Atomic collisional rate and ionization relaxation study in a helium shock wave

ATOMIC COLLISIONAL RATE AND IONIZATION RELAXATION

STUDY IN A HELIUM SHOCK WAVE

by

Satya Pal Kalra

Submitted Mareh, 1972.

ACKNOWLEDGEMENTS

The author wishes to express his thanks to Dr. G.N. Patterson, Dr. J.H. deLeeuw and the Canadian Governme~t and the Indian Government for making this work possible.

The project was carried out under the supervision of Dr. R.M. Measures His helpfuI supervision, keen interest during the course of this work and

numerous valuable suggestions are acknowledged with gratitude.

Sincere gratitude is also expressed to Dr. J.H. deLeeuw for his advice and recommendations.

The author extends his appreciation to the following individuals:

Mr. S. Garg for his expert suggestions in computer programming, Miss Roberta Dunn and Miss Madhu Kohli for their drafting of figures, Mr. J. Brandon for his

technical assistance, and Mrs. Dorothy Findlay for typing the rough draft and Mrs. Barbara Waddell for the final manuscript.

I am indebted to the "Canadian·Commonwealth Scholarship and Fellowship Commi ttee" for a research fellowship . / .

I express my great appreciation to my wife Kusum for her patience and encouragement.

This research was supported by the Defence Research Board of Canada under grant No. DRB 9550-40 and by the United States Air Force Office of Scientific Research under grant No. AF-AFOSR 68-1552A.

."

SUMMARY

A theoretical and experimental study of the initial ionization phase of an ionizing helium shock wave has been accomplished. Ionization relaxation time a~d the electron density profile behind the shock translational front, have been measured. Both of these parameters have strong dependence upon the atomic collisional excitation rate in the initial phase of the ionizing shock wave and therefore a comparison with theory led to an estimation of atom-atom collisional exçitation cross-section.

The~retical predictions for the relaxation zone of the helium shock wave, indicated only a weak coupling between the ele~tron and atom temperature during the relaxation process, contrary to the results found for hydrogen. A plasma focus driven shock tube was used to create the appropriate helium shock waves. Preliminary results were obtained using a Fabry-Perot laser interfero-meter. Simultaneous measurement of the relaxation time and the electron density profile was performed usin& a multipass Fabry-Perot laser interferometer. A close agreement between the theoretical and the experimental temporal electron density profile was obtained with our experimental(conditions (free stream pressure

0.5

Torr -

5

Torr, shock Mach number

16-26).

The mean vaLue of the 2 cross-section for the atom-atom excitation collisions was found to be

3

x 10-times that of the corresponding electron-atom excitation.collisions. Thus the effective cross-section for atom-atom excitation close to threshold is about 2 x 10- 19 cmf.

1. 2. 3.

4.

5.

TABLE OF CONTENTS Notation INTRODUCTION 1.1 I~troductory Comments 1.2 Physical Model [

THEORY OF THE· IONIZATroN RELAXATION BEHIND' 'A STRONG SHOCK

WAVE IN HELIUM

2.1 Basic Concept

2.2 Analytical Approach to Ionization·Equilibrium

2.2.1 Rate Equations

2.2.1.1 Initial Ionization Rate· Equation

2.2.1.2 Approach to the Ionization Equi+ibrium Due to

Electron Impacts

2.2.2 Electron Ener,gy Equation

2.2.2.1 Elastic Ener-gy Transfer Due to Electron Atom

Encounters

2.2.2.2 Elastic Energy Transfer Due to Electron Ion

Encounters

2.2.2.3 Creation Energy of Electrons Due to At om-At om

Ionizing Collisions 2.2.2.4

2.2.3 2.2.4 2.2.5

Energy Drain Due to Electron Ionization Encounters Conservation Equations

Initial Conditions

Formulation for Computation

EXPERIMENTAL'TECHNIQUES

3.1 Introduction

3.2 Description of the'Facility

3.3 Diagnostic Systems.

3.3.1 The Mach-Zehnder Laser' Interferometer

3.3.2 The'Fabry~Perot Laser Interferometer

3.3.2.1 Double Pass Fabry-Perot System

3.3.2.2 Details of the Experimental Set Up Alignment and

Calibration

3.3.2.3 Multipass Fabry-Perot System

3.3.3 Streak Photography with an Image Converter Camera

RESULTS'AND DISCUSSION

4.1 Experimental Procedure

4.2 Description of Interferogram-Results

4.2.1 Mach-Zehnder Interferograms

4.2.2 Fabry-Perot Interferograms

4.3 Results and their Comparison with Theory

4.4

Error Evaluation

4.5

Discussion CONCLUS IONS REFERENCES TABLES APPENDIX (A-E) PAGE v 1 1 2 2 2 2 2

3

5

9

9

10 11 12 12 13 14

1

7

17

17

.

19

'

19

19

19

21

23

24

25

25

28

28

29

29

30

31

32

35

,

.

a

~

A(p,q) A (g,p) ,A (g,c) a a A (g,p),A (g,c) e e B(p,q) c c e

c .

_ee~ d max e E E _ex E* NarATION

A nondimensional number, determining the shock trans la-tional front thickness

Absorptivity of the Fabry-Perot plate

Einstein coefficieRt for spontaneous transitions from energy level p to q

Slopes of the linear approximating curve for the cross-section of excitation and ionization (from ground level) in atom-atom inelast!c collision

Approximating slopes for the cross-section of excitation and ionization (from ground level) in electron atom in-elastic collision

Einstein coefficient for stimulated transition from energy level p to q

Denotes continuum state of the atom Velocity of light

Electron-velocity

Rate coefficients of inelastic processes ~eading to ionization in at om-at om and electron-atom collisions respectively

Rate coefficient of 3-body recombination process

Maximum deviation of the laser beam inside the external cavity

Elementary charge

Specific internal energy

Ionization potentialof a helium atom Ionization potentialof the impurity atoms Excitation potentialof a helium at om

Ionization energy of an excited electron in a helium atom Oscillator strength

Degeneracy factor for atom Degeneracy factor for ion

g g h h I. ~ ~ I s k K K

(g,p),K

(g,c) a a K (g,~},K (g,e) e e · K (c,p),K (p,c) e e K re Kph(V) K(T ) _e L _p L _ext L(w) m e m. ~ m a

~s

N , a N , e N. ~

Denotes ground state of the atom

Relative velocity of the colliding particles Plank constant

c~t-off impact parameter

Incident (input) laser intensity on the Fabry-Perot

interfero~eter

Transmitted (output) laser intensity from the Fabry~

Perot interferometer

Normalized output of the Fabry-Perot laser interferomet~

Boltzmann constant Gladstone-Dale constant

Rate constants for excitation and ionization processes in at om-at om col~isions

Rate constants for excitation and ionization processes iry electron-atom collisions

Rate constants for the processes of the collisio~al

recombination to the excited level and the collisional ionization from the excited level, due to electron impacts

