Study of the initial ionization process in a strong shock wave

,

STUDY OF TEE INITIAL IONIZATION FROCESS IN A STRONG SHOCK WAVE by

A. Be1ozerov

2

7

~.

'.

\S68

I ISnlE

I,°l: (ROOl Elft UEG'fülGBOUW KUNDE

STUDY OF THE INITIAL IONIZATION PROCESS IN A S['RONG SHOCK WAVE

Manuscript received March 1968

JUNE 1968

by

A.

Be1ozerov

UTIAS REPORT NO. 131 AFOSR 68-0828

ERRATA

is should be

Page 2, line 14 10 1-1 sec 10 n sec

Page 4, line

4

nobel noble

[d~~J

r

Eion 2

Page 6

,

eqn. for _~ Eion na _h3

Page 17, eqn. for -dS e

~

(Se + Sion) (-1 S + Sion)

d.X 2 e

•

_{Page 20, line 16, 18 12c, 12e 12e, 1'2f}

Page 29, line 3 Fig. 32 Fig. 31

Page Al, eqn. (A-3) 1 _{2 "2} na

Page Al, (A-5) 1 na

2

eqn. _{2 2}

Fig. 21 2 sec, cm/sec 2l-1 sec , cm/l-1sec

AC KNOWLEDGF,:MENTS

The author wishes to thank Dr. G. N. Patterson, Dr. I. I. Glass, the Canadian Government and the Moscow Institute for Mechanics for making

possible the author's visit to ~anada and the present work.

The'project was supervised by Dr. R. M. Measures. Ris numerous helpful suggestions and keen interest during the course of this study are gratefully acknowledged.

Special appreciation is extended to Dr.

V.V.

Zhurin from the Moscow Instit~te of Earth Physics who originated the project.

fruitful

Sincere gratirude is also expressed to Dr. J.

H.

deLeeuw for discussions and recomendations.

Thanks are likewise extended to the following individuals for their contrib~ion in support of this work: Mr. A. Jaskolka for his invaluable assistance with the computer programming, members of the machine shop for

their capable technical assistance, and Mrs. Jean Dublack for typing the manuscript.

This research was supported by the Defence Research Board of Canada and by the United States Air Force Office of Scientific Research under grant AFOSR

366-66

SUMMARY

A theoretical and experimental investigation has been made of the initial ionization_processes in a strong shock wave in hydrogen. The re-laxation length for ionization, which is principally determined by the rate of excitation, was measured and from a comparison with the theory an estimate vlas obtained for the cross-section for atom-atom exci tation collisions

Detailed theoretical calculations showed that the electron tem-perature approaches to wi thin a few percent of the atom temtem-perature in a dis-tance that is small compared to the total relaxation length for ionization. This enabled considerabl~ simplification for i t indicated that a single tempera-ture model could be used in calculating the theoretical relaxation profile over the experimenta1 range of operating conditions. An electromagnetic shock-tube, with a Philippov pi~ch to create the driver plasma, was employed to ~roduce

shock waves of the required velocity. The ionization behind the shock front was studied by means of a double frequency Mach-Zehnder interferometer, with a ruby laser and a KDP crysta1 (a second harmonie generating crystal) as the light source. A close agreement between the theoretical and experimental electron density profiles, behind the shock front, was obtained for small relaxation

lengths, when the cross-section for the atom-atom excitation collisions was

assumed to be about 2.10- 2 times that of the corresponding electron-atom exci tation collision. For lagger relaxation distances which correspond to shock speeds

less than

3.3

x 10 cm/sec i t was necessary to introduce corrections for blast ,,,ave effects in order to get good agreement with experiment for the same value of cross-section.

I. II. III.

v.

VI.

TABLE OF CONTENTS NOTATIONS INI;RODUCTION

MATHEMATICAL FORMULATION OF TEE FLOW BEHIND A STRONG SHOCK WAVE

IN HYDROGEN .

1. Easic Concept of Relaxation in Strong Shocks 2. Rate Equations for Ionization Processes

3.

Conservation Equations

4.

Electron Energy Equations

4.1

Energy Gain in Atom-Atom Collisions

4.2

Energy Transfer in Elastic Electron-Atom Encounters

4.3

Energy Transfer in Elastic Electron Ion Encounters

5.

Initial Conditions

APPARATUS, AND EXPERIMENTAL TECHNIQUE 1. General

2. Discharge Charnber and Shock Tube

3.

Capacitor Bank

4.

Experimental Technique for General Calibration of the Facilities

5.

Double Frequency Interferometer

6.

Experimental Procedure

EXPERlMENTAL RESULTS AND DISCUSSION

v 1

3

3

3

11 12 12

14

15

19

19

20 21

1-

2 23

25

27

1. General Description of Interferograms 27

2. Analysis of the Experimental Errors

28

3.

Çomparison of Experimental Results with the Theory

29

4.

Blast Wave Corrections to the Theory

29

5. Comparison of the Experimen;i<al Results with the Corrected 32 Theory CONCLUDING REMARKS REFERENCES APPENDIX FIGURES iv

33

35

NOTAT I ONS

Aa (1,z),Aa (1,2) slopes of the approximating curves for the cross section of ionization and excitation in atom-atom inelastic collision. Ae (1,z),Ae (1,2) slopes of the approximating curves for the cross section )f

ionization and excitation in electron-atom inelastic collision.

A(p,q) B(p,q) c E* 6Einit e f(v) g( q) g h K(T_{e )} k k(w) P, Aa (1,2)d· . 1 t· f t t . 1 t· 11·· Ae{1,2) lmenSl0n ess cross sec lon or a om-a om lne as lC co lS10n

Einstein coefficient for spontaneous transitions from energy level p;to q

Einstein coefficient for stimulated transitions from energy level p to q

velocity of light in vacuum

energy of ionization for hydrogen atom

excitation energy for hydrogen atom in (18, 2p) or (lS,2S) transitions

ionization energy for excited atom

dissociation energy for hydrogen molecule

average energy which is given to electron in an atom-atom ionizing collision

elementary charge distribution function ,degeneracy factor for at om

degeneracy factor for excited atom degeneracy factor for ion

relative velocity of colliding particles Planck constant

equilibrium constant Boltzmann constant

coefficient of absorbtion optical path length

N n n(p) n₈₀₀ na + ni

n

= nao

mass of hydrogen atom mass of electron refracti ve index nurnber density

number density of atoms, electrons and ions respectively number density of atoms in energy level l'

atomie number density ahead of the shock

atomie number density behind the dissociative shock

dimensionless number density of atoms and ions behind the shock

l' pressure of the gas

l' ₀₀ pressure of the gas ahead of the shock

Po pressure of the gas behind the dissociative shock Qae,Qei rate coefficients of energy transfer to electrons in

electron-atom and electron-ion elastic collisions qaa' qae ,qei dimensionless rate coefficients of energy transfer to

electrons in atom-atom ionizing cOllisions, electron-atom and electron-ion elastic cOllisions, respectively.

