The Nuclear Equation of State
and the neutron skin thickness
in nuclei
Xavier Roca-Maza
Università degli Studi di Milano e INFN, sezione di Milano
Physics beyond the standard model and precision nucleon
structure measurements with parity-violating electron scattering
Trento, August 1st-5th 2016.
1
Table of contents:
→ Brief introduction
→ The Nuclear Many-Body Problem
→ Nuclear Energy Density Functionals
→ Nuclear Equation of State: the symmetry energy
→ The neutron skin thinckess and the symmetry energy
→ The impact of the neutron skin (symmetry energy) on
nuclear and astrophyiscs observables
→ Some examples: Neutron stars outer crust, parity violating
asymmetry, pygmy states (?), GDR, dipole polarizability,
GQR, AGDR ...
→ Conclusions
2
INTRODUCTION
3
The Nuclear Many-Body Problem:
→ Nucleus: from few to more than 200 strongly interacting
and self-bound fermions.
→ Underlying interaction is not perturbative at the
(low)energies of interest for the study of masses, radii,
deformation, giant resonances,...
→ Complex systems: spin, isospin, pairing, deformation, ...
→ Many-body calculations based on NN scattering data in
the vacuum are not conclusive yet:
→ different predictions (interaction in the medium) are found
depending on the approach
→ EoS and (recently) few groups in the world are able to
perform extensive calculations for light and medium mass
nuclei
→ Based on effective interactions, Nuclear Energy Density
Functionals are successful in the description of masses,
nuclear sizes, deformations, Giant Resonances,...
4
Nuclear energy density functionals E[ρ] are commonly
derived from an effective Hamiltonian solved at first
order perturbation theory (Hartree-Fock)
Exact EDF?
E = hΨ|H|Ψi ≈ hΦ|Heff (ρ)|Φi −−−−−−−−→ E[ρ]
Kohn-Sham iterative scheme (static)
→ Determine/copy/invent/derive... E[ρ]
→ Initial guess ρ0
→ Calculate potential Veff from ρ0 (h = δE/δρ)
→ Solve single particle equation of motion (hφi = ǫi φi ) φi
A
X
→ Use φi for calculating new ρ1 =
|φi |2
i
Repeat until consistency between ρ and Veff
Runge-Gross Theorem (dynamic): exist also E[ρ(t), t]
Z
dt {hΦ(t)|i∂t |Φ(t)i − E[ρ(t), t]} = 0
Useful for the study of small perturbations of the gs ρ: GR
5
Nuclear Energy Density Functionals:
Main types of successful EDFs for the description of masses,
deformations, nuclear distributions, Giant Resonances, ...
Relativistic mean-field models, based on Lagrangians where
effective mesons carry the interaction:
Lint
= Ψ̄Γσ ΨΦσ + Ψ̄Γδ τΨΦδ − Ψ̄Γω γµ ΨA(ω)µ
− Ψ̄Γρ γµ τΨA(ρ)µ − eΨ̄Q̂γµ ΨA(γ)µ
Non-relativistic mean-field models, based on Hamiltonians
where effective interactions are proposed and tested:
long−range
eff
VNucl
= Vattractive
short−range
+ Vrepulsive
+ VSO + Vpair
→ Fitted parameters contain (important) correlations
beyond the mean-field
→ Nuclear energy functionals are phenomenological → not
