Optical Properties of Metals

V. L. Ginzburg and G. P. Motulevich

Contents

Introduction ........................................................................ 469

§ 1. Reflection of light from a metallic surface ................................ 471

§ 2. Surface impedance and the effective complex dielectric constant of a metal ........................................................................ 477

§ 3. Microtheory in the case of the normal skin effect ......................... 484

§ 4. Microtheory in the case of the anomalous skin effect ...................... 497
    a) Fundamentals of the theory of the anomalous skin effect ................. 497
    b) Results pertaining to the infrared part of the spectrum .................. 504
    c) On taking quantum effects into account ..................................... 512

§ 5. Methods of measuring $n_{\mathrm{eff}}$, $\chi_{\mathrm{eff}}$, and $A$ ........... 516

§ 6. Discussion of the available experimental data .............................. 525

Conclusion ......................................................................... 533

Cited literature .................................................................... 534

Introduction

Although the optical properties of metals have been studied for many decades, the present state of metal optics is unsatisfactory, especially from the experimental point of view. Indeed, investigation of the polarization and intensity of light of various frequencies reflected by a metallic surface at different angles of incidence makes it possible to determine two quantities characterizing the metal—the effective refractive index $n_{\mathrm{eff}}(\omega)$ and the effective absorption index $\chi_{\mathrm{eff}}(\omega)$. At the same time, any reasonably complete data on $n_{\mathrm{eff}}(\omega)$ and $\chi_{\mathrm{eff}}(\omega)$ for typical metals exist only for Au, Ag, and Cu. Moreover, these data were obtained in 1913 and cannot be regarded as reliable and accurate because of the imperfection of the metallic surfaces used. Considerably more data are available concerning the absorptivity of metals at normal incidence, $A(\omega)=1-r(\omega)$, where $r(\omega)$ is the reflection coefficient at

at normal incidence. However, the single quantity \(A(\omega)\) is completely insufficient for a complete characterization of the optical properties of a metal; for this, as has been said, one must know two quantities, i.e., \(n_{\mathrm{eff}}(\omega)\) and \(\chi_{\mathrm{eff}}(\omega)\), or two independent combinations of them, one of which may be chosen to be \(A(\omega)\).

Such an experimental situation is connected, first of all, with a very widespread misunderstanding of the tasks of metal optics from the standpoint of obtaining information that is valuable and needed for the electron theory of metals. The second and, at the same time, deeper cause of the unsatisfactory state of metal optics is that, until very recently, the necessity had not been recognized of applying, in the optical part of the spectrum in most cases, the theory of the anomalous skin effect, and not the theory of the normal (ordinary) skin effect, which is based on the use of the concept of a complex dielectric constant of the metal. This circumstance, together with the difficulty of taking into account the influence of the nonideality of the reflecting surface, impeded the comparison of theory with experiment and led in a number of cases to an incorrect interpretation of experimental data.

At present, after the theory of the anomalous skin effect in metals has been developed, significant progress has been achieved in understanding the tasks and possibilities of metal optics; illuminating these is the principal aim of the present article. In addition to the theoretical questions considered in §§ 1–4 of the article, § 5 describes a method for measuring the quantities \(n_{\mathrm{eff}}(\omega)\), \(\chi_{\mathrm{eff}}(\omega)\), and \(A(\omega)\), and § 6 presents and discusses the available experimental data.

It is also necessary to point out from the very beginning that in the present article only part of the questions relating to metal optics is touched upon. Namely, we restrict ourselves to considering the frequency region where quantum absorption (the internal photoelectric effect) does not occur; that is, we have in mind primarily the infrared part of the spectrum and do not touch upon the ultraviolet, and in most cases not even the visible, part of it. Further, only the reflection of light from a massive (i.e., sufficiently thick) metallic mirror is considered, while the case of the transmission of light through thin metallic films is not discussed. In addition, the metal is regarded as isotropic on the average, which corresponds to the case of polycrystals or, in a known approximation (in particular, in the case of the normal skin effect), to the case of single-crystal specimens of metals possessing cubic symmetry. Finally, it is assumed that the magnetic permeability of the metal is \(\mu = 1\) and that the magnetization of the metal, as well as the external magnetic field, are zero. The assumptions made are, generally speaking, rather substantial, but in the simplest and most interesting cases they may be considered fulfilled. Therefore, in the present state of metal optics there are, in our ...

view, there is every reason to restrict the range of questions discussed and to deal first with the simplest and, at the same time, most important case of reflection of light from an isotropic nonferromagnetic metallic surface.

§ 1. REFLECTION OF LIGHT FROM A METALLIC SURFACE

Let us assume, as has always been done until recently, that in considering the propagation of light a metal may be regarded as a medium with a certain complex dielectric constant

\[ \varepsilon'(\omega)=\varepsilon(\omega)-i\frac{4\pi\sigma(\omega)}{\omega} =[n(\omega)-i\chi(\omega)]^2, \tag{1.1} \]

where \(\varepsilon\) is the dielectric constant, \(\sigma\) the conductivity, \(n\) and \(\chi\) the refractive and absorption indices, and \(\omega\) the cyclic frequency. The assumption made means, in particular, that a plane electromagnetic wave propagating in the metal normally to its surface (along the \(z\)-axis) varies according to the law

\[ E(z)=E_0 e^{-i\frac{\omega}{c}\sqrt{\varepsilon'}\,z} =E_0 e^{-\frac{\omega}{c}\chi z}e^{-i\frac{\omega}{c}nz}, \tag{1.2} \]

where \(E_0\) is the field in the metal at its surface \((z=0)\), and the \(z\)-axis is directed into the depth of the metal.

The depth of the skin layer, i.e. the distance over which the field decreases by a factor of \(e\), is equal to

\[ \delta_n=\frac{c}{\omega\chi}=\frac{\lambda_0}{2\pi\chi}, \tag{1.3} \]

where \(\lambda_0=\frac{2\pi c}{\omega}\) is the wavelength of light in vacuum. The case under consideration, when the field in the metal varies according to an exponential law of the type (1.2), is the case of the ordinary, or normal, skin effect. We shall return later to the question of under what conditions the skin effect cannot be regarded as normal. For the moment let us note that even in the case of the normal skin effect the relation (1.1) between \(\varepsilon'\) and \(n\) and \(\chi\) holds only for optically isotropic media (i.e. for single crystals of cubic symmetry, for polycrystalline substances, and, of course, liquids). In optically anisotropic media, in the case of the normal skin effect, instead of \(\varepsilon'\) one may introduce the tensor \(\varepsilon'_{ik}\) (in an optically isotropic medium \(\varepsilon'_{ik}=\varepsilon'\delta_{ik}\), where \(\delta_{ii}=1\) and \(\delta_{ik}=0\) for \(i\ne k\)); the variation of the field into the depth of the metal remains exponential, and only the quantities \(n\) and \(\chi\), which are expressed rather simply through \(\varepsilon'_{ik}\), depend on the mutual orientation of the \(z\)-axis and the axes of the crystal. As already mentioned in the introduction, we shall not consider optically anisotropic metals (see, for example, \(^{1}\) § 69 and \(^{2}\) § 22); moreover, already in (1.1)

it is assumed that the magnetic permeability of the medium is \(\mu = 1\) and that its magnetization is zero. The latter assumption means that we shall not consider magneto-optical phenomena that occur upon reflection and transmission of light through magnetized ferromagnetic substances (see \(^{3}\)).

One of the main problems of metal optics is to relate the quantities measured experimentally—the intensity of the light reflected from a metal and its polarization—to \(n\) and \(\varkappa\), or, if the skin effect is not normal, to other quantities characterizing the metal. It is precisely this question, and practically this question alone, that is treated in the sections of optics courses devoted to metal optics (see \(^{1}\S 68\) and \(^{2}\S 21\)). In doing so, the skin effect is always regarded as normal, which is also assumed below in this paragraph, where, for convenience, the corresponding well-known formulas will be given.

Fig. 1.

Let a plane wave be incident from vacuum on the plane boundary of a medium with complex dielectric constant \(\varepsilon'\):

\[ \mathbf{E}_{A}=\mathbf{A}\exp i\omega\left\{t-\frac{x\sin\varphi+z\cos\varphi}{c}\right\}, \]

where the plane of incidence is chosen to be the \(xz\) plane and \(\varphi\) is the angle of incidence (Fig. 1). The reflected and refracted waves have, respectively, the form:

\[ \mathbf{E}_{R}=\mathbf{R}\exp i\omega\left\{t-\frac{x\sin\varphi'-z\cos\varphi'}{c}\right\} \]

and

\[ \mathbf{E}_{D}=\mathbf{D}\exp i\omega\left\{t-\frac{x\sin\psi+z\cos\psi}{\dfrac{c}{\sqrt{\varepsilon'}}}\right\}. \]

At the boundary (for \(z=0\)), at any instant of time the conditions

\[ E_{1t}=E_{2t},\qquad H_{1t}=H_{2t}, \tag{1.4} \]

must be satisfied, where the subscript \(t\) indicates that the tangential components of the fields \(\mathbf{E}\) and \(\mathbf{H}\) are taken. Here \(\mathbf{E}_{1}=\mathbf{E}_{A}+\mathbf{E}_{R}\), \(\mathbf{E}_{2}=\mathbf{E}_{D}\), and anal-

analogously for \(\mathbf H\). Conditions (1.4) can be satisfied for any \(x\) and \(t\) only if the phases of all waves are equal, i.e.

\[ \sin\varphi'=\sin\varphi,\qquad \sqrt{\varepsilon'}\,\sin\psi=\sin\varphi, \tag{1.5} \]

whence \(\psi'=\varphi\), and, for a complex constant \(\varepsilon'\), the angle \(\psi\) is complex. The latter means that

\[ \mathbf E_D=\mathbf D\exp i\omega\left(t-\frac{\sin\varphi}{c}x-\frac{\sqrt{\varepsilon'-\sin^2\varphi}}{c}z\right), \]

and the wave not only oscillates but also attenuates as \(z\) increases.

The use of conditions (1.4) now leads to the Fresnel formulas:

\[ D_p=\frac{2\sin\psi\cos\varphi}{\sin(\varphi+\psi)\cos(\varphi-\psi)}A_p,\qquad D_s=\frac{2\sin\psi\cos\varphi}{\sin(\varphi+\psi)}A_s, \tag{1.6} \]

\[ R_p=\frac{\operatorname{tg}(\varphi-\psi)}{\operatorname{tg}(\varphi+\psi)}A_p,\qquad R_s=-\frac{\sin(\varphi-\psi)}{\sin(\varphi+\psi)}A_s; \tag{1.7} \]

where \(A_p, R_p\), and \(D_p\) are the components of \(\mathbf A, \mathbf R\), and \(\mathbf D\) lying in the plane of incidence and indicated in Fig. 1 by arrows; \(A_s, R_s\), and \(D_s\) are the components of the same vectors perpendicular to the plane of incidence (directed along the \(y\)-axis) and indicated in Fig. 1 by circles.

Formulas (1.6) and (1.7) are usually derived under the assumption that \(\psi\) is real, but they are also valid for complex \(\psi\), as can be verified by direct calculation without using complex quantities*).

Below only the reflected wave will be considered, and moreover in the practically important case when \(A_p=A_s\) (the incident light is linearly polarized, the plane of polarization making an angle of \(45^\circ\) with the plane of incidence). In this case, as is clear from (1.7),

\[ \frac{R_p}{R_s}=-\frac{\cos(\varphi+\psi)}{\cos(\varphi-\psi)} \equiv P e^{i\Delta}\equiv \operatorname{tg}\rho\cdot e^{i\Delta}. \tag{1.8} \]

If there is no absorption and total internal reflection is impossible, i.e. \(\varepsilon'(\omega)=\varepsilon(\omega)>1\), then the angle \(\psi\) is real and \(\Delta=\pi\) for \(\varphi<\varphi_b\).

\[ \text{*) In a homogeneous medium the condition } \operatorname{div}\mathbf D=4\pi\rho \text{ leads to the relation} \]

\[ \operatorname{div}\mathbf E=0, \]

since \(\operatorname{div}\mathbf j+i\omega\rho=0\) and, consequently,

\[ \operatorname{div}\left(\mathbf D-i\frac{4\pi}{\omega}\mathbf j\right) =\operatorname{div}\varepsilon'\mathbf E=0. \]

Therefore, even in the presence of absorption, \((\mathbf k\mathbf E)=0\), where

\[ \mathbf k\left\{\frac{\omega}{c}\sqrt{\varepsilon'}\,\sin\psi,\ 0,\ \frac{\omega}{c}\sqrt{\varepsilon'}\,\cos\psi\right\}. \]

and \(\Delta=0\) for \(\varphi>\varphi_b\), where \(\varphi_b\) is the Brewster angle \((\tg\varphi_b=\sqrt{\varepsilon})\). In this case \(P=\tg\rho=1\) at \(\varphi=0\) and \(\varphi=\dfrac{\pi}{2}\), and \(\tg\rho=0\) at \(\varphi=\varphi_b\) (Fig. 2). In the general case, when \(\sigma\ne0\), at \(\varphi=0\) and \(\varphi=\dfrac{\pi}{2}\) we likewise have \(P=\tg\rho=1\), and respectively \(\Delta=\pi,\ \dfrac{R_p}{R_s}=-1\) and \(\Delta=0,\ \dfrac{R_p}{R_s}=1\). However, in the intermediate region of angles the phase \(\Delta\) varies smoothly from \(\pi\) to 0, while \(\tg\rho\) nowhere reaches zero (Fig. 3).

Fig. 2. Absorption is absent, \(\varepsilon>1,\ \sigma=0\).

Fig. 3. Absorption is nonzero \((\sigma\ne0)\).

The reflected light is then elliptically polarized (the polarization is linear only for \(\Delta=\pi\) and \(\Delta=0\), which, in the presence of absorption, occurs only at \(\varphi=0\) and \(\varphi=\dfrac{\pi}{2}\)). At a certain angle \(\varphi=\varphi_0\) the phase \(\Delta=\dfrac{\pi}{2}\), and the polarization ellipse is closest to a circle.

The angle \(\varphi_0\) is called the principal angle of incidence. For \(\Delta\ne\dfrac{\pi}{2}\) the axis of the polarization ellipse does not lie in the plane of incidence, but for \(\Delta=\dfrac{\pi}{2}\) this occurs (Fig. 4). In the latter case the quantity \(P=\tg\rho\) is minimal, and the corresponding azimuthal angle \(\rho\) is denoted by \(\rho_0\) and is called the principal azimuth*). The angles \(\varphi_0\) and \(\rho_0\), or the quantities \(\rho\) and \(\Delta\) for any angle \(\varphi\), can be determined experimentally (see § 5). Therefore it is necessary to relate all these quantities to the parameter characterizing the reflecting substance, \(\sqrt{\varepsilon'}=n-i\chi\).

From (1.8), using (1.5), we obtain:

\[ \frac{1+Pe^{i\Delta}}{1-Pe^{i\Delta}} = \frac{\sin\varphi\sin\psi}{\cos\varphi\cos\psi} = \frac{\sin\varphi\,\tg\varphi}{\sqrt{\varepsilon'-\sin^2\varphi}}. \tag{1.9} \]

*) The angles \(\rho\) and \(\rho_0\) are often denoted by \(\psi\) and \(\psi_0\).

In what follows we shall take into account that, in the spectral region of interest to us, the quantities \(|\varepsilon'|\) and \(|\varepsilon|\) are usually very large, i.e.

\[ |\varepsilon'|=n^2+\chi^2 \gg 1, \tag{1.10} \]

\[ |\varepsilon|=|n^2-\chi^2| \gg 1. \tag{1.11} \]

Under condition (1.11), and usually also under the weaker condition (1.10), the quantity \(\sin^2\varphi\) may be neglected in comparison with \(\varepsilon\), as will be done below. More precisely, the condition

\[ |\varepsilon|\gg \sin^2\varphi, \tag{1.12} \]

will be used, for which

\[ \left. \begin{aligned} \frac{1+Pe^{i\Delta}}{1-Pe^{i\Delta}} &=\frac{\sin\varphi\,\operatorname{tg}\varphi}{\sqrt{\varepsilon'}}, \\[6pt] \frac{1+iP_0}{1-iP_0} &= \\[6pt] &=\frac{\sin\varphi_0\,\operatorname{tg}\varphi_0}{\sqrt{\varepsilon'}} \\[6pt] &=\frac{\sin\varphi_0\,\operatorname{tg}\varphi_0}{n-i\chi}, \end{aligned} \right\} \tag{1.9a} \]

where \(P_0=\operatorname{tg}\rho_0\) is the value of \(P\) at \(\varphi=\varphi_0\), when \(\Delta=\dfrac{\pi}{2}\). Multiplying the second relation (1.9a) by the complex conjugate quantity, we obtain:

\[ \sin\varphi_0\,\operatorname{tg}\varphi_0 =\sqrt{|\varepsilon'|} =\sqrt{n^2+\chi^2}. \tag{1.13} \]

From (1.9a) and (1.13) it follows that

\[ \operatorname{tg}^2\rho_0 = \frac{\sqrt{n^2+\chi^2}-n}{\sqrt{n^2+\chi^2}+n}. \tag{1.14} \]

Hence, using the relation \(\operatorname{tg}2\alpha=\dfrac{2\operatorname{tg}\alpha}{1-\operatorname{tg}^2\alpha}\), we find:

\[ \operatorname{tg}2\rho_0=\frac{\chi}{n}. \tag{1.15} \]

Fig. 4.

From (1.9a), for \(\Delta\ne\dfrac{\pi}{2}\), we have \((P=\operatorname{tg}\rho)\):

\[ \frac{1-Pe^{i\Delta}}{1+Pe^{i\Delta}} \cdot \frac{1+Pe^{-i\Delta}}{1+Pe^{-i\Delta}} = \frac{1-2iP\sin\Delta-P^2}{1+2P\cos\Delta+P^2} = \]

\[ = \frac{\cos 2\rho-i\sin 2\rho\,\sin\Delta}{1+\sin 2\rho\cos\Delta} = \frac{n-i\chi}{\sin\varphi\,\operatorname{tg}\varphi}, \]

whence, equating the real and imaginary parts, we find:

\[ \frac{\cos 2\rho}{1+\sin 2\rho\cos\Delta} = \frac{n}{\sin\varphi\,\operatorname{tg}\varphi}, \qquad \frac{\sin 2\rho\sin\Delta}{1+\sin 2\rho\cos\Delta} = \frac{\varkappa}{\sin\varphi\,\operatorname{tg}\varphi} \tag{1.16} \]

or

\[ \left. \begin{aligned} n&=\frac{\sin\varphi\cdot \operatorname{tg}\varphi\cdot \cos 2\rho} {1+\sin 2\rho\cdot \cos\Delta},\\ \varkappa&=\frac{\sin\varphi\cdot \operatorname{tg}\varphi\cdot \sin 2\rho\sin\Delta} {1+\sin 2\rho\cdot \cos\Delta}, \end{aligned} \qquad \frac{\varkappa}{n}=\sin\Delta\,\operatorname{tg}2\rho. \right\} \tag{1.17} \]

From (1.16) and (1.17), putting \(\Delta=\frac{\pi}{2}\), one can immediately obtain expressions (1.13) and (1.15).

Thus, by measuring \(\varphi_0\) and \(\rho_0\), or \(\Delta\) and \(\rho\), at any angle of incidence \(\varphi\), it is indeed possible to determine \(n\) and \(\varkappa\) at the frequency \(\omega\) under investigation. The same can be achieved by measuring the reflection coefficients \(r_{\parallel}=\left|\frac{R_p}{A_p}\right|^2\) and \(r_{\perp}=\left|\frac{R_s}{A_s}\right|^2\) (see (1.7)); in this case expressions (1.7), after simple transformations in which the law of refraction \(\sqrt{\varepsilon'}\sin\psi=\sin\varphi\) is used, can be written in the form:

\[ \left. \begin{aligned} \frac{R_p}{A_p} &= \frac{ \sqrt{\varepsilon'}\cos\varphi - \sqrt{1-\dfrac{\sin^2\varphi}{\varepsilon'}} }{ \sqrt{\varepsilon'}\cos\varphi + \sqrt{1-\dfrac{\sin^2\varphi}{\varepsilon'}} }, \\[6pt] \frac{R_s}{A_s} &= \frac{ \cos\varphi-\sqrt{\varepsilon'-\sin^2\varphi} }{ \cos\varphi+\sqrt{\varepsilon'-\sin^2\varphi} }. \end{aligned} \right\} \tag{1.18} \]

The formulas for the quantities \(r_{\parallel}\) and \(r_{\perp}\), expressed in terms of \(n\) and \(\varkappa\), will be given in § 5; here we shall restrict ourselves to the case of normal incidence, when

\[ \left. \begin{aligned} r&=\left|\frac{R_p}{A_p}\right|^2 =\left|\frac{R_s}{A_s}\right|^2 = \left|\frac{\sqrt{\varepsilon'}-1}{\sqrt{\varepsilon'}+1}\right|^2 = \frac{n^2+\varkappa^2+1-2n}{n^2+\varkappa^2+1+2n}, \\[6pt] A&=1-r = \frac{4n}{n^2+\varkappa^2+1+2n}. \end{aligned} \right\} \tag{1.19} \]

Expressions (1.18) and (1.19) are exact, i.e. valid also when conditions (1.10)—(1.12) are not satisfied. Measurement of \(A\) makes it possible to relate \(n\) and \(\varkappa\), or to check independent measurements of \(n\) and \(\varkappa\).

Let us also dwell in somewhat more detail on the relation between $\varepsilon$ and $\sigma$ and $n$ and $\chi$. As follows from (1.1),

\[ \left. \begin{gathered} n^2-\chi^2=\varepsilon,\qquad n\chi=\frac{2\pi\sigma}{\omega},\\[6pt] n=\sqrt{\frac{\varepsilon}{2}+\sqrt{\left(\frac{\varepsilon}{2}\right)^2+\left(\frac{2\pi\sigma}{\omega}\right)^2}},\\[6pt] \chi=\sqrt{-\frac{\varepsilon}{2}+\sqrt{\left(\frac{\varepsilon}{2}\right)^2+\left(\frac{2\pi\sigma}{\omega}\right)^2}}, \end{gathered} \right\} \tag{1.20} \]

where the inner root is always taken with the plus sign, which ensures that $n$ and $\chi$ are real for any $\varepsilon$ (in metals, in most regions, $\varepsilon<0$). Before the outer root in (1.20) one may likewise always choose the plus sign, since the minus sign corresponds simply to replacing the axis $z$ by $-z$ (i.e., to propagation and attenuation of the wave in the opposite direction).

