Fission from saddle to exit : influence of curvature and compression energies

HAL Id: jpa-00232257

https://hal.archives-ouvertes.fr/jpa-00232257

Submitted on 1 Jan 1983

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Fission from saddle to exit : influence of curvature and

compression energies

J. Treiner, R.W. Hasse, P. Schuck

To cite this version:

Fission from

saddle

to

exit :

influence

of

curvature

and

_compression

_energies

J. Treiner

_(*),

R. W. Hasse

Institut _{Laue-Langevin, 156X,} 38042 Grenoble Cedex, France

and P. Schuck

Institut des Sciences Nucléaires, 38026 Grenoble Cedex, France

(Re~u le 19 avril 1983, revise le

_{6 juillet,}

_accepte le 11

_juillet

₁₉₈₃₎

Résumé. 2014 Nous étudions l’influence des termes

qui

vont au-delà du modèle standard goutte

liquide

sur la _pente de la vallée de fission. Une réduction

importante

de la pente est obtenue avec

des valeurs de

_l’énergie

de courbure calculées à _partir des forces effectives couramment utilisées ;

nous corroborons ainsi les calculs

microscopiques

récemment effectués par Berger et al.

Abstract. 2014 We

investigate

the influence of terms

_beyond

the standard

_Liquid

_Drop Model on the

slope

of the fission

_valley.

An

_important

reduction of the

_slope

is obtained when _using values of the

curvature energy calculated from effective forces

currently

in use, thus

corroborating

a recent

micros-copic calculation

_by

_{Berger et} al.

Classification

Physics Abstracts 21.10D - 25.85E

In a recent _paper,

_Berger,

Girod and

_Gogny,

hereafter referred to as BGG

_[1]

have

_presented

a

microscopic description

of nuclear fission based on a

_{Hartree-Fock-Bogoliubov}

_(HFB)

cal-culation of the

_potential

energy surface E =

V(ql,

q2

...)

of240pu,

where ~i, q2 ... denotes some set

of collective variables

_{characterizing}

the

_shape

of the nucleus. Some

important

features of this

potential

energy surface can be seen in

_figures

1 and 2 of BGG. In

_figure

2,

the descent from saddle

to scission is

_presented

as a function of two collective

variables,

namely

the

_quadrupole

moment

Q2

and the

_hexadecapole

moment

_Q4.

Certain

_aspects

of BGG’s results are common to other calculations : the fission

_valley

is

_{separated by}

a

_ridge

from the fusion

region, corresponding

to

fragmented configurations

with

_large

and

_{negative potential energies :}

this

_ridge

vanishes with

increasing

deformation

_(both

_valleys

have to

_{join somewhere)}

at the so-called exit

_point.

However,

figure

2 of BGG _presents a new and

striking

feature : the mean

_slope

of the fission

_valley

is much smaller than that

_currently

obtained in the

_{Liquid Drop}

Model

_{(LDM) :}

the

gain

in _energy between saddle and exit is of the order _{of 4-5 MeV in BGG’s paper}

_[2]

while it is of the order of

(*)

On leave of absence from DPT-IPN, F-91046

_Orsay

_Cedex, France.

L-734 JOURNAL DE PHYSIQUE - LETTRES

Fig.

1. - Left ~

plot of saddle and scission _energies for 240pU as functions of the curvature _{energy coefficients.}

Right : plot

of the difference

_{Esaddle - Escission}

as a function of the curvature _{energy coefficient.}

Fig.

2. -

Potential energy surfaces for 240Pu. a : LDM calculation _{(no curvature,} no

compression),

b, c, d :

correspond

to values of the curvature energy coefficient

of ac

= 5, 8, 10 MeV,

respectively,

with inclusion

of the

_compression

effect.

15-20 MeV in the LDM

_(see

_{e.g. Ref.}

_3).

Such a

_discrepancy

will have

_important

_consequences on the

_dynamics

of the fission process; in

particular

« the slow variation of the

potential

_energy

surface

_during

_(the)

evolutive

_phase

_justifies

the adiabatic

_assumption

on which our

_{interpretation}

of the scission process is based »

[1].

