On the possibility of detecting superheavy quasimolecules

Nttc%ar PIRytlct A272 (1976) 493-SO1 ; © North-.Sot7and Publlahinp Co ., Atntterdo»t Not to be repxndnoed h7 yhotoprhu or miaofllm wlthont wrletm permfaion tbom ehe publlaher

ON TIC POSSIBILITY OF DETECTING SUPERHEAVY QUASIMOLECf~ !. -.. . WOLFGANG SCHÄFER, VOLKER OBERACKER and GERHARD SOFF Institut fur Theoretische Phytü: der Johartrt Wolfgang Goethe-Unlversit6t, Frankfurt am Male, Germany Received 24 June 1976 (Revised l l August 1976) Abstract : The formation of superheavy electronic quasimolavles leads to measurable deviations from the Rutherford cross section in heavy ion collisions. For the system Z q;U-'siU with E~.s . = 800 MeV we find an average correction of 1 .5 ~ for the scattering cross section in forward directions . Additional background contributions like electronic shielding of the nucleus, vacutem polarvation, nucleâr Coulomb excitation and static deformations of the nuclei are taken into account.

1. Intr+oduction In connoctitm with the increasing interest in quantum electrodynamics of strong fields, e.g. in the induced and spontaneous positron creation in ovèrcritical fields (Zt +Z= > ZQ~ 170) t), experimental techniques for the detection of superheavy electronic olesailes have bean developed. i3aually the urnoharacterisl~c X=ray

Fig. 1 . Modification of the Rutherford hyperbola by the etFective quesimolecular potential (schematic drawing) .

radiation induced by non-adiabatic rearrangements in the electronic cloud indicates the formation of quasimolecules. Electronic transitions into the K-shell ~ " 3 ), U shell <, s) and M-shells e,') belonging to two Coulomb centres have been investigated . An X-ray independent test, first predicted by Mûller, 12afelski and Greiner 8), would be the direct measurement of the heavy ion scattering cross section. The t Work supported by the Bundesministerium fiür Forschung und Technologie (BMFT), and by the Gesellschaft file St~wwerionenforscbung (G3n . 493

494

W. SCHÄFER et al.

modification generated by the quasimolecular potential is illustrated qualitatively in fig. 1 . This potential originates from the non-linear dependence of electronic binding energies on nuclear charge and equivalently on the internuclear distance R. After a short review on calculations of scattering cross sections and electronic binding energies we present numerical results for deviations from the Rutherford cross section caused by electronic binding for the systems z 92U-z 9zU, S3I-, 9Au and iéNi- 28Ni . In sect . 4 the influence of the dominant part of vacuum polarization, the Uehling potential, on heavy ion scattering is discussed. Additional background contributions like nuclear Coulomb excitation and static nuclear deformation are considered in the next paragraph. Finally we summarize our results and give a short outlook on future calculations . 2. Heavy ion scattering cross section and electronic banding eaergjes

Bocause of the large Sommerfeld parameter q . = Z,Zzac/v ~ 1 in heavy ion collisions, a classical treatment of the scattering process is justified. The differential scattering cross section in classical approximation can be written as _dQ __ ue~.m.~~/~~.m.l pzt,z sin dia ' B~.m.

(2.~)

where L denotes the relative orbital angular momentum, depending on the c.m . scattering angle, and p = M,Mzl(M, + Mz) the reduced mass of the system . In general, the quantities 9C .~, and L are connected by the equation of motion. Provided that the interaction can be described by a central potential V(R) this leads to z(n-e~.m.). = fRo Rz

LdR

2u(E_

y(R))_L lR

(2.2)

The distance of closest approach Ro follows from the zero of the square root in eq . (2.2). By means of both equations, the scattering cross section can be calculated for any potential V(R). In particular, if we insert the Coulomb potential in eq. (2.2), this yields the wéll-known Rutherford formulâ Z1Zze zz - ( 2~z ) sm

dQxwh _

C.m.'

(2.3)

The electronic bindingenergies in the individual atoms aredetermined by numerical integration of the Dirac equation with respect to finite nuclear size and electronelectron interaction usually evaluated by a Hartreo-Fock-Slater (HFS) potential ~. If we assume adiabaticity in heavy ion collisions the electrons are described by the solutions of the two-centre Disc equation. For the one-electron case, binding energies as function of the distance R(t) between both nuclei have been calculated by

