THE CURRENT STATE OF RADIO SPECTROSCOPY*)

M. A. El’yashevich

The subject of radio spectroscopy is the study of those transitions between energy levels which correspond to the radio-frequency region of the spectrum of electromagnetic waves. In its experimental methods, based above all on the achievements of modern radio engineering, radio spectroscopy differs substantially from ordinary “optical” spectroscopy, which encompasses the ultraviolet, visible, and infrared regions of the spectrum of electromagnetic waves. However, in its theoretical foundations and in the methods of interpreting experimental data, radio spectroscopy is a typical branch of spectroscopy with the characteristic features of the latter. At the same time, radio spectroscopy opens up new possibilities for studying the structure of matter, inaccessible to optical spectroscopy, because it makes it possible to determine directly very small differences between the energy levels of atomic systems. These differences correspond to the fine and hyperfine structure of electronic spectra, to the rotational structure of spectra, to various effects of interaction within atoms and molecules, and to the splitting of the energy levels of atoms, molecules, and crystals in external magnetic and electric fields.

At present, radio-spectroscopic methods are being used successfully to study transition frequencies ranging from tens and hundreds of kilocycles to hundreds of thousands of megacycles, i.e. corresponding to wave numbers from millionths of a fraction of $\text{cm}^{-1}$ to tens of $\text{cm}^{-1}$, which covers a very wide range of wavelengths from kilometers to fractions of a millimeter. Investigations in the microwave region (the region of superhigh radio frequencies) from 300 Mc/s to 300,000 Mc/s, i.e. from $0.01\ \text{cm}^{-1}$ to $10\ \text{cm}^{-1}$, encompassing decimeter, centimeter, and millimeter waves (wavelengths from $\lambda = 1\ \text{m}$ to $\lambda = 1\ \text{mm}$), have acquired especially great importance. For this reason microwave spectroscopy is often singled out, distinguishing it from radio spectroscopy proper—

*) Presented in abbreviated form at the Ninth All-Union Conference on Spectroscopy in Tartu in July 1954.

radio-frequency spectroscopy; by the methods of the latter the range of long and short radio waves is studied, and most often the region from approximately 3 MHz to 30 MHz, i.e. from \(10^{-4}\ \text{cm}^{-1}\) to \(10^{-3}\ \text{cm}^{-1}\) (wavelengths from \(\lambda = 100\ \text{m}\) to \(\lambda = 10\ \text{m}\)). In separate investigations transitions corresponding to low frequencies of tens and hundreds of kHz (kilometer waves)\(^{1}\) have been observed; on the other hand, progress is being successfully made into the “submillimeter” region (wavelengths less than \(\lambda = 1\ \text{mm}\))\(^{2}\), which overlaps with the far infrared region (wavelengths greater than \(\lambda = 100\ \mu = 0.1\ \text{mm}\)).

A characteristic feature of radiospectroscopic methods is their high resolving power and the very great accuracy of measurement of transition frequencies. Unlike the methods of optical spectroscopy, which make it possible to determine small differences of energy levels (for example, by studying the hyperfine structure of spectral lines) only with small relative accuracy, radiospectroscopic methods make it possible to determine transition frequencies and, consequently, small differences of energy levels with an accuracy reaching millionths of a percent (\(10^{-4}—10^{-5}\%\)).

Under favorable conditions, with a small width of spectral lines\(*\), it is possible to obtain extremely high resolving powers \(\nu/\Delta\nu\) (\(\nu\) is the frequency, \(\Delta\nu\) is the minimum frequency difference of neighboring lines at which these lines are observed separately), of the order of hundreds of thousands and even millions.

Along with transition frequencies, radiospectroscopic methods make it possible, with considerable accuracy and quite rapidly, to determine the intensities and contours of spectral lines.

The beginning of the development of radiospectroscopy dates to the period before the Second World War, when absorption by ammonia molecules in the microwave region (about 20,000 MHz)\(^{3}\) was discovered and the method of magnetic resonance in molecular beams\(^{4}\) was developed, by means of which it became possible to measure the magnetic moments of the proton and the deuteron. However, the rapid development of radiospectroscopy began only after the war, and almost simultaneously in a number of directions: in 1945–1947 work on the investigation of gas absorption in the microwave region was widely expanded; in particular, the structure of the absorption bands of the ammonia molecule was studied in detail. In 1945 E. K. Zavoisky\(^{5}\) discovered the phenomenon of paramagnetic resonance in solids. Beginning in 1946, the phenomenon of ferromagnetic resonance\(^{6}\), theoretically investigated by L. D. Landau and E. M. Lifshitz in 1935\(^{7}\), began to be intensively studied. In the same year, 1946, Purcell\(^{8}\) and Bloch\(^{9}\)

\(*\) For example, for proton resonance at low temperatures (in the region of ordinary radio frequencies), for absorption lines of some molecules at low pressures (in the microwave region).

...it proved possible to observe nuclear paramagnetic resonance in solids and liquids. Finally, in 1950 Dehmelt and Krüger[^10] began the study of purely quadrupole spectra in solids. At the present time radiospectroscopy has become an extensive branch of spectroscopy, highly diverse both in the objects of investigation and in the methods employed. Hundreds of papers appear annually; a number of reviews have been written on its individual subdivisions[^11–^18], and several collections of papers devoted to radiospectroscopy[^19,^20,^6] have already appeared; the first monographs on microwave spectroscopy[^21,^22] have been published; and the important question of measuring nuclear moments by radiospectroscopic methods is treated in detail in the corresponding monographs[^23,^24]. It should be emphasized that radiospectroscopy is a complex field connected with various problems of the structure of nuclei, atoms, molecules, solids, and liquids; new directions in its development are constantly arising. In the present review questions concerning the types of spectral transitions studied by radiospectroscopic methods and the features of absorption and emission in the radio-frequency region are considered; a brief characterization of the principles of the experimental methods is given; a survey is made of some of the most important results of investigations; and certain questions of the further development of radiospectroscopy are touched upon.

I. TYPES OF TRANSITIONS STUDIED BY RADIOSPECTROSCOPIC METHODS

As is well known, the methods of optical spectroscopy are used to study electronic transitions in the visible and ultraviolet regions of the spectrum, vibrational transitions in the near infrared region, and rotational transitions for light molecules in the far infrared region of the spectrum. In the diagram of Fig. 1, which represents the scale of electromagnetic waves, the most typical examples are shown of electronic transitions (the first member of the Lyman series for the hydrogen atom, the resonance lines of mercury and sodium), vibrational transitions (corresponding to the fundamental vibrational frequencies of the OH radical and of the molecules CO, NaH, NaCl, HgJ), and rotational transitions (the transitions 1—2, 3—4, 6—7, 10—11 between the lower rotational levels of the HCl molecule).

Electronic transitions are characterized by frequencies of the order of \(10^{15}\) Hz (tens of thousands of cm\(^{-1}\)); vibrational transitions, by frequencies of the order of \(10^{13}\)—\(10^{14}\) Hz (hundreds and thousands of cm\(^{-1}\)); and rotational transitions for light molecules, by frequencies of the order of \(10^{12}\)—\(10^{13}\) Hz (tens and hundreds of cm\(^{-1}\)). All transitions between close energy levels, to which there correspond frequencies not exceeding \(10^{12}\) Hz \(= 10^{6}\) MHz, fall in the radio-frequency region of the electromagnetic spectrum. In the optical region these small differences of energy levels...

in absorption for molecules possessing permanent dipole moments.

The simplest case is the rotation of diatomic and linear polyatomic molecules. For them the rotational energy depends on a single quantum number \(N\), which determines the rotational angular momentum and takes integral values \((N=0,1,2,\ldots)\). If the molecule is regarded as a rigid body, then for the rotational energy one obtains the well-known expression

\[ E_N = hBN(N+1), \tag{5} \]

where the rotational constant

\[ B=\frac{h}{8\pi^2 I_\perp} \tag{6} \]

is inversely proportional to the moment of inertia \(I_\perp\) of the molecule with respect to an axis perpendicular to the molecular axis. For diatomic molecules \(I_\perp=\mu r_0^2\), where \(\mu\) is the reduced mass of the molecule, and \(r_0\) is the distance between the nuclei. By virtue of the selection rule \(\Delta N=\pm 1\) for dipole radiation, the differences of energy levels for possible transitions are determined by the formula

\[ \Delta E_N = E_{N+1}-E_N = 2BN(N+1) \tag{7} \]

and in absorption one obtains a series of equally spaced lines \(N\to N+1\) \((0\to1,\ 1\to2,\ 2\to3,\) etc.) with frequencies \(2B(N+1)\) \((2B,\ 4B,\ 6B,\) etc.).

Fig. 3. Rotational transitions for the OCS molecule.

When centrifugal stretching is taken into account, in formula (5) one must add the term \(-hD[N(N+1)]^2\), and correspondingly in formula (7) the term \(-4hD(N+1)^3\), which leads to a gradual decrease of the intervals between the lines; however, the coefficient \(D\) is usually very small, and centrifugal stretching becomes noticeable only at large rotational energies, when \(N\) is large. The rotational constants for molecules with molecular weights of the order of several tens amount to thousands and tens of thousands of Mc/s. For example, for CO \(B\simeq 58\,000\) Mc/s, for HCN \(B\simeq 44\,000\) Mc/s, for CH\(_3\)F \(B\simeq 25\,500\) Mc/s, for N\(_2\)O \(B\simeq 12\,500\) Mc/s, for ClCN \(B\simeq 6000\) Mc/s. Figure 3 gives a scheme of transitions for the well-studied rotational spectrum of OCS \((B=6081.5\) Mc/s); it is interesting to note that for this molecule transitions between very high rotational levels were also observed,

up to the transition \(31 \to 32\) with a frequency of \(389\,000\) MHz \((\lambda = 0.77\ \mathrm{mm})\) in the submillimeter region\({}^{2}\).

