W. Udo Schröder, 2007 Semi-Classical Reaction Theory 1

Compound-Nucleus Processes W. Udo Schröder, 2007 Classical Pot. Scattering 2 a A E cm,  Formation C* E*=E cm +Q I=  Equilibration  Compound Nucleus Decay Particle Evapor- ation  Evaporation Residues ER -ray emission Fission fragments Statistical Independence Hypothesis : All degrees of freedom equilibrated, no memory of formation, except conservation laws (momentum, energy, angular momentum,… Fission

W. Udo Schröder, 2007 Classical Pot. Scattering 3 14 N 26-n Al 12 C Fusion reaction 14 N+ 12 C leading to compound nucleus 26-n Al, emitted at <≈ 0 0 (Momentum Conservation) elastic CN decays in flight by particle evaporation (ER) or fission

Fusion Excitation Functions W. Udo Schröder, 2007 Classical Pot. Scattering 4 R.G. Stokstad et al., PRL 41, 465 (1978) P. Sperr et al., PRL37, 321(1976) 148 Sm:  2 =0 154 Sm:  2 =0.3 Deformation changes the effective barrier height  larger  fus  fus ≈  R only for E cm below and close to barrier.  Maximum L fus due to yrast limitation (nuclear centrifugal stability) ER = lowest window RR RR 0 ER F R Fusion Fission multi- nucleon Transfer Elastic/quasi- elastic Scattering Fusion -ER

ER Angular Distributions W. Udo Schröder, 2007 Classical Pot. Scattering 5 Random emission from moving CN does not change average velocity, preserves = 0 0, Sideways recoil components important for angular distributions of ERs.

Independence Hypothesis W. Udo Schröder, 2007 Classical Pot. Scattering 6 Compound nucleus reaction (formation+decay) a+A  C*  b+B Decoupled 2-step process, intermediate equilibration following fusion takes long and leads to the same asymptotic condition C*(E, I,…) Separation of cross sections: Independent probabilities of formation and decay multiply for overall reaction

(HI, xn) Excitation Functions W. Udo Schröder, 2007 Classical Pot. Scattering 7 a+A b+B C’* + n C’’* + 2n C’’’* + 3n  C* (HI, xn) cross sections E lab ( 19 F, 7n) ( 19 F, 8n) ( 19 F, 9n) Channels open successively. Statistical competition in overlap regions.

Evaporation Particles W. Udo Schröder, 2007 Classical Pot. Scattering 8 cm spectra of particles statistically emitted from CN (evaporated) are of Maxwell Boltzmann type EBEB V eff R Even for fixed E* the particles spectrum is continuous (Maxwell- Boltzmann), except for transitions to discrete spectrum at low E ER * E* CN ER neutrons protons EBEB

CN Decay Widths W. Udo Schröder, 2007 Classical Pot. Scattering 9 E* CN ER Unstable state (finite energy “line” width )  mean lifetime – Heisenberg’s UR: ·≈   ≈  / = decay probability Total production prob. of CN in reactions: Specific reaction channel    Transition probability Principle of detailed balance: #states  ·P(  #states  ·P(    #final states 

CN Decay Widths W. Udo Schröder, 2007 Classical Pot. Scattering 10 E* CN ER Principle of detailed balance: #states  ·P(  #states  ·P(   Partial decay width  : all “channels” by which C can be formed or into which it can decay Can compute total width  and partial widths   for decay to particular channel  if all formation cross sections are known, all “channels” by which C can be formed in the inverse process.

Decay Width for Neutron Emission W. Udo Schröder, 2007 Classical Pot. Scattering 11 C’ +n Density of states of CN parent at original excitation Final state density of daughter nucleus, accounting for energy lost in neutron emission C Energy spectrum of emitted neutrons depends on level density in final nucleus, non- monotonic ~E n · C’ (…- E n …)

E* Dependence of Nuclear Level Density W. Udo Schröder, 2007 Classical Pot. Scattering 12 Strongly excitation energy dependent  shape of dN/dE n Weakly dependent on E n (neglect this) Internal system of nucleons at high energies = chaotic (Fermi) gas Use statistical mechanics concepts: Entropy Constant-temperature level density (good for small |Q| Set k B =1  [T]= energy  C’ and T correspond to final nucleus+n

FG Nuclear Temperatures and Level Densities W. Udo Schröder, 2007 Classical Pot. Scattering 13 Spectrum of single neutron Spectrum of cascade of neutrons Fermi gas relations: Deviations at shell closures

Angular Distributions of CN Decay Particles W. Udo Schröder, 2007 Classical Pot. Scattering 14 Beam axis and collision trajectory define the “reaction plane.” Orbital and CN spin angular momentum have to be perpendicular to it. Random emission in reaction plane (in ), symmetry about  cm =90 0.