JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 41. NUMBER 5 MAY, 1951

Spectroscopy in Infrared by Reflection and Its Use for Highly Absorbing Substances* IVAN SIMON

Arthur D. Little, Inc., Cambridge, Massachusetts (Received January 23, 1951)

The infrared spectra of many strongly-absorbing solids and liquids can be measured directly only with difficulties, involving preparation of very thin sections, or mulling the substance with diluents, etc. In such cases, the measurement of reflection offers an advantage. If the measurements are performed at two different angles of incidence, the two optical constants, namely, the refractive index n and the index of absorption K of the studied substance can be obtained. A graphical method for obtaining n and K from the measured reflecting powers is described and due corrections for the apparatus polarization are given. An alternative method for obtaining the optical constants from a single set of measurements at normal incidence is indicated for a case of a simple resonance band. This method is based on the impedance concept of the reflecting medium. Experimental arrangement for reflection measurements, using a standard infrared spectrometer, is described. Examples of measured reflecting powers and optical constants derived from them are given for the cases of quartz, mica, and liquid carbon tetrachloride.

I. INTRODUCTION

IN the course of an investigation of the structure of glasses, a method of measurement of the optical

constants in infrared by reflection was worked out and extensively used. This method appears to be extremely useful in a study of substances which are so highly ab­sorbing that the preparation of samples for measure­ment by transmission is not feasible. This situation is met mostly in the solid state (glasses, oxides, silicates, plastics, etc.) but many liquids, too, exhibit very high values of the extinction coefficient in their absorption bands. In both cases, the reflection method yields not only the absorption coefficient but also the refractive index. Therefore, a complete analysis of the dispersion region associated with each strong absorption band is possible.

The reflection method has been known in principle since the classical investigation of the "Reststrahlen." It has been applied since then, on a few occasions, to obtain the optical constants of crystalline solids and glasses.1-3 It has not come into general use, however, as a complement or counterpart to the standard trans­mission technique.

From the experimental point of view, the method is simple, but the analysis of the data is difficult and has hitherto been cumbersome.

In order to obtain the two unknowns, n and K (re­fractive index and absorption index), the value of the reflecting power r is measured at two different angles of incidence φ. The formulas obtained from the electro­magnetic theory for the reflective power r a s a function of n, K, and φ cannot, in general, be solved explicitly. From time to time different procedures have been devised in order to analyze the reflection data in terms of n and K. Boeckner2 used polarized light, for which the general formulas are somewhat simpler, and was able to

* This research was sponsored by the Owens-Illinois Glass Company, Toledo, Ohio.

1 A. Krebs, Ann. Physik 82, 113 (1916). 2 C. Boeckner, J. Opt. Soc. Am. 19, 7 (1925). 3 Scott Anderson, J. Am. Ceram. Soc. 33, 45 (1950).

obtain n and K by numerical trial-and-error solutions. This method is extremely tedious in practice if it is necessary to analyze a large quantity of data. Tousey4

computed four sets of curves (for φ=45, 60, 75, and 85°) for the case of natural (unpolarized) light and for an n-interval of 0.6 to 3.0 and K of 0 to 2.0. Collins and Bock5 used a numerical method instead of a graphical one and computed sets of tables for polarized light (parallel and perpendicular to the plane of incidence) and for angles of incidence φ=50, 60, 70, and 80°. Unfortunately, the tables are not included in their paper.

We have computed similar tables for natural as well as polarized light, for angles of incidence φ= 20 and 70°. It has been found in practice that the graphical method is much faster than interpolation from tables and, there­fore, appropriate sets of curves were constructed by means of the tables and used almost exclusively. These diagrams are included in this paper. Moreover, correc­tion curves have been computed for a case of 30 percent polarization caused by refraction and reflection in the spectrometer.

