DISPERSION PROPERTIES OF PLASMA IN SHOCK TUBES (200–7300 MHz)

Yu. I. Tusnov

The paper presents the results of an experimental study of the frequency dependence of the complex dielectric permittivity \(\tilde{\varepsilon}\) of plasma in shock tubes. At temperatures of ionized air \(3 \div 5 \cdot 10^3\,^\circ\mathrm{K}\) and argon \(4 \div 10 \cdot 10^3\,^\circ\mathrm{K}\), in the frequency range 200–7300 MHz, satisfactory agreement was found between the measured dependences \(\varepsilon'\) and \(\varepsilon''\)—the real and imaginary parts of the complex dielectric permittivity—on frequency and the theoretical data.

The study of the electrical characteristics of ionized gases as a function of the frequency of the incident radiation is of considerable interest. Frequency variation in measurements of the dielectric permittivity of one and the same “plasma formation” makes it possible to determine the dispersion properties of such a formation and thereby to test experimentally the existing concepts of the elementary processes of interaction of electromagnetic waves with an ionized gas. In addition, it seems to us that, along with the development of studies of various methods of influence on the electrical characteristics of an ionized gas in order to improve the propagation of radio waves, a substantial contribution to solving the problem of establishing reliable radio communication with a spacecraft surrounded by a plasma sheath may also be made by an experimental investigation of the conditions of interaction of electromagnetic waves of different frequencies with a nonuniform layer of shock-heated gas moving at supersonic velocity.

Despite the abundance of experimental work connected with the application of ultrahigh-frequency methods to studies of the properties of ionized gases, there are still no more or less detailed investigations of the dielectric permittivity of plasma over a significant range of electromagnetic-wave frequencies. The scattered data on the characteristics of an ionized medium obtained at different frequencies unfortunately refer to different objects of investigation and are difficult to compare. Measurements of electrical characteristics carried out in shock tubes are determined by a very narrow frequency interval for each experiment. Moreover, the main development has been obtained only in the wavelength range \(2 \div 30\) mm.

Carrying out measurements at wavelengths from centimeter to meter is a very difficult task. The possibility of using one or another microwave method for measuring the electrical characteristics of an ionized gas

of the substance is determined by the frequency range in which the measurements are made. Thus, to carry out measurements in the wavelength range \(4 \div 150\) cm by any of the existing microwave methods [1] (resonator, waveguide, and free-space methods), it would have been necessary to create a whole series of experimental setups and to apply various kinds of techniques. Comparing the results of measurements of the electrical characteristics of the plasma under such an experimental arrangement would be very difficult.

In the present work, experiments to study the dependence of the real and imaginary parts of the complex dielectric permittivity (c.d.p.) of a plasma on the frequency of the probing radiation were carried out in a coaxial shock tube by probing with a TEM-type wave propagating toward the moving plug. The use of a shock tube with a low-pressure chamber in the form of a coaxial transmission line made it possible to carry out measurements of \(\tilde{\varepsilon}\) over a broad radio-frequency range for essentially one and the same plasma formation. As a technique allowing a sufficiently simple interpretation of the results of the interaction of an electromagnetic wave with an ionized gas, the “infinite-layer” method, described in detail in [2], was adopted in the present work.

The determination of \(\tilde{\varepsilon}\) from measurements of the complex reflection coefficient of an electromagnetic wave from a plasma moving along a coaxial tube was carried out in argon in the range of shock-wave velocity \(V = 2.0 \div 3.2\) km/sec, \((M = 6 \div 10)\), for initial pressures \(p_0 = 2.5\) and \(10\) mm Hg, and in air for \(V = 2.8 \div 4.1\) km/sec \((M = 8 \div 12)\), \(p_0 = 5\) and \(10\) mm Hg.

In view of the fact that experiments on shock tubes are difficult to reproduce, it was necessary, for each frequency of the decimeter range (the values of \(\varepsilon'\) and \(\varepsilon''\) for the frequency 7300 MHz at identical parameters of ionized air and argon were taken from [3, 4]), to obtain continuous experimental curves of the dependence of \(\varepsilon'\) and \(\varepsilon''\) (the real and imaginary parts of the c.d.p., respectively) on the Mach number at each of the frequencies used for probing. This is especially important because possible errors in reproducing the gas characteristics behind the shock-wave front may introduce substantial changes into the actual course of the dependence \(\tilde{\varepsilon}(\omega)\) and thereby make it difficult to compare the experimentally obtained regularities with the calculated ones.

