- Email: [email protected]
Ionization of heavy targets by impact of relativistic projectiles
Nuclear Instruments and Methods in Physics Research B35 (1988) loo-102 North-Holland. Amsterdam
100
Letter to the Editor IONIZATION Gustav0
OF HEAVY TARGETS
Ricardo
BY IMPACT OF RELATIVISTIC
DECO ‘)*, Pablo Daniel FAINSTEIN
PROJECTILES
‘) and Roberto Daniel RIVAROLA
‘) Institut ftir Theoretische Physik, Justus-Lit-big-Universitiit, Giessen, FRG ‘) Institute Balseiro and Centro At6mico Bariloche, 8400 San Carlos de Bariloche, Argentina ” Laboratoire des ColIisions Atomiques * * , Universite Bordeaux I, 351, Cows de la Liberation,
3,
33405 Talence, France
Received 14 July 1988
Electron ejection from atomic targets by impact of bare heavy projectiles at relativistic collision energies is studied theoretically. First-order Born calculations are presented by using initial Darwin and final Sommerfeld-Maue wavefunctions. Comparisons with other calculations and experimental data are given.
Electron ionization by impact of bare heavy ions on atomic targets at relativistic collision energies is studied theoretically in the present work. This process is related to inner shell-vacancy production. We will focus our attention on K-shell electrons. Previous calculations have been done by other authors [l-4], in order to represent existing experimental data [4,5]. In theoretical studies, the first-order Born approximation has been used in a one active-independent electron model. Different initial and final wavefunctions have been employed: semirelativistic Darwin initial and final spinors [1,2], or partial wave expansions in terms of relativistic Coulomb-Dirac spinors with defined angular momentum [3]. In the last case, the transition operator is described by a multipole expansion. We present here first-order Born calculations into the straight-line impact parameter approximation, but choosing a Darwin wavefunction
to represent the initial electron Somerfeld-Maue spinor
X,F,[
-iv,,
1,
bound state and a
-i(pr,+p-+)I
(2)
to represent the electron in a continuum state of the target. In expressions (1) and (2), Z, is the nuclear charge of the target, c is the speed of light, A, = Z,/2c, YT = Z,/u, with IJ, the final electron velocity, y, = (1 ei and cf are the initial and final electronic v;/ca)-“*, energies, S = c/( c* + e f ), (r are the 2 x 2 Pauli matrices, (Y are the 4 x 4 Dirac matrices, r, and p are the electron position vector and the final electron momentum referred to a frame fixed on the target nucleus from which the process is described, pi and pf are the initial and final projections of the spin of the electron, and xr is given by 1
(1
0
0
x1/2
=
)
x-1/2=
(
The scattering amplitude parameter p results in:
1
1 as a function
of the impact
* Absent from Instituto de Fisica Rosario, Universidad Na-
**
cional de Rosario and Consejo National de Investigaciones Cientificas y Tkcnicas, Av. Pellegrini 250, 2000 Rosario, Argentina. Equipe de Recherche no. 260, du Centre National de la
Recherche Scientifique. 0168-583X/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where /3 = u/c with u the collision velocity, y = (1 I/* I Z p is the nuclear charge of the projectile and
P-
G.R. Deco et al. / Ionization by impact of relativistic projectiles
r,’ is the position vector of the electron referred to a frame fixed on the projectile nucleus. Expression (4) includes the current-current interaction between the projectile and the target electron. By using the Fourier transform method it is possible to write the scattering amplitude under the form
f-31/2,1/2 x
=
G-1/2.x
101
1/2
=s(p,-ip,)A+iX,(B,-iB,) +$(DX-iD,) e
(13)
G’/2,- i/2 = G-1/2,1/2 x
x
=
•C iX,B,
-Sp,A
- &Dz,
(14)
e
where
with (6)
( K2 + Z+)( p
A=v,
+
(1 - iv,)Z,,
(K2+K+-2p*K-2ipZT)
where w = cr - ci, Q is the transverse momentum transfer, 4 = (rl, w/u) and Or, = e-iC~‘~/~ and a/“ = e-ic,’ ~“r @f
+ iZ,)
(15)
.
1
(K2+Z+)p,-2(p.K+iZTp)Kj
Thus, the double differential cross section as a function of the final energy ef and of the solid angle $2 of the ejected electron is obtained as
(K*+Z+--2p*K-2ipZ,)
+K,
(16)
> i ( PKj + ‘Z, Pj
D,=iv,(K’+Z$)
Expression (7) contains the contributions coming from the change of spin of the electron and also the average of the spin of the initial state. In order to facilitate the integration of expression (6), we take q to be the polar axis .z [1,2], so that
(8) with FWi,Pr=
/
drr(#‘)+@;i
GJl,“f = /drr(
eigz,
@:r)t~,@~~ eiqz.
