cern summer student program 2011 introduction to nuclear physics s. péru introduction to nuclear...

CERN Summer student program 2011Introduction to Nuclear Physics S. Péru

Introduction to Nuclear PhysicsIntroduction to Nuclear Physics

S.PÉRU

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The nucleus : a complex system

I) Some features about the nucleusdiscoveryradius, shapebinding energynucleon-nucleon interactionstability and life time

nuclear reactionsapplications

III) Examples of recent studies figures of merit of mean field approaches exotic nuclei isomers

shape coexistence

IV) Toward a microscopic description of the fission process

II) Modeling of the nucleusliquid dropshell modelmean field


Modeling of the nucleus

nucleus = A nucleons in interaction

2 challenges Nuclear Interaction inside the nuclei (unknown)

N body formalism

The liquid drop model : global view of the nucleus associated to a quantum liquid.

The Shell Model : each nucleon is independent in a attractive potential.

« Microscopic » methods ~ Hartree-Fock , BCS ,Hartree-Fock-Bogoliubov :The nuclear structure is described within the assumption that each nucleon is

interacting with an average field generated by all the other nucleons.


The nucleus is a charged quantum liquid.

Quantum : The wave length of the nucleons is large enough with respect to the size of the nucleons to vanish trajectory and position meaning.

Liquid : Inside the nucleus nucleons are like water molecules. They roll “ones over ones” without going outside the “container”.

The nucleus and its features, radii, and binding energies have many similarities with a liquid drop :

The volume of a drop is proportional to its number of molecules.

There are no long range correlations between molecules in a drop.

-> Each molecule is only sensitive to the neighboring molecules.

-> Description of the nucleus in term of a model of a charged liquid drop


The liquid drop

The binding energy of the nuclei is described by the Bethe-Weiszaker formula

Volume ≈ AR= r0 A 1/3

Equilibrium shapespherical

Density ρ0

R Rrr

Model developed by Von Weizsacker and N. Bohr (1937)It has been first developed to describe the nuclear fission.

The model has been used to predict the main properties of the nuclei such as:

* nuclear radii, * nuclear masses and binding energies,* decay out,* fission.


Bethe and Weizsacker formula

3/12

3/123/2

AaA

ZNaAZaAaAaB pacsv

Binding term : volume av

Unbinding terms : Surface as , Coulomb ac , Asymmetry aa

Paring terms+ even-even

- odd-odd0 odd-even and even-odd


Binding energy per nucleon


1) Fission fragment distributions

Liquid drop : only symmetric fissionPro

ton

n

um

ber

Neutron number

Experimental Results :

K-H Schmidt et al., Nucl. Phys. A665 (2000) 221

A heavy and a light fragments = asymmetric

fission

Two identical fragments= symmetric fission

18

Problems with the liquid drop model



2) Nuclear radii

Evolution of mean square radii withrespect to 198Hg as a function of neutron number.

Light isotopes are unstable nuclei produced at CERN by use of the ISOLDE apparatus.

-> some nuclei away from the A2/3 law

Fig. from http://ipnweb.in2p3.fr/recherche

14


Halo nuclei

I. Tanihata et al., PRL 55 (1985) 2676I. Tanihata and R. Kanungo, CR Physique (2003) 437

15



3) Nuclear masses

Existence of magic numbers : 8, 20 , 28, 50, 82, 126

Difference in MeV between experimental masses and masses calculated withthe liquid drop formula as a function of the neutron number

Fig. from L. Valentin, Physique subatomique, Hermann 1982

Neutron number

E (

MeV

)

16

Nuclear shell effects


Two neutron separation energy S2n

For most nuclei, the 2n separation energies are smooth functions of particle numbers apart from discontinuities for magic nucleiMagic nuclei have increased particle stability and require a larger energy to extract particles.

S2n : energy needed to remove 2 neutrons

to a given nucleus (N,Z)S2n=B(N,Z)-B(N-2,Z)

17


There are many « structure effects » in nuclei, that can not bereproduced by macroscopic approaches like the liquid drop model.

