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Temperature dependence of the Casimir force between real metals: problems and approach to their resolution

We investigate the Casimir force acting between real metals at nonzero temperature. It is shown that the zero-frequency term of Lifshitz formula has interpretation problem in the case of real metal described by Drude model. It happens because the scattering theory underlying Lifshitz formula is not well formulated when the dielectric permittivity takes account of dissipation. To give the zeroth term of Lifshitz formula the definite meaning different prescriptions were used recently by different authors with diversed results. These results are shown to be improper and in disagreement with experiment and the general physical requirements. We propose the new prescription which is a generalization of Schwinger, DeRaad and Milton recipe formulated earlier for ideal metals. On this base the detailed numerical and analytical computations of the temperature Casimir force are performed in configuration of two plane plates and a spherical lens (sphere) above a plate. The corrections due to nonzero temperature and finite conductivity found in the paper are in agreement with the limiting case of perfect metal and fit all experimental and theoretical requirements. Among other facts, the previous results obtained in frames of plasma model are confirmed. It appears they are the limiting case of Drude model computations when the relaxation parameter goes to zero. The comparison with the Casimir force acting between dielectric test bodies is made.


I INTRODUCTION
The H.B.G. Casimir effect [1] predicted more than fifty years ago is one of the most interesting manifestations of zero-point vacuum oscillations of quantized fields. The Casimir effect implies that there is some force acting between two uncharged bodies closely spaced in the vacuum. This effect is purely of quantum origins. There is no such force in classical physics.
Unique to the Casimir force is its strong dependence on shape, switching from attractive to repulsive as a function of the size, geometry and topology of the boundary. The force results from the alteration by the boundaries of the zero-point electromagnetic energy that pervades all of space as predicted by quantum field theory. Alternatively, the Casimir force can be described as the retarded electromagnetic interaction of atomic and molecular dipoles and was extended to forces between macroscopic dielectric bodies characterized by some dielectric constant [2].
In recent years the Casimir effect has attracted much attention because of numerous applications in quantum field theory, atomic physics, condensed matter physics, gravitation and cosmology, and mathematical physics (see monographs [3,4,5,6] and references therein). New precision experiments have been performed on measuring the Casimir force between metallic surfaces [7,8,9,10,11,12]. In Refs. [13,14,15] some promising applications of the Casimir effect were proposed for diagnostic in thin films and in nano electro-mechanical systems. Given the above reasons it is very important to understand the Casimir force between real materials including the effect of such influential factors as surface roughness, finite conductivity, and nonzero temperature.
Finite conductivity corrections to the Casimir force have long been investigated. They were calculated using plasma model [16,17,18] up to the first order in the relative penetration depth of the zero-point electromagnetic oscillations into the metal. In [19] more exact results up to the second order were obtained, and in [20] -up to the fourth order. In Refs. [21,22] the finite conductivity corrections to the Casimir force were computed using tabulated optical data for the frequency-dependent complex refractive index. For all separations between the test bodies larger than the effective plasma wavelength of the metal under study the results of [20] and [21,22] are shown to be in good agreement. The effects of surface roughness in combination with finite conductivity were investigated in detail in [23] for the configuration of a sphere above a plate used in experiments [8,9,10,11]. Roughness contributions to the Casimir effect (including origination of a lateral force) has been treated recently in [24,25,26,27].
The action of nonzero temperature on the Casimir force between dielectric semispaces was taken into account in the Lifshitz theory [2,16]. For perfect metals at nonzero temperature the Casimir force was calculated in [28,29] within the limits of quantum field theory in terms of the free energy density of vacuum. There were apparent differences between the results of [2,16] and [28,29] which were resolved in [18]. As shown by Schwinger, DeRaad and Milton [18], to obtain the case of perfect conductor from the Lifshitz theory one must take the limit of infinite dielectric permittivity before putting the frequency equal to zero in the temperature sum. The results of [2,16] adapted for the case of perfect metal and of [28,29] are then in agreement.
The temperature corrections to the Casimir force turned out to be negligible in the experiments [8,9,10,11] where the measurements were performed in the separation range a < 1 µm. However, at a > 1 µm, as in [7], the temperature corrections make large contributions to the zero-temperature force between perfect conductors (e.g., for a = 5 µm the temperature correction in configuration of a spherical lens above a plate exceeds the zero-temperature force [30]).
