A simple model for the nonretarded dispersive force between an ...

A simple model for the nonretarded dispersive force between an electrically polarizable atom and a magnetically polarizable one C. Farina,a) F. C. Santos,b) and A. C. Tortc) Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, CP 68528, Rio de Janeiro, RJ, Brasil, 21945-970

共Received 4 August 2000; accepted 22 March 2001兲 It is known that the consideration or not of retardation effects on the dispersive van der Waals force between two electrically polarizable atoms leads to power laws for the forces that differ only by one power: 1/r 8 for the retarded force and 1/r 7 for the nonretarded one. On the other hand, for the case of an electrically polarizable atom and a magnetically polarizable one, while the retarded force is still proportional to 1/r 8 , the nonretarded one is proportional to 1/r 5 . Here, employing the fluctuating dipole model for both atoms, we reobtain this quite unexpected behavior for the nonretarded force mentioned above. A physical interpretation is also given for such a result. © 2002 American Association of Physics Teachers.

关DOI: 10.1119/1.1378015兴

In the context of thermodynamics of real gases, it was recognized by van der Waals1 in the second half of the nineteenth century that attractive forces should exist between molecules 共atoms兲 even when the molecules do not posses permanent electric dipole moments. Since these forces depend on the atomic 共electric兲 polarizabilities, which in turn are closely related to the refractive index, they are referred to as dispersive van der Waals forces. When retardation effects of the electromagnetic field interaction are neglected, an approximation that can be made only for short distances, these forces are called nonretarded dispersive van der Waals forces, while the name retarded dispersive van der Waals forces is left for the cases where the retardation effects are not negligible. From experimental data, van der Waals tried to infer the form of the interaction energy between the atoms, but the correct form and explanation for the nonretarded forces had to wait for the advent of quantum mechanics and only after the paper by London2 in 1930 was the precise origin of these forces understood. The influence of retardation effects on the dispersive forces was obtained for the first time in 1948 by Casimir and Polder.3 Basically they showed that while the nonretarded force is proportional to 1/r 7 , where r is the distance between the two atoms, the retarded force is proportional to 1/r 8 . An immediate consequence of dispersive forces between two atoms is the existence of 共dispersive兲 forces also for macroscopic bodies 共for a detailed discussion see, for instance Ref. 4兲. For the case of two rarefied macroscopic bodies, the resultant dispersive force can be obtained by a direct integration of the pairwise forces. On the other hand, for nonrarefied media the nonadditivity effects must be taken into account4,5 共see also Ref. 6 for a simple discussion兲. However, though the final numerical result is affected by these effects, the pairwise integration already gives the correct dependence on the geometry. For instance, the retarded force per unit area between two parallel semi-infinite slabs of polarizable material a distance d apart is proportional to 1/d 4 , no matter whether the nonadditivities are taken into account or not. Retardation effects are automatically taken into account when the interaction energy is calculated with the aid of the zero-point energy method, introduced in 1948 by Casimir.7 In fact, the 共standard electromagnetic兲 Casimir force between two macroscopic bodies can be considered as the retarded dispersive 421

