Nuclear radii in density functional theory Witold Nazarewicz - - PowerPoint PPT Presentation

Nuclear radii in density functional theory

Witold Nazarewicz (FRIB/MSU) Neutron Skins in Nuclei MITP, Mainz,, May 17-27, 2016

• Perspective
• Theoretical strategies
• Nuclear DFT
• Radii, skins, and halos
• Uncertainty quantification and

correlation analysis

• Proton-, neutron radii, skins, and

nuclear matter properties

• Conclusions

How to explain the nuclear landscape from the bottom up? Theory roadmap

• J. Phys. G 43, 044002 (2016)

http://iopscience.iop.org/article/10.1088/0954-3899/43/4/044002

Hagen’s talk

Mean-Field Theory ⇒ Density Functional Theory

• mean-field ⇒ one-body densities
• zero-range ⇒ local densities
• finite-range ⇒ gradient terms
• particle-hole and pairing

channels

• Has been extremely successful.

A broken-symmetry generalized product state does surprisingly good job for nuclei.

Nuclear DFT

• two fermi liquids
• self-bound
• superfluid

Degrees of freedom: nucleonic densities

• Constrained by microscopic theory: ab-initio functionals provide quasi-data!
• Not all terms are equally important. Usually ~12 terms considered
• Some terms probe specific experimental data
• Pairing functional poorly determined. Usually 1-2 terms active.
• Becomes very simple in limiting cases (e.g., unitary limit)
• Can be extended into multi-reference DFT (GCM) and projected DFT

Nuclear Energy Density Functional

p-h density p-p density (pairing functional)

isoscalar (T=0) density ρ0 = ρn + ρp

( )

isovector (T=1) density ρ1 = ρn − ρp

( )

+isoscalar and isovector densities:

spin, current, spin-current tensor, kinetic, and kinetic-spin

+ pairing densities

E =

• H(r)d3r

Expansion in densities and their derivatives

NN+NNN ¡ interac+ons ¡ Density ¡Matrix ¡ Expansion ¡ Input ¡ Energy ¡Density ¡ Func+onal ¡ Observables ¡

• Direct ¡comparison ¡with ¡

experiment ¡

• Pseudo-­‑data ¡for ¡reac+ons ¡

and ¡astrophysics ¡ Density ¡dependent ¡ interac+ons ¡ Fit-­‑observables ¡

• experiment ¡
• pseudo ¡data ¡

Op+miza+on ¡ DFT ¡varia+onal ¡principle ¡ HF, ¡HFB ¡(self-­‑consistency) ¡ Symmetry ¡breaking ¡ Symmetry ¡restora+on ¡ Mul+-­‑reference ¡DFT ¡(GCM) ¡ Time ¡dependent ¡DFT ¡(TDHFB) ¡

Nuclear Density Functional Theory and Extensions

• two fermi liquids
• self-bound
• superfluid (ph and pp channels)
• self-consistent mean-fields
• broken-symmetry generalized product states

Technology to calculate observables

Global properties Spectroscopy

DFT Solvers Functional form Functional optimization Estimation of theoretical errors

0.0 0.4 0.8 1.2 1.6

momentum q (fm

−1)

0.0 0.1 0.2 q|F(q)|

N= 70 N=100 N=120

2 4 6 8 10 12

radius r (fm)

0.00 0.04 0.08 neutron density (fm

−3)

RHB/NL3 Sn

Neutron skins and halos in nuclear DFT

• S. Mizutori et al., Phys. Rev. C 61, 044326 (2000)

first ¡zero ¡of ¡F(q) ¡

Horowitz’s talk

10-12 10-10 10-8 10-6 10-4 10-2 5 10 15 20 25

Radius (fm)

0.00 0.02 0.04 0.06 0.08 0.10 2 4 6 8

Density (fm-3)

Radius (fm)

(n) ¡ (p) ¡

Skin ¡ Diffuseness ¡

150Sn ¡

(p) ¡ (n) ¡

Halo ¡

Neutron ¡& ¡proton ¡density ¡distribu+ons ¡

60 80 100 120

6 7 8 6 7 8

Sn

Neutron Number Radius (fm) Radius (fm)