The increased accuracy of the Casimir force measurements invites further investigation of the temperature corrections in case of real metals. Although from a conceptual point of view the Lifshitz theory provides a way of obtaining all the required results, the problem here is worse than it was with the case of perfect metals. In [31] it was suggested to use the plasma model in order to describe the dielectric permittivity along the imaginary frequency axis in the Lifshitz formula for the Casimir force at nonzero temperature (note that in [16,17,18,19,20] the plasma model was applied for this purpose at zero temperature only). In Refs. [32,33] the detailed calculations of the temperature Casimir force were performed in framework of the Lifshitz theory and the plasma model. It was shown that the temperature corrections are negligible at small separations where the finite conductivity corrections are very important. By contrast, at large separations finite conductivity corrections can be ignored, whereas the temperature corrections play an important role. In [32,33] the transition region between these two asymptotic regimes was also investigated where the combined effect of nonzero temperature and finite conductivity is important and should be taken into account.
It is a common knowledge that at small frequencies the dielectric permittivity ε is proportional to ω −1 as is given by the Drude model. Because of this, the Drude model is favoured over the plasma model (which implies ε ∼ ω −2 ) when calculating the Casimir force at nonzero temperature. The first attempts to calculate the nonzero temperature Casimir force between real metals based on the Lifshitz formula and Drude model were undertaken in [33,34,35,36,37]. They have led to distinct and unexpected results.
It was found that the value of the transverse reflection coefficient r 2 of the electromagnetic oscillations at zero frequency becomes indefinite when one describes the boundary made of real metal using the Drude model. In [34,35] the value r 2 = 0 has been adopted. This assumption leads to a nonphysical conclusion that the asymptotic Casimir force at high temperature in the case of real metals is two times smaller than for the case of ideal metal (without regard to the particular value of the conductivity of real metal). Also, as a result of assumption, made in [34,35], there arise large negative temperature corrections at small separations which are linear in temperature. These corrections are not only unacceptable from the theoretical point of view but are in a conflict with experimental data [38].
The other authors [36,37] assumed the value |r 2 | = 1 at zero frequency using the prescription of [18] formulated for perfect metal and well known relation by Hagen and Rubens which is valid for real photons only [39,40].
This assumption also leads to nonphysical conclusions, i.e. to linear (although positive) temperature corrections at small separations and to the absence of any finite conductivity corrections to the Casimir force for real metals starting from the moderate separations of several micrometers regardless of metal quality (note that the same assumption was accepted in the latest version of [38]).
The situation was clarified in [33] where the discontinuity of the transverse reflection coefficient as a function of frequency and photon momentum was demonstrated in the case of real metal described by the Drude model.
According to [33] to clear away the ambiguity in the zero frequency term of Lifshitz formula arising due to this discontinuity it is necessary to use an alternative representation for it [18,28]. This representation gives the possibility to redefine the zero-frequency term of Lifshitz formula in order to assign it the definite meaning for real metals in accordance with the usual physical requirements. In [33], however, the values of the Casimir force including nonzero temperature and finite conductivity in the framework of the Drude model were not computed. Thus, up to now, there is no plausible qualitative information on the Casimir effect between real metals including the temperature corrections. The need for such investigation is apparent when the experimental and technological applications of the Casimir force mentioned above are considered. It is also important that recently, the Casimir effect has been used to obtain stronger constraints on the constants of long-range interactions (including corrections to the Newtonian gravitational law) predicted by the unified field theories, supersymmetry, supergravity and string theory [30,41,42,43,44]. The reliable theoretical values of the Casimir force at nonzero temperature between real metals and the extent of their agreement with experiment are of particular interest in order to obtain the strongest constraints.
In the present paper we propose a method which allows one to attribute a definite value to the term of Lifshitz formula at zero frequency for real metals as described by the Drude model. This method avoids the above mentioned contradictions and solves the problem in a physically consistent way. The detailed computations of the Casimir force between real metals at nonzero temperature are performed for the configuration of two plane parallel plates and a sphere (lens) above a plate in a wide separation range from 0.1 µm to 10 µm. It is shown that at small separations (low temperatures) the temperature corrections are small irrespective of whether the Drude or plasma model is used. In particular, no corrections to the force arise which would be linear in temperature, such as in [34,35,36,37]. At large separations (high temperatures) there is some difference between the finite conductivity corrections to the temperature Casimir force obtained with the plasma and Drude models, although the corrections themselves are rather small. With decreasing relaxation parameter the results from both models coincide. Our results at large separations join smoothly with the increase of conductivity with the asymptotic values obtained for the perfect metal (this is not the case in [34,35] where the asymptotic force for a real metal is two times smaller than for the perfect one). Also the non-physical results of [36,37], according to which at large separations the finite conductivity corrections are absent in the case of real metals, are shown to be in error. Below the nonzero temperature Casimir force between the dielectric bodies is also computed and the distinctions between the cases of metallic and dielectric bodies are discussed.
