mean-field dynamics of nuclear collisions · 2010. 7. 21. · basic hartree-fock equations i....

Summer School V on Nuclear Collective Dynamics Feza Gürsey Institute, Istanbul, Turkey - 2010

Mean-Field Dynamics of Nuclear CollisionsMean-Field Dynamics of Nuclear Collisions

Research supported by:US Department of Energy, Division of Nuclear Physics

A. Sait UmarVanderbilt University

Nashville, Tennessee, USA


Overview: GeneralOverview: General

I. Nuclear Many-Body Problem- General introduction- N-N interaction- Effective interactions- Many-body methods- Why/when mean-field is a reliable approximation?- Basic HF-TDHF equations- Numerical Methods

II. Time-Dependent Hartree-Fock- Initialization of nuclear collisions- Examples

III. Calculation of Potential Barriers and Fusion- Traditional methods- DC-TDHF method- Examples


I. Nuclear Many-Body ProblemGeneral introduction

Two major challengesTwo major challenges

A. Interaction

- No practical first-principles theory for deriving N-N interaction- Quantum chromodynamics (QCD) – Effective field theories- Lattice QCD shows promise- n,p or quarks and gluons- Inside a nucleus – medium effects

B. Many-body Method

- Many-body wavefunction- Many-body equations- Correlations- Center-of-mass motion- Some of these solved for few nucleon systems


N-N interaction in free space (no medium)N-N interaction in free space (no medium)

I. Nuclear Many-Body ProblemN-N Interaction

Properties deduced from partial-wave analysis of scattering phase shifts and few nucleon systems (e.g. deuteron):

- Short range and attractive in the intermediate range- Hard-core (around 0.5fm) (s-wave phase shift)- Spin dependent- Spin-orbit component- Charge independence (almost!)- Tensor component - non-central

Argonne V-18: Wiringa, Stoks & Schiavilla, Phys. Rev. C 51, 38 (1995)


N-N Interaction I. Nuclear Many-Body Problem

Symmetries ofSymmetries of N-N interaction N-N interaction

Translational invariance Galilean invariance Rotational invariance Invariance under space reflections Time reversal invariance Invariance under particle interchange Isospin invariance (not exact)

V r ={V CV 1⋅2V T 1 3m r

3

m r 2 S12 r V LS 1

m r

1

m r 2 L⋅S} e

−m r

m r

Understanding: Various meson exchange theories (OBE) or chiral perturbation theoriesprovide explanation of this phenomenological N-N interaction

OBEP

π, σ

ρ, ωN N+

two mesons

2π

πρN N

baryon (p / n) baryon (p / n)

meson (π)

Okubo and Marshak, Ann. Phys. (NY) 4, 166 (1958)


Effective Interaction I. Nuclear Many-Body Problem

N-N Interaction in Medium (inside nucleus)N-N Interaction in Medium (inside nucleus)

● We usually cannot use the bare N-N interaction inside a nucleus- Hard-core short distance behavior needs renormalization- We only have the on-shell matrix elements of <k|V|k'>- Presence of other nucleons, NNN interaction, etc.

● Various theories to achieve this renormalization- G-Matrix- MB perturbation theory- Various truncation methods (model space, momentum)

● These methods can be utilized in ab initio calculations (MBtheories that start from N-N interaction)



Phenomenological N-N Interaction in MediumPhenomenological N-N Interaction in Medium

● Early nuclear structure calculations simply used a delta function

v(r)=v0 δ(r1-r2)

● Finite-range implies momentum dependence

● Delta-force gives constant. Simplest momentum-dependence whichpreserves rotational invariance, time-reversal invariance, etc.