HFB/SkP HFB/SLy4

Rrms (n) Rrms (p) Rrms (n)

H

Rskin Rhalo

0.70 – 0.85 0.55 – 0.70 0.40 – 0.55 0.25 – 0.40 0.10 – 0.25

• 0.05 – 0.10
• 0.20 – -0.05

< -0.20

Neutron Number N Proton Number Z

50 100 50 100 200 150

184 126 82 28 50 82 28 50

HFB/SLy4

neutron skin

• S. Mizutori et al., Phys. Rev. C61, 044326 (2000)

> 0.8 0.7 – 0.8 0.6 – 0.7 0.5 – 0.6 0.4 – 0.5 0.3 – 0.4 0.2 – 0.3 0.1 – 0.2 < 0.1

Neutron Number N Proton Number Z

50 100 50 100 200 150

184 126 82 28 50 82 28 50

HFB/SLy4

neutron halo proton halo

N Z

The limits: Skyrme-DFT Benchmark 2012

40 80 120 160 200 240 280

neutron number

40 80 120

proton number

two-proton drip line t w

• n

e u t r

• n

d r i p l i n e

232 240 248 256

neutron number proton number

90 110 100 Z=50 Z=82 Z=20 N=50 N=82 N=126 N=20 N=184

drip line SV-min known nuclei stable nuclei

N=28 Z=28 230 244 N=258

Nuclear Landscape 2012

S2n = 2 MeV

288 ~3,000

Erler et al, Nature 486, 509 (2012)

40 80 120 40 80 120 160 200 240 280 Proton number Z Neutron number N 28 Z=50 Z=82 N=20 28 N=50 N=82 N=126 N=184 N=258 Two-proton drip line Two-neutron drip line SDFT CDFT CDFT/SDFT overlap mic+mac GDFT

A.V. Afanasjev, et al. Phys. Lett. B (2013)

http://massexplorer.frib.msu.edu

Skyrme-DFT Skin Benchmark 2012

Erler et al, Nature 486, 509 (2012)

UNEDF0

40 80 120 160 200 240 280

neutron number

40 80 120

proton number

UNEDF1

40 80 120

proton number

SLy4 SV-min

40 80 120 160 200 240 280

neutron number

SkM*

40 80 120

proton number

SkP

Rn

• p
• 0.4 -0.2

0.2 0.4 rms

R

rms

N=258 N=184 Z=50 Z=28 Z=20 Z=82 N=82 N=50 N=28 N=20 N=126

ISNET: ¡Enhancing ¡the ¡interac+on ¡between ¡nuclear ¡experiment ¡and ¡ theory ¡through ¡informa+on ¡and ¡sta+s+cs ¡

JPG ¡Focus ¡Issue: ¡hSp://iopscience.iop.org/0954-­‑3899/page/ISNET

¡

¡

EXTRAPOLATIONS! ¡

¡

“Remember ¡that ¡all ¡models ¡are ¡ wrong; ¡the ¡prac+cal ¡ques+on ¡is ¡ how ¡wrong ¡do ¡they ¡have ¡to ¡be ¡ to ¡not ¡be ¡useful” ¡ ¡(E.P. ¡Box) ¡

Consider ¡a ¡model ¡described ¡by ¡coupling ¡constants ¡ Any ¡predicted ¡expecta+on ¡value ¡of ¡an ¡observable ¡is ¡a ¡func+on ¡of ¡ these ¡parameters. ¡Since ¡the ¡number ¡of ¡parameters ¡is ¡much ¡smaller ¡ than ¡the ¡number ¡of ¡observables, ¡ ¡there ¡must ¡exist ¡correla8ons ¡ between ¡computed ¡quan++es. ¡Moreover, ¡since ¡the ¡model ¡space ¡has ¡ been ¡op+mized ¡to ¡a ¡limited ¡set ¡of ¡observables, ¡there ¡may ¡also ¡exist ¡ correla+ons ¡between ¡model ¡parameters. ¡ ¡ p = (p1, ..., pF )

fit-­‑observables ¡ (may ¡include ¡pseudo-­‑data) ¡

χ2(p) = X

O

✓O(th)(p) − O(exp) ∆O ◆2

Objec+ve ¡ func+on ¡

Expected ¡uncertain+es ¡

Statistical methods of linear-regression and error analysis ¡

cAB = ∆A ∆B

• ∆A2 ∆B2

Product-moment correlation coefficient between two observables/variables A and B: =1: full alignment/correlation =0: not aligned/statistically independent ∆A2 = ⇥

ij

∂piA( ˆ M −1)ij∂pjA, ∂piA = ∂piA

• p0

Sta8s8cal ¡uncertainty ¡in ¡variable ¡A: ¡ Correla+on ¡between ¡variables ¡A ¡and ¡B: ¡ ∆A ∆B =

