- Email: [email protected]
The production of high-energy electrons during low energy atomic collisions in solids
Nuclear Instruments and Methods in Physics Research B 115(1996) 255-260
Beam Interactions with Materials&Atoms
ELSEYIER
The production of high-energy electrons during low energy atomic collisions in solids Mario M. Jakas * Departamento de Fkica Futuiamental y Experimental, Universidad de La Laguna. 38204 La Laguna, Tenet+,
Spain
Abstract The acceleration of electrons during low-energy atomic collisions in solids is studied. It is shown that a classical, high energy electron can be “trapped” in a sequence of head-on collisions with the ion and the target atom. Then, due to the Fermi acceleration mechanism the trapped electron may absorb energy from the center-of-mass motion up to a maximum value, above which such a sequence becomes unstable. A quantum study of the same process indicates that virtual states appear higher in the continuum of the quasi-molecule formed by the nuclei in the colliding pair, which can be used by the electron to increase its energy during the incoming part of the collision.
1. Introduction According to recent experiments [l], the interaction of slow ions with metallic surfaces may produce secondary electrons with up to a sizable fraction of the bombarding energy (Figs. la-lb). Although the low-energy part of the electron spectra is relatively well understood [2], the high energy tail in these experiments was quite unexpected. The Fermi acceleration [3] has been invoked in order to account for such observations [4,5]. In this mechanism, the between the colliding electron is temporarily “trapped” nuclei performing a series of head-on collisions with them, so it absorbs energy from the approaching nucleus up to a value that is limited by the stabilify of the collision sequence. As shown in a recent paper [4], such stability can be achieved if the electron-nuclei potential is of a screened Coulomb type and the electron energy is less than a certain value, E,,. In previous studies, however, the electron was assumed to be a classical particle. Therefore, their reliability might be questioned since the wave length of the electron is comparable to the dimension of the region within which the electron is confined. In this paper I will show that a quantum study of the same problem indicate that our previous, classical calculations were fairly accurate. A number of quasistationary states appear in place of trapping, and the life times of such states seem to be large enough to hold electrons during the incoming stage of the collision. Furthermore, such quasistationary states also have
a maximum energy in correspondence with the maximum energy for trapping in the classical counterpart. It must be mentioned that similar quantum calculations were made in the past [6-91. However, none suggested the possibility of producing high energy electrons as those in Ref. [I]. In the following sections the classical calculations will be briefly reviewed. The fundamentals of the Fermi acceleration are given in Section 2.1, whereas the possibility of trapping a classical electron in the screened Coulomb field of two nuclei are discussed in Section 2.2. The results of a quantum calculation of this problem are presented and discussed in Section 2.3. Finally, a summary and concluding remarks are offered at the end of this paper.
2. Theory 2.1. Fermi acceleration The Fermi acceleration mechanism is based on the fact that if the electron undergoes two successive head-on collisions with the projectile and the target atom, at the end its velocity will increase to 2(V, - V,); where V, and V, are the ion and target velocity before the collision. Therefore, if ut”) ’ V(“) 1 and, V:‘) are the velocities of the electron, ion and target atom after n turns of the electron around the two nuclei, one has
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PARTICLE
EMISSION
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256 electron
Jakas /Nucl.
energy
Instr. and Meth. in Phys. Res. B 115 (19961255-260
(au.)
collision sequence can be very stable because the electron can be “trapped” by the two nuclei. Prolate spherical coordinates are useful to calculate the motion of one electron in the field of two atomic nuclei [ 10,111. Suppose that two nuclei having atomic numbers Z, and Z, are located symmetrically with respect to the origin on the z-axis. Then, if r, and r2 are the distance of the electron with respect to nucleus “1” and “2”, the elliptic coordinates of such an electron become r2 -TI rl=
Ar+a --He+-Au
Au
energy
r2)
=
-e2
IkeV)
Fig. 1. Energy spectra of electrons emitted during low-energy Ar and He ions bombardment of solid Au and Mg targets, after Baragiola et al. [I].
