Full Text
NEW INSTRUMENTS AND METHODS OF MEASUREMENT
MOLECULAR GENERATOR AND AMPLIFIER
N. G. Basov and A. M. Prokhorov
1. INTRODUCTION
The investigation of the spectra of substances in the radio-wave range, carried out by a new branch of physics—radiospectroscopy—has made a valuable contribution to various fields of physics, chemistry, and technology.
The following principal results obtained by radiospectroscopic methods over the past 10 years may be listed:
-
In 1947 Lamb and Retherford discovered a shift of the levels of atomic hydrogen, and somewhat later a number of researchers discovered the anomalous magnetic moment of the electron. These fundamental experimental works gave a powerful impetus to the development of quantum electrodynamics.
-
The magnetic dipole and electric quadrupole moments, as well as the spins, of many nuclei have been measured, including those of rather short-lived radioactive isotopes (with half-lives of up to several hours). Work in this field continues to develop; in particular, work is beginning on the determination of higher moments of nuclei (magnetic octupole moments).
-
The structures of a large number of molecules, crystals, and liquids have been studied. The data obtained by radiospectroscopic methods considerably surpass in accuracy the data of all other known methods and provide a whole series of new pieces of information that are extremely necessary for the creation of a theory of the chemical bond.
-
Radiospectroscopic methods are beginning to be used for qualitative and quantitative chemical analysis of substances.
-
Radio-frequency spectral lines are beginning to be used for the creation of frequency standards.
-
Work is being carried out on the precision measurement of magnetic fields by means of proton resonance with an accuracy of \(10^{-6}\), as well as on the stabilization of a magnetic field with high accuracy.
The advances of radiospectroscopy are due to the high resolving power of radiospectroscopic methods. The high resolving power of radiospectroscopic methods is ensured chiefly by two circumstances:
-
The very small value of the natural width of spectral lines, which does not exceed \(10^{-6}\) cps for centimeter-range waves.
-
The presence, in the radio-frequency range, of sufficiently powerful monochromatic radiation sources.
Owing to the fact that sufficiently narrow spectral lines can be obtained in the radio-frequency range, the direct study of weak interaction effects has become possible.
Along with high resolving power, radiospectroscopic methods possess a sufficiently high sensitivity, despite the fact that the probability of transitions from one level to another decreases strongly with decreasing radiation frequency. The high sensitivity of radiospectroscopic methods is due to the fact that the powers of radiation sources are sufficiently large, so that limiting absorption can be achieved.
In contrast to optical spectroscopy, in the radio-frequency range one can observe the saturation effect, which consists in the fact that an increase in radiation power does not increase the absorbed power. The mechanism of the saturation effect is connected with the fact that the radiation equalizes the numbers of molecules in the upper and lower levels of the transition under consideration, which corresponds to a very high radiation temperature at the line frequency.
Usually, in the radio-frequency range, absorption spectra of substances are observed, although in principle, as follows from Kirchhoff’s law, emission spectra could also be observed. The sensitivity of radio-engineering devices is so great that they make it possible to indicate changes in the temperature of radiating bodies by several degrees. This is widely used in radio astronomy. For example, the spectrum of radiation from galactic hydrogen has been investigated by radio-astronomical methods.
Observation of resonance emission spectra of substances under laboratory conditions is very difficult, since under practically realizable experimental conditions it is impossible to get rid of the continuous spectrum of thermal radiation; therefore observation of the spectra of substances in thermodynamic equilibrium is greatly impeded.
It would seem that thermodynamically nonequilibrium systems could be used for observing spontaneous radiation, as is done in optics. However, the large lifetime of excited states in the radio-frequency range (\(\sim 10^6\) sec.) does not permit this possibility to be used. True, under certain conditions it is possible to create such nonequilibrium systems (coherent states) in which observation of spectra of spontaneous radiation may become possible1.
However, in radio spectroscopy there is the possibility of observing spectra of induced radiation. This latter circumstance is connected with the fact that, on the one hand, the lifetime in excited states is long, and, on the other hand, in the presence of powerful radiation sources the probability of induced emission can be made large.
2. ON THE WIDTH OF SPECTRAL LINES
As was noted in the introduction, in the radio-frequency range the natural width of spectral lines is negligibly small, and therefore the line width is determined by various interactions and by the Doppler effect. The narrowest spectral lines are obtained in gas spectroscopy, and therefore in the present section we shall not touch on the question of line width in other methods of observing spectra.
In gas radio spectroscopy, the width of a spectral line is determined mainly by the following factors:
- Collisions of molecules with one another and with the walls of the vessel.
- The Doppler effect.
- The saturation effect.