Am. J. Phys. 70 共4兲, April 2002

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van der Waals force between them with the nonadditivity effects included. It is worth mentioning that the Casimir effect is more general than that, since it occurs whenever a relativistic field 共not only the electromagnetic one兲 is constrained by some boundary conditions or is considered in a topologically nontrivial manifold.8 In this article, we shall address our attention to a less familiar interaction, namely, the interaction energy between an electrically polarizable atom and a magnetically polarizable one. We will be interested only in the nonretarded interaction between them. The retarded case of this kind of interaction was first studied by Feinberg and Sucher9 and also by Boyer.10 Regarding the nonretarded case, Feinberg and Sucher11 obtained a quite unexpected result, namely: while the retarded interaction is proportional to 1/r 7 共retarded force proportional to 1/r 8 兲, the nonretarded interaction is proportional to 1/r 4 共nonretarded force proportional to 1/r 5 兲, in contrast to the 1/r 6 behavior for the 共nonretarded兲 interaction energy between two electrically polarizable atoms obtained by London.2 In Ref. 11 the authors develop a quite general theory of the van der Waals interaction, which has the advantage of being a model-independent approach. Basically they show that the van der Waals interaction can be expressed in terms of measurable quantities which are related to the interactions of the individual systems with real photons. More specifically, since the van der Waals interaction can be viewed as arising from the exchange of two virtual photons between the atoms, they show that this interaction can be computed in terms of the amplitude for the emission or absorption of two real photons by each atom. Although Feinberg and Sucher’s theory11 is a very nice and general theory, it is quite difficult for any undergraduate student and far beyond the knowledge of students who have just finished their introductory courses in classical electromagnetism and quantum mechanics. However, it is not impossible to obtain the correct power law for the nonretarded force between an electrically polarizable atom and a magnetically polarizable one in a simple and accessible way. This is precisely our purpose here, to wit: for this less familiar situation, we will employ the fluctuating dipole model for the atoms 共a model that has been applied with success for the dispersive interaction between two electrically polarizable atoms5兲 and will reobtain the 1/r 5 behavior for the nonre© 2002 American Association of Physics Teachers

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tarded force. A recent QED calculation also confirms in a straightforward way this unexpected change in the power law of the force when we pass from the retarded to the nonretarded case.12 Our procedure will be based on the fluctuating dipole method as discussed, for instance, in Refs. 4 – 6. To this end we model the electric polarizable atom by an electric charge e of mass m e which oscillates along a fixed but arbitrary direction defined by the unit vector uˆe , so that the atom is effectively replaced by a fluctuating electric dipole p(t) ⫽ex e (t)uˆe , where x e measures the position of the electric charge e along the oscillation axis taking its equilibrium position as the origin. The magnetically polarizable atom will be replaced by a magnetic charge g of mass m g which, similarly to its electric counterpart, also oscillates along another fixed but arbitrary direction defined by the unit vector uˆg . Effectively we have a magnetic dipole m(t)⫽gx g (t)uˆg playing the role of the magnetically polarizable atom, with x g being the position of the magnetic charge g along the oscillation axis taking its equilibrium position as the origin. It must be emphasized here that the introduction of the magnetic charge simply simulates the fluctuating magnetic dipole moment of the magnetically polarizable atom. Each dipole is a source of an electric and a magnetic field that pervade all space. Each dipole also probes the electric and magnetic fields created by the other. The electric and magnetic fields at a point of the space identified by the position vector r created by an electric dipole p are given by13 Ep共 r,t 兲 ⫽ 关 3 共 p共 t * 兲 •rˆ 兲 rˆ ⫺p共 t * 兲兴

1 ⫹ 关 3„p˙共 t * 兲 •rˆ …rˆ ⫺p˙共 t * 兲兴 r3

1 1 ⫻ 2 ⫺ 关 p¨共 t * 兲 ⫺„p¨共 t * 兲 •rˆ…rˆ兴 2 , cr c r and Bp共 r,t 兲 ⫽

冋

册

1 1 ¨ 共 t * 兲 ⫻rˆ. p ˙ 共 t*兲⫹ 2 p cr 2 c r

共1兲

If we project these forces along the directions defined by the 共fixed兲 unit vectors uˆe and uˆg , neglect retardation effects and radiation reaction, and take into account the forces that bind the oscillating electric and magnetic charges to their respective centers of force by simulating them through Hookean forces, we end up with a coupled system of linear differential equations which reads x¨ e 共 t 兲 ⫹ ␻ 2e x e 共 t 兲 ⫽

共3兲

共4兲

e e v 共t兲 E 共 r ,t 兲 •uˆe ⫹ me g e m ec e ⫻Bg 共 re ,t 兲 •uˆe ,

x¨ g 共 t 兲 ⫹ ␻ 2g x g 共 t 兲 ⫽

共5兲

g g B 共 r ,t 兲 •uˆg ⫹ v 共t兲 mg e g m gc g ⫻Ee 共 rg ,t 兲 •uˆg .

