Transcript CERN Summer student program 2011 Introduction to

Introduction to Nuclear Physics
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S.PÉRU
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
The nucleus : a complex system
I) Some features about the nucleus
discovery
radius, shape
binding energy
nucleon-nucleon interaction
stability and life time
nuclear reactions
applications
II) Modeling of the nucleus
liquid drop
shell model
mean field
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
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
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.
Introduction to Nuclear Physics
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CERN Summer student program 2011
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
Introduction to Nuclear Physics
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CERN Summer student program 2011
The liquid drop
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.
Equilibrium shape
spherical
Density
ρ0
Volume ≈ A
R= r0 A 1/3
r
r
R
R
The binding energy of the nuclei is described by the Bethe-Weiszaker formula
Introduction to Nuclear Physics
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CERN Summer student program 2011
Bethe and Weizsacker formula
B  av A  as A
2/3
 ac Z A
2
1 / 3
N  Z 
2
 aa
A
 a p A1/ 3
Paring terms
+ even-even
- odd-odd
0 odd-even and even-odd
Binding term :
volume av
Unbinding terms :
Surface as ,
Coulomb ac ,
Asymmetry aa
Introduction to Nuclear Physics
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CERN Summer student program 2011
Binding energy per nucleon
Introduction to Nuclear Physics
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CERN Summer student program 2011
Problems with the liquid drop model
1) Fission fragment distributions
A heavy and
a light fragments
= asymmetric fission
Proton number
Liquid drop : only symmetric fission
Neutron number
Experimental Results :
Two identical fragments
= symmetric fission
Introduction to Nuclear Physics
K-H Schmidt et al., Nucl. Phys. A665 (2000) 221
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Problems with the liquid drop model
2) Nuclear radii
Evolution of mean square radii with
respect 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
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Problems with the liquid drop model
Halo nuclei
I. Tanihata et al., PRL 55 (1985) 2676
I. Tanihata and R. Kanungo, CR Physique (2003) 437
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3) Nuclear masses
E (MeV)
Difference in MeV between experimental masses and masses calculated with
the liquid drop formula as a function of the neutron number
Nuclear
shell effects
Neutron number
Existence of magic numbers : 8, 20 , 28, 50, 82, 126
Fig. from L. Valentin, Physique subatomique, Hermann 1982
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Two neutron separation energy S2n
S2n : energy needed to
remove 2 neutrons
to a given nucleus (N,Z)
S2n=B(N,Z)-B(N-2,Z)
For most nuclei, the 2n separation energies are smooth functions of particle
numbers apart from discontinuities for magic nuclei
Magic nuclei have increased particle stability and require a larger energy to
extract particles.
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CERN Summer student program 2011
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The nucleus is not a liquid drop :
Shell effects
There are many « structure effects » in nuclei, that can not be
reproduced by macroscopic approaches like the liquid drop model.
There are «magic numbers»
and so «magic»
2, 8, 20, 28, 50, 82, 126
90
140
Zr
40
50 58 Ce82......
and «doubly magic» nuclei
40
20
Ca20
208
82
Pb126 ...100
50 Sn50 ...
-> need for microscopic approaches, for which the fundamental
ingredients are the nucleons and the interaction between them
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Microscopic description of the atomic nucleus
Nucleus = N nucleons in strong interaction
The many-body problem
Nucleon-Nucleon force
unknown
(the behavior of each nucleon
influences the others)
Can be solved exactly for N < 4
For N >> 10 : approximations
Shell Model
only a small number of
particles are active
Introduction to Nuclear Physics
Different effective forces used
Depending on the method chosen
to solve the many-body problem
Approaches based on the Mean Field
• no inert core
• but not all the correlations between particles are taken
into account
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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
Introduction to Nuclear Physics
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 
 
 
 
 
Neutrons
Protons
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Shell Model
neutron
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.
nortuen
proton
“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.
Introduction to Nuclear Physics
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notorp
neutron
neutron
neutron
CERN Summer student program 2011
Shell Model
Energy (MeV)
U (r )
Schrödinger equation
r (fm)
R
0
Wave function φ and energy ε for each nucleon
Wave function ψ and nuclear binding energy E
R
Features of the nucleus in his ground state and
his excited ones
Introduction to Nuclear Physics
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r
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Shell Model : potential
Nuclear potential deduced from exp :
Wood Saxon potential
V0
V r   
r R
1  exp 

a


or
square well
or
harmonic oscillator
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
Shell Model : potential
spin orbit effect
 
S .O.  l .s
Introduction to Nuclear Physics
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CERN Summer student program 2011
Shell Model : how describe the ground state ?
-3d5/2
-2g9/2
-1i13/2
-3p1/2
-3p3/2
-2f5/2
-2f7/2
-1h9/2
126
-1h11/2
+3s1/2
+1d3/2
82
Ex: Z=10
+2d5/2
+1g7/2
+1g9/2
-1p1/2
-1f5/2
-2p3/2
-1f7/2
+1d3/2
+2s1/2
+1d5/2
-1p1/2
-1p3/2
+1s1/2
-1p1/2
-1p3/2
50
+1s1/2
Ex: N=20
8
2
+1d3/2
+2s1/2
+1d5/2
20
-1p1/2
-1p3/2
8
2
+1s1/2
28
20
8
2
For a nucleus with A nucleons you fill the A lowest energy levels, and
the energy is the sum of the energy of the individual levels
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
Shell Model : how describe excited states ?
Ex: Z=10
Ex: N=20
-1f7/2
8
2
-1p1/2
-1p3/2
+1s1/2
Ex: Z=10
+1d3/2
+2s1/2
+1d5/2
20
-1p1/2
-1p3/2
8
2
+1s1/2
Ex: N=20
-1f7/2
-1p1/2
-1p3/2
+1s1/2
8
2
Introduction to Nuclear Physics
+1d3/2
+2s1/2
+1d5/2
20
-1p1/2
-1p3/2
8
2
+1s1/2
S. Péru
Ground state
Excited state :
you make a particle-hole excitation.
You promote one particle to a higher
energy level
CERN Summer student program 2011
Beyond this “independent particle Shell Model
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
Introduction to Nuclear Physics
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CERN Summer student program 2011
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Beyond this “independent particle Shell Model
proton
Introduction to Nuclear Physics
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neutron
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Mean field approach
Main assumption
each 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.
Nuclear interaction
Neptune
FG
2 nucleons bare force
many nucleons  effective interaction
Feffective
Flibre
Soleil
Uranus
Self consistent mean field : the mean field is not fixed. It depends on the configuration.
No inert core
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
Jacques Dechargé
Jacques Dechargé
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
The phenomenological
effective finite-range Gogny force
P : isospin exchange operator
P : spin exchange operator
 r1 - r2 2 
range
 Wj  B jP - H jP - M jP P  Finite
v12   expcentral term
pj 

