CERN ISOLDE, August 2009

NUCLEAR MASS MODELS

Jirina Rikovska Stone

Oxford University, University of Tennessee

Accurate prediction of the ground-state nuclear binding energies is one of the most challenging tasks of low-energy nuclear structure theory.

- Fundamental importance to nuclear physics- Basic importance to astrophysics -

atomic masses, beta-decay transition rates etc

I. Why is it a problem?

II. What we have done so far?

III. Can we do better?

The nucleus

It exists and is

made of nucleons(most of the time)

1 2g

m mV G

rGravity

Coulomb(elmg)

attractive

1 2el

q qV C

r attractive

repulsive

Color(strong)

attractive

Van der Waals

attractiverepulsive

attractive

BondingattractiveCovalent

IonicMetallic

??

4

3s

sV krr

?????????

156 mesons, 111 baryons and

counting…

Easy…..

It is not possible to do the same in nuclei:

Many-fermion systems cannot be solved exactly at present

Two reasons:

We do not know the analytic form of the nucleon-nucleon interaction

Even if we did

Model space of A nucleons would be too large even for supercomputers

TRUNCATION OF THE PROBLEM NEEDED

Strong

Weak

Elmg

Almost everything you can think of is present in atomic nuclei…..

Repulsive nuclearforce

What do we know about atomic nuclei:

I. They behave like a structure-less drops of incompressible matter

They behave like a system of well defined clusters

They behave like a system of fermions moving collectively

4He 14C 24,26Ne 28,30Mg 32,34Si

rotation vibration

En

erg

y

En

erg

y

They generate energy through fission and fusion

They behave like a system of

correlated fermions(shell model)

They behave like a system of

independent fermions in a mean field

generated by the rest

Single-particlestates (levels)

Models of the nucleon- nucleon interaction

Free nucleon scattering

Meson exchange between point-like nucleonsMeson exchange between quarks in each nucleonsQuark model used for both mesons and nucleons

5 fm

BUT!!!!!

Free N-N interaction is significantly modified in nuclear environment (finite nuclei) in an unknown way:

Forget free N-N interactionConstruct an energy functionalwith about 10-15 parametersFitto properties of doubly-closednuclei and symmetric nuclear matter

Renormalize free N-N interaction(with 40-60 parameters)Undergo a complicated mathematicalprocessAdd more phenomenological termsFit to properties of symmetric nuclear matter

Over 60 years of trial and error in modelling of finite nuclei

Experimental masses have to be extrapolated several massunits towards the expected r-process path:

Mass formulae – usually based on the assumption that nuclei are drops of in incompressible liquid without internal structure

Mean field models - nuclei are made of individual nucleons; each nucleon moves in a mean field generated by all nucleons

(two extreme models and many others in between)

HFB

ETFSI-I

FRDM

Duflo-Zuker

U.Hager et al., PRC75, 064302 (2007)

Which one to choose?

• The liquid drop model ( by Niels Bohr)

• Treats the nucleus as a drop of incompressible fluid made of nucleon and held together by the strong nuclear force. The nucleons interact strongly with each other, just like molecules in a drop of liquid.

• Does not explain all the properties of nuclei, but does give qualitative notion how a nucleus can deform and undergo fission.

• This model was first introduced to explain the binding energy and the mass of nuclei. It also gives a physical picture of the fission processes.

ASSUMPTIONS

• This model accepts the nucleus as a sphere.• The volume of a nucleus is proportional to A .• The mass density is constant inside the nuclei,

however, it decreases rapidly zero on the surface• The binding energy per nucleon is approximately

constant (the saturation of nuclear forces)• The nuclear force is identical for every nucleon, and

is charge independent.

Binding Energy

• Volume Energy Term

• Surface Energy Term– Nucleons on the surface

are less tightly bound

3/10ArR

Ar )3/4(R)3/4(volume 30

3 ππ AaB VV 2/32

02 Ar 4πR 4π

3/2AaB SS

Coulomb Energy Term• For a liquid drop of charge Ze, binding energy Bc is:

1/3

2

CC A

Z-aB

A

NZaB

2

Symsym

Symmetry Term

• Pairing Energy Term

-3/4Pp A a B λ

nuclei odd-oddfor 1

nuclei odd-even andeven -oddfor 0

nucleieven -evenfor 1

λwhere

Semi-empirical mass formula

• Fit to experimental data to obtain:

