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Faddeev approach to electron capture
Nuclear Instruments and Methods North-Holland, Amsterdam
FADDEEV Steven
APPROACH
Research
TO ELECTRON
B43 (1989) 19-23
CAPTURE
ALSTON
Physics Department, Received
in Physics
7 February
Pennsylvania
State Unioersity,
Wilkes-Barre
1989 and in revised form 31 March
Campus, Lehman,
Pennsylvania
18627, USA
1989
The Faddeev-Watson-Lovelace scattering formalism is applied to the rearrangement problem in three-body collisions. In a second-order approximation, the transition operator is seen to generalize the second Born approximation by replacing each appearance of a two-body potential with a corresponding two-body transition operator, except for the first Born electron-nuclear potential term which remains the same. Application to electron capture in ion-atom collisions for both forward- and large-angle projectile scattering is discussed and the particular role of the internuclear potential in each case considered. Extension is made of results obtained within the second Born approximation, particularly the zero contribution of the internuclear potential at forward angles and the structure of critical-angle scattering. The treatment of initial- and final-state binding effects is also addressed. Differential capture cross sections for proton-hydrogen and proton-helium collisions in the MeV energy range are presented.
1. Introduction As evidenced by comparison with experiment [1,2], considerable progress in understanding high-energy electron capture in symmetric and asymmetric ion-atom collisions has been made in recent years along several lines involving multiple-scattering treatments [3-61 of the process. In this article the most general extension of the Born expansion of the capture transition operator, namely, the Faddeev expansion of it, is considered. By retaining all two-body potentials in the three-body problem, a more general class of collisions is capable of treatment, and the advantages of the second Born approximation (B2) such as the conceptually simple breakdown of the overall collision process into combinations of double scatterings [7] or the critical-angle behavior of the amplitude [S], e.g., the forward-angle Thomas peak [9], are retained. The Faddeev approach is quite general: The capture amplitude obtained using it is intrinsically symmetric and so geared toward symmetric and near-symmetric collisions, but it is also capable of treating asymmetric ones as well because of the inclusion of infinite numbers of expansion terms for each potential. Internuclear potentials are included allowing for large-angle projectile scattering and permitting also an explanation of the internuclear contribution in the near-forward angles. Ion-neutral collisions are treated here; ion-ion collisions will require an extended formalism [lo]. In the following, the amplitude for electron capture derived from a second-order approximation [6,11] to the Faddeev expansion of the transistion operator is discussed and used for the calculation of the Is-1s cross section in the forward directions for proton-hydrogen 0168-583X/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
and proton-helium collisions. Comparison is than made with experiment [1,2] with good to excellent agreement resulting. An approximate evaluation of the amplitude is used whereby the neglect of initial- and final-state binding effects of the captured electron leads to the introduction of a slightly generalized form of the elastic scattering amplitude in place of the two-body transistion matrix [12]. Taking advantage of the deep penetration of the potentials which follows for large impact velocities, Coulomb versions of the scattering amplitudes are employed. Consideration is given to the complete amplitude as well as the one not including the internuclear potential. In addition, a more complete treatment of the capture process is emphasized, but only forward-angle cross sections are presented. Section 2 analyses the amplitude, its approximate evaluation, and also some specific mechanisms. Section 3 presents results of numerical calculations along with some discussion and conclusions. Atomic units are used throughout.
2. Theory and analysis The multiple-scattering approach to three-body collisions used here to treat electron capture employs an expansion (denoted FWL) of the transistion operator developed variously by Faddeev, Watson, and Lovelace [ll]. The basic idea is to retain the conceptually simple decomposition of the total collision process into sums of products of two-body scatterings (with free propogation of the third particle) which is a signature of the Born expansion of the transition operator, and to represent the two-body scatterings by means of single-poten-
20
S. Alston / Faddeeu approach to electron capture
