Vacuum polarization and the nuclear mass formula

Nuclear Physics A451 (1986) 160-170 ONorth-Holland Publishing Company

VACUUM POJARIZATION AND THE NUCLEAR MASS FORMULl S.K. SAMADDAR Saha Institute of Nuclear Physics, Calcutta 700 009, India

M.M. MAJUMDAR and B.C. SAMANTA Department of Physics, Burdwan University, B&wan,

W. B., India

J.N. DE

Nuclear Physics Division, BARC, VEC Centre, Calcutta 700 064, India Received 18 April 1985 (Revised 24 June 1985) Abstract:

The effect of vacuum polarisation on the ground-state mass systematics of nuclei is considered. Its contribution to the ground-state energy of heavy nuclei is found to be a few MeV and can be simulated by using an effective nuclear radius in the Coulomb energy term of the mass formula. Its effect on the fission barriers seems to be small. The sub-barrier heavy-ion fusion cross section is found to be reduced at most by an order of magnitude with the inclusion of vacuum polarization in the interaction potential.

1. Introduction

The recent microscopic-macroscopic mass formulae lm3) include various finer details of a nucleus viewed as a two-component charged liquid drop with a diffuse surface. There are several contributions - higher-order diffuseness correction, microscopic zero-point energies, proton form factor, nuclear compressibility, Coulomb redistribution, surface redistribution, etc. - to the ground-state energy of a nucleus and each of these contributions is a few MeV at most. The contribution due to vacuum polarization (VP) is expected to be approximately equal to the fine structure constant (a = &) times the ordinary Coulomb energy which gives around 10 MeV for nuclei around uranium. It appears that several effects of this magnitude, or less as mentioned earlier, have been taken into account, though the VP contribution has not been considered. In this note we evaluate the contribution to the nuclear ground-state masses from VP. We also consider the shape dependence of this contribution in order to study its effect on fission barriers. We do not, however, make any attempt to determine the various constants of the mass formula with the inclusion of the VP term but only indicate its role on the mass formula and on the fission barriers. 160

S. K, Samahr

et al. /

Vacuum polarization

161

The small effects of VP on the sub-barrier elastic scattering of heavy ions have been considered by several authors 4*5). The sub-barrier heavy-ion fusion cross sections are very sensitive to the interaction and the effect of VP may be more important here. We have therefore studied the effect of VP on sub-barrier heavy-ion fusion cross sections which may be reduced substantially. In sect. 2 we derive the basic formulae for the VP energy of a spheroidal nucleus and the corresponding interaction energy between two nonoverlapping spherical nuclei. In sect. 3 the results for the effect of VP on the ground-state energies and on fission barriers are presented and discussed. The effect of VP on sub-barrier heavy-ion fusion cross section is also discussed in this section. The results are summarised in sect. 4.

2. Formalism We present the VP energy for a single spheroidal nucleus and the corresponding interaction energy between two nonoverlapping spherical nuclei in this section. 2.1. VACUUM

POLARIZATION ENERGY FOR A SINGLE NUCLEUS

We consider VP in (Y*and higher powers is well approximated two point charges Q,

the lowest order, as the higher-order terms are proportional to of (Yand can therefore be neglected 6). The lowest-order term by the Uehling potential ‘) which was originally proposed for and Q2 separated by a distance r. It is given by

(1) where X is the electron Compton wavelength and the function f(t) is given by

/(t)=($++&G.

(2)

It is straightforward to extend it for an arbitrary charge distribution for which the VP energy is given by

oh)

* r+*)

where the integrations over r, and

7

(3)

r,

extend over the whole charge distribution given

by p(r). For simplicity we assume that the nucleus has a sharp surface within which the charge resides. The diffuseness correction to the VP energy would be a higher-order effect and is therefore neglected. Thus the charge distribution within the nuclear volume is taken as p(r) = 3ze/(4&)

= PO)

(4

S.K. Samadahr et al. / Vacuum polarization

162

where Ze is the total charge and R, is the equivalent sharp-surface radius for a spherical dist~bution. For a spherical nucleus, the volume integrations in eq. (3) can be performed analytically and we have E VP

a,+1 = 31yZ2e2X3 Mdf f(t) 2 --a --@%-1) j t3 i 3 O 16?TR40 1 +

I

(a0+ l)exp(-2a,)} , (5)

with a, = 2R,t/h.

