MICROWAVE SPECTROSCOPY

W. Gordy *)

CONTENTS

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
II. Apparatus and experimental procedure . . . . . . . . . . . . . . . . . . . 204
    A. The waveguide and related parts . . . . . . . . . . . . . . . . . . . . 204
    B. Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
    C. Detecting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
    D. Determination of the sensitivity of the spectrometer . . . . . . . . . 231
    E. Intensity measurements . . . . . . . . . . . . . . . . . . . . . . . . 232
    F. Frequency measurements . . . . . . . . . . . . . . . . . . . . . . . 235
III. Absorption spectra of gases and vapors . . . . . . . . . . . . . . . . . . 240
    A. Atomic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
    B. Molecular spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
        1. Inversion spectra . . . . . . . . . . . . . . . . . . . . . . . . . 242
        2. Electronic spectra . . . . . . . . . . . . . . . . . . . . . . . . 243
        3. Pure rotational spectra . . . . . . . . . . . . . . . . . . . . . 247
        4. Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . 257
        5. Line shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
        6. Stark effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
        7. Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . 271
    C. Determination of molecular and nuclear properties . . . . . . . . . . 274
IV. Study of liquids and solids by means of microwaves . . . . . . . . . . . 289
V. Applications of microwave spectroscopy . . . . . . . . . . . . . . . . . . 291
    A. Stabilization of microwave generators by means of spectral lines . . . 291
    B. Spectral lines as frequency standards . . . . . . . . . . . . . . . . . 292
    C. Qualitative and quantitative analysis of gases . . . . . . . . . . . . 292

I. INTRODUCTION

One can cite a great many examples from the past in which some instrument or field of investigation, originally the subject of pure science, later found application under practical conditions. The reverse relation is encountered much more rarely. An outstanding example of this kind is the technique of microwaves, which developed almost exclusively in connection with the narrowly practical problems of radar and which is now proving—

*) Walter Gordy, Microwave Spectroscopy, Rev. of Mod. Phys. 20, 668 (1948). Translated from English.

time, extremely valuable aid to science in its striving toward the further expansion of the boundaries of knowledge of nature.

“Microwave spectroscopy,” which has now become a fairly extensive and rapidly growing branch of pure physics, owes its existence almost entirely to the creation of microwave radars. So far as we know, in the prewar literature there was only a single article devoted to microwave spectroscopy. It contained a description of the first experiments of Cleeton and Williams1, who used quasi-optical methods to detect the absorption of ammonia in the wavelength region around 1.25 cm. In recent years, however, no fewer than a hundred investigations on spectroscopy in the microradio-wave region have appeared, and the publication of a review article on this question now seems entirely timely.

It is appropriate to begin such a review with a comparison, or contrast, of optical and microwave spectroscopy. Although with both methods we obtain information of one and the same type, since both serve to measure quantized changes of energy in atoms and molecules, the experimental techniques and the apparatus used in each of them are, however, completely different. Moreover, the regions of the electromagnetic spectrum that can be effectively investigated by each of these methods do not overlap.

Although electronic devices had previously been used in optical spectroscopy as such auxiliary means as automatic recorders, microphotometers, and the like, the principal components of the spectroscope remained optical in their nature right up to the appearance of “microwave spectroscopy.” In microwave spectroscopy the dispersing element—the prism or grating—which is the heart of the optical spectroscope, becomes superfluous. This elimination of the dispersing element proves possible as a result of the use of such a fundamentally different source of radiation as a controllable electronic generator. Such a generator produces substantially monochromatic radiation of constant phase, making it possible to use an electrically tuned receiver or detector.

Thanks to the controllable source and the tuned detector, the microwave spectroscope is an instrument of exceptionally high sensitivity and resolving power. Its resolving power is, in order of magnitude, 100,000 times greater than the resolving power of the best infrared spectrometer with a grating. Measurements of wavelengths or frequencies are likewise carried out electronically and at present are made with an accuracy of up to the seventh significant figure. The indicator is usually an electronic device, namely a cathode-ray oscillograph. By applying the method of frequency modulation of the source and corresponding synchroniz-

by controlling the sweep of the oscillograph, one can obtain on the screen of the latter a visual image of the spectral lines.

Since the regions of microwave and optical spectroscopy do not overlap, the two methods do not compete, but rather supplement one another. Microwave spectroscopy can be used for the study of heavy molecules, which cannot be investigated sufficiently effectively by optical methods. It can be applied with considerably greater success than optical methods to the study of nuclear effects in molecular spectra. Indeed, by virtue of its exceptional resolving power, microwave spectroscopy promises to become one of the most powerful methods for studying nuclear spins and nuclear quadrupole moments.

The experiments of Cleeton and Williams in 1934 are of interest not only because they were the first spectral measurements in the microwave region, but also because this first spectral work at the junction between the optical and radio-frequency portions of the electromagnetic spectrum was carried out with the aid of a hybrid spectroscope, in which both optical and electronic methods were used in an essential way. The source was an electronic device, namely a magnetron with a split anode, which Cleeton and Williams constructed by proportionally reducing a 9-centimeter model supplied by the Westinghouse firm. They were able to obtain radiation with a wavelength down to 1 cm, which at that time was the lower limit of wavelengths generated by electronic devices. Their tubes were tunable within limits of about 30%. The authors used four tubes to obtain the inversion spectrum of ammonia at atmospheric pressure in the wavelength region from 1.06 to 3.8 cm. The collimating system consisted of a parabolic mirror, and although the radiation was substantially monochromatic, an echelette grating was used for wavelength measurement. No electrical amplification was employed; after detection by means of a crystal pair of pyrite—phosphor bronze, the signals were sent directly to a sensitive galvanometer. The absorbing cell with which the authors worked was probably unique of its kind in spectral measurements: it was simply a bag made of rubberized material.

Despite the promising beginning, no further successful steps in microwave spectroscopy were made until recent years.

Although the boundaries of the microwave range are not entirely distinct, it may be defined as that part of the electromagnetic spectrum lying between the region of the longest waves of the infrared portion, where optical methods must still be applied, and the region where ordinary electron tubes and high-frequency ...

radio apparatus. This is the region in which volumetric generators, volumetric wavemeters, and waveguides prove effective and convenient in size. With such typical microwave apparatus it is now possible to carry out spectral measurements within the limits of wavelengths from approximately 3 mm to 20 cm. Since molecular absorption lines generally predominate and are more strongly expressed in the region of shorter waves, the millimeter range is probably of greatest interest for microwave spectroscopy. Spectral measurements in the millimeter range were carried out by Beringer²; he used the second harmonic of centimeter generators to measure the absorption of oxygen in the wavelength region from 4.8 to 6.1 mm. We³ extended the measurement range from 6 mm to the centimeter region on one side and to 3 mm on the other.

II. APPARATUS AND EXPERIMENTAL METHOD

Despite the novelty of this branch of physics, microwave spectroscopy makes use of a very wide range of disciplines, technical methods, and apparatus. In a review article it is impossible to give anything like a complete consideration of all the questions relating to this area. Fortunately, the Proceedings of the Radiation Laboratory of the Massachusetts Institute of Technology⁴, which are currently being published, as well as some other recently issued books⁵ and articles⁵, together constitute a very complete and up-to-date source of information on the theory of microwaves and on questions of microwave engineering and apparatus, so that inclusion of all details in the review becomes unnecessary. Our task will be to describe the adaptation and modification of microwave radar apparatus for the purposes of microwave spectroscopy, to show the nature of the information required in this field, and also to indicate the sources from which this information may be obtained.

A. WAVEGUIDE AND PARTS RELATED TO IT

Waveguide. In microwave spectroscopy an ordinary rectangular waveguide is used. To cover the range indicated above, several sizes are required. The upper wavelength limit for a rectangular waveguide of given dimensions is determined by the formula for the critical wavelength as a function of the transverse dimensions of the waveguide \(a\) and \(b\):

\[ \lambda_c=\frac{2}{\sqrt{\left(\frac{m}{a}\right)^2+\left(\frac{n}{b}\right)^2}} . \tag{1} \]

This formula is valid both for transverse-electric \((TE)\) and for transverse-magnetic \((TM)\) oscillations; \(m\) and \(n\) are numbers,

MICROWAVE SPECTROSCOPY

denoting partial oscillations. Thus, for the lowest oscillation \(TE_{01}\) the critical wavelength is \(\lambda_c = 2b\), i.e., twice the larger dimension of the waveguide. The same formula is used for choosing \(a\) and \(b\) in order to cut off the higher types of oscillations and, in this way, it determines the pass band of the waveguide for the propagation of the fundamental wave. In Fig. 1 the dimensions of waveguides corresponding to the various portions of the millimeter range are given, together with an indication of the losses for the recommended upper and lower limits of each portion.

Fig. 1. Dimensions of waveguides for the millimeter region and losses in them (silver waveguide). \(H, I, J, K\)—designations of the ranges.

Following the terminology established for centimeter waves, it is convenient to designate the millimeter frequency bands covered by means of waveguides of different sizes by the indices \(H, I, J\), etc. Sections of waveguides are shown in Fig. 2.

Fig. 2. From left to right: sections of waveguides for the \(H\)-, \(I\)-, \(J\)-, \(K\)-, \(X\)- and \(S\)-ranges. Below is shown a scale in centimeters.

The attenuation of the fundamental oscillation \(TE_{01}\) in a rectangular silver waveguide, expressed as a function of its transverse dimensions \(a\) and \(b\) and of the wavelength \(\lambda\) (all quantities in cm), is given by the formula

\[ \alpha_c = 0{,}1426^{-3/2} \frac{ \dfrac{b}{2a}\left(\dfrac{2b}{\lambda}\right)^3 + 1 }{ \sqrt{ \left(\dfrac{2b}{\lambda}\right)^3 - \dfrac{2b}{\lambda} } } \quad (\text{db per meter}). \tag{2} \]

For a gold-plated waveguide these quantities must be multiplied by 1.23, for a copper one by 1.03, and for a brass one by 2.08.

Cells. Like most parts of microwave apparatus, absorption cells may be, and are, of various kinds. Most often sections of rectangular waveguide with thin mica windows are used to ensure that the cells are airtight. Since in an absorption cell there are coupling elements, T-junctions, slots, etc., it is possible to use waveguides of increased dimensions without exciting undesirable modes of oscillation. In particular, this is expedient for the millimeter-wave region, where the waveguide, owing to the small dimensions required for suppressing higher modes, greatly attenuates the fundamental wave. In Fig. 3 are shown losses for a 5-mm wave in waveguides of various dimensions, according to measurements by A. G. Smith in our laboratory.

Fig. 3. Experimentally measured, for a 5-mm wave, losses in rectangular waveguides of various dimensions. Waveguides for the J- and K-ranges are silver-plated; for the X- and S-ranges, bronze.

We have used very successfully, as an absorption cell for the wavelength region from 3 to 5 mm, a waveguide of the S-range, \(7.68 \times 3.81\ \text{cm}^2\). The losses were very small even for the shortest waves, and the presence of reflections did not create any particular difficulties. In other ranges, cells of enlarged dimensions are expedient in order to avoid the effect of saturation of the molecules by the radiation. By means of a gradually tapering section, an oversized waveguide can be connected to a small waveguide without troublesome reflections. The absorption of the gas located in a waveguide cell is \(\lambda_g/\lambda\) times greater than that of gas located in free space, for the same path length of the ray. Here \(\lambda\) denotes the wavelength in free space, and \(\lambda_g\) the wavelength in the waveguide; the latter is determined by the formula

\[ \lambda_g=\frac{\lambda}{\sqrt{1-\left(\frac{\lambda}{\lambda_c}\right)^2}}, \tag{3} \]

where \(\lambda_c\) is the critical wavelength, given by expression (1).

Often, cavity resonators are also used as absorption cells. It can be shown that the effective absorbing length of a cavity resonator is, in order of magnitude, equal to

\[ \frac{\lambda^{2}}{\pi \lambda_g} Q_L . \]

(\(Q_L\) here denotes the quality factor of the loaded resonator.) In the case of a resonator with a high quality factor this is equivalent to a very long cell. Cavity resonators are especially convenient for studying the Zeeman effect, since a magnetic field can readily be applied. It is also desirable to use them in the study of inert gases, since resonators of small volume may be employed.

Open cells, or the investigation of a gas in free space, may also be used. In the case of millimeter waves, wide-band horns for focusing and receiving radiation are obtained with dimensions that are sufficiently convenient.

For modulating lines with an electric field, a semi-open cell can be used successfully. A horn designed to focus radiation in only one dimension feeds, through a small gap or choke connection, a flat waveguide cell open at the sides. The cell then feeds a second horn connected to the receiver. In order to place a long cell of this type in a short reservoir (for convenience in pumping), a labyrinth-type construction may be used; or, as proposed by W. V. Smith, the cell may be wound into a compact spiral. Since the opposite plates of the flat waveguide are not in electrical contact, an electric field can be applied between these plates. It is to be expected that with such a system smaller losses and a more uniform field will be obtained than with the usual method of using a metal strip stretched on dielectric insulators along the middle of a rectangular waveguide. The method makes it possible to spread the energy through the cell, thereby reducing molecular saturation, despite the close spacing of the electrodes; moreover, no high-voltage source is required for modulation.

Other waveguide components. Other necessary waveguide components, such as choke or flange joints, crystal holders, directional couplers, T-junctions, and attenuators, are usually more narrow-band than the waveguide itself. Consequently, to cover the operating microwave range a considerable number of similar components of different sizes is required. Descriptions and details of the calculation of these components for certain wavelength ranges above \(1\ \text{cm}\) may be found elsewhere\(^{4,5}\). Usually, for the millimeter region, a recalculation can be made. But as the wavelength decreases, the requirements on the accuracy of manufacture of these components increase. In this connection it is often necessary to modify the calculations. Frequently, in the millimeter range it is much easier to make a good contact joint than a satisfactory

throttle connection. In the case of a round waveguide and a cylindrical resonator, one should strive to use the \(TE_{01}\) oscillation. This oscillation has the property, unique of its kind, that its attenuation actually decreases as the frequency increases. Details concerning certain essential parts as applied to the millimeter range will be given in the following sections (Figs. 4–7).

Fig. 4. Some details of microwave apparatus:

\(A\)—generator for the \(J\)-band, \(B\)—directional coupler, \(C\)—wavemeter, \(D\)—crystal detector, \(E\)—attenuator (all parts for the \(J\)-band), \(F\)—crystal converter for the \(J\)-, \(I\)-, and \(H\)-bands, \(G\)—crystal detector for the \(H\)-band.

B. SOURCES

Klystrons and their power supplies. The most widespread source of radiation is the generator with a reflecting klystron. Although this tube appeared comparatively recently\(^6\), it is at present well known to most physicists. There exists a large number of types of this tube, differing from one another in design details,

Fig. 5. Details of a crystal holder for the \(H\)-band\(^ {25}\).

Labels in the figure: holder for the \(1N26\) crystal; flange of a choke joint; short-circuiting plunger; high-frequency choke; high-frequency blocking capacitor; coaxial connector; ordinary connector of the “Sverdli” type; \(H\)-band waveguide (inside dimensions \(1.3 \times 2.8\) mm); scale, 0, 5, 10, 15, 20, 25 mm.

such as, for example, the Oxford tubes, the Shepherd-Pierce tubes, the Heer tubes, the McNally tubes, and others. They differ in such details as

arrangement of the resonator (internal or external) and the method of tuning it. However, all of them operate on one and the same principle of modulating the velocity of the electron beam and of the presence of a drift space, in which the velocity-modulated electrons are bunched, then reflected by a reflector and directed back through the same resonant cavity in such a phase that they give up their energy to the oscillating field of the resonator. A reflex klystron, having only one cavity, is considerably easier to tune than the earlier models with separate resonators for electron bunching and for energy generation. The two-cavity klystron has been successfully used as a frequency multiplier in measuring the frequencies of absorption lines. In this case, however, there was no need to retune the klystron after it had already been set to the desired frequency.

Certain features of the klystron make it an especially suitable source of radiation for spectroscopy. Although it generates relatively small powers—only a few milliwatts, in comparison with several megawatts of peak power delivered by separate magnetrons—this is not a serious disadvantage in view of the existence of saturation effects in molecules, and also because the very sensitive crystals widely used as detectors cannot detect higher powers. In addition to the advantages of simplicity and ease of operation, the klystron can be well stabilized by Pound’s method[^7] with the aid of an external resonator and can thus provide exceptionally high resolving power. The frequency of this stabilized monochromatic source can be readily varied by selecting the stabilizing resonator. Probably the most important feature of the klystron for spectroscopy is the ease with which this tube can be electrically frequency-modulated over any desired range, from a few kilocycles to thirty or forty megacycles, simply by applying an alternating potential to the reflector. When the “frequency-sweep” method is used, the oscillator need not be stabilized in order to obtain high resolving power, or, in other words, to obtain high accuracy in measuring the frequencies of absorption lines, since in this case it is possible to superpose visual markers on the images of the absorption lines on the screen of an oscilloscope. These markers may be obtained by exciting beats between the frequency-modulated source and an external monochromatic frequency used as a standard. More convenient, though less accurate, is the superposition on the investigated line of the “absorption line” of a cavity wavemeter.

At present, klystron generators can be obtained covering the microwave range approximately from 5.7 mm to 1.6 cm and from 2.7 cm to ~16 cm. The first region is one of the most interesting for molecular absorption spectroscopy. The company

Raytheon manufactures a series of klystron tubes for complete coverage of the range \(0.57—1.6\ \text{cm}\). Table I gives the corresponding characteristics of these tubes. The characteristics of other klystron generators for the region above \(1\ \text{cm}\) may be found in the literature2.

Table I

Characteristics of Raytheon klystron tubes

Type	Operating voltage of the resonator, in volts	Operating output power, in watts	Covered wavelength range, in centimeters	Waveguide-output dimensions, in centimeters
RK—2K33	1800	40	1,1—1,6	1,168×0,533 internal dim. 1,270×0,635 external »
QK—140B	2200—2500	18	0,96—1,14	0,711×0,356 internal » 0,813×0,457 external »
QK—140A	2200—2500	18	0,85—1,0	0,711×0,356 internal » 0,813×0,457 external »
QK—141	2200—2500	10	0,80—0,90	0,711×0,356 internal » 0,813×0,457 external »
QK—142	2500—3600	5	0,74—0,85	0,711×0,356 internal » 0,813×0,457 external »
QK—226	2500—3600	∼5	0,68—0,80	0,569×0,284 internal » 0,625×0,386 external »
QK—227	2500—3600	∼5	0,57—0,70	0,569×0,284 internal » 0,625×0,386 external »

Figure 6 shows the schematic diagram of a power supply constructed for Raytheon tubes by W. Bennett. This is a rectifier with electronic stabilization, calculated especially for tubes of the millimeter range. It can easily be adapted for operation with tubes in the range from 1 to \(2\ \text{cm}\). No modulating voltage is provided in this power-supply circuit, but it can without difficulty be applied to the reflector potential from an external source.

The modulating voltage depends substantially on the nature of the measurements being made. Thus, in measurements by the standing-wave method the tube is usually modulated by rectangular pulses. In the frequency-sweep method, sawtooth pulses are used.

Double modulation is often used, in which high-frequency sinusoidal oscillations are superposed on slow sawtooth oscillations, serving to change the frequency of the tube periodically in accordance with their form. After slight amplification, one may use sawtooth pulses from the same oscillograph on which the observation is being made. Usually, with this method, no stabilization is required. For sinusoidal

Fig. 6. Stabilized power supply for Raytheon QK 140, 141, 142 millimeter-wave generators.

Visible Russian labels in the schematic include: “Variac”; “Capacitances in microfarads; resistances in ohms; \(K = 1000\)”; “To the klystron”; “Modulation.”

modulation at high frequencies, commercial high-frequency generators are suitable. Rectangular pulses can be obtained from an ordinary multivibrator.

Crystal harmonic generators. For the wavelength region around 5 mm, no klystron generators are at present applicable. As energy sources for covering the range from 3 to 7 mm, crystal multipliers operating on the second harmonic of the Raytheon tubes listed in Table I have been successfully used\(^3\). Fig. 7 shows the detailed

Labels in the figure: flange of soldered joint; choke; Sperry-type connection; waveguide of the J-band (internal dimensions \(1.0 \times 3.1\ \text{mm}^2\)); locking half-wave section; waveguide of the H-band (internal dimensions \(2.8 \times 1.3\ \text{mm}^2\)); holder of the 1N28 crystal; scale \(0, 5, 10, 15, 20, 25\ \text{mm}\).

Fig. 7. Details of a harmonic converter for the \(J\)-, \(I\)-, and \(H\)-bands\(^ {25}\).

diagram of a converter used in the range from 3 to 5 mm. Behringer’s early work\(^2\) with oxygen in the 5-millimeter region was carried out with a crystal generator of doubled frequency, excited by a 1-centimeter klystron generator.

Unfortunately, there are no exact measurements for determining the efficiency of harmonic generators in the millimeter-wave region. This, of course, is explained by the fact that instruments for absolute power measurements in the millimeter region have not yet been developed. From tubes operating at a wavelength of \(6.4\ \text{cm}\), it was possible to obtain\(^9\) oscillatory power in the \(3.2\ \text{cm}\) region, the latter being \(11\text{–}14\ \text{db}\) below the input power (wavelength \(6.4\ \text{cm}\)). One might hope to obtain from a \(K\)-band generator a second harmonic that would be \(20\text{–}25\ \text{db}\) weaker than the fundamental frequency. Since with \(K\)-band klystrons one can easily have an output power of up to \(20\ \text{mW}\), it is thus possible to obtain in the 5–7 mm region an oscillatory power of the order of \(150\ \mu\text{W}\). This power is sufficient for operation of a spectrometer with a video receiver*) and is at the limit of the power required from a local hetero-

) The term “video receiver” means the connection of a crystal detector directly to a video amplifier, i.e. a wide-band amplifier with a band of approximately \(0.1\) to \(4\ \text{MHz}\). (Translator’s note.*)

suitable for operation of a superheterodyne receiver, provided that there are not too large conversion losses. For the range from 5.7 to 10 mm there are klystrons with output power up to 5 mW. Making the optimistic assumption that it is possible to obtain from them the second harmonic, whose power is only 25 db below the power of the fundamental oscillation, one may hope to have, for spectroscopy in the region from 3 to 5 mm, a power of about 15 μW. This is just what is necessary for the operation of a video receiver or a vacuum thermocouple, but in a superheterodyne receiver the conversion losses prove to be too large. With 1N26 crystals, both as a frequency multiplier and as a detector, the limiting sensitivity of the spectrometer at a wavelength of 3.4 mm is obtained of the order of \(3\cdot 10^{-4}\ \mathrm{cm}^{-1}\).

Data have been obtained\(^{10}\) indicating that welded germanium crystals, developed by G. K. Nortem\(^{11}\), considerably surpass silicon crystals for the purposes of generating both the second and the third harmonics.

Magnetrons. The use of magnetrons with a split anode was already indicated in the early work of Cleeton and Williams. Later, magnetrons were used to measure the absorption of water vapor in the \(K\)-band region\(^{12}\).

Theoretically, the sensitivity of a spectrometer increases with the power of the source, and the higher powers delivered by magnetrons suggest that they should be better than klystrons as sources. However, there are factors that hinder the realization of this advantage.

It is possible that the most serious of these is the saturation effect\(^{13}\), which limits the energy that can be absorbed by a gas at low pressure. It is, of course, possible to distribute the energy over a large volume of gas by using oversized waveguides, large volume resonators, or methods of operation in free space. But in this case the experimental difficulties also increase.

In addition, most sensitive microwave detectors are incapable of withstanding the powers delivered by magnetrons. One could use a balance at high frequency to reduce the energy incident on the detector, but here again the experimental difficulties increase. Although some types of magnetrons can be tuned, in general they are less convenient in this respect than klystrons.

For these reasons magnetrons have not yet become widely used in microwave spectroscopy. They may find their application in the future, when more practical methods are developed for overcoming the difficulties noted above.

In the shortest millimeter region, where the generation of energy is difficult, magnetrons may prove to be the most suitable sources.

Volume VI of the “Proceedings” of the radiation laboratory4 and other publications14 contain detailed and up-to-date information on all the various types of magnetrons.

B. DETECTING SYSTEMS

Superheterodyne receivers. In determining the noise factor of a receiver, the noise level is usually referred to the calculated power of Johnson noise, \(kT\Delta f\). Assuming that a signal of minimum power \(P_{\min}\), which can still be detected, produces at the output a voltage just equal to the voltage of all the noises, we shall have:

\[ P_{\min}=F_r kT\Delta f. \tag{4} \]

Here \(F_r\) is the noise factor of the receiver, including conversion losses and detector noise, as well as noise produced by the amplifier; \(k\) is Boltzmann’s constant, \(T\) is the absolute temperature, and \(\Delta f\) is the effective frequency band of the receiver noise. The purpose of a receiver in microwave spectroscopy is somewhat different from that of a radar receiver, since in microwave spectroscopy the receiver is not required to detect small values of signal energy; rather, it is necessary to detect small variations, or attenuations, against the background of a relatively large signal energy. It is possible, however, to apply balancing at high frequency so as to reduce the energy in the region of the absorption line to zero and thus make the spectral line appear in the form of a small unbalanced power \((\Delta P)'\). It is easy to show that

\[ (\Delta P)'\simeq \frac{(\Delta P)^2}{4P_i e^{-\alpha_c l}}, \tag{5} \]

where \(\Delta P\) is the power absorbed by the gas, \(P_i\) is the power at the input of the cell, \(l\) is the equivalent length of the cell, and \(\alpha_c\) is the attenuation coefficient of the cell. When the total input resistance of the receiver is matched to the source, as in the case of microwave spectroscopy, the quantity \(F_s kT\Delta f\) is added to the available noise power (where \(F_s\) is the noise factor of the source), so that

\[ (\Delta P)'_{\min}=F_r kT\Delta f+F_s kT\Delta f=FkT\Delta f, \tag{6} \]

where \(F\) is now the total noise factor of the spectrometer. Combining (5) and (6), we obtain:

\[ (\Delta P)_{\min}=\sqrt{4Fk\Delta f P_i e^{-\alpha_c l}}, \tag{7} \]

where \((\Delta P)'_{\min}\) denotes the minimum detectable unbalanced power, and \((\Delta P)_{\min}\) the magnitude of the minimum detectable absorption of energy by the gas.

