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Two-center electron-electron interaction in fast ion-atom collisions
Nuclear Instrumentsand Methods in Physics Research B53 (1991) 498-503 North-Holland
498
Two-center electron-electron
interaction
in fast ion-atom
collisions
W.E. Meyerhof and H.P. Hiilskijtter Department of Physics, Stanford University, Stanford CA 94305, USA
The interactionbetweena target electron and a projectileelectron in one-electron projectile excitation or loss is examinedon the basis of the PWBA and demonstratedexperimentally.
1. Introduction During the last decade, the subject of electron-electron correlation has received increasing attention by atomic theorists and experimenters [l]. McGuire [2] has discussed electron-electron correlations in terms of static correlations present in the initial or final atomic states and dynamic, or scattering, correlations induced by the dynamics of the collisions. Also, Stolterfoht [3] has extended these concepts to two-center systems, such
as a collision between a projectile ion and a target atom. A certain part of the Coulombic interaction between a projectile electron and a target electron is called “twocenter scattering correlation”. The simplest case to illustrate these ideas is that of two nuclei (2, Z’) in relative motion and two electrons (e, e’), as sketched in fig. 1. Each electron may go from some initial state (i, i’) to some final state (f, f’) or stay in the initial state. A necessary, but not sufficient, condition for recognizing a correlation effect is that at least two electrons change their state [3]. In the present case, this implies that correlation requires at least f + i, f’#i’.
In terms of the situation sketched in fig. 1, a dynamic scattering correlation is produced or induced by the time dependence of the internuclear vector R. The effect of such a correlation on a particular cross section
Y
z
4 %t)
i
e
l
.
(i-at)
z’
-
Cc) (b) (a) Fig. 2. Possible groupingsof the electrons in fig. 1, according to ref. [3]: (a) one-center, (b) two-center, (c) bi-centric; the origin of ri, r, is the center of charge is indicated by a cross. The Perturbingdynamical interactionsare indicatedby dashed lines.
depends on the grouping of the electrons. Stolterfoht mentions three groupings, shown in fig. 2: (a) a onecenter case in which the two electrons initially are associated with one center, (b) a two-center case in which each electron initially is associated with one center, and (c) a bi-centric case in which both electrons are associated with both centers, as in the molecularorbital (MO) model. In fig. 2, the perturbing time dependent potentials are indicated by dashed lines. Only in the two-center case is a perturbing potential, namely that between the electrons, directly related to the dynamic electron-electron (e-e) correlation. In the two other cases, the dynamic e-e correlation is induced by the electron-nuclear perturbation, as shown by McGuire [2]. The two-center case appears to be the simplest (and, hence, according to Reading [4], the least interesting) case of a scattering correlation, and we discuss it alone.
e’ (i’->f)
Fig. 1. Two-nucleus and two-electron system. The relative projectilevelocity v is indicated,as well as the transitionsi - f and i’ + f’ of the electrons e and e’. The internuclearvector R(t) changes with time and induces dynamic correlations betweenthe electrons. 0168-583X/91/$03.50
Z
z’
2. Theory: Born approximation for two-center scattering Consider fig. 2b. The cross section for elastic or inelastic electronic processes is given by [S]
u = M2/(4n2tt4)x
0 1991 - Elsevier Science PublishersB.V. (North-Holland)
/ 1(I I* dQ. fin
(1)
499
W.E. Meyerhof, H. P. Hiilskiitter/ Two-centere-e interaction
For simplicity, we ignore the slight change in projectile velocity in inelastic processes. The quantity M is the reduced mass of the system. The summation is over all relevant final states f and f’, which, in the case of ionization, implies integration over the electron kinetic energies E and e’ of the electrons e and e’. In the plane-wave Born approximation (PWBA), the scattering amplitude a is expressed in terms of the momentum transfer Aq which is the vector difference between the incident and scattered c.m. particle momentum AK,, and AK,,,. Using the relation
and ionization of hydrogenic ions [6,7]. For a given projectile process, the summation in eq. (3) is over the final states f’ of the target. This yields
Pa)
e,/ = %L%” + %X, 9 where = 8ne4/( *screen
h2vZ)~~~~,~,~‘(dq/q3)
1F(ls -+f)
X/Z’-F(ls’+ls’)(2
I2 (9b)
and q’=K;+K=
out- 2KinKout COS0,
(2)
one can conveniently replace the integration over the c.m. scattering solid angle dS2 = 28 sin 13 d8 = 27r qdq/Kk in eq. (1) by an integration over q: u = 1/(2nfi2u2)
c j 1a ) 2q dq,
(3)
fin
where v is the projectile velocity. For the case of fig. 2b, the scattering amplitude in first order PWBA is given by
[61
a - /dR
exp(iq*R)//
drdr’~,(r)~,,(r’)l/cp,(r)
X+(r’),
(4)
where the perturbing
potential
V is
V=e2[ZZ’/R-Z/IR-r’l-Z’/IR+rl +l/lR+r-r’l].