In order to study the effect of VP on fission barriers we have to evaluate E,, around the second saddle point where the nucleus is highly deformed. We consider only the spheroidal deformation, as the other complicated deformations - octupole, hexadecapole, etc. -will change the result slightly and will not affect our conclusions. The detailed shape is not important for the present purpose since it is the noncompactness of the shape compared to a sphere that matters and this is predominantly determined by the quadrupole deformation. The double volume integral of eq. (3) can be reduced to a double surface integrals) and for an axially-symmetric system, one has to perform a three-dimensional integration besides the t-integration. Here the volume function is of the Yukawa type and the corresponding surface function is available in ref. 8). We follow a more straightforward and simpler way of determining the surface functions applicable for a wider class of volume functions. This is presented in appendix A. Using cylindrical coordinates, the VP energy for a spheroidal deformation is obtained from eqs. (3) and (4) as E,=

- $&~mdtf(t)

j_;dz

j_;dz’

i2nd+

X-$[(Y-l)-(F+l)exp(-i)]p(z)p(z’) x p(~~-p(z’)cosg_jz-2’)~] I

(6)

r(z’)-p(z)cosg-(r’-r)~], where

p(2) = a(1 - zz/C2)1’2, Y= Ir,-r21=

(7)

[~2(Z)+p~(z’)-22p(z}p(z’)cos~ +z2+z’*-2zzf]

l/2

)

(8)

S. K. Samadhr

163

et al. / Vacuum polarization

and c is the semi-axis along the axis of symmetry (z-axis) with a the other semi-axis. Here b = X/2t.

2.2. INTERACTION

POTENTIAL

DUE TO VACUUM

POLARIZATION

The VP potential between two heavy ions with diffused spherical charge distributions has been calculated by Roesel et al. 9). However, for the sake of completeness, the expression for the VP potential between two spherical nuclei having sharp surfaces of radii R, and R, with separation between their centres r 2 R, + R, is given below; the geometry is shown in fig. 1. The effect of diffuseness of the charge distributions on the VP potential is negligible, as shown in ref. 9). The VP interaction potential is given by

where the charge density

p

is given by for ri< Ri,

(10)

for ri> Ri with i = 1,2. Here Z,e and Z,e are the total charges of the interacting nuclei. The volume integrations in eq. (9) can be performed and we get

v&r)=--

3aA* Z,Z,e* mdt f(t) ~exp( ga R;R;r I 1

- r/b)

Fig. 1. The configuration for two spherical nonoverlapping nuclei.

164

S. K. Samaddar et al. / Vacuum polarization

where g,(t) = b*[exp( -R/b) +R&[exp(

-exp(RJb)j

(12)

+RJb)+ exp( -RJb)]

with i = 1,2. It may be noted that as R, --, 0 and R, --, 0, (13) as expected for two point charges Z,e and Z,e separated by a distance r. 3. Results and discussion 3.1. GROUND-STATE

VACUUM POLARIZATION ENERGY

The ground-state VP energies of nuclei are calculated with the help of eq. (5) by performing the t-integration numerically using the Gauss quadrature method. The result is shown in fig. 2 by the solid line where the abscissa represents the atomic

I I

t Fig. 2. The vacuum polarization energy Evp versus atomic number Z, the mass numbers corresponding to stable nuciei. The solid line represents the exact calculation, and the dashed line is fitted to the form given by eq. (14).

S. K. Samaddar et al. / Vacuum polarization

165

numbers Z and the mass numbers A correspond to the stable nuclei. Throughout our calculation the radius parameter r0 has been taken as 1.16 fm. The calculated VP energy has been fitted to the Coulomb energy form E,

= KZ2,‘A1/3.

(14)

The constant K has been determined by a least-square fit for nuclei with 10 I 2 2 100 and it turns out to be 0.003452 MeV. The result is displayed in fig. 2 by the dashed line. We find that the fit agrees with the exact values obtained from eq. (5) within 0.15 MeV. This indicates that the VP energy is implicitly included in the Coulomb energy term of the mass formulae id3). The radius parameter r, of these mass formulae is to be interpreted as the effective radius parameter and the actual radius parameter is - 0.5% greater than this effective value. The surface energy has been taken as a,A2j3 with a2 = 4ar&, y being the surface tension constant. Since the constant a2 is varied independently, the surface energy is unaffected in the mass formula. However, the surface tension constant y will be reduced by - 1% and this is to be taken into account in studies of ion-ion interactionstO) and other situations where y is used directly.

