Radio Spectroscopy of Molecules

V. L. Ginzburg

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
§ 1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
§ 2. The inversion spectrum of ammonia . . . . . . . . . . . . . . . . . . . . . . . . . 327
§ 3. Absorption of a number of gases at atmospheric pressure . . . . . . . . . . . . 333
§ 4. Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
§ 5. Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
§ 6. Zeeman and Stark effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
§ 7. Radio spectroscopy of atoms and atomic nuclei . . . . . . . . . . . . . . . . . . 342

Introduction

Recently, the application of radiophysical methods to the study of molecular terms and the nature of collisions between molecules in a gas has been developing intensively. This new field of application of radio methods may with good reason be called radio spectroscopy of molecules. The general idea of radio spectroscopy is not new and is closely connected with the main direction in which radio waves have been used, and are being used, to study the properties of matter—namely, with measuring, as a function of frequency, the indices of absorption and refraction of radio-range waves in various bodies (superconductors, ferromagnets, ferroelectrics, polar solutions, electrolytes, etc.). Molecular radio spectroscopy likewise amounts, in essence, to measuring the refractive index and, chiefly, the absorption of radio waves in a gas. In this case, however, it is usually easy to pass from such a macroscopic quantity as the absorption coefficient to the properties of an individual molecule. In particular, the natural frequencies of molecular transitions are simply equal to the frequencies of maximum absorption in a sufficiently rarefied gas. The objects of radio spectroscopy are very diverse, since in the radio range lie the frequencies of rotational transitions of most molecules that are not too light, the frequencies corresponding to the fine structure of rotational terms (multiplet splitting, $\Lambda$-doubling, inversion doubling), and also the frequencies corresponding to transitions between sublevels of the Zeeman and Stark splittings of molecular terms. Also included here are the frequencies of transitions between sublevels of the hyperfine structure—

...the structure of atomic terms (in a magnetic field and without a field) and the frequencies of transitions between Zeeman sublevels arising when atomic nuclei with a magnetic moment are placed in a strong magnetic field.

The circumstance that radiospectroscopy has begun to develop intensively—and, in essence, has even come into being—during the last one or two years is connected to a considerable degree with the successes achieved during the war in the technology of centimeter radio waves. In many cases, however, measurements at considerably longer wavelengths are also of interest, and thus radiospectroscopy is by no means confined to the region of microwaves. The prospects opening up before radiospectroscopy, and the results already achieved by it, lead one to think that in the near future we shall witness a rapid development of this whole field.

The present review, which, so far as we know, is the first of its kind, is devoted to an exposition chiefly of the fundamental aspects and the principal results of radiospectroscopic work published in journals that had appeared in Moscow up to January 1947.

§ 1. THEORY

As has already been indicated, radiospectroscopic methods are usually based on measuring the absorption coefficient of a gas \(z\); it is therefore necessary to establish a theoretical formula for this quantity, defined by the relation \(S=S_0 e^{-zx}\), where \(S\) is the flux of radiation propagating along the \(x\)-axis at the point \(x\), and \(S_0\) is this flux at the point \(x=0\)*). From the theory of radiation it is known that, for absorption accompanied by a quantum transition of an atom or molecule from some state 1 with energy \(E_1\) to state 2 with energy \(E_2\), the absorption coefficient is equal to (see \({}^{1}\), § 12, 3):

\[ z(\nu)= \frac{8\pi^2 e^2 |x_{12}|^2 \nu_0 \Delta\nu\, N_1} {3ch\{(\nu-\nu_0)^2+(\Delta\nu)^2\}}, \tag{1} \]

where \(|\mathbf p_{12}|=e|\mathbf x_{12}|\) is the modulus of the matrix element of the dipole moment corresponding to the transition \(1\to2\), \(\nu_0=\nu_{12}=\dfrac{E_2-E_1}{h}\) is the natural frequency of the transition, \(\Delta\nu\) is the half-width of the line \(\left(z(\nu_0+\Delta\nu)=\dfrac{1}{2}z(0)\right)\), and \(N_1\) is the concentration of the molecules under consideration (for definiteness, below we shall speak of molecules) in the lower state 1.

*) The absorption index \(k\) is defined by the relation \(E=E_0 e^{-\frac{2\pi}{\lambda}kx}\), where \(E\) is the field strength and \(\lambda\) is the wavelength in vacuum; hence it is clear that \(z=\dfrac{4\pi}{\lambda}k\). We note that below the absorption coefficient always refers to the case of a plane wave propagating in an unbounded medium, and not, for example, in a waveguide.

In formula (1) it is assumed that the statistical weights of both states are equal to unity and, most importantly, induced emission is not taken into account. The latter consists, as is known, in the fact that, under the influence of the incident wave, molecules in the excited state 2 pass into state 1 with the emission of a quantum \(h\nu\) in the direction of the incident quantum of the same energy \(h\nu\). In optics, under ordinary conditions, induced emission plays no role, since the number of excited molecules (atoms) is negligibly small. On the contrary, in the radio-frequency region, even at room temperatures the numbers of atoms in states 1 and 2, which interested us above, are approximately equal, and thus induced emission (negative absorption) is of decisive importance \(^{2,3}\). Obviously, the presence of this process will lead to a decrease of the absorption coefficient measured experimentally. The ratio of the probabilities of absorption and induced emission is such that if the numbers of molecules in the upper and lower levels were related to one another as the statistical weights of these levels, then the radiation incident on the gas would not be absorbed at all (see [§ 11, 2]). In reality, in the state of thermodynamic equilibrium there are, respectively, on the lower and upper levels,

\(N_1 = Cg_1 e^{-E_1/kT}\) and \(N_2 = Cg_2 e^{-E_2/kT}\)

molecules per \(cm^3\) of gas, where \(C\) is a normalization constant and \(g\) is the statistical weight. Simultaneous allowance for both absorption and induced emission shows that the absorption coefficient observed experimentally is obtained from (1) by replacing \(N_1\) by

\[ \frac{N_1}{g_1} - \frac{N_2}{g_2} = C e^{-\frac{E_1}{kT}} \left(1 - e^{-\frac{E_2-E_1}{kT}}\right) \simeq \frac{N_1}{g_1}\frac{h\nu_0}{kT}, \tag{2} \]

where the last transformation has been made because in the radio region at room temperature one always has

\[ \frac{E_2 - E_1}{kT} = \frac{h\nu_0}{kT} \ll 1. \tag{3} \]

Indeed, for \(T \sim 300^\circ\) abs., \(kT \sim 4 \cdot 10^{-14}\), and thus even for

\[ \nu_0 = 3 \cdot 10^{11} \qquad \left(\lambda_0 = \frac{c}{\nu_0} = 1\,\text{mm}\right), \qquad \frac{h\nu_0}{kT} \sim \frac{1}{20}. \]

In addition, it should be noted that for degenerate levels one must in (1) replace \(|x_{12}|^2\) by \(\sum |x_{12}|^2\), where the summation is carried out over all degenerate sublevels of the initial and final levels\(^1\). As a result of what has been said, under condition (3), we obtain the following final formula for \(\chi\):

\[ \chi = \frac{ 8\pi^2 e^2 \sum |x_{12}|^2 \nu_0 \Delta\nu \,\dfrac{h\nu_0}{kT}\, N_1 }{ 3hc g_1 \left\{(\nu-\nu_0)^2 + (\Delta\nu)^2\right\} } = \frac{ 8\pi^2 e^2 \sum |x_{12}|^2 \nu_0^2 \Delta\nu\, N_1 }{ 3c g_1 kT \left\{(\nu-\nu_0)^2 + (\Delta\nu)^2\right\} }. \tag{4} \]

At the maximum of the line, for \(\nu=\nu_0\):

\[ \chi(\nu_0)\equiv \chi_0 = \frac{8\pi^2 e^2 \sum |\mathbf{x}_{12}|^2 \nu_0 \dfrac{h\nu_0}{kT} N_1} {3hc g_1 \Delta \nu}. \tag{5} \]

If the spectrum of the incident radiation is continuous and does not change over the width of the line, then the total energy absorbed in \(1\ \mathrm{cm}^3\) of gas is equal to

\[ W=\int \chi(\nu) S(\nu)\,d\nu = \frac{8\pi^2 e^2 \sum |\mathbf{x}_{12}|^2 \nu_0^2} {3c g_1 kT}\,S(\nu_0), \tag{6} \]

where \(S(\nu)\,d\nu\) is the energy flux in the frequency interval \(d\nu\). Above it was assumed that the transition under consideration is an electric dipole transition (\(e\mathbf{x}_{12}\) is the matrix element of the electric dipole moment)\({}^{12}\). Meanwhile, under known conditions, substantial and less intense transitions may occur; in optics such transitions are quadrupole and magnetic dipole transitions. As is known, the ratio of the intensity of quadrupole (\(I_{\mathrm{qd}}\)) and magnetic dipole (\(I_{\mathrm{md}}\)) transitions to the intensity of an electric dipole transition (\(I_{\mathrm{ed}}\)) is, in order of magnitude, equal to (the frequencies and other parameters characterizing the transition being regarded as identical):

\[ \frac{I_{\mathrm{qd}}}{I_{\mathrm{ed}}}\sim \left(\frac{a}{\lambda}\right)^2 \sim \frac{10^{-16}}{\lambda^2}, \qquad \frac{I_{\mathrm{md}}}{I_{\mathrm{ed}}}\sim \left(\frac{h}{2\pi mca}\right)^2 \sim 10^{-4}\text{—}10^{-5}, \tag{7} \]

where \(a\) is the radius of the atom, \(\lambda\) the wavelength, and \(\dfrac{eh}{4\pi mc}\) the Bohr magneton.