The width of a spectral line can be substantially reduced if the absorption of radio waves is observed not in a gas, but in a molecular beam, where there are no collisions of molecules with one another or with the walls of the vessel.
The width of the lines of a beam that is “monochromatic” in velocities is determined by the transit time \(\tau\) of the molecules through the radiation field:
\[ \Delta \nu = \frac{1}{2\pi \tau}. \tag{1} \]
In view of the fact that the velocities of the molecules in the beam are not the same, it would seem necessary also to take Doppler broadening into account. However, Doppler broadening of the lines in a beam that is not monochromatic in velocities can be eliminated if, in a volume resonator or waveguide, such types of waves are excited for which the phase velocity of the wave in the direction of propagation of the beam is equal to infinity. This follows from the fact that the frequency shift is determined by the ratio of the velocity of the molecules in the beam to the phase velocity of the waves in the direction of propagation of the beam[^1]. Let us note that the Doppler broadening of the lines of gas molecules filling a volume resonator or waveguide is equal to the Doppler broadening of gas lines in free space[^1].
Thus, the use of molecular beams makes it possible to eliminate not only line broadening associated with collisions of molecules, but also the Doppler broadening of spectral lines. It thereby becomes possible to reduce the width of spectral lines by a factor of 50–100 in comparison with the case of a gas.
3. On Molecular Beams
Molecular beams have been used for quite a long time to increase the resolving power of an apparatus. They were first used by L. N. Dobretsov and A. N. Terenin for optical investigations, and later by Rabi and others for radio-spectroscopic investigations. At present a number of methods have been developed for studying the properties of atoms, molecules, and nuclei in beams. In all these methods the intensity of the beam is indicated. Resonant transitions of the beam particles are judged from the change in the intensity of the beam incident on an indicator[^2].
Recently experiments have been carried out on the study of radio-wave absorption in molecular beams of NH$_3$[^3][^4]. Because the density of molecules in the beam is comparatively small, the absorption of the beam energy by the molecules is small, owing to which the signal associated with absorption of the radiation exceeds the noise level of the receiving device by only a few times. Therefore the method of recording radio-wave absorption has not, in this form, received sufficient development, despite the fact that this method of indication is universal, whereas beam-intensity indicators work well only for a certain group of atoms and molecules.
However, there is the possibility of greatly increasing the intensity of radio-wave absorption by disturbing the distribution of molecules over energy levels. Let us consider this in more detail.
As is known, in a thermodynamically equilibrium system having a set of levels, the spacing between which is small in comparison with the energy of thermal motion $kT$, the difference in the numbers of particles of neighboring levels amounts to only the fraction
$$ \frac{\hbar\omega}{kT} $$
of the number of particles present on one of the levels. For example, for the centimeter wavelength range this difference in the number of molecules amounts to $10^{-3}$ of the number of molecules present on a level. Since radio-wave absorption is determined precisely by this difference, the intensity of the absorption spectrum can in principle be increased by a factor of
$$ \frac{kT}{\hbar\omega}, $$
if the molecules are completely “removed” from the upper level[^1].
If the molecules are removed from the lower level, then emission lines will be observed[^1].
An increase or decrease in the number of molecules on the levels will hereafter be called sorting, and the difference between the number of molecules on the upper and lower levels of the transition under consideration will be called the number of active molecules.
4. Methods of Sorting Molecules
The simplest way is to carry out the sorting of molecules in a beam. For this purpose one may make use of the deflection of molecules (or atoms) of the beam by acting on them with inhomogeneous electric or
magnetic fields. This method was first used in the experiments of Stern and Gerlach.
For example, for diatomic molecules the projection of the electric dipole moment of the molecule onto the direction of the external electric field depends on the quantum numbers characterizing the diatomic molecule \((J, M_J)\).^5 This dependence is shown in Fig. 1.
When a beam of diatomic dipolar molecules is repeatedly passed through an electric field with a gradient directed perpendicular to the motion of the molecules of the beam, molecules in different states will undergo different deflections. By cutting out with diaphragms beams of molecules that have undergone one or another deflection, it is possible to select molecules in a definite state. To obtain appreciable deflections, fields with large gradients are necessary. Thus, for beams with sufficiently large cross-sectional dimensions one has to deal
Fig. 1.
with large fields, when the energy of interaction of the dipole moment of the molecules with the external field is greater than the spacing between the levels of the molecule. Therefore, calculations require the theory of the Stark effect in strong fields.
For sorting \(\mathrm{NH}_3\) molecules according to inversion states, a system consisting of a cylindrical quadrupole capacitor^6 proves effective (Fig. 2).
Fig. 2.
