[PDF] A simple way of understanding the nonadditivity of van der Waals





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[PDF] A simple way of understanding the nonadditivity of van der Waals

in which the nonadditivity of the van der Waals dispersion forces arises in a very transparent way was not able to explain the precise origin of the forces re-

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[PDF] A simple way of understanding the nonadditivity of van der Waals 100773_7farina11.pdf A simple wayof understanding thenonadditivity of van der Waals dispersion forces

C. Farina,

a)

F. C. Santos,

b) and A. C.Tort c)

Instituto de FõÂsica, UniversidadeFederal do Rio de Janeiro, Cidade UniversitaÂria, Ilha do FundaÄo,

Caixa Postal 68528, 21945-970 Rio de Janeiro,Brazil ~Received 1 April 1998; accepted 22 September 1998!

Using a semiclassical model for three interacting ¯uctuating dipoles we introduce a simple scenario

in which the nonadditivity of the van der Waals dispersion forces arises in a very transparent way. For simplicity, we illustrate our model in the case of nonretarded dispersion forces. The argument

can be straightforwardly generalized to the case ofNinteracting ¯uctuating dipoles. For pedagogical

reasons, we also give a brief review of some basic points concerning van der Waals forces. © 1999

American Association ofPhysicsTeachers.

I. INTRODUCTION

The name of van der Waals is inextricably related to the equation of state for real gases, proposed by him in his thesis 1 in 1873. Written for 1 mol of gas this equation reads S P1a V 2D ~V2b!5RT,~1! whereP,V, andTare the pressure, volume, and the absolute temperature, respectively, andRis the universal gas con- stant. The parametersaandbare known as van der Waals constants and their respective values can be obtained by ®t- ting this equation of state to the experimental data. The pres- ence of the constantbin~1!takes into account the ®nite size of the molecules and tells us that one individual gas mol- ecule does not have access to the total volumeV.Astothe parametera, this constant was associated by van der Waals with an attractive force between two molecules. A conse- quence of such intermolecular~attractive!forces is to dimin- ish the pressure at the walls. Although van der Waals sug- gested explicit forms for these intermolecular potentials, he was not able to explain the precise origin of the forces re- sponsible for theappearance of parametera. When we talk about van der Waals attraction, it is tacitly assumed that we are dealing with molecules~atoms!whose separation is large enough to exclude the overlapping of electronic orbitals. In this case, we can distinguish three types of van der Waals forces, to wit: orientation, induction, and dispersion van der Waals forces, which we now describe brie¯y~a more detailed discussion can be found in Refs. 2, 3, and 4!. The orientation van der Waals force corresponds to the force between two polar molecules~polar here means that the molecule has a permanent electric dipole moment, for instance a water molecule!. W. H. Keesom, in the early twenties, was the ®rst to compute the thermal average of the force between two polar molecules and found a temperature- dependent interaction energy, namely, (22/3)(p 12 p 2 2 / 4 pe 0 r 6 )(1/k B

T),~fork

B T@(p 1 p 2 /4pe 0 r 3 )), wherep 1 and p 2 are the magnitudes of the electric dipole moments of the two molecules, respectively,ris the distance between the molecules, andk B is the Boltzmann constant. The attractive nature of the orientation force is not dif®cult to understand: Although the number of attractive orientations is exactly the same as the number of repulsive ones, the attractive orienta- tions are statistically favored over the repulsive ones. This isso because the Boltzmann weight ise 2E/k B T ~and hence it diminishes with increasing energyE!, and the smaller ener- gies correspond to attractive orientations. Notice also that the orientation force decreases with increasing temperature. This is quite natural since as the temperature increases repulsive orientations tend to be as accessible as attractive orientations. Approximately at the same time, P. Debye and others rec- ognized that even if one molecule had a permanent dipole moment~or even a higher multipole moment, as for example, a permanent quadrupole moment!and the other had not, the ®rst molecule would induce a dipole moment in the second one and a resulting dipole±dipole~quadrupole±dipole,...!at- tractive force would appear. That the induction van der Waals force is also attractive is not dif®cult to understand: It suf®ces to remember a very common experience, namely, that neutral tiny pieces of dielectric material such as small pieces of paper are always attracted by a charged object~like a small bar of glass charged by friction!. This is so because the induced dipoles point in the same direction as the electric ®eld that induced them, and these relative orientations give rise to an attractive resultant force. Moreover, due to this correlation between the induced dipole and the dipole of the polar molecule, the induction force does not vanish with in- creasing temperature. There is still another important feature of induction forces: its dependence on the distancerbetween the two molecules is proportional to 1/r 7 , which can be qualitatively understood as follows: Recall that the magni- tude of the induced dipole on the second molecule (p 2 )is proportional to the magnitude of the electric ®eld created by the~permanent!dipole moment of the ®rst molecule (E 1 ), which in turn is of the formE 1 'p 1 /r 3 . The electrostatic energy between the molecules is then U ~r!'p 2 E 1 ~r!'aE 12 ~r!' ap 1 2 r 6 ,~2! where ais the molecular polarizability. SinceU(r) is pro- portional to 1/r 6 , the force has the 1/r 7 behavior mentioned above. However, neither the orientation forces nor the induction ones can explain the attraction between molecules~atoms!in all cases, as for example the attractive interaction between two noble gas atoms, since these types of forces still require at least one molecule~atom!to exhibit a permanent dipole ~quadrupole!moment.

