THE CASIMIR-POLDER INTERACTION AN ATOM WITH SPHERICAL SHELL

# The Casimir-Polder Interaction an Atom With Spherical Shell

NAIL KHUSNUTDINOV Institute of Physics, Kazan Federal University, Kremlevskaya 18
Kazan, 420008, Russia
7nail7@gmail.com
23 June 2011
###### Abstract

The Casimir-Polder and van der Waals interaction energy of an atom with infinitely thin sphere with finite conductivity is investigated in the framework of the hydrodynamic approach. We put the sphere into spherical cavity inside the infinite dielectric media, then calculate the energy of vacuum fluctuations in the context of the zeta-function approach. The energy for a single atom is obtained by rarefying media. The Casimir-Polder expression for an atom and plate is recovered in the limit of the infinite radius of the sphere. Assuming a finite radius of the sphere, the interaction energy of an atom falls down monotonic as third power of distance between atom and sphere for short distance and as seventh power for large distance from the sphere.

Casimir effect; Zeta-function; Zero-point energy.
\catchline{history}\ccode

PACS numbers: 73.22.-f, 34.50.Dy, 12.20.Ds

## 1 Introduction

The general theory of the van der Waals force was developed by Lifshits in Refs. \refciteLif56,LifPit80 in the framework of statistical physics. In the case of interaction between particle and plate it is commonly referred to as the Casimir-Polder force.[3] For small distance the potential of interaction is proportional to inverse third degree of distance from the plate. For large distance the retardation of the interaction is taken into account and the potential falls down as fourth degree of distance. The last achievements in Casimir effect have been discussed in great depth in books and reviews.[4]\cdash[6]

The van der Waals force is very important for interaction of graphene (graphite layers) with bodies[7]\cdash[10] and microparticles.[11]\cdash[13] An understanding of the mechanisms of molecule-nanostructure interaction is of importance for the problem of hydrogen storage in carbon nanostructures.[14] The microscopic mechanisms underlying the absorption phenomenon remain unclear (see, for example Ref. \refciteNec06).

In the present paper we use model of the fullerene in terms of the two dimensional free electron gas[16] which is usually called as hydrodynamical model. This model was applied and developed for the molecule in Ref. \refciteBar04, for flat plasma sheet in Ref. \refciteBorPirNes05 and for spherical plasma surface in Ref. \refciteBorKhu08. In the framework of this model the conductive surface is considered as infinitely thin shell with the specific wave number , where is surface density of electrons and is the electron mass. Since the surface is infinitely thin, the information about the properties of the surface is encoded in the boundary conditions on the conductive surface which are different for TE and TM modes. It was shown[19] that the energy of the vacuum electromagnetic fluctuations for surface shaped as sphere has a maximum for radius of sphere approximately equal to the specific wavelength of the model . What this means is the Casimir force tries to enlarge sphere with radius larger then and it tries to reduce the sphere with radius larger then . The Boyer result[20] is recovered in the limit .

At the same time it is well known[21] that the energy of electrons in graphene has linear frequency dependence whereas in framework of the hydrodynamic model the energy of electrons is quadratic in the frequency. There is also another point that the electrons in the graphene have zero or very small effective mass. To describe correctly these unusual properties of electrons in graphene the Dirac fermion model was suggested in Ref. \refciteSem84. The electrons in this model are described by Dirac action with characteristic propagation velocity as Fermi velocity and very small mass gap . This model was applied for calculation of Casimir interaction energy between graphene plate and perfect conductor plane[10] and recently for Casimir-Polder interaction energy between graphene and H, He and Na atoms.[13]

It was shown that the Casimir energy for large distance between graphene plate and perfect conductor plane[10] is decreasing by one power of the separation a faster than for ideal conductors, that is as . If the mass of gap is zero at the beginning of calculations, , they obtained standard dependence . For the case of Casimir-Polder interaction energy between graphene and atoms[13] the hydrodynamic and the Dirac models give qualitatively different results. For the large separation the energy decreases with separation as which is a typical behavior of the atom-plate interaction at relativistic separations, but the coefficients are different. In the case of H, He and Na atoms, the hydrodynamic model gives times larger coefficient than the Dirac model. There is also interesting observation about mass gap parameter: the energy does not depend on the parameter for and therefore the limit is satisfied.