● This corresponds to a v(r)

⟨k∣v∣k ' ⟩=∫d3 r e−k−k '⋅r v r11

⟨k∣v∣k ' ⟩=v0v1k2v1 k '2

v2k⋅k '

v r11=v0r11v1k '2

r11r11 k2v2

k '⋅r11 k



Skyrme InteractionSkyrme Interaction

From above

Phenomenologicaldensity dependent term

Spin-orbit term

k=∇ 1−∇ 2

2 iP=

1 1⋅ 2

2k†=−

∇ 1−∇ 2

2 i(acts to right) (acts to left)

t 0 , t 1 , t 2 , t 3 , t 4 , x0 , x1 , x2 , x3 , Parameters fitted to nuclear properties

12t11x1

P { r12 k2k†2

r12 }

t21x2P k

†⋅ r12 k

16t 31x3

P r12

Similar form obtained from local density approximation (LDA)

it 4 1 2 ⋅k†× r12 k

v12=t 01x0P r12


Many-Body Methods I. Nuclear Many-Body Problem

Many-Body TheoriesMany-Body Theories

Perturbative many-body theories Coupled cluster theory Shell-model, no-core shell-model Shell-model Monte-Carlo Density functional theory (ab-initio) Renormalization group Others.....(path integrals etc.)

Ab-initio or almost ab-initio methods

Effective methods

Non-relativistic mean-field methods (Hartee-Fock, TDHF, ATDHF) Relativistic mean-field (RMF) methods Hartree-Fock Bogoliubov (HFB, TDHFB) Generator coordinate method (configuration mixing) Gaussian overlap approximation (configuration mixing) Energy density functional theory


I. Nuclear Many-Body ProblemMany-Body Methods

Practical ApplicationsPractical Applications


Many-Body Methods I. Nuclear Many-Body Problem

Scope of the problemScope of the problem

H x1, x2, x3,, xA = E x1, x2, x3, , xA

H=∑i

A

t i∑i j

A

v ij

xi= r i , i ,i

Solve the many-body Schroedinger equation for A particles

Hamiltonian

Differential equation in 3A dimensions!

Very difficult to tackle for large A numerically

MB wavefunction must satisfy certain conditions (antisymmetry)

We still have the center of mass not fixed ∑i=1

A

r i= 0


I. Nuclear Many-Body ProblemMean-Field (NR) Method

General DiscussionGeneral Discussion

Zeroth order approximation to nuclear MB problem Assume each nucleon moves in an average effective field generated

by all the other nucleons and rarely experience hard collisions First used for describing atoms

➔ Well defined center (nucleus) ➔ Electrons primarily feel the Coulomb field of the nucleus

➔ No well defined center! ➔ Nucleons seem tightly packed! ➔ Does mean-field make sense?

208Pb

H=∑i=1

A pi2

2m−∑

i=1

AZ e2

r i

12 ∑i≠ j=1

Ae2

rij

Central potential


I. Nuclear Many-Body ProblemMean-Field (NR) Method

Why is mean-field a good approximation?Why is mean-field a good approximation?

Because of Pauli blocking the only way two nucleons can interact that results in a different final state is if at least one is excited to an unfilled level

This violates energy conservation so it can last a limited duration,

The measure is the mean-free path

Which is many times the nuclear size

E t=ℏ

=1

0


x1 , x2 , x3 ,, x A =1

A !∣1 x1 1 x2 ⋯ 1 x A 2 x1 2 x2 ⋯ 2 x A ⋮ ⋮ ⋱ ⋮

A x1 A x2 ⋯ A x A ∣≡ 1

A !det∣ x i∣

I. Nuclear Many-Body ProblemBasic Hartree-Fock Equations

WavefunctionWavefunction

MB wavefunction in the mean-field (independent particle model) is a product

x1, x2, x3,, x A =P { x1 x2 x3 x4 ⋯ x A }

Wavefunction must satisfy certain conditions e.g.

