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Nonretarded van der waals interaction between two metal spheres
CHEhW3-L
Volume 28, nbmber 2
15 September
PHYSICS L&TERS-
1974
.:
‘NONRETARDEDVANDERWAALSINTERA(3TION BETWEENTWOMETALSPHERES A.A. LUSHNiKOV and A.J. SIMONOV. Institute of Phydcol Chern~stry. ~foorcow. USSR
Karpov
Received 17 April 1974
An approximate
solution is found to an equation defting
the spectrum of surface plasmons in systems of two
equal metal spheres end two plane metal surfaces. The spectrum obtained is then used for calculation of the van der Weals interaction energy between these spheres at large and small separations and between the plane met21 SurfaCeS.
The energy U of the van der Waals interaction tween macroscopic bodies can be expressed as a change in the zero point energy of surface charge sity fluctuations which arise wilen the separation tween two interacting bodies varies from infinity
bedenbeto .Z
V(r) =
[II:
where w, are the Surface eigenfrequencies and the following system of units is adopted: tl.= ~1 = 1 (in is the electron mass). In this note we use eq. (1) for calculation of the nonretarded interaction energy between two metal spheres and between two plane metal surfaces. To make our presentation more clear we simplify the problem as far as possible by considering two equal metal spheres (or planes) interacting across a vacuum and adopting for the metal a jelly-like model in which the positive icns are smeared out into a uniform background having a step-function discontinuity at the metal surface. The condition that the sp tern exhibits collective charge density oscillations is that the equation
V(f) = 2
:
tion operator of the elkctron gas. At Iarge frequencies 3 V,/h where vF is the Fermi velocity and h is a characteristic wave length of the plasma niodes), eq. (2)may be transformed into the form [3]: (w
j&-Jn,Jq ,,f2)V(r,) d3qd’fi
(2)
should have a nontrivial sblution for some y&es of w; these being the collectivti plasma eigenfrequencies [2]. Here V(r) is the potential of the effective electric field of normal plasma modes and Q,(r, ,r2) is the. polariza-’
tihere n(r) is ihe conduction electron density. To demonstrate how eq. (3) works we consider first a simple example of two plane surfaces separated by a distance L Let the z axis be perpendicular to the surfaces. Thkn n(z) = n&-z)
+8(2-c)]
,
W
where e(z)js the step-function and 11~is the bulk density-of conduction electrons. On expanding V(r) into the two-dimensional Fourier integral for x,y V(r) = (27r)-* JV,(z)
e-iK’P d2K ,
(5)
where p = (x;~) is the vector in the xy-plane, and substituting eis. (4) and (5) into eq. (3) we obtain at O
where ti = 2-?%.1~ = (27in,B”)lD i the pI&~e surface plasmonSfre&ency [4]. The equality
,.
_.’
,.
: ,_
-. .’
(6)
_.’ :
2-x.
:,
.;:‘J’olunk 28.’ nsm’bpr. 2 _,
‘,. ._
-‘...
‘.’
:.
:...
hs~lh~~&m~ in& account in deriving eq. (6). Put,;jiJi+F d,l-in lecj. (6) we’arrive at two coupled equa..
_.,tlo,n.s-srrbr-t~e,dete~ation
..-.cJ2$j .; w2v
(r) &2[&)
;’
.. : :, ,-
$1
:
: (7)
This.rksult substitute.d into eq. (I) gives. the interaction energy .,per-kit area:
we pbtain
(1 tie-?‘/’
--,, _. 1.’
“.
e4L&lp-~~l ‘(h
(13)
.’
..
it 1~1
f
sinh~u
x L+i-
-.
=[‘drx[(l+e-1)“‘+
‘,.
..
’ where S? f (i,(p) and, Yi,&Q are normalized spherical harmomcs. On differentiating eqs. (11) and’(12) “with respect top, substituting the results into eq; (3) ‘. ah ihtroducing
(8)
&
:
T.
