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Page 1

VOLUME 1

APPLIED PHYSICS AND ENGINEERING
An International Series

Relaxation in Shock Waves

Page 2

Relaxation
in Shock Waves

Yeo V. Stupochenko
S. A. Losev

and A. I. Osipov

Translated by

Scripta Technica, Inc.

Translation Editor
Richard Shao-lin Lee

College of Engineering

State University of New York

Stony Brook, New Yorh

SPRINGER-VERLAG • Berlin • Heidelberg • New York • 1967

Page 201

192 Relaxation in Shock Waves

garded as valid for electron concentrations of 1014_1017 cm -3, correct within
10% and in good agreement with experimental results. Here Ne is then deter-
mined by comparing the contour of the observed line with a number of cal-
culated contours of the same line, obtained for different values of Ne and by
selecting a contour for calculations which is closest to the experimentally
obtained contour [1, 78]. Alyamovskiy and Kitayeva [1], using a shock
tube, have obtained an accuracy of at least 15-20% in measuring Ne• McLean
and others [442] have measured Ne by the widening of lines of ionized helium
with an error of up to 50%. Griem, however, assumes that in the study of
contours of Hp lines with Ne '" 2 x 1017 cm - 3 and T~ 18,000 oK it is possible
to obtain an accuracy in measuring Ne of not less than ± 5%. Comparison
with experimental data shows that it is possible to appreciably reduce the
error by measuring the widening of a number of lines at the same time,
determining Ne and then averaging the result thus obtained.

In the case of the quadratic Stark effect the line with")' and the shift of the
maximum Ll, according to the Weiskopf-Lindholm theory, are determined by

(13-8)

where C4 is the constant of the quadratic Stark effect, v is the relative veloc-
ity of the atom and the exciting particle, and N is the number density of the
exciting particles. These formulas, however, are valid only for small v.
Mandel'shtam and Mazing [120] have found that the experimental results
obtained in argon plasma do not conform to Eqs. (13-8). The subsequent
development of the theory of the quadratic Stark effect has made it possible
to overcome these disagreements by taking into account nonelastic collisions
of electrons with atoms and the reduction of the effect of rapid traverses by
electrons by the amount of widening (within the framework of the non-
steady-state theory due to Sobel'man-Vaynshteyn [181]). In this case the
widening and shift of the lines are determined by the parameter

Z2 (2ncLlEq)tCt
f3 = 2 '

V
(13-9)

where Z is the ionic charge, LlEq is the distance from the ion under considera-
tion to the nearest exciting ion (cm -1), so that Eqs. (13-8) now contain
instead of")' and Ll the correction factors I' (f3) and 1" (f3), respectively. Spec-
ific values of f3 and C4 for a number of lines of inert gases can be found in
[120, 116, 117], and the values of functions I' (f3) and 1" (f3) are given in [181].
Satisfactory agreement with experimental data is here noted.

The setting up of experiments for study of the Stark effect in plasma be-
hind a shock front is complicated by the need of simultaneously obtaining
(after passing of the shock wave) the dependence of the radiation on the

Page 202

Nonequilibrium Phenomena in Shock Waves 193

wavelength (over the spectrum) as well as on time with a high resolving
power. The photographic method is most frequently used in experiments
[475, 1, 442, 416]. In this case either the spectral region under study is
projected on a moving film, or the spectrum is scanned in time by moving
the image of the spectrum over a stationary film, using a disk with inclined
slits which rotates in front of the input slit. The time resolution which is thus
obtained is not too high. Thus, in experiments by Petschek, et at., [475], it
comprised about 12-24 microseconds (in terms of the natural time); the
authors have found that the measured Ne (Ne~ 1016 _1017 cm -3) corresponds
to equilibrium values. This is understood, since the resolving power in time of
the arrangement under experimental conditions is comparable with the time
needed to establish ionization equilibrium (obtained from later measure-
ments by other methods [474]).

The use of photoelectric recording methods is more promising. For this
it is necessary to divide the image of each spectral line into a number of sec-
tions and to direct each of them to the photomultiplier. A number of methods
is available for successfully overcoming this problem, such as the use of
multi component slits and mirrors, hair optics for breaking up the ray, in-
clined slits, interferometer techniques, etc. [410]. Measuring the image of
one line into seven segments has made it possible to study the behavior of the
contour of a number oflines in shock waves in helium and deuterium [305].
Malyshev and others [118] suggest the use of the Fabry-Perot standard [in-
terferometer] with a monochromator and an electron-optical amplifier for
obtaining contours of spectral lines with a high time resolution.