From (7) it is clear that in the radio region quadrupole transitions are completely negligible (the same applies, of course, to all higher multipole transitions), and only magnetic dipole radiation can play a role. In the latter case all the formulas given remain valid, but with the replacement

\[ |\mathbf{p}_{12}|^2 \equiv e|\mathbf{x}_{12}|^2 \quad \text{by} \quad |\boldsymbol{\mu}_{12}|^2, \tag{8} \]

where \(|\boldsymbol{\mu}_{12}|^2\) is the square of the modulus of the matrix element of the magnetic moment.

In considering absorption we have taken into account above only the contribution to the absorption of one transition (the transition with frequency \(\nu_{12}=\nu_0\)); moreover, formulas (1), (4), and (5) are valid, generally speaking, only if the inequalities

\[ \left. \begin{array}{c} |\nu-\nu_0|\ll \nu_0,\\ \nu_0\gg \Delta\nu \end{array} \right\} \tag{9} \]

are simultaneously satisfied.

Taking into account the role of other lines besides the one under consideration, and calculating the absorption when inequalities (9) are not satisfied (nonresonant absorption), have not yet been carried out correctly and require detailed

analysis (in this direction see,⁴ and also the results of an unpublished work by Van Vleck cited in²³). At the same time, for the determination by absorption of the proper frequency of an individual molecule, and in general for the principal radiospectroscopic purposes, the case of a gas at sufficiently low pressure is of interest, when the second of inequalities (9) is satisfied, and it is quite sufficient to restrict oneself to the frequency region where the first of these inequalities holds. Under these same conditions the frequency dependence of the absorption associated with other lines may be neglected, and thus the contribution of all these lines reduces to a constant absorption which must be added to the value of \(\chi(\nu)\) obtained above. The magnitude of this constant absorption can be determined with any reliability only from experiment (see, for example, the results of the work²³).

Of the quantities entering expression (4) for the absorption coefficient \(\nu_0\), \(|x_{12}|^2\), or \(|\mu_{12}|^2\), and \(g_1\) are completely determined by the molecule itself (this is true at not too high pressures, as is assumed). The number of molecules \(N_1\) in level 1 is equal to

\[ N_1=\delta N=\delta \frac{P}{kT}, \tag{10} \]

where \(\delta\) is the relative number of molecules in level 1, determined solely by the system of molecular levels and by the temperature \(T\), and \(N\) is the total concentration of molecules at the given pressure \(P\) (the gas is assumed ideal).

The half-width of a line \(\Delta \nu\), generally speaking, is determined by a number of causes: natural and Doppler broadenings and the broadening associated with interaction between molecules (see, for example,⁵˒⁶ and the recent work in the field of the theory of line width⁷). It is easy to see that in the radio range natural broadening may always be neglected. Indeed, the half-width associated with it is

\[ \Delta \nu_e \sim \frac{e^2}{mc^3}\nu^2 \sim \frac{10^{-2}}{\lambda_0^2}, \]

i.e., in the radio range it is less than \(1\) cps. The Doppler half-width*)

\[ \Delta \nu_d=\sqrt{\ln 2}\,\frac{\nu_0}{c}\sqrt{\frac{2kT}{M}}\sim \frac{\overline{v}}{\lambda_0}, \tag{11} \]

where \(M\) is the mass of the molecule and \(\overline{v}\) is its mean velocity. For \(\lambda_0\sim 1\) cm and room temperature \(\Delta \nu_d\sim 10^4—10^5\) cps and, as will be seen below [see (14)], is smaller than the half-width associated with collisions at pressures \(P \gtrsim 10^{-2}\) mm Hg.

The question of line broadening associated with molecular interaction belongs among the very complicated and unresolved in

*) Doppler broadening leads, as is well known, not to the dispersion form of the line (4), but to its exponential form, and thus the half-width given for orientation, of course, cannot be used in a formula of type (4).

to a sufficient degree; under the specific conditions of the radio range this question has not yet been investigated theoretically; it is also closely connected with the extension of the theory to the region where inequalities (9) are not satisfied. However, in the region where inequalities (9) are satisfied and the gas pressure is small, one may apparently assume that the broadening associated with molecular interaction has a collisional character (see ⁶). This is in any case true if the broadening is caused by the influence of a foreign gas that has no dipole moment (for example, the broadening of the inversion lines of $\mathrm{NH}_3$ as a result of increasing the pressure of air admixed with the ammonia; see § 2). If, however, the broadening is caused by a dipole gas and especially by the gas under investigation itself (i.e., broadening is observed in a gas without admixtures under the influence of an increase in the pressure of this gas), as occurs in most cases, the situation may become more complicated. Since, as indicated, this question has not been investigated, we shall restrict ourselves to considering only collisional damping in its usual interpretation ⁶. In this case, if the broadening is due to an admixture of a foreign gas, the half-width of the line (due solely to collisional damping) is equal to

\[ \Delta \nu_{\mathrm{coll}}=\frac{1}{2\pi\tau}=\frac{1}{2\pi}\,\pi d_{ab}^{2}N_b\overline{v}_{ab}, \tag{12} \]

where $\tau$ is the mean free time, $d_{ab}$ is the distance between the centers of the colliding molecules $a$ and $b$, $\overline{v}_{ab}$ is the arithmetic mean relative velocity of these molecules, and $N_b$ is the concentration of molecules of the foreign (“quenching”) gas, which in (12) is assumed to be considerably greater than the concentration of the molecules of the absorbing gas $N_a$. If the molecular masses are $M_a$ and $M_b$, then

\[ \overline{v}_{ab}=\sqrt{\frac{8kT\,(M_a+M_b)}{\pi M_aM_b}}. \]

In the case of a single gas (without admixtures), formula (12) takes the form

\[ \Delta \nu_{\mathrm{coll}}=\frac{1}{2\pi}\cdot \sqrt{2}\pi d^{2}N\overline{v};\qquad \overline{v}=\sqrt{\frac{8kT}{\pi M}}, \tag{13} \]

where $d$ is the molecular diameter effective for the given process, $N$ is the molecular concentration, and $\overline{v}$ is the arithmetic mean velocity. By definition, $\pi d^2=\sigma$ is the effective cross section for collisions leading to broadening of the line. In order of magnitude $\sigma$ is greater than or equal to the corresponding gas-kinetic cross section $\sigma_0$. The latter, at atmospheric pressure and $T\sim 300^\circ$ abs., is equal to

\[ \sigma_0=\pi d_0^2\simeq 2\cdot 10^{-15}\ \mathrm{cm}^2 \]

(for air), while

\[ \frac{1}{\tau_0}\sim 10^{10}. \]

Hence it is clear that

at atmospheric pressure of the absorbing gas *)

\[ \Delta \nu_{\mathrm{coll}} \gtrsim 2 \cdot 10^{9}. \tag{14} \]

From (12) and (13) it follows that

\[ \Delta \nu = C(T)P, \tag{15} \]

where \(C\) is some function of temperature and \(P\) is the pressure. The range of applicability of formula (15), however, is wider than that of formulas (12)—(13) themselves, owing to which the index “coll” on \(\Delta \nu\) has been omitted. If the line shape determined experimentally has the dispersion character (4) and \(\Delta \nu\) is proportional to the pressure, then one can always formally determine the effective cross section \(\sigma\) by means of (12)—(13), independently of whether these formulas themselves have a deeper basis. If relation (15) is valid, then, as is clear from (5) and (10), in the state of thermodynamic equilibrium the maximum absorption does not depend on the pressure, i.e.

\[ \chi(\nu_0)=\chi_0=f(T). \tag{16} \]

Relation (16) applies, of course, only to a pure gas. If, however, the pressure changes owing to a nonabsorbing gas, then \(N_1=\mathrm{const.}\), \(\Delta \nu \sim P\), and \(\chi_0 \sim \dfrac{1}{P}\).

