EQUATION-OF-STATE PARAMETERS

FROM THE FRDM MASS MODEL

P. Moller (LANL)

W. D. Myers, H. Sagawa (Aizu), S. Yoshida (Hosei)

Please look on my web page for more details aboutthe mass model and in the future on the symmetryenergy

http://t16web.lanl.gov/Moller/abstracts.html

1

Nuclear masses (binding energies) can to a high ac-curacy (3 MeV out of a total of 1500 MeV or so) bedescribed by an EOS-related Droplet model macro-scopic model. Volume, surface and some other coef-ficients can be determined by adjusting these to op-timize agreement with nuclear masses. However todetermine smaller contributions, such as symmetry-energy effects we need account for features of nuclearmasses that are outside the macroscopic model.

These effects are for example, microscopic “shell ef-fects”, pairing effects, Wigner energy, which is a mi-croscopic effect near N = Z nuclei but no micro-scopic interaction has been proposed, so it is routinelymodelled very phenomenologically as a macroscopicterm.

In our approach we have also introduced a finite-rangesurface energy, a charge asymmetry term, and andan exponential term.

Potential Energy of Deformation

We use the macroscopic-microscopic method introduced by

Swiatecki and Strutinsky:Epot(shape) = Ema r(shape) +Emi r(shape) (1)

The macroscopic term is calculated in a liquid-drop type

model (for a specific deformed shape).

The microscopic correction is determined in the following

steps

1. A shape is prescribed

2. A single-particle potential with this shape is generated.

A spin-orbit term is included.

3. The Schrodinger equation is solved for this deformed

potential and single-particle levels and wave-functions

are obtained

4. The shell correction is calculated by use of Strutinsky’s

method.

5. The pairing correction is calculated in the BCS or

Lipkin-Nogami method.

Etot(Z,N, shape) =

+

(

−a1 + Jδ2 −

1

2Kǫ2

)

A volume energy

+

(

a2B1 +9

4

J2

Qδ2

Bs2

B1

)

A2/3 surface energy

+ c1Z2

A1/3B3 Coulomb energy

− c2Z2A1/3Br volume redistribution energy

− c4Z4/3

A1/3Coulomb exchange correction

− c5Z2BwBs

B1surface redistribution energy

+ other terms

The other terms are:

+ a0A0 A0 energy

+ f0Z2

Aproton form-factor

− ca(N − Z) charge-asymmetry energy

+ W

(

|I|+

{

1/A , Z and N odd and equal0 , otherwise

)

+

+ ∆p +∆n − δnp , Z and N odd

+ ∆p , Z odd and N even

+ ∆n , Z even and N odd

+ 0 , Z and N even

− aelZ2.39 energy of bound electrons

+ Emicr microscopic shell+pairing

The relative deviation (ǫ) in the bulk, of the density ρ,from its nuclear matter value ρ0, the average relativedeviation in the bulk (ǫ), of the density, the bulk nuclearasymmetry δ, and the average bulk nuclear asymme-try (δ) are defined/given by, respectively

ǫ = −

1

3

ρ− ρ0ρ0

ǫ =

(

Ce−γA1/3− 2a2

B2

A1/3+ Lδ2 + c1

Z2

A4/3B4

)

/K

δ =ρn − ρp

ρbulk

δ =

(

I +3

16

c1Q

Z

A2/3

BvBs

B1

)

/

(

1+9

4

J

Q

1

A1/3

Bs2

B1

)

Before I focus on our studies of the symmetry termsI will show the model agreement with a diverse set ofnuclear-structure properties across the nuclear chart.

σ529 = 0.462 MeV σ1654 = 0.669 MeV

FRDM (1992)

New Masses in Audi 2003 Evaluation, Relative to 1989, Compared to Theory

− 20 − 15 − 10 − 5 0 5 10 15 Neutrons from β-stability

− 6

− 4

− 2

0

2

4

6

Mex

p −

Mca

lc (

MeV

)

σ217 = 0.642 MeV σ1654 = 0.669 MeV

FRDM (1992)

New Masses in Audi 1993 Evaluation, Relative to 1989, Compared to Theory

− 20 − 15 − 10 − 5 0 5 10 15 Neutrons from β-stability

− 6

− 4

− 2

0

2

4

6 M

exp

− M

calc (

MeV

)

