Proceedings of the International Workshop
Fission Dynamics of Atomic Clusters and Nuclei
Fission Dynamics of Atomic Clusters and
Proceedings of the International Workshop
Fission Dynamics of Atomic Clusters and Nuclei Luso, Portugal
1 5 - 1 9 May 2000
edited by Joao da Providencia University of Coimbra, Portugal
David M. Brink Oxford University, UK
Feodor Karpechine St. Petersburg State University, Russia
F. Bary Malik Southern Illinois University at Carbondale, USA
fe World Scientific ll
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Proceedings of the International Workshop on FISSION DYNAMICS OF ATOMIC CLUSTERS AND NUCLEI Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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PREFACE The idea of this workshop was conceived in the spring of 1999 in Coimbra while discussing the contents of a paper by one of our colleagues at Coimbra and by one of us (FK) dealing with the fission path in the decay of atomic clusters. Many of the ideas seemed to be similar to concepts used in describing the fission of atomic nuclei. Since the discovery of atomic clusters, many stimulating new experiments have been performed and ideas have been proposed to clarify observed physical properties. Notwithstanding the different terminology, the underlying physics is similar. Prior to 1999 there had already been two interdisciplinary conferences on clusters of atoms and clustering in nuclei but the broad scope of those conferences did not allow sufficient time to focus on fission dynamics. A proper understanding of this process is likely to determine the limits on the production of man made super-heavy elements in nuclear physics and also the scope and possibility of linking atomic clusters to mesoscopic atomic structures. At the beginning of this new century, it seemed to be fruitful to communicate the ideas used in the sixty-year old field of nuclear fission to scientists studying the relatively new and uncharted territory of fission of atomic clusters. At the beginning of the 21st century, the time was ripe to take stock in the past and plan a vision for the future. We were particularly encouraged when Professor John A. Wheeler, a founding father of the theory of nuclear fission and Professor Ben Mottelson, a prime mover in promoting cross-fertilization of ideas between the two areas, agreed to be patrons of such a workshop. The cross-fertilization requires an amicable atmosphere where people are ready to listen to and learn from each other rather than promoting their own viewpoint. What location could be more suitable than Luso with its serene and beautiful natural surroundings! It is located in Portugal, a country famous for being the home of many famous pioneers with a passion to discover unexplored areas of planets. The participation of Professor John A. Wheeler was a tremendous inspiration to all participants. His deep commitment to science in general and nuclear fission in particular over seven decades set the tone of the workshop. Fission has no parallel in classical physics and embodies the quantum phenomenon of meta-stable many-body complex systems. As such, the physics of fission dynamics encompasses many branches of physics. Understanding fission has led to the creation of super-heavy elements and the production of large atomic clusters. In this workshop, many new discoveries were reported: the production of new elements, the development of new techniques to measure very short halflives from less than a microsecond to longer than 1018 years, the study of cold and hot fission and of new forms of ternary fission. The many aspects of nuclear
fission on one hand, and of Coulomb explosions and other violent fissions of metal clusters and atomic systems on the other hand, represent a vast range of rich physics setting the tone of new ideas to come. Density wave instability and the consequences of the formation of a neck of low-density-nuclear-matter prior to scission put a limit to man's ability to create elements beyond a certain mass. The susceptibility of metal clusters to violent excitations discussed in the workshop provided fascinating insight into the fission of atomic clusters. The rebirth of ideas of alpha particle like structure on nuclear surface and the use of negative muons to study the fission path, originally envisioned by John A. Wheeler and noted in the workshop, will certainly generate new challenges to theorists in the future. Experiments using kaons, heavy-leptons, lambda and deltas to chart the fission path will challenge the ingenuity of experimental physicists. Judging from the conversations at the workshop and the enthusiasm of the participants, the workshop succeeded in taking stock of the past in many of these areas and projecting a vision for the future. The organizers wish to acknowledge the generous financial support received from the following institutions or organizations: Faculdade de Ciencias e Tecnologia da Universidade de Coimbra, Fundagao para a Ciencia e Tecnologia, INTAS, Fundagao Calouste Gulbenkian, Fundacao Luso-Americana para o Desenvolvimento, and Grupo Teorico de Altas Energias.
CONTENTS Preface Fission kills nuclear ropes, loops and chains
1
J. A. Wheeler Semiclassical calculation of shell effects in deformed nuclei
5
M. Brack and Ch. Aman Coupling of the collective pairing vibrations with the fission mode K. Pomorski
13
Fission studies with large detector arrays J. K. Hwang, C. J. Beyer, A. V. Ramayya, J. H. Hamilton, G. M. Ter Akopian, A. V. Daniel, J. 0. Rasmussen, S. -C. Wu, R. Donangelo, J. Kormicki, X. Q. Zhang, A. Rodin, A. Formichev, J. Kliman, L. Krupa, Yu. Ts. Oganessian, G. Chubaryan, D. Seweryniak, R. V. F. Janssens, W. C. Ma, R. B. Piercey and J. D. Cole
21
Adiabatic and non-adiabatic dynamics of ions in the formation of electron excitation widths in metal clusters A. V. Solov'yov
37
Fission, decay of nuclei and the extension of the periodic system W. Greiner
45
Fission in (the light of) a quark-gluon liquid theory of the nucleons H. Bohr
65
Statistical modeling of nuclear systematics J. W. Clark, E. Mavrommatis, S. Athanassopoulos, A. Dakos and K. Gernoth
76
Toroidal structures in light nuclei A. Arriaga
86
Neutron Halo of Fissile Nuclei V. I. Serov, S. N. Abramovich and F. F. Karpeshin
94
Search for possible isomer 2 3 2 Pa E. F. Fomushkin, S. N. Abramovich and M. F. Andreev
104
Broken symmetry in nuclear matter A. W. Overhauser and A. E. Pozamantir
108
The diverse manifestations of fission of classical and quantal droplets H. J. Krappe
117
Energy-density functional approach to fission and half-lives of superheavy elements
127
/. Reichstein and F. B. Malik Decay channels of hot nuclei and hot metalic clusters P. Frobrich
135
Cluster radioactivity A. A. Ogloblin, G. A. Pik-Pichak and S. P. Tretyakova
143
Static and dynamical properties of simple metal clusters: Analogies with atomic nuclei J. A. Alonso, M. Barranco, F. Garcias, P. -G. Reinhard and E. Suraud
163
The role of exotica in studying nuclear fission F. F. Karpeshin
192
Particle-acompanied fission M. Mutterer, Yu. N. Kopatch, D. Schwalm and F. Gonnenwein
204
Alpha-particle structure on the surface of the atomic nucleus M. Brenner
214
Superasymmetric fission
221
V. A. Rubchenya Quaternary fission F. Gonnenwein, P. Jesinger, M. Mutterer, A. M. Gagarski, G. A. Petrov, W. H. Trzaska, V. Nesvishevsky and 0. Zimmer