Indeed if the surface is almost flat the whole

_picture

of the fission process

adopted

in the

_past

_(see

_e.g. Refs.

_{3, 4)}

must be revisited : there would be

_essentially

no

_damping

because of no

_{quasi-particle}

_excitation;

moreover the fission process would be so

slow that

_tunnelling

across the

_ridge

could occur, i.e. before the system reaches the exit

_point.

It is therefore

_interesting

to locate the

_origin

of the

_discrepancy

in the calculation of the mean

slope

of the fission

_valley

between the usual LDM and BGG. Two main effects can be

_investigated

(i)

the shell and

_pairing

corrections

_which,

in the

_{Strutinsky approach [5],}

have to be added to

the LDM _part in order to _{compare with} a

_microscopic

calculation and

_(ii)

the

_{liquid drop}

_part

itself,

which is truncated in the standard

LDM,

to

_surface,

surface

_asymmetry

and Coulomb contributions. In fact some other effects could also

play a

role : for

_example,

BGG allow for a

possible left-right

asymmetry

while the LDM

_obviously

_{does not; this tends} to decrease

_slightly

the

_slope

of the fission

_{valley [2] ;}

also in the LDM

calculations,

one

_usually

uses a

parametrized

family

of

_shapes

_{whereas in BGG’s paper the}

_shapes

come out

_naturally

from a minimization of

the _energy under constraints.

Among

all these

_possible

_effects,

we do not think that the shell and

_pairing

corrections can be

convincing

candidates.

_Strutinsky

_type

calculations of the

_potential

_energy surface show that shell effects are

_important

in

explaining

the local structure of the surface

near

the saddle

_point

but do not

_change

the

_general

features of the

_landscape

towards exit

_(see

_e.g. Ref.

_3).

To our

opinion,

the drastic

_discrepancy

in the mean

_slope

of the fission

_valley

should rather be looked

at as an indication of a

_qualitative

effect

_present

_already

in the smooth

_part

of BGG’s surface

but not in the standard LDM. The

_simplest

effect one can think of is that of curvature and

com-pression energies,

which we shall now

_analyse

in details.

The choice

_usually

made for the curvature _energy coefficient _ac in the recent

_macroscopic

approaches

to nuclear masses _{is ac} = 0

(see

e.g. Refs.

6,

7;

in the

folding

model of Ref.

7,

ac = 0

by construction).

When

_starting

with a

_microscopic

effective

_interaction,

one _expects a

_positive

value

_{of ac}

because the defect of

_binding

energy at the

_surface,

which

_gives

rise to the surface

energy, is

greater

for small

_systems

(i.e.

large curvature)

than for

_large

ones. However an accurate

value

_{of ac}

cannot be extracted

_through

a

_{fitting procedure}

to calculated masses, because

large

error bars and _strong correlations to the other coefficients of the mass formula are

obviously

obtained in such a

_{procedure [8].}

Let us now

_briefly

recall how one can derive a

_{simple analytical}

formula for _ac within the framework of an

_Energy

_Density

Formalism

_(EDF).

Such an

_EDF,

in which the energy of the nucleus appears as a functional of the local densities

only,

can be constructed

by using

a

Density

Matrix

_Expansion

of the full

one-body density

matrix

_[9]

for the

_potential

_energy

_part

_together

with an Extended Thomas Fermi

_{approximation}

for the kinetic _energy

_part.

If one then

_represents

the neutron and

_proton

densities

_by

Fermi

_type

distributions,

an

_expansion

of the _energy of the nucleus in _powers of the ratio of the surface

diffuseness to the radius can be obtained

_using

the results of reference 10.

_By

this method

_simple

analytical

formulae can be

_given

for the surface _{energy as,} the surface

_symmetry

_energy

_{ass, the}

curvature _{energy ac and the} constant term

_{ao in}

the mass formula. The

expressions

are

parti-cularly

transparent

when one uses a

_simple

Fermi distribution for the

_density

_(more

elaborate

parameterizations

are considered in

Ref.11 ).