The energy of the upper inversion level of the \(\mathrm{NH}_3\) molecule increases in an electric field, while that of the lower level decreases, and
\[ \Delta E=\pm \alpha \mathcal{E}^{2}, \tag{2} \]
where $\mathcal{E}$ is the electric-field strength. Therefore, in a nonuniform electric field, forces of opposite signs will act on the molecules: molecules located at the lower level will be drawn into the region with the maximum value of the electric field, while molecules located at the upper level will be drawn into the region with the minimum value of the field. The expression for the forces acting on the molecules has the form
\[ F=\pm 2\alpha \mathcal{E}\frac{\partial \mathcal{E}}{\partial r}; \tag{3} \]
$\dfrac{\partial \mathcal{E}}{\partial r}$ is the derivative of the electric-field strength in a direction perpendicular to the motion of the molecules.
If we restrict consideration to the region near the axis of the capacitor, then the expression for the force has the following form:
\[ F=\pm \frac{d^2 k^2 M_J^2}{J^2(J+1)^2}\frac{1}{\hbar\omega_{\mathrm{inv}}}\frac{V^2}{a^4}\,r, \tag{4} \]
where $d$ is the dipole moment of the molecule, $k$ is the projection of $J$ onto the symmetry axis of the molecule, $M_J$ is the projection of $J$ onto the direction of the external field, $\dfrac{\omega_{\mathrm{inv}}}{2\pi}$ is the frequency of the inversion transition, $V$ is the potential difference between the plates of the capacitor, and $a$ is the distance from the axis of the system to the surface of the electrodes (see Fig. 2). The surfaces of the quadrupole capacitor are made in the form of equipotential surfaces of the field of a quadrupole system.
Molecules located at the upper level, passing through the quadrupole system, will perform harmonic oscillations with frequency $\omega$, determined by the relation
\[ \omega^2=\frac{|F|}{rm}, \tag{5} \]
where $m$ is the mass of the molecule.
The length of the capacitor can be approximately calculated from the root-mean-square velocity of the molecules in the beam.
There is also another method of obtaining active molecules, namely preliminary irradiation of the molecular beam with an auxiliary high-frequency field, causing resonant transitions between different molecular levels$^{7}$. Figures 3 and 4 indicate possible variants of using auxiliary radiation of frequency $\nu_{\mathrm{aux}}$ to enrich the upper level in order to obtain active molecules at level 1 relative to level 2.
In the case shown in Fig. 3, active molecules at the first level are obtained by transferring molecules from the third level by means of the high-frequency field. If the high-frequency field has sufficient power, so that the saturation effect is achieved,
then the number of active molecules is equal to
\[ \frac{1}{2}(N_3-N_1)+N_1-N_2, \tag{6} \]
where \(N_i\) is the number of molecules at the \(i\)-th level. The number of active molecules at the first level increases with increasing energy difference between
Fig. 3. Fig. 4.
the first and third levels. In this connection it should be taken into account that the number of molecules at the levels under thermodynamic equilibrium is determined by the Boltzmann factor
\[ N_i \sim e^{-\frac{E_i}{kT}}, \tag{7} \]
where \(E_i\) is the energy of the \(i\)-th level, and \(T\) is the absolute temperature.
The same considerations are valid for the case shown in Fig. 4, except that instead of an increase in the number of molecules at the first level, here there is a decrease in the number of molecules at the second level. The number of active molecules in this case is equal to
\[ \frac{1}{2}(N_2-N_3)+N_1-N_2. \tag{8} \]
This method makes it possible to obtain active molecules for low-frequency molecular transitions.
5. MOLECULAR GENERATOR
If a beam of active molecules is passed through a resonator tuned to the frequency of the spectral line, then the molecules, flying through the resonator, will radiate energy. (Initially the energy may be radiated, for example, under the influence of thermal noise.) The energy radiated by the molecules will be partly stored in the volume of the resonator and partly lost in its walls. If the losses in the walls are less than the energy radiated by the molecules (i.e., the resonator has a sufficiently high quality factor), then the energy stored in the volume resonator will increase, i.e., the resonator
will self-excite. An auto-oscillatory system with feedback is obtained, which consists in the molecules emitting energy under the influence of energy previously emitted by other molecules.
Such a molecular generator will “deliver” monochromatic oscillations. Indeed, at the initial instant of time, when the molecules radiated under the influence of thermal radiation, the intensity of which may be assumed not to depend on the frequency over the interval of the width of the spectral line, they emitted a spectrum of frequencies. The width of the spectral line of the initial radiation is determined by the time of flight of the molecules through the resonator field
\[ \Delta \omega \sim \frac{1}{\tau}. \tag{9} \]
But at subsequent instants of time a larger number of molecules will emit energy at a frequency close to the peak of the spectral line, where the radiation density in the resonator was greater, for induced radiation can occur only at the frequency of the external force. Thus the line will narrow toward its peak. In the limit, when the establishment processes are completed, monochromatic oscillations will be established in the resonator, the frequency of which will coincide with the frequency of the peak of the spectral line.