344344Am. J. Phys.67~4!, April 1999 © 1999 American Association of Physics Teachers

The explanation of the true origin of such general forces had to wait for the advent of quantum mechanics. In 1930, F.

London

5 using perturbation theory showed that even if nei- ther molecule posessed a permanent dipole moment, they still would attract each other. It was only necessary that di- pole moments could be induced in both molecules. In other words the molecules ought to be polarizable~the molecular polarizability amust be different from zero!. Since the po- larizability is related to the refractive index and dispersion, this type of force is known as dispersion force. At this point it is quite natural to ask ourselves how these induced dipoles arise in both molecules~remember that none of them had initially a permanent dipole moment!. Here, classical physics fails completely as dispersion forces have a genuine quantum origin. Modern quantum ®eld theory, considered as one of the most important scienti®c achievements of the twentieth century, tells us that even when the ``medium'' is the vacuum, there is a residual~quantized!electromagnetic ®eld E(r,t) andB"r,t) whose vacuum expectation value is zero, but whose ¯uctuations given by ^E 2 &and^B 2 &do not aver- age to zero. The vacuum ®eld will induce an instantaneous dipole moment in one molecule, which in turn, together with the vacuum electromagnetic ®eld, will also induce in the second molecule another instantaneous dipole moment. We can say then that the vacuum electromagnetic ®eld induces ¯uctuating dipoles in both atoms and the dipole±dipole in- teraction of these zero-mean but correlated dipoles corre- sponds to the van der Waals dispersive interaction~see

Milonni's book

4 for a detailed discussion on the connection between dispersion forces and the vacuum electromagnetic

®eld!.

There are two types of dispersion forces, to wit: nonre- tarded and retarded dispersion forces. Nonretarded forces oc- cur when the speed of light is taken to be in®nite, so that the interaction between two molecules is instantaneous. Re- tarded dispersion forces occur when we take into account the ®niteness of the speed of light. In this case, the dipole ®eld of the ®rst molecule, which is induced by the vacuum ¯uc- tuations, will reach the second one after the time interval r/c. Consequently, the reaction ®eld of the second molecule will be delayed at the ®rst one by the time interval 2r/c. Nonretarded forces are then good approximations when the distance between the molecules is small. As the distance be- tween the two molecules increases, retardation effects be- come more and more important. London's work 5 refers to nonretarded dispersion forces and shows that for two identi- cal polarizable atoms the interaction potential energy is given by U ~r!52 S 1 4pe 0 D 2 3\v 0 a 2 4r 6 ,~3! where in the above formula ameans the static polarization of each atom and v 0 is the transition frequency between the ground state and the ®rst excited one. Almost two decades after London's result Casimir and Polder 6 obtained for the ®rst time, after an involved perturbative quantum electrody- namics calculation, the expression for the retarded dispersion van der Waals forces between two polarizable atoms. Basi- cally, retardation effects change the 1/r 6 behavior of the in- teraction energy to 1/r 7 . However, this change in the power law for the dispersion force with increasing distance was not easy to verify experimentally and only in 1968 Tabor and