X= (co&-co&)-1/2p ..
..
‘l/Z ” .’ ” [(COShII’--oS~)(~oOS,~~~‘,-coS77’)]
.‘.O1 ,’ L:.. $.~M(Q)y&&Q’) x.4;i~ ‘x 2L+1 L=l xv=-1 ‘,.” ,’ .. ‘, -. ,-
:_:.
,eigenfriquencies: . .. ‘($)2_=oT(l f e-“‘) . I -.
=-~Af4i71~; : s wb.e& __” -‘.-
’ tr- r’+= .-__
‘:
‘.
s ‘X + y(r)1 . K -,-._, ., :.. Tne condjfjqq for ,the~&.istke of nontrivial solu‘- tjoni tif these equations defines the spectrum of :
A
:’ -.‘.
of V$l) and V;(Z):.
%,~~~Jo~+‘v(I)~-~~
.‘. _,:_, .-,::I ‘1.5Septcmb’gr I&‘4 .,_.‘. _;
”
CHEMICAL PHYSIC! LETTERS~~ :.
.[
2(COsh& - cosq’)
1
- 21 G 0.069.)
n -(14)
in agrrement with the result of ref. [5] obtkned withib the hydrodynamic model of the electron g&. -Now let us consider two equal metal spheres of ra-
In deriving eq. (14) the following
‘dius R separated by distance 1(1+2R is the distance be:ween the centers of the .spheres) and introduce bi: spherical coordinates [6]: .x_+iT=.
fl’sinq &V
..a =
codlp 7 cosq ‘:
tihere-:-m<;r<‘m,
0 &s<
(9)
cosh/.r - cosq ’
ti and 0
a
2T..Let
‘.
~&_r}has the following ..
,.
form in:
.. ; ; -. :,. ,.“ .. ._,__- :: ._-. . .: _;
-.
x*wIz)
=mi$,R)
...
;’
integral equation for X’(CL):-
.. ..
_:-- I I4 ‘,
,:
.,
,: ,, :
:,
..:
.. .,’
(15)
.:
‘q-i@>,,
.‘we ‘obtaia the foliowig
.
-, _.‘: :-. ; .’
-’
COS~)-~dpdSZ ,
‘tihere dbQ= sini dq dq. Noti, &tting p = 50 kr eq. : (14) and introducing
,The inverse dist&ce between two pnmts i and ,r’ is. &ve*.by thc~fo!loking expression;’ ‘. : .-.
[6] have
(cosh,g-- cosn) -$
d3r + n3(,osh&
Z; = ti ,cothpc be the positions of the s@here centers.. .Thenthe parameters u and & are expressed in terms ofR ‘and 1 as follows: ..C .. a ,+($.+4/+2 _. ,The e&on-density these coordinates:
->I
and
a SinhJl
equalities
been used:
: ..-.-, ‘.
..
-,
(16) ,’ ,..
‘.
Volume 28;nknber
15 September 1974
CHEMICAL PHYSICS LEl+E&
2 :.
It is &.& equation which should be solved for the determination of the $lasma eigenfrequencies in the system oftwo-equal metal spheres (see also ref. [7]). JJnfortunately, this equation is very complicated and ikso!ution cannot be. found analytic&ly at arbitrary ~0. However, at go % 1 (large sepkations) and at p. < 1 (small separations) some simplifications arise and it is possible to define the asymptotic behaviour of the plasma frequency at go % 1 and at ~6 Q 1. We begin with the case p. 9 1. Notice that the so- ‘. lutions of the equation
-(22) ,x L1f~-Z(coshpo-co .I,-(2L1+1 ..L sir&&
1
At very large separations the difference between L2I and 52 may be neglected id sinhpO(cosh~O-cosn’)-l replaced by 1. We fmd then: -g
f e-w+l)Po)
= J&l
.