II. REFLECTION OF LIGHT FROM A PLANE BOUNDARY OF A SEMI-INFINITE,

DISPERSIVE, ABSORBING MEDIUM

Fresnel Formulas

The reflecting powers defined as ratios of the reflected to the incident-light intensities are given by the Fresnel formulas

Subscript s refers to the case of light polarized perpen­dicularly to the plane of incidence, p to the case of parallel polarized light. φ is the angle of incidence, χ the angle of refraction. In the case of natural light with

4 R. Tousey, J. Opt. Soc. Am. 29, 235 (1939). 5 J. R. Collins and R. O. Bock, Rev. Sci. Instr. 14, 135 (1943).

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assumed equal amplitudes of parallel and perpendicular components, the reflecting power is

The relation between φ and χ is provided by Snell's law

in which n' denotes the refractive index. In a dispersive, absorbing medium the refractive index is complex and can be written as

where κ0 = nκ. n is the real part of the refractive index, K is the absorption index, and κ0 is the extinction coefficient. This is related to the more familiar absorption coefficient, a, of Lambert's law, Ix=I0 exp(-ax), by the equation

in which λ denotes the wavelength of the radiation in vacuum. (In view of the fluctuating usage of the terms such as absorption coefficient, extinction coefficient, etc., the nomenclature adopted here was made con­sistent with that of Jenkins and White.6)

Because of the complex refractive index, Eq. (4), the angle of refraction χ becomes complex too, and a complex cosine is therefore introduced in the usual way.

FIG. 1. Reflecting power of an absorbing medium (absorption index k) as a function of the refractive index n at two different angles of incidence, φ=20° and 70°. Natural light.

Substituting this into the formulas (1), we obtain the complex reflecting powers, the absolute values of which are (see, for example, Born7):

and

The expressions of (α2+b2) and a in terms of n, K, and φ are obtained from Eqs. (6) and (4):

6 F. A. Jenkins and H. A. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., New York, 1950).

7 Max Born, Optik (1933) (Lithoprinted by Edward Brothers (1941)).

In this way, the reflecting powers are expressed in terms of n, K, and φ. If we measure (for instance, in normal light) the reflectivity of a given substance at two different angles of incidence, we may obtain from the previous relations the unknowns n and K. However, direct solution turns out to be impractical and one has to resort to one of the numerical or graphical procedures mentioned in the Introduction.

At first, tables were computed, giving the values of rs , rp , and r at two fixed angles of incidence φ (20° and 70°) as functions of n and K, according to Eqs. (6)-(8). The total range of n and K and their increments were properly chosen so as to cover the whole region with a density sufficient for interpolation. The numerical com­putations were performed on standard IBM machines. Later, when a large number of experimental data had to be analyzed, a graphical procedure was adopted and the tables were plotted in the form of diagrams, which are reproduced in Figs. 1-3. The range covered in n (real part of the refractive index) is 0.1 to 100, in K (absorp­tion index) 0.1 to 10.

The graphical analysis proceeds as follows: The measurement yields a pair of values of reflecting powers (either in natural light or in polarized light of chosen direction) r (20°), r (70°). There exists an infinite num­ber of pairs (n, K) giving the specified value of r (20°) in the φ=20° diagram and similarly an infinite number of

facilitated by sliding a transparent sheet with a vertical line and with two horizontal bars marked at corre­sponding heights r (20°) and r (70°) over the diagrams in Figs. 1-3.

The solution is single-valued. If the experimental data are not self-consistent, it is generally impossible to find a coincident pair fitting the curves perfectly in both diagrams (20° and 70°) at the same time. The measured reflecting powers may be affected by several disturbing circumstances, such as (a) insufficient size of the sample, so that the projected area of 70° incidence is smaller than at 20°, (b) imperfection of the surface, either on a macroscopic or microscopic scale, resulting in poor image formation or diffuse scattering of radiation, and (c) discrimination of the spectrometer for a component of radiation polarized in some particular direction (appa­ratus polarization).

The effect of apparatus polarization cannot be avoided when working with natural light, and, unless an appropriate correction is made, it precludes the possi­bility of obtaining self-consistent data.

Correction for Apparatus Polarization In most of the prism spectrometers currently used, the

refracting and reflecting surfaces are vertical and the light passing through the instrument is partially polar­ized in a horizontal plane. The surface of the sample is also vertical (see Fig. 12), and the horizontal plane constitutes, therefore, a common plane of incidence. This plane is chosen as a reference in characterizing the plane of polarization of the radiation as either parallel or perpendicular.

