Symmetry energy and pure neutron systems
Stefano Gandolfi
Los Alamos National Laboratory (LANL)
Neutron Skins of Nuclei
17-27 May 2016, Mainz Institute for Theoretical Physics
www.computingnuclei.org
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
1 / 31
Neutron stars
Neutron star is a wonderful natural laboratory
Atmosphere: atomic and
plasma physics
Crust: physics of superfluids
(neutrons, vortex), solid state
physics (nuclei)
Inner crust: deformed nuclei,
pasta phase
Outer core: nuclear matter
Inner core: hyperons? quark
matter? π or K condensates?
D. Page
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
2 / 31
Homogeneous neutron matter
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
3 / 31
Outline
The model and the method
Neutron drops: energies and radii, correlations radii-skin and radii-L
Equation of state of neutron matter, three-neutron force and
symmetry energy
Neutron star structure and symmetry energy
Conclusions
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
4 / 31
Nuclear Hamiltonian
Model: non-relativistic nucleons interacting with an effective
nucleon-nucleon force (NN) and three-nucleon interaction (TNI).
H=−
A
X
~2 X 2 X
∇i +
vij +
Vijk
2m
i=1
i<j
i<j<k
vij NN fitted on scattering data. Sum of operators:
X p=1,8
vij =
Oij
v p (rij ) , Oijp = (1, ~σi · ~σj , Sij , ~Lij · ~Sij ) × (1, ~τi · ~τj )
NN interaction - Argonne AV8’.
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
5 / 31
Phase shifts, AV8’
0
60
120
Argonne V8’
SAID
1
S0
1
Argonne V8’
SAID
3
S1
100
P1
Argonne V8’
SAID
40
20
0
δ (deg)
δ (deg)
δ (deg)
80
60
40
-20
20
-20
-40
0
0
100
200
300
400
Elab (MeV)
500
-20
0
600
100
200
300
400
Elab (MeV)
500
-40
0
600
0
P0
20
P1
-40
0
Argonne V8’
SAID
100
200
300
400
Elab (MeV)
500
-30
-50
0
600
3
D1
Argonne V8’
SAID
50
500
600
3
P2
10
0
100
200
300
400
Elab (MeV)
500
-10
0
600
100
200
Elab (MeV)
300
400
8
60
-10
300
400
Elab (MeV)
20
-20
-40
0
200
Argonne V8’
SAID
δ (deg)
δ (deg)
δ (deg)
-10
-20
Argonne V8’
SAID
3
0
100
30
3
Argonne V8’
SAID
3
ε1
D2
6
-20
δ (deg)
δ (deg)
δ (deg)
40
30
4
20
-30
Argonne V8’
SAID
2
10
-40
0
100
200
300
400
Elab (MeV)
500
600
0
0
100
200
300
400
Elab (MeV)
500
600
0
0
100
200
300
400
Elab (MeV)
500
600
Difference AV80 -AV18 less than 0.2 MeV per nucleon up to A=12.
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
6 / 31
Scattering data and neutron matter
Two neutrons have
k≈
p
Elab m/2 ,
→ kF
that correspond to
kF → ρ ≈ (Elab m/2)3/2 /2π 2 .
Elab =150 MeV corresponds to about 0.12 fm−3 .
Elab =350 MeV to 0.44 fm−3 .
Argonne potentials useful to study dense matter above ρ0 =0.16 fm−3
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
7 / 31
Three-body forces
Urbana–Illinois Vijk models processes like
π
π
π
π
π
∆
∆
π
π
+
π
π
π
∆
∆
short-range correlations (spin/isospin independent).
Urbana UIX: Fujita-Miyazawa plus short-range.
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
8 / 31
Quantum Monte Carlo
Projection in imaginary-time t:
H ψ(~r1 . . .~rN ) = E ψ(~r1 . . .~rN )
ψ(t) = e −(H−ET )t ψ(0)
Ground-state extracted in the limit of t → ∞.
Propagation performed by
Z
ψ(R, t) = hR|ψ(t)i =
dR 0 G (R, R 0 , t)ψ(R 0 , 0)
dR 0 → dR1 dR2 . . . , G (R, R 0 , t) → G (R1 , R2 , δt) G (R2 , R3 , δt) . . .
Importance sampling: G (R, R 0 , δt) → G (R, R 0 , δt) ΨI (R 0 )/ΨI (R)
Constrained-path approximation to control the sign problem.
Unconstrained calculation possible in several cases (exact).
Ground–state obtained in a non-perturbative way. Systematic
uncertainties within 1-2 %.
