Decay out of A Super-deformed Decay out of A Super-deformed BandBand

Jianzhong Gu Jianzhong Gu （顾建中）（顾建中）

China Institute of Atomic EnergyChina Institute of Atomic Energy

（中国原子能科学研究院）（中国原子能科学研究院）

(Workshop on “ Relativistic Many-Body Problems for (Workshop on “ Relativistic Many-Body Problems for Heavy Heavy

and Super-heavy Nuclei ” June 8-June 27,and Super-heavy Nuclei ” June 8-June 27, 2009, Beijing)2009, Beijing)

OutlineOutline

11 Introduction to random matrix theory (RMT)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 1. Random Matrix Theory 1950s, E. P. Wigner, Ann. Math. 53 (1951) 36; 55 (1952) 7;1950s, E. P. Wigner, Ann. Math. 53 (1951) 36; 55 (1952) 7; 62(1955)548.62(1955)548.

Originally, dealing with the statistics of eigen-values and Originally, dealing with the statistics of eigen-values and eigen-eigen-

functions of complex many-body quantum systems, e. g. functions of complex many-body quantum systems, e. g. slow slow

neutron resonances.neutron resonances.

N. Bohr 1936 paper onThe Compound nucleusNature 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.

There are several useful statistical measures of spectral fluctuations

(1) Nearest-neighbor level-spacing distributions

Joint probability of the independent matrix elements

Joint probability ofthe eigen-values

Solid line: GOE

Dashed line: GUE

Dotted line: GSE

GOE

(2) Spectral Rigidity (Dyson-Mehta Statistic)

Information on correlations among level spacings, which not contained in P(s)

GUEGSE

For large L

RMT successfully describes the spectral RMT successfully describes the spectral fluctuation properties of complex atomic nuclei, fluctuation properties of complex atomic nuclei, complex atoms, complex molecules, quantum complex atoms, complex molecules, quantum dots and biological systems. 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 1. M. L. Mehta, Random Matrices, 1991 (San Diego: Academic Press).Press).

2. T. A. Brody et al., Rev. Mod. Phys. 53 (1981) 385.2. T. A. Brody et al., Rev. Mod. Phys. 53 (1981) 385. 3. T. Guhr, A. Mueller-Groeling and H. A. Weidenmueller , 3. T. Guhr, A. Mueller-Groeling and H. A. Weidenmueller , Phys. Rep. 299 (1998) 189.Phys. Rep. 299 (1998) 189.

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.

Two Breakthroughs in the Two Breakthroughs in the YearsYears

of 1983 and 1984of 1983 and 1984

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 anticommuting variables

Complex commuting variables

Two-point function (strength, cross section, conductance)

1. Generating function

2. Ensemble average

Supermatrix(Graded matrix)

For diagonal matrices, their elements Commuting;forOff diagonal ones, their elements anticommuting.

SupervectorGraded vector

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 595(2004) 231.

4. Zeros of Riemann Zeta function and quantum chaos

O. Bohigas, E. Bogomolny

2 The decay out problem

Feeding and Decay ProcessFeeding and Decay Processa beautiful double cycle between disorder and ordera beautiful double cycle between disorder and order

A nucleus Suddenly changes its shape at low spins.

Total number of paths around 40 for 152Dy,

while around 100 for 192Hg.

133Nd, path completely clear, 59Cu almost clear.

Over 300 SD bands in total up to now!

Around 200 SD bands in these mass regions

The intensities of the E2 gamma transitions within a The intensities of the E2 gamma transitions within a SD band show a remarkable feature: The intra-band E2 SD band show a remarkable feature: The intra-band E2 transitions follow the band down with practically transitions follow the band down with practically constant intensity. At some point, the transition constant intensity. At some point, the transition

intensity starts tointensity starts to drop sharplydrop sharply. . This phenomenon This phenomenon is referred to as the decay out of a SD band. is referred to as the decay out of a SD band.

It is due to the It is due to the mixingmixing of the SD state and the normally of the SD state and the normally deformed (ND) states with equal (similar) spin. The barrier deformed (ND) states with equal (similar) spin. The barrier separating the first and second minima of the deformation separating the first and second minima of the deformation potential depends on and decreases with decreasing spin. Decay potential depends on and decreases with decreasing spin. Decay out of the SD band sets in at a certain spin value for which out of the SD band sets in at a certain spin value for which penetration through the barrier is competitive with the E2 decay penetration through the barrier is competitive with the E2 decay within the SD band.within the SD band.

The decay mechanism for the rapid The decay mechanism for the rapid depopulation?depopulation?