As a result of measurements of the amplitude and phase characteristics of the interference pattern obtained ahead of the ionized gas-dynamic plug in a coaxial shock tube, curves of the dependence of \(R\) and \(\Theta\) on the number \(M\) were constructed at four frequencies \((2 \cdot 10^3,\ 1 \cdot 10^3,\ 0.45 \cdot 10^3,\ 0.20 \cdot 10^3\) MHz). These dependences are presented in Fig. 1 for argon. A monotonic increase of the value \(R\) is observed at all the frequencies studied. In argon the maximum value \(R_{\max} = 0.96\) was measured at \(p_0 = 10\) mm Hg, \(V = 2.7\) km/sec and \(f = 200\) MHz; the minimum, \(R_{\min} = 0.10\), at \(p_0 = 5\) mm Hg, \(V = 2.10\) km/sec and \(f = 2000\) MHz. The corresponding values of the VSWR (standing-wave ratio) are 50 and 1.2.

Analogous data on the reflectivity of air were as follows: \(R_{\max} = 0.87,\ V = 3.32\) km/sec, \(p_0 = 2\) mm Hg, \(f = 200\) MHz; \(R_{\min} = 0.23,\ V = 2.95\) km/sec, \(p_0 = 5\) mm Hg, \(f = 2000\) MHz. With increasing wavelength of the probing radiation, the value \(R\) for air and argon also increases. Comparison of the measurement results for different initial pressures shows that, for \(M = \mathrm{const}\), the magni-

on \(R\), for all frequencies studied, is always greater for higher initial pressures.

In the upper part of Fig. 1 the dependences of \(\Theta\) on \(M\), \(f\), and \(p_0\) are given. The curves presented indicate that the phase shift upon reflection of an electromagnetic wave from a moving semitransparent mirror increases continuously for all frequencies with increasing \(M\), approaching asymptotically \(180^\circ\), which corresponds to reflection from a metallic mirror. Both in argon and in air, the value of \(\Theta\) depends strongly on frequency, although the range of variation of \(\Theta\) in air is considerably smaller than in argon (\(120\)—\(170^\circ\)).

The experimentally measured values of \(R\) and \(\Theta\) were used to calculate the parameters \(\varepsilon'\) and \(\varepsilon''\) according to the expressions given in [2]. The method for plotting the graphs of the dependence of \(\varepsilon'\) and \(\varepsilon''\) on \(f\) was as follows. Through a point selected on the velocity scale, a straight line was drawn parallel to the ordinate axis (\(R\) or \(\Theta\)), and the points of intersection with the curves of the dependence of \(R\) and \(\Theta\) on \(V\) determined the values of

Fig. 1. Dependence of the modulus of the reflection coefficient \(R\) and the phase angle \(\Theta\) on the Mach number \(M\), the frequency of the probing radiation \(f\), for initial argon pressures \(p_0 = 2.5\) and \(10\) mm Hg: \(1\)—\(f = 200\) MHz, \(2\)—\(f = 452\) MHz, \(3\)—\(f = 1000\) MHz, \(4\)—\(f = 4000\) MHz; ○, ●—\(p_0 = 2.5\) and \(10\) mm Hg.

the quantities \(R\) and \(\Theta\) for the gas state with \(M = \mathrm{const}\) and \(p_0 = \mathrm{const}\) at different frequencies. The quantities \(\varepsilon'\) and \(\varepsilon''\) were then calculated. Using error nomograms, the relative errors in measuring \(\varepsilon'\) and \(\varepsilon''\) were determined from the values of \(R\) and \(\Theta\).

Figure 2 presents curves of the dependence of \(\varepsilon'\) and \(\varepsilon''\) on frequency for \(p_0 = 2\) mm Hg (the frequency values are plotted on a logarithmic scale). Similar curves were constructed for other initial pressures. They are not given here. Figure 3 presents the same dependences obtained for air at \(p_0 = 5\) mm Hg. A characteristic feature of the curves shown in Fig. 2 is the relatively weak dependence of the value of \(\varepsilon'\) over a very large frequency interval, \(7300 \div 2000\) MHz. At low frequencies in argon, a rapid change in the value of \(\varepsilon'\) is observed, and these changes are especially large for high values of the shock-wave velocity. In contrast to argon, in the curves shown in Fig. 3 there are essentially no regions with a weak dependence of \(\varepsilon'\) on \(f\) in the frequency range \(7300\)—\(2000\) MHz. At the same time, it should be noted that air is characterized by the presence of a so-called saturation region of the curve of the dependence of \(\varepsilon'\) on \(f\) at the low frequencies of the range studied (\(452\)—\(200\) MHz). The saturation region is especially clearly expressed for the case \(V = 3.2\) km/s in Fig. 3, where in the interval \(452\)—\(200\) MHz \(\varepsilon'\) changes from \(-1.8\) to \(-1.75\), i.e., the values of \(\varepsilon'\) are practically identical, if