• The binding energy per particle becomes MeVa

MeVa

MeV.a

MeVa

MeVa

P

sym

C

S

V

00.312

28.23

700

23.17

56.15

7/4-P2

2

Sym4/3-2

C3/1

SV A a A

NZaAZaAaa

A

)Z,A(Bλ

INFINITE (SYMMETRIC) NUCLEAR MATTER

7/4-P2

2

Sym4/3-2

C3/1

SV A a A

NZaAZaAaa

A

)Z,A(Bλ

Binding energy per particle: B(A,Z)

A=aV=-15.56 MeV

Average density:

ρ∞ =A

V= (

4

3π r0

3)−1 =~ 0.16 fm−3

FINITE RANGE DROPLET MODEL (FRDM) (Moller et al, ADNDT 59, 185 (1995); 66, 131 (1997)

Deformed shapes (quadrupole, octupole, hexadecapole, hexacontatetrapole)

Shell corrections (Strutinsky method) Calculated microscopically using single-particle energies generatedPairing correlations (Lipkin-Nogami) by folded Yukawa potential

Experimental mass minus spherical FRDM

Calculated mass minus spherical FRDM

Experimental minus calculated mass

AaB VV

SMF FRDM

MeV 56.15va

Droplet model constants: (Myers and Swiatecky, 1974)

Self-consistent mass models:

Based on simplifications of the insoluble (as yet) problem (complexity A!):Find the exact solution of the A-particle SCHROEDINGER equation:

A1,2,...,, |)(|det!

1

),...,,(),...,,()2

1( 2121

,,

rcA

rrrErrrVT

kk

AA

A

jiji

A

ii

A

AA(1)

Total wave-function Single-particle wave-function

Two-body interactionPotential energy

Kinetic energyTotal energy

HARTREE-FOCK APPROXIMATION

Assumptions: Existence of an average single-particle potential

created by all nucleons in which each nucleon moves independently from all the other nucleons present

)(rU HF

The best approximation to the ground state is represented by a wave function which is a single determinant

|)(|det!

1 r

AHF

Corresponding to the minimum energy of the system.

To obtain the single-particle wave-functions and single-particle potential :

1. Evaluate the expectation value of the total energy

2. Apply variational principle with a condition that must be normalized:

3. Solve iteratively a system of A Hartree-Fock Equations:

Starting from a trial solution

SELFCONSISTENCY

HF

A

jiji

A

iiHF VTE |

2

1|

,,

0)|)(|( 32 A

iii

i

rdrE

iiiHF rU )]([......

initiali

i

|)(|det!

1 r

AHF

),(),( NZMNMZMENZE nHR

A

iibinding

Finally, we obtain the total binding energy of the system

related to atomic mass excess M(Z,N)

and final total HF wave function which is then used to calculate otherground state properties such as root-mean square radii and deformation.

The choice of two-body nucleon-nucleon interaction in HF models:(subject too big to be discussed here in detail).

Basically functions of coordinates, angular momenta, spin and isospin and nuclear density, dependent on many adjustable parameters

In addition, the pairing interaction must be added in some empirical form dependent on several more adjustable parameters

Hartree-Fock+BCS Hartree-Fock-Bogolyubov (HFB)Pairing is added after Pairing is included inthe iteration process the iteration process

SKYRME interaction (Vautherin&Brink,1972, Stone&Reinhard,2007)DUFLO-ZUCKER pseudo-potential (Duflo&Zucker,1995)

SEPARABLE monopole interaction (see this talk)

jiV ,

Example: - The Skyrme Interaction (Chabanat et al., 1997)

Adjustableparameters

The problem of odd-A and odd-odd nuclei

HF Skyrme method assumes time-reversal symmetry TWO PARTICLES EVEN-EVEN NUCLEI – real wave-functions

When only one particle in an orbit(s)

(i) Time reversal symmetry broken (ii) Imaginary wave functions (iii) extra contributions from the force Stone&Reinhard (2007)

Fitting of the Skyrme parameters

Properties of ground states of doubly closed shell nuclei-binding energies, mean-square radii, etc

Unknown dependence of the parameters on N and Z , especiallyunreliable at regions far from stability

Some fit to properties of infinite symmetric nuclear matter(infinite medium with equal number of protons and neutronsand no coulomb force)

The parameters are correlated – in principle infinite numberof parameters sets, more or less equally good for finite nucleibut more sensitive to nuclear matter and neutron star properties(87 Skyrme parameter sets tested - Stone et al, 2003)

SELFCONSISTENT MASS MODELS OF THE BRUSSELS-MONTREAL GROUP:

I. The Extended Thomas-Fermi + Strutinsky Integral (ETFSI) models (Abousir et al, 1995)

ETF - further simplification of HF – practically removing shell correctionsSI - restoration of shell corrections in a perturbative wayQ - quenched (by brute force)

II. HF+BSC Models

III. HFB Models

All models have are based on the Skyrme interaction fitted to nuclear masses – laterversions include some other observables to the fit (see below).

