1. Nucleon DIffusion in Heavy-Ion Collisions
S. Saatci1, O. Yılmaz1, B. Yılmaz2, and Ş. Ayık3
1Physics Department, Middle East Technical University, Ankara, Turkey 2Physics Department, Ankara University, Ankara, Turkey 3Physics Department, Tennessee Techn University, Cookeville, USA
NUFRA 2015, Fifth International Conference on Nuclear Fragmentation, Antalya, Kemer
October 05, 2015

2. Outline Heavy-Ion Reactions
Nucleon Exchange and Transport Coefficients
Formalism for Nucleon Exchange
Quantal Diffusion Coefficients
Preliminary Results

3. Heavy-Ion Reactions Fusion
Small impact parameters and energies above Coulomb barrier.
Target and projectile can merge to produce one nuclei.
Deep-Inelastic Collisions
Close collisions, low energies near Coulomb barrier, Ecm ≈0.95Vc.
Colliding projectile and target ions exchange a few nucleons through the window and seperate again.

4. Nucleon Exchange and Transport Coefficients
Transport coefficients for nucleon exchange are related to macroscopic variables such as mass and charge transfer of target-like and projectile-like fragments.
The nucleon number of the projectile-like fragments or its mass distribution can be deduced from transport coefficients.
Fluctuations of collective variables play important role in processes such as deep inelastic heavy-ion collisions and heavy-ion fusion near Coulomb barrier energies.
Standard mean-field theories
Cannot describe the dynamics of fluctuations of collective motion.
Provide a good description of the average evolution, however, the width of the fragment mass distribution appears smaller.
Stochastic mean-field approximation
an ensemble of single-particle density matrices is constructed by incorporating quantal and thermal fluctuations in the initial state.
Not only the mean value of an observable, but the probability distribution of the observable is also found.
Two mechanisms for fluctuations:
collisional fluctuations generated by two-body collisions
(at high energy nuclear collisions),
one-body mechanism or mean-field fluctuations originating from quantal and thermal fluctuations in the initial state (at low energies).

5. Formalism for Nucleon Exchange
A member of single-particle density matrix of each event, λ :
Time-independent elements of density matrix determined by the initial condition, which are Gaussian random numbers with mean value
Occupied (hole) and unoccupied (particle) states
Spin-isospin
quantum number
The variance of the fluctuating part is given by :
Non-zero initial fluctuations only between particle and hole states!
ensemble average
Mean values of the single-particle occupation factors that is zero or one at zero temperature, Fermi-Dirac distribution at finite temperature
Time evolution of each wave function by its own mean-field of each event :

6. Quantal Diffusion Coefficients
Nucleon density
Continuity equation
Current density
Mass number of the projectile-like fragments:
Step function
Location of the window
Nucleon Flux:
Nucleon drift coefficient , its mean value is zero for symmetric systems
Fluctuating part of nucleon flux (quantal effect)

7. Variance of the fragment mass distribution is calculated by
Quantal and memory dependent diffusion coefficient for nucleon exchange is determined by the correlation function of the stochastic part of the nucleon flux
Diffusion coefficient
particle
hole
Variance of the fragment mass distribution is calculated by
We extended Umar’s TDHF code to calculate the time-dependent unoccupied single-particle wave functions in addition to the occupied hole states.

8. Preliminary Results
Quantal diffusion coefficient for nucleon exchange in the central collision of
t=400 fm/c t=700 fm/c t=1000 fm/c
The calculations exibit a smooth behavior as a function of time.
As expected, neutron exchange becomes larger than proton for neutron rich system.
Quantal variance of the fragment mass distributions is obtained in a stochastic approach.