If

\[ |\varepsilon|\gg \frac{4\pi\sigma}{\omega}, \tag{1.21} \]

then

\[ \left. \begin{gathered} n\approx \sqrt{\varepsilon},\qquad \chi\approx \frac{2\pi\sigma}{\omega\sqrt{\varepsilon}}\quad (\text{for } \varepsilon>0),\\[6pt] n\approx \frac{2\pi\sigma}{\omega\sqrt{-\varepsilon}},\qquad \chi=\sqrt{-\varepsilon}\quad (\text{for } \varepsilon<0). \end{gathered} \right\} \tag{1.22} \]

If, however,

\[ \varepsilon \ll \frac{4\pi\sigma}{\omega}, \tag{1.23} \]

then

\[ n\approx \chi\approx \sqrt{\frac{2\pi\sigma}{\omega}},\qquad \delta_0=\frac{c}{\omega\chi}=\frac{c}{\sqrt{2\pi\sigma\omega}}. \tag{1.24} \]

The case (1.23) and (1.24) occurs in the radio-frequency region; in this case $\operatorname{tg}2\rho_0=1$, $P_\perp=\operatorname{tg}\rho_0=0.414$, and, under the conditions (1.10) satisfied in metals,

\[ A\approx 2\sqrt{\frac{\omega}{2\pi\sigma}}. \tag{1.25} \]

§ 2. SURFACE IMPEDANCE AND THE EFFECTIVE COMPLEX DIELECTRIC CONSTANT OF A METAL

In the region of millimeter, centimeter, and even decimeter radio waves, when one has to deal with hollow resonators and waveguides, the properties of the medium (the metal) are especially often

are characterized by the surface impedance

\[ Z(\omega)=R(\omega)+iX(\omega)=\frac{4\pi}{c}\left[\frac{E_x}{H_y}\right]_0, \tag{2.1} \]

where the subscript 0 indicates that the components \(E_x\) and \(H_y\) are taken on the surface of the metal (the surface is, for definiteness, taken to be plane and coincident with the plane \(xy\)). In the case of incidence on the surface of a plane wave, evidently, \(E_x=E_{Ax}+E_{Rx}\) and \(H_y=H_{Ay}+H_{Ry}\) (see § 1), and it is easy to see (see in more detail\({}^{1}\) § 10) that \(E_x=(A_p-R_p)\cos\varphi,\ H_y=A_p+R_p,\ E_y=A_s+R_s\) and \(H_x=-A_s\cos\varphi+R_s\cos\varphi\). Hence, using (1.18), we obtain:

\[ \left. \begin{aligned} Z(\omega)&=\frac{4\pi}{c}\left[\frac{E_x}{H_y}\right]_0 =\frac{4\pi}{c}\, \frac{\sqrt{1-\dfrac{\sin^2\varphi}{\varepsilon'}}}{\sqrt{\varepsilon'}},\\[6pt] Z'(\omega)&=-\frac{4\pi}{c}\left[\frac{E_y}{H_x}\right]_0 =\frac{4\pi}{c\sqrt{\varepsilon'-\sin^2\varphi}}. \end{aligned} \right\} \tag{2.2} \]

For normal incidence

\[ \left. \begin{aligned} Z(\omega)&=Z'(\omega)=\frac{4\pi}{c\sqrt{\varepsilon'}},\\[6pt] \varepsilon'&=\varepsilon-i\frac{4\pi\sigma}{\omega} =\frac{16\pi^2}{c^2Z^2} =\frac{16\pi^2}{c^2}\, \frac{R^2-X^2)-2iXR}{(X^2+R^2)^2}. \end{aligned} \right\} \tag{2.3} \]

Expressions (2.3), as approximate ones, also hold for any \(\varphi\), if condition (1.11) or (1.10) is satisfied. Therefore, under conditions (1.10)—(1.11), which for metals are usually well satisfied, the surface impedance is, up to terms of order \(\dfrac{\sin^2\varphi}{\varepsilon'}\), a universal quantity, i.e. it does not depend on the character of the electromagnetic field in vacuum. This is connected, of course, with the fact that under conditions (1.10)—(1.11) the field decreases very rapidly into the depth of the metal and, to a good approximation, is a damped wave propagating along the \(z\)-axis independently of the character of the field in vacuum. In view of what has been said, the expressions

\[ Z(\omega)=\frac{4\pi}{c\sqrt{\varepsilon'}} =\frac{4\pi}{c}\left[\frac{E_x}{H_y}\right]_0 =-\frac{4\pi}{c}\left[\frac{E_y}{H_x}\right]_0 \tag{2.4} \]

may be regarded as approximate boundary conditions connecting the fields \(\mathbf E\) and \(\mathbf H\) in vacuum and making it possible to find these fields without computing the field in the metal. These boundary conditions are widely used at present in the theory of radio-wave propagation (see, for example,\({}^{4}\) where in § 19 it is discussed

also the question of the accuracy of conditions (2.4)). With the aid of conditions (2.4) it is easy to obtain formulas (1.13)—(1.17), as well as formulas (1.18), where one must neglect \(\sin^2\varphi\) in comparison with \(\varepsilon\), i.e. the formulas

\[ \frac{R_p}{A_p}=\frac{\sqrt{\varepsilon'}\cos\varphi-1}{\sqrt{\varepsilon'}\cos\varphi+1},\qquad \frac{R_s}{A_s}=\frac{\cos\varphi-\sqrt{\varepsilon'}}{\cos\varphi+\sqrt{\varepsilon'}}. \tag{2.5} \]

In these formulas one may, of course, everywhere replace \(\sqrt{\varepsilon'}\) by \(\dfrac{cZ}{4\pi}\).

The value of the boundary conditions (2.4), from the point of view of the subject of the present article, consists mainly in the fact that these conditions essentially retain their meaning also in the case of the anomalous character of the skin effect. The latter occurs if the mean free path of the current carriers (electrons) cannot be regarded as arbitrarily small in comparison with the depth of the skin layer \(\delta\). Therefore, under the conditions of the anomalous skin effect*) the current \(\mathbf{j}\) and the polarization \(\mathbf{P}\) at the same point (i.e. the total current \(\mathbf{j}_t=\mathbf{j}+i\omega\mathbf{P}\)) are determined not only by the field \(\mathbf{E}\) at the same point, but also by the field in the neighborhood of this point and, thus, one can no longer use the relations

\[ \mathbf{j}=\sigma\mathbf{E},\qquad \mathbf{P}=\frac{\varepsilon-1}{4\pi}\mathbf{E}\quad \text{and}\quad \mathbf{j}_t=\frac{i\omega}{4\pi}(\varepsilon'-1)\mathbf{E}, \]

nor the equations of phenomenological electrodynamics into which the quantity \(\varepsilon'\) has been introduced. In other words, the field decreases into the depth of the metal no longer according to the exponential law (1.2), but in a more complicated manner (see § 4). At the same time, however, the decrease of the field remains very rapid, since the shielding current in metals is still (as in the case when

\[ \mathbf{j}_t=\frac{i\omega}{4\pi}(\varepsilon'-1)\mathbf{E} \]

) relatively very large even in the region of the anomalous skin effect. Therefore, from physical considerations it is quite evident that in the region of the anomalous skin effect the character of reflection of waves from a metallic surface, in the sense of the dependence on the angle of incidence \(\varphi\) and of the number of parameters that must be specified in order to determine the intensity and polarization of the reflected wave, remains the same as in the case of the normal skin effect. In other words, it appears quite natural that in the region of the anomalous skin effect in metals the boundary conditions (2.4) are valid and the impedance \(Z(\omega)\) is universal (i.e. does not depend on \(\varphi\)). At the same time, however, it is no longer possible to express \(Z(\omega)\) through \(\varepsilon'(\omega)\) (i.e. \(Z\ne \dfrac{4\pi}{c\sqrt{\varepsilon'}}\)), and the impedance \(Z(\omega)\) may be regarded as a component replacing \(\varepsilon'(\omega)\)

*) The anomalous skin effect and the conditions under which it occurs will be discussed in more detail in §§ 3 and 4.

complex quantity that completely characterizes the reflection of waves from a metallic surface. In order to establish a closer correspondence with the region of the normal skin effect, it is nevertheless convenient to introduce an effective complex dielectric constant \(\varepsilon'_{\mathrm{eff}}\), which, by definition, is equal to

\[ \left. \begin{aligned} \varepsilon'_{\mathrm{eff}} &= \varepsilon_{\mathrm{eff}} - i\,\frac{4\pi\sigma_{\mathrm{eff}}}{\omega} = (n_{\mathrm{eff}} - i\chi_{\mathrm{eff}})^2 = \frac{16\pi^2}{c^2 Z^2}, \\[6pt] Z(\omega) &= \frac{4\pi}{c\sqrt{\varepsilon'_{\mathrm{eff}}(\omega)}} . \end{aligned} \right\} \tag{2.6} \]

In the case of the normal skin effect \(\varepsilon'_{\mathrm{eff}}=\varepsilon'\). In general, however, \(\varepsilon'_{\mathrm{eff}}\) coincides with the complex dielectric constant of a medium in which the skin effect is normal and which, if substituted for the metal, would exert on the field outside the metal the same action as does this metal.

To prove\(^5\) the assumption made, let us recall that, when it is possible to use the quantity \(\varepsilon'\) (i.e. in the region of the normal skin effect), at the interface \(E_z\) (in the metal) \(=\dfrac{1}{\varepsilon'}E_z\) (in vacuum) \(\sim \dfrac{E_{x,y}}{\sqrt{\varepsilon'}}\) (since in vacuum at the interface \(E_{x,y}\sim \dfrac{E_z}{\sqrt{\varepsilon'}}\), and we do not pay attention to factors of the type \(\sin\varphi\) for simplicity; see\(^4\) § 19). Further, in the metal, at the interface,

\[ \frac{\partial E_z}{\partial x} \ll i\,\frac{\omega}{c}E_z \sim i\,\frac{\omega}{c}\,\frac{E_{x,y}}{\sqrt{\varepsilon'}}, \qquad \frac{\partial E_{x,y}}{\partial x} \ll i\,\frac{\omega}{c}E_{x,y} \]

and

\[ \frac{\partial E_{x,y}}{\partial z} \sim i\,\frac{\omega}{c}\sqrt{\varepsilon'}\,E_{x,y}, \qquad \text{i.e.} \qquad \left|\frac{\partial E_z}{\partial x}\right| \sim \frac{1}{|\varepsilon'|} \left|\frac{\partial E_{x,y}}{\partial z}\right|. \]

Thus, in the metal,

\[ \left. \begin{gathered} |E_z| \ll |E_{x,y}|,\qquad \left|\frac{\partial E_z}{\partial x}\right| \ll \left|\frac{\partial E_{x,y}}{\partial z}\right|, \qquad \left|\frac{\partial E_z}{\partial y}\right| \ll \left|\frac{\partial E_{x,y}}{\partial z}\right|, \\[6pt] \left|\frac{\partial E_{x,y}}{\partial x}\right| \ll \left|\frac{\partial E_{x,y}}{\partial z}\right|, \qquad \left|\frac{\partial E_{x,y}}{\partial y}\right| \ll \left|\frac{\partial E_{x,y}}{\partial z}\right|. \end{gathered} \right\} \tag{2.7} \]

Therefore the equation \(\operatorname{rot}\mathbf{E}=-\dfrac{i\omega}{c}\mathbf{H}\) in the metal takes the form:

\[ i\,\frac{\omega}{c}H_x \simeq \frac{\partial E_y}{\partial z}, \qquad i\,\frac{\omega}{c}H_y \simeq -\frac{\partial E_x}{\partial z}. \tag{2.8} \]

Hence, taking into account that in the region of the normal skin effect, in a metal,

\[ E_{x,y}\simeq E^{(0)}_{x,y}e^{-i\frac{\omega}{c}\sqrt{\varepsilon'}z} \quad (\text{see § 1}) \quad \text{and} \]

\[ \frac{\partial E_{x,y}}{\partial z}\simeq -\,i\frac{\omega}{c}\sqrt{\varepsilon'}\,E_{x,y}, \]

we obtain the boundary conditions (2.4).

In view of what was said earlier, in a good conductor and in the case where it is impossible to use the quantity \(\varepsilon'\), the conditions (2.7) may be regarded as fulfilled; therefore one may assume that, independently of the character of the external field, a plane wave \(E_{x,y}(z)\) propagates in the metal in the approximation under consideration; equations (2.8) then, obviously, also hold. At the boundary, because of the assumed linearity and homogeneity of the problem, as well as the equivalence of the directions \(x\) and \(y\), one may put

\[ \left[\frac{\partial E_{x,y}}{\partial z}\right]_0=a(\omega)[E_{x,y}]_0, \]

where \(a(\omega)\) does not depend on the field. Hence, and from (2.8), we obtain the boundary conditions:

\[ Z(\omega)\equiv \frac{4\pi}{c\sqrt{\varepsilon'_{\mathrm{eff}}}} =\frac{4\pi}{c}\left[\frac{E_x}{H_y}\right]_0 =-\frac{4\pi}{c}\left[\frac{E_y}{H_x}\right]_0, \tag{2.9} \]

where

\[ Z(\omega)=-i\frac{4\pi\omega}{c^2 a(\omega)}. \]

The question of the accuracy of the boundary conditions (2.9) is not clear without further analysis, but it must be thought that, when the inequality

\[ |\varepsilon'_{\mathrm{eff}}|=\frac{16\pi^2}{c^2|Z|^2}\gg 1 \tag{2.10} \]

is satisfied, this accuracy is very high (inequality (2.10) is analogous to (1.10); we shall not write out the inequalities corresponding to (1.11) and (1.12)).

Let us note that, in the case of non-universality of the impedance, this concept can also be used, but under the condition that the character of the external field is fixed; finally, the question of the existence of a universal impedance independent of the character of the external field can be decided experimentally, for example by investigating the reflection of plane waves incident on the interface at different angles.

Below it will be assumed, as according to all evidence is the case in the part of the spectrum of interest to us, that the conditions (2.9) may be used for all angles of incidence. It follows from this that all results obtained using the conditions (2.4) remain valid also in the region of the anomalous skin effect, but with \(\varepsilon'\) replaced by \(\varepsilon'_{\mathrm{eff}}\).

Thus, formulas (1.13)—(1.17) and (2.5) remain valid independently of the character of the skin effect. In this case, however, the quantity \(\varepsilon'\) must be replaced by \(\varepsilon'_{\mathrm{eff}}\), and \(n\) and \(\chi\) by \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) (see (2.6); of course, in the limiting case of the normal skin effect \(\varepsilon'_{\mathrm{eff}}=\varepsilon'\), \(n_{\mathrm{eff}}=n\), and \(\chi_{\mathrm{eff}}=\chi\)). Consequently, measurements of the polarization and intensity of the reflected light, as in classical metal optics, make it possible to determine two functions characterizing the metal, \(n_{\mathrm{eff}}(\omega)\) and \(\chi_{\mathrm{eff}}(\omega)\), which, however, in the general case cannot be identified with the refractive and absorption indices \(n(\omega)\) and \(\chi(\omega)\).

Before proceeding to establish the relation between \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) and the microscopic parameters (see §§ 3 and 4), we shall dwell on certain general properties of the functions \(Z(\omega)\) and \(\varepsilon'_{\mathrm{eff}}(\omega)\). It can be shown that the quantities \(Z(\omega)\) and \(\varepsilon'_{\mathrm{eff}}(\omega)\), considered as functions of the complex variable \(\omega\), have neither poles nor zeros in the lower half-plane, nor also on the real axis, except, possibly, only at the point \(\omega=0\) (the static field)*). The proof of this assertion, which follows primarily from the requirement that the principle of causality be satisfied, may be found in\(^6\) (see also\(^4\), § 83). The absence of poles of the functions \(Z(\omega)\) and \(\varepsilon'_{\mathrm{eff}}(\omega)\) in the lower half-plane leads to the fact that integrals of a very general form, taken over a contour situated in the lower half-plane, vanish. This circumstance makes possible the derivation of definite integral relations between the real and imaginary parts of the functions \(Z(\omega)\) or \(\varepsilon'_{\mathrm{eff}}(\omega)\), i.e., between \(R\) and \(X\), or \(\varepsilon_{\mathrm{eff}}\) and \(\sigma_{\mathrm{eff}}\). Thus, with respect to \(Z(\omega)\) we have (for the derivation see\(^5\)):

\[ \left. \begin{aligned} R(\omega_2)-R(\omega_1) &= -\frac{2}{\pi}\int_{0}^{\infty} \left\{\frac{1}{\omega'^2-\omega_2^2} -\frac{1}{\omega'^2-\omega_1^2}\right\} X(\omega')\,\omega'\,d\omega', \\[6pt] X(\omega) &=\frac{2\omega}{\pi}\int_{0}^{\infty} R(\omega')\,\frac{d\omega'}{\omega'^2-\omega^2}, \end{aligned} \right\} \tag{2.11} \]

where all integrals are taken in the sense of the principal value**).

*) In the present paper the dependence \(e^{i\omega t}\) is used, and not \(e^{-i\omega t}\), as is also sometimes done. Passage to the complex plane of the variable \(\omega\) corresponds to the consideration not only of fields harmonic in time, but also of fields that increase or decrease exponentially in time.

**) The principal value of the integral

\[ \int_a^b \frac{f(\omega')\,d\omega'}{\omega'-\omega} = \lim_{\delta\to 0} \left\{ \int_a^{\omega-\delta}\frac{f(\omega')}{\omega'-\omega}\,d\omega' + \int_{\omega+\delta}^{b}\frac{f(\omega')}{\omega'-\omega}\,d\omega' \right\}. \]

Formulas (2.11) make it possible to find \(R(\omega)\) from \(X(\omega)\) and conversely, and, most importantly, to carry out various estimates. For the latter purpose the second of expressions (2.11) is especially suitable, since the quantity \(R(\omega)\) is more easily measured and, moreover, is always positive. The latter is clear from the fact that \(R\) is directly connected with the heat liberated in the metal,

\[ Q=\left(\frac{c}{4\pi}\right)^2 \cdot \frac{|H_0|^2 R}{2}, \tag{2.12} \]

where \(Q\) is the average amount of heat liberated per unit time, calculated per unit surface area (it is assumed that the wave cannot pass through the metallic layer), and \(H_0\) is the amplitude of the high-frequency field at the surface, i.e. \(|H|_0=H_0 e^{i\omega t}\).

For the real and imaginary parts of \(\varepsilon'_{\mathrm{eff}}\) we similarly have:

\[ \left. \begin{aligned} \varepsilon_{\mathrm{eff}}(\omega) &=1+8\int_0^\infty \frac{\sigma_{\mathrm{eff}}(\omega')\,d\omega'}{\omega'^2-\omega^2},\\[6pt] \sigma_{\mathrm{eff}}(\omega) &=-\frac{1}{2\pi^2}\int_0^\infty \frac{\omega'^2\{\varepsilon_{\mathrm{eff}}(\omega')-1\}\,d\omega'}{\omega'^2-\omega^2} =\\[6pt] &=-\frac{\omega^2}{2\pi^2}\int_0^\infty \frac{\varepsilon_{\mathrm{eff}}(\omega')-1}{\omega'^2-\omega^2}\,d\omega' +\frac{1}{2\pi^2}\int_0^\infty \{1-\varepsilon_{\mathrm{eff}}(\omega')\}\,d\omega', \end{aligned} \right\} \tag{2.13} \]

where, for simplicity, it is assumed that the metal is not in the superconducting state (see more fully in \(^{5}\)).

For very high frequencies (in the X-ray part of the spectrum)

\[ \varepsilon'_{\mathrm{eff}}(\omega)=\varepsilon'(\omega)=\varepsilon(\omega) =1-\frac{4\pi e^2 N_n}{m\omega^2} \tag{2.14} \]

\[ (\text{as } \omega\to\infty), \]

since the electrons behave as free ones, and in this case formula (2.14) is precisely applicable (see, for example, \(^{4}\), § 57). Here \(N_n\) is the total concentration of all electrons in the body (i.e. the sum of the concentrations of the conduction electrons and the atomic electrons). On the other hand, the first expression (2.13) at very high frequencies \(\omega\) has the form:

\[ \varepsilon_{\mathrm{eff}}(\omega)=1-\frac{8}{\omega^2}\int_0^\infty \sigma_{\mathrm{eff}}(\omega')\,d\omega', \]

whence, on comparison with (2.14), we obtain*):

\[ \frac{2m}{\pi e^2}\int_0^\infty \sigma_{\mathrm{eff}}(\omega')\,d\omega' = N_n. \tag{2.15} \]

Relation (2.15) is the so-called sum rule. As we shall see below, in the optical part of the spectrum there is, in most cases, a region where, with good accuracy,

\[ \varepsilon'_{\mathrm{eff}}(\omega)\cong \varepsilon(\omega)\cong -\,\frac{4\pi e^2 N}{m\omega^2}, \tag{2.16} \]

terms of order unity being neglected, and \(N\), by definition, is the concentration of conduction electrons. Under similar conditions, analogously to the preceding case, we obtain:

\[ \frac{2m}{\pi e^2}\int_0^{\omega_0}\sigma_{\mathrm{eff}}(\omega')\,d\omega' \cong N, \tag{2.17} \]

where \(\omega_0\) is a frequency lying in the region in which relation (2.16) is valid. The inaccuracy associated with the known uncertainty of the frequency \(\omega_0\) is immaterial in view of the approximate character of relations (2.16) and (2.17). In those cases when, for all frequencies, \(\varepsilon'_{\mathrm{eff}}(\omega)=\varepsilon'(\omega)\), in the formulas given one must, of course, understand \(\varepsilon\) and \(\sigma\) by \(\varepsilon_{\mathrm{eff}}\) and \(\sigma_{\mathrm{eff}}\).