• ij

∂piA( ˆ M −1)ij∂pjB covariance matrix

How to estimate systematic (model) error?

• Take a set of reasonable models Mi
• Make a prediction O(Mi)
• Compute average and variation within this set
• Compute rms deviation from existing experimental
• data. If the number of fit-observables is huge,

statistical error is small.

A = ¯ A ± (∆A)stat ± (∆A)syst

• 1

1 2 3 4 5 6

σ2 (10-3 fm2)

200 204 208 212 216 220 224

A

remaining

left bars: UNEDF0 right bars: SV-min

asym L C

ρΔρ

rskin

1

• 0.2

0.2 0.4 0.6 0.8 16 20 24 28 32 36 40 44 48 52

neutron number

Ca

0.04 0.08 0.12

SV-min UNEDF0

40 48 56 64 72 80 88 96

Zr

84 96 108 120 132 144 156

Er

176 192 208 224 240 256 272 288

Z=120

(a) (d) (b) (c) (e) (f) (h) (g)

Δrskin (fm)

stat

rskin (fm)

Neutron-skin uncertainties of Skyrme EDF

• M. Kortelainen et al., Phys. Rev. C 88, 031305 (2013)

Bayesian inference methods

Uncertainty Quantification for Nuclear Density Functional Theory and Information Content of New Measurements

• J. McDonnell et al. Phys. Rev. Lett 114,

122501 (2015).

The frontier: calcium isotopes

Nuclear Forces from χEFT

derived 2000 derived 2002 derived 2003 derived 2011 N2LO N3LO NLO LO NN NNN

+... +... +... +...

• ptimized simultaneously 2014

Quantified input 40

proton number

20 20

neutron number

NNLOsat

postdiction prediction

Consistency with known data Prediction

• G. Hagen et al., Nature Physics 12, 186 (2016)

0.15 0.18 0.21

Rskin (fmD

3.2 3.3 3.4 3.5

Rp (fmD a

3.4 3.5 3.6

Rn (fmD b

2.0 2.4 2.8

αD (fmn D c

• G. Hagen et al., Nature Physics 12, 186 (2016)

Nuclear charge and neutron radii and nuclear matter: trend analysis in Skyrme-DFT approach

P.-G. Reinhard and WN, PRC 93, 051303 (R) (2016)

(E, R) (E)

Nuclear charge and neutron radii and nuclear matter: trend analysis in Skyrme-DFT approach

P.-G. Reinhard and WN, PRC (R) (2016)

E/A K m*/m EGDR J L rch σch 0.2 0.4 0.6 0.8 1.0

SV-min SV-E

rn rskin EGMR κTRK ρ0 αD E/A K m*/m EGDR J L rch σch rn rskin EGMR κTRK ρ0 αD EGQR EGQR

6.16 6.20 6.24 6.28 6.20 6.24 6.28 6.32 6.36

SV-min

3.46 3.54 3.54 3.58 3.62

ρ0 K L J κTRK m*/m

48Ca 298Fl

rn (fm) (a) (b) rn (fm) rch (fm) rch (fm)

S V

• m

i n

• 0.02

0.02 0.152 0.154 0.156 0.158

rrms−rrms,0 (fm)

• 0.04
• 0.02

0.02 0.04

• 0.04
• 0.02

0.02 0.04

• 10
• 5

5 10 15 20

L (MeV)

208Pb 48Ca 298Fl

protons neutrons

ρ0 (fm-3) (a) (e) (d) (c) (b) (f)

2000 samples

0.14 0.15 0.16 27 28 0.10 0.12 0.08 0.10 0.12

• 10

10 20

L (MeV) J (MeV) rskin (fm)