Rate constant for the recombination process (to the ground state) obtained by considering the detailed - balance between the reactions

Photoionization rate ~oefficient

Equilibrium constant of Saha's equation

Thickness of the plasma slab in the observation section External-eavity-length

Normalized-line-profile Mass of electron

Mass of helium ion Mass of helium atom

Free-stream Mach number of helium shock wave

N afs N ao N P o r o R a R. ~

s

s

_pp t lab t part T

Free-stream atom density

Initial atom density behind the translational front of the shock wave

(=(N + N.»)/N

'a

Non-dimensional number

a ~ I ad

The initial pressure just behind ~he translational front Free stream pressure

Coefficient for the rate of energy transfer in electron-atom elastic collisions

Coefficient for the rate of energy transfer in electron-ion elastic colliselectron-ions

Classical radius of the electron

Rate of ionization due to atom-atom collisions Rate of ionization due to impurity atom collisions Reflectivity of the Fabry-Perot plates

Reflectivity of the folding mirrors

Effective reflectivity of the normal Fabry~Perot system Effective reflectivity of the multi-pass Fabry-Perot system

= (A (g,1)1A (g,l) collision parameter - ratio of the

a e

two approximating slopes, A (g,l) and A (g,l)

a e

Fringe-shift

Fringe-shift due to neutral density variation Rringe-shift due to electron density variation Fringe-shift across translational front

Fringe-shift corresponding to the peak to peak displace-ment of the oscillating mirror

Collisional-radiative-ionization coefficient for

helium plasma \

Laboratory relaxation time Partiele relaxation time

Tem~rat1,lI"e

u

u( \») , x x G'reek-letters aCR S(p,c) y y llatm Ö Plasma Ö _{1.n1. 1.a} ° °tO 1 p

At om , electron and ion temperatures in the non-equilibrium zone of the ionizing helium shock wave, respectively

Transmitivity of Fabry-Perot plates and observation windows respectively

Relative velocity of the gas in the non-equilibrium zone with respect to shock front

Velocity of the free-stream gas with respect to sheck front

Radiation-energy density

Distancefrom the shock front dOWllstream

Non-dimensional distance '

Degree of ionization

Collisional-radiative recombinations coefficient Photoionization rate coefficient from energy level p Specific heat rat~o

Full width at half 'maxima of a line (FWHM) Wavelength

Refractive index

,Refractive index'at atmospheric conditions

Creation energy of the electron in atom~atom ionizing

'collisions

Beam deviation due to density gradients

'Frequency

Polarizabili ty of species i

Phase at instant 'tl betMeen interfering beams Phase variation due to plasma

Initial relat~ve phase

I

Coefficient of absorption for light of frequency \) Density

•

Pfs Patm ~aa ~ae ~ei ~ aa ~ae'

w

P

fs ~. e~ Freeastream density

Density af atmospheric pressure ándtemperature (STP) Dimensionless ionization rate coefficient for atom-atom collisions

Dimensionless ionization rate coefficient for electron-atom collisions

Dimensionless recombination rate coefficient of

3-body recoIDbination collision

Dimensionless r~te coefficients of energy transfer to electron in atom-atom ionizing collisions

Dimensionless rate~oefficient for energy transfer to electrons in electr9n-atom and electron-ion elastic collisions respecti~ely:

Response time of the external cavity Cross-section

Elastie cross-section Plasma frequency

this subscript denotes initial conditions (F

f ,Tf ) ahead of the shock front . s s

1. INTRODUCTION

1.1 Introductory'Comments

In the past decade there has been a growing interest' in' obtaining a better understanding of the relaxation zone behind a strong shock wave'.' , This ,has.' been reflected' in' the published work' of' various investigators (Refs'. 1-6). Such studies have led to'an understandingof the-various-microscopic-phenomena, such as, the mechanism for transfer' of' energy' between-dif'ferent degrees of' freedom and the efficiency of the collisional' rate processes. Information' about-these microscopic processes arerequired-in severaltechnological'applications, involving a high

temperature environment, such' as, the- re~entry of a space vehicle into the atmosphere, combustion processes~ - An extensive review of the rate constants for ionization

relaxation-has been-given in'two-recentreview papers (Refs.

7-8).

Holenback andSalpeter

9

,

approachedsemi~empirically

'

the

-

problem

of evaluating the heavy particle' collisional' ionization cross~section in the low energy range with-no single parametric dependence. In addition to the theoretical difficulty, direct experimentation using low energy atomic beams to investigate heavy particle collisional excitation and ionization cross-sections is an extremely difficult experimental problem.

Shock tube studies of monatomic gases, such as Ar, Kr, Xe have been performed extensively because of their simplicity (Ref. 1-4). Recently, similar studies of neon (Ne) have also been reported (Ref. 10). Heliumhas ,not previously been accessible to such observations-due to the inherent difficulty in producing high Mach shock waves in helium using conventional shock tube. The work reported in this report has been performed with'helium employing the same pl~sma focus driven shock tube as used in-the study of the initial ionization in hydrogen by Belozerov and Measures5.

The'familiar'two step collisional ionization model has been adopted in the work-to'be-described. Theoretical calculations based on this model predicts only a wealc coupling-between the' electron-and atomic temperatures.-, contrary to the . case-of hydrogen' (Ref. -5-) -.' -Consequently-'the-considerable' simplification in the

numerical-integration allowedby' the-assumption Te

=

Ta' was not assumed in the present work. The preliminary results for this more' complex si tuation have been published in Physics of Fluids (Ref. 11).

The details of the mathematical formulation of the relaxation zone is given in Chapter 2.

The basic diagnostic techniques used were spatially resolved Mach-Zehnder interferometer, using a pulsed ruby laser as the light souree, and temporally resolved single and multi-pass Fabry-Perot interferometer, using a C. W. 'helium-neon laser as a coherent source. The detailed descrtption is pre-sented in Chapter

3.

The interpretation of the results along with some discussion forms the material of Chapter

4,

and the concluding remarks are given in Chapter

5.

1.2 Physical Model

The physical model Msed to describe the microscopie processes occurring within the shock-heated gas can be summarized as follows. A propagating strong shock wave randomsizes ~he directed energy of the gas flow so that directly be-hind the translational discontinuity the thermal energy of atoms is greatly in-creased "lithin a few elastic collisions to a val'ue defined by the Rankine-Hugoniot relations. SUbsequently, inelastic collisions between atoms results in their excitation and ionization creating a supply of free electrons at low thermal

energy. Once a substantial number of electrons are available, the relatively large electron-atom inelastic cross-section as well as the high mobility of the electrons enables electron-atom inelastic encounters to serve as the dominant mechanism for excitation and ionization. Under such conditions, the electrons and heavy partieles (atoms and ions) separately establish a Maxwellian distribution of their thermal energies, defining temperatures Te and Ta, respectively. Furthermore, since the energy gap between the continuum and the first excited level is small compared to the separation of the first excited level and ground level o.f the helium atom, it is reasonaqle to assume that the inelastic collisional rate for the free electrons is adequate to at least maintain a Boltzmann distribution at the electron temperature within the excited states of the atom.

