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Theory of positronium formation on solid surfaces
277
Surface Science 147 (1984) 277-294 North-Holland, Amsterdam
THEORY OF POSITRONIUM FORMATION ON SOLID SURFACES I. General theory and its application to positronium formation probability Akira
IS11
Department
Received
of Physics, Waseda Vnioersity, Okubo 3 - 4 -I, Shinjuku, Tokyo 160, Japan
25 April 1984; accepted
for publication
25 July 1984
We consider the micro-processes of positronium formation on surfaces with a new Hamiltonian, in which we treat the positron as a quantum particle. First we calculate the temperature dependence and the positronium work function dependence of the positronium formation probability in the case of no surface binding states. We succeeded in deriving a slightly increasing profile of the positronium fraction at low temperatures, as was found in the experiment of Lynn et al. We also calculated the probability of the positronium formation at excited levels and of the formation of the negative positronium ion. Furthermore, the positronium fraction is shown to be proportional to the length at which the surface electron density drops to half the maximum value.
1. Introduction Low energy positrons which have been implanted into a solid target in a vacuum form a unique and sensitive probe of the surface regions of the solids [l]. Of particular interest is the abundant positronium (Ps) formation in the outer surface region. We present in this paper a new general procedure for treating this attractive physical process. In the experiments a monoenergetic beam strikes a well-characterized single-crystal surface. The majority of the positrons penetrate the solid and simultaneously lose their energy rapidly within a characteristic stopping distance. The thermalized particles are fairly mobile and a large fraction of them eventually diffuse back to the entrance surface. Some materials have a negative work function for positrons, i.e., positrons are spontaneously re-emitted from the surface. A large fraction of the positrons reaching the surface may capture an electron at the surface and thus emerge as Ps rather than ef . Yet another possibility is the trapping of a positron at the surface owing to the image force. In the present paper we focus our attention on microscopic processes of Ps formation occurring on a metal surface. Nieminen and Oliva [2] have treated this problem using the Anderson-Newns model Hamiltonian. The Anderson-Newns model is well-known in the charge exchange problem, ion 0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
278
A. Isii / Posrtronium
formation
on solid surfaces. I
neutralization [3,4] and negative ion formation [5] in atom-surface scattering. However, we cannot apply the Anderson-Newns model to positrons, because infinite positron mass is assumed in the model. We present here a new Hamiltonian which is generalized from the AndersonNewns model. Using this model we correct the work of Nieminen and Oliva and can describe the temperature dependence of this process as discussed by Brako-Newns [4] in the atom-surface scattering. Moreover, we can describe the effects of surface contamination using the new model. Before proceeding to the theory, one or two remarks about the positron states in metal and vacuum should be made. In fig. 1 we show a schematic one-particle presentation of the potential energy diagram for a positron on a metal and the Ps energy level for a positron near a metal. The Ps level includes both the binding energy of Ps and electron work function. The transition rate between a metal state and a positronium state gives rise to a finite lifetime of the state, which results in a Heisenberg broadening of the level. The width of the level is also shown in the figure. In section 2 we present our model Hamiltonian and discuss the characteristic aspects of the rseonant transition in our case. Moreover, we classify metals into three types for convenience to consider various aspects of Ps formation. In section 3 we obtain a general formula for the probability of Ps formation on solid surfaces with an equation of motion method which is mathematically similar to that of Bloss-Hone and Brako-Newns. In the calculation we take approximations similar to those of Bloss-Hone and Brako-Newns. In section 4 we obtain a useful formula for the Ps fraction using a model of level width. Using the formula we calculate the Ps fraction for many metals. The fraction for excited state Ps and that for Ps negative ions are also obtained. Furthermore, we find that our theory contains only the work function of Ps. This result agrees well with the experiments of adsorbate coverage [6]. Moreover, the attenuation length of the surface electron density profile affects the Ps fraction. In section 5 we discuss these points in detail. We use the atomic units throughout the paper.
Fig. 1. Schematic
one-particle
representation
of the state for a positron
on or near a metal.
279
A. Isii / Positronium formation on solid surfaces, I
2.
Model
We associate the Ps formation with the interaction of the positron with the delocalized (broad band) electrons. A similar problem has been discussed in the field of ion neutralization in terms of a resonant process by Blandin et al. [7], Bloss and Hone [3], and Brako and Newns [4] in a many-body approach using a time-dependent Anderson-Newns Hamiltonian. But, in the case of positrons, one difficulty arises; the wave function of a positronium cannot be separated into a part for the electron and a part for the positron. Thus we cannot define a single-particle operator for the positronium state. Nieminen and Oliva have ignored-this diffkklty.
I+.( n-2) --.-._-____--__-_
Ps level (n=l )
W
level
e- Energy
C ?
Fig. 2. Illustration of the electron energy level in a metal and the effective positronium energy level for the electron. (a) A-type metals; alkali metals. (b) B-type metals: W and Cr. (c) C-type metals: Al, Cu, etc.
We start with the SchrOdinger picture in order to use the wave function of poistronium, that is a two-particle state. We see in figs. 2 and 3 the essential features of the model; the energy level structure of the electron in fig. 2, and that of the positron in fig. 3. In the solid, we have broad conduction bands, corresponding to spatially delocalized electrons, which we assume to interact with the ground state or, in some cases, with the first excited state. It is convenient to classify metals into three groups by the energy level structures as shown in figs. 2 and 3. Alkali metals are of type A. These have the positronium work functions +pps which are nearly equal to zero. The positron
be* Energy a Metal
Vacuum
_-,,~*i&ge potential ,,’ 1’
Qt
Ps level
b
A,
( n=lf
P Energy Ps(n=2)
L
f
f”,,_--’
:
Metal
C
/’
/’
_e---
_--
e
......_..... Ps(n=l) Vacuum
1 e’ Energy
/ Metal
Fig. 3. Illustration of the positron energy levels in a metal and the effective positronium energy level for the positron. (a) A-type metals: alkali metals. (b) B-type metals: W and Cr. (c) C-type metals: Al, Cu, etc.
A. Isii / Posrrronium form&on
on solid surfuces. I
281
work functions ++ are positive in this type. The metals of type B have negative work functions of large magnitude for positrons. Thus the emission of positrons occurs. Moreover, the production of Ps at excited levels (n = 2) is possible for metals of this type. The formation of negative Ps ions is also possible. The metals of the last group are Al, Cu, etc., which are used frequently in experiments. In this case the positron work functions $+ are nearly equal to zero and +r,s have large negative values. We find the surface binding state of a positron in the surface of metals of type C, which also interacts with the ground state of Ps in vacuum. This interaction causes Ps formation via e+ escaping from the surface state. The model Hamiltonian we propose, incorporating the above features, is a generalization of the Anderson-Newns model. Here we treat the impurity in the model of Brako and Newns as a quantum particle. We assign the subscript k to the conduction electrons, q to the positron in metals (including surface states) and d to the state of positronium. Therefore, the Hamiltonian may be written H = c Ekd;fdk + c Ep~a, k
rl
+ E,,djd,a$a,
+ c
(~~~d~d~~~~~
+ h.c.),
(1)
k.q
where Vkd is the interaction between the conduction electron, the positron and the positronium state, and d+ and at are creation operators of electrons and positrons, respectively. In choosing a one-electron picture of the metal we have necessarily excluded processes which depend on electron-electron interaction. We also disregard the Ps-hole interaction, which must be included in the later calculations. In the theory of ion neutralization, it is not simple to establish the expected relative probabilities for neutralization from resonant transfer from. the conduction band and from Auger effects [3]. Fortunately, without W and Cr, we have only one level of Ps to which the conduction electrons transfer. Thus we do not consider either Auger neutralization or interaction between two resonant levels of Ps. We can use these facts to examine the resonant neutralization theory because it has been difficult to measure the relative probabilities for neutralization from resonant transfer and from Auger effects in experiments of ion-surface collision.