For the more complicated case of molecules that are symmetric tops (the molecule has two different moments of inertia—the moment of inertia \(I_{\parallel}\) with respect to the axis of symmetry of the top and the moment of inertia \(I_{\perp}\) with respect to any axis perpendicular to it), the rotational energy depends on two quantum numbers: on the quantum number \(N\), which determines the total rotational angular momentum, and on the quantum number \(K\), which determines the projection of this angular momentum on the axis of symmetry of the top and takes \(2N+1\) values

\[ K = N,\ N-1,\ N-2,\ldots,\ -N. \tag{8} \]

The rotational energy is determined by the formula (if centrifugal stretching is not taken into account)

\[ E_{NK}=h\left[BN(N+1)+(A-B)K^{2}\right], \tag{9} \]

where the rotational constants

\[ A=\frac{h}{8\pi^{2}I_{\parallel}} \quad \text{and} \quad B=\frac{h}{8\pi^{2}I_{\perp}} \tag{10} \]

are expressed through the moments of inertia \(I_{\parallel}\) and \(I_{\perp}\) with respect to the axis of symmetry of the top and to an axis perpendicular to it.

For a given \(N\), according to formula (9), there are \(N+1\) levels with \(|K|=0,1,2\ldots\). By virtue of the selection rules for dipole radiation \(\Delta N=\pm 1\) and \(\Delta K=0\), the difference of the energy levels for possible transitions is determined by the formula

\[ \Delta E_{N,K}=E_{N+1,K}-E_{N,K}=2B(N+1), \tag{11} \]

which coincides with formula (7); therefore, from the rotational spectrum only the rotational constant \(B\) is determined (but not \(A\)).

The most complicated and at the same time very interesting and important general case is represented by molecules that are asymmetric tops (all three moments of inertia \(I_A, I_B, I_C\) are not equal to one another). In this case, for each value of the quantum number \(N\) one obtains \(2N+1\) levels, arranged differently depending on the mass ratios.*) Investigation of the rotational spectra of such molecules makes it possible to determine three rotational constants:

\[ A=\frac{h}{8\pi^{2}I_A}, \quad B=\frac{h}{8\pi^{2}I_B} \quad \text{and} \quad C=\frac{h}{8\pi^{2}I_C}, \tag{12} \]

*) These levels are denoted, for a given \(N\), in the order of their arrangement by the symbol \(N_{\tau}\), where the index \(\tau=-N,-N+1,\ldots,N-1,N\) (for example, \(3_{-2}\) denotes the second-highest level with \(N=3\)). The notation \(N_{K_B K_C}\) is also used, where \(K_B\) is the quantum number \(K\) for the limiting case of a prolate symmetric top \((I_A<I_B=I_C)\), and \(K_C\) is the quantum number \(K\) for the limiting case of an oblate symmetric top \((I_A=I_B<I_C)\); in this case \(\tau=K_B-K_C\), for example \(3_{-2}=3_{1,3}\).

...expressed in terms of the corresponding moments of inertia. It should be noted that even for light molecules that are asymmetric tops, a typical representative of which is the nonlinear symmetric molecule of water H$_2$O, rotational levels that are close in energy may be obtained; transitions between such levels give frequencies lying in the microwave region. In particular, for the water molecule rotational absorption lines are observed with frequencies $\nu \simeq 22200$ Mc/s ($\lambda = 1.35$ cm) and $\nu \simeq 183300$ Mc/s ($\lambda = 1.62$ mm)*).

c) Transitions due to interaction effects within molecules

Along with purely rotational transitions, for molecules an important role in the microwave region is played by effects associated with interactions of the rotational motion with the electronic and vibrational motion. The first type of interaction causes spin splitting of the rotational levels; the second type—inversion splitting, which depends on the rotational quantum numbers, $l$-doubling of rotational levels, and, in addition, the dependence of rotational constants on the vibrational state.

Fig. 4. Triplet splitting of the rotational levels of the oxygen molecule O$_2$.

The values of the quantum number $J$ are indicated for a given value of the rotational quantum number $N$.

Spin splitting of rotational levels. Owing to the interaction of the rotational moment $\mathbf{N}$ with the total spin moment $\mathbf{S}$ in diatomic molecules, each of the rotational levels is observed to split into $2S+1$ components (Hund’s case b), where $S$ is the total spin quantum number. An example is the ground triplet state ${}^3\Sigma$ of the oxygen molecule O$_2$. Each level splits into three components ($S=1$, $2S+1=3$) with values of the quantum number $J$, which determines the total angular momentum of the molecule, equal to $J=N+1$, $N$, $N-1$, where $N$ denotes the rotational quantum number. According to the selection rules $\Delta J=\pm 1$ and $\Delta N=0$, transitions $J=N-1 \to J=N$ and $J=N \to J=N+1$ are possible (Fig. 4), which gives two series of absorption lines in the region between 50000 Mc/s and 70000 Mc/s, with a maximum intensity at about 60000 Mc/s ($\lambda=5$ mm).

Inversion splitting. For nonplanar molecules two equilibrium configurations are possible, obtained one from

*) They correspond to the transitions $5_{-1}-6_{-5}$ and $2_2-3_{-2}$.

another path of inversion, i.e., reflection of all nuclei in the center of gravity. This means that the potential energy \(V\) has two identical minima separated from one another by a maximum (Fig. 5). Owing to the tunnel effect, each rotational-vibrational level is split into two, one of which is even (the wave function does not change sign under inversion), while the other is odd (the wave function changes sign under inversion). A transition between these levels is possible. Such

Fig. 5. Potential energy in the presence of two equilibrium configurations. The inversion splitting is shown on an enlarged scale. In the case of the ammonia molecule \(h\) is the distance of the N atom from the plane of the H atoms.

Fig. 6. Two equilibrium configurations of the ammonia molecule \(\mathrm{NH}_3\).

a case is realized for the pyramidal ammonia molecule \(\mathrm{NH}_3\), whose two possible configurations are shown in Fig. 6\(*\). The magnitude of the inversion splitting \(\nu_0\) in the zero vibrational state in the absence of rotation is approximately \(23\,800\) MHz \((0.74\ \mathrm{cm}^{-1}\), which corresponds to \(\lambda = 1.35\ \mathrm{cm})\) and increases according to an exponential law for rotating molecules; for small quantum numbers \(N\) and \(K\) (the ammonia molecule is a symmetric top and its energy is determined by formula (9)) this law has the form

\[ \nu = \nu_0 e^{a'N(N+1)+b'K^2}, \tag{13} \]

where \(a'\) and \(b'\) are constants.

The form of the dependence (13) corresponds to the fact that the probability of passage through the barrier increases exponentially with increasing rotational energy (the probability of passage is a quantity,

\(*\) They can be obtained from one another by reflection of the nitrogen atom in the plane containing the hydrogen atoms, which is equivalent to inversion with a rotation through \(180^\circ\).

inversely proportional to the time \(\tau\) of penetration through the barrier, and, consequently, proportional to the transition frequency \(\nu=\dfrac{1}{\tau}\).

For each rotational level its own splitting is obtained, and therefore the spectrum must consist of a large number of lines corresponding to different values of \(N\) and \(K\). The microwave absorption spectrum of ammonia, mentioned above, is such a spectrum. At present it has been investigated in considerable detail.

\(l\)-doubling of rotational levels. For linear molecules, deformation vibrations perpendicular to the molecular axis are doubly degenerate (the vibrations may occur in two mutually perpendicular planes; see Fig. 7), and this leads to the splitting of each rotational level into two sublevels, i.e., to the so-called \(l\)-doubling. The difference of the energies of the resulting sublevels is equal to

\[ \Delta E = hqN(N+1), \tag{14} \]

where

\[ q=q_0(v_s+1), \tag{15} \]

and, for a given value of the vibrational quantum number \(v_s\) of the deformation vibration, is constant. For the linear molecule HCN, \(q=225\) Mc/s, which leads to splittings of the order of tens of thousands of megacycles at rotational quantum numbers of the order of 10; the corresponding transitions are observed in the microwave region.

Fig. 7. Deformation vibrations of a linear molecule in two mutually perpendicular planes.
The vibrations are shown for the case of a linear triatomic molecule, the central atom of which is displaced relative to the end atoms perpendicularly to the molecular axis.

Dependence of rotational constants on the vibrational state. The values of the rotational constants depend on the vibrational state of the molecule, because during vibrations the moments of inertia change somewhat. This leads to somewhat different frequencies of rotational transitions for different vibrational states, and instead of a single rotational line one may observe a series of close lines, whose intensities will be proportional to the numbers of atoms in the corresponding vibrational states. In practice, an intense line will be observed for the ground vibrational state and weak satellites for the lowest excited vibrational states, for which at the given temperature the Boltzmann factor \(e^{-\Delta E_{\mathrm{vib}}/kT}\) (\(\Delta E_{\mathrm{vib}}\) is the height of the excited vibrational level) is not very small.

c) Transitions associated with nuclear spin

(hyperfine structure)

An especially important type of transitions studied by radiospectroscopic methods are transitions associated with nuclear spin. The moment of the nucleus—the nuclear spin—interacts with the electron shells, and this leads to splittings of levels, to which there usually correspond frequencies of the order of tens to thousands of Mc/s (from thousandths to tenths of \(\mathrm{cm}^{-1}\)). In optical spectra these splittings appear in the form of the hyperfine structure of spectral lines; the term “hyperfine structure” is often also used in radiospectroscopy for transitions caused by nuclear spin. The interactions of nuclear spins with electron shells may be of two types—magnetic and electrostatic.