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
9 / 31
Light nuclei spectrum computed with GFMC
-20
-30
-40
0+
1+
2+
3+
1+
2+
0+
4He 6
He
7/2−
5/2−
5/2−
7/2−
1/2−
3/2−
6Li
2+
0+
8He
Energy (MeV)
7Li
-50
-60
-70
Argonne v18
with UIX or Illinois-7
GFMC Calculations
4+
0+
2+
2+
1+
3+
1+
2+
0+
1+
4+
3+
1+
2+
4+
8Li
2+
0+
8Be
7/2+
5/2+
7/2−
7/2−
3/2−
3/2+
5/2+
1/2−
5/2−
1/2+
3/2−
9Be
1 June 2011
4+
2+
3+
2+
3+
1+
1+
3+
4+
2+
3+
2+
1+
1+
3+
10B
-80
AV18
-90
AV18
+UIX
AV18
+IL7
Expt.
0+
12C
-100
Carlson, et al., RMP (2015)
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
10 / 31
Neutron matter and the ”puzzle” of the three-body force
20
AV8’+UIX
AV8’+IL7
AV8’
Energy per Neutron [MeV]
18
16
14
12
10
8
6
0.04
0.06
0.08
0.1
0.12
0.14
0.16
-3
ρ [fm ]
Maris, et al., PRC (2013)
Note: AV8’+UIX and (almost) AV8’ are stiff enough to support observed
neutron stars, but AV8’+IL7 too soft. → How to reconcile with nuclei???
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
11 / 31
Neutron matter and the ”puzzle” of the three-body force
Hagen, et al., Nature Physics (2016)
Similar trend: very low simmetry energies, soft EOS, probably leading too
small radii in neutron stars.
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
12 / 31
Neutron drops
What are, and why to study neutron drops?
They model inhomogeneous neutron matter. Ab-initio→DFT
Neutrons are confined by an external potential:
H=−
A
X
X
~2 X 2 X
Vijk +
Vext (ri )
∇i +
vij +
2m
i=1
i<j
i<j<k
i
Vext tuned to change boundary conditions and densities.
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
13 / 31
Neutron drops, harmonic oscillator well
External well: harmonic oscillator with ~ω=5, 10 MeV. Interaction:
AV8’+UIX
1.1
10 MeV
E / ωN
4/3
1
0.9
0.8
AFDMC
GFMC
SLY4
SKM*
SKP
BSK17
ETF ξ=0.5, 1
0.7
1.1
5 MeV
E / ωN
4/3
1
0.9
0.8
0.7
0
10
20
30
40
50
60
N
Gandolfi, Carlson, Pieper, PRL (2011)
Skyrme systematically overbind neutron drops.
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
14 / 31
Neutron drops
Ab-initio calculations meant as ”experimental data”. Interaction:
AV8’+UIX
AFDMC (AV8’+UIX)
HFB (UNEDF0)
HFB (UNEDF1)
8
h̄ω =10MeV
Etot/N
4/3
(MeV)
10
6
h̄ω =5MeV
4
0
10
20
30
40
50
60
N
M. Kortelainen, et al., PRC (2012).
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
15 / 31
Neutron drops
Comparison with other functionals. QMC: AV8’+UIX, AV8’, AV8’+IL7
(from Maris, et al., PRC 2013)
1.05
QMC
1.00
E/h̄ωN 4/3
0.95
NL3
PK1
DD-ME2
PKDD
PC-PK1
DD-PC1
h̄ω = 10 MeV
0.90
0.85
0.80
0.75
0.70
0.65
0
10
20
30
40
50
Neutron Number N
Zhao, Gandolfi, arXiv:1604.01490
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
16 / 31
Neutron drops - radii
Note: tune Vext or N to have central density about 0.16(2) fm−3 . For
example, this is given by putting 20 neutrons in an external trap with
~ω=10 MeV:
0.2
_
AFDMC, N=20, hω=10 MeV, AV8’+UIX
-3
ρ(r) (fm )
0.16
0.12
0.08
0.04
0
0
Rms radius:
√
1
2
3
r (fm)
4
5
6
< r 2 > = 3.14(1) fm
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
17 / 31
Correlation between neutron drops radii and the skin thickness of
and 48 Ca:
0.40
∆rnp (208 Pb) (fm)
0.05
NL2
1
-ME
DD
2
-ME
DD
SII 99
TW
0.10
*
SKM
6
SLY
5
SLy
SKP
k6
MS
SIIISk3
M
0.15
4
SLy
0.20
K1
PC-P
1
D
PC-F
PKD
0.25
r = 0.95
0.00
Pb
NL1
Linear fit
Relativistic DFTs
Nonrelativistic DFTs
1
-PC
DD SKMP
SV
0.30
3* 3
NLN
L
