238U + 238U Collisions - CENTAUR · [2] S. Jin, K. J. Roche, A. Bulgac, and I. Stetcu, “The SLDA code: a solver for static and time- dependent superfluid local density approximation

RESEARCH POSTER PRESENTATION DESIGN © 2015

www.PosterPresentations.com

We simulate the collision of two Uranium 238 nuclei using the time dependent superfluid localdensity approximation (TDSLDA). We test out four various initial relative orientations (shownbelow), while varying the initial boost energies of the nuclei as well as setting the relative phases ofthe paring gaps to be in or out of phase. After completing the simulations we then calculate variousobservables. Note, all calculations were performed at zero impact parameter, but will eventually beextended to finite impact parameters.

SUMMARY

METHOD

Multi-nucleon transfer in case of tip to waist collisions.

RESULTS FUTURE OUTLOOK

REFERENCES[1] V. I. Zagrebaev and W. Greiner, Phys. Rev. C 87, 034608 (2013).[2] S. Jin, K. J. Roche, A. Bulgac, and I. Stetcu, “The SLDA code: a solver for static and time-dependent superfluid local density approximation (SLDA) equations in 3D coordinate space, to be submitted to arXiv and Comp. Phys. Comm. as soon as LANL grants permission (2019),”.[3] C. Simenel, Phys. Rev. Lett. 106, 112502 (2011) .

ACKNOWLEDGEMENTS

We thank Kenny Roche, Gabriel Wlazlowski, and Ionel Stetcu for various useful discussions throughout the project. We acknowledge the Department of Energy (DOE), and National Nuclear Security Administration (NNSA) for funding; and the INCITE award on Summit.

Ibrahim Abdurrahman1, Aurel Bulgac1, Shi Jin1, Kazuyuki Sekizawa2, Nicolas Schunck3

1University of Washington, 2Niigata University, 3Lawrence Livermore National Lab

238U + 238U Collisions

Theoretical framework: The quasiparticle wavefunctions

satisfy the following evolution equations

for any local energy density functional (EDF). Here we focused on the SeaLL1 NEDF [2]. These solutions can then be used to construct various densities,

�k(~r) =

0

BB@

uk"(~r)uk#(~r)vk"(~r)vk#(~r)

1

CCA

E = Ekin + Eint + Ecoul + Epair

Ekin =P

q=n,ph̄2

2mq⌧q(~r)

Ecoul = e2

2

R np(~r)np(~r0)

|~r�~r0| d3~r0 � 3e2

4 ( 3⇡ )1/3

n4/3p (~r)

Epair =P

q=n,p geff (~r)|q(~r)|2

Eint =P2

j=0(ajn05/3+bjn0

2+cjn07/3)(n1

n0)j+

Pt=0,1

�Cn�n

t nt�nt+CrJt (nt

~r·~Jt + ~st · (~r⇥ ~jt))

Note: the above interaction term is only for SeaLL1. The form changes ofwe were to use any of the Skyrme density functionals.

1

�k(~r) =

0

BB@

uk"(~r)uk#(~r)vk"(~r)vk#(~r)

1

CCA

ih̄ @@t

0

BB@

uk"(~r, t)uk#(~r, t)vk"(~r, t)vk#(~r, t)

1

CCA =

0

BB@

h""(~r, t)� µ h"#(~r, t) 0 �(~r, t)h#"(~r, t) h##(~r, t)� µ ��(~r, t) 0

0 ��⇤(~r, t) �h⇤""(~r, t) + µ �h⇤

"#(~r, t)�⇤(~r, t) 0 �h⇤

#"(~r, t) �h⇤##(~r, t) + µ

1

CCA

0

BB@


1

CCA

�E(~r)�v⇤

ks(~r)= 0

E = Ekin + Eint + Ecoul + Epair

Ekin =P

q=n,ph̄2

2mq⌧q(~r)

Ecoul = e2

2

R np(~r)np(~r0)

|~r�~r0| d3~r0 � 3e2

4 ( 3⇡ )1/3

n4/3p (~r)