(9) (IO)
Inserting expressions (1) and (2) into expressions (9) and (10) we have F’/2,‘/2 = F-
1/2.-
1/2
=A+~[~=~=+(~~-iPY)(~~+i~~)]
+iV[d4+
(~~-i~,)(~,+i~,)], (11)
Fl/2/.-
l/2
=
F-1/2,1/2
(12)
)
(K2+Z+--p-K-2ipZT)’ (17) K,=P,,
i=x,
y;
Kz =P,
- 4,
(18)
In the evaluation of the term F’/2*‘/2, contributions coming from a product of a * v operators have been neglected, because they introduce corrections of the order ( Z,/C)~. It must be noted that orders superior to (Z,/c)’ are neglected here. Then, the total cross section u can be obtained through a multiple integration over the tranverse momentum transfer, the direction of ejection of the electron and the final electron energy. If atomic targets with two K-shell electrons are considered, total cross sections obtained from eq. (7) must be multipled by a factor of 2. In these cases and in order to take into account the influence of the passive electrons, effective charges Z$ = Z, - 0.3 and experimental binding energies have been considered. In fig. 1, total cross sections are shown for impact of protons as a function of the nuclear charge Z,, for a 4.88 GeV collision energy. Present theoretical results are compared with other theoretical calculations from refs. [2] (see also ref. [l]) and [3] and with experimental data for Ta, Pt, Au, Pb and U [5]. At this high impact velocity (u = 135.21 a.u.), our model gives a very good description of the measured points. This success could be attributed to a correct representation of large angular momentum when a Sommerfeld-Maue wavefunction is employed [6]. Large angular momentum contributions are neglected in the case of using a partial wave expansion.
102
G. R. Deco et al. / ~vn~at~vn by impact of relativistic prvjkctifes
70
80
90 Zr
Fig. 1. Total cross sections for K-shell target ionization by impact of 4.88 GeV-protons, as a function of the nuclear charge Z,. Theoretical calculations: ( -) present ones; (---) from ref. [2]; (.‘*.-v) from ref. [3]. Experimental data from ref. [S].
In fig. 2, total cross sections are presented for impact of bare Ne ions on different targets, for a 670 MeV/amu collision energy. Present theoretical results are compared with calculations from refs. [3] and [4] and with experimental data extracted from ref. [4]. In ref. [4] total cross sections have been obtained in a first-order Born approximation similar to the one developed in ref. [2]. At this lower impact velocity (u = 111.29 a.u.) the representation of experimental data is poorer than in the case previously analysed. However, it must be noted that the dispersion of measured points is also greater in the present case. So new experimental data would be welcome. For the heavier targets, polarization and binding effects could play a role in the determination of total cross sections. Thus, it would be interesting to develop a higher order distorted wave theory, as previously introduced for norelativistic collisions [7,8]. This theory which works for all charges Z, and Z,, takes into .account those effects by including the projectile-electron interaction in the initial and final wave functions, and gives more complete information on the process than that obtained from the pioneering methods developed by other authors [9,10]. This approach is presently under investigation. References
Ill D.M. DavidoviE, B.L. Moiseiwistsch and P.H. Norrington, J. Phys. Bll (1978) 847.
3
121R. Anholt, Phys. Al9 (1979) 1004. (31 U. Becker, N. Grtin and B. Scheid, J. Phys. BIB (1985) 4589.
[41 R. Anholt, W.E. Meyerhof, Ch. Staller, E. Moretuoni, S.
60
70
80
90
Zr
Fig. 2. Same as that in fig. 1, but for impact of 670 MeV/amu Ne bare ions. Theoretical calculations: () present ones; (- - -) from ref. (41; (7 . . . .) from ref. [3]. Experimental data from ref. [4].
Andriamonje, J.D. Molitoris, O.K. Baker, D.H.H. Hoffman, H. Bowman, J.S. Xu, Z.Z. Xu, K. Frankel, D. Murphy, K. Crowe and J.O. Rasmussen, Phys. Rev. A30 (1984) 2234. [61 R. Anholt, S. Nagamiya, J.O. Rasmussen, H. Bowman, J.G. Ioannou and E. Rauscher, Phys. Rev. Al4 (1976) 2103. 161 V.B. Beresteiskii, E.M. Lifshitx and L.P. Pitaevskil, Teoria Cuantica Relativista (Editorial Revert&, Barcelona, 1971). [71 D.S.F. Crothers and J.F. McCann, J. Phys. B16 (1983) 3229. PI P.D. Fainstein, V.H. Ponce and R.D. Rivarola, Phys. Rev. A36 (1987) 3639. 191 G. Basbas, W. Brandt and R. Laubert, Phys. Rev. A7 (1973) 983. WI G. Basbas, W. Brandt and R. Laubert, Phys. Rev. Al7 (1978) 1655.