Magnetic interaction. For nuclei with spin different from zero \(\left(I=\frac{1}{2},\,1,\,\frac{3}{2},\,2,\,\frac{5}{2},\ldots,\right.\) where \(I\) is the nuclear spin quantum number\(\left.^*\right)\), there is a magnetic interaction of the nuclear magnetic moment with the electron magnetic moments of atoms and molecules (caused both by electron spin and by electron orbital motion). In this case, for atoms, the spin angular momentum of the nucleus \(\mathbf{I}\) combines with the electronic angular momentum \(\mathbf{J}\) into the total angular momentum of the atom \(\mathbf{F}=\mathbf{J}+\mathbf{I}\), whose magnitude is determined by the quantum number \(F\), taking the values

Fig. 8. Hyperfine structure of the ground level of the hydrogen atom.

\[ F=J+I,\ J+I-1,\ldots,\ |J-I|, \tag{16} \]

where \(J\) is the quantum number determining the magnitude of the total electronic angular momentum of the atom. \(F\) takes \(2I+1\) values for \(J\ge I\) and \(2J+1\) values for \(J\le I\).

A characteristic example of magnetic interaction is the splitting of the ground level of the hydrogen atom \(1^2S_{1/2}\). In this case \(J=\frac{1}{2}\), \(I=\frac{1}{2}\), and the quantum number \(F\) takes two values: \(F=1\) and \(F=0\) (Fig. 8).

\[ \text{*) It is often called, for brevity, the “nuclear spin,” meaning thereby that the intrinsic angular momentum of the nucleus—its spin—is determined by the corresponding value of the nuclear spin quantum number } I. \]

The magnitude of the splitting is approximately \(1420\) Mc/s \((0.047\ \mathrm{cm}^{-1})\). In general, the largest splitting is obtained for atoms having unpaired \(s\)-electrons, among which is the hydrogen atom in the ground state. For \(\mathrm{Na}^{23}\), \(\mathrm{Rb}^{87}\), and \(\mathrm{Cs}^{113}\), which also have the ground state \({}^{2}S_{1/2}\), caused by the presence of one unpaired \(s\)-electron, the magnitudes of the splitting due to the nuclear spin are \(1770\), \(6800\), and \(9200\) Mc/s \((0.059,\ 0.228,\ \text{and }0.307\ \mathrm{cm}^{-1})\), respectively. Since the magnitude of the splitting depends on the magnitude of the magnetic moment of the nucleus, the latter can in principle be calculated from the magnitude of the splitting1. However, this calculation is comparatively simple only in the case of hydrogen-like atoms, while in other cases it presents considerable difficulties, since it requires determining the value of the magnetic field produced by the electron shell at the location of the nucleus, which is easily done only for a one-electron system.

The great majority of molecules in the ground state do not possess an electronic magnetic moment (as a result of which these molecules are not paramagnetic), and therefore for them an interaction of the type described is absent. For the few paramagnetic molecules, a splitting is obtained analogous to the splitting in the case of atoms.

Electrostatic (quadrupole) interaction. For nuclei with spin exceeding \(1/2\) \((I = 1,\ 3/2,\ 2,\ldots)\), there is an electrostatic interaction of the nuclear electric quadrupole moment with the electron shells; this interaction is usually called quadrupole interaction. It arises in the presence of a nonzero gradient \(q\) of the electric field \(\mathcal{E}\) produced by the electron shells at the location of the nucleus, and is of the order of magnitude

\[ eQq, \tag{17} \]

where \(eQ\) is the quadrupole moment of the nucleus. In order that the field gradient at the location of the nucleus be nonzero, it is necessary that the potential \(V\), produced by the electrons, not possess spherical or cubic symmetry with respect to the nucleus. For atoms, the potential will not have such symmetry if, in addition to closed shells, there are electrons with azimuthal quantum number \(l \ne 0\). In the case of molecules and crystals, the potential cannot possess spherical symmetry (this can occur only approximately), and for quadrupole interaction to be possible it is sufficient that the potential not possess cubic symmetry. A very important case is that of axial symmetry of the potential acting on the nucleus. In this case \(q\) (the field gradient) will have the form, if the axis of symmetry is chosen as the \(z\)-axis,

\[ q = -\frac{\partial \mathcal{E}}{\partial z} = \frac{\partial^{2} V}{\partial z^{2}}. \tag{18} \]

The nuclear quadrupole interaction (17) usually has the same order of magnitude as the nuclear magnetic interaction, and lies within the range from tens to thousands of megacycles. Transitions between closely spaced sublevels caused by quadrupole splitting (purely quadrupole spectra) can be observed for crystals in the frequency range of tens and hundreds of megacycles. An example is provided by quadrupole transitions in organic crystals whose molecules contain the bond C—Br\({}^{79}\) (the nucleus Br\({}^{79}\) has spin \(I=3/2\)); these transitions lie in the region of about 300 Mc.

In addition to the magnetic and electrostatic interactions of nuclear moments with electron shells, magnetic interactions are possible between nuclear moments and the magnetic moments of molecules caused by rotation (rotational magnetic moments). Rotational magnetic moments have the same order of magnitude as nuclear magnetic moments, and are a thousand times smaller than electron magnetic moments. Just as in the case described above, the energy levels split into \(2I+1\) or \(2N+1\) components (cf. (16)), where the quantum number \(N\) determines the magnitude of the rotational angular momentum. However, because of the smallness of rotational magnetic moments, the magnitude of the corresponding splitting does not exceed tens or hundreds of kilocycles; it accounts for the fine structure of rotational absorption lines observed in the microwave region of the spectrum.

2. Transitions between sublevels of Zeeman splitting (magnetic resonance)

a) Transitions caused by electron magnetic moments

Let us consider the case of an atom. In a magnetic field of strength \(\mathscr{H}\), each electron level characterized by the value of the quantum number \(J\), which determines the total electron angular momentum, is split into \(2J+1\) sublevels. The energies of the sublevels in a weak field (i.e. a field that does not disturb the coupling of the moments from which the total angular momentum \(J\) is composed) are determined by the formula

\[ E_M = g\mu_{\mathrm{B}}\mathscr{H}\cdot M_J, \tag{19} \]

where the magnetic quantum number \(M_J\) takes \(2J+1\) values \((M_J=J,\ J-1,\ldots,-J)\),

\[ \mu_{\mathrm{B}}=\frac{eh}{4\pi m_e c}. \tag{20} \]

— the Bohr magneton (\(m_e\) is the electron mass), \(g\) is the Landé factor (gyromagnetic ratio), depending on the quantum numbers characterizing the given level, and in the special case of normal (Russell–Saunders) coupling of moments having the value

\[ g=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}, \tag{21} \]

where \(S\) and \(L\) are the total spin and total orbital quantum numbers (\(g=1\) for a purely orbital moment and \(g=2\) for a purely spin moment).

According to formula (19), \(2J+1\) equidistant levels are obtained (Fig. 9). According to the selection rule

\[ \Delta M_J=\pm 1 \tag{22} \]

for the magnetic quantum number, transitions between adjacent levels are possible; the frequencies of these transitions are equal to

\[ \nu=\frac{E_{M_J+1}-E_{M_J}}{h}=\frac{1}{h}\,g\mu_{\mathrm{Bor}}\mathcal{H}. \tag{23} \]

Fig. 9. Sublevels of the Zeeman splitting and transitions between them.

The magnetic quantum number \(M\) corresponds to the value of \(J\) (or \(I\) or \(N\)) equal to 3; \(2J+1=7\) values from \(-3\) to \(+3\). On the right is shown the relation between \(J\) and \(M\) from the point of view of visual representations.

The order of magnitude of the transition frequency is obtained by putting \(g=1\). For \(\mathcal{H}=1\) oersted

\[ \frac{\nu}{c}=4.7\cdot 10^{-5}\ \mathrm{cm}^{-1}; \]

\[ \nu=1.4\cdot 10^{6}\ \mathrm{cps}=1.4\ \mathrm{Mcps}. \tag{24} \]

For commonly used magnetic fields with intensities of thousands of oersteds, the frequencies lie in the microwave region. From a visual semiclassical point of view, the magnetic moment, equal to \(\mu=g\mu_{\mathrm{Bor}}J\), precesses about the direction of the magnetic field \(z\) (Fig. 10), making with it an angle \(\alpha\), for which, according to the rules of space quantization,

\[ \cos\alpha=\frac{M_J}{J}, \tag{25} \]

and the projection of the magnetic moment on the direction of the field is equal to

\[ \mu_z=\mu\cos\alpha=g\mu_{\mathrm{Bor}}J\cdot\frac{M_J}{J}=g\mu_{\mathrm{Bor}}\cdot M_J. \tag{26} \]

The transition frequency (23) coincides with the Larmor frequency \(\nu_L\) of precession of the magnetic moment \(\frac{\mu}{J}=g\mu_{\text{Bohr}}\), equal to \(\frac{1}{h}g\mu_{\text{Bohr}}\mathcal H\).