, xa ,, xb ,=− , xb ,, xa ,

, xa ,, xa ,=0

, xa ,, xb ,=0 if a=b

Slater determinant satisfies these conditions

Without loss of generality orthonormalize single-particle states

∫dxa* x b x=ab



Time-Dependent Variational MethodTime-Dependent Variational Method

Use the MB Slater determinant as a trial function for variational method

H=∑i

A

t i∑i j

A

v ij

Not too hard since single-particle states are uncorrelated and orthonormal

Use Skyrme potential whichsimplifies the exchange term

In coordinate space E can be written in terms of Hamiltonian density

E=⟨∣ H∣ ⟩=∫d 3 r {HSkyrme , , j , s ,T , J ;r HCoulomb p}

All defined in terms of single-particle states e.g.

q r =∑i=1

A

∑

i* r , , q i r , , q

q r =∑i=1

A

∑

∣∇ i r , ,q ∣2

Particle density

Kinetic energy density

S=∫t 1

t 2

dt ⟨ t ∣H−i ℏ∂t∣ t ⟩=0

Engel et. al.,Nucl. Phys. A249, 215 (1975). Chabanat et. al. Nucl. Phys. A635, 231 (1998); Nucl. Phys. A643, 441 (1998).


H S r = ℏ2

2m

12

t 0112

x02−

12

t 012 x0 [p2n

2 ]14 [ t111

2x1t211

2x2] − j 2

−14 [ t112 x1−t 212 x2] ppnn− j p

2− jn

2 −116 [3t111

2x1−t211

2x2]∇ 2

116 [3t112 x1t 212 x2] p ∇

2pn∇

2n

112

t 3[2112

x3− p2n

2 x312 ]

+14t0 x0 s

2−14t 0sn

2s p2

124

t 3 x3 s2−

124t 3

sn2s p

2

+18t1 x1t 2 x2( s⋅T−J

2 )+18t 2−t1∑

q

( sq⋅T q−J q

2 )

+1

32t 23t1∑

q

sq⋅∇2 sq−

132

t 2 x2−3 t1 x1 s⋅∇2 s

−t 42 ∑qq '

1qq ' [ sq⋅∇× j q 'q∇ ⋅J ]


Skyrme Hamiltonian DensitySkyrme Hamiltonian Density

(s,j,T) time-odd, vanish for static HF calculations of even-even nuclei non-zero for dynamic calculations, odd mass nuclei, cranking etc.

Time-odd terms come in pairs!Total is TR invariant


I. Nuclear Many-Body ProblemNumerical Methods

Time-Reversal InvarianceTime-Reversal Invariance

Without time-reversal invariance Skyrme has many extra terms- They come in pairs/products such that Hamiltonian density is time-even

Engel, Brink, Goeke, Krieger, and Vautherin, Nucl. Phys. A249, 215 (1975)

These terms are zero while fitting the force parameters- To properties of static even-even nuclei

Chabanat, Bonche, Haensel, Meyer, and Schaeffer, Nucl. Phys. A635, 231 (1998)

They are all non-zero for dynamical calculations (also for odd-A)

The extra terms are required to satisfy (no new parameters) - Galilean invariance- Local gauge invariance (Dobaczewski and Dudek, Phys. Rev. C52, 1827 (1995))

In addition, we have a time-even spin-current tensor - Not included in the past due to numerical difficulty J



Hartree-Fock EquationsHartree-Fock Equations

Static HF equations can be obtained by substituting

Depends on s.p. states! Equations has to be solved self-consistently

A coupled set of differential equations

i ℏ∂

∂ t=h {}

r , t =e−i t /ℏ r h {}=

Variation with respect to single-particle states gives TDHF equations

1. Guess a set of orthogonal single-particle states.

2. Compute densities, currents, etc.

3. Compute the Hartree-Fock potential.

4. Solve the Poisson equation for direct Coulomb contribution.

5. Perform an imaginary time step with damping.

6. Do a Gramm-Schmidt orthogonalization of all states.

7. Repeat beginning at step 2 until the convergence criteria are met.

Single-particle energies



Time-Evolution and Gradient DampingTime-Evolution and Gradient Damping

t=exp[−ih]t

t≈[1∑n=1

N−ihn

n! ] t

D E 0= [1T x

E0 ]−1

[1T y

E0 ]−1

[1T z

E0 ]−1

k1=O {

k−x0 D [E0]h

k−

k }

E0=20 MeV x0=0.05 O=Gramm−Schmidt

Formal solution for a small time-step

Numerical approximation

Damped-Gradient for static solutionUmar et. al., Comp. Phys. Comm. 63, 179 (1991).Umar et, al. Phys. Rev. C 44, 2512 (1991).