The change \(\Delta n\) of the refractive index \(n\) of the gas near the given line is equal to (it is assumed that \(n-1 \ll 1\))

\[ \Delta n = \frac{2\pi e^2 \sum |x_{12}|^2(\nu_0-\nu)\nu_0 N_1} {3kT \cdot \xi_1\{(\nu-\nu_0)^2+(\Delta \nu)^2\}} . \tag{17} \]

The quantity \(\Delta n\) in a gas is small and is of no special interest.

Let us note that, for all known objects of radiospectroscopic interest, the transition matrix elements differ from zero only if the molecule has a permanent electric or permanent magnetic moment (this statement, as is well known, is valid for all purely rotational transitions, whose frequency in fact usually lies in the radio range).

From what has been said it is clear that measurement of the absorption coefficient as a function of frequency makes it possible to determine \(\nu_0\), \(\Delta \nu\), and also

*) Let us note that the frequency \(\Delta \nu\), according to (14), itself lies in the radio region of interest for investigation. In this connection, nonresonant absorption in the frequency region \(\nu \sim \Delta \nu\) may be connected not only with the “tails” of various lines, but also with a change in the state of the molecule within the limits of the given broadened level. In other words, nonresonant absorption may be due both to collisions in which the molecule passes from one level to another, and to collisions not accompanied by such a transition but occurring with some change in the kinetic energy of the colliding particles.

one of the quantities $\sum |x_{12}|^2$, $N_1$ and $g_1$ when the other two are known. Whereas the measurement of $\nu_0$ is of predominantly spectroscopic interest, the determination of $\Delta \nu$ as a function of the pressure, temperature, and kind of “quenching” gas may serve as a method for studying molecular collisions. In the following paragraphs the corresponding experimental material will be considered.

§ 2. INVERSION SPECTRUM OF AMMONIA

The inversion spectrum of ammonia has been investigated most fully by radio-spectroscopic methods$^{8-15}$. The molecule $\mathrm{NH_3}$ is a regular pyramid (Fig. 1) and, with respect to its rotational spectrum, belongs to the type of a symmetric top. In this case, as is known,$^{16}$ the wave number $\tilde{\nu}=\nu/c$ of the rotational terms is equal to

\[ \tilde{\nu}=BJ(J+1)-(A-B)K^2,\qquad B=\frac{h}{8\pi^2cI_B},\qquad A=\frac{h}{8\pi^2cI_A}, \tag{18} \]

where $J$ and $K$ are integers ($J \geq |K|$) determining the total angular momentum of the molecule and its projection on the axis of symmetry, and $I_A$ and $I_B$ are the moments of inertia, respectively along the molecular axis and in the perpendicular direction; for $\mathrm{NH_3}$, $B=9.96\ \mathrm{cm}^{-1}$ and $A=6.29\ \mathrm{cm}^{-1}$. States with $|K|>0$ are doubly degenerate, corresponding to the two possible mutually opposite directions of the component of the total angular momentum along the molecular axis. This degeneracy, however, is removed if one takes into account the possibility of inversion of the molecule, i.e., in the case of $\mathrm{NH_3}$, the transition of the N atom through the plane formed by the H atoms into the mirror-symmetric position. The removal of the degeneracy and the splitting of the level into two close sublevels, i.e., the inversion doubling of the terms, is especially easy to understand by considering the motion of the N atom in the direction perpendicular to the plane $\mathrm{H_3}$ (such motion is very close to being realized in the normal, fully symmetric vibration of the molecule $\nu_2$). In this case the potential energy of the molecule has the form shown in Fig. 2. At the lower vibrational levels, oscillations occur about each equilibrium position approximately as they would if the other equilibrium position did not exist. However, the nonzero probability of passage of the N nucleus through the potential barrier (by virtue of the “tunnel effect”) leads to splitting of the levels. The situation here is analogous to the splitting of frequencies in the case of two weakly coupled identical pendulums. Calcul-

Fig. 1.

measurements, which have only a very approximate quantitative significance (see 14, 15, and 16, p. 220), show that the inversion splitting \(\Delta \tilde{\nu}\) increases sharply with increasing vibrational quantum number \(v\); for the vibration \(\nu_2\), at \(v=0\), \(\Delta \tilde{\nu}\simeq 0.7\ \mathrm{cm}^{-1}\), for \(v=1\), \(\Delta \tilde{\nu}\simeq 36\ \mathrm{cm}^{-1}\), and for \(v=2\), \(\Delta \tilde{\nu}\simeq 312\ \mathrm{cm}^{-1}\). In the case of other normal vibrations the splitting is considerably smaller (for example, for the vibration \(\nu_1\), \(\Delta \tilde{\nu}\simeq 1\ \mathrm{cm}^{-1}\) at \(v=1\)), which is quite understandable, since in this case the distance of the atom N from the plane \(\mathrm{H}_3\) changes only weakly.

Fig. 2

The inversion splitting depends not only on \(v\), but also on the quantum numbers \(J\) and \(K\), since the potential energy of the molecule depends on the state of rotation (the influence of the centrifugal force). Furthermore, as can be seen from the selection rules (see 16, p. 256), in the presence in the molecule of a constant dipole moment (for \(\mathrm{NH}_3\), \(p_0=1.46\cdot 10^{-18}\); see 17), radiative transitions are possible between the two levels of the inversion doublet. Thus there must exist a purely inversion spectrum of the \(\mathrm{NH}_3\) molecule. At room temperature the majority of molecules are in the lower vibrational level, and together with this only the frequencies of the inversion spectrum corresponding to this level lie in the radio range. The splitting mentioned, \(\Delta \tilde{\nu}=0.7\ \mathrm{cm}^{-1}\), corresponds to a radio-wave wavelength \(\lambda=1.43\ \mathrm{cm}\); as experiment has shown, for \(\mathrm{NH}_3\) at \(v=0\), \(\Delta \tilde{\nu}\simeq 0.8\ \mathrm{cm}^{-1}\), and thus \(\lambda\simeq 1.25\ \mathrm{cm}\).

In this example the role of radiospectroscopy is clearly seen as a method supplementing optical spectroscopy. In the optical region the inversion splitting of the ground level of the molecule could in principle be observed only as a small effect superposed on the general vibrational-rotational spectrum; in the radio region, however, only the single transition of interest to us is observed in pure form, which leads to immeasurably greater possibilities. At the same time it should be emphasized that the radio method, based on absorption measurements, is limited to the study of the ground electronic and usually also vibrational states.

The first measurements of absorption in ammonia were carried out as early as 1934 8 at atmospheric pressure. In this case the lines corresponding to different values of \(J\) and \(K\) are not resolved because of their broadening, and a single band is observed with a maximum at \(\lambda=1.25\ \mathrm{cm}\).

\((\Delta \nu = 0.8\ \mathrm{cm}^{-1})\); at the maximum \(\chi_0 = 8.3 \cdot 10^{-5}\ \mathrm{cm}^{-1}\), i.e. \(\dfrac{1}{\chi_0} = 1.2\ \mathrm{m}\). According to more recent measurements\(^{9}\), \(\dfrac{1}{\chi_0} = 1.28\ \mathrm{m}\). The absorption is half of the maximum at \(\lambda = 1\ \mathrm{cm}\) and \(\lambda = 1.5\ \mathrm{cm}\). The method of measuring \(\chi\) at high pressure (of the order of atmospheric) is in principle very simple (Fig. 3); here the matter is reduced to measuring the attenuation of the field of a wave propagating in a waveguide filled with the gas under investigation (for conversion of the absorption in a waveguide to the absorption in an unbounded medium, see \(^{9}\)). The use of a waveguide up to \(10\ \mathrm{m}\) long made it possible\(^{9}\) to measure by this method values of \(\chi\) greater than \(5 \cdot 10^{-5}\ \mathrm{cm}^{-1}\).

Fig. 3.
Labels in the diagram: Generator; Waveguide; gas under investigation; dielectric “lens”; receiver.