The selection rule (22) corresponds to oscillations in a plane perpendicular to the direction of the external magnetic field (the \(\sigma\)-components of the Zeeman splitting, circularly polarized in longitudinal observation and linearly polarized in transverse observation); transitions with \(\Delta M_J=\pm 1\) will therefore be caused by an alternating magnetic*) field of frequency \(\nu\), determined by formula (23), directed perpendicular to the external magnetic field. The frequency \(\nu\) is the frequency of magnetic resonance. From a visual point of view this permits a very simple interpretation. A perpendicular magnetic field \(\mathcal H'\) tends to tip over the magnetic moment (Fig. 10). A field oscillating with frequency \(\nu\) along some axis (for example, \(x\)) can be decomposed into two fields rotating about the direction \(z\) with the same frequency in opposite directions. When the field frequency \(\nu\) coincides with the Larmor precession frequency \(\nu_L\), the rotating field (whose angular velocity has the same sign as the angular velocity of precession) will all the time exert a tipping action on the magnetic moment and cause a change in its angle of inclination, which corresponds to the possibility of a quantum transition. In the absence of resonance the alternating magnetic field will alternately tip over and raise the magnetic moment, and transitions will not occur. Thus, at resonance the perpendicular magnetic field will not change with time relative to a coordinate system rotating with the angular velocity of precession, whereas in the absence of resonance this field will be alternating (the frequency of its relative rotation will be equal to \(\nu-\nu_L\)). The visual treatment of magnetic resonance by introducing a coordinate system rotating about the direction of the magnetic field with the Larmor frequency \(\nu_L\) is widely used, and its results can be expressed in quantum-mechanical form \(^{26}\).

Fig. 10. Precession of the magnetic moment \(\mu\) about the direction of the field.
\(\mathcal H\) is the external constant magnetic field; \(\mathcal H'\) is the alternating magnetic field perpendicular to it.

Magnetic resonance associated with electronic magnetic moments can be observed only for substances whose atoms or molecules possess such moments, and therefore for

*) Transitions between the sublevels of the Zeeman splitting correspond to magnetic dipole radiation.

paramagnetic and ferromagnetic substances. The phenomena of magnetic resonance observed in paramagnetic bodies and in ferromagnetic bodies are called, respectively, paramagnetic and ferromagnetic resonance. These phenomena are usually observed in the microwave region, at fields of hundreds and thousands of oersteds [cf. estimate (24)].

b) Transitions caused by nuclear
magnetic moments

For atoms and molecules containing nuclei that possess magnetic moments, in magnetic fields there is obtained a Zeeman splitting caused by these moments. The energies of the sublevels in a magnetic field will be determined by a formula analogous to formula (19), in which the quantum number \(J\) is replaced by the quantum number \(I\), and the Bohr magneton \(\mu_{\text{bor}}\) by the nuclear magneton \(\mu_{\text{nucl}}\):

\[ E_{M_I}=g_I \mu_{\text{nucl}} \mathcal{H}\cdot M_I, \tag{27} \]

where

\[ \mu_{\text{nucl}}=\frac{eh}{4\pi m_p c} \tag{28} \]

contains, instead of the electron mass \(m_e\) entering expression (20) for the Bohr magneton, the proton mass \(m_p\) (and is correspondingly 1837 times smaller than \(\mu_{\text{bor}}\)); \(g_I\) is the nuclear gyromagnetic ratio (nuclear Landé factor), equal for the proton itself to approximately 5.6.

The transition frequencies, taking into account the selection rule \(\Delta M_I=\pm 1\) for the nuclear magnetic quantum number \(M_I\), will be equal to

\[ \nu=\frac{E_{M_I+1}-E_{M_I}}{h}=\frac{1}{h}g_I \mu_{\text{nucl}}\mathcal{H}. \tag{29} \]

We obtain the order of magnitude by putting \(g_I=1\). For \(\mathcal{H}=1\) oersted,

\[ \frac{\nu}{c}=2.55\cdot 10^{-8}\ \text{cm}^{-1};\qquad \nu=760\ \text{cycles}. \tag{30} \]

At field strengths of thousands of oersteds, the transition frequencies fall in the range of frequencies of the order of several Mc, i.e., in the region of ordinary radio frequencies. Such transitions will be induced by an alternating magnetic field of frequency \(\nu\), determined by formula (29), directed perpendicular to the constant external magnetic field. Nuclear magnetic resonance will occur. In substances containing nuclei that possess magnetic moments, nuclear paramagnetic resonance (also called nuclear induction) will be observed in the region of ordinary radio frequencies. The intuitive interpretation based on consideration of the precession of magnetic moments is applicable in this case as well.

We have considered Zeeman splitting caused by nuclear moments, independently of Zeeman splitting caused by electronic moments. This is legitimate when the external magnetic field is sufficiently large to break the coupling of the electronic magnetic moment \(\mathbf{J}\) and the nuclear magnetic moment \(\mathbf{I}\), which leads to their addition into the total moment \(\mathbf{F}\). Owing to the weakness of this coupling, it is already broken at small magnetic-field strengths. In this case, from a visual point of view, we obtain an independent precession of the electronic and nuclear moments about the direction of the magnetic field, and the precession frequency of the first moment is large in comparison with the precession frequency of the second [cf. (23) and (29)]. Each sublevel with a given \(M_J\) turns out to be split into \(2I+1\) close components with different values of \(M_I\). Let us note that the number of components at once makes it possible to determine the nuclear spin \(I\). In fields whose strength is insufficient to break completely the coupling of the moments \(\mathbf{J}\) and \(\mathbf{I}\), phenomena will be observed analogous to the Paschen–Back effect in atomic spectra.

c) Transitions caused by rotational magnetic moments

For molecules one obtains Zeeman splitting caused by rotational magnetic moments. Transitions between neighboring sublevels of the Zeeman splitting will occur under the action of an alternating field of the corresponding resonance frequency. Rotational magnetic moments, as was already indicated above, are of the same order as nuclear magnetic moments, and therefore the frequencies of these transitions fall in the same range as the frequencies of transitions caused by nuclear magnetic moments. The energy of the Zeeman-splitting sublevels may be represented in the form

\[ E_{M_N}=g_N\mu_{\mathrm{nucl}}\mathcal{H}M_N, \tag{31} \]

where \(g_N\) is the gyromagnetic ratio (Landé factor) for rotational levels, while the transition frequencies \((\Delta M_N=\pm1)\) will be expressed by the formula

\[ \nu=\frac{1}{h}g_N\mu_{\mathrm{nucl}}\mathcal{H}, \tag{32} \]

analogous to (29).

With the simultaneous presence of nuclear magnetic moments and a rotational magnetic moment, in strong fields they will precess independently of one another, while in weaker fields, insufficient to break the coupling between the moments, a complicated pattern of splitting will be observed, also analogous, as in the case mentioned above, to the Paschen–Back effect.

3. Transitions between the Sublevels of Stark Splitting (Electric Resonance)

For nonlinear molecules having a permanent dipole moment \(\mathbf p\), in an electric field a linear Stark effect is obtained—the splitting of levels proportional to the intensity of the electric field \(\mathscr E\). In the simplest case, when the molecule is a symmetric top with a dipole moment directed along the symmetry axis of the molecule [to which corresponds the moment of inertia \(I_1\), see (10)], the energy of a sublevel with a given value of the magnetic quantum number \(M_N\), which determines the projection of the angular momentum on the direction of the field and takes \(2N+1\) values, is equal to

\[ E_{M_N}=-\frac{p\mathscr E K M_N}{N(N+1)}, \tag{33} \]

where \(K\) is the quantum number determining the projection of the angular momentum on the axis of the molecule [see (8)]. This quantum-mechanical formula of first approximation*) can be obtained from pictorial considerations, taking into account that the energy of a dipole in an electric field is equal to \(-p_N\mathscr E \cos\vartheta\), where \(p_N\) is the projection of the dipole moment on the direction of the angular momentum \(\mathbf N\), and \(\vartheta\) is the angle between \(\mathbf N\) and the direction of the field (Fig. 11). According to the rules of space quantization,

\[ p_N=p\frac{K}{N}, \qquad \cos\vartheta=\frac{M_N}{N}, \tag{34} \]

whence

\[ E_{M_N}=-p_N\mathscr E\cos\vartheta=-p\frac{\mathscr E K M_N}{N^2}, \tag{35} \]

which, when \(N^2\) is replaced by the quantum-mechanical value \(N(N+1)\), gives formula (33).

An alternating electric field, perpendicular to the constant external field \(\mathscr E\), will induce transitions for which \(\Delta M_N=\pm 1\); the corresponding transition frequency is

\[ \nu=\frac{E_{M_N}-E_{M_N+1}}{h} =\frac{1}{h}\frac{p\mathscr E K}{N(N+1)}, \tag{36} \]

*) Representing the mean value of the perturbation energy, i.e., the first approximation of perturbation theory.

and at this frequency an electric resonance will be observed. The maximum value of the splitting is obtained for \(K=N\) and is equal to

\[ \nu=\frac{\Delta E}{h}=\frac{1}{h}\frac{p\mathcal{E}}{N+1}. \tag{37} \]

For

\[ p=1\ \text{debye}\ (10^{-18}\ \mathrm{CGSE})\quad \text{and}\quad \mathcal{E}=1\ \mathrm{V/cm} \]

\[ \frac{\nu}{c}=1.7\cdot 10^{-5}\frac{1}{N+1}\ \mathrm{cm}^{-1}; \]

\[ \nu\simeq \frac{0.5\cdot 10^{6}}{N}\ \text{cps}=\frac{0.5}{N}\ \mathrm{Mc}. \tag{38} \]

Therefore, the corresponding transition frequencies, at electric-field strengths of the order of thousands of \(\mathrm{V/cm}\), may amount to thousands of megacycles and fall in the microwave region.

Along with the linear splitting there will also be quadratic splitting, proportional to \(\mathcal{E}^{2}\). It is precisely such splitting that is obtained for linear molecules\(^*\). The transition frequencies in this case, in fields of the order of thousands of \(\mathrm{V/cm}\), will not exceed tens of megacycles and correspond to the region of ordinary radio frequencies.

In the diagram of Fig. 1 the orders of magnitude of the principal types of transitions listed above are compared. The great variety of types of transitions studied by radiospectroscopic methods leads to an even greater variety of types of spectra, owing to combinations of different types of transitions, just as in the optical region, for molecules, electronic transitions are accompanied by vibrational and rotational transitions.