A New Generation TDHF CodeA New Generation TDHF Code

Unrestricted 3-D Cartesian geometryOld version: Umar et al, Phys. Rev. C44, 2512 (1991)

Basis-spline discretization for high accuracyUmar et al, J. Comp. Phys., 93, 426 (1991)

Coded in Fortran-95

Use of modern Skyrme forces with spin-orbit (SLy4, SkP, etc.)

No time-reversal symmetry assumed

Thread-safe, runs under OpenMP

Umar et. al., Phys. Rev. C 73, 054607 (2006).


Basis Splines of order M=5 with boundary conditions

f x =∑k=1

N

Bk

M xck

[Of x]∑

'

O

' f '

O

'≡∑

k=1

N

[OBk x ]

Bk '

∫a

b

f x dx ∑

f

Expand functions in B-splines

Lattice-Collocation Method

f x f

Lattice operators are given by

Umar et al, J. Comp. Phys., 93, 426 (1991)


Discrete Mathematics – Basis-Spline Collocation MethodDiscrete Mathematics – Basis-Spline Collocation Method


f x =∑k=1

N

Bk

M xck

[Of x]∑

'

O

' f '

O

'≡∑

k=1

N

[OBk x ]

Bk '

∫a

b

f x dx ∑

f

Expand functions in B-splines, discretize on collocation lattice

f x f

Solve for expansion coefficients by inverting B

f=∑

k=1

N

Bk M ck

ck=∑=1

N

[Bk ]−1 f

on lattice

Action of an operator on a function

[Of x ]=∑k=1

N

[OBkM x ]ck [Of x ]

=∑

k=1

N

[OBkM x ]

∑ '=1

N

[Bk ']−1 f

'

substitute ck

Rewrite by defining collocation operator

where

Lattice integration defined in a similar way

≡∑

k

hkckwith


Basis-Spline Collocation Method - DerivationBasis-Spline Collocation Method - Derivation


x , y , z ; t =∑ijk

Bi x B j yBk z cijkt

S=∫dt∑

V {H −[ i ℏ∑

∗ ∂

∂ t ]}

∫ d3 r 2=∑

∣∣

2

∇ =∑ '

D

' ' ∑ '

D

' ' j∑ '

D

' ' k

Expand single-particle states in B-spline basis

Discretize on the collocation lattice before variation

After variation local terms are local

Non-local terms look like (matrix-vector multiply)

∗

∗ ' ' ' =

1V

' ' '


Discretization of TDHF Equations - OutlineDiscretization of TDHF Equations - Outline

NuSTAR SeminarSummer School V on Nuclear Collective DynamicsSummer School V on Nuclear Collective Dynamics Feza Gürsey Institute, Istanbul, Turkey 2010

Presentation OverviewPresentation Overview

General Overview Presentation

1) Summary of traditional approaches for ion-ion potentials

2) Implementation of TDHF theory

3) TDHF and density constraint – dynamical potentials

4) 3-α states of 8Be

5) Potentials and fusion cross-sections for selected systems

6) Dynamical calculation of excitation energy

7) Application to superheavy formations

8) Conclusions


No ab-initio many-body theory for sub-barrier fusion applicable to heavier nuclei exists All approaches involve two prongs

a) Calculate an ion-ion barrier (usually one-dimensional, V(R))- Phenomenological (Wood-Saxon, Proximity, Folding, Bass, etc.) using frozen densities.- Microscopic, macroscopic-microscopic methods using collective variables (CHF, ATDHF, empirical methods).