Winterton

7 were able to achieve this goal.To conclude this brief review concerning van der Waals forces, it is worth emphasizing at this point that van der Waals forces~orientation, induction, and dispersion!have a great variety of theoretical and practical applications as for example: in condensation and crystallization, in structural and energetic effects in colloid chemistry and biology, in surface physics and chemistry, in the large ®eld of adhesion and its implications in washing~good detergents must dimin- ish van der Waals attraction between dirt and clothing!,inits connection with the so-called Casimir effect 8 ~retarded dis- persion force!, among others. Here we shall be concerned with dispersion forces, in par- ticular with a very important property of these forces, namely, their nonadditivity. It is a well-known fact that the retarded and unretarded dispersion van der Waals forces are nonadditive, that is, the resultant force which acts on a given ¯uctuating dipole~polarizable molecule!due to the presence of the other ¯uctuating dipoles~other polarizable molecules! is not simply given by the sum of the individual forces that each one of them would exert if all the others were absent. In other words, the force between two ¯uctuating dipoles~po- larizable molecules!is altered in a nontrivial way by the presence of a third one, see, for example, Refs. 2, 3, and 4, and references therein. The degree of nonadditivity, how- ever, may depend on the density of the medium. In fact, for rare®ed media it is possible to suppose that the forces are additive. For two semi-in®nite dielectric slabs this leads, for instance, to an attractive macroscopic force which is in- versely proportional to the cube of the distance between the slabs in the nonretarded case. If retardation effects are taken into account then one obtains a force which is inversely pro- portional to the fourth power of the distance between the slabs. Although the functional dependence of the force with the distance between the slabs is correct, the theoretical val- ues computed taking into account only pair interaction do not ®t experimental data. It was precisely this discrepancy be- tween experimental results and microscopic theories that as- sume additive van der Waals forces that led to a consider- ation of their nonadditivity. 9 The main purpose of this article is to show in a simple and transparent approach how the nonadditivity of these forces arises allowing in this way students with a very elementary knowledge of quantum mechanics to understand such an im- portant issue. For simplicity, we shall work here only with nonretarded van der Waals forces. For pedagogical reasons this article is organized as follows: In order to introduce the notation and basic ideas of the model to be employed, Sect. II is devoted to a rederivation of London's result; in Sect. III we generalize the model of Sect. II to the case of three po- larizable molecules which suf®ces to make the ®rst ``nonad- ditive'' term~in the expression for the energy of the system! appear. Section IV is left for the conclusions and ®nal re- marks.

II. LONDON'S RESULT: A SEMICLASSICAL

APPROACH

In order to introduce the basic ideas as well as to establish our notation, we shall review London's result 5 using an el- ementary approach for the dispersive nonretarded van der Waals forces between two neutral but polarizable atoms. In this model the instantaneous dipole moments exhibited by each atom will be simulated by a point chargeeoscillating harmonically with angular frequency v 0 around a ®xed point

345345Am. J. Phys., Vol. 67, No. 4, April 1999 Farina, Santos, and Tort

~``the electronic cloud around a ®xed nucleus''!. The restor- ing harmonic force linking the electron to the center of force simulates its binding to the nucleus~which remains ®xed by assumption!. It is well known that for many purposes an atom in its ground state level can be represented by a charged harmonic oscillator. 4

Hence, this assumption means

that we shall obtain the interaction energy between two ground state atoms. To neglect retardation effects in the propagation of the electromagnetic ®eld means essentially that the time interval that a light signal takes to propagate between the two molecules, which is given byDT5r/c,is much smaller than the characteristic time associated with the system, which is given byD t'1/v 0 , wherev 0 5(E 1 2E 0 )/\'(mc 2 /\)a fsc2 . Here,a fsc is the ®ne structure con- stant ( a fsc ªe 2 /(4pe 0 \c)'1/137). Since the Bohr radius a 0 can be written asa 0 5\c/a fsc mc 2 , it follows that the nonretarded regime holds ifDT!D t,orr!137a 0 . For our purposes, it suf®ces to consider these oscillations in ®xed directions~although with the relative orientations of the two instantaneous dipoles completely arbitrary!.We know that the force between induced dipoles must be attrac- tive, so it is desirable that this model reproduces this pecu- liarity for arbitrary directions. This is not obvious and we will come to this point later. Besides, we shall neglect the radiation emitted by these oscillating dipoles and conse- quently in the following discussion there will be no radiation reaction terms. This fact should cause no surprise: Recall that we are trying to obtain the interaction energy between two atoms in their ground state. In order to get an expression for the nonretarded van der Waals interaction energy be- tween these atoms in such a way that its connection with zero-point energy becomes clear, we must proceed as fol- lows: First, we write down the~coupled!classical equations of motion for both oscillating charges, taking into account the electric force that one exerts on the other; then, we de- couple these equations and compute the frequencies of the corresponding normal modes. Finally, since dispersion forces are genuinely quantum in nature, we must quantize the system of the two independent oscillators~normal modes!, compute the associated ground state energy level, and identify the desired interaction energy as the difference between this energy and the ground state energy of the de- coupled system of oscillating charges. This is equivalent to taking as the interaction energy the expression containing only the terms involving the distancerbetween the atoms.

