.

Coulomb excitation with radioactive ion beams

The Quantum Mechanics of Simple Systems

Molecular Bonding Molecular Schrödinger equation

Molecular orbitals for polyatomic systems. The molecular orbitals of polyatomic molecules are built in the same way as in diatomic molecules, the only.

Pavel Stránský 29 th August 2011 W HAT DRIVES NUCLEI TO BE PROLATE? Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México Alejandro.

Molecular Quantum Mechanics

Emergence of Quantum Mechanics from Classical Statistics.

Sensitivity of Eigenproblems Review of properties of vibration and buckling modes. What is nice about them? Sensitivities of eigenvalues are really cheap!

Le groupe des francophones Heirs of P. Bonche, H Flocard, D Vautherin, … with still two ancestors: P-H Heenen and J. Meyer and many young people: M Bender.

Determinants Bases, linear Indep., etc Gram-Schmidt Eigenvalue and Eigenvectors Misc

Review Three Pictures of Quantum Mechanics Simple Case: Hamiltonian is independent of time 1. Schrödinger Picture: Operators are independent of time; state.

Principal Component Analysis Principles and Application.

Section 8.3 – Systems of Linear Equations - Determinants Using Determinants to Solve Systems of Equations A determinant is a value that is obtained from.

Section 9.5: Equations of Lines and Planes

Section 9.4: The Cross Product Practice HW from Stewart Textbook (not to hand in) p. 664 # 1, 7-17.

Physics “Advanced Electronic Structure” Pseudopotentials Contents: 1. Plane Wave Representation 2. Solution for Weak Periodic Potential 3. Solution.

Unit 2: Properties of Matter. The Goals… Define Matter Define Matter Calculate and understand density Calculate and understand density Describe viscosity.

What is meant by the term “ground state”? It is the lowest energy state that an electron can occupy.

How much energy is contained in a 10 g piece of copper heated 15 degrees Celcius?

Lecture 20: More on the deuteron 18/11/ Analysis so far: (N.B., see Krane, Chapter 4) Quantum numbers: (J , T) = (1 +, 0) favor a 3 S 1 configuration.

Nuclear Structure and dynamics within the Energy Density Functional theory Denis Lacroix IPN Orsay Coll: G. Scamps, D. Gambacurta, G. Hupin M. Bender and.

6.852: Distributed Algorithms Spring, 2008 April 1, 2008 Class 14 – Part 2 Applications of Distributed Algorithms to Diverse Fields.

Mathematical foundationsModern Seismology – Data processing and inversion 1 Some basic maths for seismic data processing and inverse problems (Refreshement.

Atomic Models Scientist studying the atom quickly determined that protons and neutrons are found in the nucleus of an atom. The location and arrangement.

From Democritus to now….  a Greek philosopher, proposed that matter was made up of small, hard, indivisible particles, which he called atoms.  Democritus.

Chapter 5.  The scale model shown is a physical model. However, not all models are physical. In fact, several theoretical models of the atom have been.

1 MODELING MATTER AT NANOSCALES 4. Introduction to quantum treatments The variational method.

Experimental measurement of the deformation through the electromagnetic probe Shape coexistence in exotic Kr isotopes. Shape coexistence in exotic Kr isotopes.

Erosion of N=28 Shell Gap and Triple Shape Coexistence in the vicinity of 44 S M. KIMURA (HOKKAIDO UNIV.) Y. TANIGUCHI (RIKEN), Y. KANADA-EN’YO(KYOTO UNIV.)

PHYS 773: Quantum Mechanics February 6th, 2012

1 MODELING MATTER AT NANOSCALES 4. Introduction to quantum treatments Outline of the principles and the method of quantum mechanics.

Application of the Adiabatic Self-Consistent Collective Coordinate (ASCC) Method to Shape Coexistence/ Mixing Phenomena Application of the Adiabatic Self-Consistent.

Petrică Buganu, and Radu Budaca IFIN-HH, Bucharest – Magurele, Romania International Workshop “Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects”

Numerical accuracy of mean-field calculations The case of the 3-dimensional mesh scheme The Lagrange implementation P. Bonche, J. Dobaczewski, H. Flocard.

Quantum Theory and the Electronic Structure of Atoms Chapter 6.