§ 3. MICROTHEORY IN THE CASE OF THE NORMAL SKIN EFFECT

The study of light reflected from the metallic surface under consideration makes it possible, as is clear from what has been said, to determine two functions \(n_{\mathrm{eff}}(\omega)\) and \(\chi_{\mathrm{eff}}(\omega)\), which also depend on the temperature of the metal**). Establishing the connection between \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) and the microscopic characteristics of the metal is already a problem of the electron theory of metals. In this connection, in order to make any significant progress, one has to make use

*) The possibility of writing the general expression (2.13) for \(\varepsilon_{\mathrm{eff}}\) at high frequencies in the form
\[ \varepsilon_{\mathrm{eff}}(\omega)=1-\frac{8}{\omega^2}\int_0^\infty \sigma_{\mathrm{eff}}(\omega')\,d\omega' \]
also follows from comparison with expression (2.14), which follows from theory and is confirmed by experimental data.

**) The functions \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\), generally speaking, also depend on the method of treatment of the surface. We shall not, however, touch upon this point here, assuming that what is involved is a certain “ideal surface.” Such a surface is approached by a mirror whose optical properties no longer change practically upon further improvement of the surface (polishing, cleaning, etc.; see also § 5).

known model concepts, in contrast to the phenomenological approach that was possible in §§ 1 and 2. In accordance with this, below we shall use the one-electron, or gas, model customary in the theory of metals\(^{7,8}\), in which the electrons are regarded as moving in the periodic field of the lattice independently of one another. More precisely, the peculiar “interaction” associated with the Pauli principle is necessarily taken into account, and the self-consistent Coulomb interaction between electrons may also be taken into account; but a fuller and more consistent treatment of the interelectron interaction is not carried out.

The success of such a gas model\(^*\) is undoubtedly connected with the action of the Pauli principle, by virtue of which, in a degenerate electron gas, the main portion of the electrons cannot give up energy at all, and can receive energy only in the case when this energy is sufficiently large\(^ {**}\). As a result, the interelectron interaction (for example, when any two electrons approach one another, i.e., in a collision) is to a considerable extent “suppressed,” so that only electrons lying in the region of Fermi smearing can change their state. The corresponding collisions, owing to the high degree of degeneracy of electrons in metals, play only a very small role and may be regarded as a perturbation that can usually be neglected (for details see \(^{9}\)).

Fig. 5.

In the gas model, the state of the electrons is described by a distribution function depending on the momentum \(\mathbf p\), the coordinates \(\mathbf r\), and the time \(t\),

\[ f(\mathbf p,\mathbf r,t)=f_0(\mathbf p)+f_1(\mathbf p,\mathbf r,t), \tag{3.1} \]

where \(f_0\) is the equilibrium distribution function that obtains in the absence of a field, and \(f_1\) is an addition reflecting the influence of the field; in all cases encountered in metal optics this addition may be considered very small, so that the condition \(|f_1|\ll f_0\) is well satisfied. In (3.1) and below spin variables are not taken into account and are not

\(^*\) Metals in the superconducting state are not considered.

\(^ {**}\) What has just been said is immediately clear from Fig. 5, which shows the Fermi distribution function (the solid line corresponds to temperature \(T=0\); the dashed line corresponds to temperature \(T\ne0\), the smearing region, where \(f\ne1\) and \(f\ne0\), being of order \(kT\)). An electron with some energy \(W<W_0\) (\(W_0\) is the energy at the Fermi boundary) at \(T=0\) can receive only an energy \(\Delta W>W_0-W\). It cannot give up energy, nor can it receive an energy \(\Delta W<W_0-W\).

no distinction is made between momentum and quasimomentum, since taking these points into account is immaterial for what follows.

The equilibrium function \(f_0\) in a homogeneous metal having some temperature \(T\) is the well-known Fermi function

\[ f_0(\mathbf{p}) = f_0(W(\mathbf{p})) = \left[e^{\frac{W-W_0}{kT}}+1\right]^{-1}. \]

As for the function \(f_1\), in the linear approximation of interest to us it is determined by the kinetic equation

\[ \frac{\partial f_1}{\partial t}+\mathbf{v}\nabla_r f_1+e\mathbf{E}\nabla_p f_0+\frac{f_1}{\tau}=0, \tag{3.2} \]

where \(\mathbf{E}\) is the electric-field strength (the magnetic field \(\mathbf{H}\) is assumed absent), \(e\) is the electron charge, \(\mathbf{v}=\nabla_p W(\mathbf{p})\) is the electron velocity, \(\tau=\frac{l}{v}\), and \(l(\mathbf{v})\) are the time and length of the mean free path, and the usual notation has been adopted:

\[ \nabla_r=\mathbf{i}\frac{\partial}{\partial x}+\mathbf{j}\frac{\partial}{\partial y}+\mathbf{k}\frac{\partial}{\partial z},\qquad \nabla_p=\mathbf{i}\frac{\partial}{\partial p_x}+\mathbf{j}\frac{\partial}{\partial p_y}+\mathbf{k}\frac{\partial}{\partial p_z}. \]

The introduction of the mean free path, i.e. the allowance for collisions of electrons with lattice vibrations by introducing the term \(\frac{f_1}{\tau}\) in (3.2), is not always possible. However, at high (in particular, room) temperatures, and also at low temperature (in the region of the residual resistance), the adopted approximation is, generally speaking, justified; this approximation may be used as an extrapolating one also at intermediate temperatures, in consequence of which it is applied below without restrictions. Electron-electron collisions are not taken into account in (3.2), which, within the accuracy of interest to us, is quite justified both in the low-frequency region (the static case, radio frequencies) and in the infrared part of the spectrum\({}^{9}\), in any case if a kinetic equation of the type (3.2)\({}^{*}\) is applicable.

This last point, i.e. the question of the range of applicability of the kinetic equation itself of the type (3.2), requires special discussion. The point is that an equation of the type (3.2), where the action of the electric field is taken into account by the term \(e\mathbf{E}\nabla_p f_0\), is suitable only in the quasiclassical approximation, when the electron energy changes gradually. At high frequencies, however, the process of interaction of the electron with the field has a quantum character, and the equa—

\({}^{*}\) In work\({}^{10}\) the opposite opinion was expressed, namely it is asserted that allowance for the electron-electron interaction (specifically, in\({}^{10}\) the discussion concerns “electronic viscosity”) is essential in calculating the absorptive capacity of metals. However, as shown in\({}^{9}\), the method and results of work\({}^{10}\) are incorrect.

... (3.2), generally speaking, is inapplicable. The condition for the equation (3.2) to be certainly applicable can be written in the form

\[ \hbar \omega \ll kT, \tag{3.2a} \]

where \(\omega\) is the frequency of the electromagnetic field.

For \(T \sim 300^\circ\), inequality (3.2a) means that \(\omega \ll \dfrac{kT}{\hbar} \sim 4 \cdot 10^{13}\), or \(\lambda = \dfrac{2\pi c}{\omega} \gg 4 \cdot 10^{-3}\ \text{cm} = 40\ \mu\). In fact, the range of applicability of an equation of type (3.2) is apparently somewhat broader. We shall return to this question in § 4, c), while in the remaining part of this section and in § 4 we shall be based entirely on the kinetic equation (3.2), as a rule without making any further reservations.

In the problem of reflection of light from a metallic surface, all quantities in the metal may be regarded as proportional to \(e^{i\omega t}\) and as depending only on the coordinate \(z\) (see § 2); moreover, for simplicity we choose the \(x\)-axis in the direction of the field \(\mathbf E\) in the metal; under the assumption made that the metal is isotropic, the current is then also directed along the \(x\)-axis.

Then equation (3.2) takes the form \((E \equiv E_x)\):

\[ \frac{\partial f_1(z)}{\partial z} + \frac{1+i\omega\tau}{\tau v_z}\, f_1(z) = -\frac{eE(z)}{v_z}\,\frac{\partial f_0}{\partial p_x}. \tag{3.3} \]

To find \(f_1\) and the field \(E(z)\), this equation must be solved together with the field equations

\[ \left. \begin{aligned} \operatorname{rot}\mathbf H &= \frac{4\pi}{c}\rho \mathbf v + \frac{i\omega}{c}\mathbf E, & \operatorname{rot}\mathbf E &= -\frac{i\omega}{c}\mathbf H, \\ \mathbf j_t \equiv \overline{\rho\mathbf v} &= \mathbf j+i\omega\mathbf P = \frac{2e}{(2\pi\hbar)^3}\int \mathbf v f_1\,d\mathbf p, \end{aligned} \right\} \tag{3.4} \]

where \(\mathbf j\) is the conduction-current density, \(\mathbf P\) is the polarization, and the factor 2 is associated with the need to take into account the two possible orientations of the electron spin.

Eliminating the field \(\mathbf H\) from (3.4), we have:

\[ \operatorname{rot}\operatorname{rot}\mathbf E = -\Delta \mathbf E+\operatorname{grad}\operatorname{div}\mathbf E = -i\,\frac{4\pi\omega}{c^2}\mathbf j_t + \frac{\omega^2}{c^2}\mathbf E \]

or, in the case under consideration,

\[ \left. \begin{aligned} \frac{d^2E}{dz^2}+\frac{\omega^2}{c^2}E &= i\,\frac{4\pi\omega}{c^2}j_{t,x}, \\ j_{t,x} &= \frac{2e}{(2\pi\hbar)^3}\int v_x f_1\,d\mathbf p . \end{aligned} \right\} \tag{3.5} \]

Thus, the problem reduces to the simultaneous solution of equations (3.3) and (3.5).

The approximation corresponding to the classical or normal skin-effect theory is connected simply with neglecting, in equation (3.3), the first term (the term \(\partial f_1/\partial z\)). In this case we immediately obtain*):

\[ \left. \begin{aligned} f_1&=-\frac{eE\tau}{1+i\omega\tau}\,\frac{\partial f_0}{\partial p_x},\\[4pt] j_{t,x}&=\left\{-\frac{2e^2}{(2\pi\hbar)^3}\int \frac{\tau(\mathbf p)v_x}{1+\omega\tau(\mathbf p)} \frac{\partial f_0(W(\mathbf p))}{\partial p_x}\,d\mathbf p\right\}E_x\\[4pt] &=\left\{\sigma+\frac{i\omega(\varepsilon-1)}{4\pi}\right\}E_x =\frac{i\omega}{4\pi}(\varepsilon'-1)E_x, \end{aligned} \right\} \tag{3.6} \]

where \(v_x=\partial W(\mathbf p)/\partial p_x\), and the transition to the last two expressions for \(j_{t,x}\) is, in essence, the definition of the quantities \(\sigma\), \(\varepsilon\), and \(\varepsilon'\).

Substituting (3.6) into (3.5), we obtain:

\[ \frac{d^2E}{dz^2}+\frac{\omega^2}{c^2}\varepsilon' E=0, \tag{3.7} \]

whence, as was to be expected from the introduction of the quantities \(\sigma\), \(\varepsilon\), and \(\varepsilon'\), expression (1.2) is obtained

\[ E=|E|_0 e^{-i\frac{\omega}{c}\sqrt{\varepsilon'}\,z}, \]

where \(|E|_0\) is the field at the surface (i.e. at \(z=0\)). According to (1.2) and (3.6),

\[ \frac{\partial f_1}{\partial z} =-i\frac{\omega}{c}\sqrt{\varepsilon'}\,f_1 =-\frac{\omega}{c}(\chi+i\eta)f_1. \tag{3.8} \]

It is then easy to establish the conditions for the realization of the normal skin effect, i.e. the possibility of neglecting the term \(\partial f_1/\partial z\) in (3.3). Indeed, as a result of comparing the real and imaginary parts

*) In order that the current components \(j_{t,y}\) and \(j_{t,z}\) be identically zero for \(\mathbf E=(E,0,0)\), it is of course necessary that the function \(f_0(W(\mathbf p))\), or more precisely the energy \(W(\mathbf p)\), possess the corresponding symmetry. The assumption that the energy \(W\) has the necessary symmetry is already contained in the earlier assumption of the isotropy of the metal, when \(\varepsilon'_{ik}=\varepsilon'\delta_{ik}\).

from the first two terms in (3.3) we obtain the above-mentioned conditions in the form

\[ l \ll \frac{c}{\omega\chi}=\delta_n, \tag{3.9} \]

\[ \frac{v_0}{\omega}\ll \frac{c}{\omega n} \quad \left(\text{i.e. } v_0\ll \frac{c}{n}\right), \tag{3.10} \]

where the replacement of \(v_z\) (see the second term in (3.3)) by \(v_0\) and of \(\tau\) by \(\tau_0=\dfrac{l}{v_0}\) has also been made\(^*\). Conditions (3.9) and (3.10) ensure the possibility of applying the theory of the normal skin effect for calculating both the real and imaginary parts of the impedance \(Z=R+iX\). Instead of conditions (3.9) and (3.10), one sometimes writes a single inequality obtained by comparing the moduli of the first two terms in (3.3):

\[ l \ll \frac{c\sqrt{1+\omega^2\tau_0^2}}{\omega\sqrt{n^2+\chi^2}} . \tag{3.11} \]

This condition is necessary, but not sufficient, in order that both quantities \(R\) and \(X\) may be calculated in the approximation (3.6).

From (3.6) and from all that has been said above it follows that the skin effect is normal if the current and polarization at a given point are determined by the field at the same point. On the other hand, from physical considerations it is clear that the current at some point \(A\) is determined by the field in a region with dimensions of the order of the mean free path \(l\) around this point \(A\). Therefore the current at point \(A\) depends on the field only at the same point \(A\) only if this field itself changes little over distances \(\sim l\). But the field attenuates substantially at the depth of the skin layer \(\delta_n\), whence condition (3.9), which has a simple physical meaning, is obtained. The polarization \(\mathbf{P}\), in turn, is determined by the displacement of electrons under the action of the electric field. In this case, during one period \(\theta=\dfrac{2\pi}{\omega}\) an electron travels a distance \(v_0\theta=\dfrac{2\pi v_0}{\omega}\), over which the field is constant only if this distance is less than the wavelength in the metal \(\lambda=\dfrac{2\pi c}{\omega n}\). Hence condition (3.10) is obtained. Thus, fulfillment of both conditions (3.9) and (3.10) makes it possible to use the corresponding

\(^*\) In (3.9) and (3.10), and below, \(v_0\), \(\tau_0\), and \(l\equiv l_0\) are the velocity, free-path time, and mean free path at the Fermi surface; as is well known and, in particular, shown below, only these quantities enter the final expressions for the electrical conductivity, and thus the replacement made is quite legitimate. As for the replacement of \(v_z\) by \(v_0\), it can be justified only as a result of a more detailed analysis; in the region of the normal skin effect, when conditions (3.9) and (3.10) are satisfied, such a replacement is correct in order of magnitude.

by equation (3.6), which relates the current and the polarization at a certain point only to the field at that very same point.

To obtain explicit expressions for \(\sigma\) and \(\varepsilon\), it is necessary to specify the form of the function \(W(\mathbf p)\), i.e., the dependence of the electron energy on their momentum (quasimomentum). The Fermi surface \(W(\mathbf p)=W_0\), generally speaking, is very complicated and even multiply connected; moreover, this surface cannot be found at all reliably from theoretical calculations. As a result, one usually has to choose a definite form of the Fermi surface (for example, assume that it is an ellipsoid of revolution, etc.) and then determine the characteristic parameters (the ratio of the semiaxes of the ellipsoid, etc.) by comparison of theory with experiment. However, for noble and alkali metals one may usually, to a good approximation, regard the Fermi surface as a sphere, and the velocity \(v_0\) as proportional to the momentum \(p_0\):

\[ W(\mathbf p)=W_0=\frac{p_0^2}{2m_{\mathrm{eff}}}=\frac{m_{\mathrm{eff}}v_0^2}{2}, \tag{3.12} \]

where \(m_{\mathrm{eff}}\) is the effective mass and \(v_0=\left(\dfrac{\partial W}{\partial p}\right)_0=\dfrac{p_0}{m_{\mathrm{eff}}}\) is the velocity on the Fermi surface.

For multivalent metals the Fermi surface is obviously more complicated than (3.12), but for polycrystalline specimens, when the whole picture is averaged, one may think that the use of expression (3.12) is permissible in some cases. In this case, of course, the values \(m_{\mathrm{eff}}\) and \(v_0\) have the meaning of certain averaged quantities. In one way or another, the whole subsequent exposition will be based mainly on relation (3.12). Generalization of all the results to the case of more complicated Fermi surfaces is one of the principal tasks facing the theory in this field*); the first steps in this direction have already been made\(^{11-13}\).

In the case (3.12), near the Fermi surface the density of states is

\[ \frac{2\,d\mathbf p}{(2\pi\hbar)^3} = \frac{2p_0^2\,dp\,d\Omega}{(2\pi\hbar)^3} = \frac{2^{3/2}m_{\mathrm{eff}}^{3/2}W_0^{1/2}\,dW\,d\Omega}{(2\pi\hbar)^3}, \]

or, after integration over the solid angle \(d\Omega\), the density of states is

\[ 2\left(\frac{dY}{dW}\right)_0 dW = \frac{2^{7/2}\pi m_{\mathrm{eff}}^{3/2}W_0^{1/2}}{(2\pi\hbar)^3}\,dW . \]

*) The case of the anomalous skin effect is meant, since in the region of the normal skin effect the consideration of more complicated Fermi surfaces is in principle not especially difficult and has already been carried out more than once\(^{7,8,13a}\).

Further, in the general case
\(\dfrac{\partial f_0}{\partial p_x} = \dfrac{\partial f_0}{\partial W}\dfrac{\partial W}{\partial p_x} = v_x \dfrac{\partial f_0}{\partial W}\), while for an almost degenerate gas the function \(\dfrac{\partial f_0}{\partial W}\) has a delta-like character
\(\left(\dfrac{\partial f_0}{\partial W} \ne 0\right.\) only near the point \(W = W_0\) in Fig. 5, and
\(\left.\int_0^\infty \dfrac{\partial f_0}{\partial W}\,dW = f_0(\infty) - f_0(0) = -1\right)\).

Taking this into account, we immediately obtain from (3.6)

\[ j_t = -e^2 \int \frac{\tau(v)v_x^2}{1+i\omega\tau(v)} \frac{\partial f_0}{\partial W}\cdot 2\left(\frac{dY}{dW}\right)_0 dW\,\frac{d\Omega}{4\pi} = \frac{2e^2 v_0^2 \tau_0 \left(\dfrac{dY}{dW_0}\right)_0} {3(1+i\omega\tau_0)}\,E, \tag{3.13} \]

where for \(j_t\) and \(E\) the index \(x\) has been omitted, and the function \(\tau(\mathbf p)\) is assumed to depend only on \(v = \dfrac{p}{m_{\mathrm{eff}}}\) (in the case (3.12) such an assumption is quite natural).

According to (3.6) and (3.13),

\[ \left. \begin{aligned} \varepsilon &= n^2-\chi^2 = 1 - \frac{4\pi e^2\cdot \dfrac{2}{3} v_0^2 \left(\dfrac{dY}{dW}\right)_0} {\omega^2+\nu_0^2} = 1 - \frac{4\pi e^2 N}{m(\omega^2+\nu_0^2)}, \\[6pt] \sigma &= \frac{n\chi\omega}{2\pi} = \frac{e^2\cdot \dfrac{2}{3} v_0^2 \left(\dfrac{dY}{dW}\right)_0\cdot \nu_0} {\omega^2+\nu_0^2} = \frac{e^2 N\nu_0}{m(\omega^2+\nu_0^2)} = \frac{1-\varepsilon}{4\pi}\nu_0, \\[6pt] \sigma(\omega=0) &\equiv \sigma(0) \equiv \sigma_0 = \frac{e^2N}{m\nu_0} = \frac{e^2Nl}{mv_0}, \qquad \nu_0=\frac{1}{\tau_0}=\frac{v_0}{l}, \end{aligned} \right\} \tag{3.14} \]

where \(\nu_0\) is the number of collisions.

The expressions (3.14) have exactly the same form as for a free classical electron gas with electron concentration equal to \(N\). It is precisely in this sense that the conduction electrons in a metal may be regarded as “free,” while their concentration (the effective concentration of free electrons, or conduction electrons, in the metal) is by definition equal to

\[ N=\frac{2}{3}mv_0^2\left(\frac{dY}{dW}\right)_0 = \frac{8\pi}{3}\frac{m}{m_{\mathrm{eff}}} \left(\frac{m_{\mathrm{eff}}v_0}{2\pi\hbar}\right)^3, \tag{3.15} \]

where, as in (3.14), \(m = 9.1\cdot 10^{-28}\,\mathrm{g}\) is the mass of the free electron.

Sometimes the electrons in a metal are regarded as completely free, i.e., it is assumed that

\[ m_{\mathrm{eff}}=m \quad \text{and} \quad N=N_{\mathrm{f}}=\frac{8\pi}{3}\left(\frac{mv_{0}}{2\pi\hbar}\right)^{3}, \]

but such an assumption is completely unjustified. Also irrational, as has already been noted in ^{14}, is the definition of the concentration of conduction electrons as the quantity

\[ N'=\frac{8\pi}{3}\left(\frac{m_{\mathrm{eff}}v_{0}}{2\pi\hbar}\right)^{3} =\frac{m_{\mathrm{eff}}}{m}\,N, \]

which is entirely analogous to \(N_{\mathrm{f}}\). In this case

\[ \varepsilon=1-\frac{4\pi e^{2}N'}{m_{\mathrm{eff}}\left(\omega^{2}+\nu_{0}^{2}\right)} \]

and \(N'\) is no longer the effective concentration of free electrons. Most importantly, only two quantities characterizing the metal, \(N\) and \(v_{0}\), enter the expressions (3.14), and from these formulas one can still find only the ratio

\[ \frac{N'}{m_{\mathrm{eff}}}=\frac{N}{m}. \]

The quantity \(N\) introduced in (3.14) coincides with the concentration \(N\) appearing in the “sum rule” (2.17).

