Transcript Slajd 1 - Department

Kazimierz 2009
What is the best way to synthesize the element Z=120 ?
K. Siwek-Wilczyńska, J. Wilczyński, T. Cap
A model for calculating the evaporation residue cross
sections for reactions with x (1,2,….) neutrons evaporated
from the compound nucleus.
Our approach is based on the assumption:
(synthesis) = (capture) × P(fusion) × P(survive)
(fusion) = (capture) × P(fusion)

Capture cross section
(capture) - the „diffused-barrier formula” ( 3 parameters):
W. Świątecki, K. Siwek-Wilczyńska, J. Wilczyński Phys. Rev. C 71 (2005) 014602,
Acta Phys. Pol. B34(2003) 2049
 cap ( E )   R2 X  (1  erf X )  exp(  X 2 )
where :
X
E  B0
,
2w
w
E 2
erf X  Gaussian error integral.
Formula derived assuming:
• Gaussian shape of the fusion barrier distribution
• Classical expression for σfus(E,B)=πR2(1-B/E)
A 2 fit to 48 experimental near-barrier fusion excitation functions
in the range of 40 < ZCN < 98 resulted in the systematics that allow
us to predict values of the three parameters B0, w, R
(K. Siwek-Wilczyńska, J. Wilczyński Phys. Rev. C 69 (2004) 024611)
For very heavy systems, a range of partial waves contributing to CN
formation is limited (critical angular momentum for disappearing macroscopic
+ microscopic fission barrier).
We propose:
cap(subcritical l ) = cap
cap(subcritical l ) = cap(E=Bo)*Bo/Ec.m.
for E ≤ Bo
for E > Bo
 systematics
of P(fusion) = Hindrance
(synthesis) = (capture) × P(fusion) × P(survive)
hindrance = P(fusion) = σexp(synthesis)/(σ(capture)  P(survival))
Hindrance deduced from experimental xn data as a function of Coulomb
interaction parameter z and the energy excess above the mean barrier Bo.
K. Siwek-Wilczyńska, A. Borowiec and J. Wilczyński, IJMP E17 (2008) 12
Y. Ts. Oganessian et al.
 P(survive) – Statistical model (Monte Carlo method)
Partial widths for emission of light particles – Weisskopf formula
i 
mi
 2

2si  1
2
Eimax
0


 i Eimax   i
 i i
d i
*
E
where: E max  E *  E i  B  V C  P
i
rot
i
i
 
Upper limit of the final-state excitation
energy after emission of a particle i
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
The fission width (transition state method),
 fiss
1

2

E max
f

 fiss E max
K
f
0
E max
 E * ( saddle)  Erot ( saddle)  P
f
 
 E*
 dK
E*< 40 MeV
Upper limit of the thermal excitation energy
at the saddle
The integrals were calculated using very accurate analytic formulas derived
by W.J. Świątecki (W.J. Świątecki et al. Phys. Rev. C78 (2008)0 54604)
The level density is calculated using the Fermi-gas-model formula
 E   exp2 aE 
• Shell effects
included as proposed by Ignatyuk
(A.V. Ignatyuk et al., Sov. J. Nucl. Phys. 29 (1975) 255)
  shell

U Ed
a  amacro 1 
1 e

U


U - excitation energy, Ed - damping parameter

where:

 shell – shell correction energy:
δshell (g.s.),
(P. Möller et al., At. Data Nucl. Data Tables 59 (1995) 185
Muntian et al. Acta. Phys. Pol. B32 (2003) 2141)
δshell(saddle)≈ 0
amacro  0.04543 r03 A  0.1355 r02 A2 3 Bsj  0.1426 r0 A1 3 Bkj
r0  1.153 fm
Ed  18.5 MeV
B s , Bk
(W. Reisdorf, Z. Phys. A. – Atoms and Nuclei 300 (1981) 227)
( W.D. Myers and W.J. Świątecki, Ann. Phys. 84 (1974) 186)
The ratio of n/f depends very strongly on the value of Bf - Bn
Bn
Bf
Bf = (saddle mass – ground state mass)
Bn = neutron separation energy
Möller et al. – P. Möller et al., At. Data Nucl. Data Tables 59 (1995) 185,
Myers & Swiatecki – W.D. Myers and W.J. Swiatecki LBL-36803
Sobiczewski et al. –I. Muntian et al. Acta. Phys. Pol. B32 (2003) 2141)
A change of Bf-Bn by 1 MeV results in the change of n/f on each
step of the deexcitation cascade by one order of magnitude.
Comparison of the measured (black dots) cross sections for synthesis of super
heavy nuclei (Z = 114-118) with our calculations (solid red and black lines)
experiment - Yu. Ts. Oganessian et al. , Phys. Rev. C69 (2004) 021601(R), Phys Rev. C70 (2004) 064609, Phys.
Rev. C74 (2006) 044602
calculations – K. Siwek-Wilczyńska, A. Borowiec, J. Wilczyński, IJMP E18 (2009) 1073
Conclusion: the ground state masses and fission bariers of Sobiczewski et al.
are the most suitable in the range of heaviest nuclei around Z = 115 and heavier.
The heaviest produced superheavy nucleus – 48Ca + 249Cf
There is a possibility to produce 296118 in
48Ca + 251Cf
299118*
296118 +3n
294118
Can we go further?
• no target heavier than Cf !!!!
• using heavier projectiles means decreasing the formation
cross section
+ 3n
Several experimental attempts to produce nuclei with Z > 118
58Fe
+ 244Pu
302-xn120*
+ xn - no events observed
Yu. Ts. Oganessian et al. Phys. Rev. C 79 (2009) 024603
The sensitivity of the experiment corresponded to a cross
section of 0.4 pb for the detection of one event.
Other reactions to produce nucleus with Z = 120
50Ti
+ 249Cf
50Ti + 251Cf
54Cr + 248Cm
58Fe + 244Pu
64Ni + 238U
299120*
301120*
302120*
302120*
302120*
Evaporation residue cross section for Z=120
Conclusion: The most promising reaction is
54Cr
+
248Cm.
(synthesis) = (capture) × P(fusion) × P(survive)
(capture) - Does not change significantly from one system to another.
Resulting uncertainties are not large unless deeply sub-barrier
reactions are studied (e.q. cold fusion)
P(fusion) - Depends on the Coulomb parameter and excitation energy.
Theoretical (or phenomenological) predictions may results in
large uncertainties of several orders of magnitude for unexplored
heavy and symmetric systems.
P(survival) - Very strong dependence on Bf-Bn easily resulting in orders of
magnitudes differences of the cross sections.
It is very important to use well tested theoretical predictions of
both, ground state and saddle masses.
Capture and fusion cross section
E = Egs + E* - total energy
En= (Mn+MA-1) c2 = Egs + Bn
Ef – the saddle-point energy
To calculate the ratio
Γn/Γf
we need the level density of
the daugther nucleus (A-1)
ρ(E-En)
and the level density at the
saddle-point of the nucleus A
ρ(E-Ef)
E-En= Egs +E*-Egs-Bn= E*- Bn
E-Ef= Egs+E*-Ef = E*- (Ef-Egs)
„Experimental” determination of fusion hindrance
86Kr
+ 136Xe
+ 204Pb
40Ar + 180Hf
48Ca + 172Yb
70Zn + 150Nd
124Sn + 96Zr
222Th
19F
+ 197Au
30Si +186W
22Ne + 194Pt
216Ra
220Th
86Kr
220U
160
40Ar
+ 178Hf
58Fe + 160Gd
64Ni + 154Sm
124Sn + 94Zr
40Ar
+ 176Hf
86Kr + 130Xe
124Sn + 92Zr
+ 134Ba
86Kr + 138Ba
48Ca
218Th
216Th
+ 208Pb
48Ca + 209Bi
50Ti + 208Pb
50Ti + 209Bi
54Cr + 208Pb
54Cr + 209Bi
58Fe + 208Pb
58Fe + 209Bi
64Ni + 209Bi
70Zn + 208Pb
224U
 102
 103
 104
 105
 106
 107
 108
 109
 111
 112
n

f
EEn
2mn r02 2sn  1A2 / 3 0
EE
2 0 f

 nn (E  En  εn )dεn
 f (E  E f  K )dK
(1)
R. Vandenbosch & J.R. Huizenga, „Nuclear Fission” - formula (VII-3)
mn, sn, εn - mass, spin and kinetic energy of the emitted neutron
f - level density of the fissioning nucleus (at saddle)
n - level density of the daughter nucleus (A-1)
E – total energy
Ef – saddle-point energy
En - energy of the system n + (A -1) nucleus
E-En= Egs +E*-Egs-Bn= E*- Bn
E-Ef= Egs+E*-Ef = E*- (Ef-Egs)
Assuming:
(E)  exp (2 aE )

, and a =const
4mn r02 A2 / 3a f ( E*  Bn )
n

exp 2 an ( E*  Bn )  2 a f ( E*  ( E f  E gs ))
f  2 a (2 a (E*  ( E  E ))  1)
n
f
gs
f

R. Vandenbosch & J.R. Huizenga, „Nuclear Fission” - formula (VII-7)
(2)

4mn r02 A2 / 3a f ( E  En )
n

exp 2 an ( E  En )  2 a f ( E  E f )
f  2 a (2 a (E  E )  1)
n
f
f

Shell effects included using:
the energy dependent level density parameter (A.V. Ignatyuk et al., Sov. J.
 E
1  e E* E d 
a E*  aLDM 1  shell



E* 

E * - excitation energy, Ed – damping parameter
Eshell – shell correction energy,
aLDM- the LDM level density parameter
 
where:
Nucl.
Phys. 29 (1975) 255)
or
an exponentially dependent fission barrier replacing the saddle-point energy
(erroneously postulated by G. G. Adamian, N. V. Antonenko and W. Scheid, Nucl. Phys. A678, 24
(2000), and their followers)
Ef – Egs = BLDM + Bmicr exp(-E*/Ed)
independent of the
excitation energy
for super-heavy nuclei BLDM= 0
Bmicr= - Eshell(gs)
dependent on the
excitation energy

4mn r02 A2 / 3a f ( E*  Bn )
n

exp 2 an ( E*  Bn )  2 a f ( E*  ( E f  E gs ))
f  2 a (2 a (E*  ( E  E ))  1)
n
f
gs
f
an, af = const

no shell effects in (A-1) nucleus
shell effects in fission channel, only via Eshell(gs)