where M, and Mz are the masses of the ion and the target, respectively; and m is the electron mass. If one assumes that V,“) = V,, V2(‘) = 0, and u(O)= a; and also that V/“) -K uCn), the finite difference system in Eqs. (1) can be readily solved. The energy of the electron after the nth turn becomes E(“) -IT, sin’(a + n+), E(‘)/E . with Mr where + = 2 J7Gf, and (Y= arctan /--TT = M, M,/( M, + M2). and E, = ~II~,V,~. These results indicate that the electron energy can increase up to a value which is approximately equal to the maximum energy available in the collision, i.e. E,. However, for such an energy to be reached the electron should realize
k= -
R
(2)
’
where R is the distance between the nuclei. The third elliptic coordinate is the azimuth angle, +, which is measured on the “x-y”-plane. The electron-nuclei potential. Other electrons in the atoms are not allowed to participate, except to provide screening of the Coulomb attraction between the active electron and the nuclei, i.e. electron-electron interactions are neglected. In this approximation, the electron is subjected to a potential of the type, fJ(f,,
electron
’
R
r2 +rl
and
( 1 :
+
22 ;
‘p(r,,
r,),
(3)
where e is the elementary charge and cp denotes the screening function. In terms of the prolate spherical coordinates such a potential can be written as
where Z, - Z, + Z, and Z, = Z, - Z,. 2.2. Trapping of a classical electron As shown in text-books [ 101,the motion of a particle in the field given by Eq. (4) can be exactly solved provided the potential has the form
(5) Hence, the screening function cp has to be written as
(6) Using atomic units, one can write the classical Hamiltonian as
turns before ejection. That is, n = 80, 130, and 250; for He on Au, Ar on Mg, and Ar on Au, respectively. If arriving to such a large number of turns were a matter of having a succession of small impact parameters as independent stochastic processes, then the Fermi acceleration would be an extremely inefficient mechanim (see Ref. [I?]). This, however, does not seem to be the case because, as shown below (see also Ref. [4]), such a
+(4(I_q2)P;
1
+ws,rl)!
where, ps, p,, and p+ are the corresponding generalized momenta.
M.M. Jakas/Nucl.
Instr. and Meth. in Phys. Res. B Il.5 (1996) 255-260
257
To avoid unnecessary complications it is assumed that p+ = 0, that is, the e 1ectron motion is restricted to move in a plane which contains the nuclei so that, Eq. (7) becomes
+-?&I)] - 2[ZsA(S) R(S2-rl’)
.
(8)
As this Hamiltonian admits separation of variables, one can readily calculate the momenta as
(9) where v,,(C)
2 Z,RA(S) + P = - 2 E;a-1 ’
2 -TdWrl) Weff(rl) = - 2 l-n2
-10
I.0
t
- P ’
(10)
p is the separation constant, and E is the energy of the electron. Since the motion along the coordinate n is bounded by definition, the problem reduces to finding whether or not V,,(t) has a potential barrier high enough to prevent (E > O&electrons from escaping to infinity. As can be readily seen in Fig. 2, if V,,(e) has a maximum at a certain 5,, with Veff(S,, ) > 0, there will be a range of energy 0 < E s Em, in which the electron will be confined to move within 1 5 5 5 t,, where E - V,,(E,) = 0. Therefore,
Fig. 2. Effective potential V,,,(e), cf Eq. (18). Curves are for several values of p, p being the separation constant (see text). (a) p>O;(b)pO.
for the existence of trapping of the electron between the nuclei. To proceed further we need an explicit expression for the screening function. To keep the calculations as simple as possible, two identical nuclei are assumed, i.e. Z, = Z, = Z. In this case, Z, = 22, and Z, = 0. Furthermore, the screening function ‘is approximated as cp(S, T) =exp[-o(R)R(S-
E nlax = max{V&k)]r
= V&S,,,).
3.0
2.0
(11)
It must be recalled that A(t) is a positive-defined function, which becomes equal to unity at .$= 1 [ A( 1) = 11 and decreases steadily to zero with increasing 5. Hence, if p > 0, the function V,, will be negative over the entire .$-axis. Thus, the whole space becomes classically accessible and trapping will not be possible. If p < 0 two possibilities arise. (i)Casep+Z,R 0. Here V,,(k) = - 30 with 5 + 1 +, whereas V,,(c) will asymptotically approach zero from positive values with increasing 5. In this case V,,(s) will have a finite maximum at a certain &,, with 1 5 &,,, < +m. It turns out, therefore, that inequalities p < 0 and p + Z,R > 0 appear as the necessary and sufficient conditions
~)/a],
(12)
where a is the screening length that will be approximated as 0.8853/Z’/3, and a(R) a numerical factor obtained by fitting Eq. (16) to the electron-nuclei potential, U(r,, r2), assumed to have the form Z U(r,,
Z
rz) =
(‘3)
where &T)(r) is the Thomas-Fermi screening function in the Moliere approximation. The results of the fitting suggested that a(R) = (1 + R)/2. The approximation in Fq. (11) implies that B(n) = 0 and A(c)
=
2ZE; exp[ -a(R)R(t-
~)/a].
Note that Eq. (12) reproduces reasonably well the essential features of a screening function: it goes to unity at the nuclei and decreases monotonically at large distances. Note that such an approximation produces no screening along the internuclear axis. This, however, does not seem to be a serious limitation, since a bounded motion should be looked for on 5, not on the q-coordinate. III. EXCITATION/SECONDARY PARTICLE EMISSION
M.M. Jakas / Nucl. Insir. and Meth. in Phys. Res. B I15 (1996) 255-260
258
If one assumes a wave function of the form [lo] + =X(s) Y(n) exp(iA$), where A = 0, 1, . . + stands for the azimuthal quantum number, the radial and angular part of the wave function becomes
Improving the screening function in this respect will have nearly no effect on V,,(@. The maximum energy for trapping, E,,,,, can be readily obtained from Eq. (1 I). A rough estimate yields E max= 2Z/a,
(14)
which differs from the result in Ref. [4] since such previous calculations relied on an independent-nuclei approximation, whereas a molecular interaction is assumed in this paper. According to our results above, one can conceivably imagine that one electron will stay orbiting around the two nuclei as long as its energy is smaller than E,,. Therefore, if one electron becomes trapped in such a type of trajectory, the Fermi acceleration can increase the energy up to -&ax unless, of course, interactions with other electrons produce an early ejection of the electron, in which case the energy will be smaller than E,,. In this regard, E,,,, appears as an upper limit above which no electrons should be observed. 2.3. High energy quasistationary
[
+4zRA(s)+p
X=0,
$1
-n’);
- &
[
+2ER2(1
1
+4ZRB(n)-l3
Y-O,
where p is the separation constant. The most remarkable result of using Eq. (16) however, is that having assumed that B(n) = 0 the equation for the Y-wave function becomes that of a free electron. In this case the angular equation can be exactly solved [ 111. The solutions are the spheroidal wave functions, viz. Y(n) = S,,,($, where n = 0, 1, . . . is the angular quantum number, and p is determined by the energy and the quantum numbers A and n, i.e. 13A,n.The eigenvalues PA., are tabulated in Ref. [I 11, or can be obtained by a procedure described in Ref. [9]. Finally, A = 0 will be assumed, so that the radial equation becomes
In this subsection, our previous analysis will be re-formulated using quantum mechanics. The quantum Hamiltonian for one electron in the screened Coulomb field of two nuclei at rest becomes
(15)
1
2
45
-1.dX-
where
R2 de2
A=
R2(t2-
4Z%S)
LX_ 1) dt
(18)
This corresponds to the equation of one electron having total energy E with the same “effective” potential as that in the classical case, c.f. Eq. (10).
(16)
c
+ P0.n x
R2( t2 - 1)
=EX.
“,
52
1
states
cp(S,$ +=W,
-$t2-1)$-& + 2EP(
LO-
c 0
_
m
Lz
o0
1
2
1
0
2
0
1
2
3
R (a.u.1 Fig. 3. Resonance energy as a function of the internuclear (b), and 50 cc), respectively.
separation;
they were obtained for nuclei of atomic number 2, = 2, = 15 (a); 30
M.M. Jakus/Nucl. 10
z 5
Instr. and Meth. in Phys. Res. B II5 (1996) 25.7-260
111’1
0-
(a)Z=lS 12
Resonance Fig. 4. Lifetimes of the resonances
as functions
of the resonance
As in the classical case, if PeTn is negative and 4ZR + PO.” becomes greater than zero, then, the potential V,J() should have a “well” around 5 = 1 where electrons with energies greater than zero can be trapped. Analogously, the maximum energy for trapping is 2Z/a. QuasisMonary states. The occurrence of a potential well, however, does not imply that electrons can be trapped. To see whether V,, can host virtual states I used a well-documented method [ 141 based on the fact that, at large distances, the phase of the wave-function must show an increase of IT across the resonance. Therefore, the wave-function X(s) is calculated starting from 5 = 1 up to distances where the effective potential becomes negligible. Fitting the wave function at large 5 to the asymptotic expression [ 111
sin( kc + 4) X(S) = A
5
’
259
(19)
with k = R$%, the phase C$ is readily obtained. The procedure above is repeated a number of times, on a coarse sweep along the energy E. Then, if a sudden change of the phase 4 occurs, a finer sweep is carried out until the resonance energy is isolated to a pre-determined accuracy. As the amplitude of the fitted expression shows a minimum around the resonance, the resonance width Iis approximated by the half width at half maximum of Am2 around the resonance energy, and the lifetime of the resonance is obtained as 7 = l/r.
3. Results and discussions Figs. 3 and 4 show the results of calculating the quasistationary states for Z= 15, 30 and 50. In Figs. 3a-3c the resonances are shown as functions of the internuclear separation, whereas in Figs. 4a-c the corresponding life-times are displayed as functions of the energy. One can readily see that a number of resonances appear for the
energy
(a.u.)
energy. Results are in correspondence
to those in Figs. 3a-3c.
three cases. They go from lower energies up to tens, or more than a hundred in the case of Z = 50. Observe that they agree with our results in the previous section, in the sense that the higher the atomic number the larger the resonance energy. In all the cases the maximum resonance energies are lower than the predicted value, i.e. 2Z/a. However, this is not at all unexpected, since 2Z/a was obtained as a conservative upper limit rather than a least upper bound. Observe that with an increase of the nuclear separation the maximum energy of the quasistationary states increases as well. Then, the resonances are also seen going up to large energies. One, however, has to be cautious about these results since our screening function turns into a fairly bad approximation at large separations. Observe that at lower energies the resonances exhibit a rich structure. The curves have different slopes and they cross each other giving rise to a complicated pattern. At times, two resonances are seen for the same quantum number: One at small separations, whereas the other, notably less steeper, appears at large separations. The life-times of the resonances show a strong dependence on the energy, decaying as Em’; with 10 < s < 13 nearly independent of Z.
4. Summary Using both classical and quantum mechanics, it can be shown that high energy electrons can be trapped in the attraction field of two slowly approaching atoms. A maximum energy for such a trapping appears, i.e. E,,. No confinement, or alternatively no quasi-stationary state exists for energies greater than E,,, which is found to increase with the atomic number of the atoms in the colliding pair. In this regard, our results agree with the experiments in Ref. [l] in the sense that the larger the atomic number the higher the energy of the ejected electrons. However. more work is needed before a conclusive
III. EXCITATION/SECONDARY
PARTICLE
EMISSION
260
M.M. Jakas/Nucl.
Insrr. and Meth. in Phys. Res. B I15 (1996) 255-240
connection between such trapped electrons and the ejection of high energy electrons can be established. The occurrence of both the classical trapping and the high energy virtual states, nevertheless, appear to be part of an interesting phenomenon per se, and also a promising first step towards a more complete accounting for the production of high energy electrons during slow atomic collisions.
Acknowledgements This work was supported in part by the Gobiemo Atdnomo de Canarias through the Consejeria de Educacibn, Cultura y Deportes. The author is grateful to R.A. Baragiola for a careful reading of this manuscript.
References [I] R.A. Baragiola,
E.V. Alonso, A. Oliva. A. Bonnano and F. Xu, Phys. Rev. A 45 (1992) 5286. [2] R.A. Baragiola, Nucl. Instr. and Meth. B 78 (1993) 223, and references therein.
[31 E. Fermi, Phys. Rev. 75 (1949) 1169. [41 M.M. Jakas, Phys. Rev. A 52 (199.5) 866. M.M. Jakas, A. Hem&ndez-Cabrera and P. Aceituno, Mod. Phys. Lett. B 9 (1995) 299. El B. Burghardt and M. Vicanek, Nucl. Instr. and Meth. B 101 (1995) 303. 161 V.K. Nikulin, and N.A. Guschina, J. Phys. B 11 (1978) 3553. [71 E.A. Solov’ev, Sov. Phys. JETP 54 (1981) 893. [81D.I. Abramov, S.Yu. Ovchinnikov and EA. Solov’ev, Phys. Rev. A 42 (199016366. [91 Y.S. Tergiman, Phys. Rev. A 48 (1993) 88. 1101 L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon, 1976) and, Quantum Mechanics, Non relativistic theory (Pergamon, 1977). [I 11 M. Abramowitz and I.A. Stegun (Dover, New York, 1972). 1121 E.W. McDaniel, Atomic Collisions, Electron and Photons Projectiles (Wiley, 1989) p. 236. [I31 Observe that our quantum number Bn,i\ relates to A,,, in Ref. [ 111 as B,r,h = cz - A,,,, where c* = ER2/2. 1141 M.S. Child, Molecular Collision Theory (Academic Press, London, 1974) p. 50. See also, Z. Chen, I. Ben-Itzhak, C.D. Lin, W. Koch, G. Frenking, I. Gertner and B. Rosner. Phys. Rev. A 49 (1994) 3472.
Beam Interactions with Materials&Atoms
ELSEYIER
The production of high-energy electrons during low energy atomic collisions in solids Mario M. Jakas * Departamento de Fkica Futuiamental y Experimental, Universidad de La Laguna. 38204 La Laguna, Tenet+,
Spain
Abstract The acceleration of electrons during low-energy atomic collisions in solids is studied. It is shown that a classical, high energy electron can be “trapped” in a sequence of head-on collisions with the ion and the target atom. Then, due to the Fermi acceleration mechanism the trapped electron may absorb energy from the center-of-mass motion up to a maximum value, above which such a sequence becomes unstable. A quantum study of the same process indicates that virtual states appear higher in the continuum of the quasi-molecule formed by the nuclei in the colliding pair, which can be used by the electron to increase its energy during the incoming part of the collision.
1. Introduction According to recent experiments [l], the interaction of slow ions with metallic surfaces may produce secondary electrons with up to a sizable fraction of the bombarding energy (Figs. la-lb). Although the low-energy part of the electron spectra is relatively well understood [2], the high energy tail in these experiments was quite unexpected. The Fermi acceleration [3] has been invoked in order to account for such observations [4,5]. In this mechanism, the between the colliding electron is temporarily “trapped” nuclei performing a series of head-on collisions with them, so it absorbs energy from the approaching nucleus up to a value that is limited by the stabilify of the collision sequence. As shown in a recent paper [4], such stability can be achieved if the electron-nuclei potential is of a screened Coulomb type and the electron energy is less than a certain value, E,,. In previous studies, however, the electron was assumed to be a classical particle. Therefore, their reliability might be questioned since the wave length of the electron is comparable to the dimension of the region within which the electron is confined. In this paper I will show that a quantum study of the same problem indicate that our previous, classical calculations were fairly accurate. A number of quasistationary states appear in place of trapping, and the life times of such states seem to be large enough to hold electrons during the incoming stage of the collision. Furthermore, such quasistationary states also have
a maximum energy in correspondence with the maximum energy for trapping in the classical counterpart. It must be mentioned that similar quantum calculations were made in the past [6-91. However, none suggested the possibility of producing high energy electrons as those in Ref. [I]. In the following sections the classical calculations will be briefly reviewed. The fundamentals of the Fermi acceleration are given in Section 2.1, whereas the possibility of trapping a classical electron in the screened Coulomb field of two nuclei are discussed in Section 2.2. The results of a quantum calculation of this problem are presented and discussed in Section 2.3. Finally, a summary and concluding remarks are offered at the end of this paper.
2. Theory 2.1. Fermi acceleration The Fermi acceleration mechanism is based on the fact that if the electron undergoes two successive head-on collisions with the projectile and the target atom, at the end its velocity will increase to 2(V, - V,); where V, and V, are the ion and target velocity before the collision. Therefore, if ut”) ’ V(“) 1 and, V:‘) are the velocities of the electron, ion and target atom after n turns of the electron around the two nuclei, one has
* E-mail [email protected]. 0168-583X/96/$15.00
Copyright 0 1996 Elsevier
SSDIO168-583X(95)01559-0
Science B.V. All rights resewed III. EXCITATION/SECONDARY
PARTICLE
EMISSION
MN.
256 electron
Jakas /Nucl.
energy
Instr. and Meth. in Phys. Res. B 115 (19961255-260
(au.)
collision sequence can be very stable because the electron can be “trapped” by the two nuclei. Prolate spherical coordinates are useful to calculate the motion of one electron in the field of two atomic nuclei [ 10,111. Suppose that two nuclei having atomic numbers Z, and Z, are located symmetrically with respect to the origin on the z-axis. Then, if r, and r2 are the distance of the electron with respect to nucleus “1” and “2”, the elliptic coordinates of such an electron become r2 -TI rl=
Ar+a --He+-Au
Au
energy
r2)
=
-e2
IkeV)
Fig. 1. Energy spectra of electrons emitted during low-energy Ar and He ions bombardment of solid Au and Mg targets, after Baragiola et al. [I].
where M, and Mz are the masses of the ion and the target, respectively; and m is the electron mass. If one assumes that V,“) = V,, V2(‘) = 0, and u(O)= a; and also that V/“) -K uCn), the finite difference system in Eqs. (1) can be readily solved. The energy of the electron after the nth turn becomes E(“) -IT, sin’(a + n+), E(‘)/E . with Mr where + = 2 J7Gf, and (Y= arctan /--TT = M, M,/( M, + M2). and E, = ~II~,V,~. These results indicate that the electron energy can increase up to a value which is approximately equal to the maximum energy available in the collision, i.e. E,. However, for such an energy to be reached the electron should realize
k= -
R
(2)
’
where R is the distance between the nuclei. The third elliptic coordinate is the azimuth angle, +, which is measured on the “x-y”-plane. The electron-nuclei potential. Other electrons in the atoms are not allowed to participate, except to provide screening of the Coulomb attraction between the active electron and the nuclei, i.e. electron-electron interactions are neglected. In this approximation, the electron is subjected to a potential of the type, fJ(f,,
electron
’
R
r2 +rl
and
( 1 :
+
22 ;
‘p(r,,
r,),
(3)
where e is the elementary charge and cp denotes the screening function. In terms of the prolate spherical coordinates such a potential can be written as
where Z, - Z, + Z, and Z, = Z, - Z,. 2.2. Trapping of a classical electron As shown in text-books [ 101,the motion of a particle in the field given by Eq. (4) can be exactly solved provided the potential has the form
(5) Hence, the screening function cp has to be written as
(6) Using atomic units, one can write the classical Hamiltonian as
turns before ejection. That is, n = 80, 130, and 250; for He on Au, Ar on Mg, and Ar on Au, respectively. If arriving to such a large number of turns were a matter of having a succession of small impact parameters as independent stochastic processes, then the Fermi acceleration would be an extremely inefficient mechanim (see Ref. [I?]). This, however, does not seem to be the case because, as shown below (see also Ref. [4]), such a
+(4(I_q2)P;
1
+ws,rl)!
where, ps, p,, and p+ are the corresponding generalized momenta.
M.M. Jakas/Nucl.
Instr. and Meth. in Phys. Res. B Il.5 (1996) 255-260
257
To avoid unnecessary complications it is assumed that p+ = 0, that is, the e 1ectron motion is restricted to move in a plane which contains the nuclei so that, Eq. (7) becomes
+-?&I)] - 2[ZsA(S) R(S2-rl’)
.
(8)
As this Hamiltonian admits separation of variables, one can readily calculate the momenta as
(9) where v,,(C)
2 Z,RA(S) + P = - 2 E;a-1 ’
2 -TdWrl) Weff(rl) = - 2 l-n2
-10
I.0
t
- P ’
(10)
p is the separation constant, and E is the energy of the electron. Since the motion along the coordinate n is bounded by definition, the problem reduces to finding whether or not V,,(t) has a potential barrier high enough to prevent (E > O&electrons from escaping to infinity. As can be readily seen in Fig. 2, if V,,(e) has a maximum at a certain 5,, with Veff(S,, ) > 0, there will be a range of energy 0 < E s Em, in which the electron will be confined to move within 1 5 5 5 t,, where E - V,,(E,) = 0. Therefore,
Fig. 2. Effective potential V,,,(e), cf Eq. (18). Curves are for several values of p, p being the separation constant (see text). (a) p>O;(b)p
for the existence of trapping of the electron between the nuclei. To proceed further we need an explicit expression for the screening function. To keep the calculations as simple as possible, two identical nuclei are assumed, i.e. Z, = Z, = Z. In this case, Z, = 22, and Z, = 0. Furthermore, the screening function ‘is approximated as cp(S, T) =exp[-o(R)R(S-
E nlax = max{V&k)]r
= V&S,,,).
3.0
2.0
(11)
It must be recalled that A(t) is a positive-defined function, which becomes equal to unity at .$= 1 [ A( 1) = 11 and decreases steadily to zero with increasing 5. Hence, if p > 0, the function V,, will be negative over the entire .$-axis. Thus, the whole space becomes classically accessible and trapping will not be possible. If p < 0 two possibilities arise. (i)Casep+Z,R
~)/a],
(12)
where a is the screening length that will be approximated as 0.8853/Z’/3, and a(R) a numerical factor obtained by fitting Eq. (16) to the electron-nuclei potential, U(r,, r2), assumed to have the form Z U(r,,
Z
rz) =
(‘3)
where &T)(r) is the Thomas-Fermi screening function in the Moliere approximation. The results of the fitting suggested that a(R) = (1 + R)/2. The approximation in Fq. (11) implies that B(n) = 0 and A(c)
=
2ZE; exp[ -a(R)R(t-
~)/a].
Note that Eq. (12) reproduces reasonably well the essential features of a screening function: it goes to unity at the nuclei and decreases monotonically at large distances. Note that such an approximation produces no screening along the internuclear axis. This, however, does not seem to be a serious limitation, since a bounded motion should be looked for on 5, not on the q-coordinate. III. EXCITATION/SECONDARY PARTICLE EMISSION
M.M. Jakas / Nucl. Insir. and Meth. in Phys. Res. B I15 (1996) 255-260
258
If one assumes a wave function of the form [lo] + =X(s) Y(n) exp(iA$), where A = 0, 1, . . + stands for the azimuthal quantum number, the radial and angular part of the wave function becomes
Improving the screening function in this respect will have nearly no effect on V,,(@. The maximum energy for trapping, E,,,,, can be readily obtained from Eq. (1 I). A rough estimate yields E max= 2Z/a,
(14)
which differs from the result in Ref. [4] since such previous calculations relied on an independent-nuclei approximation, whereas a molecular interaction is assumed in this paper. According to our results above, one can conceivably imagine that one electron will stay orbiting around the two nuclei as long as its energy is smaller than E,,. Therefore, if one electron becomes trapped in such a type of trajectory, the Fermi acceleration can increase the energy up to -&ax unless, of course, interactions with other electrons produce an early ejection of the electron, in which case the energy will be smaller than E,,. In this regard, E,,,, appears as an upper limit above which no electrons should be observed. 2.3. High energy quasistationary
[
+4zRA(s)+p
X=0,
$1
-n’);
- &
[
+2ER2(1
1
+4ZRB(n)-l3
Y-O,
where p is the separation constant. The most remarkable result of using Eq. (16) however, is that having assumed that B(n) = 0 the equation for the Y-wave function becomes that of a free electron. In this case the angular equation can be exactly solved [ 111. The solutions are the spheroidal wave functions, viz. Y(n) = S,,,($, where n = 0, 1, . . . is the angular quantum number, and p is determined by the energy and the quantum numbers A and n, i.e. 13A,n.The eigenvalues PA., are tabulated in Ref. [I 11, or can be obtained by a procedure described in Ref. [9]. Finally, A = 0 will be assumed, so that the radial equation becomes
In this subsection, our previous analysis will be re-formulated using quantum mechanics. The quantum Hamiltonian for one electron in the screened Coulomb field of two nuclei at rest becomes
(15)
1
2
45
-1.dX-
where
R2 de2
A=
R2(t2-
4Z%S)
LX_ 1) dt
(18)
This corresponds to the equation of one electron having total energy E with the same “effective” potential as that in the classical case, c.f. Eq. (10).
(16)
c
+ P0.n x
R2( t2 - 1)
=EX.
“,
52
1
states
cp(S,$ +=W,
-$t2-1)$-& + 2EP(
LO-
c 0
_
m
Lz
o0
1
2
1
0
2
0
1
2
3
R (a.u.1 Fig. 3. Resonance energy as a function of the internuclear (b), and 50 cc), respectively.
separation;
they were obtained for nuclei of atomic number 2, = 2, = 15 (a); 30
M.M. Jakus/Nucl. 10
z 5
Instr. and Meth. in Phys. Res. B II5 (1996) 25.7-260
111’1
0-
(a)Z=lS 12
Resonance Fig. 4. Lifetimes of the resonances
as functions
of the resonance
As in the classical case, if PeTn is negative and 4ZR + PO.” becomes greater than zero, then, the potential V,J() should have a “well” around 5 = 1 where electrons with energies greater than zero can be trapped. Analogously, the maximum energy for trapping is 2Z/a. QuasisMonary states. The occurrence of a potential well, however, does not imply that electrons can be trapped. To see whether V,, can host virtual states I used a well-documented method [ 141 based on the fact that, at large distances, the phase of the wave-function must show an increase of IT across the resonance. Therefore, the wave-function X(s) is calculated starting from 5 = 1 up to distances where the effective potential becomes negligible. Fitting the wave function at large 5 to the asymptotic expression [ 111
sin( kc + 4) X(S) = A
5
’
259
(19)
with k = R$%, the phase C$ is readily obtained. The procedure above is repeated a number of times, on a coarse sweep along the energy E. Then, if a sudden change of the phase 4 occurs, a finer sweep is carried out until the resonance energy is isolated to a pre-determined accuracy. As the amplitude of the fitted expression shows a minimum around the resonance, the resonance width Iis approximated by the half width at half maximum of Am2 around the resonance energy, and the lifetime of the resonance is obtained as 7 = l/r.
