Yoritaka IwataYoritaka Iwata11 and Takaharu Otsuka and Takaharu Otsuka1,21,2

Reaction mechanism Reaction mechanism in neutron-rich nucleiin neutron-rich nuclei

11Department of Physics, University of TokyoDepartment of Physics, University of Tokyo

Advices about using TDHF code: C. Simenel (Saclay & MSU)Advices about using TDHF code: C. Simenel (Saclay & MSU)

22CNS, University of TokyoCNS, University of Tokyo

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Schrodinger equationSchrodinger equation

)],(),(),([)(| 211 trtrtrAt nnn

)(|)(| tHti t

Vm

HN

jj

j

1

22

2

Slater determinantSlater determinant

0 Hidt t

TDHF TDHF equationequation

・・・・

TDHF LagrangianTDHF Lagrangian

TDHF formalismTDHF formalism

(← time-dependent variational principle)(← time-dependent variational principle)

antisymmetrizerantisymmetrizer

(Dirac 1930, Bonche-Koonin-Negele 1976 ～ )for nuclear physicsfor nuclear physics

TDHF eq. TDHF eq. for each single particle wave functionfor each single particle wave function

'),'(),',(2

22

drtrtrrVm

i iij

it

j

jj drdrtrtrrrrrVtrrV '''''),'''(),''()''',','',(~

),',( * Antisymmetrized potential

TDHF equations for single particle wave TDHF equations for single particle wave functionfunction

From substitution, we obtainFrom substitution, we obtain

→ → One body evolutionOne body evolution

Skyrme Skyrme interactioninteraction

ji

ijkji

ij vvV )3()2(

)()1( 2100 rrPxt

)')()(()1(2

1 22121

211 krrrrkPxt

')()1( 2122 krrkPxt

')()( 21210 krrkiW

SLy4SLy4dd

SLy4dSLy4d Chabanat - Bonche - Hansel, 1995

(Skyrme 1956　～ )

size

x

y

z

3D 3D lattice lattice

Each single particle wave functionsingle particle wave function is defined on the (3+1)D lattice

Mesh size: Δx = 0.8 fm

Symmetric about z=0 plain

Spatial DiscretizationSpatial Discretization

ΔxΔx

Δx

““TDHF3D-TDHF3D-code”code”

Bonche-Grammatico-Koonin 1978 ～

spacspacee

timetime

Δt = 0.015×10-22 s

UnitUnit

of unit

Collision of “Collision of “Ca isotopesCa isotopes””

4He + **Ca →　・・・

Neutron

Proton

4He

**Ca

ReactionReaction

““Very first few moments of Very first few moments of reactionreaction””

A spot light is casting on…A spot light is casting on…

4He **Ca

　1) Initial1) Initial

2) Contact2) Contact

3) Full overlap3) Full overlap

Relative low energy collisionRelative low energy collision

The very first The very first few moments few moments

11

22

33

TimeTime

View pointsView points

Can we see scatterings according to the Pauli effect ?Can we see scatterings according to the Pauli effect ?

Is there a specific neighboring for 4 nucleon @ Is there a specific neighboring for 4 nucleon @ projectileprojectile

Accelerations in early timesAccelerations in early times

PP

pp

nn

nn projectileprojectile

during reaction (especially for neutron-rich during reaction (especially for neutron-rich case) ?case) ?

(spherical-spherical)(spherical-spherical)

4He +40Ca 　

tt = = 0.0(s)0.0(s)

initial energy30.8MeV

(E/A = 0.7MeV)

yy [fm]

xx [fm]

dt = 1.5 * 10-24s

Impact parameter = 0.0 fmImpact parameter = 0.0 fm

TDHF calculationTDHF calculation

For comparisonFor comparison

0 ||Jz

1/21/2

1/21/2

Estimated contact time = Estimated contact time = 10.0 10.0 dtdt

|Jz| start to change at |Jz| start to change at 14.0 14.0 dtdt

Protons of Protons of projectileprojectile

Neutrons of Neutrons of projectileprojectile

|Jz| becomes larger than 1/2 at |Jz| becomes larger than 1/2 at 22.0 22.0 dtdt

(sufficient to be non 1s-state)(sufficient to be non 1s-state)