Epair =P

q=n,p geff (~r)|q(~r)|2

Eint =P2

j=0(ajn05/3+bjn0

2+cjn07/3)(n1

n0)j+

Pt=0,1

�Cn�n

t nt�nt+CrJt (nt

~r·~Jt + ~st · (~r⇥ ~jt))

Note: the above interaction term is only for SeaLL1. The form changes ofwe were to use any of the Skyrme density functionals.

testvar

n(~r) =P

k,s v⇤k,s(~r)vk,s(~r),

(~r) =P

k v⇤k"(~r)uk#(~r),

⌧(~r) =P

k,s~rv⇤k,s(~r) · ~rvk,s(~r),

~s(~r) =P

k,s,s0 ~�s,s0v⇤k,s(~r)vk,s0(~r),

~J(~r) = 12i (

~r� ~r0)⇥ ~s(~r,~r0)|~r=~r0

~j(~r) = 12i

Pk,s

�vk,s(~r)~rv⇤k,s(~r)� v⇤k,s(~r)

~rvk,s(~r)

1

Boosts: To perform boosts we apply the following transformation,

where,

Above 𝜒 is positive if we are in the left half of the box, and negative if we are in the right side of the box, with a smooth transition between the two near the center and outer boundaries.

✓uk(~r, t)vk(~r, t)

◆!

✓ei� 00 e�i�

◆✓uk(~r, t)vk(~r, t)

◆

� = ±pxxh̄

Note, the plus is for the left fragment and the minus is for the right fragment.

px =q

m(ecm�ecoul)A

2


◆!

✓ei� 00 e�i�


◆

� = ±pxxh̄


px =q

m(ecm�ecoul)A

2

Other collisions:We plan to examine the collisions 238U+248Cm and 232Th+250Cf.

Above is the before and after of a preliminary 238U+248Cm collision.Mass widths of reaction products:We will be applying the Balian Veneroni prescription for calculating the mass widths. This means werun our simulation until we arrive at our final state (in most cases two well separated nuclei at 50 fm).Then we apply the following transformation to the wavefunctions,

where ε is a small real number [3].From here we run the solution backwards in time until we reach two nuclei separated by 45 fm (theinitial separation distance). Using the initial and final densities we calculate the following observable,

Post collision decay modes:We will calculate the decay products for the excited nuclei after the collision, either by using theGEMNI code or another code.Below is an example of how a result might look like for the decay of 238U with excitation energy 200MeV obtained with the GEMINI code

500 750 1000 1250 1500 1750Ecm (MeV)

-60

-40

-20

0

20

40

60

N

N ( Ecm )

YX (In Phase)YX (Out of Phase)

500 750 1000 1250 1500 1750Ecm (MeV)

-30

-20

-10

0

10

20

30

40

Z

Z ( Ecm )

YX (In Phase)YX (Out of Phase)

We plot the change in particle number for the leftfragment. Notice the reversal in the direction oftransfer between 1700 MeV and 1800 MeV.

Excitation Energies:

Other quantities:The separation distance between the two fragments, given by,

was tracked as a function of time for all runs, and used to extract the minimum separation between the two nuclei.

The interaction time is defined as the time interval when the neutron central density was larger 0.04 fm-3. This prescription does not work for ternary events.

Additional quantities were also extracted, such as the energy loss, TKE before and after the collision, maximum density, and so forth, but are not included in this presentation.

600 800 1000 1200 1400 1600 1800Ecm (MeV)

8

10

12

14

16

18

20

22

d min

(fm

)

dmin(Ecm)

XX (In Phase)XX (Out of Phase)YY (In Phase)YY (Out of Phase)YX (In Phase)YX (Out of Phase)ZY (In Phase)

-500 0 500 1000t (fm/c)

0

10

20

30

40

50

d sep (f

m)

dsep(t) for XX (In Phase)

600MEV700MEV750MEV800MEV850MEV900MEV950MEV1000MEV1100MEV1200MEV


◆!

