Time-dependent Hartree Fock with full Skyrme Forces in 3 Dimensions

Collaborators:

P.-G. Reinhard, U. Erlangen-Nürnberg

P. D. Stevenson, U. Surrey, Guildford

TopicsThe codeQualitative explorationsEnergy loss in 16O+16O:

Effect of full Skyrme and 3DThe spin excitation mechanismAccuracy of relative motion energy

TDHF in the late `70s

Computer facilities: The 3D code was run on an IBM „supercomputer“ 360/195 with 1MB of memory!

Therefore: no spin, simplified interaction: BKN or g-matrix

Really very few checks of accuracy (!?) R.Y. Cusson and J.A. Maruhn, „Dynamics of 12C + 12C in a Realistic T.D.H.F. Model“, Phys. Lett.

62B, 134 (1976). R.Y. Cusson, R.K. Smith, and J.A. Maruhn, „Time-dependent Hartree-Fock Calculation of the

16O+16O Reaction in Three Dimensions“, Phys. Rev. Lett. 36, 1166 (1976). J.A. Maruhn and R.Y. Cusson, „Time-Dependent Hartree-Fock Calculation of 12C + 12C with a

Realistic Potential“, Nucl. Phys. A270, 471 (1976). R.Y. Cusson, J.A. Maruhn, and H.W. Meldner, „Direct Inelastic Scattering of 14N+12C in a Three-

Dimensional Time-Dependent Hartree-Fock Scheme'', Phys. Rev. C18, 2589 (1978). C.Y. Wong, J.A. Maruhn, and T.A. Welton, „Comparison of Nuclear Hydrodynamics and Time-

Dependent-Hartree-Fock Results“. Phys. Lett. 66B, 19 (1977).

The New TDHF Code

Three-dimensional Skyrme-force Hartree-Fock, both static and time-dependent

Differencing based on Fast-Fourier-Transform; Grid spacing typically 1 fm All variations of modern Skyrme forces can be

treated fully Fourier treatment of Coulomb allows correct

solution for isolated charge distribution Coded fully in Fortran-95 TDHF version can run on message-passing

parallel machines

The Skyrme Energy Functional

Fourier calculation of potential for isolated charge distributions

(fictitious) empty space

The solution constructed via

with two FFToperations in the enlarged region with periodic boundary conditions fulfills the boundary condition for an isolated charge distribution in the physical region

J.W. Eastwood and D.R.K. Brownrigg, J. Comp. Phys. 32, 24 (1979)

2

4( ) ( )V k k

k

The wave functions have periodic boundary conditions, but for the Coulomb filed interaction with images must be avoided

16O+48Ca slightly below barrier

16O+48Ca slightly above barrier

16O + 48Ca, Boost 0.3 MeV / nucleon, t=0..450 fm/c

16O + 48Ca, Boost 0.3 MeV / nucleon, t=500..950 fm/c

Mass Moments in 16O+48Canormalized to initial value

0 100 200 300 400 500 600 700 800 900 1000

t [fm/c]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Q /

Qin

it

MomentQ2Q3Q4

Heavy Systems: 48Ca+208PbImportant for Superheavy Element Formation!

Does the interaction dynamics differ dramatically from light system?

12 fm initial distance 4000 time steps of 0.25 fm/c : 1000 fm/c total Initial boost just sufficient to cause interaction Needs longer times and systematic variation in boost

48Ca + 208Pb sligtly above barrier

48Ca+208Pb near barrier, t=0..450 fm/c

48Ca+208Pb near barrier, t=500..950 fm/c

Mass Moments in 48Ca+208Pbnormalized to initial value

0 100 200 300 400 500 600 700 800 900 1000

t [fm/c]

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Q /

Qin

it

MomentQ2Q3Q4

Deformed partners: 20Ne+20Ne

A curious case: 12C+16O

Energy Loss in 16O+16O

Past experience shows that relaxing symmetries increases the dissipation

Spin orbit coupling is essential for correct shell structure!

Few calculations performed at that time show increased dissipation due to relaxation of symmetries

Now examine energy loss aspects in new directions:• Accuracy• Effect of 3-D and full modern Skyrme forces• Role of time-odd parts in the s.p. Hamiltonian

Changes in results The dissipation is generally increased when

symmetries are relaxed and new degrees of freedom enter

A.S. Umar, M.R. Strayer, and P.-G. Reinhard, Phys. Rev. Lett. 56, 2793 (1986).

Translational Invariance of T.D.H.F.

A ground state nucleus with s.p. wave functions fulfilling

leads to a propagating stationary solution with a common phase factor

This solves the time-dependent equation (i.e., produces a uniformly translating nucleus), if the s.p. Hamiltonian is Galilei invariant

This is trivial for pure density dependence, but requires adding terms involving currents and spin currents to the density functional(Y. M. Engel, D. M. Brink, K. Goeke, S. J. Krieger, and D. Vautherin,

Nucl. Phys. A249, 215 (1975)).