2. THEORY OF THE IONIZATION RELAXATION BEHIND A STRONG SHOCK WAVE IN HELIUM 201 Basic Concept

A strong shock wave propagating through a gas produces rapid changes in the physical conditions of the gas. These changes result from the transforma-tion of directed motransforma-tion into random motransforma-tion. This transformatransforma-tion results in a sharp increase in the temperature of the gas. Sub.seq-qently, the atoms in the high energy tail of the new distribution are capable of suffering, collisions that result in the excitation and ionization of the atoms. Consequent1y, a nonequilibrium zone forms behind the translational front of the shock wave. Within this zone the concen-tration of ions gradually increases until equilibrium is reached. The thickness of this nonequilibrium regio~ is referred to as the relaxation length and the corresponding time as the relaxation time.

2.2 Analytical Approach to Ionization Equilibrium

A two-step collisional model was proposed by Weynman12 to account for the experimental values of the relaxation zone which could not qe explained on the basis of a single step theory, i.e., one assuming direct ionization from the ground state. This' single step theory predicted r~laxation lengths an order of magnitude greater than the experimental values 13 ,1. Furthermore, Harwell and Jahn l established that the activation energy for the atom-atom ionization reaction corresponds to excitation to the first excited level instead of the continuum. Physically this simply indicates that the probability of a collision leading to the excitation is much higher than for direct ionization and that once excited an atom is much more likely to be ionized than de-exéited.

2.2.1 Rate Equations

In the present formulation the ionization prpcess in the relaxation zone has been divided into two sections: (i) the slow ionization regime where the initial ionization process is primarily due to atom-atom cOllisions, and (ii) the fast ionization regime where electron impact dominates the approach to

.

'

the ionization equilibrium.

2.2.1.1 Initial Ionization Rate Equation

The relevant atom-atom ionization mechanisms are as follows:

(i)

.?ingle step process

He + He (ii) Two step process

He + He

*

He + He Ka,(g,c) + ~ • He + e + He Ka(~P) Ka(Pi'C) ~

*

He + He + He + e + He

Taking both processes into consfuderation one can express the electron productio~

rate as follows (Ref.15)

dN

e

dt (~-l)

where g,p,c denote the ground excited and continuum states of the helium atom. Ka (x,y) represents the appropriate atom-atom collisional rate coefficients.

Since the rate of ionization of excited atoms, Kä{P,C) dominates the de-excitation rate, Kä(p,g), Eq. 2.1 becomes

dN

e

dt (2-2)

The various rate coefficients can be related to the appropriate cross

-section in the following way.

Figure 1 illustrates the general position

and

behaviour of the cross-section in relation to the energy distribution function of the colliding species. Since within the temperature range of interest the mean thermal energy of the atoms is much less than the required threshold energy for e~citation or ionization,

a linear approximation may be use~ to represent the cross-section's dependence upon collision energy, viz.

In general the collision rate coefficient is given by

K

=

.

1

00

cr(v) f(V')

vdv

'V.i

th

so that the number of such collision events per sec. will become

K N.N.

~ J ==

N

~

.

N.i

J 00 cr(v) f(v) lP th vdv

(2-3)

(2-4)

where V

th is a relative velocity corresponding to the threshold energy, viz

2

I.l V_th

2

=

_Eth

I.l is a reduced mass and f(V) is the appropriate velocity distribution fftnction.

Substituting for,~ from, Eq. (2.3) in Eq.(2.4)and assuming the distri-bution function to be a Maxwellian, one obtains the rate coefficient 'K':

1/2 3/2 E E

K

=

A (

~

)

(kT

~

(

k~h

+ 2 ) exp (

~h

) (2-5) Using this form of expression for the relevant rate eoefficients in Eq.(2.2) we obtain:

( E ,-E

/kTè: ]

+ A ( 1) ~ + 2 ex

a g, kT~

where ~

=

m /2 and k is the Boltzmann constant. E. and E _{a .}

1 U

energies of ionization and exeitation (to the first excited A {g,l) are the approximating slopes for the ionization and

c~oss-sections, respectively.

(2-6)

are the respective level), A (g,c) and

excitatio~ collision

In addition to atomie collisions giving electron production in the initial ionization zone, there are two other processes that could potentially enhanee the rate of ionization:

,

(i) Photo-exeitation by resonance radiation (ii) Ionization due to impurity atoms.

In Appendix A.l, the mean free path for resonanee line radiation has been evaluated. For the typical experimental conditions, this mean free path is of the order of 10-

6

cm. Obviously this form of radiatio? has a penetration length

smaller than the collisiona1 relaxation length

(z

2-3 cm) and therefore its effect on the ionization relaxation ean safely be negleeted. Direct ionization in the initial ionization zone could result from the absorption of U.V. radiations by atoms (and ions). The sourees of this radiation are (i) the equilibrium zone of the shock wave and (ii) the driver plasma behind the contact surface. In Appendix A.2 it is established that the rate of photoionization is much less than the rate of eollisional ionization for the experimentally obtained yelocity range of the shock wave. Indeed, the collision parameter

's'

(which is defined as the ratio of the slopes of atom-atom collisional excitation cross-section, A (g,l) to that pf eleetron-atom eollisional excitation, A (g,p) would have to b~ less than 10-4 if the former proeess were to domi~ate iff the ereation of initial ioni-zation. The rates of ionization due to U.V. absorption of radiation (photo-ionization) and due to collisions have been eompared in Fig.2. For the range of the present experimen~al conditions and also from the experimentally estimated value

4

I

..

"

of S (see Chapter 4) it is obvious that photoionization effects are also insigni-ficant.

Moreover, comparison of the mean free path of the radiation withi~ the ionization relaxation zone reveals that the equilibrium zone is optically thick to this radiation, thus further reducing the effect on initial ionization.

The effect of impurities on the relaxation length is still a disputed question and no quantitative results are available showing the degree to which this process affects the relaxation zone. Ottinger

16 ,

experimentigg with Argon,

showed no appreciable effect of impurities up to a concentration of 12 ppm on the relaxation length. This ~oes not seem to be generally supportedl,lj.

In the case of helium gas, which has the highest ionization potential of all the elements, impurities could present a ?erious problem. However, as indicated in the discussion of Chapter

4

apd also shown in Appendix A.3, one can neglect this effect under the present conditions of operation.

From the abave one can also cOfclude that U.V. radiation from the driver plasma would be completely absorbed in the equilibrium zone, having no effect on the initiaJJ ionization zone. Therefore, in brief, the influence of radiatiop on the initional ionization can be disregarded for the experiment al conditions under investigation.