✓ei� 00 e�i�


◆

� = ±pxxh̄


px =q

m(ecm�ecoul)A

SeaLL1 formula.

E(N,Z) = avA+ asA2/3 + acZ2

A1/3 + aI(N�Z)2

A

Seperation distance.

dsep =RVR

~r⇢(~r)d3r �RVL

~r⇢(~r)d3r

2

0 500 1000 1500t (fm/c)

0

0.02

0.04

0.06

0.08

0.1

0.12

(fm

- 3)

n(t) at center for XX (In Phase)

600MEV700MEV750MEV800MEV850MEV900MEV950MEV1000MEV1100MEV1200MEV

600 800 1000 1200 1400 1600 1800Ecm (MeV)

0

500

1000

1500

t int (f

m/c

)

tint(Ecm) (Neutrons)

XX (In Phase)XX (Out of Phase)YY (In Phase)YY (Out of Phase)YX (In Phase)YX (Out of Phase)ZY (Out of Phase)

Ternary Events

Ecm (MeV) DeltaZ (fm) Orient Phase N Z A750 20 XX In phase 6.1 3.4 9.5800 20 XX In phase 7.4 4.2 11.6800 20 XX Out of phase 6.7 3.8 10.5

1300 40 XX In phase 178.9 104.5 283.41400 20 YY In phase 8.2 3.4 11.6

Ternary quasi fission:We observed several collisions where a third fragment was formed.


◆!✓ei� 00 e�i�


◆

� = ±pxxh̄


px =q

m(ecm�ecoul)A

SeaLL1 formula.


A1/3 + aI(N�Z)2

A


dsep =RVR

~r⇢(~r)d3r �RVL

~r⇢(~r)d3r


◆! e✏N̂ 0

0 e�✏N̂

!✓uk(~r, t)vk(~r, t)

◆

2


◆!✓ei� 00 e�i�


◆

� = ±pxxh̄


px =q

m(ecm�ecoul)A

SeaLL1 formula.


A1/3 + aI(N�Z)2

A

�2 = lim✏!01

2✏2 Tr (⇢0 � ⇢✏)2


dsep =RVR

~r⇢(~r)d3r �RVL

~r⇢(~r)d3r


◆! e✏N̂ 0

0 e�✏N̂


◆

2

Neutron distribution

10 12 14 16 18N

0

0.1

0.2

0.3

0.4

0.5Light decay products

0 2 4 6N

0

1

2

3

4

5

6

7

Z

0

0.05

0.1

0.15

0.2

Heavy decay products

30 40 50 60N

20

25

30

35

40

45

50

55

Z

0

1

2

3

4

5

10-3

which in turn are used to calculate the energydensity functional, various observables, suchas kinetic energies, quadrupole and octupolemoments, etc...

800 1000 1200 1400 1600Ecm (MeV)

50

100

150

200

Parti

cle

Num

ber

238 U + 238 U YX (In Phase)ZLZHNLNH


◆!✓ei� 00 e�i�


◆

� = ±pxxh̄


px =q

m(ecm�ecoul)A

SeaLL1 formula.


A1/3 + aI(N�Z)2

A

�2 = lim✏!01

2✏2 Tr (⇢0 � ⇢✏)2


dsep =RVR

~r⇢(~r)d3r �RVL

~r⇢(~r)d3r


◆! e✏N̂ 0

0 e�✏N̂


◆

Total angular momentum.

j =�1+

q1+ 4h |J2| i

h̄2

2

ih̄ @@t

0

BB@


1

CCA =

0

BB@

h""(~r, t) h"#(~r, t) 0 �(~r, t)h#"(~r, t) h##(~r, t) ��(~r, t) 0

0 ��⇤(~r, t) �h⇤""(~r, t) �h⇤

"#(~r, t)�⇤(~r, t) 0 �h⇤

#"(~r, t) �h⇤##(~r, t)

1

CCA

0

BB@


1

CCA

2


◆!✓ei� 00 e�i�


◆

✓uk�(~r, t)vk�(~r, t)

◆!✓ei� 00 e�i�

◆✓uk�(~r, t)vk�(~r, t)

◆

� = ±pxxh̄


px =q

m(ecm�ecoul)A

SeaLL1 formula.


