The physics of warm nuclei

Transcription

1 The physics of warm nuclei and of mesoscopic systems Helmut Hofmann Physik Department, TUM CLARENDON PRESS. OXFORD 2007

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3 To Helga

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5 PREFACE Ein Buch, das nicht wert ist, wenigstens zweimal gelesen zu werden, ist auch nicht wert, dass man es einmal liest. 1 Von allen Unmöglichkeiten ist es die größte, ein Buch zu schreiben, das allen gefällt. 2 Karl Julius Weber A full comprehension of present day nuclear physics is no longer possible solely on a knowledge of quantum mechanics. In the past decades nuclear physics has developed into a field which borrows methods and concepts from diverse areas in physics. For an understanding of the nuclear force, for instance, connections to Quantum Chromo Dynamics are in sight. To follow this line requires application of quantum relativistic field theory, together with requisites of elementary particle physics. In typical accelerator experiments with heavy ions, on the other hand, nuclear compounds are created with large excitation energies for which methods of statistical physics are indispensable. It is true that a similar situation has already been faced in conventional reaction theory. Different methods are necessary, however, to cope with the problem of collective motion. In many cases the latter is of a time scale much larger than that of the residual degrees of freedom. This is the realm of non-equilibrium statistical mechanics with transport equations at one s disposal. In this book we concentrate on examples which fall in this latter category. The notion warm nuclei is meant to indicate that only very moderate thermal excitations will be considered. A nuclear temperature should not only be much smaller than the Fermi energy, but will be assumed small enough so that some of those many body features are still present, which in a sense represent traditional nuclear physics. Take shell effects as one of the most prominent examples, known to disappear at about k B T 3 MeV. The study of this transition will play a major role in our discussion, in particular because of its intriguing implications to transport properties. We will, however, refrain from describing any effects which go beyond this regime. In fact for hotter systems the nucleons dynamics may no longer be associated with that in a deformed mean field. At still larger energies sub-nucleonic degrees of freedom may be excited, for which field theoretical concepts are involved, and which are beyond the scope of this book. It should be evident that concepts of statistical mechanics cannot be taken over to nuclear physics without serious forethought. This holds true both for 2 A book not worthy of being read twice is not worthy of being read once. 2 The greatest of all impossibilities is to write a book which pleases everyone.

6 viii Preface static and dynamic properties, as well as for those involving irreversible features. Most of the problems are related to the fact that the nucleus is a small, self-bound and thermally isolated system. For example, a temperature can be defined only within some inherent uncertainty, and will not stay constant in any dynamic process. This is so even in the reversible case as required by the constancy of entropy. The latter feature is intimately related to the extension of the quantal adiabaticity theorem to statistical mechanics, and thus to the difference of the susceptibilities involved. Another problem is that collective degrees of freedom are not given per se; they must be introduced obeying conditions of self-consistency with those of the nucleons. This feature is already known for the conventional case of zero thermal excitation, but it still plays an important role at finite excitations. It is predominantly here that irreversible features come into play, which for nuclear physics are not unexpected. Indeed, they have already shown up in traditional theories of nuclear reactions, like those for overlapping resonances and in relation to the nuclear compound model. A large part of the book is devoted to explaining and developing such aspects at the borders where microscopic and macroscopic physics meet. Some of them may be applied to other finite Fermi systems, like metal clusters and quantum dots or wires. In some cases one may simply dwell on analogies, like the appearance of nuclear dissipation and the absorption of electromagnetic waves in metal clusters. However, as is known, many methods have been applied outside of nuclear physics which were originally developed solely for that domain. To cover such a wide field is not an easy task. The general reader must not only be introduced to the basics of nuclear physics (and of mesoscopic systems), but also to elementary topics of the theoretical tools involved. For this reason the book is split into different parts. The first part is meant to acquaint the reader with methods aimed at an understanding of the essentials of collective motion and transport problems used in nuclear physics. In the second and third part applications to nuclear reactions and mesoscopic systems are given. The fourth part contains a thorough discussion of several theoretical methods and is kept at a more general level, but with emphasis on the specific problems discussed in the other parts. The overall level of the presentation is oriented at a general, alert reader, not necessarily a specialist in nuclear physics. In fact, part one and part four are meant to serve as self-contained introductions to the fields presented there. Hardly more than a basic knowledge of quantum mechanics is required for an easy understanding of such diverse fields as the theories of reaction, transport or linear response, as well as of the methods of Wigner transformations or functional integrals.