In the present paper the hydrodynamical model of fullerene is adopted – the infinitely thin sphere with radius in vacuum and finite conductivity. To obtain the van der Waals interaction energy between an atom and this sphere we use the following approach which is due to Lifshits (see Refs. \refciteLif56,LifPit80,BorGeyKliMos06,BlaKliMos07). We put the sphere inside the spherical vacuum cavity with radius which is inside the dielectric media with coefficients . Then we find the zero-point energy of this system by using the zeta-function regularization approach, and take the limit of the rared media with , where is the volume density of the atoms and is the polarizability of the unit atom. The interaction energy per unit atom which is situated from the sphere is found by simple formula

 Ea(s)=−limN→0∂dE(s)4πN(R+d)2,

where is the zeta-regularized energy with regularization parameter .

## 2 Matching Conditions For Two Cylinders

Let us consider a conductive infinitely thin sphere with radius in vacuum spherical cavity with radius which is inside the dielectric media with parameters . We have two concentric spheres and we should consider the boundary conditions on two spherical boundaries.

Assuming the spherical symmetry, the electromagnetic field is factorized for two independent polarizations usually called as TE and TM modes. The angular dependence is described by spherical functions and radial function subjects for radial equation

 f′′+2rf′+(ω2c2εμ−l(l+1)r2)f=0. (1)

The two independent solutions of this equation are the spherical Bessel functions , where .

At the boundary, , the matching conditions read

 n⋅[B2−B1]L = 0, n⋅[D2−D1]L=0, (2a) n×[H2−H1]L = 0, n×[E2−E1]L=0, (2b)

where is an unit normal to the sphere. We have to take into account also that inside the sphere and outside the sphere. The square brackets above denote the coincidence limit on the boundary .

The electromagnetic fields given infinitely thin conductive surface in vacuum was considered by Fetter.[16] The applications of this model for vacuum fluctuations of field see in Refs. \refciteBar04–\refciteBorKhu08. The boundary conditions on the sphere with read

 n⋅[H2−H1]R = 0, n⋅[E2−E1]R=Ωk2∇||⋅E||, (3a) n×[H2−H1]R = −iΩkn×E||, n×[E2−E1]R=0, (3b)

where and is a specific wave number on the sphere. Because of the fact that the sphere is infinitely thin we may consider the Maxwell equations in vacuum with zero right hand side and all information about sphere will be encoded in boundary conditions (3). An interesting treatment of this boundary condition is in Ref. \refciteVas09.

## 3 The Solution of the Matching Conditions

Let us represent the radial function in the following way

 f=⎧⎪ ⎪⎨⎪ ⎪⎩fin=ainjl(kr),rL (4)

where and are the spherical Bessel functions and inside the sphere, and outside the sphere for .

In this case the matching conditions (2) and (3) in manifest form read

 [rfout−rfin]R = 0, [(rfout)′r−(rfin)′r−Ω(rfin)]R = 0, [rfout−rfε]L = 0, (5) [(rfout)′r−1μ(rfε)′r]L = 0,

for TE mode, and

 [(rfout)′r−(rfin)′r]R = 0, [(rfout)−(rfin)+Ωk2(rfin)′r]R = 0, [rfout−1μrfε]L = 0, (6) [(rfout)′r−1με(rfε)′r]L = 0,

for TM mode. The solutions of these equations exist if and only if the following equations are satisfied ()

 Σ\scriptsize TE = H′(zε)Ψ\scriptsize TE−1√εH(zε)Ψ′\scriptsize TE=0, (7a) Σ\scriptsize TM = z2{H(zε)Ψ′TM−1√εH′(zε)Ψ\scriptsize TM}=0, (7b)

where , ; the prime is derivative with respect the argument, and

 Ψ\scriptsize TE(z) = J(z)+ΩkJ(x)[J(x)Y(z)−J(z)Y(x)], (8a) Ψ\scriptsize TM(z) = J(z)+ΩkJ′(x)[J′(x)Y(z)−J(z)Y′(x)]. (8b)

Here are the Riccati-Bessel functions, and . For , the result obtained in the Ref. \refciteBorKhu08 is recovered

 Σ\scriptsize TE = i{1−ΩikJ(x)H(x)}=if% \scriptsize TE(k), (9a) Σ\scriptsize TM = (9b)

for real value of .

