Quantum mechanical treatment of electron-positron excitations in heavy ion collisions with nuclear contact

4NNALS

OF PHYSICS

Quantum Excitations

151.227-261 (1983)

Mechanical Treatment in Heavy Ion Collisions

of Electron-Positron with Nuclear Contact*

ULRICH HEINZ. BERNDT MUELLER, AND WALTER GREINER lnstitut fir Theoretische Phwik der Johann Warfgng Goethe-Utliuersitiil. Postfach 1 I 19 32. DdOOO Frankfurt a. M. Il. West German? Received

May

11, 1983

A general theory is formulated of electron-positron excitations in heavy ion collisions with nuclear contact, treating the nuclear relative motion quantum mechanically. A set of coupled channel equations for the electronic occupation amplitudes is derived. which is formally very similar to the semiclassical theory based on a classical nuclear trajectory, and reduces to the latter in the JWKB approximation. The new coupled equations contain all the quantum mechanical information on the details of the nuclear scattering during nuclear contact. The importantce of this formulation for a quantitative theory of spontaneous positron creation in supercritical systems with nuclear time delay is pointed oui. The possibility of line structures in the positron spectrum, as predicted semiclassically and recently discovered experimentally, is discussed in the framework of the DWBA approximation. For light-particle scattering off a nuclear resonance. the Blair formula for vacancy production is recovered in the same approximation.

I. INTRODUCTION Recent experiments with supercritical (Z, + Z, > 173) heavy ion collisions (U + U, U + Cm, U + Th) close to the Coulomb barrier have shown pronounced structures in the measuredpositron spectra [l-3]. Such structures had been predicted to occur in collisions with prolonged nuclear contact by Rafelski, Mtiller, and Greiner several years ago [4]. Detailed quantitative calculations of this effect in the framework of the quasimolecular theory of electronic excitations in heavy ion collisions [5-71 have been carried out by J. Reinhardt et al. [El. The most promising explanation of measured structures involves a treatment of the collision process, in which the nuclear relative motion is assumedto be classical and the nuclear contact leads to the formation of a long-lived (r > 10-*’ set) superheavy nuclear molecular system with an associated classical time delay (“sticking time”) in the classical trajectory [El. Thus the phenomenon is considered to be a typical case of interference between nuclear and atomic physics processes.The peaks in the positron spectra are attributed to spontaneous positron creation in these supercritical systems, this creation mechanism being enhanced by the sticking between the two nuclei [S]. * Supported (GSI).

by Deutsche

Forschungsgemeinschaft

(DFG)

and Gesellschaft

fur Schwerionenforschung

227 0003.4916183 $7.50 All

CopyrIght Cs, 1983 by Academic Press, Inc. rights of reproduction in any form reserved.

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In this paper we develop a general quantum mechanical theory to &scribe interference between nuclear and atomic excitation processes in heavy ion collisions, Such a quantum mechanical theory has become necessary since the validity of assumptions underlying the semiclassical treatment presented in Refs. [7, 81 is not entirely clear. The reason for mistrust in the semiclassical treatment is this: Since the nuclear processes that lead to nuclear sticking involve internuclear distances close to the classical turning point of the Rutherford trajectory, the validity of the semiclassical approximation for the nuclear relative motion is questionable, Moreover, the concept of “time delay” in quantum mechanics is quite subtle, and any calculation based on the ad hoc introduction of a classical time delay should be supported by a fully quantum mechanical treatment of the nuclear reaction process, The formalism presented in this paper is completely general and in no way confined to the description of positron spectra. To give an idea of the broad range of possible applications we will give a short survey of already existing literature on interference effects between nuclear and atomic physics processes in ion-atom collisions. First theoretical suggestions of such interference effects date back to the early sixties: Eisberg et al. (91 postulated on the basis of a semiclassical model that compound nucleus formation in nuclear collisions should lead to structures in the nuclear bremsstrahlung spectrum, from which the compound nucleus lifetime could be determined. Feshbach and Yennie [lo] showed that the discussion of this special effect can be extended to a fully quanta1 description. A successful measurement of this effect has been performed by Maroni et al. [ 111. A related discussion of this subject together with the suggestion of a test using nuclear molecules formed in ‘*C-‘*C collisions can be found in [76] (see also [77]). A similar effect of compound nucleus formation on the K-shell ionization in atomic collisions was suggested by Ciocchetti and Molinari [ 121. Their semiclassical treatment was recently extended by several groups to a quantum mechanical description of this process in the distorted wave Born approximation (DWBA) [ 13-161. The theory well describes data on interference effects in K-hole production in light systems like p + ‘*Ni collisions around E&lab) = 3.15 MeV [ 131, p + **Sr near E,(lab) = 5.06 MeV [ 171 and near E,(lab) = 6 MeV [ 181. (See also [ 191.) Another measurable quantity which is influenced by nuclear reactions is the X-raq’ spectrum. Inner electronic shell vacancies created in the incoming part of the atomic collision may be filled by X-ray transitions either in the molecular regime (where the two nuclei are still sticking together, or close to each other: molecular (MO) X-rays) or in the atomic regime (where the two ions have again separated: atomic X-rays). If the nuclear compound system lives long enough, the molecular X-rays form sharp lines at the united atom energies and thus may be well separated from the atomic Xrays. The ratio of molecular to atomic X-rays was suggested by Gugelot [22] as a measure for compound nuclear lifetimes. For elastic proton scattering off light nuclei the effect was measured by Chemin et al. [23] and by Rohl et al. 1241. These groups found compound nuclear lifetimes of the order of 10P”sec. In the case of much shorter lifetimes, the ratio of molecular to atomic X-rays may be understood

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quantum mechanically as a measure for the ratio of nuclear to electronic decay widths [25] or, in the case of many strongly overlapping nuclear resonances, as a measure for the correlation width of the autocorrelation function of the nuclear scattering matrix [26, 271. That the interference between nuclear and atomic physics aspects also works in the other direction, i.e., that details of the atomic structure can modify the nuclear scattering process, was shown in an early experiment involving quasielastic (x--u scattering (281. Theoretical treatments of the effect of atomic binding on a nuclear resonance may be found in Refs. [29, 301. The main application of the theory of excitations in delayed nuclear collisions seems to be in heavy ion collisions. Besides the effect on positron spectra already mentioned [7,8], oscillations in the spectrum of &electrons emitted out of the K-shell have also been predicted as the consequence of a nuclear delay time r [3 1, 32 ]. It was suggested to determine r from the wavelength of the oscillations, thus using the 6electron spectra as an atomic clock to measure nuclear sticking times. Anholt 133 ] and U. Miiller et al. [34] showed that the total K-vacancy production rate also contains sufficient information on the nuclear reaction to determine the reaction time. First experimental results in 7.4 MeV/a.m.u. U + U have been obtained by the group of W. E. Meyerhof et al. 1351. For heavy ion collisions molecular X-ray spectra in the case of nuclear contact have been studied in the semiclassical model by Anholt [36] and recently by Kirsch et al. 1371. They show that reaction times of the order of lo-“set are required to observe distinct compound nucleus X-ray lines due to molecular transitions in superheavy systems. Experimental detection has so far not been possible because of severe background problems caused by internal conversion of nuclear transitions in the compound system [38,39]. We conclude this survey of the field of heavy ion atomic physics in collisions with nuclear contact by saying a few words as to what one can expect to learn from these complicated and often cumbersome investigations. On the side of atomic physics, i.e., electron physics, the most important result would be-if firmly establishedthe proof of the existence of spontaneous positron production, i.e., the decay of the neutral electron-positron vacuum into a charged one in supercritical fields. The change of the vacuum structure is the most fundamental process in a field theory. Its observation is therefore of utmost importance. Besides this, information may be obtained about atomic excitation amplitudes during the collision, whereas only squares of amplitudes after the collision are measured in conventional experiments. However, other data, like those of quasimolecular X-rays, also contain such information and are, in many cases, more easily measured. Perhaps equal fundamental importance of this field will emerge in providing new and independent information about nuclear reaction processes that is often not accessible otherwise. Our definite preoccupation with reactions between heavy nuclei is due to the existing boundary conditions: In reactions between light nuclei usually only a small number of channels are open which can be investigated in detail. It is

