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Influence of spatial dispersion on the interaction energy of a positronium atom and a metal surface
PHYSICA ELqEVIER
Physica B 193 (1994) 161-165
Influence of spatial dispersion on the interaction energy of a positronium atom and a metal surface aDo Nam,
D.B.
T r a n T h o a i b'*
aUniversity of Hue, Le Loi 3, Hue, Viet Nam bNational Centre for Scientific Research, Mac Dinh Chi 1, Ho Chi Minh City, Viet Nam
Received 4 January 1993; revised 3 May 1993; final revised 31 August 1993
Abstract
The influence of spatial dispersion of the medium on the interaction energy of a positronium atom and a metal surface is studied using the self-energy formalism developed by Manson and Ritchie. It is found that within the dipole approximation when spatial dispersion is taken into account the interaction energy saturates to a constant value at the surface even for positronium at rest.
1. Introduction
The interaction energy between a metal surface and a positronium atom has been studied with considerable interest [1-3]. Using a semiclassical approach Lifshitz [1] has calculated the van der Waals interaction energy between a positronium atom and a metal surface. For rather large distances between the atom and the surface Lifshitz's result showed that the interaction energy varies a s 1 / Z 3. A recent experimental work of Lynn and Welch [2] on the scattering of a positronium atom by a metal surface has led to increased activity in the field. The quantum nature of the interaction between the positronium atom and the metal surface should be addressed. Manson and Ritchie [3], using their self-energy formalism, have calculated the self-energy of a positronium atom near a metal surface, and came to the conclusion that
* Corresponding author.
the quantum recoil weakens the interaction energy from its Lifshitz's value. Manson and Ritchie have found a very interesting result: near the surface the interaction energy varies as 1 / Z instead of 1 / Z 3. We know from the calculation of the image potential of an electron near a metal surface that spatial dispersion of the metal has made the classical image potential, which also varies as 1 / Z , finite at the metal surface [4,5]. Therefore, it would be of great interest to include spatial dispersion in the calculation of Manson and Ritchie, to see whether the interaction energy of a positronium atom near the metal surface becomes finite or not. We shall employ the hydrodynamic model for the metallic electron to include spatial dispersion.
2. Interaction Hamiltonian
Let us consider a positronium atom approaching a metal occupying half-space Z~<0 with
0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved S S D I : 0921-4526(93)E0337-G
162
Do N a m . D.B. Tran Thoai ," Phvsica B 193 (19941 I0I
planar surface. The metal is described as a sentiinfinite electron gas in the hydrodynamic model [6]. An atom can be considered as a charge density p(r) consisting of a nuclear charge Ze and an electron cloud. The interaction Hamiltonian of the atom at r = (R, Z) and the metal is
V(r) = ( d r ' p ( r ' ) O ( r + r') .
(1)
Here O(r) is the interaction Hamiltonian of an clement of charge e with oscillator quanta of the metal:
O(r)
e~'~ [~,A(r)aA + ~ ( r ) ~ , l
(2i
A
where a A and h a are the creation and annihilation operators of plasmons. The index a indicates the plasmon modes and their wave vectors. ~A(r) is the electrostatic potential caused by the plasmons specified by A. An explicit expression of q~A for the hydrodynamic model of a metal occupying the half-space Z ~< 0 has been derived by Barton [6]. Like Manson and Ritchie we shall only consider the interaction with surface plasmons, for which: rn
~(r)
-
,
e N~[w;Q
-
eS,/.
Z<~O, rn e N~[°J2pQ -
- w
~p~
eC)Zl e <'u
oJ~P~]e iO"
.
P
##1
• 1[ ~ n ¢ O : = ~ - ['tO P -}- - -
~
rn
~
~
@e q- ~n-H Q ( ( ~ - +
3~
Q2+
/j2
)
Q
hP, e ~N~ = rrrn'~aJo)~Q(ZP, + Q)(P,
+ a c,)9,(Z)
(S)
q
where A is a vector which in Cartesian components is given b~ A
(iQx, iQy,
Q ~ , .
p
f dr' p(r')r' is the atomic dipole moment, DI
~(z)
T[(,,,~c ~'" - ~ope : "," ) P 0 ( - Z )
+ (,,,~t"
,o~(j)e ~"
(~1
and <#,(Z) is the Z-dependent part of the electrostatic potential caused by surface plasmons. given in Eq. (3).
