Localized diagnostic possibilities of selective excitation spectroscopy

•

LOCALlZED DIAGNOSTIC POSSIBILITIES OF SELECTIVE EXCITATION SFECTROSCOFY

by

R. M. Measures

SEPrEMBER 1967 UTIAS REPORT NO. 127

LOCALIZED DIAGNOSTIC POSSIBILITIES OF

SELECTlVE EXCITATION SPECTROSCOFY

by

R. M. Measures

Manuscript received June ~L967

S]1?I'EMBER

1'967

illIAS REPORT NO. 127

ACKNOWLEDGEMENTS

The author is grateful to Dr. G. N. Patterson, Director of the Institute for Aerospace Studies, for his interest and encouragement in the re-ported research. The author would also like to express his appreciation to Dr. J. H. de Leeuw for many stimulating discussions.

This work was supported in: part by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. AF-AFOSR-944-66.

SUMMARY

Localized measurements of the state and behaviour of a plasma are required in many experiments. The techniques available up to now have in almost all instances re lied on inserting a probe into the plasma to sample its condition over some small volume. Thomson scattering of an intense beam of radiation, by the free electrons of the plasma, has been the one exception of significance.

A new approach to the attainment of localized information about the condition of a plasma is proposed in~this report and a practical example is discussed in detail. In this alternative technique a given atomic transition, within one of the species comprising the plasma, is optically pumped to produce a localized region of enhanced population in the upper state of the transition. This gives rise to an intensification of the spontaneous emission arising

from this upper state. Such intensified emission contains information about the conditions within the pumped region of the plasma and can be detected against the background radiation from the rest of the plasma provided, the excitation temperature is small compared to the energy separation of the two states compris-ing the transition.

It is shown that for plasmas seeded with potassium localized

values of electron temperature and number densities should be attainable with the use of a pulsed ruby laser operating at a temperature of about

_6o

o

c.

Further-more because this technique involves the use of a pulsed laser,temporal as well as spatial resolution is possible.

TABLE OF CONTENTS

1. INTRODUCTION 1

2. GENERAL CONCEPI' OF SELECTIVE EXCITATION SPECTROSCOFY

3

2.1 Rate Equations and Simple Model

3

2.2 More Realistic Model for Intense Laser Beam Approximation

4

2.3

Intense Laser Beam Solution

8

2.4

Diagnostic Possibilities of Strong Beam Approximation

9

2.5 High Density Steady Solution 12

2.6

Weak Laser Beam Approximation

13

2.7

Comparison Between Classical Scattering Theory and the Weak

14

Beam Solution

2.8

Reabsorption Length of Scattered Radiation

16

3.

A PRACTICAL DIAGNOSTIC SCHEME FOR POTASSlUM

4.

3.1

Model for Potassium

17

3.2

Duration of the Intensified Emission Pulse

18

3.3

Amplitude of the Intensified Emission Pulse

19

3.4

The Number of Photons Created by Selective Excitation 21

3.5

Laser Power Requirement and Optical Depth for Resonanc.e and

23

Scattered Radiation

3.6

High Density Limit for a Potassium Plasma

24

SUMMARY REFERENCES APPENDIX 1 FIGURES

26

28

1. INTRODUCTION

In the past, detailed spatial resolution of the state of a plasma was in almost all instances determined by inserting a probe of one sort or

another into the plasma to sample its condition over a small volume. More re-cently the advent of the high power laser has enabled localized.measurements to be made on the state of a plasma by using the scattering properties of the free electrons. This latter technique has the advantage that no material probe is insérted into the plasma so that there is no possibility of contamination. Un-fortunately the extremely high power, necessary to get a meaningful scattered signal, can cause some perturbation of the plasma at the very point of observa-tion. The extremely small value of the cross-section, for Thomson's scattering of radiation from the free electrons, is responsible for the power requirements and also limits the applicability of this technique to fairly high density plasmas.