directly connected to any NN (or NNN) interaction
6
The Nuclear Equation of State: Infinite System
30
e ( MeV )
20
neutron matter
e(ρ,δ=1)
10
S(ρ)~
0
Saturation
−3
(0.16 fm , −16.0 MeV)
-10
e(ρ,δ=0)
symmetric matter
-20
0
0.05
0.1
0.15
−3
0.2
0.25
ρ ( fm )
E
E
(ρ, β) =
(ρ, β = 0) + S(ρ)β2 + O(β4 )
A
A
→ Nuclear
Matter
ρn − ρp
β=
ρ
7
The Nuclear Equation of State: Infinite System
30
e ( MeV )
20
neutron matter
e(ρ,δ=1)
10
S(ρ)~
0
Saturation
−3
(0.16 fm , −16.0 MeV)
-10
e(ρ,δ=0)
symmetric matter
-20
0
0.05
0.1
0.15
−3
0.2
0.25
ρ ( fm )
E
E
(ρ, β) =
(ρ, β = 0) + S(ρ)β2 + O(β4 )
A
A
→ Nuclear
Matter
→ Symmetric
Matter
ρn − ρp
β=
ρ
8
The Nuclear Equation of State: Infinite System
30
e ( MeV )
20
neutron matter
e(ρ,δ=1)
10
S(ρ)~
0
Saturation
−3
(0.16 fm , −16.0 MeV)
-10
e(ρ,δ=0)
symmetric matter
-20
0
0.05
0.1
0.15
−3
0.2
0.25
ρ ( fm )
E
E
(ρ, β) =
(ρ, β = 0) + S(ρ)β2 + O(β4 )
A
A
→ Nuclear
Matter
→ Symmetric
Matter
→ Symmetry energy
ρn − ρp
β=
ρ
9
The Nuclear Equation of State: Infinite System
30
e ( MeV )
20
neutron matter
e(ρ,δ=1)
10
S(ρ)~
0
Saturation
−3
(0.16 fm , −16.0 MeV)
-10
e(ρ,δ=0)
symmetric matter
-20
0
0.05
0.1
0.15
−3
0.2
0.25
ρ ( fm )
E
E
(ρ, β) =
(ρ, β = 0) + S(ρ)β2 + O(β4 )
A
A
1
E
2
2
3
= (ρ, β = 0) + β J + Lx + Ksym x + O(x )
A
2
β=
ρn − ρp
;
ρ
x=
ρ − ρ0
3ρ0
10
The Nuclear Equation of State: Infinite System
30
e ( MeV )
20
neutron matter
e(ρ,δ=1)
10
S(ρ)~
0
Saturation
−3
(0.16 fm , −16.0 MeV)
-10
e(ρ,δ=0)
symmetric matter
-20
0
0.05
0.1
0.15
−3
0.2
0.25
ρ ( fm )
E
E
(ρ, β) = (ρ, β = 0) + β2
A
A
→ S(ρ0 ) = J
1
J + L x + Ksym x2 + O(x3 )
2
d
P0
L
S(ρ) =
= 2
→
dρ
3ρ0
ρ0
ρ0
→
K
d2
= sym
S(ρ)
2
dρ
9ρ20
ρ0
ρn − ρp
β=
;
ρ
ρ − ρ0
x=
3ρ0
11
The Nuclear Equation of State: Infinite System
30
e ( MeV )
20
neutron matter
e(ρ,δ=1)
10
S(ρ)~
0
Saturation
−3
(0.16 fm , −16.0 MeV)
-10
e(ρ,δ=0)
symmetric matter
-20
0
0.05
0.1
0.15
−3
0.2
0.25
ρ ( fm )
E
E
(ρ, β) = (ρ, β = 0) + β2
A
A
1
J + L x + Ksym x2 + O(x3 )
2
The uncertainties on
density
S(ρ) around saturation
2
and Ksym
d
P
L
d
(mainly due to L) impact
0 nuclear physics
= on many
=
S(ρ)
=
→
S(ρ)
→
→ S(ρastrophysics
2
2
0) = J
dρ
3ρ0
dρ
ρ0
observables.
9ρ20
ρ0
ρ0
12
The symmetry energy and the neutron skin in 208 Pb
∆rnp ≡ hr2n i1/2 − hr2p i1/2
1 IR
Simple macroscopic model: ∆rnp ∼
L
12 J
-Rs
K
I2
Sk HF-P
R
SV -Gs
Sk
Sk
24
F-P
A1
RH
∆rnp (fm)
1
NL
G2
-T4
Sk
5
.s2
1.s
PK
3
KO
P
Sk
7
B-1
HFSkX 6
-T
Sk
II
SG
N
D1
D1
B-8
HF
k7
MS 0
9
v0
0.15
3
NL
0.2
MP
a
Sk M*
Sk 1
S
Sk
-PC
DD
SIV L0
MS SkA
M CP
old
B
E1
-M
y5
SL y4
DD -ME2
SL
DD
M*
Sk
UG
FS
0.25
kI5
2
S
NL
1
G1 -RA
NL C-F1 H
3*
NL 3
P L-S
1
N
-PK
NLK1
PC
P V2
-S
NL TM1
Linear Fit, r = 0.979
Mean Field
0.3
L ∝ pneut
0
S
0.1
0
100
50
150
L (MeV)
Physical Review Letters 106, 252501 (2011)
The faster the symmetry energy increases with density (L), the
largest the size of the neutron skin in (heavy) nuclei.
