Current Status of the Problem of the Optical Constants of Metals

M. V. Savostyanova, Leningrad

1. The optical constants of metals—the refractive index $n$ and the absorption coefficient $n\chi$—are among the most important material constants in physics: exact knowledge of them would make it possible to approach one of the fundamental questions of the structure of matter—the question of the structure of the crystal lattice. Indeed, the behavior of an absorbing crystal with respect to light waves incident upon it is wholly determined by the motion of electrons within the lattice, and any peculiarities of their interaction with the incident waves must inevitably be manifested also in the spectral course of absorption, reflection, and refraction.

Indeed, it has long been known that the spectral course of the optical constants of a number of metals is highly peculiar; it is enough to point to the ultraviolet absorption (and reflection) minimum of silver, which directly indicates some abrupt change in the mechanism of absorption on passing through this region. In recent years experimental material has accumulated which reveals ever new details in the spectral distribution of the optical properties of metals. Thus, the experiments of Wood on the alkali metals, described in detail by him in his Physical Optics, attracted general attention; the region of minimum absorption in these metals is expressed particularly sharply.¹)

Further, Smaкula $^{59,60}$, who undertook careful measurements of the absorption of metals (in thin layers), found that the absorption curve does not proceed smoothly, but in jumps (Fig. 1), and this circumstance is manifested especially sharply in the ultraviolet region (around 300 mμ). This fact points to still more subtle

¹) This discovery by Wood has not only theoretical but also practical significance: films of alkali metals, obtained between two quartz plates, are excellent selective light filters for the ultraviolet region. Such filters are manufactured at the State Optical Institute in the laboratory of Acad. S. I. Vavilov by Comrade Sverdlova.

features in the mechanism of absorption by metals—features that call to mind the discrete spectral lines of the spectra of gases and vapors and give hope that a careful study of the dispersion of metals should, in the end, lead to the same results in the field of studying the internal structure of metals as the study of the line spectra of atoms and molecules has led to, making it possible to elucidate their structure in all details.

Fig. 1. Absorption by a silver layer of thickness \(d = 3.32 \cdot 10^{-6}\) cm.

On this path, however, there stand serious difficulties, as a result of which at the present time the optical study of the properties of solids is still in its initial stage. The first of these difficulties is of a theoretical character and consists in the fact that in solids the phenomena of emission and absorption of light, which according to modern ideas are associated with a transition from one energy state to another, are extremely complicated by the interaction of individual atoms or ions with one another; here one cannot speak of the energy level of this or that electron, but only of the electronic potential levels of the whole lattice. This is a concept introduced into physics by quantum mechanics, and only by applying its methods can one hope to achieve tangible results in this direction.

Indeed, attempts to connect these two fields—metal optics and electron theory—were made long ago. Classical optics, in the investigations of its most brilliant representative—Drude—led to valuable results, now included in all textbooks, but it proved powerless to bring the problem to completion; only after, in recent times, through the works of Bloch\(^2\), Sommerfeld\({}^{61}\), and others, the electron theory of metals had been created on a quantum-mechanical basis, did it become possible to draw from it certain conclusions of an optical character.

The first attempt in this respect was the work of Kronig\({}^{34—38}\), who became interested in the above-mentioned results of Wood and McKay; after him, for a whole series of years, authors tried to expand and refine his conclusions. Thus, in 1932–1934 there appeared the works of Fujinoka\({}^{18}\), Fröhlich\({}^{17}\), Mott and Zener\({}^{48}\) (calling special attention to the physical assumptions), Seitz and Chernickovsky\({}^{54}\), Pisarenko and Khurgin\({}^{23}\); a critical survey of the results of their work was given by Wilson\({}^{70}\) (see also the articles of Herzfeld\({}^{24,25}\)).

Most of these works concern the simplest—theoreti-

...—of alkali metals; Kronig, and also Khurgin and Pisarenko, succeeded in arriving at expressions that make it possible to calculate the optical constants (\(n\) and \(n\chi\)); as we shall see below (Fig. 4), the calculated values are close to the experimental ones, but—for \(n\chi\)—only within the limits of experimental accuracy; for \(n\) the discrepancies are much greater. This is an extremely important circumstance: any theory can develop successfully only if it is based on, and accompanied by, experimental data.