To measure the degree of apparatus polarization, we illuminate the entrance slit first with parallel polarized light of intensity Ip, and again with perpendicularly polarized light of the same intensity Is (in absolute value). The corresponding responses of the instruments can be denoted Ip' and Is' and the ratio

is then a measure of the apparatus polarization. The effect of apparatus polarization can be accordingly conceived as an attenuation of the perpendicular com­ponent of the original 50-50 percent natural mixture by a factor K, namely, Is'=Ip'/K. In the formula for the reflecting power of natural light the intensities of the reflected and incident polarized components do not add up directly to give the formula (2) but have to be added as follows:

The true reflecting power r is obtained by multiplying the measured r' by a correction factor

FIG. 2. Reflecting power of an absorbing medium for light polarized parallel to the plane of incidence.

pairs (n, K) giving the value r (70°) in the φ=70° dia­gram. There is, however, one, and only one, pair of (n, K) which satisfies the given values of r (20°) and r (70°) in both diagrams simultaneously.

In practice, the task of finding this coincident pair is

FIG. 3. Reflecting power of an absorbing medium for light polarized perpendicular to the plane of incidence.

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in which p=rp /rs is the right-hand term of Eq. (8). Accordingly, k depends not only on the angle of inci­dence φ but also on n and K of the substance under test.

For the spectrometer used (Perkin-Elmer Infrared Spectrometer with rock-salt prism, Type 12-B), it was found that K= 1.36 (around v= 1000 cm-1); this corre­sponds to a percentage polarization P=30.4 percent (the "percent polarization" P may be defined as p= [Ip '-Is '/½(Ip '+Is ')]=2[(K-1)/(K+1)]). with this value the correction coefficient k was computed for both chosen angles of incidence and plotted as a function of n, with K as a parameter (Fig. 4). We can observe that the correction is maximum at Brewster's angle (p=1), which is attained when n=0.37 for 20° incidence and when n= 2.74 for 70° incidence. Below the critical angle of total reflection (n=0.34 for 20° incidence, n= 0.94 for 70° incidence, and with κ=0), no correction is required (k= 1) if the measurements are normalized relative to a metallic mirror whose reflecting power is independent of the direction of polarization.

If the correction is required for a value of the appa­ratus polarization other than 30 percent (K=1.36, Fig. 4), a plot of k vs p (with K as a parameter) is helpful (Fig. 5). In order to set up the plot of k vs n (with K as a parameter), one has merely to obtain the values of p=rp /rs from the diagrams in Figs. 2 and 3.

When a large amount of data taken with a given apparatus must be analyzed, it is convenient to set up a special diagram by dividing all ordinates in Fig. 1 by the corresponding values of the correction factor k.

Absorption and Dispersion

The following discussion will be limited to the case of a simple resonance line, treated from the point of view of the classical electromagnetic theory. It is a case of a polarizable molecule possessing an isolated vibrational mode with the natural frequency ω0. Many vibrations of common molecules can be treated in this way, in par­ticular in the liquid or solid state, because of the absence of the rotational fine structure. In order to locate the natural frequency ω0 from the data obtained by the reflection method, one has to form the product n2κ and find its maximum as a function of frequency. Only at normal incidence and in a case when n varies slowly around a mean value, n ≈1, does the maximum of the reflecting power itself coincide with the maximum of n2κ and indicate the natural frequency directly.

This is evident (for n ≈1) from the familiar formula

FIG. 4. Correction factor k as a function of n and K for 30 percent apparatus polarization.

reflection spectra taken at 70° incidence are appreciably distorted and should not be used without a complete analysis.

It is interesting to point out the difference between the quantity n2κ and the absorption coefficient a as measured by the transmission method. The relation be­tween a, n, and K, as obtained from the electromagnetic theory, is

and a is therefore proportional to nk (λ varying slowly through the resonance band) rather than to n2κ. Ac­cordingly, the maxima of a do not coincide with the maxima of n2κ and do not indicate the natural fre­quencies directly. Neither the maxima of the absorption index k nor those of the absorption coefficient a indicate the natural frequencies. It can be shown that the maxi­mum of K is displaced from the natural frequency ω0 by

which follows from Eq. (7) when φ=0. For a preliminary estimate of the location of the

natural frequencies, an inspection of the reflection spectrum taken at 20° incidence is quite satisfactory, as this is still rather close to the normal incidence. With the equipment used, it was not possible to decrease the angle of incidence below about 15°. On the other hand, the FIG. 5. Correction factor k as a function of p=rp /rs for different

percentages of apparatus polarization.

340

FIG. 6. Impedance of a loss-less resonant cir­cuit in a cartesian com­plex plane.

an amount

whereas the maximum of nk (and thus of a) by half of that amount, i.e., ∆ωnκ=½∆ωκ.