2.2.1.2 Approach to the Ionization Equilibrium Due to Electron Impacts

The inelastic collisions of electrons with atoms determine the ioni-zation rate, once a sufficient number of electrons have been produced by the initial ionization process because of its larger cross-section compared to the heavy particle collisions. Therefore the properties of the various species existing in the plasma and their respective energy levels can be described in terms of electron number density Ne and electron temperature Te. These parameters are relevant because of the domina~t role of electrons in colli-sional processes.

In order to ascertain the relative populations of the various quantum states, thermodynamic equilibrium arguments are used if full thermodynamic equilibrium obtains. Otherwise it becomes necessary to consider the detailed atomic processes. These atomic processes have recently been considered by manyauthors (Refs. 17,18,19,20). The method used is based on that originally outlined by Bates, Kingston and McWhirter 17 ,18. Their theory, though developed for hydrogen atoms or hydrogenic ions has been adapted to helium atoms having one optical electron. In a manner similar to the initial ionization regime, both single and two-step processes will be considered as leading to ionization. These basic atomic processes are:

(a)Collisional ionization:

He(p) + e

(b) Three body recombination to He(p) (inverse of (a))

(c) Collisional excitation: + He + e + e He(q) + e K (c,p) e~ He(p) + e

(d) Collisional de-excitation (inverse of 'c') He(p~ + e

(e) Photoionization:

He(p) + hv ~lP.,c)

....

p( v)

He + + e (f) Radiative recombination (inverse of 'e')

He+ + e

~(C4P)

He(p) + h v (g) Spontaneous emission

He(p) A(p,q)

....

He(q) + hv (h) Photoexcitation (absorption, inverse of 'g')

He(q) + hv

B(q,~)P(v)

He(p)

where 'p' and 'q' are bound states and crepresents the continuum state. The rate coefficients, shown above the arrow for each reaction, are defined for transitions between any two states. A and Bare the Einstein coefficients. It should be noted that stimulated emission is negligible in our case. These

reactions are shown schematically in Fig.3.

Summing the aboye reactions, one can de fine the excited-state level populations in terms of an infinite set of rate equation of the form:

- - N (p) N [ K' , (p ,

c)

+ E K' (p ,q.) ] -a e e pfqe + E Na(q)A(q,p) + [N 3K :(c,p) + e e q<p N_e2

~(c,p)J

+ [

p(v) E B(q,p) Na(q) q<p (2-7)

where the first term is the rate for coltisional processes from p to the continuum c and to any other level q. The second term represents spontaneous emission from 'p' to levels q with energy below that of p. The third term represents colli-sional population of p from all other bound levels q. The fourth term is the population by spontaneous transitions from levels q above p: and the fifth term represents collisional and radiative recombination from the continuum to level p. Finally, the last term takes account of absorption and photoionization pro-cesses, which are negligible for the optically thin plas~.

- - - -- -- - - -- -

---_._-In the zone of interest, it is reasonable to assume that there are sufficient inelastic collisions to maintain the number densities of all excited levels of this atom in thermal equilibrium with the free electrons. Therefore for a given electron temperature? any jump of a ground state electron into an excited level must be accompanied by the transition of the excited electron to the continuum, and·,'meohan.is!Ill for the recombination of a free electron must be analogous. However, the number densities of the excited levels are

fix

~d

quanti-ties and subject to change with the electron temperature, though the rate of such changes is small when compared to that of the ground levet; viz,

dN (p) dN (g)

L: a

«

a

pfg dt dt

(2-8)

when p

=

g

represents the ground level of the atom, and :implies that

-(2-9)

Following Bates et al18 one can express the rate of ionization which is determined by the rate of depopulation of the ground state as

dN e

dt

where SCR

=

SCR(~~Ne) is the collisional rad~ative ionization coefficient and

QCR

=

QCR(Te,N

e) the collisional radiative recombination coefficient.

(2-10)

Drawin and Emard20 recently calculated collisional-radiative

recombina-tion and ionizarecombina-tion coefficients QCR and SCR on the basis of a more realistic

non-hydrogen-like collisional radiation model for atomic helium. They considered

a wide range of helium plasma conditions, iiiz, opticallYvth:h-9, optically thick and optically thick towards all resonance lines, etc. Using their data (Refso

20,21) we compared the coefficients QCR and SeR for the optically thick for resonance lin,es case versus cOllisionally domd.iJ.ated case, within the range of our experimental condition

( N

z

10 -10 ,T 15 17

z

1-3

e.v.

' ,N - 10 ~ 18 cm - 3 )

e ' e a

It is obvious from Fig.4 that there is hardly any difference between these two cases. Therefore, we can safely assume th at the escaping radiations has a

negligible influence and so the helium plasma in this regime may be assumed to be àoililmsian dominated.

Since in the relaxation region we have nonuniform plasma conditions, the assumption that radiative diffusion mechanism is slower than any changes in the state of gas, has to be made in order to apply the above analysis for uni-form plasmas. This assumption is valid as the velocity of gas relative to the

shock is much higher than the upstream propagation of resonance radiation. In view of the above discussion, we can conclude that the ionization

rate, in this zone, can be writte~ in terms of collisional processes o~ly, ~iz,

NN

a e [K (g,c) e + K (g,l)] - K e re e N

3

(2-11)

where the rate coefficients

K

(g,c) and

K

(g,l) are defined by Eq.(2-5) af ter

re-e e

lating their respective cross-sections by using Eq. (2.3). The recombinatio~ coefficient

K

is obtained from the principle of detailed balance viz,

re K _re

=

[K (g,c) + K (g,l)] e e

[ ::2 ]

Eiq·

where the equilibrium va1ue of

N

2/

N

is obtained from Saha's equation

e a (2mn kT

)3/

2 e 3 e exp (-E./kT ) h J. e

[

N

2}

g. o ~ K(T) = 2

..2

N e g a a eq (2-12) (2-13)

where g. and gare the ionandatom electronic partition function, respectively.

J. a }

Substituting the rate coefficie~ts in Eq.(2.11) and using Eqs.(2.5 and 2.12), the rate of ionization for this zone can be written as

C

E 2) e-Eex/kTe

+ A ( _eg,

°

_·

ib

)

~ _kT +

e

(2-14)

Cross-section data for inelastic collisions b~tween electrons and atoms

leadi~ to excitation and ionization is available in the literature. The theo-retical work of Massey et al (Ref.22) gives good agreement with the experimental results of Schultz and Holt24 ,2? for the excitation cross-section (to 23 s , 21s, 23p and 23p excited levels) of the helium atom due to electron impact. We have used the linear approximation given in Fig.5 using the experimental data of Ref. 24. From these data the value of A (g,l) was determined, i.e.,

e

( )

-18

2/

A g,l

= 7.7

x 10 cm eV

e

of A (g,c) was obtained from Ref. 26, with the following numerical value

e

2.2.2 Electron EnergyEquation

The rate of thermal energy ga in by free electrons can be obtained by considering the free electrons being subjected to randomly directed elastic and inelastic encounters with the atoms and ions. The ionizing process invoiving

collfuions draws on the high energy electrons from the tail of the Maxwellia~

distribution. Therefore, the assumption that a Maxwellian distributio~7or the

free electrons exists at each instan~ of time must be checked. DeBoer has

shown that for our condition the rate at which electrons are elevated into the

tail of the Maxwellia~ distribution, exceeds their rate of loss due to ionization

collisions*. Thus assuming a Maxwellian distribution for the electrons and neg-lecting (a) any diffusion effects, (b) Bremsstralung radiation losses, one can

write the electron e~ergy equation as comprising three energy so~ces, viz,

(i) elastic energy transfer resulting from electron-atom

encounters,

(ii) elastic energy transfer due to electron-ion encounters, (iii) creation energy supplied by the at om-at om ionizing

en-counters and an energy loss,

(iv) due to electron inelastic collisions leading to

excita-tation and ionization.