SUPERHEAVY QUASIMOLECULES

495

Mûller, Rafelski and Greiner'o " 11) taking into account the finite extent of nuclear charge . The main correction arises from electron screening which is roughly a 10 effect for the 1s electrons with increasing portion related to electrons in higher states . For superheavy systems radiative corrections like vacuum polarization and selfenergy have to be investigated carefully since the electromagnetic coupling constant in the united system (Zl +Z2~c > 1 and the K-shell radius is smaller than the Compton wavelength of the electron
The total energy of the quasimolecule differs from the energy of both individual atoms which is defined as the sum of electronic binding energies of the occupied bound states depending non-linearly on the external electrostatic potential. The effective quasimolecular potential is just the difference between these energies s) :

with the adiabatic condition of constraints i~ = j~+k~. In figs. 2 and 3 this potential is plotted as function of the internuclear distance R for the systems 2 92U-2 92U+ saI-~ 9Au and 2éNi- 28°Ni . In a U-U collision it reaches its maximum at R x 0 with VQ, = 6.5 MeV. All electronic levels up to the 3d t state which are shown in fig. 1 of ref. ' e) are included in the calculations . Taking into account eleotrop screening corrections this value will be lowered by about 20 ~. The dependence of VQ,,, on the degree of ionization has been discussed in ref. 1 s). -~h~ can~ondiug scattering cross section dQQ,,, can be evaluated as described in sect. 2, using V =. Vc + Vim, where Vc is the ordinary Coulomb potential. In

496

W. SCHXFER et al. Yo

[MeYI

23B~~ Z3 92"-

6

43-

0

200

400

gifml

600

8011

Fig . 2 . The absolute value of the effective molecular potential Vo(R, Z Z=) as function of the internuclear distance R for the system'9z U-'9~U.

10~

10~

-

lOS

lp5

Fig. 3 . The absolute value of the effective molecular potentials VQ,,,(R, Z,, Z~) for the systems ~ 3 I-, 9 Au and =éNi--=sNi .

Fig. 4 . Percentage deviations from the Rutherford cross section in the system =9iU--=9sU with E~ .m . = 800 MeV . The influence of the quaaimolecular potential VQ,,, (broken line}, nuclear Coulomb excitation (short dashed line) and vacuum polarization (dot-dashed line) is indicated . The full torve onrresponds to the sum of all corrections to the Rutherford cross section .

figs . 4 and 5 we present the resulting ratio between dQQ,,t and the Rutherford cross section ; the projectile energies are chosen slightly below the Coulomb barrier. Our calculated deviations vary between 5.2 % at B°.m. = 5° and 0.9 ~ at B°.m. = 175'

SUPERHEAVY QUASIMOLECULES

497

err ~i.l U6 Q4

_ ~.i _I li0

Ob

etrC~.1

0.4 _

az 0

_pT

w

.____

ü i

i

_p,y _ _p,6 _pg

i

1~1

12(I

stl_~

CJ

12J

,~ i i

Fig. S. The percentage deviations from the Rutherford aces section caused by quasimolecular potentials (QM) and vacuum polarization effects (VP) forthe systems sjI-.,9 Au at E~ .~ . = 400 MeV and zeNi-zeNi with E~ .m . = 100 MeV. The notation of the curves corresponds to fig. 4.

for Z 92U-~9iU and between 2.8 ~ and 0.3 ~ for 53I-,9Au at the same angles . In the ZéNi-28Ni system we find a deviation of about 0.5 ~ at B~ .m. = 5° [ref. ")]. This significant correction of the scattering cross section at forward angles should allow an experimental verification of superheavy quasimolecules which is independent of the uncharacteristic MO X-rays. The cross section for 8~... < 5° is strongly influenced by the electron-electron interaction and the contribution of outer shell electrons (n ~ 4) where the adiabaticity argument fails and a full time-dependent Hartree-Fuck calculation would be necessary. Therefore we always restricted our evaluations to B,.a. ~ 5°. . In ords_r_to_prove that electroniç . shielding oî~~,enudeus~lcill measurement ofthe formation of gttasimolecules we considered the extreme model that a naked nucleus Z, is scattered by a shielded atomic Coulomb potential given by an asymptotic Thomas-Fermi distribution V(R)

_

Zt RZe2

exp

C

al

(3.2)

with the shielding radius a = 1 .4aoZZ # and the fu~st Bohr radius ao. The resulting percentage deviation from the Rutherford cross section is always less than 0.1 which verifies the neglect of this contribution .

498

W . SCHÄFER er al .

4. The Ü~l~g pOtCOtlal m heavy

!OR

acatteriog

The measurement of vacuum polarization in heavy ion collisions was first proposed by Rafelski and Klein ' e) and later discussed in refs. ' 9 . z°). Only the influence of the dominant first order term has been investigated . The Uehling potential for pointlike nuclei is determined by numerical integration according to m za dte-zrx (4.1) _ ~ VvP(R) ~tR C3tz + 3t°) t -1, and is shown as function of R in fig. 6. VP lMeVl 10 ____

0.01 ~ 0

~ 20

40

~ 60

v 00

Rlfm]

v 100

92~_g2~

w . 120 140

1fi0

y ~~disRU1~ R for the Fig. 6. Vacuum polariTation poterttiala of order aZs as funMion-of rti~ ~ntM ,.1 aystema . ~U-~U.~~1- ., 9 Au and 2aNi- ze Ni .

