Shell Model and residual interactions

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Short introduction

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Shell Model and residual interactions

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Pairing coupling scheme

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A two-level model

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BCS

Pairing plus Quadrupole 101

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Shape and pairing phase transitions

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Atomic nuclei constitute unique many body systems of strongly interacting fermions. Their properties and structure, are of paramount importance to many aspects of physics.

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Many of the phenomena encountered in nuclei share common basic physics ingredients with other mesoscopic systems, thus making nuclear structure research relevant to other areas of contemporary research, for example in condensed matter and atomic physics.

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These are exciting times in the field of physics of nuclei:

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Existing and planned accelerator facilities worldwide, and new detector systems with increased sensitivity and resolving power not only will allow us to answer some important questions we have today, but most likely will open up a window to new and unexpected phenomena.

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New developments in theory and computer power are shaping a path to a predictive theory of nuclei and reactions.

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• How did visible matter come into being, and how does it evolve?

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• How does subatomic matter organize themselves, and what

phenomena emerge?

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• Are the fundamental interactions that are basic to the structure of

matter fully understood?

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• How can the knowledge and technological progress provided by

nuclear physics best be used to benefit society?

Intellectual Drivers

http://www.nap.edu/catalog/13438/nuclear-physics-exploring-the-heart-of-matter

The Physics of Nuclei: Science Drivers

The Physics of Nuclei: Science Drivers

THE SYNERGY between THEORY and EXPERIMENT

• A comprehensive and quantified model of atomic nuclei does not yet exist
• In recent years, enormous progress has been made with measurements of properties of rare isotopes and developments in nuclear theory and computation
• Access to key regions of the nuclear chart constrains models and identifies missing physics
• Theory identifies key nuclei and properties to be studied

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The Ultimate Goal

The Nuclear Landscape

The Nuclear Landscape

1975

Where it all started

B. R. Mottelson,

Proceedings of the International School of Physics "Enrico Fermi, “ Course 15,

edited by G. Racah (Academic, New York,1962).

Nuclear Shell Structure

Prof. Liotta

Me

Z

N

Energy of First Excited State

Nuclear Shell Structure

Maria Goeppert-Mayer & Hans D. Jensen

1963

Maria Goeppert-Mayer, Phys. Rev. 75, 1969 (1949).

O. Haxel, Phys. Rev. 75, 1766 (1949).

Nuclear Shell Model

Nuclear shell model

Mean Field

Residual Interaction, V(1,2)

In principle if the form of the bare nucleon-nucleon interaction is known, then the properties and structures of a given nucleus can be calculated ab-initio:

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In the shell model we make the following approximation to the problem:

+ 3-body + …

“Quantality Parameter”

Fermi Liquid, quasiparticles

The Mean Field

The Mean Field

The average potential U(r_k) , experienced by all the k particles approximates the combined effects of all the two-body interactions.

We now consider the motion of the valence nucleons

( i.e. neutrons or protons that are in excess of the last,

completely filled shell) in the mean field and the effect of a

residual interaction, V(r_1, r₂) , only among them.

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This is not completely so as valence particles will tend to polarize the core.

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However, this require excitations with energy of

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Which is large compared to an average residual interaction

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and thus can be treated perturbatively.

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Note that <V>/ΔE decreases as A^2/3.

The residual interaction

Derive from the nn interaction with in-medium effects

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Determine the residual interaction from experimental data.

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Use a schematic model with a simple spatial form that captures the main ingredients of the force.

W.W.Daehnick Physics Reports 96 (1983) 317

Problem #1

The pairing coupling scheme

1/l

Correlations within a distance

r ≤ R/l

Wave function is

Short range force favors 0⁺ pairs

For I ≠ 0 the distance is ≈ IR/l

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2Δ

Even-even gap

Δ

Odd-Even mass difference

2Δ

Odd-odd to Even-even mass difference

^AZ_N

^A+1Z_N+1

^A+2Z_N+2

^A+2(Z+1)_N+1

BM Vol 1 page 170

Pair gaps from mass differences

A simple microscopic model: Two j-shells

“Control parameter”

Small X - Pairing Vibrations

Large X – Pairing Rotations

X ~ 1

To treat more realistic situations the jⁿ model has to be generalized

BCS wave function does not have definite number of particles

⇨ minimize with a constrain that fixes the average number of particles to N

D