Fig. 2. Dependence of the real \(\varepsilon'\) and imaginary \(\varepsilon''\) parts of the complex dielectric permittivity on the frequency \(f\) at \(p_0 = 2\) mm Hg in argon.
\(1 — V = 2.2\) km/sec, \(— — — = \varepsilon''\), \(\varepsilon'\);
\(2 — V = 2.4\) km/sec; \(3 — V = 2.6\) km/sec; \(4 — V = 2.8\) km/sec.

Fig. 3. Dependence of \(\varepsilon'\) and \(\varepsilon''\) on \(f\) at \(p_0 = 5\) mm Hg in air.
\(1 — V = 3.0\) km/sec, \(2 — V = 3.2\) km/sec; \(— — — = \varepsilon''\), \(\varepsilon'\).

take into account the magnitude of the error \(\Delta \varepsilon'/\varepsilon'\), which is especially large for combinations of \(R\) and \(\Theta\) characteristic of air.

Before proceeding to the interpretation of the results obtained, let us consider a number of conditions whose fulfillment is necessary for applying the “infinite layer” method [2] and for an unambiguous interpretation of the data presented.

Let us introduce the notation: \(d\) is the extent of the transition zone (from the front to the quasi-homogeneous part of the sample), \(\delta\) is the depth of penetration of the field into the gas ionized by the shock wave, \(l\) is the length of the sample (from the front to the contact surface). Then the above-mentioned conditions can be written in the form

\[ \frac{d}{\lambda} \ll 1,\qquad \delta > d,\qquad \delta < l. \]

The extent of the transition zone behind the shock-wave front in air and argon was determined from the results of measurements of the transit time of the transition zone past a microwave receiver, given in [3, 4]. The values of \(d\) in argon are collected, for ease of comparison, in Table 1. A theoretical estimate of the error in determining \(R\) and \(\Theta\), arising from the finite length of the transition layer, showed that in the measured range of values of \(\varepsilon'\) and \(\varepsilon''\) the magnitude of the error does not exceed the accuracy of measurement of \(R\) and \(\Theta\). For the estimate, the solution of the wave equation for a layer of the hyperbolic-tangent type was used.

Fulfillment of the second condition—the possibility of assigning the results of measurements of \(R\) and \(\Theta\) at different wavelengths to the quasi-homogeneous part of the sample—was checked by carrying out measurements of the depth of penetration of the field into the ionized gas. Figure 5 gives the results of such measurements in argon. For comparison, two calculated curves are given for the quantity

\[ \delta = \frac{c}{\sqrt{2\pi\omega\sigma}}, \]

where the values of \(\sigma\) were calculated from measurements of \(n_e\) and \(\nu_{\mathrm{eff}}\) (the electron concentration and effective-

Fig. 4. Dependence of the depth of penetration of the field \(\delta\) into the plasma on the Mach number \(M\), frequency \(f\), for \(p_0 = 2.5\) and \(10\ \mathrm{mm\,Hg}\) in argon.

\[ p_0 = 10.5,\ 2\ \mathrm{mm\,Hg},\quad f = 2\cdot 10^3\ \mathrm{MHz} \]

\[ \text{”}\qquad f = 1\cdot 10^3\ \mathrm{MHz} \]

\[ \text{”}\qquad f = 0.45\cdot 10^3\ \mathrm{MHz} \]

\[ \text{”}\qquad f = 0.20\cdot 10^3\ \mathrm{MHz} \]

\[ 1\text{—calculation for } f = 2000\ \mathrm{MHz} \]

\[ 2\text{— ” } f = 1000\ \mathrm{MHz} \qquad \left\{ \begin{array}{l} p_0 = 5\ \mathrm{mm\,Hg}. \end{array} \right. \]

collision frequency) for the corresponding \(p_0\) and \(V\), taken from [3, 4]. Similar dependences were also obtained in air. From Fig. 4 it is clear that, for the pressures being compared, the value of \(\delta\) measured experimentally is always greater than the calculated one. This is due to the fact that the calculation is carried out for an electrically homogeneous medium, whereas the attenuation of the microwave signal was measured mainly in the “head” part of the slug, which differs sharply in its electrical characteristics from a homogeneous section. It should also be noted that there is an extremal point in the obtained dependence of \(\delta\) on \(f\); its presence can be explained rather simply from the elementary theory of the high-frequency conductivity of plasma [5]. Similar results were also obtained for air.