(23)
Hence CD
X I(I+.e-(2L+1)&)1P
+(I _e-(2L+11b)I/2
-21
.(24)
are known: X,.,@)
= YL,&C+)
at
wiL= wi L/(;?L + 1) , (18)
where cosv coshpo - 1 cosn1 .=.-co&o _ cos’sl
% =Y,
(13)
. This result can be easily understood. Eq. (17>de_ scribes surface plasma oscillations in a single metal sphere centered at z = CJcothpo. The solid angle 52, determines the direction from the center to the surface of,the sphere. The set of functior;:.. Y&;L~) m”, serve as a basis for the.~on&ructk,l of Likrrbatiorr theory for eq. (16) at large separations. Let US expand. .X*(fL) in spherical harmonics (18):
x’_=
c X’?
L,M
Y &2,).
L
Then we obtain from eqs. (16)-(18):
The sum on the right-hand side of eq. (24) converges at1. = ~0’. Thus, at large p. the dipole term only contributes to this sum’and U(r) takes the form: U(l) = -&J,fi
e-@-Q = &+$B
&l%
(25)
Higher order corrections (-(R/b8 etc.) can r&o be obtained from eq. (21). This will be done in a more’detailed paper. A variant of perturbation theory can be used for the determination of U(fi at small separations &
00) where
:
.~Volu.me28,~umb&2 .
: :
.,: :.
‘.CHE~~i=AiPHYSlCSLETTERS.:
>,:.
‘. : The irallies &, are seen to,decrease with decreasing 1 : pd. This’fiakes one.tispect;that eq.,(26j might be. .. .I -. solved with the aid of p~krt%bat&n tiec~);, The, uh‘y perturbed &g&frequ&cies vie defme,as. i .” ,: -i ,’ (fg= wZ(l +-e-(=+‘)h) . (28)’ : : ., ,:, In order’to check whether the cop tribution of
,. ,-. ‘.~E$p+‘.
‘,’
W&=G;‘-+.B~, .L where the corrections is then: .‘.. &&g
2WOL) .
from
&[(l
+ (1 _ e’x,;/2
)
- 21 .
.., :*$y :;
..
,. “-
;:., ,,..‘, _ _:_: ,, _
..‘:_ .: ,:
-. ::
‘:.
I,.. ~ ..“.
at
..
&)
const&t
At
ISR
(33)
and U&)=-+H(R/I)
l
at
04
The-ratio of-coefficients of (R/g6 in eq. (33) and of R/Z in e;q. (34) is &I/3, whereas our calculation gives a different result, this ratio being = 17:8. Finally, it shouId.be noted that the collective fl&tuations of the conduction electron charge density give a,major contribution to the total interaction energy between metal spheres. The characteristic energy coilstam entetig.‘the expression for U(i) has the order of the bxlk plasma~frequenc~ which is known to be a0 = 1~ eV =‘10-11 erg, whereas the characteristic due ofHis of the order of 10-15-10’-12 erg for I& electric spheres..
References ‘. [ 11 H.B.G. C~imir, Pk.
Koninkl: Ned. Akid. Wetenichap. 51 (1948) 793. [7] D:Pir,es and P. Nozieres, The theory of quantum liquids, Vol. I (Benjamin, New York, 1966). [3] A.A.:Lushnikov and A.J. Simonov, Phys. Let!ersA. to be published. 141 D.M. Newns, Phys. Rev. Bl (i970) 3304..
(30)
[5] D.B. Chang, R.L Cooper, J.E. Drammond and kc. Yang, J. Chem. Phys: 59 (1973) 1232. (61 P.M. Norse and H. Feshbach, ,Methods OFtheoretic;11 physic% Part II IMcCcaw-Hill, New York, 1953). [7] D.J. Mitcheu arid B.W. Finham, J. Chem. Phys. 56 (1972) 1117. 18) H.C. HBmaker,~Phy&o4 i1937) 1058.
‘_ _..
.,,_c : .,, ,: _. ..
‘.