The meaning of the quantity n2κ = nκ0 can be obtained from the electromagnetic theory of dispersion. If the radiation of a frequency ω propagates through a medium containing N molecules per unit volume with a polariza-bility p, the complex refractive index n' changes ac­cording to

so far as the polarization can be assumed proportional to the exciting field. If the exciting field is periodic (with frequency ω) and the account is taken of the restoring force, mass, and damping, the polarizability turns out to be a function of frequency

Here, e denotes the electronic charge (in esu), ne the number of effective charges per molecule, M the reduced mass associated with the motion of charges, and ß= b/M the constant of the damping term in the equation of motion (ß being equivalent to the transition probability in the more familiar terminology of quantum theory).

By separation of the real and imaginary parts of n'2— 1 we get

FIG. 7. Impedance of a resonant circuit with small resistive compo­nent.

The maximum of n2κ occurs at ω=ω0, with

where

is the maximum value of polarizability (occurring also at ω = ω0).

The position of the maximum of n2κ, therefore, locates exactly the natural frequency of the resonance line, and its magnitude is proportional to the product of the polarizability of the molecule and the damping. The damping itself can be obtained either from the width of the n2κ-vs-ω curve at half-height (being ß= 2(ω0—ω½ )) or from the positions of maximum and minimum of the n2-vs-ω curve, which are apart, one from the other, by a frequency difference equal to ß (provided the damping is not large).

It should be also pointed out that at the long-wave side of the dispersion band, where the real part of the refractive index approaches its limiting value n∞ and where K is small, the Lorenz-Lorentz law may be applied in order to obtain the limiting value p∞ of the polariza­bility, according to the equation

The left-hand side of this equation, multiplied by the molar volume, corresponds to the so-called molar re­fraction, as frequently used in the refractometry in visible light.

The Impedance Concept of the Reflection

The method of obtaining the optical constants n and K of a given substance from Fresnel's equations by the procedure described in Section II is entirely general; this means that no assumptions were made about the nature of the elementary vibrators causing the dis­persion and absorption. If we do so and assume (simi­larly, as we did in the preceding section) that the vibrators are harmonic oscillators, we can simplify the experimental as well as the analytical procedure by adopting the concepts of the transmission line theory.8

The experimental simplification consists of the fact that reflection data at only one angle of incidence, namely φ=0, need be taken. On the other hand, the data have to be taken as function of frequency throughout the absorption band. The analysis of the data is then based on the following theory.

The reflection of a plane wave incident normally at the plane boundary, between empty space and the medium, can be treated as a case of an electromagnetic wave, propagating along a transmission line of a characteristic impedance Z0 and reflected at its termination. Z0 is, in this case, the impedance of empty space

8 J. C. Slater, Microwave Transmission (McGraw-Hill Book Company, Inc., New York, 1942).

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ε0 and μ0 being equal to 8.85 ×10 -12 farad/meter and 4 × 10-7 henry/meter, respectively. Let us now find the relationship between the impedance of the medium and its optical reflecting power.

If the transmission line is terminated by an impedance Z, the reflection coefficient R (i.e., the ratio of the reflected and incident amplitudes) is given by a formula

In a dispersive, absorbing medium, the impedance Z is complex, whereas Z0 is always real. We denote the normalized complex impedance

and the reflection coefficient is, therefore, also complex,

The square of the absolute value of R is the reflecting power, consistent with the usage in optics, |R|2 = r. Equation (13) then becomes analogous to the "optical" formula (11).

Next, we shall make use of the assumption that the impedance Z of the medium varies with frequency in a similar manner as that of a resonant circuit.

If the line were terminated by the loss-less resonant circuit, the impedance of the terminal would vary with frequency in a characteristic way, which is represented in the complex plane by a circle (Fig. 6). This corre­sponds to the reflection of light from a medium con­taining undamped molecular vibrators. If there is damping, a constant real part is added to the complex impedance Z and the resonance-circle is modified, as indicated in Fig. 7.

Equation (13) is a bilinear transformation of the com­plex variable z=z(ω) into the complex variable R(ω). Any z of the whole complex z-plane is transformed into an R, which, however, is confined to a region within a circle of unit radius in the complex R-plane (Fig. 8). The formulas for the actual transformations of the amplitude r and phase Φ into ζ and 0, respectively, can be easily found, but they can be replaced by a graphical procedure involving the use of the so-called Smith chart.9 If we have plotted the complex reflection coefficients in a polar diagram of a type shown in Fig. 9, we can obtain the amplitude ζ and phase θ of the corresponding z by superposition of a Smith chart on the R-diagram.