In order to obtain the optimum cell length and the minimum detectable absorption coefficient of the gas with a receiver of this type, one should increase the detected energy \((\Delta P)'\), and not the absorbed energy \((\Delta P)\), as Hershberger incorrectly did\({}^{15}\). We have:

\[ (\Delta P)'=\frac{1}{4}P_i e^{-\alpha_c l}\left(1-e^{-\alpha_g l}\right)^2 \simeq \frac{1}{4}P_i e^{-\alpha_c l}(\alpha_g l)^2. \tag{8} \]

Equating \(\dfrac{\partial(\Delta P)'}{\partial l}\) to zero, we find:

\[ l_{\mathrm{opt}}=\frac{2}{\alpha_c}, \tag{9} \]

which together with (8) gives:

\[ (\Delta P)'_{\max}=P_i\left(\frac{\alpha_g}{e\alpha_c}\right)^2. \tag{10} \]

Substituting this quantity into (6), we obtain

\[ (\alpha_g)_{\min}=e\alpha_c \sqrt{\frac{FkT\Delta f}{P_i}} \tag{11} \]

as the minimum detectable absorption coefficient of the gas.

Although in the analysis presented it was assumed that there is a balance at high frequency, exactly the same results are obtained for a receiver in which linear detection is employed without balance at high frequency. Detection in an ordinary superheterodyne receiver is linear. Similarly, detection in a crystal video receiver in the case of reception of strong signals is approximately linear. For them we have:

\[ V_0=GV_s, \tag{12} \]

where \(V_0\) is the output voltage, \(V_s\) is the rms signal voltage, and \(G\) is a constant including the amplifier gain and the gain (or loss) in conversion in the detector. For a small change in \(V_s\)

\[ \Delta V_0=G\Delta V_s. \tag{13} \]

The noise voltage at the receiver output \(N_0\) is equal to

\[ N_0=G\sqrt{4FkTR\Delta f}, \tag{14} \]

where \(F\) is again the total noise factor, and \(R\) is the total input resistance. To obtain the usual detectability criterion, we equate \(\Delta V_0\) and \(N_0\). Combining (13) and (14), we obtain:

\[ (\Delta V_s)_{\min}=\sqrt{4FkTR\Delta f}. \tag{15} \]

It is easy to show that

\[ \Delta V_s=\frac{R(\Delta P)}{2V_s}=\sqrt{\frac{R}{4P}}\cdot \Delta P, \tag{16} \]

which, together with (15), gives:

\[ (\Delta P)_{\min}=\sqrt{16FkT\Delta fP} =\sqrt{16FkT\Delta fP_i e^{-\alpha_c l}}. \tag{17} \]

This expression coincides, apart from the factor 2, with expression (7), obtained for the case of balancing at high frequency. The optimum cell length is again equal to \(\frac{1}{2\alpha_c}\), and

\[ (\alpha_g)_{\min}=2\alpha_c e\sqrt{\frac{FkT\Delta f}{P_i}}. \tag{18} \]

This result shows that there is no appreciable advantage in using balancing at high frequency so long as the independence of the total noise factor from the received energy is assumed. Such an assumption does not correspond to reality, except in the case of relatively weak signals; therefore balancing at high frequency will give an advantage when high-power sources are available, provided that cells of sufficiently large volume are used to prevent molecular saturation.

Although Gershberger\({}^{15}\) was the first to determine the minimum detectable absorption coefficient, his results are incorrect in order of magnitude because he equated the absorbed power \((\Delta P)_{\min}\) to \(FkT\Delta f\). Tauns and Geschwind\({}^{16}\) pointed out this error and, by means of a somewhat different method, obtained results substantially coinciding with those presented above. These authors take as the initial level of the noise voltage \(\sqrt{2kT\Delta f}\), rather than \(\sqrt{4kT\Delta f}\). They assume that the gas is admitted into and exhausted from the cell at a constant frequency, and use the fact that an amplitude-modulated oscillation is equivalent to an unmodulated carrier with sidebands, in which the signal energy is contained.

An estimate of the expected qualities of a spectrograph with a heterodyne receiver may be made under the assumption of the most readily realizable parameter values: \(\alpha_c=5\cdot10^{-4}\) neper per cm, \(P_i=10^{-5}\) W, \(f=300\) cps, \(F=40\) (or 16 dB). For these data:

\[ l_{\mathrm{opt}}=40\ \mathrm{m} \]

and

\[ (\alpha_g)_{\min}=6\cdot10^{-9}\ \frac{\text{neper}}{\text{cm}} =1.2\cdot10^{-8}\ \mathrm{cm}^{-1}. \]

In the region of the \(K\)-band one can have considerably more than \(10^{-5}\) W. Here it is assumed that a \(K\)-band waveguide is used and that the gas is investigated at such low pressure that powers of more than \(10^{-5}\) W prove ineffective because of molecular saturation. It is also assumed that a pass band has been chosen which permits the use of a convenient sweep speed of the oscilloscope. In order to obtain an estimate of the probable limits of detection, it is useful to assume idealized values of the parameters: \(P_i = 5\cdot 10^{-5}\) W, \(\Delta f = 30\) cps, \(F = 2\), \(a_c = 10^{-4}\) neper/cm. Then

\[ l_{\mathrm{opt}} = 100\ \mathrm{m} \]

and

\[ (\alpha_g)_{\min} = 3.8\cdot 10^{-12}\ \frac{\mathrm{neper}}{\mathrm{cm}} = 7.6\cdot 10^{-12}\ \mathrm{cm}^{-1}. \]

Starting from the assumed power, it is advisable to use a high-frequency balance in order to prevent excessive noise in the crystal, and a cell of large volume in order to prevent molecular saturation. In these estimates losses due to such factors as the output-cell window, mismatch of wave impedances, etc., are neglected. Spurious signals caused by high-frequency mismatch can be eliminated by a method to be described below, though not without appreciable losses.

A superheterodyne receiver is somewhat more complicated than other receivers suitable for effective use. Under conditions in which separate klystrons are used for the source and the local heterodyne, some form of automatic frequency control is necessary to maintain the tuning of the generators, especially if the “frequency-swinging” method is used. Automatic frequency-control circuits\(^{17}\), used in microwave radars, are feasible at slow sweep rates.

L. K. Levitt pointed out to us the possibility of using one and the same generator as the source and as the local heterodyne, as a result of which no automatic frequency control is required. The method can be carried out by allowing a certain part of the source energy to be reflected from a crystal modulated by an intermediate-frequency generator. Sidebands are obtained, having frequencies

\[ \nu_s + \nu_{if}\quad \text{and}\quad \nu_s - \nu_{if}, \]

where \(\nu_s\) is the source frequency, and \(\nu_{if}\) is the frequency of the indicated generator. One of these side frequencies can then be used as the energy source for the absorption cell, while part of the energy of the generator of frequency \(\nu_s\), which does not pass through the cell, is used to feed the local heterodyne. This method of stabilizing generators with the aid of crystals was used by Pound\(^{18}\).

Using the 1N23B crystal for the \(X\)-band, Pound\(^{19}\) obtained by this method of frequency conversion a conversion loss \(L=6\,db\). It seems probable that an analogous local heterodyne can be obtained by the same method also for operation in the \(K\)-band region and, possibly, also in the \(J\)-band.

As will be shown below, the enormous superiority in sensitivity of a superheterodyne receiver in radar over a simple video receiver operating with a square-law detector is not retained in microwave spectroscopy, where not small pulses of energy must be detected, but small changes of it. For this reason, and also because of the complexity of the superheterodyne receiver, which makes it too slow in searching for lines, it has not yet found wide application in microwave spectroscopy.

Crystal video receivers. A simple, fast, and sufficiently sensitive microwave spectrograph can be obtained by means of the “frequency-sweeping” method and a crystal video receiver. In this receiver the crystal detector is mounted in a waveguide holder and must be matched to the microwave line. The crystal is connected directly to the video amplifier. In practice, very narrow-band audio-channel amplifiers are used, but here we retain the term “video receiver.”

The noise arising in the crystal in the absence of a direct-current component and of high-frequency excitation exceeding \(\simeq 5\cdot 10^{-6}\,vt\), is almost entirely Johnson noise. The noise voltage \(N_0\) from the crystal at the output of a video receiver operating in the square-law region can therefore be expressed in the form\(^{20}\)

\[ N_0 \simeq G\sqrt{4kT(R+R_A)\Delta f}, \tag{19} \]

where \(R\) is the resistance of the crystal at the video frequency, and \(R_A\) is that resistance which, being connected in series with the crystal, would give a noise equivalent to the noise produced by the amplifier. Usually \(R_A\) is \(\simeq 1000\,ohm\). In the square-law region the voltage \(V_d\), detected by the crystal, is equal to

\[ V_d = S V_s^2, \tag{20} \]

where \(S\) represents the sensitivity of the detector, and \(V_s\) is the rms voltage at the input. The output voltage of the receiver is equal to

\[ V_0 = G S V_s^2, \tag{21} \]

and for small changes in \(P\)

\[ \Delta V_0 \simeq GS\cdot 2V_s\Delta V_s \simeq GS\cdot R\Delta P, \tag{22} \]

since \(G\) (the voltage amplification factor) and \(S\) are constants.

To obtain the usual detectability criterion, we equate

\[ \Delta V_0 = GS \cdot R \Delta P = G \sqrt{4kT(R+R_A)\Delta f} \]

and obtain:

\[ \Delta P=\frac{\sqrt{4kT(R+R_A)\Delta f}}{RS} =\frac{\sqrt{4kT\Delta f}}{M}, \tag{23} \]

where

\[ M=\frac{RS}{\sqrt{R+R_A}} \tag{24} \]

may be called the sensitivity coefficient of the system \(^{29}\). By definition

\[ (\Delta P)=P_i e^{-\alpha_c l}-P_i e^{-(\alpha_c+\alpha_g)l} \simeq P_i e^{-\alpha_c l}\cdot \alpha_g l . \tag{25} \]

Substituting this quantity into (23), we obtain:

\[ (\alpha_g)_{\min}=\frac{\sqrt{4kT\Delta f}}{M l P_i e^{-\alpha_c l}} . \tag{26} \]

If we assume that \(M\) does not depend on the magnitude of the received energy \(P_i e^{-\alpha_c l}\), then \(l_{\mathrm{opt}}\) can be determined from equation (26), setting

\[ \frac{d\alpha_g}{dl}=0. \]

Then we obtain:

\[ l_{\mathrm{opt}}=\frac{1}{\alpha_c} \tag{27} \]

and

\[ (\alpha_g)_{\min}=\frac{2e\alpha_c\sqrt{kT\Delta f}}{P_i M}. \tag{28} \]

For the \(K\)-band or the millimeter region there are no experimental values for \(M\). In the 3-centimeter region, sensitivity coefficients up to 55 are easily obtained. To estimate the expected result in the 1-centimeter region, let us assume reasonable values: \(M=20\), \(P_i=5\cdot10^{-6}\ \mathrm{W}\), \(\alpha_c=5\cdot10^{-4}\) neper/cm, \(\Delta f=300\) cps; then

\[ l_{\mathrm{opt}}=20\ \mathrm{m} \]

and

\[ (\alpha_g)_{\min}=3\cdot10^{-8}\ \frac{\text{neper}}{\text{cm}} =6\cdot10^{-8}\ \mathrm{cm}^{-1}. \]

In the case where it is possible to have a power greater than that which can be effectively detected, as is usually the case, it is advisable to use a cell of greater length than the \(l_{\mathrm{opt}}\) found above. Suppose that the detectable power is \(\simeq 10^6\ \mathrm{W}\), and that one can have a power considerably exceeding this

quantity, and, since molecular saturation may be neglected, \(\alpha_g\) will be minimal if

\[ l_{\mathrm{opt}} \simeq \frac{6 \ln 10 + \ln P_i}{a_c}. \]

For example, at \(P_i = 10^{-4}\) W the optimum length is equal to \(\dfrac{4.6}{a_c}\). If these data are substituted into (28), taking \(a_c = 5 \cdot 10^{-4}\ \dfrac{\text{neper}}{\text{cm}}\), \(\Delta f = 300\) Hz, \(M = 20\), we obtain:

\[ l = 92\ \text{m} \]

and

\[ (\alpha_g)_{\min} = 1.3 \cdot 10^{-8}\ \frac{\text{neper}}{\text{cm}}. \]

Thus, it turns out that, for the purposes of detecting an absorption spectrum, a simple video receiver successfully competes in sensitivity with the more complicated superheterodyne receiver. This result probably does not have general significance, since it is known that a superheterodyne receiver has much greater sensitivity for detecting radar pulses—namely, approximately \(10^5\) times greater for ordinary pulses of duration \(1\) microsecond.

The fact that \((\alpha_g)_{\min}\) is expressed as a function inversely proportional to the power would seem to suggest that the sensitivity can be increased without limit by increasing the power of the source. This does not correspond to reality, owing to the fact that the sensitivity coefficient \(M\) remains constant in the microwave region only for small values of the energy.

Miller, Greenblatt, et al.\(^{21}\) obtained a large amount of experimental data showing that the noise “temperature” of silicon crystal rectifiers at low frequencies is very large when the applied energy of a dc or high-frequency voltage exceeds several microwatts, and that it varies inversely with frequency. Throughout the entire region covered in these measurements, from 50 Hz to several kilohertz, the inverse proportionality was maintained. Such a dependence makes it impossible to suppress excessive noise by the usual method of slow scanning, so that narrow-band receivers could be used. Suppose, for example, that the frequency of the generator is varied in the region of an absorption line at such a rate that the scanning time \(d_t\) of the line is equal to \(1/1000\ \text{Hz}^{-1}\). The optimum bandwidth, giving the best signal-to-noise ratio, is equal to \(\simeq \dfrac{1}{d_t} = 1000\) Hz. If the line is traversed at a rate equal to \(1/10\) of this value, then a band of width only 100 Hz can be used. This means an increase in sensitivity by 10 dB. However, for

MICROWAVE SPECTROSCOPY

In order that the same portion of the microwave spectrum pass at a slower rate, the sweep frequency must be reduced by a factor of 10. The frequencies of the Fourier components containing the signal are reduced by the same factor. Accordingly, the lower limiting frequency of the receiver must also be lowered by a factor of 10. As a result of the rise in the noise “temperature” of the crystal with decreasing noise frequency, the noise factor increases by a factor of 10. The gain in sensitivity from reducing the receiver pass band is thereby eliminated. The advantages of a superheterodyne receiver, or of one of the similar types to be discussed in the following paragraph, become obvious in the case where it is possible to use, without fear of molecular saturation, an energy much greater than several microwatts. If, however, germanium crystals with a welded contact are available, high sensitivity can be obtained with a video receiver operating at powers much greater than several microwatts. With these crystals high sensitivity coefficients have been obtained at a quite appreciable constant applied voltage.^22

It is very interesting to know the lower energy limit in the operation of crystals in the millimeter region, in particular in the region of the shortest waves, of length 3–5 mm, where it is difficult to obtain sufficient local-heterodyne power for the operation of a superheterodyne receiver without large conversion losses. Any quantitative estimates for waves shorter than 3 cm are apparently not yet possible. A qualitative theoretical consideration leads to an approximate expression of the form^23

\[ \beta=-\frac{a}{1+b f^2}, \]

where \(\beta\) is the current sensitivity of the crystal, coinciding with the proportionality coefficient \(S\) in (20), and \(a\) and \(b\) are two other constants depending on the characteristics of the crystal. To determine the constants \(a\) and \(b\) we used two experimental points, determined for a 3-centimeter crystal by Behringer^24 at frequencies of 9300 Mc/s and 3300 Mc/s, and as a result obtained:

\[ h=\frac{2}{1+1.7\cdot 10^{-20} f^2}. \]

This relation shows that the sensitivity coefficient of a video receiver in which a crystal is used, at a wavelength of 1 cm, decreases to 15% of the value occurring at 3 cm, and to 1.4% of the same value at a wavelength of 3 mm. Better performance than that just indicated can be obtained in the millimeter range by selecting from \(K\)-band crystals those most suitable for millimeter waves. The results obtained in our laboratory^25 with 1N26 crystals operating near 3 mm are in approximate agreement with the calculation, although the quantitative

a comparison is still impossible. Sometimes a balanced two-crystal video receiver is used[^13],[^26]. One of the crystals detects the energy passing through the absorption cell, while the other detects part of the energy of the source that does not pass through the cell. The signals from the two crystals are then brought together in opposite phases and balanced by attenuators until the receiver is balanced (for the case where there is no absorption). In the frequency-sweep method this type of balance is not entirely satisfactory, owing to the difficulty of obtaining sufficiently good matching of the total resistances over a wide range of changes in the generator frequency.

With greater success in the frequency-sweep method, a simple crystal video receiver[^27] is used with an amplifier equipped with a filter that cuts off low frequencies sufficiently sharply to eliminate the influence of slow changes in amplitude and of spurious signals caused by mismatch at high frequency. In this system the gas pressure is chosen so as to make the absorption line as narrow as possible in comparison with the signal caused by random amplitude variation, or by any spurious signal caused by mismatch at high frequency. The sweep rate is chosen so that the principal Fourier components of the random amplitude variations and of the spurious signals lie in the region below the cutoff frequency, while the main part of the components of the absorption line would fall within the passband of the amplifier. In this system it is usually difficult to avoid some distortion of the signal. Although this is not very important in the case of a system used only for finding lines, this method will be unsuitable for studying the shapes of lines. One of the features of this system, which proves useful for the possibility of distinguishing absorption lines from spurious electronic disturbances, is the circumstance that the absorption-line signal can be immediately destroyed by increasing the gas pressure in the cell by a factor of 8–10, i.e., enough to broaden the line substantially and cause its principal Fourier components to fall into the region below the cutoff frequency of the filter.

Modulation methods. In order to avoid excessive low-frequency noise when silicon crystals are used in the case of detecting powers above several microwatts, and also in order to get rid of low-frequency noise from other sources such as the flicker effect in tubes[^28], it is desirable to amplify the signal at frequencies of 100 kc and above. There are two methods that make it possible to do this without the complexity inherent in a superheterodyne receiver. Both methods are fundamentally similar in that, in both, the absorption line itself is used to create high-frequency amplitude modulation of the microwave radiation, while for amplification of the signal

at the modulation frequency a narrow-band commercial receiver is used. The principle is similar to the well-known beam-chopping method, which makes it possible in infrared spectroscopy to use alternating-current amplifiers instead of direct-current amplifiers. However, at the frequencies that are desirable in the case of interest to us, the use of a mechanical chopper is impossible, and therefore the modulation must be achieved electrically. We call these two methods of electrical modulation molecular modulation and source modulation.

Molecular modulation. Soon after the Stark effect was first demonstrated in the rotational spectrum by Daikin, Good, and Coles²⁹, electrical modulation of absorption lines at low radio frequencies was introduced by Hughes and Wilson as an auxiliary detection method. It is easy to see that amplitude modulation can be achieved by this method if we imagine the source oscillator tuned to a fixed frequency at some point of a narrow absorption line. If the absorption line is shifted, the magnitude of the absorption will change, and the intensity of the radiation reaching the detector will be modulated. In practice, the frequency of the source oscillator is made to vary in the region of the absorption line at a rate small in comparison with the frequency of the molecular modulation, so that the absorption lines can be observed on the screen of a cathode-ray oscillograph in the usual way. In addition to increasing the sensitivity by eliminating low-frequency noise, this simple method also completely eliminates the troublesome problem of spurious signals caused by mismatch in the microwave line. Because only the absorption line produces amplitude modulation of the microwaves at the frequency to which the very selective receiver is tuned, spurious signals of all types are reduced to a minimum. Although a more complicated spectrum is observed as a result of Stark splitting of the lines, the additional data obtained in this case have their own value. If a voltage in the form of rectangular pulses³¹ is used for molecular modulation, the Stark splitting can be resolved and used for additional identification of the lines. It can also be used to determine the dipole moment of the molecules under study.

As in other methods, a number of difficulties are encountered. Some of the gain in sensitivity is lost because the line is split into a large number of components. Further, in the usual methods of carrying out modulation inside a waveguide cell, conducting strips are placed in the middle of it. This introduces a mismatch of the total resistance, and dielectric losses in the posts or spacers supporting the conducting strips also contribute to a reduction in sensitivity. A cell in which some of these difficulties can be avoided is described in the section on the waveguide

and the parts connected with it. To prevent direct coupling between the high-voltage modulating source and the highly sensitive receiver, tuned to the same frequency, careful shielding is necessary. Owing to the large input capacitance, it proves difficult to obtain the required high-voltage modulation with a cell of optimal length. This is true, in particular, for modulation by rectangular pulses. The chief advantage of this method in sensitivity over a simple video receiver lies in the possibility of detecting large energies without excessive noise. In order to realize this advantage, cells of sufficiently large dimensions must be used so as to eliminate molecular saturation.

Hughes and Wilson^30 report that, with the aid of Stark modulation, the 1.1 line in the inversion spectrum of N^15H₃ is detected in the presence of N^15 at the naturally occurring concentration of 0.37%.

A group at the Massachusetts Institute of Technology^32 applies Stark modulation by rectangular pulses at audio frequency and uses a superheterodyne receiver. After preliminary amplification at a high intermediate frequency of 30 Mc/s, the audio frequency is detected and amplified by a narrow-band audio amplifier. In this way, only signals from the absorption line appear on the indicator. At this low frequency the total resistance between the electrodes of the cell is large, and an electric field of several thousand volts per cm can be applied to the molecules. The Stark splitting is resolved and can be studied. Although this system lacks the advantage in sensitivity, it is used with great effectiveness. Fig. 8 shows the block diagram of this spectrometer.

A theoretical analysis of the influence of various forms and frequencies of modulation on the line shape was given by Karplus^33. For the study of sinusoidal modulation see also the works of Merritt^34, Blokhintsev^35, and Hershberger^15.

A magnetic field can also be used for frequency modulation of spectral lines. Indeed, the Zeeman effect has already been used as an auxiliary means of detection. (See the section on special methods.)

Modulation of the source. Gordy and Kessler^36, and independently of them Hershberger^15, showed that, to obtain high sensitivity without using a heterodyne receiver, high-frequency modulation of the source can be used*).

) Apparently, Wilson and his co-workers had tried still earlier to apply source modulation by rectangular pulses at a frequency of 50 kc/s without apparent success [unpublished communication, cited in R. J. Watts and D. Williams, Phys. Rev. 72*, 1122 (1947)]. Utz and Williams report that they have successfully applied this method.

On the slowly varying sawtooth voltage, used to impart to the klystron the corresponding periodic regime, an alternating voltage of much higher frequency (up to several megahertz) is superposed; this forces the generator to make rapid “excursions” into the region of greatest absorption and back, while the spectral line is gradually passed through by the slow sweep. The spectral line here acts as

Diagram labels in Fig. 8, in the order shown: source generator; directional coupler; attenuator; matched termination; absorbing cell; matched termination; branch coupler; attenuator; local oscillator; crystal mixer B; IN26; frequency standard; frequency marks, 20 Mc/s; transition from 1-cm to 3-cm waveguide; pumping; Stark electrode; rectangular-pulse generator, 6 kc/s; transition from 3-cm to 1-cm waveguide; crystal mixer A; intermediate frequency; IF amplifier, 30 Mc/s, and detector; discriminator, 300 kc/s; Stark effect; direct absorption; amplifier, 6 kc/s; signal; cathode follower; oscilloscope; signal and marks; marks of the standard frequency; sawtooth-pulse generator, 20 c/s; cross-section of the absorbing cell; 3-cm waveguide; electrode; polystyrene.

Fig. 8. Block diagram of a microwave spectrograph operating by the Stark-modulation method.^32

a discriminator, converting the frequency modulation into intensity modulation of the same frequency). After detection by means of a crystal, the signal is then further amplified by a narrow-band high-frequency amplifier tuned to the modulation frequency. It then passes through an audio-frequency amplifier with a filter that cuts off the low frequencies sufficiently sharply and plays the role of a discriminator with respect to the low-frequency oscillations arising as a result of reflections in the microwave line*).

The early results of Gordy and Kessler, reproduced in Fig. 9, show that with this system one can obtain good resolution and good sensitivity.

*) Apparently, the steep slope of the absorption line can indeed be used as a discriminator for a commercial frequency-modulation receiver operating in the microwave range.

**) As a possible variant, one may use a low-frequency Stark modulation in combination with high-frequency modulation of the source.

The line of N¹⁵H₃ shown here, exceeding the noise level by approximately 10 db, denotes a sensitivity of approximately \(10^{-7}\) neper/cm. These results were obtained with a receiver having a pass band of about 3000 cps, and with a \(K\)-band cell which, owing to molecular saturation, somewhat limited the effective power level. The distortions noticeable in these photographs are due to the low-frequency limiting filter mentioned above. To achieve this low-frequency filtration one must sacrifice at least 3 db of signal power.

Fig. 9. Upper curve: the 3.3 line of ammonia N¹⁵H₃ at its natural concentration (10.3%) in normal ammonia. (Cell length 3.6 m, pressure \(2 \cdot 10^{-3}\) mm Hg, source modulated at a frequency of 100 kc.) Lower curve: the 3.3 line of ammonia N¹⁴H₃, showing the structure of the satellites; source modulation 100 kc.^36

Specialized methods. The sensitive detecting system employed by Roberts, Beers, and Hill^37 for direct observation of the hyperfine structure of the Cs vapor spectrum in the 3-centimeter wavelength region can be used for many other purposes. In this method a Pound stabilizer is used to lock the microwave generator to the frequency of an external tunable cavity resonator containing gas or vapor at low pressure, of the order of \(10^{-2}\) mm Hg. The anomalous dispersion of the vapor in the absorption-line region slightly changes the resonant frequency of the resonator. Because the absorption line is modulated at some frequency by means of an alternating magnetic field, the klystron associated with the cavity will be frequency-modulated at this frequency. For detecting the absorption line, a frequency-modulation receiver is then used.

