- Email: [email protected]
Dynamical image charge effects on convoy electron emission from solid surfaces
Nuclear Instruments and Methods North-Holland, Amsterdam
in Physics
Research
B24/25
(1987) 139-142
139
DYNAMICAL IMAGE CHARGE EFFECTS ON CONVOY ELECTRON EMISSION FROM SOLID SURFACES Joachim
B~RGD~RFER
Ph_vsics &partment, University of Tennessee, KnoxviNe, TN 37996-1200, and Oak Ridge National Laboratory, Oak Ridge, TN 378314377, USA
USA
We analyze surface effects on the spectrum of convoy electrons emitted from solids in close association with fast highly charged ions. Three competing mechanisms have been identified: Deceleration by the surface barrier, acceleration by the repulsive dynamical image force, and energy loss due to surface plasmon generation, The resulting energy shift is significantly reduced compared to the value for the barrier height for a jellium surface. We also give estimates for the probability for nonadiabatic transitions between bound states due to the sudden perturbation by the potential step upon exit.
1. In~~uction The velocity distribution of electrons ejected into forward direction by fast highly charged ions traversing thin solid targets displays a cusp-shaped peak. This electron cusp appears at electron velocities, 4, that approximately match the projectile velocity, ur,, in both mag~tude and direction. Recent experimental evidences [l-3] indicate that the production of these “convoy” electrons is a bulk effect, that is, the strong phasespace correlation between the ion and the accompanying electron is established inside the foil and maintained over a distance X, corresponding to a mean free path (MFP) for convoy electrons. In several experiments, X, has been observed to be enhanced relative to the inelastic MFP of isotachic free electrons [3-41. The question is then posed: what is the effect of the rapid changes in the potential near the exit surface on the correlated electron-ion motion. A naive picture would suggest a velocity (or energy) shift of the convoy peak relative to the ionic velocity due to the deceleration by the surface barrier. Such an energy shift of = 10 eV for keV electrons should be observable with high-resolution electron spectrometer [5]. Changes in the velocity of the ion, on the other hand, should be entirely negligible because of the small electron/ion mass ratio (in atomic Units, l/~i~” (c: l>In the following we present a semiquantitative discussion of surface effects on convoy electrons which turn out to be significantly more complex than implied by the elementary model mentioned above. Competing mechanisms have to be considered: the deceleration by the surface barrier, the acceleration by the image field produced by the receding ion, and the energy loss by plasmon excitation. We employ in our study a (nearly) free electron gas model. The complex dynamical prob0168-583X/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics abusing Division)
lem of a correlated electron-ion
motion in the presence of surface interactions will be analyzed from two different somewhat complementary viewpoints. We first calculate the energy shift of convoy electrons using standard dynamical charge theory [6,7] where we take into account the strong phase-space correlation between the electron and the ion but treat them as free in the sence that we neglect their direct mutual interaction on the exit-surface effects. We then take the opposite view and give an estimate for nonadiabatic transitions between strongly bound states of the electron-ion system due to the sudden perturbation generated by the surface barrier.
2. Dynamical image potential inducedenergy shift In the Somerfeld model [8] for a semi-infinite electron gas, conduction electrons are bound by a potential step V(z) = I@(-r),
(I) where the unit step function is denoted by 8 and the surface is located at z = 0. The height of the barrier, I$, is given in terms of the Fermi energy er (= I/2(9~/4) ‘/‘rsp2) and the work function W by v,=
-(tr+
w).
(2)
We express the parameters of the electron gas in terms of the one-electron radius r, in a metal. A more refined model for the surface barrier has to take into account the finite width of the transition region between the bulk and the vacuum. Outside the solid (z > 0) the dominant contribution is provided by the image charge potential which is due to virtual surface plasmon excitation. Employing a plasmon-pole I. ATOMIC
PHYSICS
/ RELATED
PHENOMENA
J. Burgdiirfer / Conuoy electron emission
140
approximation to the dielectric account the surface-plasmon potential is given by [7]
VI(z) =
function and taking into dispersion, the image
exp(-2Qz) +/“dQ 0 m:(Q)
where the surface-plasmon proximated by w,(Q)
dispersion
relation
= (w,’ + oQ + /3Q2 + Q4/4)1’2,
is ap-
(4)
with o,=
r/2
(I 3 7 2r,
&,,(z,
the surface-plasmon a =
frequency.
Furthermore,
(gcFy20s
(6)
and /_I is chosen such that the surface plasmon branch and the bulk plasmon branch enter the particle-hole continuum at the same point. At large distances z + cc, eq. (3) reduces to the classical image potential -l/(42). In the limit z --) 0, eq. (3) yields a finite value which may be considered as a collective response contribution to the effective height V, of the barrier. Accordingly, the independent-particle part of the barrier height is Va V,(O). We use in our study dynamical generalizations [7] of the image potential for finite velocities u. The u-dependent image potential corresponding to the real part of the self-energy is V,(z,
(b) the interaction with the surface barrier part of which is the image potential and is dynamically modified at high speeds according to eq. (7) (c) the irreversible energy loss due to real plasmon excitation (eq. (8)) and (d) the indirect Coulomb interaction through the image charge of the multiply charged ion. The latter leads to an acceleration and is of crucial importance for an estimate of the effective velocity (energy) shift. Using the close phase space correlation between the projectile ion and the convoy electron, i.e. rP z r,, uP z u,, the image force due to the ion of charge q can be approximately written as
exd-Qz)
u) = - $/,“dQ
dQ)‘+ exp(-Qz)+-sin
u2Q2
2Qv w,(Q)
as(Q)z
(
~
V
)I (7)
The imaginary part of the self-energy describes the energy loss due to real plasmon excitations. For an outgoing trajectory parallel to the surface normal we find for the electronic energy loss
Q2
wp, = -2&&v22 dQ /
(dQ)’
+ 02Q2)’
(8)
A convoy electron associated with an emergent ion experiences competing forces upon exit: (a) the direct Coulomb interaction between the electron and the ion which does not contribute to the energy shift to first order (the Coulomb interaction enters the zero&order Hamiltonian which generates the projectile-centered Coulomb wave and defines the “cusp” states),
up) = 4;
vr(z,
c,).
(9)
The momentum outward passage
transfer to the electron is then given by
A P = jomdG’rJ
z, Dp) = - ;vr(c,
during
up).
the
(10)
In eq. (10) we have used that 1AP 1
AP=
-qV,(O,
up).
(11)
Eq. (11) differs from the expression for the energy shift in the chemisorption problem [9] where the electronic energy in the field of a fixed ion is considered. It should be noted that the linear response theory underlying eq. (7) may break down for q z+ 1 and at small distances from the surface. Summing over all contributions (b)-(d) we find for the energy shift in the lab (target) frame A<,=
Vo- VI(O)-(q-l)V,(O,
3. Numerical
v,)+
Wpl.
(12)
results
Fig. 1 shows the image charge contribution to the barrier VI(O, up) and the energy loss due to plasmon excitation, Wpl. In the static limit (up 4 0) only the surface barrier contributes while at high velocities Wpl dominates. A numerical estimate for AC, requires the knowledge of experimental, up dependent, mean charge states cl. We use an empirical formula of Betz [lo] for 4. The resulting energy shift (eq. (12)) for Oq+ with average charge q (fig. 2) shows a drastic reduction compared to the shift predicted by the static barrier height V, for aluminum and gold. In order to test the feasibility of a measurement of the shift by using projectile autoionization lines in forward and backward direction in the frame of the projectile as reference points we have transformed the energy shift into the
J. Burgdijrfer / Convoy electron emission
I
I
I
I
I
I
0
Ill
-0.4
I
’
Au AI
___----_-
_----____
,’ -0.4
-
-0.5
-
_o,6
________---------______-a
a
0.1
I
fl
s d -0.3
s
I -
-0.2
-
141
-
AU
--------~~__
,_--.
0
-0.2
-40 -20 s E0
-30
2p -40 -50
-0.3
I C,
I
2
3
4
5
6
7
0
-60
V (a. u.1 Fig. 1. Image potential Vt(r = 0, or), -; and energy loss due to surface plasmon excitation Wri, - - - as a function of the velocity up of the receding electron from an aluminum surface ( rs = 2.07).
projectile
-70
1 3
I
I
I
I
4
5
6
7
8
v (a. U.)
Fig. 2. Energy shift of a convoy electron for aluminum ( rs = 2.07) and gold targets (rs = 3.01) (a) in lab frame; (b) in projectile frame; - - shift as predicted by the static barrier height Vo.
frame (fig. 2b) using (13)
The shifts are only of the order of meV at high velocities and can easily escape detection. The present investigations suggests that the shift should depend on the target material as well as the outgoing charge state. It also should be pointed out that the absence of a shift cannot be used as an indication for exit-surface production as opposed to a bulk production of convoy electrons as long as the compensating processes discussed above are not properly taken into account.
consider the hydrogenic 1s --, 2p transition in an ion of charge state 4 during the transit through the step discontinuity. The transition amplitude is given in first Born approximation by 1 ‘1,,2p_
=-/
i
* _
dt exp[it(c,-ei)](2pm]V(t)
(14) with
(15)
4. Bound-state excitation at the surface In the preceding two sections we treated the projectile-convoy electron system as strongly phase-space correlated but essentially noninteracting (i.e., neglecting the direct Coulomb interaction). In this section we take the opposite view and considered nonadiabatic transitions between bound states of the electron-projectile system. In order to maximize nonadiabatic effects we use the step potential (eq. (1)). As an example we
]ls),
m
We note first that eq. (15) renders the selection rule Am = 0 for normal incidence and a quantization axis chosen to be parallel to the surface normal. This selection rule has important implications regarding the shape of the charge cloud formed by excitation near the surface. Transitions induced by the step discontinuity would lead to an alignment along the beam axis. We note parenthetically that this is at variance with recent experimental observations [ll]. I. ATOMIC
PHYSICS
/ RELATED
PHENOMENA
J. Burgdfirjer / Convoy electron emission
142
of eq. (14) is straightforward
The evaluation
leading
transit time T is exclusively determined by the size of the orbit& involved, eqs. (16) and (18) suggests that the transition probabi~ty
to
P Is-2pa=
X
becomes
w4
1+
+2
1+ ( wT)2
[
l-3(oT)2
vided
3 tl+(WT)2)Z
J
Ia Is -
2pg I2
large for large orbital
(20)
radii (i.e. large 7) pro-
OT GZ1 still holds. We have calculated P Is _ 2po using the parameter q to simulate large, weakly bound orbitals (fig. 3). The value q = 0.22 corresponds to an orbital radius (1s) of about the lattice constant (= 6 a.u.) and a binding energy of 0.66 eV. At high velocities up B= 1 the excitation probability at the exit surface is only of the order of a few percent even for weakly bound states. This agrees with the observation that the observed angular distribution [ll] does not display the longitudinal alignment predicted by eq. (15).
.
In eq. (16) we have introduced abbreviations for the effective transit time of the orbital through the surface step
074
that
and for the inelasticity w=cr-ei=l.
*9*.
4. Conclusions
(17b)
Obviously, the transition amplitude is maximal when the process is strongly nonadiabatic, i.e., wr < 1. In this limit, eq. 916) becomes ~l~_+O=(-i)~&r,
In our se~qu~titative survey of possible exit surface effects on the convoy electron emission from solid targets we have approached the complex dynamical problem from two opposite ends: energy shift of quasifree electron ion pairs versus excitations of bound states. The two vastly different models reveal a common trend: at high speeds surface effects produce only a minor perturbation on correlated electron-ion pairs formed in the bulk. A detailed classical Monte Carlo study of the correlated electron-ion motion in the bulk is in progress [12].
(18)
Eq. (18) has a simple intuitive interpretation in terms of the change in the phase of the wave function during the rapid passage through the surface region, !Pk(~)=qv(O)
exp( -i]vdr)
= *k(O) exp( -i const V,T),
(19) which reduces to eq. (18) to first order in 0,. Accordingly, surface effects can be neglected if Y$ < 1. Noting that for a step-shaped surface potential the
It is a pleasure to thank R. Ritchie and I. Sellin for helpful discussions. This work is supported in part by the NSF and by US DOE under contract no. DE-AC05 840R21400 with Martin Marietta Energy Systems, Inc.
References
40-3
1
I
I
I
I
I
2
3
4
5
6
7
V, (a. u.) Fig. 3. Excitation probability 1s -+ 2pa as a function of projectile velocity (Va = 0.5 a.u.) for different values of 4,
8
[l] LA. Sellin et al., in: Forward Electron Ejection in Ion Collisions, eds., K. Groeneveld, W. Meckbach and I.A. Sellin, Lecture Notes in Physics 213 (Springer, Berlin, 1984) p. 109. [2] H.D. Betz, in ref. [l] p. 115. [3] R. Schramm, P. Koschar, H.D. Betz, M. Burkhard, 0. Heil and K.-O. Groeneveld, J. Phys. B18 (1985) L507. [4] LA. Sellin et al., J. Phys. B19 (1986) L155. [S] N. Stolterfoht and Y. Yamazaki, private communication [6] R. Ray and G.D. Mahan, Phys. Lett. A42 (1972) 301. [7] P. Echenique, R.H. Ritchie, N.Barber&n and J. Inkson, Phys. Rev. B23 (1981) 6486. [8] N. Ashcroft and N. Mermin, Solid State Physics (HoltSaunders, P~ladelp~a, 1981) ch. 2; B.A. Trubnikov and Y.N. Yavlinskii, Sov. phys. JETPZS (1967) 1089. [9] J.D. Muscat and D.M. Newns, Prog. Surf. Science 9 (1978) 1. [lo] H.D. Betz, Rev. Mod. Phys. 44 (1972) 465. [ll] SD. Berry et al., J. Phys. Bl9 (1986) L149. [12] J. Burgdiirfer and C. Bottcher, to be published.
in Physics
Research
B24/25
(1987) 139-142
139
DYNAMICAL IMAGE CHARGE EFFECTS ON CONVOY ELECTRON EMISSION FROM SOLID SURFACES Joachim
B~RGD~RFER
Ph_vsics &partment, University of Tennessee, KnoxviNe, TN 37996-1200, and Oak Ridge National Laboratory, Oak Ridge, TN 378314377, USA
USA
We analyze surface effects on the spectrum of convoy electrons emitted from solids in close association with fast highly charged ions. Three competing mechanisms have been identified: Deceleration by the surface barrier, acceleration by the repulsive dynamical image force, and energy loss due to surface plasmon generation, The resulting energy shift is significantly reduced compared to the value for the barrier height for a jellium surface. We also give estimates for the probability for nonadiabatic transitions between bound states due to the sudden perturbation by the potential step upon exit.
1. In~~uction The velocity distribution of electrons ejected into forward direction by fast highly charged ions traversing thin solid targets displays a cusp-shaped peak. This electron cusp appears at electron velocities, 4, that approximately match the projectile velocity, ur,, in both mag~tude and direction. Recent experimental evidences [l-3] indicate that the production of these “convoy” electrons is a bulk effect, that is, the strong phasespace correlation between the ion and the accompanying electron is established inside the foil and maintained over a distance X, corresponding to a mean free path (MFP) for convoy electrons. In several experiments, X, has been observed to be enhanced relative to the inelastic MFP of isotachic free electrons [3-41. The question is then posed: what is the effect of the rapid changes in the potential near the exit surface on the correlated electron-ion motion. A naive picture would suggest a velocity (or energy) shift of the convoy peak relative to the ionic velocity due to the deceleration by the surface barrier. Such an energy shift of = 10 eV for keV electrons should be observable with high-resolution electron spectrometer [5]. Changes in the velocity of the ion, on the other hand, should be entirely negligible because of the small electron/ion mass ratio (in atomic Units, l/~i~” (c: l>In the following we present a semiquantitative discussion of surface effects on convoy electrons which turn out to be significantly more complex than implied by the elementary model mentioned above. Competing mechanisms have to be considered: the deceleration by the surface barrier, the acceleration by the image field produced by the receding ion, and the energy loss by plasmon excitation. We employ in our study a (nearly) free electron gas model. The complex dynamical prob0168-583X/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics abusing Division)
lem of a correlated electron-ion
motion in the presence of surface interactions will be analyzed from two different somewhat complementary viewpoints. We first calculate the energy shift of convoy electrons using standard dynamical charge theory [6,7] where we take into account the strong phase-space correlation between the electron and the ion but treat them as free in the sence that we neglect their direct mutual interaction on the exit-surface effects. We then take the opposite view and give an estimate for nonadiabatic transitions between strongly bound states of the electron-ion system due to the sudden perturbation generated by the surface barrier.
2. Dynamical image potential inducedenergy shift In the Somerfeld model [8] for a semi-infinite electron gas, conduction electrons are bound by a potential step V(z) = I@(-r),
(I) where the unit step function is denoted by 8 and the surface is located at z = 0. The height of the barrier, I$, is given in terms of the Fermi energy er (= I/2(9~/4) ‘/‘rsp2) and the work function W by v,=
-(tr+
w).
(2)
We express the parameters of the electron gas in terms of the one-electron radius r, in a metal. A more refined model for the surface barrier has to take into account the finite width of the transition region between the bulk and the vacuum. Outside the solid (z > 0) the dominant contribution is provided by the image charge potential which is due to virtual surface plasmon excitation. Employing a plasmon-pole I. ATOMIC
PHYSICS
/ RELATED
PHENOMENA
J. Burgdiirfer / Conuoy electron emission
140
approximation to the dielectric account the surface-plasmon potential is given by [7]
VI(z) =
function and taking into dispersion, the image
exp(-2Qz) +/“dQ 0 m:(Q)
where the surface-plasmon proximated by w,(Q)
dispersion
relation
= (w,’ + oQ + /3Q2 + Q4/4)1’2,
is ap-
(4)
with o,=
r/2
(I 3 7 2r,
&,,(z,
the surface-plasmon a =
frequency.
Furthermore,
(gcFy20s
(6)
and /_I is chosen such that the surface plasmon branch and the bulk plasmon branch enter the particle-hole continuum at the same point. At large distances z + cc, eq. (3) reduces to the classical image potential -l/(42). In the limit z --) 0, eq. (3) yields a finite value which may be considered as a collective response contribution to the effective height V, of the barrier. Accordingly, the independent-particle part of the barrier height is Va V,(O). We use in our study dynamical generalizations [7] of the image potential for finite velocities u. The u-dependent image potential corresponding to the real part of the self-energy is V,(z,
(b) the interaction with the surface barrier part of which is the image potential and is dynamically modified at high speeds according to eq. (7) (c) the irreversible energy loss due to real plasmon excitation (eq. (8)) and (d) the indirect Coulomb interaction through the image charge of the multiply charged ion. The latter leads to an acceleration and is of crucial importance for an estimate of the effective velocity (energy) shift. Using the close phase space correlation between the projectile ion and the convoy electron, i.e. rP z r,, uP z u,, the image force due to the ion of charge q can be approximately written as
exd-Qz)
u) = - $/,“dQ
dQ)‘+ exp(-Qz)+-sin
u2Q2
2Qv w,(Q)
as(Q)z
(
~
V
)I (7)
The imaginary part of the self-energy describes the energy loss due to real plasmon excitations. For an outgoing trajectory parallel to the surface normal we find for the electronic energy loss
Q2
wp, = -2&&v22 dQ /
(dQ)’
+ 02Q2)’
(8)
A convoy electron associated with an emergent ion experiences competing forces upon exit: (a) the direct Coulomb interaction between the electron and the ion which does not contribute to the energy shift to first order (the Coulomb interaction enters the zero&order Hamiltonian which generates the projectile-centered Coulomb wave and defines the “cusp” states),
up) = 4;
vr(z,
c,).
(9)
The momentum outward passage
transfer to the electron is then given by
A P = jomdG’rJ
z, Dp) = - ;vr(c,
during
up).
the
(10)
In eq. (10) we have used that 1AP 1
AP=
-qV,(O,
up).
(11)
Eq. (11) differs from the expression for the energy shift in the chemisorption problem [9] where the electronic energy in the field of a fixed ion is considered. It should be noted that the linear response theory underlying eq. (7) may break down for q z+ 1 and at small distances from the surface. Summing over all contributions (b)-(d) we find for the energy shift in the lab (target) frame A<,=
Vo- VI(O)-(q-l)V,(O,
3. Numerical
v,)+
Wpl.
(12)
results
Fig. 1 shows the image charge contribution to the barrier VI(O, up) and the energy loss due to plasmon excitation, Wpl. In the static limit (up 4 0) only the surface barrier contributes while at high velocities Wpl dominates. A numerical estimate for AC, requires the knowledge of experimental, up dependent, mean charge states cl. We use an empirical formula of Betz [lo] for 4. The resulting energy shift (eq. (12)) for Oq+ with average charge q (fig. 2) shows a drastic reduction compared to the shift predicted by the static barrier height V, for aluminum and gold. In order to test the feasibility of a measurement of the shift by using projectile autoionization lines in forward and backward direction in the frame of the projectile as reference points we have transformed the energy shift into the
J. Burgdijrfer / Convoy electron emission
I
I
I
I
I
I
0
Ill
-0.4
I
’
Au AI
___----_-
_----____
,’ -0.4
-
-0.5
-
_o,6
________---------______-a
a
0.1
I
fl
s d -0.3
s
I -
-0.2
-
141
-
AU
--------~~__
,_--.
0
-0.2
-40 -20 s E0
-30
2p -40 -50
-0.3
I C,
I
2
3
4
5
6
7
0
-60
V (a. u.1 Fig. 1. Image potential Vt(r = 0, or), -; and energy loss due to surface plasmon excitation Wri, - - - as a function of the velocity up of the receding electron from an aluminum surface ( rs = 2.07).
projectile
-70
1 3
I
I
I
I
4
5
6
7
8
v (a. U.)
Fig. 2. Energy shift of a convoy electron for aluminum ( rs = 2.07) and gold targets (rs = 3.01) (a) in lab frame; (b) in projectile frame; - - shift as predicted by the static barrier height Vo.
frame (fig. 2b) using (13)
The shifts are only of the order of meV at high velocities and can easily escape detection. The present investigations suggests that the shift should depend on the target material as well as the outgoing charge state. It also should be pointed out that the absence of a shift cannot be used as an indication for exit-surface production as opposed to a bulk production of convoy electrons as long as the compensating processes discussed above are not properly taken into account.
consider the hydrogenic 1s --, 2p transition in an ion of charge state 4 during the transit through the step discontinuity. The transition amplitude is given in first Born approximation by 1 ‘1,,2p_
=-/
i
* _
dt exp[it(c,-ei)](2pm]V(t)
(14) with
(15)
4. Bound-state excitation at the surface In the preceding two sections we treated the projectile-convoy electron system as strongly phase-space correlated but essentially noninteracting (i.e., neglecting the direct Coulomb interaction). In this section we take the opposite view and considered nonadiabatic transitions between bound states of the electron-projectile system. In order to maximize nonadiabatic effects we use the step potential (eq. (1)). As an example we
]ls),
m
We note first that eq. (15) renders the selection rule Am = 0 for normal incidence and a quantization axis chosen to be parallel to the surface normal. This selection rule has important implications regarding the shape of the charge cloud formed by excitation near the surface. Transitions induced by the step discontinuity would lead to an alignment along the beam axis. We note parenthetically that this is at variance with recent experimental observations [ll]. I. ATOMIC
PHYSICS
/ RELATED
PHENOMENA
J. Burgdfirjer / Convoy electron emission
142
of eq. (14) is straightforward
The evaluation
leading
transit time T is exclusively determined by the size of the orbit& involved, eqs. (16) and (18) suggests that the transition probabi~ty
to
P Is-2pa=
X
becomes
w4
1+
+2
1+ ( wT)2
[
l-3(oT)2
vided
3 tl+(WT)2)Z
J
Ia Is -
2pg I2
large for large orbital
(20)
radii (i.e. large 7) pro-
OT GZ1 still holds. We have calculated P Is _ 2po using the parameter q to simulate large, weakly bound orbitals (fig. 3). The value q = 0.22 corresponds to an orbital radius (1s) of about the lattice constant (= 6 a.u.) and a binding energy of 0.66 eV. At high velocities up B= 1 the excitation probability at the exit surface is only of the order of a few percent even for weakly bound states. This agrees with the observation that the observed angular distribution [ll] does not display the longitudinal alignment predicted by eq. (15).
.
In eq. (16) we have introduced abbreviations for the effective transit time of the orbital through the surface step
074
that
and for the inelasticity w=cr-ei=l.
*9*.
4. Conclusions
(17b)
Obviously, the transition amplitude is maximal when the process is strongly nonadiabatic, i.e., wr < 1. In this limit, eq. 916) becomes ~l~_+O=(-i)~&r,
In our se~qu~titative survey of possible exit surface effects on the convoy electron emission from solid targets we have approached the complex dynamical problem from two opposite ends: energy shift of quasifree electron ion pairs versus excitations of bound states. The two vastly different models reveal a common trend: at high speeds surface effects produce only a minor perturbation on correlated electron-ion pairs formed in the bulk. A detailed classical Monte Carlo study of the correlated electron-ion motion in the bulk is in progress [12].
(18)
Eq. (18) has a simple intuitive interpretation in terms of the change in the phase of the wave function during the rapid passage through the surface region, !Pk(~)=qv(O)
exp( -i]vdr)
= *k(O) exp( -i const V,T),
(19) which reduces to eq. (18) to first order in 0,. Accordingly, surface effects can be neglected if Y$ < 1. Noting that for a step-shaped surface potential the
It is a pleasure to thank R. Ritchie and I. Sellin for helpful discussions. This work is supported in part by the NSF and by US DOE under contract no. DE-AC05 840R21400 with Martin Marietta Energy Systems, Inc.
References
40-3
1
I
I
I
I
I
2
3
4
5
6
7
V, (a. u.) Fig. 3. Excitation probability 1s -+ 2pa as a function of projectile velocity (Va = 0.5 a.u.) for different values of 4,
8
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