Fig. 2. Dispersion curves for copper

But here we are faced with a second difficulty that is hindering further progress in the most recent metal optics: in the overwhelming majority of cases the optical constants have been determined with such low accuracy that it is insufficient even for an approximate comparison of theory with experiment. Very often there are discrepancies not only between the data of different authors, but also between different measurements in one and the same laboratory; the nature of these discrepancies can be judged from Figs. 2 and 3, where the spectral course of the optical constants for copper and silver is plotted according to the data of various authors.

The experimental skill of these authors leaves no room for doubt; the natural conclusion suggests itself—and most authors come to it—that these discrepancies are explained not by errors of measurement, but by deeper causes, conditioned by the physical state of those metal specimens on which the measurements were performed.

The author of the present review has also set himself the goal of elucidating the current state of the question of the influence of the physical nature of the specim—

of metals (chiefly their surface properties) on optical constants; for a description of research methods and of work up to 1926 we refer to detailed surveys, for example in Handbuch der Experimentalphysik (the article by Jaffe\(^{30}\)).

1. For determining the optical constants of metals, two methods are mainly used, depending on whether we are dealing with a transparent layer or with the surface of a massive mirror.

Fig. 3. Dispersion curves for silver

In the first case \(n\chi\) is determined directly by measuring transmission, while \(n\) must be calculated from the coefficient of reflection by the known formula:

\[ R = \frac{(n-1)^2 + n^2 \chi^2}{(n+1)^2 + n^2 \chi^2} \]

or determined separately.

Vernicke\(^{65,66}\), Win. Rathenau\(^{50}\), and, finally, Hagen and Rubens\(^{20}\) worked by this method. However, the greatest number of measurements of optical constants was made on massive polished metallic mirrors by the method first proposed by Jamin\(^{33}\) and then developed by Drude\(^{7,9,12}\); it is based on determining the ellipticity of light reflected from a metal. In the ultraviolet region, measurements of these quantities are made by a very elegant method proposed by Foerster\(^{64}\); another method, also making it possible to penetrate into the ultraviolet region, was used by Pfestorf\(^{49}\).

By Fogg’s method, in the period 1910–1913 in Göttingen, and also in other places, a whole series of metals and substances was investigated in the region from 226 mμ to 668 mμ, namely: steel, cobalt, copper, silver^4; steel, gold, nickel, iron, platinum, bismuth, zinc, selenium, mercury, iodine^44, chromium and manganese; mercury and tin^15; selenium, copper, nickel, iron, and cobalt^62.

All these measurements suffer, however, from a common shortcoming: they were made on polished surfaces. In this case we encounter three difficulties—one of them of a practical nature, and the other two fundamental. The first difficulty consists in the fact that only quite recently have we learned to prepare more or less perfect polished metallic surfaces; all the above-mentioned investigations were carried out using simple polishing methods (Schmirgelpapier); but even if we were actually dealing with an ideal surface, we must bear in mind that, as the latest investigations have shown, all metals, as well as other substances (glass), become covered in air with a surface film; the process of polishing consists in abrading this film, while the polishing material acts chemically, dissolving the film and thereby facilitating its separation from the surface of the metal. A polished surface differs from an unpolished one only by the absence of coarse roughnesses; the surface film, however, remains, forming again and again. Thus the question must be raised as to what, in fact, we measure on such polished surfaces—the optical constants of the metal itself, or only those of its surface film, which obviously in different cases has a different chemical composition and therefore must exert a different influence on the reflected light.

If, however, we take mirrors obtained by evaporation in vacuum or by cathodic sputtering, then here too we must reckon with the possibility of oxide films being present, formed when the vacuum is not sufficiently perfect. Laux^43 points to such possibilities; he showed that the optical constants of metallic mirrors obtained by cathodic sputtering in various gases (air, N₂, H₂) differ quite noticeably from one another (see also^63, ^4a, ^6a). Further, even in a good vacuum, it is very probable that the surface layer of the mirror will have a different structure from that of the underlying thick layer: indeed, by evaporation (or cathodic sputtering) it is possible to obtain layers which, in their optical properties (color, i.e. absorption, and reflection), differ very sharply from the layers of so-called massive metal. Thus, silver in thick layers is always bluish; but, under certain conditions, it is possible to obtain yellow, orange, and violet films. A. Ashcheulov^1 showed that these shades are caused not so much by thickness, as had earlier been supposed, as by the structure of the films; in all probability, here we have silver in colloidal form.