tial transition operators. For the ion-neutral collisions considered here, the latter implies that the infinite contributions which plague the Born series due to free-particle propagations do not arise in the FWL treatment. The collision is assumed to involve the impact of a projectile ion on a neutral target consisting of a target ion to which is bound initially an active electron. Both of the ions are taken to be bare nuclei possibly screened by nonactive electrons whose sole role in the scattering is to introduce model potentials which are modifiedCoulomb in nature. The final bound system of the electron and projectile ion is assumed to be neutral. A complete derivation and detailed analysis of the second-order approximation to the FWL transistion amplitude is given elsewhere [13]; the form obtained after integrating the free, third-particle motions and neglecting factors of order electron mass over heavyparticle masses is found to be A nw=Ae+AAn = (2n)-3/
dk, dkr6T(kr)[T,+
T,]6i(k,), (I)
with
A,,
T, = (2a)9’2 6(ki+J)fp,(kr-ki+v) +T,,(k,+u,
k,+k,-K;
6’(E)T,,(k,+k,+J,
k,-u;
+T&pu-J-ki, G;(E)T,,(k,,
j
dk, dkr @(k,)&(ki)
E,), pu; E)
ki; EC)
Go+(E)Z-,(pu+K-k,,
= (2~)-~
E,)
T,= (2a)36(kc-k,+v)7-,&-ki-J, +TTe(k,+u,
The FWL amplitude eq. (1) describes the capture of the active electron by representing it with an initial momentum distribution (wave packet) centered about the target ion which scatters through multiple collisions to form a final momentum distribution centered about the receding projectile ion. The scattering in each of the two-body collisions is expressed through a transistion operator. The partial amplitudes A, and A,, separately collect the purely electronic terms and the terms containing internuclear potentials. When A, is neglected, the resulting amplitude is identical to the one which was derived previously using a distorted-wave formalism [4] and which has been the subject of several earlier studies [5]. The following treatment of the partial amplitude A, stresses a more explicit second Born connection with its straightforward description of the forward-angle Thomas mechanism. If the transistion operators Ti (i = Pe, Te, PT) in the second-order terms of T, + T, are approximated by their first Born forms Vi and the first-order term r, is approximated by its second Born form V&l + Gz V,), then the second Born capture amplitude A,, is derived. Written in abbreviated form it is
pu; Ei) pu; E,)
k, - u; E,),
where the T,(k’, k; 6) with i = Pe, Te, PT are two-body transistion operators of momenta k’, k and off-shell energy c corresponding, respectively, to the electronprojectile, electron-target nuclear, and internuclear potentials Vi,,, V,,, and V,; the ii and 6, are the momentum-space bound-state wave functions of energies ci and
of a func-
Comparison of eqs. (1) and (2) shows that the former provides the highest extension in second order of the latter attainable in the sense that each potential, with the two exceptions noted above, is replaced by the transistion operator derived from it. Furthermore, the first-order term TpT. represents a complete summation of the pure internuclear terms in the full Born expansion. The second-order FWL approximation (and n th order ones as well) occupies a unique position then in the hierarchy of transition-operator approximations. In order to evaluate eq. (1) the small sizes of the initial and final bound-state momenta ki of order Z-r and k, of order Z, relative to the impact velocity u are used to introduce for the two-body transistion operators simplified forms recently derived by Alston [12] which are valid for the case of pure- and modified-Coulomb potentials. Specifically, one gets T(k’,
k; E) =
-27iQ(k’,
P)Q(~,
~)fc,k(E)>
(3)
with Q(k, p) = e- nv”/2F(l - iv”)[(k* - ~i~)/4p~]~” where E = p2/2m (m is a two-body reduced mass) and fk,,k( E) is the generalized form of the elastic scattering amplitude for the given potential [12]. The Sommerfeld parameter is denoted by va = Z”m/p where Za is the charge of the asymptotic, Coulomb form of the potential. Eq. (3) is equivalent to the approximation of neglecting initial- and final-state binding effects.
21
S. A&on / Faddeeu approach to electron capture
The appearance of the Q factors in eq. (3) is a consequence of the Coulomb character of the potentials asymptotically. For a Coulomb potential, the off-energy-shell T matrices involve scattering wave functions which do not contain logarithmic phase terms in their asymptotic forms whereas the wave functions in the on-shell limit do have logarithmic phase terms. An accumulation of phase leading to Q thus results when going on-shell. Since for capture, large impact velocities lead necessarily to large momentum transfers K and J, implying that the potentials are deeply probed, a further approximation is introduced whereby the amplitude f is replaced by the pure-Coulomb version f’, with scaledcharge dependence. One can write