Combining the various energy loss and gain terms we can write the

complete electro~ energy equation in the form

We will consider each of these in turn.

2.2.2.1 Elastic Energy Transfer ~ue to Electron Atom Encounters

Petschek and Byron13 arrived at the following expression for

determin-ing the rate of energy transfer due ~o elastic collisions between electrons and

atoms

. (2-16)

where uel~g) is the cross-section for elastic collisions. This cross-section

as a f~tion of egergy , has been evaluated by using the experimental results of

Golden and Bende12 and approximating their empirical curve by three linear

seg-ments (Fig.6). This leads to the foltowing expression for Qea:

* This is because the mean

2 eV and neutral density of to initial ionization time is of the order of 104.

collision time fQr helium atoms at a temperature of

the order of lOlö cm-3 ~ 0.5 x 10-3 ~sec, compared

where 4a-a J7r 0.5 0.5 m 2 2kT 3/2

T)

(kT )

~

(_e) (2: _

1 [1 _

0.5· a-

_e

m m- T asl E 1 a e e s -E./kT { e l . e 1+ _ _ 3kT e + E a

6

(:e)2 +

6

(:Te)3}

+

a a :,

( :;e)2 e

-EjkTe { (1 +

2~e+

2

(~e

_

)2

) (

"aSl- l ) }

+

(

~e)3

(1 +

3~e)

e-E/kTe

a- (E) = el; for 0 ~ E ~ 2.25 eV

=

for 2.25

<

E

<

14 eV

=

for 14 ~ E ~ 5~ eV

=

0 for E

>

50 eV a-_as1

₌

a-_{sl a} /a-

₌

1.09 E

₌

2.25 eV E

₌

27.8 a sl a-_as2

₌

_{a- ia-s a}

₌

0.75

_~

=

14.0 eV E s2

=

50.0 E

=

26.0 eV c

2.2.2.2 Elastic Energy Transfer Due to Electron-Ion Encounters

(2-17)

The rate of energy transfer in this category of 2~11isions is due to

the Coulomb interactioQ between charged particles. Landau evaluated such

energy transfer rates for differe~t temperatures. Petschek et al 13 derived

an alternative formula for the ge:deral case of an arbitrary interaction law which states 2 m e Qei

=

ffi. l.

where c is the electron velocity and Q . is the coefficient for the rate of

e el.

(2-18à)

energy transfer by electron-ion elastic collisions. In the case of the Coulomb interaction (Ref.30 )

cr(c ) e

=

2rr ( e 2 2 m c e e Log e

where h is the cutoff fumpact parameter.

2 1/2one cau approximate Q . by taking the Debye shielding length (~ (kT /

8rrN e) ) as the cutoff param~ter hand simplifying (2.18~)by substituting e e

m c 2

=

3kT within the log term. Therefore, averaging cr(c )c 3 over a Maxwellian

e e e e e e

distribution and substituting in Eq. (2.18~),results in

2 e

4

~

_{(kTa- kTe ) {9(kTe )3} + l}

Qei

=

ma (kT )3/2 Loge 8rr N e

6

(2-19)

e e

where T.~ Tand M. ~ M is assumed. l a l a

lt is worth noting that Q . can be evaluated using only fundament al

\ el

constants and does not depend on any measured cross-section. rherefore the main uncertainty in the evaluation is in the selection of the parameter 'hl. This uncertainty is estimated to be less than 20% (Ref. 13).

2.2.2.3 Creation Energy of Electrons Due to Atom-Atom lopizing Collisions

lp the electron energy equation describing the flow behind a strong shock

wave, the evaluation of the initial energy term has been open to question as no empirical information is available on the thermal energy gain by the electrons

created in atomic encounters. Belozerov and Measures

5

used the concept of

equi-partition in order to estimate this energy of creation for the free electrons and

derived an expression for the initial thermal energy of the electrons produced

just behind the translational front of a strong shock wave in hydrogene Since physically this seems a reasonable assumption, the same approach is outlined in the following paragraph.

The assumption of equipartition leads to a creation energy

E

'3

1 _{(E - Ei)} (2-20)

for the liberated electron, where E is the energy of encounter and E. is the energy

required to free the electron. This is then averaged over all such ~ollisions

assuming a Maxwellian distribution. This averaged energy per electron is evaluated for both single step and two step processes. Under these circumstances the initial

energy possessed by a~ electron is given by

E* + 3kT { A (g,c) [

E

=}

kTa E* + 2kTa 1 -

~

Aa(g,l) EexkTa/(E* + _{3kTa )}

a a

E*/kT ] }

((E + 2kT ) e a +

0.5

(E. + 2kT ))

where E* = E. - E • E* being the ionization energy of the excited electron.

1. ex

Equation (2-21) is used later, to determine initial electron temperature for computation~l purposes.

2.2.2.4 Energy Drain Due to Electron Ionization Encounters

As mentioned earlier the fast ionization process is governed by electron-atom encounters and for each ionizing event, an energy E. is removed from the free electrons. The rate of such an energy drain can be exprêssed mathematically as

Ei [

::e

1

This energy drain ean be eva1uated using the earlier derived Eq. (2-14) for [ ::eJe

2.2.3 Conservation Equations

The rate equation and electron energy equation discussed in the preceding section are expressed in terms of four parameters, viz, electron density. Ne' atom density N electron temperature Tand atom temperature T. Ad~itional relations

a e a

are required between these parameters in order to solve these equations simultan-eously. The laws of conserv~tion when applied to the flow through the shock front, along with the equation of state, provide the additional required relations. Below, we write the set of such conservation equations i~ the coordinate system moving with the non-attenuating shock wave.

Mass m N U

=

m (N + Ni) U (?-22a)

a ao 0 a a Momentum P + mN U 2

₌

P + m (N + N.)if (2-22b) 0 a ao 0 a a 1. P U 2 P U2 + 0 ₊ 0 _{E +} (2-22c) Energy E

₂

=

_(N+N.)m +

:2

0 N m ao a a 1. a

Where the index '0' refers (Fig.7) to the state of the gas just behind the translational shock front. Conditions in this region referred to as initial conditions, are determined in terms of the free stream conditions, using the gas dynamic shock wave jump conditions which are stated in the next section.