7 ACKNOWLEDGEMENTS It is a great pleasure to acknowledge the credit due to the many scientists and colleagues with whom I had closer contact over the years in which my understanding of the topics of this book has grown steadily albeit sometimes very slowly. First I should like to acknowledge the important contributions from my former collaborators R. Alkofer, G.-L. Ingold, F.A. Ivanyuk, A.S. Jensen, D. Kiderlen, A.G. Magner, R. Nix, C. Ngo, C. Rummel, R. Samhammer, P.J. Siemens, R. Sollacher and S. Yamaji for the fruits of joint publications which have been incorporated within these pages. I should also like to express my deep obligation and the great benefit I have had from discussions with Professors A. Bohr, R. Balian, M. Brack, W. Brenig, K. Dietrich, R.J. Glauber, W. Götze, D. Kusnezov, B.R. Mottelson, D. Pines, P. Ring, V.M. Strutinsky, N.G. van Kampen, M. Vénéroni, H.A. Weidenmüuller, V.G. Zelevinsky and S. Aaberg, in which I have been informed about pertinent methods and ideas which has often prevented me from choosing inappropriate paths. My last thanks go to Professor J.A. Wheeler who was the first to suggest that I take up the endeavor of writing this book, to Professor R.H. Lemmer who gave the final push to this project, and to Professor W.U. Schröder for his continued interest during the preparation period, as well as Dr. R.R. Hilton for much advice not only about questions of proper English style.

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9 CONTENTS I BASIC ELEMENTS AND MODELS 1 Elementary concepts of nuclear physics The force between two nucleons Possible forms of the interaction The radial dependence of the interaction The role of sub-nuclear degrees of freedom The model of the Fermi gas Many body properties in the ground state Two body correlations in a homogeneous system Basic properties of finite nuclei The interaction of nucleons with nuclei The optical model The liquid drop model 23 2 Nuclear matter as a Fermi liquid A first, qualitative survey The inadequacy of Hartree-Fock with bare interactions Short range correlations Properties of nuclear matter in chiral dynamics How dense is nuclear matter? The independent pair approximation The equation for the one-body wave functions The total energy in terms of two-body wave functions The Bethe-Goldstone equation The G-matrix Brueckner-Hartree-Fock approximation (BHF) BHF at finite temperature A variational approach based on generalized Jastrow functions Extension to finite temperature Effective interactions of Skyrme type Expansion to small relative momenta The nuclear equation of state (EOS) An energy functional with three-body forces The EOS with Skyrme interactions Applications in astrophysics Transport phenomena in the Fermi liquid Semi-classical transport equations 62

10 xii Contents 3 Independent particles and quasi-particles in finite nuclei Hartree-Fock with effective forces H-F with the Skyrme interaction Constrained Hartree-Fock Other effective interactions Phenomenological single particle potentials The spherical case The deformed single particle model Excitations of the many-body system The concept of particle-hole excitations Pair correlations 79 4 From the shell model to the compound nucleus Shell model with residual interactions Nearest level spacing Random Matrix Model Gaussian ensembles of real symmetric matrices Eigenvalues, level spacings and eigenvectors Comments on the RMM The spreading of states into more complicated configurations A schematic model Strength functions Time dependent description Spectral functions for single particle motion 99 5 Shell effects and Strutinsky renormalization Physical background The independent particle picture, once more The Strutinsky energy theorem The Strutinsky procedure Formal aspects of smoothing Shell correction to level density and ground state energy Further averaging procedures The static energy of finite nuclei An excursion to periodic-orbit theory (POT) The total energy at finite temperature The smooth part of the energy at small excitations Contributions from the oscillating level density Average collective motion of small amplitude Equation of motion from energy conservation Induced forces for harmonic motion Equation of motion One-particle one-hole excitations 133