The amplitude of the established oscillations is determined by the saturation effect. Indeed, if the power stored in the resonator is large, so that during the time of flight through the resonator a molecule manages to pass several times from level to level, then the width of the spectral line will be determined not by the time of flight of the molecules through the resonator field, but by the time the molecule remains on the level, i.e., the line will broaden. Thus, the saturation effect is the nonlinearity which determines the amplitude of the established oscillations in the resonator.
The fundamental possibility of creating a molecular generator was indicated by the authors of this article in a paper published in 1954.^1 Somewhat later, independently of the cited work,^1 a group of American scientists reported the creation of a molecular generator using NH\(_3\) molecules. In the paper,^6 which is a brief note, there are no theoretical considerations concerning the operation of the molecular generator.
It should be noted that the molecular generator is in many respects analogous to other auto-oscillatory systems with feedback; there are also a number of essential differences, since the molecular generator is not a classical system, because induced radiation has no classical analogue. Unlike other generators, in the molecular generator the oscillatory energy is not produced in the generator circuit, but is introduced into the generator circuit by the molecules of the beam, each of which, upon consideration, may be represented as an excited oscillatory circuit,
MOLECULAR GENERATOR AND AMPLIFIER
i.e., a molecular generator is an oscillatory system with a very large number of degrees of freedom. Therefore its operation should be considered by statistical quantum-mechanical methods. For example, the theory of the molecular generator can be constructed on the basis of the theory of dispersion, taking the saturation effect into account.
6. QUANTUM THEORY OF DISPERSION TAKING THE SATURATION EFFECT INTO ACCOUNT
The quantum theory of dispersion with allowance for the saturation effect for a gas was first developed by Korpelus and Schwinger2. The formulas obtained in2 can be derived on the basis of the work of Landau and Lifshitz3. In view of the fact that, as applied to a molecular beam, it is necessary to modify somewhat the formulas obtained in2, we shall give a consecutive derivation of the formulas of the quantum theory of dispersion*).
Let the molecule have two isolated levels \(E_m\) and \(E_n\), characterized by the eigenfunctions \(\psi_m e^{i\frac{E_m}{\hbar}t}\), \(\psi_n e^{i\frac{E_n}{\hbar}t}\). Then
\[ H_0\psi_i=E_i\psi_i, \tag{10} \]
where \(H_0\) is the Hamiltonian of the molecule.
In the presence of interaction of the molecules with an electromagnetic field of frequency \(\omega\), it is necessary to introduce into the Hamiltonian a term allowing for the interaction,
\[ H=H_0+F\cos\omega t. \tag{11} \]
For molecules possessing an electric dipole moment,
\[ F=\mathbf{n}d\boldsymbol{\mathcal E}, \tag{12} \]
where \(\boldsymbol{\mathcal E}\) is the vector of the electric-field strength, and \(\mathbf{n}\) is the unit vector directed along the dipole moment \(d\). In the presence of interaction we seek the solution in the form
\[ \psi(t)=a_m(t)\psi_m+a_n(t)\psi_n. \tag{13} \]
Using (13) and (10), after transformations we obtain the following equations3 for \(a_m\) and \(a_n\):
\[ \begin{aligned} i\hbar \frac{da_m}{dt} &= -\frac{1}{2}F_{mn}e^{i\varepsilon t},\\ i\hbar \frac{da_n}{dt} &= -\frac{1}{2}F_{nm}e^{-i\varepsilon t}. \end{aligned} \tag{14} \]
\[ \left. \begin{aligned} \\[-2.2em] \\[-0.4em] \end{aligned} \right\} \]
*) This calculation was carried out by N. G. Basov, K. V. Svidzinskii, and A. V. Oraevskii.
where \(F_{ik}=F^{x}_{ki}\) is the matrix element \(F\) corresponding to the transition between the \(i\)-th and \(k\)-th states of the molecule,
\[ \varepsilon=\omega_{mn}-\omega, \tag{15} \]
\[ \omega_{mn}=\frac{E_m-E_n}{\hbar} \]
is the resonance frequency of the molecular transition.
