2 +

Search for shape coexistence in even-even stable
molybdenum isotopes using Coulomb excitation method
Katarzyna Wrzosek
Heavy Ion Laboratory, Warsaw University
M. Zielińska, T. Czosnyka, J. Choiński, Y. Hatsukawa, J. Iwanicki, M. Kisieli ński, M. Koizumi,
M. Kowalczyk, H. Kusakari, M. Matsuda, T. Morikawa, P. J. Napiorkowski, A. Osa, M. Oshima,
L. Reissig, T. Shizuma, J. Srebrny, M. Sugawara, K. Zajac
˛
Katarzyna Wrzosek
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 1/12
Motivation
•
•
•
•
•
Shape coexistence in medium mass nuclei was extensively studied during
recent years.
One of the interesting phenomena associated with shape coexistence is
the occurence of low-lying 0+ states in even-even nuclei. In extreme
cases the second 0+ state appears to be the first excited state.
Among stable nuclei with Z > 20 only four of them have the 0+ first excited
state, namely 72 Ge, 90 Zr, 96 Zr and 98 Mo.
Coulomb excitation method allows to determine deformation parameters
of an excited nucleus independently for each level.
Shape parameters of the two lowest 0+ states in 96 Mo, 98 Mo, 100 Mo nuclei
were inferred from Coulomb excitation data.
Katarzyna Wrzosek
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 2/12
Coulomb excitation - basic facts
•
Colliding nuclei do not interact via nuclear forces - if the distance between
nuclear surfaces is at least 5 fm, the strong interaction can be neglected
(Cline’s empirical criterion).
238
U
40
•
•
•
•
Ar
Trajectories of scattered nuclei are described classically.
Energy transfer is small compared to the energy of the projectile.
The excitation strength strongly depends on charges and masses of the
collision partners, on beam energy and scattering angle.
γ-radiation emitted by the Coulomb-excited nuclei carries the information
on their internal structure.
Katarzyna Wrzosek
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 3/12
Coulomb excitation set-ups
The data were collected using two detection set-ups:
• CUDAC (Coulex Universal Detector Array Chamber) at HIL (Poland).
Ge detector
Ge detector
target
PIN-diodes array
Ge detector
• 30 particle detectors placed at backward angles.
• 3 γ-ray detectors.
• The deexcitation γ rays measured in coincidence with backscattered
particles.
•
GEMINI + LUNA detector array at JAERI (Tokai, Japan).
Katarzyna Wrzosek
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 4/12
Experiments
2096
4
2
1870
2345
2
4
1625
1628
1498
1148
2
98
Mo
0
2
6
1432
1510
1759
2
735
0
4
2
96 Mo
778
787
2
0
0
40 Ar
+ 96 Mo (HIL, Warsaw, Poland)
20 Ne + 96 Mo (HIL, Warsaw, Poland)
96 Mo + nat Pb (JAERI, Tokai, Japan)
20 Ne
+ 98 Mo (HIL, Warsaw, Poland)
84 Kr + 98 Mo (JAERI, Tokai, Japan)
136 Xe + 98 Mo (JAERI, Tokai, Japan)
Published:
M.Zielińska et al., Acta Phys. Pol. B33, 515 (2002)
M.Zielińska et al., Nucl. Phys A712, 79 (2002)
M.Zielińska et al., Acta Phys. Pol. B36, 1289 (2005)
K. Zajac,
˛ Acta Phys. Pol. B34 2241 (2003)
K. Zajac,
˛ Int. J. Mod. Phys. E13 103 (2004)
Katarzyna Wrzosek
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 5/12
Experiments
3
1908
1847
4
1136
1771
4
1463
2
0
6
1064
2
695
100 Mo
2
536
0
40
Ar + 100 Mo (HIL, Warsaw, Poland)
20
Ne + 100 Mo (HIL, Warsaw, Poland)
Published:
K.Wrzosek et al., Int. J. Mod. Phys. E14 359 (2005)
Katarzyna Wrzosek
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 6/12
Matrix elements determination
measured γ yields
particle and γ
detectors geometry
level scheme
available
spectroscopic data
minimization procedure
GOSIA
errors estimation
set of matrix elements
information
for the sum rules code
shape parameters
for individual levels
Katarzyna Wrzosek
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 7/12
Quadrupole Sum Rules method
... is based on a fact, that zero–coupled products of E2 operators are
rotational invariants. Operator products may be expressed using the
intermediate state expansion formula:
6+1
2+3
4++1
2+2
21
0+2
0+1
Katarzyna Wrzosek
hQ2 i
√ = hi |[E2 × E2]0 | ii =
5
(
X
2
1
p
hikE2ktihtkE2kf i
=
Ii
(2Ii + 1) t
2
If
0
It
)
6+1
2+3
4++1
2+2
21
0+2
0+1
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 8/12
Quadrupole Sum Rules method
To get information on triaxiality the higher order invariant is needed:
r
2 3
Q hcos 3δi = hi |[E2 × E2]2 × E2]0 | ii =
35
(
)
X
2 2 2
1
hikE2kuihukE2ktihtkE2kii
=
(2Ii + 1) t,u
Ii It Iu
6+1
2+3
4++1
2+2
21
0+2
0+1
Katarzyna Wrzosek
6+1
2++3
4+1
2+2
21
0+2
0+1
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 9/12
Shape coexistence
+
ground state 0+
1 is deformed while the excited state 0 2 is almost spherical.
This behaviour is similar to the one observed in Ge isotopes and 96 Zr.
• 96 Mo:
• 98 Mo:
the expectation value of Q2 is almost the same for the ground state and for
the first excited state.
• 100 Mo:
the preliminary results show similar trend as observed in case of the
isotope: the expectation value of Q2 is almost the same for both 0+ states.
Deformation of 100 Mo seems to be larger compared to other Mo isotopes.
98
Mo
0.6
〈Q 2〉 [e2b2]
0+1 (ground state)
+
0 2 (excited state)
preliminary
0.4
0.2
0
72Ge
Katarzyna Wrzosek
74Ge
76Ge
96Mo
98Mo
100Mo
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 10/12
Shape coexistence
•
The triaxiality seems to be the same for each of Ge isotopes. On the contrary, the
98
Mo is triaxial while the first excited state (0+
ground state (0+
1 ) of
2 ) has a prolate
shape.
〈cos(3δ )〉
1
prolate
0
0+1 state
0+2 state
−1
Katarzyna Wrzosek
72Ge
74Ge
98Mo
oblate
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 11/12
Conclusions and future plans
•
•
•
•
In case of 96 Mo and 98 Mo the resulting sets of matrix elements were rich
and precise enough to determine shape parameters of the two 0+ states.
Data collected during two experiments performed at HIL were not
sufficient for the difficult case of 100 Mo, therefore the presented
+
and
0
deformation parameters for 0+
1
2 states are preliminary.
One more experiment with 100 Mo target using the new gamma and
particle detection set-up (12 γ-ray detectors and about 50 particle
detectors) is planned at HIL in the near future.
The simulations performed point out that all doubts should be dispelled.
We expect to determine the diagonal matrix elements with a higher
accuracy.
Katarzyna Wrzosek
XII Nuclear Physics Workshop, Kazimierz Dolny, September 2005 - p. 12/12