Deexcitation mechanisms in compound nucleus reactions

Transcription

1 Deexcitation mechanisms in compound nucleus reactions Curso de Reacciones Nucleares Programa Inter-universitario de Física Nuclear Universidade de Santiago de Compostela March 2008

2 Contents Elements of equilibrium statistical mechanics Level densities in nuclei Fermi gas model Structure and pairing corrections Collective modes corrections Weisskopf model nucleon evaporation -ray evaporation cluster evaporation Transition states fission model Bohr Wheeler statistical model Kramers dynamical model Mass and charge distribution of fission fragments Multifragmentation

3 Elements of equilibrium statistical mechanics Microstates, Macrostates and equilibrium Density of states Partition function

4 Microstates Macrostates and equilibrium Microstate: Is the most complete description which is necessary to describe any system in any physical context (e. g. for a certain volume of gas, all the positions and velocities of the molecules) Thermodynamical equilibrium: Any system is in equilibrium when each and every microstate of it occurs with the same probability Macrostate: Is a subset of microstates which satisfy certain condition (e g. for a certain volume of gas, a value of pressure or temperature) Entropy: KB=1

5 Density of states System with defined energy E. The number of states up to energy E is Density of states (level density) Smoothed density of microstates: (Average number of states per unit Energy) Number of states in the interval energy: Entropy:

6 Canonical Partition Function System formed by two subsystems Density of states (Level density): Laplace Transform of density of states Thermodynamic parameters can be obtained from partition function (e. g.) Density of states can be calulated as:

7 Fermi gas model Model applied for systems of weakly interacting fermions protons and neutrons are moving quasi-freely within the nuclear volume. The binding potential is generated by all nucleons. Average number of energy levels for momenta on the range p and p+dp For the ground state, the lowest states will be filled up to a maximum momentum, pf. e. g. for protons And the Fermi energy EF

8 Fermi gas model Any configuration E can be described in terms of occupation numbers n={nk}. States with energies near the ground state are described as excitations of single nucleons to unoccupied single particle energy levels in the nuclear potential well. For higher energies several single particles may be excited simultaneously==>as the excitation energy increases there are many different ways of exciting the nucleus to a small energy region==>level densities increases with the excitation energy Assuming constant spacing between energy levels, we can calculate the level density with two different approaches Combinatorial methods Statistical Mechanics methods

9 Level densities in nuclei N=20 fermions, d=level separation Ground state Excited states Excitation energy a b c d d Possible configurations have to satisfy Pauli principle The number of available states increases rapidly with excitation energy ==> Statistical mechanics

10 Level densities in nuclei Properties excited nuclei, can statistically be described within the grand canonical ensemble, our system interchanges energy and matter with the environment. g=occupation numbers 0,1 i=all available states From partition function we get level density via the (double) Laplace inverse transform 0 and 0 are chosen to get a minimum in the integrand

11 Level densities in nuclei Level density for a Fermi gas of two types of fermions (p and n) is therefore: Level density parameter This approach is limited to excitation energies which involve single particle levels around EF. For higher excitation energies the equally spaced levels approximation is not good and the level density parameter has to be corrected due to estructure and pairing effects

12 Level densities in nuclei Realistic descriptions of level densities corrections due to surface and volume dependencies corrections due to estructure and pairing effects (back shifted Fermi gas) corrections due to collective excitations (vibrations and rotations) odd-odd nuclei even mass nuclei even-even nuclei U A. R. Junghans et al., Nuc. Phys. A 629 (1998) Excitation energy in MeV 30

13 Compound nucleus Bohr indenpendence hypothesis (N. Bohr Nature 137 (1936) 344) short mean free path of nucleons inside the nucleus multiple scatterings and energy sharing lost of memory of the entrance channel thermodynamical equilibrium all possible final states can be populated with the same probability the probability of a certain deexcitation channel is given by the number of sates excitations in the continuum single particle excitations cannot be considered at high excitation energy statistical description in terms of level densities

14 Evaporation Weisskopf model (V. Weisskopf, Phys. Rev. 52 (1937) 295) Detailed balance principle: Reversal time invariance==> Evaporation of nucleons Emmision probability can be calculated from capture probability To get the emission probability we have to calculate both densities of states Fermi gas level density

15 Energy spectrum of the emitted nucleons emission probability per energy From the definition of entropy S=ln( ) if V. Weisskopf, Phys. Today 14 (1961) 18 Maxwell-Boltzmann distribution

16 Competition between proton and neutron evaporation Isotopic distributions of evaporation residues provide information about Coulomb barrier and binding energies at haigh excitation energies

17 Evaporation of photons For photons p2= 2, so, we can write Capture corss section for photons is quite low (photons do not interact strongly)==>deexcitation probability by photon emission is quite low for energies higher than separation energies for nucleons. Statistical emission of photons competes only with particle evaporation only in giant resonances.

18 Cluster evaporation Emission of intermediate mass fragments, can be understood as a very asymmetric fission process. Statistical model can also be used taking into account that the final state configuration corresponds to the convolution of level densities of final fragments

19 Fission Is a consequence of a large-scale collective motion of nucleons inside the nucleus. It splits the nucleus in two fragments. Part of the internal excitation energy is transformed into collective motion: Deformation of nucleus (at constant volume) the surface term of binding energy opposes deformation the separation reduces Coulomb energy and favours elongation The competition of this two effects creates a potential barrier

20 Fission Fission barrier height is only few MeV-->Very sensitive to small energy variations, suh as sheel effects or dissipation-->fission is an appropriate tool for investigating structure and dynamics of nuclei. Bohr and Wheeler model is based on transition state method. The probability of fission does not depend on the level density of residual nuclei. It depends o the properties of the level densities of the compound nucleus at the saddle point Transmission through the barrier is not taken into account Shell and collective effects can be added to both level densities N. Bohr & J. A. Wheeler, Phys. Rev. 56 (1939) 426

21 Fission Bohr and Wheeler model does not take into account the mass assymetry degree of freedom-->fission probability can be calculated, but no fission fragments yields Surface energy is maximum at mass symmetry Coulomb energy is minimum at mass simmetry For light fissioning systems, surface energy dominates and is maximum at symmetry For heavy systems, coulomb energy sominates and is minimum at symmetry

22 Fission Mass and charge distribution of fission fragments will be determined by the statistical population over the mass asymmetry conditional barrier at saddle Fission yield can be calculated from the density of states above the barrier as Fission yield without shell effects J. Benlliure et al., Nuc. Phys. A 628 (1998) 458

23 Fission

24 Fission U E*=1 MeV 238 U E*=20 MeV U E*=5 MeV U E*=100 MeV Shell effects are damped with temperature N/Z ratio and assymmetry fluctiations increase with temperature

25 Fission At high excitation energies the statitiscal description of Bohr and Wheeler model overestimates the fission yields Dynamical models have to be used (coupling between collective and intrinsic degrees of freedom) Fokker-Plack or Langevin equations have to be used Dissipation coefficient has to be added H. A. Kramers, Physika VII 4 (1940) 284

26 Multifragmentation Under very high excitation energy conditions, the multifragmentation deexcitation channel is opened. It is characterized by a complete disintegration of the nucleus is several simultaneously emitted intermediate mass fragments. Description of this channel is beyond the sequential evaporation approach given.