Introducing \(b_n=a_n e^{i\varepsilon t}\) and eliminating \(a_m\) from equations (14), we obtain the following equation:
\[ \ddot b_n-i\varepsilon \dot b_n+\frac{1}{4\hbar^2}|F_{mn}|^2 b_n=0. \tag{16} \]
The solution of this equation has the form
\[ b_n=Ce^{i\alpha t}, \]
where the quantity \(\alpha\) is determined by the following equation:
\[ \alpha^2-\varepsilon\alpha-\frac{1}{4\hbar^2}|F_{mn}|^2=0, \tag{17'} \]
i.e.,
\[ \alpha_{1,2}=\frac{\varepsilon}{2}\pm \sqrt{\frac{\varepsilon^2}{4}+\frac{|F_{mn}|^2}{4\hbar^2}} =\frac{\varepsilon}{2}\pm\gamma. \tag{17''} \]
Let at the initial moment of time \(t=0\) all particles be on the upper energy level \(E_m\); then \(\psi(t)\) must satisfy the initial condition
\[ \psi(0)=\psi_m. \tag{18} \]
The normalized wave function satisfying condition (18) has the form
\[ \psi(t)= \left(\cos\gamma t-\frac{1}{2}i\frac{\varepsilon}{\gamma}\sin\gamma t\right) \psi_m e^{-i\frac{E_m}{\hbar}t} e^{i\frac{\varepsilon}{2}t} - i\frac{F_{mn}}{\hbar\gamma}\sin\gamma t\, \psi_n e^{-i\frac{E_n}{\hbar}t} e^{-i\frac{\varepsilon}{2}t}. \tag{19} \]
Using this value of \(\psi(t)\), we obtain expressions for the dielectric constant
\[ \varepsilon=\varepsilon'-i\varepsilon''=1+4\pi\chi' N-4\pi i\chi'' N, \tag{20} \]
where \(\chi\) is the polarizability of the molecule, and \(N\) is the number of active molecules in \(1\ \mathrm{cm}^3\). But
\[ d=\chi \mathcal{E}\cos\omega t. \tag{21} \]
The mean value of the dipole moment in the states with \(\psi(t)\) is equal to
\[ d=\int \psi^*(t)\,d\,\psi(t)\,dq. \tag{22} \]
Substituting (19) into (22) and carrying out the integration, we obtain:
\[ d(t)=\left(\frac{1}{2}\frac{\varepsilon}{\gamma^{2}}\sin^{2}\gamma t\,\frac{|d_{mn}|^{2}}{\hbar} -i\cos\gamma t\sin\gamma t\,\frac{|d_{mn}|^{2}}{\hbar\gamma}\right)\mathscr{E}\cos\omega t, \tag{23} \]
i.e.
\[ \chi'=\frac{1}{2}\frac{\varepsilon}{\gamma^{2}}\frac{|d_{mn}|^{2}}{\hbar}\sin^{2}\gamma t, \tag{24'} \]
\[ \chi''=\frac{1}{2}\frac{|d_{mn}|^{2}}{\hbar\gamma}\sin^{2}\gamma t. \tag{24''} \]
For a molecular beam, when \(N_0\) active molecules fly per second through the transverse section \(S\) of the resonator, the number of active molecules per unit volume inside the resonator is
\[ N=\frac{N_0}{Sv}=\frac{N_0\tau}{Sl}, \tag{25} \]
where \(l\) is the length of the resonator, \(\tau=\dfrac{l}{v}\) is the time of interaction of the beam molecules with the field.
In view of the fact that the molecular beam is not a “monochromatic” beam with respect to velocities, (24′) and (24″) must be averaged over the flight times.
If we assume that the concentration of molecules has the following distribution over flight times:
\[ dN=\frac{N}{\tau}e^{-\frac{t}{\tau}}\,dt, \tag{26} \]
then, carrying out the indicated averaging, we obtain:
\[ \varepsilon'=1-A\beta\, \frac{(\omega_{mn}-\omega)\tau} {(\omega_{mn}-\omega)^2+\dfrac{1}{\tau^2}+\beta|\mathscr{E}|^2}, \tag{27'} \]
\[ \varepsilon''=-A\beta\, \frac{1} {(\omega_{mn}-\omega)^2+\dfrac{1}{\tau^2}+\beta|\mathscr{E}|^2}, \tag{27''} \]
where
\[ A=\frac{4\pi N_0\hbar}{Sl}, \tag{28'} \]
\[ \beta=\frac{|d_{mn}|^{2}}{\hbar^{2}}. \tag{28''} \]
The assumption made about the distribution (26) of molecules with respect to flight times is unjustified for a molecular beam. Here it has
the following distribution holds:
\[ dN=Ae^{-\frac{\tau^2}{t^2}}\frac{1}{t^4}\,dt, \tag{29} \]
where \(A\) is a normalizing constant.
