Kazimierz 2011

T. Cap, M. Kowal, K. Siwek-Wilczyńska, A. Sobiczewski, J. Wilczyński

Predictions of the FBD model for the synthesis cross sections of Z = 114-120 elements based on

macroscopic-microscopic fission barriers

Nuclear Elongation

Existence of super–heavy nuclei due to shell effects

Yu. Ts. Oganessian et al.PRL 104, 142502 (2010)

249Bk(48Ca,xn) 297-xn117; x = 3, 4

X + 208Pb, 209Bi

48Ca + X

= 2.5 (+1.8,-1.1) pb

GSI

279Ds

9.5200.015 /23215 MeV

T½ = 6.9 s

bsf = 0.5

210 MeV

T½ = 0.18 s

CN

6.9 –2.3

30–15

0.32 –0.07

283Cn

= 0.72 (+0.58,–0.35) pb

3 chains

4 chains

S. Hofmann et al. , Eur. Phys. J. A32,251 (2007)

260 MeV TKET½ = 1.3 hbsf = 1.0

279Ds

275Hs

267Rf

9.54±0.06 MeVT½ = 3.8 sbsf = 0.10

9.70 / 228 MeVT½ = 0.20 s b = 0.10

9.29±0.08 MeVT½ = 0.19 sb = 1.0

8.54 / 248 MeV T½ = 1.9 minb = 0.70

283Cn CN

271Sg FLNR

48Ca +238U 286Cn + 3n

Confirmation of Dubna results at GSI

Yu.Ts. Oganessian, Phys. Rev. C 70, 064609 (2004) Yu.Ts. Oganessian, J. Phys. G 34 (2007) R165

8 events, =

48Ca +244Pu 288114 + 4n

FLNR GSI

284Cn

9.940.06

T½ = 0.8 s

216 MeV

T½ = 97 ms

CN

0.27– 0.16

30–15

31 –19

288114

284Cn

9.940.04

T½ = 0.47 s

T½ = 101 ms

CN

0.24 –0.12

50 –25

288114

Ch. Düllmann et al., PRL 104, 252701 (2010)

J.M. Gates et al. PRC 83, 054618 (2010)

12 events, = pb0.30.20.5

pb8.3

1.38.9

GSI281Ds

9.850.04

T½ = 0.97 s

T½ = 20 s

CN

0.97 –0.32

289114

285Cn

9.190.04

T½ = 30 s

20 –7

30 –10

18–2

277Hs

Tsf = 3 s

(8.67 MeV)(T = 5.7 s)

Yu.Ts. Oganessian, Phys. Rev. C 70, 064609 (2004) Yu.Ts. Oganessian, J. Phys. G 34 (2007) R165

2 chains,

48Ca + 244Pu 289114+3n

FLNR

281Ds

9.820.05

T½ = 2.6 s

212 MeV

T½ = 11.1 s

CN

1.2 –0.7

5.0 –2.7

289114

285Cn

9.150.05

T½ = 29 s 13 –7

Ch. Düllmann et al. PRL 104, 252701 (2010) J.M. Gates et al. PRC 83, 054618 (2011)

pb4.75.40.8

5 chains,

=

pb2.21.17.1

=

48Ca +248Cm 296-xn116

248Cm 296116

S. Hofmann et al., GSI Scientific Report 2010, 197

4 chains

6 chains

4 chains 1 chains 3.4 pb 0.9 pb E* = 41.0 MeV GSI-SHIP

6 chains 2 chains 3.3 pb 1.2 pb E* = 39.0 MeV FLNR

242

48Ca + 242Pu 290-xn114 +xn x = 3, 4, 5

L. Stavstera et al., Phys. Rev. Lett. 103, 132502 (2009)P. A. Ellison et al., Phys. Rev. Lett. 105, 182701 (2010)

Yu.Ts. Oganessian, Phys. Rev. C 70, 064609 (2004) Yu.Ts. Oganessian, J. Phys. G 34 (2007) R165

2n - 1 event, =0.5 (+1.0, -0.5) pb3n - 15 events, =3.6 (+2.4, -1.6) pb4n - 9 events, =4.5 (+2.5, -1.5) pb

3n - 1 event, =3.1 (+4.9, -2.6) pb 4n - 1 event, =3.1 (+4.9, -2.6) pb

5n - 1 event, =0.6 (+0.9, -0.5) pb

FLNR, DUBNA LBNL, Berkeley

Confirmation of Dubna results in LBNL

How to produce elements with Z > 118?

Essential experimental difficulties:

The use of 48Ca beams will require heavier targets of Z > 98.Possibilities: Z=99, 252Es, 254Es T1/2 = 472 d, 275.7 d

Z=100 257Fm T1/2 = 100.5 d

Alternative:

Heavier beams → much smaller cross sectionMulti-nucleon transfer reactions.

Experiments 2007/2008

GSI DubnaNumber of irradiation days : 120 days ---Excitation energy of the CN : 36.0 MeV 45.5 MeVTotal number of beam particles : ≈ 2 x 1019 7.1 x 1018 Number of detected events : 0 0Cross section limits : < 0.1 pb < 0.4 pb

First experiments to produce element of Z = 120

GSI Dubna

58Fe 244Pu

Yu. Ts. Oganessian et al. Phys. Rev. C 79, 024603 (2009)

capture

fusion

survive evaporation residue

symetric fission

fast fission

(synthesis) = (capture) × P(fusion) × P(survive)

Nucleus-nucleus collision which may lead to the formation of super-heavy nuclei

T. Cap, K. Siwek-Wilczyńska, J. Wilczyński, IJMP E 20, 308 (2011) T. Cap, K. Siwek-Wilczyńska, J. Wilczyński, PR C 83, 054602 (2011)