(4 2,,,o,7 /3:
)
'
],
5(r) = nZ.l,k
(,,I,')
(5 Ps
~%" [ p . A e x p ( i ( ) R ) ] ( D ~ ,
Comparing thc definition of thc self-energy through the energy shift demonstrated by Manson and Ritchie [7] with the expression of the energy shift in second order in perturbation theory, one gets
(3
4"rre2n --
V (r)
3. Sell-energy
where
2
parallel to the metal surface, #1 the electron density in the metal, and/3-" the coefficient of the hydrodynamical pressure term, taken to be 6,'5 of the Fermi energy E t . Regarding the atom as a ncutral chargc distribution and assuming that the interaction energy is due to the mduccd cxcilations of surface plasmons and the dipolar excitations o1 the atom we obtain for the interaction Hamiltonian an cxpression as follows:
C,z
Z/>0,
O)
105
,
(6
Q)e -
(7
Here m is the electron mass, Q the wave vector
(%1,-)
<~,,, z, .IvIo, o. ,~,,> x
E 0 + F, + e o
E, - (~o + Q)
ek + i~r
(ll))
whcre In) is the quantum surface state wittl energy E,,; II} represents the internal state of the positronium atom having energy ~,, + e~ and I~k) is the state associated with the center of mass M of the positronium atom and energy e~ :
h2(K ~-+ k : ) / 2 M .
163
Do Nam, D.B. Tran Thoai / Physica B 193 (1994) 161-165
Let us consider the case of the positronium atom with momentum hk o moving perpendicular to the metal surface. We assume that states of the electronic system in the atom have no changes by the presence of the metal. For a plane wave basis set of eigenfunctions we obtain the following expression for the self-energy:
Z(Z)-
2Mw~ 3h
e
ikoZ
2MwZp
.~(z)-
3h ~I I<°Nz)l=
x f dQ Q2
Ps oz ~o~(2P~ + Q) e
0
[ e -ez_
e s°z(
1
,s 1,o)1
~o ~ - 0
× LQ~+O~ ~] I
(16)
l
where
x i dQ
Q2
P~ Oz w~(2P~ + Q) e
So = Q2 + Q~ + Q~.
0 e
The expression of the self-energy in the dispersionless limit in the same case is
QZ+ik°z
V~MwZp
Q~ + Q~ + 2iQk o
e S,~Z(1 2~,~ \ S q - Q + i k o
1
)]
P~+Sq+iko (11)
~(z,/3=0)--
l
×
Q~ = 2Mo)~/h,
(12)
Q~ = 2MeJh 2,
(13)
Sq2 = 0 2 + Q~2 + QI2 - k~.
(14)
Now we study the limit of the dispersionless case. Within this limit the corresponding result of Manson and Ritchie [3] is obtained by letting /3 = O:
x E [(ol/~[/')l 2 i d Q I
Q 2e
Q2~ + O~
2So(-~-0_-Q)
.
(18)
Lifshitz's result is obtained from Eq. (18) by neglecting the quantum recoil effect (M = 2). Now we are interested in the saturation of the image potential at the surface, so from Eq. (16) we obtain for Z = 0:
X(Z = O) -
2MwZp 3h
I(0]/?1/)] 2 j dO 0 2
t
[ 1 ×
Q~ + Q~
Qz
0
P~ ws(2P~ + Q)
1(1
1)]
2S0 So Z Q
Ps + So
(19)
and
0
Q~ + Q~ + 2iQko
0
[eO
x •
X/2M~o 2pe_i<,z 3h
3h
× Z I<01~ll>l2 jdQ Q2e-QZ
where
~(z,/3=0)--
(17)
X/-2Mo)p
2Sq(S2--O-+ik,,/
X(Z = 0,/3 = 0) - -
6h
(15) Consider now an important case when the particle is at rest (k o = 0). Then the self-energy (11) reduces to
X Z ](0111~11)12i ,
dQ
Q2
1
S0(S0 + Q) "
(20)
0
It is clear that the integral in Eq. (20) is di-
164
1)o N a t n .