7/27/2

7/27/2

|Jz| has maximal at |Jz| has maximal at 28.0 28.0 dtdt

1s1s1/21/2

1s1s1/21/2

f f 7/27/2

f f 7/27/2

1s knock out1s knock out

1s knock out1s knock out

contactcontact

contactcontact

)(||)( tt ii Jz Each single waveEach single wave

Time evolution(by TDHF)Time evolution(by TDHF)

20

10

0

30

40

ContactContact

Composite nucleiComposite nuclei

xx [fm]yy [fm]

Time (Time (*dt [sec]))

dt = 1.5 * 10-24s

What happens in the What happens in the 1s knock out time ?1s knock out time ?

jj

jii xxx )(ˆ 2

)ˆ,ˆ( ii yxj

jjii yyy )(ˆ 2

Center-of-massCenter-of-mass

Trace of projectile (calculated result)Trace of projectile (calculated result)

t t = 0.0= 0.0

t t = 14.0= 14.0

t t = 22.0= 22.0

t t = 22.0= 22.0

t t = 28.0= 28.0contactcontact

yy [f

m]

xx [fm]

Estimated contact time = Estimated contact time = 10.0 10.0 dtdt

|Jz| start to change at |Jz| start to change at 14.0 14.0 dtdt

(sufficient to be non 1s-state)(sufficient to be non 1s-state)

|Jz| has maximal at |Jz| has maximal at 28.0 28.0 dtdt

|Jz| becomes larger than 1/2 at |Jz| becomes larger than 1/2 at 22.0 22.0 dtdt

Jz evolutionJz evolution

t t = 14.0= 14.0ScatteringScattering

2 neutrons @ He2 neutrons @ He← Scattering due to the Pauli effectPauli effect

Copy from the former Copy from the former pagepage

Highly correspondingHighly corresponding

1s neutrons @ Ca1s neutrons @ Ca

1s knock out1s knock outneutronneutronprotonproton

nucleon @ projectilenucleon @ projectile

Separated (n-p) pairs Separated (n-p) pairs alwaysalways have the same sign have the same sign of nuclear spin.of nuclear spin.

I.e. (nI.e. (n++, p, p++) ------ (n) ------ (n--, p, p--)) deuteronsdeuterons

Center-of-mass motion of projectileCenter-of-mass motion of projectile

time time Period/2Period/2

Large mean free Large mean free pathpath

Neutrons of projectile

Target neutrons

Space Space period/2period/2

Space period/2Space period/2

time time Period/2Period/2

AcceleratioAccelerationn

(for Ca)(for Ca)

Time evolution of center-of-mass Time evolution of center-of-mass velocityvelocity

timetime

Observation of the Observation of the early accelerationearly acceleration

Velocity [(2/3)* 10Velocity [(2/3)* 1099 m/s] (in lab. frame) m/s] (in lab. frame)

Ohnishi-Horiuchi-Wada 1990: via Vlasov eq. (Vlasov eq. (1616O+O+1616O)O)Previous workPrevious work

(Norenberg 1983: large mean free path via Dissipative Diabatic DynamicsDissipative Diabatic Dynamics)・・

: head-onhead-on & stable-stablestable-stable reaction study→ → we consider “we consider “non head-on”non head-on” & “ & “non-stable”non-stable” reaction reaction

4He +12C

4He +16O

Neutrons of projectileNeutrons of projectile

velocityvelocity

timetime timetime

velocityvelocity

Neutrons of projectileNeutrons of projectile

Acceleration can be seen in other targets Acceleration can be seen in other targets

Other neutronsOther neutrons Other neutronsOther neutrons

SupplementSupplement

Scattering Scattering due todue to the Pauli effect the Pauli effect

They are found in the dynamics of the lighter the lighter nucleinuclei

40Ca, 16O , 12C

4He

““Acceleration”Acceleration”

~~

Brief summary for stable-reactionBrief summary for stable-reaction

Reaction of neutron-rich Reaction of neutron-rich nucleinuclei

The previous arguments are preparations…

4He +70Ca 　

NeNeww

For the early acceleration,

nuclear reaction with unstable nuclei

NeNeww

non zero impact parameter (particular in 3D-space)