With regard to formulas (3.14) it should also be noted that they determine not the complete values of \(\varepsilon_t\) and \(\sigma_t\), but only the contribution to these quantities due to the conduction electrons. However, in the frequency range discussed here, and where quantum absorption (the internal photoelectric effect) is absent,

\[ \sigma_t=\sigma \quad \text{and} \quad \varepsilon_t=\varepsilon-1+\varepsilon_0 =\varepsilon_0-\frac{4\pi e^{2}N}{m\left(\omega^{2}+\nu_{0}^{2}\right)}, \]

where far from the threshold of the internal photoelectric effect \(\varepsilon_0\sim 1\div 10\) and, generally speaking, is much smaller than

\[ \frac{4\pi e^{2}N}{m\left(\omega^{2}+\nu_{0}^{2}\right)}. \]

Below we shall not distinguish between \(\sigma\) and \(\sigma_t\) and, omitting the index \(t\), shall write \(\varepsilon\) in the form

\[ \varepsilon=\varepsilon_0-\frac{4\pi e^{2}N}{m\left(\omega^{2}+\nu_{0}^{2}\right)}. \tag{3.16} \]

The concentration \(N\) is not known in advance; attempts to determine \(N\) from considerations that one, two, etc. conduction electrons are associated with each atom are not justified and cannot have any definite quantitative value. Therefore the magni-

the quantity \(N\) must be determined by comparing formulas (3.14) and similar ones (see § 4) with experiment. In this case an especially clear picture obtains in the region of frequencies satisfying the inequalities

\[ \omega_0^2 \gg \omega^2 \gg \gamma_0^2, \tag{3.17} \]

where \(\omega_0\) is the frequency corresponding to the boundary of quantum absorption (the internal photoelectric effect). The point is that

\[ \varepsilon_0 \sim \frac{4\pi e^2 N}{m\omega_0^2} \]

and, under condition (3.17),

\[ \varepsilon \cong -\frac{4\pi e^2 N}{m\omega^2} = -3.18\cdot 10^9 \cdot \frac{N}{\omega^2}, \tag{3.18} \]

i.e., it depends only on the frequency \(\omega\) and on the quantity \(N\) to be determined. Formula (3.18) contains no mean free path at all, and therefore one may expect that this formula will remain valid also in the case of an anomalous character of the skin effect. In § 4 it will be shown that such an assumption corresponds to reality. Moreover, formula (3.18), which has a clear classical meaning, essentially does not depend on the model assumptions used. Indeed, the classical equation of motion for the aggregate of \(N\) electrons in the absence of friction has the form

\[ mN\ddot r = eNE_0 e^{i\omega t}, \]

where the interaction between the electrons drops out by virtue of the law of conservation of momentum (or, in other words, owing to the equality to zero of all internal forces). The equation just written immediately gives

\[ P=eNr=-\frac{e^2N}{m\omega^2}E_0 e^{i\omega t}, \]

whence

\[ \frac{4\pi E}{P}\cong \varepsilon-1\cong \varepsilon = -\frac{4\pi e^2 N}{m\omega^2}. \]

In view of the classical character of this result, obtaining it by a quantum method\({}^{15}\), in our opinion, contributes little that is new. The center of gravity of the whole problem lies elsewhere, namely in the question of to what extent and in what frequency range conditions (3.17) can actually be fulfilled. It is not excluded that for some metals there does not exist, or is practically unattainable, a region in which expression (3.18) is sufficiently accurate. In this case the quantity \(N\) must be determined from more general, but at the same time less reliable, formulas (see, for example, (3.14)).

In order to estimate the accuracy of expression (3.18), it is useful to expand the aforementioned more general formulas in a series with accuracy

up to and including terms of order \(\dfrac{\nu_0^2}{\omega^2}\). In the case (3.14) we then obtain:

\[ \left. \begin{aligned} \varepsilon &= n^2-\chi^2 \simeq -\frac{4\pi e^2N}{m\omega^2}\left[1-\left(\frac{e^2N}{m\omega\sigma_0}\right)^2\right] \\ &= -\frac{4\pi e^2N}{m\omega^2}\left(1-\frac{\nu_0^2}{\omega^2}\right) = -\frac{4}{3}\beta^{-2}p^2+\frac{4}{3}\beta^{-2}p^4q^2, \\[6pt] n &\simeq \frac{1}{2}\sqrt{\frac{4\pi e^2N}{m\omega^2}}\, \frac{\nu_0}{\omega}\left(1-\frac{5}{8}\frac{\nu_0^2}{\omega^2}\right) \\ &= \frac{1}{\sqrt{3}}\,\beta^{-1}p^2q -\frac{5}{8\sqrt{3}}\,\beta^{-1}p^4q^3, \\[6pt] \chi &\simeq \sqrt{\frac{4\pi e^2N}{m\omega^2}}\left(1-\frac{3}{8}\frac{\nu_0^2}{\omega^2}\right) = \frac{2}{\sqrt{3}}\beta^{-1}p -\frac{\sqrt{3}}{4}\beta^{-1}p^3q^2, \\[6pt] A &\simeq \frac{4n}{n^2+\chi^2} \simeq \nu_0\left(\frac{m}{\pi e^2N}\right)^{1/2} \left(1-\frac{1}{8}\frac{\nu_0^2}{\omega^2}\right) \\ &= \sqrt{3}\,\beta q-\frac{\sqrt{3}}{8}\beta p^2q^3, \\[6pt] \sin\varphi_0\,\tg\varphi_0 &\simeq \sqrt{n^2+\chi^2} \simeq \left(\frac{4\pi e^2N}{m\omega^2}\right)^{1/2} \left(1-\frac{1}{4}\frac{\nu_0^2}{\omega^2}\right) \\ &= \frac{2}{\sqrt{3}}\beta^{-1}p -\frac{1}{2\sqrt{3}}\beta^{-1}p^3q^2, \\[6pt] \tg 2\varphi_0 &\simeq \frac{\chi}{n} \simeq \frac{2\omega}{\nu_0}\left(1+\frac{1}{4}\frac{\nu_0^2}{\omega^2}\right) =2p^{-1}q^{-1}+\frac{1}{2}pq, \end{aligned} \right\} \tag{3.19} \]

where the quantities used below have been introduced:

\[ \left. \begin{aligned} \beta &= \frac{v_0}{c}, \qquad q=\frac{c}{v_0\sigma_0}\left(\frac{e^2N}{3\pi m}\right)^{1/2} =\frac{2c}{\sqrt{3}\,l}\left(\frac{m}{4\pi e^2N}\right)^{1/2}, \\[4pt] p &= \frac{3\pi}{\omega}\,\frac{v_0}{c} \left(\frac{e^2N}{3\pi m}\right)^{1/2}, \qquad pq=\frac{\nu_0}{\omega}. \end{aligned} \right\} \tag{3.20} \]

For good conductors at room temperature \(\nu_0\sim 3\cdot 10^{13}\) (this value is obtained if one sets \(\sigma_0=\dfrac{e^2N}{m\nu_0}\simeq 3\cdot 10^{17}\), \(N\simeq 5\cdot 10^{22}\), or from the relation \(\nu_0=\dfrac{v_0}{l}\), taking \(v_0\sim 10^8\) and \(l\sim 3\cdot 10^{-6}\)). At helium temperatures in the region of residual resistance in very pure specimens, \(l\sim 3\cdot 10^{-3}\) and \(\nu_0\sim 3\cdot 10^{10}\).

However, in the near infrared region, for example at \(\lambda=2\mu\), \(\omega\sim 10^{15}\), and thus, even at room temperature,

\[ \frac{\nu_0^2}{\omega^2}\sim 10^{-3}. \]

On the other hand, for \(\omega\sim 10^{15}\) and \(N\sim 5\cdot 10^{22}\), the quantity

\[ \frac{4\pi e^2N}{m\omega^2}\sim 10^2 \]

and the condition \(|\varepsilon|\gg \varepsilon_0\sim 1\div 10\) is also satisfied. Therefore the impossibility of neglecting \(\nu_0^2\) in comparison with \(\omega^2\), and \(\varepsilon_0\) in comparison with \(\varepsilon\), as is necessary for applying formula (3.18), can become significant only if the boundary of the internal photoeffect already lies in the infrared part of the spectrum. We shall return to the question of the accuracy of formula (3.18) in § 4; for the present we shall confine ourselves to the remark that lowering the temperature (decreasing \(\nu_0\)) and passing to the region of lower frequencies are favorable from the point of view of the possibility of applying formula (3.18) with sufficient reliability. In the corresponding frequency region, measuring experimentally \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\), or only the single quantity

\[ \sin\varphi_0\,\tg\varphi_0 \simeq \sqrt{n_{\mathrm{eff}}^2+\chi_{\mathrm{eff}}^2}, \]

we determine the concentration of conduction electrons:

\[ N=\frac{|\varepsilon|m\omega^2}{4\pi e^2} =\frac{\left(n_{\mathrm{eff}}^2+\chi_{\mathrm{eff}}^2\right)m\omega^2}{4\pi e^2} = \]

\[ =\frac{\sin^2\varphi_0\,\tg^2\varphi_0\,m\omega^2}{4\pi e^2} =1.12\cdot 10^{21}\frac{\sin^2\varphi_0\,\tg^2\varphi_0}{\lambda^2}, \tag{3.21} \]

where \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) appear instead of \(n\) and \(\chi\), since formula (3.21), under the conditions (3.17), is also valid in the region of the anomalous skin effect; in the last expression of (3.21), \(\lambda\) is the wavelength in microns. Moreover, in (3.21) equality signs have been put in place of approximate-equality signs, since, for example, at

\[ \varepsilon\sim 10^4\varepsilon \sim n_{\mathrm{eff}}^2+\chi_{\mathrm{eff}}^2 \sim \sin^2\varphi_0\,\tg^2\varphi_0 \]

with an accuracy \(\gg \dfrac{1}{\varepsilon}\sim 10^{-4}\).

Finding the concentration of conduction electrons \(N\) by means of formula (3.21) as a result of optical measurements of the quantity \(n_{\mathrm{eff}}^2+\chi_{\mathrm{eff}}^2\) is a principal task of metal optics from the point of view of obtaining information of important fundamental significance for the theory of metals. In essence, expressions (3.18) and (3.21) may even be regarded as definitions of the quantity \(N\), which also enters into a whole series of other formulas that, however, are not so independent of particular model assumptions. The unsatisfactory state of experimental metal optics noted in the introduction will now become quite clear after we point out that modern

data that would make it possible to determine the value of \(N\) from formula (3.21) are practically nonexistent. True, for a number of metals \((\mathrm{Ag}, \mathrm{Au}, \mathrm{Cu})\) there are old data more or less suitable for this purpose. But for not a single superconductor are there in fact such data (for more detail see § 6), whereas knowledge of the value of \(N\) is of the utmost importance for the theory of superconductivity\(^{16,5}\).

Besides \(N\), formulas (3.14) contain only the collision number \(\nu_0\), which can be determined only from these same formulas. In this case the theory can be checked by determining \(\nu_0\), say, from the expression for the static electrical conductivity

\[ \sigma_0=\frac{e^2N}{m\nu_0} \]

and then comparing the measured and calculated values of \(\varepsilon\) and \(\sigma\) (or \(n\), \(\chi\), \(A\), etc.; see (3.14), (3.19)) at different frequencies. Such a comparison leads, especially at low temperatures, to a clear disagreement of formulas (3.14) with experiment (see\(^{8}\), ch. 4,\(^{17}\) and below § 6). Earlier attempts were made to explain this discrepancy by surface effects, in the sense that the surface layer possesses a smaller electrical conductivity, and the like. Recently it has become clear that the point here is simply the need to take into account the anomalous character of the skin effect, and that, generally speaking, one cannot use all formulas (3.14), (3.19) in the infrared part of the spectrum. This is immediately clear from the applicability conditions of the theory of the normal skin effect (3.9) and (3.10). Indeed, at room temperature for good conductors \(l\sim 3\cdot 10^{-6}\), and in the region (3.17) the classical skin depth is

\[ \delta_n=\frac{c}{\omega\chi}\simeq \frac{c}{\sqrt{\dfrac{4\pi e^2N}{m}}}. \]

Thus, for \(N\sim 5\cdot 10^{22}\), \(\delta_n\sim 3\cdot 10^{-6}\sim l\), i.e. condition (3.9) is not fulfilled (condition (3.10) is fulfilled, which in the present case corresponds to the possibility of using formula (3.18)). And only for poor conductors, when the mean free path \(l\) (and in some cases \(N\)) decreases, can one assume at room temperature that in the region (3.17) \(l\ll\delta_n\), and use the theory of the normal skin effect for calculating all quantities \((A, n,\) etc.)\(^*\). In the region of low temperatures, however, where the mean free path \(l\) reaches, for very pure samples, values \(l\sim 10^{-2}\div 3\cdot 10^{-3}\), there can generally be no question of using the theory of the normal skin effect.

\(^*\) It should be borne in mind that for calculating some quantities the condition \(l\ll\delta_n\) is not necessary. The latter, as already indicated, also occurs in calculating \(\varepsilon\) (see (3.18)).

§ 4. Microtheory in the Case of the Anomalous Skin Effect

a) Fundamentals of the theory of the anomalous skin effect

In the general case, when the character of the skin effect may be arbitrary (strongly anomalous, weakly anomalous, or normal), the determination of the surface impedance \(Z\) and of other quantities (for example, the field in the metal) must be carried out on the basis of solving equations (3.3) and (3.5)*), without neglecting the term \(\dfrac{\partial f_1}{\partial z}\) in (3.3). Bearing in mind in what follows the case (3.12), i.e., putting \(\mathbf p=m_{\mathrm{eff}}\mathbf v\), we write equation (3.3) in the form

\[ \frac{\partial f_1}{\partial z} + \frac{1+i\omega\tau}{\tau v_z}\,f_1 = - \frac{eE(z)}{m_{\mathrm{eff}}v_z}\, \frac{\partial f_0}{\partial v_x}. \tag{4.1} \]

The general solution of this equation is as follows:

\[ f_1 = \exp\!\left[ -\frac{(1+i\omega\tau)z}{\tau v_z} \right] \times \]

\[ \times \left\{ F(\mathbf v) - \frac{e}{m_{\mathrm{eff}}v_z}\, \frac{\partial f_0}{\partial v_x} \int_0^z E(\xi) \exp\!\left[ \frac{1+i\omega\tau}{\tau v_z} \right] \,d\xi \right\}, \tag{4.2} \]

where \(F(\mathbf v)\) is an arbitrary function of \(\mathbf v\).

We shall now denote by \(f_1^{(1)}\) and \(f_1^{(2)}\) the values of \(f_1\), respectively, for \(v_z<0\) and \(v_z>0\) (see Fig. 1; for \(v_z>0\) the electrons move into the interior of the metal, and for \(v_z<0\), toward its surface). Of course, \(f_1=f_1^{(1)}+f_1^{(2)}\). For \(v_z<0\) the function \(F(\mathbf v)\) is determined from the requirement that, as \(z\to\infty\) (in the interior of the metal), the function \(f_1\) not be infinitely large. As a result,

\[ f_1^{(1)} = \frac{e}{m_{\mathrm{eff}}v_z}\, \frac{\partial f_0}{\partial v_x} \exp\!\left[ -\frac{(1+i\omega\tau)\xi}{\tau v_z} \right] \times \]

\[ \times \int_z^\infty E(\xi) \exp\!\left[ \frac{(1+i\omega\tau)\xi}{\tau v_z} \right] \,d\xi . \tag{4.3} \]

*) In (3.5) it is assumed that \(\varepsilon_0=1\) (only the conduction electrons contribute to \(\varepsilon\)). In the more general case, in (3.5) one must replace \(\dfrac{\omega^2}{c^2}E\) by \(\dfrac{\omega^2}{c^2}\varepsilon_0E\).

For \(v_z>0\) the function \(F(\mathbf v)\), and consequently also \(f_1^{(2)}\), are determined by the reflection conditions at the metal surface (at \(z=0\)). If the reflection is specular, then

\[ f_1^{(2)}(v_x, v_y, v_z, z=0)=f_1^{(1)}(v_x, v_y,-v_z,z=0). \tag{4.4} \]

In the case of purely diffuse reflection, the electrons after reflection completely “forget” the ordered velocity they had, i.e.,

\[ f_1^{(2)}(v_x, v_y, v_z, z=0)=0. \tag{4.5} \]

Finally, if a fraction \(\rho\) of the electrons is reflected specularly, and the remaining part \((1-\rho)\) is reflected diffusely, then

\[ f_0+f_1^{(2)}(v_x, v_y, v_z, z=0)= \]

\[ =\rho\bigl[f_0+f_1^{(1)}(v_x, v_y,-v_z,z=0)\bigr]+(1-\rho)f_0. \tag{4.6} \]

The problem under discussion was solved\(^{18-20}\) in all three cases (4.4)—(4.6), with the quantity \(\rho\) assumed independent of \(\mathbf v\) (more general boundary conditions with \(\rho=\rho(\mathbf v)\) were not yet used).

As follows from comparison of theory with experiment, even the best samples reflect electrons diffusely.\(^{21,34}\) In the case of polycrystalline mirrors, which we mainly had in mind, this result is quite natural, and there is no reason to suppose that further improvement in the treatment of the surface will change the character of the reflection. Therefore, although the possibility is not excluded of the existence of samples reflecting electrons predominantly specularly,* below, unless the contrary is stated, we shall usually have in mind only the case of diffuse reflection, when \(\rho=0\). Condition (4.5) leads to the expression

\[ f_1^{(2)}=-\frac{e}{m_{\mathrm{eff}}v_z}\frac{\partial f_0}{\partial v_x} \exp\left[-\frac{(1+i\omega\tau)}{\tau v_z}z\right]\times \]

\[ \times \int_0^z E(\xi)\exp\left[\frac{1+i\omega\tau}{\tau v_z}\xi\right]\,d\xi. \tag{4.7} \]

* Such (i.e., specularly reflecting) samples would, it would seem, have to be single-crystal samples with a sufficiently well-treated surface and kept at low temperature, when the thermal motion of atoms on the surface is small. However, even under such “ideal” conditions the reflection, generally speaking, will not be strictly specular, owing to the possibility of diffraction arising when electrons are incident on the crystal surface. In this respect there are a number of possibilities that have recently been revealed in work\(^{25}\) and more fully by L. V. Keldysh.

The symmetric part of the distribution function \(f_0\) makes no contribution to the expression for the current density, which, according to (3.4), (3.5), and (3.12), has the form

\[ j_t(z)=2e\left(\frac{m_{\mathrm{eff}}}{2\pi\hbar}\right)^3 \int v_x f_1\,dv_x\,dv_y\,dv_z, \tag{4.8} \]

where the index \(x\) in \(j_t\) is omitted.

Substituting (4.3) and (4.7) into (4.8), after a number of transformations we obtain\(^{10}\):

\[ \left. \begin{aligned} j_t(z)&=\frac{e^2 m_{\mathrm{eff}}^2 v_0^2}{4\pi^2\hbar^3} \int_0^\infty K_a\!\left(\frac{z-\xi}{l}\right)E(\xi)\,d\xi,\\[6pt] K_a(u)&=\int_0^{\pi/2}\sin^3\theta\,\sec\theta\, e^{-(1+ia)|u|\sec\theta}\,d\theta\\ &=E_{i_1}\{(1+ia)|u|\}-E_{i_3}\{(1+ia)|u|\}, \end{aligned} \right\} \tag{4.9} \]

where

\[ E_{in}(u)=\int_1^\infty \frac{e^{-su}}{s^n}\,ds \qquad (\operatorname{Re}u>0),\qquad a=\omega\tau_0,\quad l=\tau_0 v_0, \]

and, in carrying out the integration, polar coordinates have been introduced,

\[ \mathbf v=\{v\sin\theta\cos\varphi,\ v\sin\theta\sin\varphi,\ v\cos\theta\}, \]

and the formula

\[ -\int_0^\infty g(v)\frac{\partial f_0}{\partial v}\,dv=g(v_0) \]

has also been used.

By virtue of (4.9), equation (3.5) takes the form\(^*\):

\[ \frac{d^2E}{dz^2}+\frac{\omega^2}{c^2}E =i\,\frac{\omega e^2 m_{\mathrm{eff}}^2 v_0^2}{\pi\hbar^3} \int_0^\infty K_a\!\left(\frac{z-\xi}{l}\right)E(\xi)\,d\xi. \tag{4.10} \]

Or, introducing the variables \(\zeta=z/l\), \(\eta=\xi/l\), and denoting \(E(\zeta l)=\Psi(\zeta)\),

\[ \frac{d^2\Psi(\zeta)}{d\zeta^2} +\frac{\omega^2 l^2}{c^2}\Psi(\zeta) =i\alpha\int_0^\infty K_a(\zeta-\eta)\Psi(\eta)\,d\eta, \tag{4.11} \]

\[ {}^* \text{Let us note that, following in this paragraph paper }{}^{19}, \text{ we sometimes use somewhat different notation compared with that adopted in }{}^{49}. \text{ In particular, in }{}^{19}\text{ the quantity }\rho\text{ is denoted by }p. \]

where

\[ \alpha=\frac{\omega e^2 m_{\mathrm{eff}}^2 v_0^2 l^3}{\pi c^2\hbar^3} =\frac{3}{2}\frac{l^2}{\delta_0^2}, \tag{4.12} \]

\[ \delta_0=\frac{c}{\sqrt{2\pi\sigma_0\omega}},\qquad \sigma_0=\frac{e^2 Nl}{m v_0},\qquad N=\frac{8\pi}{3}\frac{m}{m_{\mathrm{eff}}}\left(\frac{m_{\mathrm{eff}}v_0}{2\pi\hbar}\right)^3 . \]

The quantity \(\delta_0\) is the skin-layer depth introduced earlier (see (1.24)) in the case of the normal skin effect at radio frequencies (in (1.24) one should put \(\sigma\), not \(\sigma_0\), but under condition (1.23) this is practically the same thing).

