4 November 2010. Master 2 APIM
Le problème à N corps
nucléaire: structure nucléaire
The atomic nucleus is a self-bound quantum many-body (many-
nucleon) system
Rich phenomenology for nuclei
Mean field
Which particles?
Which are the degrees of freedom
taken into account in the nuclear many- body problem?
Which is the interaction?
STRONG INTERACTION
D. Lacroix Lecture. École Joliot-Curie 2009, Lacanau, France
Deriving the nucleon-nucleon interaction from the underlying theory of QCD is extremely difficult (so far almost no quantitative results).
Three main possible directions to deal with the nuclear interaction in low-energy nuclear physics are:
• Realistic interactions
• Effective field theories (deep links with QCD)
• Phenomenological interactions
Nuclear physics and the strong interaction
• Chadwick discovers the neutron (he
measures its mass) in 1932 (Nobel Prize in 1935): the two types of nucleons are thus known
• How do the nucleons interact among themselves?
The idea at the basis of Yukawa hypothesis:
In the case of the electromagnetic interaction the mediator of the force is the photon (zero mass). Quantum
electrodynamics: at the beginning of the 30s, it is already formulated.
Yukawa hypothesis
Yukawa introduces a particle that he calls U. He suggests that this particle is the mediator of the nuclear strong interaction
m
U≈ 200 m
e;
m
p≈ 1800 m
e15 years later the formulation of the hypothesis…
• In 1947 and 1948 the Yukawa particle is identified with the meson π : pseudoscalar (negative parity), spin 0 (the two quarks have opposite spin). The mass is ~ 140 MeV/c
2- Observed in cosmic rays - Produced at Berkeley
π mesons interact strongly with nucleons and have the correct mass that leads to the correct range of the nuclear force
• 1949: NOBEL PRIZE FOR YUKAWA !!
Yukawa potential
g -> coupling constant mπ -> mass of pion
Range
Compton wave length of pion:
60s: heavier mesons are discovered
ω and ρ mesons: mass ≈ 800
MeV
Realistic interactions
The earliest realistic potentials in the 60s: one-pion exchange potentials with adjustable parameters + a repulsive hard core
(to describe high-energy scattering data)
• 70s - 80s: Paris and Bonn potentials
• 90s: High-precision realistic potentials (starting from the 90s) : CD-Bonn, Argonne V18, Nijmegen I et II. High-precision adjustments (around 50 parameters) are done to reproduce phase shifts (free nucleon- nucleon scattering) and spectroscopy of nuclei with a small number of nucleons
The adjustment of the parameters
Scattering amplitude
Differential cross section
Total cross section
• Long range (pion)
• Intermediate range (more or less phenomenological +, eventually, exchange of mesons)
• Short range: hard core (phenomenological +,
eventually, the exchange of mesons)
Problem of the hard core: motivation to use phenomenological interactions (calculations with more sophisticated models become easier)
• Two types:
• 1) zero range (contact interaction) (Skyrme)
• 2) finite range (Gogny)
The simple idea
• Construction of an effective phenomenological force that is chosen to reproduce (at the mean-field level) global
properties of some selected nuclei (binding energies and radii) and properties related to the EoS of nuclear matter (first of all saturation point)
• Form and parameters to choose
• For instance, to have in-medium effects (Pauli principle), we need the density-dependent term
• Correlations are contained in the interaction in an effective way.
Finite-range: Gogny interaction
It has been introduced from a realistic G matrix Standard form:
Finite-range (gaussian form) Density dependent (zero-range)
Spin-orbit (zero-range)
First parametrization: D1
Déchargé, Gogny, PRC 21, 1568 (1980)
Most used: D1S (fit also on fission barriers) Berger, Girod, Gogny, NPA 502, 85c (1989)
A recent modification (towards finite-range in all terms): D1N Chappert, Girod, Hilaire, Phis. Lett. B 668 (2008), 420
CEA
Bruyères-le- châtel
and -> 14 parameters
Spin and isospin exchange operators acting on the left
acting on the right
Isospin matrices Spin matrices
Zero-range: Skyrme interaction
Standard form: 10 parameters
central non local
density dependent
spin-orbit
T.H.R. Skyrme, Phil. Mag. 1, 1043 (1956), Nucl. Phys. 9, 615 (1959)
First applications: Vautherin, Brink, PRC 5, 626 (1972) IPN Orsay
Advantage of using phenomenological
interactions. Which nuclei
can be treated?
Mean field for ground state nuclear structure (HF, HFB,..) RPA and QRPA for small amplitude
oscillations Beyond small amplitude
oscillations: time- dependent mean field for dynamics (TDHF, TDHFB,…)
Beyond-mean - field models
(correlations): GCM, particle-vibration coupling, variational multiparticle-
multihole
configuration mixing, extensions of RPA, Second RPA,…
Links with Quantum Chromodynamics (70s,
Standard Model)
Internal degrees
of freedom of the
nucleon
Links with QCD Lagrangian
(chiral symmetry)
Djalali Lecture.
Ecole Joliot- Curie 2009
The symmetry is spontaneously broken…
…Goldstone boson (pion !!! Links with realistic
interactions)…but the pion has a non-zero mass…
Djalali Lecture.
