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Surface polariton response function approach to dispersion forces between macroscopic bodies
Volume 43A, number 5
PHYSICS LETTERS
SURFACE POLARITON
9 April 1973
RESPONSE FUNCTION APPROACH TO
D I S P E R S I O N F O R C E S B E T W E E N M A C R O S C O P I C BODIES G.S. AGARWAL*
Institut fiir theoretische Physik, UniversitiitStuttgart, Germany Received 14 February 1973 A new method for the calculation of dispersion forces between macroscopic bodies is developed. The basis idea here is to use the surface polar#on response function and the fluctuation-dissipation theorem. The calculation of the Van der Waal forces between macroscopic bodies is quite an involved problem, the solution of which was first presented by Lifshitz [1 ]. In recent years simplified methods are being developed. In the method due to Van Kampen et al. [2] one calculates the interaction energy by summing over the zero point energies of all the modes. The modes which contribute to the interaction energy are only the "surface polariton modes". Another novel method based on response functions to calculate dispersion forces between atoms or molecules or between atoms and a conducting wall [3] has been developed by Mc'l:achlan [4]. Generalization of the response method to the case of macroscopic bodies has not been carried out so far. In this note we present preliminary results of our investigation. We show how the surface polariton response functions in conjunction with fluctuation-dissipation theorems may be easily used to calculate the dispersion forces between macroscopic bodies [5, 6]. We consider in this note only the case of no retardation and also we restrict ourselves to the calculating dispersion forces at zero temperature. We calculate the dispersion forces between two isotropic dielectrics characterized by the dielectric constant e-(~) and separated by vacuum. We assume that the two dielectrics occupy the volumes 0 ~< z ~< oo and - ~ ~< z <~-d respectively. Since we are ignoring the retardation effects and considering only isotropic dielectrics, the electrostatic potential • satisfies the Laplace equation V2qb = 0. We now apply a surface charge p at the plane z = 0 + and calculate the response of the medium. We express the potential in each of the domain - ~ ~< z ~< - d , - d ~< z ~< 0 and 0<~z ~<~,as
=JJ
o,z)exp(iux +ioy) d u d o ,
(1)
_oo
where
~(II1) = ~3 exp(-iWoZ) '
- ~ ~< z ~ < - d ,
(2a)
qb(II) = ~ + exp(iWoZ ) + q~_ exp(-iWoZ),
- d <~ z ~< 0 ,
(2b)
2hi @(I) = ff~l exp(iWoZ) + ~oo p e x p O w ° z ) '
0 <<.z <<.~ ,
(2c)
where w o = (u 2 + 02) 1/2 = ik, and/9 is the transform of the surface charge +~
p(x, y) =f f
p(u, o)exp(iux +ioy) du do.
(3)
_oo
The coefficients ~ . , ~1, qb3 are obtained by matching the boundary conditions at each of the planes z = 0, z = - d , viz the tangential component of the electric field and the normal componten of the electric induction be * On leave of absence from the University of Rochester, Dept. of Physica and Astronomy, Rochester N.Y., USA. 447
Volume 43A, number 5
PHYSICS LETTERS
9 April 1973
continuous across the boundary. A straightforward calculation shows that
rb(1)(u, v, z, co) = exp(iWoZ) P(U, v, co) TO)(U, V, w) ,
(4)
where we have added the argument ce for clarity and where the "surface polariton response function" r (I) is given by
r(dl)(u,u, ce)= 2~_m 1-eWo
{1-exp(2iwod)}
I
(e-1)2exp(2iWoCO-1 (e + 1)2
t
.
(5)
If the second medium were absent, then o~(~,, ~, ce) = -
I+
.
(b)
6~4Zo
On subtracting (6) from (5) we obtain the effective response function of the medium I, -87r
T(u, v, ce) re5--(~lc _
k
]'
(e+l)2 exp(2kd)- 1
(7)
*1
It is interesting to note that T(u, u, ce) contains a resonant denominator the vanishing of which gives the surface polariton dispersion relation [5] [(e+ 1)2/(e - 1) 2] exp(2kd) = 1 .