[Exp. from strongly interacting probes: ∼ 0.15 − 0.22 fm (Physical Review C 86 015803 (2012))].
13
The impact of the neutron
skin on nuclear and
astrophyiscs observables
14
The neutron skin and the parity violating asymmetry in 208 Pb
8
BHF90
v0
7,4
7
k
MS
S
D1
Linear Fit, r = 0.995
Mean Field
From strong probes
n
N an
D1 ffm 6
T os
HoSk- Kl
X 5
P
Sk SLy P
S Sk 17
A
BCSk
B- rt
L0
M
2
HF rbe Ly4 *
MS -ME
S kM
Ta
3
S
DD kMP E1 KO ro
S D-MF-P ihi
D H en Rs
A1
R
Z kPK
V
S
FSI M* 1
d
S C l
RH SV 2
SkD-P Go ka
I
D SU S
Sk -T4
s
F
G
Sk .s25 K1
Sk .s24
3 -P
1
NL PC 2
I5
PK
G
Sk -F1
PC PK1
3*
H
S
2
G1 NL
NL -SV
NL RA1 NL3
NL
,
1
TM
7,2
7
10 Apv
I
GI
7,0
6,8
0,1
0,15
0,2
∆ rnp (fm)
0,25
Physical Review Letters 106, 252501 (2011)
0,3
→ Electrons interact by exchanging a γ (couples to
p) or a Z0 boson (couples to n)
→ Ultra-relativistic electrons, depending on their
helicity (±), will interact with the nucleus seeing
a slightly different potential: Coulomb ± Weak
→ Apv ≡
L2
N
1
NL
dσ+ /dΩ − dσ− /dΩ
Weak
∼
dσ+ /dΩ + dσ− /dΩ
Coulomb
→ Input for the calculation are the ρp and ρn
(main uncertainty) and nucleon form factors for
the e-m and the weak neutral current.
→ In PWBA for small momentum transfer:
!
1/2
q2 hr2
G q2
pi
1−
Apv ≈ √F
∆rnp
3Fp (q)
4 2πα
(Calculation at a fixed q equal to PREx)
The largest the size of the neutron distribution in nuclei (∆rnp ),
the smaller the parity violating asymmetry.
[Exp. from ew probes: 0.302 ± 0.175 fm (Physical Review C 85, 032501 (2012))].
15
Isovector Giant Resonances (some considerations)
→ In isovector giant resonances neutrons and protons
“oscillate” out of phase
→ Isovector resonances will depend on oscillations of the
density ρiv ≡ ρn − ρp ⇒ S(ρ) will drive such “oscillations”
→ The excitation energy (Ex ) within a Harmonic Oscillator
approach is expected to depend on the symmetry energy:
r
d2 U
√
∝ k → Ex ∼
s
p
δ2 e
∝ S(ρ)
2
δβ
where β = (ρn − ρp )/(ρn + ρp )
Z
σγ−abs
∼ IEWSR)
→ The dipole polarizability (α ∼
2
ω=
1
m dx2
Energy
measures the tendency of the nuclear charge distribution
to be distorted, that is, from a macroscopic point of view
α∼
electric dipole moment
external electric field
16
The neutron skin and the Giant Dipole Resonance in 208 Pb
(Ex ≈ f(0.1) ∝
q
S(0.1fm−3 ) )
6.4
f(0.1)={S(0.1)(1+k}}
1/2
1/2
[MeV ]
6.2
6
5.8
5.6
5.4
5.2
5
4.8
4.6
11
12
13
14
15
16
E-1 [MeV]
Physical Review C 77, 061304 (2008)
The larger the neutron skin of 208 Pb, the faster the symmetry
energy increases with density around saturation
ρ − ρA
S(ρA ) ≈ J − L 0
, and the smaller the excitation energy of the
3ρ0
Giant Dipole Resonance (GDR).