b) Employ quantum mechanical tunneling methods for the reduced one-body problem (WKB, IWBC) Incorporate quantum mechanical processes by hand

a) Neutron transferb) Few excitations of the entrance channel nuclei (CC)

Many-Body Fusion and FissionMany-Body Fusion and Fission

General Overview Ion-Ion Potentials

R


Traditional Microscopic Methods (CHF-ATDHF)Traditional Microscopic Methods (CHF-ATDHF)

Advantages

- Fully microscopic, self-consistent description of nuclear potential energysurface (PES)

- Use same microscopic interaction used in ground state. calculation- Gives global information on collective potential (collective subspace)- Include some correlations by restoring broken symmetries

Shortcomings

- Artificial introduction of constraining operators - Collective motion not necessarily confined in constrained phase space- Static adiabatic approximation- Most energetically favorable state may require sudden rearrangement- No reason why dynamical system should move along the valley of PES- CHF calculations seldom produce the correct saddle-point



Phenomenological Fusion BarriersPhenomenological Fusion Barriers

Most information taken from ground state properties of nuclei - These may be correct prior to nuclear overlap but different after Use frozen densities (folding potentials)

- Typically Fermi densities fitted to experimental data Phenomenological heavy-ion interaction potentials

- Wood-Saxon, Proximity, Bass, double-folding- Several free parameters

Use few excited states (2-3)- Coupling potentials for excitations derived in simple models

(rigid rotor, harmonic vibrator) - B(EL) values taken from experimental data Rotating frame approximation

- To reduce the number of channels Allow for neutron transfer

- Based on Q-values



Frozen-density based approximations break down Inner part of the barrier is modified by many effects

- Neck formation- Particle transfer- Dynamical rearrangement of the densities

Actually different barriers are seen by individual states

Fusion at Low EnergiesFusion at Low Energies


Similar (but not the same) phenomenon for very heavy systems


Desired ImprovementsDesired Improvements

Explore collective dynamics in terms of mean-field dynamics- self-organizing system selects its evolutionary path by itself following the microscopic dynamics.

Develop dynamical methods for selecting constraining operators- which are not known from the outset nor from the static theory- should it be coordinate or constraint?

Go beyond the static adiabatic approximation- Explore nonlinear dynamics between single-particle degrees

of freedom and collective motion by going beyond adiabatic approximation

Diabatic states- Go beyond single determinant (shape coexistence)



Generate HF Slater determinants for each nucleus

Multiply each determinant by a boost, determined from Coulomb trajectory and the asymptotic Ecm, at the initial nuclear separation

j exp ik j⋅R jfor nucleus-j R= 1A j∑i=1

A j

r iand

Combine two determinants into a single one

TDHF

initial state final state

Initial TDHF SetupInitial TDHF Setup

Time-Dependent Hartree-Fock Dynamical Approach

TDHF equations are translationally invariant

Contains c.m. wavefunction!

Summer School V on Nuclear Collective Dynamics Feza Gürsey Institute, Istanbul, Turkey 2010

II. Time-Dependent Hartree-FockInitialization

Definition of Final StateDefinition of Final State

Coulomb trajectory inCoulomb trajectory out

TDHF

If final stage contains a single fragment – FUSION If final stage contains two fragments – DEEP INELASTIC SCATTERING Initial approach is determined by Coulomb interaction only

Validity of mean-field approximation. All or most of Ecm can betransformed to internal excitation!