The new approach proposed in this report is based upon the selec-tive excitation of a specific transition by the absorption of radiation fr om an intense laser beam. The main advantage of this concept lies in the fact that the cross-section for absorption of radiation of frequency coincident with that of an emission line, within one of the species comprising the plasma, is typically about ten orders of magnitude greater than Thomson's scattering cross-section. For each photon absorbed there is a certain probability that the atom will relax back to its original state byemission of another photon of the same fre~uency

but in a different direction to that of the exciting photon. This two step pro-cess can be thought of as akin to resonant Rayleigh scattering. Alternatively the atom may decay to some other state with the emission of a photon of differ-ent frequency from the absorbed one. This two photon process is similar to Raman scattering. Consequently in the context of this report, "scattered radia-tión" refers to the radiation that is isotropically produced by the spontaneous emission of atoms that were excited by absorption of the incident radiation. If collisions occur during the lifetime of the excited state, the atom may undergo radiationless transitions to either higher or lower states. These four main depopulation processes are schematically illustrated in Fig. 1. It will be shown later that it is collisional depopulation of the laser pumped state that limits the range of applicability of this technique.

If a laser can be found that operates at a frequency that coincides with the frequency of an emission line, then it becomes possible to selectively excite a large number of the appropriate atoms into the upper state of the transition, corresponding to this spectral line, along the path of the laser beam. This will cause an intensification of the spontaneous emission, which can be interpreted as scattering of the laser beam, from the volume exposed to the laser radiation. If this enhanced emission can be identified from the back-ground radiation emanating from the rest of the plasma useful localized informa-tion about the state of the plasma can be ascertained.

In order to produce a substantial intensification of the light emission, two conditions must be satisfied; f~rstly the population in the upper level should, prior to irradiation by laser, be very small compared to the popu-lation illn the lower level, secondly the laser intensity must be sufficient to essentially redistribute the population between the two levels. The former condition will in general be satisfied if the excit.ation temperature is small compared to the energy separation of the two levels that comprise the transition of interes.t. That is to say hv» kT, where v is the laser frequency and T is

the excitation temperature, hand k have their usual values. It is worth point-ing out that if the upper level of the laser excited transition has alternative radiative decay modes a frequency different from that of the laser can be

monitored. This means that one can discriminate against the laser rad~ation

scattered by walls, background gas and small partieles (Mie scattering).

This condition relating the laser frequency and the temperature for which this technique would be applicable immediately excludes a large class of possibilities. It implies that for any realistic plasma temperature the laser frequency must be at least in the visiblepart of the spectrum. However almost all gas laser lines that are in the visible emanate from the ion so that at the temperatures complying with the requirement, kT

«

hv, the population density of the level that can be excited in the gas, to be analyzed, by exposure to the laser (of the same gas) is negligible.

To clarify this point let me give an example. One of the best candidates for this technique might be to use the intense 48800_{A argon laser} line to investigate'an argon plasma. Unfortunately since 48800A corresponds to about 2.5eV, the maximum temperature for which this technique would be appli-cable is about 7,3000_{K. However}

_at

_{this temperature the number of ions in the}

lower level of the 48800_{A transition is so small that'the scattered radiation} would be too small to be identified against the background radiation fr om the plasma. There are also further difficulties to do with the collisional life time of this state which would prevent it from being of significant use. This aspect of the problem will be disoussed later.

The only immediate~sffiDi~Ülf]ies, in finding a chance coincidence between the frequency of a strong laser line and th at of a suitable low lying

atomie transition. Abella and Cummins (Ref. 1) demonstrated that the wavelength at which a ruby will lase can be tuned by several angstroms by adjusting the temperature at which it operates. At a temperature of about _6oo

c

the ruby will lase at 6939°A which coincides with the second member of the sharp series in potassium (62S~ - 42p~/2). This is almost an ideal combination because the power that we can get fr om the ruby laser is unequalled and the 69390A line gives us a temperature limitation of about 5000oK, which is sufficiently high to ensure a large population density in the lower level of the 6939°A transition in potass;ium. Furthermore since both levels are fairly deep, electron depopulation is minimal.