Ra (1,z),Ra(1,2) ionization and excitation rate coefficients for atom-atom inelastic collisions

saa' sae

seei 68 T

ionization and excitation rate coefficients for electron-atom inelastic collisions

distance of the blast front from the origin at _{the moment t l} distance of the fluid element with Lagrange coordinate sn from the origin atr the moment tl

dimensionless ionization rate coefficients for atom-atom and electron-atom collisions .

dimensionless recombination rate coefficient fringe shift

temperature

temperature of atoms and electrons respectively atomie temperature behind the dissociative shock

t u x x

ex

~ ~(p,z) T

e

-e*

et

~on v p time

life time of the excited atom

velocity of the gas with respect to the shock front velocity of the gas behind the dissociative shock velocity of the shock front

distance from the shock front downstream dimensionless distance

degree of ionization degree of dissociation enthalpy parameter

rate coefficient for photoionization from energy level p specific heat ratio

energy which is given to electron in an atom-atom ionizing collision

dimensionless te~perature

dimensionless temperature of ionization from the ground level dimensionless temperature for excitation

-dimensionless temperature of ionization from the first excited state

wavelength reduced mass frequency

polarizabilities of molecular and atomic hydrogen.

Lagrange coordinate of the fluid particle on the distance xn from the shock front·

density

density of the gas ahead of the shock

p(V)

(J,(V)

w

radiation density cross section

cross section for photoionization from the ground level impressed optical frequency (w

=

2nv)

.

..

I. INTRODUCTION

The fast progress being made in recent years in the development

of high speed flights stimulated the large number of investigations dealing

with the high temperature gas flows where the consideration of chemical reactions,

sueh as dissoeiation and, for even higher temperatures, ionization, is essentia1.

The subject of many of these studies was the behaviour of a gas experiencing a

radica1 change in temperature and pressure. It was shown that the time required

for the gas to readjust to the new environment (the so-called relaxation time)

depends upon the va1ues of ~he cross-sections for the reactions involved.

The cross-section for some of these reactions, for instance, the ionization by electron impact can be calculated for hydrogen and hydrogen like

types of atoms, (Refs. 1, 2 and 3). They ca~ also be obtained by direct

experi-ment using the electron beam technique, (Refs. 4,

5

and 6). The calculations of

the cross-seetion for the ionization by impact of heavy particles for energies

at 10-20 eV is eonsiderably more difficult and so far there has been no attempts to solve the problem. The direct experiment also fails due to difficulties of

obtaining the beam of neu

₁

ral particles of such low energies.

The development of the shock tube technique in recent years gave

the opportunity for indirect measurements of the above cross-sections for heavy

gases. Sueh a measurement for argon was made recently by Wong

&

Bershader (Ref.

7).

Unfortunately, for ligh~ gases the speed of the shock wave obtainable in '

conventional types of shock tubes are too low to reach the conditions neeessary

for ionization (Ref. 8), on the other hand electromagnetic shock tubes, which

were for more than 10 years the basic tool for the produetion ofbot ionized gas

samples, are unre1iable for such measurements due to some inborn difficulties like

radiation from the discharge plasma and the strong interaction of the plasma

driver with the shock heated gas, (Refs. 9, 10 and 11). Some of the above

short-comings inherent in the electromagnetic shock tubes were eliminated by Zhurin

et al (Refs.12 and 13) who used as a driver the Philippov pinch (Refs. 14 and

15). In the present work the advantages of the above type of electromagnetic

shock tube were utilized for the study of the initial ionization processes in

strong shock waves in hydrogene In particular, the cross-section for atom-atom

ionizing collisions was evaluated.

Any indirect measurements of the cross-section should be based

upon the appropriate theoretical model of the flow behind the shock wave. The

basic theory of the ionization behind the shock wave was developed in two papers

by Pe~schek and Byron (Ref. 16) and Bond (Ref. 17). They showed the importance

of the energy equa~ion for the electrons, relating changes in the free electron

energy ~o elastic and unelastic encounters. Weymann (Ref. 18) and Harwell and

J-ahn (Ref. 19) showed that the initial ionization rates due to atom-atom

collisions iqvolved a two step process of, excitation followed by ionization

from the exeited state .

The most complete theoretical model is discussed by qettinger

(Ref. 20p for the particular case of argon. He considered radiative as wel1 as

collisional processes in the rate and conservation equations. The present

theo-retica1 model which is discussed in Section 11 is very similar to the one in

Ref. 20, excepf that the initial value for the electronic temperature is

It was shown th at for the case of hydrogen for shock speeds less than

4

cm/~sec the simple model which assumes electronic and atomic temperature to be equal throughout the relaxation region leads to a negligible error in the electron number density profiles behind the shock wave. This is explained by the higher energy transfer rate between the electronic and atomic componentsof the flow for hydrogen in comparison with argon. This is the consequence of two facts, first the mass of the hydrogen atom is 1/40 that of the argon, and second, there is no Ramsauereffect for the electron-hydrogen atom elastic collisions. The latter effect reduces the cross-section for elastic electron-argon atom collisions ifr the energy range of interest.

The experimental electron density profiles behind the shock wave were recorded by the double frequency interferometric technique. The required high time resolution was achieved by using a ruby pulsed laser with a saturable cell giving pulses of about 10 ~sec duration. The K D.P. crystal mounted on the output of ruby laser transformed some of the

6943R

radiation from the ruby into near ultra violet radiation of

3472R.

By this means almost instan~aneous interferograms of the shock wave at two frequerlcies were recorded.

Section 111 gives the detailed description of the apparatus and experimental technique applied. The problems related to data analysis are dis-cussed in Section IV.

The condition of the present experiment were such that the shock speed was decreasing with the distance along the shock tube (approximately

according to the plane blast wave theory). This phenomenon would cause deviation of the behavio'llr of the gas behind the shock front from that predicted by the theory which assumes the constant shock velocity. The gas particles situated at some fixed moment of time at different distances from the shock front exper-iertced the transition through the shock at different speeds. The gas particles furthest from the shock front were processed by the highest shock speed. Although the blast wave influence was unimportant for small relaxation lengths its effect became significant for large relaxation lengths.

With appropriate corrections for blast wave behaviour the theore-tical electron density profiles agreed well with experimental results for shock speed from

2.9

to

3.5

cm/~sec and pressures from

1.5

to

3

Torr. The value of the cross- section for atom-atom excitation collisions which gives the best fit is about 2.10- 2 Ae,where Ae is the corresponding cross-section of elec~ron-atom collisions. The comparison of the theoretical and experimental results is dis-cussed in Section V.

.,.

..

'

.