U,(O = - ,FIf(R/1>6
(3i)
‘.’ :
.,
1974
:
‘,
Remembeting that at small ‘poJ cash b x 1 + $P; Fd fig= i/j? [see eq; (I 0)] ws finally get:
:_
2);;j;b2]
‘,
0..
U(r) = -(w;Rj@I)A..
‘15 September
.’ where .!J = I+ 2R and His the Hamaker large-cd small separations we have:
eq:- (29):
+.e-xjl/T
-i”[ci
,+++)j~
V(r) = (wg /.+/Q$
x.:7
‘.
(29)
S&e many t&ns (I, 3 ,uo~) contribute to this sum at PO < 1, it can be replaced by an integral c, +jU and terms of the order.ofL-l can be neglected. Then we obtain
Q)z
sinr or. w ,-,L-1. The energy U(o
(cd; + cd;; + St’/$f+ Is&-
--.
Le; us @ompare the asymptotic’dependences U(r) defm&l by eq3. (25) +(31) with those foUo&ir& from tlte well-I&awn Hamaker formula [S]:
-.
,wou:d lead to small corrections to wi the nondiago; nal matrkekments p ‘,‘sh+d be compared with 2LL at L = p-l. From‘eq. (28) the level spacing aw,/X 9 we have &$/at = p&while @“,x w&, at Mf’q and /I& e ~~,$+,ln+,. These estimations shpw that the.perturbaticz approach cannot be used for the determination of corrections to 02~ Nev&theless, these corrections are small in a particular sense (they ,. are of the order of the distance between adjacent ii envalues of eq. (26), i.e., theq’ are of the prder of : 5-1. wgL )- This circumstance allows onk to find U(r) at small 1-10.Let us write the exact eigenfrequencies iri the form:
.,I. ‘,
. .
,,,..
.-
‘. .-
.
: ,.. : -_;
: .’
., .’ ,.
.’
;‘y ..‘, ,‘: ,..,,
“, .,“.,. .“.‘....
.
., :
,,. .,
;--.;.
‘. ._ - .-
‘. ,,
Volume 28, nbmber 2
15 September
PHYSICS L&TERS-
1974
.:
‘NONRETARDEDVANDERWAALSINTERA(3TION BETWEENTWOMETALSPHERES A.A. LUSHNiKOV and A.J. SIMONOV. Institute of Phydcol Chern~stry. ~foorcow. USSR
Karpov
Received 17 April 1974
An approximate
solution is found to an equation defting
the spectrum of surface plasmons in systems of two
equal metal spheres end two plane metal surfaces. The spectrum obtained is then used for calculation of the van der Weals interaction energy between these spheres at large and small separations and between the plane met21 SurfaCeS.
The energy U of the van der Waals interaction tween macroscopic bodies can be expressed as a change in the zero point energy of surface charge sity fluctuations which arise wilen the separation tween two interacting bodies varies from infinity
bedenbeto .Z
V(r) =
[II:
where w, are the Surface eigenfrequencies and the following system of units is adopted: tl.= ~1 = 1 (in is the electron mass). In this note we use eq. (1) for calculation of the nonretarded interaction energy between two metal spheres and between two plane metal surfaces. To make our presentation more clear we simplify the problem as far as possible by considering two equal metal spheres (or planes) interacting across a vacuum and adopting for the metal a jelly-like model in which the positive icns are smeared out into a uniform background having a step-function discontinuity at the metal surface. The condition that the sp tern exhibits collective charge density oscillations is that the equation
V(f) = 2
:
tion operator of the elkctron gas. At Iarge frequencies 3 V,/h where vF is the Fermi velocity and h is a characteristic wave length of the plasma niodes), eq. (2)may be transformed into the form [3]: (w
j&-Jn,Jq ,,f2)V(r,) d3qd’fi
(2)
should have a nontrivial sblution for some y&es of w; these being the collectivti plasma eigenfrequencies [2]. Here V(r) is the potential of the effective electric field of normal plasma modes and Q,(r, ,r2) is the. polariza-’
tihere n(r) is ihe conduction electron density. To demonstrate how eq. (3) works we consider first a simple example of two plane surfaces separated by a distance L Let the z axis be perpendicular to the surfaces. Thkn n(z) = n&-z)
+8(2-c)]
,
W
where e(z)js the step-function and 11~is the bulk density-of conduction electrons. On expanding V(r) into the two-dimensional Fourier integral for x,y V(r) = (27r)-* JV,(z)
e-iK’P d2K ,
(5)
where p = (x;~) is the vector in the xy-plane, and substituting eis. (4) and (5) into eq. (3) we obtain at O
where ti = 2-?%.1~ = (27in,B”)lD i the pI&~e surface plasmonSfre&ency [4]. The equality
,.
