Hagen-Rubens relation beyond far-infrared region

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May 2010

EPL, 90 (2010) 44004 www.epljournal.org

doi: 10.1209/0295-5075/90/44004

Hagen-Rubens relation beyond far-infrared region

F. E. M. Silveira1(a) and S. M. Kurcbart2(b)

1 Centro de Ciencias Naturais e Humanas, Universidade Federal do ABC - Rua Santa Adelia, 166, Bairro Bangu,09210-170, Santo Andre, SP, Brazil2Departamento de Ciencias Naturais, Universidade Federal de Sao Joao del Rei - Campus Dom Bosco, 36301-160,Sao Joao del Rei, MG, Brazil

received 5 April 2010; accepted in final form 10 May 2010published online 8 June 2010

PACS 42.25.Bs – Wave propagation, transmission and absorptionPACS 41.20.Jb – Electromagnetic wave propagation; radiowave propagation

Abstract – An extension of Hagen-Rubens relation beyond the far-infrared region is proposedby taking into account inertial effects due to charge carriers. The influence of inertia is describedby introducing a finite relaxation time τ for the current density in the framework of a generalizedOhm’s law. A closed formula for τ as a function of metallic permittivity and radiation frequencyis derived. Our approach is applied to aluminum for which it is found that τ ∼ 6.91 fs at roomtemperature. It is also shown that the behavior of the observed absorptivity by that metal isin excellent agreement with our formulation up to the near-infrared region. The macroscopicapproach proposed here is totally independent of other microscopic formulations such as Drude’stheory of metallic conduction. Applications of our theory to related problems can lead to progressin extending the classical theory for the optics of metals.

Copyright c© EPLA, 2010

More than a century ago, investigations by Hagenand Rubens [1] established that the absorptivity A ofmonochromatic radiation (angular frequency ω) by manymetals (static electric conductivity σ), as calculated fromclassical electrodynamics,

A= 2

√2ε0ω

σ, (1)

where ε0 denotes the vacuum electric permittivity, is ingood agreement with experimental results, provided thewave energy is not higher than about 100meV. This meansthat the domain of validity of eq. (1), commonly referredto as the Hagen-Rubens (H-R) relation, is restricted tothe far-infrared region and downwards, that is, to awavelength not shorter than about 10µm. Since eq. (1)describes the observed high reflectivity by most metalsat small frequencies, the limit σ� ε0ω holds, that is, themetallic refractive index n and extinction coefficient ksatisfy the so-called Hagen-Rubens (H-R) condition [2,3],k= n� 1, where the unit denotes the vacuum refractiveindex. Therefore, precise measurements of metallic opticalconstants are required to determine the limits for consis-tency of the H-R condition with eq. (1), and even to

(a)E-mail: francisco.silveira@ufabc.edu.br(b)E-mail: samuel@ufsj.edu.br

attempt extensions of the classical approach for the opticsof metals.For the last four decades, or so, increasingly good-

quality data of optical constants have been available inthe literature for several metals within a broad regionof wavelengths [4–10]. We anticipate that, among those,the ones for aluminum [7–10], particularly at moderatedenergies, will prove useful for our purposes. Accuratedeterminations of optical constants are of great practi-cal interest in many areas of applied research. To quotea few examples, we mention the characterization of grainsize effects in sputtered metallic films [11], the technol-ogy of laser devices [12], the development of undula-tors and waveguides [13,14], and, more recently, for thenanoscale [15,16]. All the above-referred investigationsconfirm the long-wavelength constraint on eq. (1), whichis particularly apparent from imaging studies at high ener-gies (see [17] for recent results). Therefore, most attemptsto extend the classical formulation for the optics of metalshave been proposed in the realm of large frequencies (foran interesting approach, see [18]).Previous works [19,20] rediscussed the attenuation and

damping of electromagnetic fields in connection with theskin effect, on the basis of a generalized Ohm’s law,(

1+ τ∂

∂t

)�J = σ �E, (2)

44004-p1

F. E. M. Silveira and S. M. Kurcbart

where τ denotes the relaxation time of the induced currentdensity �J , due to inertial effects of charge carriers asa response to the applied electric field �E. In the limitτ → 0, the standard Ohm’s law, �J = σ �E, is recovered.By assuming that the fields vary in space and time as∼ exp(ı�κ · �x− ıωt), it has been found that the combinationof eq. (2) with the quasi-static approximation of Maxwell’sequations (the displacement current is ignored) leads tothe complex wave number

