Nuclear Equation of State (ˆ · Introduction Symmetry Energy Bayesian Inference Topology at Low ˆ...

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions Nuclear Equation of State (ρ ρ 0 ) from Reactions Pawel Danielewicz Michigan State U International Workshop on Gross Properties of Nuclei and Nuclear Excitations, Nuclear Equation of State and Neutron Stars Hirschegg, Kleinwalsertal, Austria, January 12 - 18, 2020 EOS from Reactions Danielewicz

Transcript of Nuclear Equation of State (ˆ · Introduction Symmetry Energy Bayesian Inference Topology at Low ˆ...

Page 1: Nuclear Equation of State (ˆ · Introduction Symmetry Energy Bayesian Inference Topology at Low ˆ Conclusions Nuclear Equation of State (ˆ ˆ 0) from Reactions Pawel Danielewicz

Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Nuclear Equation of State (ρ ≤ ρ0)from Reactions

Pawel Danielewicz

Michigan State U

International Workshop onGross Properties of Nuclei and Nuclear Excitations,

Nuclear Equation of State and Neutron Stars

Hirschegg, Kleinwalsertal, Austria, January 12 - 18, 2020

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Energy in Uniform MatterEA

(ρn, ρp) =E0

A(ρ) + S(ρ)

(ρn − ρp

ρ

)2+O(. . .4)

symmetric matter (a)symmetry energy ρ = ρn + ρp

E0

A(ρ) = −aV +

K18

(ρ− ρ0

ρ0

)2+ . . . S(ρ) = aV

a +L3ρ− ρ0

ρ0+ . . .

Known: aa ≈ 16 MeV K ∼ 235 MeV Unknown: aVa ? L ?

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Symmetry-Energy Stiffness: M & R of n-StarEA

=E0

A(ρ) + S(ρ)

(ρn − ρp

ρ

)2

S ' aVa +

L3ρ− ρ0

ρ0

In neutron matter:ρp ≈ 0 & ρn ≈ ρ.

Then, EA (ρ) ≈ E0

A (ρ) + S(ρ)

Pressure:

P = ρ2 ddρ

EA' ρ2 dS

dρ' L

3ρ0ρ2

Stiffer symmetry energy correlates withlarger max mass of neutron star & larger radii

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Low-ρ Matter Phases: Topological Structures

Caplan&Horowitz RMP89(17)041002

Low-ρ uniform matter unstableFig from PD PRC51(95)716

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

n↔p Symmetry Invariants: Energy & DensitiesEA

(ρn, ρp) =E0

A(ρ) + S(ρ)

(ρn − ρp

ρ

)2+O(. . .4)

symmetric matter (a)symmetry energy ρ = ρn + ρp

S(ρ) = S(ρ0) +L3ρ− ρ0

ρ0+ . . .

Unknown: S(ρ0)? L ?

Net ρ = ρn + ρp isoscalar

Difference ρn − ρp isovector

ρa = AN−Z (ρn − ρp) isoscalar

ρn,p(r) =12

[ρ(r)± N − Z

Aρa(r)

]ρ & ρa universal in isobaric chain!

Energy min in Thomas-Fermi:

ρa(r) ∝ ρ(r)

S(ρ(r))

low S ⇔ high ρa

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Relation between ρ, ρa & S(ρ)Results f/different Skyrmeints in half-∞ matterPD&Lee NPA818(09)36Isoscalar (ρ=ρn+ρp; blue) &isovector (ρa∝ρn-ρp; green)densities displacedrelative to each other.

As S(ρ) changes, ρa(r) ∝ ρ(r)S(ρ(r)) ,

so does displacement or aura

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Probing Independently 2 Densities

Jefferson LabDirect: ∼ pInterference: ∼ n

PD, Singh, Lee NPA958(17)147[after Dao Tien Khoa]

elastic: ∼ p + ncharge exchange: ∼ n − p

p

p

σ

ρn+ρp

p

n

π+

ρn−ρp

IAS

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Expectations on Isovector Aura?

ΔR

Much Larger Than Neutron Skin!

Surface radius R '√

53 〈r

2〉1/2

rms neutron skin

〈r 2〉1/2ρn − 〈r

2〉1/2ρp

' 2N − Z

A

[〈r 2〉1/2

ρn−ρp− 〈r 2〉1/2

ρn+ρp

]rms isovector aura

Estimated ∆R ∼ 3(〈r2〉1/2

ρn − 〈r2〉1/2ρp

)for 48Ca/208Pb!

Even before consideration of Coulomb effects that furtherenhance difference!

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Direct Reaction Primer

DWBA:

dσdΩ∝∣∣∣ ∫ dr Ψ∗

f U1 Ψi

∣∣∣2

Lane Potential U = U0 +4τττTTT

AU1

U0 ∝ ρ U1 ∝ ρn − ρp

It is common to assume the same geometry for U0 & U1,implicitly ρ & ρa, e.g. Koning&Delaroche NPA713(03)231

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Cross-Section Sensitivity to Isovector SurfaceKoning-DelarocheNPA713(03)231same radii R for U0 & U1!