The pressure Pand the specific internal energy E of the partially ionized gas are given by the two equations:

P = (N + N.) k( T +

ex

T ) a 1. a e

E

= 1.

₂ k _m a (T +

ex

T ) +

ex

a e (2-23) E. 1. on m (2-24) a

where,

ex

== N / (N + N.) is the degree of ionization, Tand T being the respecti ve

e a 1. a e

temperatures of the heavy and light components of the plasma.

2.2.4 Initial Conditions

Initial conditions determine ~he gas parameters just af ter the

trans-lational front. These are evaluated in terms of the free stream condition and

the shock Mach number. We use them in the form of three algebraic relations,

determined by the Rankin Hugoniot conditions (Ref. 31)

Pa Pfs T ao

T

fs

=

=

N

(r

+ 1) M 2 ao fs N_afs (r - l)M fs 2+2 2 2r M _fs -(r-l)

(r

+ 1) 2 2 [2rMfs -(r-l)][(r-l)Mfs + 2] (r+l )2 M 2 fs (2-25a) (2-25b) (2-25c)

The initial electron temperature is determined from the initial thermal energy of

the free electrons as given by Eq. (2-20) by expressing ltl equal to(3/2)kT one

eo obtains T eo

4

= 9 E* + 3kT ao E* + 2kT ao { A (g,c) 1 _~

₂

1 a r E kT / (E* + 3kT ) A a ( g ,1 ) ex ao ao E*/kT ( (E + 2kT ) e ao +

o.

5 (E. + 2kT )

l

}

ex ao l ao -' (2-25d)

The initial value of the electron density is much more difficult to

evalu-ate as it is dependent on the experimental conditions and techniques involved.

The source of such a precursor is expected to be radiation emerging from both the d

is-charge in an electromagnetic shock tube, and also from shock heated gas (Refs. 32to35).

The exp eriment

I5in helium have been performed using a similar facility as was used by Belozerov for hydrogen. He did not observe any appreciable

precursor ionization, and since, it has been estimated (see Sec. 2.2.1 and

Appendix A) that radiation from the equilibrium zone does not affect the region of

initial ionization it is reasonable to argue that it should have little effect on

the gas ahead of the shock wave. Therefore, we do not expect the initial ioni-zation measurements to be affected by any appreciable pre-ioniioni-zation.

Preheating of the gas ahead of the shock wave is also being neglected on the similar grounds moreover, the close agreement observed between the theory and the experimental results (see Chapter

4)

strengthens our justifications for the neglect of these precursor effects.

In view of the above discussion, we can take the initial electron

density to have an arbitrary value less than 1012 cm-3 . It is interesting to note that there was no significant change in the computed relaxation time for an initial

electron density in the range 1012 - 1013 cm-3 2.2.5 Formu1ation for Computation

Rewriting the rate equation for ionization,as derived in the preceding Section 2.2.1, in the following form

dN

e

dt

=

~

N 2 C a aa + N N C aeae (2-26) where the various inelastic rate coefficients C ,C ,C . can be defined as follows, (compare the above equation with Eqs.(~~6

aftä

2.Î~1).

C aa = 4(kT )3/2

(wn:)17

2 [Aa(g,C) E A (g,l) ( ~ + 2) a kT a E A (g, 1) ( ex + 2) e ' kT e C . = C /K(T) ee1 ae e E. _1_ + 2) e kT a -E _ex /kT _a

J

e -e E. _1_+ kT e -E _ex /kT _e

J

-E. /kT 1 a -E. /kT 2) e 1 .e +

K(T) is defined by Saha's equation (see Eq. (2.13».

e

(2-27)

(2-28)

(2-29)

Similary, rewriting the electron energy equation developed in Section 2.2.2:

d

dt ( 32 kT N ) e e

=

~

N 2 a E C aa + N N a e

- E. ( N N C - N 3

ç

.)

1 a e ae e ee1 (2-30)

where the e1astic rate coefficients for the energy transfer due to the electron atom encounters, Q and the electron-ion encounters Q . are given in Eqs. (2.17 and 2.19), whereaseathe creation energy for electron''tI§ given in Eq. (2.21).

Using a substantial derivative for the flowing gas and reducing it to a time independent one-dimensional relation with respect to the coordinate system travelling at the velocity of the shock front, one can write

dN

dte

=

(U

_fs

-U

_po)

N

_ao do. _dx (2-31)

where U

f is the free stream particle velocity relative to the shock front (i.e., shock velocity) and upo~ the particle flow velocity behind the shock front.

The numerical integration of the ordinary differential equations is made more convenient by reducing them into a non-dimensional form. This is accomplished by defining the following non-dimensional parameters:

T

=

T

IT

e e ao

and introducing the non-dimensional length

x

=

xl

(

(

U

f s -lJpo ' )

IN

ao eo K ( g ,1) ) .

(2-32)

(2-33)

where K (g,l) is the inelastic rate coefficient for excitation (the general form is give50in Eq. (2.5)) by electron-atom impacts, assuming the electron temperature is equal to the initial atom temperature. _,

Making use of the non-dimensional parameters defined above and the definition of the substantial derivative (Eq. 2-31) we can write both the rate equation - and the electron energy equation - into the non-dimensional form as follows:

dT

e

ëïi'

=

cp aa = cp ae = 7T

=

aa m + CX(l-CX) m

~

m

~aa ~ae - ~ei

2 ~ C _aa 2 C ae 3 2

(T

e +

T

1

.

)

-2 N K (g,l) eo -2 N K (g,l) eo C N N'3 eei ao , K (g,l) eo _E_'_ kT ' +T.)cp 1 aa ao (2-34) (2-35) (2-36a) (2-36b) (2-36c) (2-36d)

1T ae 1T • el.

=

=

kT K (g ,1) ao eo kT K (g ,1) ao eo (2-36e) (2-36f)

In addition to, the ordinary ,differential ,equ:atiohs (2.34 and 2.35) two algebraic ,

equations are formed ,in a non-dimensional form, using the' conservation equations

and the e qu.atïon of sta.te defined above. This is necessaI'y in order to evaluate all the four parameters N , _{, e}

N ,

_a

T

_{' e} ,T (or a,N , T , T ) simultaneously. These_{a a e a .} ' algebr.aic equations are:

T +T = :~ N +,( N - l ) Ë] / N2 a e N 1 + Ë+ [9/25 Ë 2 - (6/5 32/25 aT.) Ë + 1 ]1/2 à ' _{l. (2-38)} 2 - 4/5' a T. + 2/5 Ë l. Ë 2 U where = m U/kT , U

=

U fs a 0 ao 0 po

In the above equations (2.37 and 2.38), ~ and Nare expressed,as a

a

function of T , a. Substitution of T ' and

N

in terms of Tand a in the ordiriary

e a e

differential eqs. (2.34 and'2.35) followed by simultaneous numerical integration

of these equations, using anumerical prediction correct0r,technique (Ref. 36),

leads to the evaluatiln of,T and'a and'hence Tand

N:

,These non-dimensional

e a

parameters are then used to de fine N , N ,T and T . A typical integration* is

e a e a

presented in,Fig.8ajb.