(5)
Because of orthogonality of +, and +f or I#+, and +,, in any inelastic process, the first term in eq. (5) does not contribute to a in an inelastic process. We now consider projectile excitation or electron loss. Then, the orthogonality between $, and 9, eliminates also any contribution from the second term in eq. (5). Using the Bethe integral [7]
/
dR exp(iq*R)/IR-rI
=(4a/q2)
exp(iq*r), (6)
one can show that [6] a = (4a/q2)F(ls
+f)[
z’+
- F(ls’
+‘)I.
(7)
Here, we assume that each electron initially is in a 1s state and express a in terms of the form factor
F(ls -f)
= jd rexp(iq.r)~,(r)9,,(r)
(8)
and similarly for F(ls’ -f’). The form factors are real, so that the sign of the argument of the exponent is of no importance. They have been evaluated for excitation
x I F(ls
-f)
I = I F(ls’
-f’)
12.
PC)
In eqs. (9b) and (SC), q&i’ or f’) is the minimum momentum transfer needed to excite electron e from i to f and e’ from i’ to i’ or f’. For heavy particles, qmax is typically set to infinity. The first term in eq. (9a), called u,,,,, is the cross section induced by a screened target nucleus. The second term, called a,,,, is due to the electron-electron interaction: it can be traced back to the last term in eq. (5). At various times, this part of the cross section has been called “inelastic” [6,8], “incoherent” [9], “antiscreening” [lo], “two-center scattering correlation” [3], or we can call it “electron induced correction to the screened nuclear cross section”, in order to leave the subscript “corr” somewhat neutral. The partial cross section a,,, fulfills the correlation condition that two electrons make a transition, albeit as a result of a single Coulombic interaction. (This is not the case for u,,, where only one electron makes a transition.) The question now arises whether the two electrons make the transition independently or not. As pointed out by Bates and Griffing [6], the target electron must make an inelastic transition ( f ’ # i’) in order to induce the inelastic transition i +f in the projectile. This is a consequence of momentum conservation. Therefore, the transitions are not independent and u,,,, can be considered as being due to a two-center scattering correlation [3]. Before turning to experimental results, we note that the expression for a,,, can be simplified if qmin(f’) is not very dependent on f’. This occurs for Z’ < Z. Then, the summation over f’ in eq. (9) can be put inside the integral. Using closure [7], c
I F(ls’-f’)
I 2 = 1,
f’
one can rewrite
*,,, =
[6]
8ne4/(tr2u2)Jq~.(f,)(dq/q3) I F(ls -f) I2 x(1
- ]F(ls’+
1s’) 12).
(IO)
W.E. Meyerhoj H.P. Hiilskiitter / Two-center e-e interaction
500
As noted by Anholt [7], since cr,,, is due to the electron-electron interaction, there must be, in the limit of free target electrons, a threshold effect: uarr = 0, unless (nonrelati~stic~ly) (1/2)m&?
> I,
(11)
where m, is the electron mass and I the projectile 1s ionization energy. In eq. (SC), this threshold effect is effectively included in the dependence of qmin on f ‘, but is lost in eq. (10). It can be reinstated there through an appropriate multiplicate factor [ll] or by setting qmin(f’) equal to the minimum momentum transfer needed by scattering a free electron of velocity u on the projectile so that the transition i + f is induced [12].
3. Experimental results Even though eq. (9a) has been known since 1953 [6], experimentally the existence of the term a,, had not been demonstrated until very recently. In particular, the threshold effect, mentioned above, had not been clearly apparent in previous excitation or ionization measurements [13]. 3.1. Projectile excitation By scattering 4-35 MeV Li-like 05+ and F6+ ions on H, and He gas targets, Zouros, Lee, and Richard [14] have been able to detect the (2s22s)*S + (ls2~2p)~P excitation process in these ions. In this particular transition, the effective 1s --f 2p excitation cannot be produced by direct nuclear scattering because a spin flip must occur. Hence, this transition selects the pure a,, part of the cross section in eq. (9a), here due to electron-electron exchange [which is not included in eqs. (9)]. Other, more complex two-step processes can also produce the ‘S -+ 4P transition, but can be separated energetically [14]. A threshold effect, smeared out by
a1(a) PROJECTILE
1
ENERGY (MeV)
12,‘,,,,,,,,Li,l‘lfll,l(
1
10
Projectile energy [Mev/Nl
‘0
4
8
12
16
~JECTILE
x)
24
28
ENERGY (Me@
Fig. 3. Data: Cross sections for the production of ls2s2p4P states by 1s + 2p projectiie excitation in collisions of (a) OS* and (b) of F 6+ (1~~2s) projectiles with He and H2 targets vs projectile energy. Only statistical errors are shown. Total absolute error is - 30%. Calculation: electron-electron excitation (eeE) cross sections using ls2s2p4P theoretical electron impact excitation cross sections folded by the Compton profile of the target. Dashed lines, calculated eeE for H2 targets; dash-dotted lines, calculated eeE for He targets. Arrows (at 16.3 and 25.0 MeV) indicate the projectile energies corresponding to the threshold for Is + 2p electron-impact excitation in 05+ and F6+, respectively. Cross sections at lower energies are due to two-step processes (ref. [14]).