-> need for microscopic approaches, for which the fundamental ingredients are the nucleons and the interaction between them

There are «magic numbers» 2, 8, 20, 28, 50, 82, 126

and so «magic»

and «doubly magic» nuclei ......100208405050126822020 SnPbCa

......CeZr 82140

58509040

The nucleus is not a liquid drop : Shell effects

19


Nucleus = N nucleons in strong interaction

Nucleon-Nucleon forceunknown

Different effective forces used Depending on the method chosen to solve the many-body problem

The many-body problem(the behavior of each nucleoninfluences the others)

Can be solved exactly for N < 4

For N >> 10 : approximations

Shell Modelonly a small number of particles are active

Approaches based on the Mean Field• no inert core• but not all the correlations between particles are takeninto account

20

Microscopic description of the atomic nucleus


Quantum mechanics

Quantum mechanics

Nucleons are quantum objects :Only some values of the energy are available : a discrete number of states

Nucleons are fermions : Two nucleons can not occupy the same quantum state : the Pauli principle

Neutrons Protons

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Shell Model

neutron

proton

neutron

“In analogy with atomic structure one may postulate that in the nucleus the nucleons move fairly independently in individual orbits in an

average potential …” , M. Goeppert Mayer, Nobel Conference 1963.

neutronneutron

Model developed by M. Goeppert Mayer in 1948 :The shell model of the nucleus describes the nucleons in the nucleus

in the same way as electrons in the atom.


Schrödinger equation

Wave function φ and energy ε for each nucleon

Wave function ψ and nuclear binding energy E

Features of the nucleus in his ground state and his excited ones

U (r )

Energy (MeV)

0r (fm)R

rR

Shell Model


Nuclear potential deduced from exp :Wood Saxon potential

orsquare well

orharmonic oscillator

a

RrV

rVexp1

0

Shell Model : potential


spin orbit effect

slOS

...

Shell Model : potential


+1s1/2

-1p3/2

-1p1/2

+1d5/2

+2s1/2

+1d3/2

+3s1/2

-1h11/2

+1g7/2

+2d5/2

-1h9/2

+1g9/2

-1f7/2

-2p3/2

-1f5/2

-1p1/2

-2f7/2

+1d3/2

-1i13/2

-2f5/2

-3p3/2

-3p1/2

-2g9/2

-3d5/2

2020

5050

8282

126126

2828

For a nucleus with A nucleons you fill the A lowest energy levels, andthe energy is the sum of the energy of the individual levels

+1s1/2

-1p3/2

-1p1/2

22 88

Ex: Z=10

+1s1/2

-1p3/2

-1p1/2

+1d5/2

+2s1/2

+1d3/2

88

2020

Ex: N=20

22

88 22

Shell Model : how describe the ground state ?


+1s1/2

-1p3/2

-1p1/2

22 88

Ex: Z=10

+1s1/2

-1p3/2

-1p1/2

+1d5/2

+2s1/2

+1d3/2

88

2020

Ex: N=20

22

-1f7/2

+1s1/2

-1p3/2

-1p1/2

22 88

Ex: Z=10

+1s1/2

-1p3/2

-1p1/2

+1d5/2

+2s1/2

+1d3/2

88

2020

Ex: N=20

22

-1f7/2

Ground state

Excited state : you make a particle-hole excitation. You promote one particle to a higher energy level

Shell Model : how describe excited states ?


Satisfying results for magic nuclei : ground state and low lying excited states

Problems :

• Neglect of collective excitations• Same potential for all the nucleons and for all the configurations• Independent particles

•Improved shell model (currently used):The particles are not independent : due to their interactions with the other particles they do not occupy a given orbital but a sum of configurations having a different probability.-> definition of a valence space where the particles are active

26

Beyond this “independent particle Shell Model

CERN Summer student program 2011Introduction to Nuclear Physics S. Péru 26

Beyond this “independent particle Shell Model

proton neutron


Mean field approach Main assumptioneach particle is interacting with an average field generated by all the other particles : the mean field.The mean field is built from the individual excitations between the nucleons.

Self consistent mean field : the mean field is not fixed. It depends on the configuration. No inert core

Nuclear interaction

2 nucleons bare forcemany nucleons effective interaction

Soleil

Uranus

Flibre

FG FeffectiveNeptune


Jacques Decharg

é

Jacques Dechargé


21

2

21

211221.12

212103

jjjj

2

1

2

2112

2121

.