One

_gets

~n

_particular

for the curvature _energy

where a is the surface

diffuseness, a

the surface

_tension,

_Pom and _rnm the saturation

_density

and the nuclear radius constant,

respectively,

of infinite nuclear _matter,

_Vso

a coefficient related to

the

_strength

of the

_spin-orbit

interaction

_{(F~ ~ -}

200

_{MeV. fm6), P}

the coefficient of the Weiz-sackcr term in the ETF functional and m* the effective mass in nuclear matter. We want to stress

the fact

_{that ac}

_appears as the sum of two terms of the same

_{positive sign,}

so that it cannot vanish for some

_particular

values of the

_parameters

of the effective interaction

_giving

rise to our EDF.

Indeed the numerical values obtained

_{for ac in}

reference 11 lie all in the _range

_(10-14)

MeV,

effec-L-736 JOURNAL DE _{PHYSIQUE -} LETTRES

tive interaction : it is related to

_physical

_quantities

that are well determined

_(notice

that the dominant term in the r.h.s. of eq. 1 is the first

one),

so that _any realistic effective interaction should

_reproduce

them. Our values

_{of ac}

_agree with

_previous

_{calculations,}

_e.g. the Thomas Fermi calculations of reference

12,

the ETF calculations of reference

13,

and with the estimate

_given

in reference 14 for the

_Skyrme

_type

interactions SIII and SkM

_{(9-10 MeV),}

obtained from a fit to

calculated masses. Note that the coefficient of the

~4~-term

in the mass formula is not to be identified with the curvature _energy; it contains a

_compression

term

(where

K is the

_{incompressibility}

modulus in infinite nuclear

_{matter) arising}

from the central

compression

of the

_density

due to the surface tension. This term, which lies in the range

-

(2

...

3)

A

1/3 _MeV

depending

on the values of

_K,

varies with the _square of the surface of the nucleus when it is deformed. It thus

_plays

an

_important

role in the calculation of the

_potential

energy

surface,

as we shall see below.

Concerning

the isovector

_properties,

we shall consider

_only

the surface

_symmetry

_energy, which has been calculated for

_Gogny’s

force

_[17]

in references

11,15.

We shall

_neglect

the

curva-ture

_symmetry

_{energy, acs} for which no

simple expression

has been derived so far. Let _us,

_however,

mention that for the whole set of

_Skyrme

_type

interactions studied in reference

_13,

which cover a rather wide _range of

_{isospin properties,}

_acs has been found

_large

and

_positive

_(in

the _range 20 ... 120

MeV)

so that

including

it would enhance the effect of the curvature energy.

Finally

the Coulomb _energy will be treated as in the standard LDM : diffuseness and

_exchange

corrections are known to be

_{shape-independent [6]}

and the Coulomb redistribution _energy in the

_droplet

model is

_{significantly}

smaller than the curvature and

_{compression energies [16].}

As

_already

_mentioned,

we want to

_investigate

here a

qualitative

and

_large

effect so that we

_keep

only

the

_leading

terms

_beyond

the LDM.

The set of _parameters used in the

_following

are the ones calculated for

_Gogny’s

force :

and we shall

_study

the

_potential

_energy surface as a function of the curvature _{energy ac}

_which,

as we stated

_above,

lies in the _{range 0} MeV

_(LDM)

to about 10 MeV. For the

_Gogny

_force,

ac has not been calculated but we

_rely

on

_equation

1 to

_predict

that it has to be in the same

range’

of values than found for the

_Skyrme

forces studied in reference 10.

For the

_family

of

_{shapes underlying}

we use the

symmetric

Lawrence

_shapes

in

cylindrical

coordinates

_[16,18]

and. convert the

_elongation

and constriction _{parameters zo, Z2}

_by

nonlinear

transformations to

.,

Let us now

proceed

with the discussion of our results. A

rough

_way of

evaluating

the mean

slope

of the fission

_valley

is to calculate the difference AE =

Esad - Eseis

between saddle and

scission

_energies.

The

_{corresponding plots}

are shown in

_figure

1. On the vertical axis

_(ac

_==~0)

are also

_given

the LDM

points

obtained when

neglecting

both curvature and

_compression

energies.