Using the conformal properties of the bilinear trans­formation, we observe that any circle in the z-plane is transformed into a circle in the R-plane and vice versa. An impedance changing with frequency in such a way that it describes a circle in the z-plane makes, therefore, the complex reflection coefficient describe another circle in the R-plane. The frequency scale along these circles is, of course, not linear.

In this way we derive the values of z from the given values of R, and it remains to translate them into

FIG. 8. Complex re­flection coefficient R as given by the bilinear transformation of z in a cartesian complex plane.

optical constants n and K. Expressing the complex index of refraction as

and comparing it with z=ζeiθ by means of the relation

we obtain finally

As a practical example of the application of this theory, we have analyzed the reflectivity data (taken from Fig. 14) of crystalline quartz in the region of the main peak. This is actually a more complicated case, as there are two characteristic frequencies close to each other. The problem was treated as follows (Fig. 10): The whole broad main peak was taken as one characteristic resonance, and the corresponding R-circle was drawn through the two points given by the maximum of R at resonance and the (estimated) "background reflectivity" far-off resonance. The wave-number scale was estab­lished along the circle according to the data from Fig. 14. At the wave number, corresponding to the dip (1060 cm-1), another resonance circle was constructed, using the depth of the dip, measured from the top of the main peak, as diameter. The values of the complex impedance Z were then read off this diagram by superimposing the coordinate lattice of the Smith's chart, and the values of n and K were eventually calcu­lated by means of the formula (14). The result is plotted in Fig. 11. The agreement with the results of the rigorous method (Fig. 14) is fairly good, in particular if we take into account the estimates which have been made in drawing the circle diagram of Fig. 10. It is remarkable

FIG. . 9. Complex re­flection coefficient R cor­responding to an impe­dance, changing accord­ing to Fig. 6.

9 P. H. Smith, Electronics 12, 29 (1939).

342 I V A N Š I M O N Vol. 41

FIG. 10. Complex reflection coefficient of quartz (cut perpendicu­larly to the optic axis) plotted in Smith's diagram.

that we arrive at a correct result at all by considering the whole broad reflection peak as a resonant vibration. We shall see later, from the rigorous analysis, that the actual characteristic frequency lies way off the middle of the main peak of reflectivity.

As a limitation of the impedance method, it should be pointed out that it is not strictly correct to simulate a spatially periodic structure, such as crystalline medium by a lumped constant circuit having an impedance dia­gram of circular shape. A rigorous treatment using circuits with distributed constants would probably lead to more complicated resonance curves than circles, but the graphical method would then become rather difficult to use.

III. EXPERIMENTAL TECHNIQUE The apparatus for measurement of reflectivities was

built around the modified Perkin-Elmer infrared spec­

trometer Model 12B equipped with a dc amplifier and recorder system designed in this laboratory. All of the data so far measured were taken with a rock-salt prism in a wave-number interval from about 700 to 3000 cm-1.

The auxiliary radiation source (globar) and optical system required for measurements of reflectivities at different angles of incidence were added to the spec­trometer according to Fig. 12. The built-in source S1 was shut off, the first 45° mirror M1 removed, and the spherical mirror M2 was tilted sufficiently, by means of a set of micrometer screws, to bring the light beam reflected from the sample to a proper alignment with the spectrometer optical system. A new globar source S2, plane mirror M4, and spherical mirror M3 were added and mounted on an arm which could be rotated around a vertical axis coinciding with the reflecting plane of the sample. The mirrors M2 and M3 had the same focal distance, and the sample was mounted in the center of curvature of the mirror M3 as well as that of the mirror

FIG. 11. Optical constants of quartz, derived from the measured reflecting powers (Fig. 14) by means of the impedance method.

FIG. 12. Modification of the Perkin-Elmer infrared spectrometer, Model 12-B, for reflection measurements. Part A is a rotatable arm with the auxiliary globar source S2 and the mirrors M3 , M4. Reflection sample is at S.

M2. The distances S2—M3—S and S—M2-entrance slit were also all equal to the radius of curvature (or twice the focal length). An unmagnified image of the source was thus formed in the plane of the sample at any angle of incidence.