For the case of a non-attenuating shock wave, the relaXation ,zone

be-hind the shock translational front moves as a control volume of,width SA (Fig.9)

because of the fact 'that an observer sitting on a shock wave front at S would

see properties of the SA zone as time independent. 'There~ore the laboratory

relaxation time t

lab is

t

1ab

= relaxation length / shock

~elocity

Thus using the above relation one can express the 'relaxatiènlength in terms of the relaxation time.

An alternative procedure for obtaining therelaxation td:m~";]jm·.a labonae'

tory coordinateis to integrate Eqs. (2.26 and 2.30) taking time as an independent '

* Flow chart 'of computational'procedureis given in Fig. 36.

16

variable, thus determining particle relaxation time, t part given by (Fig.9)

where

t

lab

=

SB

=

P'B - P'S

=

t part P'p/Tan8

(U

f s - U ) p

PP'

=

U t and Tan8

=

U

P part fs

Using conservation of mass equation, one can write

t part Therefore t lab is

(2-40)

(2-41)

This relation is justified for the low ionization case as P

lp

fs does not change

very much. For the general case

Pfs , dt

p

(2-42)

Therefore the particle time thus computed theoretically, can be transferred to the laboratory time, for a comparison with the experiment.

3. EXPERIMENTAL TECHNIQUES

3.1 Introduction

Two principal techniques, -viz, the Mach Zehnder interferometer and the Fabry~Perot interferometer were used independently to monitor the variation in the '

refractive index in the nonequilibrium zone of the strong helium shock wave. Qualitative information regarding the planarity of the shock wave and the uni-formity of the plasma in the zone of interest were obtained from the streak pictures, using a high speed image converter camera.

3.2 Description of the Facility

Conventional diaphragm shock tubes even with improved driving techniques are still unable to drive strong ionizing shocks into light gases, such as hydrogen

or helium. Therefore, a novel form of shock tube using a plasma focus for the

creation of high Mach number shock waves has been used. In the present facility,

a Philippov discharge 37 has been used to produce the high velocity shock waves in both hydrogen and helium.

The detailed description of the plasma focus driven shock tube facility is given elsewhere (Ref.15) so that only a brief description of this facility, including the minor modification for use with helium is presented.

The facility includes two basic components: (1) an energy source, a capacitor bank with an appropriate triggering arrangement device, (2) a discharge

chamber and shock tube with an efficient high vacuum system.

The capacitor bank consists of 7 sets, of three, S.51~F 20 KV capacitors, connected in parallel with a triggering spark-gap and a high voltage power supply (Fig.lO). A high voltage triggering pulse, needed to trigger each of the 7 main spark gaps is generated by discharging 300 feet of cable (charged to 25 KV) through a master gap, In this way the cable is used; as a very fast O. 2 ~F capaci-tor. A separate trigger control unit and a low current power supply were used

(manually or by a synchronized pulse) to trigger master gap. To ensure a simul-taneous discharge of all 21 capacitors the jitter time of the system had to be maintained below 20 nanoseconds, (Ref.15). In order to increase the efficiency of the discharge, a large number of capacitors, switches and cables were connected in a parallel configuration, which reduced the net inductance of the system,

thereby the time constant. The total inductive load of th

3

system was about 42.5 nH corresponding to the natural frequency of 5.7 x 10 c/sec,

The stainless steel discharge chamber (Fig.ll) has a central electrode (S" x 7" dia.) of stainless steel separated from the outer chamber (9" x 10" dia.) by a cylindrical ceramic insulator . There is one inlet (l/S" dia.) for the test gas. The outlet (1.5" dia.) on the axis of the discharge chamber connects to the stainless steel shock tube of the square cross section (1-1/4" x 1-1/4") which has 3 interchangeable sections of approximately 20" long. The two are high quality glass plates with anti-reflection coating for 632SoA. It is worth mentioning that a minor modification to the discharge chamber improved substan-tially the reproducibility of the system and higher Mach number shock waves in helium. A conducting stainless ring was attached to the inner electrode so as to reduce the insulating gap between the two electrodes.

The other end of the shock tube was connected through a dump chamber to a vacuum system. Due to the high ionization potentialof helium impurity plays an important role. Therefore, it was 'necessary to have a very good vacuum system. The double wall liquid nitrogen cold-trap and speedvac (type 200) oil diffusion pump, proved quite appropriate for the present facility. Another cold-trap at the inlet was used, especially to trap any H

20 impurity in the ultra high pure (UHP) helium gas. The present facility is schematically shown in

Fig. 12.

As shown by Philippov37, the formation of the discharge has 3 basic phases. This is explaine.d schematically in Fig. 13. During the first phase, the current sheet formation follows the path of least inductance which is near the surface of the insulator and then it expands. This current sheet then rapidly moves towards the axis of the chamber as the current reaches its maximum. This

is the second phase. In the so-called, compression phase the current sheet becomes conical and so forms a focal point on the axis of the chamber near the curved

inner electrode. A dense plasma is momentarily created at this focal point with a temperature of 1 keV and a pressure close to 2,000 atm. The sudden expansion of this high pressure region acts as a piston and is responsible for the formation of the strong shock waves. One advantage of this discharge configuration is that the gas contaminated with insulated material during its first expanding phase does not take part in the compression phase.

The reproducibility of the shock speed with pressure at a fixed capacitor bank voltage is in general found to be poor at low pressure as shown in Fig. 14 consistent with the observations made with hydrogen (Ref. 15).

3.3 Diagnostic Systems

The rate of ionization of a fluid element behind the translational front depends on the velocity by which it is being processed, One of the most powerful techniques used to follow the history of this process utilizes the phase variation

induced in an electromagnetic wave due to the change of concentration of the various species. In the following section, three different arrangements are discussed, which were used to monitor the ionization relaxation in the non-equilibrium zone of a strong shock wave in helium. In principle they are very similar, each using interference effects, but the degree of sensitivity is quite different.

3.3.1 The Mach-Zehnder Laser Interferomete~ '

A Mach-Zehnder interferometer was used to measure the variation of the refractive index and hence of the density, using a Q-spoiled ruby laser as a coherent light source. This versatile interferometer has been used by several workers (a) to study relaxation phenomena (Refs. 38,39), (b) to measure

polari-zability of the atomic species (Refs. 40,41) and (c) to monitor the electron density profiles in shock wave (Refs. 2,5,42,43,45,46). In general, double fre-quency interferometers are essential for monitoring electron and neutral den-sity variations but in the present system this requirement is not essential as the neutral density contribution to the refractive index is negligible (see Chapter

4)

.

A schematic diagram of the arrangement used in this work is shown in Fig. 15.