The recent use of non-linear dielectrics (Ref. 2) to extend the spectrum of frequencies at which a strong, coherent, monochromatic beam of radia-tion can be attained, should substantially improve the range of constituents and conditions for which this technique of selective excitation spectroscopy could be applied. Although second, third harmonie generation and stimulated Raman cells are useful, the most exciting possibility is the tunable parametrie oscillator

2. GENERAL CONCEPr OF SELECTlVE EXCITATION SPECTROSCOFY

2.1 Rate Equations and Simple Model

The rate of change of the population density, n(s), of the IISIl

level within an atom in an optically thin plasma is given by the expressionJ

dn(s) dt n(c)[ n(q)K(q,s)+

L

n(r)X(r,s) ] q>s r<s + n(c)2[K(C,S)+R(C,st

I

n(q)A(ci

~

s)

q>s o· 'ot,:#-- n(c)n(s)[x(s,q)+

L

K(s,r) ]-n(s)[n(c)x(s,c) q>s r<s +

I

A(s,r) ] r <s where:

K(q,s) is the collisional de-excitation coefficient for transitions from level q to level s.

X(r,s) is the collisional excitation coefficient for transitions from level r to level s.

A(s,r) is the Einstein spontaneous transition probability for transitions from level s to level r.

X(s,c) is the collisional excitation coefficient for transitions from level s to the continuum.

K(c,s) is the collisional de-excitation coefficient for transitions from the continuum to level s.

R(c,s) is the radiative recombination coefficient for optical transitions from the continuum to level s.

n(c) is the free electron number density.

Photo excitation and photoionization have been neglected because of our assumption of an optically thin plasma. However, in a region exposed to laser radiation, of wavelength coincident with that of the transition from level p to level s, two additional terms must be taken into account. These are

n(p)B(p,s)! It(V)L(V)dV and n(s)B(s,p)! It(V)L(V)dV

Where the first term represents the rate of populating the s level by absorption of the laser radiation by atoms in the p level, and the second ierm represents the rate of depopulating the s level by stimulated emission. I (v) is the laser intensity and L(v) is the absorption line profile. B(p,s) and B(s,p) are the respective . I 0 transition probabilities for absorption 0 and stimulated emission.

With sufficiently intense laser radiation it is possible that the population densities of the s and p levels depart considerably from their steady values. An asterisk is used to indicate the perturbed population densities. In reality not only will the population in the s and p levels be disturbed but also adjacent levels. However, it is the purpose of this report to attempt to indicate the expected value of the enhanced radiation from the region excited by the laser beam. In order to accomplish this certain assumptions will have to be made to make the problem tractable.

To allow us some physical insight into the problem let us con-sider the limiting situation where the laser is so intense that the two terms discussed above completely dominate the rate equations for the levels pand s. Under these circumstances we may write for the perturbed population densities n*(s) and n*(p) the rate equations,

that is or dn*( s) dt dn*(p) dt [ n* ( p ) B ( P , s) n*(s)B(s,p) ]

J

I~(v)L(v)dv

[n*(s)B(S,P) - n*(p)B(p,s) ]

J

I~(v)L(v)dV

dn*(s) dt + dn*(p) dt

o

n*(s) + n*(p) nes) + nep) (2.1.2) (2.1.3) (2.1.4) (2.1.5) where nes) and nep) are the unperturbed

levels prior to the laser radiation. A

population densities in the s and p steady state is reached when

~:~~~

=

iliillgicl

~

=grsy

(2.1.6)

that is to say when stimulated emission balanees absorption. Equation (2.1.6) indicates that the laser effectively couples the s and p levels to the point where their populations are related by the ratio of their degeneracies. Now equation (2.1.6) can be combined with (2.1.5) to yield

n())

=

(~(p)

( )

{l

+

n~p~

} (2.1.7) ns gp +gs , ns

In general g(p) and ges) are comparable in value so that the perturbed population density in the s level, n*(s), is substantially increased over its value, prior to the laser initiation, providing the temperature was such that nes)

«

nep). This will mean that the intensity of the spectral lines arising from the spon-taneous radiative relaxation of the s level will be considerably enhanced in the region of the plasma exposed to the laser radiation.