4 1
10
Projectile energy [MeV/Nl
Fig. 4. Electron loss cross section for (a) Li2+ +H,, (b) Liz+ + He. In each case, the solid line gives eq. (9a) evaluated with the
the Anholt approximation (ref. [ll]). The dotted line gives cr,,,,. Low-energy data from ref. 1241.
the momentum distribution in the target, is also expected in excitation, as in eq. (lx), but now I represents the ‘5 -+ ‘P excitation energy. Fig, 3, taken from Zouros et al. f14], shows the experimental results. The curves, computed by the impulse approximation for Hz and He targets, have been scaled to the data (by factors 1.7-2.3) 1251.The expected threshold effect is clearly present in the data. The cross section rising at lower energies is due to two-step
processes.
Measurements by HiikkiStter et at. [15,X] at StanLi2+, CSf, and 07+ ford with =O.g- to 35MeV/N ions scattered on H, and He targets have determined the Is ionization cross section of these hydrogenic ions. Figs. 4 to 6 show the experimental results. In each case, the full curve give the total ionization cross section [u,~ of eq. (9a)] calculated with the PWBA
0
1
2 Propstile
e~rgy
c
3
4
3
4
IM~v/N]
3.w*5 (b)
4se+5 4.oe+s 3.5#5 3.oe+5. '..
2.se+s2.oe+5-
0
.'4 -.. I
0
IA+5
2 Projectile
f ?&A<
energy IMeV/N]
Fig. 6. Same as fig. 4, but for 0’* 0 Projectile
3c+5J 0
(ref. [X6]>.
4 energy
[MeViNl
f I Projectile
2
3
1
energy [MeV/Nl
Fig. 5. Same as fig. 4, but for C5” (ref. [16]):The dashed line gives the pure nuclear cross section.
(with no scaling factors). The dotted line gives uK-reen. The dashed line is the ionization cross section, called uaorn, due to the target nuclei alone, obtained by setting all F(ls’ -+ **. ) = 0 in eqs. f9b) and (SC). The Anholt approximation [ll] to u,,,, has been used throughout. The molecular effect due to the use of Hz, rather than H, targets has been included by using the H, form factor proposed by Hubbell et al. [17]. A molecular form factor using the Weinbaum wavefunction [lg] has also been tried f19]. The difference between the experimental ionization cross section and Oscreenis equal to the “experimental” value for u,,,,, i.e. the two-center scattering correlation effect [S]. Alternatively, one can emphasize the difference between uBom and the experimental cross section, which is brought out strikingly by taking ratios of the experimental cross sections with Ii, and He targets (fig. 7). Here, also, the threshold effect is clearly apparent. Recently, similar experiments were made lOO-
WE. Meyerhof; HI? Hiilskijtter / Two-center e-e interaction
502
0.00
I
1
Projectile energy [MeV/N)
10
100
Projectileenergy IM~v/N]
0.3
0.4
-_---1_-
0
1
2
3
4 10
1 Projectile energy IMeVlNl
loo
IO00
Projectileer%zrqy~M~v~~~ Fig. 8. One-electron
1.0
F
ict
loss cross section for AI??+ +H,, caption for fig. 4).
He (see
“pure” as for Is ionization, because one must use 151 appropriate screening constants for the nuclear charge and make adjustments for the experimental subsbell
0.4 -I0
I
z
3
4
Fig. 7. Ratio of electron loss cross sections for Hz and for He targets. (a) Li2*, (b) C’+, (c) O’+. Solid lines from eq. (9a); dashed line for pure nuclear cross section. The a,,,, ratio is omitted.
and 380-MeV/N Ac?~+ ions impinging on H, and He gas targets [XI] (figs. 8 and 9). In this case, single-dectron loss was measured, which implies preda~Rant1~ ionizatiun of the M shell. The cnfculations are not as
Fig. 9. Ratio of one-electron
and for At?*
toss cross sections for Aus * + Hz + He (see caption of fig. 7).