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then possible to determine reaction times directly by measuring the resonance width or, in the case of overlapping resonances, by performing a fluctuation analysis. This is different for reactions of heavy nuclei where the number of open channels is immense and one can only measure cross-sections that are averaged over many individual final states. The situation is complicated by the fact that many highly excited final states decay by particle or photon emission long before they can be experimentally detected. But even if a well-defined final state could be selected, the vast density of levels combined with the limited energy resolution of accelerators would probably make even a fluctuation analysis impossible, Theoretical attemps to describe the physics of reactions between heavy nuclei have generally taken account of this situation by treating the reaction in terms of a few macroscopic or “collective” variables, and have often called upon classical and/or statistical concepts to describe the reaction dynamics. In such gross descriptions of the reaction all direct (i.e, quantum mechanical) information about the reaction time is lost and one has to rely on indirect ways to obtain some knowledge from consistency checks (deflection functions, energy loss, etc.). Here, the independent information provided by atomic excitations can be of tremendous help. It is particularly valuable, because the interpretation does not rely on specific models for the nuclear reaction. One may say that atomic excitations are a (nuclear) modelindependent “clock” for nuclear reaction processes. In the special case of positron emission, it has been shown [S] that this clock may be sensitive to reaction channels forming a minute part of the total cross-section and thus might even be a tool to establish entirely new reaction channels, such as the formation of long-lived giant composites. i The paper is organized as follows. In Section II we formulate the scattering problem in a general way and reduce it to a set of coupled differential equations for the nuclear relative motion. The boundary conditions with which these equations have to be solved are discussed in Section III. In Section IV we derive coupled channel equations for the occupation amplitudes and compare them with the JWKBapproximation, General expressions for the computation of cross-sections are also derived in that section. In Section V the coupled channel equations are solved in DWBA for the amplitudes for K-hole production and for positron creation. A discussion of the limits of applicability of this approximation is included. In Section VI we generalize the semianalytical form for the excitation amplitudes beyond the Born approximation. In Section VII, we summarize our results. II. THE COUPLED EQUATIONS FOR THE NUCLEAR RELATIVE MOTION

We face the following problem of stationary scattering theory: our total system is an eigenstate with eigenvalue E of the total energy: ’ We use the expression “giant nuclearsystems”or “giant nuclear composites” for systems with proton numbers Z =: 160-190, whereas the adjective “superheavy” is reserved for nuclear systems around Z = 1 IO-1 14 or 126.

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H,,, YE@,r, x>= E’Y,(R, r, x). R is the center of collective possibility

internuclear distance, t mass (CMN) system, coordinates) (Fig. 1). of nuclear excitations.)

(2-l)

is the set of electronic coordinates in the nuclear and x are internal nuclear coordinates (possibly (To be as general as possible, we include the The Hamiltonian of the total energy is given by

+ t' H&(r,R)+ i=l

231

&N

+ 2 H;,m,'(x,R)[+ H&x, r,R)]. rn=l

(2.2)

P is the relative momentum of the two nuclei; ,u is their reduced mass, M, is their combined mass; V,-(R) is the Coulomb potential between the nuclei; V,(R) is some nuclear optical potential to be specified later; H$,, is the two-center Dirac Hamiltonian for a single electron i [42,43); the next term is the energy contained in a possible collective motion of the complete electron cloud with respect to the CMN [44 1; Vee(rij) is the interaction between electron i and electron j (rii = Ipi - rjl). The second to last term is the internal nuclear Hamiltonian, summed over all degreesof freedom for internal motion. The last term in brackets, Hradr is the Hamiltonian for electromagnetic radiation; it depends on nuclear and electronic coordinates since it

FIG. 1. Definition of the combined center of mass. (X. between the two nuclei defines the laboratory system is given nuclear center of mass CMN.

coordinates used in this paper. A. B denote the two nuclei, CMN their Y, 2) denote the laboratory fixed coordinate system, The distance vector R the z-axis of the rotating coordinate system. Its orientation with respect to by angles B, 4. r, denotes the location of electron i with respect to the The internal coordinates x of the nuclei are not shown in this figure.

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contains coupling terms of the electromagnetic potential to the nuclear and electronic currents [41, 451. Hrad has to be taken into account for the calculation of X-ray spectra [37, 41, 461, bremsstrahlung [47], and internal conversion [45] of nuclear excitations into electrom excitations. These latter processes will not be considered in this paper, and we will from now on neglect radiation degrees of freedom. To solve the problem ((2.1), (2.2)) in full beauty is a difficult task, and we will not attempt that in this paper. Instead, after completion of the setup of the problem in this section, we will confine ourselves to electronic excitations only, beginning from Section III. The multidimensional problem (2.1) is reduced to a one-dimensional one in the nuclear relative coordinate R by expanding YY,(R, r, x) into a basis set, which diagonalizes the electronic and internal nuclear motion 1781. For the electronic states in undercritical systems it has proven convenient [43, 481 to use stationary twocenter Dirac eigenstates in the frame rotating with the internuclear distance vector R [42, 43, 491,

He&,RI(~n~(ry RI= E,(R)~,,a@, RI,

(2.3)

where H,,(r, R) is obtained from

5 H$,(r,

R) + &--

(2.4) N

i=l

by transformation into the rotating frame [44]. (In (2.3) it is the main quantum number, and I is the projection of the electron angular momentum onto the internuclear axis R.) Usually the second term in (2.4), which is an O(m/p) correction to the kinetic energy of the electron, is neglected, and the electron+lectron interaction is treated in the Hartree-Fock approximation [37, 41, 501. For overcritical systems the negative energy continuum contains one or more resonances from dived electronic bound states [ 5 11. These make the use of exact twocenter eigenstates awkward for_practical computations. It is then more convenient [6] to project a quasibound state &(r, R), which approximates the resonance due to the dived state, out of the positron energy continuum and to orthogonalize this continuum with respect to this quasibound state, yielding modified positron states hJr, R). Instead of the two-center eigenstates (2.3) one then uses t_his projected basis for negative electron energies [6]. The coupling matrix element (& IH,,I dEP), which is now non-zero since the projected basis states are not eigenstates of H,,, is interpreted as the matrix element of the spontaneous decay of a hole in the dived state into a continuum positron with energy E, [6]. Whenever we write #,n(r, R) we mean either the states (2.3) for subcritical or the projected states for supercritical systems. E,(R) then simply is defined as @“A IHeI I $nd* As a basis for internal nuclear excitations we introduce [52] eigenstates to

ff&,

R) x&, RI = q#) x,(x, RI,

(2.5)

ATOMIC

where H&x, R) frame. We may also eigenstates of the in Refs. 144, 53,

COLLISIONS

is obtained from C”,=,

WITH

H$‘(x,

NUCLEAR

233

CONTACT

R) by transformation

to the rotating

separate the dependence on the direction of R by finding the angular part of P2/2,u in the rotating frame. They are derived, e.g., 541 and found to be given by

(2.6) where B and $ are the angles of R in the laboratory frame. The rotation matrices [ 55 1 in (2.6) are interpreted as those responsible for a rotation of a state of total angular momentum J and projection 1 onto the internuclear axis R (the latter is given completely by the sum of the projections on R of the intrinsic electronic and nuclear angular momenta) into the laboratory frame, where the total angular momentum has projection M onto the beam axis. Putting all this together, the expansion of the total wavefunctions reads

Comparing

this to the usual channel expansion in nuclear reaction theory [56] Y,(R, r, X) = C C, Tc(R, r, x) - 1 U,,,FP,,(R, r. x) i 1 c C’

(2.8)

with ingoing channel functions rc and outgoing channel functions P,, we realize the following identifications: (i)

Our channel quantum numbers are given by the index vector (JMnaL}.

(ii) The C, E C itA determine the ingoing part of the total wavefunction are hence given by the boundary conditions characterizing the beam.

and

(iii) The scattering matrix, which is proportional to U,,,, is determined by the outgoing part of the wavefunction. Our problem will be to isolate the latter out of F;,,(R) which contains both in- and outgoing contributions. Inserting the expansion (2.7) into (2.1) and projecting out the electronic, nuclear and angular momentum basis states ((Z$,,A )X$,A) = lS$(@, $)X$,,(B, 4) dl2 =

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HEINZ,MijLLER,

AND GREINER

’ independent of 1) one finds 1441 the following the F:,,(R):

~JJ ) 4uM

d2 z-

J(J+

!

I)-12 R2

++,(R)-M+

set of coupled equations for

V,(R)-

W)I[

x C%&,,(R)

+

*a*.

v-9)

The dots denote terms like Coriolis coupling and translation effects [44] which we do not write down here explicitly but which are known and can be included 144, 521. The sums on the right hand side go over all channels ($1’) # (nal), whereas the diagonal terms have been included on the left hand side. At this point a short comment on the ranges of the different coupling terms seems in order. The longest ranged forces are the internuclear Coulomb potential V,(R) (whose monopole part is given by Z,Z,e’/R) and the angular momentum barrier [A’ -&It 1)1/R’. They determine the asymptotic behaviour of the FJ,,.l’s (see Section III). The nuclear optical potential and the pseudopotential due to the (possible) R-dependence of the nuclear internal eigenenergies E,(R) are of range R,, where R, is a typical nuclear distance (10-20 fm). So are the matrix elements of H nuc, as well as the radial coupling between intrinsic nuclear states. A slightly larger scale is defined by the spontaneous decay matrix elements (nn] H,, Imd’): They exist only if one or more of the bound electron states have become overcritical. In 9zU + ,,Cm collisions, e.g., this happens for the molecular lsa-state at R N 45 fm. The radial coupling between molecular electron states and the pseudopotential E,(R) - E,,(oo) due to the R-dependence of the electronic binding energies are quite long range, however. They vanish only when the system has again separated into two different atoms or ions, with a corresponding distance scale of the order of 10.000 fm. This is how far any numerical solution of (2.9), in principle, has to be extended. In practice, only inner electron shells can be considered, and the computation is usually stopped at R = 3000-4000 fm [5, 40, 41, 431, where these shells can already be considered atomic.