A somewhat more specialized, but exceptionally sensitive, method is the atomic-beam method, used by Lamb and Retherford^38 in the well-known experiment with hydrogen at microwave frequencies. In the future, undoubtedly, certain other varieties of the atomic and molecular beam method will be adapted for microwave frequencies.^39 Similarly, sensitive radio-frequency methods,^40 recently deve...

obtained for detecting nuclear resonance in solids and liquids may later find application in the microwave region.

A very effective method for studying the Zeeman effect in gases is described in the work of K. K. Jen$^{40a}$. A hollow resonator of high quality factor is used as the cell. The gas pressure is such that the absorption line is narrow in comparison with the resonance curve of the resonator. To increase the sensitivity, the above-described method of high-frequency modulation of the source is employed. The minimum detectable signal, by Jen’s definition, is equal to

\[ \alpha_{\min}=\sqrt{\frac{4kTN\Delta f}{P_0}}\cdot \frac{2\pi}{Q\lambda}, \tag{29} \]

where \(P_0\) is the supplied power, \(N\) is the noise factor of the receiver, \(\Delta f\) is the pass band of the receiver noise, \(\lambda\) is the wavelength in free space, and \(Q\) is the ordinary quality factor of the resonator. Fig. 10 shows the block diagram of this instrument and gives photographs of spectral lines as they appear on the screen of a cathode-ray oscilloscope.

Thermal detectors. Thermocouples and bolometers have not yet found wide application in microwave spectroscopy as detectors. Nevertheless, in view of their ability to detect relatively large powers in comparison with crystals and because of their possibly greater sensitivity than crystals to the shortest millimeter waves, they deserve consideration. A very serious drawback, particularly of thermocouples, is their considerable inertia. Unlike crystals, the lower limit of sensitivity (of the order of a microwatt) of thermal detectors of microwave radiation did not advance far during the war, despite the fact that fairly sophisticated thermistor bridges were developed for measuring powers of the order of milliwatts in the \(K\)-, \(X\)-, and \(S\)-bands. Research and design work on thermal detectors for microwave spectroscopy should be of great importance, particularly in the millimeter-wave region, where the effectiveness of crystals decreases rapidly with increasing frequency. Although receivers equipped with thermal detectors are considerably worse than crystal heterodyne receivers for detecting small values of the energy of centimeter waves, as is required in radiolocation, theoretical consideration shows that, for detecting small changes in comparatively large energies, as is required for observing an absorption spectrum, thermal detectors prove comparable in sensitivity with crystals. If we assume, as we did for crystals, that thermal detectors can be matched to the microwave line and that the limiting factor is Johnson noise, then it becomes possible to carry out

Fig. 10. a — diagram of a microwave spectrograph for studying the Zeeman effect; b and c — oscillograms of the Zeeman splitting of a spectral line in a gas located in a volume resonator\(^{40a}\).

Labels in the diagram:

• a)
• Electromagnet
• Coils
• To pump and manometer
• Gas-filled resonator
• Probe of the flowmeter sensor
• Mica window
• Crystals
• Radio-location receiver
• Cathode oscillograph
• Frequency-modulation receiver, 30–100 MHz
• Microwave generator, 0.1–10 m
• 2nd klystron
• Low-frequency generator, 100 Hz
• 1st klystron
• Attenuator
• Attenuator
• Attenuator
• Sawtooth sweep voltage
• b)
• c)

of simple analysis. The thermoelectromotive force \(V\) developed in the thermocouple as a result of the action of the power \(P\) is equal to

\[ V = SP, \tag{30} \]

where \(S\) is the sensitivity of the thermocouple. For small changes in \(P\),

\[ \Delta V = S \Delta P . \tag{31} \]

We assume here that a sufficiently long counting time is taken so that \(S\) may be regarded as constant. Equating \(\Delta V\) to the Johnson-noise voltage, from (31) we obtain:

\[ \Delta P_{\min} = \frac{\sqrt{4 k T (R_t + R_A)\Delta f}}{S}, \tag{32} \]

where \(R_t\) is the resistance of the thermocouple, and \(R_A\) is the resistance which would produce thermal noise equal to the noise arising in the galvanometer or amplifier connected to the thermocouple.

Combining (32) and (25), we obtain:

\[ (\alpha_g)_{\min} = \frac{\sqrt{4 k T (R_t + R_A)\Delta f}} {S I P_i e^{-\alpha_c l}}, \tag{33} \]

which, in the case of the optimum cell length \(l_{\mathrm{opt}} = \frac{1}{\alpha_c}\), gives:

\[ (\alpha_g)_{\min} = \frac{2 e \alpha_c \sqrt{kT(R_t + R_A)\Delta f}} {S P_i}. \tag{34} \]

If we denote

\[ M = \frac{S}{\sqrt{R_t + R_A}}, \]

then this expression coincides with (28), where \(M\) now represents the sensitivity coefficient of the receiver with the thermocouple. The resistances and sensitivities of a large number of microwave thermocouples are given in Volume IX of the Proceedings of the Radiation Laboratory\(^{41}\). For one of the vacuum types, having a power limit of \(5\) mW, \(S\) is equal to \(20\) mV/mW and \(R\) is equal to \(38\) ohms. If this thermocouple is used with an amplifier system having the same resistance, then very small \(\alpha_g\) can be detected, provided that the gas absorption is distinguishable from power fluctuations caused by high-frequency reflections. The value of \(M\) for this case is equal to \(2\), and, taking for the \(K\)-band reasonable values \(\Delta f = 30\) cps, \(P_i = 5 \cdot 10^{-3}\) W, \(\alpha_c = 10^{-4}\) neper/cm, we obtain:

\[ (\alpha_g)_{\min} = 2 \cdot 10^{-11}\ \text{neper/cm}. \]

This is quite comparable with the value \(3.8 \cdot 10^{-12}\) neper/cm obtained for a superheterodyne receiver with an idealized noise factor and with the other parameters equal to those taken here.

Considerable work with thermocouples and bolometers has been carried out for the purpose of detecting powers of the order of fractions of a microwatt in the infrared region. These data can be used for

estimates of the minimum detectable value \(a_g\) that should be expected in the microwave region under the condition of effective absorption by the thermocouple of the incident radiation. In the case of quadratic detectors one may assume that the minimum value \(\Delta P\) that can be detected is comparable with the minimum detectable energy in that energy interval for which the sensitivity remains constant. Thus,

\[ (\Delta P)_{\min} \simeq P_i e^{-\alpha_c l}\cdot (a_g)_{\min}=P_{\min} \tag{35} \]

or

\[ (a_g)_{\min}=\frac{P_{\min}}{P_i e^{-\alpha_c l}} =\frac{e a_c P_{\min}}{P_i}, \]

if

\[ l=l_{\mathrm{opt}}=\frac{1}{a_c}. \tag{36} \]

Here \(P_{\min}\) denotes the minimum power detectable by means of the given thermocouple or bolometer. For ordinary thermocouples and bolometers operating in the infrared region, the minimum detectable power is approximately \(10^{-8}\) W \(^{42,43}\). For \(P_i=5\cdot 10^{-5}\) W and \(a_c=5\cdot 10^{-4}\) neper/cm we shall have

\[ (a_g)_{\min}=2.7\cdot 10^{-7}\ \text{neper/cm}. \]

Both bolometers and thermocouples have succeeded in detecting \(^{43}\) powers of the order of \(10^{-10}\) W. Thus the estimate made is apparently too low. If Andrews’s superconducting bolometer is not counted, dc thermocouples appear to be somewhat more sensitive than bolometers. However, for microwave spectroscopy this advantage in sensitivity is probably of secondary importance in view of the much smaller inertia of bolometers. With bolometers one may, if desired, use modulation with a frequency greater than a thousand. Thus the Stark-modulation method can be applied so that the indicator does not respond to changes in energy not caused by gas absorption. Recently \(^{44,40}\) thermocouples have appeared with sufficiently small inertia for use with transformers and ac amplifiers. Special low-frequency amplifiers have been designed \(^{45}\) for working with them.

Apparently, it is possible to construct a microwave spectrometer with a bolometric detector, with good stability and high sensitivity, and in doing so to use a narrow-band locking amplifier \(^{46}\) and Stark modulation of the molecules to neutralize changes in the intensity of the beam. It would be still better to stabilize the source generator by coupling it to a tunable cavity resonator \(^{7}\), and to employ automatic recording. As a possible variant one may use the method of frequency sweeping in combination with a cathode oscillograph with slow sweep and afterglow. Deli and Cezernland \(^{47}\) developed a system for demonstration on a screen

of the oscilloscope of the infrared absorption spectrum, using a bolometer with a response time of 0.01 sec.

Otler and Beker\({}^{12}\) successfully used thermocouples to measure the absorption of water vapor in the region of \(1.32\ \text{cm}\). The measurements were carried out with water vapor at various partial pressures in air at one atmosphere. A magnetron was used as the source. The water vapor was contained in an approximately cubical chamber of volume \(15.8\ \text{m}^3\). The thermocouple system used for detection had 360 junctions. The thermocouples were distributed throughout the chamber, and the junctions were alternately covered with absorbing material. The measurements consisted in determining the change in \(Q\) (the quality factor) of the chamber due to absorption by the water vapor and in calculating the absorption coefficient from this change. To attain a uniform radiation density in the chamber, an “oscillation mixer” in the form of an electric fan was used, which “stirred” a large number of types of oscillations in the chamber. It was assumed that the steady-state thermocouple current was proportional to the quality factor \(Q\) of the chamber. The theory of this spectroscope was considered by Lamb\({}^{48}\).

G. DETERMINATION OF THE SENSITIVITY OF THE SPECTROMETER

The methods used to measure the noise factor of radio-location receivers may be applied to monitoring the operation of a microwave spectrometer. In the region of the \(K\)-band, thermistor bridges\({}^{49}\) are applicable for measuring energy, as are combined attenuators\({}^{50}\), by means of which the measured power is reduced by a definite number of times to a value considerably below that which can be measured directly. With their aid one can measure the input power required in order to have at the output \(F_r kT\Delta f\) or, for video receivers, \(\sqrt{4kT\Delta f/M}\), and thus obtain \(F_r\) or \(M\). Similar methods are applicable to receivers with thermal detectors. \(P_r\) is readily measured by a thermistor bridge. Knowing this quantity, one can calculate \((a_\sigma)_{\min}\), expected for the given spectrometer. Sources of high-frequency noise and of intermediate-frequency noise are also suitable for convenient determination of noise factors\({}^{51}\).

If the noise factor of the amplifier at the intermediate frequency \(F_{if}\), the noise “temperature” of the crystal \(t_c\), and the conversion loss \(L\) of the crystal are measured independently, then they may be used to calculate the noise value of the heterodyne receiver from the formula

\[ F_r = 10 \lg L + 10 \lg (t_c + F_{if} - 1). \tag{37} \]

For an intermediate frequency of \(30\ \text{Mc/s}\), the conversion loss \(10 \lg L\) in the case of a typical 1N26 crystal is \(\approx 7\ \text{db}\) when operating in the \(K\)-band region, and \(t_c\) is approximately equal to 2. A good value of the noise factor at intermediate frequencies is 2 or 3 db. Hence \(F_r \approx 12\ \text{db}\). If a balanced mixer is not used,

excluding the noise of the local heterodyne, one can obtain a value greater by 3–8 db^52. The noise of the source must also be taken into account. In a system without high-frequency balancing it may raise the total noise by several decibels. The increase in the noise figure caused by the local heterodyne or by the source noise is quite different for different types of klystron oscillations, even if the power is maintained at one and the same level. The fraction of the noise due to the klystron is minimal for the average frequency of oscillations of the given type; for lower oscillation frequencies it is less significant than for higher ones^52. Since the conversion losses are not associated with the noise of the local heterodyne or of the source, these latter noises may be included in the total receiver noise by increasing \(t_c\) in (36) by a suitable amount.

The most direct method for determining the total sensitivity, or the sensitivity coefficient \(M\), of a microwave spectroscope of any type is to measure the signal-to-noise ratio for weak spectral lines with a known absorption coefficient. For the 3.3 line of the inversion spectrum of \(\mathrm{N}^{14}\mathrm{H}_3\), many investigators^13,26,53 have obtained a value of the absorption coefficient of about \(7 \cdot 10^{-4}\ \mathrm{cm}^{-1}\). The intensity of the \(\mathrm{N}^{15}\mathrm{H}_3\) lines, in the case of the naturally occurring concentration of \(\mathrm{N}^{15}\), equal to 0.37%, ensures, in the region of the \(K\)-band, the production of weak signals whose intensity can be calculated from measurements of the \(\mathrm{N}^{14}\mathrm{H}_3\) lines. The \(\mathrm{N}^{14}\mathrm{H}_3\) lines in the range between 7 mm and 1 cm are convenient for calibration purposes, since in general they amount to less than one percent of the 3.3 line^54. The absorption coefficients of all measured ammonia lines have been calculated; the results are given in Table II (see p. 244), together with the frequencies. Other molecules may be used in a similar way if their line widths have been measured, since the absorption produced by them can then be calculated theoretically. For example, the most intense JCN lines with \(\mathrm{C}^{13}\) at its normal concentration of 1% scarcely rise above the noise level of existing spectrometers, while the \(\mathrm{JC}^{12}\mathrm{N}^{15}\) lines are approximately another three times weaker. Thus, small but known in advance concentrations of naturally occurring isotopes such as \(\mathrm{S}^{34}\), \(\mathrm{N}^{14}\), and \(\mathrm{C}^{13}\) form useful calibrated “attenuators” many decibels below the absorption value of the corresponding most abundant isotope.

D. INTENSITY MEASUREMENTS

Absolute intensities. In the regions of resonant absorption, measurements have been made of the absorption coefficients of oxygen^2, water vapor^47, and ammonia^13,15; the absorption of some other gases has been measured in nonresonant regions^55. All these measurements were made at fixed frequencies (the constant-frequency method) at relatively high gas pressures. Under these condi-

conditions the problem is similar to the measurement of ordinary losses in a microwave transmission line or in a hollow resonator, except that in the present case it is also necessary to take into account the influence of the cell windows and of dielectric effects in the gas. Consequently, all the information concerning absorption measurements that has accumulated in the course of the development of radar may be useful[^49].

Measurement in a waveguide cell. One of the simplest methods is the measurement of gas absorption in a waveguide cell. A detector whose “response” is proportional to the energy is placed at the output of the cell, and after it a linear amplifier is used. Readings on the indicator are taken before and after the gas has been admitted into the cell. The absorption coefficient is equal to

\[ \alpha=\frac{10\lambda}{L\lambda_g}\lg\frac{d-\Delta d}{d} \quad \text{db per unit length}, \]

where \(d\) is the deflection, or reading on the indicator, before admission of the gas, and \(\Delta d\) is the change caused by the gas. A balanced detector is most often used. A change in the wavelength in the cell, due to the change in dielectric constant resulting from admission of the gas, can cause errors if reflections occur in the line. These and other extraneous effects are considered in detail by Beringer[^2], and also by Bleany and Penrose[^56]. The need for a square-law detector and a linear amplifier is eliminated if a calibrated microwave attenuator is used to return the indicator reading after admission of the gas to its value before admission. The absorption of the gas will then be equal to the difference between the attenuator settings. This method was used by Good[^51]. Townes[^13], for determining \(\alpha_g\), measured the change in standing-wave coefficient that occurs as a result of gas absorption in a shortened waveguide cell. In order to avoid possible errors of the detector and amplifier, he used a calibrated attenuator, which brought the indicator readings to the same value at the maximum and at the minimum. The accuracy attainable by this method is estimated by Townes at approximately \(2\cdot10^{-5}\) neper/cm.

Method of the tuned resonator. For measuring the absolute absorption of \(\mathrm{NH}_3\), Bleany and Penrose[^53] used a tuned cavity resonator. At a given wavelength the absorption coefficient is equal to

\[ \alpha_g=\frac{2\pi}{\lambda}\left(\frac{1}{Q_1}-\frac{1}{Q_0}\right), \tag{38} \]

where \(Q_0\) and \(Q_1\) are the effective \(Q\)-factors of the resonator before and after admission of the gas*).

*) A simple check of this equation is possible. By definition,

\[ Q=\frac{2\pi\nu W}{-\dfrac{dW}{dt}}, \]

\(Q_0\) and \(Q_1\) are measured by detuning the resonator. According to the estimate of Bleaney and Penrose, by this method, without frequency stabilization, one can measure values of \(\alpha\) of the order of \(5\cdot 10^{-4}\ \mathrm{cm}^{-1}\). A similar but more sensitive method, used by the same authors, is the measurement of the energy, by means of a weakly coupled crystal detector, at the top of the resonance curve of the resonator in the presence and in the absence of gas. Under the indicated conditions

\[ \frac{Q_1}{Q_0}=\sqrt{\frac{d_1}{d_0}}, \tag{39} \]

where \(d_1\) and \(d_0\) are the indicator readings in the presence and in the absence of gas inside the volume resonator. Substituting (39) into (38), we obtain:

\[ \alpha=\frac{2\pi}{\lambda Q_1}\left(1-\sqrt{\frac{d_1}{d_0}}\right) = \frac{2\pi}{\lambda Q_0}\left(\sqrt{\frac{d_0}{d_1}}-1\right). \tag{40} \]

The sensitivity depends first of all on the constancy of the generator power. By this method Bleaney and Penrose were able to detect absorption down to \(2\cdot 10^{-6}\ \mathrm{cm}^{-1}\). Recently Weidner\(^{57}\), using the same method, achieved a sensitivity of \(6\cdot 10^{-8}\ \mathrm{cm}^{-1}\) in the region of \(4^{1}/_{2}\)-centimeter waves.

The use of Pound’s frequency stabilizer should yield an improvement in this type of measurement with hollow resonators. The use of the frequency-swinging method will also be effective. In this case the resonance curve of the resonator may be obtained on the screen of a cathode oscilloscope. If the gas pressure is such that the absorption line is very broad in comparison with the resonance curve of the volume resonator, the visible changes of this curve after admitting the gas can be used to calculate the absorption coefficient by means of the equations written above.

Other methods. We have already pointed out the use of a large resonance chamber by Becker and Autler\(^{12}\). This rather laborious method is distinguished by good sensitivity. Other methods, developed specifically for measurements of gas absorption in the atmosphere\(^{58}\), we shall not consider here. However, the principle of ampli—

where \(W\) is the stored energy, and \(dW/dt\) is the rate of loss of energy. By the definition of \(\alpha_g\):

\[ \alpha_g=-\frac{1}{W}\cdot\frac{dW}{dx} = -\frac{1}{W}\cdot\frac{dW}{dt}\cdot\frac{dt}{dx} = -\frac{1}{W}\cdot\frac{dW}{dt}\cdot\frac{1}{c}. \]

Combining both expressions, we have:

\[ Q_g=\frac{2\pi\nu}{\alpha_g c}=\frac{2\pi}{\alpha_g\lambda}. \]

Further,

\[ \frac{1}{Q_1}=\frac{1}{Q_0}+\frac{1}{Q_g} = \frac{1}{Q_0}+\frac{\alpha_g\lambda}{2\pi}, \]

which after rearrangement coincides with (37).

... with phase synchronization (phase-lock-in amplifier), used in Dicke’s wide-band microwave radiometer[^58], can be adapted to various kinds of laboratory instruments that measure absorption spectra. This principle had in fact already been used by Beringer[^2] in the above-mentioned measurements of oxygen absorption.

Relative intensities. The methods just described for obtaining absolute values of absorption coefficients are inapplicable for measurements at low pressures of the order of \(10^{-3}\) mm Hg or lower. At such pressures the width of the absorption lines is a fraction of a megacycle, and even for the roughest measurements an exceptional stability of the generator frequency would be required. Usually the study at low pressures is carried out using a frequency-modulated source and observing the lines on the screen of a cathode-ray oscillograph. With the aid of some of the methods described in the section on detecting systems, narrow and extremely weak absorption lines can be detected, but these methods do not directly give absolute values of \(\alpha_g\). It is possible, however, to obtain the relative intensities of a group of lines localized in one spectral region by comparing the heights of different lines under the same conditions of power, receiver gain, and sensitivity. If the lines differ greatly in intensity, the gain factor may be varied with the aid of a calibrated attenuator. Difficulties arise in monitoring the applied power and the sensitivity of the receiver. The usual control instruments (monitors) are themselves sensitive to changes in frequency. But most often the conditions change little within several megacycles, and, for closely spaced lines, an exceptionally accurate comparison can be made. The accuracy obtained is usually sufficient to allow the different components of the hyperfine structure to be identified by comparison with theoretical intensities. The best results are obtained when a receiver with high fidelity of reproduction is used. High-frequency Stark modulation, or modulation of the source, greatly increases the uncertainty of the relative intensities.

E. FREQUENCY MEASUREMENTS

For many problems in microwave spectroscopy, a simple cavity wavemeter, allowing rapid counting, is quite suitable. Nevertheless, when in determining the structure of polyatomic molecules one has to deal with different isotopes of one and the same element, the accuracy of determining molecular dimensions depends, among other conditions, on the relative accuracy with which the difference in the frequencies obtained for the different isotopes is established. These differences are often of the order of 100 Mc. Thus...

whereas, if an ordinary cavity wavemeter were used for measurements, the structural determinations would be so inaccurate as to be almost useless, while with the aid of the frequency standards described in the following paragraphs these differences can be readily measured with an accuracy of up to \(0.1\%\). A similar need for precise measurements arises in determining nuclear quadrupole coupling factors, which also depend on relatively small differences in frequency. If a frequency standard is unavailable, simpler methods used for measuring frequency differences may often prove effective.

Resonance cavity wavemeters. For the wavelength region above \(1\ \text{cm}\), various cavity wavemeters have been developed, with accuracy limits of approximately from \(\pm 3\) to \(\pm 10\) Mc. Details of their construction and various information about them may be found elsewhere\(^{59,60}\). The coaxial type is self-calibrating and can be made tunable over a very wide frequency range. The cylindrical resonator operating in the \(TE_{01n}\) oscillation mode is simpler in construction, especially for the millimeter range; owing to its higher \(Q\), it gives better accuracy in comparison both with the coaxial type and with the cylindrical resonator in the \(TE_{11n}\) mode.

The frequencies of the various types of oscillations of a cylindrical resonator are given by the formula

\[ f=\frac{c}{2}\sqrt{\left(\frac{2x_{lm}}{\pi D}\right)^2+\left(\frac{n}{L}\right)^2}, \tag{41} \]

where \(c\) is the velocity of light in free space, and \(D\) and \(L\) are, respectively, the diameter and length of the cylinder. For \(TM\) oscillations, \(x_m\) is the \(m\)-th root of the Bessel function \(J_l(x)=0\), and for \(TE\) oscillations it is the \(m\)-th root of \(J'_l(x)=0\). Tables of roots of Bessel functions, as well as oscillation patterns, very useful in the calculation of cavity wavemeters, may be found in various works\(^{59,60}\). Since in \(TE_{01n}\) oscillations there are no axial currents, in order to obtain a high \(Q\) there is no need to provide chokes in the plungers and to fit a tightly fitted piston. This simplifies the construction and makes it possible to suppress undesirable oscillations simply by means of a loosely fitted piston. Usually a damping ring is placed behind the piston, contributing to the suppression of these oscillations. For the case of oscillations of type \(TE_{01n}\), equation (41) becomes

\[ f=\frac{c}{2}\sqrt{\left(\frac{7.6634}{\pi D}\right)^2+\left(\frac{n}{L}\right)^2}. \tag{42} \]

This expression can be used for calibration purposes, which in most cases is sufficiently accurate for the identification of absorption lines already measured by more precise methods. These

spectral lines can then be used to obtain a more precise calibration of the wavemeter, or the latter can be calibrated by means of the frequency standard described in the following paragraph. Since extraneous oscillations are not completely suppressed, wavemeters of this type should be used with some caution. Figure 11 shows a transverse section of a cylindrical cavity wavemeter for the \(TE_{01n}\) oscillation and gives the dimensions of the cylinder for various regions of the millimeter range.

Fig. 11. Cross section of a cavity wavemeter for the \(TE_{01n}\) oscillation. Cylinder diameter: 14.33 mm for the upper part of the \(J\)-band (wavelength \(\sim 8.1\)—\(11.0\) mm), 11.91 mm for the \(I\)—\(J\) region (wavelength \(\sim 7.0\)—\(8.5\) mm), 9.52 mm for the upper part of the \(I\)-band (wavelength \(\sim 5.5\)—\(7.1\) mm).

In the case of a silver cylindrical resonator operating in the \(TE_{01n}\) mode, the quality factor \(Q\), which determines the sharpness of tuning, may be approximately determined from the equation

\[ Q = \frac{ 2.73 \cdot 10^{6} \left[ 1 + 0.168 \left( \frac{D}{L} \right)^{2} n^{2} \right]^{\frac{3}{2}} }{ \sqrt{f} \left[ 1 + 0.168 \left( \frac{D}{L} \right)^{3} n^{2} \right] }, \tag{43} \]

where \(f\) is given in megahertz. This quantity will be somewhat reduced by the presence of the coupling window. The accuracy of the wavemeter depends on the half-width of its resonance curve, which is equal to

\[ \Delta f = \frac{f}{Q_{\mathrm{eff}}}, \tag{44} \]

where \(Q_{\mathrm{eff}}\) is the effective quality factor, or the quality factor in the loaded state. Thus, for a specified accuracy, the required value of \(Q\) increases with frequency. The \(TM_{11n}\) oscillation, which is the expression of \(TE_{01n}\), can reduce \(Q\) for the latter if cross-coupling occurs. (For measures to prevent this, see \(^{59}\).) Wavemeters that we have constructed for the range from 5 to 10 mm have experimentally measured quality factors of about 30,000.

