IS THE NUCLEAR LARGE AMPLITUDE COLLECTIVE DYNAMICS ADIABATIC OR NON

ADIABATIC ?W. Brodziński, M. Kowal, J. Skalski

National Centre for Nuclear Research (Warsaw)P. Jachimowicz

University of Zielona Gora

Motivation = an inherent limitations of both: hot & cold fusion reactions:

Hot (well- deformed radioactive actinides (Act.) targets are used and compound nucleus is quite excited )

• attempts of going beyond the reactions Act. + 48Ca by using heavier projectiles like 50Ti, 54Cr, 58Fe, and 64Ni gave no results so far.

• all heavier actinides with Z>98 live to short that one could perform target with them.

Cold (magic nuclei as targets are used with projectiles heavier than 40Ar and Compound system is in this case only weakly heated and is cooled down via emission of just one or two neutrons )

• produced nuclei lies belong to the far “island of stability” of superheavy elements.

• to produce more & more heavier nuclei the mass and charge of projectile should be increased but it pulls an increase of the Coulomb repulsion what drastically reduces the cross sections.

CROSS SECTIONS ~ FEMTOBARNS

Introduction• Done: calculations for masses, shapes, fission

barriers, Q-alpha etc… for even – even actinides• Conclusions: quite good agreement with exp (masses &

Qalpha: rms = 0.3 MeV, BfA rms=0.5 MeV, BfB=0.7MeV

• Hope 1 : with the same set of the parameters one can predict properties of SHE

+• Hope 2: one can extend calculations on odd systems

• General Hope:K-isomers or high–K ground states of odd & odd-odd nuclei - a

chance for longer half-lives of SHE

Microscopic-macroscopic method with a possibility of many various deformations

)()()( micromacrotot EEE •

• Calculated energy: )0()( macrotot EEE• lexponentia Yukawa)( macroE• BCS pairing Saxon -Woods)( microE

=> ~L => ~R => ~A | - spliting energy

The effect of intruder states lying sclose to the Fermi level is most apperent in the heavier nuclei

SPHERICAL HARMONICS PARAMETRIZATION

136 144 152 160 168 176 184 192

98100102104106108110112114116118120122124126 min

20

-0.50-0.45-0.40-0.35-0.30-0.25-0.20-0.15-0.10-0.0500.050.100.150.200.250.30

N

Z

SDOOBLATE

PROLATE

SPHERICAL

Ground state shapes, even-even

20

:0.5

3.2

SDO

Axis ratio

Micro-macro results

In contrast to many Skyrme forces, Woods-Saxon micro-macro model gives lower barriers and mostly oblate ground states for Z>=124,126 (no magic gap for 126 protons).

P. Jachimowicz, M. Kowal, and J. Skalski, PRC 83, 054302 (2011).

Fission barriers calculated using micro-macro model (e-e nuclei)

Performance for even-even actinides:

1-st barriers, 18 nuclei

rms : 0.5 MeV

2-nd barriers, 22 nuclei

rms : 0.69 MeV

Even-even SH nuclei: barries decrease for Z>114 The highest barrier for Z=114, N=178

P. Jachimowicz, M. Kowal, and J. Skalski, PRC 85, 084305 (2012).M. Kowal, P. Jachimowicz and A. Sobiczewski, PRC 82, 014303 (2010) .

Statistical parameters for different macro – micro calculations of first (in parentheses) and second barriers.

162 164 166 168 170 172 174 176 178 180 1820

1

2

3

4

5

6

7

8

9

10

Z=112

B

f [M

eV]

N

HN FRLDM RMF SkM*

164 166 168 170 172 174 176 178 180 1820

1

2

3

4

5

6

7

8

9

10

Z=114

Bf [

MeV

]

N

HN FRLDM RMF SkM*

168 170 172 174 176 178 180 182 184 1860

1

2

3

4

5

6

7

8

9

10

Z=120

Bf [

MeV

]

N

HN FRLDM RMF SkM*

HN – Woods-SaxonFRLDM – P. Moller et al.SkM* - A.Staszczak et al.RMF – H.Abusara et al.