3.2. EFFECT ON FISSION BARRIERS

We have already seen that the VP energy is effectively hidden in the Coulomb energy term of the mass formula. However, one has to find how this energy changes with nuclear deformation in order to investigate its effect on fission barriers. We define the shape functions for vacuum polarization (Gg) and for Coulomb energy (G:)

as G;(B) = &,(B)/‘&&

= 1) >

(154

05b) where fl= c/a and E, is the Coulomb energy. The shape function Gl is evaluated with the help of eqs. (5) and (6).The Coulomb shape function is given by il) GC

=

s

1 (1 - c2)1’3 In 1-f E

2

E

l--E’

where the eccentricity E = d-/c. In fig. 3 the shape functions Gz and Gz versus deformation p are displayed. We note that Gg falls somewhat faster compared to Gz as expected, because of an extra exponential damping in the VP term compared to the Coulomb energy term. Thus the fission barriers which occur I*) around fi - 3.0 for elements with Z = 98 are overestimated a little (= 0.1 MeV) for no~clusion of the VP term explicitly. It may be noted that to ascertain the complete role of VP on the fission barriers, the various constants in the mass formula are to be recalculated with explicit inclusion of this term.

166

S. K. Samaddar et al. / Vacuum polarisation

I

1.0

I

I

I

1.5

2.0

2.5

3

P Fig. 3. The shape functions

for Coulomb

(Gz) and vacuum text.

polarization

(G!)

energies as discussed

in the

In passing, it may be noted that the stability of superheavy nuclei is reduced by VP. Here the stability is very sensitive to a slight variation of the potential energy and the role of VP may not be negligible. 3.3. SUB-BARRIER

HEAVY-ION

FUSION

The sub-barrier heavy-ion fusion cross sections are very sensitive to the shape of the barrier and the ion-ion potential has to be determined accurately. The onedimensional barrier penetration model underestimates the fusion cross sections 13) and various reaction mechanisms - zero-point motion’4), dynamic deformation15), coupling to inelastic and transfer chaMels16), effect of charge polarizationl’), etc. - have been considered earlier. All the above-mentioned mechanisms cause enhancement of sub-barrier fusion cross sections. The effect of vacuum polarization on the ion-ion potential increases the barrier height and the fusion cross sections are reduced. In fig. 4 we plot the mutual vacuum polarisation interaction energy against the surface separation S = r - R, - R, between the nuclei lo9Ag, ‘09Agwith the help of eq. (11). We find an interaction energy of = 1 MeV for touching configurations (S = 0). The sub-barrier heavy-ion fusion cross section has been calculated in the onedimensional model with and without the VP interaction energy. For the nuclear

S. K. Sumuddar et al. /

I 0

167

Vucuum polurizution

I

I

I

1

1

2

3

4

S (fm)

Fig. 4. The vacuum

potential VvP for the system to9Ag + to9~41g for spherical ions with surface separation S.

polarization interaction nonoverlapping

and

1.0 -1 10

1O-2

103 z s

,o-4

b' lo-!

12

16'

I

I 225

III

I1

I

I

230

I

I

I

I 235

I

II

t “B

ECM( MeV) Fig. 5. The sub-barrier heavy-ion fusion cross sections for the system to9Ag + losAg. The solid and dashed lines represent calculations respectively with and without inclusion of the vacuum polarization interaction. The arrow indicates the position of the barrier VB.

168

S. K. Samaahkr et al. / VUCUUWI polarization

interaction, the nuclear proximity potentiallO) has been taken. In fig. 5 the fusion cross section ur is plotted against the centre-of-mass energy for the ‘09Ag+ ?Ag system. The arrow on the abscissa indicates the barrier height V, without the VP energy. We find that VP suppresses the fusion cross section for this system by approximately an order of magnitude. This may be considered as the maximum suppression of sub-barrier heavy-ion fusion due to VP as the system chosen is close to the heaviest one that undergoes fusion.