232
Bremsstrahlung emission in alpha-decay D. M. Brink
248
A model for particle induced fission
256
F. B. Malik
On the half-lives of trinuclear molecules
266
A. Sdndulescu, F. Carstoiu, I. Bulboacd and W. Greiner Fusion-fission reactions of heaviest nuclei. Synthesis of superheavy elements with Z = 114 and 116
275
Yu. Ts. Oganessian, M. G. Itkis and V. K. Utyonkov Puzzling results on nuclear shells from the properties of fission channels
302
K. -H. Schmidt, A. R. Junghans, J. Benlliure, C. Bockstiegel, H. -G. Clerc, A. Grewe, A. Heinz, M. De Jong and S. Steinhduser Emission of light charged particles from fission fragments of uranium nuclei induced by slow negative muons and pions, protons with energy of 153 and 1000 MeV and negative pions at 1700 MeV G. E. Belovitzkiy and 0. M. Shteingrad
311
Possible explanation of the difference in nuclear fission induced by the intermediate energy protons and neutrons V. E. Bunakov, L. V. Krasnov and A. V. Fomichev
318
Scission configurations in cold fission S. Mi§icu, P. Quentin and N. Pillet
324
1
FISSION KILLS NUCLEAR ROPE, LOOPS, AND CHAINS John Archibald Wheeler Joseph Henry Laboratories Princeton University Princeton, NJ 08544-0609 and Center for Theoretical Physics University of Texas Austin, TX 78712 For the spoken version of this talk, I brought with me, and passed around, a sample of graphite from mankind's first nuclear reactor, the so-called CP-1. Saturday, December 2000 marks the 58th anniversary of that day when Fermi put into operation under the West Stands of the University of Chicago's Stagg Field the forerunner of hundreds of reactors that today supply electricity for millions of our fellow humans all around Magellan's globe. The very mention of the name "Fermi" reminds me of the adventurous quality of the lunches that he and I and half a dozen others used to share at the Metallurgical Laboratory in Chicago during World War H To motivate the participants, Fermi attached a standard fee of one dime, ten cents, to each item discussed. Point out to him that his pronunciation "foonk-she-own" differed from the standard "funk-shun" and see the others around the table nod agreement and all watch as he paid out the dime. "Look out the window," he'd say by way of return challenge, "and tell me why today the clouds have that fantastically regular pattern as if they were a pile of pipes laid out side by side on a field." We struggled mightily with that question as we sat around the table, but did not come up with a thoroughly satisfying response. Fermi then gave us a little lecture on the Rayleigh-Taylor theory of hydrodynamic instability. Then we passed our dimes to Fermi.
2
"Now for a new challenge, Professor Fermi," I said. "Someday, surely, the world will have nuclear rope unbelievably finer and stronger than a strand of silk. When your car breaks down, the tow truck, even if it is a block ahead of you, will run a strand of this nuclear ropefromthe truck to the bumper of your car. But on the long tow tripfromyour breakdown spot to the automobile repair shop, that invisible nuclear rope, finer than gossamer, will slice in two every car coming in from the side and perhaps maim their drivers and passengers. Laws must be made to force the tow truck operator to hang red flags all along the length of that nuclear rope." Neither Fermi nor anyone around the table seemed alarmed by the danger I had sketched. I sensed that Fermi felt that nuclear rope was not a likely development. Today, let me present to you a stronger conclusion: Spontaneous nuclear fission will quickly destroy the rope, breaking it into thousands of tiny fragments—and destroy also other materials that one might imagine building out of nuclear matter—destroy nuclear rope, nuclear doughnuts, and nuclear chains, in a time less than the time for an electron to move across an atom. I am truly unhappy to have to accept this negative assessment of the building potential of nuclear matter. Build connections, bring people together, and in that way add to the health and happiness of mankind. History shows us the benefits of roads, rails, canals, tunnels, and bridges. Visit the tunnel (or "Chunnel") that connects England with the European Continent. Here and now (I said in Portugal), let me pass around a piece of chalk that I gotfromthe diggings of that tunnel. Marvel at the thousands of people, cars, and lorries that zoom aboard trains each day under the water, going one way or the other.
3
Whoever thrills at this sight, as I do, let him join in beating the drum for more bridges that draw the world closer together. Bridge builders have shown us how to bundle wires together to make cables and how to use long cables to support a long bridge. Anybody who brings an observant eye to New York knows some of its great bridges and how they are supported by wire rope cables. Look at the Brooklyn Bridge, a pioneer (1883) in capitalizing on that new form of support. Later came New York's Bronx-Whitestone Bridge and its better known cousin, the George Washington Bridge But all those bridges belong to the last millennium. Today we, as creatures of a new millennium, have wider ambitions for what bridges can be and do for the world. Each of us will have his own ideas. For a starter, consider spanning the Gates of Hercules at Gibraltar to unite Morocco and the Iberian peninsula. Or link two great industrial powers, Japan and South Korea, via a two-way connection across the Pusan Strait to the Island of Tsushima. And let trainsfromLondon and Paris and Moscow cany precious perishable cargo to San Francisco, Toronto, and New York via Siberia and Alaska, across the International Dateline, the Bering Strait, and such an island as little Diomede. Don't let anybody say NO to such prospects merely because fission kills nuclear rope. And why shouldn't it? A thin jet of water, we all know, breaks up spontaneously into droplets, their size governed by the balance between the destabilizing action of surface tension and the inertial drag of density. Niels Bohr, in his student days in Copenhagen, had analyzed such breakup theoretically and experimentally, providential preparation for his later contribution to nuclear fission. We do not need the teaching of
4
Bohr to tell us thatfissionbreaks a rope of water, but his work on nuclearfissionmakes us realize why nuclear rope breaks up spontaneously. I have gonefromfissionto rope,fromrope to cables built of many wire ropes and to the bridges that have been built of such cables, then on to the human need for longer bridges and, therefore, stronger cables. Fortunately, recent years have seen a new development that promises rope stable against fission and able to withstand stresses much higher than even the strongest metals can bear. I refer to the exciting new development, carbon nanotubes, discovered in 1991. Single wall carbon nanotubes (SWCNT) are, in effect, graphite sheets rolled up into tiny cylinders. Some specific defect-free forms of SWNCT show remarkable mechanical properties and metallic behavior. Hundreds of such "metal" tubes have been bundled side by side to make carbon "ropes" more than ten times as strong per unit area as steel or carbon-reinforced polymers. So we can shed a tear for the instability of nuclear rope but smile at the strength of carbon nanotubes, and wonder what marvel may come next.