The obtained results are in agreement with the performed experiments. They establish the theoretical basis for new precise experiments on measuring the Casimir force.
The paper is organized as follows. In Sec. II the general formalism for the Casimir force between real metals at nonzero temperature is presented for the configuration of two plane parallel plates. Here special attention is paid to the indefinite character and discontinuity of the zeroth term of the Lifshitz formula in case of the Drude model. The representation of this term is given which is in accordance with the general physical requirements. In Sec. III the same is done for the configuration of a sphere (lens) above a plate. Sec. IV is devoted to the numerical and analytical computations of the temperature Casimir force for the configuration of two plane parallel plates.
Both the asymptotics of low and high temperatures are considered and also a transition region between them. Sec. V contains the results of analogous computations for the configuration of a sphere (lens) above a plate. In Sec. VI the case of dielectrical test bodies is considered and the Casimir force at nonzero temperature is found. In Sec. VII reader will find conclusions and discussion. Appendices I and II contain some details of the mathematical calculations.

II TEMPERATURE CASIMIR FORCE BETWEEN TWO PLANE PARALLEL PLATES
The original Lifshitz derivation [2,16] of the temperature Casimir force between two semispaces was based on the assumption that the dielectric materials can be considered as continuous media characterized by randomly fluctuating sources. The correlation function of these sources, situated at different points, is proportional to the δ-function of the radius-vector joining these points. The force per unit area acting upon one of the semispaces was  [45,46,47]. In this manner the allowed surface modes and a harmonic oscillator free energy of each mode can be determined. The total renormalized free energy is obtained by the summation over all modes and subtraction of the free energy of vacuum. The nonzero temperature Casimir force is finally given by the negative derivative of the renormalized free energy with respect to the distance between semispaces (at zero temperature this procedure is presented in detail in [22,47]).
The modern derivation of Lifshitz formula [48] is based on the temperature field theory in Matsubara formulation. In case of a static Casimir effect the system is in thermal equilibrium. To describe it at nonzero temperature one should take the Euclidean version of the field theory with a field periodic in the time variable within the interval β =h/(k B T ), where T is temperature, k B is the Boltzmann constant. To find the free energy in case of two semispaces the scattering problem on the z-axis perpendicular to the boundary planes is considered. An electromagnetic wave coming from the left in the dielectric is scattered on the air gap and there exist a transmitted and a reflected wave. Finally the free energy and the Casimir force is expressed in terms of the scattering coefficient on the imaginary axis. Calculating this coefficient for the problem of two semispaces (notated by an index ss) with a frequency dependent dielectric permittivity ε(ω) separated by a gap of width a one finally obtains the Casimir force in the form [48] where r 1,2 are the reflection coefficients with parallel (perpendicular) polarization respectively given by Here k ⊥ is the momentum component lying in the boundary planes, k ⊥ = |k ⊥ |, ω = iξ, and the following notations are used Introducing a new variable p according to where and the prime near the summation sign means that the zeroth term is taken with the coefficient 1/2.
Note that the representation of Eq. (5) for the nonzero temperature Casimir force has a disadvantage as the l = 0 term in it is the product of zero by a divergent integral. This is usually eliminated [49] by introducing the variable whereξ l is the dimensionless frequency. In terms of new variables Eq. (5) is where Both changes of variables (4) and (7) are, however, singular at l = 0. In fact (8) can be obtained from (1) by the regular change of variable Because of this, the equivalent representations of Lifshitz formula (1) and (8) are preferred as compared to (5).
To calculate the Casimir force at nonzero temperature between real metals one should substitute the appropriate values ε(iξ l ) into Eq. (1) or (8). In some frequency range ε(iξ l ) can be found by the use of optical tabulated data for the complex refractive index for the metal under consideration tabulated in, e.g., [50] (this was done in [21,22]). But in any case the optical data should be extrapolated outside the region where they are available in the tables to smaller and larger frequencies. This can be performed by the use of plasma model function or more exact Drude model one where ω p is the plasma frequency and γ is the relaxation frequency. For some metals, the model dielectric functions with appropriate values of ω p and γ can be reliably used throughout the whole spectrum. Calculations of this type were performed in [32,33,34,35,36,37] with different results. As shown below, the reason for this is the discontinuity of the transverse reflection coefficient r 2 at zero frequency with respect to relaxation parameter which must be made continuous in a physically reasonable way in order to describe real metals by the Lifshitz formula.