FRDLM & RMF also perform well in actinides!

Comparison of various models: some must be wrong.

Heaviest even-even fissioning nuclei:112, 170 0.8 ms (old calc. 71 ms) 112, 172 97 ms (old calc. 4 s)114, 172 130 ms (old calc. 1.5 s) (for Z=114, the local minimum in barrier at N=168)

Old calculation: Smolańczuk, Skalski, Sobiczewski (1995)

K. Siwek-Wilczyńska, T. Cap, M. Kowal, A. Sobiczewski, and J. Wilczyński, Phys. Rev. C 86, 014611 (2012).T. Cap, K. Siwek-Wilczyńska, M. Kowal, and J. Wilczyński,Phys. Rev. C 88, 037603 (2013).

Second minima in actinides,

Max diff = ~800 KeV!

M. K

owal

and

J. S

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ki,P

RC

82,

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303

(201

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P. J

achi

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.

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.

Max diff = ~4 MeV!

SHE masses (including odd & odd-odd)

• A fit to exp. masses Z>82, N>126, • number of nuclei: 252• For odd and odd-odd systems there are 3

additional parameters – macroscopic energy shifts (they have no effect on Q alpha).

>>Predictions for SHE:88 Qalpha values, Z=101-118, 7 differ from exp. by more than 0.5 MeV; the largest deviation: 730 keV (blocking).

Slight underestimate for Z=108;Overestimate: Z=109-113

P. Jachimowicz, M. Kowal, and J. Skalski, PRC 89, 024304 (2014)

Statistical parameters of the fit to masses in the model with blocking in separate groups of even-even, odd-even, even-odd and odd-odd heavy nuclei:

The same but for the method without blocking.

Q alpha

204 nuclei in the fit region

blocking q.p.method

mean 326 keV 225 keV error

rms 426 keV 305 keV

88 nuclei Z=101-118

mean 217 keV 196 keVerror

rms 274 keV 260 keV

105 107 109 111 113 115 117

8

9

10

11

12

13

14

A=270

A=286

A=282

A=278

A=274

A=290

A=294

Q [

MeV

]

Z

Blocking method Experimantal data Goriely et al. Blocking method Moller et al. (fixed K-number)

105 107 109 111 113 115 117-3-2-10123456789

101112

A=270

A=274

A=278

A=282A=286

A=290

A=294

log 10

[T 1/

2 (s)

]

Z

Blocking method Experimantal data Goriely et al. Blocking method Moller et al. (fixed K-number)

102 104 106 108 110 112 114 116 118 120

7

8

9

10

11

12

13

14

15

A=267

A=271A=275

A=279

A=283

A=287

A=291

A=295

Q [

MeV

]

Z

Blocking method Experimantal data Goriely et al. Blocking method Moller et al. (fixed K-number)

102 104 106 108 110 112 114 116 118 120-4-3-2-10123456789

10

A=291

A=267A=271

A=275

A=279

A=283 A=287

A=295

log 10

[T 1/

2 (s)

]

Z

Blocking method Experimantal data Goriely et al. Blocking method Moller et al. (fixed K-number)

High-K states: a chance for longer half-lives.

< Candidates for high-K g.s. in odd or odd-odd SHN in the W-S model

In even-even systems one should block high-Kclose-lying orbitals,like:9/2+ and 5/2- protonsbelow Z=108 or 11/2- and 9/2+ neutronsbelow N=162