f,c.k(E) =fk,,,c(E) =(22/lk’-k12)
e2100[(k’+k)/lk’-k1]2’Y, (4)
The phase ua equals argr(1 + iv) and v = Zm/p with Z the charge. Equation (4) has been combined with eq. (1) previouly, but without the proper and necessary treatment of eq. (3) included. Assuming an impulse approximation, earlier calculations in ref. [6] neglected one of the Q factors in each use of eq. (3). It is now known [3,12] that such a neglect is unjustified and it is not done here. In order to actually evaluate the approximation to the second-order FWL amplitude obtained when eqs. (3) and (4) are inserted into eq. (l), the dependence of A, on ki and k, is retained only in &, &, and the energy-difference factors of Q and neglected elsewhere [14]; consequently, a closed-form expression is derivable. One finds explicitly A, = CL,N,(~~U~)‘(““-~“~) x ( ZP [
,l/zz+ip=,(
“2
+
p)
-2+iG
_L,N,z,2i’8~~“~~-2+‘4’;
x
( u2 _
tz,
Z+dml-“‘kp’A’]
711’222
[
pivam(
J-2-Ziv,
u2 +
z;>
-2+iG
_L,N,z,~IV;Z:“‘~~-~+,~“;
x
(u2
_
z;)-l-i(++)
1
K-2-2iup,),
NP = r(1/2
+ 2iu,)r(l
+ ivt)/I’(2
A, = C( Z,Z,/J2K2n3)
Xexp{i[(2
L,L,
v+ - vr + 2v+ - v,)ln(4u2)
+ 2v, ln( K) + 2vr ln( J)] x / dk, dk,(kf
- 2q-‘-I”‘(
x(gi,2)l+i(G+~C)
} kf - 2~~)~~~‘“’ (5b)
Eq. (5b) is evaluated by performing the angular integrals over k, and k, analytically and the radial integrals over ki and k, numerically. Terms of order (Z,/U)~ and (Z,/O)~ have been neglected in deriving eqs. (5). In contrast to previous approximate treatments of the partial amplitude A, [S], an advantage of eq. (5b) is that use of eqs. (3) and (4) exposes more clearly the salient features of the collision. When the Coulomb and off-shell natures of eqs. (5) are neglected, implying that all six Sommerfeld parameters are set to zero, the expression for the B2 amplitude in the same approximation is derived. As a consequence of this connection and the specific forms of the factors which become unity on equating the parameters to zero, it follows [13] that the pronounced structure (peaks or valleys) [8] exhibited by the B2 amplitude at specific critical scattering angles exists also in the FWL amplitude, and at the same angles, although the shape and height or depth of the structures will be altered. This conclusion follows even though infinite numbers of terms involving each of the potentials have been summed in the FWL amplitude. Consideration of eq. (5a) when the Sommerfeld parameters are set to zero shows that, because of cancellation, A, is of order (Z,/IJ)~ or (Z,/U)~, which means, within the accuracy of the present analysis, that a zero contribution of the internuclear potential to the capture amplitude is found. This behaviour of the second-Born approximation to A, was first observed some time ago [15]. In the FWL case, however, the contribution of A,, is not zero to leading order because the factors multiplying the common velocity and momentum-transfer dependences are not the same in the different terms and thus do not cancel. In other words, the coherent addition of the double-scattering, partial amplitudes of A, with the single-scattering, nuclear amplitude is not zero when considering a proper treatment (by use of transition matrices) of the two-body collisions.
(5a)
where L, = exp( +avt
In A,, the ki and k, dependence in G$ is also retained except for the ki. k, term; a closed-form expression is not easily found. One has instead
+ ivap)2, + iv:),
and similarly for L,, NT, L,, and N,, and C= 25(ZpZr)5/2. In L, and N,, the parameters v& and vpT appear with opposite signs.
3. Results and discussion Results of calculations for proton-helium and proton-hydrogen collisions are presented in figs. 1 and 2.
S. A&on / Faddeev approach to electron capture
22
p +
00
02
He,
5 42
04 06 aLaB (mmd)
MeV
08
10
Fig. 1. The FWL Is-1s capture cross section for p-He collisions convoluted with the beam profile is compared with the experimental data taken from ref. [l] which includes capture to
all final states. Also shown are the unconvolutedcross sections dividedby a factor of 10: solid curve, FWL; long dash, FWL withoutinternuclearterms (A, only); and short dash, B2.
A comparison of the full FWL curve, i.e., the one derived from A, + A,, with experiment shows excellent agreement for the helium target [l] where the data has much better statistics and generally good agreement for
p + H, 5 0 MeV
00
02
04
06
08
10
@LAB (m-ad)
Fig. 2. The FWL Is-1s capture cross section for p-H collisions convoluted with the beam profile is compared with the experimental data taken from ref. [2] which includes capture to all final states. Also shown are the unconvoluted cross sections divided by a factor of 100: solid curve, FWL; long dash, FWL without internuclear terms (A, only); and short dash, B2.