κ=

√µσ

τ

[√sinϕ (1− sinϕ)

2+ ı

√sinϕ (1+ sinϕ)

2

],

(3)where µ denotes the material magnetic permeability and�J lags �E in time by the dephasing angle ϕ= tan−1(ωτ).For small frequencies, the condition ωτ � 1 holds, termsof O(ω2τ2) can be neglected, and eq. (3) approaches

κ=

√µσω

2

[(1− ωτ

2

)+ ı(1+ωτ

2

)]. (4)

In the limit τ → 0, the classical wave number, κ=√ıµσω,is recovered.The present letter proposes an extension of the classical

H-R relation, eq. (1), beyond the far-infrared region, inthe framework of the generalized Ohm’s law, eq. (2).Particularly, a very simple closed formula for the inertialrelaxation time τ is derived. The theory is then applied toaluminum and it is found that the behavior of its observedabsorptivity at normal incidence is in excellent agreementwith the extended formulation up to the near-infraredregion.We start by ascribing to the system (radiation +

metal) the usual transverse dispersion relation [2,3] κ2 =ω2µ0ε0(εr+ ıεi), where µ0 denotes the vacuum magneticpermeability and εr+ ıεi represents the metallic relativepermittivity. Since we restrict our approach to moderatedwavelengths, we assume µ∼ µ0, that is, the linear mediaare supposed to be non-magnetic (the unique materialsfor which µ appreciably differs from µ0 at low energies arethe ferromagnets that, however, are not linear). Therefore,eq. (3) leads to

εr+ ıεi = ıστ

ε0

(cos2 ϕ

tanϕ+ ı cos2 ϕ

). (5)

By neglecting terms of O(ω2τ2), eq. (5) approaches

εr+ ıεi = ıσ

ε0ω(1+ ıωτ) . (6)

In the limit τ → 0, one gets εr+ ıεi = ıσ/ε0ω, that is, themetallic relative permittivity becomes a purely imaginarynumber (its real part vanishes), as observed for mostmetals at small frequencies.A much interesting result of the present theory is that it

provides a simple closed formula for the inertial relaxation

time. As one may easily check, eq. (5) leads to

τ =−εr�εiε, (7)

where, to comply with experimental practice, use has beenmade of Einstein’s relation ε= �ω for the photon energyε, with � denoting the normalized Planck’s constant. Thismeans that the inertial relaxation time is fully determinedfrom the metallic permittivity and photon energy. Weemphasize that our macroscopic formulation is totallyindependent of other microscopic approaches, such asDrude’s theory of metallic conduction. Within the latterscheme, the relaxation time of the current density isthe averaged interval between two successive collisions offree electrons with (essentially) stationary ions (classicalcollisional regime). On the other hand, our relaxationtime is due to inertial effects of charge carriers, andthis can be rigorously justified in the realm of extendedirreversible thermodynamics (see [19,20], and referencestherein).Now, we calculate the complex refractive index. It can

be determined from the complex relative permittivity [2,3],n+ ık=

√εr+ ıεi. Then, the former quantity may be

promptly computed from the complex wave number, κ=ω√µ0ε0(n+ ık). Equation (3) leads to

n+ ık=

√στ

ε0

[cosϕ

√1− sinϕ2 sinϕ

+ ı cosϕ

√1+ sinϕ

2 sinϕ

].

(8)Interestingly enough, we see that, when inertial effects ofcharge carriers are taken into account, the classical H-Rcondition, k= n� 1, must be extended to k� n� 1. Byneglecting terms of O(ω2τ2), eq. (8) approaches

n+ ık=

√σ

2ε0ω

[(1− ωτ

2

)+ ı(1+ωτ

2

)]. (9)

In the limit τ → 0, one gets k= n=√σ/2ε0ω. Therefore,by requiring the classical H-R condition, k= n� 1, to besatisfied, the observed high reflectivity by most metals atsmall frequencies, σ� ε0ω, is described.Finally, we derive an extended formula for the H-R

relation. The absorptivity at normal incidence can bedetermined from the metallic optical constants [2,3],

A=4n

(n+1)2+ k2

, (10)

where the unit denotes the vacuum refractive index. Then,if the extended H-R condition, k� n� 1, is satisfied,eq. (8) leads to

A= 2

√ε0

στ

√2 sinϕ

1+ sinϕ. (11)

That is the sought extended H-R relation. By neglectingterms of O(ω2τ2), eq. (11) approaches

A= 2

√2ε0ω

σ

(1− ωτ

2

). (12)

44004-p2

Hagen-Rubens relation beyond far-infrared region

Fig. 1: (Colour on-line) Absolute value for the ratio of the real(negative)-to-imaginary (positive) parts of the complex rela-tive permittivity for aluminum as a function of the angularfrequency. The squared points are plotted according to tabu-lated values for that metal extracted from [10]. The straightline is drew from linear regression (least squares) of the plot-ted points. The slope of the straight line gives the inertialrelaxation time for aluminum at room temperature, τ ∼ 6.91 fs(see eq. (7)).

