aim to compare our model predictions with the measured (Dubna and GSI) evaporation cross sections for the 48Ca + 204-208Pb reactions.

Calculations of fusion-evaporation cross sections in the 48Ca + 208Pb and 48Ca +206Pb

reactionsK. Siwek-Wilczyńska, I. Skwira- Chalot, J. Wilczyński

Kazimierz 2005

in future to predict cross sections for the synthesis of super-heavy nuclei in cold and hot fusion reactions.

A collision of two heavy nuclei

Overcoming the interaction barrier

Fission

fusion

CN

„Fast fission”

ER

nn

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

For moderately heavy, asymmetric systems ( ZCN <

100 ) P(fusion) ≈ 1

(evaporation residue) ≈ (capture) × P(survive) ≈ (fusion) × P(survive)

Kazimierz 2004 - Results of calculations for two reactions:

16O + 208Pb and 12C + 236U.

For these two systems we know:• experimental evaporation-residue cross sections• experimental fusion cross sections• experimental fission barriers (saddle energies).

P(survival)

N - number of cascades which end at the ground state of a given final nucleus Ntot - the total number of generated deexcitation cascades.

totN

NsurviveP )(

The deexcitation cascade is determined at each step by branching ratios

The survival probability is calculated using the Monte Carlo method.

,tot

j

where: j = fission, n, p, d, t, , etc.

tot is the sum of all partial decay widths, including fission.

max

0*

max

2212

iE

iiii

iiii

i dE

Es

m

Partial widths for emission of light particles – Weisskopf formula

PVBEEE Cii

iroti *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 a compound nucleus in inverse process

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

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

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.) (Möller et al., At. Data

Nucl.

Data Tables 59 (1995) 185), δshell(saddle)≈ 0

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)

our calculations: diffused-barrier formula

2n 3n 4n

5n

K.Siwek-Wilczyńska, I Skwira, J. Wilczyński Phys. Rev. C 72 (2005) 004600

Experimental evaporation–residue cross sectionsT. Sikeland et al.,Phys. Rev. 169 (1968) 1000

Experimental fusion cross Sections T. Murakami et al.Phys. Rev. C 34 (1986) 1353

Experimental fusion cross sections Morton et al.Phys. Rev. C 60 (1999) 044608

Experimental evaporation–residuecross sections V.I. Zagrebaev et al.,Phys. Rev. C 65 (2001) 014607

A 2 fit to 48 experimental near-barrier fusion excitation functions

in the range of 40 < ZCN < 98 allowed for the systematics of the

three parameters B0, w, R

(K. Siwek-Wilczyńska, J. Wilczyński Phys. Rev. C 69 (2004) 024611)

How to predict capture cross section ?

The „diffused-barrier formula” ( 3 parameters):

W. Świątecki, K. Siwek-Wilczyńska, J. Wilczyński Acta Phys. Pol. B34(2003)2049; Phys. Rev. C 71 (2005) 014602

2)exp()1()( 22

E

wXXerfXREcap

integral.errorGaussian2

: 0

Xerfw

BEXwhere

Formula derived assuming:• Gaussian shape of the fusion barrier distribution• Classical expression for σfus(E,B)

data:

• σfission Yu. Ts. Oganessian,

private comunication

º σfission R. Bock et al.,

Nucl. Phys. A 388 (1992) 334

The same method used for superheavy nuclei

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

Z = 102 • experimental evaporation–residue cross sections for xn channels

• experimental symmetric and asymmetric fission (capture) cross sections

σ(capture) ≈ σfission

data : ● Yu.Ts. Oganessian et al., Phys. Rev C64 054606 (2001) ● A.V. Belozerov et al., Eur. Phys. J A16 447 (2003) ● H.W. Gäggeler et al., Nucl. Phys. A502 561c (1989)

● A.V. Yeremin et al., JINR Rapid Commun. 6 21 (1998)

calculations: σ(capture) P(survival)

ZCN = 102

experimental data:

● Yu.Ts. Oganessian et al.,

Phys. Rev. C64 054606 (2001)

● A.V. Belozerov et al.,

Eur. Phys. J. A16 447 (2003)

calculations :

σ(capture) P(survival)

ZCN = 102

ZCN = 104

data:

● F.P. Heßberger et al., Z. Phys. A359 (1997) 415

calculations:

σ(capture) P(survival)

◦ 1n ● 2n◊ 3n

● σ(capture) P(survival)

● data

P(fusion) = σexp.(synthesis)/(σ(capture) P(survival))

Summary

• Standard statistical model calculations with shell effects in the level density accounted for by Ignatyuk formula, and zero shell energy at the saddle were used to calculate cross sections for 1n, 2n, 3n and 4n channels in 48Ca + 204 - 208Pb reactions.

• The fusion probabilities reflecting the dynamical hindrance can be deduced empirically from measured evaporation residue cross sections for xn channels.

• These results can be used for empirical verification of theoretical models of the fusion hindrance process.