If we put ξ l = 0 in Eqs. (2), (3) and use the plasma model dielectric function (11) the result is [32,33] In the case of the Drude model dielectric function (12) it holds which leads to [35] r 2 1 (0, k ⊥ ) = 1, r 2 2 (0, k ⊥ ) = 0 (15) for any k ⊥ = 0. Note that (15) is valid for arbitrary small γ and arbitrary large ω p . Thus the second equation from (15) is in contradiction with the limiting case of ideal metal r 2 2 (0, k ⊥ ) = 1 which follows from (13) in the limit ω p → ∞. The results obtained using (15) are in conflict with the known results for ideal metal [18,29]. What is more important, the second equation of (15) does not approach the second equation of (13) when the relaxation frequency γ goes to zero. In the limit γ → 0 one still has r 2 2 (0, k ⊥ ) = 0, not (13), although lim γ→0 ε D = ε p in accordance with Eqs. (11), (12). In Secs. IV, V the discontinuity of the zero frequency term of the Lifshitz formula with respect to the relaxation parameter will be discussed in detail when comparing the continuous modification of this term as suggested below. Exactly the same results, as in (13), (15), are obtained if one uses the variables (ξ, y) instead of (ξ, k ⊥ ) and the representation (8) (7) was performed in order to calculate the divergent integral in Eq. (5). We show below, that this postulate, though it works good for the perfect metal, is not justified for real metals of finite conductivity.
In actual truth, the transverse reflection coefficient r 2 from Eq. (2) or (9) has a discontinuity as a function of two continuous variables (ξ and y for instance) in the case of the Drude dielectric function [33]. If one puts ξ = 0 in (9) from the very beginning the result is r 2 2 (0, y) = 0 in agreement with (15). If, however, one approaches the point (ξ = 0, y = 0) along the directionξ = ky in the (ξ, y)-plane then r 2 2 (ξ, y) → 1. The distinguishing feature of the Drude model is that for an arbitrarily small y there exists k such that r 2 2 (ky, y) takes any value in between zero and unity. By contrast, in the case of plasma model both reflection coefficients are continuous, and r 2 2 (ky, y) does not depend on k but is determined only by the value of y.
Of even greater concern is that the transverse reflection coefficient at zero frequency is discontinuous with respect to the relaxation parameter γ.
As is seen from the above discussion, there is a serious unresolved issue concerning the value of the zero frequency term of the Lifshitz formula for real metals. In fact the scattering problem, which forms the basis of the Lifshitz formula, is meaningful only for nondissipative media (this is the case for plasma model or for dielectric materials). The case ε = ∞, as it is for metals at zero frequency, is especially complicated when the Drude model is used taking account of dissipation. The latter leads to the violation of the unitarity condition and thereby the zeroth term of Lifshitz formula becomes indefinite and must be redefined. For ideal metals of infinite conductivity this issue was resolved by the Schwinger, DeRaad, and Milton prescription [18] demanding that the limit ε → ∞ should be taken before setting ξ = 0 (which is equivalent to the use of Eq. (16)). Let us find out, in what way the problem of zero-frequency term of the Lifshitz formula can be solved for real Drude metals in a physically satisfactory way.
For this purpose let us use the representation of the Lifshitz formula in terms of continuous frequency variable. Such a representation was suggested in [28] and used in [18,29] for the other purposes. According to the Poisson then it follows We apply this formula to Eq. (8) using the identification whereξ l = τ l and are the even functions of l.
Note that in the representation (22) the l = 0 term gives the force at zero temperature. It is useful also to change the order of integration in Eq.
where t ≡ T ef f /T .
Let us isolate the zero-frequency term of the usual Lifshitz formula (8) with discrete frequencies in representation (23). In this a way it will be expressed in terms of the integrals with respect to continuous variablesξ, y.
For this purpose we write out separately the terms of Eq. (23) with l = 0, integrate all the other terms by parts with respect toξ, change the order of summation and integrations, and use the formula [51] ∞ l=1 sin(lzt) where A(z) is the integer portion of z. The result of these transformations is Here the first term is the zero-temperature force. The second and the third contributions in the right-hand side of Eq. (25) can be transformed using the definition of the function A(z) and representation of f ss in terms of the sum of parallel and transverse modes in accordance with (20). Taking into account that f (1) ss and its derivative are continuous functions (as distinct from f (2) ss ) one obtains finally after the cancellation of zero-temperature contribution (see Appendix I for details) In this equation all the terms with l ≥ 1 coincide with those in Eq. (8) and the term with l = 0 is given by The obtained representation for the zero-frequency term of the Lifshitz formula is well suited for solving the problem formulated above. In terms of dimensionless variables (ξ, y) the plasma and Drude dielectric functions take the form A completely different type of situation occurs in the double integral from the right-hand side of Eq. (27). In fact, by the use of (9), (20) one obtains where Substitution of ε = ε p from Eq. (28) into (29), (30) leads to so that in the case of the plasma model the double integral in (27) vanishes.