共6兲

where ␻ e and ␻ g are, respectively, the natural frequencies of the electric and magnetic dipoles. If the dipoles are not very far from each other 共rⰆ137a 0 , where a 0 is Bohr’s radius, see, for instance Ref. 6兲, but are far enough to guarantee that 兩 x e 兩 , 兩 x g 兩 Ⰶr, the dominant terms lead to x¨ e 共 t 兲 ⫹ ␻ 2e x e 共 t 兲 ⫽F 共 r 兲 x˙ g 共 t 兲 , x¨ g 共 t 兲 ⫹ ␻ 2g x g 共 t 兲 ⫽H 共 r 兲 x˙ e 共 t 兲 ,

共7兲

where

共2兲

In the same way, the Lorentz force acting on the magnetic charge is given by g Fg 共 rg ,t 兲 ⫽gBe 共 rg ,t * 兲 ⫺ vg 共 t 兲 ⫻Ee 共 rg ,t * 兲 , c

共ii兲

F共 r 兲ª

where r is position vector of the point of observation taken the electric dipole at the origin, r⫽ 兩 rជ 兩 and t * ⫽t⫺r/c is the retarded time. The fields generated by a magnetic dipole can be obtained by means of the substitutions p→m, E→B, and B→⫺E. 13 The Lorentz force acting on the electric charge e, whose position is identified by the vector re at a given instant of time t, is e Fe 共 re ,t 兲 ⫽eEg 共 r e ,t * 兲 ⫹ ve 共 t 兲 ⫻Bg 共 re ,t * 兲 . c

field created by the fluctuating magnetic dipole m(t) ⫽gx e uˆ g 共with analogous interpretations for Be and Bg 兲. In a first approximation, re and rg can be considered respectively as the position of the electrically polarizable atom and the position of the magnetically polarizable one, so that 兩 re ⫺rg 兩 ⫽r is the distance between the atoms.

eg 共 uˆ ⫻rˆ 兲 •uˆ e , m e cr 2 g eg

H共 r 兲ª

eg 共 uˆ ⫻rˆ 兲 •uˆ g , m g cr 2 e eg

共8兲

since rˆ ge ª(re ⫺rg )/r⫽⫺rˆ eg . Equations 共7兲 form a system of coupled differential equations that can be solved in the usual way, i.e., we try a solution of the form x i (t) ⫽C i exp (⫺i␻t), where C i is a complex constant with i ⫽e,g, thereby obtaining the following algebraic equation: ⍀ 2 ⫺„␻ 2e ⫹ ␻ 2g ⫺F 共 r 兲 H 共 r 兲 …⍀⫹ ␻ 2e ␻ 2g ⫽0,

共9兲

where ⍀ª ␻ . If we set F(r)H(r)⫽0, we can easily verify that the roots of the above equation are real and given by ⍀ 1 ⫽ ␻ 2e and ⍀ 2 ⫽ ␻ 2g as expected. For F(r)H(r)⫽0, the roots of Eq. 共9兲 satisfy the simple relations 2

⍀ 1 ⫹⍀ 2 ⫽ ␻ 2e ⫹ ␻ 2g ⫺F 共 r 兲 H 共 r 兲

共10兲

⍀ 1 ⍀ 2 ⫽ ␻ 2e ␻ 2g .

共11兲

and

where rg is the position vector of the magnetic charge g. Some comments are in order here:

Let us introduce the auxiliary quantity

共i兲

where ␻ 1 and ␻ 2 are two real positive roots of the original algebraic equation of the fourth degree for ␻. Using the simple properties given by Eqs. 共10兲 and 共11兲 we find

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In the previous equations, Ee and Eg correspond, respectively, to the electric field created by the fluctuating electric dipole p(t)⫽ex e uˆ e and to the electric Am. J. Phys., Vol. 70, No. 4, April 2002

Sª 共 ␻ 1 ⫹ ␻ 2 兲 2 ⫺ 共 ␻ e ⫹ ␻ g 兲 2 .

