Deexcitation mechanisms in compound nucleus reactions

Transcription

Deexcitation mechanisms in
compound nucleus reactions
Curso de Reacciones Nucleares
Programa Inter-universitario de Física Nuclear
David Pérez Loureiro
Universidade de Santiago de Compostela
March 2008
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
Elements of equilibrium
statistical mechanics

Microstates, Macrostates and equilibrium

Density of states

Partition function
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
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:
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:
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
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

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

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

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

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) 635
0
10
20
Excitation energy in MeV
30
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
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
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
Competition between proton and
neutron evaporation

Isotopic distributions of evaporation residues provide information about
Coulomb barrier and binding energies at haigh excitation energies
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.
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
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
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
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
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
Fission
Fission
U E*=1 MeV
238
U E*=20 MeV
238
238
238
U E*=5 MeV
U E*=100 MeV
Shell effects are damped with temperature
N/Z ratio and assymmetry fluctiations increase with temperature
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
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.

Deexcitation mechanisms in compound nucleus reactions

Transcription

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