If, however, ε = ε D is substituted into (29), (30) one discovers a discontinuity inγ. Actually, as it follows from (29), for anyξ ≥ 0 and y = 0. On the other hand, To conclude, the presence of the double integral in the right-hand side of Eq. (27) renders it unsuitable for real metals described by the Drude model.
To remedy the situation let us notice that under the increase of ε it holds Consequently, the discontinuous term can be deleted by the use of Schwinger, DeRaad and Milton-type prescription [18] which is that the limit of dielectric permittivity to infinity must be performed before setting frequency to zero when considering the zero-frequency term of the Lifshitz formula. For ideal metals the use of such a prescription is necessary in order to achieve agreement between the Lifshitz formula for dielectrics and field theoretical results obtained for a perfect metal, i.e. for boundaries with the Dirichlet boundary condition. As it is advocated here, for real metals the analogous prescription must be used in a more restricted way to eliminate the discontinuity in the zero-frequency term.
With the above prescription the Casimir force acting between two plane parallel plates at nonzero temperature is given by

URATION OF A SPHERICAL LENS (SPHERE) ABOVE A PLATE
In this section the most important results of Sec. II are adapted for a lens (sphere) above a semispace (plate) which is the configuration most often used in experiments [7,8,9,10,11]. Although there is no fundamental derivation of the Casimir force acting between a lens and a plate the proximity force theorem [52] gives the possibility to express it in terms of the Casimir free energy density for the configuration of two plane parallel plates. This latter is given by [32,33,48] where the reflection coefficients and other notations are introduced in Eqs. (2), (3). Note that the Casimir force between plates from Eq. (1) can be obtained as F ss = −∂E ss /∂a.
According to the proximity force theorem [52] the Casimir force acting between a lens (sphere) of radius R and a plate is Index sl here stands for semispace-lens. This formula is valid with rather high accuracy of about a/R [53] which is usually a fraction of a percent.
Introducing the dimensionless variablesξ l and y from Eqs. (7), (10) we rewrite Eq. (37) in the form This result is in direct analogy with Eq. (8) for two plates. There is no need to use singular changes of variables as discussed in the previous section.
The term of Eq. (38) with l = 0 suffers exactly the same ambiguity as the zeroth term of Eq. (8) if one uses the Drude model (12) to describe the dependence of a dielectric permittivity on frequency. This ambiguity lies in the fact that the quantity r 2 2 (ξ, y) has a discontinuity at a point (0,0) as a function of two variables and is discontinuous also at the point of zero relaxation parameter (see the detailed discussion in Sec. II). Because of this, the zeroth term of Eq. (38), when applied to real metals, must be modified in the same way as it was done for configuration of two plane plates. For this purpose we apply the Poisson summation formula (18) and rewrite Eq. (38) in the form analogical to Eq. (22) where Evidently, the zeroth term of Eq. (39) gives us the Casimir force at zero temperature. The terms with l ≥ 1 represent the temperature correction.
After changing the order of integration in Eq. (39) one obtains Using Eq. (24) and performing exactly the same transformations as in Sec. II and Appendix I, the Casimir force between a plate and a lens takes the form Here all terms with l ≥ 1 are exactly the same as in Eq. (38), whereas the zeroth term is given by This representation of the zero-frequency term is in direct analogy with Eq. (27) for two plates. Like in (27), no discontinuity is contained in the single integral with respect to y. As to the double integral in (43), it is discontinuous with respect to the relaxation parameterγ at a pointγ = 0. In fact, it follows from Eq. (40) ∂f (2) sl (ξ, y) ∂ξ = 2r 2 (ξ, y) e y − r 2 2 (ξ, y) where the derivative of the transverse reflection coefficient was calculated in which means the discontinuity of the double integral in (43) as a function of γ.
This discontinuity leads to the nonphysical results (see Secs. IV, V). It can be deleted by taking the limit of the dielectric permittivity to infinity prior to setting the frequency equal to zero in accordance with prescription used in Sec. II. Then due to (30), (44) ∂f (2) sl (ξ, y) ∂ξ and the discontinuous term in (43) disappears.
Using this prescription, one finally obtains the expression for the Casimir force acting between a plate and a spherical lens (sphere) at nonzero temper- The calculations of the Casimir force on the base of Eq. (48) are contained in Sec. V. The obtained results are compared with those obtained without the use of the above prescription [34,35] or using another prescription [36,37].