≃213.08

3/2 =

Ecm−EB

Fermi-GasFermi-Liquid


II. Time-Dependent Hartree-FockExamples

16O+28O at Ecm = 43 MeV, SLy5, b = 7.5 fm

σfusion

= 1916 mb



16O+28O at Ecm = 43 MeV, SLy5, b = 7.6 fm



Deformed Nuclei – Orientation DependenceDeformed Nuclei – Orientation Dependence

Entrance channel Coulomb excitation

=Alignment

⟨J f∥Q2∥J i⟩Diagonal and off-diagonal

Quantum mechanical calculation Umar, Oberacker, Phys. Rev. C 74, 124606 (2006)

R



Deformed Nuclei – Effect on FusionDeformed Nuclei – Effect on Fusion

V(R)

R

Distribution of barrier heights depending on orientation

Fusion cross-section fus Ec.m. =∫0

1d cos fus Ec.m. ;

Can lead to enhancement of fusion cross section by orders of magnitude

Generalized application in TDHF Umar, Oberacker, Phys. Rev. C 74, 124606 (2006)



116

8O + 22

10Ne (alignment 1), E

cm= 95 MeV, b=0 fm



116

8O + 22


cm= 95 MeV, b=0 fm



116

8O + 22


cm= 95 MeV, b=5 fm



64Ni+132Sn, Ecm = 176 MeV, b = 4 fm, Deep-Inelastic

L initial /ℏ=76.5 L final /ℏ=60.7 E final=142.0 MeV



64Ni+132Sn, Ecm = 176 MeV, b = 5 fm, Deep-Inelastic

L initial /ℏ=95.6 L final /ℏ=93.0 E final=172.3 MeV


Project unhindered TDHF evolution onto the dynamical PES- system selects its evolutionary path by itself- constrains all collective degrees of freedom

A method to extract internal excitation energy while holding the instantaneous neutron and proton densities constrained. TDHF provides the dynamical densities for calculating V(R)

Density-ConstraintDensity-Constraint

ρTDHF

(r,t)

QuasiStatic Energy Surface

E*(R(t))

Cusson, Reinhard, Maruhn, Strayer, Greiner, Z. Phys. A320, 475 (1985)

Density Constraint and TDHF Dynamical Approach

, Z. Phys. A320, 475 (1985) , Z. Phys. A320, 475 (1985)


ImplementationImplementation

H H Q

H H∫d3 rqr

qr

H H∑i

iQ

i

Generalize the ordinary method of constraints

- for a single constraint

- for a set of constraints

- for density constraint

Works as accurately as a single constraint

- numerical method for steering the solution to TDHF density is given in:


1. Cusson, Reinhard, Maruhn, Strayer, Greiner, Z. Phys. A 320, 475 (1985)2. Umar, Strayer, Cusson, Reinhard, Bromley, Phys. Rev. C 32, 172 (1985)


Ion-Ion PotentialIon-Ion Potential

E TDHF=T RVE* V=ETDHF−T R−E*=E DC

V RE DC R−E A1−E A2

Subtract binding energies

EDC

contains the binding energies of the two nuclei

Asymptotically correct (no normalization needed):

EDC Rmax =E A1E A2

V Coulomb Rmax V Rmax =VCoulomb Rmax

Total energy in terms of the excitation energy is:

Umar, Oberacker, Phys. Rev. C 74, 021601(R) (2006)

Conserved quantity



Comparison to Empirical Fusion PotentialsComparison to Empirical Fusion Potentials

Double folding:M3Y effective NN interactiondensities from electron scattering

DC-TDHF potential contains no parameters and normalization

Energy dependence:For light systems energy dependence is small


Coulomb tails always accurate to 50-150 keV


Dynamical Effective MassDynamical Effective Mass

E c.m.=12

M R R2V R

DCTDHFTDHF Typical CHF type peak

Because we are over the barrier!