On the imaginary axis we obtain

 Σ\scriptsize TE = 1√εel(zε)Φ′\scriptsize TE−e′l(zε)Φ% \scriptsize TE, (10a) Σ\scriptsize TM = z2{el(zε)Φ′% \scriptsize TM−1√εe′l(zε)Φ\scriptsize TM}, (10b) Φ\scriptsize TE = sl(z)+Qxsl(x)[sl(z)el(x)−sl(x)el(z)], (10c) Φ\scriptsize TM = sl(z)−Qxs′l(x)[sl(z)e′l(x)−s′l(x)el(z)], (10d)

where , , and

 sl(x)=√πx2Il+1/2(x), el(x)=√2xπKl+1/2(x) (11)

are the Riccatti-Bessel spherical functions of the second kind. For we obtain

 Σ\scriptsize TE=f\scriptsize TE(ik), Σ\scriptsize TM=z2f\scriptsize TM(ik) (12)

in accordance with Ref. \refciteBorKhu08.

## 4 The Energy

Within the limits of approach,[24] the expressions for TE and TM contributions in regularized zero-point energy read ()

 E\scriptsize TE(s) = −ℏccosπsπμ2s∞∑l=1ν∫∞0dkk1−2s∂klnΣ\scriptsize TE, (13) E\scriptsize TM(s) = −ℏccosπsπμ2s∞∑l=1ν∫∞0dkk1−2s∂klnΣ\scriptsize TM, (14)

where the integrand functions are given by Eqs. (10).

Let us consider now the rared media with , where is polarizability of the atom and the density of the dielectric matter . In this case the Casimir energy is expressed in terms the energy per unit atom by relation

 E(s)=N∫∞dEa(s)4π(R+r)2dr+O(N2). (15)

From this expression it follows that

 Ea(s)=−limN→0∂dE(s)4πN(R+d)2. (16)

By virtue of the fact that the Casimir energy is zero for an atom in vacuum () without boundaries, we define the interaction energy by the following relation

 EΩ=lims→0{Ea(s)−limΩ→0Ea(s)}. (17)

With this definition we integrate by part over and arrive with the final formula ()

 EΩ=−ℏcΩπL2∞∑l=1ν∞∫0dkα(iω)⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩s2l(x)e2l(z)f\scriptsize TE(ik)+s′2l(x)e′2l(z)+s′2l(x)e2l(z)ν2−14z2f\scriptsize TM(ik)⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭, (18)

where the Jost functions on the imaginary axes read

 f\scriptsize TE(ik) = 1+Ωksl(x)el(x), (19) f\scriptsize TM(ik) = 1−Ωks′l(x)e′l(x). (20)

To perform computations one needs an expression for the atomic dynamic polarizabilities of hydrogen. It was shown[25] that the polarizabilities can be represented with sufficient precision in the framework of the single-oscillator model

 α(iω)=g2aω2+ω2a, (21)

where () and for hydrogen atom.

Let us consider different limits.

1) In the limit of perfect conductivity, , which we call the Boyer limit, we obtain

 EB=−ℏcπL2∞∑l=1ν∫∞0dkkα(iω)⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩s2l(x)e2l(z)sl(x)el(x)−s′2l(x)e′2l(z)+s′2l(x)e2l(z)ν2−14z2s′l(x)e′l(x)⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭. (22)

2) The limit of infinite radius of sphere, , with fixed distance, , between the surface of sphere and an atom requires more machinery. In this case we change the variable of integration in Eqs. (18) and (22) and use the uniform expansion for Bessel functions.[26] In the limit of , the integrands in above both expressions have the same form and the main contribution to the energy comes from the first term of uniform expansion,

 E=−limR→∞ℏcg2πc2(R+d)2∞∑l=1ν3∫∞0dyyy2ν2+q2e−2ν[η(u)−η(y)]ut(u), (23)

where , and .

Next, the sum over we represent in the following integral

 ∞∑l=1ν3e−2νδy2ν2+q2a=14qay∫∞027+17e−2(t+δ)+5e−4(t+δ)−e−6(t+δ)e3(t+δ)(e−2(t+δ)−1)4sin2qatydt. (24)

Assuming this expression we interchange the limit and integrals over and and obtain

 E=−3ℏcα(0)8πd4S, (25)

where

 S=13∫∞0dte−t⎧⎪⎨⎪⎩1+t1+t24v2+t(1+t24v2)2⎫⎪⎬⎪⎭, (26)

and . Let us consider large distance, , between the plate (sphere of infinite radius) and an atom, . In the limit of we obtain that and therefore the Casimir-Polder energy,

 E=−3ℏcα(0)8πd4, (27)

is recovered. For small distances, , we change the variable and take the limit of . In this case we obtain that and the energy has the form ,

 E=−ℏcα(0)ka8d3, (28)

as should be the case. The plot of the as function of variable is shown in Fig. 1.

