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Model independent determination of quadrupole deformation
parameters from Coulomb excitation measurements
XVIIth Nuclear Physics Workshop, Kazimierz Dolny 2010, September 25th
** Symmetry and symmetry breaking in nuclear physics **
Julian Srebrny( Heavy Ion Laboratory, University of Warsaw)
OUTLINE
•
Introduction: K. Kumar-idea, D. Cline – the method development and realisation
•
Formulae derivation, expectation value of
quadrupole deformation Q and triaxiality cos3δ
•
How does it really work - 104Ru example.
Nothing is easy : vibrational energy but shapes?
•
Typical stiff axially symmetric rotor 168Er
•
Transitional nuclei and important role of triaxiality 186-192Os and 194Pt
•
Low lying 0+ states - 72-76 Ge and 96-100Mo
•
Higher order invariants -
•
SUMMARY: The information about charge deformation.
The quality of collective quadrupole model descriptions.
Nuclear microscope –T. Czosnyka.
degree of stiffness or softness in Q or cos3δ
A result of Coulomb excitation experiment is the set of electromagnetic matrix
elements.
It can be 20 ÷ 60 ME for stable beam experiments.
mainly E2 collective transitional and diagonal matrix elements:
B(E2; i → f )
spectroscopic quadrupole moment
very often signs can be determined, not only absolute values
< f II E2 II i >
< i II E2 II i >
Comparing the list of experimental E2 matrix elements with model values exhibits
neither the uniqueness nor the sensitivity of the data to the collective model parameters.
Quadrupole collectivity produces strong correlations of the E2 matrix elements and
the number of significant collective variables is much lower
than the number of matrix elements.
The information about charge deformation parameters can be obtained using rotationally
invariant products of the quadrupole operators that relate the reduced E2 matrix
elements with the quadrupole deformation parameters
K. Kumar, Phys. Rev. Lett. 28 (1972) 249.
D. Cline, Annu. Rev. Nucl. Part. Sci. 36 (1986) 683.
• The two basic quadrupole invariants are formed of the
quadrupole operator tensorM(E2) in the following way
- where [··· × ···]L stands for the vector coupling to angular momentum L.
- invariants are denoted here up to coefficients as Q2 and Q3 cos 3δ,
in order to have a correspondence with collective coordinates,
< Q2 >
is an overall quadrupole deformation parameter
< cos 3δ >
is a triaxiality parameter
- since the components of M(E2,µ) with different µ’s commute with each
other the expectation values of the E2 invariants can be related to the
reduced E2 matrix elements by making intermediate state expansions:
ΣIR><RI
= 1
since the components of M (E2,µ) with different µ’s commute with each other
the expectation values of the E2 invariants can be related to the reduced
matrix elements by making intermediate state expansions:
- S denotes state S and at the same time the spin of state S alone;
R and T denotes intermediate states and their spins;
- having the experimental values of the reduced E2 matrix elements,
the expectation values of the basic quadrupole invariants
<S|Q2|S> and <S|Q3cos3δ IS> for a given state S
can be extracted from the experimental data.
4 phonon multiplet
3 phonon
2 phonon
1 phonon
Nuclear Physics A 766 (2006) 25–51
J. Srebrny, T. Czosnyka, Ch. Droste, S.G. Rohozinski,L. Próchniak, K. Zajac, K. Pomorski,
D. Cline, C.Y. Wu, A. Bäcklin, L. Hasselgren , R.M. Diamond , D. Habs, H.J. Körner,
F.S. Stephens, C. Baktash, R.P. Kostecki
β
≈ 0.28
≈ 0.26
≈ 0.21
106-110
128
similar behaviour
Pd ,
Xe
114
only
Cd looks like real vibrator
approximation:
3
3/2
< Q cos3δ > = < Q2 >
< cos3δ >
168
Er the centre of the rare earth region
+
rigid axially symmetric rotor E(2 ) = 80 keV
β ≈ 0.33 ,d ≈ 9°
similar results for
182,184
W and
174-178
Hf
prolate – oblate transitional nuclei Z= 76( Os), 78(Pt)
•
triaxial rotor, stable quadrupole deformation
and triaxiality – δ ≈ 20°
B
o
g
u
m
i
ł
a
B
a
s
a
j
Maximal triaxiality: d close to 30°
by adding 2 protons (
192
194
Os –
Pt) deformation
has jumped from prolate to oblate
prolate – oblate transitional nuclei Z= 76( Os), 78(Pt)
+
+
very low second 0 , close to first 2
72Ge:
0+(691 keV), 2+(834 keV)
in Ge: ground state - deformed and triaxial
excited state - spherical
in Mo: complicated picture,
see review talk of Katarzyna Wrzosek
The new generation of RIA: few order increase of intensity will allow on
comprehensive study of many new nuclei
74,76
The only results from radioactive beam experiments( SPIRAL):
Kr.
