Inner-shell ionization and the Z13 and barkas effects in stopping power

Nuclear Instruments and Methods in Physics Research B4 (1984) 227-238 North-Holland, Amsterdam

INNER-SHELL George

IONIZATION

AND THE 2;

AND BARKAS EFFECTS IN STOPPING

227

POWER

BASBAS *

Fysisk Institut, Odense University, Denmark

The basic evidence for departure of energy loss phenomena from the Z: scaling law of the plane-wave Born approximation is recounted for bare projectiles with atomic number Z,. The several theoretical treatments of higher order contributions to stopping power theory are characterized. Even though there are strong suggestions that an appropriately cut-off contribution from distant collisions of order Z: and the Bloch term of order 2: are sufficient to account for the data with light ions, the status of experiment and theory does not provide a definite conclusion about the nature of the Z: term which contributes to the Barkas effect. The role of higher-order terms in inner-shell ionization as the primary origin of the Barkas effect is explained. An example of how inner-shell physics contributes higherrorder effects through a polarized K-shell wave function is given,

1. Introduction The Barkas effect is the difference in the range of heavy particles of opposite charge under otherwise equal conditions. For example, the range of Z- is measured to be 3% greater than the range of X+ at a velocity of 0.14~ [l]. Such charge-dependent effects are found in related phenomena; for example, II+ projectiles lose 14% more energy than do a- projectiles at a velocity of 0.05~ [2]. (This implies a greater range for the negative pion than for the positive one, as in the case of X* .) These sign-dependent effects diminish as the velocity increases. Since in the plane-wave Born approximation the energy loss of heavy charged particles is proportional to the square of the bare-projectile charge Zie, it cannot account for differences between oppositely charged particles. In this paper the atomic number Z, carries the sign of the charge. The next term in the Born series is proportional to Z,s and so has the required characteristic for describing sign-dependent phenomena. Barkas emphasized the significance of such a term. Focus on the Z: term as the source of the sign dependence proceeds under the tacit assumption that the Born series exists for these collisions and that other terms proportional to odd powers of Z, are negligible in the velocity regimes under study. (At low velocities the Born series is generally assumed to fail and other methods of calculating the consequences of collisions are required.) So far the theoretical descriptions which have evolved to account for the Barkas effect calculate a Z: term, but not necessarily in the formal context of a Born series.

* Permanent address: Physical Review Letters, P.O. Box 1000, Ridge, NY 11961, USA

0168-583X/84/.$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

Bloch [3] calculated a negative term to include with the Bethe stopping power formula. An even function of Z, with a high-velocity leading term of order 214, it alone cannot account for the Barkas effect. It is not the result of a Born series expansion but is expected to be subsumed in the Z: and higher even-powered terms of such an expansion. The observation of the Barkas effect signaled the start of a general study of the departure of energy loss phenomena from the Zf scaling law of the plane-wave born approximation. Andersen et al. [4] launched the era of the Z: effect in the stopping of light ions, bare and positive, with highly accurate measurements of the energy loss of hydrogen and helium ions which penetrated thin foils of aluminum and tantalum at the same velocity. In both targets they found the energy loss of helium ions to be greater than four times that for hydrogen ions in the velocity range corresponding to 2.5-7 MeV/u. The larger stopping cross section for helium nuclei implied by this result is consistent with the Barkas effect: The greater range of negative particles requires the coefficient of an assumed Zf term in the stopping power formula to be positive if there are no other higher order terms. Departures from the Z: behavior of the Bethe stopping power formula are now identified as the Barkas effect even though oppositely charged projectiles are usually not involved. These discrepancies are alternatively called Z: effects because the several theoretical accounts that have been developed to address this energy loss phenomenon treat terms to this order. Since the Z: term can be seen to arise from distortions of the target electron motion or wave function during the collision, it is also referred to as a polarization effect. In general the study by experiment of these developments takes place at velocities corresponding to a few

G. Basbas / The 2: and Barkas effects

228

MeV/u or more. These collisions are regarded as fast because the projectile velocities are above the mean orbital velocities of the target electrons. For a fixed projectile velocity this condition will be violated for inner-shell electrons if the target atomic number Z, is high enough. In these cases the collisions, even if not fast, are regarded as weak because the large charge of the target nucleus dominates the interaction with the inner-shell electrons over the ionizing interaction with the projectile as long Z, ci~ Z,. Accordingly the theories are constructed under these conditions. The main burden of the present paper is to emphasize that information and understanding of the Barkas effect is to be found in other atomic collision phenomena besides the stopping of charged particles. In particular departures from the Z: scaling law abound in inner-shell vacancy production. Much data exist to document it and a large theoretical effort has been brought to bear on its analysis. An important leader in this area has been Werner Brandt whose provocative views and inspired physical intuition have informed its modem development. In particular his marshalling of the Coulomb deflection effect [5,6] and binding [7] and polarization [8] effects has helped to ripen our comprehension of inner-shell vacancy production and make the Linz Workshops, for example, such interesting and fruitful activities. In sec. 2 a brief characterization of the theories and calculations of the 2: effect in stopping powers will be presented in order to highlight the interesting problems and controversies that have arisen and which might benefit from study of related inner-shell processes. Section 3 discusses the status of stopping power theory and experiment. Section 4 recalls how inner-shell excitation contributes to the stopping phenomenon and shows how the inner-shell Z: effect is propagated into the stopping power as the Barkas effect. Section 5 presents a report on an attempt to incorporate Z,-dependent effects into K-shell ionization by use of a trial wave function rather than a Born series to obtain binding and polarization effects. Section 6 outlines some possibilities for demonstrating the Z: effect in inner shells. Conclusions are given in sec. 7.