11. MAIl.'HEMATICAL FORMULATION OF THE FLOW BEHIND A STRONG SHOCK WAVE IN HYDROGEN 1. Basic Concept of Relaxation in Strong Shocks

A shock wave when propagating into a gas transforms the gas from the low temperatpre, low pressure state in front of the shock into the high tem-perature, high pressure state behind it. For an idealized case this trans forma-tions takes place in an infinitesimal region cal\ed the shock front. Since al~

the changes in the state of a real gas can only be achieved by collisions between gas particles, in practice there will be a certain finite distance of approach to equilibrium state behind the shock. For moderate shock speeds only a few collisions between particles are sufficient to re ach an equilibrium since only translational degrees of freedom of the gas molecules are involved. With in-crease of the shock speed the kinetic energy of the particles becomes high enough to exciteother than translational degrees of freedom, such as vibration of atoms in the molecule, then dissociation of molecules and finally ionization of the atoms.

The probability, or the cross-section, for the latter processes is much smaller than that for the translational degrees of freedom so that many

collisions are required before ionizational equilibrium is reached. Schematically the different relaxation regions and their relative lengths as a function of shock velocity are shown on Fig. la, lb, resp. The data on Fig. lb was taken from

Ref. 21 and Ref. 22 for vibrational and dissociational relaxation lengths re-spectively, and from the present paper for the ionizational relaxation. A com-parison of relaxation lengths shows th at for certain intervals of shock velocities it is possible to consider the different relaxation processes independently. For example, ionization can be considered separately from dissociation for shock Mach numbers, Ms

<

'27.

2. Rate Equations for Ionization Processes

As a result of the previous discus sion we can assume that ioniza-tion behind the strong shock in hydrogen effectively starts af ter dissociaioniza-tion is compleyed. With this assumption there will be no difference between hydrogen and any monatomic gas like argon in the description of the ionization process except for the initial conditions which in the case of hydrogen will correspond to a fully dissociated gas behind the shock.

Following the established procedure mdealing with multiple and com-peting processes of excitation de-excitation, ionization, recombination etc., (Refs. 7,16,17 and 20), which lead to ionizational equilibration we separate the whole relaxation region into

3

subzones each being dominated by some principle ionization mechanisms. First is the zone of the initial ionization, then the zone where the primary reaction is the ionization by electron impact and lastly, the third zone where recombination becomes important. We now consider each of them in more detail.

The First Zone

In this region the most important ionizations prJcesses are the atom-atom collisions , photoionization and at,om-impuri ty colli si ons • The atom-atom process may involve two possible mechanisms:

a) a single step ionization

+

H+H~H +e+H

b) a two step ionization with an intermediate, excited, state

H + H ~ H* + H H* + H ~ H+ + e + H

from the ground state is greater than ionization from the ground state and that the probability for ionization of an excited state is more favourable than de-excitation.

The microwave study of the ini tial ionization in nobel gases by Harwell and Jahn (Ref. 19) showed that the threshold energy carresponds to the first excited state rather than f'rom ground to continuum transition. Since the probability of subsequent ionization of an excited state is several orders of magnit ude larger than that for atoms in the ground state, the ini tial step of the two step process becomes the rate-controlling one* and we can write the following expression for the rate of electron production by both a) and b) mechanisms

( 1.1)

The derivation of this formula is given in Appendix. In order to relate the above rate coefficients which describe the macroscopie behaviour of gas with the corresponding parameters, cross-sections ,on the microscopie level further assumptions about the flow conditions and the cross-sections themselves have to be made. Figure 2 shows the standard type of dependenee of the cross-section on the energy of the gas partieles. Since the mean thermal energy of the gas partieles is much less than the thre-shold energy for the range of gas temperatures which we are considering, a linear approximation for this dependenee near the threshold is justified.

( 1.2) (where Eo is the threshold potential)

From the definition of a cross-section the number of ionization (excitation) collisions per cm

3

per sec is equal to

00

R nln2 = nln2

J

cr(v) v f(v) dv V_o

"There v o:is tbe ~lative speed corresponding to the threshold energy ~2

2

=

Eo ' where ~ is the reduced mass

f(v) is the distribution function

nl, n2 are number densi ties of reacting species.

(1. 3)

Substituting the approximate expression (1.2) into (1.3) and assuming a Maxwellian velocity distribution we can perform the integration which results in the

following well known expres sion

* Strictly speaking we should consider also the excitation into higher excited levels but as it will be shown later during the description of the electron-atom inelastic collisions their contribution to the reaction is negligible.

4

..

(1.4 )

ma

where ~

=

~ and k is Boltzmann's constant.

Upon substitution of the above formula equation (1.1) becomes

E.

_E*

. 3/2 -~

[~Ja

=

~a2

~~":~ [Aa(l,Z)~~n

+

2)~

kT

+ Aa (1,2).

(~;

+

2)

e -kT

]

(1.5)

here Eion and E* are respectively energles of ionization and excitation to the

first excited state, Aa(l,z) and A(1,2) are the approximating slopes of cross

section for ionizatiofr and excitation correspondingly •

Similar expression can be written for the rate of the electron

production in. the atom - impurity atom inelastic collisions. ~he influence of

impurities on the concentrations which are expected in the present experiment

is small (as it will be shown in Section

V),

and we disregard it in the following

theoretical analysis.

The l~_processes to be considered in the first zone are

photo-excitation by the resonance radiation

H + h v ~ H* , where h v = E* and direct photoionization by ultraviolet continuum

H + hv ~ H+ + e, _{where hv ::: Eion}

Resonauce radiation becomes an important mechanism of excitation of atoms when

~he mean free path of it is comparable with the collisional relaxation length. The mean free path for line radiation, xabs, depends upon the br8adening of the

line Ref. 23

( 1.6)

where ro

=

2.82 x 10-13cm is the classical radius of the electron

f(1,2) is the absorption oscillator strength for resonance transition L(w) is the normalized line profile tabulated for different Tand ne

by Griem in Ref. 23.

For the conditions; T

=

104oK, n path of the Lyman line radiatio~ width of the line is equal

=

1017cm-3, n(l)

=

1018 cm-3 the mean free at a frequency which cJrresponds to the half

The collisional relaxation length for electron-atom collisions under the same conditions is of the order of 10- 1 cm and for atom-atom c011isions it is even longer. Therefore, the Lyman-a line radiation is always in equilibrium with the gas and can be neglected (Ref. 24)*

Two sources of vacuum ultraviolet radiation appear to be most

important for the process of direct photoionization they are;the equilibrium region behind the shock and the driver plasma behind the contact surface. If u(v)dv

is the amount of radiation energy in the interval v to v + dv and ~l(V) is the cross-section for photoionization from the ground level, then the rate of photoionization is given by, see Ref. 24

~dv)c dv (1. 7)

where c is the speed of Light, and hv 1 = Eion.