For example, superposed on the rotational transitions studied in the microwave region are transitions caused by nuclear moments, which leads to a complicated structure of the observed rotational lines; in a magnetic field the Zeeman effect is observed for this structure. In an electric field the linear and quadratic Stark effect is observed for rotational lines, which becomes more complicated in the presence of quadrupole interaction. Superposed on transitions between the sublevels of Zeeman splitting caused by electronic moments are transitions between the sublevels of Zeeman splitting caused by nuclear moments.

\(^*\) A linear molecule may be regarded as a symmetric top with \(K=0\) [cf. (9) and (5)]; for it the energy, determined in the first approximation by formula (33), vanishes, and the energy must be determined in the second approximation, proportional to \(\mathcal{E}^{2}\).

II. ABSORPTION AND EMISSION IN THE RADIO-FREQUENCY REGION

A characteristic feature of the transitions between closely spaced energy levels studied by the methods of radiospectroscopy is that, along with absorption, stimulated emission plays the principal role, while spontaneous (self-acting) emission is practically absent. Only in observations of certain cosmic sources of radio emission is spontaneous emission detected. Along with the well-studied continuous spectra of radio emission of cosmic sources, in the latter there is observed the emission line 1420 MHz \((\lambda = 21\ \mathrm{cm})\), corresponding to the transition between the sublevels of the hyperfine structure of the ground level \(1^2S_{1/2}\) of the hydrogen atom, which was considered above (see Fig. 8). The absence of spontaneous emission under ordinary conditions of laboratory investigations is connected with the fact that, owing to the proportionality of the probability of this emission to the third power of the frequency \((\nu^3)\), for small frequencies its probability is very small in comparison with the probability in the optical region. For example, a decrease in frequency in the microwave region, corresponding to \(\lambda = 5\ \mathrm{mm}\) \((6\cdot 10^{10}\ \mathrm{cps} = 60\,000\ \mathrm{MHz})\), by \(10^4\) times in comparison with the frequency in the visible region, corresponding to \(\lambda = 0.5\ \mu\) \((6\cdot 10^{14}\ \mathrm{cps})\), leads to a decrease of \(\nu^3\) by \(10^{12}\) times. Therefore spontaneous emission in radiospectroscopic investigations of terrestrial sources may be disregarded. On the contrary, stimulated emission plays a very substantial role, since for closely spaced levels their populations are almost identical not only at ordinary but also at low temperatures; because of this the number of processes of stimulated emission (transitions from the upper level to the lower, see Fig. 12) differs only very little from the number of absorption processes (transitions from the lower level to the upper).

Fig. 12. Scheme of absorption and stimulated emission.
\(E_1\) and \(E_2\) are the energies of the lower and upper levels; \(N_1\) and \(N_2\) are the populations of the lower and upper levels.

Let us consider in more detail the processes of absorption and stimulated emission. The energy of radiation of frequency

\[ \nu = \frac{E_2 - E_1}{h}, \]

absorbed in a small interval of time \(dt\), is equal to

\[ h\nu N_1 B_{12}\rho(\nu)\,dt, \tag{39} \]

where \(\rho(\nu)\) is the radiation density, \(N_1\) is the number of particles in the state with energy \(E_1\) (at the lower level) contained in \(1\ \mathrm{cm}^3\), and \(B_{12}\) is the probability of absorption. The energy of radiation emitted over

that same interval of time, is, if spontaneous emission is neglected,

\[ h\nu N_2 B_{21}\rho(\nu)\,dt, \tag{40} \]

where \(N_2\) is the number of particles in the state with energy \(E_2\) (at the upper level), contained in \(1\ \mathrm{cm}^3\), and \(B_{21}\) is the probability of stimulated emission. Since \(B_{21}=B_{12}\) *), we obtain for the difference between the absorbed and emitted energies

\[ h\nu (N_1-N_2)B_{12}\rho(\nu)\,dt. \tag{41} \]

Since the decrease in intensity of the absorbed beam over a length element \(dx\) is equal to

\[ -\alpha\rho(\nu)\,dx=-\alpha\rho(\nu)c\,dt, \tag{42} \]

where \(c\) is the velocity of light, the absorption coefficient is

\[ \alpha=\frac{h\nu}{c}(N_1-N_2)B_{12}. \tag{43} \]

The ratio of the populations of the levels in thermal equilibrium is determined by the ratio of the Boltzmann factors

\[ \frac{N_2}{N_1} = \frac{e^{-\frac{E_2}{kT}}}{e^{-\frac{E_1}{kT}}} = e^{-\frac{E_2-E_1}{kT}} = e^{-\frac{h\nu}{kT}}. \tag{44} \]

Under the conditions of radiospectroscopic investigations the ratio

\[ \frac{h\nu}{kT} = \frac{\frac{\nu}{c}}{\frac{kT}{hc}} \tag{45} \]

is very small. At ordinary temperatures the thermal energy, expressed in \(\mathrm{cm}^{-1}\), is \(kT/hc=200\ \mathrm{cm}^{-1}\) (\(1^\circ\) corresponds to \(0.70\ \mathrm{cm}^{-1}\)), and even for a frequency of \(10\ \mathrm{cm}^{-1}\) (\(300\,000\ \mathrm{Mc}\)) the ratio (45) is only 0.05. Therefore, approximately, one may put

\[ N_1-N_2 = N_1\left(1-\frac{N_2}{N_1}\right) = N_1\left(1-e^{-\frac{h\nu}{kT}}\right) \simeq \]

\[ \simeq N_1\left(1-1+\frac{h\nu}{kT}\right) = N_1\frac{h\nu}{kT}, \tag{46} \]

and consequently for the absorption coefficient \(\alpha\) we obtain the final formula

\[ \alpha = \frac{h\nu}{c}\cdot\frac{h\nu}{kT}\cdot N_1B_{12} = \frac{(h\nu)^2}{ckT}B_{12}. \tag{47} \]

*) The probabilities of the direct and reverse processes are equal; here, for simplicity, we consider nondegenerate levels, for which the statistical weights \(g_1=g_2=1\); for degenerate levels \(g_1B_{12}=g_2B_{21}\).

Thus, the absorption coefficient is inversely proportional to temperature. In comparison with the usual absorption coefficients in the optical region, it is very small, and its determination becomes possible only thanks to the great sensitivity of radiospectroscopic methods for measuring energy absorption.

It is important to note that at large radiation densities \(\rho(\nu)\), thermal equilibrium between the populations of the combining levels is disturbed. As \(\rho(\nu)\) increases, the number of particles \(N_2\) in the state \(E_2\) approaches the number of particles \(N_1\) in the state \(E_1\), and in the limit saturation sets in, at which the absorbed energy \(aW\) no longer depends on the energy \(W\) of the incident radiation beam, i.e., the absorption coefficient decreases inversely proportional to \(W\). Because of the presence of the saturation effect, which at sufficiently large \(W\) is observed experimentally, in radiospectroscopic investigations one uses electromagnetic-wave generators that are not too powerful; in particular, in studies in the microwave region one usually uses klystrons, and not the more powerful magnetrons.

The magnitude of the absorption probability \(B_{12}\) is proportional to the square of the matrix element of the corresponding moment (electric dipole or magnetic dipole) for the transition under consideration and depends on the type of transition. For rotational transitions we have ordinary (electric) dipole radiation; the transition probabilities are comparatively large and, despite the factor \(\dfrac{h\nu}{kT}\) in expression (47) for the absorption coefficient, this coefficient turns out to be sufficiently large. Therefore absorption in the microwave region is readily observed in the gas phase; in a number of cases it can be detected even at very small concentrations of the molecules being studied. For transitions between the levels of Zeeman splitting, i.e., for magnetic resonance, and also for purely quadrupole transitions, we have magnetic-dipole radiation*; the transition probability is considerably smaller (by \(10^4\)—\(10^6\) times), and therefore absorption is usually observed in the solid or liquid phase; only in individual cases (for the paramagnetic gases \(\mathrm{O}_2\), \(\mathrm{NO}\), and \(\mathrm{NO}_2\)) has it been possible to observe magnetic resonance also for absorption in gases. Transitions between sublevels of fine and hyperfine structure for atoms are likewise magnetic-dipole transitions. As for electric quadrupole radiation, its probability for the radio-frequency region is very small not only in comparison with the probability of ordinary electric-dipole radiation, but also in comparison

* Quadrupole splitting is caused by electrostatic interaction, while the transitions themselves between the levels are associated with a change in the projection of the magnetic moment of the nucleus (with a change in the quantum number \(M_I\)).

with the probability of magnetic dipole radiation, since it contains the additional factor \(\nu^2\), which rapidly decreases as the frequency decreases.

Of great interest is the question of the width of spectral lines in the radio-frequency region. The natural line width, as a rule, is very small*). The Doppler width usually also plays no role, and the main factor determining the width of spectral lines in the case of gases is broadening as a result of collisions. In solids and liquids, in the study of paramagnetic resonance, the line width is determined by various types of interaction of magnetic moments (spin–spin interaction, spin–lattice interaction, exchange interaction). In general, depending on the experimental conditions and the choice of objects for radiospectroscopic investigation, the line width may be highly varied and, in the most favorable cases, amounts to negligible fractions \((10^{-6}—10^{-7})\) of the frequency of the line being studied.