W.E. Meyerhof,
binding energies in the PWBA expressions theless, there is good agreement between experiment.
H. P. Hiilskiitter
[21]. Nevertheory and
4. Discussion The examples given in section 3 provide evidence for two-center scattering correlation [3] (fig. 2b). This correlation is less subtle than the expected one-center or bi-centric correlation effects [1,4], which may be static, i.e. present in the initial and final states, or dynamic, i.e. induced by the electron-nuclear perturbation [2]. Examples of correlation effects in various electron capture processes have been discussed by Tanis and coworkers [22] and Stolterfoht et al. [23]. The concept of correlation provides a unifying picture for the system shown in fig. 1. Nevertheless, the present authors believe that the name one gives to a process is not as important as understanding its basic nature.
Acknowledgements Comments of J. McGuire on this paper are very much appreciated. This work was supported in part by the National Science Foundation grant PHY 86-14650.
/ Two-center e-e interaciion
161 D.R. Bates and G. Griffing,
Proc. Phys. Sot. (London) A66 (1953) 961; A67, 663 (1954): A68, 90 (1955). [71 H. Bethe, Ann. Physik (Leipzig) 5 (1930) 325. PI G. Gillespie and M. Inokuti, Phys. Rev. A22 (1980) 2430. I91 Yong-Ki Kim and M. Inokuti, Phys. Rev. 165 (1968) 39. [lOI J.H. McGuire, N. Stolterfoht and P.R. Simony, Phys. Rev. A24 (1981) 97. Pll R. Anholt, Phys. Lett. 114A (1986) 126. PI H.M. Hartley and H.R.J. Walters, J. Phys. B20 (1987) 1983. P31 T.N. Tipping, J.M. Sanders, J. Hall, J.L. Shinpaugh, D.H. Lee, J.H. McGuire and P. Richard, Phys. Rev. A37 (1988) 2906. P41 T.J.M. Zouros, D.H. Lee and P. Richard, Phys. Rev. Lett. 62 (1989) 2261. W.E. Meyerhof, E.D. Dillard and N. 1151 H.P. Htilskbtter, Guardala, Phys. Rev. Lett. 63 (1989) 1938. WI H.P. Htilskiitter, Ph.D. Thesis, Stanford University (1990), unpublished. v71 J.H. Hubbell, W.J. Veigele, E.A. Briggs, R.T. Brown, D.T. Cromer and R.J. Howerton, J. Phys. Chem. Ref. Data 4 (1975) 471. [181 S. Weinbaum, J. Chem. Phys. 1 (1933) 593. Quiang Dai, W.E. Meyerhof and J.H. P91 H.P. Hiilskotter, McGuire, submitted to Phys. Rev. A. but not yet accepted. Lw H.P. Hiilskiitter et al.. submitted to Phys. Rev. A. but not yet accepted. PI B.-Y. Choi, Phys. Rev. A7 (1973) 2056. PI J.A. Tanis, G. Schiwietz, D. Schneider, N. Stolterfoht, W.G. Graham, H. Altevogt, R. Kowallik, A. Mattis, B. Skogvall, T. Schneider and E. Szmola, Phys. Rev. A39 (1989) 1571.
~231N. Stolterfoht,
References [l] J.F. Reading [2] [3]
[4] [5]
and A.L. Ford, Comments At. Mol. Phys. 23 (1990) 301 and references given therein. J.H. McGuire, Phys. Rev. A36 (1987) 1114. N. Stolterfoht, in: Spectroscopy and Collisions of Few Electron Ions, eds. V. Florescu and V. Zoran (World Scientific, Singapore, 1989) p. 342, Physica Scripta 42 (1990) 192. J.F. Reading, private communication. E. Merzbacher and H.W. Lewis, Handbuch der Physik, vol. 34 (Springer, Berlin, 1958) p. 166.
503
K. Somer. D.C. Griffin, C.C. Havener, M.S. Hug. R.A. Phaneuf. J.K. Swenson and F.W. Meyer, Nucl. Instr. and Meth. B40/41 (1989) 28. 1241 M.B. Shah and H.B. Gilbody (to be published): see also M.B. Shah. T.V. Goffe and H.B. Gilbody, J. Phys. Bll (1978) 6233. v51 Nofe added in proof: The scaling factor in ref. [14] has been eliminated with a recalibration of the effective solid angle of the metastable 4P beam. See P. Richard, in: X-ray and Inner-Shell Processes, ed. T.A. Carlson, M.O. Krause and S.T. Manson (Am. Inst. Phys., New York. 1990) p. 315.