4. Summary It has been observed that the VP energy increases appro~ately as Z2/.41/3 and so this is implicitly contained in the Coulomb energy term of the mass formula. This means that the actual radius parameter is - 0.5% larger than the effective radius parameter obtained without inclusion of the VP energy. This reduces the surface energy per unit area by - 1%. The fission barriers are also reduced a little being = 0.1 MeV for a nucleus with 2 = 98, and therefore the possibility of the existence of superheavy nuclei decreases if the VP energy is taken into account. It is further found that the barrier height for the ion-ion potential is increased due to VP, thereby reducing the sub-barrier fusion cross sections by an order of magnitude for the heaviest systems which undergo fusion. One of the authors (B.C.S.) gratefully acknowledges partial financial support from the Department of Atomic Energy, India.

Appendix A The conversion of a single or double volume integral into a single or double surface integral is presented in a simple and straightforward way in this appendix. Consider first a single volume integral of the form

(A-1) where z-l’12 = lri - r,[ and the integration extends over a vohnne V, enclosed by the surface S,. We want to find a function F(rJ that satisfies the relation

(4 where v2 denotes differentiation with respect to r,. Using the divergence theorem, the volume integral can be converted into a surface integral given by (A-3)

S. K. Samaddar et al. / lfacuum polarization

169

From eq. (A.2) we have -G. f

g+$F=

(A-4)

12

Solving eq. (A.4) we get Fh2)

=

-

+/.“‘&f F-12

(d

dr,, 3

(A-5)

0

assuming r3F( r) --) 0 as r --) 0 which will be the case for most of the physical situations of interest. Now we consider the case of the double volume integral. Consider a double volume integral of the form

(A.61 Replacing one volume integral by a surface integral and interchanging the order of integrations, we have J=~ds2.~~dr,r,,F(r,,),

(A-7)

where F( r12) is given by eq. (A.5). Next we convert the second volume integral into a surface integral in the same way and finally we obtain

where G(r) = - -$jbdrirdrr2f(r)

(A.9)

provided r3G(r) --, 0 as r + 0. We note that we have similar restrictions on the behaviour of the function at the origin as in ref. *). However, there is no restriction regarding its behaviour at infinity contrary to that of ref. 8, where f(r) must go to zero as r + cc in order that its Fourier transform exists. Following ref. 8, the four-dimensional integral represented by eq. (A.@ can be reduced to a three-dimensional integral if the volume of integration possesses cylindrical symmetry. References 1) 2) 3) 4) 5)

W.D. Myers and W.J. Swiatecki, Ann. of Phys. 84 (1974) 186 P. Mijller and J.R. Nix, Nucl. Phys. A361 (1981) 117 P. Mailer and J.R. Nix, At. Nucl. Data Tables 26 (1981) 165 G. Baur, F. R&e1 and D. Trautmann, Nucl. Phys. A288 (1977) 113 M.S. Hussein, V.L.M. Franzin, R. Franzin and A.J. Baltz, Phys. Rev. C30 (1984) 184

170

S. K. Samadhr

et al. / Vacuum polarization

6) J. Blomqvist, Nucl. Phys. B48 (1972) 95

7) 8) 9) 10) 11) 12) 13) 14)

15) 16) 17)

E.A. Uehling, Phys. Rev. 48 (1935) 55 K.T.R. Davies and J.R. Nix, Phys. Rev. Cl4 (1976) 1977 F. Roesel, D. Trautmann and R.D. Viollier, Nucl. Phys. A292 (1977) 523, and references therein J. Blocki, J. Randrup, W.J. Swiatecki and C.F. Tsang, Ann. of Phys. 105 (1977) 427 R.A. Lyttleton, The stability of rotating liquid drop masses (Cambridge Univ. Press, 1953) p. 35 S.G. Nilsson et al., Nucl. Phys. A131 (1969) 1 M. Beckerman et al., Phys. Rev. Lett. 45 (1980) 1472; M. Beckerman et ul., Phys. Rev. C23 (1981) 1581 S. Landowne and J.R. Nix, Nucl. Phys. A368 (1981) 352; W. Reisdorf et al., Phys. Rev. Lett. 49 (1982) 1811; R. Pengo et al., Nucl. Phys. A411 (1983) 255 H. Esbensen et al., Nucl. Phys. A411 (1983) 275 C.H. Dasso, S. Landowne and A. Winther, Nucl. Phys. A405 (1983) 381; A407 (1983) 221 A.K. Banejee, B.C. &manta and S.K. Samaddar, Phys. Lett. 153B (1985) 213