III. PRINCIPLES OF EXPERIMENTAL METHODS OF RADIOSPECTROSCOPY

The features of absorption and emission in the radio-frequency region of the spectrum considered in the preceding section are very important in the analysis of the experimental methods of radiospectroscopy, which are quite varied. These methods may be divided into two groups: methods of radio-wave absorption in a certain volume of the substance under study, and methods of resonance in atomic and molecular beams. We shall discuss only the fundamental principles of these methods, a detailed description of which is available both in review articles \(^{13,15,16,20,21,23,24}\) and especially in the original literature.

1. Absorption methods

a) Investigation of the absorption of gases by means of microwave spectrographs with waveguides

The basic scheme of a microwave spectrograph is shown in Fig. 13. The microwave generator \(G\) is usually a klystron. The absorbing chamber is a waveguide into which the gas under study is introduced. The microwaves that have passed through the waveguide are received by a crystal detector \(D\), amplified by an amplifier \(U\), and recorded by an oscillograph or a mechanical recorder \(P\). The curve obtained when the frequency is varied

*) An exception is provided by excited electronic states, an example of which is the two-quantum state of the hydrogen atom \((n=2)\).

recording of the spectrum makes it possible to determine the frequencies of the absorption lines, their contours, and intensities. The waveguide may be placed in a magnetic field for investigation of the Zeeman effect (Zeeman spectrograph), and an electric field may be created in it for investigation of the Stark effect (Stark spectrograph).

Fig. 13. Diagram of a microwave spectrograph.
\(G\)—microwave generator (klystron), \(D\)—detector, \(U\)—amplifier, \(P\)—recording device.

Modern microwave spectrographs, whose varied designs are used by various investigators20, 21, 22, make it possible to detect absorption coefficients of the order of \(10^{-9}\), and to achieve high resolving power and great measurement accuracy down to the region \(\lambda = 1—2\ \text{mm}^{27}\).

b) Investigation of magnetic resonance in liquids and solids

For the study of magnetic resonance, the substance under investigation is placed between the poles of an electromagnet, which creates a uniform constant magnetic field, and at the same time a radio-frequency field is applied (whose magnetic component is perpendicular to the constant magnetic field). The energy of the radio-frequency field is partially absorbed by the substance owing to transitions between the sublevels of Zeeman splitting (\(\sigma\)-transitions with \(\Delta M = \pm 1\)).

Fig. 14. Diagram for investigating paramagnetic and ferromagnetic resonance in the microwave region.
\(G\)—microwave generator (klystron), \(P\)—resonant cavity, \(D\)—detector, \(U\)—amplifier, \(P\)—recording device, \(N, S\)—poles of the electromagnet.

For the investigation of paramagnetic and ferromagnetic resonance in the microwave region (see the diagram in Fig. 14), the investigated

the substance is usually placed in a resonant cavity \(P\), situated between the poles of an electromagnet, to which electromagnetic waves generated by a klystron \(G\) are supplied by means of a waveguide. Detection, amplification, and recording are carried out by the same methods as in the case of studying the absorption of gases in the microwave region.

For the study of nuclear paramagnetic resonance (nuclear induction), a radio-frequency generator is used, producing electromagnetic oscillations in the required frequency range; these are fed to a magnetic coil \(K\), inside which the substance under investigation is located; the coil is placed between the poles of an electromagnet (Fig. 15). Various methods are used for detection:

a) the radio-frequency bridge method \(^{8}\)—the coil is introduced into one of the arms of the bridge, and the change in absorption in this arm is measured directly.

b) Bloch’s two-coil method \(^{9}\)—with the aid of a second coil, whose axis is perpendicular both to the constant magnetic field and to the axis of the first magnetic coil producing the alternating magnetic field, the change in the magnetization of the substance caused by the field of the first coil is measured; in the second coil a current is induced owing to the change in the induction flux associated with the precession of the magnetic moments (hence the name “nuclear induction”).

Fig. 15. Scheme for studying nuclear paramagnetic resonance.
\(K\)—coil inside which the substance under investigation is placed, \(N, S\)—electromagnet (the remaining parts of the apparatus are not shown).

c) methods in which changes in the parameters of the circuit containing the coil with the substance cause a change in the mode of high-frequency oscillations \(^{28}\).

d) methods in which short radio-frequency radiation pulses are used—the pulse method of Torrey \(^{29}\) and Hahn’s echo method (two successive pulses) \(^{30}\).

In the study of magnetic resonance, both paramagnetic and ferromagnetic, as well as nuclear (nuclear induction), as a rule the frequency of the generated oscillations is kept unchanged, while the intensity of the magnetic field produced by the electromagnet is varied; at certain intensities of the magnetic field, when the frequency of Larmor precession \(\nu_L\) becomes equal to the transition frequency between adjacent sublevels of the Zeeman splitting, resonance is observed. With such a method, changes in the mode of electromagnetic oscillations caused by a change in the frequency of the generated oscillations are eliminated. The spectrum is recorded on the scale of magnetic-field intensities, from which it is easy to pass to the frequency scale.

c) Investigation of Pure Quadrupole Transitions in Solids

The investigation of pure quadrupole transitions that have been observed in crystals is carried out in the short-wave region by methods analogous to those used in the study of nuclear paramagnetic resonance. The difference is that there is no constant magnetic field, and the sample of the substance under study is subjected only to a radio-frequency field, the magnetic component of which is perpendicular to the axis of symmetry of the crystal. The internal electrostatic fields of the crystal cause a splitting proportional to the magnitude of the quadrupole moment of the nucleus being studied.

To obtain spectra, the frequency of the oscillation generator is varied, analogously to the way this is done in studying gas absorption in the microwave region.

2. Methods of Molecular and Atomic Beams

a) Methods of Magnetic Resonance in Molecular and Atomic Beams

The principle of the method, whose scheme is shown in Fig. 16, consists in the following: the magnetic moments of the molecules or atoms forming the beam are oriented by a homogeneous constant magnetic field, and the perpendicular radio-frequency magnetic field superposed on it causes transitions ($\Delta M = \pm 1$, see Fig. 9) between the sublevels of the Zeeman splitting (as in the study of magnetic resonance by absorption

Fig. 16. Scheme of the investigation of magnetic resonance in molecular and atomic beams.

$I$ — beam source, $P$ — particle receiver, $M_0$ — electromagnet creating a constant homogeneous field $H$, $BB$ — high-frequency circuit creating an alternating magnetic field $H'$, $M_1$ and $M_2$ — electromagnets creating constant inhomogeneous fields with oppositely directed gradients. The solid line shows the path of particles whose magnetic moments are oriented by the field in a definite way; the dotted line is the path of particles whose magnetic moments have undergone reorientation under the action of the alternating field $H'$.

by gases).

in solids and liquids, see above). One measures the change in the intensity of a beam, initially focused on the receiver of molecules or atoms, when the radio-frequency field is switched on. The beam is usually focused by passing it through two inhomogeneous constant magnetic fields with oppositely directed gradients and parallel to a homogeneous constant magnetic field, which is switched on between them, as shown in Fig. 16^13; arrangements are also possible in which focusing is achieved merely by the appropriate placement of diaphragms^31. In the absence of the radio-frequency field, a definite flux of particles reaches the receiver; when the radio-frequency field is switched on, all particles whose projection of magnetic moment has changed in the transitions caused by this field will be deflected differently by the second inhomogeneous field and will no longer reach the receiver, which will lead to a decrease in the beam intensity. The decrease in beam intensity is usually recorded as a function of the magnetic-field strength, i.e., one obtains a spectrum on the magnetic-field-strength scale, as in the case of studies of magnetic resonance by absorption in solids and liquids.

The method considered has very high sensitivity. This is connected with the fact that the change in the beam intensity is determined by the sum of the number of transitions in absorption \((M \to M + 1)\) and the number of transitions in induced emission \((M \to M - 1)\) (since both kinds of transitions remove particles from the original beam), and not by the difference of these numbers, as in absorption methods.

For molecules and atoms that do not possess electronic magnetic moments (in particular, for linear molecules in the \({}^1\Sigma\) state and for atoms in the \({}^1S_0\) state), the method described is used to study the Zeeman effect in the region of ordinary radio frequencies (several MHz or several tens of MHz), due to the magnetic moments of nuclei. Molecular beams are especially widely used for this purpose, since most molecules in the ground state do not possess electronic moments, in contrast to atoms and ions, only a few of which have an electronic moment equal to zero, and which therefore usually have paramagnetic properties. Examples of molecules with an electronic moment equal to zero are the molecules \(\mathrm{H}_2\), \(\mathrm{D}_2\), and \(\mathrm{HD}\). The study of the Zeeman effect, due to the magnetic moments of the nuclei H and D (proton and deuteron), by the method of magnetic resonance in molecular beams has made it possible to determine the corresponding moments with very high accuracy. The accuracy of measurement can be further increased by creating an alternating magnetic field not along the entire path of the molecular beam in the constant field, but only in sections where the molecules enter this

field and emerge from it[^32]. Let us also note that the magnetic moment of the neutron can likewise be determined by the method of magnetic resonance in a neutron beam—according to a scheme analogous to the scheme of Fig. 16, with the difference that, in order to orient the neutron spins, the neutron beam is passed through two pieces of magnetized iron[^33].

For atoms, and also for molecules possessing electronic moments, the ordinary Zeeman effect is studied by the method of magnetic resonance in a beam, usually in the microwave region; in this case the nuclear moments, magnetic and quadrupole, determine the structure of the observed lines.