Transform effect to V(R)

d R= M R

12 dR

M R=2E c.m.−V R

R2



Formation of Formation of 1212C In the Universe (also C In the Universe (also 1616O)O)

Density Constraint and TDHF Triple Alpha Reaction

8Be has 10-16s lifetime and not found in nature In stars due to 4He abundance small amount of 8Be always present 4He+8Be combine to form resonant state of 12C (Hoyle state) Excited state decays to ground state via an intermediate state Use TDHF to study the dynamics of this process

See Movie

Umar, Maruhn, Itagaki, OberakerPhys. Rev. Lett. 104, 212503 (2010)


Dynamics of TransitionDynamics of Transition

Density Constraint and TDHF Triple Alpha Reaction


64Ni+132Sn, Ecm = 176 MeV, SLy5, b = 3 fm

6464Ni + Ni + 132132Sn FusionSn Fusion

At low energies fusion x-section orders of magnitude larger than CC Prediction (Liang et al., PRL 91, 152701 (2003))

Qzz(n) = -0.85 b, Qzz(p) = -0.59 b

Density Constraint and TDHF Application to Fusion

3D HF (Sly5) gives oblate deformation for 64Ni:


Barrier for β=0o agrees with empirical No parameter/normalization in TDHF VB = 155.81 MeV RB = 12.12 fm

Barrier for β=90o lower VB = 150.13 MeV RB = 12.87 fm

6464Ni + Ni + 132132Sn Limiting BarriersSn Limiting Barriers

Umar, Oberacker, Phys. Rev. C 74, 061601(R) (2006)Umar, Oberacker, Phys. Rev. C 76, 014614 (2007)



6464Ni + Ni + 132132Sn Complete Set of BarriersSn Complete Set of Barriers

Umar and Oberacker, Phys. Rev. C 76, 014614 (2007)



6464Ni + Ni + 132132Sn Fusion Cross-SectionSn Fusion Cross-Section

Exp. DataJ.F. Liang et al.,PRL 91, 152701 (2003)PRC 75, 054607 (2007)

f Ec.m.=∫0

1

dcosP Ec.m. ,

Use IWBC Average over orientations


Umar and Oberacker, Phys. Rev. C 76, 014614 (2007)


6464Ni + Ni + 6464Ni FusionNi Fusion

(a) outer turning point(b) inner turning point(c) reorientation of the core(d) Density at the minimum

Turning points for Ecm = 86 MeV

((ββ11=90=90oo, , ββ22=90=90oo))

Interesting neutron rich identical system



6464Ni + Ni + 6464Ni Limiting BarriersNi Limiting Barriers

Umar, Oberacker, Phys. Rev. C 77, 064605 (2008)

Variation with Euler angle αi is negligible!



6464Ni + Ni + 6464Ni Fusion Cross-SectionNi Fusion Cross-Section

Exp. DataC.L. Jiang et al.,PRL 93, 012701 (2004)

Problem at low energies (CC) Compression potential Mişicu, Esbensen, PRL 96, 112701 (2006) Modify inner turning point Ichikawa, Hagino, Iwamoto, PRC 75, 064612 (2007)



Conjecture - Skin versus Core OrientationConjecture - Skin versus Core Orientation

Neutron rich systems have extended outer skins Orientation of core in 64Ni perpendicular to outer surface Top of the barrier primarily determined by outer surface Lower energies imply larger overlaps for inner turning point Ambiguity in which orientation to choose in angle averaging



Umar, Oberacker, Phys. Rev. C 77, 064605 (2008)

For Low Energies use Core OrientationFor Low Energies use Core Orientation



1616O + O + 208208Pb Fusion Cross-Section Pb Fusion Cross-Section

Umar, Oberacker, EPJA 39, 243 (2009)



Excitation EnergyExcitation Energy

E*=E c.m.Qgg

Umar, Oberacker, Maruhn, Reinhard, Phys.Rev. C 80, 041601(R) (2009)

TDHF and Heavy-Ion Dynamics Excitation Energy

Excitation energy is an important indicator of reaction dynamics

Superheavy formations are very sensitive to excitation energy

Indicator of temperature for compound configurations

Traditional knowledge based on initial and final reaction products:

Time evolution of excitation energy could tell us about the survival of intermediate configurations