498
Two-center electron-electron
interaction
in fast ion-atom
collisions
W.E. Meyerhof and H.P. Hiilskijtter Department of Physics, Stanford University, Stanford CA 94305, USA
The interactionbetweena target electron and a projectileelectron in one-electron projectile excitation or loss is examinedon the basis of the PWBA and demonstratedexperimentally.
1. Introduction During the last decade, the subject of electron-electron correlation has received increasing attention by atomic theorists and experimenters [l]. McGuire [2] has discussed electron-electron correlations in terms of static correlations present in the initial or final atomic states and dynamic, or scattering, correlations induced by the dynamics of the collisions. Also, Stolterfoht [3] has extended these concepts to two-center systems, such
as a collision between a projectile ion and a target atom. A certain part of the Coulombic interaction between a projectile electron and a target electron is called “twocenter scattering correlation”. The simplest case to illustrate these ideas is that of two nuclei (2, Z’) in relative motion and two electrons (e, e’), as sketched in fig. 1. Each electron may go from some initial state (i, i’) to some final state (f, f’) or stay in the initial state. A necessary, but not sufficient, condition for recognizing a correlation effect is that at least two electrons change their state [3]. In the present case, this implies that correlation requires at least f + i, f’#i’.
In terms of the situation sketched in fig. 1, a dynamic scattering correlation is produced or induced by the time dependence of the internuclear vector R. The effect of such a correlation on a particular cross section
Y
z
4 %t)
i
e
l
.
(i-at)
z’
-
Cc) (b) (a) Fig. 2. Possible groupingsof the electrons in fig. 1, according to ref. [3]: (a) one-center, (b) two-center, (c) bi-centric; the origin of ri, r, is the center of charge is indicated by a cross. The Perturbingdynamical interactionsare indicatedby dashed lines.
depends on the grouping of the electrons. Stolterfoht mentions three groupings, shown in fig. 2: (a) a onecenter case in which the two electrons initially are associated with one center, (b) a two-center case in which each electron initially is associated with one center, and (c) a bi-centric case in which both electrons are associated with both centers, as in the molecularorbital (MO) model. In fig. 2, the perturbing time dependent potentials are indicated by dashed lines. Only in the two-center case is a perturbing potential, namely that between the electrons, directly related to the dynamic electron-electron (e-e) correlation. In the two other cases, the dynamic e-e correlation is induced by the electron-nuclear perturbation, as shown by McGuire [2]. The two-center case appears to be the simplest (and, hence, according to Reading [4], the least interesting) case of a scattering correlation, and we discuss it alone.
e’ (i’->f)
Fig. 1. Two-nucleus and two-electron system. The relative projectilevelocity v is indicated,as well as the transitionsi - f and i’ + f’ of the electrons e and e’. The internuclearvector R(t) changes with time and induces dynamic correlations betweenthe electrons. 0168-583X/91/$03.50
Z
z’
2. Theory: Born approximation for two-center scattering Consider fig. 2b. The cross section for elastic or inelastic electronic processes is given by [S]
u = M2/(4n2tt4)x
0 1991 - Elsevier Science PublishersB.V. (North-Holland)
/ 1(I I* dQ. fin
(1)
499
W.E. Meyerhof, H. P. Hiilskiitter/ Two-centere-e interaction
For simplicity, we ignore the slight change in projectile velocity in inelastic processes. The quantity M is the reduced mass of the system. The summation is over all relevant final states f and f’, which, in the case of ionization, implies integration over the electron kinetic energies E and e’ of the electrons e and e’. In the plane-wave Born approximation (PWBA), the scattering amplitude a is expressed in terms of the momentum transfer Aq which is the vector difference between the incident and scattered c.m. particle momentum AK,, and AK,,,. Using the relation
and ionization of hydrogenic ions [6,7]. For a given projectile process, the summation in eq. (3) is over the final states f’ of the target. This yields
Pa)
e,/ = %L%” + %X, 9 where = 8ne4/( *screen
h2vZ)~~~~,~,~‘(dq/q3)
1F(ls -+f)
X/Z’-F(ls’+ls’)(2
I2 (9b)
and q’=K;+K=
out- 2KinKout COS0,
(2)
one can conveniently replace the integration over the c.m. scattering solid angle dS2 = 28 sin 13 d8 = 27r qdq/Kk in eq. (1) by an integration over q: u = 1/(2nfi2u2)
c j 1a ) 2q dq,
(3)
fin
where v is the projectile velocity. For the case of fig. 2b, the scattering amplitude in first order PWBA is given by
[61
a - /dR
exp(iq*R)//
drdr’~,(r)~,,(r’)l/cp,(r)
X+(r’),
(4)
where the perturbing
potential
V is
V=e2[ZZ’/R-Z/IR-r’l-Z’/IR+rl +l/lR+r-r’l].
