Van der Waals interaction between an atom with spherical plasma shell

# Van der Waals interaction between an atom with spherical plasma shell

Nail R. Khusnutdinov111e-mail: 7nail7@gmail.com Institute of Physics, Kazan Federal University, Kremlevskaya 18, Kazan, 420008, Russia
###### Abstract

The van der Waals interaction energy of an atom with infinitely thin sphere with finite conductivity is investigated in the framework of the hydrodynamic approach. Thin sphere models the fullerene. 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 interaction energy for a single atom is obtained from this expression in the limit of the rare 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. Numerically, the interaction energy is obtained to be for hydrogen atom placed on the surface of the sphere with parameters of fullerene . We show also that the polarizability of fullerene is merely cube of its radius.

###### pacs:
73.22.-f, 34.50.Dy, 12.20.Ds

## I Introduction

The general theory of the van der Waals force was developed by Lifshits in Refs. Lif56 (); 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 CasPol48 (). 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 Mil01 (); BorMohMos01 (); BorKliMohMos09 (); KliMohMos09 ().

The van der Waals force is very important for interaction of graphene (graphite layers) with bodies BogOveHylLunBruJen00 (); HulHylLun01 (); RydDioJacShrHylSimLanLun03 (); JunGarDobGod04 (); KleiHylSch05 (); DobWhiRub06 (); BorGeyKliMos06 (); BorFiaGitVas09 () and microparticles DinNakKas04 (); BonLam04 (); BonLam05 (); BlaKliMos07 (); ChuFedKliYur10 (). An understanding of the mechanisms of molecule-nanostructure interaction is of importance for the problem of hydrogen storage in carbon nanostructures DilJonBekKiaBetHeb97 (). The microscopic mechanisms underlying the absorption phenomenon remain unclear (see, for example review Nec06 ()).

In the present paper we use model of the fullerene in terms of the two dimensional free electron gas Fet73 () which is usually called as hydrodynamical model. This model was applied and developed for the molecule in Refs. Bar04 (); Bar05 (), for flat plasma sheet in Ref. BorPirNes05 () and for spherical plasma surface in Ref. BorKhu08 (). 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. In the Ref. BorKhu08 () it was shown 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 Boy68 () is recovered in the limit .

At the same time it is well known GeiNov07 () 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. Sem84 (). 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 in Ref. BorFiaGitVas09 () and recently in Ref. ChuFedKliYur10 () for Casimir-Polder interaction energy between graphene and H, He and Na atoms.

It was shown that the Casimir energy for large distance between graphene plate and perfect conductor plane BorFiaGitVas09 () 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 ChuFedKliYur10 () 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.

There is another approach for van der Waals interaction based on the density-functional theory HohKoh64 (); KohSha65 () and the local-density approximation KohSha65 () which has proved to be a very useful tool for calculating the ground-state properties of atoms, molecules, and solids. In framework of density-functional theory a number of studies of van der Waals interaction has been made RapAsh91 (); AndLanLun96 (); HulAndLun96 (); DobDin96 (); KohMaiMak98 (); DobWan00 (); RydDioJacSchHylSimLanLun03 (); DioRydSchLanLun04 (); DobWhiRub06 (); LeeMurKonLunLan10 (). The main problem in this theory is to find approximations for the exchange-correlation energy. The density-functional theory describes cohesion, bonds, structures, and other properties very well for dense molecules and materials. The theory fails to describe the interactions at sparse electron densities. The solution of this point by introducing the non-local correlations may be found in Refs. RydDioJacSchHylSimLanLun03 (); DioRydSchLanLun04 (); LeeMurKonLunLan10 ().

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. Lif56 (); 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 .

The paper is organized as follows. In Sec. II we derive the boundary conditions for electromagnetic field on the infinitely thin conductive sphere as well as on the boundary of the cavity. Section III is devoted to the construction of solutions satisfying the boundary conditions. The expression for the van der Waals energy is found in Sec. IV and it is analyzed in the limits of infinite radius of the sphere and for short and large distances between atom and sphere. Section V contains the numerical calculations of the interaction energy between hydrogen atom and the infinitely thin sphere with parameters of the fullerene . In the last section VI we discuss results obtained.

## Ii Maxwell’s equations and matching conditions

Let us consider a conductive infinitely thin sphere with radius in vacuum spherical cavity with radius which is inside the dielectric media with parameters (see fig. 1).

We have two concentric spheres and we should consider the boundary conditions on two spherical boundaries.