A1/3 + aI(N�Z)2

A

�2 = lim✏!01

2✏2 Tr (⇢0 � ⇢✏)2


dsep =RVR

~r⇢(~r)d3r �RVL

~r⇢(~r)d3r


◆! e✏N̂ 0

0 e�✏N̂


◆


j =�1+

q1+ 4h |J2| i

h̄2

2

ih̄ @@t

0

BB@


1

CCA =

0

BB@


0 ��⇤(~r, t) �h⇤""(~r, t) �h⇤

"#(~r, t)�⇤(~r, t) 0 �h⇤

#"(~r, t) �h⇤##(~r, t)

1

CCA

0

BB@


1

CCA

2


◆!✓ei� 00 e�i�


◆

✓uk�(~r, t)vk�(~r, t)

◆!✓ei� 00 e�i�

◆✓uk�(~r, t)vk�(~r, t)

◆

� =", #

� = ±pxxh̄


px =q

m(ecm�ecoul)A

SeaLL1 formula.


A1/3 + aI(N�Z)2

A

�2 = lim✏!01

2✏2 Tr (⇢0 � ⇢✏)2


dsep =RVR

~r⇢(~r)d3r �RVL

~r⇢(~r)d3r


◆! e✏N̂ 0

0 e�✏N̂


◆


j =�1+

q1+ 4h |J2| i

h̄2

2

ih̄ @@t

0

BB@


1

CCA =

0

BB@


0 ��⇤(~r, t) �h⇤""(~r, t) �h⇤

"#(~r, t)�⇤(~r, t) 0 �h⇤

#"(~r, t) �h⇤##(~r, t)

1

CCA

0

BB@


1

CCA

2

Nuclear Matter Dynamics in Real Time and the Heaviest Elements in Nature, Lead PI: Bulgac

were created and added to the periodic table in 2016, filling its 7th row.

FIG. 5. Evaluated cross section for theformation of primary fragments in reactions48C+248Cm and 238U+248Cm [115].

Naturally the next step is the element 119 or 120,however, experimentalists do need a guideline of howto create new, still heavier elements, and thus reli-able theoretical predictions are mandatory. The syn-thesis can be divided into three steps: 1) capture,2) compound nucleus formation (thermalization), and3) evaporation (cooling by particle emissions or fis-sion). To date the first stage has been usually describedby the semi-classical coupled-channels approach [116],the second stage requires to use stochastic Langevinmodel[117], and finally the statistical treatment isneeded for the description of the third stage. Micro-scopic time-dependent Hartree-Fock (TDHF) approachhas recently been utilized to investigate the first andthe second stages [118]. However, e↵ects of pairing(neutron superfluidity and/or proton superconductiv-ity) in the formation of SHEs have never been investi-gated to date, which may have significant importanceas described below. The pairing correlations can bedescribed, using top-tier supercomputers, applying thenewly developed theoretical framework, developed byus [26] ).

We also plan to study the multi-nucleon and theenergy transfer in low and medium energy heavy-ion collisions in order to produce SHE. The latestslew of SHE have been produced in multi-nucleon transfer (MNT) reactions with 48Ca or 50Nior other medium size nuclei projectiles on heavy elements, using either the cold or hot fusionsscenarios [2, 3, 119]. As Fig. 5 demonstrates, the cross sections for SHE in such reactions is small,but it can become much larger in collisions of very heavy nuclei [115, 120–122]. The theory ofMNT reactions for collisions of nuclei with time-dependent Hartree-Fock (TDHF) approximationis in pretty good shape [123], but an extension to superfluid nuclei, namely TDSLDA, is definitelydesired, since the pairing correlations may play an important role.