III. BOUNDARY CONDITIONS FOR THE INGOING WAVE Mainly for the purpose of easier presentation, from now on we are going to neglect excitations of intrinsic nuclear degrees of freedom. Although the inclusion of the latter will lead to considerable practical complications, we do not anticipate any prin-

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235

cipal difficulties to arise therefrom. Here, the nuclear interactions will be globally taken into account via the nuclear optical potential V,(R) (which may be complex). This is sufficient for the purpose of this paper, namely to relate the quantum mechanical and semiclassical pictures in the case of nuclear sticking. The latter may well be simulated by a pocket in the potential V,(R) 157, 581. Of course, we are aware that a complete understanding of electron and positron spectra in collisions with nuclear contact will require a thorough treatment of the intrinsic nuclear excitations. This must await future efforts. As mentioned in the last section, the characteristics of the beam will determine the coefficients CFi in Eq. (2.7). We require that the beam wavefunction asymptotically becomes an electron wavefunction in a state with definite main quantum number n and projection ,L?of its angular momentum on rhe beam axis [59], multiplied by a “plane” Coulomb wave for the nuclear relative motion: ‘Pinc(r, R) P, = K,Zt is the asymptotic

(3.1)

R-m

relative momentum in channel n,:

PO = pn,,= v%@

- %o(m 1);

PO =P,“;

(3.2)

,D,, is the reduced mass of the relative motion in channel n. In our case, where we neglect nuclear excitations, in particular nucleon exchange, ,U is independent of n. The effect of electron exchange on the reduced mass can be taken into account via electron translation factors, leading to a modification of the radial coupling which can be easily included [44]. Hence we neglect the channel dependence of ,u, and the only channel dependence of the asymptotic momentum P, is due to the asymptotic electron binding energy s,(co). This, however, is important as we will show below. v,, is the Sommerfeld parameter in channel n:

‘?” = The normalization

Z, Z,e* fiv, ;

V”S+

P o-oP

rl no’

(3.3)

of (3.1) is such that v.

I

) ylinc12 R* d.0 d3r

corresponds to unit flux. The electron state (D~~(c,R) is related to states with projection L onto the internuclear axis via

(3.4)

good angular momentum

(3.5)

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where the pnoJA are projections of the molecular angular momentum:

states onto good total electronic

c ICnj,(R)j2 = 1.

(3.6b)

Single electron bound states asymptotically angular momentum j,; in that case

become atomic eigenstates with good

(3.7)

For continuum states a decomposition as in (3.5) is necessary also for R -+ co [44]; the same will be true in general for many electron states, due to different possibilities to couple the individual electron angular momenta toj and 1. The plane Coulomb wave can be expanded into partial waves [60]: iLtl

Yinc(r, R) -1 R-m ,&,

-? (2L + q2P,(cos L%,

X [H:(K,R)

The H,’ are in-/outgoing fWC$)

0

- fZL+(KoR)]

spherical Coulomb -R-rm exp[fi(K,R

B)& ~~&r, R).

(3.8)

waves with the asymptotics

- q,, In 2K,R - Ln/2 + a:)]

(3.9)

where a: is the Coulomb phase shift (3.10)

01 = arg T(L + 1 + iv,).

To determine the C$y from (3.8), we have to recouple angular momenta from L and p to J, M, and A. Equating the ingoing parts of (3.8) and (2.7) at large R and projecting with rpk,(r, R) and R,$(e, 4) we obtain (Ii1 is the ingoing part of FJ,,) .Lfl

C;TZ;,(R)

+ R-cc

-

'0

-eH;(KoR)

&L;,(2L+1)z2

K,

(3.11) Using (2.6) as well as (oPmcj,A1c~,,~,(~,)~,) = 6,,oSjj~S,,,

and P,(cos 8) = Dk,(O, 8, o),

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and evaluating the integral over three D-matrices on the right hand side by standard methods [6 11, we find

(3.12)

The origin of the factor 6,,0 is obvious. The factor 6,, shows that, due to the azimuthal symmetry of the incident wave function for the nuclear relative motion, the orbital angular momentum of the nuclear trajectory does not contribute to the projection M of the total angular momentum in the lab system. The Clebsch-Gordan coefficients count the possibilities of coupling different values for electronic angular momentum j and orbital angular momentum L, where the projections in the lab and in the rotating frame are given, to a given value J of total angular momentum. The asymptotic behaviour of the Ii,(R) is given by the long range structure of the differential equations (2.9). As already pointed out at the end of Section II, at large values of R the Coulomb potential and the angular momentum barrier dominate. Neglecting the electronic contribution AZh2/2,uR2to the latter (the corresponding energy in a U + U collision at the touching radius is of the order of 1 keV x AZ, i.e., much smaller even than electronic binding energy effects), the Ii,(R) become independent of L and asymptotically ingoing spherical Coulomb waves with momentum P, and (possibly half integer) angular momentum quantum number J:

cl@)+H;(K,R). R-02

Making use of the asymptotic form of the Coulomb G,,s-factor, (3.12) is seen to yield

(3.13) waves (3.9) and employing

the

(3.14) This fixes the channel weights Ciy in the expansion (2.7) such that the ingoing wave has the correct asymptotics (3.1). For illustration we give the explicit expressions for the case that the ingoing wave contains a K-shell electron (or hole) in the heavier ion (which, without excitations, would become an electron or hole in the molecular 1s orbital during the collision). Since the K-shell of the heavier system is a state with good electron angular

HEINZ, MijLLER,

238

AND GREINER

momentum j = l/2, the sum in the expansion (3.6) breaks down. Hence in (3.14) A = f l/2, ,E= f l/2 and L = J f l/2 are possible. We find (3.15a) cJ.

- 112 =

(3.15b)

no.1/2

The ingoing wave in this case has the partial wave expansion ‘J’;‘1/2,,&

R) + ~+a

ft

-

.J i,jffY(KoR)

&Jz,2l

x

1 (J-

-

(Jt

e

$3

$3

KoR

a)no.,,2(r,R)D:,2.1,2(0,

don,.-,,2(rrR)DJL,2,,,2(0,

0, cp)

@a)/

+

(3.16)

Comparing this expansion in terms of channel functions defined in the frame rotating with the internuclear axis with the corresponding partial wave expansion (3.8) in terms of channel functions defined in the non-rotating CMN system, we seethat we have to pay for the easier convergence of the coupled equations (2.9) in the rotating frame the price of a considerably more complicated partial wave expansion. Whereas in (2.9) the expansion coefficients are independent of the electron spin, this is obviously no longer the case in (3.16). To clearly exhibit this additional complication in any quasimolecular system (compared to systems which can be treated in an atomic basis [ 141) is one of the reasonsfor presenting the calculation of the correct ingoing boundary conditions in such detail.

IV. OCCUPATION AMPLITUDES AND DIFFERENTIAL CROSS SECTIONS

In this section we reformulate the set of coupled differential equations for the nuclear relative motion (2.9) in terms of a set of coupled channel equations for socalled “modulating functions” or “occupation amplitudes.” There are basically two reasons for following this way: (1) Due to the large momentum contained in the nuclear relative motion, the solutions F;,(R) of (2.9) are rapidly oscillating functions with a wavelength of the order of one tenth of a Fermi. Hence direct numerical integration of (2.9) would require a very short steplength and, since the integration must go out to distances of the order of 10,000 fm, it would be practically impossible. This makes an analytical extraction of a rapidly oscillating factor absolutely necessary. If done cleverly [63,

ATOMIC

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641, the resulting differential equations for weakly R-dependent and may be integrated will discuss the question to what extent the can be done in “Born approximation” expansion” (see Section V).

NUCLEAR

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the modulating functions are relatively with a much larger steplength. We also integration for the modulating functions [ 14, 651 or “first order perturbation

(2) The second reason is that the procedure we will follow 163, 641 is formally very similar to the one used for a JWKB-approximation of the nuclear relative motion 144, 53 1, leading to a classical trajectory R(t). Therefore, the treatment presented here will allow a direct discussion of the essential differences and similarities between the semiclassical and fully quantum mechanical solution of the problem. In practical computations it may allow for a matching procedure between the low-J partial wave region (to be treated exactly in this representation) and high-l partial waves (to be treated in the JWKB approximation). We start by rewriting (2.9) in a more compact form: d’ 2 dR

I

- U,(R)

i

R,(R)

=X(R).