j 1

    
 t 3 1  x0 Ps  r1  r2  r1  r2  Density dependent term

  


 iWls 12. r1  r2  12 . 1   2  Spin orbit term
2
e2
 1  2t1 z 1  2t 2 z   
r1  r2
Coulomb term
Finite range for pairing treatment
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
Mean field approach : Hatree-Fock method
Hartree-Fock equations
  2 2

  U HF ( )i ( xi )   ii ( xi )

2
M


(A set of coupled Schrodinger equations)
For more formalism see
“The nuclear many body problem”,
P. Ring and P. Schuck
Single particle wave functions
Hartree-Fock potential
Self consistent mean field :
the Hartree Fock potential depends on the solutions
(the single particle wave functions)
-> Resolution by iteration
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
Resolution of the Hatree-Fock equations
Trial single particle wave function
i ( xi )
Effective interaction
Calculation of the HF
potential
U HF ( )
  2 2

Resolution of the HF equations 
  U HF ( )i ( xi )   ii ( xi )
 2M

New wave functions
i ( xi )
Test of the convergence
Jacques Dechargé
Calculations of the properties of the nucleus in its ground state
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
Hatree-Fock method : deformation
We can “measure” nuclear deformations as the mean values of the
Q̂
mutipole operators
q 
 Qˆ  

Spherical Harmonic
If we consider the isoscalar axial quadrupole operator
We find that:
Qˆ 20  r 2 Y20
http://www-phynu.cea.fr
Most of the nuclei are deformed in their ground state
Magic nuclei are spherical
Introduction to Nuclear Physics
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CERN Summer student program 2011
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Constraints Hatree-Fock-bogoliubov calculations
We can impose collective deformations and test the response of the nuclei:
  qi  Hˆ   iQˆi  N Nˆ  Z Zˆ qi   0
i
with
qi  Nˆ ( Zˆ ) qi   N (Z)
qi  Qˆ i qi   q i
Where ’s are Lagrange parameters.
Introduction to Nuclear Physics
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Constraints Hatree-Fock-bogoliubov calculations :
results
g.s deformation predicted
with HFB using the Gogny
force
http://www-phynu.cea.fr
Introduction to Nuclear Physics
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CERN Summer student program 2011
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Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
Constraints Hatree-Fock-bogoliubov calculations :
What are the most commonly used constraints ?
What are the problems
with this deformation ?
Introduction to Nuclear Physics
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Constraints Hatree-Fock-bogoliubov calculations :
Potential energy landscapes
Deformations pertinent for fission:
Elongation
Asymmetry
…
Introduction to Nuclear Physics
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Evolution of s.p. states with deformation
154Sm
New gaps
Introduction to Nuclear Physics
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CERN Summer student program 2011
Hatree-Fock-bogoliubov calculations with blocking
Particle-hole excitations  one (or two, three, ..) quasi-particles curves
Introduction to Nuclear Physics
S. Péru
CERN Summer student program 2011
Beyond mean field …
Introduction of more correlations : two types of approaches
Random Phase Approximation (RPA)
Generator coordinate Method (GCM)
Coupling between HFB ground state
and particle hole excitations
Introduction of large amplitude
correlations
Give access to
a correlated ground state and to the excited states
Individual excitations and collective states
Introduction to Nuclear Physics
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CERN Summer student program 2011
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Beyond mean field … with GCM
(GCM+GOA 2 vibr. + 3 rot.)
= 5 Dimension Collective Hamiltonian
5DCH
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CERN Summer student program 2011
Beyond mean field … with RPA
Dipole
Monopole
S. Péru, J.F. Berger, and P.F. Bortignon, Eur. Phys. Jour. A 26, 25-32 (2005)
Introduction to Nuclear Physics
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CERN Summer student program 2011
The nuclear shape : spectrum ?
«vibrational» spectrum
+
Spherical nuclei
6
4+
E  J
2+
0+
Deformed nuclei
«rotational» spectrum
6+
4+
2+
E  J(J +1)
0+
Introduction to Nuclear Physics
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CERN Summer student program 2011
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Angular velocity of a rotating nucleus
For a rotating nucleus, the energy of a level is given by* :
With J the moment of inertia
We also have
so
With
To compare with a wash machine: 1300 tpm
* Mécanique quantique by C. Cohen-Tannoudji, B. Diu, F. Laloe
Introduction to Nuclear Physics
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Modeling the nuclei:
Summary
• 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
Introduction to Nuclear Physics
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CERN Summer student program 2011
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