5
SEMICLASSICAL CALCULATION OF SHELL E F F E C T S IN D E F O R M E D NUCLEI
Institute
M. B R A C K , C H . A M A N N for Theoretical Physics, University of D-93040 Regensburg, Germany
Regensburg
We summarize recent work in which the shell effect, which causes the onset of the mass asymmetry in nuclear fission, could be explained semiclassically in the framework of the periodic orbit theory. We also present new results for the inclusion of a spin-orbit interaction in the semiclassical calculation of the level density.
1
Introduction
The explanation of the asymmetry in the fragment mass distributions of many actinide nuclei has long been a stumbleblock in fission theory. Shortly after the advent of the shell model, Lise Meitner 1 related the mass asymmetry to shell structure, in particular to magic nucleon numbers in one of the fragments. Indeed, since the classical liquid drop model (LDM) favours symmetric shapes, 2 the mass asymmetry had to be a quantum shell effect. Johansson 3 attempted to calculate fission barriers by summing the single-particle energies of a Nilssontype shell model including left-right asymmetric deformations. However, due to their wrong average behaviour at large deformations, he could not obtain finite barriers. Only after the introduction of the ingenious shell-correction method by Strutinsky 4 this became possible, launching large-scale investigations of the shell structure in fission barriers. 5 ' 6 ' 7 The resulting picture of the doublehumped fission barrier was nicely confirmed in many detailed experiments. 8 One interesting outcome was that the onset of the left-right asymmetry of the nuclear shapes starts quite early during the fission process: in nuclei like 240 Pu, already the outer fission barrier is unstable against octupole-type deformations. 9 This quantum shell effect could be related to a few specific diabatic single-particle states which were particularly sensitive to the asymmetric deformations.10 Today, we know that symmetric and asymmetric fission modes may coexist in the same nucleus. 11 Of course, the mass distributions of the fission fragments can only be predicted from a dynamical theory involving inertial parameters. 12 But interestingly enough, the most probable experimental mass ratios were found to be roughly equal to those of the nascent fragments obtained statically at the asymmetric outer barrier. 6,13 The onset of the mass asymmetry in nuclear fission was thus established as a quantum shell effect due to specific single-particle states in the deformed average potentials of the nucleons, which could not be understood in the classical liquid drop model.
6
However, new developments have taken place over the last 25 years to describe quantum shell effects semiclassically in the framework of the periodic orbit theory (POT). Through the so-called trace formula, the oscillating part Sg (E) of the level density of a quantum system can be related to properties of the periodic orbits of the corresponding classical system. This approach was systematically developed by Gutzwiller;14 an alternative derivation valid for billiard systems (i.e., particles enclosed in a cavity with ideally reflecting walls) was given by Balian and Bloch. 15 Strutinsky et al.16'17 generalized the Gutzwiller theory and used it for the semiclassical calculation of shellcorrection energies in various shell-model potentials (neglecting, however, the spin-orbit interaction). In particular, they successfully explained the systematics of nuclear ground-state deformations in terms of the leading shortest periodic orbits. 17 Similarly, the onset of the fission mass asymmetry could recently be explained semiclassically.18 For a comprehensive presentation of the POT, the trace formula and its further refinements, and its applications to finite fermion systems, we refer to a recent monograph. 19 In Sect. 2 we summarize the recent work on the fission asymmetry done by one of us (M.B.) in collaboration with P. Meier, S. M. Reimann, and M. Sieber. 18,20 ' 21 In Sect. 3 we present some new semiclassical results for the level density of a three-dimensional deformed potential, including the spin-orbit interaction which so far has been omitted in applications of the POT to the calculation of shell structure in nuclear physics. 2
Semiclassical explanation of the onset of m a s s asymmetry
In our recent semiclassical investigation of the fission mass asymmetry, 18 ' 20 we have approximated the nuclear mean field by a cavity with constant volume and reflecting walls, neglecting spin-orbit, Coulomb, and pairing interactions and considering only one kind of particles. We use the parameterization (c, h, a) of Ref.6 to define the boundary of the cavity. Here 2c is the length of the nucleus along the symmetry axis in units of the radius i? 0 of the spherical cavity (given by c = l , /i=a=0), h is the neck parameter, and a / 0 yields left-right asymmetric shapes. In 2 4 0 Pu, e.g., the maximum of the symmetric outer barrier is found6 near c = 1.53, h — a = 0. The semiclassical trace formula for the shell-correction energy 8E reads 16,19
SE-J2AV(EF)(~) po
COS
\
SPO(EF)
'po ,
(1)
po /
The sum in (1) is taken over all periodic orbits (po), but the gross-shell effects are dominated by the shortest periodic orbits of the system. Sp pdq