II. RESEARCH OBJECTIVES AND MILESTONES

The development and validation of the TDSLDA has proved to be surprisingly fruitful. Thisnew theoretical framework for the description of non-equilibrium phenomena in fermionic super-fluids has been formulated and validated in cold atoms, neutron star dynamics, nuclear fission andqualitatively phenomena have been uncovered. The theoretical framework has been extended evenfurther by including for the first time in a quantum mechanical formulation dissipation and fluctu-ations, which are crucial to obtain a full picture of such complex quantum processes. The rathercomplex mathematical structure of this theoretical framework has been successfully implementedon leadership computing facilities.

The goal of the proposed work is to advance our ability to describe and predict complex non-stationary quantum phenomena using state-of-the-art microscopic approaches rather than phe-nomenological models, and increase the predictive power of our theoretical tools. TDSLDA andleadership class computing capabilities will be used to study and gain deep insight into the real-time dynamics of nuclear systems with a strong focus on induced nuclear fission. TDHF became

Page 8

Orientations:

From top to bottom the orientations are labeled XX, YY, YX, ZY, where X,Y,Z denote the largest moment of inertia of each colliding nuclei along the collision axis OX.

Motivation:A big motivation for the collision ofheavy elements is to produce super-heavyelements (SHEs) and neutron rich nucleiclose to the neutron drip line via multi-nucleon transfer (MNT) reactions.As evidence, examine the followingfigure

which shows that the probability of producing super heavynuclei is far greater in the case of colliding two heavynuclei when compared to colliding one heavy and one lightnuclei.In our project we will also focus on extending traditionalapproaches such as TDHF or TDHF+BCS to TDSLDA tosee if pairing plays a role in these types of collisions.

Acquired from source [1].

0 1000 2000 3000 4000 5000t (fm/c)

50

100

150

200

250

300

N, Z

N, ZCen(t)

NZ

Orient XXEcm = 1300 MeV

z = 40 fm

1000 1500 2000 2500 3000 3500 4000t (fm/c)

100

120

140

160

180

200

N, Z

N, ZCen(t)

NZ


z = 40 fm

1000 1050 1100 1150 1200t (fm/c)

4

5

6

7

8

9

N, Z

N, ZCen(t)

NZ


z = 20 fm

0 200 400 600 800 1000 1200t (fm/c)

0

50

100

150

200

250

N, Z

N, ZCen(t)

NZ


z = 20 fm

We typically observe small ternary fragments, but sometimes very large ones can form.

Examples:Displayed are thenumber densitiesneutrons (upper half)and protons (lowerhalf) as a function oftime for runs atcenter of massenergies 800 MeV(left panel) and 1300MeV (right panel)respectively, with thepairing field betweenthe two nuclei inphase. In both casesthe final time was1080 fm/c.

800 1000 1200 1400 1600Ecm (MeV)

0

200

400

600

800

E* (MeV

)

238 U + 238 U YX (In Phase)E*

L

E*H

800 1000 1200 1400 1600Ecm (MeV)

0

0.5

1

1.5

2

2.5

3

E* /A (M

eV)

238 U + 238 U YX (In Phase)E*

L

E*H

Space discretization: ∆x = ∆y = ∆z = 1.25, Nx = Ny = 24 , Nz = 64, Lx = Ly = 30 fm , Lz = 80 fm, pcutoff = 𝜋ℏ/∆x ~ 500 MeV/c, # of PDES = 16 Nx Ny Nz = 589,824, # time steps per run ~ 30,000,∆t = 0.03 fm/c, wall time per run ~ 3 hrs,GPUs per run = 720.

The simulations were run on Oak Ridge’s Summit supercomputer.

Documents

238U + 238U Collisions - CENTAUR · [2] S. Jin, K. J. Roche, A. Bulgac, and I. Stetcu, “The SLDA code: a solver for static and time- dependent superfluid local density approximation