For the 1.2-centimeter range, cylindrical cavity wavemeters using oscillations of the type \(TE_{11n}\) were constructed.\(^{59}\) They are self-calibrating and have the additional advantage that extraneous oscillations can be eliminated by restricting the diameter of the resonator. Because of the presence of the chokes required in the plunger, they are, for millimeter waves, more complicated in design. The attainable \(Q\) is also somewhat lower than for the \(TE_{0n}\) oscillation.

Wavemeters can conveniently be connected to a microwave line by means of directional couplers or T-junctions. Usually there must be an attenuation of several decibels between the wavemeter and the generator in order to avoid pulling the generator frequency. In the frequency-modulation method, the “peak” of the wavemeter is superposed on the absorption line appearing on the oscilloscope screen.

Standard frequency multipliers. Precise frequency standards for the centimeter-wave region have been created, obtained by multiplying standard low frequencies maintained with an accuracy exceeding \(1:10^{7}\). Although these standards were created for calibrating secondary standards such as cavity wavemeters, one of them is used for the direct measurement of spectral lines.\(^{61}\) Descriptions of these standards are available in the literature.\(^{62,63,64}\)

One of the systems\(^{64}\) was developed for obtaining standards in the millimeter region and for direct measurements of absorption lines. The initial frequency is supplied by a temperature-compensated 10 Mc/s generator, controlled by a station transmitting standard frequencies (a WWV station). With the aid of a Lissajous figure, continuous monitoring is achieved, as shown in Fig. 12. The frequency of 10 Mc/s is multiplied up to 270 Mc/s with ordinary electron tubes. To convert 270 Mc/s into 2970 Mc/s, a Sperry 2K47 klystron multiplier is used. This output frequency is multiplied by the crystal a certain number of times up to the desired region. In this way strong markers are obtained, separated from one another by a distance of 2970 Mc/s.

The crystal is also modulated by a frequency of 270 Mc/s and, in addition, by a frequency of 90 Mc/s applied in parallel to the klystron. The side frequencies produced by this modulation give markers covering the microwave region at intervals of 90 Mc/s.

A calibrated ordinary radio receiver, also controlled by the WWV radio station, is used for interpolation between the 90 Mc/s markers in the following way. The marker oscillations are mixed with the oscillations of the source in a suitable microwave crystal mixer. The receiver is tuned to the beat frequency between the oscillations of the source generator irradiating the cell and the standard oscillations nearest to the absorption line. In the frequency-sweep method it is easy, by mixing the two signals, to superpose the “peaks” from the receiver on

...absorption line appearing on the oscilloscope screen (see Fig. 12). The receiver tuning range covers 90 MHz, so that readings can be taken from markers located on both sides of the spectral line.

Fig. 12. Block diagram of a microwave spectrometer^64.

Since the two readings made in this way must complement one another to 90, a constant check on the accuracy is ensured. For identification of the markers, cavity wavemeters are used.

The NH\(_3\) lines in the millimeter range were measured^54 with the aid of the system described, down to the line 16.16, corresponding to 39,941.5 MHz. At these frequencies intense markers were still obtained. It may be thought that power sufficient for operation can be obtained at least up to the 20th harmonic*) of the klystron frequency at 2970 MHz, so that direct measurements down to 5 mm appear feasible. If the second harmonic is used to irradiate the cell, then the first harmonic is measured at the output of the source generator and the frequency is doubled. With this method, accurate measurements are possible in the wavelength region shorter than 5 mm.

With the aid of the standard described, spectral lines can be measured with an accuracy of about 50 kHz at a frequency of 25,000 MHz and about

*) In another standard^61, harmonics of the crystal up to the 35th are used, giving markers at a frequency of 32,256 MHz.

kHz at a frequency of 40,000 MHz. Good and Coles\(^{63}\) report an accuracy of 0 kHz for the NH\(_3\) lines, measured in the region of 25,000 MHz. For descriptions of other frequency standards see \(^{65}\).

Measurement of frequency differences. Daly and others\(^{66}\) proposed a method for obtaining an image of absorption lines by frequency-modulating the source generator with a calibrated tunable low-frequency generator. This method was used for the precise measurement of small intervals between closely spaced absorption lines, such as the hyperfine components of the inversion spectrum of ammonia.

Carter and Smith\(^{67}\) proposed using a secondary generator of undamped oscillations, tuned in the range covered by the source generator under periodic variation of its frequency. They found that interference of the two generators should produce sharp signals. By modulating the secondary generator, side frequencies are produced, giving a grid of marks of variable and known density. These methods are simpler to implement than standard frequency multipliers, and can be used for the precise measurement of small frequency differences.

For measuring large differences in frequency, a convenient method\(^{68}\) is the use of a calibrated receiver measuring the beat frequency between two generators, one of which is chosen as a fixed generator with constant frequency. This method can be used for absolute frequency measurements, provided that the fixed generator is set to the frequency of a known absorption line. However, to obtain high accuracy the fixed generator must be stabilized at the frequency of this known line.

III. ABSORPTION SPECTRA OF GASES AND VAPORS*)

A. ATOMIC SPECTRA

The limits from 0.05 to 3.5 cm\(^{-1}\), approximately indicating (in waves per centimeter) the region of accessible microwaves, encompass energy differences between many terms of both the fine and hyperfine structure of a large number of atoms. For example, as a result of the interaction of nuclear magnetic moments with the magnetic field of the valence electrons, the ground states of Na\(^{23}\), Rb\(^{87}\), and Cs\(^{133}\) are split into doublets with level differences of 0.059, 0.228, and 0.307 cm\(^{-1}\), respectively. The level differences of the doublet fine structure \(2^2P_{1/2} - 2^2P_{3/2}\) for H and Li are respectively 0.365 and 3.338 cm\(^{-1}\). The advantages possessed by microwave methods over optical ones with respect to the accuracy of measurement of such intervals need not be emphasized. However, experi-

*) See also the review by V. L. Ginzburg, UFN 31, no. 3, 320 (1947). (Ed.)

MICROWAVE SPECTROSCOPY

mental difficulties of observing atomic spectra in the microwave region are very great. How, in individual cases, these difficulties can be overcome is illustrated by the observations of the hyperfine structure of cesium by Roberts, Beers, and Hill^37 and by the classical experiments of Lamb and Retherford with hydrogen^68. In the experiments of Roberts, Beers, and Hill it was possible to measure 14 components of the Cs transitions. The method they employed was described in the preceding section.

The results obtained by Lamb and Retherford are of outstanding importance, for they revealed the first clear discrepancy between experiment and the contemporary quantum theory. According to Dirac, levels with the same \(n\) and \(j\) must be degenerate, i.e., the level \(2^2 S_{1/2}\) must coincide with the level \(2^2 P_{1/2}\), etc. On the contrary, Lamb and Retherford showed that the state \(2^2 S_{1/2}\) is in fact higher than the state \(2^2 P_{1/2}\) by approximately \(1000\) MHz.

Although their experiments were performed in the microwave region, Lamb and Retherford did not directly measure the transition from the level \(2^2 P_{1/2}\) to the level \(2^2 P_{3/2}\), corresponding to the wavelength \(2.74\) cm. They used the atomic-beam method with a detector sensitive only to atoms in an excited state. The detector registered atoms in the metastable state \(2^2 S_{1/2}\). When these atoms passed through weak microwave radiation of the corresponding frequency, some of them were raised to the state \(2^2 P_{3/2}\), from which they could then rapidly descend to the ground state \(1^2 S_{1/2}\). This led to a decrease of the indicator current, since a smaller number of excited atoms reached the detector. Measurements were made with components of the Zeeman splitting of the transition \(2^2 S_{1/2} \longrightarrow 2^2 P_{3/2}\) in a weak field, and by extrapolation the energy difference in the absence of a field was determined.

Bethe^70 showed that the anomaly in the fine structure of hydrogen may be the result of the interaction of the electron with the radiation field. This possibility had been indicated by Schwinger, Weisskopf, and Oppenheimer^70. Starting from such an interaction, Bethe predicted for the \(2S\) level a shift of \(1050\) MHz and a negligibly small shift for the \(2P\) level, which is in good agreement with the results of the measurements of Lamb and Retherford. According to Bethe’s theory, this electromagnetic shift of levels increases with \(Z\) and decreases rapidly with \(n\). Thus, the shift of the \(2^2 S_{1/2}\) level for \(\mathrm{He}^+\) should be much greater than for hydrogen. (According to Bethe’s prediction, greater by a factor of 13.) The shift of the \(3S\) level should be smaller than the shift of the \(2S\) level of the same atom. This conclusion may serve to test the validity of Bethe’s theory*). Apparently Lamb and Retherford propose to investigate \(\mathrm{He}^+\).

*) After this section had been written, a note by Fowle appeared (G. R. Fowle, Phys. Rev. 74, 219 (1948)), reporting that the predicted shift of the \(S\)-level of ionized helium had been observed in the optical spectrum.

B. MOLECULAR SPECTRA

1. Inversion spectra

a) Ammonia. The inversion spectrum of ammonia is not only the first spectrum investigated in the microwave region, but also the microwave spectrum most thoroughly studied theoretically. As is well known, ammonia molecules absorb microwave radiation owing to their ability to “turn inside out.” Two equilibrium positions of the nitrogen atom, situated on both sides of the plane determined by the three hydrogen atoms, are separated by a potential barrier of about \(0.38\ \mathrm{cm}^{-1}\). As a result of quantum-mechanical resonance the nitrogen atom can penetrate through the barrier (“tunnel effect”). This phenomenon was considered theoretically \(^{71}\) and investigated in the infrared region of the spectrum with respect to its influence on the purely rotational spectrum \(^{72}\). The inversion spectrum of ammonia consists of several separate lines produced by molecules in different rotational states. This fine structure was discovered by H. S. Hoo \(^{73}\), and explained theoretically by Sheng, Barker, and Dennison \(^{74}\), Good \(^{76}\), and, later, by many other investigators.

The formula for the fine structure obtained by Sheng, Barker, and Dennison \(^{74}\) contains too few terms to obtain good agreement with microwave-spectroscopic data. To calculate the fine structure of the inversion spectrum of ammonia, Good \(^{76}\), and later Simonson and Gordy \(^{54}\), used a fourth-degree formula,

\[ \nu=\nu_0+A J(J+1)+B K^2+CJ^2(J+1)^2+ \]
\[ +DJ(J+1)K^2+EK^4, \tag{45} \]

used by Slavsky and Dennison \(^{77}\) for the calculation of purely rotational spectra of nonrigid symmetric molecules. The work of the latter investigators contains measurements in the millimeter-wave region for lines up to \(J=16,\ K=16\), and a formula for the fine structure of the lines in the special form:

\[ \nu\ (\text{in } \mathrm{MHz})=23{,}787-151.3\,J(J+1)+211.0K^2+ \]
\[ +0.5503J^2-1.531J(J+1)K^2+1.055_5K^4. \tag{46} \]

This formula agrees with the results of measurements to an accuracy of up to \(25\ \mathrm{MHz}\). Thus satisfactory agreement should also hold for terms of still higher order. The observed positions of the fine-structure lines are given in Table II. Strandberg, Kyhl, Wentink, and Hillger \(^{61}\) noted an anomaly in the positions of the lines \(K=3\). This anomaly was explained by Nielsen and Dennison \(^{78}\) on the basis of \(K\)-splitting of these levels.

The Van Vleck–Weisskopf formula\(^{79}\), specialized for the inversion spectrum of ammonia, takes, for individual lines at the resonance frequency and temperature \(T = 300^\circ\mathrm{C}\), the form very close to

\[ a = 18\nu^2 [K^2(2J+1)/J(J+1)]\, g \times \]

\[ \times \exp[-0.0475J(J+1)+0.0174K^2], \tag{47} \]

where \(g=1\) for \(K=1, 2, 4, 5,\ldots\), \(g=2\) for \(K=3, 6, 9,\ldots\), and \(\nu\) is measured in MHz.

In this specialization, the nuclear quadrupole hyperfine splitting of the lines is neglected, and it is also assumed that the half-width of the lines \(\Delta \nu\) is 13 MHz for a pressure of 1 mm of mercury. Bleaney and Penrose\(^{75}\) obtained empirical confirmation that the half-width of the lines varies as \([K^2/J(J+1)]^{1/3}\). This variation, however, may be neglected here. In the transformation made above it was assumed that

\[ \mu_{ij}^{\,2} = (1.44 \cdot 10^{-18})^2 [K^2/J(J+1)]. \]

Equation (47) was used to determine the absorption coefficients of NH\(_3\) given in Table II.

The hyperfine structure of N\(^{14}\)H\(_3\), the line shape, and other properties of the NH\(_3\) spectrum will be discussed in the next section.

b) Other molecules. Although the inversion spectrum has been studied so thoroughly only for NH\(_3\), it is clear that this type of spectrum may be observed in the microwave region for some other molecules as well. In the infrared region, Sezerbland, Lee, and Wu\(^{80}\) observed a splitting of 2.4 cm\(^{-1}\) for the excited vibrational state \(\nu_2\) of the PH\(_3\) molecule. It will give rise to a microwave spectrum in the wavelength region around 4.2 mm. At room temperature only about one percent of the molecules are in the state \(\nu_2\). Nevertheless, with slight heating this spectrum could be made readily observable. Similarly, it would be possible to observe the inversion spectrum for the state \(\nu_2\) of ND\(_3\), which, on the basis of infrared studies\(^{81}\), should be expected in the wavelength region around 4.1 mm. The population of the \(\nu_2\) state of ND\(_3\) is about three percent. Inversion-type spectra of other molecules\(^{81}\), such as H\(_2\)O\(_2\), may also be well revealed in the microwave region.

2. Electronic spectra

Oxygen. Unlike most microwave absorption spectra, the oxygen spectrum\(^{82}\) arises to a greater extent as a result of magnetic, rather than electric, coupling with the radiation field. Owing to its symmetry the O\(_2\) molecule has no permanent electric dipole moment, but it does have in its own

Table II

Microwave frequencies and absorption coefficients of ammonia

$J$	$K$	Frequencies in Mc/s: Good and Coles *)	Frequencies in Mc/s: Strandberg et al. **)	Frequencies in Mc/s: Simons and Gordy ***)	Absorption coefficient in $cm^{-1}\cdot 10^6$ ****)
			$^{14}\mathrm{NH}_3$
1	1	23 694,49	23 694,48		140
2	2	23 722,63	23 722,59		320
2	2	23 098,79	23 098,78		62
3	3	23 870,13	23 870,09		720
3	2	22 834,17	22 834,02		130
3	1	22 234,53	22 234,49		30
4	4	24 139,41	24 139,38		390
4	3	22 688,29	22 688,73		340
4	2	21 703,36	21 703,32		77
4	1	21 134,29	21 134,46		14
5	5	24 532,98	24 532,90		370
5	4	22 653,00	22 653,00		170
5	3	21 285,27	21 285,32		150
5	2	20 371,46	20 371,51		28
6	6	25 056,02	25 056,06		320
6	5	22 732,43	22 732,47		150
6	4	20 994,61	20 994,63		54
6	3	19 757,57	19 757,55		63
7	7	25 715,17	25 715,11		260
7	6	22 924,94	22 924,88		240
7	5	20 804,83	20 804,76		56
7	4		19 218,52		26
8	8			26 518,91	190
8	7	23 232,24	23 232,16		10
8	6	20 719,21	20 719,19		80
9	9			27 478,00	230
9	8	23 657,48	23 657,44		58
9	7	20 735,44	20 735,47		26
10	10			28 604,73	83
10	9	24 205,29	24 205,20		72
10	8	20 852,51	20 852,50		16
11	11			29 914,66	55
11	10	24 881,90			19
11	9	21 070,70	21 070,76		17
12	12			31 424,97	50
12	11	25 695,23			11
12	10		21 391,55		4,6
13	13			33 156,95	16
13	12			26 655,00	12
14	14			35 134,44	8,8
14	13			27 772,52	2,8

Continuation of Table II

\(J\)	\(K\)	Frequencies in Mc/s	Frequencies in Mc/s	Frequencies in Mc/s	Absorption coefficient in \(\text{cm}^{-1}\cdot 10^6\)
		Good and Coles *)	Strandberg et al. **)	Simmons and Gordy ***)	****)
\multicolumn{6}{c}{\(\mathrm{N}^{14}\mathrm{H}_3\)}
15	15			37 385,18	8,3
15	14			29 061,14	1,3
16	16			34 941,54	2,0
\multicolumn{6}{c}{\(\mathrm{N}^{15}\mathrm{H}_3\)}
1	1	22 624,96			6,0
2	2	22 649,85			12,8
2	1	22 044,28			2,3
3	3	22 789,41			27,1
3	2	21 783,98			5,0
3	1	21 202,30			1,1
4	4	23 046,10			14,7
4	3	21 637,91			12,8
4	2	20 682,87			2,9
5	5	23 421,99			13,5
5	4	21 597,86			6,4
5	3	20 272,04			5,7
6	6	23 922,32			11,7
6	5	21 667,93			5,6
7	7	24 553,42			9,0
7	6	21 846,41			8,9
8	8	25 323,51			6,8
8	7	22 134,89			0,38
9	8	22 536,26			2,2
10	9	23 054,97			2,7

) W. E. Good and D. K. Coles, Phys. Rev. 71, 383 (1947).
) M. W. P. Strandberg, R. Kyhl, T. Wentink, Jr. and R. E. Hillger, Phys. Rev. 71, 326 (1947).
) J. W. Simmons and W. Gordy, Phys. Rev. 73, 713 (1948).
**) For the isotope concentrations occurring under natural conditions.

in the ground state \({}^{3}\Sigma\) a permanent magnetic dipole moment equal to two Bohr magnetons. The resonance absorption observed in the region of wavelengths of the order of \(5\) mm corresponds not to transitions between different rotational levels, but to transitions between terms of the fine structure of these rotational levels. This fine structure appears as a consequence of the interaction between the rotational moment of the molecule, characterized by the quantum number \(K\), and the total electronic spin moment, defined by the quantum number \(S\). Since in a \(\Sigma\)-state the electronic orbital moment is absent, the quantum number of the total moment \(J\) has the form

\[ J=K+S,\ K+S-1,\ldots,\ K-S. \]

The selection rules are \(\Delta J=\pm 1\) and \(\Delta K=0\). For \(\mathrm{O}_2\), \(S=1\), and, consequently, three values of \(J\) are possible for each given value of \(K\). Therefore each rotational level \(K\) is split into a triplet. The intervals between the components of the triplet depend on the nature and magnitude of the coupling of \(S\) with the molecular axes \(^{83}\).

The formula for the intervals between the lines of the superfine structure, derived by Kramers \(^{84}\) and corrected and supplemented by Schlapp \(^{83}\), gives, for microwave frequencies,

\[ \begin{aligned} \nu\ (\text{in } \mathrm{cm}^{-1}) &= -(2K+3)B+\lambda+ \left[(2K+3)^2B^2+\lambda^2-2\lambda B\right]^{\frac12} -\mu(K-1) \\ &\quad \text{for } J=K+1\to K, \end{aligned} \tag{48} \]

and

\[ \begin{aligned} \nu\ (\text{in } \mathrm{cm}^{-1}) &= (2K-1)B+\lambda- \left[(2K-1)^2B^2+\lambda^2-2\lambda-2\lambda B\right] +\mu K \\ &\quad \text{for } J=K-1\to K. \end{aligned} \]

Here \(K=1,3,5,7,\ldots\), \(B=\dfrac{h}{8\pi^2 c I_B}\), \(I_B\) is the moment of inertia of the molecule, and \(\lambda\) and \(\mu\) are coupling coefficients whose values must be determined experimentally. From optical data they have the following values: \(B=1.43777\ \mathrm{cm}^{-1}\), \(\lambda=1.985\ \mathrm{cm}^{-1}\), and \(\mu=0.00837\ \mathrm{cm}^{-1}\) \(^{82,83}\).

The formulas given above predict two series of closely spaced lines. All of them, except one, should lie in the region of wavelengths of the order of \(5\) mm. The line \(K=1\), corresponding to the transition \(J=K-1\to K\), should have a wavelength close to \(2.5\) mm. Beringer \(^{2}\) measured the absorption of oxygen*) in the region from \(4.8\) to \(5.9\) mm at such high pressures that the structure could not be resolved. Measurements of the lines at lower pressures

) See U. R. L. Lamont, Phys. Rev. 74*, 353 (1948).

now entirely feasible. They could be used to obtain more accurate values of the intervals between fine-structure components and of the parameters \(\lambda\) and \(\mu\).

Van Vleck\(^{82}\) calculated the absorption of microradio waves by oxygen, both in the resonant and in the nonresonant regions, and compared the results of the calculations with the measurements of Beringer\(^{2}\) et al. The agreement between theory and experiment was found to be satisfactory. He took \(\Delta \nu = 0.02\ \text{cm}^{-1}\) as the most probable value for the half-width of the lines at atmospheric pressure. The maximum absorption of pure oxygen measured by Beringer in the wavelength region of the order of \(5\ \text{mm}\) at atmospheric pressure is about \(67\ \text{db/km}\), or \(7.7 \cdot 10^{-5}\ \text{nepers/cm}\).

Although oxygen is the only known example of this type of spectrum appearing in the microwave region, the possibility is not excluded that similar absorption may fall within the accessible microwave region of the spectrum also in the case of other molecules having unpaired electrons, for example \(S_2\), SO, or \(Se_2\).

3. Pure rotational spectra

a) Diatomic and linear polyatomic molecules

The rotational frequencies of diatomic molecules are given, with accuracy satisfactory for our purposes, by the formula\(^{85}\)

\[ \nu\ (\text{in } \text{cm}^{-1}) = 2B(J+1) - 4D(J+1)^3, \tag{49} \]

where \(J = 0, 1, 2, \ldots;\quad B = \dfrac{h}{8\pi^2 c I},\quad D = \dfrac{4B^3}{\omega^2}\), \(I\) is the moment of inertia and \(\omega\) is the natural frequency of vibration.

For \(J\) less than approximately 3, the second term in (49) can usually be neglected. For JCl, for example, at \(J=4\) the second term amounts to only \(80\ \text{kHz}\), but at \(J=10\) it already reaches \(1.6\ \text{MHz}\). Sufficiently accurate values of \(D\) can be obtained from microwave measurements of rotational spectra only if rotational lines with \(J\) greater than approximately \(J=10\) can be measured. Thus, some vibrational frequencies can be determined from rotational spectra.

In the case of diatomic molecules, measurement of the position of only one rotational line corresponding to the lowest value of \(J\) makes it possible to determine accurately the internuclear distance for the ground vibrational state. If measurements can also be made on molecules in an excited vibrational state, equilibrium values of the parameters can be obtained. In the first approximation\(^{86}\)

\[ B_n = B_e - \alpha \left( n + \frac{1}{2} \right), \]

where \(n\) is the vibrational quantum number and \(a\) is a small constant. If, as is usually the case, the ground and first excited states are used, then

\[ B_e=\frac{1}{2}(3B_0-B_1),\quad a_e=B_0-B_1. \]

The only diatomic molecule whose purely rotational spectrum has been investigated in the microwave region is iodine monochloride. The discovery of this spectrum was first reported by Weidner \(^{87}\), who observed, in the wavelength region of the order of \(4.5\ \mathrm{cm}\), the transition \(J=0\to 1\). Later it was studied in greater detail by Townes, Merritt, and Wright \(^{87}\) in the wavelength region of the order of \(1.2\ \mathrm{cm}\). The molecular and nuclear constants determined by them are given in Tables IV–IX.

From the microwave rotational spectra of diatomic molecules it is also possible to determine the mass ratios of different isotopes. The difference of the frequencies corresponding to different isotopes in the ground vibrational state, corrected for nuclear quadrupole and nuclear magnetic perturbations, is \(^{88}\)

\[ \Delta\nu_r=\nu_r-\nu_r^i=2B_e(J+1)(1-\rho^2)-a_e(1-\rho^3)(J+1)- \]
\[ -4D(1-\rho^4)(J+1)^3, \tag{50} \]

where \(B_e\), \(a_e\), and \(D\) refer to the isotope corresponding to \(\nu_r\), and \(\rho=\left(\frac{\mu}{\mu_i}\right)^{1/2}\). Thus, from (50) the value of \(\rho\) can be obtained and, consequently, the mass ratio determined. In addition, if lines with low \(J\) are used, the last term in (50) may be neglected. Applying microwave methods to JCl, Townes, Merritt, and Wright \(^{80}\) obtained for the mass ratio \(\mathrm{Cl}^{35}/\mathrm{Cl}^{37}\) the value \(0.945\,980\,1\pm0.000\,005\,0\). This agrees with the value given by Aston \(^{89}\): \(0.945\,980\,6\pm0.000\,030\,0\), but not with the later value published by Mattauch \(^{90}\): \(0.945\,944\,1\pm0.000\,006\,5\). There is no doubt that in the future the values of the mass ratios of many other isotopes will be determined in a similar way.

The rotational frequencies of linear polyatomic molecules are given by the same equation (49). The value of the parameter \(D\) cannot always be obtained from vibrational frequencies; however, in those cases where \(D\) is not negligibly small, its value can be determined from measurements of microwave rotational spectra. The equilibrium values \(B\), in first approximation \(^{91}\), are given by the equation

\[ B_v=B_e-\sum_i a_i\left(n_i+\frac{d_i}{2}\right), \]

where the summation extends over all vibrations, with degenerate vibrations counted only once. \(n_i\) denote the vibrational quantum numbers, \(d_i\) the degrees of degeneracy of the vibrations.

and \(\alpha_i\) are small constants corresponding to various types of vibrations. Thus, the value of \(B_e\) can be obtained by measurements with molecules in \(m+1\) different vibrational states, where \(m\) is the number of types of vibrations inherent in the given molecule. As a result of the interaction between vibrations and rotations, which occurs in the case of degenerate types of vibrations associated with a change in valence bonds, a doubling of the rotational lines arises, called \(l\)-doubling. The distance between the components of the doublets increases with increasing \(J\) and is determined by the relation\(^{92}\)

\[ \Delta \nu = 2q(J+1), \]

where \(q\) is a small constant whose value was found theoretically by Nielsen and Shaffer\(^{93}\).