[5] Recording the Optical Radiation

As was noted in the preceding section, the intensity of the continuous
radiation produced by free-free and free-bound electronic transitions in
quasi-neutral plasma is proportional to the square of the electron number
density. It is known that. the radiation intensity in this case depends also on
the electron temperature (see, for example [22]). However, in the region
where Ne is sufficient for observing the radiation the electron temperature
does not vary appreciably (see [25]). Hence for the change in I, the radiation
intensity, with time behind a shock front we have

1 dI 2 dNe

I dt Ne dt
(13-10)

The arrangement for measuring the distribution of the radiation intensity
behind a shock front is similar to that described in the preceding section.
Particular attention must here be given to reducing the level of light scat-

Page 402

Subject Index 393

resonance, diffusion of, 302
short-wave, 93
ultraviolet, absorption of, 143,248

Radio-frequency method, 98
Random molecule wandering, stochastic

process of, 275
Random walks, theory of, 279
Rankine-Hugoniot relationships, 211
Refraction index, study of changes in, 121
Relaxation process, kinetics of, 348

study of, 71
(see also Shock Waves)

Resonance method, 188, 189
Reynolds number, 78, 79
Ribbon lamps, tungsten, 175
Riemann invariant, 40
Rosen's variational principle, 212, 213
Rotational equilibrium, 35

relaxation, 215
Roth's experiment, 250
Rozhdestvenskiy's hooks method, 99, 131
Runge-Kutta method, 113,318

Saha's equation, 51
Sakurai's method, 2 I 3
Schrbdinger equation, 228, 229, 232, 254

steady-state, 232
Schumann-Runge bands, 141,312

region, 346
system, 143, 172,311

Semiconductor receivers, germanium, 173
Shock adiabatic, 7
Shock front, flow inhomogeneity behind, 72

study of light reflection from, 1 17
Shock-front curvature, measurements of,

83
problem of, 83

Shock tubes, 37,49
gasdynamic flows in, 61
nonequilibrium phenomena in, 72
pulse-type magnetohydrodynamic, 172
shock-wave propagation in, 72, 80
theory of, 37, 39

Shock-tube chambers, filling of, 52
operation, techniques of, 52

two-diaphragm arrangement of, 61
Shock waves, basic properties of, 3

dimensions, 64
dissipative processes in, 8

formation and acceleration of, 64
motion of, 17

gas radiation in, 162
increasing speed of, 52
method of obtaining strong, 49
nonequilibrium phenomena in, 38, 104

kinetics in, 38
study of, 99,100,306

relaxation processes in, 206
theoretical analysis of, 206

structure of strong, 211
study of formation and propagation of,

38
thermal ionization kinetics in, 189

theory of, 3, 364
velocity of, 40

measuring, 91
Single-component system, thermal disso-

ciation in, 281
Single-photon energy transfer, concept of,

246,253,260
Small disturbances, propagation of, 359
Sobel'man-Vaynshteyn unsteady-state

theory, 192
Sodium salts, 100

heating of, 100
Spectral absorption analysis, methods of,

138,161
line reversal, method of, 176

Spectrometer, time-of-flight, 200
Speed of sound, isentropic, 1 I
Stark effect, 180, 191

basic theory of, 192
in plasma, study of, 192
quadratic, 192
use of, 191

Statistical thermodynamics, expression for,
280

theorem on, 350
Steric factor, 33, 275
Structureless particles, 13

Targets, 137
chromium, 137
copper, 137
tungsten, 137

Temperature-dependent coefficients, 243
Temperature measurement, spectral

methods of, 179

Page 403

394 Relaxation in Shock Waves

Tepler's schlieren systems, 46, 94, 97, 121,
122,124,125

operating principle of, 121
Test gas, preparation of, 100
Thermal conductivity, coefficient of, 9

diffusion processes, contribution of, 182
dissociation kinetics, 271

process, 272
theory,286,325

ionization, mechanism of, 301
of air, 342
process of, 272
study of, 342

pickups, 94
Thermoanemometer, defined, 197

probe, 197
use of, 95, 96

Thermodynamic equilibrium, 349
parameters, 349
temperature, 350

Thermodynamics of irreversible processes,
methods of, 352

Thyratron arrangements, 96
systems, actuation time of, 96

Time-measuring devices, chronographic, 96
Transition probabilities, averaging value

of,236
qu?:-:tum theory of, 231
quantum-mechanical problem of cal-

culating, 224
semiclassical theory of, 228

Transitional layer, boundary of, 2
structure of, 2

Translational energy, transfer of, 23

Transport phenomena, theory of, 225
Transverse waves, effect of, 85
Trimpi-Cohen theory, 67
Tubes, 100

Ultrahigh frequencies, method of, 183
Ultrasonic waves, dispersion and absorp-

tion method of, 215
vibrations, propagation of, 245

Valency schemes, method of, 263
Van der Waals-type forces, 308
Vl'vilov's experiments, 140
Vibration excitation, process of, 115
Vibrational energy, relaxation equation

for, 257
photons, transfer of, 313
relaxation, study of, 36, 226, 263
theory of, 31
transfer, probability of, 253

Volt-ampere characteristics, 180

Water vapor, kinetics of decomposition of,
161,162

Weiskopf-Lindholm theory, 192
Widom's calculation, 218
Wien's radiation law, 175
Wigner's distribution function, energy

representation of, 242
Wray's recommendations, 326, 327

Xenon, ionization kinetics of, 188
Xenon lamp, arc-discharge, 158
X-ray radiation, use of, 136

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