(8)
We now use the above response function to calculate the Van der Waals forces. For this purpose we make use of the fluctuation-dissipation theorem [6] according to which the symmetrize correlation function is given by
dOi¢(r r', t - t') = ½( (Ai(r, t), A/(r', t') } ) =
i f dce~ij(r, r', ce)
exp { - i c e ( t - t'))
dPij(r,r' , co) : h coth(~/3ceh) X})(r, r', co),
(9)
where/3 = 1/k B T and X}~(r,r', co) describes the "response" of the variable A i i.e. ". j (r,. r , . t - .t') A. t - . t') .: Xi](r , r , t - g') , 3 (A i ( r , )t ) / 8 fl" (r,' t ' ) - 21Xi
( 1O)
with f] standing for external forces and r/is the unit step function. If the variables A i and A/are even under time reversal, then it follows that Xq(ce) is an odd function of ce, and a simple analysis shows that at zero temperature ¢1
(T = 0) we have oo
o~ij(r,r', O) =
dcexij(r,r', ice).
( 1 1)
0 In our case (1 1) should be used with great care as we are dealing with a finite medium and we have translational invariance in x and 3' directions only. The interaction Hamiltonian between a surface charge and the electrostatic potential may be written as Hex , =
fp(r')rI,(r')d3r ' = (27r) 2 ff O(u,v)q~(u,v) du dr.
(12)
Hence the external parameter for our problem is -(2rr) 20(u, v). Applying the relation (11) mode by mode we have ( {qS(r, t), crp(r, t) })z=0+ 448
f)J
h dce du doT(u, u, ice), 4rr3 0 -~
(13)
Volume 43A, number 5
PHYSICS LETTERS
9 April 1973
where T is given by (7). The fluctuation correlation for E is given by
~PEE(r,r,O)iz=O+-
ti
j)f dw
47r3 0
dudok T(u,v, iw),
(14)
-~
where k 2 = u 2, ky2 = o 2, k 2 = k 2. From the boundary conditions and (14) we have for the fluctuation-correlations for the field at a point just outside the medium i.e. at z = O-
~PEE(r,r,O)[z=O_-
h f dw f f dudoK2T(u,v, iw), 4~'3 0
where K 2 =
(151
_,o
u2,K2y = °2, K2z = k2e(iw) • The
dispersion force is obtained by calculating the stress and is equal to
1 F = -~n (¢bzz- ~ xx - ,~yy)z=O-"
(16)
On using (7), (15) and (16) we obtain •*
["['x2dxdw[l "(iw!+ l l 2 eX-1
F=167r2d3d~
[[e0w)-lj
]-1
(171
'
which is precisely Lifshitz formula [ 1] (3.11, which we have obtained by using the response function (7). For the case when the medium I has the dielectric el(W ), the medium III a dielectric constant e3(w ) and are separated by a medium of dielectric constant e2(w ), then it is easily shown that the response function is 8rre2
T(u, o, w I = - k(e21_,2 )
[ (e3 +e2) ('1 +e2) [~e3_,2)(, l_e2)
~, ,~
exp(zKa)- IJq -I .
(181
On using (18) and the above procedure one obtains the formula (4.1 5) of Dzyaloshinskii et al. [I]
?fx 2 F - 161r2d3a0a
] [" ( e 3 + e 2 ) ( ' l + ' 2 ) x d x d w [ ( e 3 _ e 2 ) ( e 1 _ ` 2 ) e -- 1j
.
(19)
The application of the present method to the calculation of dispersion forces for the case of more complicated geometries as well as to cases with retardation effects and temperature effects included, will be,considered in a series of papers.