At low pressure, when individual lines of the inversion spectrum are resolved, it is necessary to use more sensitive setups\(^{10,11,12,8,11}\); for example, a kind of “bridge” or null method is used, consisting in the following: the radio wave from the generator is split into two streams and travels along two waveguides into a device that makes it possible to measure the difference in the intensities of the two waves. In one of the waveguides there is the gas under investigation, and in the other a calibrated attenuator; in this way, by making the instrument indicate zero difference voltage, one can determine the absorption in the gas. The difference voltage may also be fed to an oscillograph. Another method\(^{10}\) is based on measuring the damping of a resonator filled with gas. These methods made it possible to measure values of \(\chi\) greater than approximately \(2 \cdot 10^{-6}\ \mathrm{cm}^{-1}\). The frequency dependence of \(\chi\) is recorded by changing the frequency of the generator.

Fig. 4.
Vertical axis: absorption coefficient. Horizontal axis: \(\mathrm{cm}^{-1}\).

On the instrumental side of the measurements we shall not dwell in greater detail. At reduced pressure the individual lines are resolved and a spectrum is obtained, the principal part of which is shown in Fig. 41. An oscillographic recording of the absorption coefficient for the line \(J=K=3\) at various pressures is presented in Figs. 5 and 62.

\(p=5\times10^{-2}\,\mathrm{mm\,Hg}\)    \(p=2.5\times10^{-2}\,\mathrm{mm\,Hg}\)

Fig. 5.

The presence of fine structure, which in principle could be due to the presence of magnetic and quadrupole electric moments in the nuclei, noticeable in Fig. 6, was not found in another work3.

\(p=1.5\times10^{-2}\,\mathrm{mm\,Hg}\)    \(p=9\times10^{-3}\,\mathrm{mm\,Hg}\)

Fig. 6.

Measurement of the positions of the absorption maxima led to the following empirical dependence for the wave number of the various lines2:

\[ \tilde{\nu} =0.79347-0.005048\,(J^{2}+J)+0.007040\,K^{2} \]
\[ +0.00001546\,(J^{2}+J)^{2} -0.00004260\,(J^{2}+J)K^{2} +0.0002920\,K^{4}. \tag{19} \]

The accuracy of the experiments does not permit real significance to be attached to all the significant figures quoted. However, the agreement between the result (19) and the values given in two other works1,3 is, on the whole, good \((\tilde{\nu}=0.7940-0.00505\,(J^{2}+J)+\ldots\) in 1 and \(\tilde{\nu}=0.79357-\)

\(-0.005016(J^2+J)+\ldots B^{12}\). Theoretical formulas for the frequency of the splitting^14,15,12 cannot give an accuracy comparable with experiment, since the magnitude of the splitting (the distance between the levels of the inversion doublet) is determined primarily by the character of the potential curve in Fig. 2, which is not known in detail; thus, calculations without any special adjustment give the coefficient \(-0.00330\) for the term with \((J^2+J)\) in (19), instead of the measured value \(0.0050\). Thus, conversely, measurements of \(\nu\) make it possible in principle to refine the value of the potential energy.

As is clear from Figs. 5 and 6, the absorption at the maximum \(z_0\) depends on the pressure, in contradiction with relation (16). Here, however, the point apparently is that under the experimental conditions thermodynamic equilibrium is disturbed^12, on the assumption of which the formulas of § 1 were obtained. The disturbance of equilibrium is connected with the fact that, at sufficiently high power of the radio radiation, so many molecules are “thrown” onto the upper excited level that they do not have time to “fall” back, and thus formula (2) becomes invalid. As a result, \(z_0\) depends on the power of the absorbed radio waves, and only at sufficiently low power, depending on the pressure, is the condition of equilibrium observed and should the value of \(z_0\) not depend on the pressure.

Fig. 7.

And indeed, if the above is taken into account, it follows from experiment that relation (16) is valid^12 (Fig. 7). Formula (15) is also observed experimentally^12, at least at not very low pressures.

Fig. 8.

Calculation of the effective cross section from the measured values \(\Delta \nu\) leads to the conclusion that this cross section for \(\mathrm{NH_3}\) is 10–20 times greater than the gas-kinetic one^10,12 (\(d=9\text{–}14\ \text{Å}\), whereas \(d_0=3\ \text{Å}\)). Such a result is quite understandable, since for a noticeable change in the momentum of molecules they must approach one another much more closely than is required to transfer one of them from one of the rotational or inversion levels to another. In

in this case is essential that a collision of two identical dipole molecules occurs (in the present case, molecules of $\mathrm{NH}_3$). If, however, an $\mathrm{NH}_3$ molecule collides with an argon atom ${}^{12}$ or with an oxygen molecule ${}^{9}$, then the cross section appearing in formula (12) proves to be of the order of the gas-kinetic one. The significantly greater effectiveness, for the broadening of lines, of $\mathrm{NH}_3-\mathrm{NH}_3$ collisions as compared with $\mathrm{NH}_3-\mathrm{O}_2$ collisions is evident from Fig. 8 ${}^{9}$ for the absorption coefficient of mixtures of $\mathrm{NH}_3$ with $\mathrm{O}_2$. From what has been said, the great possibilities opened up by the study of molecular collisions by measuring the widths of absorption lines are clear.

Formula (4) for $z(\nu)$ with respect to the line shape (the dispersion distribution) is in agreement with experiment (see Fig. 7). As for comparison of the absolute value $z_0$, measured experimentally, with formula (5), there is not sufficient clarity here. For example, in ${}^{12}$ the formula for $z$ is erroneously overestimated by a factor of two and the calculation of $|x_{12}|^2$ was apparently carried out incorrectly. However, as to order of magnitude, formulas (4)—(5) agree with experiment. It should be noted that comparison of the calculated and measured values of $z_0$ is valuable in that it makes it possible to determine whether the condition of equilibrium is satisfied and, if equilibrium is present, to determine $|x_{12}|^2$, if this quantity is unknown. In doing this, the nonresonance absorption must be taken into account in $z_0$ (see § 1). Besides $\mathrm{NH}_3$, the inversion spectrum can be investigated in the isotopic molecules $\mathrm{NH}_2\mathrm{D}$, $\mathrm{NHD}_2$, and $\mathrm{ND}_3$. The corresponding measurements are not yet known. Another possible object is $\mathrm{PH}_3$; calculations, of a very approximate character (${}^{16}$, p. 220), show that in this case $\Delta \nu \sim 1.5 \cdot 10^{-4}\ \mathrm{cm}^{-1}$ (in the lower vibrational state). Such a splitting corresponds to a wavelength $\lambda \sim 60\ \mathrm{m}$, and thus a different method is needed than in the case of $\mathrm{NH}_3$; let us also note that, as is clear from (4), $z \sim \nu_0^2$, and thus, other conditions being equal, $z$ falls strongly with increasing wavelength of the absorbed radio waves. Inversion doubling may also occur in the molecule $\mathrm{H}_2\mathrm{O}_2$, if it is non-planar (see ${}^{16}$, pp. 224 and 302). A splitting of lines very closely related to inversion doubling occurs when there are several energy minima upon rotation of one part of a molecule relative to another through different angles; this occurs, for example, for the molecule $\mathrm{CH}_3\mathrm{OH}$ ($p_0 = 1.68 \cdot 10^{-18}$; in the ground state $\Delta \nu \approx 1.5\ \mathrm{cm}^{-1}$; see ${}^{16}$, p. 225). In complex organic molecules, doublings of the indicated type may apparently occur very often. Another, less interesting case of doubling is the so-called $\Lambda$-doubling (${}^{18}$, p. 173), which occurs in the case of diatomic molecules whose projection of the orbital moment on the molecular axis, i.e. the value $\Lambda$, differs from zero ($\Pi$, $\Delta$, etc. states). In the ground electronic state $\Lambda \ne 0$ only for a limited number of molecules ($\mathrm{CCl}$, $\mathrm{OH}$, $\mathrm{NO}$, $\mathrm{CH}$, $\mathrm{SiH}$, etc.)

The distance between the split sublevels lies in the radio region, and its measurement is one of the tasks of radiospectroscopy.

§ 3. ABSORPTION OF A SERIES OF GASES AT ATMOSPHERIC PRESSURE

Apart from \(NH_3\), at reduced pressure absorption was studied only in \(H_2O\) and OCS (see \(^{19,20}\) and §§ 4, 6). At the same time, at atmospheric pressure the absorption and refractive index of a whole series of gases\(^{2}\) were measured at \(\lambda = 1.25\ \text{cm}\) with an apparatus of the type shown in Fig. 3\(^*\). The corresponding results are given in Table 1.