Note that the

_compression

term is

_negative,

so that for small curvature

_energies

the saddle

_point

_{energy is smaller than the LDM} value

_(2.6

MeV

_{for ac}

= 0 but with

compression

effect,

instead of 6.5

_MeV).

The saddle

_point

energy then increases

with ac

and reaches the value

11.2 MeV

_{for ac}

= 10 MeV.

The scission energy shows the same behaviour

(indeed

it is enhanced because the deformations

are

_{larger) :}

inclusion of the

_compression

effect alone

_{brings Escis}

down to - 51

_MeV,

_compared

to the LDM value of - 33

_MeV;

inclusion of a curvature _energy of 10 MeV then raises

_Escis

up to - 3 MeV. The net result on AE can be seen on the

_right

_part

of

_figure

2. Around the value

ac = 6 MeV

compression

and _curvature effects cancel

approximately

and _{one recovers} the

LDM value of - 38

MeV,

so that it is

_only

above this value that one _may

_expect

the

_potential

energy

landscape

to

_{really change.}

This is a _very

_interesting

_result,

because it shows that if the

curvature _energy were not

_larger,

one would in fact be entitled to

_rely

on the

LDM,

thus

neglecting

two effects which would almost

_compensate.

_{Around ac}

= 10

MeV,

however,

AE is reduced

by

a factor of about 3 with

_respect

to

_{AELDM !}

This schematic

_analysis

is confirmed

_by

the results of the more detailed calculations

_presented

in

_figure

2. Here we show the

_{equipotential}

lines from saddle to exit

_{corresponding}

to 4 different

cases : the LDM case

_(no

_compression)

and 3 values

_{of a~ :}

_5,

8 and 10

_MeV,

_including

the

com-pression

energy. For ac = 5 MeV _{one sees} that the _mean

slope

of the fission

valley

has _not

changed

much from the LDM case. For _ac = 8

_MeV,

the

_slope

decreases while the fission

_valley

_gets

narrower.

_{For ac}

= 10

MeV,

the difference in energy between saddle and exit is _{now ~} 4 MeV

instead of ~ 14 MeV in the LDM and the width of the

_valley

is still smaller -

a feature which

is as well in

_{semi-quantitative}

_agreement

when

_comparing

with BGG’s result as is the overall

shape

of the

_landscape

in

_figure

2d

_[2].

In

fact,

the distance between saddle and exit is now

larger

than in the

_microscopic

calculation so that the

_slope

is even smaller than in BGG’s paper.

How-ever, the different terms

_neglected

in our

_{simple approach}

_may have a

relatively large

effect on a

_slope

which has been reduced

_by

a factor of ~ 3. In

_particular

we have

_neglected

the

_possible

role of the constant term in the mass formula. The different

_components

of this term as well as

their

_shape

_dependence

can be derived from the

_{leptodermous expansion.}

One can show that

they

are all

_{negative [11]}

and will tend to

_slightly

reduce the effect of the curvature _{energy. Shell}

effects also will have a

_{relatively larger importance}

on a flat surface than on a

_steep

one.

Neverthe-less we think that none of the

_missing

corrections can

_produce

a

_change

_comparable

to that of the curvature _energy.

The

_general

features discussed above are not

_typical

of the

_Gogny

force : one can start with different values of the coefficients of the mass formula and check that the same effect on the mean

slope

is obtained when

_adding

curvature and

_{compression energies.}

But of course the whole

potential landscape

may be shifted downward or

_upward

in _energy

_depending

on the values of

the coefficients. It is then

_interesting

to discuss the value of the saddle

_point

energy

Esad

obtained with the

_Gogny

force,

in relation with the

_{corresponding}

mass formula coefficients. The

micro-scopic

HFB calculation

_{gives JEs~ ~17}

MeV for

_{symmetric left-right configurations}

and

_{Esaa ’~}

13 MeV when this constraint is released

_[19].

