Dissipative nuclear dynamics

Transcription

1 Dissipative nuclear dynamics Curso de Reacciones Nucleares Programa Inter universitario de Fisica Nuclear Universidad de Santiago de Compostela March 2009 Karl Heinz Schmidt

2 Collective dynamical properties Nuclear ground state: Minimum potential energy Small amplitudes: Parabolic potential, harmonic oscillator Large amplitudes: dynamical instabilities (Coulomb, phase transitions) irreversible processes, nuclear reaction channels (fission, multifragmentation,..)

3 Elongation > Fission Dynamical instability of quadrupole oscillations, driven by Coulomb force. Nucleus falls apart in two fragments

4 Expansion > Multifragmentation Density dependent potential: U U 0 c ρ ρ o 2 Entropy: S 2 a E with a V 2 /3 (Fermi gas) Thermal expansion

5 Multifragmentation Similarity of nuclear forces (between nucleons) and Van der Walls forces (between molecules) Nuclear liquid gas pase transition. Dynamical instability in density degree of freedom. Nucleus falls apart in liquid and gas phase. Liquid phase decays in many pieces (multifragmentation). Analogy to boiling water (heating, lowering pressure)

6 Collective dynamics An isolated system with a number of collective degrees of freedom: classical equation of motion under influence of potential energy and inertia. Concept for equation of motion: Potential energy + kinetic energy = constant total energy. x i 1 =x i p i m Δt Successive numerical calculation in finite time steps. (More complex when m is x dependent.) p i 1 = p i du dx Δt Powerful method to determine the dynamical path with complex potentials and inertias. But: The collective nuclear degrees of freedom do not form an isolated system. The nucleus has many more degrees of freedom, e.g. The singleparticle excitations of the nucleons in the independent particle picture. > The above equations of motion are incomplete.

7 Collective and intrinsic degrees of freedom Collective variables define the global parameters of the nuclear system Deformation parameters define the shape Density parameter defines the volume Intrinsic degrees of freedom (single particle nucleonic excitations) form the microstates consistent with specific values of global parameters. Intrinsic excitations are assumed to be in thermodynamical equilibrium. Description of nuclear reactions, which involve collective motion: thermodynamics with global conditions that evolve with time. Extension of the statistical model of nuclear reactions with explicit consideration of collective motion. Coupling between collective and intrinsic degrees of freedom Dissipation (governed by the dissipation coefficient: = [time constant of energy transfer towards equilibrium] 1 ) E coll t =E equil coll E init coll E equil coll e t

8 Dissipative dynamics Coupling between collective and intrinsic degrees of freedom: Friction (slowing down the collective motion by transfer of energy from collective to intrinsic degrees of freedom) Dissipation (more general, includes transfer of energy between intrinsic and collective degrees of freedom in both directions) Analogy: Brownian motion Concept for equation of motion: kinetic energy in collective motion + potential energy in collective motion + thermal energy of intrinsic excitations = constant total energy. Variation of thermal energy (heat): Q = S * T

9 Langevin equations (simplified) x i 1 =x i p i m Δt p i 1 = p i T ds dx Δt β p i Δt β mt Δt Γ Phase space Friction Fluctuation (driving force) Since the collective degrees of freedom are coupled to a heat bath, the driving force is not du/dx, but T ds/dx. Friction and fluctuation are connected by the fluctuation dissipation theorem (Einstein). They have common origin: The coupling of collective degrees of freedom to the heat bath. Temperature T and entropy S are defined by the thermodynamical properties of the nucleus.

10 Numerical calculation Potential energy Trajectory

11 Origin of dissipation Classical: one body dissipation (wall and window formula) two body dissipation (collisions between neighbouring nucleons) Quantum mechanical: Landau Zener effect (diabatic or adiabatic crossing of levels)

12 One body dissipation (I) Dissipation mechanism when the shape of a nuclear system is changing. Swiatecki, 1980

13 One body dissipation (II) Dissipation mechanism when two nuclei with different velocities are connected by a window. (Deep inelastic collisions, late stage of fission.) Swiatecki, 1980

14 Two body dissipation F friction = A dv dx Typical kind of friction in viscous fluids (honey)

15 Landau Zener (dynamics of level crossings) Transition probability P 12 : P 12 =1 exp V 2 h f v Δf = f 1 - f 2 (slopes) V = interaction v = velocity Neutron levels on the fission path (Mirea 2007) red line: Fermi energy

16 Combination of dissipative nuclear dynamics with statistical decay (evaporation) > fission Transient effect: delayed onset of fission 2 dimensional Langevin calculation. Gontchar et al., 1999

17 Combination of dissipative nuclear dynamics with statistical decay (evaporation) > fusion Fusion fission and fusion quasifission trajectories (Aritomo 2005)

18 Expansion spinodal decay P v diagramme liquid gas (spinodal) instability Dynamical calculations of Nörenberg and Rozmej (2000)

19 Summary The microscopic description of nuclear dynamics is very complex. An elegant and powerful way, exploiting the tools of thermodynamics, is the model of dissipative nuclear dynamics. Only a limited number of degrees of freedom are treated explicitly. The rest is contained in a heat bath. The motion of the collective variables is driven by entropy, slowed down by friction, and activated by the temperature of the heat bath. ( > Langevin equation or Fokker Planck equation) Modern simulation codes are based on numerical solutions of the Langevin equations of motion. Applications cover a wide field of nuclear reactions (fission, quasifission, fusion, deep inelastic reactions, multifragmentation,...). The compound nucleus concept is still used for intrinsic excitations and decay, however with time dependent global conditions.

20 Further reading Introduction into nuclear dynamics: W. J. Swiatecki, "Three Lectures on Macroscopic Aspects of Nuclear Dynamics", Prog. Part. Nucl. Phys. 4 (1980) 383 Review on stochastical models of nuclear dynamics: Y. Abe et al., On stochastic approaches of nuclear dynamics, Phys. Rep. 275 (1996) 49 Special volume on nuclear multifragmentation and nuclear phase transition: F. Gulminelli et al., Challenges in nuclear dynamics and thermodynamics, Eur. Phys. J. A 30 (2006) 1 3 (and other articles of that volume) Dissipative nuclear dynamics, application to fission: Gargi Chaudhuri: "Study of Dissipative Dynamics in Fission of Hot Nuclei Using Langevin Equation", arxiv:nucl th/ th/ Elaborate dynamical calculation on dissipative nuclear reactions including shell effects: Y. Aritomo, "Fusion hindrance and roles of shell effects in superheavy mass region", Nucl. Phys. A 780 (2006) 222

21 The Fokker Planck equation The Fokker Planck equation describes the evolution of the probability distribution of the system in space, momentum and time. t P x, p;t p x P x, p;t T m S x p P x, p;t = p [ p P x, p ;t ] T m 2 P x, p ;t 2 p It is the integral form of the equation of motion of a dissipative system, compared to the Langevin equations. Analytical solutions only exist for simple cases.