IV COMPUTATIONS OF THE CASIMIR FORCE BE-TWEEN TWO PLATES MADE OF REAL METAL
We start the numerical computation of the Casimir force with Eq. (35) using the Drude and plasma dielectric functions (12). For the sake of definiteness, we consider the case of aluminum whose plasma frequency and relaxation parameter can be chosen as follows [50] ω p ≈ 12.5 eV ≈ 1.9 × 10 16 rad/s, γ ≈ 0.063 eV ≈ 9.6 × 10 13 rad/s.
The most descriptive quantity illustrating the temperature correction to the Casimir force is Here the upper index f near the quantity F ss from Eq.  [34,35]. In accordance with this approach, as was discussed in Sec. II, one should substitute f (2) ss (0, y) = 0 instead of f (2) ss (y, y) into Eq. (35). As is seen from Fig. 1 (curve 1), in the case of real metals the relative temperature correction is rather small at small separations and monotonically increases with the increase of separation remaining positive as it should be from general thermodynamical considerations. In contrast to this, the temperature correction, computed on the basis of [34,35], is negative in a wide range of separations and turns into zero at a separation a ≈ 6.3 µm.
For larger distances it becomes positive. At small distances it is rather large by the modulus and increases linearly with distance (temperature). This behavior is in radical contradiction with both the case of ideal metal and a metal described by the plasma model dielectric function considered in detail in [32,33]. At the same time the results obtained here are in good agreement with both limiting cases of an ideal metal and metal described by the plasma model.
Note that in the original paper [35] not δ T (F ) but the correction factor was plotted as a function of separation a, i.e. the ratio between the Casimir force for real metals computed at T 0 = 300 K and the Casimir force for ideal metals at zero temperature. According to the results of [32] in the case of plasma model this correction factor is approximately equal to the product of the correction factors through finite conductivity (at zero temperature) and through nonzero temperature (for the ideal metal). Thus the resulting negative temperature corrections, which are unacceptable from the thermodynamical point of view, were not clearly evident in [35].
In Table 1 the values of the relative temperature corrections are given at several distances between the plates in the case of Drude model (present paper, column 2), plasma model (present paper, column 3), for the ideal metal (column 4), computed on the base of [35] in frames of Drude model (column 5), and computed on the base of [36,37] in frames of Drude and plasma models (columns 6 and 7 respectively). We remind that in [36,37] all computations were performed by Eq.  [36,37] (columns 6, 7) are also significantly different from our corresponding results of columns 2, 3. According to [36,37] there exist large temperature corrections at small separations which are linear in temperature. Now let us discuss the finite conductivity corrections to the Casimir force at nonzero temperature. They can be characterized by the quantity where, as above, at T 0 = 300 K. For comparison the corresponding results at zero temperature T 0 = 0 are given by the short-dashed curve. The long-dashed curve is obtained at T 0 = 300 K in the framework of the approach of [36,37]. In In contrast to this, the finite conductivity correction computed on the basis of [36,37] turns into zero at one and the same separation not only in different models but also for one and the same model with different values of parameters, i.e. regardless of the quality of metal under consideration. It is easily seen, that the separation value, at which the finite conductivity corrections of [36,37] vanish, is determined not by the conductivity properties of a metal but by the smallness of terms with l ≥ 1 in the Lifshitz formula which are of order exp(−2πT /T ef f ). This means that the anzats used in [36,37] to redefine the zeroth term of Lifshitz formula (see the above discussion in Sec. II) is unjustified.
It is of interest to consider the dependence of our results on the relaxation parameter γ of Drude model. In Fig. 4 the relative temperature correction computed by Eqs. (35), (50) is plotted with the decreasing of γ at a separation a = 2 µm, and at a temperature T 0 = 300 K (solid curve). In the same figure the dashed curve represents the result obtained using the plasma model. It is clearly seen that with the decrease of γ by 3 orders of magnitude the results computed in the Drude model join smoothly with the results of the plasma model as it should be from general considerations. Note that this is not the case in the approach of [34,35] where the results obtained with the Drude model do not join the results of the plasma model with decreasing γ.
The reason is that, different definitions of the zero-frequency term of Lifshitz formula are used in [34,35] and in the present paper (see Sec. II).
As to the papers [36,37], there is the smooth transition between the results obtained in Drude and plasma models. However, as was indicated in Expanding the quantity f ss defined in (20) up to the first order in powers of δ 0 /a one obtains where F 0 ss (a) = F 0 ss (a, 0) ≡ −π 2h c/(240a 4 ) is the zero-temperature Casimir force between ideal metals. The first summation in (54) is exactly the temperature correction in the case of ideal metals (see, e.g., [18]). The second summation takes into account the effect of finite conductivity combined with nonzero temperature.