36 72Kr36

Scale 0.20 (MeV)

0.00 0.10 0.20 0.30 0.40 0

20

40

60

3

4

4

5

5

5A

xial

Asy

mm

etry

γ

Spheroidal Deformation ε2

Number of Minima

Depth > 0.20 MeV, Eex < 2.0 MeV, ε2 < 0.45

1 2 3 4 5

0 20 40 60 80 100 120 140 160 Neutron Number N

0

20

40

60

80

100

120 P

roto

n N

umbe

r Z

140Gd

0+2+

328.6

4+836.2

6+1464.0

8+2139.7

10+2796.8

GS band

2+713.4

3+4+

1281.55+

6+1882

7+2412

γ band

128Te

0+

2+743.22

4+1497.04

Vibrational band

238U

0+2+

44.91

4+148.41

6+307.21

8+518.3

Rotational band

222Ra

0+

2+111.12

4+301.42

Positive-parity band

1-242.13

3-317.27

Negative-parity band

Typical Level Spectra for 3 Major Types of Deviation from Spherical Symmetry

Spherical Spheroidal

Reflection Asymmetric Axially Asymmetric

44108Ru64

Scale 0.20 (MeV)

(4)

0.00 0.10 0.20 0.30 0.40 0

20

40

60

3

4

4

5 5

6

6

78

Axi

al a

sym

met

ry,

γ

Spheroidal deformation, ε2

40104Zr64

Scale 0.20 (MeV)

(1)

0.00 0.10 0.20 0.30 0.40 0

20

40

60

3

4

5

6

6

7

7

Axi

al a

sym

met

ry,

γ

Spheroidal deformation, ε2

38 98Sr60

Scale 0.20 (MeV)

(2)

0.00 0.10 0.20 0.30 0.40 0

20

40

603

45

5

6

6

7

7

Axi

al a

sym

met

ry,

γ

Spheroidal deformation, ε2

36 70

Kr34

Scale 0.20 (MeV)

(3)

0.00 0.10 0.20 0.30 0.40 0

20

40

60

4

55

5

6

Axi

al a

sym

met

ry,

γ

Spheroidal deformation, ε2

Krypton Equilibirium-Point Shapes

Oblate ground state Prolate minimum Axially asymmetric minimum Axially asymmetric saddle

− 0.55 − 0.50 − 0.45 − 0.40 − 0.35 − 0.30 − 0.25 − 0.20 − 0.15 − 0.10 − 0.05 − 0.01

∆Eγ (MeV) Effect of Axial Asymmetryon Nuclear Mass

0 20 40 60 80 100 120 140 160 Neutron Number N

0

20

40

60

80

100

120 P

roto

n N

umbe

r Z

0+2+

4+

6+

8+

10+

12+

GS band

2+3+

4+5+

6+

7+

γ band0+2+

4+

6+

(8+)

GS band

2+3+

4+(5+)

γ band

0+2+

(4+)

GS band

2(+)(3+)

γ band

108Ru

140Gd194Pt

Mass-model error without γ correction for 71 nuclei with |∆Eγ| > 0.2 MeV

σ = 0.577 MeV µ = −0.452 MeV

50 100 150 200 Nucleon Number A

− 2.0

− 1.5

− 1.0

− 0.5

0.0

0.5

1.0

1.5

Mex

p −

Mth (

MeV

) Mass-Model Error with γ correction for 71 nuclei with |∆Eγ| > 0.2 MeV

σ = 0.381 MeV µ = −0.109 MeV

− 1.5

− 1.0

− 0.5

0.0

0.5

1.0

1.5

2.0

Mass-model error without ε3 correction for 78 nuclei with |∆Eε3| > 0.2 MeV

σ = 1.160 MeV µ = −1.032 MeV

50 100 150 200 250 Nucleon Number A

Mass-model error with ε3 correction for 78 nuclei with |∆Eε3| > 0.2 MeV

σ = 0.385 MeV µ = −0.168 MeV

HFB2 (Goriely) HFB8 (Goriely) FRLDM (1992) FRDM (1992) OTHER exp. RIKEN exp. (2004)