2. Calculations of the 2: tern in stopping power theory The nonrelativistic monly expressed as

_$-+~Z2L’

stopping

cross section

S is com-

4wZ2e4 mu2

where dE/dx is the stopping power, N and Z, are the target atomic density and number, u the projectile veloc-

ity, m the electron mass and L the stopping number. The structure of L is generally represented in the form L = L, + Z,L, L,=ln

+ ZfL,

+

... )

2mv2

I --C(u)/Z,,

where I is a mean excitation energy of the target, C(u) is the inner-shell correction and L, and L, are the coefficients of the Zf and Zf terms. Higher order terms in Z, from the implied Born series expansion are omitted. [If L, is taken to be given by the Bloch theory, then L, = - 1.202( v,,/v)~ is the high-velocity limiting form with u,, = e’/A.] The various L, depends on target properties but only on the velocity of the projectile, and not its mass or charge. The size of the L, and L, terms are typically several percent of L,. In the following survey of the calculations of the L, term, in chronological order, there is no intent to give a thorough account of the literature cited or to give a complete or a critical review of the work in this area. The purpose here is to outline what appears to this author to be the salient points of the various theoretical approaches to the Z: effect in order to orient the reader to the general situation and circumstances currently surrounding stopping power theory. The reader is referred to the original literature for detailed accounts of the various researches. The first calculation of the L, term in inelastic energy loss phenomena was made by Ashley, Ritchie and Brandt [9]. This seminal work remains the baseline for quantitative and qualitative insight about the Z: correction to the Bethe stopping power formula. They make a classical nonrelativistic calculation of the energy absorbed by a classical harmonic oscillator in the presence of a time-dependent Coulomb force which represents the passage of the projectile on a straight-line trajectory. The Z: term is obtained from the leading order of a large impact-parameter expansion of the excitation force. The expression is not valid for impact parameters which are not larger than the size of the amplitude of vibration of the harmonic oscillator. Ashley, Ritchie and Brandt assert that collisions at small impact parameters cannot transfer energy in amounts proportional to Z:. They reason that close collisions occur with such strength that the binding of the electron (whether as harmonic oscillator or in an atom) is negligible. Under this condition the target electron scatters like a free particle under the influence of the Coulomb potential of its interaction with the projectile. Since the Rutherford cross section which describes such a collision is exactly proportional to Z: to all orders, there can be no contribution of order Z: to the energy transfer A E(b). According to this argument small impact parameters need not be included in the integration of the Z: term over impact parameter b

G. Basbas / The Z,! and Barkas effects

to obtain

the stopping

cross section:

S = 2$AE(b)hdb =2?r j-mAE2(b)bdb+2?rj-mdE3(b)bdb. 0

(3)

4

Subscripts 2 and 3 denote the Z: and Z: terms. (Reference is omitted here to usual finite and nonzero upper and lower limits of the integration over impact parameters for the Z: term.) Integration over the Z: term can begin at some suitable minimum impact parameter b,. The value of b, is not given by the energy transfer calculation. It is characteristic of the size of the region in which the bound electron moves. Ashley, Ritchie and Brandt chose it in conjunction with a method of representing a real atom by the harmonic oscillator model for the bound electron. Their elementary result, before application to an atom, is given at high velocities by

where w is the oscillator frequency. With this Z: term, a standard statistical approach to the atom, and judicious choice of b, Ashley et al. account quantitatively for the excessive stopping of alpha particles over protons measured by Anderson et al. The cursory discussion above highlights two related features of the theory of the Zf effect whose scrutiny has motivated subsequent considerations of this problem. One is the appropriate value of b, and the other is the applicability of the reasoning which leads to the conclusion that small impact parameters do not contribute to the Z: term. Some subsequent treatments discussed below regard the electron as free (unbound) in an effort to infer contributions from close collisions. They incorporate the elastic energy transfer to the electron into the calculation of an electronic stopping power. This is distinct from the elastic scattering between projectile and target nucleus that contributes to stopping in a velocity regime much lower than the one under consideration. Jackson and McCarthy [lo] studied Z: corrections to energy loss and range. Following an unpublished calculation of Fermi, they exploit the relativistic version of Rutherford scattering, viz., Mott scattering which yields a Z: term, to arrive at contributions to elastic energy transfer from all impact parameters. Most comes from small impact parameters. This term vanishes in the nonrelativistic limit and is generally not significant in the velocity region where most atomic collision experiments have previously been carried out. Jackson and McCarthy also develop a relativistic version of the Ashley, Ritchie and Brandt Z: calculation of inelastic energy transfer under otherwise equal conditions. They make a choice for b, which is different from the one

229

Ashley et al. make. They set b, equal to the radius of the quantum mechanical harmonic oscillator which they take as a measure of the vibration amplitude of the target electron. The choice of b, is not given by either calculation of the energy transfer, only the need to make one. Andersen [ll] observes that this choice of b, also gives a good fit to the original data [4]. The Hill and Merzbacher treatment [12] is cast in terms of quanta1 time-dependent perturbation theory which describes the target electron as a quantum mechanical harmonic oscillator. They elucidate the formal techniques required for a systematic development and evaluation of higher order Born terms. The contributions of the various matrix elements and intermediate states are illustrated by taking advantage of the stringent selection rules for the harmonic oscillator. By considering the dipole and quadrupole terms of the interaction potential energy in the large impact parameter limit they retrieve precisely the formula of Ashley et al. which was obtained entirely by classical means. Their quantitative contribution to the calculation of the Z: term does not go beyond the result of Ashley et al. Hill and Merzbacher, however, find it not clear that at all but the vety highest impact velocities the close collisions can be safeIy ignored and state that the unimportance of close collisions cannot be taken for granted. They introduce and discuss the Bloch term, and note that since it is an even function of Z, it cannot produce a Z: term. By introducing properties of the medium in which energy loss occurs, in particular the free electron gas, Lindhard [13] argues that because of the screening of the projectile charge the close collisions (small impact parameters) are mediated by scattering from a Yukawa-like potential rather than a pure Coulomb potential. As a result there can be a term in the cross section of order Z: at small impact parameters, as Dalitz has shown for potential scattering [14]. Lindhard estimates that the contribution from the close collisions is about the same as that from the distant ones. This would produce a total Z: term whose value is twice that of Ashley et al. (or Jackson and McCarthy) and which would fail to agree with the Z: term inferred from the measurements. Lindhard points out, however, that the Bloch term, of opposite sign to the L, term, has the same strength as the Z: term of Ashley et al. (or Jackson and McCarthy) in the velocity range of interest even though it is of higher order in Z,. By including the Bloch term a description of the Barkas effect which agrees with the data would be recovered. To take into account the remark of Lindhard to include the Bloch term in analyzing the Barkas effect, Ritchie and Brandt [15] modify their original choice of b, and secure agreement with the original and new [16] data without appeal to the contribution from small impact parameters described by Lindhard. There has been no published account of the details