There is no reason to suppose that even on a “massive” layer obtained by the methods indicated above, under certain conditions

on the surface there is not formed an upper layer that is “crystalline” connected with the substrate, but rather a metal in a finely divided form, possessing constants quite different from those of the massive metal.

Finally, thirdly, we must make one more remark, concerning both massive polished mirrors and layers obtained in vacuum: by optical constants we understand certain material constants, whose physical meaning is connected with the crystalline structure of the metal. But in the mirrors and layers mentioned above we obviously have a microcrystalline structure; for each crystallite we have, in the general case, two values of the optical constants, as is the case, for example, for any anisotropic absorbing crystals. In measurements on mirrors obtained by one method or another, we observe only certain mean values, which to a large extent may depend on the method of preparation of the mirror.

These three circumstances cannot fail to influence the results. Indeed, as was indicated, considerable discrepancies are obtained not only between the data of different authors, but also between those of one and the same author on different mirrors; on the contrary, fresh surfaces, even when measured by different methods, give reproducible and agreeing results.^67

Various investigators have drawn attention to this circumstance; thus, already Conroy^4, according to Stokes’s indication, explained certain inconsistencies in his measurements by insufficiently perfect polishing of his mirrors.

However, only Drude^10 came close to elucidating the influence of polishing and of surface layers; in his classical work, the results of which have not lost their freshness even up to the present time, he sets forth, on the basis of theoretical considerations, the proposition that transition layers present on a mirror, when of sufficient thickness, strongly diminish the principal angle of incidence $\varphi$; the principal azimuth $\psi$ meanwhile remains almost unchanged. As regards the influence of imperfect polishing, it turns out that, in reflection from a mirror covered with scratches uniformly in all directions, the azimuth is smaller than in reflection from a smooth mirror. Drude attempted to establish the concept of the “normal state,” i.e., one in which it would be possible to obtain correct values of the optical constants. Such a mirror must satisfy three conditions: 1) be free of surface films, 2) be free of scratches, and 3) be plane. The difficulties in preparing mirrors applicable in practice lie in the simultaneous fulfillment of the first two conditions.

Drude’s experimental data, obtained, unfortunately, for only one wavelength ($\lambda = 589\,\mathrm{m}\mu$), refer to this rarely attainable normal state.

However, his data in many cases prove to be far more reliable than the newer ones, since Drude was very

was attentive to the physical state of the surface of his mirrors. Thus, he made measurements as quickly as possible after the mirrors had been prepared; likewise he took care that his evaporated layers should be sufficiently continuous; his remark that “a system of metallic particles separated from one another by distances comparable with the wavelength may have essentially different optical properties than a continuous layer” is nothing other than the most recent conception of the colloidal (granular) nature of thin metallic layers.

In the works of the Göttingen school of Focht listed above, the experimental skill in observations was raised to a high level; as regards the physical nature of the surface, far less attention was paid to it; in many cases mirrors were used that had been polished elsewhere (by the firm of Winkel or Schmidt and Haensch) and a considerable interval of time elapsed before the measurements. As indicated, this could not but affect the absolute values of the constants. Indeed, according to the latest data (Lowery) ^42, the refractive index for a copper mirror changes by 16% in 30 hours, and the value \(n\chi\) by 22%.

Only in the last 10–15 years has serious attention begun to be paid to the state of the surface. Unfortunately, however, we still have only scattered works.

The question of studying surface layers and their influence on optical constants was first seriously posed by Haußschild ^21 in Wiener's laboratory in 1912–1914. Haußschild developed a theory of the influence of surface layers on the state of polarization of reflected light and, on its basis, gave a method of approaching the “true” values under continuous variation of the layer thickness. No experimental results, however, were obtained, since the work was interrupted in 1914 and apparently was not resumed; in Lauch's dissertation ^43, which appeared several years later (1923) from the same Leipzig laboratory, the question of the state of the surface of mirrors (obtained by cathodic sputtering) is considered from another point of view and does not constitute the main content of the work.

A significant advance in the study of the state of the surface, which also led to a refinement of the measurements, was brought about by the successes of polishing technique.

In 1926 there appeared a detailed investigation by Pfestorf ^49 with careful measurements of the optical constants of a number of metals, carried out in Foersterling’s laboratory in Jena.