As a rule, the methods of molecular and atomic beams make it possible to determine the structure of levels and sublevels for the ground electronic state. It is possible, however, to work also with metastable atoms whose lifetime is longer than the time of their flight through the apparatus; in this case the atoms in the beam can be excited by electron impact. Such a method was applied to the investigation of the fine structure of the levels of the hydrogen atom[^31] in the state \(n = 2\); the metastable level is \(2^2S_{1/2}\) (see Fig. 2), the transition from which to the ground level \(1^2S_{1/2}\) is forbidden. The fine structure of the analogous state of the helium ion \(\mathrm{He}^{+}\) was also investigated[^34].

b) The method of electric resonance in molecular beams

Molecules that have permanent dipole moments and form a molecular beam can be oriented in a constant electric field, while a radio-frequency electric field superposed on it will induce transitions between the levels of Stark splitting[^35]. This method is analogous to the method of magnetic resonance in molecular and atomic beams and makes it possible to determine the dipole moments of molecules with high accuracy.

IV. RESULTS OF RADIOSPECTROSCOPIC INVESTIGATIONS

Despite the short period of development of radiospectroscopy, a large number of important results have been obtained by its methods, relating to a wide variety of questions of the structure of matter; there is no doubt that in the coming years the field of application of radiospectroscopy will continue to expand rapidly. Some of the results obtained, which appear to be the most important, are listed below, and certain questions of the further development of radiospectroscopy are also touched upon.

1. Properties of Nuclei

Radiospectroscopic methods are at present used as the principal methods for determining nuclear spins and the associated magnetic (dipole) moments \(\mu\) and quadrupole (electric) moments \(eQ\). In accordance with theoretical ideas\({}^{36}\), even-even nuclei (even ordinal number \(Z\) and even mass number \(A\)) have no spin; even-odd and odd-even nuclei (\(Z\) and \(A\) of different parity) have half-integral spin and possess a magnetic moment, and for \(I>1/2\) a quadrupole moment; finally, odd-odd nuclei (\(Z\) and \(A\) odd) have integral spin (\(I \geqslant 1\)) and possess both a magnetic and a quadrupole moment.

Very accurate determinations of the magnetic moments of nuclei have been carried out by the method of magnetic resonance in molecular beams. The accuracy of these data amounts to thousandths of a percent \((10^{-5}—10^{-3}\%)\).

With very high accuracy, values of the magnetic moments of a large number of nuclei have also been obtained by the method of nuclear induction in solids and liquids\({}^{37}\).

Usually the magnetic moments of nuclei are compared with the magnetic moment of the proton, as a quantity that can be determined from a number of experiments with great accuracy and serves as a natural standard. The precession frequency of the proton in a magnetic field is widely used for determining the strength of the magnetic field (calibration of the magnetic field by proton resonance).

For the magnetic moment of the proton, by comparing the precession frequency with the cyclotron frequency of its motion (with the number of proton revolutions along a circular trajectory in a magnetic field), measured with the aid of an “omegatron,” one obtains the value\({}^{38}\)

\[ \mu_p=(2.79268 \pm 0.00006)\,\mu_{\text{nuc}} . \tag{48} \]

By the analogous method of the “inverse cyclotron” one obtains the value\({}^{39}\)

\[ \mu_p=(2.7924 \pm 0.0002)\,\mu_{\text{nuc}}, \tag{49} \]

in good agreement with value (48). Let us note that the data of the nuclear-induction method require the introduction of certain small corrections.

Nuclear magnetic moments can also be determined, although with lower accuracy (of the order of 1%), from the structure of rotational absorption lines of gases in the microwave region. An important advantage of this method is that, owing to its high sensitivity, it is applicable to the determination of the magnetic moment of radioactive nuclei present only in small concentrations.

The quadrupole moments of nuclei have been determined both from analysis of the Zeeman splitting when it is observed by the method

nuclear magnetic resonance in molecular beams, and also from the structure of paramagnetic-resonance lines in solids. By the first method the quadrupole moment of the deuteron was determined; it is equal to \(0.00273 \cdot 10^{-24}\ \text{cm}^2\).

However, most data on the quadrupole moments of nuclei have been obtained by studying the structure of rotational lines in the absorption of gases in the microwave region and by studying purely quadrupole spectra in crystals. In all cases what is found directly from experiment is not the quadrupole moment of the nucleus \(eQ\), but the magnitude of the quadrupole interaction, determined by formula (17). The accuracy of the determination of \(eQ\) depends, for the experimentally found value of \(eQq\), on the accuracy of the calculation of the gradient \(q\) of the electric field acting on the nucleus; in the simplest cases this calculation can be carried out with sufficient reliability. The usual order of magnitude of the quadrupole moments of nuclei \(Q\), in units of \(10^{-24}\ \text{cm}^2\) (in barns), is hundredths for light nuclei, tenths for heavier nuclei, and several units for the heaviest nuclei.

Knowledge of the spins, magnetic moments, and quadrupole moments of nuclei is of very great interest for the development of the theory of the atomic nucleus, and therefore work on determining these quantities by radiospectroscopic methods is very topical.

Determination with high accuracy of the rotational constants of molecules from the microwave absorption spectra of isotopic molecules makes it possible to find, with high accuracy exceeding \(10^{-5}\), the ratio of isotope masses. For example, for the ratio of the masses of the isotopes \(\mathrm{Cl}^{35}\) and \(\mathrm{Cl}^{37}\), from the spectra of the molecules \(\mathrm{FCl}^{35}\) and \(\mathrm{FCl}^{37}\) one obtains the value\(^ {40}\)

\[ \frac{m_{\mathrm{Cl}^{35}}}{m_{\mathrm{Cl}^{37}}}=0.945977 \pm 0.000004; \tag{50} \]

and from the spectra of the molecules \(\mathrm{JCl}^{35}\) and \(\mathrm{JCl}^{37}\) one obtains the value\(^ {41}\)

\[ \frac{m_{\mathrm{Cl}^{35}}}{m_{\mathrm{Cl}^{37}}}=0.945980 \pm 0.000005. \tag{51} \]

The agreement of the values, as shown by comparison of (50) and (51), is very good.

2. Properties of Atoms

A very important task is the determination, by radiospectroscopic methods, of precise values of the electronic magnetic moments of atoms. These moments can be determined with high accuracy by the method of magnetic resonance in atomic beams. Of special interest is the determination of the spin magnetic moment of the electron. According to the most accurate data, obtained by the method of magnetic resonance in atomic beams\(^ {42}\),

the magnetic moment of the electron, expressed in Bohr magnetons, is equal to

\[ \mu_e=(1.001146\pm0.000012)\,\mu_{\text{Bohr}}, \tag{52} \]

whereas according to Dirac’s theory it should be exactly equal to \(\mu_{\text{Bohr}}\) (gyromagnetic ratio \(g=2\)). The deviation of the value (52) from \(\mu_{\text{Bohr}}\) (the anomaly of the electron magnetic moment) is due to interaction effects, taken into account by the methods of quantum electrodynamics; the theoretical calculation gives for \(\mu_e\) the value[^43]

\[ \mu_e=\left(1+\frac{a}{2\pi}-2.973\,\frac{a^2}{\pi^2}\right)\mu_{\text{Bohr}} =1.0011454\,\mu_{\text{Bohr}}, \tag{53} \]

in the computation of which corrections of order \(a\) and \(a^2\) have been taken into account, where \(a\) is the fine-structure constant. The experimental value (52) is thus in agreement with the theoretical value (the agreement in the sixth digit after the decimal point is, of course, accidental).

Another very important result of radiospectroscopic investigations of atoms is the data from very precise measurements of the fine structure of the two-quantum level \((n=2)\) of the hydrogen atom, carried out with an atomic beam of metastable hydrogen atoms by Lamb’s magnetic-resonance method[^31]. With high accuracy the shift of the \(S\)-level was determined; as a result of a very careful analysis of the experimental data, taking into account a number of small corrections, the following values were obtained for the shift \(\Delta_S\) of the level \(2^2S_{1/2}\) relative to the level \(2^2P_{1/2}\):

for the hydrogen atom H

\[ \Delta_{SH}=1057.77\pm0.10\ \text{MHz}, \tag{54} \]

for the deuterium atom D

\[ \Delta_{SD}=1059.00\pm0.10\ \text{MHz}. \tag{55} \]

The most accurate calculation by the methods of quantum electrodynamics gives[^44]:

for the hydrogen atom H

\[ \Delta_{SH}=1057.19\pm0.16\ \text{MHz}, \tag{56} \]

for the deuterium atom D

\[ \Delta_{SD}=1058.49\pm0.16\ \text{MHz}. \tag{57} \]

The discrepancy between the experimental and theoretical data is very small and amounts to only \(0.05\%\). This serves as confirmation of the correctness of the calculation methods used, which is especially important, since in these calculations the theoretically insufficiently justified method of “renormalization” is applied in order to eliminate the divergences arising in the modern theory (the infinite mass of the electron, the infinite energy of zero oscillations of the electromagnetic field, etc.). The very good agreement between theory and experiment, both for the shift of the \(S\)-level and for the anomaly of the electron magnetic moment, shows that, despite the lack of rigor of the calculation methods used, the theory is on the right path.

It should be noted that the difference of the experimental values of the shift of the \(S\)-levels of deuterium and hydrogen, equal to

\[ \Delta_{SD}-\Delta_{SH}=1.23\pm0.20\ \text{Mc}, \tag{58} \]

and due to the difference between the nuclei of the light and heavy atoms of hydrogen, is in agreement with the theoretical value of this difference, equal to

\[ \Delta_{SD}-\Delta_{SH}=1.30\ \text{Mc}. \tag{59} \]

The shift of the \(S\)-level has also been measured for the \(\mathrm{He}^{+}\) ion in the two-quantum state. It was found to be\({}^{34}\):

\[ \Delta_{S\mathrm{He}}=14020\pm100\ \text{Mc}, \tag{60} \]

which differs somewhat from the theoretical value (by \(1.2\%\)).