Farina, Santos, and Tort

共12兲

422

共13兲

S⫽⫺F 共 r 兲 H 共 r 兲 . Hence, from 共12兲 and 共13兲 we can write

␻ 1⫹ ␻ 2⫽共 ␻ e⫹ ␻ g 兲

冑

1⫺

F共 r 兲H共 r 兲 . 共 ␻ e⫹ ␻ g 兲2

共14兲

Upon quantization of the 共independent兲 normal modes and assuming that 兩 S 兩 Ⰶ( ␻ e ⫹ ␻ g ) 2 , the ground state energy of the system is given by ប E 0⫽ 共 ␻ 1⫹ ␻ 2 兲 2 ⬇

2 Q eg ប បe 2 g 2 . 共 ␻ e⫹ ␻ g 兲⫹ 2 4共 ␻ 1⫹ ␻ 2 兲m em gc 2 r 4

共15兲

Identifying the van der Waals interaction energy U(r) between the two atoms as the shift in the zero-point energy of the system caused by the electromagnetic interaction between them, we get U 共 r 兲 ⫽E 0 ⫺

ប 共␻ ⫹␻g兲 2 e

冉 冊冉

ប ␻ e␻ g ⫽ 4 c2

共16兲

冊

2 ␻ e ␻ g ␣␤ Q eg , 4 ␻ e⫹ ␻ g r

共17兲

冉 冊冉 ␻ e␻ g c2

冊

2 ␻ e ␻ g ␣␤ Q eg ˆreg . ␻ e⫹ ␻ g r5

共18兲

As already mentioned, the last equation displays the correct dependence of the nonretarded dispersive force on the distance between the atoms for the problem at hand, in agreement with Refs. 11 and 12. The 1/r 4 behavior of 共15兲 关or the 1/r 5 behavior of 共18兲兴 can be traced back to the nearfield approximation of the electromagnetic fields involved, which contribute to the problem with terms varying with 1/r 2 and to the fact that terms stemming from the vector products in the equations of motion are zero. Take, for instance, the magnetically polarizable atom. There are two forces acting on it: the first one is due to the magnetic field created by the fluctuating electric dipole of the other atom, which does not have the typical 1/r 3 behavior of a static zone field; the second one is due to the cross product between the magnetic charge velocity ( vជ g ) and the electric field generated by the fluctuating electric dipole (Ee ). However, though Ee exhibits the static zone term 共proportional to 1/r 3 兲, this term disappears from the calculation when projected onto the fixed 共but arbitrary兲 direction of motion of the magnetic charge g. An analogous explanation holds for the electric polarizable atom. We conclude, then, that this is a very peculiar result whose origin lies in the fact that one of the atoms is only electrically polarizable and the other only magnetically polarizable. Had we started with two magnetically polarizable atoms, we would have obtained a nonretarded dispersive interaction energy also proportional to 1/r 6 . 423

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ACKNOWLEDGMENT One of us 共CF兲 wishes to acknowledge the partial financial support of CNPq 共the National Research Council of Brazil兲. a兲

where we have written the last term so that the electric and magnetic polarizabilities ␣ ⫽e 2 /m e ␻ 2e and ␤ ⫽g 2 /m g ␻ 2 can be identified. The quantity Q eg ª(uˆg ⫻rˆeg )•uˆe is a spatial orientation factor to be properly averaged. The force between the electrically polarizable atom and the magnetically polarizable one is repulsive 共for ␤ ⬎0兲 and varies with the inverse fifth power of the separation. For instance, the force on the magnetically polarizable atom is given by F共 r 兲 ⫽ប