2.2 More Realistic Model for Intense Laser Beam Approximation

The results discussed above would only be strictly valid for very short times af ter the laser initiation, assuming the laser beam's temporal

variation was a stepfunction, because all loss mechanisms were neglected. In reality the increase in the population density of the upper level (the s level) results in a corresponding increase of the atoms lost from the s level by

collisions and radiative transitions to levels other than the p level. However, there is no comparable increase in the rate of populating either of the two

levels.~so that invariably the total population distributed between the s and p levels will decrease. But since the population of both levels are locked together by the intense laser radiation this means that the value of n*(s) will decline fr om the value indicated by equation (2.1.7).

* In actual fact collisiortal depopulation of the s level will perturb the population density of the neighboring levels. The inclusion of this affect would make the problem extremely complex and is at the present time neglected as a rough estimate has indicated that the increase of the neighboring popula-tion densities would do no more than reduce the loss from the s level by a few per .cent.

A more realistic model which takes account of the col+isional and radiative depopulating processes will now be considered. Furthermore, we neglect most of the populating processes and so evaluate the most stringent re-quirements for attaining a substantial intensification of spontaneous emission from the s level. The only exception to this is the inclusion of a term which represents populating the p lev~l by collisional excitation of the ground state. This process is only significant if the p level is a resonance state • . A sche-matic representation of the model to be studied is illustrated in Fig. 2.

With the above assumptions we may write for the rate equation that describes the temporal behaviour of the perturbed population density, n*(s), óf the s level;

- n*(s)[ n(c)D(s) + A(s)

l

~2.2.1)

where n*(s)n(c)D(s) is a loss term arising from collisional depopulation of the s level.

W~ define

D(s)

==L

X(s,q) +

L

K(s,r) + X(s,c) (2.2.2)

q>s 1'<s

as the total collisional depopulation rate coefficient for level s, and

A(s)

==L

A(s,r) (2.2.3)

1'<s

the total spontaneous emission probability for the s level. Likewise the rate equation that describes the temporal behaviour of the perturbed population density, n*(p), of the p le~ 1 can be written as

dn:~p)

=

[n*(s)B(s,p)-n*(p)B(P,S) ]

J

I$(V)L(v)dV - n*(p) [n(c)D(p)+A(P) ] +

+ n(l)n(c)X(l,p) (2.2.4)

The important additional term, n(l)n(c)X(l,p), represents a source term for the p level and arises from collisional excitation of the ground state. n(l) being the population density of the ground state, assumed unchanged by the laser. This is justified if the temperature is such that n(p)« n(l). Implicit in the above discussion is the fact that only electron-atom collisions are significant. This is reasonably justified if the degree of ionization is more than a few tenths per cent.

In order that we may obtain a reasonably simple analytic solution we further assume that the temporal behaviour of the laser intensity may be well approximated by a stepfunction, viz.

I(v,t)

=

{I~V)

t

>

0 (2.2.5)

t

<

0

Equations (2.2.1) and (2.2.4) may be reduced into nondimensional form by

initiation) va1ues, viz. and introducing 8

==

n«())

n ·s , P

==

n«())

n p g

==.gicl

g[S)

Tl

-S

-C

==

gE

==

n~p~

_{n s} n{c}D{sL + A{s} ~

B(p,s)Jr (v)L(v)dv

n{c}D~p} + A{p)

:e

B(p,s)Jr (v)L(v)dv

n(l)n{c}X(l,p} ~

n(p)B(p,s)Jr (v)L(v)dv

then equations (2.2.1)and (2.2.4) become

d8 dT

=

gEP - (g + Tl)8 dP 8 (1 +

OP

+ C - =--dT E (2.2.6) (2.2.7) (2.2.8)