If we assume that the plasma in the equilibrium region behind the shock is

optically thick, therr under the condition E. »kT the density of radiation for which hv ~ _{Eion »kT can be written in thêOPorm}

u(v) _exp

(-~;)

and the cross-section for photoionization

can be approximated by

O -18 2

~1

=

7.9

10 cm

With the above simplifications the expression for the rate of photoionization becomes

kT

Figure 3 shows the comparison of [dne/dt] and the collisional ionization rate [dne/dt] as a function of the shock velgcity for different values Aa

=

Aa(1,2)/Ae(1,~)

_{where Ae (1,2) is the known cross-section for} excita-tion by electron impact, which will be defined later. As it follows from the comparison th~ radiation from the equilibrium might influence the initial ioniza-tion rate if A

<

10-. However, further discussion will show that the value of Aa is of th: order of 10- 2 so that this type of radiation can be neglected.

The mean free path of the radiation for na

=

1018 (which is equivalent to neutral number density behind the ionizing shock propagating in

*

It will be shown later in the discus sion of the second zone th at there is no diffusion of the Lyman-a radiation upstream towards the shock front.

hydrogen with initial pressure Poo ~ 2 Torr)is equal

The experimental results wh~ch will be discussed later show that the si ze of the equilibrium reg ion is of the order of 1 cm so that the assumption that theplasma is optically thick is justified even for the Lyman continuum.

It also shows that the influence of the other souree of ultra-violet radiation, the driver plasma, can be disregarded because the radiation

from this source will be nearly completely absorbed in the equilibrium region. Summing up we conclude that for the range of experime~tal

condi-tions under investigation, the influence of radiation on the initial ionization

process can be neglected. The Second and Third Zones

Evenutally the number of electrons b ecomes suffic iently high that the inelastic collisions of these partieles with atoms start: to dominate the ionizafion process. Again, we consider both direct ionization and ionization via

an intermediate excited state as was done for the first zone. The excitation and ionization processes will have their counter-parts, corresponding to recom-bination and de-excitation reactions. The basic processes are consequently ionization and three body recombination

H(p)

+ e + e + e

collisional excitation and de-excitation R(p,q)

H(p)

+ e

-

R(:,P)

H(q) + e

We should also consider the following radiative ionization and excitation

mechanisms:

photoionization and radiative recombination

~(p,z) p(v) +

H(p)

+ hv ~ H + e ~(z,p)

H(p) A(p,q) ~ ~ B(q,p)p(v) H(q) + hv

The rate of change of the number density of excited state p is expressed in terms of the above rate coefficient in the form

~ + q<pA(p,q) + + p(v )

~(p,z)}

+ p,z ~ ~ ne qfP n(q) R(q,p) + q>p n(q) A(q,p) +

(1.9)

~ + ~ p(v ) n(q) B(q,p) + ne3 R(z,p) + n 2 ~(z,p) qrP q,p e p = 1.2 ....

coliisional-radiative processes which are described by the above system of equations were treated by Bates and McWhirter Ref. 25. Tt follows from the results of their investigation that one can reduce the system of equa-tions

(1.9)

to only two rate equations for the population of the ground and the first excited state. The population of all but the first excited states are very smal1.

n(p)« (n( 1) + n(2)

J

p

f

1,2 (1.10 )

and their quasi-equilibrium value is reached almost instantaneously without number densities of free electrons being appreciably altered. Thereafter the rates at which excited states are pr<Xluced and destroyed by collision and radiative processes are much greater than the rates at which the number densi-ties of these states change. That can be mathematically expressed as

I-

dn(p) « dn(2) +

dt dt

Pf 1,2

dn(l)

dt (1.11)

The quasi-equilibrium value of the population of the qth state of atom is re-lated by a modified Saha equations with the electron number densi ty at each particular moment of time

n(q)

=

exp ( E(q,Z))

k Te ( 1.12)

where g+ and g(q) are degeneracy factorsfor ion and excited atom respectively. The rate of ionization is determined now by only two reactions

dne dt = -

r

dn(l) _dt + 8 dn(2)J dt ( 1.13) '.

Bates and McWhirter (Ref. ~5) showed that for the most cases of practical

inter-est the population of the 2nd level also reaches the quasi-equilibrium value

in a very short time interval (which is even smaller than the time obtained in Appendix for similar atom-atom processes). Therefore af ter this time the

rate of ionization is determined by the rate of depopulation of the ground

state and can be expressed in the following form (Ref. 25). dne

dt

dn(l)

dt

where S

=

S(T~,ne) is the collisional-radiative ionization coefficient and

a

=

_{a(Te,ne )} lS the collisional-radiative recombination coefficient.

(1.14 )

The values of S and a are tabulated by Bates and McWhirter

(Ref. 25) for different types of plasmas: optically thin, optically thick, in all the lines or in Lyman lines only etc.

4

It was shown before that

tre

hydrogen plasma behind the shock wave

for T

=

10 oK, na

=

1018 cm-3, ne

=

101

7

cm- 3 is optically thick in the Lyman-a

line The remaining lines are very Stark broaden under the above conditions and their optical depths are comparable or even bigger than the size of plasma

behind the shock, for example, the mean free path for ~ (Xabs)HR ~ 3 cm.

So we consider our plasma as being optically thin for all, but tne Lyman lines.

The difference between the rate S nel) and

a

ne for this case and the rate for

the case of a fully collision dominated plasma is plotted in Fig.

4.

The

difference in ionization rate coefficient S nel) for two cases is so small that

it cannot be seen on the diagram. The difference in the recombination rate

a

ne is substantial only for very small electron number densities where the

magnitude of the recombination rate coefficient is much smaller than that for

ionization rate. For the electron number densi ties near the equilibrium where

a

ne ~ S nel) the difference in recombination rate for both plasmas is also

small. So we can conclude that the inf'luence of the escaping line rad;i.a\ion

in the temperature range from 8 x 1030_{K to 16 x 103}0_{K and for na} ~ 101ö cm- 3

is negligible and the hydrogen plasma in the 2nd and 3rd zones can be treated as collision dominated plasmas.

The results of the analysis by Bates and McWhirter are strictly

speaking applicable only to uniform plasmas. For a nonuniform case like a flow in the relaxation region behind the shock a further assumption,that the

radiative transfer mechanisms are slower than any changes in the state of gas,

has to be made. It follows from Fig. 5 that the resonance radiation can pro-pagate upstream into the flow reg ion behind the shock wave if

xabs t

life

>

vsh - vp

where t life is the lifetime of the excited state and vsh - vp

=

u2 is the

velocity of gas relative

(

0

the shock.