In the limit of low temperatures T ≪ T ef f one has from (54) neglecting terms exponentially small in 2πT ef f /T [33] For δ 0 = 0 (perfect conductor) Eq. (55) turns into the well known result [18] demonstrating that the first nonzero temperature correction is of the fourth power in T /T ef f . For T = 0 the first-order finite conductivity correction to the Casimir force [17,18] is reproduced from (55). Note that the first correction of mixing finite conductivity and finite temperature is of order (T /T ef f ) 3 . More significantly, note that there are no temperature corrections of order (T /T ef f ) k with k ≤ 4 in the higher order conductivity correction terms (δ 0 /a) i from the second up to the sixth order [33].
In the Drude model the analytical results can be obtained in the hightemperature limit T ≫ T ef f . It is easily seen that at high temperatures (large separations) only one term of Eq. (35) with l = 0 contributes the result, the other terms with l ≥ 1 being exponentially small in the parameter 2πT /T ef f . The situation here is exactly the same as for the ideal metal [18].
As a consequence, Eq. (35) can be rewritten in the following form It is seen from Eq. (56) that in the high-temperature limit only the perpendicular reflection coefficient r 2 gives rise to the finite conductivity corrections to the Casimir force (the same is valid in plasma model also).
After the straightforward calculations up to the first orders in small parameters δ 0 /a, γ/ω p (see the details in Appendix II) the result is obtained, where I 1 is a function slowly depending on the effective relaxation parameter (space separation) and given by At high temperatures, which are considered here, the Casimir force acting between the ideal metals is where ζ(3) ≈ 1.202 is the Riemann zeta function.
With the above value of relaxation frequency for Al one has I 1 ≈ 1.16 at a = 5 µm and I 1 ≈ 0.99 at a = 10 µm. In Appendix II (Fig. 8) the functional dependence of I 1 onγ is plotted. As is seen from (57), the high-temperature Casimir force depends on both plasma frequency and relaxation frequency. Note that in the framework of the approach used in [34,35] the high temperature Casimir force between real metals is given by F ss (a) = F 0 ss (a, T )/2 (see Eq. (59)), i.e. two times smaller than for ideal metal, regardless of the conductivity properties of a real metal used. As to the approach used in [36,37], the asymptotic behavior at high temperatures coincides with Eq. (59), i.e. it is the same as the case for ideal metals in both plasma and Drude models. Once again, the real properties of a metal do not influence the result.

MADE OF REAL METAL
The configuration of a spherical lens (or a sphere) above a semispace (plate) was found to be the most suitable for the precision measurements of the Casimir force [7,8,9,10,11,12]. In these experiments the finite conductivity corrections have been demonstrated [8,9,10,11,12] and the sensitivity for the detection of the temperature corrections is close to being achieved [7]. We start with the relative temperature correction where large by modulus corrections linear in temperature to the ideal zerotemperature Casimir force are present. According to [38] such corrections are in contradiction with the experimental data of [7]. On the basis of our Eq. (48) such corrections do not arise.
In Table 2  The results of columns 6, 7, although positive, also correspond to the presence of linear temperature corrections at small separations.
The relative finite conductivity correction at a temperature T 0 can be described by In Fig. 6 this quantity is plotted as a function of separation in the case of the Drude model at T 0 = 300 K (solid curve is our result computed by Eq. (48), the long-dashed curve is computed on the basis of [36,37]). The dependence of the conductivity correction on separation at T 0 = 0 is shown by the shortdashed curve. In Fig. 7 the analogous results computed using the plasma model are presented. As with the case of two plane plates (Sec. IV), Drude model leads to larger finite conductivity corrections than the plasma model.
At separations larger than 4 µm the conductivity correction computed using [36,37] turns into zero with both Drude and plasma models (the long-dashed curves in Figs. 6, 7). Once more, this property is determined by the artificial modification of the zeroth term of Lifshitz formula used in [36,37] and does not depend on the particular characteristics of a real metal.
In the same way as for two plane plates our results for a lens above a plate, computed using the Drude model, join smoothly when γ → 0 with the results computed with the plasma model. This is not the case in the approach of [34,35].
We come now to the perturbative analytical results which can be obtained for the configuration of a lens above a plate. Here, the plasma model can be used. Introducing the new variable v =ξ/y instead ofξ one obtains The expansion of f sl up to first order in the small parameter δ 0 /a is f sl (v, y) = 2 ln 1 − e −y + 2 y e y − 1 Substitution of Eq. (63) into Eq. (60) leads to result . For the case of low temperatures T ≪ T ef f Eq. (64) leads to [33] F sl (a) ≈ F 0 sl (a) The corrections to (65) are exponentially small in the parameter 2πT ef f /T .