230

G. Basbas / The Zf and Barkas effects

of the calculation of inelastic energy transfer to order 2: for small impact parameters (but see Sung and Ritchie [17]). Esbensen [18] has isolated a contribution of order Z: to the energy deposition in an electron gas described by a dielectric response function. Contributions to this term from large momentum transfer are consistent with Lindhard’s estimate of contributions from close collisions. (See parethetical paragraph below.) Morgan and Sung [19] study the problem in the formalism of a quantum mechanical plane-wave Born series expansion. In order to deal with the sums over intermediate states in the second Born amplitude they use a closure approximation which introduces a parameter they evaluate as an average excitation energy of the intermediate states (not the mean excitation energy of stopping power theory). The sum over all excitations of the resulting Z: term does not yield to sum rules as in the case of the Bethe calculation of the Z: term. To proceed Morgan and Sung make the dipole approximation in order to deal with the atomic form factors which are generally unknown. This restricts their result for the Zf term to the small momentum transfers that are usually considered to include only contributions associated with distant collisions (large impact parameters). Their calculation thus uses a cut-off parameter as an upper limit to the integration over momentum transfer. Morgan and Sung chose it to be b;‘. Although similar in form, their results give distinctly larger values (by about a factor 2), than the values for the 2: term obtained by Ashley et al. (Impact parameter b and momentum transfer q are conjugate Fourier transform variables, as the connection between first-order time-dependent perturbation theory and the plane-wave Born approximation shows explicitly [20]. Consequently integration over b of the impact-parameter dependent probability amplitude gives the q dependence of the cross section. A small fixed value of q will generally allow for contributions from a wide range of impact parameters, not just large ones. A large fixed value of q restricts contributions to those of small impact parameters only. For the integration over q to obtain the impact-parameter dependence of the probability amplitude, the connection between q and b is the same as that described above with q and b interchanged. Contributions to close collisions (for example b = 0) are not automatically restricted to be only from large values of q. It is not true in general that b = l/q. At best this relationship gives the approximate upper limit to the integration over one variable to transfer to the other. These remarks apply to all the calculations and bear recollection when pondering the question of the contribution of a particular range of impact parameters to the Z: term. For example, Morgan and Sung include contributions from small impact parameters too in their calculation of the Zi’ term for small momentum transfer, but their significance cannot be determined.)

Arista [21] has calculated Z: corrections to elastic energy losses in scattering from an exponentially screened Coulomb potential by taking over results of the Dalitz [14] analysis of this potential. Relativistic Z: terms are included. Arista’s work is cast in terms of scattering angle (momentum transfer) and inferences are drawn about the relative contributions from distant and close collisions. By comparing his Zf result with the Bethe formula he concludes that some, but not all, of the contributions from distant collisions are included. He reproduces the Fermi correction applied by Jackson and McCarthy. In the nonrelativistic limit Arista obtains a close collision contribution to the Z: term because of the screening (cf. Lindhard [13] and Esbensen [18]). His expression is similar in form to the result from an inelastic collision with an harmonic oscillator at large impact parameters [9,10]. If a response frequency, w, of the target is introduced through the screening via the adiabatic distance v/w then the charof L, is recovered. Arista acteristic v- 3 dependence states that this approach is less appropriate for distant than for close collisions because of the effective cut off for distant excitations made through the adiabatic parameter. Sung and Ritchie [17] present a calculation of the energy lost by an ion passing through an electron gas. They developed a Z: term in a quanta1 formalism and treat separately the momentum transfers to distant and to close collisions. For distant collisions they obtain the same analytical result, apart from a multiplicative factor of 8/9, as the semiclassical treatment of Esbensen [18]. Their formalism shows that the interaction is screened for small momentum transfers. By taking into account the dispersion law dictated by the dielectric response function they obtain for large momentum transfers a nonzero but small contribution to L, of order ve4. Their use of a nonzero Fermi velocity for the target electrons is crucial to this result. Sung and Ritchie find in their calculation a confirmation of the idea that contributions from close collisions are unimportant. They attribute the nonnegligible result of others [13,18,21] for large momentum transfer to the use of a static screened Coulomb interaction which is valid only for low momentum transfer. 3. Discussion of the status of the Barkas effect This brief exposition shows the richness of the approaches to the issues surrounding a Z: effect. Andersen [ll] surveys higher-order Z, effects in stopping powers and concludes that the Lindhard program to include close collisions and the Bloch term is confirmed for light projectiles by the energy-loss data obtained via the accurate experimental methods currently available. He consideres what might be learned from heavy ion bombardment and remarks that the present theory is

231

G. Basbas/ The Z: and Barkaseffects

unlikely to be applicable because of the complications stemming from considerations of electronic structure and charge-state which attend these systems. Sung and Ritchie [17], on the other hand, find no significant contributions from close collisions and give arguments against the conclusions of Lindhard and Esbensen. Only a proper choice of the impact-parameter cut-off is needed for the distant-collision contribution to reach agreement with measured values of the stopping power [IS] (with the Bloch term included). The casual observer of these circumstances will conclude that both programs (those with and those without contributions from close collisions) obtain agreement with measurements. When the impact-parameter cut-off chosen by Jackson and McCarthy is used in the distant-collision calculation of Ashley et al., an equal contribution is required for close collisions. What is missing from the analyses is an explicit calculation for small impact parameters for inelastic energy loss to a single target (atom or harmonic oscillator). Results here would elucidate the considerations that enter in determining when a close collision takes place as if the target electron were unbound. Such a calculation might be capable of being joined to the large impact parameter Z: term to obtain a complete description without a parameter like b,. The data that have been brought to bear to date do not ‘prove’ there is a Z: effect, with or without a Bloch term. If stopping power data for hydrogen, helium and lithium projectiles are fit to a three-term polynomial in Z:, Z: and Zf, then values for their coefficients will emerge [16] whether or not such terms are correct (see below). The existence of a minimum impact-parameter cut-off for contributions from distant collisions and the debate over the contributions from close collisions create descriptions of the Barkas effect which the present data do not distinguish. Accurate stopping power data for more than three projectiles are needed to test the fit to the three-term polynomial. Perhaps the Z, dependence can not be described in terms of a polynomial expansion. The possibility that other theoretical methods are needed merits investigation. The reader is directed to the review by Inokuti [22] of the Bethe theory which gives the primary description of the atomic collision processes that are relevant to the phenomena under discussion. The addenda to the review contain a brief survey and commentary on the departure of stopping power from the Z: law of the Bethe theory. This evaluation of the status of the theories of this effect and the significance of the data remains of interest. Andersen et al. [16] fit a three-term polynomial in Z, [cf. eq. (2)] to their data and infer values for L, and L, with uncertainties of about 25%, as well as values for L,. This approach has the advantage of being independent of theoretical estimates of inner-shell corrections,