Here U,(R) is the sum of all non-derivative terms on the 1.h.s. of (2.9); a denotes the set of quantum numbers (JMnR); R,(R) = Cf’yFi,(R); and X(R) is the sum of all interchannel couplings on the r.h.s. of (2.9). (Please note that U,(R) contains the pseudopotential E,(R) - e,(co) due to the R-dependent electronic molecular binding energy. Because of its relatively long range the latter causes modifications of U,(R) from a pure Coulomb potential already at distances very much larger than the nuclear radius. From a practical point of view it may prove more convenient, however, to work with pure Coulomb functions in this region. It is then necessary to add this molecular pseudopotential as a diagonal coupling term to X(R) on the r.h.s. of (4.1) [66]; see also Section VI.) The modulating functions or occupation amplitudes will be defined by expanding the solutions R, of the full problem in terms of solutions r, of the homogeneous problem, where X(R) is set equal to zero:

Regular solutionsf,

of (4.1) are defined by their asymptotic

behaviour

f,(R) -R-r%(F.,W,,R)+ e2i”JIH,+W,R))3

[64] (4.3)

where FJ = (i/2)(H; - HT) is the regular Coulomb function, and e2”T is a phase shift factor created by the “optical potential” Y,(R) + E,(R) - E,(CO) and the electronic contribution to the angular momentum barrier. L2A2/2,uR2. 59/151/l

Ih

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In- and outgoing solutions h: of (4.2) behave asymptotically

e(R) -R+CC

H:(K,R)

like

e*is;

(4.4)

With the help of these solutions of (4.2) the solutions of the full problem (4.1) can be written as

R,(R)= b, x b,(R)f,@) + a,+(R) h,+(R)1

(4Sa)

x [b,(R)h,(R)+ b,+(R) h,f(R)l.

(4Sb)

The extraction of J-dependent factors like in (4.5) is completely arbitrary and just for convenience (see (4.21)). It is possible since the equations (2.9) are diagonal in the conserved quantum numbers J and M. By (4.5) we introduce for each function R, two new functions (a,, u,‘) or (b;, bz), respectively. (2.9) is not sufficient to determine all of these functions independently; we have to impose an additional relationship between a, and CZ,’ (b; and bz). We choose the relation

f,g+h,i$$O

for all

R,

(4.6)

which is equivalent to

h-adR+h$+o. db,

(4.7)

The expansion ((4.5b), (4.7)) is very similar to the JWKB-approximation where one expands into in- and outgoing JWKB-solutions of (4.2): exp(fi/h

S;(R)

[44, 531

k h/4),

with S;(R) = IR e(R’) RO (P’,(R))2=2p(E-~,(R)-

dR’; V,(R)-

V,(R))-J(J;f)h2.

(4.8b)

ATOMIC

COLLISIONS

WITH

NUCLEAR

241

CONTACT

However, the JWKB-representation of h ,’ becomes bad near the classical turning point R,. This is critical if we consider resonance scattering: suppose V,(R) + V,(R) contains a pocket like in Fig. 2, and the beam energy is just such that it is possible to hit a resonance in this pocket. Then it is not possible to define a unique classical turning point, since with some probability the nuclei tunnel through the barrier and fall into the pocket. All this information is contained in the solutions r,(R) of (4.2) where it shows up as a resonant structure in the excitation function. (Equivalently, the phase shift in (4.3) goes through 7r/2 when a resonance in the pocket is hit.) In particular, a resonance occurs exactly at the “right” energy: since the molecular binding energy e,,(R) - E,(cL)) is taken into account in (4.2), a resonance only occurs if the available energy at the radius of the pocket, R,, i.e., E - cn(Rp) - V,(R p) J(J + 1) h2/(2pRg), corresponds exactly to the resonance energy. Since such resonances may be narrow [.57] it is important to keep track of such apparently small contributions as c,(R) (which can exceed 1 MeV). This critical dependence of r,(R) on E,(R) also invalidates another step in the semiclassical approximation: the definition of a common trajectory R(t) for all channels (nn) at a given value of J (different angular momenta J lead to different trajectories R,(t) since they classically correspond to different impact parameters). Here, even if E and J are kept fixed, different channels (an) correspond to different values of the available energy at nuclear distances, leading to either off- or onresonance scattering and hence, even classically, to different nuclear trajectories. From the representation (4.5) the following boundary conditions for the modulating functions follow: (i)

Regularity

of R,(R)

requires R,(R

= 0) = 0, hence

a,i(R =O)=O. (ii)

(4.9)

From (4.6) and f,(O) = 0, lh,+(O)l = 00 we conclude 2

(R = 0) = 0.

FIG. 2. A schematic picture of the internuclear Coulomb + optical two form a potential pocket. R, denotes the position of the pocket.

(4.10)

potential in the case where

the

242

HEINZ,

(iii) From the asymptotic follows that (a = (Jj&A))

MtiLLER.

AND

GREINER

form of the ingoing wave

a”,L1,(R+ 00) = cJ,,,,~s

[see (3.13).

(3.14) 1 it

2(2L + 1) u+

1

(-)L-JiJ

L

(4.11) (iv) Finally. az(R + co) is related asymptotically behaves like (see (2.7), (2.8))

R;?(R)

to

the

T-matrix:

Since

R ,(R )

R +a#

x

\I7

2(2L + Us1

1)

(-)“-J

j>

X F,(K,R)k,, + 70 \’ %,,&,+ (K,$)Ii ,

(4.12)

we find (4.13)

where cz,ndO is the partial T-matrix element for a transition from channel (J&l,) to channel (J,,&). The sum over A0 comes from the fact that the incident wave contains a sum over different values for the magnetic quantum number. Of course, relation (4.13) holds whatever values for a$(m) are prescribed by the ingoing boundary conditions. Since the differential equations (2.9) are linear, we may determine the relevant T-matrix elements by solving (2.9) with boundary conditions a’J,u,(m)=6 nno6,,0-far all 1, instead of (4.11); the T-matrix is then given via the corresponding g,‘.p( co):

From this the outgoing amplitudes for any arbitrary ingoing boundary condition a?;(m) can be obtained via (4.13). To calculate from (4.9~(4.13) the corresponding boundary conditions for b,i(R) is an easy exercise which we leave for the reader. Inserting the ansatz (4.5) together with the auxiliary relation (4.6) into (4.1) now

ATOMIC

COLLISIONS

WITH

NUCLEAR

CONTACT

243

yields a set of coupled channel equations for the amplitudes an, a:, and b,$, respectively. Defining p 3 (J/imA’);

a =_ #id);

D,, E (n,i i?/:laRIm’A)

for all

J,p;

H~os~(n~~He,/mi.‘)

for all

J,,L;

for all

J, ,& 1;

P, G P,

(4.15)

these equations read

+ ID,&:

h,t ’ - H,,h,+ h; ] a; ):

+ IDd3f2G'

-H,,f,h;lqil

(4.16)

(4.17)

and for the bz: db-9-z-g dR

’

2 ([Dnahnthil’-H,oh,+h,]b8 0 5#Q

+ ID,& db+ -AL-& dR

h;’

- H,,h,t

h; ] b,t 1;

(4.18)

’

\7 {[D,,h,h,‘-H,,h,h,lbJ n 4#n

+ I&h,

h; ’ - H,,h,h;j

b;}.