The change of the scattering cross section caused by the additional potential (4.1) is given by the dot-dashed lines in figs. 4 and 5. The percentage deviations vary between 0.03 ~ and 0.7 ~ for scattering angles 5° 5 8~.m. 5 175° in the discussed systems and are therefore always covered by the quasimolecular efixts in the z 92U-z 9iU system . However for small impact parameters both contributions are of the same order; for s3I-,9Au and 2éNi-28Ni collisions vacuum polarization is the dominant of%ct for B~,m. > 40° and B~, m. > 20°, respectively . Equivalent numerical results are obtained by Rafelski z~ for the systems z He-BZPb, 80- 82Pb and 90~-92U" ~ exclusive measurement of vacuum polarization is not possible in the superheavy z92U--z92U system .

SUPERHEAVY QUASIMOLECULES

499

5. Nuclear Coulomb excitation snd static deformaion of the nuclei Coulomb excitation in heavy ion collisions causes energy and angular momentum transfer to the scattered nuclei . Consequently, the trajectories of the ions depend on the nuclear states that have been excited during the collision. In a consistent semiclassical theory the resulting change of the scattering cross section can be evaluated by solving simultaneously the time-dependent Schrödinger equation z') (s H(~, R(t)>'4(~, t) = iti ~(~, t), ät

.l )

and the equation of orbital motion given by Ehrenfest's theorem (s .2)

Here, ~ denotes the internal nuclear degrees of freedom and
(s.4)

where v, and vf are the asymptotic relative velocities in initial and final states, respeotively . The velocity of depends on the mean excitation energy
R~(~~) = Ro~fl + E ~~, Yi.(Q~)l ;

t = 1, ~,

(s .s)

500

W . SCHÄFER et af.

the Coulomb interaction may be written as zs) V~R, acl~, a~z

)

=

ZZez 1 Rz

+

3ZZez SR3

z ~

z

~ Rô~:,Yi~~x,

=-z ~ =i

~PR~

(s.6)

where all integrals arising have been expanded up to first order in the quadrupole deformation parameters azp. The first term in eq. (s.6) is the usual monopolemonopole .part, while the second term describes the monopole-quadrupole interaction . The surface variables with respect to the lab system az ,~ may be transferred to the intrinsic variables ao and az by means of the rotation matrices z aZ ~,

z~ = E ~~.(s~~ ~~_2

(s .~)

where 9k are the three Euler angles connecting both coordinate systems. For nuclei with an axially symmetric equilibrium deformation ao = ß°, az = 0, the static Coulomb interaction amounts to

Z i Z zez 3Z1Zzez z z i~ z z~ z* o VdR, ~ou , ~o) _ + ~ Ror~o ~ ~~o(~k )Yz~(Srt, ~e)~ (s.8) R SR3 t=i ~,=-z

The potential Vc no longer remains in the .case of deformed nuclei ; therefore, the scattering cross section cannot be determined by the simple formulas given in sect . 2. Since the monopole-quadrupole interaction in eq. (5.8) is proportional to R -3 one may expect, however, that the deviations from Rutherford scattering causod by quasimolecular binding at large distance R - or equivalently small ion angles - will not be disturbed. Additional corrections to the scattering cross section arise due to the nuclear polarization potential Vn,r which was discussed e.g. for d-U collisions in ref. z'). Since Vn,p^' R_e it contributes only to backward scattering whereas quasimoleculat effects dominate at forward angles. The method of calculation described in ref. za) cannot be applied for U-U collisions since the assumption that the frequencies of the virtual excitations are large compared to the inverse collision time is not fulfilled. We find for the period of virtual excitation Tro, = 1 .3 x 10 - z° sec for E~ x300 keV whereas T~ .. 10- z 1 sec. 6. Conclasioes The solid lines in figs . 4 and s show the sum of all deviations discussed from the Rutherford aces section as function of the heavy ion scattering angle 9° .m.. For B~.m. < s0° the dominant part arises from the quasimolecular potential. Since these deviations are rather large, relative measurements of the scattering cross section should allow an experimental verification of the formation of superheavy eloctronic quasimolecules. The significance of this X-ray independent proof becomes evident from the fad that quasimolecular spectroscopy is prevented in systems like U-U by the y-rays following nuclear Coulomb excitation z`) .

SUPERHEAVY QUASIMOLECULES

50 1

Vacuum polarization can be measured separately in the collision of some lighter ions e.g. Ni-Ni ; this is impossible, however, in the superheavy system U-U. Future calculations should include relativistic two-centre Hartree-Fock binding energies ; nuclearand electronic bremsstrahlung aswell as ionization during thecollision should also be taken into account. We are grateful to Prof. W. Gnomon for stimulating this work and to Prof. H. MWler for many critical and helpful discussions. We would like to thank W. Betz formaking his numerical calculations of the two~entre Dirac energies available to us. References

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