The integro-differential equation (4.10)—(4.11), or the analogous equation for \(\rho\ne 0\), are the basic ones in the theory of the anomalous skin effect. Unfortunately, its solution, especially in the case under consideration and, in general, when \(\rho\ne 1\), is rather complicated and cumbersome; moreover, final results are obtained only after numerical integration. Therefore, below we shall have to confine ourselves to giving the results and a number of remarks (for detailed calculations see \(^{19,20}\)).

First of all, we note that in order to find the surface impedance there is no need to investigate the general expression for the field \(E(z)=\Psi(\zeta)\), since according to (2.8) and (2.9)

\[ Z=\frac{4\pi}{c}\left[\frac{E_x}{H_y}\right]_0 \quad \text{and in the metal} \quad H_y=-\frac{c}{i\omega}\frac{\partial E_x}{\partial z}, \]

whence

\[ Z=-i\frac{4\pi\omega}{c^2} \left[\frac{E_x}{\dfrac{\partial E_x}{\partial z}}\right]_0 =-i\frac{4\pi\omega l}{c^2}\frac{\Psi(0)}{\Psi'(0)}, \tag{4.13} \]

and it is necessary to know only the quantity \(\Psi'(0)=\left[\dfrac{d\Psi}{d\zeta}\right]_0\), since the field \(\{E\}_0=\Psi(0)\) is prescribed*).

Next, the transition to the case of the normal skin effect (see § 3) can easily be carried out directly in relations (4.9) and (4.10), since for \(l\to 0\) the function \(K_a\left(\dfrac{z-\xi}{l}\right)\) is very sharply peaked and, by virtue of (3.14) and (3.15),

\[ j_t(z)= \frac{e^2 m_{\mathrm{eff}}^2 v_0^2}{4\pi^2\hbar^3} E(z)\int_0^\infty K_a\left(\frac{z-\xi}{l}\right)d\xi = \frac{4e^2 m_{\mathrm{eff}}^2 v_0^2 l}{3(1+i\omega\tau_0)\,4\pi^2\hbar^3}E(z) \]

\[ = \frac{e^2 Nl}{m v_0(1-i\omega\tau_0)}E(z) = \left(\sigma+i\omega\frac{\varepsilon-1}{4\pi}\right)E(z), \tag{4.14} \]

which is in agreement with (3.6).

\[ \text{*) In solving equation (4.10)—(4.11), in addition to specifying the value } \Psi(0), \]
one also takes into account that \(\Psi(\infty)=0\) (the field does not penetrate into the depth of the metal).

The expression for the field \(E(z)\) obtained as a result of solving equation (4.10) is very complicated, and the dependence of \(E\) on \(z\) is, of course, not simply exponential. As an illustration, we note that even in the simpler case of purely specular reflection \((\rho=1)\) we have:

\[ E(z)=\Psi(\zeta)=-\frac{2\Psi'(0)}{\pi}\int_{0}^{\infty} \frac{\cos \zeta u\,du}{u^2-\dfrac{\omega^2 l^2}{c^2}+i\alpha s(u)}, \]

\[ s(u)=\int_{-\infty}^{+\infty} K_a(v)e^{-ivu}\,dv. \tag{4.15} \]

For large values of \(\zeta\),

\[ \Psi(\zeta)=C_1 e^{-\beta \zeta}+C_2\frac{e^{-\zeta}}{\zeta^2}, \tag{4.16} \]

where \(C_1\) and \(C_2\) are constants and \(\beta\) is a complex parameter depending on \(\alpha\).

For an arbitrary value of the parameter \(\alpha\) (see (4.12)), the resulting expressions for \(E(z)\) and \(Z\) are not only cumbersome and nontransparent, but may also be very sensitive to the assumptions made in deriving them. For example, in the initial kinetic equation (3.2) the so-called collision integral, reflecting the effect on the distribution function of collisions of electrons with the lattice, was written in the form \(\dfrac{f_1}{\tau}=\nu f_1=\dfrac{v}{l}f_1\). As already indicated in § 3, such an approximate expression for the collision integral at low and high temperatures is usually justified in the theory of electrical conductivity. However, in the case under consideration of the anomalous skin effect, this approximation is, generally speaking, not always valid near the boundary of the metal, where in the case of diffuse reflection the distribution function changes sharply depending on the angles in velocity space\({}^9\). The circumstance noted, as well as the possible influence of the anisotropy of the mean free path, becomes immaterial in the limiting case of a sharply anomalous skin effect, when

\[ \alpha=\frac{3}{2}\frac{l^2}{\delta_0^2}\gg 1. \tag{4.17} \]

This case is also (not counting the classical limit \(l\to 0\)) the simplest from the point of view of carrying all calculations through to the end, and at the same time very important in practice, since it is usually realized throughout the entire range of helium temperatures.

In this case, at radio frequencies, at least for good conductors, the condition*) is also satisfied

\[ \omega\tau_0 \ll 1 \quad (\text{i.e. } \omega \ll \nu_0). \tag{4.18} \]

In the limiting case (4.17)—(4.18), i.e. in the radio range under conditions of a strongly pronounced anomalous skin effect, with diffuse reflection of electrons from the boundary,

\[ Z \equiv Z_\infty = R_\infty + iX_\infty = \left( \frac{\sqrt{3}\pi\omega^2 l}{c^4\sigma_0} \right)^{1/3} (1+\sqrt{3}i). \tag{4.19} \]

If, however, the reflection is purely specular, the same formula is obtained, but with the additional factor \(8/9\). Thus, at radio frequencies the character of the reflection of electrons from the boundary has little effect on the result, although the experimental data make it possible in this case also to give a definite preference to formula (4.19), i.e. they indicate the diffuse character of the reflection.

For comparison we shall also give here the expression for \(Z\) in the case of the normal skin effect under condition (4.18), equivalent to condition (1.23):

\[ |\varepsilon| = \left| \frac{4\pi e^2 N}{m(\omega^2+\nu_0^2)} - 1 \right| \ll \frac{4\pi\sigma}{\omega} = \frac{4\pi e^2 N}{m(\omega^2+\nu^2)}\cdot\frac{\nu_0}{\omega}. \tag{1.23a} \]

In this case (see (1.24))

\[ \left. \begin{aligned} Z_n = R_n + iX_n &= \frac{4\pi}{c\sqrt{\varepsilon'}} = \frac{4\pi(n+i\varkappa)}{c(n^2+\varkappa^2)} \simeq \\[3pt] &\simeq \frac{2\pi}{c\varkappa}(1+i) \simeq \left(\frac{2\pi\omega}{c^2\sigma_0}\right)^{1/2}(1+i) = \frac{2\pi\omega\delta_0}{c^2}(1+i), \\[3pt] \delta_0 &= \left(\frac{c^2}{2\pi\omega\sigma_0}\right)^{1/2}. \end{aligned} \right\} \tag{4.20} \]

If expression (4.19) is also written in the form

\[ Z_\infty = \frac{2\pi\omega\delta_\infty}{c^2}(1+\sqrt{3}i), \tag{4.19a} \]

then

\[ \delta_\infty = \left(\frac{\sqrt{3}\,c^2 l}{8\pi^2\omega\sigma_0}\right)^{1/3} = \left(\frac{\sqrt{3}\,l}{4\pi\delta_0}\right)^{1/3}\delta_0. \tag{4.21} \]

For fairly typical values at low temperatures,
\(l=10^{-3}\,\text{cm}\) and \(\sigma_0=10^{20}\) CGSE, at
\(\omega=6\cdot10^{10}\left(\lambda=\dfrac{2\pi c}{\omega}\simeq3\,\text{cm}\right)\),

*) The question of with what accuracy, in the case of the anomalous skin effect at radio frequencies, the quantity \(\omega\tau_0\) may be neglected in comparison with unity is analyzed in detail in article \(^{24}\).

\(\delta_0=5\cdot 10^{-6}\ \mathrm{cm},\quad \alpha=6\cdot 10^4\) and \(\delta_\infty=3\delta_0=1.5\cdot 10^{-5}\ \mathrm{cm}\). Consequently, not only condition (4.17), but also the condition \(\alpha'=\dfrac{3l^2}{2\delta_\infty^2}\gg 1\) are very well satisfied. The normal skin effect in the radio range occurs only for frequencies \(\omega \ll \dfrac{c^2}{3\pi\sigma_0 l^2}\) (i.e., under the condition \(\alpha \ll 1\); see (3.9)); with the above values of \(l\) and \(\sigma_0\) this means that \(\omega \ll 10^5\) (\(\lambda \gg 1\ \mathrm{km}\)).

From formulas (4.19) and (4.21) it is clear that, in the limiting case under consideration, the mean free path \(l\) drops out of the expressions for \(Z_\infty\) and \(\delta_\infty\), since (see (3.14))

\[ \frac{\sigma_0}{l}=\frac{e^3N}{mv_0}. \tag{4.22} \]

In connection with the fact that only the quantity \(l\) depends at all appreciably on temperature, this result means that the quantity \(Z_\infty\) should not depend on temperature, which is also confirmed experimentally \(^{21,22}\). It is also characteristic that

\[ X_\infty=\sqrt{3}\,R_\infty, \tag{4.23} \]

whereas \(X_n=R_n\) (see (4.20)).

Measurements of the quantity \(Z_\infty\), or of the quantity \(R_\infty\) alone, make it possible to determine the ratio \(\sigma_0/l\), and hence also \(l\), since the static electrical conductivity \(\sigma_0\) is measured independently. Measurements of \(\sigma_0/l\) also make it possible to determine the velocity at the Fermi surface \(v_0\):

\[ v_0=\frac{e^2N}{m\left(\dfrac{\sigma_0}{l}\right)} =2.53\cdot 10^8\frac{N}{\left(\dfrac{\sigma_0}{l}\right)}, \tag{4.24} \]

where the concentration \(N\) must be determined from independent optical measurements.

To find \(v_0\) one may proceed differently, using the known expression \(^{7,8}\) for the electronic part of the heat capacity of metals, which can be measured at low temperatures:

\[ C^{(e)}=\gamma T,\qquad \gamma=\frac{2\pi^2}{3}k^2\left(\frac{dV}{dW}\right)_0 =\frac{\pi^2k^2N}{mv_0^2}, \tag{4.25} \]

where \(k=1.38\cdot 10^{-16}\ \dfrac{\mathrm{erg}}{\mathrm{degree}}\) is Boltzmann’s constant. According to (4.25),

\[ v_0=\pi k\sqrt{\frac{N}{m\gamma}} =1.44\cdot 10^{-2}\sqrt{\frac{N}{\gamma}}. \tag{4.26} \]

Comparison of expressions (4.24) and (4.26) may serve to check the theory. For the same purpose the relation

\[ \left. \begin{aligned} \frac{\left(\dfrac{\sigma_0}{l}\right)^2}{\gamma N} &= \frac{e^4}{\pi^2 \hbar^2 m} = 3.1 \cdot 10^{20}\ \mathrm{CGSE}, \\[6pt] N &= 3.2 \cdot 10^{-21}\, \frac{\left(\dfrac{\sigma_0}{l}\right)^2}{\gamma}, \end{aligned} \right\} \tag{4.27} \]

which is obtained\(^{16,23}\) from (4.22) and (4.25) and contains the quantities \(\gamma\), \(N\), and \(\dfrac{\sigma_0}{l}\), measured by independent methods\(^*\), is also suitable.

The parameters \(N\), \(v_0\), and \(l\), not to mention \(\varepsilon_0\), are the only quantities appearing in the problem in the approximation used. The other parameters are expressed through these three, so that, for example, the effective mass \(m_{\mathrm{eff}}\) is equal to (see (3.15)):

\[ m_{\mathrm{eff}} = \sqrt{\frac{3(2\pi\hbar)^3 N}{8\pi m v_0^3}} = 6.15 \cdot 10^{-27} \sqrt{\frac{N}{v_0^3}} . \tag{4.28} \]

Let us recall that in the region of the normal skin effect only two quantities, \(N\) and \(\nu_0=\dfrac{v_0}{l}\), can be determined from experiment (to find the velocity \(v_0\), one may, however, use measurements of \(\gamma\); see (4.25)). In the region of the anomalous skin effect, however, as was said, the velocity \(v_0\) is determined without using data on the electronic heat capacity.

b) Results pertaining to the infrared part of the spectrum

We now turn to the results of the theory of the anomalous skin effect in the optical (mainly infrared) part of the spectrum (see \(^{17-19,14}\) and, in particular, \(^{20}\)). Here we shall restrict ourselves to the most important case, when

\[ \omega \tau_0 \gg 1 \quad (\text{i.e. } \omega \gg \nu_0), \tag{4.29} \]

\[ |\varepsilon| \gg \varepsilon_0 \sim \frac{4\pi e^2 N}{m\omega_0^2} \quad (\text{i.e. } \omega \ll \omega_0). \tag{4.30} \]

\(^*\) From (4.25) and (4.27) we obtain:

\[ v_0 = \frac{\pi^2 \hbar^2}{e^2}\, \frac{\dfrac{\sigma_0}{l}}{\gamma} = 8.15 \cdot 10^{-13} \frac{\dfrac{\sigma_0}{l}}{\gamma}. \tag{4.27a} \]

and

\[ \frac{a}{(\omega\tau_0)^3} = \frac{-\dfrac{3}{2}\dfrac{l^2}{v_0^2}} {\left(\omega\dfrac{l}{v_0}\right)^3} = \frac{3\pi e^2 N v_0^2}{m c^2 \omega^2} \ll 1. \tag{4.31} \]

Conditions (4.29) and (4.30) essentially coincide with condition (3.17) and, as indicated in § 3, can be satisfied in a very broad range of frequencies, including the entire infrared part of the spectrum. Condition (4.31) is more stringent than condition (4.29), in which \(\tau_0\) increases as the temperature is lowered. For \(N\sim 5\cdot 10^{22}\) and \(v_0\sim 10^8\), inequality (4.31) has the form

\[ \omega^2 \gg 10^{27} \left(\lambda=\frac{2\pi c}{\omega}\ll 100\,\mu = 10^{-2}\ \mathrm{cm}\right). \tag{4.32} \]

In addition, one must not forget an inequality of type (3.2a), which restricts the application of the kinetic equation used; however, this aspect will be discussed only somewhat later (see § 4, c).

If inequalities (4.29)—(4.31) are satisfied, we have\(^*\):

\[ \varepsilon_{\mathrm{eff}} = n_{\mathrm{eff}}^2-\varkappa_{\mathrm{eff}}^2 \simeq \]

\[ \simeq -\underline{\frac{4}{3}\beta^{-2}p^2} + \beta^{-2}p^4 \left( \frac{301}{720} + \frac{5}{2\sqrt{3}}q + \underline{\frac{4}{3}q^2} \right), \tag{4.33} \]

\[ n_{\mathrm{eff}} \simeq \frac{\beta^{-1}p^2}{\sqrt{3}} \left(\frac{\sqrt{3}}{4}+q\right) - \]

\[ -\beta^{-1}p^4 \left( \frac{86}{945} + \frac{16\ln 2}{315} + \frac{14}{15\sqrt{3}}q + \frac{3}{4}q^2 + \underline{\frac{5}{8\sqrt{3}}q^3} \right), \tag{4.34} \]

\[ \varkappa_{\mathrm{eff}} \simeq \underline{\frac{2}{\sqrt{3}}\beta^{-1}p} - \beta^{-1}p^3 \left[ \frac{4}{15\sqrt{3}} + \frac{1}{2}q + \underline{\frac{\sqrt{3}}{4}q^2} \right], \tag{4.35} \]

\[ A\simeq \frac{cR}{\pi} = \frac{4n_{\mathrm{eff}}}{n_{\mathrm{eff}}^2+\varkappa_{\mathrm{eff}}^2} \simeq \sqrt{3}\beta \left(\frac{\sqrt{3}}{4}+q\right) - \]

\[ -\beta p^2 \left[ \frac{16\ln 2}{105} + \frac{8723}{80640} + \frac{83\sqrt{3}}{192}q + \frac{3}{4}q^2 + \underline{\frac{\sqrt{3}}{8}q^3} \right], \tag{4.36} \]

\[ \text{—} \]

\(^*\) These formulas are given in \(^{20}\), where the quantities \(\beta\), \(q\), and \(p\) used here and in \(^{14}\) (see (3.20)) are denoted respectively by the bold letters \(\boldsymbol{\nu}\), \(\boldsymbol{\sigma}^{-1}\), and \(\boldsymbol{\nu}^{-1}\).

\[ \frac{cX}{\pi}=\frac{4\chi_{\mathrm{eff}}}{n_{\mathrm{eff}}^{2}+\chi_{\mathrm{eff}}^{2}} \simeq \underline{2\sqrt{3}\,\beta p^{-1}}+ \]

\[ +\beta p\left[\frac{83\sqrt{3}}{480}+\frac{3}{4}q+\underline{\frac{\sqrt{3}}{4}q^{2}}\right], \tag{4.37} \]

\[ \sin\varphi_{0}\tan\varphi_{0}\simeq \sqrt{\,n_{\mathrm{eff}}^{2}+\chi_{\mathrm{eff}}^{2}\,}\simeq \]

\[ \simeq \underline{\frac{2}{\sqrt{3}}\beta^{-1}p} -\beta^{-1}p^{3}\left[\frac{211}{960\sqrt{3}}+\frac{3}{8}q+\underline{\frac{1}{2\sqrt{3}}q^{2}}\right], \tag{4.38} \]

\[ \tan 2\varphi_{0}\simeq \frac{\chi_{\mathrm{eff}}}{n_{\mathrm{eff}}} \simeq \frac{2p^{-1}}{\dfrac{\sqrt{3}}{4}+q} \left[ 1+p^{2}\left( \frac{83}{960}+\frac{\sqrt{3}}{8}q+\frac{1}{8}q^{2} +\right.\right. \]

\[ \left.\left. +\frac{ \dfrac{16\ln 2}{105\sqrt{3}}+\dfrac{8723}{80640\sqrt{3}}+\dfrac{83}{192}q+\dfrac{\sqrt{3}}{4}q^{2}+\underline{\dfrac{1}{8}q^{3}} }{ \dfrac{\sqrt{3}}{4}+q } \right) \right]. \tag{4.39} \]

In these formulas the parameters \(\beta\), \(q\), and \(p\) are defined according to (3.20), and the underlined terms are those obtained in the same approximation in the case of the normal skin effect (see (3.19)*). In this case

\[ -\frac{4}{3}\beta^{-2}p^{2}=-\frac{4\pi e^{2}N}{m\omega^{2}},\qquad pq=\frac{\nu_{0}}{\omega} \quad\text{and}\quad p^{2}=3\pi\frac{e^{2}Nv_{0}^{2}}{mc^{2}\omega^{2}}, \]

i.e., under conditions (4.29)—(4.31), \(pq\ll 1\) and \(p^{2}\ll 1\). Therefore in (4.33)—(4.39) the terms of higher degree with respect to \(p\) are small in comparison with the first terms (the terms of lowest degree with respect to \(p\)). It is clear from the formulas that, in the expressions for \(\varepsilon_{\mathrm{eff}}\), \(\chi\), \(\dfrac{cX}{\pi}\),

*) It is useful to recall that, by virtue of (1.19), (2.9), and what was said in § 3,

\[ 1-A\simeq \left| \frac{\dfrac{4\pi}{c}-Z}{\dfrac{4\pi}{c}+Z} \right|^{2}, \]

whence, for small \(\left(\dfrac{cX}{\pi}\right)^{2}\) and \(\left(\dfrac{cR}{\pi}\right)^{2}\), we obtain:

\[ A\simeq \frac{cR}{\pi} \left\{1-\frac{1}{16}\left(\frac{cX}{\pi}\right)^{2}-\ldots\right\} -\frac{1}{2}\left(\frac{cR}{\pi}\right)^{2}-\ldots \]

Under conditions (4.29) and (4.30), with sufficient accuracy,

\[ A=\frac{cR}{\pi}=\frac{4n_{\mathrm{eff}}}{n_{\mathrm{eff}}^{2}+\chi_{\mathrm{eff}}^{2}} \]

(see also (1.19) and (3.19)), as was also adopted in (4.36).

OPTICAL PROPERTIES OF METALS

and \(\sin\varphi_0, \tg\varphi_0\), the higher (principal) terms coincide with the corresponding terms for the normal skin effect (see (3.19)). For

\[ q=\frac{c}{v_0\delta_0}\left(\frac{e^2N}{3\pi m}\right)^{1/2} =\frac{c}{v_0}\frac{\nu_0}{\left(\dfrac{3\pi e^2N}{m}\right)^{1/2}}\gg 1, \tag{4.40} \]

i.e.

\[ l\gg \frac{c}{\left(\dfrac{3\pi e^2N}{m}\right)^{1/2}}, \tag{4.40a} \]

expressions (4.33)—(4.39) already pass completely into (3.19). This is quite understandable, since condition (4.40a) coincides with the condition for the realization of the normal skin effect (3.9), because in the region (4.29)

\[ \delta_n=\frac{c}{\omega\chi}\simeq \frac{c}{\omega\sqrt{\dfrac{4\pi e^2}{m\omega^2}}} = \frac{c}{\sqrt{\dfrac{4\pi e^2N}{m}}} \quad\text{(see (3.19) and (4.35)).} \]

What has been said substantiates the assertion made in § 3, namely that in the region (3.17) formula (3.18) is usually valid to a good approximation also in the case of the anomalous skin effect. To estimate the accuracy of this formula,

\[ \varepsilon=-\frac{4\pi e^2N}{m\omega^2} \tag{3.18} \]

we indicate that for typical values \(v_0\sim 10^8\), \(N\sim 5\cdot 10^{22}\), \(p^2\sim \dfrac{10^{27}}{\omega^2}\sim 10^4\lambda^2\), and for \(\lambda\sim 10\mu=10^{-3}\,cm\), \(p^2\sim 10^{-2}\); therefore, according to (4.33), for \(q\ll 1\) expression (3.18) is valid with an accuracy of \(\sim 0.3\%\). This conclusion is not changed at room temperature, when for typical metals \(q\sim 1\). The application of formula (3.18) is therefore limited to the case when the boundary of the internal photoeffect \(\omega_0\) already lies in the infrared part of the spectrum (see also § 4, c). In this case, in order to satisfy condition (4.30), it is necessary to lower the frequency, which may lead to a contradiction with conditions (4.31) and (4.32)\(^*\) and to an inaccuracy of formula (3.18).