Table 1

The accuracy of the method used is such that it does not make it possible to detect reliably an absorption to which a value of \(\chi\) less than \(0.5 \cdot 10^{-4}\ \text{cm}^{-1}\) corresponds. With this accuracy, at the wavelength \(1.25\ \text{cm}\) no absorption was observed in the following gases: hydrogen, oxygen, nitrogen, carbon monoxide, carbon dioxide, ethylene, acetylene, nitrous oxide, methane, ethane, propane, propylene, \(n\)-butane, isobutane, butene-1, butene-2, isobutylene, butadiene, and trimethylamine. All the absorbing gases about which we were able to find information have a permanent dipole moment, whereas most of the nonabsorbing ones do not. However, the character of the transitions with which the absorption is associated has not yet been considered in individual cases. The presence of noticeable absorption in a large number of substances at an arbitrarily chosen wavelength \(\lambda = 1.25\ \text{cm}\) even causes some surprise. It should be borne in mind, however, that at atmospheric pressure the lines are so broad that absorption can be noticeable at a large distance from the line maximum. The author points out\(^{9}\), among other things,

\(^*\) The refractive index is determined by measuring the wavelength in the gas, which, as is known, is equal to \(\dfrac{\lambda}{n}\), where \(\lambda\) is the wavelength in vacuum.

that the absorption in CH$_3$F, CH$_3$Cl, and CH$_3$Br is approximately 10 times greater than expected, and also makes the assumption that in these substances there occurs some previously unnoticed splitting of terms of the inversion-doubling type, of which there can be no doubt.

Thus, in one way or another, both a theoretical analysis of the question (an approximate calculation of $\chi_0$ and $\gamma_0$ for a number of substances) and further experimental work (measurement of $\chi$ at low pressure and at different frequencies) are urgently needed. There is already an indication in the literature of an as yet unpublished theoretical analysis of the absorption spectrum of a number of gases$^{21}$.

§ 4. WATER

From the standpoint of a whole series of applications of microwaves, and above all their use for radiolocation, the absorption of radio radiation in air is of special importance. The composition of air near the surface of the earth includes, in appreciable quantities, the molecules and atoms N$_2$, O$_2$, H$_2$O, CO$_2$, H$_2$, Ar, Ne, and He. Of all these molecules and atoms, only the H$_2$O molecule possesses a permanent electric dipole moment, and only the O$_2$ molecule possesses a permanent magnetic moment. Since the radiation transitions of rotational type that are important in the radio region can occur only in the presence of permanent moments, it is clear that only water vapor and oxygen can absorb in air*). For this reason, the absorption of microwaves in water$^{22,23,19}$ and in oxygen$^{24}$ has been subjected to very detailed investigation. In this paragraph we shall dwell on the absorption of water.

The H$_2$O molecule belongs to the type of an asymmetric top; its moments of inertia are $I_A = 1.023 \cdot 10^{-40}$, $I_B = 1.94 \cdot 10^{-40}$, and $I_C = 2.944 \cdot 10^{-40}$, while the permanent electric moment is directed along the axis to which the mean moment of inertia corresponds, and is equal to $p_0 = 1.84 \cdot 10^{-18}$ (see $^{25,16}$). The H$_2$O molecule has a whole series of intense absorption lines lying in the region of wavelengths shorter than 1 mm, i.e. in the terminology now in use, in the infrared region. However, there are also allowed transitions corresponding to longer wavelengths. Thus, according to the data given in $^{25}$ (see p. 189), the allowed transition $(+--6_{-5}) \leftrightarrow (--+5_{-1})$ corresponds to the wavelength $\lambda = 1.27$ cm ($\nu = 0.78$), the transition $(---3_{-2}) \leftrightarrow (+++2_2)$ to the wavelength $\lambda = 1.62$ mm, and the transition $(--+10_{-7}) \leftrightarrow (+--9_{-3})$ to the wavelength $\lambda \simeq 1$ mm. The first of the lines cited, lying in a readily accessible and interesting region, has been studied in detail.

In the case of pure water vapor, measurements can be made by means of the methods$^{10,11,12}$ mentioned in § 2, as was done$^{19}$. The corresponding data, however, are still very incomplete and

* The discussion, of course, concerns only pure air, without condensed water vapor, dust, etc.

of them will be discussed below. The practically important case of water vapor in air at atmospheric pressure has been studied in detail ^{22, 23}. In this case the value \(z\) is very small, since the value \(\Delta \nu\) is mainly proportional to the air pressure, while \(N_1\) is proportional to the partial pressure of the water vapor. One of the methods used ^{22} consisted in measuring the quality factor of a large resonator (volume \(15\ m^3\)) as a function of the pressure of the water vapor ^{22, 26}. The results of the measurements are given in Table 2, where \(\rho\) is the amount of water in grams per cubic meter of air (atmospheric pressure, \(T = 318^\circ\) abs.); the value \(\rho = 0\) refers to the case when \(\rho\) is small and approaches zero. The values \(\frac{\chi}{\rho}\) are given in the table in \(\frac{db}{km}\) (decibels per kilometer) per \(\frac{g}{m^3}\). To convert the values in \(\frac{db}{km}\) to \(cm^{-1}\), or, as is written in the Anglo-American literature, to \(\frac{\text{nepers}}{cm}\), one must multiply the value in \(\frac{db}{km}\) by \(0.2303 \cdot 10^{-5}\). The data of Table 2 are also shown in Fig. 9. The value of \(\tilde{\nu}_0\) in the experiment is equal to \(0.744\ cm^{-1}\) \((\lambda_0 = 1.34\ cm)\), whereas from the optical data the value \(\tilde{\nu}_0 = 1.27\ cm^{-1}\) followed (see above); here, however, there is no contradiction, since the accuracy of the optical data in this case is not great. At the maximum:

\[ \frac{z_0}{\rho} = 0.025\ \frac{db}{km}\ \text{per}\ \frac{g}{m^3} = 5.8 \cdot 10^8\ cm^{-1}\ \text{per}\ \frac{g}{m^3}. \]

Fig. 9.

Usually in air \(\rho \sim 10\) (this corresponds to \(\sim 1\%\) water molecules in the air in comparison with the total number of air molecules), and thus \(z_0 \simeq 6 \cdot 10^{-7}\ cm^{-1}\), i.e. the intensity of the wave decreases by a factor \(e\) over a path of \(\simeq 17\ km\). The line width in this case is so considerable that the absorption is large over a wide frequency interval near the maximum (see Fig. 9). For \(\rho = 0\), \(\Delta \tilde{\nu} = \frac{\Delta \nu}{c} = 0.087\ cm^{-1}\); for \(\rho = 50\), \(\tilde{\nu}_0 = 0.742\) (instead of \(0.744\ cm^{-1}\) at \(\rho = 0\)) and \(\Delta \tilde{\nu} = 0.107\ cm^{-1}\). The dependence of \(\frac{\chi}{\rho}\) and \(\Delta \tilde{\nu}\) on \(\rho\) is explained by the role of collisions between molecu-

Table 2

Values of \(\dfrac{z}{\rho}\)

cules of water, which, evidently, increases with increasing \(\rho\). These collisions have a larger effective cross section than the collision of water molecules with air molecules. From the measured values of \(\tilde{\Delta}\), it follows that \(\sigma_{\mathrm{H_2O,H_2O}} = 4.7\,\sigma_{\mathrm{H_2O,air}}\). This result is quite analogous to that discussed in § 2 for ammonia. The authors\(^{22}\) compare the measured values of \(\dfrac{z}{\rho}\) with those calculated by Van Vleck in an unpublished work, and give the following theoretical formula:

\[ \frac{z}{\rho} = C_{1}\tilde{\nu}^{2} \left\{ \frac{\Delta\tilde{\nu}} {(\tilde{\nu}-\tilde{\nu}_{0})^{2}+(\Delta\tilde{\nu})^{2}} + \frac{\Delta\tilde{\nu}} {(\tilde{\nu}+\tilde{\nu}_{0})^{2}+(\Delta\tilde{\nu})^{2}} \right\} + C_{2}\tilde{\nu}^{2}\Delta\tilde{\nu}. \tag{20} \]

As for the values of \(C_{1}\) and \(C_{2}\), which are not given, \(C_{1}\) can be determined by comparing (20) with (4) under condition (9), when the second term in the braces is not substantial (see also\(^{4}\)). In (20) the term proportional to \(C_{2}\) should take into account the nonresonant absorption associated with the presence of absorption lines in the infrared region. As is clear from Fig. 10 and from the more detailed analysis\(^{22}\), the term with \(C_{1}\) in (20) approximately corresponds to experiment, whereas the value of \(C_{2}\) calculated by Van Vleck is erroneous and, as it turns out, in order to obtain agreement with experiment it must be multiplied by a coefficient approximately equal to 5. In accordance with § 1, this indicates that nonresonant absorption can at present be reliably obtained only from experiment. The question of whether the use of a formula of the type

for (20), a real advantage in comparison with using formula (4) with an added constant term remains unclear; it seems to us that such an advantage hardly exists, at any rate from the theoretical point of view.