Z N Omega(n) Omega(p) K113 173 5/2+ 7/2- 6-112 173 15/2- 15/2-114 170 11/2+ 11/2+ 169 5/2+ 9/2- 7- 163 13/2- 3/2- 8+110 163 13/2- 13/2-109 All 11/2+ > 11/2 169 9/2+ „ 10+ 161 „ „ „ 159 „ „ „ 163 13/2- „ 12-115 163 „ 13/2- 157 11/2- 11/2-107 163 13/2- 5/2- 9+ 157 11/2- „ 8+116 163 13/2- 13/2- 157 11/2- 11/2-117 157 11/2- 9/2+ 10- 151 9/2- 9/2+ 9-118 157 11/2- 11/2-119 157 11/2- 7/2- 9+ 151 9/2- 7/2- 8+ 149 7/2+ 7/2- 7-101 157 11/2- 1/2- 6+

Possible Q-alpha hindrance: the 14- SD oblate ground state in parent. The G.S. to G.S. transition inhibited; SDO to SDO has smaller Q.

protons

neutrons

156 158 160 162 164 166 168 170-5-4-3-2-1012345

log 10

[T 1/

2 (s)]

N

GS->GS GS->EXZ=109

156 158 160 162 164 166 168 170 172 174-0.20.00.20.40.60.81.01.21.41.61.82.02.2

Q [

MeV

]

N

Z=113 Z=111 Z=109 Unique blocked orbitals may hinder

alpha transitions. The effect of a reduced Q alpha for g.s. -> excited state (left panel) on the life-times (below)according to the formula by Royer.

8.07.06.0

5.04.0

3.0

2.0

1.001.0

2.0

7.0

0

8.0

3.0

-1.0

9.0

4.0

10.0

-2.0

5.0

11.0

-3.0

6.0

12.0

7.0

-4.0

13.0

8.0

14.0

-5.0

9.0

15.0

-6.0

16.0

-1.0

-2.0-3.0

-4.0-5.0

-6.0

-2.0

-7.0

-1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.2

-0.1

0.0

0.1

0.2

20

40272Mt

G.S. configuration:P:11/2+ [6 1 5]N:13/2- [7 1 6]

9.0

7.0

6.0

5.0

4.0

3.0

2.0

8.0

1.0

8.0

9.0

0

10.0

9.0

7.0

11.012.0

6.0

13.0

14.0

5.0

10.0

15.0

10.011.0

4.0

16.0

12.0

17.0

12.0

-1.0

-2.0

-3.0

-4.0-5.0

-6.0-7.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.2

-0.1

0.0

0.1

0.2

272Mt

Fixing the g.s. configuration rises the barrier by 6 MeV. Even if configuration is not completely conserved, a substantial increase in fission half-life is expected.

Mass parameter for odd system

• In the diabatic scienario one can imagine that blocked state lies higher in energy than the g.s.

=> Negative values of mass parameter!• Around the crosing region two states are close together

=> Mass parameter explode!• Due to pairing even far awy from crossing one can imagine that on both side of

fermi level quasiparticles have practicaly the same energies (if slopes of the levels are similar)

=> Mass parameter explode!

even

odd

2Δ

Landau – Zener effect

Diabatic ADiabatic

To CROSS or NOT to CROSS ?

IS THE LASD ADIABATIC OR NON ADIABATIC ?

W. Nazarewicz, Nucl. Phys A 557 (1993)

 

 

Q 

 

Central Open Questions/Problems:

• What are the conditions for the many – body system to give up its quantum characteristic ?

• How to calculate the stability of odd nuclei (fission & alpha half – lives) ?

• Problem of mass parameters for odd systems!• How many colective variables is enough ?

Status of third minimum in actinides:

Theory:

Experiment:

Shallow minima (0.5 MeV or less )

Deep minima(3 - 4 MeV)

self-consistent models

mac-mic modelP. Moller et, al.

mac-mic modelS. Ćwiok et, al.

Blons et, al.(231,232,233Th)

Debrecen-Munich(232,234,236U)

?

?

?