α-decay of 278113

101 103 105 107 109 111 113 115

153 155 157 159 161 163 165 167 Neutron Number N

Proton Number Z

7

8

9

10

11

12

13

14

Ene

rgy

Rel

ease

Qα

(MeV

)

125X 180

Scale 1.00 (MeV)

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0

20

40

60

-20

-15

-10

-5

-5

-5

0

0

0 Axi

al A

sym

met

ry

γ

Spheroidal Deformation ε2

Dubna exp. Muntian et al. (2003) FRDM (1992) FRLDM (1992)

α-Decay Chain from 291116

104 106 108 110 112 114 116

163 165 167 169 171 173 175 Neutron Number N

Proton Number Z

7

8

9

10

11

12 E

nerg

y R

elea

se Q

α (M

eV)

Dubna exp. Muntian et al. (2003) FRDM (1992) FRLDM (1992)

α-Decay Chain from 293116

106 108 110 112 114 116 118

167 169 171 173 175 177 179

⇐ ε2>0.1 ε2<0.1 ⇒

Dubna exp. Muntian et al. (2003) Sobiczewski (2010) FRDM (1992) FRLDM (1992)

α-Decay Chain from 294117

105 107 109 111 113 115 117

165 167 169 171 173 175 177 Neutron Number N

Proton Number Z

7

8

9

10

11

12

Ene

rgy

Rel

ease

Qα

(MeV

)

Now let us address the determination of the symme-try term parameters. To do this we have in successivesteps improved the FRDM(1992), the current interim“table” is FRDM(2011a). Once we have more consis-tently recalculated the “beyond-mean-field” zero-pointvibrational-energy corrections we are planning to sub-mit FRDM(2012) for publication.

FRDM enhancements

OptimizationThe search for optimum FRDM macroscopic pa-rameters has been improved.Accuracy improvement: 0.01 MeV

New exp. mass data baseWe agree better with the new mass data baseAccuracy improvement: 0.04 MeV

Full 4D energy minimizationSearch for minimum energy versus ǫ2, ǫ3, ǫ4, ǫ6,full 4D in steps of 0.01.Accuracy improvement: 0.02 MeV

Axial asymmetryResults in correct gs assignments in SHE regions,and mass improvements.Accuracy improvement: 0.01 MeV

L variationAccuracy improvement 0.02 MeV

1

FRDM-2011a, L var., Adj. to AME03, Comp. to AME11 Discrepancy (Exp. − Calc.)

− 2.00 − 1.50 − 1.00 − 0.50

0.00 0.50 1.00 1.50

σ2149 = 0.5700 MeV σ154 = 0.5618 MeV

|∆E | (MeV)

0 20 40 60 80 100 120 140 160 Neutron Number N

0 10 20 30 40 50 60 70 80 90

100 110 120

Pro

ton

Num

ber

Z

FRDM(1992) adj. to AME1989 Comp. to AME 2011 Discrepancy (Exp. − Calc.)

− 2.00 − 1.50 − 1.00 − 0.50

0.00 0.50 1.00 1.50

σ1654 = 0.669 MeV σ671 = 0.528 MeV

|∆E | (MeV)

0 10 20 30 40 50 60 70 80 90

100 110 120

Pro

ton

Num

ber

Z

FRDM1992 Compared to FRDM2007b Difference (FRDM1992− FRDM2007b)

− 2.00 − 1.50 − 1.00 − 0.50

0.00 0.50 1.00 1.50

|∆E | (MeV)

0 20 40 60 80 100 120 140 160 Neutron Number N

0 10 20 30 40 50 60 70 80 90

100 110 120

Pro

ton

Num

ber

Z

FRDM1992 Compared to FRDM2011a Difference (FRDM1992− FRDM2011a)

− 2.00 − 1.50 − 1.00 − 0.50

0.00 0.50 1.00 1.50

|∆E | (MeV)

0 20 40 60 80 100 120 140 160 Neutron Number N

0 10 20 30 40 50 60 70 80 90

100 110 120

Pro

ton

Num

ber

Z

When you introduce new effects, such as the density-symmetry term with coefficient L and the exponentialterm you obviously get better fits to data because youhave additional parameters. How do you know the val-ues are realistic?

We rely on the following (strong) test. We adjust themodel to a 1989 mass data set of 1654 masses, thentest it(s extrapolation behavior) on 529 new massesmeasured up to 2003.

