1. Università degli Studi and INFN, MIlano
Density Functional Theory for stable and exotic nuclei (plus extensions) LECTURE III
Gianluca Colò
Università degli Studi and INFN, MIlano
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams

2. Università degli Studi and INFN, MIlano
Outline (II+III)
Properties of finite nuclei within DFT (continued): ground-state properties like masses, radii, deformations …
Brief account of nuclear superfluidity and link with clustering.
The time-dependent DFT and applications to the nuclear collective motion.
How to extract (some) information on the nuclear Equation of State.
Astrophysical applications: neutron stars.
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

3. Università degli Studi and INFN, MIlano
Motivation
We would like the same theory to describe both ground state and excited states.
In this lecture, I will restrict myself to collective excitations of spherical nuclei (vibrations – yet in a broad sense).
How can DFT be applied to excited states ?
Will this study provide some kind of general information ?
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

4. Formal background: the RG theorem
The Hohenberg-Kohn theorem (HK) guarantees the existence of a functional E that provides the exact g.s. for a many-fermion system:
Runge-Gross theorem guarantees an exact functional also exists in the time-dependent case:
To be more precise, this theorem establishes a one-to-one correspondence:
Therefore, the time-dependent density also determines all properties of the system (total energy etc.)
Problems… Also so-called “memory” problem.
E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984).
R. Van Leeuwen, Phys. Rev. Lett. 80, 1280 (1998).
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

5. The time-dependent HF with density dependent forces (I)
In practice, everybody uses so far the same functional used for the g.s. and the so-called “adiabatic” approximation.
Slater determinant
1-body density matrix
Skyrme (SEDF)
Gogny (GEDF)
Relativistic MF or HF
(CEDF)
local functionals (evolved from Veff ÷ δ(r1-r2))
non-local from Veff having Gaussian shape
covariant functionals
(Dirac nucleons exchanging effective mesons)
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

6. The time-dependent HF with density-dependent forces (II)
Reminder of the static case
Direct solution of the TD case
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

7. The time-dependent HF with density-dependent forces (III)
The Fourier transform will give the energy of the mode(s).
From: P. Stevenson (U. Surrey)
GDR (Giant Dipole Resonance): excited by e.g. a photon beam impinging on a nucleus.
E≈10-20 MeV implies that λel.field = hc / E >> R (nuclear dimension).
Consequently, the e.m. field in the nuclear region can be considered constant (dipole approximation).
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

8. RPA (Random Phase Approximation)
The so-called Random Phase Approximation(*) should be better called linearization of the equation of motion.
In fact, to derive it we truncate the equation of motion at first order in the density variation.
(*) The name comes from plasma physics, cf. eikr…
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

9. Università degli Studi and INFN, MIlano
We can take the previous equation and express it on a basis (particle-hole, p-h) + =
RPA equation.
X and Y are forward and backward amplitudes.
Probability amplitude of a given configuration.
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

10. Università degli Studi and INFN, MIlano
Nuclear excitations
Single-particle: one nucleon from below to above the Fermi surface.
Collective: vibrations (spherical nuclei), rotations (deformed nuclei).
Figure by M. Harakeh
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

11. Low-lying dipole strength vs. threshold effect
Increasing (N-Z)/A, we expect decoupling of neutron and proton excitations, and low-lying neutron strength.
Favoured in light nuclei ↔ low angular momenta.
From. T. Nakamura
After its discovery in light nuclei, low-lying dipole strength has been identified also in medium-heavy systems.
“core”
excess neutrons
A. Klimkiewicz et al., PRC 76, (R) (2007).
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

12. Example: Giant Monopole Resonance
Breathing mode:
This is a clear example in which the knowledge of a specific nuclear excitation may shed light on a more general property: the (in)compressibility of nuclear matter.
Supernova SN1987a
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

13. Charge-exchange transitions
Some transitions may be inside the allowed β-decay window.
However, most of them require external energy so that they are induced by charge-exchange reactions, like (p,n) or (3He,t).
Z+1,N-1
Z,N
Z-1,N+1
(n,p)
(p,n)
The study of their properties can greatly improve our knowledge of the charge-changing part of the nuclear H.
At the same time the matrix elements of the transitions involved in double-β decays must be known to extract the neutrino mass. p n
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

14. Gamow-Teller resonanceZ N
Unperturbed GT energy related to the spin-orbit splitting
Highest and lowest particle-hole transitions in the picture
RPA GT energy related also to V in στ channel
Osterfeld, 1982:
Using empirical Woods-Saxon s.p. energies, the GT energy is claimed to determine g0’
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

15. Università degli Studi and INFN, MIlano
Taken from : H. Sakai, talk at IInd Topical Workshop on Modern Aspects in Nuclear Structure, Bormio, February 2014
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

16. Università degli Studi and INFN, MIlano
Correlations between observable vibrational modes and properties of nuclear matter (equation of state)
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