Figure 1: Right: Semiclassical deformation energy 5E(c, a) (h=0) in a perspective 3D plot (from Ref.18). The arrows 'symmetric' and 'asymmetric' show two alternative fission paths. Left: shapes along the asymmetric path. Planes of the leading periodic orbits are shown by vertical lines (solid for stable and dashed for unstable orbits). are the action integrals, Tpo = dSpo/dE the periods, and apo are phases related to the number of conjugate (or focal) points along the periodic orbits. The amplitude Apo of an orbit depends on its stability and its degeneracy; it includes a factor that exponentially damps contributions from orbits with large periods Tpo. All quantities in (1) are evaluated at the Fermi energy Ep. In Fig. 1 we show a perspective 3D plot of the deformation energy 5E(c, a) (h = 0) obtained from Eq. (1). The missing effect of the spin-orbit interaction was compensated by adjusting the only parameter of the model, EF, such that the isomer minimum is found at c = 1.42, h = a = 0, corresponding to the quantum result for 2 4 0 Pu in the realistic Strutinsky calculations. 6 We see that the semiclassical model correctly yields the instability of the outer barrier against the asymmetry, and a realistic value (~ 0.13) of the parameter a at the asymmetric saddle point. Inclusion of the LDM energy, which varies very little in the deformation region shown here, does not change this result qualitatively. To the left in Fig. 1 we show the nuclear shapes at three points (A,B,C) along the adiabatic fission path. The vertical lines indicate the planes perpendicular to the symmetry axis in which the shortest periodic orbits are found. These are simply the polygons with p reflections inscribed in the circular cross section of the cavity with these planes. A fast convergence of 8E with increasing p, i.e., with the length of the periodic orbits, was found in Ref.20 (p = 2 and 3 were essentially sufficient); other periodic orbits were also shown to be negligible. We thus obtain a simple physical explanation of the onset of the mass asymmetry of fission in terms of very few classical periodic orbits. The valley
8
of steepest descent through the deformation energy surface is simply given 18 by the requirement that the action of the leading periodic orbits be stationary: SSpo = 0. We have also shown 21 that the diabatic states which cause this asymmetry effect quantum-mechanically have their probability maxima exactly on the planes containing the leading periodic orbits, and that a semiclassical quantization of the quasi-regular motion near these planes reproduces the energy levels of the diabatic quantum states surprisingly well. Our semiclassical picture is only qualitative. It is not meant to replace the Strutinsky (or Hartree-Fock) type calculations, but to give an intuitive physical understanding of the mass asymmetry using classical mechanics. Of course, one has to question if the neglect of the spin-orbit interaction can be justified since it is known to crucially affect the shell structure. Locally, its effect could here be simulated (like in Ref.17) by adjusting the Fermi energy. But a semiclassical calculation including spin-orbit effects is certainly desirable. 3
Inclusion of spin-orbit interaction in P O T
We present now some new results including a spin-orbit interaction in the semiclassical trace formula. The cavity model of the previous section leads to problems with the standard (Thomas) form of the spin-orbit interaction which contains the gradient of the local potential. We therefore use for V(r) a 3-dimensional deformed harmonic-oscillator potential and add a generalized spin-orbit term which in the spherical limit is proportional to 1 - cr: H=^-p2
+ V(r) + KB{r,p)-a
(2)
with nr) = y
£
w?r?,
B(r,p) = W ( r ) x p .
(3)
i=x,y,z
In Eq. (2), a = {(7x,ay,az) are the Pauli matrices. We allow for different frequencies u>i to mimic the shell-model potential of a deformed light nucleus. The quantum-mechanical eigenvalues of (2) were obtained by an exact diagonalization in the basis of V(r) up to a maximum energy of ~ 150ftwj;. A first clue to the classical behaviour of this system is gained from a Fourier transform of the oscillating part 6g (E) of the level density from the energy to the time domain. To emphasize the gross-shell structure, the level density is Gaussian averaged over an energy range 7 < TioJi. Due to the energy scaling behaviour 22 of the Hamiltonian (2), the peaks in the Fourier spectrum lie at the time periods Tpo of the classical orbits that dominate Sg (E). In Fig. 2 we show the Fourier spectra obtained for a case with irrational frequency ratios, both without and with spin-orbit interaction. For K = 0, we clearly see three
Figure 2: Squared Fourier amplitudes (in arbitrary units) of 5g (E) for the anisotropic harmonic oscillator (2) with frequencies uix = 1, w„ = 1.12128, UJZ = 1.25727, both without (left) and with spin-orbit interaction (right). Energy eigenvalues up to ~ 50 hcjx were included; the Gaussian averaging range was 7 = 0.2 huix. main peaks which correspond exactly to the periods T{ = 2n/uJi of the only three periodic orbits of this system, which are the librations along the principal axes. 19 At twice these periods, three small peaks corresponding to their second harmonics are visible. For K = 0.1, a more complicated peak structure is observed, which will be explained in the following. The difficulty in the semiclassical description is that the Pauli operators in (2) act on a two-component spinor and couple its components in a nontrivial way. Littlejohn and Flynn 23 have developed a theory whose basic idea is to diagonalize the Hamiltonian (2 x 2) matrix locally in phase space after its Wigner transformation, and to expand both Hw(r,p) and the diagonalizing unitary matrix in powers of K. Two problems arise hereby. First, on those surfaces in phase space where B(r, p) = 0, the eigenvalues become degenerate and the method breaks down (so-called mode conversion). Second, one obtains gauge-transformation dependencies of the diagonal matrix. This problem can in principle be solved by the introduction of non-canonical coordinates. 23 A more heuristic solution, proposed by Frisk and Guhr, 24 leads to the determination of the dynamics to lowest order in K by the two classical Hamiltonians (4) ^(r'p)=2^p2+y(r)±'c|B(r,P)l whose combined periodic orbits are to be used in the trace formula. The firstorder h corrections to the energy, in Ref.23 denoted by \%erry and \%N, are to be included perturbatively, i.e., only their contributions to the actions have to be included in the trace formula. This ad hoc prescription has recently been justified through a relativistic trace formula derived from the Dirac equation by Bolte and Keppeler. 25 From the non-relativistic reduction of their result, the use of the Hamiltonians (4) and the above treatment of XBerry and A^ w can, indeed, be obtained in the "large-spin" limit. 25
10
The equations of motion for the Hamiltonians H^ with (3) become ri=
Pi
pi = -Jfn
± tijk |B(r,p)| -1 (B,-w£r fc ± eijk I B C r . p ) !