3. Calculation Within the above model, we can proceed to solve the problem in the Schrodinger picture by two-particle wave functions. We define the wave function of the total system as:
A. fsii / Posi~ronium formationon solid surfaces.i
282
where Cq.k(t) is the product of the one-particle wave function of positron (4) and electron (k), and Cd(t) is the wave function of Ps including coordinates of the positron and the electron. I/c) is a product of the positron state and the Fermi sea having one hole in state k. Our solution is similar to the equation of motion method of Bloss and Hone. The state It) agrees with the Schriidinger equation i(d/dt)\r) = H(t), from which we derive the equations for C, (t ) and C,,n ( t ).
where nk is the Fermi distribution Cd{ 2) = C,(t)
exp(iE&,
and where we have defined
.
(5) (6)
c,,.(t)=C~,,(t)expIi(E,+E,)t], k,(t)
= Y.&>
exp[i($
(7)
+ Ek - G&].
These equations have the same mathematical form as that of Bloss and Hone. We can rewrite the equation of Cd(t) using the integral form of eq. (4),
Following Bbss and Hone, we also assume that the time variation of V,,(t) depends little on the value of q and k; instead it characterizes the motion of the positronium, so that V,, (t ) can be factorized as v,,(t)
(9)
= Y&(t)-
Then the sums over k and q in the second term of eq. (8) take the form
=A(t)6(t-T),
(10)
= A&(t)12.
(11)
where A(t)=
&v@)~IK,I~)E~~E 4
This is similar to the approximations used by Bless and Hone, where the band is assumed to have no important structure and to have an infinite bandwidth. Within these approximations the equation of motion for Cd(t) becomes &(1,
= -Ill’&
dd(f)
-ix
c v&(t) k
4
cq,k(r,>*
02)
A. Isii / Positronium jbrmation
283
on solid surfaces. I
Eq. (12) may be solved to give C;(t)
= e-rCrO*‘) 2;,(t,)
- ix Cn,l:r k rl
eda
Thus we obtain the positronium production momentum ~7and the electron of momentum
e-‘~‘.‘)
Cq;,.,(f,).
(13)
probability for the positron k as follows,
of
ICd(t)12 = (e-‘(‘o~‘) ed( + c c k
q
c k’
)3nkv&v,‘d
c$k(tO>
cq’,k&l>
q’ 2
Ff7.f)
X
X
I*0TdT V&U(T)
where we defined
r(tr,
ei(E,iE,-Epqf7
e ff7.*) ei(E,+E~-Eps)~ +
c_cs
(14)
t2) as:
r(f,,t,)=A~~r2d~I~(1-)12.
f
(15)
1
The first term of eq. (14) describes the decay of the initial state to equilibrium state in a solid where the system is initially in the positronium state. This term is negligible because positronium states do not exist inside the solid. The last two terms are also negligible for the same reason In the rest of this paper we concern ourselves with the situation where the second term in eq. (14) is the dominant one. Our aim is to calculate the expectation value
= (IGW12)?
h&))
(16)
where the bracket means averages over the positron and the Eventually t will be put to + cc as the measurements of (nd) the positronium has left the solid. From now on we shall take t, assume that in the initial state the metal was in equilibrium at From the initial condition we obtain F
~(c;k(-co)
where tively,
cq,k(-m>>
=.fF@k,
Th&p
f, and fr, are the Fermi and Boltzmann
electron states. are made after = - cc and we temperature T.
(17)
T)t
distribution
functions,
respec-
fF(E, T) = (I+ exp[(E+ +-)/kT]}ml,
(18)
f,(E,
(19)
T)=(nkT)-3’2exp[-(E+cp,)/kT].
284
A. lsii / Positronium formation
on solid surfaces. I
Thus we obtain the expectation value of the occupation same approximation as that of eq. (10): @L&o))
=c
C%(Krl12)fF(~~~ k 4
co
X
IJ
T)f”&
Eq. (20) is an generalization positrons.
using
the
T)
2
!.
dr t{(T) e -1.(7.cc) ei(E+G?--ll‘p,)T
--3c
(n,(m)>
(20)
of the results of Brako and Newns to the case of
4. Results 4.1. Exponential
model
Before discussing the results of the above theoretical calculation, let us first discuss the static situation of a positronium at a given distance from the metal surface. When the level of a positronium is situated within the conduction band of a metal, resonant electron transition between the metal and the positronium will take place. In this theory Ps-metal interaction is considered as a perturbation of the pure Ps and metal states. This method was worked out by Gadzuk [8] for the chemisorption of alkalis on metal surfaces. This interaction gives rise to a shifting SE and a broadening A of the Ps level, which we calculate in the appendix. In the appendix we derive the level shift and the transition matrix element related to the level broadening as follows: SE=O,
(2la)
]Ti:,,12Cce-+-t’+
(21b)
It is very complex to calculate eq. (11) by substituting eq. (21b). It was concluded by Nieminen and Oliva [2] that the emerging positron is emitted into a Gaussian angular distribution around the surface normal with a rather narrow angular spread. Thus we assume there that the emerging positronium is also emitted into a narrow cone around the surface normal. Therefore we can discuss the motion of the positronium only on the surface normal direction. Since we consider here theoretical problems and qualitative properties of positronium formation at solid surfaces, we assume simply that A =A,exp(-2s/a),
(22)
A. Isii / Positronium formation on solid surfaces. I
where a is radius of positronium in vacuum, i.e. two times the Bohr corresponds to the fact that the electron density distribution outside drops to half the maximum value at approximately a (or two times radius) from the nominal surface. It seems to be a reasonable value with the density profile obtained by Lang and Kohn [9]. Substituting (21a) and (22) into eq. (20) we obtain the following the Ps production probability
X/dw~~f,(w,T)~sech[(E,,-E-w)a/2o],
radius. It the metal the Bohr compared form for
(23)
where we assume the velocity of the positronium constant o defined as: v2=E+o-E
285
normal
to the surface to be a
(24)
PS’
Eq. (23) gives exact results in finite temperature cases. But numerical integration of this integral is not so simple because of drastic exponential decays in the integrand. It is a surprising fact that A, cancels out, so that we have no need to calculate the transition matrix elements in detail. It is worth noting the momentum conservation for the surface parallel direction. The momentum conservation for the parallel direction should be included in the transition matrix element and is smeared by the average in eq. (11). Since the amplitude of the transition matrix element cancels out, we can neglect the momentum conservation when we pay no attention to the angular distribution of re-emitted positroniums. 4.2. Production probability
of Ps in ground state
We take the sech in eq. (23) to be unity, because it is nearly equal to unity where the kinetic energy of Ps is not so large compared with the work function of Ps. It is valid for slow Ps which we discuss here. We neglect thermal smearing of the Fermi surface and take fr( E, T) to be a step function. This approximation is valid where ]$J~,] is quite large in comparison to thermal energies. Using the approximation above, we find that when (pp, -C 0, (nd(Oc))
= ( -2Gps/\/2?Tkr)
e-+p~‘2krK1(
-GPs/2kT),
(25)
where K, is the modified Bessel function of the second kind. The above formula depends only on +ps and not on c$+ and $_ individually. From this fact we can suggest that the Ps production probability is not sensitive to surface contaminants which change ++ and c#_ but do not change +p, [lo]. The effects of surface contaminants found in previous experiments [6] may be caused by the modification of the local electron density near the surface.
286
A. Isii / Positronium formation
We may easily get the low temperature
on solid surfaces. I
limit as follows,
Using eq. (24), we show first in fig. 4 the #rS dependence of the Ps fraction at T= 300 K. As we are not yet including surface states of positrons in the present theory, this fraction cannot be compared to the experimental data of metals of type C. The magnitudes of the fractions for metals of type C like Al or Cu (&,, = -0.095) [6] are, however, compatible with experiments at the temperatures where there is no positronium coming from surface binding states. Moreover our result is very different from the calculation of Nieminen and Oliva. It comes from their wrong choice of the velocity (see appendix). In fig. 5, we show the temperature dependence of Ps fractions for (a) Rb and Cs, and (b) cases of small @)pS.From these figures we find that the Ps fraction is I -
c ii
0
0
0.1
0.2
0.3
0.4
0.5
- (be (au)
Fig. 4. Ps formation
0
0
probability
as a function
of Ps work function
at T = 300 K. Atomic
I
2oD
400
600
T(K)
T(K) Fig. 5. Ps fraction (b) Ideal material
units.
as a function of temperature T for various and -0.0001. of cpps = -0.0005
metals. Atomic
units.