Therefore the shape of the spectral line in the resonator through which the molecular beam passes differs from a Lorentzian.
7. THEORY OF THE MOLECULAR GENERATOR
The theory of the molecular generator was given in paper \(^{11}\). In the present section we shall obtain formulas determining the generation frequency \(\omega\) of a molecular generator using an isolated spectral line of frequency \(\omega_s\). At the same time formulas will be obtained for the amplitude of the established oscillations of the electric field \(\mathcal E\) in the generator resonator, tuned to the frequency \(\omega_0\) and having quality factor \(Q\).
The posed problem reduces to considering the behavior of a cavity resonator filled with a medium with negative losses near the frequency \(\omega_s\). In solving this problem we neglect Doppler broadening of the spectral line, assuming that one can always choose such a type of oscillation in the cavity resonator for which Doppler broadening is absent.
The indicated medium is characterized by the complex dielectric constant (27).
The equation for the electric-field strength in the resonator may be written in the form
\[ \frac{d^2}{dt^2}\mathcal E+\frac{\omega_0}{Q}\frac{d}{dt}\mathcal E+\frac{\omega_0^2}{\varepsilon}\mathcal E=0. \tag{30} \]
For simplicity we assume that \(\mathcal E\) does not depend on the coordinates \(x, y, z\) of the resonator, since formulas (27′) and (28″) were derived under the assumption that the field \(\mathcal E\) is uniform over the cross section of the resonator. If the field \(\mathcal E\) depends on the coordinates of the cross section, it is necessary to derive new formulas taking into account the fact that the magnitude of the saturation effect is different for different points inside the resonator. Such an allowance cannot give anything essentially new, and reduces to introducing a certain effective field. We seek the stationary state in the form
\[ \mathcal E=\mathcal E_0 e^{i\omega t}. \tag{31} \]
Substituting (31) into (30) and setting the real and imaginary parts equal to zero, we obtain:
\[ -\omega^2+\frac{\omega_0^2\varepsilon'}{(\varepsilon')^2+(\varepsilon'')^2}=0, \tag{32′} \]
\[ \frac{\omega}{Q}+\frac{\omega_0^2\varepsilon''}{(\varepsilon')^2+(\varepsilon'')^2}=0. \tag{32″} \]
MOLECULAR GENERATOR AND AMPLIFIER
Taking into account expressions (27′) and (27″), these equations can be rewritten in the form
\[ \frac{Q^2}{\left(Q\frac{\omega}{\omega_0}\right)^2+1} = 1-A\beta \frac{(\omega_2-\omega)\tau} {(\omega_2-\omega)^2+\frac{1}{\tau^2}+\beta|\mathcal{E}|^2}, \tag{33′} \]
\[ \frac{Q\frac{\omega_0}{\omega}} {\left(Q\frac{\omega}{\omega_0}\right)^2+1} = A\beta \frac{1} {(\omega_2-\omega)^2+\frac{1}{\tau^2}+\beta|\mathcal{E}|^2}. \tag{33″} \]
Eliminating from (33′) and (33″) the amplitude of the oscillations \(\mathcal{E}\), we obtain the following equation for the frequency of the established oscillations:
\[ \omega^3+\omega\omega_0^2\left(\frac{\omega_0\tau}{Q}+\frac{1}{\omega^2}-1\right) -\frac{\omega_0^3\omega_2\tau}{Q}=0. \tag{34} \]
From equation (34) it is evident that
\[ \Delta=\frac{|\omega_2-\omega_0|}{\omega_2}\ll 1. \tag{35} \]
Solving the equation to terms of first order with respect to \(\Delta\), we find the generation frequency
\[ \omega=\omega_2\left(1+\frac{2Q}{\omega_0\tau}\frac{\omega_0-\omega_2}{\omega_2} -\frac{1}{Q\omega_2\tau}\right). \tag{36} \]
As is seen from (36), the generation frequency does not coincide with \(\omega_2\) even in the case when the resonator is tuned exactly to the frequency of the spectral line \((\omega_0=\omega_2)\). For example, for a molecular generator for which \(\omega_0\tau=2\cdot10^7\), \(Q=1000\),
\[ \frac{\omega_0-\omega_2}{\omega_2}=5\cdot10^{-6} \]
\[ \frac{2Q}{\omega_0\tau}\frac{\omega_0-\omega_2}{\omega_2}=5\cdot10^{-10}, \]
\[ \frac{1}{Q\omega_2\tau}=5\cdot10^{-11}. \]
Calculations show that such a molecular generator can be used as an absolute frequency standard with an accuracy of \(\sim 10^{-9}\).