References [1] E.M. Lifshitz, Soy. Phys. JETP 2 (1956) 73; I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii, Advances in Physics 10 (1961) 165. [2] N.G. Van Kampen, B.R.A. Nijboer and K. Schram, Phys. Lett. 26A (1968) 307; See also E. Gerlach, Phys. Rev. B4 (1971) 393. [3] H.B.G. Casimir and D. Polder, Phys. Rev. 73 (1948) 360. [4] A,D. McLachlan, Proc. Roy. Soc. 271 (1963) 387; Mol. Phys. 6 (1963) 423. [5] The use of bulk polariton response functions is discussed at length in a recent review article by A.S. Barker and R. Loudon, Rev. Mod. Phys. 44 (1972) 18; The surface polariton response function has also been mentioned by A.S. Barker, Phys. Rev. Lett. 28 (1972) 892. [6] P.C. Martin, Measurement and correlation functions (Gordon and Breach, N.Y. 1968); § B; our notation is the same as that of this reference. 449
PHYSICS LETTERS
SURFACE POLARITON
9 April 1973
RESPONSE FUNCTION APPROACH TO
D I S P E R S I O N F O R C E S B E T W E E N M A C R O S C O P I C BODIES G.S. AGARWAL*
Institut fiir theoretische Physik, UniversitiitStuttgart, Germany Received 14 February 1973 A new method for the calculation of dispersion forces between macroscopic bodies is developed. The basis idea here is to use the surface polar#on response function and the fluctuation-dissipation theorem. The calculation of the Van der Waal forces between macroscopic bodies is quite an involved problem, the solution of which was first presented by Lifshitz [1 ]. In recent years simplified methods are being developed. In the method due to Van Kampen et al. [2] one calculates the interaction energy by summing over the zero point energies of all the modes. The modes which contribute to the interaction energy are only the "surface polariton modes". Another novel method based on response functions to calculate dispersion forces between atoms or molecules or between atoms and a conducting wall [3] has been developed by Mc'l:achlan [4]. Generalization of the response method to the case of macroscopic bodies has not been carried out so far. In this note we present preliminary results of our investigation. We show how the surface polariton response functions in conjunction with fluctuation-dissipation theorems may be easily used to calculate the dispersion forces between macroscopic bodies [5, 6]. We consider in this note only the case of no retardation and also we restrict ourselves to the calculating dispersion forces at zero temperature. We calculate the dispersion forces between two isotropic dielectrics characterized by the dielectric constant e-(~) and separated by vacuum. We assume that the two dielectrics occupy the volumes 0 ~< z ~< oo and - ~ ~< z <~-d respectively. Since we are ignoring the retardation effects and considering only isotropic dielectrics, the electrostatic potential • satisfies the Laplace equation V2qb = 0. We now apply a surface charge p at the plane z = 0 + and calculate the response of the medium. We express the potential in each of the domain - ~ ~< z ~< - d , - d ~< z ~< 0 and 0<~z ~<~,as
=JJ
o,z)exp(iux +ioy) d u d o ,
(1)
_oo
where
~(II1) = ~3 exp(-iWoZ) '
- ~ ~< z ~ < - d ,
(2a)
qb(II) = ~ + exp(iWoZ ) + q~_ exp(-iWoZ),
- d <~ z ~< 0 ,
(2b)
2hi @(I) = ff~l exp(iWoZ) + ~oo p e x p O w ° z ) '
0 <<.z <<.~ ,
(2c)
where w o = (u 2 + 02) 1/2 = ik, and/9 is the transform of the surface charge +~
p(x, y) =f f
p(u, o)exp(iux +ioy) du do.
(3)
_oo
The coefficients ~ . , ~1, qb3 are obtained by matching the boundary conditions at each of the planes z = 0, z = - d , viz the tangential component of the electric field and the normal componten of the electric induction be * On leave of absence from the University of Rochester, Dept. of Physica and Astronomy, Rochester N.Y., USA. 447
Volume 43A, number 5
PHYSICS LETTERS
9 April 1973
continuous across the boundary. A straightforward calculation shows that
rb(1)(u, v, z, co) = exp(iWoZ) P(U, v, co) TO)(U, V, w) ,
(4)
where we have added the argument ce for clarity and where the "surface polariton response function" r (I) is given by
r(dl)(u,u, ce)= 2~_m 1-eWo
{1-exp(2iwod)}
I
(e-1)2exp(2iWoCO-1 (e + 1)2
t
.
(5)
If the second medium were absent, then o~(~,, ~, ce) = -
I+
.
(b)
6~4Zo
On subtracting (6) from (5) we obtain the effective response function of the medium I, -87r
T(u, v, ce) re5--(~lc _
k
]'
(e+l)2 exp(2kd)- 1
(7)
*1
It is interesting to note that T(u, u, ce) contains a resonant denominator the vanishing of which gives the surface polariton dispersion relation [5] [(e+ 1)2/(e - 1) 2] exp(2kd) = 1 .