the hydrogen target [2]. The theoretical cross sections have been convoluted with the experimental beam profiles [16]. Also shown in the figures are unconvoluted curves derived from the electronic amplitude A, alone and from the B2 approximation to it. In the numerical calculations for helium, the values assumed for the charges are Z, = 1.0, Zg = 1.0, Z, = 1.6875, Z+ = 1.0, Z,Z, = 1.6875, and ZtZ+ = 1.0. The small differences seen between the three unconvoluted curves in the large forward-peak region reflects the dominance of the first Born amplitude whose contribution is identical in all three theories. In the secondary Thomas peak (0.47 mrad) region, the FWL and A, curves agree well but the B2 peak differs considerably from them, especially in the helium case. It can be shown within the B2 approximation [17] that the Thomas peak (or hump) results from a local maximum in the imaginary part (for s states) of the amplitude, which represents scattering via on-energy-shell intermediate states. In a classical and, consequently, on-shell treatment of the capture process a similar peak in the amplitude is found [9]. A comparison of the imaginary parts of the A, and B2 amplitudes [13] shows indeed that the FWL treatment of this component of the scattering is modified. Two other notable features are apparent in these figures. One is the very much less accentuated dip in the B2 cross sections relative to the FWL ones. Indeed, for helium, only a shoulder appears. Since the dip or shoulder results from a partial cancellation in A, of the real first Born term (for Is-1s captures) with the real part of the second-order term, one sees that in the FWL amplitude much more cancellation takes place. As the real part of the second-order contributions of the B2 amplitude can be shown to represent scattering via off-energy-shell intermediate states [17], which have no classical analogue, it follows that the FWL treatment of this part of the intermediate scattering is much better according to the data. The inclusion in the calculation of the “off-shell” Q factors of eq. (3) is the reason for the improvement [ 131. The second feature is the enhanced FWL cross sections over those of the pure electronic term A, and those of the B2 approximation in the angular region beyond the local maximum. The helium data supports this increase very well while the hydrogen data do not really extend to high enough angles. The agreement of the A,, amplitude with the data beyond the Thomas peak supports the discussion of the previous section which concluded that A, gives the first explicit representation of the mechanism by which the internuclear potential contributes to the capture of the electron. The eikonal transformation of the electronic ampltiude [18] has previously been derived as a general description of and perscription for including the effects of the internuclear potential. Since the second Born approximation
S. Alston / Faddeev approach to electron capture is zero, the FWL theory is the first one to show specifically how the internuclear contribution as described by the eikonal transformation arises because incomplete cancellation occurs among the three terms in A, when they are represented at a more complete level, that is, when the true Coulomb nature of the problem is accounted for. The large differences between the FWL and A, curves in the vicinity of the local minimum similarly result from cancellation of the second-order terms with the first Born term and thus are expected to expose non-leading-order components of the second-order terms which, however, are not treated as accurately as the leading-order components. The convoluted A, cross sections (not shown) give agreement comparable to the FWL ones in the forward-peak and Thomas-peak regions but not as good agreement with experiment in the minimum region. In summary, it has been shown that a Faddeev expansion through second order of the transistion operator for electron capture leads to Is-1s differential cross sections in good agreement with the experimental data. A new explanation of how the contribution of the internuclear potential arises and fits within a time-independent scattering formalism has been put forward. Finally, a direct and explicit relationship of the FWL theory to the second Born theory has been derived: A simple picture of capture involving double scatterings is maintained while a much more exact treatment of each of the scatterings is employed. to A,
The calculations reported here were performed on the IBM 3090 mainframe at the Pennsylvania State University.
[l] E. Horsdal-Pedersen, CL. Rev. Lett. 50 (1983) 1910.
PI H. Vogt, R. Schuch,
E. Justiniano, M. Schulz, and W. Schwab, Phys. Rev. Lett. 57 (1986) 2256. 131 J. Macek and S. Alston, Phys. Rev. A26 (1982) 250; S. Alston, Phys. Rev. A38 (1988) 3124; J. Macek and X.Y. Dong, Phys. Rev. A38 (1988) 3327; J.S. Briggs, J. Phys. BlO (1977) 3075. [41 K. Taulbjerg and J.S. Brigs, J. Phys. B16 (1983) 3811. [51 S.H. Hsin and M. Lieber, Phys. Rev. A35 (1987) 4833; J.H. McGuire, N.C. Deb, N.C. Sil, and K. Taulbjerg, Phys. Rev. A35 (1987) 4830. [‘31 M.J. Roberts, J. Phys. B20 (1987) 551. [71 R.M. Drisko, Ph.D. Thesis, Carnegie Institute of Technol-
PI [91
[lOI illI
[12] [13] [14]
[15] [16]
[17] [18]
References Cocke,
and M. Stockli,
Phys.