I).B.
Tran Thoai
v c r g e n t , while the e x p r e s s i o n in Eq. (19) has a linitc w d u e . T h e s u m m a t i o n in Eqs. (16) and (18) was a c c o m p l i s h e d using f o r m u l a e for oscillation s t r e n g t h s by S u g u i r a 181, for b o t h discrete a n d c o n t i n u u m e l e c t r o n i c states of a h y d r o g e n like atom.
. Phv.sica B 193 4 1 9 9 4 )
1(~1 I o 3
0, •J
f~ / /
4. Numerical calculations and discussions I n this section we c a r r y out n u m e r i c a l calculations for the s e l f - e n e r g y of a p o s i t r o n i u m a t o m and c o m p a r e o u r results with those of p r e v i o u s a u t h o r s . We will focus on the case when thc p o s i t r o n i u m a t o m is at rest. In Fig. 1 we plot the i n t e r a c t i o n e n e r g y Z(Z) v e r s u s the d i s t a n c e Z of the p o s i t r o n i u m a t o m from the m e t a l surface, as c a l c u l a t e d with spatial d i s p e r s i o n ( E q . (16)) a n d w i t h o u t d i s p e r s i o n ( E q . (18)). T h e d e n s i t y p a r a m e t e r r of the m e t a l is 3. N e a r the m e t a l surface, spatial dispersion causes a s t r o n g r e d u c t i o n of the i n t e r a c t i o n e n e r g y a n d a c t u a l l y m a k e s it finite at the m e t a l
;&
-4~
I'ig 2. Sell cncrg.x as a I n n c t l o n ol the c l c c m m gas dcn,,ilx p a l a n l c l c r for Z I/\ Solid c u r x c : s c l l - c n c r g } i n c h t d i n ? s p a t i a l d i s p c r s i ( m : l o n g d a s h e d c u t x c : w i t h o u t spatial dispel s h i n : , , h o r t - d a s h c d c u r x c : l.il'shitz's rcsull.
'2,
38-
Fig. 2 we plot the s e l f - e n e r g y at Z - I A v e r s u s the e l e c t r o n gas d e n s i t y p a r a m e t e r r,. T h e d e p e n d e n c e of the s e l f - e n e r g y on r, is s t r o n g e r w h e n r~ gets smaller.
/
26-
,,
['
'
/j
.~. -2~
,"
3-
06~
/' ;
-4-~'
2
/," /"
/ I i
7
i •o
In
/
-:
E
surface (solid curve). The Lifshitz's result v l ( Z ) (short-dashed curve) is also shown in Fig. 1.
-~
~--T¸
0 4--
/r -
2-
c - ~ 0
i
2
z
.......
"5
(,g)
£
t"ig. 1. Sclf-cncrg~ o f l h c positronium atom as a tunclhm o f
Iqg. 3. Ratio o l the self cncrg.~ to lhc scmicla~,sical Lil,dfit/
the distance Z from thc surface for electron-gas densilx parameter r = 3. Solid curve: self-energy including spatial dispersion: long-dashed curve: without spatial dispersion: short-dashed curxe: Lifshitz's result.
v a l u e a~, a f u n c t i o n o f the d i s t a n c e Z f r o m the ~,urfac¢. Solid c u r v e : s e l l - e n e r g y i n c l u d i n g spatial d i s p e r s i o n : l o n g - d a s h e d c u r v e : w i t h o u t s p a t i a l d i s p e r s i o n s . T o p : t' - 2 : b o t t o m ' r 5.