The absence of reflection from the second boundary of the slug was checked by measuring the length of the slug \(l\) in a coaxial shock tube, obtained with an electrostatic probe together with a piezoelectric pressure transducer, which determined the position of the beginning of the slug. Table 1 gives data on the values of \(d\), \(\delta\), and \(l\) for several experiments. Using the table, it is easy to analyze the fulfillment of all three conditions for the selected experiments.

It follows from the table that the conditions \(\dfrac{d}{\lambda} \ll 1\), \(l > \delta\), \(\delta > d\) are satisfied for the following curves of the dependence of \(\varepsilon'\) and \(\varepsilon''\) on frequency: in argon for \(p_0 = 2\) mm Hg—\(V = 2.4;\ 2.6;\ 2.8\) km/s; \(p_0 = 5\) mm Hg—\(V = 2.4;\ 2.6\) km/s; \(p_0 = 10\) mm Hg—\(V = 2.6\) km/s.

Table 1

In air, for \(p_0 = 2\) mm Hg—\(V = 3.2;\ 3.4\) km/s; \(p_0 = 5\) mm Hg—\(V = 3.0;\ 3.2\) km/s. Thus, the results of these experiments can be subjected to further consideration and compared with theory.

The calculation of the complex dielectric permittivity of an isotropic plasma on the basis of the kinetic equation leads to the known expressions [5]:

\[ \varepsilon' = 1 - \frac{4\pi e^2 n_e}{m\left(\omega^2 + \nu_{\mathrm{eff}}^2\right)} \cdot K_1\!\left(\frac{\omega}{\nu_{\mathrm{eff}}}\right), \]

\[ \varepsilon'' = \frac{4\pi e^2 n_e}{m\left(\omega^2 + \nu_{\mathrm{eff}}^2\right)} \cdot \frac{\nu_{\mathrm{eff}}}{\omega} \cdot K_2\!\left(\frac{\omega}{\nu_{\mathrm{eff}}}\right). \]

The form of the functions \(K_1\) and \(K_2\) is defined in [5].

The values of \(n_e\) and \(\nu_{\mathrm{eff}}\) needed for calculation by these expressions for air and argon were taken from [3, 4]. Figures 5 and 6 show curves of the dependence of \(\varepsilon'\) and \(\varepsilon''\) on frequency, compared with calculations according to the elementary and kinetic theories. For the case represented

in Fig. 5, the initial data for the calculation, taken from [3, 4], were \(n_e \simeq 2\cdot 10^{11}\ \mathrm{cm}^{-3}\) and \(\nu_{\mathrm{eff}} \simeq 1.4\cdot 10^{10}\ \mathrm{s}^{-1}\). The calculated curves for argon (Fig. 6) are given for \(n_e = 0.5\cdot 10^{11}\ \mathrm{cm}^{-3}\), \(\nu_{\mathrm{eff}} \simeq 2\cdot 10^3\ \mathrm{s}^{-1}\).

Fig. 5. Dependence of \(\varepsilon'\) and \(\varepsilon''\) on \(f\). \(p_0=2\) mm Hg. \(V=3.2\) km/s. Air.
” — — — Elementary theory.
” — · — · — Kinetic theory.

Fig. 6. Dependence of \(\varepsilon'\) and \(\varepsilon''\) on \(f\). Argon.
\(p_0=2\) mm Hg. \(V=2.6\) km/s.

LITERATURE

1. V. E. Golant. Journal of Technical Physics XXX, issue 11, 1960.
2. Yu. I. Tusnov, V. V. Krylov. Thermophysics of High Temperatures, 1966, 6.
3. V. V. Krylov, O. I. Popyrin, Yu. I. Tusnov. Thermophysics of High Temperatures, 1968, No. 5.
4. Yu. I. Tusnov. All-Union Conference on the Physics of Low-Temperature Plasma. Abstracts of Reports. Kiev, “Naukova Dumka,” 1966.
5. V. L. Ginzburg. Propagation of Electromagnetic Waves in Plasma. FM, 1960.