and L, in general. It is not a definitive test of the presence of the L, and L, terms because this method postulates their existence and then obtains their values from a solution of three equations (one for each projectile) and three unknowns (L,, L, and L2). This type of analysis can be used to establish any three-parameter function of Z,. A more revealing test can be made when data for more projectiles with different Z, values are available. As the values of Z, increases the effect of electron capture grows and may have to be taken into consideration in such an experiment. The overdetermined set of equations for L, and L, which results will give an indication of the adequacy of describing the data with L, and L, terms. A fit to a higher order polynomial would be expected to produce zero values for L, and L, terms, etc., if L, and L, suffice. If L, is taken as given by theory then the present data overdetermine L, and L,. This analysis would introduce into the considerations the concerns about values for I and the inner-shell corrections that attend the theory of L,. Porter and Jeppesen [23] have addressed the issues of close-collision contributions and the value of the impact-parameter cut-off b,. They fit the stopping power formula, eqs. (1) and (2), to data for protons and alpha particles stopping in rare gases by varying b, and a multiplicative strength parameter, 5, for the L, term. (Contributions from only the distant collisions correspond to t = 1; values of .$ > 1 signal additional contributions, presumably from close collisions.) They find no consistent pair of values for b, and t to describe all the data, even when the mean excitation energy I is varied. Andersen [ll] remarks that these difficulties may stem from the choice of inner-shell corrections used to obtain the fits to the data. Additional effects may require consideration: Sigmund [24] has made estimates of shell corrections to L, and L,. Next: the Z; term. It is unlikely that the leading Bloch correction gives the complete Zf term. Its potent strength in some data analysis might be weakened (or strengthened) by evaluation of the Zf Born series term. Continual appeal to higher terms would raise doubts about the usefulness of the expansion in the physical region of interest.

4. Connection between stopping power and inner-shell ionization: Inner-shell corrections and the 2: effect The stopping cc

cross section can be organized

by shells:

.... GK - Eo)%,o+f(EnL-Eo)o,Lo+ =ngo( K

*I.

(5)

G. Basbas / The 2: and Barkas effects

232

where E, denotes the energies of the unperturbed atom. the sum (or inThe summation index nk constraints tegration) to only those states with a K-shell vacancy, etc. The excitation cross section is an integral over the momentum change q of the projectile:

make q, small enough to satisfy q,,r < 1. Under these circumstances the oscillating exponential factor in the form factor can be expanded and approximated by a few terms:

e, do,, -ddq, %o = / 4. dq

]F,,(q)12=]?

(6)

where q, = (E, - E,)/tiu and qw = 2mv/h. The Z: term is found in a Born series expansion of du,,/dq, or of the corresponding differential cross sections for each shell. In the following development it is shown that the leading contribution to Z: term comes from inner shells. In the nonrelativistic plane-wave Born approximation [superscript (0)] the momentum-transfer cross section is

(7) where

is the atomic form factor and n and 0 denote the excited and ground unperturbed atomic states. The sum over j is made to include all the target electrons. The powerful sum rule obtained by Bethe,

E (En- Eo)lElo(q)12 =gz29

(9)

n-o

is crucial for the development of the stopping power formula [25]. To use it the division into shells is temporarily abandoned. The lower limit of the q integration, since it depends on n, interfers with the immediate application of the sum rule. Some advantage can be gained by division of the q integration into two parts and by use of the sum rule on the n-independent part to obtain an exact intermediate result in the nonrelativistic plane-wave Born approximation

2 +&z,&

1 .

4'

(10)

If the range of q in the remaining integral can be restricted to the small values which satisfy qr -SC1, an expansion of the form factor can be made. [Here r is the largest relevant value for the position coordinate of the atomic electrons in the ground state. Values much larger than one atomic unit are never relevant to the integration over coordinate space to calculate the form factor, eq. (8). The point is that r is bounded and satisfaction of the inequality lies entirely with q.] This inequality requires the projectile velocities to be high enough to

(n]I]O)+iq? J=l

(nIr,lO> J=l

=2

+f(iq)2

C (n[zfjO)

+ ... 12.

(11)

j-l

The first term is zero because of the orthogonality of the atomic states (n = 0 does not contribute), the second term is proportional to the dipole matrix element, etc. When only the first nonvanishing term is retained, as in Bethe’s calculation, the result is the dipole approximation [25]. Because of the complex arithmetic the next correction term to the dipole-approximation form factor is of order q4. It is at this point where the special contribution of inner shells enters the stopping power in the form of inner-shell corrections. The dipole approximation fails when q,,r Q: 1 is no longer satisfied. Since q, is largest for the inner shells the inequality fails first in the K shell for a given projectile velocity. The inner-shell correction appears through the term of order q4. If that term becomes important others may be also and a term by term evaluation is not necessarily the way to compute inner-shell corrections. For the purpose at hand it will suffice to study the q4 term in some detail to elucidate the emergence of the Z: effect when the second Born term is included. First we complete the derivation of the Bethe stopping power formula in the dipole approximation with the assistance of the sum rule, eq. (9), in the dipole limit, viz.,

Ef,=l,

(12)

n-o

where f, is the oscillator n.