For the Ly_{man line xabs}

~

_{10- 5 cm, t life}

~ 10-

9

sec and for

typical u2

=

3.105 cm/sec the diffusion of the radiation upstream is not possible because

(1.15 )

104 cm/sec

which is smaller than u2'

Consequenvly since in our case radiative transfer is also unim-portant the ionization rate will in zones 2 and 3 be determined only by

collisional processes and instead of the system of equations (1.9) the follow-ing rate equations can be used

[ dneJ _dt = [R (l,z) _e + Re (1,2)] n n - R (z,l) n _{e a e e}

3

e

(1.16) The rate coefficients R(l,z) and R(1,2) can be related with

appropriate cross-sections Ae(l,z) and Ae (1,2) by the equation (1.4). The recombination rate coefficient Re(z,l) can be expressed in terms of Re(l,z) by the principle of . detailed balance

(1.17) where the equilibrium constant K(T_{e ) can be written in the form}

K(T ) =

~

e g(l) ( 1.18)

Upon the substitution of the expression (1.4) and (1.18) equation (1.16) becomes

E*

kT

e e

]

-3

[EkT

i e on + 2J Ae(l,z) 2h -,,2 m 2 e (1.19)

The cross-section data for electron at om inelastic collisions is available from many sources. Fite and Brackmann (Ref. 4) experimentally measured cross-section for (ls, 2p) and(ls, z) transitions, Lichten and Schultz (Ref. 5)

and Stebbings et al (Ref. 6) obtaided the experimental value for the forbidden (ls, 2s) transition. Figures

6

shows values of cross-sections for (ls,2p) and (ls,2s) transitions from the above references together with the linear approximation which was used in the present calculation, for which

where E is in ev

(1.20 )

For the cross-section of the (ls,z) transition we used the following approximation ( 1.21) 10

Equations

(1.19)

aad

(1.5)

give the rate of ionization in hydro-gen gas as a function of

4

parameters: electronic and atomic number densities ne , na and the temperature of the electrons and atoms Te,Ta . The laws of

con-servation applied to the flow through the shock front and the energy balance between the electronic and atomic components of the flow supply the additional relations between ne,na,Te,T_a• These relations are discussed in the next two paragraphs.

3.

Conservation Equations

Conservation equations for the gas flow behind the strong non-attenuating shock wave can be written in the coordinate system moving with the shock in the following form:

a) Mass b) Momentum c) Energy E o + = E + p (n +n. )ma _a l. (1.22) (1.23) (1.~4 )

where index "0" refers to the state of gas behind the dissociative shock (behind the region 2 on Fig. la).

Using the two temperature equations of state in the form p

=

(n _a + n.) keT + a Te)

l. a (1.25)

and the expression for the specific internal energy of partially ionized gas in the form

k

_(12.6)

(T + aT) +

a e

ne

where a

=

is degree of ionization we obtain af ter some algebraic na + ni

manipulations the following two equations relating na,a,T a and Te 1 + c

+~9

c2 _

C~

-

~~

_{a 9ion}

J

_c ÏÏ

=

25

4

2 2 -5 a 9ion + -5 c ( 1.27) 9a + _{a ge}

=

1

[ n

+

(n -

l)c] ÏÏ2 ( 1.28)

here we introduced the di mensi onle s s parameters

Ta Te na + n· _l.

and

9 = - g

e = n

The above conservation equations apply to the whole flow and do not provide us with the information concerning the distribution of energies between components of this flow;atoms and electrons. The energy equation for the electronic component is considered in the next paragraph.

4.

Electron Energy Equation

The competing processes of energy losses in inelastic encounters with atoms and energy gain in elastic collisions with atoms and ions

determine the law governing the change of electron energy. With the assumption that there is no diffusion of electrons with respect to ions and atoms and neglecting Bremsstrahlung radiation losses the rate of change of electron energy in unit time is expressed in the form*

[d neJ

.6E init

<It

a +

(1.29)

We consider each term on the R.H.S. of the equation separately.

4.1 Energy Gain in Atom-Atom CollisionB

This term describes the energy which electrons possess when they are produced in atom-atom ionizing colli si ons and determines the initial con-ditions for the electronic temperature. In order to calculate 6Einit we should know the energy distribution between three particles atom, ion and electron re-sulting from each of the atom-atom ionizing encounters. Unfortunately, there is no available data on this account either theoreticalor experimental. The most logical assumption one can make in this case is the assumption of equi-partition of energy between the resulting par~icles.

If the energy of relative motion of the two atoms before a collision was ~g2/2 the resulting energy of each electron under the above assumption would be

1

3

[

~

2 - Eion

]

The energy received by all the electrons per cm3 per sec as a result of ioniza-tion by atom-atom collisions is

*

Tt is necessary to ment ion that the requirement of a Maxwellian distribution for the electron velocities has to be maintained at each moment, which is not obvious when there is'a continuous drain of highly energetic electrons far ionization. A certaln time is required to fill the tail of the Maxwellian dis-tribution. Fort~ately, as it is shown in Ref.

47,

the time required to popu-late the high energy tail in the distribution is small in comparison with the relaxation time for ionization.

and the average energy 00 na2

J

<Ya(g) Ee(g)g f(g) dg 80 per electron is 00

J

<Ya(g) go 6Einit

=

--~~oor---J

_go (1.30 )

Assuming as always a Maxwellian distribution and approximaying the cross-section in the adopted manner the expression (1. 30) can be integrated which results in the following formula for initial electron energy

2 Eion +

3

k Ta

6E in i t;:

'3

kTa E. + 2 k T

lon a

(1. 31) If we assume also the Maxwellian distribution for the electrons the initial temperature of the electrons which corresponds to the initial electron energy is determined by ~he expression

T eo

4

Eion +

3

k To

9

To E_ion + 2 _{k To} (1. 32)

In the limit of very high temperatures k T_o

»

_{Eion initial}

electron temperature is

and in the limit of very low temperature k ~o

«

_{Eion '} which is very close to

our case

The above results can also be obtained from quite a different approach which uses ~he liner approximation _{to the probability p(E e ) of} the

electron bei~g produced with an energy Ee in the interval 0 ~ Ee ~ ~g2/2 - _Eion

such that

where the constant c is determined by normalization requirement

~g2

J

2 1

o

An expression similar to (1.31) can be readily obtained if we consider also the ionization from the first excited level. The final form of the ini~ial electron energy where both direct ionization and ionization from

2 E!on +

3

k Ta

E. _~n~ ·t

=

-

₃

k T _{a E-l!" +} 2 k T _a

~on

1

2

AJ"l,2) .

~

_(_E_i_on __ -__

E~ï~o~n~)

__ k __

T~a~

____________

~---}

(E!on +

3

·

kTa) [(E* + 2 k Ta)e

~i~:

+

~

_{(E ion}+ 2 kTa) ]

(1.33)

where E-l!" = E - E* is the ionization potentialof the first excited level.