(8)
We now use the above response function to calculate the Van der Waals forces. For this purpose we make use of the fluctuation-dissipation theorem [6] according to which the symmetrize correlation function is given by
dOi¢(r r', t - t') = ½( (Ai(r, t), A/(r', t') } ) =
i f dce~ij(r, r', ce)
exp { - i c e ( t - t'))
dPij(r,r' , co) : h coth(~/3ceh) X})(r, r', co),
(9)
where/3 = 1/k B T and X}~(r,r', co) describes the "response" of the variable A i i.e. ". j (r,. r , . t - .t') A. t - . t') .: Xi](r , r , t - g') , 3 (A i ( r , )t ) / 8 fl" (r,' t ' ) - 21Xi
( 1O)
with f] standing for external forces and r/is the unit step function. If the variables A i and A/are even under time reversal, then it follows that Xq(ce) is an odd function of ce, and a simple analysis shows that at zero temperature ¢1
(T = 0) we have oo
o~ij(r,r', O) =
dcexij(r,r', ice).
( 1 1)
0 In our case (1 1) should be used with great care as we are dealing with a finite medium and we have translational invariance in x and 3' directions only. The interaction Hamiltonian between a surface charge and the electrostatic potential may be written as Hex , =
fp(r')rI,(r')d3r ' = (27r) 2 ff O(u,v)q~(u,v) du dr.
(12)
Hence the external parameter for our problem is -(2rr) 20(u, v). Applying the relation (11) mode by mode we have ( {qS(r, t), crp(r, t) })z=0+ 448
f)J
h dce du doT(u, u, ice), 4rr3 0 -~
(13)
Volume 43A, number 5
PHYSICS LETTERS
9 April 1973
where T is given by (7). The fluctuation correlation for E is given by
~PEE(r,r,O)iz=O+-
ti
j)f dw
47r3 0
dudok T(u,v, iw),
(14)
-~
where k 2 = u 2, ky2 = o 2, k 2 = k 2. From the boundary conditions and (14) we have for the fluctuation-correlations for the field at a point just outside the medium i.e. at z = O-
~PEE(r,r,O)[z=O_-
h f dw f f dudoK2T(u,v, iw), 4~'3 0
where K 2 =
(151
_,o
u2,K2y = °2, K2z = k2e(iw) • The
dispersion force is obtained by calculating the stress and is equal to
1 F = -~n (¢bzz- ~ xx - ,~yy)z=O-"
(16)
On using (7), (15) and (16) we obtain •*
["['x2dxdw[l "(iw!+ l l 2 eX-1
F=167r2d3d~
[[e0w)-lj
]-1
(171
'
which is precisely Lifshitz formula [ 1] (3.11, which we have obtained by using the response function (7). For the case when the medium I has the dielectric el(W ), the medium III a dielectric constant e3(w ) and are separated by a medium of dielectric constant e2(w ), then it is easily shown that the response function is 8rre2
T(u, o, w I = - k(e21_,2 )
[ (e3 +e2) ('1 +e2) [~e3_,2)(, l_e2)
~, ,~
exp(zKa)- IJq -I .
(181
On using (18) and the above procedure one obtains the formula (4.1 5) of Dzyaloshinskii et al. [I]
?fx 2 F - 161r2d3a0a
] [" ( e 3 + e 2 ) ( ' l + ' 2 ) x d x d w [ ( e 3 _ e 2 ) ( e 1 _ ` 2 ) e -- 1j
.
(19)
The application of the present method to the calculation of dispersion forces for the case of more complicated geometries as well as to cases with retardation effects and temperature effects included, will be,considered in a series of papers.
References [1] E.M. Lifshitz, Soy. Phys. JETP 2 (1956) 73; I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii, Advances in Physics 10 (1961) 165. [2] N.G. Van Kampen, B.R.A. Nijboer and K. Schram, Phys. Lett. 26A (1968) 307; See also E. Gerlach, Phys. Rev. B4 (1971) 393. [3] H.B.G. Casimir and D. Polder, Phys. Rev. 73 (1948) 360. [4] A,D. McLachlan, Proc. Roy. Soc. 271 (1963) 387; Mol. Phys. 6 (1963) 423. [5] The use of bulk polariton response functions is discussed at length in a recent review article by A.S. Barker and R. Loudon, Rev. Mod. Phys. 44 (1972) 18; The surface polariton response function has also been mentioned by A.S. Barker, Phys. Rev. Lett. 28 (1972) 892. [6] P.C. Martin, Measurement and correlation functions (Gordon and Breach, N.Y. 1968); § B; our notation is the same as that of this reference. 449