Denoting byx

i the displacement of the electron attached to atomifrom its equilibrium position and assuming that ux i u is much smaller than the distancerbetween the atoms, the electric ®eld created, for instance by the ®rst oscillating charge at the position of the second one, can be considered as the ®eld of an electric dipolep W 1 (t)5ex 1 (t)mà 1 , wheremà 1 is a unit vector which determines the ®xed~but otherwise arbi- trary!direction of oscillation of the ®rst instantaneous dipole ~recall that there is no monopole term because the atom is neutral and the oscillating charge is just a way of simulating its ¯uctuating dipole!. Denoting bywthe angular frequency of the oscillating dipole, this ®eld is given by, 10 E W 1 ~P 2 ,t!5e 4pe 0 x 1 ~t2r/c! H @3~rÕmà 1 !rÃ2mà 1 # 3 S 1 r 3 2ik r 2D 1k 2 ~rÃ3mà 1 !3rà r J ,~4!wherek5 v/c. In the above equation, we identify clearly the near ®eld electrostatic contribution~terms proportional to 1/r 3 !and the radiation ®eld~the term proportional to 1/r!. Since we are interested in nonretarded dispersion forces, which means small distances between the atoms, the domi- nant term turns out to be the near ®eld contribution. Also, x 1 (t2r/c) can be approximated byx 1 (t). The component of the relevant electric force exerted by the dipolee 1 x 1 m à 1 upone 2 along the allowed direction of oscillations ofe 2 ~determined bymà 2 !is then (e 1 5e 2 5e) e 2 E W 1 ~P 2 ,t!•mà 2 5e 2 x 1 ~t! 4pe 0 r 3 @3~rÕmà 1 !~rÕmà 2 !2mà 1 •mà 2 #. ~5! Hence, the equations of motion for these oscillating charges are simply xÈ 1 ~t!1v 02 x 1 ~t!5K 12 x 2 ~t!, ~6!xÈ 2 ~t!1v 02 x 2 ~t!5K 12 x 1 ~t!, where for convenience we de®ned K 12 5Q 12 4pe 0 r 3S e 2 m D , Q 12

5@3~rÕmÃ

1 !~rÕmà 2 !2mà 1 •mà 2 #,~7! wheremis the mass of the electron. The factorQ 12 is the spatial orientation factor. The above equations describe a coupled system with two vibrational degrees of freedom, which can be decoupled in the usual way by introducing normal coordinates, say h 1 andh 2 , as follows: h 1 ~t!5x 1 ~t!1x 2 ~t!,~8! h 2 ~t!5x 1 ~t!2x 2 ~t!.~9!

The resulting~decoupled!equations are simply

hÈ 1 1v 12 h 1

50,~10!

hÈ 2 1v 22
h 2

50,~11!

where the normal frequencies are given by v 6 5~v 02 7K 12 ! 1/2 5v 0S 17K 12 2v 02 2K 122
8v 0 4 7K 123
16v 0 6 25K
124
128v
0 8 1O S K 125
v 0 10 DD , ~12! where we assumed here thatK 12 !v 02 ~see the discussion in Sec. IV!. The quantization of this system is immediate since it consists of two independent harmonic oscillators with fre- quencies v 1 andv 2 , respectively. The corresponding ground state energy is then E5\ 2 ~v 1 1v 2 ! 5\v 0H 12K 122
8v 0 4 25K
124
128v
0 8 1O S K 126
v 0 12 DJ .~13! Substituting~7!into~13!and identifyingU(r) with the r-dependent terms we obtain to lowest order in powers of K 12 /v 02 ,