After

_correcting

for

_{zero-point energies, Esad}

is

brought

down to 8

_MeV,

which is still too

_{high compared}

to the

_experimental

value 5 MeV. The value found in the

_present

work

_Esad

= 11.2 MeV

_(for

_ac = 10

MeV)

reflects this

discre-pancy. A

possible explanation

of this could of course be that the inclusion of further correlations

brings

the barrier down to the correct

_experimental

value.

However,

one _{may also}

_speculate

compa-L-738 JOURNAL DE _{PHYSIQUE -} LETTRES

rable to the values used in the

_macroscopic

_approaches

of references

6, 7,

but in these latter

cases the curvature _energy is zero so that the surface _energy is somehow renormalized. On the other

_hand,

one would _expect a smaller surface _energy when curvature _energy is

_present.

Another and more subtle _aspect which could

_explain

the

_discrepancy

in the barrier

_height

is related to the

isospin properties

of the interaction. The

_comparison

between

experimental

and calculated masses

in different series of

_isotopes

_(Ti,

Ni,

Zr)

seems to indicate that

the

surface _{symmetry energy ass}

might

be

_(in

absolute

_value)

too small

_(see

2nd of

_{Ref. 17).}

The same

conclusion can be drawn

from studies of the Giant

_Dipole

Resonance

[20],

which favour a value of the ratio of surface

symmetry

energy to volume

_symmetry

_energy of 2 ... 3 instead of 1.3 in the case of the

Gogny

force.

_Increasing

_{I ass}

_I

_by

_e.g. 25 MeV

_{(and readjusting consequently}

the volume

_symmetry

energy)

would

_decrease

the _{effective surface energy in}

24opu

_{by ~}

1

MeV,

which would be

enough

to

_bring

the calculated fission barrier to the

_experimental

one. The same kind of

readjust-ment on _ass

_(it

is indeed a small

readjustment)

has been made in the case of SkM interaction

_[21]

which lead to the Modified SkM force

_{[22] ;}

this last interaction

_{reproduces correctly}

the

240pU

barrier

_height.

In conclusion we have shown in the

_present

_{investigation}

that the inclusion of a curvature term in the LDM calculation can

_{drastically change}

the

_slope

of the

_potential

_energy surface from saddle to exit and scission. The effect of

_compression

dominates for values of the curvature

energyac up to 5-6

MeV,

leading

to a

slight

increase of the mean

_slope;

the

_opposite

is true for

ac > 6 MeV. The values

_{of ac}

obtained in an EDF

_approach

exhibit little

_dependence

on the

effective interaction used and lie around 10 MeV. Such a value reduces the

_{slope by}

a factor

of ~

_3,

which

_gives

a surface in

_{semi-quantitative}

_agreement with the

_{completely microscopic}

calculations of BGG in

24 OPu.

_Arguments

are

_given

that the too

_high

fission barrier

_might

be due to a too small surface

_symmetry

_energy. The

_present

_study

calls for more detailed

investi-gations

concerning

the

_{liquid drop}

_parameters

to all

_orders,

inclusion of shell corrections

_{[3, 23]}

and

_dynamical

calculations

_allowing

for

_{tunnelling through}

the

_ridge.

This

_question

has been

brought

up

by

recent fission

_{experiments [24]}

at the

_high

flux reactor in Grenoble

_(ILL).

As a

matter of

_{fact the fission processes} at _very

_high

kinetic

_energies

seem to

_support

the idea that no

quasi particle

excitations are _present in the

_fragments

and that

_they

are bom

in’compact

configu-rations. A small number of

_{quasiparticle}

excitations

_during

the

descent from saddle to exit has also been found in a recent calculation

_by

Nifenecker et al.

_[25].

These different _aspects seem

to corroborate the idea of a weak

slope

of the

_potential

energy surface

beyond

the barrier towards

scission,

and it could

_imply

_{that the fission process is} a frictionless motion of collective variables

in a

_superfluid

_system

at least up to the

_point

of the more or less fast neck

_rupture.

Acknowledgments.

’ ~ " , , °

J. F.

_Berger

and D.

_Gogny

are

especially

_acknowledged

for their constant and fruitful

interest

I

in this work. We are also thankful for numerous discussions with M.

_Asghar,

F.

Gonnenwein,

C.