(a) CS and Rb.
A. Isii / Positronium formation
287
on solid surfaces. I
sensitive to changes of temperature in the case of small Ps work function, and we can observe it for Rb and Cs. In the case of Al we find the slightly increasing profile which was revealed in the experiments of Lynn et al. 1111. But the relationship between this fact and the problem of internal reflection of positrons based on the model of Nieminen and Oliva is not clear at this time. 4.3. Production
of Ps (n = 2) and Ps -
Since evidence of excited state Ps (Ps*) formation from metal surfaces and of positronium negative ion (Ps-) formation has been found recently, it is necessary and important to present a theory to calculate the probabilities of Ps* and Ps- production on metal surfaces. One purpose of calculating them is to find convenient sources of Ps* and Ps-, because positronium as a leptonic atom serves as a unique system to test the predictions of quantum electrodynamics [ 11. Since we expect that the most important feature of Ps* and Ps- compared with Ps is the position of their energy level, we calculate the formation probability of Ps* and Ps- using our theory and assuming that eqs. (21a) and (22) are also valid for Ps* and Ps-. Thus systems are characterized mainly by the work functions #pps. and (pr,-. The work functions (pi,,* and (pi,- are defined in the usual manner: 9
Ps*(n)
=
++
+
+_
-
0.25/n’,
(27)
(pps- = (p, + 2+_ - 0.25 - 0.012. For metals
(28)
of type B, e.g. W and Cr, we have the following
BWI, (pp,. = 0.0085 (W), 0,007 (Cr);
&,~ = -0.028
(W),
+O.OOll
respective
values
(Cr).
Before discussing the results of their formation yield, we must first derive an approximate formula for the Ps formation probability for positive work function. Using the same approximation as that of eq. (25), we obtain (nd(cc))
= /m
e-*ps/kT l*
ds imps
+ 1) e-‘.
(29)
We find simple analytic results if kT/+,, < 1. The limit should be reasonably appropriate to the cases discussed here. We find that
In the case of Ps*, we.get fps. = 1.44 x 10s5 for W, which is very small in comparison to the experimental value of Shoepf et al. [13], 1 X 10p3. There are
four reasons for this discrepancy: the experimental uncertainty of the positron work function, the neglect of thermal smearing of the Fermi surface, deviation from the exponential model for level broadening A, and d-electron effects which were also ignored in the theory of Bloss-Hone and Brako-Newns. We can dispose of the second reason through numerical integration of eq. (23). To dispose of the third reason, we must introduce a new function for A(t) in solving eq. (20) numerically. As for the fourth reason, Kawai and Ohtsuki 1141 pointed out that in the field of ion neutralization and negative ion production the localization of electron density of states is important. This problem is very important and essential, but very difficult to solve. Next we discuss the fraction of Ps-. We take a model similar to that used by Van Wunnik in his theory of negative hydrogen ion formation. We obtain fPs - 0.1 if #I~~= - 0.028 for W. For Cr, we use a new work function value [12] of 40.0011 and we obtain a probability of about 0.7%. It is rather larger than that of the expe~mental value, < 0.1% 1121. The discrepancy may result from the experimental accuracy of the positron work function, or from the assumption of a, the attenuation length of the electron mode at the surface. It seems probable that the prediction of our theory becomes a convenient guide to produce Ps- sources using positron-surface interaction. 5. Discussion Let us consider the formation and the results of our theory. First we consider the theoretical formulation. The Hamiltonian (1) corresponds to one of the generalized Anderson-Newns model in which impurities are treated as quantum particles. This Hamiltonian itself can be applied to many aspects of positron-solid surface interaction. In future calculation we must include many-body effects, e.g. positron-hole and electron-hole interaction, and the relaxation effects which are well-known in photoemission. Nonetheless, the above theoretical formulation is very simple and easy to extend. For example, we can add a positron surface state as one of ai. We will discuss this problem later. The approximations used to derive eq. (22) are available for solids of semi-infinite electron gas. Substitution of l/a for the average of k, corresponds to the fact that the attenuation length of surface electron states is of the order of u. It is compatible with the calculations of the surface density profile obtained by Land and Kohn. Thus, the above approximation is suitable for the present calculation. Moreover, we find from eq. (23) that the positronium formation probability is proportional to the attenuation length of the surface electron states or the surface electron density profile. Thus if the surface electrons are pressed into the bulk by the negative charge of oxide contamination, the Ps fraction decreases as we have observed in previous experiments [15]. On the other hand,
A. Isii / Positronium formation
on solid surfaces. I
289
the Ps fraction increases with alkali metal atoms and others. In particular, it is confirmed by measurements of surface dipole that sulfur contamination extends the surface density profile to a vacuum. It is confirmed by experiments that the experimental value of fps increases with increasing sulfur coverage [6]. Thus it is expected that it will be possible to make direct measurements of variations of surface density profiles due to various surface contaminations. Our formula (23) depends on only +r, and does not depend on $+ and +with increasing surface conseparately. Since +rs remains almost constant taminations [6], we reach the conclusion that fps is insensitive to adsorbate coverage. This agrees well with experimental results. In experiments we find slight variations of fPs with adsorbate coverage. One of the sources of the variation is the change of surface electronic density profile due to adsorbate coverage. Another reason may be changes in the electronic density of states due to surface contaminations. Moreover, it is first necessary to derive the temperature dependence of the Ps formation probability itself. Probably this effect and the temperature dependence of +rs itself [16] are both causes of the slight variation of fPs observed by Lynn et al. [ll]. Next we discuss the various results of Ps production probability presented in the above section. Unfortunately, there have been no experiments on metals of type A and B. Nonetheless, we can state the following conclusions: A-type metals: fPs is 0.05-0.1. Since they have no surface state and no possibility to produce Ps* or Ps-, it seems to be the simplest example of resonant transfer theory. In addition the temperature dependence of fps is strong for these metals. Moreover, since the attenuation length of the surface electronic density profile is very large for these metals, the fPs of them is very sensitive to surface contaminations. Thus we can check our theory when we compare the above theoretical results with experimental values. Therefore, we have pointed out that experiments on positron-alkali metal surface interaction are necessary and important to confirm our theory. B-type metals: fps is about 0.6 for Cr and W. It is particular interesting that we find in Cr and W finite probabilities of formation of Ps* or Ps-. Thus we can expect to have a source of Ps* and Ps- in vacuum using the mechanism. However, it is necessary to study the level broadening of Ps* and Ps- near the metal surfaces in detail. For Ps-, it is also necessary to examine the model of negative ion production used here that was presented by Van Wunnik for proton-surface interaction [5]. C-type metals: For these metals there have been many experiments. However, since surface trapping states exist on the surfaces, it is difficult to compare the experimental data with the predictions of our theory. For Al or Cu, it seems to be reasonable when we compare the theoretical values with the experimental values of low temperatures where there are no Ps escaping from the surface state.
290
A. Isli / Posrtronium formation
on solid surfaces. I
We must add one more theoretical discussion. In eq. (23) we used the Boltzmann dist~bution for positrons in metal. This is a simple model of the thermalized positron. In more accurate calculations we must find more accurately the momentum distribution of positrons on the surface. We cannot use the diffusion mode1 of Nieminen and Oliva, because they derive only the spatial distributions of positrons in solids.
6. Conclusions The principal aim of this paper is to solve the problems of positronium formation on solid surfaces as direct resonance charge exchange. Although our calculations are based on a rather idealized model, they give a reasonably good account of the calculations of the positronium formation probability. Our calculations agree with experiments where there are no positroniums escaping from the surface state. The positronium fraction increases with increasing work function of positronium or increasing temperatures. The above result is very different from that of Nieminen and Oliva. Further we have succeeded in explaining, by usig the present theory, that the positronium fractions are insensitive to surface contamination or variations of the surface dipole. The slight change of positronium fractions due to adsorbate coverage is due to the change of the attenuation length of the surface electron states. Furthermore, we calculate the formation probability of excited state positronium and positronium negative ion with a simple model. We need experimental data on A-type and B-type metal. In particular we need experiments on alkali metals in order to examine our theory in detail. These studies are also available for theoretical developments of ion neutralization on solid surfaces. In the subsequent paper we calculate the velocity distribution function of emitted positronium using the present theory.