The ruby laser (Model TRG 104) used to obtain interferograms produces a Q-switched output pulse of 0.8-1 Joule. A narrow band filter at 6943Ao and two stops placed at the focal points of the beam were used to eliminate plasma radi-ation. High speed polaroid (type

47)

film was used for recording the inter-ferograms . The photograph of the arrangement is shown in Fig. 16. A supple-mentary diffused light source (mercury lamp) was used to check the alignment before each run to ensure best contrast and horizontal position of the fringes .

The aperture of the interferometer was 3 cm x

5

cm and it was very well isolated from the floor vibrations.

A synchronizing pulse from the laser head power supply was used to trigger the capacitor bank, by controlling the prism inclination with a cali-brated micrometer attached to the laser head; a set delay was provided so as to Q-switch the laser at the time of arrival of the shock wave in the observation

section, The triggering time of the laser pulse was recorded, using a photo-diode, in order to know the position of the shock front at the time of obser-vation.

The velocity of the shock wave was monitored by a double slit photo-multiplier arrangement (Fig. 17) at the observing section of the window. The far end of the window was used as a reference arm for the M.Z. interferometer.

3.3.2 The Fabry-Perot Laser Interferometer 3.3.2,1 The Double Pass Fabry-Perot System

Results obtained by the M.Z. interferometer (see Chapter

4),

were not sensitive enough to reveal the growth of the initial electron density and the

location of the shock translational front. Hence, a more sensitive diagnostic technique was sought. A Fabry-Perot laser interferometer is both simple and has twice the sensitivity of a Mach Zehnder interferometer and so was employed next.

The Fabry-Perot laser interferometer is essentially a development of the well known Fabry-Perot etalon. A standing wave is created between two parallel, partially reflecting plane mirrors, forming an optical cavity. The resonant condition for optical radiation along the axis is (Ref. 45),

IlL

=

n.

~

(3.1)

where

'11'

is the refractive index of the medium

'L'

is the length of the medium in the direction of the laser beam, À is the radiation wave length and n is an integer.

Physically the above equation reveals that if the optical path length of the cavity is altered by means of either moving one of the mirrors or by

introducing density changes inside the cavity, output intensity of the resonator is modulated for a maximum to a minimum everytime the optical path length changes by À/2. The maximas and minimas correspond to the bright and dark fringes in a spatially resolved interferometer. Therefore, the term fringe shift will be used here for the observed temporal intensity modulation.

6Sh~ce the introduction of the laser interferometer by Ashby and

Jephcott 4 , in 1963, several modifications have been made, extending the use-fulness of the device (Refs. 48-51). The frequency resp onse

60f the laser inter-ferometer (Ashby-Jephcott type) was not better than 0.5 x 10 resonance per second. It has been found that for resonanee modulation greater than 1 MHz the modulation of the laser reduces to

<

5% compared to

>

50% within the range of response (Ref. 52). Garardo, et al 53-studied the response time of the laser interferometer and showed that in an external cavity mode the laser interfero-meter is capable of detecting resonanees in excess of 50 per microseconds. They also argued regarding the maximum response time obtainable with the laser cavity mode and the external cavity mode. In the latter case, the response time is given by:

T =

ext

L

ext _(3.2)

Where Lext is the external cavity length T

2, T3 are the actual trans-mission coefficient of the mirrors, M2' M3 in the 3-mirror laser interferometer

system (Fig.18), L is the fractional loss per pass through the external cavity. The bracketed term in the denominator, representing loss, can be taken as 1%;

in a very low loss cavity or 90% in a high loss cavity. The time constants for each of these cavities are 3.3 x 10-7 sec and 3.7 x 10-9 sec respectively (for L

=

100 cm long cavity). For a smaller cavity, the response will be still

f~~~er

.

On the other hand, there is a very weak dependenee of laser cavity

response to external cavity losses, viz, even 100% loss in the external cavity; would change the response time by only a factor of 2.

In the present studies the ionization rate behind the heligm shock wave was eXPgcted to be such that the fringing rate would vary for 2 x 10 sec-l to 20 x 10 sec-l Therefore, in view of the above discussion~ a modified form of

laser interferometer, in the external cavity mode, was appropriate. A schematic view of the system employed in the present studies is presented in Fig. 19. If the laser output through mirror M

2 is constant, the transmitted intensity will

attain itsmaximum, when the laser output is resonantly supported in the external

cavi ty.. Unfortunately, the laser output will vary due to change in the optical

length in the external cavity unless some form of isolation is incorporated be-tween mirrors Mand M. A high loss medium bebe-tween two mirrors could give a good

degree of

isola~ion

but consequently increases the difficulty in investigating

the high luminous plasmas because of the much reduced intensity. In a directional

isolator a combination of a À/B wave plate and a polarizer can be employed which

'transmits the laser beam only in one direction (M2~M3) and greatly attenuates in

the other direction (M3~M2)' Unfortunately, the plasma to be investigated needs

an interferometric response much faster than the laser cavity could respond which would keep laser cavity almost decoupled with the external cavity during the

observatión ::. period of the transient plasma evente Therefore,

in the present configuration, a signal transmitted through the reference cavity

can easily be related to the resonant support of fields in the external cavity.

It is worthwhile to ment ion that in the three mirror laser interferometers, al-though the relative value of the transmitted intensity IT' is higher, but it completely removes the isolation between the external cavity and the laser (e.g.

Fig.1B) . Also the transmitted intensity would not be maximum when the laser

out-put is resonantly supported in the external cavity because this is also the

con-dition for the minimum laser output. Therefore, it would retain some

inter-mediate value, thus introducing difficulty in interpreting the observed signal

of the reference cavity. Although such a system was used by Bristow52 , he managed

to get a high degree of isolation by using a polarizer inside the external cavity.

In general, the laser has several spatial and temporal modes and if

they are simultaneously allowed to enter the interferometer, they will be modulated

in an uncoo~dinated manner; resulting in aweak and noisy output signal. The

laser employed (Spectra Physics Model 120) was initially checked and set for

operation in the TEM mode and hence a particular spatial mode was predetermined.

00

Readjustment of this mode was possible by fine adjustments provided with the_laser.

The separation between the various temporal mode present in a laser depends on the length L of the laser cavity i.e., mode separation is equal to C/2L, where C is the velocity of light. Therefore to modulate all the temporal modes in the external cavity simultaneously, one should have the external cavity length as an

integral multiple of the laser cavity length.

In the present studies, the optical length of the external cavity was

initially set the same as the laser cavity length (within ~ 0.25 cm) , This

re-sulted into much stronger output signal by eliminating expected noise due to several temporal modes of the laser.

3.3.2.2 Details of the Experimental Setup Alignment and Calibration

Although two different configurations were used in the Fabry-Perot type of interferometer, i.e., the double pass system and the multipass system, the procedure for the experimental setup alignment and calibration is essentially

the same. The schematic of the 'multipass' system is shown in Fig. 20.

The Fabry-Perot plate with the first surface coating (flatness À/50) were m0unted on the two massive holders provided with adJustment controls ofaxial