Do Nam, D.B. Tran Thoai / Physica B 193 (1994) 161-16,5
T h e r a t i o of the s e l f - e n e r g i e s f r o m Eqs. (16) a n d (18) to the Lifshitz's v a l u e [1] for r~ = 2 and r~ = 5 is p l o t t e d in Fig. 3. W e n o t e that the i n t e r a c t i o n e n e r g y ,Y(Z) as given by Eq. (16) to t h e Liftshitz's result is significantly s m a l l e r t h a n t h e c o r r e s p o n d i n g r a t i o in the a p p r o x i m a t i o n of M a n s o n a n d R i t c h i e for all values of Z. F o r e x a m p l e , at Z :: 2 A o u r result gives ~ ' / ~ L = 0.44 for r~ = 2 a n d Z / Z L = 0 . 3 3 for r s = 5, while the r a t i o o b t a i n e d in the l i m i t / 3 = 0 is 0.99 a n d 0.96, respectively. In s u m m a r y , we have e v a l u a t e d the influence o f s p a t i a l d i s p e r s i o n on the i n t e r a c t i o n e n e r g y of a p o s i t r o n i u m a t o m a n d a m e t a l surface. W e h a v e f o u n d that the effect of spatial d i s p e r s i o n a n d t h e q u a n t u m recoil effect give an i n t e r a c t i o n e n e r g y o f the p o s i t r o n i u m a t o m that is finite at t h e surface, giving a result d r a m a t i c a l l y different f r o m t h a t f o u n d by Lifshitz, w h o used semiclassical c o n s i d e r a t i o n s .
165
Acknowledgement W e a r e grateful to the r e f e r e e for his sugg e s t i o n s a b o u t the p r e s e n t a t i o n of o u r p a p e r .
References [1] E.M. Lifshitz, Sov. Phys. JETP 2 (1956) 72. [2] K.G. Lynn and D.O. Welch, Phys. Rev. B 22 (1981))99. [3] J.M. Manson and R.H. Ritchie, Phys. Rev. B 29 (1984)
11184. [4] D.B. Tran Thoai, E. Zeitler and Do Nam, Phys. Stat. Sol. B 158 (1991)) 557. [51 J. Mahanty, K.N. Pathak and V.V. Paranjape, Phys. Rev. B 33 (1986) 2333. [6] G. Barton, Rep. Phys. Prog. 42 (1979) 963. [7] J.M. Manson and R.H. Ritchie, Phys. Rev. B 24 (1981) 4867. [8] Y. Suguira, J. Phys. Radium 8 (1927) 133.
Physica B 193 (1994) 161-165
Influence of spatial dispersion on the interaction energy of a positronium atom and a metal surface aDo Nam,
D.B.
T r a n T h o a i b'*
aUniversity of Hue, Le Loi 3, Hue, Viet Nam bNational Centre for Scientific Research, Mac Dinh Chi 1, Ho Chi Minh City, Viet Nam
Received 4 January 1993; revised 3 May 1993; final revised 31 August 1993
Abstract
The influence of spatial dispersion of the medium on the interaction energy of a positronium atom and a metal surface is studied using the self-energy formalism developed by Manson and Ritchie. It is found that within the dipole approximation when spatial dispersion is taken into account the interaction energy saturates to a constant value at the surface even for positronium at rest.
1. Introduction
The interaction energy between a metal surface and a positronium atom has been studied with considerable interest [1-3]. Using a semiclassical approach Lifshitz [1] has calculated the van der Waals interaction energy between a positronium atom and a metal surface. For rather large distances between the atom and the surface Lifshitz's result showed that the interaction energy varies a s 1 / Z 3. A recent experimental work of Lynn and Welch [2] on the scattering of a positronium atom by a metal surface has led to increased activity in the field. The quantum nature of the interaction between the positronium atom and the metal surface should be addressed. Manson and Ritchie [3], using their self-energy formalism, have calculated the self-energy of a positronium atom near a metal surface, and came to the conclusion that
* Corresponding author.
the quantum recoil weakens the interaction energy from its Lifshitz's value. Manson and Ritchie have found a very interesting result: near the surface the interaction energy varies as 1 / Z instead of 1 / Z 3. We know from the calculation of the image potential of an electron near a metal surface that spatial dispersion of the metal has made the classical image potential, which also varies as 1 / Z , finite at the metal surface [4,5]. Therefore, it would be of great interest to include spatial dispersion in the calculation of Manson and Ritchie, to see whether the interaction energy of a positronium atom near the metal surface becomes finite or not. We shall employ the hydrodynamic model for the metallic electron to include spatial dispersion.