23
ogy (1955); see S. Alston, Phys. Rev. A38 (1988) 6092 for recent work and a list of refs.; R. Shakeshaft and L. Spruch, Phys. Rev. A29 (1984) 605. J.S. Briggs, Nucl. Instr. and Meth. BlO/ll (1985) 574. L.H. Thomas, Proc. R. Sot. (London) Ser. All4 (1927) 561; R. Shakeshaft and L. Spruch, Rev. Mod. Phys. 51 (1979) 369. D.P. Dewangan and J. Eichler, J. Phys. B18 (1985) L65. C.J. Joachain, Quantum Collision Theory (North-Holland, Amsterdam, 1975) Ch. 19; S. Alston, in Abstracts of Contributed Papers, 14th Int. Conf. on the Physics of Electronic and Atomic Collisions, Palo Alto (1985) eds. M.J. Coggiola, D.L. Huestis, and R.P. Saxon (North-Holland, Amsterdam, 1985) p. 516. S. Alston, Phys. Rev. A38 (1988) 636. S. Alston, Phys. Rev. A to be submitted. The ki integration of the first-order term Tm is performed to order ( Z,/V)~ and ( Z,/U)~ as if two separate peaks at k, = 0 and k, = o existed. K. Dettmann and G. Leibfried, Z. Phys. 218 (1969) 1. J.H. McGuire, M. Stockh, CL. Cocke, E. Horsdal-Pedersen, and N.C. Sil, Phys. Rev. A30 (1984) 89, and R. Schuch (private communication). J.H. McGuire, P.R. Simony, O.L. Weaver, and J. Macek, Phys. Rev. A26 (1982) 1109. R. McCarroll and A. Salin, J. Phys. Bl (1968) 163.
FADDEEV Steven
APPROACH
Research
TO ELECTRON
B43 (1989) 19-23
CAPTURE
ALSTON
Physics Department, Received
in Physics
7 February
Pennsylvania
State Unioersity,
Wilkes-Barre
1989 and in revised form 31 March
Campus, Lehman,
Pennsylvania
18627, USA
1989
The Faddeev-Watson-Lovelace scattering formalism is applied to the rearrangement problem in three-body collisions. In a second-order approximation, the transition operator is seen to generalize the second Born approximation by replacing each appearance of a two-body potential with a corresponding two-body transition operator, except for the first Born electron-nuclear potential term which remains the same. Application to electron capture in ion-atom collisions for both forward- and large-angle projectile scattering is discussed and the particular role of the internuclear potential in each case considered. Extension is made of results obtained within the second Born approximation, particularly the zero contribution of the internuclear potential at forward angles and the structure of critical-angle scattering. The treatment of initial- and final-state binding effects is also addressed. Differential capture cross sections for proton-hydrogen and proton-helium collisions in the MeV energy range are presented.
1. Introduction As evidenced by comparison with experiment [1,2], considerable progress in understanding high-energy electron capture in symmetric and asymmetric ion-atom collisions has been made in recent years along several lines involving multiple-scattering treatments [3-61 of the process. In this article the most general extension of the Born expansion of the capture transition operator, namely, the Faddeev expansion of it, is considered. By retaining all two-body potentials in the three-body problem, a more general class of collisions is capable of treatment, and the advantages of the second Born approximation (B2) such as the conceptually simple breakdown of the overall collision process into combinations of double scatterings [7] or the critical-angle behavior of the amplitude [S], e.g., the forward-angle Thomas peak [9], are retained. The Faddeev approach is quite general: The capture amplitude obtained using it is intrinsically symmetric and so geared toward symmetric and near-symmetric collisions, but it is also capable of treating asymmetric ones as well because of the inclusion of infinite numbers of expansion terms for each potential. Internuclear potentials are included allowing for large-angle projectile scattering and permitting also an explanation of the internuclear contribution in the near-forward angles. Ion-neutral collisions are treated here; ion-ion collisions will require an extended formalism [lo]. In the following, the amplitude for electron capture derived from a second-order approximation [6,11] to the Faddeev expansion of the transistion operator is discussed and used for the calculation of the Is-1s cross section in the forward directions for proton-hydrogen 0168-583X/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
and proton-helium collisions. Comparison is than made with experiment [1,2] with good to excellent agreement resulting. An approximate evaluation of the amplitude is used whereby the neglect of initial- and final-state binding effects of the captured electron leads to the introduction of a slightly generalized form of the elastic scattering amplitude in place of the two-body transistion matrix [12]. Taking advantage of the deep penetration of the potentials which follows for large impact velocities, Coulomb versions of the scattering amplitudes are employed. Consideration is given to the complete amplitude as well as the one not including the internuclear potential. In addition, a more complete treatment of the capture process is emphasized, but only forward-angle cross sections are presented. Section 2 analyses the amplitude, its approximate evaluation, and also some specific mechanisms. Section 3 presents results of numerical calculations along with some discussion and conclusions. Atomic units are used throughout.