Acknowledgements I wish to thank Professor Y.H. Ohtsuki, M. Kato, S. Shindo and R. Kawai for their useful discussions. I also thank Professor C. McFarland for his kind advice on my English.
Appendix. bvel
shift and level broadening of positronium
We discuss the matrix elements of positronium-metal interaction based on the work of Gadzuk 181 who considered the quantum mechanics of atom-solid
A. Isii / Positro~jum~ormatiQn
an solid surfaces. I
291
surface interaction in a general way. Let us illustrate the situation by considering positronium as a pair of a positron and an electron (see fig. 6). Into the positronium system we introduce the relative coordinates r and the coordinates of the center of mass R = (0, 0, s) as follows. r=r+-r_,
(A.1)
R = (r+ + r-)/2.
(A.2)
The wave function above coordinates, 9(R,
of the Is state of positronium
can be presented
using the
r) =
tA.3)
where h*K */4m = mu*, L a normalization constant of plane wave, and the positronium radius a is two times the Bohr radius a,. In the calculation, the positronium Is wave function is fitted to the exact positronium wave functions in a vacuum. There are four image interactions and the Coulomb attraction between the positron and the electron: VCoulomb= -e2/r, ylmage = - &
_
(A.4) - -$
-t+
e2 2)22+ + rl
+
e2 212s_ f rl’
(A.8
Since the positronium is allowed to interact with the metal via an interaction given by the above potential yrnager there is a shift in the atomic level relative to the Fermi level of the metal and the vacuum potential resulting in a perturbed eigenvalue, The first-order shift with the zero-order wave function used, is given by 6E = (ls~Vi,,,,,,~ls).
(A-6)
Fig. 6. Electric image model for the interaction between a positronium and a metal surface.
A. Isii / Positronium formation
292
on solid surfaces.
I
The coordinate system used in the calculations of the energy shift is one with the origin, z = 0, at the center of mass of the positronium and with z increasing positively away from the metal. In this system, shown in fig. 6, the distance of the positron and the electron from the metal surface is given by z+=s+rcos~/2 respectively. counterpart
and The distance is given by
z_=s--rcos8/2, between
the positron
and the electron
image, and the
122, + rl= 12z_ + rl= 2s.
(A.7)
This is a very good approximation for small values of r since u0 - 1 A and the classical picture of the image interaction is valid only in the region separated from the metal by a few A or more. Combining this idea with eqs. (A.3) and (A.4) allows eq. (A.6) to be written as E=zj
1
L
t?*
dss-
&lLdSjr_dij&
+ A)(
f + Izl) ee*“““,
u where z = r cos 8/2 and we use the equalities of a large S, we find that SEaZj
1
obtained
by Gadzuk.
In the case
Lds ?. (I s
In the limit of infinite
(A.9) L, we obtain
the following
important
conclusion, (A.10)
6E=O. We can see from this result that the level shift of the image potential is negligible small. it forms a striking ion. The difference comes from the nonlocality of positronium. Next we consider the level broadening of positronium From Gadzuk’s discussion we find an approximate matrix element as: qr = (Is1 -
(A.8)
positronium due to the contrast to that of the the wave function of near a metal surface. form of the transition
e*/rlu+, u,),
(A.ll)
where u, and uj are the wave functions solid, respectively. As done by Gadzuk, 1 u = Pei(k,,-x+k,,.v) ’ k,L3,/*
1 u _ Ci(kl,X+hz,_v) J k,L3,/*
[(k’?, + k,,)
2k’
ik?,:31e
)
e
of the positron and the electron in a the single electron wave functions are ik;,=m+ (k;, _ k3,) e-%=
],
for z_ < 0,
(A.12)
for zP > 0,
(A.13)
293
where the following
nomenclature
P+i-=h2k,2/2m,
has been adopted:
k,Z=k:,+k;,+kj,,
k’+k:;+k$
kj=:k;+k$ (A.14)
L, is the length of the large but finite cubic metal. The indices 1, 2 denote the x, y directions on the plane of the surface and 3 denotes the z direction that is normal to the surface. Note that k., and k, are both imaginary numbers. The integrals in the matrix element (A.ll) have been restricted to the region outside the metal. Substituting (A.3), (A.13) in eq. (A.ll) and writing the matrix element as an integral yields
-2e2k$,
r,,=
Jdr+ u+(J.+) k,L;J7la’ dx dy
X
Using T,=:
e”k,,“_‘l,.,,‘~~-“’
the same techniques --c
00 &_ eik,,-_ J0
dr+ tl+(r,>
e-iK+K,
as Gadzuk
e-i(K,-kt).r+
fA.1.5)
we obtain
,-i(Kz-k,).y+
e-iKz-i/2
f X
*&_
eitk~-K,/2t:-
Ge-F,;_
f0
-it],
(A.16)
where F2 = a-’ + (K,/2 - k,)* + (K,/2 - Ic,)~ and c is the constant in front of the above integral. Since the wave function of Ps (A.3) extends in all directions in a vacuum, there is no surface effect in eq. (A.16). In order to include the effect of the
dependence on the distance between the positronium and the metal surface, we assume that ~(R.r)ae-iK’K[6(I+~-~R,)-~(-~++-~R,)].
(A.17)
Thus we find that when cp, is negative, [7YfiZa e--2@31s,
(~38)
and when $+ is positive, (A.19) where I+ has the z-dependence eelq3tr+. From the above result we find that the level broadening A has the form of simple exponential decay. Since s
in the above equation is the spatial coordinate of Ps, we conclude that Nieminen and Oliva made a wrong choice for s: they treat s as the coordinate of e+.
294
A.
Isii
/
Positronium formation on solid surfaces. I
References [l] A.P. Mills, in: Proc. 6th Intern. Conf. on Positron Annihilation, Arlington. TX. 1982, Eds. P.G. Coleman, S.C. Sharma and L.M. Diana (North-Holland, Amsterdam, 1982). [2] R.M. Nieminen and J. Oliva, Phys. Rev. B22 (1980) 2226. [3] W. Bless and D. Hone, Surface Sci. 72 (1978) 277. [4] R. Brako and D.M. Newns, Surface Sci. 108 (1981) 253. [5] J.N.M. van Wunnik, PhD Thesis, Universiteit van Amsterdam (1983); J.N.M. van Wunnik, R. Brako, K. Makoshi and D.M. Newns, Surface Sci. 126 (1983) 618. [6] A.P. Mills, Jr., in: Lecture Notes International School of Physics “Enrico Fermi”. 83rd session, Varenna, Italy, 1981, Eds. W. Brandt and A. Dupasquier. [7] A. Blandin, A. Nourtier and D. Hone, J. Physique 37 (1976) 369. [S] J.W. Gadzuk, Surface Sci. 6 (1967) 133, 159. [9] N.D. Lang and W. Kahn, Phys. Rev. B3 (1971) 1215. [lo] C.A. Murray, A.P. Mills, Jr. and J.E. Rowe, Surface Sci. 100 (1980) 647. [ll] K.G. Lynn, P.J. Schultz and I.K. MacKenzie, Solid State Commun. 38 (1981) 473. [12] R.J. Wilson and A.P. Mills, Jr., in ref. [l]. [13] D.C. Schoepf, S. Berko, K.F. Canter and A.H. Weiss, in ref. [l]. [14] R. Kawai and Y.H. Ohtsuki, Nucl. Instr. Methods B2 (1984) 414. R. Kawai, private communication. 1151 A.P. Mills, Jr., L. Pfeiffer and P.M. Platzman, Phys. Rev. Letters 51 (1983) 1085. [16] P.J. Schultz and K.G. Lynn, Phys. Rev. B26 (1982) 2390.