2. Interaction Hamiltonian
Let us consider a positronium atom approaching a metal occupying half-space Z~<0 with
0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved S S D I : 0921-4526(93)E0337-G
162
Do N a m . D.B. Tran Thoai ," Phvsica B 193 (19941 I0I
planar surface. The metal is described as a sentiinfinite electron gas in the hydrodynamic model [6]. An atom can be considered as a charge density p(r) consisting of a nuclear charge Ze and an electron cloud. The interaction Hamiltonian of the atom at r = (R, Z) and the metal is
V(r) = ( d r ' p ( r ' ) O ( r + r') .
(1)
Here O(r) is the interaction Hamiltonian of an clement of charge e with oscillator quanta of the metal:
O(r)
e~'~ [~,A(r)aA + ~ ( r ) ~ , l
(2i
A
where a A and h a are the creation and annihilation operators of plasmons. The index a indicates the plasmon modes and their wave vectors. ~A(r) is the electrostatic potential caused by the plasmons specified by A. An explicit expression of q~A for the hydrodynamic model of a metal occupying the half-space Z ~< 0 has been derived by Barton [6]. Like Manson and Ritchie we shall only consider the interaction with surface plasmons, for which: rn
~(r)
-
,
e N~[w;Q
-
eS,/.
Z<~O, rn e N~[°J2pQ -
- w
~p~
eC)Zl e <'u
oJ~P~]e iO"
.
P
##1
• 1[ ~ n ¢ O : = ~ - ['tO P -}- - -
~
rn
~
~
@e q- ~n-H Q ( ( ~ - +
3~
Q2+
/j2
)
Q
hP, e ~N~ = rrrn'~aJo)~Q(ZP, + Q)(P,
+ a c,)9,(Z)
(S)
q
where A is a vector which in Cartesian components is given b~ A
(iQx, iQy,
Q ~ , .
p
f dr' p(r')r' is the atomic dipole moment, DI
~(z)
T[(,,,~c ~'" - ~ope : "," ) P 0 ( - Z )
+ (,,,~t"
,o~(j)e ~"
(~1
and <#,(Z) is the Z-dependent part of the electrostatic potential caused by surface plasmons. given in Eq. (3).
(4 2,,,o,7 /3:
)
'
],
5(r) = nZ.l,k
(,,I,')
(5 Ps
~%" [ p . A e x p ( i ( ) R ) ] ( D ~ ,
Comparing thc definition of thc self-energy through the energy shift demonstrated by Manson and Ritchie [7] with the expression of the energy shift in second order in perturbation theory, one gets
(3
4"rre2n --
V (r)
3. Sell-energy
where
2
parallel to the metal surface, #1 the electron density in the metal, and/3-" the coefficient of the hydrodynamical pressure term, taken to be 6,'5 of the Fermi energy E t . Regarding the atom as a ncutral chargc distribution and assuming that the interaction energy is due to the mduccd cxcilations of surface plasmons and the dipolar excitations o1 the atom we obtain for the interaction Hamiltonian an cxpression as follows:
C,z
Z/>0,
O)
105
,
(6
Q)e -
(7
Here m is the electron mass, Q the wave vector
(%1,-)
E 0 + F, + e o
E, - (~o + Q)
ek + i~r
(ll))
whcre In) is the quantum surface state wittl energy E,,; II} represents the internal state of the positronium atom having energy ~,, + e~ and I~k) is the state associated with the center of mass M of the positronium atom and energy e~ :
h2(K ~-+ k : ) / 2 M .
163
Do Nam, D.B. Tran Thoai / Physica B 193 (1994) 161-165
Let us consider the case of the positronium atom with momentum hk o moving perpendicular to the metal surface. We assume that states of the electronic system in the atom have no changes by the presence of the metal. For a plane wave basis set of eigenfunctions we obtain the following expression for the self-energy:
Z(Z)-
2Mw~ 3h
e
ikoZ
2MwZp
.~(z)-
3h ~I I<°Nz)l=
x f dQ Q2
Ps oz ~o~(2P~ + Q) e
0
[ e -ez_
e s°z(
1
,s 1,o)1
~o ~ - 0
× LQ~+O~ ~] I
(16)
l
where
x i dQ
Q2
P~ Oz w~(2P~ + Q) e
So = Q2 + Q~ + Q~.