-1
Bku?rj),
^ ^ , . - Bku2iPj) •
hhk^
{x,y,z}
(5)
This is a non-linear system of six equations, and the search for periodic orbits is not easy. We have determined them by a Newton-Raphson iteration, employing the stability matrix that enters the amplitudes in the trace formula.26 Special care must be taken at mode-conversion points where B(r, p) = 0 and hence the right-hand sides of (5) diverge. Some of the periodic orbits, however, can be found more easily. This follows from the fact that the three planes rk = pk — 0 (k = x,y,z) in phase space are invariant under the Hamiltonian flow. The equations of motion for these two-dimensional orbits are almost harmonic: h -
Pi 2
T eijk sign(.B*) wjr,-,
Pi = -w rj =F eijk sign(JBjfe) uJfpj .
i,j,ke
{x, y, z}
(6)
Harmonic solutions of (6) can be obtained analytically by ignoring the factors sign(J3jfe) that make the equations nonlinear, and by accepting only solutions with the correct constant values of sign(Bk). In total, we find six doublydegenerate planar solutions with different frequencies Wy (i ^ j) given by -i 1/2
\ (j\ + J] + 2/cVH2 ± VK 2 - u,?)a+ 8«V"| K2 + "?))
(7)
These solutions have the form of ellipses, sketched in the left part of Fig. 3, which for «; —• 0 shrink to librations along the r; axes with the original frequencies Wj. An important fact for these orbits is that Bk ^ 0 for all K ^ 0. It turns out that the frequencies £>^, whose dependence on K is displayed in the right-hand part of Fig. 3, for K = 0.1 closely agree with the peak positions found in Fig. 2. We cannot resolve all six main peaks; a slightly better resolu-
0.2
0.3
Figure 3: Left: A schematic plot of a pair of the two-dimensional periodic solutions of (6). Right: Frequencies uif- (7) of the six planar orbits versus spin-orbit coupling parameter K (using the same parameters u>i as in Fig. 2.)
11
'4i(M|| H twi 1 K= 0
K = 0.1
~m
lnHI
A »
0
10
20
E/T,cox
30
40
Figure 4: Oscillating part f, (e.g., the tiny peak at T = 10.1) can be explained by genuine three-dimensional orbits. 26 We now calculate the oscillating part 6g (E) of the level density by including the lowest harmonics of the six planar periodic orbits in the trace formula. The ^-correction terms for the planar orbits give a constant shift A5 t ^ = (AB erry + XNN)T^ = Knwl/wfj of the actions at all energies. 26 The result for Sg (E) obtained with K = 0.1 is shown in Fig. 4 by the dotted lines. The quantum-mechanical results are shown by the solid lines. The Gaussian averaging range was here 0.5 Hu)x in order to suppress the contributions from second and higher harmonics. For comparison, we also show the curves for K = 0 which exhibit a completely different shell sctructure. A good agreement is found also in the presence of the spin-orbit interaction, except for low energies where semiclassical approximations usually are worst. This shows that the planar orbits are the most important ones for the gross-shell structure. The numerical calculation of the three-dimensional orbits and their properties is not difficult, as long as they are not crossing the mode-conversion surfaces. A study of their influence on the finer details of the shell structure is in progress. 26 Acknowledgment This work was supported in parts by the Deutsche Forschungsgemeinschaft.
References 1. 2. 3. 4. 5. 6. 7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
23. 24. 25. 26.
L. Meitner, Nature 165, 561 (1950); Arkiv Fysik 4, 383 (1952). see, e.g., S. Cohen and W. J. Swiatecki, Ann. Phys. (N. Y.) 19, 67 (1962). S. A. E. Johansson, Nucl. Phys. 22, 529 (1962). V. M. Strutinsky, Nucl. Phys. A 95, 420 (1967); ibid. A 122, 1 (1968). S. G. Nilsson et al., Nucl. Phys. A 131, 1 (1969). M. Brack et al., Rev. Mod. Phys. 44, 320 (1972). J. R. Nix, Ann. Rev. Nucl. Sci. 22, 65 (1972). For an early review of experimental data on fission barriers, see S. Bj0rnholm and J. E. Lynn, Rev. Mod. Phys. 52, 725 (1980); for newer data, see several contributions to this conference. P. Moller and S. G. Nilsson, Phys. Lett. B 31, 283 (1970); H. C. Pauli, T. Ledergerber, and M. Brack, Phys. Lett. B 34, 264 (1971). C. Gustafsson, P. Moller, and S. G. Nilsson, Phys. Lett. B 34, 349 (1971). see, e.g., P. Moller and A. Iwamoto, Phys. Rev. C 61, 047602 (2000); V. V. Pashkevich, this conference. see, e.g., J. Maruhn et al, in Physics and Chemistry of Fission 1973 (IAEA Vienna, 1974), Vol. I, p. 569. see also W. Greiner et al., this conference. M. C. Gutzwiller, J. Mat. Phys. 12, 343 (1971), and earlier references quoted therein. R. Balian and C. Bloch, Ann. Phys. (N.Y.) 69, 76 (1972). V. M. Strutinsky, Nucleonica 20, 679 (1975); V. M. Strutinsky and A. G. Magner, Sov. J. Part. Nucl. 7, 138 (1976). V. M. Strutinsky et al, Z. Phys. A 283, 269 (1977). M. Brack, S. M. Reimann, and M. Sieber, Phys. Rev. Lett. 79, 1817 (1997). M. Brack and R. K. Bhaduri: Semiclassical Physics (Addison and Wesley, Reading, 1997). M. Brack et al, in Similarities and Differences between Atomic Nuclei and Clusters, eds. Y. Abe et al. (AIP New York, 1998), p. 17. M. Brack, M. Sieber, and S. M. Reimann, in Nobel Symposium on Quantum Chaos (2000), ed. K.-F. Berggren; to be published in Physica Scripta. Since H is purely quadratic in the components of p and r, the classical dynamics governed by Eq. (4) becomes independent of the energy E after scaling coordinates and momenta by a factor proportional to \/E. G. Littlejohn and W. G. Flynn, Phys. Rev. A 44, 5239 (1991). H. Frisk and T. Guhr, Ann. Phys. (NY.) 221, 229 (1993). J. Bolte and S. Keppeler, Ann. Phys. (N.Y.) 274, 125 (1999). Ch. Amann and M. Brack, to be published.