346346Am. J. Phys., Vol. 67, No. 4, April 1999 Farina, Santos, and Tort

U~r!5E223

1 2 \v 0 ~14! 52
S 1 4pe 0 D 2 Q 122
\v 0 a 2 8r 6 ,~15! where in the previous equatione 2 /mv 02 was identi®ed with the classical static polarizability aof an atom~considered as a charged harmonic oscillator!. Now, we can understand why this model~with ®xed di- rections!works: We know that the total quantum mechanical energy of two independent oscillators always diminishes when coupling terms between them of the form given by Eq. ~6!are introduced. This fact becomes very clear if we look at

Eq.~12!~notice that the ®rst corrections to

v 1 andv 2 have opposite signs, but the second-order corrections have the same sign and are both negative!. Of course the semiclassical model presented here could not yield London's result with all numerical factors. How- ever, it must be clear that our purpose in this section was simply to introduce the ¯uctuating oscillator model. Any- way, London's result can be obtained if we think of aas the quantum static polarization and v 0 as the angular frequency associated with the transition between the ground state and the ®rst excited one and changeQ 122
by its corresponding quantum mechanical evaluation which gives the factor 2

33, see for instance Refs. 11 and 12.~Quantum mechani-

cally, ais given by (2/3\)(ud W u/v 0 ), whereud W uis the dipole moment amplitude between the ground state and the ®rst excited one.!

III. NONADDITIVITY OF VAN DER WAALS

DISPERSION FORCES

The standard discussions of the nonadditivity of dispersion van der Waals forces are in general rather involved for stu- dents with an elementary knowledge of quantum theory~the interested reader can ®nd a detailed discussion on this sub- ject in the excellent book by Langbein 2 !. It is also possible to introduce the nonadditive effects in connection with the ~quantized!vacuum electromagnetic ®eld. For instance, in order to discuss the nonadditivity of van der Waals forces,

Milonni

4 considered the interaction energy of an~induced! oscillating dipolep

Wwith the quantized source-free electro-

magnetic ®eld~vacuum ®eld!. In the quantization of the ®eld, or in other words, in computing the mode functions for the ®eld, the presence of all the other atoms~considered also as induced oscillating dipoles!must be taken into account. After a rather involved calculation it is shown that nonaddi- tive terms contribute to the expectation value of the energy of the induced dipolep W i at pointrW i in the quantized source- free electric ®eldE(r i ,t), namely,^E i &52 1 2 ^p i •E(r i ,t)&. These terms are three-body contributions of third order~as- sociated with the tripleti,j,l!of the form 4 ^E ijl &} a 3 r ij3 r jl3 r il

3,~16!

where ais the static polarizability of the atoms, considered identical in the formula above. This kind of term had already been derived before through standard perturbation theory and studies concerning its magnitude and sign date from the early forties. 13 It is precisely this kind of term that we want to derive here

using an elementary approach, namely, a simple generaliza-tion of the model used in Sec. II. Hence, in order to discuss

the nonadditivity of nonretarded van der Waals dispersion forces in a simple way we shall introduce a third ¯uctating dipole~polarizable atom!interacting with the other two. Us- ing the same assumptions as in Sec. II, the equations of mo- tion now read xÈ 1 ~t!1v 02 x

1~t!5K

12 x 2 ~t!1K 13 x 3~t!, xÈ

2~t!1v

02 x 2 ~t!5K 12 x 1 ~t!1K 23
x 3 ~t!,~17! xÈ 3 ~t!1v 02 x 3 ~t!5K 13 x 1 ~t!1K 23
x 2~t!, where in an obvious notation the symmetrical coef®cients K ij de®ne the coupling constants between the three oscillat- ing charges and are given by K ij 5Q ij 4pe 0 r ij3 S e 2 m D withQ ij

53~mÃ

i •rà ij !~mà j •rà ij !2mà i •mà j .~18! As before, the above coupled equations describe a system of three~vibrational!degrees of freedom that can also be de- coupled by introducing normal coordinates. Since we need only compute the normal frequencies and recalling that when the system vibrates in one of its normal modes all particles oscillate with the same frequency, we can substitute into~17! x i 5C iexp$ivt%which yields S v 02 2v 22K
12 2K 13 2K 12 v 02 2v 2 2K 23
2K 13 2K 23
v 02 2v 2 D S x 1 x 2 x 3 D 5 S 0 0 0 D .~19! In order to avoid trivial solutions we must require that the above transformation have no inverse, which is equivalent to setting the determinant of the above matrix to zero. This assumption simply gives ~v 02 2v 2 ! 3 2~K 12 2 1K 132
1K 232
!~v 02 2v 2 !22K 12 K 13 K 23
50.
~20!