_.’
,.
: ,_
-. .’
(6)
_.’ :
2-x.
:,
.;:‘J’olunk 28.’ nsm’bpr. 2 _,
‘,. ._
-‘...
‘.’
:.
:...
hs~lh~~&m~ in& account in deriving eq. (6). Put,;jiJi+F d,l-in lecj. (6) we’arrive at two coupled equa..
_.,tlo,n.s-srrbr-t~e,dete~ation
..-.cJ2$j .; w2v
(r) &2[&)
;’
.. : :, ,-
$1
:
: (7)
This.rksult substitute.d into eq. (I) gives. the interaction energy .,per-kit area:
we pbtain
(1 tie-?‘/’
--,, _. 1.’
“.
e4L&lp-~~l ‘(h
(13)
.’
..
it 1~1
f
sinh~u
x L+i-
-.
=[‘drx[(l+e-1)“‘+
‘,.
..
’ where S? f (i,(p) and, Yi,&Q are normalized spherical harmomcs. On differentiating eqs. (11) and’(12) “with respect top, substituting the results into eq; (3) ‘. ah ihtroducing
(8)
&
:
T.
X= (co&-co&)-1/2p ..
..
‘l/Z ” .’ ” [(COShII’--oS~)(~oOS,~~~‘,-coS77’)]
.‘.O1 ,’ L:.. $.~M(Q)y&&Q’) x.4;i~ ‘x 2L+1 L=l xv=-1 ‘,.” ,’ .. ‘, -. ,-
:_:.
,eigenfriquencies: . .. ‘($)2_=oT(l f e-“‘) . I -.
=-~Af4i71~; : s wb.e& __” -‘.-
’ tr- r’+= .-__
‘:
‘.
s ‘X + y(r)1 . K -,-._, ., :.. Tne condjfjqq for ,the~&.istke of nontrivial solu‘- tjoni tif these equations defines the spectrum of :
A
:’ -.‘.
of V$l) and V;(Z):.
%,~~~Jo~+‘v(I)~-~~
.‘. _,:_, .-,::I ‘1.5Septcmb’gr I&‘4 .,_.‘. _;
”
CHEMICAL PHYSIC! LETTERS~~ :.
.[
2(COsh& - cosq’)
1
- 21 G 0.069.)
n -(14)
in agrrement with the result of ref. [5] obtkned withib the hydrodynamic model of the electron g&. -Now let us consider two equal metal spheres of ra-
In deriving eq. (14) the following
‘dius R separated by distance 1(1+2R is the distance be:ween the centers of the .spheres) and introduce bi: spherical coordinates [6]: .x_+iT=.
fl’sinq &V
..a =
codlp 7 cosq ‘:
tihere-:-m<;r<‘m,
0 &s<
(9)
cosh/.r - cosq ’
ti and 0
a
2T..Let
‘.
~&_r}has the following ..
,.
form in:
.. ; ; -. :,. ,.“ .. ._,__- :: ._-. . .: _;
-.
x*wIz)
=mi$,R)
...
;’
integral equation for X’(CL):-
.. ..