13
COUPLING OF T H E COLLECTIVE PAIRING V I B R A T I O N S W I T H T H E FISSION MODE* K. POMORSKI Department of Theoretical Physics, M. Curie-Sklodowska University ul Radziszewskiego 10, 20-031 Lublin ,Poland E-mail:
[email protected] Spontaneous fission and alpha-decay half-lives of heaviest nuclei with atomic number 100 < Z < 114 are calculated on the basis of the deformed Woods-Saxon potential and the finite range liquid drop model. The cranking as well as the generator coordinate method were used to obtain the collective inertia tensor and the 'zero-point' corrections to the fission barriers. The fission probability was evaluated within the WKB approximation along the least action trajectory to fission. We have examined the influence of the dynamics in the pairing degrees of freedom on the fission life-times as well as the effect of the higher even-multipolarity shape parameters and the role of the reflection asymmetry.
1
Introduction
Nuclear fission is a rather old problem but still fascinating both for experimentalists and theorists. Theory of this process seems to be very simple at the first sight of view, but when one would like to get the quantitative agreement with the experimental data or clarify new events, then usually arise a lot of questions. Not all of them have an explicit answer. Much progress has been made recently in the synthesis of very heavy nuclides. Deformed superheavy nuclei with proton numbers from Z=108 to Z=114 have been discovered through reactions of cold-fusion at GSI-Darmstadt x and hot-fusion in Dubna 2 . Last year in Berkeley it was performed an experiment 3 leading to the synthesis of the element Z=118. After these spectacular successes of the experimental groups a more accurate calculation of the lifetimes of the heaviest nuclei is needed. In the present paper we are going to show that the coupling of the fission mode with the pairing and shape vibrations is important when describing the spontaneous fission. We discuss this problem within the cranking model as well as in the generator coordinate method (GCM). The GCM theory leads to the quantal collective hamiltonian when assuming the Gaussian overlap approximation (GOA) of the generator functions 4 . One has to stress that * INVITED LECTURE ON THE INTERNATIONAL WORKSHOP ON FISSION DYNAMICS OF ATOMIC CLUSTERS AND NUCLEI, LUSO, PORTUGAL, 15-20 MAY, 2000
14
contrary to the cranking approximation or the time dependent Hartree-Fock theory in the GCM method one avoids an arbitrary quantization of classical Hamilton function to get the Schrodinger equation in the collective (i.e. generator) coordinates space. 2
The model
The crucial point for our investigation is the proper choice of collectives (i.e. generator) coordinates. In order to describe simultaneously shape and pairing vibrations and their coupling we have chosen the pairing gaps (A p , A n ) , the gauge angles (p,4>n) and the deformations parameters (Pi) as collective coordinates. We have taken the product of the BCS type wave functions for protons and neutrons as the generator function | q > : \q>=\l3i;Ap,(j>p,An,n>=
J J exp{-i(NT
- NT)<j>T} \ BCS >T .
(1)
r=p,n
This choice of the generator function ensures also the approximate particle number projection 5 . The microscopic many-body hamiltonian is taken in the following form: H = flbOSf) - \ 5 3 XijFfFi
+ #pair - X(N-
< N >),
(2)
where the deformed single-particle mean field hamiltonian (Ho) and the pairing hamiltonian (Hpaii) are the sums of the proton and neutron parts. The two-body long-range residual interaction is taken here in the local approximation:
Mb) = H U=/3o " < P I H I P >lft=A, •
(3)
The strength Xij is determined by the self-consistency condition: (X
%
=
-Q^1
IA=A> •
(4)
Following the GCM+GOA method 4 we get the collective hamiltonian in the form:
^on = ~^Y:^(B-^
+
V(q)
(5)
15
Here 7 stands for the determinant of the overlap width tensor:
^
=
(6)
^4 * -
The collective inertia tensor B is given by the equation:
(*-')« = 5 E(7-')..««14*41 «>»-2+) and 497.0 (4+—>2+) keV transitions in 100Zr. Four examples are shown in Figs. 6 to identify 110Ru(or 108 Ru), 104Mo(or 108Mo), 140Xe and
31
Figure 7: Coincidence spectrum double-gated on 376.7 and 457.4 keV transitions in 140 Xe. See 151.0 ( 102 Zr) and 212.0 ( 100 Zr). 100
Zr, respectively. But the identification of the gamma-transitions belonging to these partner fragments is not clear in those spectra. From the 7 — 7 — 7 cube we could clearly eatablish coincidence for iooZr_i42Xe a n d i02 Zr _i40 Xe A l s 0 ; b y double gating on the 376.7 and 457.3 7-rays in 140Xe (Fig. 7), we can see clearly the zero neutron channel 102Zr and probably the 100Zr 2n channel which is weaker by a factor of 5-10 if present. The identification of several isotopes related with the 10Be emission is made by the observation of two or three transitions in coincidence belonging to each isotope and from the 7 - 7 — 7 cube. All isotopes and the related 7 transitions identified in the present work are tabulated in Table 2. In Table 2, partner fragments pertaining to the cold (neutronless) channel are shown some of which are confirmed as noted. Quadrupole deformations for each isotopes are taken from Refs. [14,15]. From these examples, we can see that the statistics of the coincident spectrum with a single gate on the lowest gamma tran-
32
sition does not depend on the statistics of the gated peak shown in Fig. 1 because of complexity of the gamma-ray multiplicity and the enhanced population of the lowlying levels in the 10Be SF. Two fragment pairs, 138Xe-104Zr and 136Te-106Mo with no neutrons emitted show 7 rays produced from both of pair fragments in the 10Be gated coincidence spectrum with a single 7 gate. In other words, the 171.6 keV transition of 106Mo is observed in the coincidence spectrum with a single gate on the 606.6 keV transition of 136Te. Also, the 140.3 and 312.5 keV transitions in 104Zr are observed in the coincidence spectrum with a single gate on the 588.9 keV transition of 138 Xe. For a single gate set on the 588.9 keV transition (2+—>-0+) in 138Xe, the coincidence spectrum is shown in Fig. 8. The 4+-S-2+ and 2+->-0+ transitions in 104Zr and the 483.8 and 482.9 keV doublet transitions (6+->-4+ and 4+->2+) in 138Xe show up clearly. To find the real peaks coincident with both the 588.9 and 483.8 keV transitions we set the "AND" gate of 588.9 and 483.8 keV transitions as shown in Fig. 8. This logical "AND" gate takes arithmetic minimum of two spectra for each channel in the Radware program [16]. Then only three transitions of energies 140.3 keV (2+-»0 + ) and 312.5 keV (4+^2+) in 104Zr and 482.1 keV (6+—»4+) transition in 138Xe show up clearly in Fig. 8. Although the three peaks in Fig. 4c contain only two counts, the background is less than 0.01/channel in Fig. 8. The 140.3 and 312.5 keV transitions do not exist in the level scheme of 138Xe. Since the 7 — 7 matrix is gated by 10Be particles, 140.3 and 312.5 keV transitions belong to the partner nucleus 104Zr. However, 109.0 and 146.8 keV transitions in 103Zr ( 10 Be+ln channel) and the 151.8 and 326.2 keV transitions in 102Zr ( 10 Be+2n channel) do not show up clearly. In another case of 136Te-106Mo, also, the 1 0 Be+ln and 10 Be+2n channels are not observed. This could be caused by the larger feeding to ground state but more likely by small yields in the 10 Be+n and 10 Be+2n channels. The hot fission mode can excite the fragments up to higher level energies than the cold fission.