(5)
Because of orthogonality of +, and +f or I#+, and +,, in any inelastic process, the first term in eq. (5) does not contribute to a in an inelastic process. We now consider projectile excitation or electron loss. Then, the orthogonality between $, and 9, eliminates also any contribution from the second term in eq. (5). Using the Bethe integral [7]
/
dR exp(iq*R)/IR-rI
=(4a/q2)
exp(iq*r), (6)
one can show that [6] a = (4a/q2)F(ls
+f)[
z’+
- F(ls’
+‘)I.
(7)
Here, we assume that each electron initially is in a 1s state and express a in terms of the form factor
F(ls -f)
= jd rexp(iq.r)~,(r)9,,(r)
(8)
and similarly for F(ls’ -f’). The form factors are real, so that the sign of the argument of the exponent is of no importance. They have been evaluated for excitation
x I F(ls
-f)
I = I F(ls’
-f’)
12.
PC)
In eqs. (9b) and (SC), q&i’ or f’) is the minimum momentum transfer needed to excite electron e from i to f and e’ from i’ to i’ or f’. For heavy particles, qmax is typically set to infinity. The first term in eq. (9a), called u,,,,, is the cross section induced by a screened target nucleus. The second term, called a,,,, is due to the electron-electron interaction: it can be traced back to the last term in eq. (5). At various times, this part of the cross section has been called “inelastic” [6,8], “incoherent” [9], “antiscreening” [lo], “two-center scattering correlation” [3], or we can call it “electron induced correction to the screened nuclear cross section”, in order to leave the subscript “corr” somewhat neutral. The partial cross section a,,, fulfills the correlation condition that two electrons make a transition, albeit as a result of a single Coulombic interaction. (This is not the case for u,,, where only one electron makes a transition.) The question now arises whether the two electrons make the transition independently or not. As pointed out by Bates and Griffing [6], the target electron must make an inelastic transition ( f ’ # i’) in order to induce the inelastic transition i +f in the projectile. This is a consequence of momentum conservation. Therefore, the transitions are not independent and u,,,, can be considered as being due to a two-center scattering correlation [3]. Before turning to experimental results, we note that the expression for a,,, can be simplified if qmin(f’) is not very dependent on f’. This occurs for Z’ < Z. Then, the summation over f’ in eq. (9) can be put inside the integral. Using closure [7], c
I F(ls’-f’)
I 2 = 1,
f’
one can rewrite
*,,, =
[6]
8ne4/(tr2u2)Jq~.(f,)(dq/q3) I F(ls -f) I2 x(1
- ]F(ls’+
1s’) 12).
(IO)
W.E. Meyerhoj H.P. Hiilskiitter / Two-center e-e interaction
500
As noted by Anholt [7], since cr,,, is due to the electron-electron interaction, there must be, in the limit of free target electrons, a threshold effect: uarr = 0, unless (nonrelati~stic~ly) (1/2)m&?
> I,
(11)
where m, is the electron mass and I the projectile 1s ionization energy. In eq. (SC), this threshold effect is effectively included in the dependence of qmin on f ‘, but is lost in eq. (10). It can be reinstated there through an appropriate multiplicate factor [ll] or by setting qmin(f’) equal to the minimum momentum transfer needed by scattering a free electron of velocity u on the projectile so that the transition i + f is induced [12].
3. Experimental results Even though eq. (9a) has been known since 1953 [6], experimentally the existence of the term a,, had not been demonstrated until very recently. In particular, the threshold effect, mentioned above, had not been clearly apparent in previous excitation or ionization measurements [13]. 3.1. Projectile excitation By scattering 4-35 MeV Li-like 05+ and F6+ ions on H, and He gas targets, Zouros, Lee, and Richard [14] have been able to detect the (2s22s)*S + (ls2~2p)~P excitation process in these ions. In this particular transition, the effective 1s --f 2p excitation cannot be produced by direct nuclear scattering because a spin flip must occur. Hence, this transition selects the pure a,, part of the cross section in eq. (9a), here due to electron-electron exchange [which is not included in eqs. (9)]. Other, more complex two-step processes can also produce the ‘S -+ 4P transition, but can be separated energetically [14]. A threshold effect, smeared out by
a1(a) PROJECTILE
1
ENERGY (MeV)
12,‘,,,,,,,,Li,l‘lfll,l(
1
10
Projectile energy [Mev/Nl
‘0
4
8
12
16
~JECTILE
x)
24
28
ENERGY (Me@
Fig. 3. Data: Cross sections for the production of ls2s2p4P states by 1s + 2p projectiie excitation in collisions of (a) OS* and (b) of F 6+ (1~~2s) projectiles with He and H2 targets vs projectile energy. Only statistical errors are shown. Total absolute error is - 30%. Calculation: electron-electron excitation (eeE) cross sections using ls2s2p4P theoretical electron impact excitation cross sections folded by the Compton profile of the target. Dashed lines, calculated eeE for H2 targets; dash-dotted lines, calculated eeE for He targets. Arrows (at 16.3 and 25.0 MeV) indicate the projectile energies corresponding to the threshold for Is + 2p electron-impact excitation in 05+ and F6+, respectively. Cross sections at lower energies are due to two-step processes (ref. [14]).