Measurements on pairs of water without air, of which only a brief report has appeared \({}^{19}\), led to the value \(\nu_0 = 22\,309 \pm 5\) Mc/sec (\(\tilde{\nu}_0 = 7436\)) for \(\mathrm{H_2O}\) and to the value \(\nu_0 = 22\,237 \pm 5\) Mc/sec for the same line in \(\mathrm{HDO}\)*).

Absorption in air, caused by the presence of water vapor, has also been investigated \({}^{27}\) by means of a microwave radiometer (microradiometer). The principle of the method is that, according to Kirchhoff’s theorem, a gas emits precisely those frequencies which it absorbs; therefore measurement of the thermal radiation of the atmosphere by means of a microradiometer makes it possible to determine the absorption of radio waves in the atmosphere (we shall not dwell in detail on the theory of the method, see \({}^{23}\)). The results of this work are in agreement with those given above. We note that the sensitivity of the microradiometer is so high that, despite the impossibility of obtaining intense radio emission from molecules at \(\lambda > 1\) mm (see \({}^{28}\)), the study of emission radio spectra of molecules is entirely possible and may even be of applied interest (the creation of frequency standards in the millimeter and centimeter range \({}^{28}\)).

Fig. 10.

§ 5. OXYGEN

The oxygen molecule, \(\mathrm{O_2}\), is one of the few which in the ground electronic state have a magnetic moment (among diatomic molecules, besides \(\mathrm{O_2}\), only the molecule \(\mathrm{NO}\) has a magnetic moment; among polyatomic molecules, \(\mathrm{NO_2}\), \(\mathrm{ClO_3}\), and some others). The moment of the \(\mathrm{O_2}\) molecule is equal to two magnetons, i.e. \(e\hbar/mc\) (more precisely, the spin of the \(\mathrm{O_2}\) molecule is equal to \(\hbar\), since the mean magnetic moment \(\leq e\hbar/mc\)). The ground term of the molecule has the structure \({}^{3}\Sigma_g^{-}\), and its rotational levels are cha—

*) In note \({}^{19}\) there is some confusion in the table; we proceed from the fact that the figures attributed to HDO in reality refer to \(\mathrm{H_2O}\).

are characterized by the quantum number of the total angular momentum \(J=1,2,3\ldots\) and by the rotational quantum number \(K=0,1,2,\ldots\). To all values of \(K\), except \(K=0\), there correspond three terms with \(J=K,\ J=K\pm1\). The multiplet structure for small \(K\) belongs to an intermediate type between Hund’s cases \(a\) and \(b\); for large \(K\) the case \(b\) may be regarded as applying. The energy of the terms is determined by the formula\({}^{29}\)

\[ \left. \begin{aligned} \tilde{\nu}_{k+1} &=\nu_k+(2K+3)B-\lambda-\bigl[(2K+3)^2B^2+\lambda^2-2\lambda B\bigr]^{\frac12} +\mu(K+1),\\ \nu_k &=BK(K+1)=\frac{h^2}{8\pi^2 I}\,K(K+1),\\ \nu_{k-1} &=\nu_k-(2K-1)B-\lambda\\ &\quad-\bigl[(2K-1)^2B^2+\lambda^2-2\lambda B\bigr]^{\frac12}-\mu K, \end{aligned} \right\} \tag{21} \]

where

\[ B=1.438\ \mathrm{cm}^{-1},\quad \lambda=1.985\ \mathrm{cm}^{-1}\quad \text{and}\quad \mu=-0.008\ \mathrm{cm}^{-1}. \]

The selection rules for magnetic dipole radiation are as follows: \(\Delta J=0,\pm1\), while the parity, sign, and symmetry of the term do not change\({}^{18}\). In the case of the \(\mathrm{O}_2\) molecule, terms with even \(K\) are antisymmetric, negative, and even, while for odd \(K\) the terms are symmetric, positive, and odd\({}^{18}\).

It follows from what has been said that only transitions with \(\Delta K=0,\pm2,\pm4\ldots\) are possible. To all lines with \(|\Delta K|\ge2\) there correspond wavelengths

Fig. 11.

\[ \lambda<1.6\ \mathrm{mm}, \]
and these lines are at considerable distances from one another. On the contrary, in the case \(\Delta K=0\) there is an entire band consisting of lines with \(\Delta J=\pm1\) (transitions from the term \(J=K\) to the terms with \(J=K\pm1\), and conversely), and situated in the region

\[ \lambda\sim0.5\ \mathrm{cm}\quad (\tilde{\nu}\sim2\ \mathrm{cm}^{-1}). \]

The point is that the frequency corresponding to the indicated transitions in general depends only very weakly on \(K\)—this is evident from Fig. 11 (see\({}^{30}\) and formula (21), and also Fig. 1 in\({}^{29}\)). Practically the wavelengths of almost all the lines of this band lie in the interval

\[ 0.45<\lambda<0.55\ \mathrm{cm}. \]

At room temperature the excita-

a whole series of rotational terms of the molecule \(O_2\)*), and therefore in experiment the entire band should appear at once, which at atmospheric pressure should not be resolved into separate lines. Absorption in \(O_2\) was discovered and investigated only in one paper \(^{24}\), and only at atmospheric pressure. The measurements were made in a waveguide; the generator was a one-centimeter klystron, the frequency of which was doubled by a crystal detector. The values of \(\chi\) are given in Table 3.

Table 3

At the maximum, for \(\lambda \simeq 0.5\) cm, \(\chi = 14.8 \dfrac{\mathrm{db}}{\mathrm{km}} = 3.4 \cdot 10^{-5}\ \mathrm{cm}^{-1}\) for air and \(\chi = 70.5 \dfrac{\mathrm{db}}{\mathrm{km}} = 1.6 \cdot 10^{-4}\ \mathrm{cm}^{-1}\) for pure \(O_2\) (at atmospheric pressure). To speak of the width of the lines in the case when they are not sufficiently resolved has no special meaning; as a rough estimate one may take \(\Delta \tilde{\nu} \sim 0.02—0.05\ \mathrm{cm}^{-1}\); these figures in essence refer rather to the width of the whole band (see \(^{24}\)). From what has been said it is clear that, in air, radio radiation with \(\lambda = 0.5\) cm will decrease in intensity by a factor of \(e\) over a path of approximately 300 m.

For finding the values of \(\tilde{\nu}_0\) and \(\Delta \tilde{\nu}\) for the various lines of the band, measurements at reduced pressure are, of course, necessary. However, already from paper \(^{24}\) it follows that collisions of \(O_2\) molecules with one another and with nitrogen molecules \(N_2\), with respect to broadening, are approximately equivalent, as indeed was to be expected.

*) At \(T = 300^\circ\) the largest number of \(O_2\) molecules is at the level with \(K = 8\), and in this case \(N_{K=8} \simeq 0.07N\), where \(N\) is the total number of molecules (see \(^{18}\)).

In the centimeter region the absorption of oxygen is small: over the entire thickness of the atmosphere in the vertical direction²³ \(\int \chi\,dh = 0.07\) db at \(\lambda = 1.00\ \mathrm{cm}\); \(\int \chi\,dh = 0.04\) at \(\lambda = 1.25\ \mathrm{cm}\) and \(\int \chi\,dh = 0.03\) at \(\lambda = 1.50\ \mathrm{cm}\); thus even at \(\lambda = 1.00\ \mathrm{cm}\), \(\chi \sim 0.01\ \mathrm{db/km}\). As indicated in § 1, a reliable theory of such nonresonance absorption does not yet exist.

In conclusion to this paragraph we make one remark concerning isotopic molecules. Molecules with identical nuclei, for example the molecules \(O^{16}O^{16}\), \(N^{14}N^{14}\), etc., as is known, cannot have a permanent electric dipole moment. This does not apply, however, to isotopic molecules, such as the molecules \(O^{16}O^{17}\), \(O^{16}O^{18}\), \(N^{14}N^{15}\), and so on. The moments of such molecules must be considerably smaller than the ordinary one (\(P_0 \sim 10^{-18}\)), but if these moments were even \(\sim 10^{-20}\), then the absorption would be of the same order as that due to the presence of a magnetic moment, which is just \(\sim \dfrac{e\hbar}{2mc} \sim 10^{-20}\). In reality \(P_0\) for isotopic molecules is evidently much less than \(10^{-20}\) (this question, as far as we know, has been investigated neither experimentally nor theoretically). Since in air the molecules \(O^{16}O^{17}\), \(O^{16}O^{18}\), and \(N^{14}N^{15}\) are present in amounts of approximately \(2\cdot 10^{-2}\%\), \(8\cdot 10^{-2}\%\), and \(0.6\%\), it is clear that the absorption associated with isotopic molecules has no practical significance. However, measurement of radio absorption is one of the methods that can be used for measuring such small dipole moments as may be expected in the present case. The usual methods are entirely inapplicable here, and the most effective one is apparently the molecular-beam method³¹.

