Transcript 第三次工業革命

Decay out of A Super-deformed Band
Jianzhong Gu （顾建中）
China Institute of Atomic Energy
（中国原子能科学研究院）
(Workshop on “ Relativistic Many-Body Problems for Heavy
and Super-heavy Nuclei ” June 8-June 27, 2009, Beijing)
Outline
1 Introduction to random matrix theory (RMT)
2 The decay out problem
3 A fully solution
4 Chaoticity dependence of decay out intensity
5 Overview theoretical activities of the decay
out problem
6 Microscopic understanding of the decay out
problem
7 Summary and outlook
1. Random Matrix Theory
1950s, E. P. Wigner, Ann. Math. 53 (1951) 36; 55 (1952) 7;
62(1955)548.
Originally, dealing with the statistics of eigen-values and eigenfunctions of complex many-body quantum systems, e. g. slow
neutron resonances.
N. Bohr 1936 paper on
The Compound nucleus
Nature 137:344.
Photograph of Niels Bohr’s wooden toy
model for compound-nucleus scattering
The typical level spacing 10 eV
and the Width 1 eV
Slow neutron resonance cross section on thorium 232 and
uranium 238 nuclei, Phys. Rev. C 6 (1972) 1854.
The nuclear states are so dense that it is a hopeless task to explain
the individual states, and a statistical approach is called for.
Standard statistical
mechanics
One considers an ensemble of
identical physical systems, all
governed by the same
Hamiltonian but differing in
initial conditions, and
calculates thermo-dynamic
functions by averaging over
this ensemble.
Wigner proceeded differently: He
considered ensembles of dynamic
systems governed by different
Hamiltonians with some common
symmetry property.
A New Statistical Mechanics
This novel statistical approach
focuses attention on the generic
properties which are common to
all members of the ensemble and
determined by the underlying
fundamental symmetries.
Each of the
elements is a
random
variable.
They are
statistically
independent of
each other.
Gaussian Ensembles
Gaussian Orthogonal Ensemble (GOE):
with time reversal invariance
Gaussian Unitary Ensemble (GUE): without time reversal invariance
Gaussian Symplectic Ensemble (GSE): With time reversal invariance and
with spin-orbit coupling
for the GOE and
GUE, respectively.
Joint probability of
the independent
matrix elements
Joint probability of
the eigen-values
There are several useful statistical measures of spectral fluctuations
(1) Nearest-neighbor level-spacing distributions
Solid line: GOE
Dashed line: GUE
Dotted line: GSE
(2) Spectral Rigidity (Dyson-Mehta Statistic)
Information on correlations among level
spacings, which not contained in P(s)
For large L
GOE
GUE
GSE
RMT successfully describes the spectral fluctuation
properties of complex atomic nuclei, complex atoms,
complex molecules, quantum dots and biological
systems.
Histogram for the nearest-neighbor spacing distribution
for the nuclear data ensemble, plotted versus the level
spacing in units of its mean value. The solid line labeled
“Poisson” would apply to an integrable system.
Taken from O. Bohigas et al., Nucl. Data for Sci. and
Tech., Ed. K. Boeckhoff (1983), P. 809.
R. Haq, A. Pandey and
O. Bohigas, Phys. Rev.
Lett. 48 (1982) 1086
1. M. L. Mehta, Random Matrices, 1991 (San Diego: Academic Press).
2. T. A. Brody et al., Rev. Mod. Phys. 53 (1981) 385.
3. T. Guhr, A. Mueller-Groeling and H. A. Weidenmueller , Phys. Rep.
299 (1998) 189.
Two Breakthroughs in the Years
of 1983 and 1984
Breakthrough 1
Bohigas-Giannoni-Schmit Conjecture
The fluctuation properties of generic quantum
systems with (without) time reversal symmetry,
which in the classical limit are fully chaotic, coincide
with those of the GOE (GUE).
A generic link between RMT and fluctuation properties of
classically chaotic quantum systems with few degrees of
freedom.
Phys. Rev. Lett. 52 (1984) 1.
Spectral fluctuations for
desymmetrized Sinai’s
billiards as specified in the
upper right-hand corner of
(a). 740 levels have been
included in the analysis.
This conjecture was
proved recently (Phys. Rev.
Lett. 98 (2007) 044103)
Breakthrough 2
Supersymmetry functional integrals Originally
developed for disordered solids, proved also useful for
problems in RMT.