4 1
10
Projectile energy [MeV/Nl
Fig. 4. Electron loss cross section for (a) Li2+ +H,, (b) Liz+ + He. In each case, the solid line gives eq. (9a) evaluated with the
the Anholt approximation (ref. [ll]). The dotted line gives cr,,,,. Low-energy data from ref. 1241.
the momentum distribution in the target, is also expected in excitation, as in eq. (lx), but now I represents the ‘5 -+ ‘P excitation energy. Fig, 3, taken from Zouros et al. f14], shows the experimental results. The curves, computed by the impulse approximation for Hz and He targets, have been scaled to the data (by factors 1.7-2.3) 1251.The expected threshold effect is clearly present in the data. The cross section rising at lower energies is due to two-step
processes.
Measurements by HiikkiStter et at. [15,X] at StanLi2+, CSf, and 07+ ford with =O.g- to 35MeV/N ions scattered on H, and He targets have determined the Is ionization cross section of these hydrogenic ions. Figs. 4 to 6 show the experimental results. In each case, the full curve give the total ionization cross section [u,~ of eq. (9a)] calculated with the PWBA
0
1
2 Propstile
e~rgy
c
3
4
3
4
IM~v/N]
3.w*5 (b)
4se+5 4.oe+s 3.5#5 3.oe+5. '..
2.se+s2.oe+5-
0
.'4 -.. I
0
IA+5
2 Projectile
f ?&A<
energy IMeV/N]
Fig. 6. Same as fig. 4, but for 0’* 0 Projectile
3c+5J 0
(ref. [X6]>.
4 energy
[MeViNl
f I Projectile
2
3
1
energy [MeV/Nl
Fig. 5. Same as fig. 4, but for C5” (ref. [16]):The dashed line gives the pure nuclear cross section.
(with no scaling factors). The dotted line gives uK-reen. The dashed line is the ionization cross section, called uaorn, due to the target nuclei alone, obtained by setting all F(ls’ -+ **. ) = 0 in eqs. f9b) and (SC). The Anholt approximation [ll] to u,,,, has been used throughout. The molecular effect due to the use of Hz, rather than H, targets has been included by using the H, form factor proposed by Hubbell et al. [17]. A molecular form factor using the Weinbaum wavefunction [lg] has also been tried f19]. The difference between the experimental ionization cross section and Oscreenis equal to the “experimental” value for u,,,,, i.e. the two-center scattering correlation effect [S]. Alternatively, one can emphasize the difference between uBom and the experimental cross section, which is brought out strikingly by taking ratios of the experimental cross sections with Ii, and He targets (fig. 7). Here, also, the threshold effect is clearly apparent. Recently, similar experiments were made lOO-
WE. Meyerhof; HI? Hiilskijtter / Two-center e-e interaction
502
0.00
I
1
Projectile energy [MeV/N)
10
100
Projectileenergy IM~v/N]
0.3
0.4
-_---1_-
0
1
2
3
4 10
1 Projectile energy IMeVlNl
loo
IO00
Projectileer%zrqy~M~v~~~ Fig. 8. One-electron
1.0
F
ict
loss cross section for AI??+ +H,, caption for fig. 4).
He (see
“pure” as for Is ionization, because one must use 151 appropriate screening constants for the nuclear charge and make adjustments for the experimental subsbell
0.4 -I0
I
z
3
4
Fig. 7. Ratio of electron loss cross sections for Hz and for He targets. (a) Li2*, (b) C’+, (c) O’+. Solid lines from eq. (9a); dashed line for pure nuclear cross section. The a,,,, ratio is omitted.
and 380-MeV/N Ac?~+ ions impinging on H, and He gas targets [XI] (figs. 8 and 9). In this case, single-dectron loss was measured, which implies preda~Rant1~ ionizatiun of the M shell. The cnfculations are not as
Fig. 9. Ratio of one-electron
and for At?*
toss cross sections for Aus * + Hz + He (see caption of fig. 7).