•

(2.2.9) (2.2.10) (2.2.11) (2.2.12) (2.2.13) This pair of coupled first order 1inear differentia1 equations in Pand 8 can easi1y be rearranged into two uncoup1ed second order 1inear

differe ntiàL!_ equations

+ G dS + HS - gEC dT

o

d 2 P G dP + HP _ ( ) dT2 + dT g + Tl C

=

0 where G

=

(1 +

g

+

Tl

+

S)

and H

=

(gS

+

Tl

+

TlS)

(2.2.14) (2.2.15) (2.2.16) The solutions of these equations are derived in Appendix 1, to avoid comp1icating this ,sec.tîi..on, and can be put in the form

where w~ and W2 are the roots of the equation, w2 + Gw + H 0 and are given in Appendix 1.

Also, U

=

gEC Hand N

=

(g + ~)

The power radiated per unit volume per unit solid angle per unit frequency. interval in the line corresponding to spontaneous transitions from level s to level p is

j(v)

=

~(s)

A(s,p)L(v) (2.2.19)

• prior to the laser initiation. If j*(v) is the corresponding value of the power radiated from the region exposed to the laser radiation then

~7f))

=

S (2.2.20)

assuming that the line profile, L*(v), is the same as it was prior to the laser initiation, L(v).

It is worth'digressing slightly at this point to consider under what conditions the line profile of the~.l!ntensified emission is the same as it was prior to the laser initiation. For a laser the spectral intensity distribu-tion can be reasonably well approximated by

I.e ( v) = \ ' Q, 5 ( v - v) ( 2 • 2 • 21 )

L

m m

where the Q,m is the total intensity in the mth mode of the laser at frequency vm' Under these circumstances

(2.2.22) It is sufficient for our purpose to restrict our attention to studying the effect of one of these modes. It is convenient to consider the situation where the frequency of the laser coincides with the centre of the absorption profile. If the density of the atoms is low enough that the energy levels have only their natural width then according to Heitler (Ref.

4)

the scattered profile will be a "dispersion profile" with a half intensity width equal to at most the sum of the widths of the energy levels.

At higher densities the energy levels are likely to be broadened by the influence of neighborilug),L.l:; atoms. In this case the scattered radiation has the same collision broadened profile as the self emission line. If Doppler broadening dominates then the light scattered by the moving atoms is not

mono-chromatic but reflects the velocity distribution of the atoms. This can be seen by the following argument. ~he frequency as seen by an atom moving at velocity v relative to the incident monochromatic laser beam is shifted by an amount 6v, where

6v v

=

v c

If this shifted frequency falls within the absorption profile of the atom then the atom will absorb a photon. However, there is no correlation between the atoms motion parallel to the laser radiation and its motion along the line of

observation. Thus if the atom decays by emitting a photon in the direction of observation this, so called, scattered photon will have a frequency shifted by an amount depending upon the atom motion in this direction. Consequently the

scattered radiation will have the same doppler profile as the self emission line.

2.3 . Intense Laser Beam Solution

It is the strong intensity limit that is of interest to us and the solution under these circumstances can be obtained by assuming th at ,

B(p,s) !re(v)L(V)dV

»

n(c)D(s) + A(s) B(p,s)f re(V)L(V)dV

»

n(c)D(p) + A(p) and (2.3.1) viz., 1'] SK 1

t

«

1

Then the roots w).. and w2 can be simplified by expansion (see Appendix 1) to

In which case wl. (1 + g) (g~ + T)) 1 + g .(A.1.15) -

r(~)~] {(~\..}

r(gE-g)

~

.

]

{

l.,

~

S - L\g+l -

g~+1']

exp -\l+g)' i- l\g+l - (g+1)2 exp

-(l+g)T~ g~+1']

. (2.3.2) and EP

=r(~)

_

.