Initiated by K. B. Efetov,
K. B. Efetov, Supersymmetry and theory of disordered
metals, Adv. Phys. 32 (1983) 53; Supersymmetry in
disorder and chaos, Cambridge University Press
(1997).
For quantum field theory:
F. A. Berezin, Introduction to super-manifolds, Reidel,
Dordrecht ( 1987)
For nuclear physics:
J. J. M. Verbaarschot, H. A. Weidenmueller and M. R.
Zirnbauer, Phys. Rep. 129 (1985) 367.
Grassmann Variables
Grassmann variables are anticommuting
Supersymmetry Technique
One-point function (level density, transition amplitude)
Generating function
Complex commuting variables
Complex anticommuting variables
Two-point function (strength, cross section, conductance)
1. Generating function
2. Ensemble average
Supermatrix
(Graded matrix)
Supervector
Graded vector
For diagonal matrices,
their elements
Commuting;for
Off diagonal ones, their
elements
anticommuting.
Supertrace
(graded trace)
3. Hubbard-Stratonovich transformation
J. Hubbard Phys. Rev. Lett., 3 (1959) 77
4. Saddle point approximation
The problem reduces to a
zero-dimensional sigma
model,
Which is integrable.
Recently,
1. Does a more realistic stochastic modelling of many-body
systems yield the same results as the RMT predictions?
T. Asaga, L. Benet, T. Rupp and H. A. Weidenmueller,PRL 87
(2001) 01061, Ann. Phys. (N. Y.)
292 (2001) 67.
2. Ground-state dominance of 0+?
PRL 80 (1998) 2749. Many groups studied this problem.
3. Nuclear mass and quantum chaos
O. Bohigas and P. Leboeuf, Phys. Rev. Lett. 88(2002)92502
J. Hirsch, V. Velazquez and A. Frank, Phys. Lett. B
4. Zeros of Riemann Zeta function and quantum chaos
O. Bohigas, E. Bogomolny
595(2004) 231.
2 The decay out problem
Feeding and Decay Process
a beautiful double cycle between disorder and order
A nucleus Suddenly changes its shape at low spins.
Over 300 SD bands
in total up to now!
Around 200 SD
bands in these
mass regions
Total number of paths around 40 for 152Dy,
while around 100 for 192Hg.
133Nd, path completely clear, 59Cu almost clear.
The intensities of the E2 gamma transitions within a SD
band show a remarkable feature: The intra-band E2
transitions follow the band down with practically constant
intensity. At some point, the transition intensity starts to
drop sharply. This phenomenon is referred to as the
decay out of a SD band.
It is due to the mixing of the SD state and the normally deformed
(ND) states with equal (similar) spin. The barrier separating the first
and second minima of the deformation potential depends on and
decreases with decreasing spin. Decay out of the SD band sets in at a
certain spin value for which penetration through the barrier is
competitive with the E2 decay within the SD band.
The decay mechanism for the rapid depopulation?
E. Vigezzi, R. A. Broglia and T. Dossing, Nucl. Phys.
A 520 (1990) 179c; Phys. Lett. B 249 (1990) 163.
The theoretical description of the mixing between SD and
ND states uses a statistical model for the ND states first
proposed by Vigezzi et al.
The ND states to which the SD state is coupled, lie several
MeV above the ground state. The spectrum of these states
is expected to be rather complex. The ND states can be
described in terms of random--matrix theory, more
precisely, by the Gaussian Orthogonal Ensemble (GOE) of
random matrices.
The results of this approach have been used to analyze
experimental data. The formula actually used by Vigezzi et
al. is not really derived from the statistical model. It is
rather based on physically plausible and intuitive
reasoning.
3 A fully solution
The Basic Picture
J. Z. Gu and H. A. Weidenmueller, Nucl. Phys. A 660 (1999) 197.
exactly treated the model analytically and numerically
The Hamiltonian H
of the system is a
matrix of dimension
K+1 and has the
form (j,l=1,…K)
Using the
supersymmetry
approach
developed in
Phys. Rep. 129
(1985) 367
Valid for A=150, 190 mass regions
Comparison with the approach by Vigezzi et al
J. Z. Gu and H. A. Weidenmueller, Nucl. Phys. A 660 (1999) 197.
J. Z. Gu, Int. J. Mod. Phys. E 17 (Supplement ) (2008) 292.
H. A. Weidenmueller et al., Rev. Mod. Phys. 81 (2009) 539.
GW model has been used to analyzed experimental data.