Points of concern: Many ad hoc features, not included in the self-consistent calculation e.g. Wigner energy, rotation correction, vacuum polarization etc. treatment of odd-A and odd-odd nuclei symmetry restrictions inadequate performance in nuclear matter and cold non-rotating neutron stars

Review of work before 2005 (HFB-9): J.Phys.G: Nucl.Part.Phys. 31, R211-230 (2005)

Rms errors for models BSk9 – BSk13

Samyn 2003 HFB 2Goriely 2004 HFB 4-7Samyn 2004 HFB 8Goriely 2005 HFB 9Samyn 2005 manyGoriely 2006 HFB 10-13Goriely 2007 HFB 14

The latest:The new HFB-14 model, that is fitted to the fission data through adjustment to a vibrational term in the phenomenological collective correction.

= 0.729 MeV

From 2149 measured masses of nuclei with Z,N >8 with 24 variable parameters of theBsk14 force.

Compare with FRDM = 0. 633 MeV (P.Moller – private communication)

rms

rms

Examples of parameter variations for BSk1 – BSk9 Skyrme parametersNOTE THE VERY SMALL CHANGES!!!

BSk17 !

DUFLO-ZUCKER MASS FORMULA, PRC 52, R23, (1995)

Based on existence of a pseudo-potential , ready for use in shell model calculation

Consisting of a monopole term and higher multipole terms, which can be derived from realistic nucleon-nucleon interaction.

When only dominant terms in the potential are selected and their contribution to total energy calculated, an interesting mass formula is generated – the semi-empirical mass formula is the asymptotic limit – promise of good extrapolation properties.

Fits to 1751 binding energies with rms error of 0.375 MeV

N=82 shell and shell quenching – importance for r-process

FOR

AGAINST?

What now?

Call for better understanding of the nuclear physics

Two possible examples

N=82 N=126

Prediction of matter flows during the r-process and creation of heavy elements is CRITICALLY

dependent on nuclear physics input:

neutron separation energies

weak decay rates

neutron capturecross sections.

O.Sorlin and M-G.PorquetProg.Part.Nucl.Phys.61, 602 (2008)(also courtesy of D. Lunney)

Waiting point approach (K.L.Kratz)(static model)

Courtesy H. Schatz

The astrophysical Segre chart

Courtesy Kratz and Schatz

Approximate results of dynamicalmodel – extreme pink

First suggestions of increasing deformation and disappearance of shell gaps:

J.Dobaczewski et al. PRC 53, 2809 (1995) SPHERICAL HFB + SkP interaction

(improved r-process abundances but did not contain deformation)

Dobaczewski et al, Prog.Part.Nucl.Phys. 59, 432 (2007) Chen et al., Phys.Lett.B 355, 37 (1995)

Nuclear mass formula with Bogolyubov-enhanced shell quenching – ETFSI-Q (includes deformation)

Pearson et al, PLB 387, 455 (1996)

No quenching

quenching

Main outcome:

1. Reduced deformability

2. Smoothing of two-neutron-separation energies as a function neutron number

3. Filling in the troughs in the abundance curve.

Measure of the magnitudeof a shell gap

Δn (N ,Z ) = S2n (N ,Z ) − S 2n (N + 2,Z )

Neutron number

ETFSI-1

ETFSI-Q

Steep fall and strong variation with neutron number – NO quenching

Smoother fall and variationwith neutron number– WITH quenching

ETOTAL =EMEAN FIELD + ESHELL CORRECTION + EPAIRING

Model ofthe effectiveN-N interaction

EMICRO

EMICRO' =q(N,Z)EMICRO

q(N,Z) =1+ expaZ+b⎛

⎝⎜⎞⎠⎟

⎧⎨⎩

N +cZ+d} deformation independent

four free parameters

Strutinsky NP A95,420;122,1Lin PRC 2, 871

V(r)

r

bare N-N

Effective interactions used in HFB and ETFSI-Q models:

SkP and SkCS4 – complicated functions of density and 10-13 correlated parameters fitted to properties of doubly-closed shell nuclei

NO GUARANTEE they are applicable at the place of r-process

Two worrying features: - predict collapsing neutron matter at about 3 ρsat

- do not predict existence of stable neutron stars Stone et al., PRC68, 034324, 2003

Closer to home: effective nucleon mass m* = 1 for both SkP and SkSC4:

COMPRESSION of single-particle levels near the Fermi surface

Experimental data on nuclei around Z=50, N=82 and below

Dworschak et al., PRL 100, 072501 (2008)

Chamel et al., NPA 812, 72 (2008)

GSI report 2003

Blue dashed – ETFSI-QBlack solid HFB14

HFB16

AIP Conf. Proc. Vol 990, 309 (2008)

HFB14ETFSI-Q

Is the reduction of the shell gap the only thing that made the ETFSI-Q work OR is the smoothing out the S2n – N dependence equally important?