(4.19)

Primes always denote derivatives with respect to R. Thanks to the structure of the auxiliary relation (4.6), these equations, although ingoing and outgoing amplitudes are coupled, are only of first order. As can be seen from (4.18) and (4.19) the coupling between in- and outgoing amplitudes takes place via very rapidly oscillating terms proportional to hz hi or hihi, whereas the ingoing and outgoing amplitudes are coupled among themselves via slowly oscillating terms proportional to h;h,+. In situations where the nuclear relative motion is semiclassical it can be shown 144, 671 that the former average themselves away along the trajectory, such that the equations for b; and b,+ decouple, leading to an ingoing and an outgoing branch of the classical trajectory, respectively, in the framework of the JWKB approximation. The remaining differential equations for the b-‘s and b+‘s only contain slowly oscillating coupling terms and may be integrated with a quite

244

HEINZ,

MijLLER,

AND

GREINER

large steplength (of the order of several Fermis or larger). Starting outwards from a sufficiently large value R, (Z?, > 100 fm) it should even be possible to do the integration of the equations in the “Born approximation” [65] or first order perturbation theory” (see the following section). Whether or not one is allowed to throw away the rapidly oscillating coupling terms depends on the degree to which the system behaves semiclassically. One is certainly not allowed to do so near the classical turning point, i.e., for R of nuclear dimensions. There the full set of equations (4.16)/(4.17) or (4.18)/(4.19) has to be solved or, directly, the Schrodinger equation (2.9). (Practically, the latter may be even more convenient, see Ref. [63].) Iteration schemes for the numerical solution of (4.16), (4.19) have been suggested in Refs. [63, 641. We will not dwell on this point in this paper, but postpone it until numerical results are available. Here, we want to draw attention to one more detail in the structure of the coupled channel equations for the amplitudes. Our comment concerns the influence on the ampitudes of scattering through a nuclear resonance. The resonance shows up as a large amplitude of h,i at nuclear distances if the available energy in channel a at those distances corresponds to the resonance energy. For fixed beam energy E, only for a few channels a this available energy will lie within the resonance width, whereas for all the others it will not. For the few channels a “on resonance” the time derivative of the amplitude at nuclear distances is given by matrix elements multiplied with large amplitude wavefunctions, i.e., they change (and may grow) rapidly. For all other channels the amplitudes change comparatively slowly. If energetic conditions are such that just the spontaneous decay matrix element for a K-hole into a continuum positron is favoured in this way, a positron resonance (“resonance” since the resonance-amplification will only work for a few positron energies se) due to spontaneous positron production seems a possible consequence of the nuclear resonance. In this way results obtained in semiclassical computations with an artificial sticking time [8] could be supported by a fully quantum mechanical computation involving nuclear resonance scattering. As this argument shows the possibility for a qualitative reproduction of the semiclassical result is clearly given in the structure of the equations, although quantitative statements are only possible after a quantum mechanical calculation has been completed. In the next section we will try to put the qualitative side of this argument on more solid grounds, using the Born approximation. Before we proceed to that calculation, however, let us note a few expressions required to derive the cross sections from the amplitudes. Since the ingoing wave was normalized to unit flux (see Eq. (3.4)), the cross section into channel (nL) is directly given by the outgoing flux. The latter is obtained by summing over J and M (due to the 6,, in the amplitudes the sum over M breaks down) the outgoing waves in all channels (.MrzL), integrating over internal (electronic) coordinates and multiplying with the surface element R*dQ and the asymptotic outward velocity v, = P,/,u. From (4.3)-(4.5) we obtain the following expression for the angular distribution:

ATOMIC

COLLISIONS

WITH

NUCLEAR

CONTACT

245

Using the asymptotic form (4.4), (3.9) of h,f. and Eq. (2.6) we find

(4.21)

The corresponding expression in terms of the a,, a,: is obtained b:?(m) by (see (4.5)) a++) nA

+ 6,,. (e2ib:-g

u$(m).

by replacing (4.22)

For n # n,, b,+,JL’(a) and uip (cry) contain the same information (see also (4.13)). The expression (4.21) for the differential cross section has the familiar form [68] except for the fact that the Legendre polynomials P,(cos 0) have been replaced by Wigner-matrices d:,(B), reflecting the effect of electron spin. For systems that behave semiclassically, the J dependence of the amplitudes bip(co) is very weak compared to that of the d:,-functions, since the initial conditions (4.11) and the classical trajectories are weakly J-dependent. Hence, for not too small values of J, the sum in (4.21) usually may be evaluated by the saddle point method using the analytic large J-behavior of the Legendre polynomials [69, 701. The saddle point J, relates the classical scattering angle to the trajectory impact parameter, and the result is (4.23)

where bJS(co) is the amplitude for the saddle point value of the angular momentum, J, = P,Z,Z2e2/(2E) cot(0/2). In the case of resonance scattering the amplitudes are expected to be rapidly Jdependent, thus invalidading the saddle point approach. It may even happen that the amplitudes for a few J-values, for which resonances are hit, dominate the angular distribution at almost all angles (except at very forward angles, where Rutherford scattering is always dominant). As a consequence, a numerical evaluation of the angular momentum sum in (4.21) cannot be avoided in general [ 7 11. If the nuclear resonance is narrow and confined to one J-value, J,, the angular distribution may be represented by a single ~d~~(~)~‘. In that case we get the simple expression (4.24)

246

HEINZ,

MijLLER,

AND

GREINER

Please note the A, ,U dependence of this expression, reflecting the influence of electron angular momentum. If J, is sufftciently low, the ii and L dependence of such an angular distribution might be measurable, using electron polarized beams and measuring the electron polarization in the outgoing channel. It must be noted, however, that for such an investigation the amplitudes b,+(co) must be computed under inclusion of the Coriolis coupling since the latter mixes states of different electron spin projections.

V. THE DISTORTED

WAVE

BORN APPROXIMATION

Summarizing the result of the last section we see that, to compute cross-sections (4.21) for electronic transitions (n&ii)+ n,?), we have to solve (4.18)/(4.19) for the amplitudes b,+,J’(m) with boundary conditions b,-,J”(co) = (i/2) a:?(~) given by Eq. (4.11). For inelastic scattering (n, # n) b,+,JL’(a) = aLy(a~); then we can use the relation (4.13) between a,‘(co) and the T-matrix and reduce the problem to a computation of the latter, i.e., the solution of the coupled equations (4.18)/(4.19) for ~~~(~0) with boundary conditions b”,-,J(oo)= dnnoSllo. (These &do not depend on ,L? whence we dropped this index with the &) As shown in Ref. (651, the distorted wave Born approximation to the solution of (4.18)/(4.19) is obtained by dropping the rapidly oscillating terms (thus decoupling the equations for b; from the ones for bz), and integrating the remaining equations over R, keeping ?heamplitudes on the r.h.s. constant and equal to their initial values. Integrating the equation for 5; (corresponding semiclassically to the ingoing branch of the nuclear trajectory) from R = co to R, = 0 (in practice one would choose a finite, but infinitesimally small R, to avoid the singularities of the in- and outgoing waves hz at R = 0), keeping the amplitudes on the r.h.s. equal to b”,-(GO), we obtain

Similarly, integrating the equation for b”,’ from R, to co, keeping the amplitudes on the r.h.s. equal to EL(R,,), yields

&co)

= &R,)

+f

t] b”,+(R,) jm [D,,h,h; RO a 5+a

’ -Ha&h;

] dR.

(5.2)

The two solutions have to be matched at R, by ba’(R,) + &R,)

= 0,

(5.3)

ATOMIC

COLLISIONS

WITH

NUCLEAR

247

CONTACT

which is obtained from (4.9) and (4.5). Inserting the boundary 8nnodllo E Jaao (we define a = (Jnn), 0 = (Jn,,&)) in (5.1),

condition

g;(co)

=

we see that the distorted wave Born approximation corresponds to letting the matrix elements, that are responsible for transitions from one channel to another, act just once between R = co and R = R,. States empty at R = 03 can be occupied by a single transition out of states occupied at R = co, but a second transition from there on to a third state is not allowed. This warrants the name “first order perturbation theory” for this approximation, since only one-step transition processes are included, but multistep processes are not taken into accent. The approximation certainly is good only if the second term in (5.4) is small compared to one. (See, however, Ref. 1401.) If we insert (5.3) and (5.4) into (5.2). in order to confine ourselves consistently to one-step excitations we have to drop terms which contain the transition matrix elements twice. This leads to

- ff,&~,thO_

+ h,h,+)

J

dR.

In general, an evaluation of this DWBA expression still requires a detailed knowledge of the in- and outgoing wavefunctions for the nuclear relative motion h,i; in particular, if the excitation process (like the spontaneous positron production mediated by the matrix elements H,,) occurs at nuclear distances where the details of the behaviour of the hz are crucial. For the process of spontaneous positron production in supercritical collisions with nuclear contact. the nuclear and atomic physics aspects appear to be inextricably intertwined in Eq. (5.5). Thus a reliable determination of the positron spectrum even in DWBA requires detailed knowledge of the nuclear physics that determines the distorted wavefunctions h:(R). Below we will discuss an approximation, which separates the nuclear physics from the electronic excitations and has generally been used in DWBA calculations so far 114-16, 25-271. Whereas it is supposed to be sufficiently good for K-hole probabilities and there successfully describes several light ion experiments [ 13, 14, 17. 18 1, its validity when applied to positron spectra has still to be checked. Still. the qualitative arguments brought forward in the last section to point at the chance of obtaining structures in the positron spectrum from Eqs. (4.18)/(4.19) in collisions with resonance scattering continue to hold true for the much simpler expression (5.5). There is hope that a shcematic model can be constructed in which (5.5) can be evaluated [72]. In the remainder of this section we will show how the DWBA-results on K-hole