I. First of all let us consider a spherical boundary with radius . Inside the sphere we have vacuum, and outside – the dielectric media with . Assuming the spherical symmetry, the electromagnetic field is factorized for two independent polarizations usually called as TE and TM modes. The Maxwell equations with oscillatory time dependence read

 rot \!E−iωcB = 0, div \!B=0, (1a) rot \!H+iωcD = 0, div \!D=0, (1b)

where we should use the material equations and . For TE mode to be obtained we express from the first equation (1a) and substitute it into the second Eq. (1b)

 B\scriptsize TE=−icωrot \!E\scriptsize TE, △E\scriptsize TE−ω2c2μεE\scriptsize TE=0. (2)

For TM mode to be obtained we express from second equation (1b) and substitute it into the first Eq. (1a)

 E\scriptsize TM=icωμε% rot \!B\scriptsize TM, △B% \scriptsize TM−ω2c2μεB% \scriptsize TM=0. (3)

We next expand solutions over spherical functions, , and obtain the following expressions for TE and TM polarizations,

 B\scriptsize TElm = −icωrot \!E% \scriptsize TElm, E\scriptsize TElm=f(kr)LYlm, (4) E\scriptsize TMlm = icωμεrot \!B\scriptsize TMlm, B\scriptsize TMlm=f(kr)LYlm, (5)

where . In the standard spherical vector basis, , we obtain in manifest form the modes we need

 E\scriptsize TElm = (0,ifsinθ∂φYlm,−if∂θYlm), B\scriptsize TElm = (cfωrl(l+1)Ylm,c(rf)′ωr∂θYlm,c(rf)′ωrsinθ∂φYlm), B\scriptsize TMlm = (0,ifsinθ∂φYlm,−if∂θYlm), E\scriptsize TMlm = −cεμ(fωrl(l+1)Ylm,(rf)′ωr∂θYlm,(rf)′ωrsinθ∂φYlm), (6)

where the function obeys the following radial equation

 f′′+2rf′+(ω2c2εμ−l(l+1)r2)f=0. (7)

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, (8a) n×[H2−H1]L = 0, n×[E2−E1]L=0, (8b)

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 .

II. The electromagnetic fields given infinitely thin conductive surface in vacuum was considered by Fetter in Ref. Fet73 (). The applications of this model for vacuum fluctuations of field see in Refs. Bar04 (); Bar05 (); BorPirNes05 (); BorKhu08 (). The electrons of conductivity on the sphere produce current and the Maxwell equations read

 rot \!E−iωcH = 0, div \!H=0, (9a) rot \!H+iωcE = 4πJ, div \!E=4πρ, (9b)

where , . Taking into account the equation of continuity and the Newton equations we obtain the following expressions for density and of charge and current on the boundary:

 (10)

where the superscripts indicate the vector components parallel to the surface ; and are the charge and mass of electron, and is a surface density of charge.

As a consequence of the charge and current obtained above, the boundary conditions on the sphere with read

 n⋅[H2−H1]R = 0, n⋅[E2−E1]R=Ωk2∇||⋅E||, (11a) n×[H2−H1]R = −iΩkn×E||, n×[E2−E1]R=0, (11b)

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 (9) in vacuum with zero right hand side and all information about sphere will be encoded in boundary conditions (11). An interesting treatment of this boundary condition is in Ref. Vas09 ().

## Iii The solution of the matching conditions

Let us represent the radial function in the following way

 f=⎧⎪ ⎪⎨⎪ ⎪⎩fin=ainjl(kr),rL (12)

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

In this case the matching conditions (8) and (11) in manifest form read

 [rfout−rfin]R = 0, [(rfout)′r−(rfin)′r−Ω(rfin)]R = 0, [rfout−rfε]L = 0, (13) [(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, (14) [(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

 1√μεH(zε)Ψ′\scriptsize TE−1μH′(zε)Ψ%TE = 0, (15a) −1√μεH(zε)Ψ′\scriptsize TM+1εH′(zε)Ψ\scriptsize TM = 0, (15b)

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)], (16a) Ψ\scriptsize TM(z) = J(z)+ΩkJ′(x)[J′(x)Y(z)−J(z)Y′(x)]. (16b)

Here are the Riccati-Bessel functions, and . Therefore the functions we need (see next section) to obtain the spectrum of the energy read (we set )

 Σ\scriptsize TE = H′(zε)Ψ\scriptsize TE−1√εH(zε)Ψ′\scriptsize TE, (17a) Σ\scriptsize TM = H(zε)Ψ′TM−1√εH′(zε)Ψ\scriptsize TM. (17b)