~on ion

4.2

Energy'Transfer in Elastic Electron-Atom Encounters

The rate of energy transferred to the electrons ~y atoms during the elastic collisions was computed by Petschek and Byron,Ref. 16.

00

J

g3 rr n (g) f (g) dg

e,(l e (1.

34)

o

where rre2 (g) is the cross-section of the elastic collisions. The experimental data (Ref. 26), theoretical curve (Ref. 27) and approximating curve

r

rr

[l-~J

E

S

ER

R ER

(rrd)app= (1. 35)

0 E

>

ER

are plotted on Fig.

7.

14

ev .

where

The values of rrR and ER were taken to be rrR =

1.78 10-15

cm2 and Upon substituting (rr n) the expression

(1.34)

becomes

e,(l app

ER TR =

k

+ 2 + 3

~~

]

The last term in the square bracket can be omitted in calcula-tions as being much smaller than unity.

4.3 Energy ~ransfer in Elastic Electron Ion Encounters

The energy transfer from ions to electrons is determined by the

Coulomb cross-section. The resulting equation for energy ga in in these collisions

is

Q. (1. 37)

el

where A and

tnA

is of the order of 10.

Now we have all the necessary information for the equation (1.29)

which completes the system of equations (1.19), (1.5), (1.27), (1.28~for the ionizational relaxation behind the strong shock in hydrogen. In the next

para-graph we will discuss the ipitial conditions for this system.

5. Initial Conditions

In case of no precursor effects the initial conditions for the ionization relaxation would correspond to the parameters of the gas behind the shock wave with complete dissociation providing that the shock is strong enough. The Rankine-Hugoniot relation for a dissociating gas can be written in the form

(Ref. 28) Po 2 (3 - 1 = Poo Po

ç

I' ₁ M 2 _s ((3-1) = Poo 2 (3 - 1 T 0 2 1'1 MS2 ((3 - 1) T 2(2(3-1)2 00 7+3CX d E vib 1 - CX CXd Ed· (3 = _2(1+cx d lS d) + - - + -k T k To 1 + CXd l+cx d 0

where index

"00"

refers to conditions ahead of the shock.

For the strong shocks with the degree of dissociation CXd ~ 1 the equation for (3 becomes

and system (1.38) can be rewritten in the form

where T

=

o = 2f3 - 1

k (t3 - 2.5)

MSJ2'Yl k Too E_dis

The solution of this system of equations for the range of shock speeds of interest is given on Fig.

8.

The initial electron temperature Teo can be determined from expres sion (1.33)

4

E~ +3kT T lon 0 eo ==

"9

To E* + 2 k T ion 0 ( E _ion - E* _ion ) k T 0 1 Aa(l,z)

'2

A_{a (1,2)} x (1.

39)

(E~

rl + 3 kT) [(E* + 2 kT) e 10 0 0 E~ lon + kT o

~

(E_{ion +} 2 _{k To) ]} } (1.40 ) The problem of evaluation of the initial electron number density is more complex and is related directly to the experimental conditions. More-over it was shown experimentally as well as theoretically (Ref.

9,

10,

29,

30

and 31) that there exists an appreciable number of free electrons ahead of the shock wave. The source of these precursor electrons in most cases was believed tb be the radiation from the discharge in electro-magnetic shock tubes and radiation from the shock heated gas.

A simple microwave transmission experimental (see Fig. 9) was performed to evaluate the level of precursor ionization. ThI3resu~ts of this experiment showed that the electron density was less than 10 cm- ,which corresportds to the degree of ionization being less than 10- 5. However, the poor spatial resolution in the microwave experiment, which was estimated to be of the order of 1 cm (8 mm wavelength) did not permit conclusions regarding

a thin gas layer immediately in front of the shock. More over , ~he microwave experiment was not capable of showing the presence of preheating of the gas, which would involve an excitation of atoms rather than their ionization.

As it was shown in the first paragraph of this section the influence of the radiation from the equilibrium region in the initial ioniza-tion zone is hegligible, aresult which would also apply to the layer immed-iately ahead of the shock, and so there is no reason_to expect the electron concentrations in this layer to be more than 1013 cm

3.

As for the preheating of the gas ahead of the shock, the existance of which was strongly argued by Wiese, Griem et al (Ref. 10,

29)

we have only indirect evidences that in the present experiments this effect was of no im-portance. First of all, the experiments of Wiese, Griem et al, where they found the preheat ing, were conducted in T-tubes~adistance of several cm from the discharge, the radiation from which is believed to be the source of the

preheati~g. The distance between the observation point and discharge chamber in the present shock tube is about 10 times longer (65 cm) so that one would expect that the ultraviolet radiation would be considerably weakened before i t reaches the observation point. Second, as a result of the preheating Wies·e, Griem et al found astrong desperancy between the experimentally measured C8n-ditions behind the shock and those predicted from the Rankine-Hugoniot rela-tions. It will be shown later that the present experiments give a reasonable agreement with theory taking into account no pre-heating effects.

As a result of the above discussion we conclude that the pre-cursor effects are small _{and initial conditions for na' ne , Ta are determined} by the state of gas behind the dissociative shock and an initial electron num-ber density of not more than

1013

cm-

3.

Calculations of the relaxation length performed for different neo in the interval from

0

to

1013

cm-

3

did not show any noticeable changes in the results for the range of the shock velocities of interest (from

2.8

cm/~sec to

3.6

cm/~sec).

At this point it is convenient to give the complete set of equa-tions describing the relaxation regio~. In order to make the comparison of theory and experiment easier the spatial derivatives are introduced into the rate equations instead of time derivatives in the following manner

=

u _(1.41)

where u is the velocity of ~he gas relative

rO

the shock and x is the distance from the shock front to any poi~t in the relaxation region at a given moment of time.

The system of equations

(1.19), (1.5), (1.27), (1.28), (1.29)

can b~ written in the following final form saa

e

_a +

a

e

_e

(1-a)2

+

S

_ae

a(l-a)-= ~

Cri

+

Cri -

l)c ]

n

2

1

+ c

+~9

c2

-

(~

-

32

25

25

a n = 2 - - ae

4

₅

_ion + -

2

5

daJ

d'X ( 1.42)

eiO~

c c

where the source functions s and q s

=10

A

(l 2)

J

2<ne ÏÏ2.Ja aa 2 a ' ma a are 8* + 2 8a 8*+ 2 [ Aa~l,z) 8 ion+ 28 a x 1 + Aa 1,2)· 8* + 28 a

=

n

2

_.Jë

e 8* + _{2 8e}

exp (8*

:~

_.[1 + Ae(l2z ) 8ion + 28e

sae _{8* + 2 Ae Cl}

_;2)

_{8* + 2 8} 3 8* Ae (l,z) 3 ( 8 h:m + 28e) h nao e 11 _{(8* + 2)8e} s

₌

( 7TffiekT o)

3/

2 AeC1,2) eei

=

{~

8a 8!on+

8~

+2 8

3

8a [1 _ 21 lon a where

A

(1,2) a Aa(1,2) AeCl;2) Aa(l,z) A_aCl,2) . 1 8 2" (8 - 8 ) e a e 8* + 2 8 - 8 a e n

~

+ ... ]

and

x

is a nondimensional distance defined by the expression x

x

e . exp ( 81:n ) ]

where the denominator is the relaxation distance of electron-atom collisions for an electron temperature being equal to the initial atomie temperature. This non-dimensional distance was very convenient for computations since it does not depend strongly upon the initial conditions and therefore the step size of intégration established for one initial condition can be used for any

other.