This constant was determined for some molecules directly from microwave measurements\(^{94,95}\). Agreement with the values obtained on the basis of theoretical calculations proved to be only approximate.

Since linear polyatomic molecules, generally speaking, are characterized by \(n-1\) independent molecular-structural parameters, their structure cannot be determined from only one moment of inertia, as is done in the case of diatomic molecules. To a very good approximation, however, internuclear distances remain unchanged when a given atom is replaced by various isotopes. By carrying out measurements with molecules in \(n-1\) different isotopic combinations, it is possible to determine the structure of the molecule completely. Owing to the difference in zero-point vibrational energies for different isotopes, the average internuclear distances in the ground vibrational state for molecules containing different isotopes do not coincide completely. In most cases, however, this difference does not cause errors in structural determinations exceeding a few thousandths.

If hyperfine structure or other splitting is neglected, then the absorption coefficient for an individual microwave rotational line at the resonance frequency turns out to be very close to the value

\[ \alpha(\text{in } cm^{-1}) = \frac{8\pi^3 N F_\nu \nu^3 |\mu_J|^2}{3ckT\Delta \nu} \cdot \frac{g_I g_J}{Q_r} \exp\left(-\frac{E_r}{kT}\right). \tag{51} \]

This expression is applicable to linear molecules, both symmetric and asymmetric. Here:

\(N\) — number of molecules in \(1\ cm^3\) of the absorbing medium;

\(\nu\) — frequency of the rotational line in \(sec^{-1}\);

\(|\mu_J|^2\) — square of the matrix element for the moment corresponding to the transition (averaged over all orientations of \(J\));

\(c\), \(k\), and \(T\) — the speed of light, Boltzmann’s constant, and the absolute temperature, respectively;

\(\Delta\nu=\dfrac{1}{2\pi\tau}\) — the half-width of the line, i.e. half its width between the points corresponding to half the maximum intensity;

\(\tau\) — the mean time of flight between collisions that perturb the molecular vibrations;

\(F_v=\dfrac{g_i}{Q_v}\exp\left[-\dfrac{\omega_i hc}{kT}\right]\) — the fraction of molecules in a definite vibrational state; \(g_i\) — the statistical weight of the state; \(\omega_i hc\) — the excess energy relative to the fundamental vibrational state; \(Q_v\) — the vibrational distribution function;

\(\dfrac{g_I g_J}{Q_r}\exp\left[-\dfrac{E_r}{kT}\right]\) — the fraction of molecules in the lower rotational state of the transition; \(g_I g_J\) — the weight of the state and \(Q_r\) — the rotational distribution function.

For asymmetric diatomic and linear polyatomic molecules, which are of interest here,

\[ g_n=1,\qquad g_J=2J+1,\qquad |\mu_J|^2=\mu^2\frac{J+1}{2J+1}, \]

\[ Q_r=\frac{kT}{hcB},\qquad E_r=hcBJ(J+1), \]

and the relation given above takes the form

\[ \alpha=\frac{4\pi^3 NF_\nu hc\, \nu\mu^2}{3(kT)^2\Delta\nu} \exp\left[-\frac{hcBJ(J+1)}{kT}\right], \tag{52} \]

where \(\alpha\), \(B\), \(\nu\), and \(\Delta\nu\) are expressed in \(\mathrm{cm}^{-1}\), and \(\mu\) (the dipole moment) in electrostatic units. In this expression \(J\) refers to the lower state. From kinetic theory it follows that the mean free-flight time \(\tau\) is proportional to \(\sqrt{T}\) and inversely proportional to the pressure. It has been well established experimentally that, under conditions where molecular collisions are the main cause of line broadening and the pressures do not exceed 10 cm of mercury, the half-width of the line is proportional to the pressure. For this case we may write:

\[ \Delta\nu=(\Delta\nu)_1 P_{\mathrm{mm}}\sqrt{\frac{300}{T}}, \tag{53} \]

where \((\Delta\nu)_1\) is the half-width of the line at a pressure of 1 mm of mercury and a temperature of \(300^\circ\mathrm{K}\). It is assumed here that all collisions are equally effective in interrupting molecular vibrations, independently of the speed of the molecules. The number of molecules in 1 \(\mathrm{cm}^3\) of absorbing thickness is given by the relation

\[ N=9.68\cdot 10^{18}\frac{P_{\mathrm{mm}}}{T}. \]

With the aid of these expressions, the equation obtained above is reduced to...

to a form convenient for calculations:

\[ \alpha = \frac{76.7 \cdot 10^{-3} F_{\nu}\mu^2\nu^3} {T^{5/2}(\Delta\nu)_1} \cdot \exp\left[-0.72\,\frac{J\nu}{T}\right], \tag{54} \]

where \(\mu\) is expressed in Debye units, while \(\nu\), \((\Delta\nu)_1\), and \(\alpha\) are, as before, in \(\mathrm{cm}^{-1}\). Some of the values of the half-width of the lines are contained in Table IV. The equations given show the advantages of studying molecules at low temperatures and high frequencies, owing to the stronger absorption under these conditions.

A certain number of linear molecules have already been investigated in the microwave region of the spectrum. The results obtained are collected in Tables VI and VII.

Molecules of the symmetric-top type. The rotational energy of a nonrigid molecule of the symmetric-top type was found by Slawsky and Dennison\(^{77}\), and in Herzberg’s notation\(^{96}\) has the form

\[ \frac{E_r}{hc} = BJ(J+1)+(A-B)K - \]

\[ - D_J J^2(J+1)^2 - D_{JK}J(J+1)K^2 - D_KK^4, \tag{55} \]

where

\[ A=\frac{h}{8\pi^2 c I_A} \]

(\(I_A\) is the moment of inertia with respect to the axis of symmetry),

\[ B=\frac{h}{8\pi^2 c I_B} \]

(\(I_B\) is the moment of inertia with respect to an axis perpendicular to the axis of symmetry), and \(D_J\), \(D_{JK}\), and \(D_K\) are distortion coefficients—constants, extremely small in comparison with \(A\) and \(B\).

Applying the usual Bohr relations for the selection rules

\[ \Delta J=+1,\quad \Delta K=0, \]

we obtain the rotational frequencies

\[ \nu_r\;(\text{in } \mathrm{cm}^{-1}) = 2B(J+1)-4D_J(J+1)-2D_{JK}(J+1)K^2, \tag{56} \]

\[ J=0,\;1,\;2,\ldots \]

For the lower rotational states the last terms may be neglected, and

\[ \nu_2 = 2B(J+1)=1677929\,\frac{J+1}{I_B}, \]

where \(\nu_2\) and \(B\) are expressed in \(\mathrm{MHz}\), and \(I_B\) in \(\mathrm{g\cdot cm^2}\cdot 10^{-40}\), and where the value \(h=6.6242\cdot 10^{-27}\ \mathrm{erg/sec}\) has been adopted.

The expression given above does not include perturbations of the nuclei by frequent collisions. This, however, can be taken into account by adding to it the corresponding term, which will be given in the section devoted to hyperfine structure. Likewise, it ceases to be strictly valid for molecules in excited vibrational states. Formulas for excited vibrational states have been obtained. For their discussion see \(^{97}\).

The absorption coefficient of a given rotational line at the resonance frequency can be calculated with the aid of formula (51) for the following parameter values (which differ from the parameter values in the case of linear molecules)\(^{98,99}\):

\[ |\mu_J|^2=\mu^2\frac{(J+1)^2-K^2}{(J+1)(2J+1)}, \]

\[ E_r=hcBJ(J+1)+(A-B)K^2, \]

\[ g_J= \begin{cases} 2(2J+1) & \text{for } K\ne 0,\\ 2J+1 & \text{for } K=0. \end{cases} \]

For the case of molecules with three identical atoms arranged at the corners (of the type PF\(_3\) or CH\(_3\)Cl),

\[ G_I= \begin{cases} \dfrac{1}{3}(2I+1)(4I^2+4I+3) & \text{for } K \text{ divisible by } 3,\\[6pt] \dfrac{1}{3}(2I+1)(4I^2+4I) & \text{for } K \text{ not divisible by } 3, \end{cases} \]

\[ Q_r=\frac{(2I+1)^3}{3}\sqrt{\frac{\pi}{B^2A}}\left(\frac{kT}{hc}\right)^3, \]

where \(I\) is the nuclear spin of identical atoms arranged at the corners.

Microwave rotational spectra of a number of molecules of the symmetric-top type have been studied. The results obtained are collected in a table of molecular and nuclear constants. Since only values of \(I_B\) are obtained from the measurements, for a more complete determination of the molecular structure one resorts to the use of molecules with different isotopic composition.

The structure of methyl halides for a long time remained a subject of speculation. The most reliable estimates of the C—Hal distances (with an error of about three percent) were made with the aid of electron diffraction. However, by this route, owing to the weak scattering of electrons by hydrogen, neither the valence angles nor the C—H distances can be determined. Accurate values of \(I_A\) and \(I_B\) likewise cannot be obtained from infrared rotation-vibrational spectra. But, using the Johnson and Dennison relation\(^{100}\)

\[ \sum \Delta \nu_i=\left(\frac{6}{I_A}-\frac{7}{I_B}\right)\frac{h}{8\pi^2c}, \]

one can express one moment through the other by measuring the splitting of lines in vibrational bands corresponding to the three fundamental mutually perpendicular vibrations of the molecule.

If \(I_B\) is now determined exactly from the microwave rotational spectrum, then, with the aid of the formula given above, it can be ex-

determined \(I_A\). By means of the indicated procedure and by microwave measurements on molecules of various isotopic composition, a complete determination of the structures of all four methyl halides was carried out \({}^{101,102}\). As was to be expected, the C—H distances in these molecules proved not to be identical, but to vary monotonically from F to J. The C—H distances are arranged in a sequence opposite to the sequence of increase of the bond force constants. With the exception of C—F, the C—Hal bonds are longer than expected. They are considerably longer than the covalent radii of single bonds, if one uses the value of the radius of C proposed by Pauling \({}^{103}\), equal to \(0.77\,\text{Å}\). The agreement becomes better if one uses the measured radius of C, equal to \(0.79\,\text{Å}\), proposed by the author \({}^{104}\) on the basis of consideration of the work of Mulliken, Rieke, and Brown \({}^{105}\) on hydrocarbons. The exceptionally short C—F distance in methyl fluoride, equal to \(1.385\,\text{Å}\), indicates that the bond has to a considerable extent the character of a double bond, corresponding to a structure of the type

\[ \begin{array}{c} \mathrm{H}-\\ \mathrm{H}-\mathrm{C}=\mathrm{F}^{+}.\\ \mathrm{H}/ \end{array} \]

The Shoemaker–Stevenson \({}^{106}\) correction for the ionic character of the bond \((0.09|\chi_1-\chi_2|)\) can explain some decrease of the C—F distance in comparison with the sum of the radii, but for the other halogen-substituted compounds this correction substantially worsens the agreement with the measured values. Until the complete determination of structures depends in part on data drawn from infrared spectra, some improvement in the accuracy of determining the C—H distance and the valence angles can be achieved by microwave measurements on molecules containing additional isotopic combinations. Along this path, a complete determination of structures can be made relying solely on microwave data. Such measurements are already being planned by J. U. Simmons.

Methyl cyanide \({}^{107}\), methyl isocyanide \({}^{107}\), methyl acetylene \({}^{108}\), and boron carbonyl \({}^{109}\), as has been shown, have symmetrical structures. Other methods had already indicated the symmetry of the structure, but a weak asymmetry could not be detected by the method used earlier. In a preliminary communication from our laboratory \({}^{110}\) it was noted that additional lines for the \(J=1 \to 2\) transition in \(\mathrm{CH_3NC}\) possibly indicate a weak asymmetry of the structure. It has now been proved that these additional lines are produced by molecules in excited vibrational states. Some of the molecular parameters have been determined, and they can be found in Tables VI and VII.

Further work is developing in the direction of studying the named compounds with molecules containing various isotopes of carbo-

genus. There is hope that the structure of each of these compounds will soon be fully determined.

The results obtained for these and other molecules of the symmetric-top type are collected in Tables VI and VII. Nuclear effects will be considered below.

Molecules of the asymmetric-top type. The rotational energy for molecules of the asymmetric-top type was expressed by Wang^111 in the form

\[ \frac{E_r}{hc}=\frac{1}{2}(B+C)J(J-1)+\left[A-\frac{1}{2}(B+C)\right]W_\tau, \tag{57} \]

where

\[ A=\frac{h}{8\pi^2 c I_A},\qquad B=\frac{h}{8\pi^2 c I_B},\qquad C=\frac{h}{8\pi^2 c I_C}, \]

and where \(I_A, I_B, I_C\) are the three principal moments of inertia, not coinciding with one another. \(W_\tau\) is here the analogue of \(K^2\) for the symmetric top. However, in contrast to \(K^2\), \(W_\tau\) is not restricted to integral values, but in the general case is a complicated algebraic function of the moments of inertia. To each value of \(J\) there correspond \(2J+1\) values of \(W_\tau\), denoted by the quantum number \(\tau\), which takes integral values lying between \(-J\) and \(+J\). Algebraic expressions for \(W_\tau\) as functions of the parameter

\[ b=\frac{C-B}{2A-(B+C)} \]

were obtained for values of \(J\) up to \(J=10\) by Nielsen^112 and for \(J=11\) by Randall, Dennison, Ginsburg, and Weber^113. These equations were reproduced for values of \(J\) up to \(J=6\) by Herzberg^114. The degrees of these equations increase with \(J\). Therefore an exact solution is possible only for very low values of \(J\).

King, Hainer, and Cross^115 compiled tables intended to facilitate many laborious calculations connected with the analysis of molecules of the asymmetric-top type. They were originally compiled for infrared spectra and, in order to become convenient for investigations in the microwave region, require considerable extension. Nevertheless, they are very useful in their present form. The authors expressed the rotational energy in the form

\[ \frac{E_r}{hc}=\frac{1}{2}(A+C)J(J+1)+\frac{1}{2}(A-C)E_\tau^J, \tag{58} \]

where \(A\) and \(C\) have their usual meanings, and the third moment of inertia is expressed as an implicit function \(E_\tau^J\). \(E_\tau^J\) is expanded as a function of \(x\), where

\[ x=\frac{2B-(A+C)}{A-C}. \]

In Table I of the cited work^115 are given the values of \(E_\tau^J\) for 11 values of the parameter \(x\), lying between \(-1\) (symmetric top) and \(0\) (completely asymmetric top), and for \(J\) up to \(J=12\). Inter-

MICROWAVE SPECTROSCOPY

interpolation*) between the values \(E_\tau^J\) given in the table is usually insufficiently accurate for microwave spectra.

Table II of the cited work gives the expansion of \(E_\tau^J\) in a series in powers of \(x\):

\[ E_\tau^J=\sum A_\tau x^n . \]

The expansion is carried out only up to the second order and, consequently, is applicable only for very small values of \(x\), i.e., for the most asymmetric types of molecules. Table III in the same work gives \(E_\tau^J\) in the form of a power series in \(\delta\):

\[ E_\tau^J=\sum B_\tau \delta^n, \]

where

\[ \delta=\frac{x+1}{2}=\frac{B-C}{A-C}. \]

The expansion is carried out to the third order. It is applicable only to very weakly asymmetric rotators, i.e., for very small \(\delta\), and for comparatively small values of \(J\), since for high \(J\) the series converges slowly.

Recently Golden\(^{116}\) developed a method that in a number of cases makes it possible to determine exactly the levels of an asymmetric rotator for high \(J\). He showed that the solutions of Mathieu’s equation (which have already been tabulated) can be transformed in such a way that they give, to a good approximation, the energy-level values for large \(J\) corresponding in the limiting case of a symmetric top to small \(K\). To obtain exact values of the levels, perturbation methods are used. The corresponding principle was also applied\(^{117}\) to obtain satisfactory solutions for a weakly asymmetric rotator at large \(J\), corresponding in the limiting case of a symmetric top to large \(K\). In addition to other difficulties, centrifugal distortion for states with large \(J\) greatly increases the difficulty of interpreting the spectra of an asymmetric rotator. This is discussed in Golden’s second communication\(^{118}\).

Apart from the laboriousness of the problem of obtaining numerical solutions, the problem of correctly determining the levels is very complex. The Stark effect\(^{119}\) acquires outstanding importance in this connection.

The selection rules for a slightly asymmetric rotator can be obtained directly from the identification of the levels with the corresponding \(K\) for the limiting symmetric rotator. A discussion of various cases may be found in the papers of Herzberg\(^{120}\) and of Cross, Hainer, and King\(^{121}\).

Cross, Hainer, and King also compiled tables\(^{121}\) that make it possible to calculate easily the intensities of the lines of asymmetric rotators. Pri-

) A method for carrying out rapid interpolation with the aid of punched cards is described by G. W. King, P. C. Cross, and G. B. Thomas, J. Chem. Phys. 14*, 35 (1936).

the equation (51) given above can be used together with these tables to find the absorption coefficients, if one sets

\[ \mu f=\mu^{2}|\Phi|^{2}, \]

where \(\mu\) is the constant dipole moment of the molecule and where, in the notation of the authors,

\[ |\Phi|^{2}=\sum_{X,Y,Z}\sum_{M''}\sum_{M'}\left|(\Phi_{F_g}A)^{J'',\,\tau'',\,M'';\,J',\,\tau',\,M'}\right|^{2}. \]

Numerical values of this function are given in the cited investigations for \(J\) up to \(J=12\) and for five different values of \(\chi\) (the degree of asymmetry). These authors\(^{122}\) calculated microwave absorption coefficients specifically for \(\mathrm{H_2O}\), \(\mathrm{HDO}\), \(\mathrm{D_2O}\), and similar molecules.

Up to now only two complete structural analyses of asymmetric molecules from microwave spectra have been published (see Table VII). However, owing to the abundance of comparatively simple asymmetric molecules, there is no doubt that in the near future a great deal of work will be done in this direction. The study of some other molecules of this type is already being carried out, and certain preliminary results are collected in Tables VI and VII. The large amount of labor required for the analysis of an asymmetric rotator is partly compensated by the abundance of the information obtained. Since all moments of inertia can be determined from purely rotational spectra alone, it is possible, for example, to determine the structure of triatomic molecules without resorting to various isotopic combinations.

Internal rotations. Molecules in which one part can rotate relative to another part are of interest. One may expect that those among them in which one of the rotating parts has a dipole moment with a component perpendicular to the axis of rotation, and a sufficiently large moment of inertia, will have microwave internal rotational spectra.

So far not a single free internal rotational spectrum in the microwave region has been published, although spectra have been published which are apparently caused by hindered internal rotations of the OH group in \(\mathrm{CH_3OH}\) and the \(\mathrm{NH_2}\) group in \(\mathrm{CH_3NH_2}\).

The theory of internal rotations was developed for the case of a symmetric top by Nielsen\(^{124}\) (see also Koehler and Dennison\(^{125}\)). This theory is also applicable to a slightly asymmetric rotator. It predicts two series of lines, given in Herzberg’s notation\(^{126}\) as

\[ \nu=A_1-B+2BK\pm 2A_1K_1, \tag{59} \]

where

\[ B=\frac{h}{8\pi^{2}cI_B},\qquad A_1=\frac{h}{8\pi^{2}cI_{A_1}}, \]

where \(I\) is the moment of inertia with respect to the axis of symmetry of the internal

of the rotating part possessing a dipole moment, and where \(K_1\) is the absolute value of the quantum number for this rotation. For symmetric molecules, such as \(\mathrm{H_3C—CF_3}\), one cannot expect internal rotational spectra; nevertheless, internal rotations can be detected indirectly from centrifugal effects in the ordinary rotational spectrum.

Series of lines for \(\mathrm{CH_3OH}\) and \(\mathrm{CH_3NH_2}\) were discovered by Gerhberger and Turkevich \(^{123}\) in the region around \(1.2\ \mathrm{cm}\). Series of \(\mathrm{CH_3OH}\) lines were studied by Dailey \(^{31}\) and Coles \(^{127}\). The lines were interpreted by Burkhard and Dennison*) as arising as a result of hindered internal rotations of the \(\mathrm{OH}\) group, corresponding to the transitions \(J = 2 \to 2\) and \(K = 2 \to 1\). Lines corresponding to free internal rotations of \(\mathrm{OH}\) should fall in the infrared region of the spectrum \(^{128}\). However, the energy of hindered rotations \(^{125}\), which may be negative, is sufficient for the lines to appear in the region around \(1\ \mathrm{cm}\), provided only that the barrier height has the corresponding magnitude. The barrier height chosen by Burkhard and Dennison by fitting is \(320\ \mathrm{cm}^{-1}\), somewhat lower than previous determinations from infrared spectra \(^{128}\). They treated the molecule as an asymmetric top with hindered internal rotations. By a suitable choice of three parameters in their formula they were able to calculate the positions of ten lines observed near \(1.2\ \mathrm{cm}\), with an average error of only \(0.18\ \mathrm{MHz}\). Some features of the methyl-alcohol spectrum remained unexplained. One of them is the correct interpretation of the group of lines in the region near \(6\ \mathrm{mm}\), observed by Edwards, Gilliam, and Gordy \(^{129}\), which apparently correspond to the transitions \(J = 0 \to 1\) and \(K = 0 \to 0\).

4. Hyperfine structure

Nuclear quadrupole interaction. Good \(^{76}\) discovered a distinct hyperfine structure in the inversion spectrum of ammonia. The observed satellite structure was later interpreted by Coles and Good \(^{130}\) and by Dailey, Kyhl, Strandberg, Van Vleck, and Wilson \(^{66}\) as due to the interaction of the quadrupole moment of the \(\mathrm{N}^{14}\) nucleus with the molecular field. Since that time, a distinct hyperfine structure due to nuclear quadrupole interaction has been found in the microwave spectra of a large number of molecules. Indeed, it is clear that such a hyperfine structure can confidently be expected for all molecules having atoms with nuclear spin greater than \(\dfrac{1}{2}\). Nuclear quadrupole interaction had, of course, previously been discovered in atomic spectra and, with the aid of the radio-frequency method

*) A report on this work was presented by D. M. Dennison at the Ohio State Symposium on Molecular Spectroscopy, June, 1948. The theory has not yet been published.

studies of molecular beams for some diatomic molecules. The theory of this interaction, developed by Casimir \({}^{131}\), was applied to molecules of the symmetric-top type by Coles and Good \({}^{130}\) and by Van Vleck \({}^{66,132}\). The final formula for the interaction energy has the form

\[ E_Q=eQ\frac{\partial^2 V}{\partial z^2} \left(\frac{3K^2}{J(J+1)}-1\right) \frac{\frac{3}{4}C(C+1)-I(I+1)J(J+1)} {2(2J+3)(2J-1)I(2I-1)}, \tag{60} \]

\[ C=F(F+1)-I(I+1)-J(J+1), \]

\[ F=J+I,\quad J+I-1,\ldots,|J-I|, \]

where \(I\) is the spin of the interacting nucleus, \(e\) is the electron charge, \(Q\) is the nuclear quadrupole moment, and \(\dfrac{\partial^2 V}{\partial z^2}\) is the divergence along the molecular axis of the molecular field at the interacting nucleus. The corresponding selection rules are:

\[ \Delta J=\pm 1,\quad \Delta K=0,\quad \Delta F=0,\pm 1. \]

The formula is also applicable to linear molecules if one puts \(K=0\). It is applicable to molecules having only one interacting nucleus.

The formula given above differs somewhat from the formula obtained in the original work. In the literature it occurs in various forms because of the existence of different definitions of the unit quadrupole coupling \({}^{133}\). The form given above is at present apparently the generally accepted one, and the quadrupole-interaction factors defined earlier have all already been transformed in accordance with it. As an illustration of the theory, we have drawn in detail the diagram of energy levels and the theoretical spectrum of \(\mathrm{CH_3J}\) \({}^{127}\) for the transition \(J=1\to2\) (Fig. 13). For comparison the observed spectrum is given. Some of the levels here are accidentally degenerate. The second-order effects mentioned below were taken into account in order to bring the observed and calculated spectra into agreement. However, these effects, although large in comparison with the experimental errors, are too small to be noticeable on a graph of this kind. As a further illustration, Figs. 14–16 show the line \(1,1\ \mathrm{N}^{14}\mathrm{H}_3\) and selected portions of the spectra of JCN and HCN.

Second-order effects. It was found \({}^{87,134,139}\) that the nuclear quadrupole interaction of the first order does not completely explain the hyperfine structure of a number of molecules. In the cases of JCH and \(\mathrm{CH_3J}\) deviations of the order of megacycles were discovered. However, the theory extended by Bardeen and Townes \({}^{134}\) to the second order proved to be in agreement with the observations. Table III gives the observed and calculated second-order effects

Fig. 13. Scheme of the hyperfine structure for a molecule of the symmetric-top type. Transition \(J=2 \to 3\), \(C^{12}H_3J^{127}\) (see \({}^{101}\)).