U1(r) ∝ U01

1 + exp r−Ra

R → R + ∆R1

charge-exchange csoscillations grow

Elastic

Charge Exchange

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Simultaneous Fits to Elastic & Charge-Change: 48CaDifferent radii for densities/potentials: Ra = R + ∆R

PD/Singh/Lee NPA958(17)147EOS from Reactions Danielewicz

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Simultaneous Fits to Elastic & Charge-Change: 92ZrDifferent radii for densities/potentials: Ra = R + ∆R

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Thickness of Isovector Aura6 targets analyzed, differential cross section + analyzing power

Colored: Skyrme predictions. Arrows: half-infinite matterThick ∼ 0.9 fm isovector aura! ∼Independent of A. . .

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Difference in Surface Diffuseness

Colored: Skyrme predictions. Arrows: half-infinite matterSharper isovector surface than isoscalar!

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Bayesian InferenceProbability density in parameter space p(x) updated asexperimental data on observables E , value E with error σE , getincorporated

Probability p is updated iteratively, starting with prior ppriorp(a|b) - conditional probability

p(x |E) ∝ pprior(x)

∫dE e

− (E−E)2

2σ2E p(E |x)

For large number of incorporated data, p becomes independentof pprior

In here, pprior and p(E |x) are constructed from all Skyrme intsin literature, and their linear interpolations. pprior is madeuniform in plane of symmetry-energy parameters (L,S0)

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Likelihood f/Symmetry-Energy Slope

E∗IAS - from excitations to isobaric analog states

in PD&Lee NPA922(14)1

Oscillations in prior of no significance- represent availability of Skyrme parametrizations

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Likelihood f/Neutron-Skin Values

Sizeable n-Skins

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Likelihood f/Energy of Neutron Matter

E∗IAS - from excitations to isobaric analog states

in PD&Lee NPA922(14)1

Some oscillations due to priorEOS from Reactions Danielewicz

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Modest-E Collisions Probe Low-ρ EOSBoltzmann eq simulations of central Au + Au collisions: ρ-contours

PD et al arXiv:1910.10500 different energies & centralities

Late equilibration in expanding low-ρ matter that cannot stay uniform⇒ Ring formation: Threshold phenomenon at E/A & 60 MeV/u

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Page 20: Nuclear Equation of State (ˆ · Introduction Symmetry Energy Bayesian Inference Topology at Low ˆ Conclusions Nuclear Equation of State (ˆ ˆ 0) from Reactions Pawel Danielewicz

Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Modest-E Collisions Probe Low-ρ EOSBoltzmann eq simulations of central Au + Au collisions: ρ-contours

PD et al arXiv:1910.10500 different energies & centralities

Late equilibration in expanding low-ρ matter that cannot stay uniform⇒ Ring formation: Threshold phenomenon at E/A & 60 MeV/u

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Boltzmann vs Fluctuating Dynamics(a) - Boltzmann, (b)-(d) - Brownian Motion model - extra ρ fluctuations

10

0

-10

x (

fm)

0-1010

-20

20z (fm)0

10-10

y (fm)

10

0

-10

x (

fm)

0-1010

-20

20z (fm)0

10-10

y (fm)

(a) (b)

10

0

-10

x (f

m)

20100-10-20

z (fm)

(c)10

0

-10

x (f

m)

100-10

y (fm)

(d)

z>0

(c) & (d) - average over events, similar to Boltzmann!

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Experimental Detection: Flow

Expansion in the finalstate maps structuresfrom configurationonto velocity space:

(a)2nd & (b)1st ordermoments vs rapidity &

(c) rapidity distribution

(a)

(b)

(c)

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Isovector Aura

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

Rings at Onset of Collective Expansion

10

0

-10

x (fm

)

0-1010

-20

20z (fm)0 10

-10 y (fm)

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Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

ConclusionsSymmetry-energy polarizes densities, pushing isovector densityout to low isoscalar densityFor large A, isovector-isoscalar surface displacement expectedroughly independent of nucleus and dependent on LSurface displacement studied in comparative analysis of data onelastic scattering and quasielastic charge-exchange reactionsAnalysis produces thick isovector aura ∆R ∼ 0.9 fm!

Symmetry & neutron energies are stiff!L = (70 – 100) MeV, S(ρ0) = (33.5 – 36.5) MeV at 68% level

Matter diving into low-ρ region undergoes detectable spinodaldecomposition into rings w/stones

PD, Lee & Singh NPA818(09)36, 922(14)1, 958(17)147 + in progressPD, Lin, Stone & Iwata arXiv:1910.10500DOE DE-SC0019209

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Page 26: Nuclear Equation of State (ˆ · Introduction Symmetry Energy Bayesian Inference Topology at Low ˆ Conclusions Nuclear Equation of State (ˆ ˆ 0) from Reactions Pawel Danielewicz

Introduction Symmetry Energy Bayesian Inference Topology at Low ρ Conclusions

ConclusionsSymmetry-energy polarizes densities, pushing isovector densityout to low isoscalar densityFor large A, isovector-isoscalar surface displacement expectedroughly independent of nucleus and dependent on LSurface displacement studied in comparative analysis of data onelastic scattering and quasielastic charge-exchange reactionsAnalysis produces thick isovector aura ∆R ∼ 0.9 fm!

Symmetry & neutron energies are stiff!L = (70 – 100) MeV, S(ρ0) = (33.5 – 36.5) MeV at 68% level

Matter diving into low-ρ region undergoes detectable spinodaldecomposition into rings w/stones

PD, Lee & Singh NPA818(09)36, 922(14)1, 958(17)147 + in progressPD, Lin, Stone & Iwata arXiv:1910.10500DOE DE-SC0019209

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