Surface Science 147 (1984) 277-294 North-Holland, Amsterdam
THEORY OF POSITRONIUM FORMATION ON SOLID SURFACES I. General theory and its application to positronium formation probability Akira
IS11
Department
Received
of Physics, Waseda Vnioersity, Okubo 3 - 4 -I, Shinjuku, Tokyo 160, Japan
25 April 1984; accepted
for publication
25 July 1984
We consider the micro-processes of positronium formation on surfaces with a new Hamiltonian, in which we treat the positron as a quantum particle. First we calculate the temperature dependence and the positronium work function dependence of the positronium formation probability in the case of no surface binding states. We succeeded in deriving a slightly increasing profile of the positronium fraction at low temperatures, as was found in the experiment of Lynn et al. We also calculated the probability of the positronium formation at excited levels and of the formation of the negative positronium ion. Furthermore, the positronium fraction is shown to be proportional to the length at which the surface electron density drops to half the maximum value.
1. Introduction Low energy positrons which have been implanted into a solid target in a vacuum form a unique and sensitive probe of the surface regions of the solids [l]. Of particular interest is the abundant positronium (Ps) formation in the outer surface region. We present in this paper a new general procedure for treating this attractive physical process. In the experiments a monoenergetic beam strikes a well-characterized single-crystal surface. The majority of the positrons penetrate the solid and simultaneously lose their energy rapidly within a characteristic stopping distance. The thermalized particles are fairly mobile and a large fraction of them eventually diffuse back to the entrance surface. Some materials have a negative work function for positrons, i.e., positrons are spontaneously re-emitted from the surface. A large fraction of the positrons reaching the surface may capture an electron at the surface and thus emerge as Ps rather than ef . Yet another possibility is the trapping of a positron at the surface owing to the image force. In the present paper we focus our attention on microscopic processes of Ps formation occurring on a metal surface. Nieminen and Oliva [2] have treated this problem using the Anderson-Newns model Hamiltonian. The Anderson-Newns model is well-known in the charge exchange problem, ion 0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
278
A. Isii / Posrtronium
formation
on solid surfaces. I
neutralization [3,4] and negative ion formation [5] in atom-surface scattering. However, we cannot apply the Anderson-Newns model to positrons, because infinite positron mass is assumed in the model. We present here a new Hamiltonian which is generalized from the AndersonNewns model. Using this model we correct the work of Nieminen and Oliva and can describe the temperature dependence of this process as discussed by Brako-Newns [4] in the atom-surface scattering. Moreover, we can describe the effects of surface contamination using the new model. Before proceeding to the theory, one or two remarks about the positron states in metal and vacuum should be made. In fig. 1 we show a schematic one-particle presentation of the potential energy diagram for a positron on a metal and the Ps energy level for a positron near a metal. The Ps level includes both the binding energy of Ps and electron work function. The transition rate between a metal state and a positronium state gives rise to a finite lifetime of the state, which results in a Heisenberg broadening of the level. The width of the level is also shown in the figure. In section 2 we present our model Hamiltonian and discuss the characteristic aspects of the rseonant transition in our case. Moreover, we classify metals into three types for convenience to consider various aspects of Ps formation. In section 3 we obtain a general formula for the probability of Ps formation on solid surfaces with an equation of motion method which is mathematically similar to that of Bloss-Hone and Brako-Newns. In the calculation we take approximations similar to those of Bloss-Hone and Brako-Newns. In section 4 we obtain a useful formula for the Ps fraction using a model of level width. Using the formula we calculate the Ps fraction for many metals. The fraction for excited state Ps and that for Ps negative ions are also obtained. Furthermore, we find that our theory contains only the work function of Ps. This result agrees well with the experiments of adsorbate coverage [6]. Moreover, the attenuation length of the surface electron density profile affects the Ps fraction. In section 5 we discuss these points in detail. We use the atomic units throughout the paper.
Fig. 1. Schematic
one-particle
representation
of the state for a positron
on or near a metal.
279
A. Isii / Positronium formation on solid surfaces, I
2.
Model
We associate the Ps formation with the interaction of the positron with the delocalized (broad band) electrons. A similar problem has been discussed in the field of ion neutralization in terms of a resonant process by Blandin et al. [7], Bloss and Hone [3], and Brako and Newns [4] in a many-body approach using a time-dependent Anderson-Newns Hamiltonian. But, in the case of positrons, one difficulty arises; the wave function of a positronium cannot be separated into a part for the electron and a part for the positron. Thus we cannot define a single-particle operator for the positronium state. Nieminen and Oliva have ignored-this diffkklty.
I+.( n-2) --.-._-____--__-_
Ps level (n=l )
W
level
e- Energy
C ?
Fig. 2. Illustration of the electron energy level in a metal and the effective positronium energy level for the electron. (a) A-type metals; alkali metals. (b) B-type metals: W and Cr. (c) C-type metals: Al, Cu, etc.
We start with the SchrOdinger picture in order to use the wave function of poistronium, that is a two-particle state. We see in figs. 2 and 3 the essential features of the model; the energy level structure of the electron in fig. 2, and that of the positron in fig. 3. In the solid, we have broad conduction bands, corresponding to spatially delocalized electrons, which we assume to interact with the ground state or, in some cases, with the first excited state. It is convenient to classify metals into three groups by the energy level structures as shown in figs. 2 and 3. Alkali metals are of type A. These have the positronium work functions +pps which are nearly equal to zero. The positron
be* Energy a Metal
Vacuum
_-,,~*i&ge potential ,,’ 1’
Qt
Ps level
b
A,
( n=lf
P Energy Ps(n=2)
L
f
f”,,_--’
:
Metal
C
/’
/’
_e---
_--
e
......_..... Ps(n=l) Vacuum
1 e’ Energy
/ Metal
Fig. 3. Illustration of the positron energy levels in a metal and the effective positronium energy level for the positron. (a) A-type metals: alkali metals. (b) B-type metals: W and Cr. (c) C-type metals: Al, Cu, etc.
A. Isii / Posrrronium form&on
on solid surfuces. I
281
work functions ++ are positive in this type. The metals of type B have negative work functions of large magnitude for positrons. Thus the emission of positrons occurs. Moreover, the production of Ps at excited levels (n = 2) is possible for metals of this type. The formation of negative Ps ions is also possible. The metals of the last group are Al, Cu, etc., which are used frequently in experiments. In this case the positron work functions $+ are nearly equal to zero and +r,s have large negative values. We find the surface binding state of a positron in the surface of metals of type C, which also interacts with the ground state of Ps in vacuum. This interaction causes Ps formation via e+ escaping from the surface state. The model Hamiltonian we propose, incorporating the above features, is a generalization of the Anderson-Newns model. Here we treat the impurity in the model of Brako and Newns as a quantum particle. We assign the subscript k to the conduction electrons, q to the positron in metals (including surface states) and d to the state of positronium. Therefore, the Hamiltonian may be written H = c Ekd;fdk + c Ep~a, k
rl
+ E,,djd,a$a,
+ c
(~~~d~d~~~~~
+ h.c.),
(1)
k.q
where Vkd is the interaction between the conduction electron, the positron and the positronium state, and d+ and at are creation operators of electrons and positrons, respectively. In choosing a one-electron picture of the metal we have necessarily excluded processes which depend on electron-electron interaction. We also disregard the Ps-hole interaction, which must be included in the later calculations. In the theory of ion neutralization, it is not simple to establish the expected relative probabilities for neutralization from resonant transfer from. the conduction band and from Auger effects [3]. Fortunately, without W and Cr, we have only one level of Ps to which the conduction electrons transfer. Thus we do not consider either Auger neutralization or interaction between two resonant levels of Ps. We can use these facts to examine the resonant neutralization theory because it has been difficult to measure the relative probabilities for neutralization from resonant transfer and from Auger effects in experiments of ion-surface collision.