17
The symmetry energy and the Pygmy Dipole Resonance
(Pygmy: low-energy excited state appearing in the dipole response of N , Z nuclei)
Physical Review C 81, 041301 (2010)
The larger the neutron skin in 208 Pb, the faster the symmetry
energy increases with density, the larger is the energy (E) times
the probability (P) of exciting the Pygmy state (EWSR = E × P)
WARNING: we lack of a clear understanding of the physical
reason for this correlation
18
−2
3
10 αD J (MeV fm )
Dipole polarizability and the neutron skin in 208 Pb
10
9
8
7
r=0.97
FSU
NL3
DD-ME
Skyrme
SV
SAMi
TF
Macroscopic model:
→ Using the dielectric theorem: m−1 moment can
be computed from the expectation value of the
Hamiltonian in the constrained (D dipole
operator) ground state H ′ = H + λD
→ Assuming the Droplet Model (heavy nucleus):
1 L
where
αD ≈ αbulk
1+
D
5 J
πe2 Ahr2 i
bulk
αD ≡
(Migdal first derived)
54
J
6
50.12
0.16
0.2 0.24
∆rnp (fm)
0.28
0.32
exp
→ L≈
αD − αbulk
D
αbulk
D
5J
Physical Review C 85 041302 (2012); 88 024316 (2013); 92, 064304 (2015)
By using the Droplet Model
" one can also find:
surf
πe2
5 ∆rnp −∆rcoul
np −∆rnp
αD J ≈
Ahr2 i 1+
54
2
hr2 i1/2 (I−IC )
#
For a fixed value of the symmetry energy at saturation, the
larger the neutron skin in 208 Pb, the larger the dipole
polarizability.
19
IV-IS GQRs and the neutron skin in 208 Pb
6
-1
4
5 h2 Vsym hr2 i
4 2m (hω0 )2 hr4 i
and EDF calculations, one can
deduce
Vsym ≈ 8(S(ρA ) − Skin (ρ0 ))
Skin (ρ0 ) ≈ εF0 /3 (Non-Rel)
S(ρA ) ≈ J − L
εF
ρ0 − ρA
≈ 0
3ρ0
3
4
3
1+
B(E2;IV) (10
EIV
x = 2hω0
fm MeV )
Within the Quantum Harmonic
Oscillator approach
s
2
0
0
10
20
E (MeV)
30
2 A2/3 IV 2
E
−
2
EIS
+1
x
x
8ε2F0
The larger the neutron skin in 208 Pb, the smallest the difference
between the IS and IV excitation energies in GQRs.
20
E1 transitions in CER and the neutron skin in 208 Pb
AGDR
-
-
-
IVSGDR
0 ,1 ,2 ,T0-1
+
1 ,T0-1
+
0 ,T0
10.0
∆L=1
GT
IAS
Exp. Data from Ref. [56]
Exp. Data from Ref. [53]
∆L=1,∆S=1
∆S=1
-
1 ,T0
strong
(p,n)
+
1 ,T0
E1
M1
TZ=T0-1
Daughter nucleus
+
EAGDR-EIAS(MeV)
-
1 ,T0-1
9.5
9.0
8.5
0 ,T0
TZ=T0
Target nucleus
8.0
0.12
0.16
0.20
Rn-Rp(fm)
0.24
0.28
Phys. Rev. C 92, 034308 (2015)
AGDR (∆Jπ = 1− with ∆L = 1 and ∆S = 0) is the T0 − 1
component of the
charge-exchange
of the GDR.