100
Letter to the Editor IONIZATION Gustav0
OF HEAVY TARGETS
Ricardo
BY IMPACT OF RELATIVISTIC
DECO ‘)*, Pablo Daniel FAINSTEIN
PROJECTILES
‘) and Roberto Daniel RIVAROLA
‘) Institut ftir Theoretische Physik, Justus-Lit-big-Universitiit, Giessen, FRG ‘) Institute Balseiro and Centro At6mico Bariloche, 8400 San Carlos de Bariloche, Argentina ” Laboratoire des ColIisions Atomiques * * , Universite Bordeaux I, 351, Cows de la Liberation,
3,
33405 Talence, France
Received 14 July 1988
Electron ejection from atomic targets by impact of bare heavy projectiles at relativistic collision energies is studied theoretically. First-order Born calculations are presented by using initial Darwin and final Sommerfeld-Maue wavefunctions. Comparisons with other calculations and experimental data are given.
Electron ionization by impact of bare heavy ions on atomic targets at relativistic collision energies is studied theoretically in the present work. This process is related to inner shell-vacancy production. We will focus our attention on K-shell electrons. Previous calculations have been done by other authors [l-4], in order to represent existing experimental data [4,5]. In theoretical studies, the first-order Born approximation has been used in a one active-independent electron model. Different initial and final wavefunctions have been employed: semirelativistic Darwin initial and final spinors [1,2], or partial wave expansions in terms of relativistic Coulomb-Dirac spinors with defined angular momentum [3]. In the last case, the transition operator is described by a multipole expansion. We present here first-order Born calculations into the straight-line impact parameter approximation, but choosing a Darwin wavefunction
to represent the initial electron Somerfeld-Maue spinor
X,F,[
-iv,,
1,
bound state and a
-i(pr,+p-+)I
(2)
to represent the electron in a continuum state of the target. In expressions (1) and (2), Z, is the nuclear charge of the target, c is the speed of light, A, = Z,/2c, YT = Z,/u, with IJ, the final electron velocity, y, = (1 ei and cf are the initial and final electronic v;/ca)-“*, energies, S = c/( c* + e f ), (r are the 2 x 2 Pauli matrices, (Y are the 4 x 4 Dirac matrices, r, and p are the electron position vector and the final electron momentum referred to a frame fixed on the target nucleus from which the process is described, pi and pf are the initial and final projections of the spin of the electron, and xr is given by 1
(1
0
0
x1/2
=
)
x-1/2=
(
The scattering amplitude parameter p results in:
1
1 as a function
of the impact
* Absent from Instituto de Fisica Rosario, Universidad Na-
**
cional de Rosario and Consejo National de Investigaciones Cientificas y Tkcnicas, Av. Pellegrini 250, 2000 Rosario, Argentina. Equipe de Recherche no. 260, du Centre National de la
Recherche Scientifique. 0168-583X/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where /3 = u/c with u the collision velocity, y = (1 I/* I Z p is the nuclear charge of the projectile and
P-
G.R. Deco et al. / Ionization by impact of relativistic projectiles
r,’ is the position vector of the electron referred to a frame fixed on the projectile nucleus. Expression (4) includes the current-current interaction between the projectile and the target electron. By using the Fourier transform method it is possible to write the scattering amplitude under the form
f-31/2,1/2 x
=
G-1/2.x
101
1/2
=s(p,-ip,)A+iX,(B,-iB,) +$(DX-iD,) e
(13)
G’/2,- i/2 = G-1/2,1/2 x
x
=
•C iX,B,
-Sp,A
- &Dz,
(14)
e
where
with (6)
( K2 + Z+)( p
A=v,
+
(1 - iv,)Z,,
(K2+K+-2p*K-2ipZT)
where w = cr - ci, Q is the transverse momentum transfer, 4 = (rl, w/u) and Or, = e-iC~‘~/~ and a/“ = e-ic,’ ~“r @f
+ iZ,)
(15)
.
1
(K2+Z+)p,-2(p.K+iZTp)Kj
Thus, the double differential cross section as a function of the final energy ef and of the solid angle $2 of the ejected electron is obtained as
(K*+Z+--2p*K-2ipZ,)
+K,
(16)
> i ( PKj + ‘Z, Pj
D,=iv,(K’+Z$)
Expression (7) contains the contributions coming from the change of spin of the electron and also the average of the spin of the initial state. In order to facilitate the integration of expression (6), we take q to be the polar axis .z [1,2], so that
(8) with FWi,Pr=
/
drr(#‘)+@;i
GJl,“f = /drr(
eigz,
@:r)t~,@~~ eiqz.