It is seen that for the perfectly conducting lens and plate the first nonzero After some transformations (see Appendix II) one arrives at the result where I 2 is defined by The high-temperature Casimir force acting between the lens and the plate made of ideal metal is For example, I 2 ≈ 0.519 at a = 5 µm and I 1 ≈ 0.434 at a = 10 µm (using the data of (49) for Al). The dependence of I 2 onγ is plotted in Appendix II (Fig. 8). The asymptotic results of Eq. (67) coincide with the results of numerical computations for a > 5 µm. The choice of the model describing dielectric properties of a metal at large separations is rather important. At a = 10 µm the conductivity correction is δ c (F D sl ) ≈ 0.68% using the Drude model and δ c (F p sl ) ≈ 0.32% using plasma model, i.e. more than two times smaller. In analogy with two plates, the high-temperature Casimir force calculated on the basis of [35] is two times smaller than in (69) regardless of the conductivity properties of the metal. In [36,37] the high-temperature behavior for the real metals coincides with (69) obtained for the ideal metal.
So, the actual properties of a particular metal are not reflected.

VI TEMPERATURE CASIMIR FORCE BETWEEN THE DIELECTRIC TEST BODIES
Here we briefly discuss the temperature Casimir force between dielectrics.
The application of Lifshitz formulas (1) or (37) for the case of dielectric surfaces is direct. No additional prescriptions, such as the ones used above or their generalizations are needed to obtain the final result matching the general physical requirements. However, when the formalism developed for dielectrics is compared with that for metals, the origin of the above difficulties becomes clear.
The dielectric permittivity of dielectrics can be modeled, e.g., by the Mahanty-Ninham relation [46,54] Here ω e ∼ 2 × 10 16 Hz gives the main electronic absorption in the ultraviolet, ε 0 is the static dielectric constant. At small ξ ≪ ω e one has ε(iξ) ≈ ε 0 . In fact, frequencies giving large contributions to the Lifshitz formulas (1) or (37) in the micrometer separation range are much smaller than ω e . Because of this, ε(iξ) can be approximately replaced by ε 0 . Below we use in all computations ε 0 ≈ 7 which corresponds to the sheet of mica.
In reality, the zeroth term of Lifshitz formula for dielectrics is also discontinuous as in the case of metals. To illustrate this statement we put l = 0 in (3) and obtain the following values of the reflection coefficients defined in Eq. (2) These values do not depend on k ⊥ . Therefore they are preserved in the limit k ⊥ → 0. At the same time, if we put k ⊥ = 0 from the very beginning one obtains  Table I). At a = 1 µm and a = 10 µm there are δ T (F ss ) = 1.99 × 10 −2 , respectively, δ T (F ss ) = 2.50 for dielectrics. At a ≥ 1 µm the temperature correction for dielectrics is larger than for ideal or real metals at the same separation. For the configuration of a dielectric lens above a disk at separations a = 0.1, 1 and 10 µm the temperature correction is, respectively, 2.39 × 10 −4 , 6.94 × 10 −2 , and 4.25.
The relationship of these values with those computed for the real and ideal metals (see Table II) is the same as in the case of two plane plates.

VII CONCLUSIONS AND DISCUSSION
As demonstrated above, the computation of the Casimir force between real metals at nonzero temperature is a complicated theoretical problem. The first contradictions between Lifshitz theory [2,16] applied to ideal metals and calculations based on quantum field theory [28,29] were revealed in the sixties. They were resolved by Schwinger, DeRaad and Milton [18] by the use of special prescription modifying the zero-frequency term of Lifshitz formula.
After the use of this prescription the results of [2,16] in application to ideal metal became agreed with those of [28,29].
A new stage in the solution of the problem has been started only recently.
It was motivated by the increased accuracy of the Casimir  These computations did not use any modification of Lifshitz formula. In [33] the problems arising in frames of Drude model were also formulated and the way to their solution was directed.
In the present paper we propose the new prescription modifying the ze- In the nearest future one should expect the experimental registration of the temperature Casimir force. This will be the final answer in the discussion on the subject what is the temperature dependence of the Casimir force.
Meanwhile it is necessary to proceed with the more detailed elaboration of the microscopic theory of dispersion forces based on quantum field theory at nonzero temperature in Matsubara formulation. There is a real possibility that the above phenomenological prescription, concerning the zero-frequency contribution to Lifshitz formula, will be rigorously derived, at least as a good approximation, in terms of the scattering theory in dissipative media. The final solution of this problem seems to be of large importance taking into account the prospective role of dispersion forces in both fundamental and applied science.
In Fig. 8 the dependence of I 2 onγ is shown by the curve 2. The values of I 2 decrease from 0.6455 atγ = 1 until 0.3787 atγ = 10.       The dashed line is for plasma model.