3. Calculation Within the above model, we can proceed to solve the problem in the Schrodinger picture by two-particle wave functions. We define the wave function of the total system as:
A. fsii / Posi~ronium formationon solid surfaces.i
282
where Cq.k(t) is the product of the one-particle wave function of positron (4) and electron (k), and Cd(t) is the wave function of Ps including coordinates of the positron and the electron. I/c) is a product of the positron state and the Fermi sea having one hole in state k. Our solution is similar to the equation of motion method of Bloss and Hone. The state It) agrees with the Schriidinger equation i(d/dt)\r) = H(t), from which we derive the equations for C, (t ) and C,,n ( t ).
where nk is the Fermi distribution Cd{ 2) = C,(t)
exp(iE&,
and where we have defined
.
(5) (6)
c,,.(t)=C~,,(t)expIi(E,+E,)t], k,(t)
= Y.&>
exp[i($
(7)
+ Ek - G&].
These equations have the same mathematical form as that of Bloss and Hone. We can rewrite the equation of Cd(t) using the integral form of eq. (4),
Following Bbss and Hone, we also assume that the time variation of V,,(t) depends little on the value of q and k; instead it characterizes the motion of the positronium, so that V,, (t ) can be factorized as v,,(t)
(9)
= Y&(t)-
Then the sums over k and q in the second term of eq. (8) take the form
=A(t)6(t-T),
(10)
= A&(t)12.
(11)
where A(t)=
&v@)~IK,I~)E~~E 4
This is similar to the approximations used by Bless and Hone, where the band is assumed to have no important structure and to have an infinite bandwidth. Within these approximations the equation of motion for Cd(t) becomes &(1,
= -Ill’&
dd(f)
-ix
c v&(t) k
4
cq,k(r,>*
02)
A. Isii / Positronium jbrmation
283
on solid surfaces. I
Eq. (12) may be solved to give C;(t)
= e-rCrO*‘) 2;,(t,)
- ix Cn,l:r k rl
eda
Thus we obtain the positronium production momentum ~7and the electron of momentum
e-‘~‘.‘)
Cq;,.,(f,).
(13)
probability for the positron k as follows,
of
ICd(t)12 = (e-‘(‘o~‘) ed( + c c k
q
c k’
)3nkv&v,‘d
c$k(tO>
cq’,k&l>
q’ 2
Ff7.f)
X
X
I*0TdT V&U(T)
where we defined
r(tr,
ei(E,iE,-Epqf7
e ff7.*) ei(E,+E~-Eps)~ +
c_cs
(14)
t2) as:
r(f,,t,)=A~~r2d~I~(1-)12.
f
(15)
1
The first term of eq. (14) describes the decay of the initial state to equilibrium state in a solid where the system is initially in the positronium state. This term is negligible because positronium states do not exist inside the solid. The last two terms are also negligible for the same reason In the rest of this paper we concern ourselves with the situation where the second term in eq. (14) is the dominant one. Our aim is to calculate the expectation value
= (IGW12)?
h&))
(16)
where the bracket means averages over the positron and the Eventually t will be put to + cc as the measurements of (nd) the positronium has left the solid. From now on we shall take t, assume that in the initial state the metal was in equilibrium at From the initial condition we obtain F
~(c;k(-co)
where tively,
cq,k(-m>>
=.fF@k,
Th&p
f, and fr, are the Fermi and Boltzmann
electron states. are made after = - cc and we temperature T.
(17)
T)t
distribution
functions,
respec-
fF(E, T) = (I+ exp[(E+ +-)/kT]}ml,
(18)
f,(E,
(19)
T)=(nkT)-3’2exp[-(E+cp,)/kT].
284
A. lsii / Positronium formation
on solid surfaces. I
Thus we obtain the expectation value of the occupation same approximation as that of eq. (10): @L&o))
=c
C%(Krl12)fF(~~~ k 4
co
X
IJ
T)f”&
Eq. (20) is an generalization positrons.
using
the
T)
2
!.
dr t{(T) e -1.(7.cc) ei(E+G?--ll‘p,)T
--3c
(n,(m)>
(20)
of the results of Brako and Newns to the case of
4. Results 4.1. Exponential
model
Before discussing the results of the above theoretical calculation, let us first discuss the static situation of a positronium at a given distance from the metal surface. When the level of a positronium is situated within the conduction band of a metal, resonant electron transition between the metal and the positronium will take place. In this theory Ps-metal interaction is considered as a perturbation of the pure Ps and metal states. This method was worked out by Gadzuk [8] for the chemisorption of alkalis on metal surfaces. This interaction gives rise to a shifting SE and a broadening A of the Ps level, which we calculate in the appendix. In the appendix we derive the level shift and the transition matrix element related to the level broadening as follows: SE=O,
(2la)
]Ti:,,12Cce-+-t’+
(21b)
It is very complex to calculate eq. (11) by substituting eq. (21b). It was concluded by Nieminen and Oliva [2] that the emerging positron is emitted into a Gaussian angular distribution around the surface normal with a rather narrow angular spread. Thus we assume there that the emerging positronium is also emitted into a narrow cone around the surface normal. Therefore we can discuss the motion of the positronium only on the surface normal direction. Since we consider here theoretical problems and qualitative properties of positronium formation at solid surfaces, we assume simply that A =A,exp(-2s/a),
(22)
A. Isii / Positronium formation on solid surfaces. I
where a is radius of positronium in vacuum, i.e. two times the Bohr corresponds to the fact that the electron density distribution outside drops to half the maximum value at approximately a (or two times radius) from the nominal surface. It seems to be a reasonable value with the density profile obtained by Lang and Kohn [9]. Substituting (21a) and (22) into eq. (20) we obtain the following the Ps production probability
X/dw~~f,(w,T)~sech[(E,,-E-w)a/2o],
radius. It the metal the Bohr compared form for
(23)
where we assume the velocity of the positronium constant o defined as: v2=E+o-E
285
normal
to the surface to be a
(24)
PS’
Eq. (23) gives exact results in finite temperature cases. But numerical integration of this integral is not so simple because of drastic exponential decays in the integrand. It is a surprising fact that A, cancels out, so that we have no need to calculate the transition matrix elements in detail. It is worth noting the momentum conservation for the surface parallel direction. The momentum conservation for the parallel direction should be included in the transition matrix element and is smeared by the average in eq. (11). Since the amplitude of the transition matrix element cancels out, we can neglect the momentum conservation when we pay no attention to the angular distribution of re-emitted positroniums. 4.2. Production probability
of Ps in ground state
We take the sech in eq. (23) to be unity, because it is nearly equal to unity where the kinetic energy of Ps is not so large compared with the work function of Ps. It is valid for slow Ps which we discuss here. We neglect thermal smearing of the Fermi surface and take fr( E, T) to be a step function. This approximation is valid where ]$J~,] is quite large in comparison to thermal energies. Using the approximation above, we find that when (pp, -C 0, (nd(Oc))
= ( -2Gps/\/2?Tkr)
e-+p~‘2krK1(
-GPs/2kT),
(25)
where K, is the modified Bessel function of the second kind. The above formula depends only on +ps and not on c$+ and $_ individually. From this fact we can suggest that the Ps production probability is not sensitive to surface contaminants which change ++ and c#_ but do not change +p, [lo]. The effects of surface contaminants found in previous experiments [6] may be caused by the modification of the local electron density near the surface.
286
A. Isii / Positronium formation
We may easily get the low temperature
on solid surfaces. I
limit as follows,
Using eq. (24), we show first in fig. 4 the #rS dependence of the Ps fraction at T= 300 K. As we are not yet including surface states of positrons in the present theory, this fraction cannot be compared to the experimental data of metals of type C. The magnitudes of the fractions for metals of type C like Al or Cu (&,, = -0.095) [6] are, however, compatible with experiments at the temperatures where there is no positronium coming from surface binding states. Moreover our result is very different from the calculation of Nieminen and Oliva. It comes from their wrong choice of the velocity (see appendix). In fig. 5, we show the temperature dependence of Ps fractions for (a) Rb and Cs, and (b) cases of small @)pS.From these figures we find that the Ps fraction is I -
c ii
0
0
0.1
0.2
0.3
0.4
0.5
- (be (au)
Fig. 4. Ps formation
0
0
probability
as a function
of Ps work function
at T = 300 K. Atomic
I
2oD
400
600
T(K)
T(K) Fig. 5. Ps fraction (b) Ideal material
units.
as a function of temperature T for various and -0.0001. of cpps = -0.0005
metals. Atomic
units.