In the calculations we assumed tha~ the ratio of Aa(1,z)/Aa(1,2) is unity. Since this ratio is multiplied by e-8*ion/8a the second term in the square brackets in the expres sion for saa is smaller than the first one and the above assumption is not important. The computer program for the system

18

(1.42) and the calculations were done with the assistance of

Mr.

A. Jaskolka.

Solution of the system for one particular shock speed and initial pressure is shown in Fig. 10. It indicates that the electron temperature practically coincides with the atom temperature everywhere except for a very small region near x

=

O. This result is quite different from the similar solution for

argon shown in Fig.ll, where electron temperature is substantially smaller than atomic temperature for most of the relaxati0It region. As we mentioned earlier this disperancy between hydrogen and argon can be explained by the higher rate of energy transfer to the electronic component from the hydrogen atoms in com-parison with the energy transfer from argon atoms.

This convenient aspect of the hydrogen plasma allows us to make

substantial simplificatio~ of the eq~ations. With the assumption that Ta = Te every

wher~ in the relaxation region the equation for the electron temperature

be-comes redundant and therefore can be omitted. The solution of the remaining system can be done by a direct integration

n

da

saa (1_a)2 + sae a(l-a) - seeiQ3

n:

+

(n:

-

1) c

n2 ( Ha)

1 +

+

\

0

---C

-2-_--(--§-_----3-

2-

a

--

e

----)

--c /25 5 25 ion c

where in the expression for s the electronic temperature should be set equal to the atomic temperature.

Comparison of the solution of the system (1.43) with the exact solution shows (see Fig. 10) that they practically coincide. This finding made the analysis of the experimental data much simpler.

The comparison of the theoretical and experimental results will be discussed later in Section V. In the next Section we will describe the ex-perimental set up.

lIl. APPARATUS AND EXPERIMENTAL TECHNIQUE 1. General

Theoretical estimations showed that in hydrogen shock speeds greater than 2.5 cm/~sec _(Ms

>

20) are required if complete dissociation and a measurable electron concentration is to he obtained behind the shock. The

conve~tional diaphragm shock tube even with many improveme~ts is still unable

One may separate the methods, of accelerating a plasma by passing astrong pulsed current through a gas, into three groups according to the physical processes involved in the acceleration.

1. Expansion of the cloud of electron gas heated by an electric field when a current is passed through gas in a T-tube, Fig. 12a, (Ref. 33 and 34). The number of heavy particles is increased by ambipolar acceleration (ions)and

charge transfer (neutral particles).

2

.

Plasma acceleration using electrodynamic force, this is accom-plished by the selection of the pulsed current loop and electrode system of the device. Among this type we can mention the following two devices: T-tube with the return strap Fig. 12b, proposed by Kolb (Ref. 35), artd the so-called plasma IIrail

ll

source Fig. 12c, and its further development-coaxial electro-magnetic shock tube of Patrick (Ref. 36), Fig. 12d.

3. Utilization of the effect of a pinch compression for the creation of a high pressure region similar to the high pressure chamber of the diaphragm shock tube, Fig. 12c. 'Üne end of this IIchamber

ll

has an aperture , so that gas compressed in the chamber by t:re pinch effectflows through this aperture with a very high velocity. The conical tube, Fig. 12e, of Josephson (Ref. 37) and the present arrangement, proposed by Zhurin (Ref. 12 and 13) which uses a Philippov discharge chamber (Refs. 14 and 15) for this purpose, are developments of this type of device.

Several advantages of the latter device such as the product ion of high speed shock waves at large distances from the ch~mber for relatively high initial pressures (up to 10 Torr) , and the much sma11er contamination of the discharge plasma with impurities from the walls (which is due to the particular development of the discharge in Philippov's chamber) were reasons why we used the last type of electromagnetic shock tube.

The present arrangement, like all electromagnetic shock tubes, has two basic components: - (i) a discharge chamber and a shock tube with an

appropriate vacuum system, (ii) a capacitor bank with a triggering device. We consider them separately in the subsequent paragraphs.

2. Discharge Chamber and Shock Tube

The discharge chamber, which is shown in Fig. 13, consists of a stainless steel inner electrode (8 11 x 711 dia.) insulated from the outer stain-less steel charnber (9" x 1211 dia.)' by the cylindrical ceramic insulator. The energy stored in the capacitor bank is supplied to the inner electrode through 49 coaxial cables . 'The outer chamber has two apertures: - one 1/811 dia. for a test gas inlet (on the side wall) and a second, on the axis of the chamber,

1. 511 dia. where the shock tube is connected to it. The shock tube itself is a square cross-section (1-&11 x 1-&11) stainless steel tube made in three inter-changeable sections each about 20 11 long,in order to observe the shock behaviour at different distances from the discharge chamber. The test section has two 1011 x 211 X 3/811 observation windows. The general behaviour of the shock tube

was studied using a cylindrical 1-1/511 dia. pyrex tube in place of the stainless steel tube.

The end of the shock tube opposite to the discharge chamber is connected throVgh a dump chamber to the vacuum system as it is shown in Fig. 14.

The general view of the facility is shown in Fig. 14a.

3. Capacitor Bank

~he efficiency of the discharge chamber depends upon the time con-stant of the discharge. Consequently, a large number of capacitors, switches, and cab les connected in parallel have been used in order to decrease inductance of the system. A schematic illustration of the facility is shown in Fig. 15.

The capacitor bank includes four main units, they are: (1) Master

gap trigger unit (2) Master gap with 300 ~eet of 40.p.~ type of cable and low current power supply (3) Main energy storage unit consisting of 7 sets of three 8.5 ~F, 20 kv, capacitors co~ected in parallel, with spark gaps and high

current power supply (4) Gontrol U~it.

The sequence of operations of the bank is outlined as follows:

A

pulse produced by closing a manual button switch (or some

synchronizing pulse) fires the thyratron in master gap trigger unit. Discharge of a 500 ~ ~F capacitor through the closed thyratron produces a pulse which triggers the master gap. The 300 feet cable which is used as a very fast capacitor (with 0.2 ~F capacitance) being discharged through the closed master gap supplies each of the 7 main spark gaps with ahigh voltage triggering pulse. The breakdown of the main spark gaps closes the load circuit and all the

energy which was stored in the capacitor bank goes to the inner electrode of t;he dis charge chamber.