248

HEINZ,

MijLLER,

AND

GREINER

production by Blair and Anholt [ 141 follow under additional approximations from our general theory. (In a similar way related approaches in Refs. [ 15, 16, 25-273 can be shown to be contained in our theory.) We consider subcritical systems, in which the matrix elements H,, vanish. The Khole cross section as a function of the nuclear scattering angle is then obtained by summing over all excited states ,the cross-section for scattering an electron via radial coupling out of the initially occupied K-shell into an excited state: da, -= d.0

. t

du(K + n/l) d.0 ’

(5.6)

(nl)#K

The Kii(co) entering into the right hand side are given by (now denoting the K-shell by the index 0) (5.7) Following the procedure in Refs. [ 14, 15, 251, we assume that practically all of the K-holes are produced outside the regime where the nuclear scattering takes place. That means that in (5.7) we can choose R, larger than nuclear dimensions, and that in the integrand the nuclear relative wavefunctions can be replaced by their asymptotic forms. We will see that under these assumptions the nuclear and atomic physics processes factorize, and that we obtain from our theory an expression for the K-hole probability similar to that of Blair and Anholt, which well describes the existing data [13, 14, 17, 181. Besides the fact that we use relativistic electron wavefunctions, and correspondingly electron spin here plays an important role, the main difference between (5.7) and the corresponding equation in the paper by Blair and Anholt is that we use a molecular basis and the excitations are mainly due to radial coupling matrix elements D aO, whereas Blair and Anholt use an atomic basis and the excitations are due to potential coupling matrix elements of given multipolarity (the higher multipoles corresponding to the Coriolis coupling in a molecular picture). This has several consequences: (i) Our integrals in (5.7) contain derivatives of the wavefunctions relative motion, whereas those in Ref. [ 141 do not.

for nuclear

(ii) Electron spin does not enter into the angular momentum balance in Ref. [ 141, since non-relativistic electron wavefunctions are used; however, the angular momentum mediated by the considered potential multipole coupling does enter. (iii) In our case electron spin does enter into the angular momentum balance via its magnetic projection. The radial coupling, however, is a scalar operator. Therefore, it does not contribute to the angular momentum balance. Notwithstanding

these differences,

the evaluation

of (5.8) continues

very much

ATOMIC

COLLISIONS

WITH

NUCLEAR

249

CONTACT

along the same lines as in Ref. [ 141. The integral in (5.7) is evaluated using the asymptotic behaviour of the nuclear relative wavefunctions: h&f(R)+

exp{ +I’(K,R

dh$i, + fiK,

dR

- fin In 2K,R - Jn/2 + a:)); - q. In 2K,R - J7r/2 + i$)},

exp(fi(K,R

(5.8)

(5.9)

where now #j, S’j are supposed to contain both Coulomb and nuclear phase shifts in channels (Jn), (Jn,), respectively, and K, = P,/A. With this we find for (5.7)

where [ 73 J B, = B,, = [m D,,(R) -R0

(5.11)

eiq(R”R dR

and q(~) . R s (K, -K,) z(K,-K,)R

R - q,, In 2K,R + rl, In 2K,R +

ZJ2e2(Ko - 4) 2E

ln 2K R 0 *

(5.12)

Equation (5.10) is very similar to Eq. (14) of Ref. [ 141. With the help of the gz(co) which form the T-matrix we derive the amplitudes bi (co) corresponding to physical boundary conditions (see (4.11 j-(4.14)):

In the step from the first to the second line use was made of the fact that, since there is only radial coupling, the T-matrix is independent of A and p and proportional to hAA (all 1 and ,u- d ependence is in the boundary condition b;if(co) [74]). The cross-section (5.6) correspondingly is given by (5.14)

250

HEINZ,

MtiLLER,

AND

GREINER

where A ,,A(e) = 1 (-)“-”

i’+ ‘(2L + 1) e2iSJ”d~~(B)

LJ

(5.15) IS tven by a similar expression, the phase shift Sp replaced by Sy. Since the K-shell asymptotically becomes a state of good angular momentum in (5.15), and the sum over j breaks down. But still the 1 j = l/Z and p dependence of the di,-(@)-functions generally prohibits an analytical summation of Eq. (5.15). Progress can be made in cases where this A and p dependence is unimportant. For example, if we consider the limit of non-relativistic electron wavefunctions. the K-shell corresponds to j = 0, ,L = 0 (no electron spin) and, since there is only radial coupling which does not change the electron angular momentum projection, A = 0. Then the Clebsch-Gordan coefficients in (5.15) yield a,.,. and and

A,,,(@

'

C,J,(

g'

co)

=

sj,

I/*

A ,,(8) = i 1 (2L + 1) P,-(cos 0) ezisP L

= i(2iK,)f(O, i.e., proportional to the nuclear scattering energy E - Qco)). Similarly,

E - E&

amplitude in channel n, (with asymptotic

A*,,(B) = i(2iK,)f(O, The cross-section

(5.16)

(5.17)

E - E,).

comes out as

da --L=If.(e,E--60)~2$? nl

n

,nA-~.f’e’E -“I B,*, PII f(e, E - %I

‘.

(5.18)

The obvious interpretation is that B,, is the amplitude for a transition on the outgoing part of the trajectory where the nuclear scattering took place at the energy E - E, corresponding to the ingoing channel n,, while BzA is the amplitude for a transition on the ingoing part of the trajectory, followed by a nuclear scattering at the energy E - E, corresponding to the outgoing channel n. This interpretation clearly exhibits what we called a factorization of the scattering process into a nuclear and an atomic physics part. Expression (5.18) is nice since it allows for a simple connection to the concept of a classical sticking time [ 141. Writing (for not too rapidly varying f(& E)) (5.19)

ATOMIC

with ho,,, = E,(W) - Ed, delay [ 75 1

COLLISIONS

WITH

NUCLEAR

251

CONTACT

and recalling the quantum mechanical definition

of time

r

rQM(& E) = -ih &

(which

lnf(f3, E),

is again only valid if rQM is small enough, (5.19) can be written

as

The resulting form of the cross-section (5.18) is formally equivalent to the semiclassical expression first derived by Ciocchetti and Molinari [ 12). In a final remark we wish to match up this result to the conjecture at the end of the last section, namely that “resonance amplification” could be responsible for structures in the electronic excitation spectra. The strong variation of the phase shift across a resonance, found responsible for the phenomenon of “time delay” in this DWBA-calculation. is directly related to the resonant structure of the h: leading to the suspected rapid change in the occupation amplitudes across the resonance. We will conclude this section by trying to use the same procedure to calculate the positron spectrum in supercritical collisions. A priori for the spontaneous positrons nuclear and atomic physics aspects will not factorize so obviously as above, but we will introduce a few additional, very crude assumptions under which such a factorization again takes place. The final expression will look identical to (5.18), only the amplitudes B,, being modified as to include the spontaneous positrons. This will show that at least qualitatively the same mechanism can be responsible for structures in K-hole probabilities and positron spectra, with its obvious and close relationship to the classical concept of a nuclear sticking time. Work on a model which allows to evaluate Eq. (5.5) for positron production with less restrictive approximations is in progress [ 721. We now denote by the index 0 an electron state in the negative energy continuum with given energy. This state is characterized by the energy E,, which a positron will have when it is created by forming a hole in this state, and by its electron angular momentum projection & : 0 = (JE,&). This state is filled in the incoming channel. We compute the number of holes created in it by excitation into bound electron states and positive energy continuum states, which all are supposed to be initially empty. The inclusive double-differential positron cross-section is given by an expression like

(5.6): (5.22)

252 but (in DWBA)

HEINZ,

MijLLER,

AND

GREINER

the amplitudes entering the r.h.s. are solutions of (5.5):

- If,,@,+ h, + h, h,t)

1

dR.

(5.23)

We will split up the integral into two regimes by introducing a matching radius R, (see Fig. 3). As before, R, will be chosen so small that practically all the dynamical positrons created via the radial coupling D,, are excited at R > R,. Speaking semiclassically, such a matching radius will be of the order of a few times the classical turning point. A second scale is given by the onset of spontaneous positron production, i.e., the diving point R, of the K-shell. If R, is smaller than the thus determined R m, we will use R, as the matching radius. Thus (5.23) reads

The first crude approximation consists of using the asymptotic This approximation relative wavefunctions h * for R > R,.

form of the nuclear is only good for

FIG. 3. A schematic picture of radial vs. spontaneous coupling in a semiclassical model (adapted from Ref. 161). R is the two-center distance R > R,. where R, is the classical turning point (semiclassically associated with time t = 0). R, is the diving point. R, the matching radius defined in the text. For the radial excitation the semiclassically relevant combination d < 8/8R) is drawn which shows explicitly that, semiclassically at least, most of the radial coupling occurs at distances R B R,. This semiclassical picture is the motivation for our choice of R, in the text.