17. Università degli Studi and INFN, MIlano
Nuclear matter
Nuclear matter is an idealized UNIFORM system of neutron and proton having constant density.
The Coulomb interaction among protons must be taken out !
It is analogous to the uniform electron gas for condensed matter physicists.
WE STICK TO
T = 0 !
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

18. EoS from density functionals
It is very easy (analytic or semi-analytic) to extract the equation of state or the pressure from a functional. Actually we call E/A = equation of state.
We have already learnt how to calculate the energy density. Example:
In a uniform system:
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

19. Università degli Studi and INFN, MIlano
Kinetic part
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

20. Symmetric nuclear matter
In symmetric matter, besides the saturation point, we can characterise the EoS by the so-called incompressibility:
In the 1990s, the hope to extract this quantity directly from data faded away. We have to use theory.
EGMR
S. Shlomo, V. Kolomiets, GC, Eur. Phys. J A30 (2006) 23.
G.C., Phys. Part. Nucl. 39, 557 (2008). K∞
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

21. Asymmetric nuclear matter
Nuclear matter EOS
Symmetric matter EOS
Symmetry energy S
Expansion around ρ0= 0.16 fm-3
SATURATION
POINT of SNM
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

22. Symmetry energy from IV vibrations
Neutrons and protons oscillate in opposition of phase.
Promising observables to extract the properties of the symmetry energy.
Problems:
the nucleus is not a homogeneous system, it has a shell structure, and there is isoscalar/isovector mixing.
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

23. Different attempts from our group
MEASURABLE QUANTITY A
EoS PARAMETER B
IVGDR PRC 77, (R) (2008)
PDR PRC 81, (R) (2010)
J = 32.3 ± 1.3; L = 64.8 ± 15.7
Dipole polarizability PRC 88, (2013)
J = 31 ± 2; L = 43 ± 16
IVGQR PRC 87, (2013)
J = 32 ± 1; L = 37 ± 18
Anti-analog dipole PRC 94, (2016)
J = 33.3 ± 2.1; L = 98.8 ± 23.6
ALL NUMBERS IN MeV
The dipole polarizability ⍺D provides a sound correlation (cf. droplet model).
The low-lying pygmy dipole gives a large contribution to ⍺D.
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

24. Università degli Studi and INFN, MIlano
Global status
J.M. Lattimer, J. Lim, Ap. J. 771, 51 (2013)
C.H. Horowitz et al., J. Phys. G 41 (2014)
J = 31.6 ± 2.7 MeV
L = 58.9 ± 31.6 MeV
B.A. Li, NuSYM, June 2016
There is a certain degree of consistency between different constraints from nuclear structure, heavy-ion (HI) reactions and astrophysics (neutron stars).
Realistic assessment of errors ? Hidden model dependence ?
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

25. Correlations and effect of the fitting protocol
SLy5 with the constraint on the
neutron EoS made weak …in addition, neutron skin fixed !
X. Roca-Maza et al., JPG 42, (2015)
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

26. Università degli Studi and INFN, MIlano
SAMi functional
Binding energies of 40,48Ca, 90Zr, 132Sn and 208Pb
Charge radii of 40,48Ca, 90Zr and 208Pb
Ab initio calculation of neutron matter between 0.07 fm-3 and 0.4 fm-3
Spin-orbit splittings of 1g and 2f proton levels in 90Zr and 208Pb, respectively
g0 and g0’ restricted around 0.15 and 0.35, respectively
X. Roca-Maza, G.C., H. Sagawa, PRC 86, (R) (2012).
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

27. Università degli Studi and INFN, MIlano
Neutron stars (I)
Cf. exercise in books (K. Heyde, Shapiro-Teukolski …): a bound system of neutrons can exist due to the gravitational force if the number of neutrons is large enough (≈ ).
Neutron stars observed. Some properties are known (masses and, to a lesser extent: radii, thermal properties, B …).
Structure is complicated.
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

28. Hydrostatic equilibrium
Classical gravity (Newton)
General relativity corrections (TOV)
One has to input P(ρ) from the nuclear EoS.
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

29. Università degli Studi and INFN, MIlano
J. Rikovska Stone, J.C. Miller, R. Koncewicz, P.D. Stevenson, M.D. Strayer
Group I: satisfactory Group II: not so good Group III: bad !
The trend of E/A is correlated with the relation mass/radius. If BE matter is not “stiff” enough, one does not get large enough masses…
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

30. The context: different approaches to nuclear structure
Best wishes for your research to bloom on the tree of nuclear physics !
Question at the root: how do forces work into nuclei and how are nuclei organized ?
The context: different approaches to nuclear structure
The role of nuclear Density Functional Theory (DFT)
Nuclear excitations (Giant Resonances, GRs) and their link with the nuclear EoS
Neutron stars
Fit of new functionals (SAMi)
Tensor terms
Ground-state properties, superfluidity
THEORY in close connection with EXPERIMENT
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano