Download Image Potential and Charge-Transfer Phenomena in Atom (Ion

CHINESE JOURNAL PHYSICS
VOL. 26, NO. 4
OCTOBER 1988
Image Potential and Charge-Transfer
Phenomena in Atom (Ion)-Surface Collisions
Y.C.Cheng( $$j~@>andK.C.Lin( j$z@)
Department of Physics, National Taiwan University, Taiwan 10 764
(Received October 11,1988)
The image potential between an ion and a metallic surface is
calculated via a second-order perturbation approach for various metallic
electron densities varying from rs = 2 to 6. The results are compared
with the classical image potential -e2/4z. A critical distance zc, for
each rs, is determined such that for z < zc the classical result is no
longer valid. The importance of the image potential and zc in the
theory of charge transfer in atom (ion)-surface collisions is discussed.
I. INTRODUCTION
Charge exchange between a metal surface and an atom, or ion, has been observed in
many ion- and atom-surface scattering experiments. For example, if the incoming particles
are He+ or H’, then the scattered beam may contain He and He+ or H’, H and H: The distributions of the scattered particles among the various charged states are strongly dependent
on the velocity of the incoming particles and the nature of the surface.’ Therefore experiments of this kind are important in many techniques of surface analysis, such as ion-beam
scattering spectroscopy (ISS), neutral-beam scattering spectroscopy (NSS), and secondaryion mass spectroscopy (SIMS).
In molecule-surface scattering experiments, there may be involved a short-lived charge
transfer between a neutral molecule and a surface. An incoming neutral molecule may temporary become a negatively charged ion when it is close enough to the surface, and then
becomes neutral again as it reflects away from the surface. The reflected neutral molecule
becomes highly vibrational excited because of its temporary transformation to a negative
ion near the surface.2 Such processes are important in the study of the dynamics of
molecular processes at solid surfaces and may also be related to other charge-transfer phenomena such as sputtering, chemiluminescence and stimulated desorption, etc.*
To understand charge-transfer phenomena it is necessary to know the multi-potential212
Y.C.CHENGAND K.C.LIN
213
energy surfaces involved in the scattering process.
If we denote A as a neutral atom (or
molecule) and A+ and A- as its positively and negatively charged ions respectively, then the
two most frequently encountered charge-transfer processes are written as: (a) A (incoming)
+ A + A+ + A- (outgoing), and (b) A+ (incoming) + A+ + A + A- (outgoing). In process (a),
a neutral atom (or molecule) may emerge as a charged (either positively or negatively) ion
by exchanging an electron with the surface. Alternatively the emerging atom (molecule)
may remain neutral but a short-lived charge exchange occurred when the atom (molecule)
was in close contact with the surface as we have mentioned above. In process (b), a positively charged ion may obtain one, or two, electrons from the surface to become a neutral atom
(molecule) or a negatively charged ion respectively. The distributions of the scattered
particles among the various charged states are strongly dependent on the potential energy+ of
each charged state near the surface.
For process (a), the relevant potentials U are:
U,(a) = U,(z) + U,(z) ,
(1)
U,(a) = U,(z) + U,(z) + IO - W ,
(2)
U,-(a) = U,(z) + U,(z) + W - A0 ,
(3)
where z is the distance from the atom (ion) to the surface. U,, VW, and U, are the
repulsive, van der Waals, and image potentials respectively. W is the work function of the
metal. IO and A,, are the ionization energy and the electron affinity of the free atom
(molecule) respectively. For process (b), the relevant U are:
U,+(b) = U,(z) + U,(z) ,
(4)
U,(b) = U,(z) + U,(z) + W - IO 7
(9
U,-(b) = UR(z) + U , ( z ) + 2 W - IO - Ao.
(6)
We have assumed that U,, is the same for all A, A+ and A-and for both processes. The same
assumption is made for U,. The velocity dependence of these potential energies is small except for velocities much larger than the Fermi velocity of the metal considered.3 It is easily
seen that the following relation holds for all A, A’ and A-:
U(b) = U(a) + W-IO.
(7)
It is therefore sufficient to consider the potentials in process (a) only. In what follows, all
the U are referred to process (a).
In the charge-transfer process (a) the important quantities to be considered are the
energy differences
214
IMAGE POTENTIAL AND CHARGE-TRANSFER PHENOMENA IN 4TOM (ION)-SURFACE COLLISIONS
AU,
- U,+ - U, = (I, + U,) - w - I(z) - w.
AU2 = U,- - U, = W - (A, - U,) = W - A(z).
(8)
(9)
Here we neglect U, because of its smallness in comparison with other quantities. we have
defined the position-dependent ionization energy I(z) and electron affinity A(z) as
I(z) = IO +
A(z) = Ao -
.
up
(10)
2
U,(z)
.
(11)
These are the two important quantities in the study of charge-transfer phenomena.
In the classical electrostatic theory, the image potential is simply -e2 /4z. This relation
gives a general feature that as an atom (or molecule) moves close to a metal surface, the
ionization energy decreases and the electron affinity increases. This behavior is shown
schematically in Fig. 1, in which zt is the classical turning point of the incoming atom (molecule). From the figure it is easily seen that if W < A(z,) then it is energetically favorable
for the neutral atom to become negatively charged by acquiring an electron from the metal
as it moves close to the surface. On the other hand if W > I(z,), fromation of a positively
charged ion is favoured.
Energy
t
FIG. 1
A schematic diagram of the position-dependent ionization energy I(z) and electron affinity A(z)
in an atom-surface collision. EF and W are, respectively, the Fermi energy and the work function
of the metal, and zt is the classical turning point.
From the above analysis, we see that the exact z-dependence of the image potential
U,(z) and the classical turning point zt are two important quantities in the theory of chargetransfer phenomena. It is known that the classical image potential is a good approximation
Y. C. CHENG AND K. C. LIN
215
only for large z. For small z, quantum effects are important. However, no existing dynamical theories114 of the charge transfer phenomena have considered this point. It is the
main purpose of this paper to give a detailed calculation of U,(z), which is valid down to z =
0 for a point charge, for various metals with electron densities ranging from rs = 2 to rs = 6.
These results are applicable to any positive or negative ion.
It is also important to calculate zt, which is mainly determined by the repulsive
potential U, (z), with U,(z) a small correction. We use the effective medium theory’ to
estimate U, The van der Waals potential Uw (z) can be calculated via the same method as
for the image potential U,(z). As an example, we calculate the van der Waals potential
between an H atom and a metal surface, and then discuss the possibility of forming H- for
the H- surface scattering.
II. THE ION (ATOM)-SURFACE INTERACTION
.
-_ __.., -
There have been many quantum-mechanical, or semi-classica16, calculations of the
image potential for a charged particle near a metallic surface3p7-12. In most approaches the
image potential is obtained by calculating the self-energy of the charged particle in the
presence of a surface, which usually require the solution of the Schrodinger equation for
the coupled system. Attempts have been made to simplify the calculation of the selfenergy11712. In particular, the second-order perturbation approach, formulated recently by
Annett and Echenique12, seems to be the most appealing. It is physically transparent and
mathematically simple. Another advantage of this approach is that the image potential and
the van der Waals potential can be formulated in a unified way. Therefore, in this section
we use this approach to calculate the image potential and the van der Waals potential. The
surface response function also plays an important role in the theory of ion (atom)-surface
interactions. For simple metals, surface plasmons are the most important elementary excitation on the surface to be considered. We therefore use the surface plasmon-pole
approximation to approximate the surface response function”Y’3.
We consider a system consisting of a metal surface which occupies the half-space z < 0,
and an atom (or ion) located somewhere outside the surface z > 0, in which z is the coordinate perpendicular to the surface. We calculate the atom-surface interaction via a perturbative approach. The unperturbed system consists of the isolated surface and a free atom
(ion). These are initially in their respective ground state, except that the atom (ion) as a
whole may move with a finite constant velocity. The perturbation is due to the Coulomb
coupling V between the surface and the atom:
V = J 1 d3 rd3 r
P,(T) Ps(“r’ )
l;*-_;fl
’
(12)
where p,(T) and p,(y) are respectively the charge-density operators for the atom (ion) and
~_.,
_.
.
216 IMAGEPO~N~ALANDCHARGE-TRANSFERPHENOMENAINATOM(ION)-SURFACECOLL~S~ONS
the surface. The atom (ion) charge-density operator is
p,(f) = N&Tt) -
E,
&(;-yJ,
(13)
where 2 andTi are respectively the position coordinates of the nucleus and the ith electron N is the charge on the nucleus, and there are N’ electrons. We use atomic units
throughout.
The first-order perturbation is zero because the surface is everywhere neutral in its
ground state. The lowest-order interaction energy U is therefore due to the second-order
perturbation
I < 00’ I V I nn’ > I2
>
n,n’ ~0 + Eel - en - En’
u= c
.
(14)
where I nn’ > denotes a state with the atom in its nth (unperturbed) excited state (with
energy E,,) and the surface in its nth (unperturbed) excited state (with energy E,!), and
I 00’ > is the unperturbed initial state. By using the following expansion, with ;‘= G,, J),
t
r = &, z’), and {a vector parallel to the surface,
1
G-3
I
=
dZ
I-7
q
.+ -+
+,
exp[lq . (r,, - r,, 1 - q(z - z’)l,
z-z’>0
(1%
and the translational invariance parallel to the surface
waif,,, 8 P,&, z’) = ~~(0, z> psG;, -y,,, z’) ,
(16)
we rewrite Eq. (14) as
_ do
dZq 2n l<Ol~In>lZ
u=-CJ, TI--_
n
(2n)Z q
En - E0 + w
ImU&
~11 .
(17)
Here
p;t = I d2 r,, Jo dz exp(-qz + iq - r,,)P,(T),
(18)
and D(;, w) is the surface dielectric function which depends only on the surface; we have
L_-
Y.C.CHENG AND K.C.LIN
211
-+
271
Im [D(<, o)] = - J d* r,, J”_ dz JT_ dz’ exp(qz + qz’ - i;‘q - r,,)
q
X
hod,,, z, z’,
w)l, ,
(19)
where
Imki;,, -3;,, z, z',
w>l = 2 < 0’ i p,(T) I n’ > < n’ I p,(Y’) 1 0’ >
xl’
x nG(E, -E, -0)
(20)
is the imaginary part of the density-response function.
Eq. (17) is the basis of our calculation. In order to calculate U it is necessary to know
<Olp,ln> and Im[Dt4,41. It is customary to calculate U as a function of Z, the distance
between the nucleus and the surface. Therefore one usually treats the nuclear motion perpendicular to the surface classically in order to avoid difficulties arising from the uncertainty principle. The parallel motion can be treated quantum-mechanically by a plane
wave with a parallel wave vector;. We write the initial atomic (ionic) state 1 0 > as
(21)
where z0 is the nucleus-surface distance and tiO (yl . . . yNf, 3) is the ground state free
atomic (ionic) wave function centered at 2 = (it,,, Z). The nth excited state I n > can be
written in a similar form as
1X’, zb, n> =exp(i%‘,g,,)s(Z - zb)$.(Yr . . .TNs,it).
(22)
From (21) and (22) we have
<OiP;f/Il> 3 <?;,z,,Olp,
lX;‘,zb,n>
= exp(-qz,)6 (z,, - zb>6 6 -I+ q)[NS,
-
N’ N’
2
j=l
II dri,, dzi exp(-qzj + i; - fj ,,> $,* (yl . . . rNS.
+
0)
i=l
x $,G,, .T;N., O>l
)
(23)
where we have transformed the atomic (ionic) wavefunctions Go and $n to be centered at
the origin. In the evaluation of the integral in (23), we note that the atomic (ionic) wave
function $ 0 is appreciable only for a small region near the origin and that the contribution
from large q is small because of the factor exp(-qzj); therefore we can expand the exponential term in the integrand
_--.
218 IMAGEPOTENTIALANDCKARGE-TRANSFERPHENOMENAINATOM(IO~~-SURFACECOLLISIONS
exp(-qzj + i< * Yj ,,) = 1 + (-qzj + i< .Tj ,,) + .
.