PK1
SKI5
H
NL-S
0.35
208
(a)
0.30
6
SLY
4
SLy P
SK
0.15
5
SLy
0.20
k6
MS I
SII3
k
MS
∆rnp (48 Ca) (fm)
Linear fit
Relativistic DFTs
Nonrelativistic DFTs
NL2
NL1
K1
PC-P
1
*
PC-F D
NL3L3
PKD
N
PK1 SKI5
H
NL-S
1
-ME
DD
2
-ME
DDII
1
S KMP
-PC
S
DD TW99
SV
*
SKM
0.25
0.10
0.05
0.00
2.7
r = 0.97
2.8
2.9
3.0
3.1
3.2
3.3
(b)
3.4
3.5
R(N = 20, h̄ω = 10 MeV) (fm)
Zhao, Gandolfi, arXiv:1604.01490
Green points: antiprotonic atoms, pion photoproduction, electric dipole
polarizability
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
18 / 31
Neutron drops
Radii: AFDMC calculations with various Hamiltonians, compared to what
might be extracted from experiments:
3.2
AV8′ +UIX
N2 LO(D2,E1)
AV8′
N2 LO(1.0)
N2 LO(1.2)
Radii (fm)
3.0
AV8′ +UIX
JISP16
2.8
N = 20
h̄ω = 10 MeV
2.6
2.4
2.2
208
Pb Antiprotonic atom
208
Pb π photoproduction
208
Pb E1 polarizability
48
N = 14
h̄ω = 10 MeV
N =8
h̄ω = 15 MeV
Ca E1 polarizability
2.0
Zhao, Gandolfi, arXiv:1604.01490
Note, N = 14 has central density of about 0.12 fm−3 , N=8 we don’t
know yet...
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
19 / 31
Neutron drops
Radius of 20 neutrons in a ~ω = 10 MeV harmonic trap as a function of
L (from the EOS)
3.5
Linear fit
DFT
QMC
3.4
Radii (fm)
3.3
3.2
3.1
3.0
2.9
r = 0.93
2.8
2.7
0
20
40
60
80
100
120
140
L (MeV)
Zhao, Gandolfi, arXiv:1604.01490
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
20 / 31
Neutron matter equation of state
Why to study neutron matter at nuclear densities?
EOS of neutron matter gives the symmetry energy and its slope.
Assume that NN is very good - fit scattering data with very high
precision.
Three-neutron force (T = 3/2) very weak in light nuclei, while
T = 1/2 is the dominant part. No direct T = 3/2 experiments.
Why to study symmetry energy?
Theory
Esym, L
Neutron stars
Stefano Gandolfi (LANL)
Experiments
Symmetry energy and pure neutron systems
21 / 31
What is the Symmetry energy?
symmetric nuclear matter
pure neutron matter
Symmetry energy
E0 = -16 MeV
Nuclear saturation
0
ρ0 = 0.16 fm
-3
Assumption from experiments:
ESNM (ρ0 ) = −16MeV ,
ρ0 = 0.16fm−3 ,
Esym = EPNM (ρ0 ) + 16
At ρ0 we access Esym by studying PNM.
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
22 / 31
Neutron matter
We consider different forms of three-neutron interaction by only requiring
a particular value of Esym at saturation.
Energy per Neutron (MeV)
120
100
different 3N
different 3N:
80
V2π + αVR
V2π + αVRµ
(several µ)
60
40
Esym = 33.7 MeV
V2π + αṼR
V3π + αVR
20
0
0
0.1
0.2
0.3
0.4
0.5
-3
Neutron Density (fm )
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
23 / 31
Neutron matter
Equation of state of neutron matter using Argonne forces:
Energy per Neutron (MeV)
100
80
Esym = 35.1 MeV (AV8’+UIX)
Esym = 33.7 MeV
Esym = 32 MeV
Esym = 30.5 MeV (AV8’)
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
-3
Neutron Density (fm )
Gandolfi, Carlson, Reddy, PRC (2012)
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
24 / 31
Neutron matter and symmetry energy
From the EOS, we can fit the symmetry energy around ρ0 using
Esym (ρ) = Esym +
L ρ − 0.16
+ ···
3 0.16
70
AV8’+UIX
65
L (MeV)
60
55
Esym=33.7 MeV
50
45
Esym=32.0 MeV
40
35
AV8’
30
30
31
32
33
34
35
36
Esym (MeV)
Gandolfi et al., EPJ (2014)
Tsang et al., PRC (2012)
Very weak dependence to the model of 3N force for a given Esym .