E. Vigezzi, R. A. Broglia and T. Dossing, Nucl. E. Vigezzi, R. A. Broglia and T. Dossing, Nucl. Phys. A 520 (1990) 179c; Phys. Lett. B 249 Phys. A 520 (1990) 179c; Phys. Lett. B 249

(1990) 163.(1990) 163. The theoretical description of the mixing between The theoretical description of the mixing between

SD and ND states uses a statistical model for the SD and ND states uses a statistical model for the ND states first proposed by Vigezzi et al.ND states first proposed by Vigezzi et al.

The ND states to which the SD state is coupled, lie The ND states to which the SD state is coupled, lie several MeV above the ground state. The spectrum several MeV above the ground state. The spectrum of these states is expected to be rather complex. of these states is expected to be rather complex. The ND states can beThe ND states can be

described in terms of random--matrix theory, more described in terms of random--matrix theory, more precisely, by the Gaussian Orthogonal Ensemble precisely, by the Gaussian Orthogonal Ensemble (GOE) of random matrices.(GOE) of random matrices.

The results of this approach have been used to The results of this approach have been used to analyze experimental data. The formula actually analyze experimental data. The formula actually used by Vigezzi et al. is not really derived from the used by Vigezzi et al. is not really derived from the statistical model. It is rather based on physically statistical model. It is rather based on physically plausible and intuitive reasoning.plausible and intuitive reasoning.

3 A fully solution

The Basic PictureThe Basic Picture

J. Z. Gu and H. A. Weidenmueller, Nucl. Phys. A 660 J. Z. Gu and H. A. Weidenmueller, Nucl. Phys. A 660 (1999) 197.(1999) 197.

exactly treated the model analytically and numericallyexactly 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 Using the supersymmetrsupersymmetr

y approach y approach developed in developed in

Phys. Rep. 129 Phys. Rep. 129 (1985) 367(1985) 367

Valid for A=150, 190 mass regions

Comparison with the approach by Comparison with the approach by Vigezzi et alVigezzi 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.GW model has been used to analyzed experimental data.

For instance,For instance,

R. Kruecken et al., Phys. Rev. C 64 (2001) 064316.R. Kruecken et al., Phys. Rev. C 64 (2001) 064316.

A. Dewald et al., Phys. Rev. C 64 (2001) 054309.A. Dewald et al., Phys. Rev. C 64 (2001) 054309.

A. N. Wilson et al., Phys. Rev. C 71 (2005) 034319. A. N. Wilson et al., Phys. Rev. C 71 (2005) 034319.

C. J. Chiara et al., Phys. Rev. C 73 (2006) 021301(R)C. J. Chiara et al., Phys. Rev. C 73 (2006) 021301(R)

They support GW model.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 We conclude that the decay-out intensity less depends on the degree of chaoticity of less depends on the degree of chaoticity of the normal deformed states, putting the normal deformed states, putting Aberg’sAberg’s

conclusion into question!conclusion into question!

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.

5 Overview theoretical activities of the decay out problem

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 What we have done : Established the relations between the observables.observables.

Microscopically study the decay out process.Microscopically study the decay out process.

Such investigations could be of help to understand nuclear collective Such investigations could be of help to understand nuclear collective motions motions

(shape coexistence, decay out problem ) in a microscopic manner.(shape coexistence, decay out problem ) in a microscopic manner.

6 Microscopic understanding of the decay out problem

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-2

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

I=0

I=16

I=22

Ener

gy(M

eV)

Deformation 2

Zr80

I=0

Zr82

Deformation 2

I=22

I=16I=16

I=22

I=0

Zr84

Deformation 2

Together with Wenhua Zou,Yuan Tian, Shuifa Shen, Bangbao Peng and Zhongyu Ma

A=80 mass region

Nuclei at the Limits, ANL, July 26, 2004, by C. J. Nuclei at the Limits, ANL, July 26, 2004, by C. J. Chiara from Chiara from

Washington U, also Phys. Rev. C 73 (2006) Washington U, also Phys. Rev. C 73 (2006) 021301(R)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 194Hg

024681012141618

16.7014.3212.9511.066.9942.640

6.779E-51.863E-76.953E-109.688E-11

47.1646.5230.9930.0215.407.486

6.597E-41.121E-44.030E-71.202E-7

3.2132.4231.8941.085

4.780E-12.056E-17.824E-22.110E-22.044E-41.145E-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. R. Kruecken et al., Phys. Rev. C 64 (2001) 064316).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!Thank you!

Stone Flower Cave, which is close to the CIAE