"
!
r
EAGDR − EIAS ≈ 5
hc
5 J 1+γ
3 I αH Z mhr2 i1/2
εF∞
3
I−
1−
3J
2
∆Rnp − ∆Rsurf
np
mAGDR
ε
0
(EIVGDR − ε)
EAGDR − EIAS ≈
∆EC
mIVGDR
0
hr2 i1/2
−
3
I
7 C
#
The larger the neutron skin in 208 Pb, the smaller the excitation
energy of the IVGDR (as we have seen) and consistently the
smaller the difference between the excitation energies of AGDR
- IAS
21
Relevance of the neutron star crust on the star evolution and
dynamics (brief motivation)
→ The crust separates neutron star interior from the
photosphere (X-ray radiation).
→ The thermal conductivity of the crust is relevant
for determining the relation between observed
X-ray flux and the temperature of the core.
→ Electrical resistivity of the crust might be
important for the evolution of neutron star
magnetic field.
→ Conductivity and resistivity depend on the
structure and composition of the crust
→ Neutrino emission from the crust may significantly contribute to total neutrino losses from stellar interior
(in some cooling stages).
→ A crystal lattice (solid crust) is needed for modelling pulsar glitches, enables the excitation of toroidal
modes of oscillations, can suffer elastic stresses...
→ Mergers (binary systems that merge) may enrich the interstellar medium with heavy elements, created by a
rapid neutron-capture process.
→ In accreting neutron stars, instabilities in the fusion light elements might be responsible for the phenomenon
of X-ray bursts
Source: Pawel Haensel 2001
22
The neutron skin in 208 Pb and the structure and composition
of a neutron star outer crust
→ span 7 orders of magnitude in denisty (from
ionization ∼ 104 g/cm to the neutron drip
∼ 1011 g/cm)
→ T ∼ 106 K ∼ 0.1 keV → one can treat nuclei and
electrons at T = 0 K
80
contribution becomes important, it is
energetically advantageous to lower its electron
fraction by e− + (N, Z) → (N + 1, Z − 1) + νe
and therefore Z ↓ with constant (approx) number
of N
→ As the density continues to increase, penalty
energy from the symmetry energy due to the
neutron excess changes the composition to a dif
ferent N−plateau
pF e
Z
Z
where (A0 , Z0 ) = 56 Fe26
≈ 0 −
A
A0
8asym
→ The Coulomb lattice is made of more and more
neutron-rich nuclei until the critical neutron-drip
density is reached (µdrip = mn ).
N=50
50
Sr
40
Kr
Sn
Se
Kr
Ni
20
80
Composition
→ As the density increases, the electronic
N=82
Protons
Neutrons
60
30
→ At the lowest densities, the electronic
contribution is negligible so the Coulomb lattice
is populated by 56 Fe nuclei.
FSUGold
70
Cd
Pd
Ru
Mo
Zr
Sr
neutron-rich nuclei (ions) embedded in a
relativistic uniform electron gas
Composition
→ it is organized into a Coulomb lattice of
90
NL3
N=82
70
60
N=50
50
Sr
40
N=32
30
Kr
Mo Zr
SrKr
Se
Ge Zn
Ni
Fe
20
10
-4
10
-3
10
-2
11
10
-1
0
10
10
1
3
ρ(10 g/cm )
Physical Review C 78, 025807 (2008)
The larger the neutron skin of
208
Pb (L ↑), the more exotic the
composition of the outer crust.
[M(N, Z) + mn < M(N + 1, Z)]
23
Some available constraints on J and L from terrestrial
experiments and astrofisical observations
24
CONCLUSIONS
25
Conclusions: EoS around saturation
→ The isovector channel of the nuclear effective interaction is
not well constrained by current experimental
information.
→ Many observables available in current laboratories are
sensitive to the symmetry energy. Problems: accuracy and
model dependent analysis. Systematic experiments may
help.
→ Exotic nuclei more sensitive to the isovector properties
(due to larger neutron excess). Problems: more difficult to
measure, accuracy and model dependent analysis.
Systematic experiments may help.
→ The most promissing observables to constraint the
symmetry energy are the neutron skin thickness and the
dipole polarizability in medium and heavy nuclei.
26
Thank you for your
attention!
27