33
f
(o)Gatad on 5sa9( ™Xe) keV and gated on "Be
I?
H I I Hi Hill I I I lllllB I HIM III llllllll 111 llll
II ] I 1
H , LiiiyiiiiiluiHkJLjL^JJ,L Jl l "
LJ
II
I
U_
(b)GatBd on i a i ^ ^ X e ) VeV and gated on °Be
" i l l II
•
I I
U1LJ
l_l_
(c) " A W gats of 5M.9 and 48M( ^Xe) keV.
II I H I I I
I
I I I 300
I
III 500
III 700
900
1100
1300
E„ (keV)
Figure 8: (a)Coincidence spectrum gated on the 10 Be particle and 588.9 keV (138Xe) transition, (b)Coincidence spectrum gated on the 10 Be particle and 483.8 keV ( 138 Xe) transition, (c)Coincidence spectrum with "AND" gate of 588.9 and 483.8 keV ( 138 Xe) transitions, and gated on 10 Be. This logical "AND" gate takes arithmetic minimum of two spectra for each channel in the Radware program [16].
Therefore, SF yields of the 10 Be+n and 10Be-t-2n channels have to be smaller than the neutronless (cold) 10Be SF yield. The present results indicate that the cold (neutronless) process is dominant in the ternary SF accompanying a heavy third particle such as 10Be with high kinetic energy. In our work, we are gating only on the high kinetic energy part of the 10Be particles. The 104Zr isotope is highly deformed with a /32 value of around 0.4 [15,16] and the 138Xe nucleus is very spherical. Therefore, the 10Be particle seems to be emitted from the breaking of 148 Ce= 138 Xe+ 10 Be at scission which would enhance the 10Be kinetic energy. Increased deformation at the scission point increases excitation energy for the third ternary particle and two heavy frag-
34
Table 2: Fragments identified from the coincidence relationship between 7-rays and 10 Be ternary particle. * : identified in 7 — 7 — 7 data and ** : LCP-7 — 7 data.
Identified Isotopes (fc [14,15]) i{j°Zr (0.321 ) ™2Zr (0.421 ) l^Zr (0.381 ) *°4Mo (0.325 ) (or ™8Mo) IfMo (0.353 ) \l°Ru (0.303 ) (or i°8Ru) 6 ^ Te (0.000 ) IfXe (0.0309 ) ^°Xe (0.1136 )
Observed 7 rays (keV) 212.6, 352.0, 497.0 151.8, 326.2 140.3, 312.5 192.0, 368.5 171.6 with 606.6 (136Te) 240.7, 422.2 606.6, 424.0 588.9, 483.8, 482.1 376.7, 457.4, 582.5
Partner isotopes (ft [14,15]) If Xe (0.145])* £f Xe (0.1136 )* \fXe (0.0309 )** ^ 8 Te (0.000 ) (or ^ T e ) \fTe (0.000 )** *g2Sn (0.000 ) (or ^ S n ) 6 J° Mo (0.353)** *°4Zr (0.381 )** ^ 2 Zr (0.421 )*
ments. Therefore the possibility of observing the exciated levels in both the fragments increases when both of them are deformed at scission point such as 104Zr(deformed)—148Ce(138Xe+10Be) (deformed). Actually, the neutronless binary fission yield for 148Ce—104Zr pair is as high as 0.05(3) per 100 SF of 252Cf [17]. These cases are very similar to the one we reported earlier for the pair 96Sr (spherical shape) and 146Ba (deformed shape) [5]. In the a teranry fission we see the cold, zero, neutron fission but 2n and 3n channels are much stronger. However, for the cold 10Be ternary SF pairs identified from the 7 — 7 matrix gated on 10Be charged particles and the 3D data, we find the zero neutron channel clearly much stronger than In and 2n. This is a very unique discovery in the study of the cold (zero neutron) fission processes.
35
6
Acknowledgment
Research at Vanderbilt University and Mississippi State University is supported in part by the U.S. Department of Energy under Grants No. DE-FG05-88ER40407 and DE-FG05-95ER40939. Work at Idaho National Engineering and Environmental Laboratory is supported by the U.S. Department of Energy under Contract No. DE-AC07- 76ID01570. Work at Argonne National Laboratory is supported by the Department of Energy under contract W-31109-ENG-38. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
J.H. Hamilton et al, J. Phys. G20 (1994) L85 G.M. Ter-Akopian et al., Phys. Rev. Lett. 77 (1996) 32 G.M.Ter-Akopian et al., Phys. Rev. C55 (1997) 1146 A.V. Ramayya et al., Heavy elements and related new phenomena, Volume /World Scientific (1999) 477 A.V. Ramayya et a l , Phys. Rev. Lett. 81 (1998) 947 N. Schultz, (Private communication). Yu.U. Pyatkov et al., Nucl. Phys. A624 (1997) 140 R. Donangelo et al., Int. J. Mod. Phys. E7 (1998) 669 J.H. Hamilton et al., Proc. Nuclear Structure 98, Gatlinburg (1999) 473 S.-C. Wu et al, Phys. Rev. C62 (2000) 041601 D.C. Biswas et al., Eur. Phys. J. A7 (2000) 189 Y.X. Dardenne et al., Phys. Rev. C54 (1996) 206 A. Sandulescu et al., Int. J. Modern Phy. E7 (1998) 625 S. Raman et al., Atomic Data and Nucl. Data Tables 36 (1987) 1 P. Moller et al., Atomic Data and Nucl. Data Tables 59 (1995) 185 D.C. Radford, Nucl. Instr. Meth. Phys. Res. A361 (1995) 297
36
17. J.H. Hamilton et al., Prog, in Part, and Nucl. Phys. 35 (1995) 635
37
A D I A B A T I C A N D N O N - A D I A B A T I C D Y N A M I C S OF I O N S I N T H E F O R M A T I O N OF ELECTRON EXCITATION W I D T H S I N METAL CLUSTERS A. V. SOLOV'YOV ° Institute for Theoretical Physics, Frankfurt am Main University, Robert-Mayer Str. 8-10, D60054 Frankfurt am Main, Germany E-mail:
[email protected] The dynamical jellium model for metal clusters, which treats simultaneously the vibrational modes of the ionic jellium background in a cluster, the quantized electron motion and the interaction between the electronic and the ionic subsystems beyond the adiabatic approximation, is applied for the calculation of widths of electron excitations in metal clusters caused by multiphonon transitions and the investigation of their temperature dependencies. The decay time and the energy relaxation time of electron excitations in metal clusters is estimated.