2. Theory and analysis The multiple-scattering approach to three-body collisions used here to treat electron capture employs an expansion (denoted FWL) of the transistion operator developed variously by Faddeev, Watson, and Lovelace [ll]. The basic idea is to retain the conceptually simple decomposition of the total collision process into sums of products of two-body scatterings (with free propogation of the third particle) which is a signature of the Born expansion of the transition operator, and to represent the two-body scatterings by means of single-poten-
20
S. Alston / Faddeeu approach to electron capture
tial transition operators. For the ion-neutral collisions considered here, the latter implies that the infinite contributions which plague the Born series due to free-particle propagations do not arise in the FWL treatment. The collision is assumed to involve the impact of a projectile ion on a neutral target consisting of a target ion to which is bound initially an active electron. Both of the ions are taken to be bare nuclei possibly screened by nonactive electrons whose sole role in the scattering is to introduce model potentials which are modifiedCoulomb in nature. The final bound system of the electron and projectile ion is assumed to be neutral. A complete derivation and detailed analysis of the second-order approximation to the FWL transistion amplitude is given elsewhere [13]; the form obtained after integrating the free, third-particle motions and neglecting factors of order electron mass over heavyparticle masses is found to be A nw=Ae+AAn = (2n)-3/
dk, dkr6T(kr)[T,+
T,]6i(k,), (I)
with
A,,
T, = (2a)9’2 6(ki+J)fp,(kr-ki+v) +T,,(k,+u,
k,+k,-K;
6’(E)T,,(k,+k,+J,
k,-u;
+T&pu-J-ki, G;(E)T,,(k,,
j
dk, dkr @(k,)&(ki)
E,), pu; E)
ki; EC)
Go+(E)Z-,(pu+K-k,,
= (2~)-~
E,)
T,= (2a)36(kc-k,+v)7-,&-ki-J, +TTe(k,+u,
The FWL amplitude eq. (1) describes the capture of the active electron by representing it with an initial momentum distribution (wave packet) centered about the target ion which scatters through multiple collisions to form a final momentum distribution centered about the receding projectile ion. The scattering in each of the two-body collisions is expressed through a transistion operator. The partial amplitudes A, and A,, separately collect the purely electronic terms and the terms containing internuclear potentials. When A, is neglected, the resulting amplitude is identical to the one which was derived previously using a distorted-wave formalism [4] and which has been the subject of several earlier studies [5]. The following treatment of the partial amplitude A, stresses a more explicit second Born connection with its straightforward description of the forward-angle Thomas mechanism. If the transistion operators Ti (i = Pe, Te, PT) in the second-order terms of T, + T, are approximated by their first Born forms Vi and the first-order term r, is approximated by its second Born form V&l + Gz V,), then the second Born capture amplitude A,, is derived. Written in abbreviated form it is
pu; Ei) pu; E,)
k, - u; E,),
where the T,(k’, k; 6) with i = Pe, Te, PT are two-body transistion operators of momenta k’, k and off-shell energy c corresponding, respectively, to the electronprojectile, electron-target nuclear, and internuclear potentials Vi,,, V,,, and V,; the ii and 6, are the momentum-space bound-state wave functions of energies ci and
of a func-
Comparison of eqs. (1) and (2) shows that the former provides the highest extension in second order of the latter attainable in the sense that each potential, with the two exceptions noted above, is replaced by the transistion operator derived from it. Furthermore, the first-order term TpT. represents a complete summation of the pure internuclear terms in the full Born expansion. The second-order FWL approximation (and n th order ones as well) occupies a unique position then in the hierarchy of transition-operator approximations. In order to evaluate eq. (1) the small sizes of the initial and final bound-state momenta ki of order Z-r and k, of order Z, relative to the impact velocity u are used to introduce for the two-body transistion operators simplified forms recently derived by Alston [12] which are valid for the case of pure- and modified-Coulomb potentials. Specifically, one gets T(k’,
k; E) =
-27iQ(k’,
P)Q(~,
~)fc,k(E)>
(3)
with Q(k, p) = e- nv”/2F(l - iv”)[(k* - ~i~)/4p~]~” where E = p2/2m (m is a two-body reduced mass) and fk,,k( E) is the generalized form of the elastic scattering amplitude for the given potential [12]. The Sommerfeld parameter is denoted by va = Z”m/p where Za is the charge of the asymptotic, Coulomb form of the potential. Eq. (3) is equivalent to the approximation of neglecting initial- and final-state binding effects.