Pears

on

et

al,

PLB

387,

455 (

1996)

Courtesy B.Pfeiffer

ETFSI-Q - cyan

D-Z green

Grawe et al., Rep.Prog.Phys. 70, 1525

DUFLO-ZUCKER MASS FORMULA – should be tried in r-process models!

Systematic analysis of the effect of static and dynamic quadrupole correlations on S2q (q=N,Z) in the context of mean field methods

Bender et al., PRC 78, 054312 (2008)

Inclusion of beyond mean field correlations has a smoothing effect onN or Z dependence of two-nucleon separations energies and affects

shell structure

The effective density-dependent separable NN interaction

PRC 63,054309(2001)

SMO1 and SMO2: Deformation of light nuclei in Ne-Mg-Si PLB 545,291

(2002))

Shapes of N=Z and proton-rich nuclei, S2p Camerino

2001, Legnaro 2003

Shapes of neutron rich Tungsten isotopes PRC 72, 047303 (2005)

Nuclear matter and Neutron stars PRC 65, 064312 (2002)

SMO3 Binding energies of even-even nuclei Charge density distributions 16O – 208Pb r.m.s charge radii of Ca and Cd isotopes Q b value 130Cd – 130In -b decay

Selfconsistent Hartree-Fock + BCS model with axial symmetry

Additional terms:

Vsurf dependent on derivatives of density

Vpair density dependent BCS pairing Bender et al, Eur.Phys.J A8,59 (2000)

Vspin-orbit one-body spin-orbit term

Vcoul standard form with exchange term in Slater approximation

Separability: particle degrees of freedom are separated: F(x1) F(x2)

1 2 1 2( , , , ) attractive repulsiveV r r V V both attractive and repulsive terms have the same form but differ in parameters

1 2 1 2 1 2( ) ( )[1 ( ) ( )]unlike likeV W f r r a V b V ρ ρ

31/f d r ρ ,a r

Parameter fitting for SMO3:

Density dependent nuclear matter properties + neutron stars:Uniform (infinite) medium consisting of nucleons

Volume terms: Wx, ax, bx

Isospin terms: ax (unlike nucleons), bx (like nucleons)

Finite nuclei: (assuming the above parameters do not change)

Surface term: d (256Fm,14O, charge density distributions)

Isospin term UNLIKE ax (N,Z)

ax(nuclear matter) as the upper limit

Spin-orbit term: c (16O, 40Ca, 48Ca, 208Pb)

Pairing: Vp, Vn

CALCULATIONS OF GROUND STATE BINDING ENERGIES:

We fit experimental ground-state binding energies using HF+BCS technique

Keeping all the global parameters constant but the strength of the pn interaction.(aa = -0.35 for infinite matter)

12O 256Fm

Systematic behaviour sensitive to shell closures and shapes

34.008.0 aa

N-Z dependence of the variable strength of the isospin-isospin interactionBetween unlike nucleons (pn interaction)

(Stone&Walters, 2008, Stone&Moszkowski, in preparation)

Preliminary results for even-even Cd isotopes

Neutron skin

Quadrupole deformation

Discrepancies between experimental and calculated binding energiesare mainly due to minor problems with convergence of the HF method

CAN BE IMPROVED

Preliminary results for even-even Sn

All calculations include new data on Mo, Ru and Sn and beyond

Neutron skin

1.All current theories of the atomic nucleus are based on

a large number of parameters fitted to the same data set.

2. The physical content is mainly lost and the predictive

power very small and unreliable

3. New theories should justify degrees of freedom used

(nuclear, subnuclear) and use physical, not fitted

parameters.

4. This may not be possible unless a fundamentally new

idea is developed………nature is simple………….

Conclusions I

Conclusions II -personal view

1. Selfconsistent mass models of HFB – Skyrme type do not have chance of improvement

2. FRDM has reached its limit – P. Moller, private communication

3. Other global models not mentioned in this talk, including the ones based on relativisitic mean field theories do not have the quality

required for astrophysical applications (Lalazissis&Raman, 1999, (Geng et al, 2005)

4. We will try to develop further and understand better the SMO model which so far is capable of calculation of all known ground state binding energies to better then 10 keV. It is essential to explore its predictive power.

5. It may be useful to look into the controlled way of local corrections……before a truly microscopic calculation is possible.