It must also be noted once more that formulas (4.33)—(4.39) refer to the case of diffuse reflection of electrons from the surface of the metal. In the case of specular reflection of electrons, in contrast to the situation occurring in the radio range (see the end of § 4, a), one obtains, generally speaking, substantially different results. Thus

\(^*\) It should be noted that formulas can be obtained which are also suitable when conditions (4.31) and (4.32) are not satisfied. Therefore the use of this condition is not necessary.

for specular reflection:

\[ \varepsilon_{\mathrm{eff}} \approx -\frac{4}{3}\beta^{-2}p^2+\beta^{-2}p^4\left(-\frac{16}{45}+\frac{4}{3}q^2\right), \tag{4.41} \]

\[ A \approx \sqrt{3}\,\beta q+\beta p^2\left(\frac{2}{3}+\frac{2}{\sqrt{3}}q-\frac{\sqrt{3}}{8}q^3\right). \tag{4.42} \]

In the case of \(\varepsilon_{\mathrm{eff}}\), the difference lies in the small terms, which are in any case neglected in passing to the basic formula (3.18). But in the case of \(A\), under specular reflection the first term coincides with that occurring for the normal skin effect, and in the region of the anomalous skin effect it differs substantially from the expression obtained for diffuse reflection. In this latter case, for a sharply pronounced anomalous skin effect \((q \ll 1)\), neglecting terms \(\sim p^2\), we have (see (4.36)):

\[ A=A_d\approx \frac{3}{4}\beta=\frac{3}{4}\frac{v_0}{c}. \tag{4.43} \]

This expression does not depend on the static electrical conductivity \(\sigma_0\), and consequently not on temperature, whereas for the normal skin effect, likewise neglecting terms \(\sim p^2\),

\[ A=A_n=\sqrt{3}\beta q=\frac{1}{q_0}\left(\frac{e^2N}{\pi m}\right)^{1/2} =v_0\left(\frac{m}{\pi e^2N}\right)^{1/2} =\frac{v_0}{l}\left(\frac{m}{\pi e^2N}\right)^{1/2}. \tag{4.44} \]

Expression (4.44) decreases strongly when the temperature is lowered, since \(\sigma_0\) increases thousands of times in going from room temperature to helium temperature. In experiment, however, such a sharp decrease in the absorptivity \(A\) is not observed (see § 6), and, at least qualitatively, formula (4.36) is confirmed. A test of the theory is then possible by determining the parameters \(N\) and \(v_0\) from other measurements (\(N\) from measurements of \(\varepsilon_{\mathrm{eff}}\) and \(v_0\) from measurements of the electronic part of the heat capacity or \(\sigma_0/l\); see § 4,a)) and by comparing with experiment the values of \(A\) computed using these parameters. In the limiting case of the anomalous skin effect and for \(p^2\ll 1\), for this purpose, if one does not speak of taking into account the quantum effects discussed below, it is convenient to use the simple expression (4.43). In addition to the absorptivity \(A\), the expressions for \(n_{\mathrm{eff}}\) and \(\tan 2\rho_0\) also depend strongly on the character of the reflection of the electrons from the metal surface and, above all, on whether the skin effect is normal or anomalous (see (4.34) and (4.39)). This is quite understandable, since there are altogether two independent quantities \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\), while the quantities determined (in the case under consideration) by the coefficient \(\chi_{\mathrm{eff}}\) depend only weakly on \(l\) and, consequently, on the character of the skin effect and of the reflection of electrons from the surface. Among the number

such quantities include \(\varepsilon_{\mathrm{eff}}\), \(\dfrac{cX}{4\pi}\), and \(\sin\varphi_\psi \tan\varphi_\psi\) (see (4.33), (4.37) and (4.38). The same quantities that are determined by the value of \(n_{\mathrm{eff}}\) are, generally speaking, sensitive to the value of \(l\) and may differ appreciably depending on the character of the skin effect and of electron reflection from the surface; such quantities include \(A\) and \(\tan 2\varphi_\psi\).

The question of comparing the theory with experiment and the problems arising in this area will be discussed in § 6 and in the conclusion. Here we wish to dwell further on the physical meaning of the results obtained and on an elementary derivation of the limiting formula (4.43), as well as of the limiting formula obtained from (4.42) under the very stringent condition \(q \ll p^2\):

\[ A \equiv A_s = \frac{2}{3}\beta p^2 = \frac{2\pi e^2 N}{m\omega^2}\frac{v_0^3}{c^2}. \tag{4.45} \]

In the optical part of the spectrum the distance traversed by an electron during the time of one oscillation (during one period), of order \(\dfrac{v_0}{\omega}\), is considerably smaller than the skin-layer depth \(\delta=\dfrac{c}{\omega\chi}\), even in the case when \(l \gg \delta\). Therefore, in the first approximation, the electron oscillations take place in a homogeneous field and, in computing quantities that do not vanish in the absence of collisions, the skin effect may be regarded as normal\(^*\). Such quantities include, first of all, the dielectric constant \(\varepsilon\), which far from the boundary of quantum absorption (i.e., under condition (4.30)) for \(l\to\infty\) is equal to \(\varepsilon=-\dfrac{4\pi e^2N}{m\omega^2}\). Therefore, for \(l\to\infty\) (i.e., practically for \(\omega\gg\nu_0=\dfrac{v_0}{l}\)) the field in the metal is, to a good approximation, as follows:

\[ \left. \begin{aligned} E &= E_0 e^{-z/\delta}\cos(\omega t+\gamma),\\ \delta &= \frac{c}{\omega\chi}=\delta_n=\frac{c}{\left(\dfrac{4\pi e^2N}{m}\right)^{1/2}}, \end{aligned} \right\} \tag{4.46} \]

where, for convenience in what follows, the field is regarded as real and \(\gamma\) is some arbitrary phase.

\[ \text{*) The condition } \frac{v_0}{\omega}\ll \delta=\delta_n \text{ coincides, for } \omega\tau_0\gg 1,\text{ with condition (3.11), meaning that the modulus of the term } \frac{\partial f_1}{\partial z} \text{ in the kinetic equation (3.3) is small compared with the term } \frac{1+i\omega\tau}{v\tau}f_1 \text{ taken into account in the theory of the normal skin effect.} \]

In (4.46), \(E_0\) is the electric field at \(z=0\), equal to the sum of the tangential components of the fields in the incident and reflected waves at \(z=0\); assuming, for simplicity, that normal incidence is meant, we have:

\[ E_0=\frac{2i\omega\delta_n}{c}\,E_A, \tag{4.47} \]

where \(E_A\) is the amplitude of the electric field of the incident wave*).

Thus, the attenuation of the field in the metal is determined by the value of \(\varepsilon\), which in turn, in the frequency range under consideration, is almost independent of the mean free path \(l\) and, owing to the condition \(\frac{v_0}{\omega}\ll \delta \approx \delta_n\), is the same both for \(l\ll \delta_n\) and for \(l\gg \delta_n\).

The losses in metals or, more specifically, the absorptivity \(A\), on the contrary, depend essentially on the relation between the mean free path \(l\) and the skin-layer depth \(\delta\). Indeed, in the case of the normal skin effect (\(l\ll \delta\)) the value of \(A\) is inversely proportional to \(l\) [see (4.44)] and formally, as \(l\to\infty\), \(A\to 0\). In fact, as \(l\) increases, formula (4.44) is no longer applicable, since the condition \(l\ll\delta\) ceases to be fulfilled and the calculation must be carried out in another approximation. In this case, even for \(l\to\infty\), i.e. in the absence of collisions, there will be a certain absorption different from zero, determined by formulas (4.43) and (4.45). These limiting formulas are obtained from the general expressions for \(A\) when \(q\ll 1\) (i.e. \(l\to\infty\)), but may also be derived as the result of a very simple calculation\(^{18}\), which will be reproduced here in somewhat more detail than in \(^{18}\).

Fig. 6.

Putting \(l\to\infty\), i.e. neglecting collisions, consider an electron moving from the interior of the metal toward its boundary with velocity \(\mathbf v_i=(v_{xi},0,v_{zi})\) at \(z=\infty\) (Fig. 6). If reflection at the boundary is specular and the field \(\mathbf E=0\), then after reflection the velocity of the electron is equal to \(\mathbf v_t=\mathbf v_i'=\{v_{xi},0,-v_{zi}\}\).

*) By virtue of the boundary condition (2.9)

\[ E_0=\frac{H_0}{\sqrt{\varepsilon'_{\mathrm{eff}}}}, \]

where the magnetic field \(H_0\approx 2H_A\), since the reflection is almost complete. Therefore

\[ E_0=\frac{2H_A}{\sqrt{\varepsilon'_{\mathrm{eff}}}} =\frac{2E_A}{\sqrt{\varepsilon'_{\mathrm{eff}}}} \approx \frac{2E_A}{\sqrt{\varepsilon}} \approx \frac{2iE_A}{\chi} =\frac{2i\omega\delta_n}{c}\,E_A \]

(in the incident wave propagating in vacuum, \(E_A=H_A\)).

If, however, there is a field \(\mathbf E \ne 0\) in the metal, from the equation of motion \(\dfrac{d\mathbf v}{dt}=\dfrac{e}{m}\mathbf E\) we obtain*):

\[ \mathbf v_i(z=\infty)=\mathbf v_i' + \frac{e}{m}\int_{-\infty}^{+\infty}\mathbf E(t)\,dt, \tag{4.48} \]

where \(\mathbf E(t)\) is the field at the point occupied by the electron at the moment \(t\) (for \(z=\infty\), \(t=\pm\infty\); for \(z=0\), \(t=0\)). In a weak field the electron trajectory may, in practice, be regarded as the same as in the absence of the field, whence, in field (4.46),

\[ E(t)=E_0 e^{-\frac{v_i |t|\cos\theta}{\delta_n}}\cos(\omega t+\gamma), \]

where \(\theta\) is the angle indicated in Fig. 6. The mean energy acquired by the electron under the action of the field over the entire transit time is equal to

\[ \Delta W_s=\overline{\frac{m v_i^2}{2}}-\frac{m v_i^2}{2} =\frac{m}{2}\left[\frac{e}{m}\int_{-\infty}^{+\infty}E(t)\,dt\right]^2, \tag{4.49} \]

where the averaging is carried out over the phase \(\gamma\), as a result of which the term linear with respect to \(E\) vanishes and is not written out (for different electrons the phase \(\gamma\), of course, is different, and therefore averaging is necessary).

If the reflection is diffuse, then after reflection the electron forgets its ordered velocity, and

\[ \Delta W_d=\frac{m}{2}\left(\frac{e}{m}\right)^2 \overline{\left[ \left(\int_{-\infty}^{0}E(t)\,dt\right)^2 + \left(\int_{0}^{\infty}E(t)\,dt\right)^2 \right]}. \tag{4.50} \]

To find the energy loss per unit surface area per unit time, one must multiply expressions (4.49) or (4.50) by the electron flux

\[ dI=Nv\left(\sin\theta\cos\theta\,\frac{d\theta}{2}\right)\frac{3v^2\,dv}{v_0^3}, \]

incident on a unit area on the metal surface (in the expression given, \(v\cos\theta=v_z\), \(\dfrac{\sin\theta\,d\theta}{2}\) is the element of solid angle, and

*) For simplicity we regard the electrons as perfectly free, so that

\[ m_{\mathrm{eff}}=m \quad\text{and}\quad N=\frac{8\pi}{3}\left(\frac{m v_0}{2\pi\hbar}\right)^3. \]

\[ \int_0^{v_0} \frac{3v\,dv}{v_0^3}=1, \]

from which the normalization of the electron distribution function is clear. Carrying out now the elementary calculations and neglecting terms of order \(\left(\dfrac{v}{\delta_n}\right)^2\) in comparison with \(\omega^2\), we obtain:

\[ \begin{aligned} A &= \frac{ \displaystyle \int_0^{\pi/2}\int_0^{v_0} \{\rho\,\Delta W_s+(1-\rho)\Delta W_d\}\,dI }{ \displaystyle E_A^2\frac{c}{8\pi} } = \\[4pt] &= \rho\,\frac{2\pi N e^2 v_0^3}{m\omega^2 c^3} +(1-\rho)\,\frac{3}{4}\,\frac{v_0}{c} =\rho A_s+(1-\rho)A_d, \end{aligned} \tag{4.51} \]

where \(\rho\) is the fraction of electrons reflected specularly, and \(\dfrac{E_A^2 c}{8\pi}\) is the mean energy flux in the incident wave. For \(\rho=0\) or \(\rho=1\), expression (4.51) goes over, respectively, into (4.43) and (4.45), as it should. We note that

\[ \frac{A_d}{A_s} = \frac{3}{2}\, \frac{m\omega^2 c^2}{4\pi e^2 N v_0^2} = \frac{3}{2} \left(\frac{\omega\delta_n}{v_0}\right)^2 \gg 1, \tag{4.52} \]

and, as was already emphasized, in this case the difference between specular and diffuse reflection is very substantial (for \(v_0\sim 10^8\), \(A_d\sim 0.3\%\) and is independent of \(\omega\); for the same \(v_0\), \(N\sim 5\cdot 10^{22}\), and \(\omega\sim 10^{15}\), i.e. \(\lambda\sim 2\,\mu\), the limiting value \(A_s\sim 2\cdot 10^{-4}\%\); in the case of the normal skin effect, for the same values and \(l\sim 10^{-3}\), according to (4.44) \(A_n\sim 2\cdot 10^{-3}\%\)). The calculation given clearly shows that even for an arbitrarily large mean free path there is absorption (i.e. \(A\ne 0\)), which is connected with the work of the field on an electron reflected from the surface of the metal. If the mean free path cannot be regarded as infinitely large and terms containing \(l\) must be taken into account, the calculation becomes much more complicated; here one must use the kinetic equation, and, in particular, results are obtained that have already been discussed earlier.

c) On taking quantum effects into account

All the consideration carried out above (in §§ 3 and 4) was based on the use of the kinetic equation, which is, in essence, classical in character. Meanwhile, as was already pointed out in § 3, with increasing frequency it is necessary to take quantum effects into account.

even in the case when the frequency of the light is lower than that corresponding to the threshold of the internal photoelectric effect (the internal photoelectric effect corresponds to the transition of electrons into another band). Indeed, from the quantum point of view, the absorption of electromagnetic waves in a metal (in the absence of the internal photoelectric effect) occurs in the second approximation of perturbation theory, i.e., it is effected as it were in two stages: a photon with energy $\hbar\omega$ is absorbed by an electron, which then emits a phonon (a quantum of the lattice excitation energy), or, conversely, first a phonon is emitted and then a photon is absorbed; in addition, the possibility of photon emission and phonon absorption must be taken into account. In this case, if the condition (3.2a) is satisfied,

\[ \hbar\omega \ll kT, \tag{3.2a} \]

then an electron that has absorbed a photon $\hbar\omega$ not only does not leave the smearing region of the Fermi distribution, but may also be regarded as acquiring energy in a practically continuous manner (this last assertion is especially valid if, in addition, $\hbar\omega \ll k\theta$, where $\theta$ is the Debye temperature; the energy $k\theta$ is the maximum possible phonon energy; see $^{7,8}$). In this case, evidently, the method of the kinetic equation used above is fully applicable. If, however, condition (3.2a) is not satisfied, then, generally speaking, the calculation must be carried out by a quantum method. Although this circumstance has been known for a very long time (see, for example, $^{7}$, § 53), the corresponding investigation has still not been carried out with any completeness, and only very recently have two notes appeared $^{26,26a}$ on this subject (moreover, in $^{26a}$ the discussion is mainly of semiconductors, while in $^{26}$ no details of the calculations are given and there are unclear points). We are therefore deprived of the possibility of illuminating in any detail the important question of the influence of quantum effects on the reflection of light from a metallic surface and shall confine ourselves to a few remarks.

The condition (3.2a) for the certain validity of the results set forth in §§ 3, 4, obtained by the method of the kinetic equation, is very stringent. Thus, at room temperature $T \sim 300^\circ \mathrm{K}$ the condition (3.2a) means that

\[ \lambda = \frac{2\pi c}{\omega} \gg \frac{2\pi c\hbar}{kT} \sim 40\,\mu; \]

at $T \sim 2^\circ \mathrm{K}$ we already have $\lambda \gg 0.5\ \mathrm{cm}$, i.e., the kinetic-equation method can be used not even over the entire radio range. In fact, however, the situation is apparently not so unfavorable from the point of view of the possibility of using the theory of the anomalous skin effect, which does not take quantum effects into account. Thus, if one uses the results of the quantum calculation $^{26a}$ (see formula (9) of the cited work), one can be convinced that for $\hbar\omega < 15\,kT$ the influence of the quantum corrections does not exceed 10%.

and only with a further increase of \(\omega\) do these corrections become large). At \(T \sim 300^\circ\), the equality \(\hbar \omega \sim 15 kT\) corresponds to a wavelength \(\lambda \sim 2 \div 3\,\mu\), and thus, over almost the entire infrared and certainly in the far-infrared parts of the spectrum, the quantum corrections may be neglected. From this point of view, the necessity, already repeatedly emphasized above, of working precisely in the far-infrared part of the spectrum is also clear, so that the picture may be clearest and the measurement results more readily amenable to theoretical treatment. At the same time, at low temperature \((T \ll \theta)\) the situation is more complicated than at high temperature, since here, on the one hand, quantum effects must play an important role) and, on the other hand, the anomalous character of the skin effect is especially significant. Meanwhile, a joint consideration of both these aspects has not yet been carried out at all, and the construction of a quantum theory of the absorption of light in metals under conditions of the anomalous skin effect is, in our view, one of the most important and interesting tasks for further theoretical research in the field of metal optics**).

Until the corresponding consideration has been carried out, it is difficult to be fully confident that, even in the limiting case of a sharply expressed anomalous skin effect, the losses associated with the reflection of electrons from the boundary of the metal can simply be added to the quantum losses. If, nevertheless, one proceeds in this way, as is done in \(^{26}\), then, for example, to expression (4.43)

\[ A_d=\frac{3}{4}\frac{v_0}{c} \]

one must add the quantum losses \(A_k\), which in \(^{26}\) are calculated under the condition \(\hbar\omega \gg k\theta\) (since low

*) At \(\hbar\omega \cong 20 kT\) the corrections reach 20%. At \(\dfrac{\hbar\omega}{kT} \gg 1\), i.e. practically at \(\dfrac{\hbar\omega}{kT} \gtrsim 40 \div 50\), the absorption of energy according to the quantum formula differs from that obtained classically by the factor

\[ \frac{1}{8}\sqrt{\frac{\pi \hbar\omega}{kT}}; \]

this factor, for example, at \(\dfrac{\hbar\omega}{kT}=50\) is equal to 1.57. At the same time, for \(T \sim 300^\circ\mathrm{K}\), \(\hbar\omega \sim 50 kT\) for \(\lambda \sim 1\,\mu\), i.e. only at the boundary between the infrared and visible parts of the spectrum.

**) It should be borne in mind here that an especially sharply expressed quantum case occurs when not only the condition \(\hbar\omega \gg kT\) is satisfied, but also the considerably more stringent low-temperature condition \(\hbar\omega \gg k\theta\), where \(\theta \sim 10^2 \div 10^3\) is the Debye temperature.

***) This problem should apparently be solved using the quantum kinetic equation (see, for example, \(^{61}\)). Let us note that in the work \(^{62}\), which appeared after the present article had already been submitted for publication, the results set forth at the end of § 46 and in \(^{18}\) were obtained by a quantum method (in this case the electrons in the metal are regarded as completely free and only the influence of the metal surface is taken into account).

temperatures, the condition $\hbar\omega \gg kT$ is then satisfied automatically). In view of the fact that in $^{26}$ there are certain ambiguities, and the details of the calculations are absent, we shall not give here the expressions for $A_k$, confining ourselves to the statement that the expression obtained in $^{26}$ for $A_k$ does not depend on $\omega$ and, for certain quite typical values of the parameters, $A_k = 0.63 A_d$, i.e. in general $A_k \sim A_d$. It may be thought that for $\hbar\omega \lesssim k\theta$, i.e. $\lambda \gtrsim 30\,\mu$, the quantum losses decrease substantially and the losses associated with the anomalous character of the skin effect become dominant. Therefore the transition to the far infrared part of the spectrum is substantial in this case as well.

Let us note, moreover, that for $\hbar\omega \gg kT$, especially at low temperatures, one must also take into account the influence of electron–electron collisions, which in the region $\hbar\omega \ll kT$ play practically no role.$^9$ From this point of view the results obtained in $^{26}$ also cannot be considered reliable.

Finally, let us note that in the question most important, in our opinion, of finding by the optical method the concentration $N$ of conduction electrons, allowance for quantum effects over wide limits does not alter the conclusions drawn earlier. This is explained by the fact that allowance for quantum effects (in the absence of the internal photoeffect), at least qualitatively, is equivalent to an increase in the “friction” of the electrons, i.e. to an increase in the number of collisions $\nu$. At the same time, in the infrared part of the spectrum $(\lambda > 1 \div 10\,\mu)$ the absorption, when quantum effects are taken into account, increases in any case by no more than several times (see above). Therefore condition (3.17), $\omega^2 \gg \nu_0^2$, generally speaking, remains valid as before, and the use of the limiting formula (3.18), relating $\varepsilon$ and $N$ practically, does not turn out to be connected with new substantial restrictions*).