The desired eigenfrequencies

v i ,i51,2,3, are then the posi- tive roots of this equation. After quantization the ground state energy of the three ¯uctuating dipoles is given by E5\ 2 ~v 1 1v 2 1v 3 !.~21!

De®ning

bªv 02 2v 2 , we rewrite Eq.~20!in the form b 3 2~K 122
1K 132
1K 232
!b22K 12 K 13 K 23

50~22!

and Eq.~21!in the form E5\ v 0 2 HS 12 b 1 v 02 D 1/2 1 S 12 b 2 v 02 D 1/2 1 S 12 b 3 v 02 D 1/2 J ,~23! where b i ªv 02 2v i 2 ,i51,2,3 are the roots of Eq.~22!. Next, making a Taylor expansion of each term in Eq.~23!in pow- ers of b i /v 02 , we get E5\ v 0 2 H 321
2v 02 ~b 1 1b 2 1b 3 ! 21
8v 04 ~b 1 2 1b 2 2 1b 3 2 !21 16v 06 ~b 1 3 1b 2 3 1b 3 3 ! J .~24!

347347Am. J. Phys., Vol. 67, No. 4, April 1999 Farina, Santos, and Tort

It is interesting to observe that all the terms on the right- hand side~RHS!of Eq.~24!are symmetrical functions of the roots of Eq.~22!and consequently, it will be not necessary to determine explicitly each root b i , since these functions can be expressed directly as functions of the coef®cients of the algebraic equation. For a general algebraic equation of the third degree, namely, a 0 x 3 1a 1 x 2 1a 2 x1a 3

50,~25!

we write below the basic combinations of its roots as x 1 1x 2 1x 3 52a
1 a 0 , x 1 x 2 1x 1 x 3 1x 2 x 3 51a
2 a 0 ,~26! x 1 x 2 x 3 52a
3 a 0 .

Comparing Eqs.~22!,~25!, and~26!, we can write

b 1 1b 2 1b 3

50,~27!

b 1 b 2 1b 1 b 3 1b 2 b 3 52~K
122
1K 132
1K 232
!,~28! b 1 b 2 b 3 52K
12 K 13 K 23
.~29! What we must do now is to use the above equations in order to obtain the expressions for the sum of the squares of the roots ( b 12 1b 2 2 1b 3 2 ) as well as the sum of the cubes of the roots ( b 13 1b 2 3 1b 3 3 ) as required by Eq.~24!.

Taking then the square of~27!we get

b 12 1b 2 2 1b 3 2 522~b
1 b 2 1b 1 b 3 1b 2 b 3 ! 52~K
122
1K 132
1K 232
!,~30! where we used Eq.~28!. For the sum of the cubes of the roots, we take as a starting point the identity b 13 1b 2 3 1b 3 3 1b 1 2 b 2 1b 2 2 b 1 1b 1 2 b 3 1b 3 2 b 1 1b 2 2 b 3 1b 3 2 b 2

50,~31!

which follows immediately from Eq.~27!if we multiply both sides by ( b 12 1b 2 2 1b 3 2 ). On the other hand, using Eq.~27!again, we can write b 1 b 2 ~b 1 1b 2 1b 3 !50!b 12 b 2 1b 2 2 b 1 52b
1 b 2 b 3 , b 1 b 3 ~b 1 1b 2 1b 3 !50!b 12 b 3 1b 3 2 b 1 52b
1 b 2 b 3 , ~32! b 2 b 3 ~b 1 1b 2 1b 3 !50!b 22
b 3 1b 3 2 b 2 52b
1 b 2 b 3 , so that b 12 b 2 1b 2 2 b 1 1b 1 2 b 3 1b 3 2 b 1 1b 2 2 b 3 1b 3 2 b 2 523b
1 b 2 b 3 . ~33! Substituting the previous equation into~31!and also using