_:-- I I4 ‘,
,:
.,
,: ,, :
:,
..:
.. .,’
(15)
.:
‘q-i@>,,
.‘we ‘obtaia the foliowig
.
-, _.‘: :-. ; .’
-’
COS~)-~dpdSZ ,
‘tihere dbQ= sini dq dq. Noti, &tting p = 50 kr eq. : (14) and introducing
,The inverse dist&ce between two pnmts i and ,r’ is. &ve*.by thc~fo!loking expression;’ ‘. : .-.
[6] have
(cosh,g-- cosn) -$
d3r + n3(,osh&
Z; = ti ,cothpc be the positions of the s@here centers.. .Thenthe parameters u and & are expressed in terms ofR ‘and 1 as follows: ..C .. a ,+($.+4/+2 _. ,The e&on-density these coordinates:
->I
and
a SinhJl
equalities
been used:
: ..-.-, ‘.
..
-,
(16) ,’ ,..
‘.
Volume 28;nknber
15 September 1974
CHEMICAL PHYSICS LEl+E&
2 :.
It is &.& equation which should be solved for the determination of the $lasma eigenfrequencies in the system oftwo-equal metal spheres (see also ref. [7]). JJnfortunately, this equation is very complicated and ikso!ution cannot be. found analytic&ly at arbitrary ~0. However, at go % 1 (large sepkations) and at p. < 1 (small separations) some simplifications arise and it is possible to define the asymptotic behaviour of the plasma frequency at go % 1 and at ~6 Q 1. We begin with the case p. 9 1. Notice that the so- ‘. lutions of the equation
-(22) ,x L1f~-Z(coshpo-co .I,-(2L1+1 ..L sir&&
1
At very large separations the difference between L2I and 52 may be neglected id sinhpO(cosh~O-cosn’)-l replaced by 1. We fmd then: -g
f e-w+l)Po)
= J&l
.
(23)
Hence CD
X I(I+.e-(2L+1)&)1P
+(I _e-(2L+11b)I/2
-21
.(24)
are known: X,.,@)
= YL,&C+)
at
wiL= wi L/(;?L + 1) , (18)
where cosv coshpo - 1 cosn1 .=.-co&o _ cos’sl
% =Y,
(13)
. This result can be easily understood. Eq. (17>de_ scribes surface plasma oscillations in a single metal sphere centered at z = CJcothpo. The solid angle 52, determines the direction from the center to the surface of,the sphere. The set of functior;:.. Y&;L~) m”, serve as a basis for the.~on&ructk,l of Likrrbatiorr theory for eq. (16) at large separations. Let US expand. .X*(fL) in spherical harmonics (18):
x’_=
c X’?
L,M
Y &2,).
L
Then we obtain from eqs. (16)-(18):
The sum on the right-hand side of eq. (24) converges at1. = ~0’. Thus, at large p. the dipole term only contributes to this sum’and U(r) takes the form: U(l) = -&J,fi
e-@-Q = &+$B
&l%
(25)
Higher order corrections (-(R/b8 etc.) can r&o be obtained from eq. (21). This will be done in a more’detailed paper. A variant of perturbation theory can be used for the determination of U(fi at small separations &
00) where
:
.~Volu.me28,~umb&2 .
: :
.,: :.
‘.CHE~~i=AiPHYSlCSLETTERS.:
>,:.
‘. : The irallies &, are seen to,decrease with decreasing 1 : pd. This’fiakes one.tispect;that eq.,(26j might be. .. .I -. solved with the aid of p~krt%bat&n tiec~);, The, uh‘y perturbed &g&frequ&cies vie defme,as. i .” ,: -i ,’ (fg= wZ(l +-e-(=+‘)h) . (28)’ : : ., ,:, In order’to check whether the cop tribution of
,. ,-. ‘.~E$p+‘.
‘,’
W&=G;‘-+.B~, .L where the corrections is then: .‘.. &&g
2WOL) .
from
&[(l
+ (1 _ e’x,;/2
)
- 21 .