§ 6. ZEEMAN AND STARK EFFECTS

It is easy to see that, in quite attainable external magnetic and electric fields, the splitting of molecular terms (i.e., the Zeeman and Stark effects) corresponds to radio frequencies. Indeed, in the case of the Zeeman effect¹⁸,²² the frequency corresponding to the splitting is

\[ \nu \lesssim \frac{eH}{4\pi mc}=1.4\cdot H\ \mathrm{MHz}, \tag{22} \]

where \(H\) is the strength of the external magnetic field in gauss. In the case of the linear Stark effect, which occurs for molecules of the symmetric-top type³³,

\[ \nu \lesssim \frac{P_0 E}{h}\sim E\ \mathrm{MHz}, \tag{23} \]

Radiospectroscopy of Molecules

where \(E\) is the electric-field strength in volts per centimeter. Finally, in the case of the quadratic Stark effect[^33],

\[ \tilde{\nu}=\frac{P_0E^2}{h^3}\,8\pi^2 I \sim E^2 a, \tag{24} \]

where \(I\) is the moment of inertia of the molecule \((I\sim 10^{-38}\text{—}10^{-40})\), and \(E\) is measured in volts per cm.

From what has been said it is clear that radiospectroscopy can prove to be an extremely effective method for studying the Zeeman and Stark effects of molecular terms (of course, the question may chiefly concern purely rotational terms). In this case both the rotational transitions themselves in a field and the transitions between the sublevels of Zeeman and Stark splitting can be investigated.

In the case of the \(\mathrm{O}_2\) molecule, the Zeeman splitting, which depends on \(J\) and \(K\), on the frequency scale is always less than or equal[^2] to the frequency

\[ \nu_0=\frac{eH}{2\pi mc}= \]

\[ =2.8\,H\ \text{megacycles}; \]

in a sufficiently strong field \(\nu=\nu_0\). Experimentally, the Zeeman effect for molecules has not been investigated by the radio method. The absorption of radio waves in the atmosphere associated with this effect for \(\mathrm{O}_2\) molecules in the earth’s magnetic field proves to be insignificant[^2],[^3].

The molecular Stark effect has been investigated by the radio method for the OCS molecule[^20]. This molecule is linear and thus belongs to the rotor type, whose term energy has the form

\[ \left. \begin{aligned} \tilde{\nu} &= B_0J(J+1)\ \text{cm}^{-1}; \qquad B_0=\frac{h}{8\pi^2 cI};\\ \text{for radiative transitions}\quad \Delta J&=\pm 1. \end{aligned} \right\} \tag{25} \]

Experimentally, at reduced pressure, using the method described in2, a line was observed corresponding to the transition \(J=1 \longrightarrow J=2\). It was found that \(\tilde{\nu}_0=0.8107\ \text{cm}^{-1}\), whence \(I=1.379\cdot 10^{-38}\). In an electric field, in the case of a rotor, the quadratic Stark effect occurs; moreover, the level \(J=1\) splits into the sublevels \(J=1,\ m=0\) and \(J=1,\ m=\pm 1\), while the level \(J=2\) splits into the sublevels \(J=2,\ m=0;\ J=2,\ m=\pm 1\), and \(J=2,\ m=\pm 2\), with energies (see[^33], p. 34; instead of \(p\) in[^33] we write \(m\)):

\[ \left. \begin{aligned} \tilde{\nu}_{1,0} &= \left(2+\frac{2}{20}\left(\frac{P_0E}{hcB_0}\right)^2\right)B_0; &\quad \tilde{\nu}_{1,1} &= \left(2-\frac{1}{20}\left(\frac{P_0E}{hcB_0}\right)^2\right)B_0;\\ \tilde{\nu}_{2,0} &= \left(6+\frac{1}{42}\left(\frac{P_0E}{hcB_0}\right)^2\right)B_0; &\quad \tilde{\nu}_{2,1} &= \left(6+\frac{1}{84}\left(\frac{P_0E}{hcB_0}\right)^2\right)B_0;\\ \tilde{\nu}_{2,2} &= \left(6+\frac{1}{42}\left(\frac{P_0E}{hcB_0}\right)^2\right)B_0, \end{aligned} \right\} \tag{26} \]

where the first subscript of \(\tilde{\nu}\) corresponds to the value of \(J\), and the second to the value \((m)\).

In experiment 20 the constant electric field was parallel to the electric field of the radio wave, owing to which only transitions with \(\Delta m=0\) were possible, i.e., from the level \((1,1)\) to the level \((2,1)\) and from the level \((1,0)\) to the level \((2,0)\). Thus, in the electric field the line of the OCS molecule under consideration should split into two, with a difference of wave numbers equal to

\[ \Delta \nu = -(\nu_{2,1}-\nu_{1,1})-(\nu_{2,0}-\nu_{1,0}) = \left(\frac{3}{20}-\frac{1}{84}\right)\frac{8\pi^2 I}{h^2 c} P_0^2 E^2 . \tag{27} \]

In a field \(\sim 1000\ \dfrac{\mathrm{V}}{\mathrm{cm}}\), \(\Delta \nu \sim 5\cdot 10^6\ \mathrm{Hz}\) (Fig. 12). The transition \((1,1)\to(2,1)\) is doubly degenerate, whereas the transition \((1,0)\to(2,0)\) is nondegenerate; therefore the first of the indicated lines should be twice as intense as the second, which is indeed the case.^20

a

b

c

Fig. 12.

With the aid of (27) one can determine the dipole moment of the molecule \(P_0\), which proved to be \(P_0=0.72\cdot 10^{-18}\), whereas dielectric measurements led to the value \(P_0=0.65\cdot 10^{-18}\).

The example considered, which is as yet the only one investigated experimentally, clearly shows the possibilities opening up in the field of radio-spectroscopic investigation of the Stark effect in molecules (in the case cited, the values of \(I\) and \(P_0\) of the OCS molecule were measured).

§ 7. RADIO SPECTROSCOPY OF ATOMS AND ATOMIC NUCLEI

Above we have been speaking of the radio spectroscopy of molecules. However, with no less justification, perhaps, one may also speak of the radio spectroscopy of atoms and atomic nuclei. We shall touch here on this important question only very briefly and solely in order to emphasize its connection with the material set forth in the preceding sections. If an atom in its ground state possesses an electronic moment (orbital, spin, or both simultaneously), and at the same time the nucleus has a magnetic moment different from zero, then the ground term undergoes hyperfine splitting. Examples are furnished by the nuclei \(\mathrm{Na}^{23}\), \(\mathrm{K}^{39}\), and \(\mathrm{Cs}^{133}\), whose ground state is the state \({}^{2}S_{1/2}\). In these cases, since the spin of the shell is \(1/2\), the term splits into two sublevels characterized by the quantum number of the total (electronic plus nuclear) angular momentum of the atom

\[ F=i\pm 1/2, \]

where \(i\) is the nuclear spin (in units of \(\dfrac{h}{2\pi}\)). The distance between

these sublevels on the frequency scale is equal to \(^{34,35}\): for \(\mathrm{Na}^{23}\) \(\Delta\nu = 1.770\cdot 10^9\); for \(\mathrm{K}^{39}\) \(\Delta\nu = 4.60\cdot 10^8\), and for \(\mathrm{Cs}^{133}\) \(\Delta\nu = 9.1926\cdot 10^9\) cps. The wavelength in these cases is respectively equal to 17 cm, 65 cm, and 3.26 cm. Measurement of the hyperfine splitting and of the influence on this splitting of a magnetic field (the Zeeman effect of the hyperfine structure) has been carried out in a number of cases by the well-known Rabi resonance method (for a review see \(^{34}\)) and makes it possible to determine the spin and magnetic moment of the nucleus.