Eq.~29!, we obtain

b 13 1b 2 3 1b 3 3 56K
12 K 13 K 23
.~34! Therefore, substituting Eqs.~27!,~30!, and~34!into Eq. ~24!we ®nally obtainE'\ v 0 2 H 321
4v 04 ~K 12 2 1K 132
1K 232
!23 8v 06 K 12 K 13 K 23J
, ~35! which implies an interaction energy given by U ~r 12 ,r 13 ,r 23
!5E23 2\ v 0 '2\ v 0 8 HS a 4pe 0 D 2 S Q 122
r 12 6 1Q 132
r 13 6 1Q 232
r 23
6 D 13 2 S a 4 pe 0 D 3 Q 12 Q 13 Q 23
r 123
r 13 3 r 23
3 J .~36! The ®rst three terms on the RHS of~36!represent pairwise additive contributions to the interaction potential between the three oscillators. The last term on the RHS of~36!is a three- body contribution to the interaction and therefore it spoils the additivity. This can be seen as follows: If the force that acts on one of the dipoles, say number one, is computed from ~36!, then the result will be different from the sumF W 21
1F W 31
, whereF W 21
(F W 31
) is the force that dipole number two ~three!would exert on dipole number one in the absence of dipole number three~two!. In order to connect the three- body contribution appearing in~36!with that written in Eq. ~16!, it suf®ces to set a5e 2 /mv 02 and identify this constant with the polarizability at the transition frequency v 0 5(E 1 2E 0 )/\between the ground and the ®rst excited state, which is the principal transition for this system. After this we will have the kind of term appearing as the lowest order nonadditive contribution to van der Waals forces in the case ofNinteracting ¯uctuating dipoles in Milonni's calculation. 4 Notice also that this term, depending on the geometrical ar- rangement of the oscillators, may yield an attractive or a repulsive contribution. 13

IV. FINAL REMARKS AND CONCLUSIONS

In this paper we applied the ¯uctuating dipole model to the case of three~polarizable!atoms in order to explain on elementary grounds how the nonadditivity of dispersion van der Waals forces arises. The application of this model to the case ofNinteracting ¯uctuating dipoles is straightforward and yields correctly many interesting results as for example: ~i!There will not be any linear term inK ij in the expression forUin analogy with what happens forN52@see Eq.~14!# and forN53@see Eq.~36!#.~ii!Although the cubic term in K 12 is absent from the expression forUin theN52 case@see

Eq.~14!#, cubic terms inK

ij will appear forN>3, giving rise to the lowest order nonadditive terms. These terms cor- respond to the three-body contribution to the interaction van der Waals energy.~iii!Higher order nonadditive terms also appear, namely, four-body contributions,...,N-body contribu- tions. TheN-body term is of the form ( a/4pe 0 r 3 ) N ,aandr being a typical atomic polarizability and a typical inter- atomic distance, respectively. Consequently, theN-body term is a/4pe 0 r 3 times smaller than the (N21), term. Hence, in order to make sense the model must satisfy the condition a/4pe 0 r 3 !1. In fact, substituting typical values @recall that a5e 2 /mv 02 and thatv 0 can be thought of as the transition frequency between the ground state and ®rst ex- cited one, that is, v 0 is a few eV~s!per\#we obtain for an

348348Am. J. Phys., Vol. 67, No. 4, April 1999 Farina, Santos, and Tort

intermolecular distancer'10 Å thata/4pe 0 r 3 '10 23
!1. It is worth emphasizing that this value forris compatible with an earlier assumption, namely,r!137a 0 . We would also like to emphasize the importance of the nonadditive terms in the evaluation of the force between macroscopic objects like spheres, cylinders, or slabs. Al- though the dependence of this force on the relevant distance involved in each case is correct even if we take into account only pair interactions, numerical values will ®t correctly the experimental data only when nonadditivity is not neglected. Finally, we cannot conclude this article without mention- ing that retarded dispersion forces between macroscopic ob- jects~with all nonadditive contributions!correspond to the so-called electromagnetic Casimir force between them, which since Casimir's seminal paper 8 can be computed di- rectly from the ¯uctuations of the con®ned~vacuum!elec- tromagnetic ®eld subjected to the appropriate boundary con- ditions. The Casimir effect was veri®ed experimentally for the ®rst time by Sparnaay 14 with a poor accuracy and re- cently by Lamoreaux, 15 and Mohideen and Roy 16 with an excellent agreement between theory and experimental data, but that is another story.