(9) (IO)
Inserting expressions (1) and (2) into expressions (9) and (10) we have F’/2,‘/2 = F-
1/2.-
1/2
=A+~[~=~=+(~~-iPY)(~~+i~~)]
+iV[d4+
(~~-i~,)(~,+i~,)], (11)
Fl/2/.-
l/2
=
F-1/2,1/2
(12)
)
(K2+Z+--p-K-2ipZT)’ (17) K,=P,,
i=x,
y;
Kz =P,
- 4,
(18)
In the evaluation of the term F’/2*‘/2, contributions coming from a product of a * v operators have been neglected, because they introduce corrections of the order ( Z,/C)~. It must be noted that orders superior to (Z,/c)’ are neglected here. Then, the total cross section u can be obtained through a multiple integration over the tranverse momentum transfer, the direction of ejection of the electron and the final electron energy. If atomic targets with two K-shell electrons are considered, total cross sections obtained from eq. (7) must be multipled by a factor of 2. In these cases and in order to take into account the influence of the passive electrons, effective charges Z$ = Z, - 0.3 and experimental binding energies have been considered. In fig. 1, total cross sections are shown for impact of protons as a function of the nuclear charge Z,, for a 4.88 GeV collision energy. Present theoretical results are compared with other theoretical calculations from refs. [2] (see also ref. [l]) and [3] and with experimental data for Ta, Pt, Au, Pb and U [5]. At this high impact velocity (u = 135.21 a.u.), our model gives a very good description of the measured points. This success could be attributed to a correct representation of large angular momentum when a Sommerfeld-Maue wavefunction is employed [6]. Large angular momentum contributions are neglected in the case of using a partial wave expansion.
102
G. R. Deco et al. / ~vn~at~vn by impact of relativistic prvjkctifes
70
80
90 Zr
Fig. 1. Total cross sections for K-shell target ionization by impact of 4.88 GeV-protons, as a function of the nuclear charge Z,. Theoretical calculations: ( -) present ones; (---) from ref. [2]; (.‘*.-v) from ref. [3]. Experimental data from ref. [S].
In fig. 2, total cross sections are presented for impact of bare Ne ions on different targets, for a 670 MeV/amu collision energy. Present theoretical results are compared with calculations from refs. [3] and [4] and with experimental data extracted from ref. [4]. In ref. [4] total cross sections have been obtained in a first-order Born approximation similar to the one developed in ref. [2]. At this lower impact velocity (u = 111.29 a.u.) the representation of experimental data is poorer than in the case previously analysed. However, it must be noted that the dispersion of measured points is also greater in the present case. So new experimental data would be welcome. For the heavier targets, polarization and binding effects could play a role in the determination of total cross sections. Thus, it would be interesting to develop a higher order distorted wave theory, as previously introduced for norelativistic collisions [7,8]. This theory which works for all charges Z, and Z,, takes into .account those effects by including the projectile-electron interaction in the initial and final wave functions, and gives more complete information on the process than that obtained from the pioneering methods developed by other authors [9,10]. This approach is presently under investigation. References
Ill D.M. DavidoviE, B.L. Moiseiwistsch and P.H. Norrington, J. Phys. Bll (1978) 847.
3
121R. Anholt, Phys. Al9 (1979) 1004. (31 U. Becker, N. Grtin and B. Scheid, J. Phys. BIB (1985) 4589.
[41 R. Anholt, W.E. Meyerhof, Ch. Staller, E. Moretuoni, S.
60
70
80
90
Zr
Fig. 2. Same as that in fig. 1, but for impact of 670 MeV/amu Ne bare ions. Theoretical calculations: () present ones; (- - -) from ref. (41; (7 . . . .) from ref. [3]. Experimental data from ref. [4].
Andriamonje, J.D. Molitoris, O.K. Baker, D.H.H. Hoffman, H. Bowman, J.S. Xu, Z.Z. Xu, K. Frankel, D. Murphy, K. Crowe and J.O. Rasmussen, Phys. Rev. A30 (1984) 2234. [61 R. Anholt, S. Nagamiya, J.O. Rasmussen, H. Bowman, J.G. Ioannou and E. Rauscher, Phys. Rev. Al4 (1976) 2103. 161 V.B. Beresteiskii, E.M. Lifshitx and L.P. Pitaevskil, Teoria Cuantica Relativista (Editorial Revert&, Barcelona, 1971). [71 D.S.F. Crothers and J.F. McCann, J. Phys. B16 (1983) 3229. PI P.D. Fainstein, V.H. Ponce and R.D. Rivarola, Phys. Rev. A36 (1987) 3639. 191 G. Basbas, W. Brandt and R. Laubert, Phys. Rev. A7 (1973) 983. WI G. Basbas, W. Brandt and R. Laubert, Phys. Rev. Al7 (1978) 1655.