In the resonance method, as is well known, a molecular or atomic beam is used, which is defocused as a result of the absorption, by the molecules or atoms composing it, of radio radiation of the corresponding frequency. The Rabi method may therefore with considerable justification be regarded as radiospectroscopic in the sense of this term in which it is used above. Recently, however, methods have been developed for measuring the spin and magnetic moment of nuclei (and also the hyperfine structure and its Zeeman effect) which are directly adjacent to those discussed in §§ 2 and 4 in connection with the measurement of the absorption coefficient in gases. Between the sublevels of the hyperfine splitting radiative transitions are possible, which evidently belong to the type of magnetic dipole radiation (just as in the molecule \(\mathrm{O}_2\); see § 5). The frequency corresponding to these transitions can in principle be measured, for example, from the change in the damping or the natural frequency of a resonator filled with the gas under investigation. Such a method was applied in the case of cesium (the change in the natural frequency of the resonator was measured) \(^{35}\). As a result, the splitting \(\Delta\nu\) of the \(\mathrm{Cs}^{133}\) nucleus was determined (see above) and its spin, which proved to be \(7/2\) (the Zeeman effect of the hyperfine splitting was observed; the pressure of the Cs vapor was \(4\cdot 10^{-2}\) mm Hg). The possibility of applying the same method to other atoms is undoubted, which opens up great possibilities (isotopic analysis by means of an apparatus in principle incomparably simpler than the Rabi apparatus).

Another physical method for measuring the magnetic moment of nuclei consists in the following \(^{36}\). The substance under investigation is placed in a strong magnetic field, which completely breaks the coupling of the magnetic moment of the nucleus with the electron shell. The nucleus can then be in various energy states, differing in the value of the projection of its magnetic moment on the direction of the field. In other words, we obtain a Zeeman splitting of the state of the nucleus in the field. In the case when the spin of the nucleus is \(1/2\) (proton), there are only two sublevels corresponding to the magnetic moment oriented along the field and against the field. Between the different sublevels, transitions of magnetic-dipole type are possible, which can be detected from the change in the damping or in the natural frequency of a resonator or of a tuned section of coaxial cable \(^{36*}\). This method was successfully applied—

* A related method for detecting absorption in a solid associated with the electronic magnetic moment, see in \(^{37}\).

... for measuring the magnetic moment of the proton, and not a gas but a solid substance containing hydrogen (paraffin) was used. The possibility of using solid or liquid substances in measuring a nuclear moment is connected with the fact that the binding energy of the moment of the nucleus with the lattice, or in general with neighboring atoms, is already destroyed in a field of the order of a few gauss, and thus in a strong field the nuclei form, as it were, a “gas,” i.e. an ensemble of independent particles.

For a particle with spin \(1/2\) and magnetic moment \(\mu_0\), the frequency corresponding to the transition between the two possible orientations of the moment in the field is equal to

\[ \nu=\frac{2\mu_0 H}{h}=1.5\cdot 10^3\mu_0 H\ \text{cps}, \tag{28} \]

where in the last expression \(\mu_0\) is measured in nuclear magnetons.

\[ \frac{eh}{4\pi Mc}=5\cdot 10^{-4}\quad (M\text{ is the proton mass}). \]

It follows from (28) that for \(\mu_0 \sim 1\), even in a field \(H\sim 10^4\), \(\nu\sim 10^7\) and \(\lambda\sim 30\ \text{m}\). For the proton, measurements by Rabi’s method give \(\mu_0=2.7896\); by the radio method\({}^{36}\) the value \(\mu_0=2.75\) was obtained \((H=7100,\ \nu=28.9\ \text{MHz})\). Thus the method under consideration is associated with the use of radio radiation with wavelengths of meters and tens of meters. In addition to measuring the magnetic moment of nuclei, the study of “nuclear absorption” in a magnetic field makes it possible to determine the relaxation time for the establishment of equilibrium between the nuclear moments and the lattice\({}^{36}\).

In another work\({}^{38,39}\) a somewhat different, very effective radio method for measuring the magnetic moments of nuclei was developed and used, but we shall not dwell on this method here. The question of the radiospectroscopy of nuclei is so important and interesting that it should be considered in detail independently of the main topic of this article.

CITED LITERATURE

1. V. Heitler. Quantum Theory of Radiation. GTTI (1940).
2. V. L. Ginzburg. DAN, 35, 302 (1942).
3. V. L. Ginzburg. Izv. AN SSSR (ser. phys.), 7, 96 (1943).
4. J. H. Van Vleck and V. F. Weisskopf. Rev. Mod. Phys. 17, 227 (1945).
5. V. F. Weisskopf. Phys. Zschr. 34, 1 (1933).
6. V. Weisskopf. UFN, 13, 552 (1933).
7. H. M. Foley. Phys. Rev. 69, 616 (1946).
8. C. E. Cleeton and N. H. Williams. Phys. Rev. 45, 234 (1934).
9. W. D. Hershberger. Journ. Appl. Phys. 17, 495 (1946).
10. W. Bleaney and R. P. Penrose. Nature, 157, 339 (1946).
11. W. E. Good. Phys. Rev. 70, 213 (1946).
12. C. H. Townes. Phys. Rev. 70, 665 (1946).
13. B. Beaney and R. P. Penrose. Phys. Rev. 70, 775 (1946).
14. L. N. Hadley and D. M. Dennison. Phys. Rev. 70, 780 (1946).
15. H. Sheng, E. F. Barker and D. M. Dennison. Phys. Rev. 60, 786 (1941).

1. G. Herzberg. Infrared and Raman Spectra of Polyatomic Molecules. N. Y. (1945).
2. G. Stuard. Structure of Molecules. Kharkov (1937).
3. G. Herzberg. Molekülspektren u. Molekülstruktur. Leipzig (1939).
4. C. H. Townes and F. R. Merrit. Phys. Rev. 70, 558 (1946).
5. T. W. Dakin, W. E. Good and D. K. Coles. Phys. Rev. 70, 560 (1946).
6. R. M. Heiner, G. W. King and P. C. Gross. Phys. Rev. 70, 108 (1946).
7. G. E. Becher and S. H. Autler. Phys. Rev. 70, 300 (1946).
8. R. H. Dicke, R. Beringer, R. L. Kyhl and A. B. Vane. Phys. Rev. 70, 340 (1946).
9. R. Beringer. Phys. Rev. 70, 53 (1946).
10. D. M. Dennison. Rev. Mod. Phys. 12, 175 (1940).
11. W. E. Lamb. Phys. Rev. 70, 308 (1946).
12. R. H. Dicke. Rev. Sci. Instr. 17, 268 (1946).
13. V. L. Ginzburg. Izv. AN SSSR, (physical series) 11, No. 1—2 (1947).
14. R. Schlapp. Phys. Rev. 51, 342 (1937).
15. H. A. Kramers. Zschr. f. Phys. 53, 422 (1929).
16. H. K. Hughes. Phys. Rev. 70, 570 (1946).
17. F. H. Crawford Jr. Rev. Mod. Phys. 6, 90 (1934).
18. P. Debye and G. Zakk. Theory of the electrical properties of molecules. ONTI (1936).
19. J. B. M. Kellogg and S. Millman. Rev. Mod. Phys. 18, 323 (1946).
20. A. Roberts, J. Beers and A. S. Hill. Phys. Rev. 70, 112 (1946).
21. E. M. Purcell, H. C. Torrey and R. V. Pound. Phys. Rev. 69, 37 (1946).
22. E. Zavoisky. ZhETF 15, 253, 344 (1945); 16, 603 (1946).
23. F. Bloch. Phys. Rev. 70, 460 (1946).
24. F. Bloch, W. W. Hansen and M. Packard. Phys. Rev. 70, 474 (1946).

AUTHOR’S NOTE AT PROOFREADING

After this review had been written, the following articles appeared, with which the author did not have the opportunity to become more closely acquainted:

1. J. E. Walter and W. D. Hershberger. Journ. Appl. Phys. 17, 814 (1946).
   Absorption of microwaves in gases. II.

2. D. E. Coles and W. E. Good. Phys. Rev. 70, 979 (1946).
   The Stark and Zeeman effect of the inversion spectrum of NH₃.

3. B. P. Dailey, R. L. Kyhl, M. W. P. Strandberg, J. H. Van Vleck and E. B. Nilson. Phys. Rev. 70, 984 (1946).
   Hyperfine structure of the microwave spectrum of NH₃ and the existence of a quadrupole moment in the N¹⁴ nucleus.

4. E. M. Purcell, R. V. Pound and N. Bloembergen. Phys. Rev. 70, 986 (1946).
   Nuclear magnetic resonance absorption in gaseous hydrogen.

5. The same authors, Phys. Rev. 70, 988 (1946).
   Resonance absorption and nuclear magnetic moments in monocrystalline CaF₂.

6. C. H. Townes, A. H. Holden and F. R. Merrit. Phys. Rev. 71, 64 (1947).
   Rotational spectrum of some linear molecules in the region \(\lambda \sim 1\ \text{cm}\).

1. Reference 10. ↩↩↩

2. Reference 11. ↩↩↩

3. Reference 12. ↩↩