Ecole Joliot- Curie 2009
All nucleons have positive parity
… and the pion has non-zero mass because…
…the symmetry is explicitly broken… non-zero mass of quarks !! Broken symmetry scale: 1 GeV (see quark masses)
Hard scale =>
1 GeV, chiral symmetry
breaking scale Effective field theories: choice of the relevant degrees
of freedom at the scales we are interested in
|q| ~ Ml << Mh
q -> soft scale
It diverges at large q. Introduce a cut-off Λ
Leading order (LO)
Next-to-leading order (NLO)
A natural hierarchy (that is intuitive) can be formally established between 2-body, 3-body,…,N-body forces
Leading order Next-to-leading order
Next-to-next-to-leading order
Q=q/Λ
Nuclear matter
Ideal and infinite system composed by nucleons:
Symmetric matter (protons and neutrons with equal densities)
Asymmetric matter Neutron matter
Nuclei
Neutron stars
Saturation density ρ0 0.16 fm-3 ->
∼ 2.5 ּ 1014 g/cm3
After evolution (thermonuclear
reactions) of massive stars (M≥ 8 M๏) (∼107 years) -> supernova explosion . Slow
cooling for millions of years.
Typical values:
R ∼ 10 km M ∼ 1.4 M๏
EXOTIC NUCLEI
Mass measurements of
neutron stars in binary systems
Mass (M๏)
0.5 1 1.5 2 2.5 3 3.5
Equation of state (EOS): E/A as a function of the density
Isospin effects and density dependence No equilibrium point in pure neutron matter
Symmetric matter
From symmetric to neutron matter. Yp=Z/A
Equilibrium point:
saturation density
Equation of State with different models
Symmetric nuclear matter Pure neutron matter
stiff
soft
48
Ca density profiles. Spherical nucleus. Hartree-Fock calculations
with a Skyrme interaction
Neutron density Proton density
R (fm) R (fm)
Density
N = 28 Z = 20
Rich phenomenology for nuclei
Mean field
Light nuclei
Some properties of light nuclei
Borromean nuclei
• Borromean Nuclei: 3-body systems. Example:
– 11Li = 9Li+n+n
• Generalization to n-body systems. Example :
– 10C = 4He+4He+p+p
• Mixing of complex correlations
Halo densities
Some applications of mean-
field-based models. The basic
method for many-body systems
Masses Separation energies Drip lines?
Densities
Single particle spectra Energies and
occupation probabilities?
Ground state of nuclei. Mean field: individual degrees of freedom
Mean field
calculations (HFB) with a Skyrme
force
DRIP LINE ? Strongly
model dependent!!!
Binding energy (top) and two-
neutron separation energy (bottom)
48
Ca density profiles. Spherical nucleus. Hartree-Fock with
Skyrme
Neutron density Proton density
R (fm) R (fm)
Density
Last neutron single-particle Hartree-Fock states
Ca isotopes
Can we
predict single- particle
energies and spectroscopic factors? We need to go
beyond mean field (see next Lecture).
Why?
Grasso, Yoshida, Sandulescu, Van Giai, PRC 74, 064317 (2006)
1f5/2 2p1/2
Single-particle and collective degrees of freedom couple:
beyond mean field
These correlations also affect the excited states …
Effects of particle-
vibration coupling on the single-particle
spectrum
Neutron states in 208Pb
Bernard, Van Giai, Nucl. Phys. A 348 (1980), 75
Particle-vibration coupling
E. Litvinova, et al.
208Pb 132Sn
At mean field level we can include pairing correlations to treat superfluid nuclei (BCS
superfluidity -> Cooper pairs)
• Energy gap in excitation spectra
• Odd-even effect in binding energy
• Moments of inertia
No isotopes (Z=102)
Duguet et al. arXiv:nucl-th/0005040v1
Sn = E (Z,N)-E(Z,N-1)
Odd-even effect
Excitations. Collective
degrees of freedom
Two-neutron 0+ addition mode QRPA in particle-particle channel
Response function for 124Sn
Khan, Grasso, and Margueron
Excitation modes (small amplitudes)
Quadrupole mode QRPA in
particle-hole channel
Response function for 22O
Khan, Sandulescu, Grasso, and Van Giai, PRC 66, 024309 (2002)
Pair transfer mode
Khan, Grasso, Margueron, submitted PRC
Dynamics. Beyond small amplitude
oscillations. Time dependent approaches for the dynamics
Time-dependent HF
Collision Dynamics of
Two 238U Atomic Nuclei Golabek and Simenel, PRL 103, 042701 (2009)
Neutron stars. Very exotic nuclear systems
Nuclear structure / Astrophysics
Picture of the crust of a neutron star
Baym, Bethe, Pethick, NPA 175 (1971), 225
Outer crust: crystal of nuclei in an electron sea
Drip point for neutrons
∼ 4 ּ 1011g/cm3
Inner crust: crystal of nuclear clusters in an electron sea and in a gas of superfluid neutrons
ρ ≈ 0.5 ρ0
ρ
Inner crust: 0.001 ρ0 ≤ ρ ≤ 0.5 ρ0
β-stability condition -> µe = µn - µp Saturation density
ρ0 = 0.16 fm-3
∼ 2.5 ּ 1014 g/cm3
Wigner-Seitz cell model
J.W. Negele and D. Vautherin, Nucl. Phys. A 207, 298 (1973)