F. C HAPPERT N. P ILLET, M. G IROD AND J.-F. B ERGER CEA, DAM, DIF THE D2 GOGNY INTERACTION F. C HAPPERT ET AL., P HYS. R EV. C 91, (2015)

Preequilibrium Reactions Dr. Ahmed A.Selman. The exciton model was proposed by Griffin in 1966 in order to explain the nuclear emission from intermediate.

Electrons in Atoms Chapter Wave Nature of Light  Electromagnetic Radiation is a form of energy that exhibits wavelike behavior as it travels through.

MODULE 13 Time-independent Perturbation Theory Let us suppose that we have a system of interest for which the Schrödinger equation is We know that we can.

Chapter 3 Postulates of Quantum Mechanics. Questions QM answers 1) How is the state of a system described mathematically? (In CM – via generalized coordinates.

Lecture 16: Image alignment

Molecular Bonding Molecular Schrödinger equation

Matrices and vector spaces

oblate prolate l=2 a20≠0, a2±1= a2±2= 0 Shape parameterization

"How to create a 1qp state and what does it mean?

Beyond mean-field methods: why and how

The Electronic Structure of Atoms

Postulates of Quantum Mechanics

Electrons in Atoms Chapter 5.

Time-Dependent Perturbation Theory

The Nuts and Bolts of First-Principles Simulation

Chapter 4 Two-Level Systems.

Solving Quadratic Equations by Factoring

UWC Beyond mean-field: present and future

Quadratic Equations.

Cluster and Density wave --- cluster structures in 28Si and 12C---

Chapter 5 1D Harmonic Oscillator.

Nuclear shapes for the critical point

Social Practice of the language: Describe and share information

Algebra 2 – Chapter 6 Review

Aim: What is the nature of the roots of a quadratic equation?

Introductory Quantum Mechanics/Chemistry

Linear Vector Space and Matrix Mechanics

QM2 Concept Test 10.1 Consider the Hamiltonian

Presentation transcript:

Triaxial Projected Configuration Mixing 1.Collective wave functions? 2.Old results on Zr 3.Few results on 24 Mg 4.Many questions

First triaxial calculations: P. Bonche, H. Flocard, J. Meyer J. Dobaczewski, J. Skalski New developments: M. Bender

Configuration Mixing Starting point: set of    wave functions |  >, non-orthogonal: New set of wave functions: The unknown f   are solutions of the HW equation:

The f’s are non orthogonal, ill-behaved, …. Change of basis, using the overlap matrix, defining its square root: Very nice but not used directly!

First, diagonalisation of the overlap I: And then Last summation restricted to a limited number of eigenvalues It is this equation that is solved!

The collective wave function is And the eigenstates of the Hamiltonian is: Neither g nor f are the overlap Meaning of oblate, prolate, triaxial …. after configuration mixing?

Projection of triaxial map: Triaxial minimum? lost of the meaning of q after projection! no orthogonality of wave functions!

z=symmetry axis the maps for the other orientations have no simple interpretations

Q=125 fm 2,   mean-field configuration) z= longest intermediate smallest axis Spectra obtained after projection of the lowest configuration: three possible orientations Same results AFTER K-mixing

Spectroscopic properties of the min configuration before and after K-mixing compared to the Davidoff rotor model

Configuration mixing: comparison between different bases: 1.purely prolate 2.axial 3.purely triaxial 4.triaxial + a few prolate configurations We are not using a hamiltonian but a density functional generalized for non-diagonal matrix elements One must avoid pathologies: possible problems determined by projecting on N and Z with 9 and 29 points Triaxial region close to the oblate axis. No oblate points mixed with triaxial points.

Small eigenvalues of the norm kernel indicate redundancy in a basis small eigenvalues (10 -2 ) = not much information

All the GCM calculations: axial (prolate+oblate) purely triaxial (35 keV lower than axial) triaxial + prolate (160 keV lower than triaxial) Triaxial correlations described by configuration mixing of axial configurations! Cut in the Q,  plane: and GCM calculations

increase of energy for excited states due to the correlations in the ground state! Spectra in 3 bases No vectors in common!

Very careful about language: « the nucleus is triaxial after projection on J » ! Analysis of phenomenological models (clever but with the hands) Sign of triaxiality or K-bands?