ACKNOWLEDGMENTS

The authors are indebted to M. V. Cougo-Pinto, A. N. Vaidya, V. Mostepanenko, and G. L. Klimchitskaya for en- lightning discussions. One of us~CF!wishes to acknowledge the partial ®nancial support of CNPq~the National Research

Council of Brazil!.

a!

Electronic mail: farina@if.ufrj.br

b!

Electronic mail: ®ladelf@if.ufrj.br

c!

Electronic mail: tort@if.ufrj.br

1 J. D. van der Waals, ``Over de continuiteit van den gas-en vloeistoftoe- stand,'' Dissertation, Leiden, 1873. 2 Dieter Langbein,Theory of Van der Waals Attraction, Springer Tracts in Modern Physics, Vol. 72~Springer-Verlag, Berlin, 1974!. 3 H. Margenau and N. R. Kestner,Theory of Intermolecular Forces~Perga- mon, New York, 1969!. 4 P. W. Milonni,The Quantum Vaccum: An Introduction to Quantum Elec- trodynamics~Academic, New York, 1994!. 5 F. London, ``Zur Theorie und Systematik der Molecularkrafte,'' Z. Phys.

63, 245~1930!.

6 H. B. G. Casimir and D. Polder, ``The In¯uence of Retardation on the London-van der Waals Forces,'' Phys. Rev.73, 360±372~1948!. 7 D. Tabor and R. H. S. Winterton, ``Direct measurement of normal and retarded van der Waals forces,'' Nature~London!219, 1120±1121~1968!. 8 H. B. G. Casimir, ``On the Attraction Between Two Perfectly Conducting Planes,'' Proc. K. Ned. Akad. Wet.51, 793±795~1948!. 9 E. M. Lifshitz, ``The Theory of Molecular Attractive Forces between Sol- ids,'' Sov. Phys. JETP2, 73±83~1956!. 10 J. D. Jackson,Classical Electrodynamics~Wiley, New York, 1975!, 2nd ed., p. 395. 11 P. W. Milonni and P. L. Knight, ``Retardation in the resonant interaction of two identical atoms,'' Phys. Rev. A10, 1096±1108~1974!; ``Retarded interaction of two nonidentical atoms''ibid.,11, 1090±1092~1975!. 12 P. W. Milonni, ``Semiclassical and Quantum-Electrodynamical Approach in NonRelativistic Theory,'' Phys. Rep., Phys. Lett.25C, 1±81~1976!. 13 B. M. Axilrod and E. Teller, ``Interaction of the van der Waals Type Between Three Atoms,'' J. Chem. Phys.11, 299±300~1943!. 14 M. J. Sparnaay, ``Measurements of Attractive Forces between Flat

Plates,'' Physica~Amsterdam!24, 751±764~1958!.

15 S. K. Lamoreaux, ``Demonstration of the Casimir Force in the 0.6 to 6mm

Range,'' Phys. Rev. Lett.78, 5±8~1997!.

16 U. Mohideen and A. Roy, ``A precision measurement of the Casimir force from 0, 1 to 0, 9 mm,'' hep-ph/9805038.

PROPER USE OF THE SEMICOLON

With the war ended, Purcell could hardly suppress thoughts of returning to academia. The problem was, he had agreed to stay on and help grind out the Rad Lab Series. While never doubting the task's importance, he nevertheless found it hard to concentrate on writing when the long-suppressed world of research beckoned. It would have been bad enough just writing. But the stultifying bureaucracy associated with the series heightened his angst. Drafts had to be fed to a cadre of nitpicky English teachers who hammered away at the physicists' syntax and structure. They never let up. As the writing slipped further and further behind schedule, a ¯ood of memos arrived about editorial principles. One day Purcell received a two-page, single-spaced bulletin on the proper use of the semicolon... . Robert Buderi,The Invention that Changed the World~Simon and Schuster, New York, 1996!, p. 258.

349349Am. J. Phys., Vol. 67, No. 4, April 1999 Farina, Santos, and Tort


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