Thus, the results set forth earlier (and, in particular, the results of § 4,б), which, strictly speaking, are valid only under condition (3.2a), have, in general, a considerably wider range of applicability. Namely, in the infrared part of the spectrum, for ab-

*) For a free electron gas without collisions the classical formula

\[ \varepsilon = 1 - \frac{4\pi e^2 N}{m\omega^2} \]

is applicable under the condition $\hbar\omega \ll mc^2$, i.e. even in the X-ray part of the spectrum

\[ \left(\hbar\omega = mc^2 \ \text{for} \ \lambda = \frac{2\pi c}{\omega} = \frac{2\pi\hbar}{mc} = 2.4 \cdot 10^{-10}\,\text{cm}\right). \]

This result is connected with the fact that the value of $\varepsilon$ is determined by the scattering of radiation, while the scattering of light by free electrons is determined by the classical Thomson formula precisely under the condition $\hbar\omega \ll mc^2$. In view of what has been said, the range of applicability of the limiting formula (3.18)

\[ \varepsilon \approx -\frac{4\pi e^2 N}{m\omega^2} \]

is not directly restricted by condition (3.2a).

in the absence of the internal photoeffect, the formulas of § 4.6 for \(\varepsilon_{\mathrm{eff}}\), \(\chi_{\mathrm{eff}}\), \(\sin\varphi\,\tg\varphi_0\), and \(X\), practically remain valid; the formulas for \(A\), \(n_{\mathrm{eff}}\), and \(\tg 2\rho_0\) are certainly correct in order of magnitude, and in the far infrared part of the spectrum are valid even to within a few percent. Moreover, by determining experimentally the function \(A(\omega)\), \(n_{\mathrm{eff}}(\omega)\), etc. over a wide interval of frequencies \(\omega\), one can reveal the role of quantum effects and, within certain limits, verify all the formulas given in § 4.6).

§ 5. METHODS OF MEASURING \(n_{\mathrm{eff}}\), \(\chi_{\mathrm{eff}}\), AND \(A\)

Let us turn to the consideration of the principal methods for the experimental determination of the optical constants of metals. As was already indicated in the introduction, we shall exclude from consideration here methods based on the study of the properties of light that has passed through thin metallic layers (see \(^{27-30}\)).

The experimental methods for determining the optical constants of metals can be divided into 3 groups.

1. Let us first consider the question of measuring the absorption coefficient \(A = 1 - r\) or the reflection coefficient \(r\) for normal incidence of light.

Measurement of the coefficient \(A\) is carried out by the calorimetric method. The experimental scheme is shown in Fig. 7. The radiation falls normally on a thick metallic mirror \(M\). In good thermal contact with the mirror are a thermometer \(T\) and a heating coil \(H\). The change in the temperature of the metal under the action of the incident radiation is observed. In addition to the change in temperature, it is necessary to know, with good accuracy, the intensity of the incident light and the heat capacity of the system. The intensity of the incident light is determined from the rise in temperature when the plate of the metal under study is replaced by a black (well-absorbing) plate. The heat capacity of the system is determined from the rise in temperature when a definite power is released in the heating coil (the incident radiation is then absent). The accuracy of determining the coefficient \(A\) by this method was brought to 5% \(^{31}\). The coefficient \(A\) was determined for various metals in the temperature interval from room temperature to \(1.95^\circ\mathrm{K}\). The most accurate measurements were carried out in works \(^{31-34}\).

Fig. 7. Experimental scheme for measuring the coefficient \(A\): \(M\)—the metallic mirror under investigation, \(T\)—thermometer, \(H\)—heating coil. The arrows indicate the direction of the incident radiation.

Measurement of the coefficient \(r\) is carried out by comparing the intensity of the light reflected from the mirror under investigation with the intensity of the light reflected from a standard mirror

(mirrors of Ag) or from a prism of total internal reflection. The greatest number of measurements was carried out in the works \(^{35}\). The scheme of the corresponding experiment is shown in Fig. 8. Light from the source \(N\), by means of the spherical mirror \(L\), is directed at a small angle onto the plane mirror \(M\) under investigation. After being reflected from \(M\), the light falls on the slit \(S\) of a mirror spectrometer with a fluorite prism and then on a vacuum thermocouple. By carrying out measurements once with the mirror under investigation and a second time with a reference mirror placed in the same position as the mirror \(M\), the coefficient \(r\) is obtained. This method was somewhat modified in the work \(^{36}\), in which the intensity of the light reflected from the mirror under investigation is compared with the intensity of blackbody radiation. Measurements of the coefficient \(r\) were carried out in the visible and infrared regions. The accuracy of measuring \(r\) in \(^{35}\) reached 0.2–0.5%, which in the infrared region leads to an error in determining the coefficient \(A\) for good conductors of the order of 10–25%.

Fig. 8. Experimental scheme for measuring the coefficient \(r\): \(N\)—light source, \(L\)—spherical mirror, \(M\)—mirror under investigation (may be replaced by a mirror of Ag), \(S\)—spectrograph slit.

The principal shortcoming of these methods is that in this way it is impossible to obtain all the optical constants of the metal, since the experiment gives only one relation between \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\).

Another substantial shortcoming of the works under discussion is the method of preparing the metallic mirrors. Since this shortcoming also applies to all the works that will be discussed below, we shall dwell on the methods of preparing metallic mirrors in somewhat greater detail. The following methods of preparing mirror surfaces are used: mechanical polishing, chemical deposition, galvanic deposition, vacuum evaporation, casting on glass, and electrolytic polishing.

At the present time, electrolytic polishing must apparently be recognized as the best method for preparing metal surfaces free from harmful surface layers. This is evidenced by experiments on measuring the specific electrical conductivity of metals at radio frequencies \(^{37}\).

Mechanical polishing leads to the formation of a surface layer whose structure and properties differ from the structure and properties of the bulk metal. The presence of such a layer has been detected in a number of experiments. Thus, in the already mentioned work \(^{37}\), the formation of a surface layer during mechanical polishing of copper was established. The author estimates the thickness of the layer as \(2 \cdot 10^{-5}\) cm; the specific resistance of this surface layer is approximately

10 times greater than the specific resistance of the bulk metal. In work \(^{38}\) the influence of the surface layer, produced by mechanical polishing, on the optical constants of copper was investigated. Removal of the surface layer by electropolishing led to a decrease in the value of \(n_{\mathrm{eff}}\) by a factor of 1.45 and to an increase in \(\chi_{\mathrm{eff}}\) by a factor of 1.3. The authors estimate the thickness of the surface layer affecting the optical constants at \(1.3\cdot 10^{-3}\) cm. In work \(^{33}\) the coefficient \(A\) was measured for copper polished electrolytically and mechanically. It was found that electrolytic polishing reduced the value of the coefficient \(A\) by a factor of 2.4. Thus, mechanical polishing does not make it possible to obtain a surface characterizing the bulk metal.

In chemical deposition, as well as in galvanic deposition and in vacuum evaporation, comparatively thin layers of metal are obtained; moreover, the structure of the metal in these layers may differ from the structure of the bulk metal. Thus, when the optical constants of copper in the visible region of the spectrum, obtained on mirrors evaporated in vacuum, were compared with the optical constants of copper polished electrolytically and mechanically, it was found \(^{38}\) that the optical constants of evaporated copper lie between the optical constants of electrolytically polished copper and the optical constants of mechanically polished copper, agreeing more closely with the latter. In addition, it should be noted that, as found in work \(^{39}\), evaporation in vacuum of Ag, Au, and Cu onto glass and fused quartz revealed a dependence of the optical constants obtained on the nature of the substrate and on the rate of evaporation even for layer thicknesses of 300 m\(\mu\). When the values of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) in the infrared region for copper deposited galvanically and evaporated in vacuum were compared, as carried out in work \(^{40}\), it was found that these values differ by a factor of 1.5–2, which the authors explain by the insufficient thickness of the layer of galvanically deposited copper. Thus, data obtained by means of these methods of surface treatment also cannot be regarded as pertaining to the bulk metal.

The method of casting metal onto glass or onto fused quartz is suitable for Sn, Pb, and Bi. It can apparently give constants pertaining to the bulk metal if special precautions are taken to protect the metal from oxidation. Probably it is best to carry out such casting in vacuum. A comparison of the results obtained with such treatment of the metal surface with the results obtained by electrolytic polishing has not yet been made.

Of all the works that will be discussed in this paragraph, the method of electrolytic polishing was used only in three works \(^{33,34,38}\). In work \(^{34}\) only one constant, \(A\), was measured for copper in the wavelength interval 0.5–4 \(\mu\); in work \(^{33}\)

This same quantity \(A\) was measured for a number of metals, but only for one wavelength, \(14\,\mu\), with very crude monochromatization of the light. In work\({}^{38}\), both constants \(n_{\mathrm{eff}}\) and \(\varkappa_{\mathrm{eff}}\) were measured for copper in the visible region of the spectrum. But for copper in this region quantum absorption plays an essential role; therefore the corresponding data cannot be used by us to calculate the concentration of conduction electrons in copper.

1. The second group of methods for measuring optical constants is connected with determining the reflection coefficients for oblique incidence of light on a reflecting metallic surface\({}^{41-44}\). The reflection coefficients of light polarized perpendicular and parallel to the plane of incidence will be expressed as follows (see (1.18)):

\[ \left. \begin{aligned} r_{\perp} &=\left(\frac{R_s}{A_s}\right)^2 =\frac{a^2+b^2-2a\cos\varphi+\cos^2\varphi} {a^2+b^2+2a\cos\varphi+\cos^2\varphi},\\[6pt] r_{\parallel} &=\left(\frac{R_p}{A_p}\right)^2 =\frac{a^2+b^2-2a\sin\varphi\,\tg\varphi+\sin^2\varphi\,\tg^2\varphi} {a^2+b^2+2a\sin\varphi\,\tg\varphi+\sin^2\varphi\,\tg^2\varphi} \cdot r_{\perp}. \end{aligned} \right\} \tag{5.1} \]

In the case of unpolarized light \(r_0=\dfrac{1}{2}(r_{\perp}+r_{\parallel})\); in (5.1)

\[ 2a^2= \left[(n_{\mathrm{eff}}^2-\varkappa_{\mathrm{eff}}^2-\sin^2\varphi)^2 +4n_{\mathrm{eff}}^2\varkappa_{\mathrm{eff}}^2\right]^{1/2} + \]

\[ +\left[(n_{\mathrm{eff}}^2-\varkappa_{\mathrm{eff}}^2)-\sin^2\varphi\right], \]

\[ 2b^2= \left[(n_{\mathrm{eff}}^2-\varkappa_{\mathrm{eff}}^2-\sin^2\varphi)^2 +4n_{\mathrm{eff}}^2\varkappa_{\mathrm{eff}}^2\right]^{1/2} - \]

\[ -\left[(n_{\mathrm{eff}}^2-\varkappa_{\mathrm{eff}}^2)-\sin^2\varphi\right]. \]

These relations can be used to determine \(n_{\mathrm{eff}}\) and \(\varkappa_{\mathrm{eff}}\). It is clear that two independent measurements are needed. In different works these measurements were carried out in different ways. Thus, in work\({}^{41}\) the coefficient \(r_0\) was measured for two angles of incidence. In work\({}^{42}\) the coefficients \(r_{\perp}\) and \(r_{\parallel}\) were measured for one angle of incidence. In works\({}^{43,44}\) the ratio \(\dfrac{r_{\parallel}}{r_{\perp}}\) was measured for two angles of incidence. In all cases \(n_{\mathrm{eff}}\) and \(\varkappa_{\mathrm{eff}}\) had to be determined from a system of two equations, which cannot be solved explicitly. Therefore the solution is usually carried out graphically. For a given angle of incidence, a family of curves
\(r=f(n_{\mathrm{eff}},\varkappa_{\mathrm{eff}},\varphi)=\mathrm{const}\) is calculated. To determine \(n_{\mathrm{eff}}\) and \(\varkappa_{\mathrm{eff}}\), it is necessary to use at least two families of curves, i.e.
\(r_{\parallel}(n_{\mathrm{eff}},\varkappa_{\mathrm{eff}})=\mathrm{const}\) and
\(r_{\perp}(n_{\mathrm{eff}},\varkappa_{\mathrm{eff}})=\mathrm{const}\), or
\(r_0(n_{\mathrm{eff}},\varkappa_{\mathrm{eff}},\varphi_1)=\mathrm{const}\) and
\(r_0(n_{\mathrm{eff}},\varkappa_{\mathrm{eff}},\varphi_2)=\mathrm{const}\), etc. In each family a curve is chosen corresponding to

measured value of the coefficient \(r\). The intersection of these two curves from two different families gives \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\). To obtain good accuracy in determining \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) by this method, it is desirable to take the angles of incidence of light \(\varphi\) on both sides of the principal angle of incidence \(\varphi_0\). But for metals with high conductivity in the infrared region this requirement cannot be fulfilled, because the principal angle of incidence \(\varphi_0\) is very close to \(90^\circ\). If, however, angles \(\varphi\) appreciably smaller than \(\varphi_0\) are used, then all the coefficients \(r\) become close to 1 and the accuracy of the determination of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) drops sharply. In the works listed above \(^{41-44}\), good conductors were not investigated. The accuracy of the determination of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) for the substances investigated in these works was brought to 4%.

1. The third group of methods for determining the optical constants of a metal is based on the analysis of elliptically polarized light obtained after reflection of linearly polarized light from a metallic mirror. This group of methods is, from our point of view, the most promising for determining the constants of metals in the infrared region.

An ellipse is determined by two parameters. By choosing pairs of parameters convenient for each given case, which determine the elliptical polarization of the reflected light, one can find \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\). These methods can be used to carry out measurements in the ultraviolet, visible, and infrared regions of the spectrum. For measurements in the ultraviolet region the Focht method is usually used \(^{45-48}\). The arrangement of the method is given in Fig. 9. The exit slit of the quartz

Fig. 9. Diagram of Focht’s method for determining the optical constants of a metal in the ultraviolet and visible regions of the spectrum.

monochromator \(S\) is the source of monochromatic radiation. Lens \(L_1\) makes the light beam parallel. The light then passes through a Nicol polarizer \(\Pi\), oriented at an angle of \(45^\circ\) to the plane of incidence of the light on the mirror \(M\), falls on the investigated mirror \(M\), is reflected from it, passes through

two quartz compensators \(K_1\) and \(K_2\), through a Nicol analyzer \(A\), crossed with the polarizer \(P\), and falls on the photographic plate \(F\). The lens \(L\) gives an image of the plane \(B\) in the plane \(F\). The compensator \(K_1\) is a Babinet compensator; the compensator \(K_2\) is a pair of quartz wedges, one of which rotates the plane of polarization to the right and the other to the left. The compensator \(K_2\) is placed perpendicular to the compensator \(K_1\). The phase difference between the \(s\)- and \(p\)-components of the light is produced, first, by reflection from the metal and, second, by the compensator \(K_1\). At those places where the total path difference is equal to \(0\), \(\pm \pi\), \(\pm 2\pi\), etc., we shall have linearly polarized light, whose direction of polarization is determined by the azimuth \(\rho\) (see (1.8)). Thus, at the exit of the compensator \(K_1\) a system of equidistant vertical lines will be observed, corresponding to linear polarization of the light. The compensator \(K_2\), which rotates the angle of polarization of the light, is arranged so that the angle of rotation changes from bottom to top. The final direction of the plane of polarization depends, first, on the azimuth \(\rho\), and, second, on the position of the compensator \(K_2\). Therefore, after the light has passed through the compensator \(K_2\), along the lines of linear polarization there will be points at which the direction of linear polarization of the light is perpendicular to the analyzer. These points correspond to zero intensity of the light that has passed through the analyzer \(A\). In the plane \(F\) a system of black points is obtained, lying at the nodes of two orthogonal systems of parallel straight lines. The position of these points depends on \(\Delta\) and \(\rho\). In the measurements, the positions of these points are compared with the positions of the black points obtained when linearly polarized light passes through the compensators. In this way \(\Delta\) and \(\rho\) are measured for the given angle of incidence \(\varphi\). Fock’s method is also applicable in the visible region. However, if the measurements are carried out only in the visible region, one usually dispenses with the compensator \(K_2\). By rotating the analyzer until dark bands appear, one finds both directions of the restored linear polarization. The angle of rotation of the analyzer from the appearance of one system of dark bands, corresponding to total phase shifts \(0, 2\pi, 4\pi\), etc., to the appearance of the other system of dark bands, corresponding to total phase shifts \(\pi, 3\pi, 5\pi\), etc., is equal to \(2\rho\). The position of the compensator gives \(\Delta\). This method was used in works \(^{49,50,38}\). Having measured the values of \(\Delta\) and \(\rho\), for the given angle of incidence \(\varphi\) one determines \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}^{*)}\).

\({}^{*)}\) In the visible and ultraviolet parts of the spectrum, the approximate formulas (1.17), used for determining \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\), may already be insufficiently accurate. In this case, finding \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) from the measured \(\Delta\) and \(\rho\) is possible without further analysis only under the assumption of the normal character of the skin effect, when \(n_{\mathrm{eff}} = n\) and \(\chi_{\mathrm{eff}} = \chi\). In such-

Unfortunately, the data presented in these works cannot be used by us: first, because of the unsatisfactory method of preparing metallic mirrors; second, because, in all the metals investigated, in the ultraviolet and visible regions there lie absorption bands associated with bound electrons, and quantum absorption cannot be reliably separated from absorption associated with conduction electrons.

Fig. 10. Scheme of the Försterling and Fredericks method for determining the optical constants of a metal in the infrared region.

As for measurements of the optical constants of metals in the infrared part of the spectrum, which are of particular interest to us here (see above), methods using compensation of the phase shift are practically unsuitable because of the absence, in the infrared region, of suitable birefringent crystals. In three works\(^{39,40,52}\), which will be discussed below, instead of measuring \(\Delta\) and \(\rho\) at a given \(\varphi\), the principal angle of incidence \(\varphi_0\) and the principal azimuth \(\rho_0\) were measured. The most significant work in this area may be considered work\(^{40}\). Let us examine it in somewhat more detail. The scheme of the method is given in Fig. 10. The exit slit of the monochromator \(Q\), with an NaCl prism, serves as the source of monochromatic infrared radiation. Further, a parallel beam

under certain conditions, \(n\) and \(\varkappa\) are calculated by the exact formulas\(^{51}\)

\[ \begin{gathered} \tg Q = \sin \Delta \tg 2\rho, \qquad \cos 2P = \cos \Delta \sin 2\rho, \qquad n^2 = \frac{s^2 \sin 2Q}{2 \tg \dfrac{\chi}{2}},\\[6pt] s = \tg P \sin \varphi \tg \varphi, \qquad \tg \chi = \frac{s^2 \sin 2Q}{s^2 \cos 2Q + \sin^2 \varphi}, \qquad \varkappa = n \tg \frac{\chi}{2}, \end{gathered} \tag{5.2} \]

where \(Q, P, s\), and \(\chi\) are auxiliary quantities. Of course, under condition (1.10), expressions (5.2) reduce to (1.17).

monochromatic infrared light of wavelength \(\lambda\) passes through a polarizer \(P\)*), consisting of one selenium mirror and two metallic mirrors, and falls on the metallic mirror \(M\) under investigation. After reflection from the metallic mirror, the beam is divided into two equal parts. One part falls on the analyzer mirror \(A_1\), the other on \(A_2\). The analyzers \(A_1\) and \(A_2\) are set perpendicular to one another. After the analyzers the radiation falls on two bolometers \(B_1\) and \(B_2\).

The measurements were carried out as follows: a certain angle of incidence \(\varphi\) was fixed. The polarizer \(P\) was set at an angle of \(45^\circ\) to the plane of incidence of the light on the mirror \(M\). The analyzers \(A_1\) and \(A_2\) were set at angles of \(\pm 45^\circ\) to the plane of incidence. The wavelength \(\lambda\) was varied until the energies incident on the bolometers \(B_1\) and \(B_2\) proved equal (the bolometric bridge was balanced). For the wavelength thus found, the angle of incidence was the principal angle of incidence, i.e. \(\varphi=\varphi_0\). Indeed, only in this case did the principal axes of the ellipse coincide with the \(s\)- and \(p\)-directions, and only in this case was the bolometric bridge balanced. Next, for the given wavelength \(\lambda\), the analyzers \(A_1\) and \(A_2\) were set parallel and perpendicular to the plane of incidence. The polarizer was rotated until the bolometric bridge was balanced. The reading on the polarizer gave the principal azimuth \(\rho_0\). Knowing the principal angle of incidence \(\varphi_0\) and the principal azimuth \(\rho_0\), it is easy, by formulas (1.13) and (1.15), to find \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\). The authors of work\(^{40}\) succeeded in measuring the optical constants of metals with high conductivity (Ag, Au, Cu) in the infrared region up to \(\lambda=4 \div 5\,\mu\). For larger wavelengths the values of \(\varphi_0\) become so close to \(90^\circ\) that measurements are impossible. The corresponding data are so far the only ones giving the optical constants of metals with high conductivity in the region up to \(4 \div 5\,\mu\). Unfortunately, the methods of preparing metallic surfaces used in this work make it doubtful that the optical constants obtained pertain to the bulk metal. The authors studied thin films prepared either by chemical deposition (Ag), or by galvanic deposition (Cu, Au), or, finally, by cathodic sputtering (Cu, Pt). The shortcomings of these methods of preparing reflecting surfaces have already been discussed above. In two other works mentioned\(^{39,52}\), measurements were carried out down to shorter wavelengths than in\(^{40}\) (\(\lambda_{\max}=1\,\mu\) in\(^{39}\) and \(\lambda_{\max}=2.25\,\mu\) in work\(^{52}\)); the preparation of the reflecting surfaces was likewise unsatisfactory.

The method of Försterling and Frederickse\(^{40}\) is modified in a recently published work\(^{55}\). The authors proposed determining the principal

*) Methods of fabricating polarizers for the infrared region are indicated in works\(^{53,54}\).

angle of incidence $\varphi_0$ and the principal azimuth $\rho_0$ in the following way (Fig. 11). With uniform rotation of the polarizer at a certain frequency, the light reflected from the metallic mirror and passed through the analyzer is modulated at twice the frequency. The light then falls on a bolometer with a time constant smaller than the modulation period. The resulting electrical signal is fed to a resonant amplifier. If one now selects such an angle of incidence and such a position of the analyzer that the signal vanishes, then $\varphi=\varphi_0$ and $\rho=\rho_0$.

