NUCLEON DIFFUSİON İN HEAVY-ION COLLİSİONS

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, KemerOctober 05, 2015

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

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.

Nucleon Exchange and Transport Coefficients

• 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).

• Two mechanisms for fluctuations:

• 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.

Formalism for Nucleon Exchange

ˆ ( , , ) ( , ; ) 0 ( , ; )a i a jij

r r t r t i j r t A member of single-particle density matrix of each event, λ :

Spin-isospin quantum number

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 ˆ 0a ij ji j n

The variance of the fluctuating part is given by :

1(0) ' (0) ' 1 1

2a b ab ii jj i j j ii j j i n n n n

ensemble average

Mean values of the single-particle occupation factors that is zero or one at zero temperature, Fermi-Dirac distribution at finite temperature

( , ; ) ( ) ( , ; )i r t h t r tj a jt

Time evolution of each wave function by its own mean-field of each event :

Non-zero initial fluctuations only between particle and hole states!

Quantal Diffusion Coefficients

*

* *

( , ) ( , ) ( , )

( , ) ( , ) 0( , ) ( , ) ( , ) ( , ) ( , )

2

j ji iij

j i i j jiij

r t r t r t

r t j r ttj r t r t r t r t r t

im

Nucleon density

Current density

Continuity equation

x

y

Mass number of the projectile-like fragments:

0 ( ) ( , , , )pA dxdydz x x x y z t

Step function Location of the window

,0( , , , ) v ( ) v ( ) v ( ) v ( )quantalTDHF

p x A A A AdA dydz j x y z t t t t t

dt

Nucleon Flux:

Nucleon drift coefficient , its mean value is zero for symmetric systems

Fluctuating part of nucleon flux (quantal effect)

02

* *

0

( ) ' v ( ) v ( )

1 ' ( ) ( ) ( ) ( )

2 2

tAA A A

tph ph ph ph

ph

D t dt t t

dt A t A t A t A tm

2

0( ) 2 " ( ")

t

AA AAt dt D t

* *

0( ) ( )ph p x h h x p

xA t dydz t

Diffusion coefficient

Quantal and memory dependent diffusion coefficient for nucleon exchange is determined by the correlation function of the stochastic part of the nucleon flux

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.

Quantal diffusion coefficient for nucleon exchange in the central collision of 24 24O + O

• 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.

t=400 fm/c t=700 fm/c t=1000 fm/c

Preliminary Results