02 : β ≈ 0.6
d ≈ 40°
E. CLEMENT et al.
Higher order invariants allow to measure a softness of Q
2
and cos3δ
the need of longer excitation pass:
3 intermediate states for σ( Q2) and 5 intermediate states for σ(cos3δ)
SUMMARY
1. Model independent analysis of Coulomb Excitation experiment
(GOSIA) combined with non energy weighted Sum Rules
- powerful tool for quadrupole deformation parameters determination
2. Summation over double, triple or higher products of E2 matrix elements
allowed to measure in model independent way expectation values of
quadrupole deformation parameters.
3. In the future by more complicated excitation paths degree of softness
or stiffness in particular state
4. Nowadays possible mainly for stable nuclei. We got information
for more than 20 cases, including transitional nuclei.
5. Tools are ready for RIA of the new generation
6. Nuclear microscope- Tomasz Czosnyka
main authors
D. Cline, T. Czosnyka,
NSRL
Rochester
TAL
Uppsala
P. J. Napiorkowski, M. Zielinska, K. Wrzosek- Lipska, K. Hadynska-Klek, J.S.
HIL
Warsaw
D. Diamond, F. Stephens
LBL
Berkeley
C. Baktash,
BNL
Brookhaven
C.Y.Wu B. Kotlinski, R. W. Ibbotson, J.S
L. Hasselgren, A. Backlin, C. Fahlander, L.-E. Svensson, A. Kavka
E. Clement
S. G. Rohozinski
GANIL
UW,
L. Prochniak
UMCS
≈ 0.16
Rochester-Warsaw-Uppsala-Berkeley-…
Nuclear Physics A 766 (2006) 25–51
J. Srebrny, T. Czosnyka, Ch. Droste, S.G. Rohozinski,
L. Próchniak, K. Zajac, K. Pomorski, D. Cline, C.Y. Wu,
A. Bäcklin, L. Hasselgren , R.M. Diamond , D. Habs,
H.J. Körner, F.S. Stephens, C. Baktash, R.P. Kostecki
<f II E2 II i >
B(E2; i→f )
<i II E2 II i > spectroscopic quadrupole moment
98
Mo
Magda Zielińska PhD Thesis, Warsaw University 2005
Nucl. Phys. A712 (2002) 3
0.28
0.01
0.29 ±
0.02
------------------------------------
0.10
0.09
0.06
0.25 ±
0.03
-0.03
0.02
-0.01 ± 0.01
----------------------------------------------------------------
0.11
-0.04
0.02
0.09 ± 0.03
Contribution of various matrix elements to the final result
for
< 22+|Q2| 22+ >
invariant in 104Ru
contribution to the invariant [e2b2]
the component
<22+ II E2 II 2g+> <2g+ II E2 II 22+>
0.113
<22+ II E2 II 31+ > < 31+II E2 II 22+>
0.298
<22+ II E2 II 42+ > < 42+ II E2 II 22+>
0.251
<22+ II E2 II 22+ > < 22+ II E2 II 22+>
0.077
total of 4 contributions = 0.739
all contributions = 0.76(8)
SUMMARY
● thanks to GOSIA and model independent analysis we
got sets of 20-50 E2 matrix elements for many
transitional nuclei
● thanks to the Sum Rules we experimentally deduced
the shapes of many nuclei in their ground and excited
states in a model independent way:
nuclear microscope (de Broglie wavelength 0.5 fm
much smaller than radius of nucleus)
● stringent test of sophisticated microscopic collective
Q + P models, otherwise impossible
the nuclear spectroscopy
- physics of many body quantum system with finite fermions number
quantum dots, molecular clusters, ......, ....., .....
Vdef - the quadrupole deformation potential,
the dynamical variables:
β, γ - two Bohr shape deformation parameters,
Ω
- three Euler angles,
Q + P microscopic calculations of potential and all
the inertial functions, starting from the Nilsson model
Nuclear Physics A 766 (2006) 25–51
J. Srebrny, T. Czosnyka, Ch. Droste, S.G. Rohozinski,
L. Próchniak, K. Zajac, K. Pomorski, D. Cline, C.Y. Wu,
A. Bäcklin, L. Hasselgren , R.M. Diamond , D. Habs,
H.J. Körner, F.S. Stephens, C. Baktash, R.P. Kostecki