_Signarbieux,

H. Nifenecker and C. Guet.

References ,

[1]

BERGER, J. F., GIROD, M. and GOGNY, D., J.

_Physique

Lett. 42

₍₁₉₈₁₎

L-509.

[2]

BERGER, J. F., Private Communications and to be

_published.

[3]

BRACK, M., DAMGAARD, J., JENSEN, A. _{S., PAULI,} H. _{C., STRUTINSKY,} V. M., WONG, C. Y., Rev. Mod. Phys. 44

₍₁₉₇₂₎

320 ;

[6]

MYERS, W. D., SWIATECKI, W. J., Ann. _Phys. 84 ₍₁₉₇₄₎ 186.

STRUTINSKY, V. M., J. _Exp. Theor. _Phys. 45 ₍₁₉₆₄₎ 1891.

[7]

KRAPPE, H. J., NIX, J. R., SIERK, A. _{J., Phys.} Rev. C 20 ₍₁₉₇₉₎ 992. [8] VON GROOTE, H., HILF, E., Nucl. _Phys. A 129 ₍₁₉₆₉₎ 513. [9] NEGELE, J. W., VAUTHERIN, D., Phys. Rev. C 5 ₍₁₉₇₂₎ 1472. [10] KRIVINE, H., TREINER, J., J. Math.

_Phys.

22 ₍₁₉₈₁₎ 2484. [11] TREINER, J., KRIVINE, H.,

IPNO/Th

82-18, Orsay Preprint.

[12]

MYERS, W. D., SWIATECKI, W. J., Ann. _Phys. 55 ₍₁₉₆₉₎ 395.

[13] CHU, Y. H., JENNINGS, B. K., BRACK, M.,

Phys.

Lett. 68B (1977) 407. [14] BRACK, M., GUET, C., HÅKANSSON, H. B., to be

_published.

[15] FARINE, M., COTE, J., PEARSON, J. M., Nucl.

_Phys.

A 338 ₍₁₉₈₀₎ 86. [16] HASSE, R. W., Ann.

_Phys.

68 ₍₁₉₇₁₎ 377.

[17] GOGNY, D., Proc. Int. Conf. on Nuclear

_Physics,

München 73, ed. J. de Boer and H. J. _Mang

(North-Holland).

DECHARGE, J., GOGNY, D., Phys. Rev. C 21 ₍₁₉₈₀₎ 1568.

[18]

LAWRENCE, J. N. P., Phys. Rev. 139 ₍₁₉₆₅₎ B-1227.

[19] BERGER, J. _F., GIROD, M., in

_Physics

and

_Chemistry

of fission 1979, vol. 1 _{(IAEA, Vienna, 1980) p. 265.} [20] KRIVINE, H., SCHMIT, C., TREINER, J.,

Phys.

Lett. 112B (1982) 281 ;

LIPPARINI, E., STRINGARI, S.,

Phys.

Lett. 112B (1982) 421.

[21] KRIVINE, H., TREINER, J., BOHIGAS, O., Nucl.

_Phys.

A 336 ₍₁₉₈₀₎ 155.

[22] BARTEL, J., QUENTIN, P., BRACK, M., GUET, C., HÅKANSSON, H. B., Nucl. _Phys. A 386

₍₁₉₈₂₎

79. [23] LOUNIS, L., FELLAH, M., FAID, B., ATHIMENE, H., International Conference on Nuclear Data for

Science and

Technology, Antwerp

1982.

[24]

SIGNARBIEUX, C., MONTOYA, M., RIBRAG, M., MAZUR, C., GUET, C., PERRIN, P., MAUREL, N. _M., J.

_Physique

Lett. 42 ₍₁₉₈₁₎ L-437.

ARMBRUSTER, P., QUADE, U., RUDOLPH, K., CLERC, H. G., MUTTERER, M., PANNICKE, J., SCHMITT, C.,

THEOBALD, J. P., ENGELHARDT, W., GÖNNENWEIN, F., SCHRADER, H., Proceedings of the 4th

Inter-national Conference on Nuclei far from

_{Stability Helsingor,}

1981.