~]

ex

{_(~L}

'

:

+r(~\ ~

]exJ-(l+g)T}+

g~!C

g L\g+l

g~+1']

g p \g+l)' L\ g+i;,(g+1)2

~l

g 1'] (2.3.3) The temporal behaviour of S and P is schematically illustrated in figure 3. It is clear that for a step-like pulse of intense laser radiation the enhanced power radiated j*(v) due to selective excitation reaches a value of (gE+l)/(g+l)j(v) in a time,

1 + g

(2.3.4) then decays as a result of collisional and radiative depopulation of the coupled s and p levels in a time,

[g~S~

+'(Pt

J

t

However, the same peak value of S, and therefore j*(v) results on using this value of time in equation

(2.3.2)

provided tr

«

tf (which is equivalent to condition

(2.3.1)),

viz.,

j*(.v)

màx

[g::~J

j(v)

That is to say the peak value of the enhanced signal is only dependent upon the ratio of the population densities in the p and s levels prior to the laser initiation, i.e., j*{v) màx j(v)

~~~~

+ 1

ffiit

+ 1

It is worth mentioning that up to this point we have made no assumptions about local thermodynamic equilibrium (LTE). Equation

(2.3.7)

indicates that in gener al to get a large value for jfu~~)Jj(v) the ratio n(p)/n(s)

sho~ld be large. This occurs only if the excitation temperature is small com-pared to the energy separation of the s and p levels. That is to say, hv

»

kT, is the requirement for considerable intensification of the spontaneous emission from the slevel.

If prior to exposure to the laser radiation the plasma was in

LTE then n(p) and n(s) were related by a Boltzmann's distribution, viz.,

and so j*{v) màx j(v) exp

G~;)

+ 1

(2.3.8)

a function only of the excitation temperature. Under such conditions we can ascribe an upper limit to the temperature in ter~s of the wave-length, of the

laser radiation, for a minimum value of j*~)/j(v). If we require j*~)/j(v) >

250

:then from

(2.3.9)

m / m

-(2.3.10)

i.e. À T

<

2.4 Diagnostic Possibilities of Strong Beam Approximation

In practice the ratio of the intensified spontaneous emission

to the background radiation will be diluted by a factor depending upon the

ratio of the volume exposed to the laser beam,V*, to the remainder of the plasma volume within the field of view, V. Let the total intensity observed in the

line of interest, just prior.to the laser initiation, be - • • ., ,.I

I =

JJ

j

('

v )

d v dV (2.4.1) or we can write, I =- jV where

j

= ~

JJj(V

)

dVdV

the mean emission coefficient

in the field of view.

The peak value of the total intensity observed during the laser' irradiation

1* =

J

J

j*(v)dvdV*

+

.

JJ

j(v)dvdV

i.e. _1*

₌

_{j*V* +}

I (2.4.2)

where we assu~e that the region localized by the laser beam and the line of

observation,

V*

,

is sufficiently small that it can be taken out of the integràl. That is we assume j*(v) is constant over V* and define

j*

=

J

j*(v)dv (2.4.3)

The second term should in fact have a small contribution

V*J

j(v)dv subtracted

but this is negligible if j*(v) » j(v). Combining equations (2.4.1) and (2.4.2 we can w rite; or we can define an observable f3

=

j*V*

'JV

IJ. =: 1* - I (2.4.4) (2.4.5)

that can be readily monitoredo Moreover, the ratio of this signal from two volume

elements along the laser beam gives effectively the ratio of their intensified

emission coefficients, viz.,

.*

J 1

"*

J 2 (2.4.6)

From equation (2.3.7) we can relaté the ratio of the peak signals to the ratio

of the population densities in the p level prior to exposure to the laser

and since n(p)/n(s»>l,

I

If tTE applies

"*

J 1 max

"*

J 2 max nep)

=

nz~(p)

exp

(-~;)

where n is the total number density of the species of interest

excitation partition function, a weak function of temperature.

(2.4.7)

(2.4.8)

and ZeT) is the

Thus

J.llmax ....J ' {hV

exp

-J.l2max k

(~2

-

~)}