Knowing Esym or L useful to constrain 3N! (within this model...)
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
25 / 31
Neutron matter and neutron star structure
TOV equations:
G [m(r ) + 4πr 3 P/c 2 ][ + P/c 2 ]
dP
=−
,
dr
r [r − 2Gm(r )/c 2 ]
dm(r )
= 4πr 2 ,
dr
J. Lattimer
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
26 / 31
Neutron star structure
EOS used to solve the TOV equations.
3
2
2.5
sa
Cau
lity:
.9 (G
R>2
M/c
)
error associated with Esym
1.97(4) MO•
M (MO• )
2
ρ0
=3
al
ntr
ρ ce
32
1.5
1.4 MO•
ρ
=l 2 0
a
ρ centr
1
33.7
0.5
35.1
0
8
Esym= 30.5 MeV (NN)
9
10
11
12
13
14
15
16
R (km)
Gandolfi, Carlson, Reddy, PRC (2012).
Accurate measurement of Esym put a constraint to the radius of neutron
stars, OR observation of M and R would constrain Esym !
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
27 / 31
Neutron stars
2.5
2.5
0.0012
4U 1608-52
2.5
0.0016
EXO 1745-248
4U 1820-30
0.0016
0.0014
0.001
2
2
2
0.0014
0.0012
0.0012
0.0006
0.001
1.5
0.0008
1
1
M (M )
1.5
M (M )
M (M )
0.0008
1.5
0.001
0.0008
1
0.0006
0.0006
0.0004
0.0004
0.5
0.5
2
4
6
8
10
12
14
16
18
0.0002
0
0
0
2
4
6
8
R (km)
10
12
14
16
18
0
0
2.4
0.0006
1.4
1.8
0.5
1.8
1.6
0.4
1.4
0.3
1.4
1.2
1
0.0002
1
0.1
1
R (km)
18
0
0
0.25
0.2
16
18
1.6
1.2
14
16
0.4
2
0.0004
12
14
2.2
2
1.2
10
12
0.3
M (M )
0.0008
1.6
10
2.4
0.6
0.001
1.8
8
0.35
2
8
6
×10-3
0.45
0.7
ω Cen
2.2
0.0012
6
4
R (km)
×10-3
0.8
0.0014
M13
2.2
0.8
2
R (km)
2.4
M (M )
0.0002
0
M (M )
0
0
0.0004
0.5
0.0002
0.8
6
8
10
12
14
16
18
0
0.8
0.2
0.15
0.1
X7
6
8
R (km)
0.05
10
12
14
16
18
0
R (km)
Steiner, Lattimer, Brown, ApJ (2010)
Neutron star observations can be used to ’measure’ the EOS and
constrain Esym and L. (Systematic uncertainties still under debate...)
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
28 / 31
Neutron star matter
Neutron star matter model:
α
β
ρ
ρ
ENSM = a
+b
,
ρ0
ρ0
ρ < ρt
(form suggested by QMC simulations),
and a high density model for ρ > ρt
i) two polytropes
ii) polytrope+quark matter model
Neutron star radius sensitive to the EOS at nuclear densities!
Direct way to extract Esym and L from neutron stars observations:
Esym = a + b + 16 ,
Stefano Gandolfi (LANL)
L = 3(aα + bβ)
Symmetry energy and pure neutron systems
29 / 31
Neutron star matter really matters!
2.5
Quarks, no corr. unc.
Esym= 33.7 MeV
8
Fiducial
M (Msolar)
Probability distribution
No corr. unc.
6
Esym=32 MeV
2
Quarks
Polytropes
Quark matter
1.5
1
Pb (p,p)
4
0.5
Masses
0
8
HIC
2
9
12
13
14
15
32 < Esym < 34 MeV , 43 < L < 52 MeV
IAS
40
11
R (km)
PDR
20
10
60
80 100
L (MeV)
Steiner, Gandolfi, PRL (2012).
Uncertainties on observations not included!
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
30 / 31
Summary
Neutron drops provide an “artificial” useful system to calibrate
density functionals
Radii of neutron drops correlated to the skin thickness of nuclei and
to L
Three-neutron force is the bridge between Esym and neutron star
structure.
Neutron star observations becoming competitive with experiments.
For the future: use more dense neutron drops to predict L at different
densities?
Acknowledgments:
Joe Carlson (LANL)
Pengwei Zhao, Steve Pieper (ANL)
Sanjay Reddy (INT)
Andrew Steiner (UT/ORNL)
Stefano Gandolfi (LANL)
Symmetry energy and pure neutron systems
31 / 31