11 Contents xiii 6.2 The collective response function Collective response and sum rules for stable systems Generalization to several dimensions Mean field approximation for an effective two-body interaction Isovector modes Rotations as degenerate vibrations Microscopic origin of macroscopic damping Irreversibility through energy smearing Relaxation in a Random Matrix Model The effects of collisions on nucleonic motion Damped collective motion at thermal excitations The equation of motion at finite thermal excitations The strict Markov limit The collective response for quasi-static processes An analytically solvable model Temperature dependence of nuclear transport The collective strength distribution at finite T Diabatic models T-dependence of transport coefficients Rotations at finite thermal excitations Transport theory of nuclear collective motion The locally harmonic approximation Equilibrium fluctuations of the local oscillator Fluctuations of the local propagators Quantal diffusion coefficients from the FDT Fokker-Planck equations for the damped harmonic oscillator Stationary solutions for oscillators Dynamics of fluctuations for stable modes The time-dependent solutions for unstable modes and their physical interpretation Quantum features of collective transport from the microscopic point of view Quantized Hamiltonians for collective motion A non-perturbative Nakajima-Zwanzig approach 217 II COMPLEX NUCLEAR SYSTEMS 8 The statistical model for the decay of excited nuclei Decay of the compound nucleus by particle emission Transition rates Evaporation rates for light particles Fission The Bohr-Wheeler formula 229

12 xiv Contents Stability conditions in the macroscopic limit Pre-equilibrium reactions An illustrative, realistic prototype A sketch of existing theories Comments Level densities and nuclear thermometry Darwin-Fowler approach for theoretical models Level densities and Strutinsky renormalization Dependence on angular momentum Microscopic models with residual interactions Empirical level densities Nuclear thermometry Collective motion of large scale at finite thermal excitations Global transport equations Fokker-Planck equations Over-damped motion Langevin equations Probability distribution for collective variables Transport coefficients for large scale motion The LHA at level crossings and avoided crossings Thermal aspects of global motion Dynamics of fission at finite temperature Transitions between potential wells Transition rate for over-damped motion The rate formulas of Kramers and Langer Escape time for strongly damped motion A critical discussion of time scales Transient- and saddle-scission times Implications from the concept of the MFPT Inclusion of quantum effects Quantum decay rates within the LHA Rate formulas for motion treated self-consistently Quantum effects in collective transport, a true challenge Heavy ion collisions at low energies Transport models for heavy-ion collisions Commonly used inputs for transport equations Differential cross sections Fusion reactions Micro- and macroscopic formation probabilities 320

13 Contents xv 13.4 Critical remarks on theoretical approaches and their assumptions Giant dipole excitations Absorption and radiation of the classical dipole Nuclear dipole modes Extension to quantum mechanics Damping of giant dipole modes 332 III MESOSCOPIC SYSTEMS 15 Metals and quantum wires Electronic transport in metals The Drude model and basic definitions The transport equation and electronic conductance Quantum wires Mesoscopic systems in semiconductor heterostructures Two dimensional electron gas Quantization of conductivity for ballistic transport Physical interpretation and discussion Metal clusters Structure of metal clusters Optical properties Cross sections for scattering of light Optical properties for the jellium model The infinitely deep square well Energy transfer to a system of independent Fermions Forced energy transfer within the wall picture Energy transfer at finite frequency Fermions inside billiards Wall friction by Strutinsky smoothing 364 IV THEORETICAL TOOLS 18 Elements of reaction theory Potential scattering The T-matrix Phase shifts for central potentials Inelastic processes Generalization to nuclear reactions Reaction channels Cross section The T-matrix for nuclear reactions Isolated resonances 383