.., :*$y :;
..
,. “-
;:., ,,..‘, _ _:_: ,, _
..‘:_ .: ,:
-. ::
‘:.
I,.. ~ ..“.
at
..
&)
const&t
At
ISR
(33)
and U&)=-+H(R/I)
l
at
04
The-ratio of-coefficients of (R/g6 in eq. (33) and of R/Z in e;q. (34) is &I/3, whereas our calculation gives a different result, this ratio being = 17:8. Finally, it shouId.be noted that the collective fl&tuations of the conduction electron charge density give a,major contribution to the total interaction energy between metal spheres. The characteristic energy coilstam entetig.‘the expression for U(i) has the order of the bxlk plasma~frequenc~ which is known to be a0 = 1~ eV =‘10-11 erg, whereas the characteristic due ofHis of the order of 10-15-10’-12 erg for I& electric spheres..
References ‘. [ 11 H.B.G. C~imir, Pk.
Koninkl: Ned. Akid. Wetenichap. 51 (1948) 793. [7] D:Pir,es and P. Nozieres, The theory of quantum liquids, Vol. I (Benjamin, New York, 1966). [3] A.A.:Lushnikov and A.J. Simonov, Phys. Let!ersA. to be published. 141 D.M. Newns, Phys. Rev. Bl (i970) 3304..
(30)
[5] D.B. Chang, R.L Cooper, J.E. Drammond and kc. Yang, J. Chem. Phys: 59 (1973) 1232. (61 P.M. Norse and H. Feshbach, ,Methods OFtheoretic;11 physic% Part II IMcCcaw-Hill, New York, 1953). [7] D.J. Mitcheu arid B.W. Finham, J. Chem. Phys. 56 (1972) 1117. 18) H.C. HBmaker,~Phy&o4 i1937) 1058.
‘_ _..
.,,_c : .,, ,: _. ..
‘.
U,(O = - ,FIf(R/1>6
(3i)
‘.’ :
.,
1974
:
‘,
Remembeting that at small ‘poJ cash b x 1 + $P; Fd fig= i/j? [see eq; (I 0)] ws finally get:
:_
2);;j;b2]
‘,
0..
U(r) = -(w;Rj@I)A..
‘15 September
.’ where .!J = I+ 2R and His the Hamaker large-cd small separations we have:
eq:- (29):
+.e-xjl/T
-i”[ci
,+++)j~
V(r) = (wg /.+/Q$
x.:7
‘.
(29)
S&e many t&ns (I, 3 ,uo~) contribute to this sum at PO < 1, it can be replaced by an integral c, +jU and terms of the order.ofL-l can be neglected. Then we obtain
Q)z
sinr or. w ,-,L-1. The energy U(o
(cd; + cd;; + St’/$f+ Is&-
--.
Le; us @ompare the asymptotic’dependences U(r) defm&l by eq3. (25) +(31) with those foUo&ir& from tlte well-I&awn Hamaker formula [S]:
-.
,wou:d lead to small corrections to wi the nondiago; nal matrkekments p ‘,‘sh+d be compared with 2LL at L = p-l. From‘eq. (28) the level spacing aw,/X 9 we have &$/at = p&while @“,x w&, at Mf’q and /I& e ~~,$+,ln+,. These estimations shpw that the.perturbaticz approach cannot be used for the determination of corrections to 02~ Nev&theless, these corrections are small in a particular sense (they ,. are of the order of the distance between adjacent ii envalues of eq. (26), i.e., theq’ are of the prder of : 5-1. wgL )- This circumstance allows onk to find U(r) at small 1-10.Let us write the exact eigenfrequencies iri the form:
.,I. ‘,
. .
,,,..
.-
‘. .-
.
: ,.. : -_;
: .’
., .’ ,.
.’
;‘y ..‘, ,‘: ,..,,
“, .,“.,. .“.‘....
.
., :
,,. .,
;--.;.
‘. ._ - .-
‘. ,,