ATOMIC

COLLISIONS

WITH

NUCLEAR

253

CONTACT

the classical turning point, which may not be well satisfied here. Nevertheless, we use this approximation to get a rough estimate. Then the first and the last term on the r.h.s. of (5.24) yield an expression analogous to (5.10), with II,., given by

R,,, 3 R,,,

B,, = fin ei?(R).R D,,(R) t -R”

+ #H,,(R)

1

0

(5.25)

dR.

The second term on the r.h.s. of (5.24) is treated in the following differential equation (4.2) for the h *, we find for it

way [ 141: Using the

where hw,,(R) = E,(R) - e,(R). This expression can be evaluated if the radial dependence of the spontaneous decay matrix element and of the transition energy w,,(R) is neglected. Having in mind the picture of the pocket in the internuclear potential (Fig. 2) we set these quantities constant and equal to their values at R,, the position of the pocket. Hence we obtain i

hH,,(R,)

Rm dR [h,h,t ” +h,+h,“-h,th,“-h,h;“J. i0

Kc2 2wLo(R,)

(5.27)

The remaining integral can be evaluated by partial integration. Since at R = 0 h,t = h; and dhz/dR = dh;/dR, the surface integral vanishes at the lower limit. The upper limit is again evaluated using the asymptotics of the hz, yielding i(pu

“0) 2p,

By assumption @R,) amplitudes reads

AHa’J(R~)

{ie-i~(Rm)Rmei(b~~~~

_

iei”(R,)Rmei(bj-S;)

1.

(5.28)

w%o(R,)

. R, < 1, so that the final expression

for the occupation

pa+PofiH,o(R,) 2po

,uw,o(R,)

)

(5.29) with B, from Eq. (5.25). We conclude that our assumption of constant values for the decay matrix element and the transition energy allowed us to factorize nuclear and atomic physics also in the amplitude for spontaneous positron creation. Under this assumption the cross-section (5.22) again takes the form (5.18), with B,,i in (5.25) replaced by B

nA

+ plz + PO wio(R,) 2PO

Pi,,

(5.30) '

254

HEINZ, MtiLLER,

AND GREINER

The effect of a nuclear resonance now is two-fold: As the positron energy E, A E, is varied, E - c0 varies across the resonance, leading to a resonance in the nuclear scattering amplitude ]f(t9, E - &,,)I’ in (5.13). But at the same time f(t9. E - e,,) exhibits its resonant behavior in the excitation probability given by the second factor (under the sum) in (5.18): the interference between in- and outgoing channel changes characteristically as E - E, L E -E, is varied across the resonance. Semiclassically this effect was shown to be able to produce the measured structures in the positron spectra [S]. Via the relationship between the variation in f(t9, E - co) and the classical sticking time ((5.19), (5.20)) we seethat, at least to the extent to which this crude model is applicable, things come out similar in our quantum mechanical treatment.

VI. BEYOND THE BORN APPROXIMATION In this Section we want to derive a generalization of the cross section formula (5.18) which does not rely on the Born approximation, but still allows for a similarly simple interpretation in terms of excitations along the ingoing and outgoing branches of the nuclear trajectory multiplied by suitable nuclear scattering amplitudes. To this end we once again return to the coupled channel equations which we now, however, write in a slightly different form which is more suitable for our further analysis. These modified coupled channel equations are obtained by expanding the solutions R, to the problem (4.1) not into the set of functions r, defined in (4.2). but into pure Coulomb waves,

\ d2 /dR2-

J(J+ 1) 2/l R2 +piE-&,k-

V,(R)][

H,i(K,,R)=O,

(6.1)

with Ki = 2,@r2 (E - s,(co)). The corresponding expansion reads

X (c,(R)H,(R)+c,t(R)H,t(R)}.

(6.2)

with the obvious notation Hz(R) = H,’ (K,R) for a E (JMnL). Analogously to (4.6) and (4.7) we require the auxiliary condition

(6.3) With these definitions we obtain for the c,’ the coupled channel equations

ATOMIC

dc* id V ---2_~~--gHJH+c* dR ~

COLLISIONS

WITH

NUCLEAR

CONTACT

255

aa

+ .. .

(6.4)

where the ellipsis stands for the rapidly oscillating terms proportional to Hz Hi and Hi Hi coupling the ingoing to the outgoing amplitudes and vice versa. AV, contains the additional potentials from Eq. (2.9): AV,(R)

= $

[e,(R) - ~,(a)

+ V,(R)]

-$.

(6.5)

Now we remember the definition of the matching radius R, (see Fig. 3). For R > R,, the system behaves semiclassically, hence we may neglect the rapidly oscillation terms in this regime (and, if we wish, we may there replace the Hz by the JWKBwaves (4.8). In that case we can eliminate the first term on the r.h.s. of (6.4) by a suitable phase transformation on the amplitudes c,’ : c:(R)

= F:(R)

explf

ix,(R)];

)‘L’AV,(R’) X,(R)=& fl ‘X

H,(R’)

(6.6a) Ht(R’) 0

dR’.

(6.6b)

For the ?,’ we then have for R > R,

Let us now look at the regime R < R m. There our presentation will be clearer if we use the two-center basis (2.3). In this basis there is no spontaneous coupling H,,n. For the radial coupling D,, we assume as in Section V that it may be neglected for R < R,. In the projected electron basis @,,Athis assumption corresponds to neglecting the radial coupling for R < R, and setting the spontaneous coupling constant and equal to its value at R,. Since we here are not interested in the Hz behaving asymptotically at R,, we may justify this assumption by choosing R, sufficiently small. (However, R, should be larger than the range of nuclear interactions.) Calling the amplitudes in the eigenstate bases (2.3) A: and 2,‘) respectively. we thus find that the 2: do not change any more for R < R, : xz(R,,,)

=x,‘(O).

(6.8)

256

HEINZ,

At R = 0, however, regularity for R,(R):

the 2,’

MijLLER,

AND

are connected K,‘(O)

GREINER

to the 2;

through

the condition

= -K,(O).

of

(6.9)

Putting (6.6a), (6.8), and (6.9), together we find JL(R,,,)

= -exp = -e*‘“*A

i

-2ijRmAVa(R’)H;(R’)H,‘(R’)dR’ 0

I

; (R ,),

A,(R,) (6.10)

i.e., the outgoing amplitude at R, is related to the ingoing one via a simple phase shift factor, created by the nuclear potential and the molecular binding energies at R
(6.11)

Summarizing the discussion up to this point we conclude the following: For a definite transition i -+ f (J fixed) we obtain the asymptotic amplitudes by first solving (6.4) or (6.7) for all intermediate states n for AiJ(R,) with initial condition i; then computing AiJ(R,) from (6.10); and finally again solving (6.4/7) with all these initial conditions AT=‘(R,) for AfJ(co) and summing over n. Symbolically we write this as aLf (co, co) = -c

Al::@, R,) n x e*i6,(E-&,(R,))Ant:(R,,

co)

(6.12)

This formula is a very useful generalization of Eq. (5.5). It was also written down in [79]; a similar expression for the semiclassical formalism is contained in [8]. Equation (6.12) is only valid when using the eigenstate basis (2.3). When using the projected basis [6], one has to apply the projection procedure at R = R,. The result reads

257

ATOMlCCOLLlSlONSWITHNUCLEARCONTACT

where the projection matrix U is given by [so]: urn, = hn,

(m, n bound or upper continuum states); (E positron state); (6.14)

4&L) = 2 il (E-E’-iq)(E-s,,(&)++(E))] (E. E’ positron states); WI

= 2~

I(kIK, l@,s)lz.

The cross section now reads (6.15) where we neglected the slight channel dependenceof the Coulomb phasesand of the asymptotic momenta AK, as well as the dependence on the electron angular momentum (see Section V). It is easy to check that from (6.15) one recovers the cross section in Born approximation, (5.10), by applying perturbation theory to the amplitudes in (6.15):

and neglecting terms of second and higher order in the AC*. In other words, (6.15) is a generalization of the Born expression to the case, where the amplitudes are solved for to all orders in the excitation matrix elements, by using the full set of coupled channel equations (6.4/7) instead of the perturbative result (5.5) 1401.

VII. SUMMARY AND CONCLUSIONS In this paper we developed a general quantum mechanical formalism for the description of electron-positron excitations in heavy ion collisions with nuclear contact. The framework is universally applicable in the sensethat not only all kinds of electronic excitations (K-hole production, b-electron and positron spectra and, in principle, molecular X-ray spectra) can be described, but also nuclear processes leading to excitation of the two nuclei and hence to modifications of the nuclear scattering between the two ions can be included. Hence the theory is well suited for a treatment of heavy ion collisions with nuclear contact and the associatedinterference between nuclear and atomic physics excitation processes.