For instance,
R. Kruecken et al., Phys. Rev. C 64 (2001) 064316.
A. Dewald et al., Phys. Rev. C 64 (2001) 054309.
A. N. Wilson et al., Phys. Rev. C 71 (2005) 034319.
C. J. Chiara et al., Phys. Rev. C 73 (2006) 021301(R)
They support GW model.
4 Chaoticity dependence of
decay out intensity
Aberg once concluded
that the enhancement of
the decay out of the SD
band is due to the onset
of Chaos (S. Aberg,
Phys. Rev. Lett. 82
(1999) 299.)
How does the degree of the chaoticity
affect the decay out intensity ?
We conclude that the decay-out intensity
less depends on the degree of chaoticity of
the normal deformed states, putting Aberg’s
conclusion into question!
5 Overview theoretical activities of the decay out problem
Decay out of a SD band continues to receive considerable theoretical
attention.
Sargeant et al. derived the formulae for the energy average and variance
of the intraband decay intensity . They are strictly valid when the ND
states are well overlapped.
A. Sargeant, M. Hussein, M. Pato et al., Phys. Rev. C 65 (2002) 024302.
Stafford et al. calculated the decay out intensity based on a so-called
two-level model (C. Stafford and B. Barrett, Phys. Rev. C60 (1999)
051305) where only one ND state is involved in the decay out process .
This approach could be valid when the coupling between the SD state
and ND
states is rather weak, namely spreading width is small. Very recently
this approach was used to analyze the data in the 190 mass region (D.
Cardamone, B. Barrett and C. Stafford, Phys. Lett. B 661 (2008) 233) .
However, in the 190 mass region, the decay from the SD to the normal
states is spread over many different available paths. This means the SD
state are coupled to many ND states. Therefore it is difficult to
understand how the single ND state model is able to account for the data
in the 190 mass region. In addition, we notice that the decay out intensity
based on this model depends on the same ratios as those appearing in
the Vigezzi model. The decay out intensity, therefore, is independent of
the value of the fine–structure constant, which is not physically plausible.
The two-level model was generalized by Dzyublik and Utyuzh (Phys. Rev.
C 68 (2003) 024311) a few years ago. They considered infinite
equidistant ND states in their calculations.
Shimizu et al. studied the decay out problem by using the
cranked Nilsson-Strutinsky model (Nucl. Phys. A 682 (2001)
464c; 696 (2001) 85). This model allows one to calculate the
action for the superfluid tunnelling through the potential barrier
separating the SD and ND potential wells. It predicts the
dependence of the action on the spin of the state for which
decay out of the SD band occurs. The action is related to the
spreading width. Nevertheless, the large overestimation of the
spreading width by this model has not been understood .
A cluster model was suggested to study the decay out process
by Adamian et al.( Phys. Rev. C 67 (2003) 054303; 69 (2004)
054310), in which a collective Hamiltonian depends only on a
special degree of freedom (mass asymmetry coordinate) and
determines the contribution of each cluster component to the
total wave function of a nucleus.
Mixing of a collective state with its complicated background states is ubiquitous.
Multi-phonons of nuclear giant resonances (J. Z. Gu and H. A. Weidenmueller,
Nucl. Phys. A 690 (2001) 382).
J/Psi suppression (J. Z. Gu, H. S. Zong, Y. X. Liu and E. G. Zhao, Phys. Rev. C 60
(1999)035211 ).
Life evolution (X. L. Feng, Y. X. Li, J. Z. Gu et al., J. Theor. Bio. 246 92007) 28 ).
For more examples, you are referred to H. A. Weidenmueller et al., Rev. Mod.
Phys. 81 (2009) 539.
What we have done : Established the relations between the observables.
Microscopically study the decay out process.
Such investigations could be of help to understand nuclear collective motions
(shape coexistence, decay out problem ) in a microscopic manner.