0 e
The expression of the self-energy in the dispersionless limit in the same case is
QZ+ik°z
V~MwZp
Q~ + Q~ + 2iQk o
e S,~Z(1 2~,~ \ S q - Q + i k o
1
)]
P~+Sq+iko (11)
~(z,/3=0)--
l
×
Q~ = 2Mo)~/h,
(12)
Q~ = 2MeJh 2,
(13)
Sq2 = 0 2 + Q~2 + QI2 - k~.
(14)
Now we study the limit of the dispersionless case. Within this limit the corresponding result of Manson and Ritchie [3] is obtained by letting /3 = O:
x E [(ol/~[/')l 2 i d Q I
Q 2e
Q2~ + O~
2So(-~-0_-Q)
.
(18)
Lifshitz's result is obtained from Eq. (18) by neglecting the quantum recoil effect (M = 2). Now we are interested in the saturation of the image potential at the surface, so from Eq. (16) we obtain for Z = 0:
X(Z = O) -
2MwZp 3h
I(0]/?1/)] 2 j dO 0 2
t
[ 1 ×
Q~ + Q~
Qz
0
P~ ws(2P~ + Q)
1(1
1)]
2S0 So Z Q
Ps + So
(19)
and
0
Q~ + Q~ + 2iQko
0
[eO
x •
X/2M~o 2pe_i<,z 3h
3h
× Z I<01~ll>l2 jdQ Q2e-QZ
where
~(z,/3=0)--
(17)
X/-2Mo)p
2Sq(S2--O-+ik,,/
X(Z = 0,/3 = 0) - -
6h
(15) Consider now an important case when the particle is at rest (k o = 0). Then the self-energy (11) reduces to
X Z ](0111~11)12i ,
dQ
Q2
1
S0(S0 + Q) "
(20)
0
It is clear that the integral in Eq. (20) is di-
164
1)o N a t n .
I).B.
Tran Thoai
v c r g e n t , while the e x p r e s s i o n in Eq. (19) has a linitc w d u e . T h e s u m m a t i o n in Eqs. (16) and (18) was a c c o m p l i s h e d using f o r m u l a e for oscillation s t r e n g t h s by S u g u i r a 181, for b o t h discrete a n d c o n t i n u u m e l e c t r o n i c states of a h y d r o g e n like atom.
. Phv.sica B 193 4 1 9 9 4 )
1(~1 I o 3
0, •J
f~ / /
4. Numerical calculations and discussions I n this section we c a r r y out n u m e r i c a l calculations for the s e l f - e n e r g y of a p o s i t r o n i u m a t o m and c o m p a r e o u r results with those of p r e v i o u s a u t h o r s . We will focus on the case when thc p o s i t r o n i u m a t o m is at rest. In Fig. 1 we plot the i n t e r a c t i o n e n e r g y Z(Z) v e r s u s the d i s t a n c e Z of the p o s i t r o n i u m a t o m from the m e t a l surface, as c a l c u l a t e d with spatial d i s p e r s i o n ( E q . (16)) a n d w i t h o u t d i s p e r s i o n ( E q . (18)). T h e d e n s i t y p a r a m e t e r r of the m e t a l is 3. N e a r the m e t a l surface, spatial dispersion causes a s t r o n g r e d u c t i o n of the i n t e r a c t i o n e n e r g y a n d a c t u a l l y m a k e s it finite at the m e t a l
;&
-4~
I'ig 2. Sell cncrg.x as a I n n c t l o n ol the c l c c m m gas dcn,,ilx p a l a n l c l c r for Z I/\ Solid c u r x c : s c l l - c n c r g } i n c h t d i n ? s p a t i a l d i s p c r s i ( m : l o n g d a s h e d c u t x c : w i t h o u t spatial dispel s h i n : , , h o r t - d a s h c d c u r x c : l.il'shitz's rcsull.