Introduction

strength

for excitation

to state

of

lnI=

F f,In(E,-E,), (13) n-o where Z is an average excitation energy which depends only on properties of the target atom, gives

s(O) =

--.--Z, mu2 4aZ:e4

48Z2e4 =zZ, mv2

[ ln&+lny lnF-:oOq:]. [

4. 4 + “ZO 2 0jq’4”% (14)

1

G. Basbas / The 2: and B&as effects

Where the term of order q,’ appears the standard formula shows the general inner-shell correction (a sum over all shells) in the form C(u)& [cf. eq. (2)]. Note that not only is the Bethe formula obtained through the planewave Born approximation, there is an additional high-velocity limit in the form of the dipole approximation. The correction term to the dipole approximation, of order q,” = [(E, - Eo)/tiv]z, shows explicitly how the size of the correction depends on the shell: (.I?,,- E,) is much larger for inner shells than outer ones. Thus corrections which arise from this term are called innersheil corrections. They correct the In 0’ term to order I_-‘. Since (E,* - E,) > I,, the ioni~tion energy of shell s, the controf parameter can be written (Is/tu)‘. (Excitation to the narrow range of excited bound states just below the continuum is of no consequence here.) If 1, F fmoj is taken to define v, as an effective orbital velocity of s-shell electrons, the leading inner-shell correction has a control parameter proportional to ( u,/o)~. As the projectile velocity increases the importance of the outer shells diminishes [( u,~o)~ becomes small] and the primary corrections to the dipole approximation come from inner shells. Inner-shell corrections are calculated from ionization cross sections in the form of the shell stopping cross section, which expresses the full cont~bution to the total stopping cross section from ehch shell,

05) where duj’)/dW is the differential ionization cross section, in the Born appro~mation, for shell s and Wis the energy transfer. (Excitation to the narrow range of bound states below the continuum is neglected to simplify the discussion.) The notation is changed but the integral is equivatent to the sum over a given shell in eq. (5). In eq. (5) the ~fferenti~ cross se&on is exact in principle (no Born appro~mation~. Inner-shell corrections are obtained in the context of the Born approximation. It is in eqs. (5) and (15) that inner-shell collision physics enters stopping power theory (and the phenomenon); and it is through inner shells that the 2: effect first manifests itself in stopping powers to produce a Barkas effect:

This schematic formula i&&rates the Born series development for the shell cross section. The function T,(W) is independent of Z, and represents the interference term of the first and second Born amplitudes. This formulation is related to the definition of L,, cf. eq. (2),

233

where the sum is over all shells and the integral is understood to be extended to include sums over excited bound states if they contribute. The leading contribution to da,‘O)/dW is found in the dipole appro~mation. The next leading term is the inner-shell correction, as discussed above. The innershell correction applies to the first term (of order Z:) in eq. (16). The leading contribution to the second term also comes from inner shells (see below); it is of order Z: and has a physical origin in inner shells which is different from the physical origin of the ‘inner-shell corrections’. In the same sense that the correction C(u), eq. (2), to the dipole approximation is an inner-shell correction, the 2: correction to the Bethe stopping power formula for an atom is an inner-shell correction. Such shell cont~butions are not inner-shell corrections to the ZT term, they are the 2: term. The following discussion elucidates this role of the inner shells in the 2: effect with atomic targets. This is illustrated by examining the distant-collision part of L, which is calculated by all inelastic treatments to be proportions to e2w/u3 (see eq. (4)). The illustration adopts the point of view that each inner-shell electron acts like an harmonic oscillator. (Indeed this point of view is used to combine the elementary L, term, with which we now deal here, with a statistical description of the atom to obtain an averaged L, term for the contribution of the entire atom to the stopping power [9,15].) Since fro represents the energy transfer (analogous to I&, - Z&E,), it can be taken as proportional to v,“. The essential velocity dependence of L, is (u~/u)(u~,/o)~, Just as the (u,/u)~ controI parameter constraints the correction to the dipole appro~mation to be an inner-shell correction, so the relative velocity dependence of the Z: term, a high-velocity effect, identifies its primary origin in the inner shells. Standard inner&e11 corrections are of order Zf. The Z: term is identified here as an inner-shell correction only for pedagogical reasons. Continued reference to the Z! term as an inner-shell correction may confuse two different physical wncepts and create misunderstanding. The exposition here is heuristic and made only for the purpose of this section which is to emphasize that in inner shells lies the origin of the Barkas effect. We conclude that one can I& about the 2: effect or Barkas effect in stopping powers by studying the Z, dependence of inner-shell w&ion phenomena. In particular, at high projectile velocities the K shell is the leading contributor to the higher-order Z, dependence of the stopping power. Perhaps by narrowing the arena of consideration to the K shell one can make an ex-

234

G. Basbas / The 2: and Barkas effects

amination of the issues surrounding the 2: term with a detail that has not been possible previously. (Historically it was the application, by Werner Brandt, to the K shell of the Z: term he helped develop for stopping power theory that gave the original explanation of the Z, dependence of inner-shell vacancy production. The suggestion now is to reverse the direction of application.) The large impact parameter studies of the harmonic oscillator are analogous at their elementary level to a K-shell program since they deal with a single target electron rather than an entire atom. The primary theoretical treatments make direct contact with single-electron events in the laboratory. Collision experiments with inner-shell vacancy production can single out electron processes while stopping power measurements do not. For example, experiments with K-shell electrons are likely to be free of plasma screening effects which have been argued to produce a Z: term. If screening effects do exist they might be more easily treated in the well known K shell than in the many-bodied plasma. The K shell allows simplifications of theory and access to detail in laboratory measurements that could prove beneficial. The absence of a comprehensive theory need not preclude experiments with inner-shell electrons which will elucidate the phenomena. Consider coincidence experiments which isolate the small impact parameter region in K-shell ionization and mimic the studies of stopping power by surveying the Z, dependence by use of different projectiles: Do the excitation processes in close collisions follow the Zf law of Rutherford? (Do collisions at small impact parameters proceed as if the electrons were free? How does inner-shell vacancy production in close collisions depend on the target atomic number?) Does the intensity of electrons ejected from inner shells show departures from a Z: law? Cross sections for inner-shell photoionization are intimately related to the differential energy-transfer ionization cross sections for proton bombardment. Proper comparison might reveal the Z, dependence beyond the Z: law.