W.E. Meyerhof,
binding energies in the PWBA expressions theless, there is good agreement between experiment.
H. P. Hiilskiitter
[21]. Nevertheory and
4. Discussion The examples given in section 3 provide evidence for two-center scattering correlation [3] (fig. 2b). This correlation is less subtle than the expected one-center or bi-centric correlation effects [1,4], which may be static, i.e. present in the initial and final states, or dynamic, i.e. induced by the electron-nuclear perturbation [2]. Examples of correlation effects in various electron capture processes have been discussed by Tanis and coworkers [22] and Stolterfoht et al. [23]. The concept of correlation provides a unifying picture for the system shown in fig. 1. Nevertheless, the present authors believe that the name one gives to a process is not as important as understanding its basic nature.
Acknowledgements Comments of J. McGuire on this paper are very much appreciated. This work was supported in part by the National Science Foundation grant PHY 86-14650.
/ Two-center e-e interaciion
161 D.R. Bates and G. Griffing,
Proc. Phys. Sot. (London) A66 (1953) 961; A67, 663 (1954): A68, 90 (1955). [71 H. Bethe, Ann. Physik (Leipzig) 5 (1930) 325. PI G. Gillespie and M. Inokuti, Phys. Rev. A22 (1980) 2430. I91 Yong-Ki Kim and M. Inokuti, Phys. Rev. 165 (1968) 39. [lOI J.H. McGuire, N. Stolterfoht and P.R. Simony, Phys. Rev. A24 (1981) 97. Pll R. Anholt, Phys. Lett. 114A (1986) 126. PI H.M. Hartley and H.R.J. Walters, J. Phys. B20 (1987) 1983. P31 T.N. Tipping, J.M. Sanders, J. Hall, J.L. Shinpaugh, D.H. Lee, J.H. McGuire and P. Richard, Phys. Rev. A37 (1988) 2906. P41 T.J.M. Zouros, D.H. Lee and P. Richard, Phys. Rev. Lett. 62 (1989) 2261. W.E. Meyerhof, E.D. Dillard and N. 1151 H.P. Htilskbtter, Guardala, Phys. Rev. Lett. 63 (1989) 1938. WI H.P. Htilskiitter, Ph.D. Thesis, Stanford University (1990), unpublished. v71 J.H. Hubbell, W.J. Veigele, E.A. Briggs, R.T. Brown, D.T. Cromer and R.J. Howerton, J. Phys. Chem. Ref. Data 4 (1975) 471. [181 S. Weinbaum, J. Chem. Phys. 1 (1933) 593. Quiang Dai, W.E. Meyerhof and J.H. P91 H.P. Hiilskotter, McGuire, submitted to Phys. Rev. A. but not yet accepted. Lw H.P. Hiilskiitter et al.. submitted to Phys. Rev. A. but not yet accepted. PI B.-Y. Choi, Phys. Rev. A7 (1973) 2056. PI J.A. Tanis, G. Schiwietz, D. Schneider, N. Stolterfoht, W.G. Graham, H. Altevogt, R. Kowallik, A. Mattis, B. Skogvall, T. Schneider and E. Szmola, Phys. Rev. A39 (1989) 1571.
~231N. Stolterfoht,
References [l] J.F. Reading [2] [3]
[4] [5]
and A.L. Ford, Comments At. Mol. Phys. 23 (1990) 301 and references given therein. J.H. McGuire, Phys. Rev. A36 (1987) 1114. N. Stolterfoht, in: Spectroscopy and Collisions of Few Electron Ions, eds. V. Florescu and V. Zoran (World Scientific, Singapore, 1989) p. 342, Physica Scripta 42 (1990) 192. J.F. Reading, private communication. E. Merzbacher and H.W. Lewis, Handbuch der Physik, vol. 34 (Springer, Berlin, 1958) p. 166.
503
K. Somer. D.C. Griffin, C.C. Havener, M.S. Hug. R.A. Phaneuf. J.K. Swenson and F.W. Meyer, Nucl. Instr. and Meth. B40/41 (1989) 28. 1241 M.B. Shah and H.B. Gilbody (to be published): see also M.B. Shah. T.V. Goffe and H.B. Gilbody, J. Phys. Bll (1978) 6233. v51 Nofe added in proof: The scaling factor in ref. [14] has been eliminated with a recalibration of the effective solid angle of the metastable 4P beam. See P. Richard, in: X-ray and Inner-Shell Processes, ed. T.A. Carlson, M.O. Krause and S.T. Manson (Am. Inst. Phys., New York. 1990) p. 315.