CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Introduction to Nuclear PhysicsIntroduction to Nuclear Physics
S.PÉRU
2/3
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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.
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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.
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Binding energy per nucleon
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Halo nuclei
I. Tanihata et al., PRL 55 (1985) 2676I. Tanihata and R. Kanungo, CR Physique (2003) 437
15
Problems with the liquid drop model
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
21
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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.
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Nuclear potential deduced from exp :Wood Saxon potential
orsquare well
orharmonic oscillator
a
RrV
rVexp1
0
Shell Model : potential
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
spin orbit effect
slOS
...
Shell Model : potential
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
+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 ?
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
+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 ?
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Jacques Decharg
é
Jacques Dechargé
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
g.s deformation predicted with HFB using the Gogny force
http://www-phynu.cea.fr
35
Constraints Hatree-Fock-bogoliubov calculations : results
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
What are the problems with this deformation ?
38
Constraints Hatree-Fock-bogoliubov calculations : What are the most commonly
used constraints ?
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Deformations pertinent for fission:ElongationAsymmetry
…
39
Constraints Hatree-Fock-bogoliubov calculations : Potential energy landscapes
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
154Sm
Evolution of s.p. states with deformation
New gaps
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Hatree-Fock-bogoliubov calculations with blocking
Particle-hole excitations one (or two, three, ..) quasi-particles curves
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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 …
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Beyond mean field … with GCM
(GCM+GOA 2 vibr. + 3 rot.) = 5 Dimension Collective Hamiltonian
5DCH
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Beyond mean field … with RPA
MonopoleMonopole DipoleDipole
S. Péru, J.F. Berger, and P.F. Bortignon, Eur. Phys. Jour. A 26, 25-32 (2005)
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
Spherical nuclei «vibrational» spectrum
0+
2+
4+
6+
Deformed nuclei «rotational» spectrum
0+
4+
6+
2+
E J
E J(J+1)
44
The nuclear shape : spectrum ?
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
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
47
Angular velocity of a rotating nucleus
CERN Summer student program 2011Introduction to Nuclear Physics S. Péru
• 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