(a) CS and Rb.
A. Isii / Positronium formation
287
on solid surfaces. I
sensitive to changes of temperature in the case of small Ps work function, and we can observe it for Rb and Cs. In the case of Al we find the slightly increasing profile which was revealed in the experiments of Lynn et al. 1111. But the relationship between this fact and the problem of internal reflection of positrons based on the model of Nieminen and Oliva is not clear at this time. 4.3. Production
of Ps (n = 2) and Ps -
Since evidence of excited state Ps (Ps*) formation from metal surfaces and of positronium negative ion (Ps-) formation has been found recently, it is necessary and important to present a theory to calculate the probabilities of Ps* and Ps- production on metal surfaces. One purpose of calculating them is to find convenient sources of Ps* and Ps-, because positronium as a leptonic atom serves as a unique system to test the predictions of quantum electrodynamics [ 11. Since we expect that the most important feature of Ps* and Ps- compared with Ps is the position of their energy level, we calculate the formation probability of Ps* and Ps- using our theory and assuming that eqs. (21a) and (22) are also valid for Ps* and Ps-. Thus systems are characterized mainly by the work functions #pps. and (pr,-. The work functions (pi,,* and (pi,- are defined in the usual manner: 9
Ps*(n)
=
++
+
+_
-
0.25/n’,
(27)
(pps- = (p, + 2+_ - 0.25 - 0.012. For metals
(28)
of type B, e.g. W and Cr, we have the following
BWI, (pp,. = 0.0085 (W), 0,007 (Cr);
&,~ = -0.028
(W),
+O.OOll
respective
values
(Cr).
Before discussing the results of their formation yield, we must first derive an approximate formula for the Ps formation probability for positive work function. Using the same approximation as that of eq. (25), we obtain (nd(cc))
= /m
e-*ps/kT l*
ds imps
+ 1) e-‘.
(29)
We find simple analytic results if kT/+,, < 1. The limit should be reasonably appropriate to the cases discussed here. We find that
In the case of Ps*, we.get fps. = 1.44 x 10s5 for W, which is very small in comparison to the experimental value of Shoepf et al. [13], 1 X 10p3. There are
four reasons for this discrepancy: the experimental uncertainty of the positron work function, the neglect of thermal smearing of the Fermi surface, deviation from the exponential model for level broadening A, and d-electron effects which were also ignored in the theory of Bloss-Hone and Brako-Newns. We can dispose of the second reason through numerical integration of eq. (23). To dispose of the third reason, we must introduce a new function for A(t) in solving eq. (20) numerically. As for the fourth reason, Kawai and Ohtsuki 1141 pointed out that in the field of ion neutralization and negative ion production the localization of electron density of states is important. This problem is very important and essential, but very difficult to solve. Next we discuss the fraction of Ps-. We take a model similar to that used by Van Wunnik in his theory of negative hydrogen ion formation. We obtain fPs - 0.1 if #I~~= - 0.028 for W. For Cr, we use a new work function value [12] of 40.0011 and we obtain a probability of about 0.7%. It is rather larger than that of the expe~mental value, < 0.1% 1121. The discrepancy may result from the experimental accuracy of the positron work function, or from the assumption of a, the attenuation length of the electron mode at the surface. It seems probable that the prediction of our theory becomes a convenient guide to produce Ps- sources using positron-surface interaction. 5. Discussion Let us consider the formation and the results of our theory. First we consider the theoretical formulation. The Hamiltonian (1) corresponds to one of the generalized Anderson-Newns model in which impurities are treated as quantum particles. This Hamiltonian itself can be applied to many aspects of positron-solid surface interaction. In future calculation we must include many-body effects, e.g. positron-hole and electron-hole interaction, and the relaxation effects which are well-known in photoemission. Nonetheless, the above theoretical formulation is very simple and easy to extend. For example, we can add a positron surface state as one of ai. We will discuss this problem later. The approximations used to derive eq. (22) are available for solids of semi-infinite electron gas. Substitution of l/a for the average of k, corresponds to the fact that the attenuation length of surface electron states is of the order of u. It is compatible with the calculations of the surface density profile obtained by Land and Kohn. Thus, the above approximation is suitable for the present calculation. Moreover, we find from eq. (23) that the positronium formation probability is proportional to the attenuation length of the surface electron states or the surface electron density profile. Thus if the surface electrons are pressed into the bulk by the negative charge of oxide contamination, the Ps fraction decreases as we have observed in previous experiments [15]. On the other hand,
A. Isii / Positronium formation
on solid surfaces. I
289
the Ps fraction increases with alkali metal atoms and others. In particular, it is confirmed by measurements of surface dipole that sulfur contamination extends the surface density profile to a vacuum. It is confirmed by experiments that the experimental value of fps increases with increasing sulfur coverage [6]. Thus it is expected that it will be possible to make direct measurements of variations of surface density profiles due to various surface contaminations. Our formula (23) depends on only +r, and does not depend on $+ and +with increasing surface conseparately. Since +rs remains almost constant taminations [6], we reach the conclusion that fps is insensitive to adsorbate coverage. This agrees well with experimental results. In experiments we find slight variations of fPs with adsorbate coverage. One of the sources of the variation is the change of surface electronic density profile due to adsorbate coverage. Another reason may be changes in the electronic density of states due to surface contaminations. Moreover, it is first necessary to derive the temperature dependence of the Ps formation probability itself. Probably this effect and the temperature dependence of +rs itself [16] are both causes of the slight variation of fPs observed by Lynn et al. [ll]. Next we discuss the various results of Ps production probability presented in the above section. Unfortunately, there have been no experiments on metals of type A and B. Nonetheless, we can state the following conclusions: A-type metals: fPs is 0.05-0.1. Since they have no surface state and no possibility to produce Ps* or Ps-, it seems to be the simplest example of resonant transfer theory. In addition the temperature dependence of fps is strong for these metals. Moreover, since the attenuation length of the surface electronic density profile is very large for these metals, the fPs of them is very sensitive to surface contaminations. Thus we can check our theory when we compare the above theoretical results with experimental values. Therefore, we have pointed out that experiments on positron-alkali metal surface interaction are necessary and important to confirm our theory. B-type metals: fps is about 0.6 for Cr and W. It is particular interesting that we find in Cr and W finite probabilities of formation of Ps* or Ps-. Thus we can expect to have a source of Ps* and Ps- in vacuum using the mechanism. However, it is necessary to study the level broadening of Ps* and Ps- near the metal surfaces in detail. For Ps-, it is also necessary to examine the model of negative ion production used here that was presented by Van Wunnik for proton-surface interaction [5]. C-type metals: For these metals there have been many experiments. However, since surface trapping states exist on the surfaces, it is difficult to compare the experimental data with the predictions of our theory. For Al or Cu, it seems to be reasonable when we compare the theoretical values with the experimental values of low temperatures where there are no Ps escaping from the surface state.
290
A. Isli / Posrtronium formation
on solid surfaces. I
We must add one more theoretical discussion. In eq. (23) we used the Boltzmann dist~bution for positrons in metal. This is a simple model of the thermalized positron. In more accurate calculations we must find more accurately the momentum distribution of positrons on the surface. We cannot use the diffusion mode1 of Nieminen and Oliva, because they derive only the spatial distributions of positrons in solids.
6. Conclusions The principal aim of this paper is to solve the problems of positronium formation on solid surfaces as direct resonance charge exchange. Although our calculations are based on a rather idealized model, they give a reasonably good account of the calculations of the positronium formation probability. Our calculations agree with experiments where there are no positroniums escaping from the surface state. The positronium fraction increases with increasing work function of positronium or increasing temperatures. The above result is very different from that of Nieminen and Oliva. Further we have succeeded in explaining, by usig the present theory, that the positronium fractions are insensitive to surface contamination or variations of the surface dipole. The slight change of positronium fractions due to adsorbate coverage is due to the change of the attenuation length of the surface electron states. Furthermore, we calculate the formation probability of excited state positronium and positronium negative ion with a simple model. We need experimental data on A-type and B-type metal. In particular we need experiments on alkali metals in order to examine our theory in detail. These studies are also available for theoretical developments of ion neutralization on solid surfaces. In the subsequent paper we calculate the velocity distribution function of emitted positronium using the present theory.