208Pb 48Ca 298Fl

(a) (e) (d) (c) (b) (f)

20 40 60 80 100

SV-E f i x ρ0 fix L fix ρ0+L

20 40 60 80 20 40 60 80 20 40 60 20 40 60

208Pb 48Ca 298Fl

Δrrms,n (10-3 fm) Δrrms,ch (10-3 fm)

100

Δrskin (10-3 fm) Δσn(10-3 fm) Δσch(10-3 fm) fix ρ0+J fix J f i x ρ0 fix L fix ρ0+L fix ρ0+J fix J (a) (e) (d) (c) (b) SV-E

Coupled Cluster informing DFT and DFT informing Coupled Cluster

• We explored various trends of charge and neutron radii with nuclear

matter properties.

• There exist, at least within the Skyrme-DFT theory, only two strong

correlations:

• one-to-one relation between charge radii in finite nuclei and ρ0:

Rch↔︎ ρ0

• one-to-one relation between neutron skins in finite nuclei and L:

Rskin↔︎ L

• By including charge radii in a set of fit-observables, as done for the

majority of realistic Skyrme EDFs, one practically fixes the saturation density.

• The relation Rn↔︎ ρ0 is much weaker than that for Rch, so by

constraining the saturation density alone does not help significantly reducing the uncertainty on neutron (and mass) radii.

• The Rn↔︎Rp relation is fairly complex: various trends are possible when

moving along a trajectory in a parameter space.

N2LOsat ¡describes ¡low-­‑energy ¡NN ¡and ¡Nuclei ¡

• A. ¡Ekström ¡et ¡al. ¡Phys. ¡Rev. ¡C ¡91, ¡051301(R) ¡(2015) ¡
• Order-­‑by-­‑order ¡op+miza+on ¡(here: ¡NN ¡and ¡NNN ¡in ¡N2LO) ¡
• Constrained ¡by ¡data ¡on ¡few-­‑body ¡systems ¡and ¡light ¡nuclei ¡ ¡
• Focus ¡on ¡low-­‑momentum ¡physics ¡

Instead of conclusions…

WN, J. Phys. G 43, 044002 (2016) http://iopscience.iop.org/article/10.1088/0954-3899/43/4/044002

Observation Experiment Theory Prediction

BACKUP

"It is exceedingly difficult to make predictions, particularly about the future” (Niels Bohr)

Outlook: Looking into the crystal ball 10 nuclear structure theory greatest hits for the next 10–15 years

• J. Phys. G 43, 044002 (2016) http://iopscience.iop.org/article/10.1088/0954-3899/43/4/044002
• 1. We will describe the lightest nuclei in terms of lattice QCD and understand the QCD origin of

nuclear forces.

• 2. We will develop a predictive framework for light, medium-mass nuclei, and nuclear matter from

0.1 to twice the saturation density. Ab initio methods will reach heavy nuclei in the next decade.

• 3. We will develop predictive and quantified nuclear energy density functional rooted in ab initio
• theory. This spectroscopic-quality functional will properly extrapolate in mass, isospin, and

angular momentum to provide predictions in the regions where data are not available.

• 4. We will provide the microscopic underpinning of collective models that explain dynamical

symmetries and simple patterns seen in nuclei. In this way, we will link fundamental and emergent aspects of nuclear structure.

• 5. By developing many-body approaches to light-ion reactions and large-amplitude collective

motion, we will have at our disposal predictive models of fusion and fission.

• 6. By exploring quantum many-body approaches to open systems, we will understand the

mechanism of clustering and explain properties of key cluster states and cluster decays.

• 7. By taking advantage of realistic many-body theory, we will unify the fields of nuclear structure

and reactions.

• 8. We will achieve a comprehensive description, based on realistic structural input, of nuclear

reactions with complex projectiles and targets, involving direct, semi-direct, pre-equilibrium, and compound processes.

• 9. We will carry out predictive and quantified calculations of nuclear matrix elements for

fundamental symmetry tests in nuclei and for neutrino physics.

• 10. By taking the full advantage of extreme-scale computers, we will master the tools of uncertainty
• quantification. This will be essential for enhancing the coupling between theory and experiment

—to provide predictions that can be trusted.