which is an extremely sensitive function of temperature and is independent of density. This means that an accurate excitation temperature profile could be established, along the laser beam and by swee~ing the beam through the plasma, a detailed picture of the temperature distribution could be eva~uated.

If the population density of the s and p states are not related

-by simple Boltzmann law, then only a relative temperature profile can be ascer-tained from equation (2.4.7). A more detailed look at this situation will, be taken later when potassium is considered as apossibility. Once an approximate value for the temperature has been established an estimate for the loc al value of the electron number density can be obtained from the time

t _f

=

for the pulse of intensified spontaneous emhsion to fall to l/e of its peak value. This assumed that the temporal behaviour of the laser could be repre-sented by a stepfunction. However, except for a possible correction factor, it is probably of the correct form for any fast rising la,ser pulse that has a duration sufficient for condition' (2.3.1) to be satisfied for a time longer than this decay time.

The values of the total collisional depopulation rate coefficients, D(.s) and D(p), are not well known as they involve a knowledge of the excitation

cross-sections for excited states. However, a calibration experiment could be performed, under known conditions of electron number density and temperature, that would enable this technique to be used to evaluate the local values of electron number density in transient, non-uniform plasmas. Polarization effects (Ref. , ) should be carefully taken into account when absolute values of the excitation temperature are r~quired.

In the above argument it has been indicated that from the intensity and duration of the pulse of intensified spontaneous emission localized values of the electron number density and temperature may be determined. It mayalso be possible to obtain localized estimates of the macroscopie velocity, the mag-netic field and the translat~onal temperature by a study of the spectral compo-sition of the intensified emission. Unfortunately the low value of light and

the high resolution may render a direct spectral analysis impractical. Under these circumstances it may be possible to ascertain localized values for, the

macro-scopie velocity and the magnetic field under certain restricted conditions by alternative means.

If the frequency of a single mode laser can be varied then it should be possible to evaluate the local velocity in the following manner. The laser frequency is set to the centre of the absorption profile, in which case there is a maximum in the intensified signal (sometimes referred to as the scattered signal) in a motionless plasma or if the motion is at right angles to the laser beam, If the mot ion of the plasma is in the x-direction and the laser beam is' in the y-direction then there should be a large intensified signa1. The frequency of the laser is then shifted by a suitable known amount ~v in which case the intensified signal falls appreciabl, . The beam is then rotated, in the

x;y plant~, tJ.!.:.éç!U~h ·J.n angle ex UIltil a tna.ximum in the intensified signal is again

a.chieved. lJn,:lt;!r these G()udi ti·.::mH,

u ==

llV

.

....

a

..---

~

c 2 6vA

\vhere b:),'A i s the half lntensi ty 1vidth af the absorption line.

(2.4.10)

'l'he rne.gnetic field ca.n a180 be estimated by tuning the laser

fre-quenc.y gO tb.at i t oyerla.ps fir st one Zeeman component then another. This

re-quire:s thn.t thf~ scpara.tioll ui' the Zeema.n components is at least twice the line widt.l1 an/1 ;'::0 giV'es a lOI'er limit to the ma.gnitude of the fields that can be

de-termined by this methode SG.lv~m8.tic illustrations of these techniques are shown

:Î.!:1 F'lg.

he