The amplitude of the established oscillations can be determined from (33′). It is equal to
\[ |\mathcal{E}|^2= \frac{ A\beta\tau^2\frac{1}{Q}\frac{\omega}{\omega_0} \left[\left(Q\frac{\omega}{\omega_0}\right)^2+1\right] -(\omega_2-\omega)^2\tau^2-1 } {\xi\tau^2}. \tag{37} \]
If we assume that \(\omega_0=\omega_2\simeq\omega\) and \(\frac{1}{Q}\ll1\), then from (37) we have the fol-
giving the expression for the threshold of self-excitation of the generator:
\[ A\beta \tau^{2} Q > 1, \tag{38'} \]
i.e.
\[ \frac{4\pi N_{0}}{Sl\hbar}\, |d_{mn}|^{2} Q\tau^{2} \gg 1 . \tag{38''} \]
Expression (38″) was obtained by us earlier in work\(^1\).
If \(A\beta \tau^{2}Q \gg 1\), then the amplitude of the established oscillations is written in the form
\[ \mathscr{E}_{0}^{2}=\frac{4\pi N_{0}\omega_{2}}{Sl}\,Q . \tag{39} \]
In this case the molecular beam radiates into the resonator the maximum possible power
\[ W_{\max}=\frac{1}{2}N_{0}\hbar\omega_{2}. \tag{40} \]
8. MOLECULAR AMPLIFIER OF MICROWAVE POWER
If the conditions for self-excitation in the molecular generator are not satisfied, then such an instrument can be used as an amplifier of microwave power\(^{6,11,12}\). This instrument is analogous to a regenerative receiver.
Let us find the value of the power gain coefficient of such an amplifier. The molecular amplifier is described by the same equation as the molecular generator; only to the right-hand side of equations (30) it is necessary to add an external force associated with the amplified power:
\[ \frac{d^{2}}{dt^{2}}\mathscr{E} + \frac{\omega_{0}}{Q}\frac{d}{dt}\mathscr{E} + \frac{\omega_{0}^{2}}{\varepsilon}\mathscr{E} = \omega^{2} B e^{i\omega t}. \tag{41} \]
The magnitude of the amplitude of the external force \(B\) is determined by the power \(W_{\mathrm{in}}\) supplied to the resonator of the amplifier. Let us denote the power delivered by the amplifier by \(W_{\mathrm{out}}\). Then the power gain coefficient of the amplifier is equal to
\[ k=\frac{W_{\mathrm{out}}}{W_{\mathrm{in}}}. \tag{42} \]
In the absence of the molecular beam, the quantity \(k\) is the coefficient of power transmission through the resonator. In order to find the amplification (transmission) coefficient, we shall calculate the field in the resonator in two cases: 1) in the presence of the molecular beam \(\mathscr{E}_{1}\) in the resonator, and 2) in the absence of the beam \(\mathscr{E}_{2}\). Then the power gain coefficient will be equal to the ratio of the squares of the fields in the resonator in these two cases, multiplied by the coefficient of transmission through the resonator in the absence of the molecular beam \(k_{0}\):
\[ k=k_{0}\left|\frac{\mathscr{E}_{1}}{\mathscr{E}_{2}}\right|^{2}. \tag{43} \]
From equation (41) we obtain the following value for \(\mathscr{E}_1\):
\[ |\mathscr{E}_1|^2 = \frac{B}{ \left[ \left(\frac{\omega_0}{\omega}\right)^3 \frac{\varepsilon'}{(\varepsilon')^2+(\varepsilon'')^2} -1 \right]^2 + \left[ \left(\frac{\omega_0}{\omega}\right)^2 \frac{\varepsilon''}{(\varepsilon')^2+(\varepsilon'')^2} + \frac{1}{Q}\frac{\omega_0}{\omega} \right]^2 }. \tag{44} \]
The quantity \(\mathscr{E}_2\) is obtained from (44) if \(\varepsilon'=1\), \(\varepsilon''=0\). In the case of resonance \(\omega_1=\omega_2=\omega\), the amplification coefficient has its maximum value. It is equal to
\[ k=\frac{k_0}{1-A\beta^3 Q}. \tag{45} \]
From (44) one can calculate the magnitude of the pass band of the amplifier \(\Delta\). By the pass band of the amplifier we shall mean such a detuning of the frequency relative to the resonant frequency at which the amplified power decreases by a factor of two.
For \(k<Q\)
\[ \Delta^2 \approx \frac{2}{\tau^3}\,\frac{k_0}{k}. \tag{46} \]
Equation (46) shows that the pass band of the amplifier decreases with increasing amplification coefficient of the amplifier, which also occurs for an ordinary regenerative receiver.