Calculated spectrum

Observed spectrum

\(29\,600 \quad 29\,900 \quad J \to \quad 30\,200 \quad 30\,400\ \text{Mc/s}\)

for one of the transitions of \( \mathrm{CH_3J} \), borrowed from the communication of Gordy, Simmons, and Smith\(^{101}\). It is obvious that second-order effects can often have an appreciable magnitude, since in many molecules the nuclear interaction is large enough to produce this effect. The second-order interaction energy, in the form in which it was obtained by Bardeen and Townes, is represented by the expression\(^{134}\)

\[ E_Q=\sum_{J'} \frac{(IJFM_F|H_Q|IJ'FM_F)}{E_{J'}-E_J}, \tag{61} \]

where \(J'\) may differ from \(J\) by

Fig. 15. The theoretically calculated and observed hyperfine structure of the \(F_1 \to F_1+1\) line corresponding to the rotational transition \(J=7\to 8\) of the molecule \(\mathrm{J}^{127}\mathrm{C}^{12}\mathrm{N}^{14}\) in the wavelength region of about \(5.61\) mm.

Fig. 14. Hyperfine structure of line 1,1 of the molecule \(\mathrm{N}^{14}\mathrm{H}_3\); \(J=1,\ K=1^{54}\).

Fig. 16. Hyperfine structure of the rotational transition \(J=0\to 1\) of the molecule \(\mathrm{HC}^{12}\mathrm{N}^{14}\) in the wavelength region of about \(3.38\) mm\(^{25}\).

1 or 2. Physically, this expression represents the interaction between levels with different \(J\), but identical total angular momenta \(F\) and identical \(M_F\). The formulas for the numerical determination of the squares of the matrix elements on a computing machine have the form

\[ (IJFM_F|H_Q|IJ+1FM_F)^2 = \]

\[ = \left[ 3eQ\frac{\partial^2 V}{\partial z^2} \frac{K}{8I(2I-1)J(J+2)} \right]^2 \left[ 1-\frac{K^2}{(J+1)^2} \right] \times \]

\[ \times \frac{[F(F+1)-I(I+1)-J(J+2)]^2}{(2J+1)(2J+3)} \times \]

\[ \times (I+J+F+2)(J+F-I+1)(I+F-J)(J+I-F+1), \tag{62} \]

\[ (IJFM_F|H_Q|IJ+2FM_F)^2 = \]

\[ = \left[ 3eQ\frac{\partial^2 V}{\partial z^2} \frac{1}{16I(2I-1)(2J+3)} \right]^2 \left[ 1-\frac{K^2}{(J+1)^2} \right] \left[ 1-\frac{K^2}{(J+2)^2} \right] \times \]

\[ \times \frac{1}{(2J+1)(2J+5)} (F+I+J+5)(F+I+J+2)(J+I-F+2) \times \]

\[ \times (J+I-F+1)(J+F-I+2)(J+F-I+1) \times \]

\[ \times (I+F-J)(I+F-J-1). \tag{63} \]

Table III

Second-order effects in the hyperfine structure of the CH₃J line corresponding to the transition \(J=1 \to J'=2,\ K=0 \to K=0\) in the ground vibrational state. Frequencies are given in Mc/s relative to the strongest line, corresponding to the transition \(F=\dfrac{7}{2}\to F=\dfrac{9}{2},\ K=0\to K=0\);

\[ eQ\,\frac{d^{2}V}{dz^{2}}=-1934\ \text{Mc/s}. \]

[Borrowed from W. Gordy, J. W. Simmons and A. G. Smith, Phys. Rev. 74, 243, (1948).]

\(F\)-transition	Experiment	First-order theory	Experiment minus first-order theory	Second-order theory
\(3/2 \to 1/2\)	\(+\,74.33\)	\(+\,74.59\)	\(-\,0.26\)	\(-\,0.27\)
\(3/2 \to 3/2\)	\(-\,174.47\)	\(-\,174.06\)	\(-\,0.41\)	\(-\,0.37\)
\(3/2 \to 5/2\)	\(-\,448.04\)	\(-\,450.35\)	\(+\,2.31\)	\(+\,2.19\)
\(5/2 \to 3/2\)	\(+\,406.47\)	\(+\,406.14\)	\(+\,0.33\)	\(+\,0.40\)
\(5/2 \to 5/2\)	\(+\,132.72\)	\(+\,129.85\)	\(+\,2.87\)	\(+\,2.95\)
\(5/2 \to 7/2\)	\(+\,32.73\)	\(+\,33.15\)	\(-\,0.42\)	\(-\,0.39\)
\(7/2 \to 5/2\)	\(-\,273.04\)	\(-\,276.29\)	\(+\,3.25\)	\(+\,3.35\)
\(7/2 \to 7/2\)	\(-\,373.04\)	\(-\,372.99\)	\(-\,0.05\)	\(0\)
\(7/2 \to 9/2\)	\(0\)	—	—	—

Levels situated both above and below a given level interact with it. However, since the matrix element is symmetric with respect to \(J\) and \(J'\), the formulas given exhaust everything needed for the calculations. The vanishing matrix elements are small if \(J'\) differs from \(J\) by more than 2.

Quadrupole coupling due to two nuclei. The case of two nuclei belonging to one and the same molecule and giving rise to quadrupole interaction was also considered by Bardeen and Townes \(^{136}\). If the interaction energy of one of the nuclei \((E_1)\) is large in comparison with the interaction energy of the other nucleus \((E_2)\), the latter may be regarded as a perturbation of the former, and the energy is expressed as

\[ E_Q = E_1(F_1) + \sum_{F_2} c\,(I_1F_2)^2\,E_2(F_2), \tag{64} \]

where

\[ F_1=J+I_1,\quad J+I_1-1,\ldots,\ |J-I_1| \]

and

\[ F_2=J+I_2,\quad J+I_2-1,\ldots,\ |J-I_2|. \]

\(E_1(F_1)\) and \(E_2(F_2)\) are obtained by substituting \(F_1\) and \(F_2\) into formula (60). The transformation coefficients \(c(F_1F_2)\) are given by Bardeen and Townes for the cases

\[ \text{1) }\ I_1=1;\quad I_2 \text{ and } J \text{ arbitrary,} \]

\[ \text{2) }\ I_1=\frac{3}{2};\quad I_2 \text{ and } J \text{ arbitrary.} \]

Total angular momentum

\[ F=J+I_1+I_2,\quad J+I_1+I_2-1,\ldots,\ J-I_1-I_2 = \]

\[ =F_1+I_2,\quad F_1+I_2-1,\ldots,\ F_1-I_2. \]

Thus, for \(J\geq I_1+I_2\), each \(J\)-level is split into \((2I_1+1)(2I_2+1)\) hyperfine-structure levels. Since, by assumption, \(E_1(F_1)\) is large in comparison with \(E_2(F_2)\), each level \(F_1\) may be regarded as split into \((2I_2+1)\) sublevels.

The summation in the formula given extends over all values of \(F_2\) for given \(F\) and \(F_1\).

It was found that first-order theory retains sufficient accuracy for \(\alpha<0.1\) or \(1/\alpha<0.1\), where \(\alpha\) is the ratio of the quadrupole coupling constants of the two nuclei. For ClCN \(\alpha=0.05\), and the deviations from first-order theory are entirely negligible. Smith, Ring, Smith, and Gordy\({}^{137}\) applied the theory to JCN and \(N_2O\), where the ratios of the coupling constants of the central and terminal atoms are 0.0015 and 0.261, respectively. Quadrupole effects due to two atoms belonging to one and the same molecule are illustrated in Fig. 17.

Quadrupole coupling in the case of an asymmetric top. The theoretical treatment of the nuclear quadrupole interaction in the case of molecules of the asymmetric-top type was carried out by Bragg\({}^{138}\) and by Naito and Feld\({}^{139}\). The matrix elements of the coupling operator \(F(I,J)\) are the same as in the case of linear and diatomic molecules, but the factor \((3\cos^2\theta-1)_{\mathrm{cp}}\) is different

and includes two parameters of the molecular bond, \(\dfrac{\partial^{3}V}{\partial z^{3}}\) and \(\dfrac{\partial^{3}V}{\partial x^{2}}-\dfrac{\partial^{3}V}{\partial y^{3}}\), where \(x, y\), and \(z\) are coordinates relative to the principal molecular axes, and \(V\) is the potential of the interacting nuclei. The details of the calculations have not yet been published.

Fig. 17. Quadrupole splitting caused by two nuclei: hyperfine structure of the transition \(J=1 \to 2\) of the molecule \(\mathrm{C}^{135}\mathrm{C}^{13}\mathrm{N}^{14}\). The upper curve shows the splitting due to \(\mathrm{C}^{135}\) with unresolved splitting due to \(\mathrm{N}^{14}\). The lower curves show the splitting, owing to its existence, due to the influence of the \(\mathrm{N}^{14}\) nucleus, in resolved form.

Intensity. The relative intensity of the various components of the hyperfine structure of a given rotational transition can be determined from the weights \(2F+1\) of the upper and lower hyperfine-structure levels by methods usually applied in atomic spectroscopy.

The corresponding formulas for computations have the form

\[ \left. \begin{aligned} I_{+}&=\frac{1}{F}\,Q(F)Q(F-1)\\ &\Delta F=+1,\\ I_{0}&=\frac{2F+1}{F(F+1)}\,P(F)Q(F)\\ &\Delta F=0,\\ I_{-}&=\frac{1}{F}\,P(F)P(F-1)\\ &\Delta F=-1 \end{aligned} \right\} \quad \Delta J=+1, \tag{65} \]

pure rotational spectrum.

\[ \left. \begin{aligned} I_{0}&=\frac{2F+1}{F(F+1)}\,R^{2}(F)\\ &\Delta F=0,\\ I_{\pm}&=\frac{1}{F}\,P(F)Q(F-1)\\ &\Delta F=\pm 1 \end{aligned} \right\} \quad \Delta J=0, \tag{66} \]

spectrum of the inversion type,

where

\[ P(F)=(F+J)(F+J+1)-I(I+1), \]

\[ Q(F)=I(I+1)-(F-J)(F-J+1), \]

\[ R(F)=F(F+1)+J(J+1)-I(I+1). \]

Here \(J\) and \(F\) in these formulas refer to the upper state. For values of \(J\) up to 6 and values of \(I\) up to \(7/2\), the numerical values of the relative intensities have been tabulated in various places\(^{141,142}\). For molecules of the symmetric-top type, the relative intensities corresponding to different values of \(K\) for a given transition \(J\) must first be determined by means of the Dennison theory (outlined above in the section on the symmetric rotator), since \(K\) does not enter into the formulas given. These formulas are usually applied in identifying various hyperfine-structure lines.

However, present-day methods for measuring relative intensity in the microwave region are not sufficiently accurate for testing the theory.

Magnetic interaction. Investigations by Simmons and Gordy\(^{54}\) of the inversion spectrum of ammonia revealed that nuclear quadrupole interaction alone is unable properly to explain the hyperfine structure. In this case the rotational levels prove to be so widely split by ...

in comparison with the nuclear quadrupole splitting for a second-order interaction of the type discussed above, that it cannot be ignored. As a possible cause of this additional perturbation, the interaction of the nuclear magnetic moment of nitrogen with the magnetic field produced by the rotation of the molecule was proposed. Quantitative calculations by Yucha^143 and Henderson^144 showed that this explanation is quite possibly correct. The energy of such an interaction, as established by Henderson, is

\[ \Delta E=\left(\frac{aK^2}{J(J+1)}+b\right)[F(F+1)-J(J+1)-I(I+1)], \tag{67} \]

where \(a=0.0011\) and \(b=0.0057\). The quadrupole coupling \(eQ\,\dfrac{\partial^3 V}{\partial z^3}\), used to obtain better agreement, is \(4.10\).

Henderson and Van Vleck^145 developed a theory of the coupling of electron spins in rotating polyatomic molecules. This theory proves especially useful in cases where molecules with unpaired electrons—such as \(\mathrm{NO_2}\), \(\mathrm{NO}\), and \(\mathrm{ClO_2}\)—are investigated in the microwave region. Henderson noted that the theory can also be applied to the case in which it is not electron spins, but nuclear spins, that interact with electronic orbital moments.

5. Line shape

a) Natural width. The natural width of lines, due only to spontaneous radiation, is so small in the microwave frequency region that it is entirely insignificant in comparison with other factors determining the line width. For the \(K\)-band it is of the order of about \(10^{-8}\) c/s.

b) Pressure broadening. The most significant of the factors affecting the line shape, when the pressure is not too low, are molecular collisions. The most general expression for the shape of lines broadened as a result of collisions was recently obtained by Van Vleck and Weisskopf^79. This expression for the molecular absorption coefficient in the microwave region (where \(h\nu \ll kT\)) has the form

\[ \alpha=\frac{4\pi^3\nu N}{3ckT}\, \frac{\sum_i\sum_j |\mu_{ij}|^2\,\nu_0 f(\nu_0,\nu)\exp\left(-\frac{W_i}{kT}\right)} {\sum_j \exp\left(-\frac{W_j}{kT}\right)}. \tag{68} \]

The line form factor in this equation is

\[ f(\nu_0,\nu)=\frac{\nu}{\pi\nu_0} \left[ \frac{\Delta\nu}{(\nu_0-\nu)^2+\Delta\nu^2} + \frac{\Delta\nu}{(\nu_0+\nu)^2+\Delta\nu^2} \right]. \tag{69} \]

For frequencies close to the resonance frequency, \(\nu_0\), it is transformed into the ordinary Lorentz expression

\[ f(\nu_0,\nu)=\frac{1}{\pi}\frac{\Delta \nu}{(\nu-\nu_0)^2+\Delta \nu^2}. \tag{70} \]

At resonance \(f(\nu_0,\nu)\) becomes \(1/\pi\Delta\nu\), and the absorption equation assumes the form used for calculating absorption coefficients at the maximum. Van Vleck\(^{82}\) noted that for very broad lines the equation becomes similar in form to the Debye expression for resonance absorption. The theory was confirmed for pressures of the order of atmospheric by Beringer’s\(^{2}\) studies on oxygen absorption. However, the results of Bleaney and Penrose\(^{56}\), relating to \(\mathrm{NH_3}\) and confirming the theory for pressures of about \(10\ \mathrm{cm}\) Hg, showed that for pressures of the order of \(60\ \mathrm{cm}\) Hg there are distinct deviations from the theory. Measurements with \(\mathrm{NH_3}\) at pressures varying from 10 to \(538\ \mathrm{cm}\) Hg, in the wavelength region from 0.86 to \(3.2\ \mathrm{cm}\), were made by Weingarten\(^{146}\). In agreement with Bleaney and Penrose he found that at high pressures the resonance frequency is shifted toward lower frequencies and that the line width ceases to be directly proportional to the pressure. In particular he found that the line width remains substantially constant in the pressure interval from 76 to \(228\ \mathrm{cm}\) Hg. Preliminary results of Weidner\(^{57,147}\), concerning JCl at pressures from 2 to \(20\ \mathrm{mm}\) Hg, show that absorption in the wings of the lines exceeds that predicted by Van Vleck and Weisskopf\(^{79}\) by a factor close to 5.

Table IV gives values of the half-width of lines for a number of molecules. They may be used together with equation (51) to calculate the peak intensity of lines in the pressure region for which the line width varies linearly with pressure. As far as we know, no significant deviations from the linear law at intermediate pressures, i.e. approximately from \(\sim 10^{-1}\) to \(\sim 10^2\ \mathrm{mm}\) Hg, have yet been reported. The half-width of lines can also be used in the kinetic theory for calculating the “optical cross section” or collision diameter. The optical diameter is proportional to \(M^{1/4}(\Delta\nu)^{1/2}\), where \(M\) is the mass of the colliding particles (or \(\dfrac{Mm}{m+M}\) for unlike molecules). The large difference between the optical diameters of molecules is of interest. The optical diameter of JCN is more than 7 times greater than the optical diameter of \(\mathrm{O_2}\). This difference is, of course, to a significant extent due to the large dipole moment of JCN. Bleaney and Penrose\(^{147a}\) carried out a systematic study of the influence of admixtures of various nonpolar gases on the line width of the inversion spectrum of \(\mathrm{NH_3}\). Some of their results are summarized in Table V. It is noted—

Table IV

Line half-width

Molecule	\(\Delta\nu\) in MHz	Pressure	Temperature	\(\Delta\nu\) for a pressure of 1 mm Hg*)
\(\mathrm{O_2}\)	600—1000 a)	1 atm	\(27^\circ\) C	0.8—1.3
\(\mathrm{O_2}\)	6.65 б)	1.2 mm Hg	\(-15^\circ\) C	5.5
JCl	2400 в)	1 atm	Room	3.2
HCN	25 г)	1 mm Hg	\(27^\circ\) C	25
ClCN	\(25 \pm 4\) д)	1 mm Hg	Room	25
BrCN	\(21 \pm 3\) д)	1 mm Hg	»	21
JCN	\(2.0 \pm 3\) д)	0.1 mm Hg	\(0^\circ\) C	20
\(\mathrm{H_2O}\)	0.72 е)	0.103 mm Hg	Room	7
\(\mathrm{H_2O}\)	6.10 ж)	0.5 mm Hg	»	12—20
\(\mathrm{NH_3}\)	(depending on the values of \(J\) and \(K\))
\(\mathrm{NH_3}\)	29.2 з)	1 mm Hg	»	29.2

) Reduced to a pressure of 1 mm Hg on the assumption of a linear dependence of \(\Delta\nu\) on pressure.
a) R. Beringer, Phys. Rev. 70, 53 (1946).
б) C. H. Townes, F. R. Merritt and B. D. Wright, Phys. Rev. 73, 1334 (1948).
в) R. T. Weidner, Phys. Rev. 72, 1268 (1947).
г) A. G. Smith, W. Gordy, J. W. Simmons and W. V. Smith (prepared for publication).
д) C. H. Townes, A. N. Holden and F. R. Merritt (private communication).
е) C. H. Townes and F. R. Merritt, Phys. Rev. 70, 558 (1946).
ж) B. Beaney and R. P. Penrose, Proc. Phys. Soc. 5, 418 (1947).
з) C. H. Townes, Phys. Rev. 70*, 665 (1946).

there is a rough correlation between the collision diameters and the polarizability.

c) Collisions with the walls of the vessel. Collisions with the walls of the vessel become a significant factor in line broadening at such pressures when the mean free path of the molecules becomes comparable with the dimensions of the vessel. It is easy to show that for

Table V

Some data on the broadening of ammonia lines caused by collisions of NH$_3$ molecules with molecules of nonpolar gases
[V. Bleaneu and R. P. Penrose, Proc. Phys. Soc. 60, 540 (1948)].

Gas admixed to NH$_3$	Polarizability in $10^{-24}\ \text{cm}^3$	Effective diameter of molecules for collisions leading to broadening of the NH$_3$ lines (in Å): measured	Effective diameter of molecules for collisions leading to broadening of the NH$_3$ lines (in Å): calculated from gas-kinetic theory
Helium	0.21	2.35	3.20
Hydrogen	0.78	3.50	3.58
Nitrogen	1.72	6.4	4.09
Oxygen	1.51	4.85	4.02
Argon	1.74	4.6	4.01
Carbon disulfide	8.6	7.5	—
Ammonia	—	13.8	4.4

of a long rectangular waveguide cell with cross-sectional dimensions $a$ and $b$

\[ 2\Delta\nu=\frac{2}{3\pi}\left(\frac{a+b}{ab}\right)\left(\frac{2RT}{M}\right)^{\frac12} =1.54\cdot10^3\left(\frac{a+b}{ab}\right)\left(\frac{T}{M}\right)^{\frac12}\ \text{cps}, \tag{71} \]

or, for a volume resonator of volume $V$, having wall surface area $S$,

\[ 2\Delta\nu=\frac{S}{3\pi V}\left(\frac{2RT}{\pi M}\right)^{\frac12} =7.7\cdot10^2\frac{S}{V}\left(\frac{T}{M}\right)^{\frac12}\ \text{cps}, \tag{72} \]

where $2\Delta\nu$ is the line width, measured between the points at which the intensity is equal to one half of the maximum, and caused only by collisions with the walls, $M$ is the molecular weight, and $T$ is the absolute temperature. For NH$_3$ in a $K$-band waveguide at room temperature the half-width of the lines is 113 kcps.

c) Doppler broadening. The broadening of lines caused only by the Doppler effect is determined by the expression

\[ 2\Delta\nu=72\cdot10^{-8}\left(\frac{T}{M}\right)^{\frac12}\ \text{cps}, \tag{73} \]

where $T$ is the absolute temperature and $M$ is the molecular weight. For

N¹⁴H₃ and JCN at \(\nu = 24\,000\) Mc/s and \(T = 300^\circ\)K, \(2\Delta\nu\) is, respectively, 70 and 30 kc/s.

d) Saturation effects. Saturation effects of molecules under resonant irradiation were first observed in the microwave region by Townes¹³ and by Bleaney and Penrose¹⁴⁸. The principal result of the saturation effect consists in a decrease of the absorption coefficient and a broadening of the absorption lines. These effects, which are a consequence of the disturbance of statistical equilibrium, have been confirmed several times¹⁴⁹–¹⁵¹ and theoretically explained by Karplus and Schwinger¹⁵² and, less precisely, by others. The broadening caused by saturation is illustrated in Fig. 18, obtained by Carter and Smith¹⁵⁰.

Fig. 18. Broadening due to saturation. Line 3,3 NH₃.
a — input power \(1.5\cdot 10^{-6}\) W, b — input power \(150\cdot 10^{-6}\) W¹⁵⁰.

The saturation effect becomes noticeable when the incident power is sufficient for the number of molecules undergoing transitions from the lower of the two states under consideration to the upper to be comparable with the number of molecules returning to the ground state. Since collisions with other molecules are the main factor leading to relaxation of molecules, the effect becomes appreciable only at comparatively low pressures. The limiting power that can be absorbed is evidently determined by the power consumed as a result of thermal relaxation:

\[ P_{\max} = \frac{1}{2}\,\frac{N_1 - N_2}{\tau}\,h\nu, \tag{74} \]

where \(N_1\) and \(N_2\) are the populations of the two levels under consideration, \(\tau\) is the mean relaxation time, and \(h\nu\) is the energy difference between the states. For a purely rotational spectrum,

\[ P_{\max} = \frac{1}{2}\, \frac{N_J\left(1-e^{-\frac{h\nu}{kT}}\right)h\nu}{\tau} \simeq \frac{1}{2}\,\frac{N_J(h\nu)^2}{kT\tau}, \tag{75} \]

where \(N_J\) is the population of the lower state. The absorption coefficient, if it is expressed through the absorption coefficient in the absence of saturation \(\alpha_0\), has, according to Carter and Smith¹⁵⁰, the form

\[ \alpha = \frac{\alpha_0}{ \left(\frac{\nu-\nu_0}{\Delta\nu_0}\right)^2 + 1 + \frac{AP}{(\Delta\nu_0)^2} }. \tag{76} \]

The line width with allowance for saturation, \(2\Delta\nu\), expressed through the line width in the absence of saturation, \(\Delta\nu_0\), is equal to

\[ 2\Delta\nu = 2\Delta\nu_0 \sqrt{1+\frac{AP}{(\Delta\nu_0)^3}}, \tag{77} \]

where

\[ A=\frac{8\pi(\mu_{ij})^2}{3ch^2} \]

and \((\mu_{ij})^2\) is the square of the matrix element of the moment corresponding to the given transition, averaged over the Zeeman components. Karplus and Schwinger\(^{152}\) showed that one should average the absorption produced by the individual Zeeman components, and not the squares of the matrix elements.

e) Influence of external fields. The shape of absorption lines may, of course, also be affected by external magnetic or electric fields. The influence of some types of modulation on the shape of lines was established theoretically by Karplus\(^{33}\). See also the works of Blokhintsev\(^{35}\) and Townes and Merritt\(^{34}\).

Fig. 19. Stark effect for a rotational line. Transition \(J=1\to2\) of the OCS molecule. Top curve—in the absence of a field; middle—at a field strength of 750 V/cm; bottom—at a field strength of 1070 V/cm \(^{29}\).

6. Stark Effect

Already at an early stage in the development of microwave spectroscopy, Daikin, Good, and Coles\(^{29}\) observed the influence of an electric field on absorption lines. Fig. 19 is a reproduction of the original photograph and convincingly demonstrates the power of the microwave method for investigating the Stark effect in purely rotational spectra. The use of this effect as an auxiliary means for detecting lines has been mentioned. It is also applicable to the identification of lines and provides the possibility of measuring the dipole moments of gas molecules, both in excited and in ground states, with sufficient accuracy.

The theory of the Stark effect in rotational spectra\(^{153}\) was developed many years ago, but its effective application became possible only with the advent of microwave spectroscopy. For linear molecules in a \(\Sigma\)-state, the first-order effect does not appear. The second order of the rotational perturbation energy for the ground vibrational state is given by the expression

\[ W_{JM}^{(2)}= \frac{4\pi^2\mu^2 E^2}{h^2}\cdot \frac{J(J+1)-3M^2}{J(J+1)(2J-1)(2J+3)}, \]

where \(E\) is the strength of the applied electric field, \(\mu\) is the mole-

molecular dipole moment and \(|M|=J, J-1,\ldots,0\). For electric dipole transitions the usual selection rule \(\Delta J=\pm 1\) is applicable. \(\Delta M=0\), when the applied field is parallel to the electric vector of the microwaves, and \(\Delta M=\pm 1\), when it is perpendicular to this vector. Usually the case \(\Delta M=0\) is observed (a \(\pi\)-type transition), since parallel fields are more convenient to use in a waveguide than perpendicular ones. The Stark effect for some linear molecules has already been investigated. The dipole moments determined in this way are collected in Table VIII.

For molecules of the symmetric-top type the Stark perturbation energy of the first and second orders is given by the relations

\[ W_{JM}^{(1)}=\frac{-\mu E M k}{J(J+1)}, \]

\[ W_{JM}^{(2)}= \frac{4\pi^{2} I \mu^{2} E^{2}}{h^{2}} \left\{ \frac{(J^{2}-M^{2})(J^{2}-K^{2})}{J^{3}(2J-1)(2J+1)} - \frac{\big[(J+1)^{2}-M^{2}\big]\big[(J+1)^{2}-K^{2}\big]} {(J+1)^{3}(2J+1)(2J+3)} \right\}, \]

where the selection rules for linear molecules are applicable with the additional requirement \(\Delta K=0\). As far as we know, the Stark effect for molecules of the symmetric-top type in the microwave region of the spectrum has not yet been investigated.