1
Introduction.
In this paper the influence of the dynamics of ions on the motion of delocalized electrons in metal clusters is considered on the basis of the dynamic jellium model suggested in 1 and further advanced in 2 . This model generalizes the static jellium model 3 ' 4 ' 5 ' 6 , which treats the ionic background as frozen, by taking into account vibrations of the ionic background near the equilibrium point. The dynamic jellium model treats simultaneously the vibrational modes of the ionic jellium background, the quantized electron motion and the interaction between the electronic and the ionic subsystems. This paper gives a brief survey of the ideas developed in many more details in 1 , z . An example of the effect, originating from the interaction of the ionic vibrations with delocalized electrons, is the broadening of electron excitation lines. There are two mechanisms of the electron excitation line broadening, namely adiabatic and non-dynamic ones. The dynamic jellium model 1 allows one to calculate widths of the electron excitations in metal clusters caused by these two mechanisms and investigate their temperature dependence. The adiabatic mechanism is connected with the averaging of the electron excitation spectrum over the temperature fluctuation of the ionic background in a cluster. This phenomenon has been studied in a number of papers 7 ' 8 ' 9 ' 1 0 ' 1 1 . The mechanism of dynamic or non-adiabatic electron excitation line broaden°On leave from: A.F.Ioffe Physical-Technical Institute, Russian Academy of Sciences, Politechnicheskaya 26, 194021 St. Petersburg, Russia E-mail:
[email protected] 38
ing has been considered for the first time in 1. Numerically it was further advanced i n 2 . This mechanism originates from the real multiphonon transitions between the excited levels of electrons. Therefore the dynamic line-widths characterize the real lifetimes of the electronic excitations in a cluster. The adiabatic broadening mechanism explains the temperature dependence of the photo-absorption spectra in the vicinity of the plasmon resonance via the coupling of the dipole excitations in a cluster with the quadrupole deformation of the cluster surface. The photo-absorption spectrum was calculated within the framework of deformed jellium model using either the plasmon pole approximation 7 ' 8 or the local density approximation 9,10 ' 11 ' 12,13 . The interest to the problem of the electron excitation line-widths formation in metal clusters was stimulated by numerous experimental data on photoabsorption spectra, most of which were addressed to the region of dipole plasmon resonances 14 ' 15 ' 16 . The non-adiabatic line-widths characterize the real lifetimes of cluster electron excitations. The information about the non-adiabatic electron-phonon interactions in clusters is necessary for the description of electron inelastic scattering on clusters 17>18, including the processes of electron attachment 1 9 ' 2 0 and bremsstrahlung 2 1 ' 2 2 , the problem of cluster stability and fission. The non-adiabatic line-widths determined by the probability of multiphonon transitions are essential for the treatment of the relaxation of electronic excitations in clusters and the energy transfer from the excited electrons to ions, which occurs after the impact- or photo-excitation of the cluster. Following 1 ' 2 , in this work we elucidate the role of the volume and the surface vibrations of the ionic cluster core in the formation of the electron excitation line-widths and demonstrate that the volume and surface vibrations provide comparable contributions to the adiabatic line-widths, but the surface vibrations are much more essential for the non-adiabatic multiphonon transitions than the volume ones. 2
Dynamical jellium model.
Let us consider the quantized motion of the delocalized electrons and the oscillatory motion of ions near the equilibrium points in one approach. The Hamiltonian of the entire cluster in this case can be represented as a sum of the ionic Hamiltonian Hi, which includes the kinetic energy plus the electrostatic energy of the ionic background, and the Hamiltonian He of the quantized motion of electrons in the frozen field of ions: H = Hi(q)+He(q,r).
(1)
39
Here q and r represent sets of the ionic and the electronic coordinates of the cluster respectively. Let us solve the Schrodinger equation, HVX = EXVX,
(2)
with the Hamiltonian (1). The wave function of the entire cluster can be expressed as a sum of products of the electron, ^„(g,r), and the phonon, $An(n(q) = en(q) — £o(o) denotes the electron excitation energy. For a small displacement of the ionic background, when the energy shifts of the excited electron levels are smaller than the inter-level energy distances, the dependence of the excitation energies upon the ionic coordinate q can be described by the linear term of the expansion hu)n(q) over q : M , ) = b„ + ^ g j W , (6) a a
where V"( ' = dHe(q, r)/dqa action and u>„ = w n (0).
plays a role of operator of electron-phonon inter-
40
The terms at the right hand side of equation (5) are responsible for the so-called non-adiabatic coupling. In the adiabatic (or the so-called BornOppenheimer) approximation these terms are neglected. In this case, equation (5) reduces to the set of non-coupled equations describing the oscillations of the ions in the cluster. The electron-phonon spectrum in this approximation is of the form:
* = «f + ?(«W'r- + 5»-^)-
(7)
The corresponding wave functions read *A(0,r) = ^
o )
(r)n*"-(O.
(8)
a
where $JV„ (<j£J are eigen functions of the shifted oscillators, which describe the phonon excitations. This approximation is valid if the shifts of the electron energy levels are much smaller compared to the inter-level energy distances: f>(