Using Zeiss’s new (secret) polishing material, which he characterizes as “excellent,” Pfestorf was already able to separate the influence of surface layers—arising, for example, under the action of water—from the influence of polishing: a) surface layers begin to have a noticeable effect only under a more or less prolonged action of external agents and b) their influence is in general much weaker than the influence of imperfections in polishing. Hence a very important practical conclusion: reliable values of the optical constants can be obtained,

if well-polished mirrors are used and the measurements are made as rapidly as possible.

Essentially the same results were obtained quite recently (1932–1936) by Lovera \(^{40-42}\) and his collaborators.

They also used first-class mirrors obtained by two new methods: a) by Jacquet’s \(^{31,32}\) method of “electrolytic polishing,” in which the metal serves as the anode in an electrolytic bath (orthophosphoric acid); as a result, all films are removed from its surface and the pure metal is exposed, and b) by the method of evaporating metals in a vacuum, developed by the Vickers firm. Here too the principal factors noted by other authors are manifested: the influence of the interval of time between the preparation of the mirror and the beginning of the measurements and, in particular, the influence of the degree of polishing. They found still another factor exerting a very strong influence on the optical constants: changes in the surface layer of the metal arising, apparently, during the mechanical treatment of the surface (stresses) that preceded the electrolytic polishing. These changes extend to a considerable depth—up to \(0.03\) mm. Electrolytic polishing could remove this layer as well, but there will nevertheless always remain uncertainty as to how far our data characterize the truly undistorted lattice of the metal. It is therefore quite natural that we must arrive at the idea of developing a technique that would make it possible to do without polishing. Here we are faced with two alternatives: either to work in a high vacuum, or to use the natural cleavage planes of single crystals.

Fig. 4. Optical constants of potassium: • — data of Ives and Briggs, ○ — Dencken’s data, 1st mirror, + — Dencken’s data, 2nd mirror, — — — calculated according to Kronig

The first attempt to carry out measurements in vacuum was made by Gaushield \(^{21}\); as indicated, it was not developed further. Such a method, however, proves to be unavoidable for the alkali and alkaline-earth metals.

Here, in the first place, the most recent studies of Ives and Briggs \(^{26,27,27a}\), carried out by Fogt’s method, should be mentioned. The chief value of their results, in comparison with some previous ones (Dencken \(^{14}\)), lies in the fact that they were obtained also for the ultraviolet region, which, according to Wood’s data mentioned above, is the most interesting for the alkali metals.

On the Optical Constants of Metals

The curves for potassium are given in Fig. 4, where data of other authors are also marked (Denken\(^ {14}\)), as well as (the dotted curve) data calculated according to Kronig\(^ {36}\). Strange as it may seem, in this case of an alkali metal so sensitive to oxidation, the agreement between the numbers of Ives and Briggs, on the one hand, and Denken (for one of the mirrors), on the other, is much better than between the data of different authors, for example, for copper or silver (Figs. 2 and 3). This indicates that the method of obtaining unpolished layers does indeed have its advantages—even with the existing weak points in the technique of Ives and Briggs, in which—as also in Denken’s case—the alkali metal was clamped between two plates (one of quartz and the other of glass), and the reflecting surface, consequently, was inevitably “poisoned” by contact with the quartz.

This defect has been eliminated in the recently published work of O’Brien\(^3\). What is essential in his investigation is above all that he brought out of more than a century of oblivion a very old method for determining optical constants—the Brewster method, much simpler than the later polarimetric methods, including Fort’s method. The Brewster method consists in the fact that the phase difference, which produces elliptical polarization, between the mutually perpendicular components of the electric vector reflected from a mirror is greatly increased if multiple reflection from two parallel mirrors is used; for its measurement there is no need to use such an instrument as the Babinet compensator employed in Fort’s method—a less bulky Nicol prism is quite sufficient, and it can be placed in vacuum together with the mirrors. This is an extremely important circumstance, eliminating the errors that arise from stresses in the windows of the vacuum vessel, from which the above-mentioned data of Ives and Briggs are not free.

Fig. 5. Methods of measuring optical constants. A — Brewster’s method, B — vacuum modification of Brewster’s method (O’Brien).