For a theoretical discussion of the Stark effect in the inversion spectrum of ammonia, we refer the reader to the work of Yauch\({}^{154}\). The Stark effect of an asymmetric rotor was considered by Golden and Wilson\({}^{119}\).

7. Zeeman effect.

The splitting of absorption lines by a magnetic field in the microwave region of the spectrum was first carried out by Coles and Good\({}^{130}\) with the lines of the inversion spectrum of ammonia. The work was continued by K. K. Jen\({}^{40a}\), who, by improving the experimental technique and the interpretation of his results, demonstrated on several molecules the significance of the Zeeman effect in microwave spectroscopy.

The energy of the interaction with an external magnetic field of a molecule having molecular factor \(g\) and a single nucleus bound to the molecular axis, as established by Jen, is determined by the expression

\[ \Delta W=-M\mu_{0}H\left(\alpha_{J}g_{\mathrm{mol}}+\alpha_{I}g_{N}\right), \tag{78} \]

where

\[ \alpha_{J}=F(F+1)+J(J+1)-I(I+1), \]

\[ \alpha_{I}=\frac{F(F+1)+I(I+1)-J(J+1)}{2F(F+1)}, \]

\[ M=F,\ F-1,\ F-2,\ldots,-F, \]

\(\mu_{0}\) is the nuclear magneton, \(g_{\mathrm{mol}}\) is the factor \(g\) for the molecule along \(J\), \(g_{N}\) is the factor \(g\) of the nucleus under consideration, and \(H\) is the external magnetic field.

Neglecting higher-order interactions, we find that this energy is simply added to the energy of vibration, rotation, and nuclear quadrupole interaction. The positions of the absorption lines

are determined by using the corresponding selection rules. The theory is applicable to a large number of molecules of interest.

As Jen noted, the case considered above is completely analogous to the nuclear Zeeman effect in atomic spectra, except that here there is no fixed ratio between \(g_{\text{mol}}\) and \(g_N\). Indeed, this formula is analogous to that first obtained by Back and Goudsmit[^155] for the Zeeman effect in the hyperfine structure of atoms. A simple derivation of it may be found in[^156].

The selection rules are: \(\Delta M=\pm 1\) for \(H\) parallel to the vector \(E\) of the radiation field, \(\Delta M=0\) for \(H\) perpendicular to the vector \(E\) of the radiation field; the intensity rules are the same as for atomic spectra.

There is, however, an interesting difference in the applications of the Zeeman effect in microwave molecular spectra and in optical atomic spectra. In the latter case \(g_I \ll g_J\), whereas in molecular spectra \(g_N \gg g_{\text{mol}}\) is often the case. In general, the field necessary to produce a resolvable splitting in optical spectra must be so strong that the \(IJ\)-coupling is broken and the Back–Goudsmit effect is usually observed (analogous to the Paschen–Back effect in fine structure). In contrast, in the microwave region effects of extremely weak fields can be observed. With the aid of microwaves it is possible to detect very small magnetic moments of molecules in the ground \({}^1\Sigma\)-state. Of interest is the case of a molecule for which \(g_N=0\). For it

\[ \Delta W=\mu_0 g_{\text{mol}}HM, \tag{79} \]

whence, for a parallel field \((\Delta M=\pm 1)\), we obtain a doublet splitting:

\[ \Delta \nu=\pm \frac{\mu_0 g_{\text{mol}}H}{h}. \tag{80} \]

In a field of \(1670\) gauss Jen found a broadening of the rotational lines of \(\mathrm{SO}_2\), but was unable to resolve the predicted splitting. A field ten times stronger will undoubtedly make it possible to study a larger number of molecules of this type.

Molecules for which \(g_N\ne 0\) and \(g_{\text{mol}}\ne 0\), but for which the coupling between \(I\) and \(J\) is negligibly small, may likewise be considered with the aid of the simple equation (79). Here, of course, \(I\) precesses about \(H\), but this does not cause noticeable changes in the rotational frequencies because of the weak coupling with \(J\). An example of this type of molecule is \(\mathrm{N}^{15}\mathrm{H}_3\), investigated by Jen. A doublet splitting was observed, corresponding to the indicated assumption concerning \(g_{\text{mol}}\).

An important case is that in which \(g_{\text{mol}}\simeq 0\) and one nucleus is coupled to the molecular axis by means of its nuclear quadrupole moment. In this case the splitting of the hyperfine-structure levels is given by (78), where \(g_{\text{mol}}\) is set equal to zero. Fig. 20, reproduces-

![Diagram with labels: a) energy levels and Zeeman splitting for \(CH_3Cl\); b) Zeeman-effect spectra, theory and experiment, with transition labels and field strengths.]

Fig. 20a. Diagram of the energy levels
and Zeeman splitting for \(CH_3Cl\).

\(I(Cl^{35}) = 3/2\)
\(I(Cl^{37}) = 3/2\)

\(g(Cl^{35}) = 0.547\)
\(g(Cl^{37}) = 0.454\)

\(g_{\mathrm{mol}} \ll g(Cl)\)

Fig. 20b. Zeeman effect. Transition \(J = 0 \to 1\) of the molecule \(CH_3Cl^{35}\)
in the absence and in the presence of a magnetic field.

\(I(Cl^{35}) = 3/2\)
\(g(Cl^{35}) = 0.547\)
\(g_{\mathrm{mol}} \ll g_N(Cl^{35})\)

introduced from Jen’s work, illustrates in detail the microwave Zeeman effect for molecules of this type. Although the nuclear \(g\)-factors for both nuclei \(Cl^{35}\) and \(Cl^{37}\) are already accurately known, this example illustrates the way in which the method can be applied in other cases to determine unknown nuclear magnetic moments. Since a cell of small volume can be used for the measurements, the method is applicable to radioactive nuclei. For example, the nuclear \(g\)-factor of \(J^{131}\) could be determined by measurements on \(CH_3J^{131}\).

Although ammonia, methyl chloride, and sulfur dioxide are the only molecules that have been studied so thoroughly, these few, well-chosen objects of study have shown that the Zeeman effect in microwave spectroscopy will evidently become the subject of many investigations.

V. DETERMINATION OF MOLECULAR AND NUCLEAR PROPERTIES

Structure of molecules. Tables VI and VII collect the values of \(B\) and molecular dimensions determined up to the present. Certain factors affect the accuracy of the determination of molecular structures. When molecules with different isotopic combinations are used, differences in zero-point energy may lead to an increase in errors. This circumstance was pointed out by Strandberg, Wentink, and Hill \(^{157}\) and emphasized by Townes, Holden, and Merritt \(^{95}\). Errors due to these causes vary very strongly from molecule to molecule and are difficult to estimate, since they depend on the anharmonicity of the vibrational potential function, as well as on differences in the zero-point energies of the isotopes. A measure of accuracy may be, however, the degree of internal agreement of interatomic distances determined from several different isotopic combinations. Deviations of about one percent have been found, but usually the agreement is considerably better.

Dipole moments. The dipole moments of gases can be accurately measured by means of Stark modulation of absorption lines, as indicated above. Moments determined in this way are given in Table VIII. Since Stark splitting of the rotational lines of molecules in excited vibrational states can be observed, changes in dipole moments caused by vibrations can also be observed. Dipole moments may also be measured from line intensities, but the accuracy obtained in this case is not as great as in the Stark method.

Nuclear bonds. Table IX gives the coupling factors determined up to the present,
\(eQ \dfrac{d^2V}{dz^2}\).
Since \(eQ\) remains constant, differences in the couplings of a given nucleus in different molecules indicate differences in the electronic structure of the molecules.

Table VI

Pure rotational spectra

Molecules	\(J\)-transition	Observed frequencies in Mc/s	\(B_0\) in Mc/s	Literature references
		Linear molecules
\(J^{127}Cl^{35}\)	\(0 \to 1\)	Hyperfine structure not resolved		a
\(J^{127}Cl^{35}\)	\(3 \to 4\)	21 lines between 27 194.75 and 27 357.73 \(H\)*)	\((R_e = 3122.300)\)	b
\(J^{127}Cl^{37}\)	\(0 \to 1\)	Hyperfine structure not resolved		a
\(HC^{12}N^{14}\)	\(0 \to 1\)	Triplet near 88 671 \(H\)	44 336	c
\(C^{135}C^{12}N^{14}\)	\(1 \to 2\)	17 lines between 23 862.57 and 23 981.60 \(H\)	5970.820	d
\(C^{135}C^{12}N^{14}\)	\(2 \to 3\)	4 lines between 35 805.05 and 35 835.75 \(H\)	5970.823	e
\(C^{137}C^{12}N^{14}\)	\(1 \to 2\)	5 lines between 23 372.72 and 23 4?2.47 \(H\)	5847.260	d
\(C^{137}C^{12}N^{14}\)	\(2 \to 3\)	4 lines between 35 067.97 and 35 091.87 \(H\)	5847.243	e
\(C^{135}C^{13}N^{14}\)	\(2 \to 3\)	4 lines between 35 615.88 and 35 649.58 \(H\)	5939.795	e
\(C^{135}C^{13}N^{14}\)	\(2 \to 3\)	34 889.05 \((F = 7/2 \to 9/2)\) \(H\)	5814.710	e
\(Br^{79}C^{12}N^{14}\)	\(2 \to 3\)	10 lines between 24 583.00 and 24 884.57 \(H\)	4120.190	d
\(Br^{79}C^{12}N^{14}\)	\(3 \to 4\)	5 lines between 32 801.56 and 32 956.78 \(H\)	4120.224	e
\(Br^{81}C^{12}N^{14}\)	\(2 \to 3\)	11 lines between 24 465.87 and 24 717.19 \(H\)	4096.760	d
\(Br^{81}C^{12}N^{14}\)	\(3 \to 4\)	7 lines between 32 643.10 and 32 913.52 \(H\)	4096.797	e
\(Br^{79}C^{13}N^{14}\)	\(3 \to 4\)	4 lines between 32 581.71 and 32 601.53 \(H\)	4073.355	e
\(Br^{81}C^{13}N^{14}\)	\(3 \to 4\)	4 lines between 32 392.56 and 32 409.12 \(H\)	4049.608	e

*) For footnotes and literature references to the table, see p. 281.

Continuation of Table VI

Molecules	\(J\)-transition	Observed frequencies in Mc/s	\(B_0\) in Mc/s	Literature references
\(J^{127}C^{12}N^{14}\)	\(4 \to 5\)	9 lines between 31 848,77 and 32 386,29 \(H\)	3225,555	д
\(J^{127}C^{13}N^{14}\)	\(4 \to 5\)	6 lines between 31 718,28 and 31 793,46 \(H\)	3177,035	д
\(O^{16}C^{13}S^{31}\)	\(1 \to 2\)	24 325,92	6081,480	е, ж
\(O^{16}C^{13}S^{31}\)	\(1 \to 2\)	24 355,49 \((v_2 = 1,\ l = 1)\)		ж
\(O^{16}C^{13}S^{31}\)	\(1 \to 2\)	24 355,50 \((v_2 = 1,\ l = 1)\)		г
\(O^{16}C^{13}S^{31}\)	\(1 \to 2\)	24 380,70 \((v_2 = 1,\ l = 2)\)		ж
\(O^{16}C^{13}S^{31}\)	\(1 \to 2\)	24 381,07 \((v_2 = 1,\ l = 2)\)		г
\(O^{16}C^{13}S^{31}\)	\(2 \to 3\)	36 488,82	6081,470	ж
\(O^{16}C^{13}S^{31}\)	\(3 \to 4\)	48 651,64	6081,455	ж
\(O^{16}C^{13}S^{31}\)	\(4 \to 5\)	60 814,08	6081,408	ж
\(O^{16}C^{12}S^{33}\)	\(1 \to 2\)	4 lines between 24 013,01 and 24 032,75 \(H\)	6005,053	з
\(O^{16}C^{13}S^{34}\)	\(1 \to 2\)	23 731,33		е
\(O^{16}C^{13}S^{34}\)	\(3 \to 4\)	47 462,40		ж
\(O^{16}C^{13}S^{32}\)	\(1 \to 2\)	24 247,82		ж
\(O^{16}C^{13}S^{32}\)	\(1 \to 2\)	24 217,69	6061,955	ж
\(O^{16}C^{13}S^{32}\)	\(1 \to 2\)	24 275,25 \((V_2 = 1,\ l = 1)\)	6061,923	г
\(O^{16}C^{13}S^{32}\)	\(1 \to 2\)	24 301,05 \((V_2 = 1,\ l = 2)\)		ж
\(O^{16}C^{13}S^{34}\)	\(1 \to 2\)	23 646,92	5911,730	г

Continuation of Table VI

Molecules	\(J\)-transition	Observed frequencies in Mc/s	\(B_0\) in Mc/s	Literature references
\(\mathrm{O}^{16}\mathrm{C}^{14}\mathrm{S}^{32}\)	\(1 \to 2\)	24 173	6 043,25	н
\(\mathrm{O}^{16}\mathrm{C}^{14}\mathrm{S}^{32}\)	\(1 \to 2\)	24 197 \((V_3 = 1,\ l = 1)\)		н
\(\mathrm{O}^{16}\mathrm{C}^{14}\mathrm{S}^{32}\)	\(1 \to 2\)	24 224 \((V_2 = 1,\ l = 2)\)		н
\(\mathrm{N}^{14}\mathrm{N}^{14}\mathrm{O}^{16}\)	\(0 \to 1\)	25 123,03 \((F = 1 \to 1)\)	12 561,64
\(\mathrm{N}^{14}\mathrm{N}^{14}\mathrm{O}^{16}\)		25 123,28 \((F = 1 \to 2)\) Н
\(\mathrm{N}^{14}\mathrm{N}^{14}\mathrm{O}^{16}\)		25 123,64 \((F = 1 \to 0)\)
\(\mathrm{N}^{15}\mathrm{N}^{14}\mathrm{O}^{16}\)	\(0 \to 1\)	24 271,53 \((F = 1 \to 1)\)	12 137,31
\(\mathrm{N}^{15}\mathrm{N}^{14}\mathrm{O}^{16}\)		24 274,61 \((F = 1 \to 2)\) Н
\(\mathrm{N}^{15}\mathrm{N}^{14}\mathrm{O}^{16}\)		24 274,73 \((F = 1 \to 0)\)
OCSe	\(2 \to 3\)	Several lines observed between 23 000 and 24 500		л

Molecules of the symmetric-top type

Molecules	\(J\)-transition	Observed frequencies in Mc/s	\(B_0\) in Mc/s	Literature references
\(\mathrm{PF}_3\)	\(1 \to 2\)	31 279,60	7 819,90)	м
\(\mathrm{PF}_3\)	\(2 \to 3\)	46 918,90		м
\(\mathrm{AsF}_3\)	\(1 \to 2\)	12 lines around 23 500 Н	5 883,0	н
\(\mathrm{HCF}_3\)	\(1 \to 2\)	41 394,95	41 394,96	о

Continuation of Table VI

Molecules	\(J\)-transition	Observed frequencies in MHz	\(B_0\) in MHz	Literature references
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{F}\)	\(0 \to 1\)	51 071,69	25 535,85	о
\(\mathrm{C}^{13}\mathrm{H}_3\mathrm{F}\)	\(0 \to 1\)	49 724,73	24 862,37	о
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{Cl}^{35}\)	\(0 \to 1\)	26 570,77 \((F = 3/2 \to 1/2)\)	13 292,89	п
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{Cl}^{35}\)	\(0 \to 1\)	26 589,49 \((F = 3/2 \to 5/2)\ H\)	13 292,89	п
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{Cl}^{35}\)	\(0 \to 1\)	26 601,57 \((F = 3/2 \to 3/2)\)	13 292,89	п
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{Cl}^{37}\)	\(0 \to 1\)	26 164,5 \((F = 3/2 \to 1/2)\)	13 088,19	п
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{Cl}^{37}\)	\(0 \to 1\)	26 179,30 \((F = 3/2 \to 5/2)\ H\)	13 088,19	п
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{Cl}^{37}\)	\(0 \to 1\)	26 191,13 \((F = 3/2 \to 3/2)\)	13 088,19	п
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{Br}^{79}\)	\(1 \to 2\)	12 lines between 38 128,40 and 38 417,09 \(H\)	9 568,100	п
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{Br}^{81}\)	\(1 \to 2\)	12 lines between 38 006,47 and 38 247,77 \(H\)	9 531,743	п
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{I}^{127}\)	\(1 \to 2\)	17 lines between 29 598,95 and 30 179,71 \(H\)	7 501,250	п
\(\mathrm{C}^{13}\mathrm{H}_3\mathrm{I}^{127}\)	\(1 \to 2\)	11 lines between 28 069,99 and 28 687,21 \(H\)	7 119,040	п
\(\mathrm{B}^{10}\mathrm{H}_3\mathrm{CO}\)	\(1 \to 2\)	8 lines between 35 917,61 and 35 920,16 \(H\)	8 979,90	р
\(\mathrm{B}^{11}\mathrm{H}_3\mathrm{CO}\)	\(1 \to 2\)	7 lines between 31 627,24 and 34 629,32 \(H\)	8 657,21	р
\(\mathrm{C}^{13}\mathrm{H}_3\mathrm{C}^{12}\mathrm{C}^{13}\mathrm{H}\)	\(2 \to 3\)	15 lines between 51 260 and 51 461	8 544	с
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{C}^{12}\mathrm{N}^{14}\)	\(1 \to 2\)	11 lines between 36 942,15 and 36 793,64 \(H\)	9 198,845	т
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{N}^{14}\mathrm{C}^{13}\)	\(1 \to 2\)	6 lines between 40 210,27 and 40 424,49	10 052,79	т

Continuation of Table VI

Molecules	\(J\)-transition	Observed frequencies in Mc/s	\(B_0\) in Mc/s	References
\(\mathrm{C}^{12}\mathrm{H}_3\mathrm{N}^{14}\mathrm{C}^{13}\)	\(1 \to 2\)	\(38\,782,20\) \(38\,783,21\)	\(9\,693,802\)	т
\(\mathrm{CH}_3\mathrm{CF}_3\)	\(1 \to 2\)	\(20\,741\)	\(5\,185\)	у
Molecules of the asymmetric-top type	Molecules of the asymmetric-top type	Molecules of the asymmetric-top type	Molecules of the asymmetric-top type	Molecules of the asymmetric-top type
\(\mathrm{H}_2\mathrm{O}\)	\(5_{-1} \to 6_{-5}\)	\(22\,235,42\)		ф
\(\mathrm{HDO}\)	\(5_{3,3,0} \to 5_{3,2,1}\)	\(22\,307,67\)		ф
\(\mathrm{SO}_2\)	\(13_{2,12} \to 12_{3,9}\)	\(20\,420\)	\(I_c^0 = 95,14 \cdot 10^{-40}\)	х
\(\mathrm{SO}_2\)	\(6_{1,5} \to 5_{2,4}\)	\(23\,413\)	\(I_b^0 = 81,16 \cdot 10^{-40}\)	х
\(\mathrm{SO}_2\)	\(9_{1,9} \to 8_{2,6}\)	\(24\,037\) or \(24\,033\)	\(I_a^0 = 13,78 \cdot 10^{-40}\)	х
\(\mathrm{SO}_2\)	\(7_{2,6} \to 8_{1,7}\)	\(25\,392\)		х
\(\mathrm{SO}_2\)	\(3_{1,3} \to 4_{0,4}\)	\(29\,460\)		х

Continuation of Table VI

Molecules	\(J\)-transition	Observed frequencies in MHz	\(B_0\) in MHz	Literature references
\(\mathrm{HNC}^{13}\mathrm{S}^{32}\)	\(1 \to 2\)	23 461	\(\dfrac{1}{2}(B+C)=5\,866{,}0\)	ц
\(\mathrm{DNC}^{12}\mathrm{S}^{32}\)	\(1 \to 2\)	21 897	\(\dfrac{1}{2}(B+C)=5\,474{,}3\)	ц
\(\mathrm{HNC}^{18}\mathrm{S}^{32}\)	\(1 \to 2\)	23 389	\(\dfrac{1}{2}(B+C)=5\,817{,}3\)	ц
\(\mathrm{DNC}^{13}\mathrm{S}^{2}\)	\(1 \to 2\)	21 839	\(\dfrac{1}{2}(B+C)=5\,459{,}8\)	ц
\(\mathrm{HNC}^{12}\mathrm{S}^{34}\)	\(1 \to 2\)	22 915	\(\dfrac{1}{2}(B+C)=5\,728{,}8\)	ц
\(\mathrm{CH_3OH}\)	\(0 \to 1\)	7 lines between 47 840 and 48 010	\(\dfrac{1}{2}(B+C)=24\,353{,}85\)	ч
\(\mathrm{CH_3OD}\)	\(0 \to 1\)	47 346
\(\mathrm{CH_3OD}\)	\(0 \to 1\)	47 266
\(\mathrm{CH_3OD}\)	\(0 \to 1\)	47 052	\(\dfrac{1}{2}(B+C)=23\,673\)
\(\mathrm{CH_3NH_2}\)	\(0 \to 1\)	45 324,21		ч
\(\mathrm{CH_3NH_2}\)	\(0 \to 1\)	45 324,94	\(\dfrac{1}{2}(B+C)=22\,662{,}12\)	ч

End of Table VI

*) The letter \(H\) means that hyperfine structure has been observed.

a) R. T. Weidner, Phys. Rev. 72, 1268 (1947); 73, 254 (1948).
b) C. H. Townes, F. R. Merritt and B. D. Wright, Phys. Rev. 73, 1334 (1948).
c) A. G. Smith, W. Gordy, J. W. Simmons and W. V. Smith (prepared for publication).
d) C. H. Townes, A. N. Holden and F. R. Merritt (private communication).
e) A. G. Smith, H. Ring, W. V. Smith and W. Gordy, Phys. Rev. 74, 370 (1948).
f) T. W. Dakin, W. E. Good and D. K. Coles, Phys. Rev. 70, 560 (1946).
g) M. W. P. Strandberg, T. Wentink, Jr. and R. L. Kyhl, Tech. Report No 59, Research Lab. of Electronics, M. I. T., May 13, 1948.
h) C. H. Townes and S. Geschwind (private communication).
i) A. Roberts, Phys. Rev. 73, 1405 (1948).
k) D. K. Coles, E. S. Elyash and J. C. Gorman, Phys. Rev. 72, 973 (1947).
l) M. W. P. Strandberg and T. Wentink, Jr., Bull. Am. Phys. Soc. 23, No 2, 17 (1948).
m) O. R. Gilliam, H. D. Edwards and W. Gordy (prepared for publication).
n) B. P. Dailey, K. Rusinow, R. G. Shulman and C. H. Townes, Bull. Am. Phys. Soc. 23, No. 3, 53 (1948).
o) O. R. Gilliam, H. D. Edwards and W. Gordy (prepared for publication).
p) W. Gordy, J. W. Simmons and A. G. Smith, Phys. Rev. 74, 243 (1948).
r) W. Gordy, H. Ring and A. B. Burg (prepared for publication).
s) H. Ring and W. Gordy (prepared for publication).
t) M. Kessler, H. Ring and W. Gordy (prepared for publication).
u) F. Edgel and A. Roberts (private communication).
f) S. Golden, T. Wentink, Jr., R. Hillger and M. W. P. Strandberg, Phys. Rev. 73, 92 (1948).
x) B. P. Dailey, S. Golden and E. B. Wilson, Jr., Phys. Rev. 72, 871 (1947).
ts) C. I. Beard and B. P. Dailey, J. Chem. Phys. 15, 762 (1947).
ch) H. D. Edwards, O. R. Gilliam and W. Gordy (prepared for publication).

Table VII

Molecular structures determined by means of microwave spectroscopy

Molecule	Distance in the ground state (in Å)	Angle
Linear molecules	Linear molecules	Linear molecules
HCNa)	C—H = 1.059 (presumably) C—N = 1.157
ClCNб, в)	C—Cl = 1.630 C—N = 1.163
BrCNб, в)	C—Br = 1.789 C—N = 1.160
JCNб, в)	C—J = 1.993 C—N = 1.159
OCSв, г, д)	C—O = 1.161 C—S = 1.560
N₂Oе)	N—N = 1.126 N—O = 1.191
Molecules of the symmetric-top type	Molecules of the symmetric-top type	Molecules of the symmetric-top type
PF₃ж)	P—F = 1.546 ± 0.008	∠FPF = 101° ± 3° (presumably)
AsF₃з)	As—F = 1.712 ± 0.006	∠FAsF = 100° ± 5° (presumably)
HCF₃и)	C—F = 1.322	∠FCF = 111° (presumably)
CH₃Fк)	C—H = 1.111 (presumably) C—F = 1.381 C—H = 1.111	∠HCH = 110°6′
CH₃Clк)	C—Cl = 1.779 C—H = 1.109	∠HCH = 110°0′
CH₃Brк)	C—Br = 1.936 C—H = 1.104	∠HCH = 110°15′
CH₃Jк)	C—J = 2.139 C—H = 1.100	∠HCH = 110°58′
CH₃NCі)	C—N = 1.426 N—C = 1.167 C—H = 1.093 (presumably)	∠HCH = 109°28′ (presumably)

End of Table VII

Molecule	Distance in the ground state (in Å)	Angle
BH$_3$CO$^{\mathrm{m)}}$	B—H = 1.20 (presumably) C—O = 1.13 (presumably) B—C = 1.540	$\angle$HBH = 113°52′ $\angle$OSO = 119.5°
Molecules of the asymmetric-top type	Molecules of the asymmetric-top type	Molecules of the asymmetric-top type
SO$_2^{\mathrm{n)}}$ HNCS$^{\mathrm{o)}}$	S—O = 1.433 H—C = 1.2 ± 0.1 N—C = 1.21 ± 0.01 C—S = 1.57 ± 0.01	$\angle$HNC = 112° ± 10°

a) A. G. Smith, W. Gordy, J. W. Simmons and W. V. Smith (prepared for press).
b) A. G. Smith, H. Ring, W. V. Smith and W. Gordy, Phys. Rev. 74, 370 (1948).
c) C. H. Townes, A. N. Holden and D. K. Coles (private communication).
d) T. W. Dakin, W. E. Good and D. K. Coles, Phys. Rev. 71, 640 (1947).
e) M. W. P. Strandberg, T. Wentink, Jr. and R. L. Kyhl, Tech. Report No. 59, Research Lab. of Electronics, M. I. T., May 13 (1948).
f) D. K. Coles, E. S. Elyash and J. Gorman, Phys. Rev. 72, 973 (1947).
g) O. R. Gilliam, H. D. Edwards and W. Gordy (prepared for press).
h) B. P. Dailey, K. Rusinow, R. G. Shulman and C. H. Townes, Bull. Am. Phys. Soc. 23, No. 3, 53 (1948).
i) O. R. Gilliam, H. D. Edwards and W. Gordy (prepared for press).
k) W. Gordy, J. W. Simmons and A. G. Smith, Phys. Rev. 74, 243 (1948).
l) M. Kessler, H. Ring and W. Gordy (prepared for press).
m) W. Gordy, H. Ring and A. B. Burg (prepared for press).
n) B. P. Dailey, S. Golden and E. B. Wilson, Jr., Phys. Rev. 72, 871 (1947).
o) C. I. Beard and B. P. Dailey, J. Chem. Phys. 15, 762 (1947).