3) Let us analyze the energy for large () and small () distances between the sphere and an atom for finite and . In the case of large distance, , of an atom from the shell we use Eq. (18). We change integrand variable , next take limit , and then we take the integral over . The main contribution comes from the first term with :

 EΩ ≈ −3ℏcα(0)8πd4SΩ, (29a) SΩ = R3d3{7Q3(3+Q)+463F(a)}, (29b) F(a) = 8a223∫∞0y4+2y3+5y2+6y+33y2+2a2e−2ydy, (29c)

where . The first term in above expression (29b) comes from TE mode and second – from TM polarization. The function increases monotonically from zero for small () to unity for large (). In the case of the function . Therefore, in the limit of , the energy as should be the case.

Assuming a finite conductivity, , and large distance we obtain that

 SΩ=R3d3{7Q3(3+Q)+463}

and we arrive with expression

 EΩ≈−ℏcα(0)R38π(3+Q)d7(53Q+138). (30)

Taking into account the Casimir-Polder interaction energy of two atoms with polarizations and ,

 E=−234πℏcα(0)αf(0)d7, (31)

we observe that the sphere with finite conductivity has static polarizability

 αf=53Q+13846Q+138R3. (32)

For small distances we obtain that

 E=−ℏcα(0)ka8d3 (33)

as should be the case, because close to the sphere we observe flat surface.

## 5 Numerical Analysis

For simplicity we extract as a factor the Casimir-Polder expression for the interaction energy of an atom with plate,

 EΩ,B=−3ℏcα(0)8πd4SΩ, (34)

and we will numerically calculate the dimensionless quantity .

Let us consider the interaction energy between hydrogen atom and molecule . For this molecule[17] we have: , and . The polarizability of hydrogen atom within the single-oscillator model reads[27, 9, 12] () and where . Therefore, .

Taking into consideration all the numerical values of parameters we represent the energy for this system in the following form

 EΩ(eV)=−0.0156d4(nm)SΩ(qa,r), (35)

where the energy is measured in and the distance is measured in nanometres. The numerical simulations for the function are shown in Fig. 2 and the energy in Fig. 3.

The radius of the hydrogen atom is . For this minimal distance, , we have numerically . In the case of plate with hydrogen atom we obtain .

For large distances we obtain from Eq. (30)

 EΩ(eV)≈−0.0095d7(nm). (36)

This expression approximates the exact one with error starting with distance .

## 6 Conclusion

In the foregoing, we have obtained the analytic expression for the Casimir-Polder (van der Waals) energy for a system which contains an atom or microparticle and infinitely thin sphere with finite conductivity which models a fullerene. We used the zeta-regularization approach and for renormalization we used a simple physically reasonable condition – the energy should be zero for an atom alone without a sphere. The conductive sphere with radius is characterized by the only parameter with dimension of wave number, where is the surface density of electrons. The limit corresponds to the ideal case considered by Boyer.[20] The microparticle is characterized by the only parameter, polarizability .

The expression obtained reproduces in the limit the Casimir-Polder result for an atom and plate (see Eqs. (25)-(28)). For small distances we have dependence and far from the plate we obtain due to retardation. For finite radius of the sphere we have different behavior of the energy. Close to the sphere, and , we have the same dependence as in the Casimir-Polder case and far from the sphere we obtained dependence given in Eq. (30). This expression is valid for and .

Application to the molecule with hydrogen atom is plotted in Fig. 3. For closest distance atom from the fullerene, which is radius of hydrogen atom , the energy is which is two times smaller then for the case of hydrogen atom with plate. Away from the fullerene (in fact larger then ) the energy falls down as (see Eq. (36)) which is in three orders of magnitude faster then for the Casimir-Polder case. This dependence corresponds to the Casimir-Polder interaction atoms for large distance. Taking into account this analogy we obtain the polarizability of fullerene ()

 αf=53Q+13846Q+138R3≈R3=4⋅10−29m3.

This expression is close to that calculated in Ref. \refciteFowLazZan91 where the authors obtained .

## Acknowledgments

The author would like to thank V. Mostepanenko and G. Klimchitskaya for stimulation of these calculations and M. Bordag for discussions. This work was supported by the Russian Foundation for Basic Research Grant No. 11-02-01162-a.

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