6 Microscopic understanding of the decay out problem
A=80 mass region
28
80
26
82
Zr
Zr
84
Zr
24
22
I=22
20
I=22
I=22
18
Energy(MeV)
16
14
I=16
I=16
12
I=16
10
8
6
4
2
I=0
-2
-0.6
-0.4
-0.2
0.0
I=0
I=0
0
0.2
0.4
Deformation 2
0.6
0.8
-0.6
1.0
-0.4
-0.2
0.0
0.2
0.4
Deformation 2
0.6
0.8
-0.6
1.0
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Deformation 2
Together with Wenhua Zou,Yuan Tian, Shuifa Shen, Bangbao Peng and Zhongyu Ma
Nuclei at the Limits, ANL, July 26, 2004, by C. J. Chiara from
Washington U, also Phys. Rev. C 73 (2006) 021301(R)
The method of calculation of the angular momentum projected
potential energy surfaces (AMPPES). The Hamiltonian of the
PSM (K.Hara and Y.Sun, Int. J. Mod. Phys. E 4 (1995) 637)
does not contain the Coulomb interaction of protons which
is indispensable for the potential energy surfaces. To
remedy this shortcoming of the PSM and compute the AMPPES
we combine the PSM with relativistic Hartree-Bogoliubov
(RHB) theory (D.Vretenar,A.Afanasjev,G.Lalazissis et al.,
Phys. Rep.409 (2005)101; Y.Tian,Z. Ma and P. Ring, Phys
Lett. B (in Press). We first calculate the PES with zero
angular momentum based on the RHB theory with the NL3
effective interaction for the RMF effective Lagrangian and
Gogny D1S effective pairing interaction (J.Berger,M.Girod
and D.Gogny, Nucl. Phys. A 428 (1984) 32c; Y. Tian and Z.
Ma,Chin. Phys. Lett. 23 (2006) 3226. Then we calculate the
PES with a given angular momentum in the framework of the
PSM. Finally, the energy difference between the PSM
calculated PES with a non-zero angular momentum and that
with zero spin is added to the RHB calculated PES, and a
new PES is formed. Those new PES together with the RHB
calculated PES form a group of the PES with given angular
momenta.
Together with Bang-Bao Peng, Shuifa Shen,Wenhua Zou
Bandheads here are taken from Eur.
Phys. J. A 33 (2007) 237. A HFB
approach based on the D1S Gogny
force.
Experiments: for A=190 mass region,
I=8-10 hbar at decay out points.
Calculations: the barrier gets thin and
low at such spins.
Table 1 Tunneling width (in units of eV)
J
190Hg
192Hg
0
2
4
6
8
10
12
14
16
18
16.70
14.32
12.95
11.06
6.994
2.640
6.779E-5
1.863E-7
6.953E-10
9.688E-11
47.16
46.52
30.99
30.02
15.40
7.486
6.597E-4
1.121E-4
4.030E-7
1.202E-7
194Hg
3.213
2.423
1.894
1.085
4.780E-1
2.056E-1
7.824E-2
2.110E-2
2.044E-4
1.145E-6
The tunneling width could be identical to
the spreading width, which share the same
order of magnitude as those predicted by
the GW model (for instance, R. Kruecken
et al., Phys. Rev. C 64 (2001) 064316).
Together with Bang-Bao Peng, Shufa Shen,Yuan Tian, Wenhua Zou and Zhongyu Ma
Super-Heavy Nuclei, Possible
7 Summary and outlook
A fully analytical solution to the problem of decay out of a super-deformed band.
The decay out intensity is less dependent on the degree of chaoticity of the
normal deformed states.
A new method of the potential energy surface calculation has been developed and
used to understand the decay out problem. The sudden decay is mainly due to
the barrier lowering, not the degree of the chaoticity, putting Aberg’s conclusion
into question.
Random matrix theory is a useful tool, let us penetrate the nucleus through
deterministic many-body theories and random matrix theory as well.
Future work
Decay out of a super-deformed band for super-heavy nuclei.
Warm bands (S. Leoni, et al., Phys. Rev. Lett. 101 (2008) 142502 ).
Shape coexistence, including the decay out problem based on the SCC method.
J. Z. Gu and M. Kobayasi, Commun. Theor. Phys. 47 (2007) 309.
J. Z. Gu and M. Kobayasi , Science in China Series Ｇ：Physics,Mechanics &
Astronomy (in press).
Sagawa, Yoshida, Zeng, Gu, Zhang, PRC 76 (2007) 034327
Thank you!
Stone Flower Cave, which is
close to the CIAE