As a consequence of the repulsive force given by 共18兲, two macroscopic bodies, one made of an electrically polarizable material and the other made of a magnetically polarizable one, will repel each other. Due to such a difference in the power law between the retarded and nonretarded regimes of the dispersive van der Waals forces discussed above 共three powers in the separation between the atoms兲, maybe this result is of some relevance from the experimental point of view. Recall that the less abrupt change in the power law of the dispersive van der Waals force between two electrically polarizable molecules as the distance between them increases (1/r 7 →1/r 8 ) has already been verified experimentally sometime ago by Tabor and Winterton.14 As a final comment, it is worth mentioning that repulsive Casimir forces between a perfectly conducting plate and an infinitely permeable one can be found in the literature.15–18

Electronic mail: [email protected] Electronic mail: [email protected] c兲 Electronic mail: [email protected] 1 J. D. van der Waals, ‘‘Over de continuiteit van den gas-en vloeistoftoestand,’’ dissertation, 1873. 2 F. London, ‘‘Zur Theorie und Systemakik der Molecularkrafte,’’ Z. Phys. 63, 245–279 共1930兲. 3 H. B. G. Casimir and D. Polder, ‘‘The influence of retardation on the London-van der Waals forces,’’ Phys. Rev. 73, 360–372 共1948兲. 4 Dieter Langbein, Theory of van der Waals Attraction, Springer Tracts in Modern Physics, Vol. 72 共Springer-Verlag, Berlin, 1974兲. 5 P. W. Milonni, The Quantum Vaccum: An Introduction to Quantum Electrodynamics 共Academic, New York, 1994兲. 6 C. Farina, F. C. Santos, and A. C. Tort, ‘‘A simple way of understanding the nonadditivity of van der Waals dispersion forces,’’ Am. J. Phys. 67, 344 –349 共1999兲. 7 H. B. G. Casimir, ‘‘On the attraction between two perfectly conducting plates,’’ Proc. K. Ned. Akad. Wet. 51, 793–795 共1948兲. 8 V. M. Mostepanenko and N. N. Trunov, The Casimir Effect and its Applications 共Clarendon, Oxford, 1997兲. 9 G. Feinberg and J. Sucher, ‘‘General form of the retarded van der Waals potentials,’’ J. Chem. Phys. 48, 3333–3334 共1968兲. 10 T. H. Boyer, ‘‘Recalculations of Long-Range van der Waals Potentials,’’ Phys. Rev. 180, 19–24 共1969兲; ‘‘Asymptotic Retarde van der Waals Forces Derived from Classical Electrodynamics with Classical Electromagnetic Zero-Point Radiation,’’ Phys. Rev. A 5, 1799–1802 共1972兲. 11 Gerald Feinberg and Joseph Sucher, ‘‘General Theory of the van der Waals Interaction: A Model-Independent Approach,’’ Phys. Rev. A 2, 2395–2415 共1970兲. 12 C. Farina, F. C. Santos, and A. C. Tort, ‘‘The non-retarded dispersive force between an electrically polarizable atom and a magnetically polarizable one,’’ hep-th/0007190, submitted for publication. 13 J. D. Jackson, Classical Electrodynamics, 2nd ed. 共Wiley, New York, 1975兲, p. 395. 14 D. Tabor and R. H. S. Winterton, ‘‘Direct measurement of normal and retarded van der Waals forces,’’ Nature 共London兲 219, 1120–1121 共1968兲. 15 Timothy H. Boyer, ‘‘Van der Waals forces and zero-point energy for dielectric and permeable materials,’’ Phys. Rev. A 9, 2078 –2084 共1974兲. 16 V. Hushwater, ‘‘Repulsive Casimir force as a result of vacuum pressure,’’ Am. J. Phys. 65, 381–384 共1997兲. 17 M. V. Cougo-Pinto, C. Farina, and A. Tenorio Leite, ‘‘␨-function method for repulsive Casimir forces,’’ Braz. J. Phys. 29, 371–374 共1999兲. 18 F. C. Santos, A. Teno´rio, and A. C. Tort, ‘‘Zeta function method and repulsive Casimir forces for an unusual pair of plates at finite temperature,’’ Phys. Rev. D 60, 105022/1–9 共1999兲. b兲

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