1,2

2,0

2,8

2,8

3,6

4,4

5,26,0

3,6

6,87,6

4,4

5,2

4,4

6,0 6,8

5,2 6,0

0,4

5,26,8

2,0

1,2 0,4

1,2

-0,4

0,6 0,7 0,8 0,9 1,00,0

0,2

0,4

0,6

3

2

Esh+YpE

234Umin(

4,

5

6

7

8)

1,02,0

3,0

4,05,0

6,0

7,0

8,09,0

10

3,02,04,0

1,0

5,006,00

2,01,0

0,6 0,7 0,8 0,9 1,0 1,1 1,20,0

0,2

0,4

0,6

0,8

1,0

Esh+LSD

234Umin(

4,

5

6

7

8)

3

2

B

A

• The dipole deformation b1 is omitted there, as corresponding to a shift of the origin of coordinates which leaves energy (always calculated in the center of mass frame) invariant. However, this is true only for weakly deformed shapes. For large elongations, b1 acquires a meaning of a real shape variable.

2 1.1 4 0.3 6 0.1 8 0.1 3 0.3 5 0.1 7 0.1a32 0.0a52 0.0a42 0.0a44 0.0a22 0.0

1 0.0

1 0.6

R_, _ : 1 1 SphericalHarmonicY1, 0, , 0 2 SphericalHarmonicY2, 0, , 0 4 SphericalHarmonicY4, 0, , 0 6 SphericalHarmonicY6, 0, , 0 8 SphericalHarmonicY8, 0, , 0 3 SphericalHarmonicY3, 0, , 0 5 SphericalHarmonicY5, 0, , 0 7 SphericalHarmonicY7, 0, , 0

IIIrd minima – type: A

• minima with larger octupole deformations (A) have quadrupole moments Q=170 b, disturbingly close to the scission region.

• minima (A) are just intermediate congurations on the scission path, whose energy was calculated erroneously because of limitations of the admitted class of shapes.

One can nd continuous 8D paths start ing at the supposed IIIrd minimum and leading to scission, along which energy decreases gradually.

M. Kowal, J. Skalski, PRC 85, 061302(R) (2012)

IIIrd minima – type: B

1.52.0

2.53.0

4.0

4.5 4.0 3.5

3.0

2.5

2.0

5.03.55.5

4.0

6.0

3.0

6.5

1.57.0

2.5

7.51.0

2.0

1.0

0.6 0.8 1.0 1.2 1.40.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

232U

20

30

3.0

2.5

2.0 1.5

3.5

4.01.0

5.0

3.53.0

1.0

0.52.5

0.6 0.8 1.0 1.2 1.40.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

232U

20

30

1 2 3

4 5 6

0.35 0.00 0.55 1.50 0.00 0.350.10 0.35 0.20 0.20 0.15 0.15

1 0.0 1 0.2

IIIrd minima – type: B

P. Jachimowicz, M. Kowal, J. Skalski,

PRC 87, 044308 (2013)

MODIFY FUNNY-HILLS PARAMETRIZATION

eg. : Krzysztof Pomorski, Johann Bartel, Int. J. Mod. Phys. E, Vol. 15, No. 2 (2006) 417.

MODIFY FUNNY-HILLS PARAMETRIZATION

232Th

c × h × α × η = 251 904 gird points

0.5 MeV

MODIFY FUNNY-HILLS PARAMETRIZATION

236U

c × h × α × η = 251 904 gird points

no third minimum

THREE QUADRATIC SURFACE PARAMETERIZATION

THREE QUADRATIC SURFACE PARAMETERIZATION

eg. : Peter M̈oller et al., Phys. Rev. C, 79 (2009) 064304.

236U

286Fl

● Heights of the first and second fission barriers as well as the excitation of the second minimum are convergent with those three classes of nuclear shape parameterizations.

● We found the similar depth of the third minimum of the order of several hundred of keV for all tested parameterizations. This is still in a sharp contrast with the experimental status of the III-rd minima in those nuclei. Their experimental depth of 3–4 MeV contradicts all realistic theoretical predictions.

● Among all investigated parameterizations the most effective and efficient (the smallest amount of required dimension without apparent loss of the described shapes) is the modify Funny – Hills prescription.