In Fig. 5 a schematic of the apparatus is shown—on the left according to Brewster, on the right according to O’Bryan; the modification introduced by the latter is evident from the drawing. O’Bryan obtained data for a whole series of metals whose investigation by the method of polarized mirrors was hindered by the necessity of obtaining them in sufficient quantity—Be, Mg, Ca, Sr, Ba, Al, Ge, La, Ce, Mn; unfortunately, his data were obtained only for the visible region; moreover, the third of the factors mentioned above, which affect the optical properties of the surface, also remains in force here—its microcrystallinity.

Indeed, it is hardly probable that the individual crystallites in the surface layer of a mirror obtained by any method would all be oriented identically; rather, it should be assumed that their axes are arranged in all possible directions. This should be especially marked in the hexagonal system; here we must have a dependence of the optical constants on the angle of incidence. This circumstance was first pointed out by Tull^62; Schulz and Hannemann^57,58 attempted to consider it theoretically. A number of investigators, in fact, encountered such a phenomenon experimentally; thus, Laux observed on mirrors obtained by cathodic sputtering a certain anisotropy, manifested in the fact that at different angles of incidence different values of the refractive index were obtained, which, evidently, should be attributed to the different positions of the axes of the particles in the layer; a similar kind of “double refraction” of metallic layers has long been known in the literature (Kundt^39, Dessau^5).

The only way to avoid these phenomena is to work on cleavages of single crystals. The first measurements upon reflection from cleavage planes of single crystals were made (on crystals of stibnite) by Drude^8, who also gave a theory. More systematic investigations on the faces of single crystals were carried out in America by Zieg^55 (1922) and his collaborators; thus, Skinner^56, Wulff^68, and Van Dyke^13 investigated the constants of Se, Te, Miller^46—Zn and Bi, and finally Graber^19—Mg and Zn (and in another laboratory—Dix and Raus^6). These authors mainly used Fort’s method, employing the crystal ellipsometer constructed by Wulff, which makes it possible to work with very narrow light beams—a circumstance of exceptional importance in view of the small dimensions of the single crystals used. The optical constants in all these cases have different values for the two principal positions of the crystal (the principal axis $\parallel$ or $\perp$ to the plane of incidence).

Graber’s results differ greatly from the data of other authors. It should be borne in mind that the method of these measurements is still not free from shortcomings. Thus, first, Graber worked on fractures of large metallic pieces, choosing on these surfaces randomly situated faces (lateral faces) of individual single crystals, in most cases very small ones. An even more serious drawback of Graber’s measurements is that they were carried out in air: it is quite evident that the influence of atmospheric conditions on fresh fractures must be much stronger

less than on such surfaces which are already covered with a film. It is possible that Graeber’s discrepancies with other authors should be ascribed precisely to this circumstance.

Summarizing all that has been said above concerning measurements of optical constants, we arrive at the conclusion that: a) the values of material constants so important for physics as \(n\) and \(n\chi\) are at present (1937) known only with a certain, far from high, degree of approximation; b) the problem of determining them must be posed anew, on a broad, perhaps international, scale; c) measurements must be preceded by a careful study of the physical state of the surface of the metal under investigation; d) since the most reliable data are apparently those obtained on fresh cleavages of single crystals in the highest vacuum, a method must first be developed for obtaining such single crystals, sufficiently large for these purposes.

In the introduction we pointed only to the role which exact knowledge of the optical constants of metals must play in the further development of quantum-mechanical metal optics; there are also other areas of physics where accurate data are already urgently required: the very interesting theory of the selective photoelectric effect, developed by Ives and Fry\(^{28,29}\) and explaining selectivity exclusively by the optical state of the electron-emitting surface, bases its calculations on the optical constants of this layer. The need to have at hand data for the alkali metals—which chiefly exhibit the selective photoelectric effect—compelled Ives himself to undertake their measurement.

Further, we have the extensive field of the optics of colloidal solutions of metals: the varied coloration (i.e., absorption) of a number of colloidal solutions (gold, silver, sodium) can, as is known, be calculated in advance from the formulae of the optics of turbid media (see the works of Mie\(^{45}\), Fajk\(^{46}\), Savost’yanova\(^{51}\)). In some cases the result—the position and form of the absorption band—proves to be very sensitive even to such changes in the optical constants which, it would seem, from the point of view of experimental technique are quite legitimate (see the articles of Savost’yanova\(^{52,53}\)). In these cases as well it is urgently necessary to have more accurate data at hand.

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