isotope dependence of the superheavy nucleus formation cross section

Isotope dependence of the superheavy nucleus formation cr

oss section

LIU Zu-hua(刘祖华）(China Institute of Atomic Energy)

• For sufficiently heavy systems, after contact and neck formation the system finds itself outside rather than inside the conditional saddle-point of which is located in the asymmetric fission valley. As a result, automatic fusion no longer takes place after contact. Therefore, for such heavy systems, the formation cross section ( ) is taken to be the product of three factors: the cross section for the projectile and target to stick ( ), the probability for the sticking composite nucleus to reach the compound nucleus configuration ( ), and the probability for the latter to survive fission ( ).

ER

s

qfP1survP

The sticking cross section• The sticking or capture cross sections were calculated using the f

ormalisms of the neutron flow model and the coupled-channels approach , respectively. For the system of 58Fe+208Pb, at the separation s=3 fm, the neutron flow model predicts that neutrons can flow freely between the projectile and target. It is at this separation that one might expect the loss of equilibrium against neck formation to take place. In the calculations of the coupled-channels approach, the static quadrupole deformation of

• for 58Fe, and the inelastic excitations of Eex=2.6146 MeV and Eex=3.1977 MeV for 208Pb were taken into account.

199.02

-8 -4 0 4 8 12 16 20-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0V

n(r)

(MeV

)

r (fm)

208Pb 58Fe

D=14.7 fm

neutrons(2n)/2

212 214 216 218 220 222 224 226 228 23010-1

100

101

102

103

58Fe+208Pb

stic

k (m

b)

Ec.m.

(MeV)

one-dimension coupled-channel neutron flow

208 212 216 220 224 228 232100

101

102

103

stic

k (m

b)

Ec.m.

(MeV)

58Fe+208Pb

56Fe+206Pb

54Fe+204Pb

The probability to reach the compound nucleus configuration

• As a result of neutron flow the N/Z value rapidly reaches an equilibrium distribution, meanwhile the system, originally in the fusion valley, is injected into the asymmetric fission valley.

• The dynamic process of the composite nucleus in the asymmetric fission valley can be described by a diffusion process analogous to the motion of a swarm of Brownian particle suspend in a viscous fluid at temperature T in the presence of repulsive parabolic potential:

2/, 2max yxxyayxVaf

• x=s denotes the relative length between the effective surfaces of the approaching nuclei with locating the maximum in , and represents the neutron number of the light nucleus.

yxmax

afV Ny

• The probability distribution can be calculated with a two-variable Smoluchowski eqation:

where and are given by tyxWyxLyxL

t

tyxWyx ,,,,

),,(

xL yL

2

2

,,x

DyxDx

yxL xxxx

2

2

,,y

DyxDy

yxL yyyy

• We assume that the diffusion coefficients,

and are constants, i.e. ,

here and are proportional to the dissipation acting in the degrees of freedom and , respectively.

xxD yyD

xxx kTD /

yyy kTD /

x y

x y

The macroscopic deformation energy in the asymmetric fission valley of a nucleus idealized as a uniformly charged drop. The shape of the drop is parameterized by two spheres with radii R1 and R2 connected smoothly by a portion of a hyperboloid. Three variables specify a given shape: elongation, asymmetry, and neck size.

The macroscopic deformation energy is approximated by a parabolic potential:

2max2

1xxya

xVaf

• The potential governing the neutron flow can be written as

• where , , and are the masses of projectile, target and the corresponding nuclei after neutron transfer, respectively. The potentials and are calculated with the double-folding model.

TpTppTTpMMMMxVxVxyV '''',

pM TM 'pM 'T

M

pTV ''TpV

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

28 32 36 40 44-10

0

10

20

30

40

70Fe

64Fe

58Fe

52Fe

266Hs

Vaf(s

) (M

eV)

s (fm)

a)

V(y

,x)

(MeV

)

N

s=0.5fm s=2.4fm

b)

• The drift coefficient contains the driving force, .

• The drift coefficient is proportional to the driving force in the y-direction,

xD

xx xxyaD /)( max

yD

yeqy yyxbD /)(

In Smoluchowski eqation we introduced a parameter to indicate the different time scales. It is well established that in low energy heavy-ion collision, the N/Z equilibrium happens on a time scale much faster than other collective motions, such as the change in the overall length of the configuration.

In the limit , which is consistent with the assumption that y will decays very rapidly to an equilibrium value , The two-variable Smoluchowski equation

will be reduced to a one-variable Smoluchowski equation:

The operator has the form:

1

txWLtxWt

,, 0,0

0,0L

2

2

0,0x

DxDx

L xxx

dyxyyxDxD xx ,, 0

• is the eigenfunctions of the operator for

xy,0 xyLy , 0n

kT

yyxb

kT

xbxy eq

2exp

2,

2

0

• The solution of the one-variable Smoluchowski equation turns out to be a Gaussian. The probability to reach the compound nucleus configuration is equal to the area under the distribution’s tail in the region of negative values x

erfcPcomp 2

1

• , and erfc is the error function complement, equal to (1-erf). At time , the composite nucleus injects the asymmetric fission valley at . Then the barrier heights to be overcome by the diffusion Brownian particles are

kTB /0t

0x

yxxyayB max02

1

dyxyyBB 00 ,

212 214 216 218 220 222 224 226 228 230 232 23410-4

10-3

10-2

10-1

Pco

m (

%)

Ec.m. (MeV)

58Fe+208Pb

56Fe+206Pb

54Fe+204Pb

10-4

10-3

10-2

10-1 co

mp (

mb)

58Fe+208Pb

56Fe+204Pb

54Fe+204Pb

a)

Survival probability

The probability for the compound nucleus to survive fission is given by

.

The , are the neutron and fission disintegration widths, and is the probability to survive fission for the daughter nucleus after the compound nucleus emitting one neutron.

PPfn

nsurv

n fP

212 214 216 218 220 222 224 226 228 230 232 23410-10

10-9

10-8

10-7

10-6

10-5

10-4

58Fe+208Pb

56Fe+206Pb

54Fe+204Pb

Psu

rv

Ec.m.

(MeV)

• Effect of the N/Z value equilibrium

210 215 220 225 2300.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

E

R (

nb)

Ec.m.

(MeV)

PbFe 20858

evaporation cross sections

• the evaporation cross sections ( ) for the systems 54Fe+204Pb, 56Fe+206Pb, and 58Fe+208Pb

ER

210 212 214 216 218 220 222 224 226 228 230 23210-4

10-3

10-2

10-1

100

101

102

103 E

R (

pb)

54Fe+204Pb 56Fe+206Pb 58Fe+208Pb exp. Data

Ec.m.

(MeV)

summary

• The neutron flow results in the N/Z equilibrium distribution and neck formation. The system, originally in the fusion valley, is injected into the asymmetric fission valley. This dynamical process is studied on the basis of the two-variable Smoluchowski eqation. In the present study, we have focused our attention on the isotope dependence of the cross section for the superheavy nucleus formation by means of making a comparison among the reaction systems of 54Fe+204Pb, 56Fe+206Pb, and 58Fe+208Pb.

• By the comparison, we find there are three important factors infecting the final results: (1) the Coulomb barrier in the entrance channel; (2) the height of the conditional saddle-point in the asymmetric fission valley, which is dependent of the asymmetry of the di-nucleus system after reaching the N/Z equilibrium distribution; (3) the neutron separation energy of the compound nucleus. The formation cross section is found to be very sensitive to the conditional saddle-point height and the neutron separation energy. Thus we have shown the importance of the use of the neutron-rich projectile and target.