(24)
The integrals arising from the first term in the expansion (24) and the first term in (23)
together give the monopole contribution to the atom (ion)- surface interaction. The integral
arising from the second term (quantities in the parentheses) in (24) gives the dipole contribution. If the atom has a net charge, i.e., a charged ion N # N’, then the monopole term
is usually much larger than the dipole term and one keeps only the monopole term to abtain
the image potential as a function of z, the nucleus-surface distance,
U,(z)
.
=
d’q 1 e-h’Im[D(q, w)l
- dw
-Q’ j,, - - 77 s (277) q w-;* ;,, + q2 /2M ’
(25)
where M is the mass of the ion, u,, = k/M, and Q = N - N’ is the net charge of the ion. If
the atom is neutral, i.e., Q = 0, then the monopole term is zero and the lowest-order interaction is the dipole term. It is commonly called the van der Waals potential
U,(z)=
-zf,+ki*l
n
WnO
O
77
e
-2qz
q2WWq,
41
(277) q w +wno -$;,, +q2/2M
’
(26)
w h e r e f,, is the dipole oscillator strength for a transition from the atomic state 0 to state
n with frequency wnO.
In the perturbative approach we neglect the effect due to overlap of the electron
clouds between the atom (ion) and the surface. The overlap gives rise to the strong Pauli
repulsive potential U, , which is much more difficult to calculate. To approximate U, we
use the effective medium theory 5 , which has been developed for the He-surface interaction.
This theory is best for He-like atoms (ions) and is only a rough approximation for atoms
(ions) of other types.
In the next section we give the numerical results and discussion follows.
III. NUMERICAL RESULTS AND DISCUSSIONS
In evaluating the integrals (25) and (26) one needs to know Im[D(:,w)]. Because a
detailed calculation of D(:, w) would require the knowledge of the eigenstates of the
surfacer4, it is not an easy task. In practice one usually makes some approximations.
For simple metals the surface response to an external charge is dominated by the surface
plasmon oscillations. We have the following approximation’1~13
YI
Y. C. CHENG AND K. C. LIN
a’0
Im[D(;;, o)] =” - S(w -o&q))
2 o,(q)
219
(27)
for w > 0. Here w,(q) is the surface-plasmon dispersion relation which we assume to have
the form”
1
w,(q)* = w; +cuq+pq2 +4 94,
(28)
with wi = 3/2r3 s4ar3/3
’ s = l/ne; ne is the bulk electron density. The coefficient cx is determined by the static image-plane position, and fl is determined by requiring that the surfaceplasmon line w,(q) joins the single-particle continuum at the same point as the bulkplasmon line does. The values of a and p for 2 < rs Q 6 have been calculated by Annett and
Echenique and are listed in Ref. 13.
Using these values we evaluate the image potnetial U,(z) from Eq. (25) for 2 < rS < 6.
The results are plotted in Fig. 2(a) through 2(e). We note that: (1) there is little u,, dependence up to u,, = 0.5 a.u. (- lo8 cm/set); and (2) the results are appliable to any ion (M >
1835) because the term qz /2M in the denominator is negligibly small. Because our purpose
L
(b)
IO
FIG. 2 The image potential. The full curve is calculated from Eq. (25) and the dashed curve is the classical
result -l/42. Atomic units are used. (a) rs = 2, (b) rs = 3, (c) rs = 3, (d) rs = 5 and (e) rs = 6.
~__
220
IMAGEPOTEhl-IALANDCHARGE-TRANSFER
PHENOMENAINATOM (ION)-SURFACECOLLISIONS
is to compare with the classical electrostatic result - 1 j4.z. we plot the classical values in
dashed curves for comparison. It is easily seen that for small z, the classical result deviates
substantially from the quantum-mechanical result (25). It is important to find the critical
distance z = zC such that for z < z, the classical image potential fails. The values of zC for
2 Q rS < 6 are listed in Table I. We see that for metals with smaller electron density (larger
rS), the quantum correction is more important (zC is larger).
TABLE 1 The critical distance zC vs. rS, both in atomic units.
Another important quantity is the classical turning point zt . If zt > zC, then the
classical image potential is a good approximation for all z > zt. But if zt < zC then the
quantum effect is important for zt < z < zC. The classical turning point zt is determined by
the potential between the incoming particle and the surface. We distinguish two cases as in
the Introduction:
(1) The incoming particle is a positive ion A+. The relevant potential is
U,(z) = U,(z) + U,(z)
.