258

HEINZ,

MtiLLER,

AND

GREINER

In this paper the theory was developed with a focus on K-hole and positron production in supercritical collisions with nuclear contact. This is why for the atomic physics aspects we concentrated on two excitation mechanisms: the radial coupling between molecular electronic states which is predominantly responsible for K-holes and dynamically created positrons, and the spontaneous decay of dived states which creates the spontaneous positrons in supercritical collisons. Coriolis coupling, electron translation effects, etc. have been neglected here, but can be easily included in future calculations. With respect to nuclear processeswe wanted to include the fundamental phenomenon of sticking, but did not intend a detailed description of nuclear excitation processes.Therefore we here neglected internal nuclear degreesof freedom, but included the possibility of nuclear contact and inelastic nuclear scattering via a (complex) nuclear optical potential. For the latter we particularly considered a model where this nuclear optical potential contains a pocket (57, 58 ] leading to nuclear scattering resonances. This model is not intended as a realistic picture to describe the observed structures in measured positron spectra [l-3], but as a toy model that provides the context for the identification of possible quantum mechanical mechanismsfor the creation of such structures. In a first step (Sect. II) the full scattering problem was reduced to a set of coupled differential equations for the channel wavefunctions of the nuclear relative motion, Eq. (2.9). This was done by projecting out the electronic and internal nuclear eigenstates as well as the eigenstates for the orientation of the internuclear axis, leaving a dependenceon only one coordinate, the two center distance R. Boundary conditions for the ingoing parts of these channel wavefunctions (Eq. (3.14)) were derived in Section III. If the Coriolis coupling is neglected (in a molecular basis this is the only electronic excitation mechanism which changes the electron angular momentum projection), these boundary conditions contain all the information on electron spin dependence. In a next step (Sect. IV) the set of coupled equations (2.9) was transformed into a set of coupled channel equations for the “occupation amplitudes” (see Eqs. (4.16)-(4.19)). This was effected by expanding the channel wavefunctions into a set of inand outgoing wavefunctions for the nuclear relative motion which correspond to elastic scattering in the given channel. Thereby most of the rapidly oscillating behaviour of the channel wavefunctions was extracted. For this reason, the rather slowly varying “occupation amplitudes” are also called “modulating functions” 16.51. This procedure was shown to be very similar to the one applied in the JWKBapproximation [44, 53). The resulting insights in the applicability and limits of the semiclassical approximation were discussed in Section IV. As a byproduct of this discussionthere appeared a possibility to simplify quantum mechanical computations by matching them to JWKB-solutions at large nuclear distances R and for large total angular momenta J. An analysis of the coupled channel equations (4.16~(4.19) revealed one possible mechanism for the appearance of structures in positron spectra: the resonance ampkjkation contained in the in- and outgoing nuclear relative wave functions if the two nuclei scatter through a resonance. Although a quantitative analysis of the effect

259

ATOMIC COLLISIONS WITH NUCLEARCONTACT

of this resonance amplification is not yet available, we found in Section V in a DWBA-calculation a directly related phenomenon, namely the strong variation of the nuclear phase shift over the resonance, to be responsible for structures in K-hole production probabilities. The paper is completed by a compilation of formulae to derive cross-sectionsfrom the occupation amplitudes (Eq. (4.21)) (and their semiclassical analogue) and by a derivation of the DWBA-limit in Section V. There we show how previous DWBAtreatments are contained in our theory. In particular we rederive an expression for Khole probabilities given by Blair and Anholt [ 141, but now for systemswhere one has to use molecular electronic wavefunctions. We point out that under crude approximations a similar procedure works, at least qualitatively, for the calculation of positron spectra in supercritical collisions, leading to expressions for the crosssection formally quite similar to the semiclassicalcalculation of positron spectra [ 8 ]. Work on a schematic model for this latter process, involving different and more accurate approximations to our general theory, will be published separately I72 1.

ACKNOWLEDGMENTS We gratefully Reus, and Dr.

acknowledge interesting M. Rhoades-Brown.

discussions

with

Professor

H. Feshbach,

Dr.

J. Reinhardt.

T. de

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GREINER

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ATOMIC

COLLISIONS

WITH

NUCLEAR

CONTACT

261

52. See, e.g., J. Y. PARK, W. SCHEID, AND W. GREINER, Phys. Rev. C 6 (1972), 1565; G. TERLECKI. W. SCHEID, H. J. FINK, AND W. GREINER, Phys. Rec. C 18 (1978). 265; D. HAHN, G. TERLECKI. AND W. SCHEID, Nucl. Phvs. A 325 (1979), 283. 53. C. GAUSSORGUES,C. LESECH, F. MASNOU-SEEUWS, R. MCCARROLL, AND A. RIERA, J. Phys. B 8 (1975). 239. 54. J. H. VAN VLECK, Phys. Rer. 33 (1929). 467. 55. A. R. EDMONDS. “Angular Momentum in Quantum Mechanics.” Princeton Univ. Press, Princeton, N. J., 1957. 56. M. A. PRESTON.“Physics of the Nucleus,” Chap. 16, Addison-Wesley, Reading, Mass., 1965. 57. M. SEIWERT. N. ABOU-EL-NAGA. V. OBERACKER, J. A. MARUHN. AND W. GREINER. “GSI Annual Report 1981,” GSI, Darmstadt, 1982; N. ABOU-EL-NAGA, J. MARUHN, 4ND W. GREINER. On the possible existence of giant nuclei, University of Frankfurt preprint. 1983. 58. M. J. RHOADES-BROWN. V. E. OBERACKER. M. SEIWERT. AND W. GREINER, 2. Phw. A 310 (1983). 287. 59. Experimentally only the angular momentum projection in the lab frame. not the one in the rotating frame can be fixed. 60. C. J. JOACHAIN. “Quantum Collison Theory.” Eq. (6.66) ff., North-Holland. Amsterdam. 1975. 6 I. A. R. EDMONDS, “Angular Momentum in Quantum Mechanics,” Chaps. 4.3, 4.6. Princeton Univ. Press, Princeton, N. J., 1957. 62. Our (j&O (n) is identical with (j&O (jLJ1) in Ref. 155 1. 63. K. ALDER, F. ROESEL. AND R. MORF. Nucl. Phys. A 284 (1977). 145. 64. M. ICHIMURA, M. IGARASHI, S. LANDOWNE, C. H. DASSO. B. S. NILSSON. R. A. BROGLIA. AND A. WINTHER, Phys. Lett. B 61 (1977). 129. 65. M. J. RHOADES-BROWN. M. B. MACFARLANE, AND S. C. PIEPER. Phys. Rer;. C 21 (1980), 24 17. 66. We thank Mark Rhoades-Brown for a discussion of that point. See also Ref. 1651. 67. N. FROGMANAND P. D. FR~MAN, “JWKB Approximation,” North-Holland, Amsterdam. 1965. 68. See. e.g.. Ref. 156 Eqs. (16~(37)). or Ref. 169 Eq. (12)). 69. C. GAUSSORGUES.C. LESECH, F. MASNOU-SEEUWS, R. MCCARROLL. AND A. RIERA, J. Phys. B 8 (1975), 253. 70. M. S. CHILD, “Molecular Collision Theory,” Academic Press, New York, 1974. 71. In this case convergence may be improved and computer time saved by using Pad& approximants (Mark Rhoades-Brown, private communication). 72. U. HEINZ. B. MUELLER,AND W. GREINER, in preparation. 73. Please note that, when comparing (5.11) and (5.18) with the corresponding expressions in Ref. 1I4 1, one has to correct for a relative factor i in the definition of the amplitudes B, (in Ref. [ 141 called b.,). This explains the relative minus sign in (5.10) vs the relative plus sign in the corresponding Eq. (14) in 114). Our convention here agrees with the one chosen in the semiclassical theory of Ref. 18). 74. The same is true if the spontaneous coupling H,, is included which again is diagonal in 1. Inclusion of the coriolis coupling, however, introduces d-dependence in the T-matrix. 75. See, e.g., A. BGHM, “Quantum Mechanics,” Chap. XVIII, Springer-Verlag. New York, 1979. 76. J. REINHARDT, Diploma Thesis. J. W. Goethe University, Frankfurt a. M., 1975. unpublished. 77. W. GHEINER. in “Quantum Electrodynamics of Strong Fields,” NATO Advanced Study Institutes Series B (W. Greiner. Ed.), p. 1, Plenum, New York. 1982. 78. K. SMITH, B. MUELLER. AND W. GREINER. J. P&s. B 8 (1975). 75. 79. T. TOMODA. Preprint HPI-H- 1983.V8, Heidelberg. 80. J. RAFELSKI. B. MOLLER, AND W. GREINER. Nucl. Phys. B 68 (1974). 585.