For , the result obtained in the Ref. BorKhu08 () is recovered

 Σ\scriptsize TE = i{1−ΩikJ(x)H(x)}=if% \scriptsize TE(k), (18a) Σ\scriptsize TM = −i{1−ΩikJ′(x)H′(x)}=−if\scriptsize TM(k), (18b)

for real value of , and for imaginary axis we obtain from above expressions the Jost functions in imaginary axis:

 Σ\scriptsize TE = i{1+Ωksl(x)el(x)}=if% \scriptsize TE(ik), (19a) Σ\scriptsize TM = −i{1−Ωks′l(x)e′l(x)}=−if\scriptsize TM(ik), (19b)

because , and , where

 sl(x)=√πx2Il+1/2(x), el(x)=√2xπKl+1/2(x) (20)

are the Riccatti-Bessel spherical functions of the second kind. For the problem with to be avoided we multiply for

 Σ\scriptsize TE = −i{H′(zε)Ψ\scriptsize TE−1√εH(zε)Ψ′% \scriptsize TE}, (21a) Σ\scriptsize TM = −iz2{H(zε)Ψ′% \scriptsize TM−1√εH′(zε)Ψ\scriptsize TM}. (21b)

On the imaginary axis we obtain

 Σ\scriptsize TE = 1√εel(zε)Φ′\scriptsize TE−e′l(zε)Φ% \scriptsize TE, (22a) Σ\scriptsize TM = z2{el(zε)Φ′% \scriptsize TM−1√εe′l(zε)Φ\scriptsize TM}, (22b) Φ\scriptsize TE = sl(z)+Qxsl(x)[sl(z)el(x)−sl(x)el(z)], (22c) Φ\scriptsize TM = sl(z)−Qxs′l(x)[sl(z)e′l(x)−s′l(x)el(z)], (22d)

where , and . For we obtain

 Σ\scriptsize TE=f\scriptsize TE(ik), Σ\scriptsize TM=z2f\scriptsize TM(ik) (23)

in accordance with Ref. BorKhu08 ().

## Iv The energy

Within the limits of approach suggested in Ref. BorEliKirLes97 (), 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, (24) E\scriptsize TM(s) = −ℏccosπsπμ2s∞∑l=1ν∫∞0dkk1−2s∂klnΣ\scriptsize TM, (25)

where the integrand functions are given by Eqs. (22). The summations in these expressions begin with because for the electromagnetic modes (6) are zero.

The derivative of the regularized energy with respect to the distance () may be found by interchanging the derivative and summation with integration. In manifest form it reads

 ∂dE(s) = −ℏccosπsπμ2s∞∑l=1ν∫∞0dkk1−2s∂k{k(1−ε)√ε[G−1\scriptsize TE+G−1% \scriptsize TM]},

where

 G\scriptsize TE = 1√εΦ′% \scriptsize TEΦ\scriptsize TE−e′l(zε)el(zε)=Σ\scriptsize TEel(zε)Φ\scriptsize TE, G\scriptsize TM = −Φ′\scriptsize TMΦT\scriptsize TM−1√εe′l(zε)el(zε)Φ′% \scriptsize TMΦ\scriptsize TMe′l(zε)el(zε)+ν2−14z2√ε=−Σ\scriptsize TMz2[e′l(zε)Φ′\scriptsize TM,z+el(zε)Φ\scriptsize TMν2−14z2√ε].

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). (26)

From this expression it follows that

 Ea(s)=−limN→0∂dE(s)4πN(R+d)2, (27)

and in manifest form we obtain the interaction energy per unit atom

 Ea(s)=−ℏcμ2scosπsπ(R+d)2∞∑l=1ν∫∞0dkk1−2s∂k{kα(iω)G\scriptsize TE+kα(iω)G\scriptsize TM}, (28)

where

 G\scriptsize TE = Σ\scriptsize TEel(z)Φ% \scriptsize TE=f\scriptsize TE(ik)el(z)Φ%TE, G\scriptsize TM = −Σ\scriptsize TMz2[e′l(z)Φ′\scriptsize TM,z+el(z)Φ% \scriptsize TMν2−14z2]=−f% \scriptsize TM(ik)e′l(z)Φ′\scriptsize TM,z+el(z)Φ\scriptsize TMν2−14z2.

With definitions of the functions and we have the following relations

 Φ\scriptsize TE=sl(z)f\scriptsize TE(ik)−Ωks2l(x)el(z), Φ\scriptsize TM=sl(z)f\scriptsize TM(ik)+Ωks′2l(x)el(z).