5. Polarization in K-shell ionization A beginning study of the Z, dependence of inner-shell phenomena which could give rise to a Barkas effect in stopping powers is presented here. Since the theory [26,27] in this section was developed for a calculation of K-shell ionization cross sections to account for basic collision experiments, the formulation is not developed in the context of a Born series and a Z: term is not obtained explicitly. (Neither is one obtained by experiment.) The Z, structure that results is complicated but may contain Z: and 2: terms in a series expansion, or if fit by a polynomial. Since the upper limit of the projectile velocity range in the ionization experiments is

not large (but it does overlap with the lower range of the stopping power data) this theory contains adiabatic aspects which may distinguish it from the theories developed for stopping powers. The discussion of the Z: or polarization [12] effect in stopping powers, and how collisions with inner-shell electrons give a primary contribution, leads to the consideration of the polarization effect in K-shell ionization. Historically a description of the K-shell polarization effect [8] was obtained from the Z: term calculated for stopping powers [9]. A Z: contribution to the excitation probability was inferred from the energy transfer formula and then by Ansatz made to combine with the previously developed term for the binding effect. Recently a polarized K-shell wave function, featuring a correlation term with a charged particle at a fixed arbitrary distance from the the target nucleus, was constructed [26] to use in adiabatic semi-classical calculations [27] of K-shell ionization by projectiles at velocities less than the mean velocity of the target electron. The one free parameter, which sets the strength of the correlation of the target electron with the projectile, is chosen to minimize the ground-state expectation value of the total Hamiltonian, viz., the sum of the unperturbed atomic Hamiltonian plus the potential energy of interaction with the projectile. The polarized wave function in atomic units is q,+,,(r;

R)=N(R)[e-Z1iR-ri+f(R)]

\k(Z,),

(18)

where R and r are the projectile and electron coordinates, f(R) is the variational parameter and ‘k( Z,) is the unperturbed ground state atomic wave function with unit normalization. The factor N(R) is chosen to give unit normalization of ‘k,, (r ; R). The variational principle guarantees that f(R) [and N(R)] will assume values to make qpO,( r; R) exact in the united (R = 0) and separated (R = w) atom configurations. This theorem, obtained because the trial wave function has the exact form under those limits, is strictly true at R = 0 only when 9(Z,) is hydrogenic, and nearly true for a nonhydrogenic JI(Z,), for example a Hartree-Fock *k( Z,). (In the nonhydrogenic case the separated-atom form of qpO, (r ; R) is exact.) The united atom form is qk,,( r; 0) = N(0)e--Zl’~( Z,). This is a reasonable approximation to the exact united-atom wave function *k( Z, + Z,) because of the general hydrogenic character of K shells. This point is important to note because as long as this approximation is acceptable, the polarized wave \k,,,( r; R) can be used in conjunction with a variety of forms for ‘k(Z,). The theorem for exact forms is not obtained for linear combinations of atomic orbitals. For a hydrogenic unperturbed K-shell wave function, *(Z,)

= 2Z2/2e-Z2’/&.

09)

G. Basbas / The Z,! and Barkas effects

f( R ) and N( R ) can be determined analytically in a straightforward manner from the variational principle. Tedious manipulations produce ungainly but readily programmable formulas. When q(Z,) is in numerical form it is necessary to compute f(R) and N(R) numerically for each value of R for a given Z, and Z,. As an alternative in this case, the values of f(R) and N(R) obtained from a hydrogenic 9( Z,) might prove a suitable approximation that does not preclude the use of a numerical @(Z,) elsewhere in the calculation, for example in the determination of form factors. The following discussion shows how use of the polarized wave function leads naturally to the usual atomic form factors. A typical calculation for an atomic collision process will require the matrix element

where f denotes the final unperturbed electron state of interest, q specifies the electronic state from which the transition takes place, and I/=

-Z,e2/(R

- r]

(21)

is represented in the form of a Fourier integral. If e denotes an unperturbed atomic state, then (f]e*“]*) is the atomic form factor F(q). When q is qrO, (r ; R) a noteworthy computational attribute of the polarized wave function emerges: The atomic form factor is recovered. All previous routines to calculate and employ it in the evaluation of the matrix element can still be used:

= -Z,e2

I

-e

dq

-ig.R

w3

X4-n

(22) The following theorem obtains: In the context of the transition matrix element expressed in terms of the Fourier transform of the (Coulomb) interaction potential energy the prescription to introduce polarization is (23) The atomic form factor is unaffected. Recall that in principle the values of f(R) and N(R) depend on the particular choice of *(Z,); in practice this may not be important and the analytical formulas obtained from a hydrogenic !P(Z,) might suffice. The only increase in complexity of calculation is the additional R dependence introduced via f(R) and N(R). If the calculation without the polarization effect proceeds

235

by numerically integrating over R (or time and impact parameter), it is trivial in principle to include the polarization effect in the computer code; a change of one statement [cf. eq. (23)] will suffice once the simple routines to calculate f( R) and N(R) are supplied. With regard to the Z: effect discussed in sec. 2, and the contribution at small impact parameters made possible by screening the interaction potential, we note the appearance of the Yukawa potential in the expression for the matrix element, eq. (22): v~-=IIR-~

-Z,e'

= -e-=,lR-~l (R-t-1

The correlation term in the polarized wave function is in the form of a K-shell wave function centered about the projectile. Accordingly it self-screens the interaction of the projectile with the target K shell. A conclusion that, because the interaction is apparently weakened by the screening, the cross sections are reduced, is premature (and false). Inspection of the matrix element, eq. (22), reveals two ‘interaction’ terms, one screened and one weighted by (positive) f(R). The terms add. If there were only a screening term, then the conclusion above is warranted. Whether the other term compensates for the reduction due to screening is not obvious upon inspection because of the weight of f(R) [as well as the presence of N(R)]. The previous explanation [8] of the effect of polarization on the ionization cross sections argued that because the electron is drawn away from the target nucleus by the projectile its binding to the K shell is weakened; in consequence the ionization cross section is increased. This prediction for the cross section is reproduced by use of the polarized wave function. The explanation that polarization weakens the binding energy, however, now seems glib. The binding energy given in applying the variational principle increases over the value for the unperturbed atom. The polarization of the wave function is not reflected in a decrease of the binding energy. In fact the binding energy as a function of R is precisely the same (by numerical comparison) as the one obtained by Land [27] in making the “binding correction” (an increase in binding) via a trial wave function of the form \kbinding= 2[ Z( R)]3’2e-Z(R’r/&,

(25)

where Z(R) is the variational parameter. This wave function, also exact in the united and separated atom limits, is not correlated to the projectile in the sense that there is no polarization distortion term. Always isotropic, the wave function merely tightens about the target nucleus at it unites with the projectile. There are several conclusions to be drawn from the use of the polarized wave function: (1) Polarization does not effect binding. Distortions in the wave function are not correlated to changes in the binding energy.