Fig. 11. Diagram of the modulation method for determining $\varphi_0$ and $\rho_0$ in the infrared region: 1—source and collimator, 2—motor, 3—polarizer, 4—specimen, 5—analyzer, 6—monochromator and detector, 7—amplifier.

These same authors, in paper $^{56}$, proposed another method for determining the optical constants of metals. The method consists in analyzing the ellipse obtained as a result of the reflection of linearly polarized light from a metallic mirror, using only one analyzer (without modulating the light). By rotating the analyzer, one can find the directions corresponding to the maximum and minimum light intensities. These directions are precisely the directions of the principal axes of the ellipse. By measuring the ratio of the minimum intensity to the maximum, one obtains the ratio of the axes of the ellipse, equal to

\[ \sqrt{\frac{I_{\min}}{I_{\max}}}. \]

These data are sufficient for determining $\Delta$ and $\rho$. Indeed, let the major axis of the ellipse make an angle $\alpha$ with the $s$-direction (see Fig. 4). Denote the ratio of the minor axis of the ellipse to the major axis by

\[ \tg \xi=\frac{OB}{OA}. \]

The azimuth $\rho$ determines the ratio

\[ \frac{OD}{OC}=\tg \rho=\frac{|R_p|}{|R_s|}. \]

Then $\Delta$ and $\rho$ are related to $\alpha$ and $\xi$ by the relations (see $^{60}$):

\[ \tg \Delta=\pm \frac{\tg 2\xi}{\sin 2\alpha}, \qquad \cos 2\rho=\pm \cos 2\xi \cos 2\alpha . \tag{5.3} \]

In papers $^{55,56}$ the results of some measurements of the optical constants for Sb and Bi are given, illustrating the operation of the methods described. No systematic data on the measurement of the optical constants of metals by means of the described setups are presented.

Both of these methods are unsuitable for measuring the optical constants of good conductors in the infrared region $\lambda>4\div 5\,\mu$. The method proposed in paper $^{55}$ is unsuitable because the principal angle of incidence in this case is very close to $90^\circ$, and its determination is ...

with the necessary accuracy becomes impossible. The method proposed in work \(^{56}\) is also unsuitable, since in this case \(\Delta\) is very close to \(180^\circ\), and \(\rho\) to \(45^\circ\). Therefore a very small error in setting the polarizer will lead to large errors in the determination of the optical constants.

Taking into account that the threshold of the internal photoelectric effect for most metals lies in the region \(0.5 \div 9\,\mu\), the necessity of determining the optical constants in the region \(\lambda > 5 \div 10\,\mu\) is clear. The same also follows from the considerations set forth in §§ 3, 4. For work in this region none of the methods used up to now is suitable. In developing an appropriate method it will be necessary to apply multiple reflection of light from the metal, as a result of which the determination of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) is in principle entirely possible.

The second conclusion that follows from consideration of the experimental works is that it is necessary to use such methods of preparing metallic mirrors as would give constants pertaining to the bulk metal. Such methods may apparently be regarded as electrolytic polishing and casting of the metal onto polished fused quartz or glass, provided special measures are taken to protect the metal from oxidation. In every case, however, a special experimental investigation is necessary in order to make sure that the surface is sufficiently good for it to be possible to speak of obtaining constants for the bulk metal.

§ 6. DISCUSSION OF THE AVAILABLE EXPERIMENTAL DATA

As is clear from what was set forth in § 5, there is no completely reliable information on \(n_{\mathrm{eff}}\), \(\chi_{\mathrm{eff}}\), and \(A\) in the infrared region. Therefore our task is limited to presenting and comparing the most reliable of the available data.

1. The values of the absorption coefficient \(A\) at normal incidence are given in Tables I—VI.

Table I

Experimental values of the coefficient \(A\) (in percent) for copper

\(T\) (°K)	\(2\,\mu\)	\(3\,\mu\)	\(4\,\mu\)	\(2—11\,\mu\)	\(8—18\,\mu\)	\(14\,\mu\)	Method of surface treatment	Literature
293	—	—	—	1,17	1,12	—	Mechanical polishing	32
90	—	—	—	0,86	0,82	—	Same	32
4,2	—	—	—	—	—	0,62	Electropolishing	33
4,2	0,39	0,39	0,39	—	—	—	»	34

Table II

Experimental values of the coefficients \(A\) (in percent) for silver

\(T\) (°K)	\(1.0\,\mu\)	\(1.5\,\mu\)	\(2.0\,\mu\)	\(2.5\,\mu\)	\(3.3\,\mu\)	\(5.0\,\mu\)	\(7.0\,\mu\)	\(10.0\,\mu\)	\(13.0\,\mu\)	Material used	Literature
290	1.32	1.05	0.92	0.92	0.92	0.89	0.90	0.87	0.81	Technical foil, thickness \(14\,\mu\)	17
85	1.05	0.86	0.81	0.78	0.79	0.68	0.68	0.62	0.59	Technical foil, thickness \(14\,\mu\)	17

Table III

Experimental values of the coefficient \(A\) (in percent) for gold

\(T\) (°K)	\(1.0\,\mu\)	\(1.5\,\mu\)	\(2.0\,\mu\)	\(2.5\,\mu\)	\(3.3\,\mu\)	\(5.0\,\mu\)	\(7.0\,\mu\)	\(10.0\,\mu\)	\(13.0\,\mu\)	Material used	Literature
290	1.74	1.31	1.18	1.13	1.04	1.01	1.03	1.05	0.95	Foil, thickness \(10\,\mu\)	17
85	1.22	0.92	0.81	0.80	0.76	0.73	0.73	0.78	0.64	Foil, thickness \(10\,\mu\)	17

Table IV

Experimental values of the coefficient \(A\) (in percent) for platinum

\(T\) (°K)	\(1.0\,\mu\)	\(1.5\,\mu\)	\(2.0\,\mu\)	\(2.5\,\mu\)	\(3.3\,\mu\)	\(5.0\,\mu\)	\(7.0\,\mu\)	\(10.0\,\mu\)	\(13.0\,\mu\)	Material used	Literature
290	21.0	20.0	19.0	11.9	5.60	6.25	3.20	2.86	2.82	Foil, thickness \(10\,\mu\)	17
85	18.5	17.9	18.7	11.8	3.68	2.09	1.69	1.77	1.61	Foil, thickness \(10\,\mu\)	17

Table V

Experimental values of the coefficient \(A\) (in percent) for bismuth

\(T\) (°K)	\(\lambda=1.06\,\mu\)	\(1.71\,\mu\)	\(3.06\,\mu\)	\(3.96\,\mu\)	\(5.24\,\mu\)	\(6.75\,\mu\)	\(8.02\,\mu\)	\(9.38\,\mu\)	\(10.49\,\mu\)	\(12.03\,\mu\)	\(14\,\mu\)	Surface treatment method	Literature
\(\sim 290\)	46.0	41.7	31.4	28.3	23.3	19.7	16.8	13.0	13.0	13.1	—	Casting	35
4.2	—	—	—	—	—	—	—	—	—	—	1.24	Electropolishing	33

Table VI

Experimental values of the coefficient \(A\) (in percent) for aluminum

\(T\) (°K)	\(\lambda=1.06\,\mu\)	\(1.71\,\mu\)	\(3.06\,\mu\)	\(3.96\,\mu\)	\(5.24\,\mu\)	\(6.75\,\mu\)	\(8.02\,\mu\)	\(9.38\,\mu\)	\(10.49\,\mu\)	\(12.03\,\mu\)	\(14\,\mu\)	Surface treatment method	Literature
\(\sim 290\)	26.2	19.2	11.7	8.6	6.2	4.8	3.1	2.6	3.1	2.7	—	Mechanical polishing	35
4.2	—	—	—	—	—	—	—	—	—	—	1.11	Electropolishing	33

For electropolished lead at \(T = 4.2^\circ\mathrm{K}\), it was found that
\(A = 1.15\%^{33}\).

1. The values of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) at room temperature are given in Tables VII–XII.

Table VII

Experimental values of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) and calculated values of \(N\) for evaporated copper\({}^{40}\)

\(\lambda\) (\(\mu\))	\(n_{\mathrm{eff}}\)	\(\chi_{\mathrm{eff}}\)	\(A \cdot 10^{2}\), calculated by (1.19)	\(N \cdot 10^{-22}\), calculated by (3.21)
0.81	0.43	4.2	9.2	3.03
1.03	0.43	5.6	5.1	3.30
1.27	0.37	7.1	2.8	3.45
1.53	0.53	8.0	3.2	3.12
1.77	0.73	9.5	3.1	3.21
2.28	0.68	11.4	3.5	2.79
3.05	0.90	14.6	1.66	2.46
3.25	1.08	16.4	1.58	2.83
3.59	1.39	18.9	1.53	3.15
4.20	1.92	22.8	1.45	3.31
5.1 (?)	2.98	28.4	1.45	3.53

Table VIII

Experimental values of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) and calculated values of \(N\) for chemically deposited silver\({}^{40}\)

\(\lambda\) (\(\mu\))	\(n_{\mathrm{eff}}\)	\(\chi_{\mathrm{eff}}\)	\(A \cdot 10^{2}\), calculated by (1.19)	\(N \cdot 10^{-22}\), calculated by (3.21)
0.65	1.19	3.42	5.8	3.42
0.75	0.16	4.57	2.9	4.22
0.94	0.15	5.62	1.8	3.95
1.15	0.23	7.18	1.7	4.20
1.47	0.36	8.85	1.8	4.65
2.10	1.00	14.3	1.9	5.19
2.89	1.39	19.0	1.5	4.85
3.38	2.06	22.9	1.5	5.12
4.04	2.98	28.8	1.4	5.63
4.37	4.34	32.6	1.5	6.27

Table IX

Experimental values of \(n_{\mathrm{eff}}\) and \(\varkappa_{\mathrm{eff}}\) and calculated values of \(N\) for galvanically deposited gold \(^{40}\)

\(\lambda\) (µ)	\(n_{\mathrm{eff}}\)	\(\varkappa_{\mathrm{eff}}\)	\(A\cdot 10^{2}\), calculated by (1.19)	\(N\cdot 10^{-22}\), calculated by (3.21)
0.67	0.28	5.1	4.1	5.30
0.87	0.21	5.4	2.7	4.63
1.07	0.25	7.1	1.9	4.95
1.41	0.36	9.4	1.6	5.06
1.69	0.40	11.4	1.2	5.07
2.11	0.50	14.3	1.0	5.14
2.87	0.73	16.5	1.1	3.63
3.14	0.80	18.9	0.89	4.12
3.50	0.96	22.6	0.75	4.77
4.13	1.60	28.8	0.76	5.39
4.83	1.83	33.0	0.67	5.14

Table X

Experimental values of \(n_{\mathrm{eff}}\) and \(\varkappa_{\mathrm{eff}}\) for evaporated platinum \(^{40}\)

\(\lambda\) (µ)	\(n_{\mathrm{eff}}\)	\(\varkappa_{\mathrm{eff}}\)	\(A\cdot 10^{2}\), calculated by (1.19)
1.00	3.42	6.3	23
1.52	4.71	8.3	19
1.97	5.92	9.8	16
3.29	7.50	12.2	14
3.41	8.82	13.8	12
4.65	10.9	15.5	11

Table XI

Experimental values of \(n_{\mathrm{eff}}\) and \(\varkappa_{\mathrm{eff}}\) and calculated values of \(N\) for tin cast onto glass \(^{48}\)

\(\lambda\) (µ)	\(n_{\mathrm{eff}}\)	\(\varkappa_{\mathrm{eff}}\)	\(A\cdot 10^{2}\), calculated by (1.19)	\(N\cdot 10^{-22}\), calculated by (3.21)
0.325	0.980	2.396	40.6	7.83
0.361	0.905	1.931	49.2	4.44
0.425	0.701	2.487	30.8	4.62
0.468	0.703	3.124	22.3	5.70
0.502	0.780	3.575	19.6	6.35
0.588	0.121	4.512	18	7.25
0.668	1.418	5.007	18.3	7.05

The values \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) were used to calculate the concentration of free electrons \(N\) by formula (3.21). As was indicated in §§ 3, 4, the use of this formula is fully legitimate only if

Table XII*)

Experimental values of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) and calculated values of \(N\) for bismuth cast on glass and mechanically polished\({}^{44}\)

\(\lambda\) (\(\mu\))	\(n_{\mathrm{eff}}\)	\(\chi_{\mathrm{eff}}\)	\(A \cdot 10^{2}\), calculated by (1.19)	\(N \cdot 10^{-22}\), calculated by (3.21)
1.0	4.6	5.0	33	2.4
1.9	6.5	3.7	37	1.65
2.7	8.0	2.7	36	1.10
3.45	8.6	1.15	37	0.71
5.0	8.2	1.5	38	0.31
5.7	7.6	2.2	39	0.22

the wavelength \(\lambda\) is greater than the wavelength \(\lambda_{0}\) corresponding to the limit of the internal photoelectric effect. The values of \(\lambda_{0}\), according to \({}^{17}\), are given in Table XIII. In addition, of course, the inequality \(\varepsilon \gg 1\) must be satisfied.

Table XIII

Upper limit of the internal photoelectric effect in metals

Metal	Cu	Ag	Au	Pt	Sn	Al
Upper limit of the internal photoelectric effect \(\lambda_{0}\) . . .	\(0.6\,\mu\)	\(0.4\,\mu\)	\(0.5\,\mu\)	\(\sim 7\,\mu\)	\(\sim 9\,\mu\)	\(\sim 8\,\mu\)

It is clear from Table XIII that the data for Cu, Ag, and Au in the region \(1 \div 5\,\mu\) may be used to calculate \(N\), with the reservation that there is no certainty that these values of \(N\) pertain to massive metal. The data for Pt and Sn, however, cannot be used with any confidence for calculating \(N\). Nevertheless, Sn is a superconductor, and obtaining even approximate data concerning Sn is of interest; therefore, for orientation, Table XI gives the values of \(N\) calculated by formula (3.21).

*) The table was compiled from the data of the graph given in \({}^{44}\).

In Table XIV are given the results of processing data on the optical constants, electronic heat capacity, and surface impedance at radio frequencies for various metals.

Table XIV

Metal	\(N_a\cdot 10^{-22}\)	\(N\cdot 10^{-22}\) by (3.21)	\(\gamma\cdot 10^{-3}\ \mathrm{erg}\cdot \mathrm{cm}^{-3}\cdot \mathrm{grad}^{-1}\) according to \(^{57,58,59}\)	\(v_0\cdot 10^{-8}\ \mathrm{cm}\cdot \mathrm{sec}^{-1}\) by (4.26)	\(\dfrac{\sigma(0)}{l}\cdot 10^{-22}\) in CGSE according to \(^{22}\)	\(N\cdot 10^{-22}\) by (4.27)	\(v_0\cdot 10^{-8}\ \mathrm{cm}\cdot \mathrm{sec}^{-1}\) by (4.24)	\(v_0\cdot 10^{-8}\ \mathrm{cm}\cdot \mathrm{sec}^{-1}\) by (4.27a)
Cu	8.5	3.3	1.02	0.825	13.9	6.1	0.60	1.11
Ag	5.9	5.7	0.65	1.35	8.3	3.4	1.75	1.04
Au	5.9	5.1	\(\sim 0.65\)	\(\sim 1.3\)	7.6	\(\sim 2.9\)	1.70	\(\sim 1.0\)
Sn	3.7	6.9 (?)	1.03	1.2 (?)	8.6	2.3	2.0 (?)	0.68
Al	6.0	5.4	1.46	0.88	5.5	0.67	2.5	0.31

In column 1 of Table XIV the concentrations of atoms \(N_a\) are given. In column 2 are given the values of the electron concentration \(N\), calculated by formula (3.21) (the values indicated are the averages for the three largest \(\lambda\) in Tables VII, VIII, IX, and XI). In column 3 are given the values \(\gamma=\dfrac{C_e}{T}\), where \(C_e\) is the electronic heat capacity of the metals according to the data of \(^{57,58,59}\). In column 4 are given the values \(v_0\), calculated by formula (4.26). In column 5 are given the values \(\dfrac{\sigma(0)}{l}\) according to the data of \(^{22}\). In column 6 are given the values of \(N\), calculated by formula (4.27). In columns 7 and 8 are given the values \(v_0\), calculated by formulas (4.24) and (4.27a). It should be borne in mind, however, that the latter formulas, which include the quantity \(\dfrac{\sigma(0)}{l}\), are less reliable than formulas (3.21) and (4.26). This is connected with the fact that in experiments in the radio-frequency region it is not \(\dfrac{\sigma(0)}{l}\) that is measured, but \(R=\operatorname{Re} Z\), and the transition to \(\dfrac{\sigma(0)}{l}\) is carried out by means of a formula (see \(^{16,19,20,22}\)) depending on the assumption about the character of reflection and on other factors. In this connection formula (4.27) should be used not so much for determining \(N\), as for testing the theory of the anomalous skin effect from independently measured values of \(N\) and \(\gamma\) (if the theory is correct, then the values of \(N\) from (3.21) and from (4.27) must, of course, coincide). The data presented in Tab-

... XIV, unfortunately, cannot be used for such a verification, since all the quantities \(N\), \(\gamma\), and \(\dfrac{\sigma}{l}\) were obtained on different specimens with different treatment of their surfaces, etc. Therefore the data of Table XIV should be regarded only as approximate and as having chiefly an illustrative character.

Considering the data for the coefficient \(A\), it should be noted that in the principal term of the formulas for \(A\) in the general case there enters the quantity \(q\), which depends on the static electrical conductivity \(\sigma_0\); this means that the quantity \(A\) depends strongly on impurities, on the presence of stresses, on surface treatment, etc. Therefore special care is required in the experiment here. Of all the available data, the most reliable should be considered the recent measurements\({}^{34}\), made with pure, annealed, and electrolytically polished copper at a temperature of \(4.2^\circ\mathrm{K}\) in monochromatic light. For the coefficient \(A\) the value \(0.39\%\) was obtained. Calculation by formula (4.36), using the values of \(N\) and \(v\) from Table XIV and \(\sigma_0\) from \({}^{17}\) (given below in Table XV), leads to the value \(A = 0.21\%\). The discrepancy exceeds the experimental errors. The reason for the discrepancy probably lies in the need to take quantum effects into account at the temperature of liquid helium and \(\lambda \lesssim 4\mu\) (see § 4,c).

As for the remaining data, for a number of reasons indicated in § 5, they cannot be used for an exact quantitative comparison of experiment with theory. But even a rough comparison clearly indicates the necessity of taking into account the anomalous character of the skin effect in diffuse reflection of electrons from the surface. The corresponding comparison of experimental and theoretical data is given in Table XV.

Table XV

Experimental and theoretical dependences of the coefficient \(A\)
(in percent) on temperature for copper

\(T\)	\(A\) experimental	\(A\) theoretical, according to the theory of the normal skin effect	\(A\) theoretical, according to the theory of the anomalous skin effect: diffuse reflection of electrons	\(A\) theoretical, according to the theory of the anomalous skin effect: specular reflection of electrons	\(\sigma_0 \cdot 10^{-17}\) CGSE according to \({}^{17}\)
Room temperature	1.17	0.31	0.52	0.32	5.2
Liquid air	0.86	0.058	0.265	0.059	28.3
Liquid helium	0.39	0.0020	0.21	0.0025	820

The experimental data for room temperature and the temperature of liquid air are taken from work \(^{32}\), and for the temperature of liquid helium—from work \(^{34}\). The theoretical values were calculated from formulas (3.19), (4.36), and (4.42), using the data for \(N\) and \(v_0\) given in columns 2 and 4 of Table IV and the data for \(\sigma_0\) given in work \(^{17}\).

As is clearly seen from Table XV, the anomalous character of the skin effect under diffuse reflection of electrons from the metal surface, as follows also from the theory, manifests itself especially strongly at low temperatures. As for the discrepancies in Table XV between the experimental data and the theory of the anomalous skin effect with allowance for the diffuse character of electron reflection, no special significance should yet be attached to them (in the case of room temperature and the temperature of liquid air it is especially important that the measurements were carried out with an insufficiently good surface; in the case of measurements at the temperature of liquid helium, the discrepancy is probably mainly due to the neglect of quantum effects, which may be eliminated by passing to a considerably longer-wavelength region of the spectrum).

CONCLUSION

As a result of all that has been set forth, it is clear that the study of the optical properties of metals by measuring the quantities \(n_{\mathrm{eff}}(\omega)\), \(\chi_{\mathrm{eff}}(\omega)\), and \(A(\omega)\) opens up a whole series of possibilities that are of unquestionable interest for the physics of metals. The most important is the determination, by an optical method—which in this respect is the most direct and reliable—of the concentration of conduction electrons \(N\). Such measurements must be carried out for a large number of metals and, in particular, for metals that can be in the superconducting state. The experimental data available in this field are very few and unreliable. In addition to determining \(N\), it is necessary, from measurements of the electronic part of the heat capacity or of the surface impedance at radio frequencies, to find the velocity of the electrons on the Fermi surface. After this, and in a certain respect independently of it, measurements in the infrared part of the spectrum can be used to test the theory of the anomalous skin effect, to clarify the character of the reflection of electrons from the metal surface, etc. A further task of the theory, which can develop only in close contact with experiment, is, on the one hand, the generalization of the available results to the case of anisotropic Fermi surfaces and, in general, to the case of polyvalent metals. On the other hand, theory faces the no less important problem consisting in a detailed allowance for the quantum effects discussed in § 4c).

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*
*  *

Note added in proof

In recently published works 63, 64, methods for measuring $n_{\mathrm{eff}}$ and $x_{\mathrm{eff}}$ in the infrared region are discussed in detail. In this connection the authors arrive, in general, at the same conclusion as in § 5 of the present article (the most effective procedure is the measurement of $\Delta$ and $\tg \rho$; in determining the optical constants of good conductors it is necessary to use multiple reflection). In 64 data are given on the optical constants of evaporated aluminum, which we have used in Table XIV (columns 2, 4, and 7 in the last line).