3. Results and discussions Figs. 3 and 4 show the results of calculating the quasistationary states for Z= 15, 30 and 50. In Figs. 3a-3c the resonances are shown as functions of the internuclear separation, whereas in Figs. 4a-c the corresponding life-times are displayed as functions of the energy. One can readily see that a number of resonances appear for the
energy
(a.u.)
energy. Results are in correspondence
to those in Figs. 3a-3c.
three cases. They go from lower energies up to tens, or more than a hundred in the case of Z = 50. Observe that they agree with our results in the previous section, in the sense that the higher the atomic number the larger the resonance energy. In all the cases the maximum resonance energies are lower than the predicted value, i.e. 2Z/a. However, this is not at all unexpected, since 2Z/a was obtained as a conservative upper limit rather than a least upper bound. Observe that with an increase of the nuclear separation the maximum energy of the quasistationary states increases as well. Then, the resonances are also seen going up to large energies. One, however, has to be cautious about these results since our screening function turns into a fairly bad approximation at large separations. Observe that at lower energies the resonances exhibit a rich structure. The curves have different slopes and they cross each other giving rise to a complicated pattern. At times, two resonances are seen for the same quantum number: One at small separations, whereas the other, notably less steeper, appears at large separations. The life-times of the resonances show a strong dependence on the energy, decaying as Em’; with 10 < s < 13 nearly independent of Z.
4. Summary Using both classical and quantum mechanics, it can be shown that high energy electrons can be trapped in the attraction field of two slowly approaching atoms. A maximum energy for such a trapping appears, i.e. E,,. No confinement, or alternatively no quasi-stationary state exists for energies greater than E,,, which is found to increase with the atomic number of the atoms in the colliding pair. In this regard, our results agree with the experiments in Ref. [l] in the sense that the larger the atomic number the higher the energy of the ejected electrons. However. more work is needed before a conclusive
III. EXCITATION/SECONDARY
PARTICLE
EMISSION
260
M.M. Jakas/Nucl.
Insrr. and Meth. in Phys. Res. B I15 (1996) 255-240
connection between such trapped electrons and the ejection of high energy electrons can be established. The occurrence of both the classical trapping and the high energy virtual states, nevertheless, appear to be part of an interesting phenomenon per se, and also a promising first step towards a more complete accounting for the production of high energy electrons during slow atomic collisions.
Acknowledgements This work was supported in part by the Gobiemo Atdnomo de Canarias through the Consejeria de Educacibn, Cultura y Deportes. The author is grateful to R.A. Baragiola for a careful reading of this manuscript.
References [I] R.A. Baragiola,
E.V. Alonso, A. Oliva. A. Bonnano and F. Xu, Phys. Rev. A 45 (1992) 5286. [2] R.A. Baragiola, Nucl. Instr. and Meth. B 78 (1993) 223, and references therein.
[31 E. Fermi, Phys. Rev. 75 (1949) 1169. [41 M.M. Jakas, Phys. Rev. A 52 (199.5) 866. M.M. Jakas, A. Hem&ndez-Cabrera and P. Aceituno, Mod. Phys. Lett. B 9 (1995) 299. El B. Burghardt and M. Vicanek, Nucl. Instr. and Meth. B 101 (1995) 303. 161 V.K. Nikulin, and N.A. Guschina, J. Phys. B 11 (1978) 3553. [71 E.A. Solov’ev, Sov. Phys. JETP 54 (1981) 893. [81D.I. Abramov, S.Yu. Ovchinnikov and EA. Solov’ev, Phys. Rev. A 42 (199016366. [91 Y.S. Tergiman, Phys. Rev. A 48 (1993) 88. 1101 L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon, 1976) and, Quantum Mechanics, Non relativistic theory (Pergamon, 1977). [I 11 M. Abramowitz and I.A. Stegun (Dover, New York, 1972). 1121 E.W. McDaniel, Atomic Collisions, Electron and Photons Projectiles (Wiley, 1989) p. 236. [I31 Observe that our quantum number Bn,i\ relates to A,,, in Ref. [ 111 as B,r,h = cz - A,,,, where c* = ER2/2. 1141 M.S. Child, Molecular Collision Theory (Academic Press, London, 1974) p. 50. See also, Z. Chen, I. Ben-Itzhak, C.D. Lin, W. Koch, G. Frenking, I. Gertner and B. Rosner. Phys. Rev. A 49 (1994) 3472.