Only a relatively small part of the sample surface was covered by the image of the globar, and the available surface was usually much larger than the projected area of the source even at high angles of incidence; in this way it was assured that all of the incident radiation was reflected to the entrance slit. An angle-dividing mecha­nism kept the sample at the proper angle, in respect to the incident beam, when the arm was rotated. The complete arrangement can be seen in the photo­graph (Fig. 13).

The samples are inserted in a holder and are pressed against three adjustable screws by means of a flat spring. This arrangement makes it possible to adjust the sample

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to alignment with the whole optical system. In order to get a reference standard, an aluminized plane mirror is inserted in place of the sample at the beginning and at the end of each experimental run. The obtained re-flectivities are, therefore, relative values with respect to aluminum. The absolute reflecting power of the alumi-nized mirror has not been determined directly, but according to reference,10 it was assumed to be R=0.97 (and constant throughout the indicated wavelength region).

For reflection measurements on anisotropic substances (crystals) in polarized light, a holder was designed, allowing the sample to be rotated around a normal to its surface. Using a suitable cut of the crystal, the reflec­tivity at a desired orientation of the chosen crystallo-graphic axis with respect to the plane of polarization could be observed. This proved to be extremely useful in resolving some of the fundamental vibrations in crystalline substances; in the course of the present investigation, crystalline quartz has been studied in this way in considerable detail.

As a polarizer, a pile of six AgCl sheets (0.5-mm thick­ness each) inclined at 70° was used in front of the entrance slit of the spectrometer.11

The reflection method has been used also for liquids. In order to avoid the complication with changing the plane of incidence from horizontal to vertical, which would be required for a horizontal level of the liquid sample, the vertical surface of the liquid was preserved by keeping it in a cell with a rock-salt window. The presence of the third medium (rock salt with the refractive index n≈1.49 at 1000 cm-1) precluded the direct use of the graphical method, described in Section II.

As an alternative, the reflection measurements were performed on a surface of a free-flowing liquid. The liquid was permitted to flow slowly past a ground sur­face of a vertical glass plate. A glass plate roughened with a No. 60 alundum shows a reflecting power of only a few percent, even at the angle of incidence of 70°. Moreover, in the neighborhood of any strong absorption bands, the penetration depth in the sample,

FIG. 13. The auxiliary arm for reflection measurements on the Perkin-Elmer spectrometer.

contains an important dispersion region in infrared, which has been already studied on several occasions.2, 12

The curve for reflecting power at φ= 20° has been used for setting up the impedance diagram in Fig. 10.

From the curves of n2κ (upper part of Fig. 14), we can locate the three characteristic frequencies at 795, 1045,

is so small that practically no radiation reaches the back surface.

IV. EXAMPLES OF EXPERIMENTAL RESULTS Only a few examples will be given here to show the

possible applications of the method. In Fig. 14, the measured reflective powers of crystal­

line quartz, cut perpendicular to the optical axis, are plotted in the wave-number interval from 700 to 1400 cm-1, and the optical constants derived by the graphical analysis are displayed at the same time. This interval

10 International Physical Tables 5, 253 (1929). 11 R. Newman and R. S. Halford, Rev. Sci. Instr. 19, 270 (1948).

FIG. 14. Reflecting powers and optical constants of quartz (cut perpendicularly to the optic axis).

12 A. Schaefer and F. Matossi, Das Ultrarote Spektrum (1930), lithoprinted by Edward Brothers (1943), p. 314 ff.

344 IVAN S I M O N Vol. 41

FIG. 15. Optical constants of mica (muscovite), derived from reflecting powers measured in natural light (full lines) and in parallel-polarized light (dash lines).

and 1160 cm-1. This example shows, in a striking manner, that maxima in reflecting power do not necessarily coincide with the positions of the charac­teristic vibrations. Previously, attempts have been made to make fundamental vibration assignments in quartz on the basis of the 1190, 1111, and 800 cm-1

frequencies13 derived from the reflection maxima al­though, actually, no such vibrations exist at the first two frequencies.