/( ) i tr e

22 ( { }) ( ) ( )

2V r r

m

exp[ ]k

ik r r tm

The time-odd spin-orbit terms in themean-field Hamiltonian

In the Skyrme energy functional Galilei invaraince requires adding terms like

and similar terms with different isospin dependence. This leads to contributions in the mean-field Hamiltonian like

with

The spin-orbit part of these contributions was usually neglected (and is negligible for giant-resonance-type calculations)

Determination of relative motion energy

Find minimum of density alongaxis of largest moment of inertia

If density is low enough, definedividing plane

Determine c. m. distance Rof fragments and ist time derivative

Get relative motion kinetic energy from

for central collisions

Point-charge Coulomb energy agrees with full calculation toabout 0.02 MeV

Accuracy in „trivially“ conserved quantities: total energy 0.1 MeV, particle number 0.01

22 1 2

2cm

Z Z eE R

R

Initial Relative Motion Energy

2 4 6 8 10 12 14 16

R [fm]

20

40

60

80

100

120

140

Ecm

[M

eV]

time-odd termswithwithout

Omission of time-odd l*s terms leads to translational noninvariance of surprisingly strong consequences!

Importance of Time-Odd L*S-Termsin Central 16O+16O Collisions

75 100 125 150

Ecm [MeV]

0

10

20

30

40

50

60

70

80

90

Efi

na

l [M

eV

] Sly6

75 100 125 150

Ecm [MeV]

0

10

20

30

40

50

60

70

80

Efi

na

l [M

eV

] SkI3

75 100 125 150

Ecm [MeV]

0

10

20

30

40

50

60

70

80

Efi

na

l [M

eV

] SkI4

75 100 125 150

Ecm [MeV]

0

10

20

30

40

50

60

70

80

90

Efi

na

l [M

eV

] SkM*

The Mechanism

L*S Energy in Central Collision

0 50 100 150 200

-6

-4

-2

0

2

4

6

8

10

12

od

d-o

dd

l*s

en

erg

y [M

eV

]

t [fm/c]

0 50 100 150 200

-6

-4

-2

0

2

4

6

8

10

12

od

d-o

dd

l*s

en

erg

y [M

eV

]

t [fm/c]

Impact Parameter Dependence

0 50 100 150 200

-8

-6

-4

-2

0

2

4

6

8

10

12

SkM*

Ecm

=100 MeV

Od

d-o

dd

sp

in o

rbit

en

erg

y [M

eV

]

t [fm/c]

b [fm] 0.0 1.6 3.2 4.8 6.4 8.0

Force dependence of reactions: a dynamic test for Skyrme forces

J. A. Maruhn, K. T. R. Davies, M. R. Strayer, Phys. Rev. C31 1289 (1985)

0 1 2 3 4 5 6

t [10-21s]

6.5

7

7.5

8

8.5

9

9.5

Rrm

s [f

m]

ForceSkVISkVSkIVSkIIISkII

86Kr + 139La

Elab = 370 MeV

Comparison with previous results

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

0

10

20

30

40

50

60

70

80

90

Efin

al [M

eV

]

Ecm

[MeV]

old, no l*s old, with l*s SkI3 SkI4 Sly6 SkM*

Noncentral Results at Ecm=100 MeV

-1 0 1 2 3 4 5 6 7 8 9 10

0

20

40

60

80

100

Efin

al [M

eV

]

b [fm]

SkM*, odd l*s SkM*, no odd l*s SkI3, odd l*s old, no l*s old, with l*s

Late-time behavior shows severe problem!

12 12.5 13 13.5 14 14.5 15 15.5

R [fm]

25

26

27

28

29

25

Ecm

[M

eV]

50

51

52

53

54

5555

56

57

58

59

6060

50

55

60

Ecm

[M

eV]

time-odd termswith

without

Use new plot

10 12 14 16 1826

27

28

29

30

31

32

124

125

incoming

Ecm

[M

eV]

R [fm]

small mesh: 242x32

larger mesh: 322x48 absorption layer

Problem : Pairing

Without pairing, the deformations are still not quantitative and the moments of inertia will be wrong, unless pairing is destroyed rapidly (?)

“Old” calculations did not include pairing, because the BCS formalism with state-independent pairing matrix element produces interaction even between separated fragments

Newer formulations of pairing generate matrix elements from a force such as a -pairing

The solution of the time-dependent Hartree-Fock Bogolyubov problem therefore may have to be attempted

Conclusions

The use of full Skyrme forces brings surprising new effects and problems.

The is a new energy loss mechanism involved with a „spin-twist excitation“

There are problems with a continued loss of relative motion energy for separated fragments, possibly due to cross-boundary interactions. More computational expense may be needed or one has to live with 3 MeV uncertainty.

The energy loss appears to stabilize for several forces It will be interesting to see how these effects persist in heavier

systems.