(4)
Here U, is the repulsive (positive) potential and U, is the attractive (negative) potential. In
general UA+ will be negative for all z except for very small z, of the order of 1 a.u. or
smaller. Therefore we invariably have zt < zC in this case, and the quantum correction of
the image potential plays an important role. The exact value of zt depends on the ion and
the surface considered.
(2) The incoming particle is a neutral atom A. The relevant potential is
U,(z) = u,(z) + $, (‘)
’
(1)
Here the van der Waals potential Uw is attractive, but its magnitude is much smaller than
the image potential U, for small z. At large z at which U, is negligible, U, experiences a
weak attractive potential due to U, As the atom moves closer to the surface, the repulsive
term becomes more important. Eventually at moderate z (of the order of 4 a.u. or larger)
U, becomes repulsive. In the effective medium theory 5 of the He-surface interaction, the
repulsive potential U, is proportional to the surface electron density ne(z). We use this
Y. C. CHENG AND K. C. LIN
221
approximation to estimate the repulsive potential U, of other neutral atom-surface inter-
action. For rS = 3, U, becomes positive for z < 4 a.u. (see the following example). As
n,(z) is smaller for larger rS, we are almost certain to reach the conclusion that for a surface
with rS > 4, we have zt < zC (cf. Table I) and quantum corrections of the image potential
are important.
For rS Q 3, we should have a more quantitative calculation to determine whether zt >
zC or zt < zC. As an example we consider H colliding with a surface with rS = 3. The
relevant potentials are plotted in Fig. 3, in which U, is approximated by the effective
medium theory5, Uw is calculated from Eq. (26), and Utot is the sum of U, and U,. Although the van der Waals potential U, is usually weak, it produces both quantitative and
qualitative effects on Utot . It produces a weakly bound state at z around 5 a.u. and it reduces
FIG. 3
The H-surface interaction potential with surface electron density rs = 3. U, is the repulsive potential approximated by the effective medium theory and Uw is the van der Waals potential calculated
from Eq. (26). Utot = U, t Uw is the total interaction potential.
the classical turning point zt by about 1 a.u. irrespective of the incoming energy. For a
thermal atom (at 300 K) with normal incidence, we have zt = 2.3 a.u. From Table I, we
have zt < zC, as zC = 3.5 a.u. for rS = 3. The position-dependent electron affinity is A(z) =
A,, - U,(z) = 0.75 + 3.26 = 4.01 eV for H at z = zt = 2.3 a.u.. From Fig. 1, we see that for
metals with work function W < A(z,), it is energetically favorable to transfer an electron
from the metal to the atom so as to form a negative ion if the atom is close enough to the
surface. For metallic lithium rS = 3.28 a.u. and W = 2.32 eV, and metallic magnesium rS =
2.65 a.u. and W = 3.66 eV. These metals (rS is close to 3 a.u.) satisfy the criterion W <
A(z,) and therefore favour the formation of H- ions in H-surface collisions. However a
-
222
IMAGE POTEhl-IAL AND CHARGE-TRANSFER PHENOMENA IN ATOM (IONXXRFACE COLLISIONS
quantitative calculation of the probability of ion formation for the scattered beam is a much
more complicated problem. It involves the tunneling problem and also that at large distance
A(z) > W such that there is a possibility of re-neutralization. In studying this phenomenon.
one commonly solves the time-dependent Anderson-Newns model’,4. This is not the subject of this paper and we discuss it no further.
In conclusion, we have calculated the image potential for metals with electron density
rs ranging from 2 to 6. These results are compared with the classical electrostatic image
potential -e2 /4z. We find that for z < zc? quantum corrections of the image potential become important. The critical distance zc is a rs-dependent quantity and its values are tabulated in Table I. We also find that the van der Waals potential is important in determining the
classical turning point zl. Both zc and zt are important in charge-transfer phenomena in
atom (ion)-surface scatterings.
IV. ACKNOWLEDGEMENT
.
This work is supported in part by the National Science Council of the Republic of
China through Contract No. NSC 77-0208-M002-28.
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