Taking into consideration these expressions we express above formulas in slightly different form,

 G−1\scriptsize TE = el(z)sl(z)−Qxs2l(x)e2l(z)f\scriptsize TE(ik), G−1\scriptsize TM =

by separating the terms which have no dependence on the parameter . By virtue of the fact that the Casimir energy is zero for an atom in vacuum () without boundaries, we subtract the terms with and define the interaction energy by the following relation

 EΩ=lims→0{Ea(s)−limΩ→0Ea(s)}. (29)

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

 EΩ=−ℏcΩπ(R+d)2∞∑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)⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭, (30)

where the Jost functions on the imaginary axes read

 f\scriptsize TE(ik) = 1+Ωksl(x)el(x), (31) f\scriptsize TM(ik) = 1−Ωks′l(x)e′l(x). (32)

To perform computations one needs an expression for the atomic dynamic polarizabilities of hydrogen. In was shown in Ref. JohEps67 () that the precise expression for the atomic dynamic polarizability of hydrogen is given by the -oscillator formula

 α(iω)=10∑k=1g2k,aω2+ω2k,a, (33)

where are the oscillator strengths and are the eigenfrequencies. All these parameters may be found in Refs. JohEps67 (); BlaKliMos05 (). It was shown in Ref. BlaKliMos05 () that the polarizabilities can be represented with sufficient precision in the framework of the single-oscillator model

 α(iω)=g2aω2+ω2a, (34)

where () and for hydrogen atom.

One can see from the expression (30) that the energy is negative because the integrand is positive for arbitrary radius of the sphere, the wave number and arbitrary position of atom. The same observation was noted in Ref. JheKim95 () for ideal case. Let us consider different limits.

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

 EB=−ℏcπ(R+d)2∞∑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)⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭. (35)

2) The limit of infinite radius of sphere, , with fixed distance, , between the surface of sphere and an atom requires more machinery. One cannot merely interchange the limit and summation and integration in above expressions 30 and 35 because in this case the integrand has no dependence on the and the series is divergent. Indeed, in the limit of infinite radius of sphere

 2sl(x)el(z)|R→∞ = +e−kd, 2sl(x)el(x)|R→∞=+1, 2s′l(x)e′l(z)|R→∞ = −e−kd, 2s′l(x)e′l(x)|R→∞=−1, 2s′l(x)el(z)|R→∞ = +e−kd,

and the sum over is divergent,

 EΩ=−ℏcΩ2π(R+d)2∞∑l=1ν∫∞0dkα(iω)e−2kd1+Ω2k→∞. (36)

In order to obtain the correct expression for the energy in the limit we change the variable of integration in Eqs. 30 and 35

 EΩ = −ℏcΩπ(R+d)2∞∑l=1ν2∫∞0dkα(iων)⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩s2l(νx)e2l(νz)f\scriptsize TE(ikν)+s′2l(νx)e′2l(νz)+s′2l(νx)e2l(νz)1−14ν2z2f\scriptsize TM(ikν)⎫⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪⎭, (37) EB = −ℏcπ(R+d)2∞∑l=1ν3∫∞0kdkα(iων)⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩s2l(νx)e2l(νz)sl(νx)el(νx)−s′2l(νx)e′2l(νz)+s′2l(νx)e2l(νz)1−14ν2z2s′l(νx)e′l(νx)⎫⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪⎭, (38)

and use the uniform expansion for Bessel functions (see Ref. AbrSte70 ()). We obtain the following expressions

 EΩ = −ℏcΩπ(R+d)2∞∑l=1ν2∫∞0dkα(iων)e−2ν[η(z)−η(x)]{xzt(x)t(z)4w+1+t2(z)4pxzt(x)t(z)+…}, (39) EB = −ℏcπ(R+d)2∞∑l=1ν3∫∞0dkkα(iων)e−2ν[η(z)−η(x)]{zt(z)2+1+t2(z)2zt(z)+…}, (40)

where , , , and . 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), (41)

where and we changed variable . Here the single-oscillator model for polarizability (34) was taken into account.

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. (42)

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

 E=−3ℏcα(0)8πd4S, (43)

where

 S=13∫∞0dte−t⎧⎪⎨⎪⎩1+t1+t24v2+t(1+t24v2)2⎫⎪⎬⎪⎭, (44)

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, (45)

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, (46)

as should be the case. The plot of the as function of variable is shown in Fig. 2.

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. (30). 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, (47a) SΩ = R3d3