14 xvi Contents Overlapping resonances T-matrix with angular momentum coupling Energy averaged amplitudes The optical model Intermediate structure through doorway resonances Statistical theory Porter Thomas distribution for widths Smooth and fluctuating parts of the cross section Hauser-Feshbach theory Critique of the statistical model Density operators and Wigner functions The many body system Hilbert states of the many body system Density operators and matrices Reduction to one- and two-body densities Many body functions from one-body functions One and two-body densities The Wigner transformation The Wigner transform in three dimensions Many-body systems of indistinguishable particles Propagation of wave packets Correspondence rules The equilibrium distribution of the oscillator The Hartree-Fock method Hartree-Fock with density operators The Hartree-Fock equations The ground state energy in HF-approximation Hartree-Fock at finite temperature TDHF at finite T Transport equations for the one-body density The Wigner transform of the von Neumann equation Collision terms in semi-classical approximations The collision term in Born approximation The BUU and the Landau-Vlasov equation Relaxation to equilibrium Relaxation time approximation to the collision term A few remarks on the concept of self-energies Nuclear thermostatics Elements of statistical mechanics Thermostatics for deformed nuclei Generalized ensembles Extremal properties 446

15 Contents xvii 22.2 Level densities and energy distributions Composite systems A Gaussian approximation Darwin-Fowler method for the level density Uncertainty of temperature for isolated systems The physical background The thermal uncertainty relation The lack of extensivity and negative specific heats Thermostatics of independent particles Sommerfeld expansion for smooth level densities Thermostatics for oscillating level densities Influence of angular momentum Linear response theory The model of the damped oscillator A brief reminder of perturbation theory Transition rate in lowest order General properties of response functions Basic definitions Basic properties Dissipation of energy Spectral representations Correlation functions and the fluctuation dissipation theorem Basic definitions The fluctuation dissipation theorem Strength functions for periodic perturbations Linear response for a Random Matrix Model Linear response at complex frequencies Relation to thermal Green s functions Response functions for unstable modes Equilibrium fluctuations of the oscillator Susceptibilities and the static response Static perturbations of the local equilibrium Isothermal and adiabatic susceptibilities Relations to the static response Linear irreversible processes Relaxation functions Variation of entropy in time Time variation of the density operator Onsager relations for macroscopic motion Kubo formula for transport coefficients Functional integrals Path integrals in quantum mechanics Time propagation in quantum mechanics 520

16 xviii Contents Semi-classical approximation to the propagator The path integral as a functional Path integrals for statistical mechanics The classical limit of statistical mechanics Quantum corrections Green functions and level densities Periodic orbit theory The level density for regular and chaotic motion Functional integrals for many-body systems The Hubbard-Stratonovich transformation The high temperature limit and quantum corrections The Perturbed Static Path Approximation (PSPA) Properties of Langevin- and Fokker-Planck equations The Brownian particle, a heuristic approach Langevin equation Fokker-Planck equations Cumulant expansion and Gaussian distributions General properties of stochastic processes Basic concepts Markov processes and the Chapman-Kolmogorov equation Fokker-Planck equations from Kramers-Moyal expansion The master equation Non-linear equations in one-dimension Transport equations for multiplicative noise Properties of the general Fokker-Planck equation The mean first passage time Differential equation for the MFPT The multi-dimensional Kramers equation Gaussian solutions in curvilinear coordinates Time dependence of first and second moments for the harmonic oscillator Microscopic approach to transport problems The Nakajima-Zwanzig projection technique Perturbative approach for factorized coupling 580 V AUXILIARY INFORMATION 26 Formal means Gaussian integrals Stationary phase and steepest decent The δ-function Fourier and Laplace transformations 589

17 Contents xix 26.5 Derivative of exponential operators The Mori product Spin and Isospin Second quantization for Fermions Natural units in nuclear physics 594 References 595