1

PPM- PH-PBW r-r

-exp

rr

ett

rriW

rrrrPxt

pv

zz

ls

s

j j

W

P : isospin exchange operatorP : spin exchange operator

Finite range for pairing treatment

The phenomenologicaleffective finite-range Gogny force

Finite range central term

Density dependent term

Spin orbit term

Coulomb term


Mean field approach : Hatree-Fock method

Hartree-Fock equations

(A set of coupled Schrodinger equations)

)()()(2

22

iiiiiHF xxUM

Hartree-Fock potential

Single particle wave functions

Self consistent mean field : the Hartree Fock potential depends on the solutions (the single particle wave functions)

-> Resolution by iteration

For more formalism see “The nuclear many body problem”,

P. Ring and P. Schuck


Trial single particle wave function

Calculation of the HF potential

Resolution of the HF equations

New wave functions

Test of the convergence

Effective interaction

)( ii x

)( ii x

)( HFU

)()()(2

22

iiiiiHF xxUM

Resolution of the Hatree-Fock equations

Calculations of the properties of the nucleus in its ground state

Jacques Dechargé


We can “measure” nuclear deformations as the mean values of themutipole operators

http://www-phynu.cea.fr

Q̂

Q

qˆ

If we consider the isoscalar axial quadrupole operator We find that:

Most of the nuclei are deformed in their ground state

Magic nuclei are spherical

202

20ˆ YrQ

Spherical Harmonic

34

Hatree-Fock method : deformation


We can impose collective deformations and test the response of the nuclei:

0 ˆˆˆ ˆi

i ii qZNiq ZNQH

(Z) N )ˆ(ˆ ii qq ZN

iq ˆ ii qiq Q

with

Where ’s are Lagrange parameters.

36

Constraints Hatree-Fock-bogoliubov calculations


g.s deformation predicted with HFB using the Gogny force

http://www-phynu.cea.fr

35

Constraints Hatree-Fock-bogoliubov calculations : results


What are the problems with this deformation ?

38

Constraints Hatree-Fock-bogoliubov calculations : What are the most commonly

used constraints ?


Deformations pertinent for fission:ElongationAsymmetry

…

39

Constraints Hatree-Fock-bogoliubov calculations : Potential energy landscapes


154Sm

Evolution of s.p. states with deformation

New gaps


Hatree-Fock-bogoliubov calculations with blocking

Particle-hole excitations one (or two, three, ..) quasi-particles curves


Introduction of more correlations : two types of approaches

Random Phase Approximation (RPA)

Coupling between HFB ground stateand particle hole excitations

Generator coordinate Method (GCM)

Introduction of large amplitudecorrelations

Give access to a correlated ground state and to the excited states

Individual excitations and collective states

42

Beyond mean field …


Beyond mean field … with GCM

(GCM+GOA 2 vibr. + 3 rot.) = 5 Dimension Collective Hamiltonian

5DCH


Beyond mean field … with RPA

MonopoleMonopole DipoleDipole

S. Péru, J.F. Berger, and P.F. Bortignon, Eur. Phys. Jour. A 26, 25-32 (2005)


Spherical nuclei «vibrational» spectrum

0+

2+

4+

6+

Deformed nuclei «rotational» spectrum

0+

4+

6+

2+

E J

E J(J+1)

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The nuclear shape : spectrum ?


so

With

To compare with a wash machine: 1300 tpm

For a rotating nucleus, the energy of a level is given by* :

With J the moment of inertia

We also have

* Mécanique quantique by C. Cohen-Tannoudji, B. Diu, F. Laloe

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Angular velocity of a rotating nucleus


• Macroscopic description of a nucleus : the liquid drop model

• Microscopic description needed: the basic ingredients are the nucleons and the interaction between them.

• Different microscopic approaches : the shell model, the mean field and beyond

• Many nuclei are found deformed in their ground states

• The spectroscopy strongly depends on the deformation 48

Modeling the nuclei:Summary

cern summer student program 2011 introduction to nuclear physics s. péru introduction to nuclear...

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