From the experimental point of view, it should be noted that quartz shows unusually high values of K and n in the studied dispersion region, so that it is absolutely impossible to obtain the optical data reported here by any transmission experiment. This is also true about most of the silicates and glasses. The next example, namely mica (brown muscovite), is of the same kind. This should illustrate how the results agree when ex­periments are performed once with natural radiation and the other time with polarized radiation. The reflec­tion curves are quite different in the case of unpolarized and parallel-polarized radiation (Fig. 15, lower part), but the optical constants n and K, obtained by means of the diagrams in Figs. 1 and 2, agree fairly well. It has to be pointed out that even a slight change in the values

13 E. K. Plyler, Phys. Rev. 33, 48 (1929).

of the measured reflecting powers affects the result of the graphical analysis to a great extent. This is par­ticularly true in the case of perpendicularly-polarized radiation where the curves (Fig. 3) are very steep. In fact, for this particular case of the mica sample, the data obtained with perpendicularly-polarized radiation were not good enough (as concerns their self-consistency) to be analyzed by means of the diagram in Fig. 3. This may also be partly due to the fact that the analysis derived in Section II is rigorously applicable only to isotropic media. Whereas crystalline quartz can probably be con­sidered as fairly "isotropic" along the direction of the optical axis, this may not be the case for mica viewed along the normal to its natural cleavage plane.

As an example of a liquid, the reflection curves and the derived optical constants of carbon tetrachloride (CCl4 ) in the region of the strong absorption band around 800 cm-1 are shown in Fig. 16. The reflection curves show a very marked effect of "self-reversal" owing to the presence of a layer of dense vapor on the surface of the liquid. This turns out to be an inherent difficulty with volatile liquids, the vapor of which has a strong absorption. This effect can be minimized by blowing off the vapor, and a correction for a surface free of vapor can be obtained by observing the reflection on the liquid interface in a cell (rock-salt window).

V. CONCLUSIONS The reflection method, as described here, was worked

out primarily for the purpose of studying the optical properties of solid substances whose absorption in infrared is so high that their spectra cannot be studied by the transmission method. The reflection method also offers the advantage that the sample does not need to be prepared in the form of a very thin section or does not have to be powdered and mulled in a diluting medium, which is often undesirable. The further advantage is that both optical constants n and K can be obtained from

FIG. 16. Reflecting powers and optical constants of liquid carbon tetrachloride. Lower diagram—full lines: reflection curves for liquid with a layer of stagnant vapor; dash lines: reflection curves with vapor partially removed and eventually entirely avoided by use of a rock-salt cell.

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the measurements as well as the quantity n2κ, which is a measure of the polarizability.

The method has proved its usefulness not only in the specific problem of the glasses, for which it was originally intended, but it has also shown promising results with a wide variety of solids, such as silicates, silicones, semi­conductors, high polymers, and various organic crystals, as well as many liquids. These results will be published shortly in separate papers.

In its present form, the reflection method is much less sensitive than the transmission method. With substances having small values of K and having n ≈ 1, the reflective powers are, in general, small (see Figs. 1-3), and the signal-to-noise ratio in the spectrometer-recorder system is low. For this reason, it was not found practical in our case to expand the computed tables and curves in the region below κ=0.2 and in the vicinity of n= 1; for this reason, we were not able to determine the absorption indexes with a sufficient accuracy below a value of about 0.2. It has to be pointed out that such low values of K still correspond to a very high absorption from the point of view of the transmission method. If we compute the percent transmissivity T=(Ix /I0 )×100 by means of Eq. (5) or (12), we obtain, for example, with κ=0.2,

n=1.5, λ=10μ, and thickness of the sample x=10μ, a percent transmission of only 2.3 percent.

ACKNOWLEDGMENTS

This research was started as a part of an extensive investigation of the study of the structure of glasses by means of infrared spectroscopy sponsored by the Owens-Illinois Glass Company, Toledo, Ohio. We are indebted to the Director of the General Research and Develop­ment Division of the Owens-Illinois Glass Company for permission to publish this paper and to the members of the staff of the Division for their cooperation in pre­paring the samples.

The possible application of the reflection method to the study of glasses originated from a previous investiga­tion of Dr. H. O. McMahon,14 to whom the author is greatly indebted for many stimulating discussions. Thanks are also extended to Dr. R. M. Hainer for effective help and guidance in the infrared-spectroscopy study and to Dr. Gilbert W. King and the staff of the Computing Laboratory of Arthur D. Little, Inc., for help in computing the tables.

14 H. O. McMahon, J. Opt. Soc. Am. 40, 376 (1950).