Equation (12) shows that eq. (1) overestimates the metal-lic absorptivity, even at small frequencies. In other words,inertial effects due to charge carriers contribute to decreasethe value of A as predicted by the classical H-R relation(see fig. 2). In the limit τ → 0, eq. (1) is recovered.Let us now apply the just developed approach to

aluminum (σ∼ 35.7 MS/m at room temperature). Thatmetal has been selected because a large number ofmeasurements for its absorptivity at normal incidencehas been performed within a broad region of frequencies.The data for aluminum have been extracted from [10](results are accurate to better than ∼10%). By requiringthe extended H-R condition, k� n� 1, to be satisfied,one finds that the presented formulation may be appliedto that metal within the energy interval ∼40–200meV(wavelength within ∼5–30µm). This means that theproposed theory can reproduce the behavior of theobserved absorptivity at normal incidence by aluminumup to the near-infrared region.Figure 1 shows the absolute value for the ratio of the real

(negative)-to-imaginary (positive) parts of the complexrelative permittivity for aluminum as a function of theangular frequency (∼50–300Trad/s, consistently with theabove-referred energy-wavelength interval). The squaredpoints have been plotted according to tabulated values forthat metal extracted from [10]. The straight line has beendrawn from linear regression (least squares) of the plottedpoints. The slope of the straight line gives the inertialrelaxation time for aluminum at room temperature, τ ∼6.91 fs (see eq. (7)).Figure 2 shows the absorptivity, A, by aluminum as a

function of the time dephasing angle, ϕ (∼ π/8− 3π/8 rad,

Fig. 2: (Colour on-line) Absorptivity, A, by aluminum as afunction of the time dephasing angle, ϕ. The squared points areplotted according to tabulated values for that metal extractedfrom [10]. The dashed line gives the prediction for A due tothe classical H-R relation, eq. (1), and the continuous line, thesame prediction due to the extended H-R relation, eq. (11).As is apparent, the behavior of the observed absorptivity atnormal incidence by aluminum is in excellent agreement withour theory up to the near-infrared region.

again, consistently with the above-referred energy-wavelength interval). To calculate ϕ, use has been madeof the previously obtained result for the inertial relaxationtime, τ ∼ 6.91 fs, for that metal at room temperature. Thesquared points have been plotted according to tabulatedvalues for aluminum extracted from [10]. The dashed linegives the prediction for A due to the classical H-R rela-tion, eq. (1), and the continuous line, the same predictiondue to the extended H-R relation, eq. (11). To draw bothcurves, use has been made of the previously mentionedvalue of the static electric conductivity, σ∼ 35.7MS/m,for that metal at room temperature. As is apparent, thebehavior of the observed absorptivity at normal incidenceby aluminum is in excellent agreement with our theoryup to the near-infrared region.The present letter has extended the Hagen-Rubens

relation beyond the far-infrared region (see eq. (11)). Ithas been found that the inertial relaxation time τ of thecurrent density can be fully determined from the metallicpermittivity and radiation frequency (see eq. (7)). Thetheory has been then applied to aluminum and the inertialrelaxation time for that metal at room temperature hasbeen estimated, τ ∼ 6.91 fs (see fig. 1). It has been shownthat the behavior of the observed absorptivity at normalincidence by aluminum is in excellent agreement with ourtheory up to the near-infrared region (see fig. 2).Applications of our theory to other systems can lead

to advances in extending the classical theory for theoptics of metals. An interesting possibility is the analysisof electronic correlations in metallic transport. Indeed,within Landau’s theory of Fermi liquid, those processesare described by assuming an effective mass, as well asan effective scattering time, for charge carriers (for recentdevelopments, see [21,22]).

44004-p3

F. E. M. Silveira and S. M. Kurcbart

∗ ∗ ∗The authors are grateful to M. Assuncao for helpful

discussions.

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