Investigation of the fine structure of the two-quantum state of the hydrogen atom makes it possible to determine with great accuracy also the doublet splitting \(2^{2}P_{3/2}-2^{2}P_{1/2}\). The most accurate result has been obtained for deuterium

\[ \frac{E\left(2^{2}P_{3/2}\right)-E\left(2^{2}P_{1/2}\right)}{h} =10971.58\pm0.20\ \text{Mc}, \tag{61} \]

The value found for the doublet splitting in deuterium makes it possible to obtain a very accurate value of the fine-structure constant \(\alpha\) and is taken into account in determining the system of exact values of the fundamental atomic constants\({}^{45}\).

With very high accuracy it has been possible to measure, by the method of magnetic resonance in atomic beams, also the hyperfine structure of the ground state for the atoms of hydrogen and deuterium\({}^{46}\). The magnitude \(\nu\) of the splitting of the ground state \(1^{2}S_{1/2}\) is:

\[ \text{for the hydrogen atom } \mathrm{H}\quad \nu_{\mathrm{H}}=1420.4051\pm0.0002\ \text{Mc}, \tag{62} \]

\[ \text{for the deuterium atom } \mathrm{D}\quad \nu_{\mathrm{D}}=327.38424\pm0.00008\ \text{Mc}. \tag{63} \]

The results listed above were obtained by the magnetic-resonance method. In principle, transitions between levels of the fine structure, due to multiplet splitting, and transitions between levels of the hyperfine structure, due to nuclear moments, can also be observed by the method of absorption of gases in the microwave region; however, this requires a further increase in the sensitivity of microwave spectrographs. The small absorption coefficients are connected with the fact that the corresponding radiation of the atoms is magnetic dipole radiation.

3. Properties of molecules\({}^{21}\)

For molecules, the most direct method of obtaining data for moments of inertia and, consequently, internuclear distances (i.e., distances between nuclei) is by the investigation of purely rotational ...

rotational spectra of gases in the microwave region. The applicability of this method is limited by the fact that only dipolar molecules possess such spectra*). At the present time the rotational spectra of a large number of molecules, especially halide compounds, have been investigated. Among the molecules studied there are diatomic and linear polyatomic molecules, and nonlinear molecules that are both symmetric and asymmetric tops. The usual accuracy in determining internuclear distances is several thousandths or one or two hundredths of an Å. At the same time the values of valence angles are also obtained. To increase the number of parameters determined from experiment—which for linear molecules is equal to one, and for nonlinear molecules does not exceed three (the three rotational constants for molecules that are asymmetric tops)—the study of isotopic molecules is widely used. Data on bond lengths and values of valence angles are naturally of considerable interest. Comparison of these data for different molecules makes it possible to draw a number of conclusions about the causes determining the difference in the values of analogous bond lengths and valence angles in different molecules (for example, the different lengths of the C—C bond for its different positions in the molecule).

From data on the Stark splitting of rotational levels, obtained by studying microwave absorption spectra of gases and by electric resonance in molecular beams, the values of the dipole moments of molecules can be determined with considerable accuracy (especially by the latter method). The greatest amount of data has been obtained for molecules of halide compounds.

Specific data obtained by radiospectroscopic methods are data on intramolecular and intracrystalline fields, on the basis of determining the magnitude of the quadrupole interaction. Knowing the value of the quadrupole moment of the nucleus, one can, according to formula (17), find the values of the quantity \(q\), i.e. the gradients of the fields produced by the electronic shells of molecules. It is significant that the ratio of the quadrupole interactions \(eQq_1\) and \(eQq_2\) for a given nucleus in different molecules gives the ratio of the gradients. The quantity \(q\) depends directly on the character of the bond. For an ionic bond, when a spherically symmetric filled electron shell is formed around the nucleus, \(q = 0\); \(q\) will increase as one passes from an ionic bond to a covalent one. This is indeed well

*) One may hope that it will be possible to study also nonpolar molecules in excited vibrational states, in which a small dipole moment arises owing to a violation of the symmetry of the molecule [47]. For this it is necessary to use the most sensitive microwave spectrographs.

is justified experimentally for molecules in which the bond is effected by \(p\)-electrons.

Also specific are radiospectroscopic data on the rotational magnetic moments of molecules, obtained by studying the Zeeman effect on rotational lines\(^{48}\). For hydrogen the rotational magnetic moment was determined in a study of magnetic resonance in molecular beams containing the molecules \(\mathrm{H}_2\), \(\mathrm{D}_2\), and \(\mathrm{HD}\). The values of the rotational magnetic moments show that the outer electrons of a molecule, during its rotation, cannot be regarded as rigidly bound to its skeleton. Further investigation of rotational magnetic moments is of interest from the point of view of studying the properties of chemical bonds.

In addition to the data listed above, which give a general characterization of molecules, radiospectroscopic methods make it possible to obtain information on various more particular properties of individual types of molecules. Thus, the investigation of microwave absorption made it possible to study in great detail the inversion splitting for the ammonia molecule \(\mathrm{NH}_3\), to measure the \(l\)-doubling for linear polyatomic molecules, and to determine the spin splitting of the levels of paramagnetic diatomic molecules, in particular \(\mathrm{O}_2\). Data have been obtained on the centrifugal stretching of polyatomic molecules at large values of the rotational quantum number.

Alongside the data obtained by determining the positions of the energy levels of molecules on the basis of the observed frequencies of spectral lines, important information can be obtained by measuring the intensities and widths of the corresponding lines. The intensities of lines are determined by transition probabilities, proportional to the squares of the matrix elements of the dipole or magnetic moment, and by the populations of the levels. The transition probabilities, especially the relative ones, are known or can be calculated for a number of cases. From the measured ratio of the intensities of rotational lines one can, using the ratios of the probabilities of rotational transitions (reliably determined theoretically, on the basis of taking symmetry properties into account), find the ratio of the populations. Measurement of the ratio of the intensities of rotational lines for excited states corresponding to rotational vibrations, to the intensities of rotational lines for the ground vibrational state, made it possible (taking into account the Boltzmann distribution of molecules over levels) to approximately calculate the values of the frequencies of rotational vibrations for the molecules \(\mathrm{CH}_3\mathrm{CF}_3\), \(\mathrm{CH}_3\mathrm{SiH}_3\), \(\mathrm{CH}_3\mathrm{SiF}_3\), \(\mathrm{CF}_3\mathrm{SF}_5\). These frequencies turned out to be equal to 230, 183, 140, and \(93.5\ \mathrm{cm}^{-1}\), respectively; from them were calculated the heights of the potential barrier hindering internal rotation (under the assumption of a definite form of this barrier); they turned out to be equal, respectively, to 1200, 460, 410, and \(220\ \mathrm{cm}^{-1}\).

The width of microwave absorption lines of molecules in gases is determined basically, as also for optical spectra, by broadening as a result of collisions, as was mentioned above.

For the best-studied case of line broadening in the inversion spectrum of NH$_3$ as a function of pressure, the observed regularities can be explained theoretically$^{49}$. For collisions of the dipole molecule NH$_3$ with molecules that do not possess dipole moments, the width will depend on the quadrupole moment of the electron shells (the molecular quadrupole moment, which should not be confused with the nuclear quadrupole moment); from the line width these quadrupole moments can be determined, as was done for the molecules N$_2$, NO, CO, CO$_2$, and others. The quadrupole moments of molecules have an order of magnitude of $10^{-25}$—$10^{-26}$ cm$^2$.

4. Properties of Solid and Liquid Bodies$^{21,50}$

A great many important conclusions about the properties of solid and liquid bodies can be drawn from studies of nuclear induction, paramagnetic resonance, and ferromagnetic resonance.

The study of nuclear induction is closely connected with the problem of nuclear paramagnetism, the investigation of which at low temperatures was begun by B. G. Lazarev and A. V. Shubnikov even before the war$^{51}$ and to which much attention is now being devoted; in particular, the question of realizing parallel orientation of nuclear moments both by cooling to very low temperatures and by other methods is of considerable interest$^{52}$.

The study of paramagnetic resonance has made it possible to obtain a number of data on the splitting of the levels of ions of transition and rare-earth elements in crystals, on organic free radicals, on ions of alkali metals in solutions, which exhibit very interesting properties, and on the conduction electrons in metals. For crystals it is possible to determine relaxation times for the spin—lattice interaction.

The study of ferromagnetic resonance has yielded results of great significance for the theory of ferromagnetism$^{6}$. Ferromagnetic resonance is observed both in ferromagnetic and in antiferromagnetic bodies.

Characteristic of paramagnetic and ferromagnetic resonance is the fact that the value of the splitting factor (Landé factor) for the Zeeman effect, determined by this method, differs from the value determined from gyromagnetic experiments. This difference is due to the different interactions in paramagnetic and ferromagnetic substances. Therefore one distinguishes the “spectroscopic splitting factor” (denoted by $g$), determined by the method of paramagnetic and ferromagnetic

resonance, and the “gyromagnetic ratio” (denoted by \(g'\)), determined by other methods. For ferromagnetic substances, in qualitative agreement with the theory, \(g>2\) (for iron 2.12–2.17), and \(g'<2\) (for iron \(g'=1.93\)), whereas for a free electron \(g=g'=2\).

In addition to the investigation of the structure of matter, the application of radiospectroscopic methods is of great importance for solving other problems\(^{21}\), including applications in electronics (stabilization of oscillations, stabilization of magnetic fields, the use of microwave spectral lines as frequency standards\(^*\)), applications in radio astronomy, and analytical applications. The last area of application is especially important. Radiospectroscopic methods are only beginning to be used for analytical purposes; however, there is no doubt that radiospectroscopy opens new prospects for the development of spectral analysis. The very high accuracy, sensitivity, and speed of recording achieved by radiospectroscopic methods are extremely valuable for various analytical applications.

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