21
S. A&on / Faddeeu approach to electron capture
The appearance of the Q factors in eq. (3) is a consequence of the Coulomb character of the potentials asymptotically. For a Coulomb potential, the off-energy-shell T matrices involve scattering wave functions which do not contain logarithmic phase terms in their asymptotic forms whereas the wave functions in the on-shell limit do have logarithmic phase terms. An accumulation of phase leading to Q thus results when going on-shell. Since for capture, large impact velocities lead necessarily to large momentum transfers K and J, implying that the potentials are deeply probed, a further approximation is introduced whereby the amplitude f is replaced by the pure-Coulomb version f’, with scaledcharge dependence. One can write
f,c.k(E) =fk,,,c(E) =(22/lk’-k12)
e2100[(k’+k)/lk’-k1]2’Y, (4)
The phase ua equals argr(1 + iv) and v = Zm/p with Z the charge. Equation (4) has been combined with eq. (1) previouly, but without the proper and necessary treatment of eq. (3) included. Assuming an impulse approximation, earlier calculations in ref. [6] neglected one of the Q factors in each use of eq. (3). It is now known [3,12] that such a neglect is unjustified and it is not done here. In order to actually evaluate the approximation to the second-order FWL amplitude obtained when eqs. (3) and (4) are inserted into eq. (l), the dependence of A, on ki and k, is retained only in &, &, and the energy-difference factors of Q and neglected elsewhere [14]; consequently, a closed-form expression is derivable. One finds explicitly A, = CL,N,(~~U~)‘(““-~“~) x ( ZP [
,l/zz+ip=,(
“2
+
p)
-2+iG
_L,N,z,2i’8~~“~~-2+‘4’;
x
( u2 _
tz,
Z+dml-“‘kp’A’]
711’222
[
pivam(
J-2-Ziv,
u2 +
z;>
-2+iG
_L,N,z,~IV;Z:“‘~~-~+,~“;
x
(u2
_
z;)-l-i(++)
1
K-2-2iup,),
NP = r(1/2
+ 2iu,)r(l
+ ivt)/I’(2
A, = C( Z,Z,/J2K2n3)
Xexp{i[(2
L,L,
v+ - vr + 2v+ - v,)ln(4u2)
+ 2v, ln( K) + 2vr ln( J)] x / dk, dk,(kf
- 2q-‘-I”‘(
x(gi,2)l+i(G+~C)
} kf - 2~~)~~~‘“’ (5b)
Eq. (5b) is evaluated by performing the angular integrals over k, and k, analytically and the radial integrals over ki and k, numerically. Terms of order (Z,/U)~ and (Z,/O)~ have been neglected in deriving eqs. (5). In contrast to previous approximate treatments of the partial amplitude A, [S], an advantage of eq. (5b) is that use of eqs. (3) and (4) exposes more clearly the salient features of the collision. When the Coulomb and off-shell natures of eqs. (5) are neglected, implying that all six Sommerfeld parameters are set to zero, the expression for the B2 amplitude in the same approximation is derived. As a consequence of this connection and the specific forms of the factors which become unity on equating the parameters to zero, it follows [13] that the pronounced structure (peaks or valleys) [8] exhibited by the B2 amplitude at specific critical scattering angles exists also in the FWL amplitude, and at the same angles, although the shape and height or depth of the structures will be altered. This conclusion follows even though infinite numbers of terms involving each of the potentials have been summed in the FWL amplitude. Consideration of eq. (5a) when the Sommerfeld parameters are set to zero shows that, because of cancellation, A, is of order (Z,/IJ)~ or (Z,/U)~, which means, within the accuracy of the present analysis, that a zero contribution of the internuclear potential to the capture amplitude is found. This behaviour of the second-Born approximation to A, was first observed some time ago [15]. In the FWL case, however, the contribution of A,, is not zero to leading order because the factors multiplying the common velocity and momentum-transfer dependences are not the same in the different terms and thus do not cancel. In other words, the coherent addition of the double-scattering, partial amplitudes of A, with the single-scattering, nuclear amplitude is not zero when considering a proper treatment (by use of transition matrices) of the two-body collisions.
(5a)
where L, = exp( +avt
In A,, the ki and k, dependence in G$ is also retained except for the ki. k, term; a closed-form expression is not easily found. One has instead
+ ivap)2, + iv:),
and similarly for L,, NT, L,, and N,, and C= 25(ZpZr)5/2. In L, and N,, the parameters v& and vpT appear with opposite signs.
3. Results and discussion Results of calculations for proton-helium and proton-hydrogen collisions are presented in figs. 1 and 2.
S. A&on / Faddeev approach to electron capture
22
p +
00
02
He,
5 42
04 06 aLaB (mmd)
MeV
08
10
Fig. 1. The FWL Is-1s capture cross section for p-He collisions convoluted with the beam profile is compared with the experimental data taken from ref. [l] which includes capture to
all final states. Also shown are the unconvolutedcross sections dividedby a factor of 10: solid curve, FWL; long dash, FWL withoutinternuclearterms (A, only); and short dash, B2.