'2,
38-
Fig. 2 we plot the s e l f - e n e r g y at Z - I A v e r s u s the e l e c t r o n gas d e n s i t y p a r a m e t e r r,. T h e d e p e n d e n c e of the s e l f - e n e r g y on r, is s t r o n g e r w h e n r~ gets smaller.
/
26-
,,
['
'
/j
.~. -2~
,"
3-
06~
/' ;
-4-~'
2
/," /"
/ I i
7
i •o
In
/
-:
E
surface (solid curve). The Lifshitz's result v l ( Z ) (short-dashed curve) is also shown in Fig. 1.
-~
~--T¸
0 4--
/r -
2-
c - ~ 0
i
2
z
.......
"5
(,g)
£
t"ig. 1. Sclf-cncrg~ o f l h c positronium atom as a tunclhm o f
Iqg. 3. Ratio o l the self cncrg.~ to lhc scmicla~,sical Lil,dfit/
the distance Z from thc surface for electron-gas densilx parameter r = 3. Solid curve: self-energy including spatial dispersion: long-dashed curve: without spatial dispersion: short-dashed curxe: Lifshitz's result.
v a l u e a~, a f u n c t i o n o f the d i s t a n c e Z f r o m the ~,urfac¢. Solid c u r v e : s e l l - e n e r g y i n c l u d i n g spatial d i s p e r s i o n : l o n g - d a s h e d c u r v e : w i t h o u t s p a t i a l d i s p e r s i o n s . T o p : t' - 2 : b o t t o m ' r 5.
Do Nam, D.B. Tran Thoai / Physica B 193 (1994) 161-16,5
T h e r a t i o of the s e l f - e n e r g i e s f r o m Eqs. (16) a n d (18) to the Lifshitz's v a l u e [1] for r~ = 2 and r~ = 5 is p l o t t e d in Fig. 3. W e n o t e that the i n t e r a c t i o n e n e r g y ,Y(Z) as given by Eq. (16) to t h e Liftshitz's result is significantly s m a l l e r t h a n t h e c o r r e s p o n d i n g r a t i o in the a p p r o x i m a t i o n of M a n s o n a n d R i t c h i e for all values of Z. F o r e x a m p l e , at Z :: 2 A o u r result gives ~ ' / ~ L = 0.44 for r~ = 2 a n d Z / Z L = 0 . 3 3 for r s = 5, while the r a t i o o b t a i n e d in the l i m i t / 3 = 0 is 0.99 a n d 0.96, respectively. In s u m m a r y , we have e v a l u a t e d the influence o f s p a t i a l d i s p e r s i o n on the i n t e r a c t i o n e n e r g y of a p o s i t r o n i u m a t o m a n d a m e t a l surface. W e h a v e f o u n d that the effect of spatial d i s p e r s i o n a n d t h e q u a n t u m recoil effect give an i n t e r a c t i o n e n e r g y o f the p o s i t r o n i u m a t o m that is finite at t h e surface, giving a result d r a m a t i c a l l y different f r o m t h a t f o u n d by Lifshitz, w h o used semiclassical c o n s i d e r a t i o n s .
165
Acknowledgement W e a r e grateful to the r e f e r e e for his sugg e s t i o n s a b o u t the p r e s e n t a t i o n of o u r p a p e r .
References [1] E.M. Lifshitz, Sov. Phys. JETP 2 (1956) 72. [2] K.G. Lynn and D.O. Welch, Phys. Rev. B 22 (1981))99. [3] J.M. Manson and R.H. Ritchie, Phys. Rev. B 29 (1984)
11184. [4] D.B. Tran Thoai, E. Zeitler and Do Nam, Phys. Stat. Sol. B 158 (1991)) 557. [51 J. Mahanty, K.N. Pathak and V.V. Paranjape, Phys. Rev. B 33 (1986) 2333. [6] G. Barton, Rep. Phys. Prog. 42 (1979) 963. [7] J.M. Manson and R.H. Ritchie, Phys. Rev. B 24 (1981) 4867. [8] Y. Suguira, J. Phys. Radium 8 (1927) 133.