Acknowledgements I wish to thank Professor Y.H. Ohtsuki, M. Kato, S. Shindo and R. Kawai for their useful discussions. I also thank Professor C. McFarland for his kind advice on my English.
Appendix. bvel
shift and level broadening of positronium
We discuss the matrix elements of positronium-metal interaction based on the work of Gadzuk 181 who considered the quantum mechanics of atom-solid
A. Isii / Positro~jum~ormatiQn
an solid surfaces. I
291
surface interaction in a general way. Let us illustrate the situation by considering positronium as a pair of a positron and an electron (see fig. 6). Into the positronium system we introduce the relative coordinates r and the coordinates of the center of mass R = (0, 0, s) as follows. r=r+-r_,
(A.1)
R = (r+ + r-)/2.
(A.2)
The wave function above coordinates, 9(R,
of the Is state of positronium
can be presented
using the
r) =
tA.3)
where h*K */4m = mu*, L a normalization constant of plane wave, and the positronium radius a is two times the Bohr radius a,. In the calculation, the positronium Is wave function is fitted to the exact positronium wave functions in a vacuum. There are four image interactions and the Coulomb attraction between the positron and the electron: VCoulomb= -e2/r, ylmage = - &
_
(A.4) - -$
-t+
e2 2)22+ + rl
+
e2 212s_ f rl’
(A.8
Since the positronium is allowed to interact with the metal via an interaction given by the above potential yrnager there is a shift in the atomic level relative to the Fermi level of the metal and the vacuum potential resulting in a perturbed eigenvalue, The first-order shift with the zero-order wave function used, is given by 6E = (ls~Vi,,,,,,~ls).
(A-6)
Fig. 6. Electric image model for the interaction between a positronium and a metal surface.
A. Isii / Positronium formation
292
on solid surfaces.
I
The coordinate system used in the calculations of the energy shift is one with the origin, z = 0, at the center of mass of the positronium and with z increasing positively away from the metal. In this system, shown in fig. 6, the distance of the positron and the electron from the metal surface is given by z+=s+rcos~/2 respectively. counterpart
and The distance is given by
z_=s--rcos8/2, between
the positron
and the electron
image, and the
122, + rl= 12z_ + rl= 2s.
(A.7)
This is a very good approximation for small values of r since u0 - 1 A and the classical picture of the image interaction is valid only in the region separated from the metal by a few A or more. Combining this idea with eqs. (A.3) and (A.4) allows eq. (A.6) to be written as E=zj
1
L
t?*
dss-
&lLdSjr_dij&
+ A)(
f + Izl) ee*“““,
u where z = r cos 8/2 and we use the equalities of a large S, we find that SEaZj
1
obtained
by Gadzuk.
In the case
Lds ?. (I s
In the limit of infinite
(A.9) L, we obtain
the following
important
conclusion, (A.10)
6E=O. We can see from this result that the level shift of the image potential is negligible small. it forms a striking ion. The difference comes from the nonlocality of positronium. Next we consider the level broadening of positronium From Gadzuk’s discussion we find an approximate matrix element as: qr = (Is1 -
(A.8)
positronium due to the contrast to that of the the wave function of near a metal surface. form of the transition
e*/rlu+, u,),
(A.ll)
where u, and uj are the wave functions solid, respectively. As done by Gadzuk, 1 u = Pei(k,,-x+k,,.v) ’ k,L3,/*
1 u _ Ci(kl,X+hz,_v) J k,L3,/*
[(k’?, + k,,)
2k’
ik?,:31e
)
e
of the positron and the electron in a the single electron wave functions are ik;,=m+ (k;, _ k3,) e-%=
],
for z_ < 0,
(A.12)
for zP > 0,
(A.13)
293
where the following
nomenclature
P+i-=h2k,2/2m,
has been adopted:
k,Z=k:,+k;,+kj,,
k’+k:;+k$
kj=:k;+k$ (A.14)
L, is the length of the large but finite cubic metal. The indices 1, 2 denote the x, y directions on the plane of the surface and 3 denotes the z direction that is normal to the surface. Note that k., and k, are both imaginary numbers. The integrals in the matrix element (A.ll) have been restricted to the region outside the metal. Substituting (A.3), (A.13) in eq. (A.ll) and writing the matrix element as an integral yields
-2e2k$,
r,,=
Jdr+ u+(J.+) k,L;J7la’ dx dy
X
Using T,=:
e”k,,“_‘l,.,,‘~~-“’
the same techniques --c
00 &_ eik,,-_ J0
dr+ tl+(r,>
e-iK+K,
as Gadzuk
e-i(K,-kt).r+
fA.1.5)
we obtain
,-i(Kz-k,).y+
e-iKz-i/2
f X
*&_
eitk~-K,/2t:-
Ge-F,;_
f0
-it],
(A.16)
where F2 = a-’ + (K,/2 - k,)* + (K,/2 - Ic,)~ and c is the constant in front of the above integral. Since the wave function of Ps (A.3) extends in all directions in a vacuum, there is no surface effect in eq. (A.16). In order to include the effect of the
dependence on the distance between the positronium and the metal surface, we assume that ~(R.r)ae-iK’K[6(I+~-~R,)-~(-~++-~R,)].
(A.17)
Thus we find that when cp, is negative, [7YfiZa e--2@31s,
(~38)
and when $+ is positive, (A.19) where I+ has the z-dependence eelq3tr+. From the above result we find that the level broadening A has the form of simple exponential decay. Since s
in the above equation is the spatial coordinate of Ps, we conclude that Nieminen and Oliva made a wrong choice for s: they treat s as the coordinate of e+.
294
A.
Isii
/
Positronium formation on solid surfaces. I
References [l] A.P. Mills, in: Proc. 6th Intern. Conf. on Positron Annihilation, Arlington. TX. 1982, Eds. P.G. Coleman, S.C. Sharma and L.M. Diana (North-Holland, Amsterdam, 1982). [2] R.M. Nieminen and J. Oliva, Phys. Rev. B22 (1980) 2226. [3] W. Bless and D. Hone, Surface Sci. 72 (1978) 277. [4] R. Brako and D.M. Newns, Surface Sci. 108 (1981) 253. [5] J.N.M. van Wunnik, PhD Thesis, Universiteit van Amsterdam (1983); J.N.M. van Wunnik, R. Brako, K. Makoshi and D.M. Newns, Surface Sci. 126 (1983) 618. [6] A.P. Mills, Jr., in: Lecture Notes International School of Physics “Enrico Fermi”. 83rd session, Varenna, Italy, 1981, Eds. W. Brandt and A. Dupasquier. [7] A. Blandin, A. Nourtier and D. Hone, J. Physique 37 (1976) 369. [S] J.W. Gadzuk, Surface Sci. 6 (1967) 133, 159. [9] N.D. Lang and W. Kahn, Phys. Rev. B3 (1971) 1215. [lo] C.A. Murray, A.P. Mills, Jr. and J.E. Rowe, Surface Sci. 100 (1980) 647. [ll] K.G. Lynn, P.J. Schultz and I.K. MacKenzie, Solid State Commun. 38 (1981) 473. [12] R.J. Wilson and A.P. Mills, Jr., in ref. [l]. [13] D.C. Schoepf, S. Berko, K.F. Canter and A.H. Weiss, in ref. [l]. [14] R. Kawai and Y.H. Ohtsuki, Nucl. Instr. Methods B2 (1984) 414. R. Kawai, private communication. 1151 A.P. Mills, Jr., L. Pfeiffer and P.M. Platzman, Phys. Rev. Letters 51 (1983) 1085. [16] P.J. Schultz and K.G. Lynn, Phys. Rev. B26 (1982) 2390.