I will show that adding the new terms greatly improvesextrapolation accuracy . For example, if we do notinclude the exponential term, adjust to the 1989 database we obtain an error of 0.5041 MeV for the 529nuclear masses measured between 1989 and 2003.When we include the exponential term, adjust to the1981 data base we obtain a smaller error 0.4091 MeVfor the 529 that were not included in the adjustment.

And, a slightly different test, in a model with the expo-nential term and density-symmetry effects included,we obtain little increase in accuracy if the 529 new nu-clei are included in the adjustment (about 0.002 MeVimprovement only).

Another strong test is how well a diverse set of nu-clear properties across the chart of the nuclides aremodeled. we describe extremely well in a consistentmodel, not just masses, but also deformations, includ-ing the occurrence of octupole and axial asymmetryshapes, shape coexistence, ground-state spins, de-cay properties of superheavy elements 40 nucleonsaway from last data in adjustment data base.

All our results which include sensitivity studies andstudies of how well the model extrapolates (I like bet-ter the word “applies”) to nuclei outside the region ofadjustment is in the table in the next slide. We con-sider three regions of nuclei

1. AME1989 1654 nuclei

2. AME2003 2149 nuclei

3. New nuclei in (2) relative to (1) 529 nuclei

The region adjusted to is the first number in the sec-ond column (A/C, A for adjusted) the second numberis the region for which the mean deviation and sigmadeviation are given.

Model A/C a1 a2 J Q L µth σth;µ=0

(MeV) (MeV) (MeV) (MeV) (MeV) (MeV)

(92) 1/1 16.247 22.92 32.73 29.21 0 0.0156 0.6688(92) 1/3 0.1755 0.4617(92) 1/2 0.0607 0.6314

(92)-a 1/1 16.245 23.02 32.22 30.73 0 0.0000 0.6614(92)-a 1/3 0.0174 0.4208

(92)-a 1/2 0.0114 0.6180(92)-b 1/1 16.286 23.37 32.34 30.51 0 0.0000 0.6591(92)-b 1/3 0.0031 0.4174(92)-b 1/2 0.0076 0.6157(06)-a 2/2 16.274 23.27 32.19 30.64 0 0.0000 0.6140

(07)-b 2/2 16.231 22.96 32.11 30.83 0 0.0000 0.5964(11)-b 2/2 16.231 22.95 32.10 30.78 0 0.0001 0.5863(11)-c 1/1 16.251 23.10 32.31 30.49 0 −0.0003 0.6300(11)-c 1/3 0.0144 0.3874(11)-d 1/1 16.142 22.39 32.98 27.58 85.95 0.0000 0.6092

(11)-d 1/3 0.0545 0.3929(11)-d 1/2 0.0228 0.5719(11)-a 2/2 16.147 22.44 32.51 28.54 70.84 −0.0004 0.5700(11)-a 2/3 0.0214 0.3768

FURTHER OBSERVATIONS

• If we adjust to region 1 with no exponential termand calculate σth for extrapolation to region (3)we obtain σth = 0.5041 MeV, with exponentialterm we obtain 0.4091 MeV.

• With Wigner term and constant set to zero we ob-tain σth = 0.7335 MeV.

• With charge-asymmetry term set to zero we ob-tain σth = 0.9142 MeV.

• If the original 1970 spin-orbit strength choice isused a poorer mass model is obtained with σth =

0.6909 MeV for L = 0, with L varied we obtainL = 39 MeV and σth = 0.6879 MeV, that isonly a marginal 0.003 MeV improvement in ac-curacy rather than 0.0163 MeV improvement withthe more accurate shell corrections that we nowcalculate (see table).

FRDM with 1970 Spin-Orbit and Range

σth = 0.691 MeV

0 20 40 60 80 100 120 140 160 Neutron Number N

− 5

− 4

− 3

− 2

− 1

0

1

2

3

4

0

FRDM-2007b

σth = 0.596 MeV

− 4

− 3

− 2

− 1

0

1

2

3

4

5

Mas

s D

evia

tion

(Exp

. − C

alc.

) (M

eV)

R E S U L T

From our mass studies we determine the following val-ues for the symmetry-energy constant J and density-symmetry constant L:

J = 32.5± 0.5 MeV

L = 70± 15 MeV