236

G. Basbas / The Z/ and Barkas effects

(2) The increase in the ionization cross section due to polarization arises from the changes in the wave function and its influence on the matrix element. (3) This work and the work of Land [27] constitute a functional variation to obtain the minimum energy. That two different wave functions give the same energy suggests a functional minimum. (4) The well known limitation of variational wave functions is explicit here: The ability of a trial wave function to produce the minimum energy may have no bearing on its suitability for other purposes. The two variational wave functions described here have distinctly different physical content, viz., their descriptions of a correlation with the projectile. Success at minimizing the energy does not indicate which to use in calculating the ionization cross section. Whether the trial wave function employed here is physically realistic as a description of the higher-order-Z, perturbing effect of the projectile on the ionization process cannot be determined from the variational principle. Comparison with results of experiment and sensible physical judgement are needed. (5) The physical character of the polarized wave function bestowed by the energy-minimum principle through the determination off(R) is reasonable in light of the resulting strength of the correlation with the projectile. The value of f(R) is zero for R = 0 and increases monotonically with R to produce cross sections with intelligible physical properties. One of the attributes of the polarized wave function described here, eq. (18), is that it can be used to calculate cross sections, as Land has done (271. The Z, dependence of the cross section is clear but the analytical structure is opaque: no Z: term is explicit. The K-shell ionization cross sections calculated with the polarized wave function, and including the binding effect (and others), show that polarization improves the ability of the theory to describe the measured cross section. Comparison of the cross section for ionization from the state given by the polarized K-shell wave function with the treatment by Brandt and coworkers [8] does make indirect comment on the Z: term under discussion in stopping power calculations: (1) The effect of polarization in the two calculations [8,26] agree for K-shell ionization of titanium by protons in the 0.3-2.4 MeV energy range. This increases the confidence in the cross section values obtained via the Ansatz for a polarized binding energy employed In the earlier theory [8]. (2) In K-shell ionization the polarization effect obtained from the Z: term in stopping power employs a minimum impact-parameter cut-off. At the low velocities for which only small impact parameters contribute to K-shell ionization, the polarization effect calculated with a cut-off disappears rather abruptly. Calculations

with the polarized wave function show the effect of polarization to continue smoothly into the low velocity regime. Even at the lowest velocities where only small impact parameters can contribute, polarization enhancement of the cross section persists in the form of the multiplicative factor [N(0)12 which represents the increase in electron density at the origin in the united atom limit. The study of the polarization effect at high velocities via the trial wave function is currently restricted by the failure of the adiabatic description of collisions to pass to the impulse limit at higher velocities; the polarized wave function does not stay ‘rigid’ at the separated atom limit as required to produce the unperturbed atomic wave functions called for in the plane-wave Born approximation. Yet the stopping power data span projectile velocities and targets which can place in the adiabatic regime the inner-shell electrons which are regarded here as the primary source of Z: corrections. A final point to be made in this discussion is how to calculate a polarization effect for negatively charged projectiles (antipolarization). Replacement of the K-shell state bound to the projectile (correlation term) by a repulsive Coulomb-scattered wave function produces a logical analogy. This approach is fraught with the difficulty of managing such complicated functions and the fact that such a correlation term would not disappear in the separated atom limit. A more tractable procedure is to make the correlation term negative. This is equivalent to changing f( R) to -f(R). [This gives a new formula for f( R) via the energy minimization principle, not - 1 times the values for a positive charge.] Where the polarized wave function for protons, for example, shows a cusp-like exponential peak on the broad shoulder of the target K-shell wave function at c = R, the one for antiprotons would show its reflection about the shoulder, viz., a cusp-like dip. A decrease in binding is expected to occur in passing to the united atom limit, and from the correlation in the wave function (antipolarization) a reduction in the ionization cross section. (The binding decrease will increase the cross section.)

6. Search for Zf and Barkas effects in inner-shell ionization There is now promise that a bona fide Barkas effect (in the original sense of oppositely charged projectiles with the same mass) might be detected in K-shell ionization. The advent of antiprotons at the CERN accelerator facility, and in particular the development of the ELENA low-energy electron-cooled storage ring, means that beams of these particles will become available at low energies (perhaps down to 500 keV) for atomic physics experiments. Circumstances are ripe for the detection of differences in the X-ray or Auger-electron

G. Basbas / The 2: and Barkas effects

production by protons and antiprotons, analogous to the situation that confronted Barkas in Z* range differences and B * energy-loss differences. Enormous differences are predicted [6] for such experiments but in an energy range which is not currently accessible with antiprotons. When antiprotons are available in the range of several MeV a test can be made of the Z: effect that may be more stringent than that provided by stopping power measurements. In this energy range it should be possible to select targets, for example aluminum or argon, which do not evince deflection and binding effects but for which the polarization effect [8,26] gives rise to departures from the Z: law. We are in a position to exploit the phenomenon because of the insight gained from the development of the Z! term to describe stopping powers, extended to particles of different positive charges, and because of a well documented basis for understanding the Z, structure of inner-shell ionization. Besides the extensive experimental and theoretical work on the Z, dependence of inner-shell vacancy production by a range of positive light ions, we have Reading’s work, reported in the last workshop 1281, on excitation of hydrogen by p, p, e * , n * and p * . Werner Brandt’s last paper (61 predicts a huge excess of innershell vacancy production by antiprotons over protons in the energy range of planned CERN beam. The dominant contributor to this difference is the Coulomb attraction of the target nucleus for the antiproton, although ‘anti’-binding effects contribute too, as Reading has also noted. Higher energy experiments, with antiprotons, near the cross section peak are also of especial interest. As Reading has pointed out antiprotons do not capture electrons. Measurements of K-shell ionization directly to the target continuum, free of the contamination of charge capture to projectile bound or continuum states, will provide definitive data for studying the polarization effect without the competition from electron capture. [Of course a final state repulsive interaction between ejected electron and antiproton is possible. How would this manifest itself? Will backward-running convoy electrons be discovered? (No.)] Comparison (by ratios) of measurements of X-rays (or Auguer electrons) produced by antiprotons with those produced by protons at the same velocity near the cross section peak would reveal a Barkas effect in inner-shell ionization. A test of the polarized wave function would be provided, and insight into the problems of stopping power analysis obtained. Two requirements in such a program are the need to discriminate carefully against electron capture by the protons, and the solution of the theoretical problem created by use of adiabatic states at high velocities.