4He +70Ca 　

tt = = 0.0(s)0.0(s)

Initial energy

(E/A = 0.7MeV)

yy [fm]

xx [fm]

dt = 1.5 * 10-24s

Impact parameter = 0.0 fmImpact parameter = 0.0 fm

TDHF calculation of neutron-rich TDHF calculation of neutron-rich nucleinuclei

51.8MeV

Total densityTotal density

Different contact time for N Different contact time for N & P& P

Neutron densityNeutron density

Proton densityProton density

7

7

Contact time for N & P is Contact time for N & P is differentdifferent

Estimated contact time = Estimated contact time = 7.0 7.0 dt dt for Nfor N

Estimated contact time = Estimated contact time = 8.0 8.0 dt dt for Pfor P

dt = 1.5 * 10-24s

20

10

0

30

Composite nucleiComposite nuclei

x x [fm]yy [fm]

Already contactedAlready contacted

Passing Passing throughthrough

Time (Time (*dt [sec]))

Observation of the early accelerationObservation of the early acceleration

timetimetimetime

Early acceleration in stable-unstable Early acceleration in stable-unstable collisioncollision

Velocity [(2/3)* 10Velocity [(2/3)* 1099 m/s] m/s] Velocity [(2/3)* 10Velocity [(2/3)* 1099 m/s] m/s]

AccelerationAcceleration AccelerationAcceleration

which is found in the motion of lighter nuclei

Neutrons of projectileNeutrons of projectile Protons of projectileProtons of projectile

Different scattering for Different scattering for N N andand P P inside “the inside “the neutron skin”neutron skin” dt = 1.5 * 10-24s

20

10

0

30

Composite nucleiComposite nuclei

xx [fm]

Already contactedAlready contacted

Time (Time (*dt [sec]))

Passing Passing throughthrough

yy [fm]

Trace of nucleon @ He (calculated result)Trace of nucleon @ He (calculated result)

t t = 0.0= 0.0

yy [f

m]

xx [fm]

Center of mass motion = Trace of neutron @ Center of mass motion = Trace of neutron @ HeHe

neutronneutron

protonprotont t = 30.0= 30.0

t t = 10.0= 10.0

t t = 20.0= 20.0

t t = 7.0= 7.0

t t = 10.0= 10.0

magnifymagnify

yy [f

m]

xx [fm]

t t = 5.0 = 5.0 ～～ 10.010.0passing neutron skinpassing neutron skin

neutronneutron

protonprotonneutron skin of Ca targetneutron skin of Ca target

t t = 20.0= 20.0

nucleon @ projectilenucleon @ projectile

PP

pp

nn

nn projectileprojectile

Early state of 4 nucleons in projectileEarly state of 4 nucleons in projectile

pp++ nn++

pp-- nn--

rather distant correlationrather distant correlation

Description of projectileDescription of projectile

No significant difference for “t = 13.0 to 20.0”.No significant difference for “t = 13.0 to 20.0”.→→It is due to the Pauli effect between originally 4+4 1s-nucleons, than from other nucleonsIt is due to the Pauli effect between originally 4+4 1s-nucleons, than from other nucleons

t t = 13.0= 13.0

neutronneutronprotonproton

(it does not mean weak)(it does not mean weak)

IndexIndex : sign of Jz: sign of Jz

yy [f

m]

xx [fm]

neighboring correlationneighboring correlation

Deuteron neighboring pictureDeuteron neighboring picture(n(n++, p, p++) ------ (n) ------ (n--, p, p--))alwaysalways

4He +40Ca 　

xx [fm]

TDHF calculation of non-zero impact TDHF calculation of non-zero impact parameterparameter

Impact parameter =Impact parameter = 　　 4.518 fm4.518 fm

Initial energy

(E/A = 0.7MeV)

30.8MeV

xx [fm]

(Almost the radius of (Almost the radius of 4040Ca)Ca)

tt = = 0.0(s)0.0(s) xx [fm]

yy [f

m]

yy [f

m]