Density constraint makes this calculation possible via TDHF


Excitation Energy via TDHFExcitation Energy via TDHF

E*t =ETDHF−E coll t , j t

ETDHF t =∫ d 3 r H r , t

Excitation energy

Total TDHF energy (conserved)

E coll t =Ekin t , j t E DC t Collective energy

Kinetic energyE kin t , j t =m2 ∫ d3 r

j2 t t

Ecoll t =E kin t , j t V Rt E A1E A2

In terms of ionion potential:

Umar, Oberacker, Maruhn, Reinhard, Phys.Rev. C 80, 041601(R) (2009)



Example Study for Example Study for 4040Ca+Ca+4040Ca SystemCa System

Capture point Collective kinetic energy

Examine E* at the capture point



Compare with the Traditional DefinitionCompare with the Traditional Definition



Dynamics of Heavy SystemsDynamics of Heavy Systems

TDHF and Heavy-Ion Dynamics Superheavy Systems


Factors Influencing Superheavy FormationsFactors Influencing Superheavy Formations

Excitation energy - high excitation at the capture configuration – quasi-fission - high excitation of compound nucleus - fusion-fission Nuclear deformation and alignment Shell effects Mass asymmetry in the entrance channel Impact parameter dependence ......

ER=capture⋅PCN⋅P survival

Capture in ionion potential pocket

Survive FF processForm compound system



Cold and Hot Fusion of Heavy SystemsCold and Hot Fusion of Heavy Systems

70Zn+208Pb 48Ca+238U (β=45o)

Heavy nuclei exhibit a very different behavior in forming a composite system

capture≈ER≈ fusion

Light-Medium Mass Systems Heavy Systems

- Fission and quasi-fission negligible- Simple V(R) for composite system

capture=QFFFER

- Quasi-fission dominant- Di-nuclear composites common- A multi-stage V(R)- QF may masquerade as DI



Excitation Energies at Capture PointExcitation Energies at Capture Point

Eexp* =E c.m.Qgg Experimentally cited excitation energy

We calculate excitation energy as a function of R(t)



Excitation EnergiesExcitation Energies


Should be alignment averaged: E*Ec.m.=∫

0

1

dsin P E*Ec.m. ,


PotentialsPotentials

- Cores remain distinct- Nucleons exchanged- b>0 deep-inelastic

- Cores join- Capture



Cross-SectionsCross-Sections

Angle average 238U alignment: - significantly reduces x-section

f Ec.m.=∫0

1

d sin P Ec.m. ,

- x-section falls rapidly for β>10o

- sin(β) multiply small angles - P(β) is in the range 0.4-0.6


Experimental data (private communication):1. Yu. Ts. Oganessian, Phys. Rev. C 70, 064609 (2004)2. Yuri Oganessian, J. Phys. G: Nucl. Part. Phys. 34, R165 (2007)


Cross-Sections (tentative)Cross-Sections (tentative)


Experiments:1. S. Hofmann et al., Rev. Mod. Phys., 72, 733 (2000)2. S. Hofmann et al., Eur. Phys. J. A 14, 147 (2002)

Could not find data for capture x-section

Calculated capture x-section (reproducing one σ

ER x-section

value.)

G. Giardina, S. Hofmann, A.I. Muminov, and A.K. Nasirov, Eur. Phys. J. A 8, 205 (2000)


ConclusionsConclusions

There is mounting evidence that TDHF dynamics give a gooddescription of the early-stages of low-energy HI collisions

We have developed powerful methods for extracting more informationfrom the TDHF dynamical evolution (V(R), M(R), E*, etc.)

Although heavy systems pose a greater challenge, such microscopiccalculations may provide an insight into these collisions

Effort needed to incorporate deformation and scattering information in to the Skyrme parametrization

Microscopic Potentials based on TDHF Summary

mean-field dynamics of nuclear collisions · 2010. 7. 21. · basic hartree-fock equations i....

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