A comparison of the full FWL curve, i.e., the one derived from A, + A,, with experiment shows excellent agreement for the helium target [l] where the data has much better statistics and generally good agreement for
p + H, 5 0 MeV
00
02
04
06
08
10
@LAB (m-ad)
Fig. 2. The FWL Is-1s capture cross section for p-H collisions convoluted with the beam profile is compared with the experimental data taken from ref. [2] which includes capture to all final states. Also shown are the unconvoluted cross sections divided by a factor of 100: solid curve, FWL; long dash, FWL without internuclear terms (A, only); and short dash, B2.
the hydrogen target [2]. The theoretical cross sections have been convoluted with the experimental beam profiles [16]. Also shown in the figures are unconvoluted curves derived from the electronic amplitude A, alone and from the B2 approximation to it. In the numerical calculations for helium, the values assumed for the charges are Z, = 1.0, Zg = 1.0, Z, = 1.6875, Z+ = 1.0, Z,Z, = 1.6875, and ZtZ+ = 1.0. The small differences seen between the three unconvoluted curves in the large forward-peak region reflects the dominance of the first Born amplitude whose contribution is identical in all three theories. In the secondary Thomas peak (0.47 mrad) region, the FWL and A, curves agree well but the B2 peak differs considerably from them, especially in the helium case. It can be shown within the B2 approximation [17] that the Thomas peak (or hump) results from a local maximum in the imaginary part (for s states) of the amplitude, which represents scattering via on-energy-shell intermediate states. In a classical and, consequently, on-shell treatment of the capture process a similar peak in the amplitude is found [9]. A comparison of the imaginary parts of the A, and B2 amplitudes [13] shows indeed that the FWL treatment of this component of the scattering is modified. Two other notable features are apparent in these figures. One is the very much less accentuated dip in the B2 cross sections relative to the FWL ones. Indeed, for helium, only a shoulder appears. Since the dip or shoulder results from a partial cancellation in A, of the real first Born term (for Is-1s captures) with the real part of the second-order term, one sees that in the FWL amplitude much more cancellation takes place. As the real part of the second-order contributions of the B2 amplitude can be shown to represent scattering via off-energy-shell intermediate states [17], which have no classical analogue, it follows that the FWL treatment of this part of the intermediate scattering is much better according to the data. The inclusion in the calculation of the “off-shell” Q factors of eq. (3) is the reason for the improvement [ 131. The second feature is the enhanced FWL cross sections over those of the pure electronic term A, and those of the B2 approximation in the angular region beyond the local maximum. The helium data supports this increase very well while the hydrogen data do not really extend to high enough angles. The agreement of the A,, amplitude with the data beyond the Thomas peak supports the discussion of the previous section which concluded that A, gives the first explicit representation of the mechanism by which the internuclear potential contributes to the capture of the electron. The eikonal transformation of the electronic ampltiude [18] has previously been derived as a general description of and perscription for including the effects of the internuclear potential. Since the second Born approximation
S. Alston / Faddeev approach to electron capture is zero, the FWL theory is the first one to show specifically how the internuclear contribution as described by the eikonal transformation arises because incomplete cancellation occurs among the three terms in A, when they are represented at a more complete level, that is, when the true Coulomb nature of the problem is accounted for. The large differences between the FWL and A, curves in the vicinity of the local minimum similarly result from cancellation of the second-order terms with the first Born term and thus are expected to expose non-leading-order components of the second-order terms which, however, are not treated as accurately as the leading-order components. The convoluted A, cross sections (not shown) give agreement comparable to the FWL ones in the forward-peak and Thomas-peak regions but not as good agreement with experiment in the minimum region. In summary, it has been shown that a Faddeev expansion through second order of the transistion operator for electron capture leads to Is-1s differential cross sections in good agreement with the experimental data. A new explanation of how the contribution of the internuclear potential arises and fits within a time-independent scattering formalism has been put forward. Finally, a direct and explicit relationship of the FWL theory to the second Born theory has been derived: A simple picture of capture involving double scatterings is maintained while a much more exact treatment of each of the scatterings is employed. to A,
The calculations reported here were performed on the IBM 3090 mainframe at the Pennsylvania State University.
[l] E. Horsdal-Pedersen, CL. Rev. Lett. 50 (1983) 1910.
PI H. Vogt, R. Schuch,
E. Justiniano, M. Schulz, and W. Schwab, Phys. Rev. Lett. 57 (1986) 2256. 131 J. Macek and S. Alston, Phys. Rev. A26 (1982) 250; S. Alston, Phys. Rev. A38 (1988) 3124; J. Macek and X.Y. Dong, Phys. Rev. A38 (1988) 3327; J.S. Briggs, J. Phys. BlO (1977) 3075. [41 K. Taulbjerg and J.S. Brigs, J. Phys. B16 (1983) 3811. [51 S.H. Hsin and M. Lieber, Phys. Rev. A35 (1987) 4833; J.H. McGuire, N.C. Deb, N.C. Sil, and K. Taulbjerg, Phys. Rev. A35 (1987) 4830. [‘31 M.J. Roberts, J. Phys. B20 (1987) 551. [71 R.M. Drisko, Ph.D. Thesis, Carnegie Institute of Technol-
PI [91
[lOI illI
[12] [13] [14]
[15] [16]
[17] [18]
References Cocke,
and M. Stockli,
Phys.
23
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