Deuteron neighboring pictureDeuteron neighboring picture

Center of mass motionCenter of mass motion

For comparisonFor comparison

neutronneutronprotonproton

Velocity [(2/3)* 10Velocity [(2/3)* 1099 m/s] m/s]

timtimee

Neutrons of projectileNeutrons of projectile

accelerationacceleration

L-S force dominantL-S force dominant

smallsmall

4He +70Ca 　

tt = = 0.0(s)0.0(s)

yy [fm]

xx [fm] dt = 1.5 * 10-24s

TDHF calculation of neutron-rich TDHF calculation of neutron-rich nucleinuclei

Impact parameter =Impact parameter = 　　 6.668 fm6.668 fm

Initial energy

(E/A = 0.7MeV)

51.8MeV

The The samesame

xx [fm]

(Almost the radius of (Almost the radius of 7070Ca)Ca)

Different contact time for N Different contact time for N & P& P

20

10

0

30

xx [fm]

40

14

14

Contact time for N & P is Contact time for N & P is differentdifferent

Estimated contact time = Estimated contact time = 14.0 14.0 dt dt for Nfor N

Estimated contact time = Estimated contact time = 15.5 15.5 dt dt for Pfor P

Neutron densityNeutron density

Proton densityProton density

dt = 1.5 * 10-24s

Time (Time (*dt [sec]))

Early accelerations are clearly weakened, when Early accelerations are clearly weakened, when

timetimetimetime

Velocity [(2/3)* 10Velocity [(2/3)* 1099 m/s] m/s] Velocity [(2/3)* 10Velocity [(2/3)* 1099 m/s] m/s]

It is mainly due to that Pauli effectPauli effect is not so effective

0b

relative to the case of head-on collision (full overlap case).

In this neutron-rich case,In this neutron-rich case,

Neutrons of projectileNeutrons of projectile Protons of projectileProtons of projectile

NeutronNeutron ProtonProton

we can say that there is we can say that there is no accelerationno acceleration for projectile any more !! for projectile any more !!

dt = 1.5 * 10-24s

20

10

0

30

xx [fm]

Time (Time (*dt [sec]))Impact parameter =Impact parameter = 　　 6.668 fm6.668 fm

40

neutronneutronprotonproton

yy [f

m]

xx [fm]

t t = 0.0= 0.0

Center-of-mass motionCenter-of-mass motion

““Brand new” different scatteringBrand new” different scattering

t t = 24.0= 24.0Neutron skinNeutron skin

t t = 30.0= 30.0

t t = 35.0= 35.0

Di-neutron & di-proton neighboring pictureDi-neutron & di-proton neighboring picture

Isospin-difference dominantIsospin-difference dominant

nucleon @ projectilenucleon @ projectile

PP

pp

nn

nn projectileprojectile

Early state of 4 nucleons in projectileEarly state of 4 nucleons in projectile

It is due to the Neutron rich effectIt is due to the Neutron rich effect (← unbalance between N& P) (← unbalance between N& P)

pp++ nn++

pp-- nn--

neighboring correlationneighboring correlation

rather distant correlationrather distant correlation

Description of projectileDescription of projectile

neutronneutronprotonproton

IndexIndex : sign of Jz: sign of Jz

yy [f

m]

xx [fm]

t t = 24.0= 24.0

t t = 30.0= 30.0

Di-neutron & di-proton pictureDi-neutron & di-proton picture

SummarySummary Relative large early accelerations are seen mainly in head-onhead-on collisions.

Frequently found states of projectile in the very early timeFrequently found states of projectile in the very early time

00 Contactable or notContactable or not

4 nucleons of 4 nucleons of projectileprojectile

Impact parameter and neutron-richness dependence can be seen

bb[fm][fm]

nnnn / n/ nz z ( = neutron richness of target )( = neutron richness of target )

Di-neutron & di-proton pictureDi-neutron & di-proton picture

Deuteron pictureDeuteron picture

Di-neutron & di-proton pictureDi-neutron & di-proton pictureDeuteron pictureDeuteron picture

protonprotonneutronneutron

Near the “drip line”Near the “drip line”

Deuteron pictureDeuteron picturePauli scattering

(large acceleration)““stable line”stable line”

small acceleration

→ → Large acceleration is due to the Pauli effect (with full overlap) Large acceleration is due to the Pauli effect (with full overlap)

in the neighboring property neighboring property of projected 4 nucleons.

Single centerSingle center