W. Świątecki, K. Siwek-Wilczyńska, J. Wilczyński Phys. Rev. C 71 (2005) 014602,

Acta Phys. Pol. B34(2003)

20493 parameters: B0, w, R obtained from 2 fit to 48 experimental

near-barrier fusion excitation functions for 40 < ZCN < 98 (K. Siwek-Wilczyńska, J. Wilczyński Phys. Rev. C 69 (2004) 024611)

2)exp()1()(

)1()12()(

22

2max

0

22

E

wXXerfXRE

lTlE

cap

llcap

Xerfw

BEXwhere ,

2: 0

(survive)(fusion)PP1)(2lπs)σ(synthesi xn

lmax

0l

2 l

lmax – calculated from the capture cross section.

semiempirical formula

Gaussian error function

This formula derived assuming:• Gaussian shape of the fusion barrier distribution• Classical expression for σfus(E,B)=πR2(1-B/E)

FBD

λ= (d1+ d2)/(R1 + R2) - neck parameter

ρ = r/(R1 + R2) – relative distance

Δ = (R1 - R2)/(R1 + R2) – asymmetry parameter

R

1

R2

r

d2 d1

Pl (fusion)

J. Błocki, W. J. Świątecki, Nuclear Deformation Energies, Report LBL 12811 (1982)

CN

Smoluchowski Diffusion equation for the parabolic potential

Pl (fusion) = ½(1-erf√H(l )/T)

H - the barrier opposing fusionT - the temperature of the

fusing system

max

0 *

max

22 12inE

inininin

ininnn

in dE

Es

m

Partial widths for neutron emission – Weisskopf formula

PBEEE ininrotiin

*)1(

maxwhere:

The fission width (transition state method), E*< 40 MeV

max

0*

max

2

1 fEffiss

fiss dKE

KE

Upper limit of the final-state excitation energy after emission of a particle i

PsaddleEsaddleEE rotf )()(*max Upper limit of the thermal excitation energy at the saddle

i – cross section for the production of the compound nucleus in the inverse process

mi, si , εi - mass, spin and kinetic energy of the emitted particle

ρ, ρi – level densities of the parent and daughter nuclei

Pxnl (survive) for xn reaction

The level density is calculated using the Fermi-gas-model formula aEE 2exp

included as proposed by Ignatyuk(A.V. Ignatyuk et al., Sov. J. Nucl. Phys. 29 (1975) 255)

dEUshell

macro eU

aa 11

• Shell effects

where: U - excitation energy, Ed - damping parameter

shell – shell correction energy, δshell (g.s.),δshell

(saddle)

MeVEd 5.18

jk

jsmacro BArBArAra 31

0322

03

0 1426.01355.004543.0 fmr 153.10

Bs , Bk ( W.D. Myers and W.J. Świątecki, Ann. Phys. 84 (1974) 186)

,

(W. Reisdorf, Z. Phys. A. – Atoms and Nuclei 300 (1981) 227)

xnxnfxn

xnn

fn

nn

fn

nlxn PPPsurviveP

)........1()1()( 222

21

11

1

P<1n-probability that after emission of the first neutron the excitation energy is smaller than the threshold for second chance fission or 2n emission.

nfn

nln PsurviveP 1

11

11 )(

threshold

Neutron energy spectrum

nfn

nn

fn

nln PPsurviveP 2

22

21

11

12 )1()(

1n reaction channel

2n reaction channel

xn reaction channel

P<

(1 - P<)

To calculate the survival probability we need to know (for all nuclei in the xn deexcitation cascade):• ground state masses,• fission barriers,• shell correction energies and deformations (in the ground state and saddle).

Those values were calculated using the Warsaw macroscopic-microscopic model including the nonaxial shapes.

M. Kowal, Jachimowicz, A. Sobiczewski, Phys. Rev. C82 (2010) 014303M. Kowal (unpublished) A. Sobiczewski (unpublished) .

(survive)(fusion)PP1)(2lπs)σ(synthesi xn

lmax

0l

2 l

R

1

R2

r

d2 d1

sinj

Systematics of the sinj parameter from the fit to the maximum values of the experimental cross sections for 2n, 3n, 4n and 5n channels in 48Ca + X reactions (complete set of existing data)

The systematics of the sinj used to predict cross sections for Z = 119

The reaction 48Ca +252Es predicted to havemeasurable cross section

These values of the evaporation residue cross sections are much smaller (at least one order of magnitude) than previously published predictions.

V.Zagrebaev and W. Greiner, Phys. Rev. C78 (2008) 034610K. Siwek-Wilczynska, T. Cap, J. Wilczynski, IJMP E19 (2010) 500

Z = 120

The largest cross section for producing the element 120 is expected for the reaction Ti + Cf. Maximum value, of several femtobarns, is however below possibilities of present experiments.

V.Zagrebaev and W. Greiner, Phys. Rev. C78 (2008) 034610Z. H. Liu, Jing-Dong Bao, Phys. Rev C80 (2009) 054608K. Siwek-Wilczynska, T. Cap, J. Wilczynski, IJMP E19 (2010) 500A. Nasirov et al. IJMP E20 (2011) 406

SUMMARY

A modified, l-dependent version of the Fusion by Diffusion model was applied to calculate synthesis cross sections of superheavy nuclei of Z =114 – 120 in hot fusion reactions.

Fission barriers and ground state masses calculated with the Warsaw macroscopic-microscopic model (including nonaxial shapes) were applied. Good agreement with experimental cross sections was obtained.

This allowed us to use the same theoretical input to predict cross sections for synthesis of elements Z = 119 and 120.

M. Kowal, P. Jachimowicz, A. Sobiczewski, PRC 82,014303 (2010)

P. Möller et al., PRC 79, 064304 (2009)