23-l

7. Summary Barkas and coworkers established that heavy particles with opposite charge lose energy at different rates when penetrating matter. If this difference is assumed to arise from a term of order Zf in the stopping power formula, then the data indicate its coefficient is positive. Andersen and coworkers made accurate measurements of the energy loss in thin foils by bare hydrogen, helium and lithium ions at the same velocity. They established the departure from the Zf scaling law of the Bethe stopping power formula. The energy loss rate for these projectiles is higher than the 2: prediction; this also implies a positive coefficient for a 2: term (but not that one exists). The components of the theoretical description that control its comparison with the data are the minimum impact-parameter cut-off for the contribution of energy loss distant collisions, the contribution from close collisions and the Bloch term of order 2; which is currently taken to define L,. These components can be arranged in two ways (with and without contributions from close collisions) to obtain agreement with the value for L, inferred from the measurements. The data, which do not unequivocally establish L, and L, terms, cannot distinguish between the two theoretical possibilities. Sung and Ritchie [17] have given an analytical development to confirm the original argument [9] against contributions from close collisions. If this description, which contains only contributions from distant collisions to L,, is embraced, the theoretical value for L, still depends on the cut-off parameter b, which can be determined only by appeal to measurements (which themselves are not decisive). A theoretical treatment following the lead of Sung and Ritchie to obtain a description of the L, term which refers to all impact parameters without the need of a parameter to be determined from experiment is required to advance the knowledge of the stopping phenomenon. More extensive measurements are demanded to define the higher order terms. Energy loss data for an extended range of projectile atomic numbers as well as experiments on inner-shell processes, for example antiproton vacancy production or the Z, dependence of ionization as a function of impact parameter, will help provide decisive information about higher order terms in the basic atomic collision physics. The hospitality of the Fysisk Institut at Odense University, gratefully enjoyed, helped make this study possible. Financial support from NORDITA is appreciated. Many thanks are due to P. Sigmund for criticism, encouragement and inspiration.

238

G. Basbas / The Z,! and Barkas effects

References [l] W.H. Barkas, J.N. Dyer and H.H. Heckman, Phys. Rev. Lett. 11 (1963) 26. [2] H.H. Heckman and P.J. Lindstrom, Phys. Rev. Iett. 22 (1969) 871. [3] F. Bloch, Ann. Physik 16 (1933) 285. [4] H.H. Andersen, H. Simonsen and H. Sorensen, Nucl. Phys. Al25 (1969) 171. [5] W. Brandt, R. Laubert and I. Sellin, Phys. Rev. 151 (1966) 56. [6] W. Brandt and G. Basbas, Phys. Rev. A 27 (1983) 578. [7] G. Basbas, W. Brandt and R. Laubert, Phys. Rev. A 7 (1973) 983. [8] G. Basbas, W. Brandt and R. Laubert, Phys. Rev. A 17 (1978) 1655. [9] J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. B 5 (1972) 2393. [lo] J.D. Jackson and R.L. McCarthy, Phys. Rev. B 6 (1972) 4131. [ll] H.H. Andersen, Phys. Ser. 28 (1983) 268. [12] K.W. Hill and E. Merzbacher, Phys. Rev. A 9 (1974) 156. [13] J. Lindhard, Nucl. Instr. and Meth. 132 (1976) 1. (141 R.H. Dalitz, Proc. Roy. Sot. London, Ser. A 206, (1951) 509. [15] R.H. Ritchie and W. Brandt, Phys. Rev. A 17 (1978) 2102. [16] H.H. Andersen, J.F. Bak, H. Knudsen and B.R. Nielsen, Phys. Rev. A 16 (1977) 1929.

[17] CC. Sung and R.H. Ritchie, Phys. Rev. A 28 (1983) 674. [18] H. Esbensen, Thesis, University of Aarhus, Denmark (1976) unpublished. [19] S.H. Morgan and CC. Sung, Phys. Rev. A 20 (1979) 818. [203 D.H. Madison and E. Merzbacher, in Atomic inner-shell processes, vol. 1, ed., B. Crasemann (Academic Press, New York, 1975) pp. l-72. [21] N.R. Arista, Phys. Rev. A 26 (1982) 209. [22] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297; Addenda: M. Inokini, Y. Itikawa and J.E. Turner, Rev. Mod. Phys. 50 (1978) 23. [23] L.E. Porter and R.G. Jeppesen, Nucl. Instr. and Meth. 204 (1983) 605. [24] P. Sigmund, Phys. Rev. A 26 (1982) 2497. [25] For a derivation of the sum rule see: Intermediate quantum mechanics, H.A. Bethe and R. Jackiw, 2nd ed. (W.A. Benjamin, New York, 1968) p. 304. For a standard derivation and detailed discussion of the stopping power formula see U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1. [26] G. Basbas and D. Land, to be published; see also G. Basbas, in the report of the NYU Workshop, Hawaii (1982). [27] D.J. Land, M.D. Brown, D.G. Simons and J.G. Brennan, Nucl. Instr. and Meth. 192 (1982) 53. [28] J.F. Reading, A.L. Ford and M. Martir, Nucl. Instr. and Meth. 192 (1982) 1.