Table VIII

Dipole moments of gases from microwave data

Molecule	Dipole moment (in \(CGSE \times 10^{-18}\))	Method
JCl\(^{a)}\)	0.65	Intensity
ClCN\(^{b)}\)	\(2.54 \pm 0.25\)	»
OCS\(^{c)}\)	\(0.732 \pm 0.007\) for \(O^{16}C^{13}S^{33}\)	Stark effect
OCS\(^{c)}\)	\(0.722 \pm 0.007\) for \(O^{16}C^{13}S^{32}\)	»
OCS\(^{d)}\)	\(0.752 \pm 0.007\). Ground vibrational state	»
OCS\(^{d)}\)	\(0.728 \pm 0.007\). \(v_1\)-state	»
H\(_3\)O\(^{d)}\)	\(1.94 \pm 0.06\)	»
HDO\(^{e)}\)	\(1.78 \pm 0.06\)	»
NH\(_3\)\(^{f)}\)	1.5	»

a) C. H. Townes, F. R. Merritt and B. D. Wright, Phys. Rev. 73, 1334 (1948).
b) C. H. Townes, A. N. Holden and F. R. Merritt (private communication).
c) M. W. P. Strandberg, T. Wentink, Jr. and R. L. Kyhl, Tech. Report No. 59, Research Lab. of Electronics, M. I. T.; see also T. W. Dakin, W. E. Good and D. K. Coles, Phys. Rev. 70, 560 (1946).
d) T. Wentink, Jr., M. W. P. Strandberg, and R. Hillger, Bull. Am. Phys. Soc. 23, No. 2, 18 (1948).
e) S. Golden, T. Wentink, Jr., R. Hillger and M. W. P. Strandberg, Phys. Rev. 73, 92 (1948).
f) M. W. P. Strandberg, T. Wentink, Jr., R. F. Hillger, G. H. Wannier and M. L. Deutsch, Phys. Rev. 73, 188 (1948).
g) D. K. Coles and W. E. Good, Phys. Rev. 70, 979 (1946).

From Table IX it is seen that considerable differences occur. For example, the bond \(N^{14}\) varies from \(-0.27\) Mc to \(-4.7\) Mc. When more data have been collected, the correlation of the values

\[ \frac{\partial^{3}V}{\partial z^{3}} \]

with other molecular properties will undoubtedly shed appropriate light on the nature of the chemical bond. Attempts have already been made to relate

\[ \frac{\partial^{3}V}{\partial z^{3}} \]

to the magnitude of \(s\)-\(p\) hybridization\(^{158,159}\).

Nuclear quadrupole moments. For the determination of nuclear quadrupole moments from microwave data, the factor

\[ \frac{\partial^{2}V}{\partial z^{2}} \]

must be determined from other sources. Sufficiently accurate calculations of this quantity cannot be carried out in any case, with the exception of the simplest hydrogen molecule. For heavy atoms bonded by a simple covalent \(p\)-bond,

Table IX

Nuclear quadrupole interaction

Molecule	Atom	\(eQ\dfrac{\partial^2 V}{\partial z^2}\) (in MHz)	Literature references
\(\mathrm{NH_3}\)	\(\mathrm{N}^{14}\)	\(4.08\) *)	а
\(\mathrm{NH_3}\)	\(\mathrm{N}^{14}\)	\(4.10\) *)	б
\(\mathrm{NH_3}\)	\(\mathrm{N}^{14}\)	\(4.26\) *)	в
\(\mathrm{NH_3}\)	\(\mathrm{N}^{14}\)	\(4.08\)	г
\(\mathrm{NNO}\)	\(\mathrm{N}^{14}\)	\(-0.27\) central atom	д
\(\mathrm{NNO}\)	\(\mathrm{N}^{14}\)	\(-0.84\) terminal atom	д
\(\mathrm{NNO}\)	\(\mathrm{N}^{14}\)	\(-1.03\) terminal atom	е
\(\mathrm{HCN}\)	\(\mathrm{N}^{14}\)	\(-4.7\)	ж
\(\mathrm{ClCN}\)	\(\mathrm{N}^{14}\)	\(-3.63\)	з
\(\mathrm{BrCN}\)	\(\mathrm{N}^{14}\)	\(-3.83\)	з
\(\mathrm{JCN}\)	\(\mathrm{N}^{14}\)	\(-3.80\)	е
\(\mathrm{CH_3CN}\)	\(\mathrm{N}^{14}\)	\(-4.67\)	и
\(\mathrm{CH_3NC}\)	\(\mathrm{N}^{14}\)	\(<|0.5|\)	и
\(\mathrm{CH_3NH_3}\)	\(\mathrm{N}^{14}\)	\(<|\sim 1|\)	к
\(\mathrm{BH_3CO}\)	\(\mathrm{B}^{17}\)	\(-1.55\)	л
\(\mathrm{BH_3CO}\)	\(\mathrm{B}^{10}\)	\(-3.30\)	л
\(\mathrm{AsF_3}\)	\(\mathrm{As}^{75}\)	\(-235\)	м
\(\mathrm{OCS}\)	\(\mathrm{S}^{33}\)	\(-28.5\)	н
\(\mathrm{JCl}\)	\(\mathrm{Cl}^{35}\)	\(-82.5\)	о
\(\mathrm{ClCN}\)	\(\mathrm{Cl}^{35}\)	\(-83.5\)	п
\(\mathrm{ClCN}\)	\(\mathrm{Cl}^{35}\)	\(-83.2\)	з
\(\mathrm{CH_3Cl}\)	\(\mathrm{Cl}^{35}\)	\(-75.13\)	р
\(\mathrm{ClCN}\)	\(\mathrm{Cl}^{37}\)	\(-65.0\)	п
\(\mathrm{ClCN}\)	\(\mathrm{Cl}^{37}\)	\(-65.7\)	з
\(\mathrm{CH_3Cl}\)	\(\mathrm{Cl}^{37}\)	\(-59.93\)	р
\(\mathrm{BrCN}\)	\(\mathrm{Br}^{79}\)	\(+686.0\)	п
\(\mathrm{BrCN}\)	\(\mathrm{Br}^{79}\)	\(+686.5\)	з
\(\mathrm{CH_3Br}\)	\(\mathrm{Br}^{79}\)	\(+577.0\)	р
\(\mathrm{BrCN}\)	\(\mathrm{Br}^{81}\)	\(+573.0\)	п
\(\mathrm{BrCN}\)	\(\mathrm{Br}^{81}\)	\(+573.5\)	з
\(\mathrm{CH_3Br}\)	\(\mathrm{Br}^{81}\)	\(+482\)	р
\(\mathrm{JCN}\)	\(\mathrm{J}^{137}\)	\(-2420\)	с, п, з
\(\mathrm{CH_3J}\)	\(\mathrm{J}^{137}\)	\(-1934\)	р
\(\mathrm{JCl}\)	\(\mathrm{J}^{137}\)	\(-2920\)	о

) For the conversion of these values into this form, see V. T. Feld, Phys. Rev. 72*, 1116 (1947).

а) D. K. Coles and W. E. Good, Phys. Rev. 70, 979 (1946).
б) B. P. Dailey, R. L. Kyhl, M. W. P. Strandberg, J. H. Van Vleck and E. B. Wilson, Jr., Phys. Rev. 70, 984 (1946).
в) R. J. Watts and D. Williams, Phys. Rev. 72, 263 (1947).
г) J. W. Simmons and W. Gordy, Phys. Rev. 73, 713 (1948).
д) D. K. Coles, E. S. Elyash and J. G. Gorman, Phys. Rev. 72, 973 (1947).

Continuation of Table IX

e) A. G. Smith, H. Ring, W. V. Smith and W. Gordy, Phys. Rev. 73, 663 (1948).
zh) A. G. Smith, W. Gordy, J. W. Simmons and W. V. Smith (in preparation).
z) C. H. Townes, A. N. Holden and F. R. Merritt (private communication).
i) H. Ring, H. D. Edwards, M. Kessler and W. Gordy, Phys. Rev. 72, 1262 (1947).
k) H. D. Edwards, O. R. Gilliam and W. Gordy (in preparation).
l) W. Gordy, H. Ring and A. B. Burg (in preparation).
m) B. B. Dailey, K. Rusinow, R. G. Shulman and C. H. Townes, Bull. Am. Phys. Soc. 23, No. 3, 53 (1948).
n) C. H. Townes and S. Geschwind, Phys. Rev. 74, 626 (1948).
o) C. H. Townes, F. R. Merritt and B. D. Wright, Phys. Rev. 73, 1334 (1948).
p) A. G. Smith, H. Ring, W. V. Smith and W. Gordy, Phys. Rev. 74, 370 (1948).
r) W. Gordy, J. W. Simmons and A. G. Smith, Phys. Rev. 74, 243 (1948).
s) J. Barden and C. H. Townes, Phys. Rev. 73, 627 (1948).

Townes\(^{159}\) suggested that the formula*)

\[ \frac{8e\Delta\nu}{15ZRa_0^2a^3}, \]

is applicable, when used to determine nuclear quadrupole moments from atomic spectra. Here \(\Delta\nu\) is the splitting of the fine-structure lines of the atom in the corresponding \(p\)-state. This assumption, when it can be applied, is only approximate. Experience shows that the degree and type of bonding depend to a high degree on the interaction. When the nature of the bond is known, one can estimate a correction for the degree of \(s\)-\(p\)-hybridization, etc. Table X collects some quadrupole moments determined by means of the atomic-orbital approximation from microwave data for various molecules.

Even if a numerical determination of the moments proves impossible, it is usually possible to determine their sign. This gives qualitative information about the nuclear structure. Positive quadrupole moments indicate that the nucleus is elongated along the spin axis, i.e. it is prolate. Negative quadrupole moments indicate that the nucleus is compressed along the spin axis, i.e. it is oblate.

Nuclear spin. Microwave spectroscopy is an excellent method for determining nuclear spins. For spins exceeding \(1/2\), the hyperfine structure caused by the nuclear

*) See H. A. Bethe and R. F. Bacher, Rev. Mod. Phys. 8, 226 (1936).

Table X

Nuclear quadrupole moments (in \(10^{-26}\ \mathrm{cm}^2\))

Atom	Microwave values\(^*\)	Other methods
\(N^{14}\)	\(\sim +2^{a)}\)
\(S^{33}\)	\(\sim -5^{b)}\)
\(Cl^{35}\)	\(-6.6^{c)}\), \(-6.0^{d)}\)	\(7.921 \pm 0.5^{e)}\)
\(Cl^{37}\)	\(-5.2^{c)}\), \(-4.7^{d)}\)	\(6.189 \pm 0.5^{e)}\)
\(Br^{79}\)	\(+28^{c)}\), \(+24^{d)}\)
\(Br^{81}\)	\(+23^{c)}\), \(+19^{d)}\)
\(J^{127}\)	\(-75^{c)}\), \(-59^{d)}\)	\(-45^{f)}\)

\(^*\) With the atomic-orbital approximation for \(\dfrac{\partial^2 V}{\partial z^2}\), C. H. Townes, Phys. Rev. 71, 909 (1947).
\(a)\) C. H. Townes, Bull. Am. Phys. Soc. 23, No. 3, 52 (1948).
\(b)\) C. H. Townes and S. Geschwind, Phys. Rev. 74, 626 (1948).
\(c)\) Determined from the quadrupole coupling of cyano-halide compounds. C. H. Townes, A. N. Holden and F. R. Merritt (private communication) and A. G. Smith, H. Ring, W. V. Smith and W. Gordy, Phys. Rev. 74, 370 (1948).
\(d)\) Determined from the quadrupole coupling of methyl-halide compounds, W. Gordy, J. W. Simmons and A. G. Smith, Phys. Rev. 74, 243 (1948).
\(e)\) L. Davis, R. T. Feld, C. W. Zabel and J. R. Zacharias, Phys. Rev. 73, 525 (1948); L. Davis and C. W. Zabel (private communication).
\(f)\) K. Murakawa, Zeits. f. Physik 114, 651 (1939).

quadrupole interaction, provides identification of the spin. The character of the hyperfine structure depends so strongly on the spin that here the possibility of error is completely excluded, provided only that the structure is fully resolved. A spin smaller than unity can be determined in the usual way from the alternation of intensities of rotational lines, if the atom can be observed in a molecule possessing sufficient symmetry. The identification is less certain in the absence of nuclear quadrupole effects. Table XI gives spins that were either first reliably determined or confirmed by the microwave method. In most cases there was no doubt concerning the preliminary values, which were subsequently confirmed. It is of interest, however, that the spin values of \(Cl^{35}\) and \(Cl^{37}\), determined from optical spectra as \(\dfrac{5}{2}\), were corrected by the microwave method

Table XI

Nuclear spins measured or confirmed by methods of microwave spectroscopy

Atom	I *)	II **)	Literature references
B$^{10}$	1		a
B$^{11}$	3/2		a
N$^{14}$		1	b
S$^{33}$	3/2		c
As$^{75}$		3/2	g
F$^{19}$		1/2	d
Cl$^{35}$	3/2		e
Cl$^{37}$	3/2		e
Br$^{79}$		3/2	e
Br$^{81}$		3/2	e
J$^{127}$		5/2	zh

*) First accurately measured.
**) A previous value confirmed. For some values of spins 0 and 1/2, confirmations were obtained from the impossibility of detecting hyperfine structure in the microwave region of the spectrum.

a) W. Gordy, H. Ring, A. B. Burg (prepared for publication).

b) D. K. Coles and W. E. Good, Phys. Rev. 70, 979 (1946); B. P. Dailey, R. L. Kuhl, M. W. P. Strandberg, J. H. Van Vleck and E. B. Wilson, Jr., Phys. Rev. 70, 984 (1946).

c) C. H. Townes and S. Geschwind, Phys. Rev. 74, 626 (1948).

g) B. P. Dailey, K. Rusinow, R. G. Shulman and C. H. Townes, Bull. Am. Phys. Soc. 23, No. 3, 53 (1948).

d) O. R. Gilliam, H. D. Edwards and W. Gordy (prepared for publication). Confirmed from the alternation of intensities of the PF$_3$ lines for different $K$.

e) C. H. Townes, A. N. Holden, J. Bardeen and F. R. Merritt, Phys. Rev. 71, 644 (1947).

zh) W. Gordy, A. G. Smith and J. W. Simmons, Phys. Rev. 72, 249 (1947); W. Gordy, W. V. Smith, A. G. Smith and H. Ring, Phys. Rev. 72, 259 (1947).

at \(\frac{3}{2}\). In the near future, without doubt, a larger number of spin values will be determined. Since measurements can be made with gases at pressures of the order of \(10^{-4}\) mm Hg in a resonator of volume \(10\ \mathrm{cm}^3\) and less, this method acquires exceptional importance for determining the spins of radioactive and rare nuclei.

IV. STUDY OF LIQUIDS AND SOLIDS BY MEANS OF MICROWAVES

In the present review it is not intended to give a detailed consideration of liquids and solids. This section is intended primarily to draw attention to certain remarkable applications of microwaves in this field. Quite recently the Faraday Society\(^{161}\) held a meeting on dielectrics, at which a complete survey was given of work on the dielectric properties of liquids, solids, and solutions. (See also\(^{162, 163, 164}\).)

In the microwave region many liquids and solutions have an absorption maximum and corresponding regions of anomalous dispersion, caused by the orientation of molecular dipoles in the radiation field. Such loss maxima occur at frequencies corresponding to

\[ \nu = \frac{1}{2\pi\tau}, \]

where \(\tau\) is the relaxation time of the molecular dipole. According to Debye’s theory\(^{165}\), the loss factor

\[ \operatorname{tg}\delta = \frac{(\varepsilon + 2)^2}{\varepsilon}\, \frac{8\pi^2\mu^2 cN\nu\tau}{27kT\,[1 + (2\pi\nu\tau)^2]}, \tag{81} \]

where \(\delta\) is the loss angle (the complement of the phase angle), \(\varepsilon\) is the dielectric constant of the solution, \(\mu\) and \(c\) are the dipole moment and concentration of the dissolved substance, \(\nu\) is the frequency of the radiation, \(\tau\) is the relaxation time of the molecules of the dissolved substance, and \(N\), \(k\), and \(T\) are Avogadro’s number, Boltzmann’s constant, and the absolute temperature. The maximum of \(\operatorname{tg}\delta\) occurs at \(\tau = \frac{1}{2\pi\nu}\). Thus, by measuring the frequency corresponding to the absorption maximum, the relaxation time can be calculated. Similarly, by measuring \(\operatorname{tg}\delta\) for different frequencies or temperatures, \(\varepsilon\), \(\mu\), and \(\tau\) can be determined from Debye’s equation. It is not known how well this equation corresponds to reality. For solutions of nitrobenzene, bromobenzene, chloroform, acetone, and benzophenone in benzene, Jackson and Powles\(^{166}\) obtained good agreement with the theory. The work of Whiffen and Thompson\(^{167}\) revealed indisputable anomalies for some solutions. Cripwell and Sutherland\(^{168}\) used the Debye equation to calculate, from their microwave data, the dipole moments of several liquids, including nitrobenzene, nitromethane, methyl acetate, acetone, and methyl cyanide. Agreement with values obtained by other methods was found to be quite satisfactory. Со-

According to Eyring’s theory,¹⁶⁹ the activation energy \(\Delta E\) and the relaxation entropy \(\Delta S\) are related to \(\tau\) by the relation

\[ \frac{1}{\tau}=\frac{kT}{h} e^{\frac{\Delta S}{R}} e^{-\frac{\Delta E}{RT}} . \tag{82} \]

The relaxation time is also directly related to the internal viscosity.

When a dielectric material has resonance frequencies, the loss tangent, according to Fröhlich,⁷⁹ is equal to

\[ \operatorname{tg}\delta=\frac{\Delta\varepsilon}{2\varepsilon_s} \left( \frac{2\pi\nu\tau}{1+(2\pi\nu+2\pi\nu_0)^2\tau^2} + \frac{2\pi\nu\tau}{1+(2\pi\nu-2\pi\nu_0)^2\tau^2} \right). \tag{83} \]

Here \(\varepsilon_s\) is the static dielectric constant, \(\Delta\varepsilon\) is the change in \(\varepsilon_s\) caused by the oscillating dipoles, and \(\nu_0\) is the resonance frequency. In addition to the possibility of the existence of microwave resonance frequencies in solids, resonance absorption is possible in certain liquids and solutions, arising as a result of oscillations in molecular chains bound together by weak intermolecular bonds, such as the hydrogen bond. Of interest is the transfer of a proton from molecule to molecule or from a group of molecules to a group along a molecular chain. This “seeping through” (“tunneling through”) of a proton from one potential well to another may in some cases cause an increase of absorption in the microwave region of the spectrum.

Magnetic resonance absorption. Owing to the necessity of using exceptionally high magnetic-field methods, the method of nuclear magnetic resonance proposed by Purcell, Torrey, and Pound¹⁷⁰ was not used in the microwave region. However, an analogous method, but applicable to free electron spins in paramagnetic substances, was developed by Zavoisky¹⁷¹ and by Cummerow and Halliday.¹⁷² In these experiments a paramagnetic salt was placed in a magnetic field, and absorption lines were observed in the microwave region corresponding to the Larmor precession frequencies of the resultant electron spin of the paramagnetic ion. The vector of orbital angular momentum is effectively quenched by the internal electric field, leaving only the spin vector \(S\), which precesses around the external magnetic field. This produces a Stark splitting of the lines, depending on the relative orientation of the external magnetic field and the axes of the crystal. Such crystalline Stark splitting was observed by Bagguley and Griffiths and other investigators¹⁷³ and was considered theoretically by Kittel and Luttinger.¹⁷⁴ Stark splitting removes the degeneracy of the energy levels of free ions and makes magnetic dipole transitions observable in the microwave region even in the absence of an external field.

Bleaney and Penrose¹⁷⁵ studied the crystalline Stark splitting of paramagnetic resonance absorption in chromic э

ammonia as a function of temperature. They found that the Stark splitting decreases linearly with decreasing temperature until the temperature reaches approximately \(80^\circ\) K. At this point a sharp discontinuity occurs, expressed in a strong increase of the splitting at lower temperatures. They suggested that the cause of the discontinuity is a disturbance of the uniformity of the crystalline field.

The absorption lines discovered by Zavoisky \(^{171}\) and by Kämmerer and Holidei \(^{172}\) were much sharper than expected. The narrowness of the lines was explained by Gorter and Van Vleck \(^{176}\) on the basis of exchange interaction between electron spins.

Magnetic resonance effects for ferromagnetic materials were observed by Griffiths \(^{177}\) and were confirmed by Yager and Bozorth \(^{178}\). Griffiths found that the observed frequencies in ferromagnetic materials were several times higher than those predicted for the Larmor frequencies of electron spins. Kittel \(^{179}\) showed that this anomalous effect is due to the existence of an induced internal field. For the case of plane surfaces he proposed replacing, in the Larmor theorem, \(H\) by the geometric mean value \((BH)^{1/2}\). The predicted frequencies

\[ \omega_\nu=\sigma(BH)^{1/2}, \tag{84} \]

where \(\sigma\) is the magneto-mechanical ratio for the electron spin, were found to be in good agreement with Griffiths’ observations. Later Kittel \(^{180}\) showed that the correct form of the Larmor theorem depends on the shape of the ferromagnetic material. For a small sphere the usual form \(\omega_\nu=\sigma H\) gives the correct frequencies.

It is evident that magnetic resonance absorption in solids in the microwave region of the spectrum provides a new powerful method for measuring closely spaced energy levels of paramagnetic substances and for studying the structure of certain crystals.

V. APPLICATIONS OF MICROWAVE SPECTROSCOPY

A. STABILIZATION OF MICROWAVE GENERATORS BY MEANS OF SPECTRAL LINES

Stabilization of microwave generators by electrically coupling them to spectral lines was first achieved by Smith, García de Quevedo, Carter, and Bennett \(^{181}\). The possibility of using spectral lines for stabilization had earlier been indicated by Pound \(^{7}\). The use of spectral lines for stabilization was recently described by Hershberger and Norton \(^{182}\). Details of the method of coupling the generator to the absorption line will not be given here. In one of the stabilizers the anomalous dispersion in the region of an absorption line is used, and in principle it

is similar to Pound’s cavity stabilizer. The degree of stabilization depends, among other factors, on the intensity of the absorption line. In favorable cases it appears possible to stabilize the frequency of a klystron much better than to one part in a million. The impossibility of using crystals for stabilizing a generator in the microwave region, as is done for radio-frequency frequencies, makes it necessary to seek some other forms of stabilization. The use of sharp, invariable gas absorption lines has obvious advantages.

B. SPECTRAL LINES AS FREQUENCY STANDARDS

Absorption lines of gases at low pressures, since they have been accurately measured, provide convenient frequency standards for calibrating wavemeters or for measuring other lines by the already described method of determining beat frequencies.

Tables II and VI give the frequencies of the lines of several readily available and convenient gases, covering the wavelength region from 5 mm to 1.5 cm. We plan to extend precise measurements of spectral frequencies to the 3–5 mm region, with the aim of creating similar standards in this region as well.

C. QUALITATIVE AND QUANTITATIVE ANALYSIS OF GASES

One of the applications that presents very great difficulties at the outset, but that will be of exceptionally great importance in the future, is the use of microwave spectroscopy in the qualitative and quantitative analysis of polar gases and vapors. The broad industrial use of infrared spectroscopy for these purposes is an illustration of what may be expected here. Infrared spectroscopy has the advantage that some nonpolar molecules, such as CO₂, can also be observed. However, there is a very large group of molecules for which the microwave method can be applied. Wherever it can be used, it gives hope of the possibility of considerably surpassing the methods of infrared spectroscopy. Its superiority is due to the much greater resolving power, which makes it possible clearly to identify and accurately to measure line intensities. When the absorption width is brought down to 50 kc, it becomes improbable that the lines of any other gases could fall within so narrow a frequency interval, and the line will be detected as an isolated line. When questions of identification arise, there are usually other lines in the same region that can be conveniently measured. The method gives the same confidence in identifying molecules as line emission spectra give for identifying elements. In many cases we

in investigations this method has been used to determine contamination of samples. Before the method can find wide application, however, it is necessary to measure and catalogue a large number of substances. The various groups of researchers now engaged in microwave spectroscopy are already making considerable progress in this enormous task.

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1. Cleeton and Williams. ↩

2. 8. ↩