A successful operation of the shock tube requires a simultaneous energy transmission from all the capacito~s to the discharge chamber. This

restricts the tolerable jitter time of the spark gaps to less than the double

transit time of the cable which connects the bank to the load (20 nsec in the

present set up). The spark gaps used we,re of conventional design (Ref. 38), Fig.

16

and appeared to meet this requirement, providing that they operate in the optimum mode (Ref. 39),namely the voltage on the capacitors, v g , and the triggering voltage, Vtr(t), are such that the potential difference between the trigger pin and anode,vg + Vtr(t), can exceed the sparking potentialof the gap, vo' while Vtr(t) is insufficient to break down the cathode-pin gap. We found experimentally that when the gaps are operating in the fast mode the dis-tance between the electrodes was required to be 7.5 mmo This limits the

voltage on the capacitor bank from 12.5 k~ ~o 20 kv with the rate of rise of the triggering voltage Vtr(t) to be 0.2 kv/nsec.

The master gap which provides the above fast rising pulse has the design similar to the main gap, Fig. 17. "The coaxial geometry and the cable as the capacitor were used to minimize the inductance of the circuit.

The initial calibration of the bank performance was made with a dummy load (a single turn steel coil with'22 nhenry inductance) representing

the bank voltages above 13 kv spark gaps fired simultaneously, Fig. 18a, for the lower bank voltage there was a considerable delay in firing, Fig .18b. The natural frequency of the system with the dummy load was measured to be 5.7 x

103 G/sec, that corresponds to the inductance of the system of 43.5 nhenry.

4. Experimental Technique for General Calibration of the Facilities

The following aspects of the general performance of the present electromagnetic shock tube were studied.

1. The final collapse phase of the discharge in the Philippov chamber. 2. The shock velocity as a function of initial pressure and voltage

on the bank, and the reproducibility of the shock speed for diff-erent initial conditions.

3. The formation of a plane shock front in the tube for different initial conditions at different distances from the chamber. We discuss them in succession:

1. According to Philippov (Refs. 14 and 15) formation of tre discharge in the chamber has three phases. During the first one a current sheet is

formed near the surface of the insulator, this being the path of least inductance. The subsequent motion of the current sheet is indicated in Fig. 19. In the

first stage a rapid expansion takes place soon af ter the formation of the curr-ent sheet. The motion of currcurr-ent sheet towards the axis of the chamber starts when the current nearly reaches its maximum. This is the second phase. ane of

the advantages of this discharge configuration is that the gas which partici-pates in the first phase of the discharge and which is contaminated by the insulator material does not take part in the compression phase. The conical shape of the current sheet in this compression phase gives rise to a focal point on the axes of the chamber near the curved inner electrode. The result-ing very hot and dense plasma has a temperature of about 1 kev and a pressure of about 2000 atm. In the final phase this compressed plasma acts as a piston which produces a strong shock wave in the shock tube. Obviously, the symmetry of the dis charge especially at its second and third stages would determine the behaviour of the shock wave, its strength and reproducibility. Figure 20 shows the streak pictures of the formation of the focal point in the chamber obtained wi th an image converter camera when the shock tube was replaced by a flat window with a horizontal slit. The sl'eed of the luminosL ty front as calculated from the picture was about

2

x 10

7

cm/~sec. Though in most of the runs the focal point was formed near the axis of the chamber there were an

appreciable number of runs where it appeared to be 1-2 cm off the axes showing the irreproducibility of the performance.

2. A streak picture of the propagation of the luminous front in the round glass shock tube and the corresponding curve indicating the velocity as a function of distance are illustrated in Fig. 21. The time dependence of the distance travelled by the luminosity front "x" agrees well with the law

x = const.t2/ 3 (for large x) which is the law of propagation of a one-dimensional blast wave. "x-t" diagrams of the posi tion of a luminous front for a number of

runs is presented on the logarithmical scale on Fig. 22, the inclination of the curves af ter the distance of 20-30 cm from the chamber approaches the blast wave

theory value of 2/3. This fact will be used later for the explanation at the deviation of experimental relaxation length from the theoretical value.

Dependence of the shock velocity on the initial pressure and the

bank voltage were studied for stainless steel shock tube at the distance 35 cm

from the dis charge chamber, Fig. 23. Shock speed was measured from the signals

of two photodiodes SD-100 placed 5 cm apart. For more accurate velocity mea-surements photomultiplier was used instead of photodiodes.

The reproducibility of the shock speed was rather poor for pressures

lower than 2 Torr for all the bank voltages. Some improvement was accomplished

by a slight change in the geometry of the discharge chamber. Introduction of

a cylindrical sleeve which fits the ceramiç insulator and is attached to the

inner electrode reduced the effective breakdown distance and resulted in a more symmetrical format ion of the discharge and therefore more reproducible perform-ance.

3. The planari ty of the shock front was studied by the Schlieren

technique using a Q-spoiled pulse of a ruby laser ( ~ ~Onsec long) as a light

source in order to obtain the required fast time resolved pictures and to

over-come the luminosity of the shock. A selection of ~he Schlieren pictures are

shown in Fig. 24. _{They indicate that below a certain velocity (Ms}

=

60 for

P ₀₀

=

.5

Torr) a plane shock front is formed and that for lower velocities the

shock front becomes separated from the driver plasma. At the distances of

some 60 cm from the discharge chamber the separation is sufficiently large to

study the relaxation processes.

5.

Double Frequency Interferometer

The Q-spoiled ruby laser has also proved to be a very effective light source for the Mach-Zender interferometer. The coherence and small beam divergence permit interference over long optical paths and allow

con-siderable relaxation in the quality and precision of adjustment of the optical components of the interferometer. The spectral purity of the radiation makes it feasible to use narrow band pass filters to block unwanted radiation from the highly luminous phenomena such as the high temperature plasma behind the shock, while the small beam divergence al18ws the use of a very small stop at the focus of the fringe forming lens as an additional means of discriminating against such radiation. Furthermore, the Q-spoiled ruby laser readily provides a time resolution of the order of 20 nsec or less for the study of transient

events. ,.

Alpher and White (Ref. 40) first pointed out that it is highly

de-sirable in the interferometric study af a plasmas to make simultaneous

measure-ments at two wavelengths. Since the refra~tivity of the free electrons is

highly frequency dependent compared with the almost constant refractivity of the neutrals the electron and neutral number densities can be easily obtained

from two interferograms. An obvious exten~ion to the use of the ruby laser is

to make simultaneous measurement at both the fundament al laser wavelengths

694~ and its second harmonic of 347~,genèrated by passing the beam through

a nonlinear optical medium. This technique was successfully used by Ramsden