In order for the amplifier to operate in the linear region (far from the saturation effect), the following condition must be satisfied:
\[ \beta \tau^2 |\mathscr{E}_1|^2 \ll 1. \tag{47} \]
A molecular amplifier has a small noise coefficient. This is connected with the fact that the probability of spontaneous emission of molecules is small. Therefore the noise of the amplifier is determined by thermal electrical fluctuations in the volume resonator. The molecular beam will amplify both the useful signal and these fluctuations, but the signal-to-noise ratio during amplification will remain unchanged. Fluctuation of the density of the molecules of the beam in the volume resonator will change the amplification coefficient. Therefore the amplifier has a noise coefficient close to 1.
Lowering the temperature of the resonator will lead to a reduction of the noise level.
9. ON THE OBSERVATION OF INDUCED RADIATION IN RAREFIED GASES
The presence of a molecular beam is not an obligatory condition for observing induced emission of molecules. Sorting of molecules by energy states can also be performed in a rarefied gas. For this purpose one may use, for example, a method of deflecting molecules in an inhomogeneous electri-
...electric or magnetic field, as well as irradiation of the gas by an auxiliary high-frequency field.
In such systems there is no need for the injection and pumping out of molecules, since each of the molecules, after emission, can be excited again. Thus, there is the possibility of creating an “sealed” system, including a sealed molecular generator.
The use of sealed systems is especially expedient for low-frequency generators, where the gas consumption for obtaining molecular beams becomes large.
To create a sealed molecular generator on NH$_3$, one can use quadrupole capacitors for sorting, which should be placed in the immediate vicinity of the resonator (Fig. 5). If the entire system is enclosed in a vessel with gas at such a pressure that the mean free path of the molecules is greater than the dimensions of the system, then predominantly molecules located at the upper inversion level will enter the resonator. The number of active molecules entering the resonator will be smaller than in the case of a molecular beam, so that, in order to obtain the self-excitation regime, resonators with a high quality factor are necessary. The necessary values of the quality factor can be obtained, for example, by means of regeneration$^{12}$.
Fig. 5.
The method of sorting by means of an auxiliary high-frequency field can be used, for example, to create a molecular generator employing molecular transitions between the lines of the hyperfine structure of CH$_3$I molecules. To obtain active molecules it is necessary to make use of differences between rotational levels.
CONCLUSION
Let us consider possible applications of the molecular generator.
-
The molecular generator can be used as an absolute standard of frequency (time) of high accuracy, not less than $10^{-9}$. The absolute accuracy is determined by the accuracy of tuning the resonator frequency to the frequency of the spectral line. The quoted accuracy was obtained under the assumption that the resonator frequency can be determined with an accuracy of 0.5% of the pass band of the resonator. It was also assumed that $\omega_0 \tau = 2 \cdot 10^7$.
-
In view of the fact that it is possible to create such systems in which the natural frequency of the resonator over small intervals...
…of a time interval would change very little, the molecular generator can be used as a master oscillator possessing very high stability.
It may be assumed that the molecular generator will make it possible to detect experimentally the influence of the Earth’s gravitational field on the rate of the passage of time.
- The method of the molecular generator makes possible an accurate measurement of the frequencies of spectral lines and, consequently, makes possible the measurement of weak interaction effects in molecular spectra. For example, it permits the determination of quadrupole moments of nuclei. In particular, it is possible to determine the electric hexadecapole moment of a number of nuclei.
Thus, the molecular generator and amplifier, as well as methods for obtaining beams of active molecules, open up new fundamental possibilities for solving both scientific and practical problems.
These problems, of course, are not exhausted by the applications listed in this article.
References
- N. G. Basov, A. M. Prokhorov, ZhETF 27, 431 (1954).
- Estermann, UFN 32, 89 (1947).
- H. R. Johnson, M. W. P. Strandberg, Phys. Rev. 85, 503 (1952).
- M. W. P. Strandberg, H. Dreicer, Phys. Rev. 94, 1393 (1954).
- H. K. Hughes, Phys. Rev. 72, 614 (1947).
- J. P. Gordon, H. J. Zeiger, C. H. Townes, Phys. Rev. 95, 282 (1954).
- N. G. Basov, A. M. Prokhorov, ZhETF 28, 249 (1955).
- R. H. Deicke, Phys. Rev. 93, 99 (1954).
- R. Karpus, J. Shwinger, Phys. Rev. 73, 1020 (1948).
- L. Landau and E. Lifshitz, Quantum Mechanics, Part I, pp. 170–172, 1948.
- N. G. Basov, A. M. Prokhorov, DAN 101, 47 (1955).
- N. G. Basov, V. G. Veselago, M. E. Zhabotinskii, ZhETF 28, 242 (1955).