2 0"',)

:-'rh!2 int..ensifif'cl emi 5sian from the small volume selectively exci ted

by th~ la.sel' bt:a.m (;.'l.n be put in the form

" .... 11~' ( S )

wher<~ .) = -:.:T -)~

n,s

j*

=

~

n(s) S A(s,p)

1.13 gi' .... ~ll by equation (2.3.2)

F:("o'TI tlw temp:.).":'",.J. beha.vi ou.'t· of S i t is clear that, al though the peak val ue of j*

is indq)~nd.ent ,)f th~ e.Lc(;t;l'Ql1 à.erwity (praviding condition (2.3.1) is valid) , . tl..le dlJ.r.(;I.ti.0XJ cf the t r'un15ü;ut pulse aSRacÎfl.ted with the switching on of the

ÜJ.·· .. (!).' is a strong funetion o:f n(c). In fa.ct it appears fram later considerations

thaI. i1' L'l'}'!; 1R to he est,::i.bli"lwd then. the duration of this transient pulse becomes

uegUc;ible a.nd we rwve 't,CJ Wi)rk. yJith the steady soluti on. That is to say in high

(}::m:.;ity j)J.~'.f~mD.:3,

or

wlüch C:J.II. bc p~lt into the fnnn

(~:

)

n(s) A(s,p)

1\[0101 i'.1 UÜf; hi:.0h d~:m;;ity l imit n(c)D(p»> A(p) and n(c)D(s) »A(s) so that we

It is worth noting that as with the low density high intensity solution the intensified emission is independent of the laser power, providing it is sufficient to satisfy condition

(2.3.1).

However, in this case éiènlJif LTB applies the collisional depopulation rate coefficients enter into the determina-tion of j*. Nevertheless the observadetermina-tion expressed by

(2.4.6)

would still enable the relative temperature distribution within a plasma to be evaluated.

A basic premise of this work has been that n(l) can be regarded as a constant and unperturbed by the laser radiation. This is justified if n(l)

»

n(p). This implies that the temperature of the plasma is small compared to the energy separation of the p level and the ground state. A condition

similar to that required for large signal. Thus ,~;if' . ·tt,', is: reasonable to assume that n( 1) can be regarded as constant over the volume of the plasma, J 1 . I .• :i. it

i:hen the intensified spontaneous emission~is again only a function of temperature and so with calibration will provide localized temperature measurements.

2.6

Weak Laser Beam Approximation

The terms weak and strong as applied to the laser beam intensity, in the context of this report, refer to whether the laser radiation is .capable of redistributing the population density between the lower and upper levels of the transition selectively excited. If the intensity is insufficient to cause the population density of the lower level to diverge appreciable from its steady (or equilibrium) value then we must consider a new model for the interaction. This model will be a closer representation of the state of affairs when conditions

(2.3.1)

can no longer be assumed to be valid.

Under these cirumstances we may write the perturbed population density of the upper level, n*(s), as a sum of the equilibrium value n(s) arid a small additional population density N(s) arising from selective excitation of the s-p transition, viz.

n*(s)

=

n(s) + N(s)

(2.6.1)

and as stated

n*(p) ::::: n(p)

In this case the rate equation for the s level when exposed to the laser radiation dn*( s) dt n(c)1) n(k)E(k,s) - n*(s)' D(s,k) ]

+

I

n(q)A(q,s)

~s

Çfs

q>s + n(p)B(p,s)! I$(V)L(V)dV n*(s)

I

A(s,r) 1'<s

(2.6.2)

can be rewritten in the form,

Since

where

dn( s) dt

~ts) ~

n(p)B(p,S)!I$(V)L(V)dV - N(s) [n(e)D(s) + A(s)]

(2.6.3)

o

= n(c{)' n(k)E(k,s)-n(s) ) ' D(s,k) ] +

I

n(q)A(q,s)-n(s) ) ' A(s,r)

Çfs

k1s q>s

kso

J.

) ' n(k)E(k;,s) ==

L

n(q)K(q,s)

+L

n(r)X(r,s) + n(c) [K(C,S) + R(c,s) ]

.fis q>s r<s

D(s)=

~

D(s,k)='

L

X(s,q) +

L

K(s,r) + X{s,c)

and A(s) =

L

A~s:r)

q>s r<s

r<s Equation

(2.6.3)

can be put in the non-dimensional form

dY = W _ Y dT

(2.6.6)

(2.6.7)

by normalizing the additional population density, in the s level, with respect to the steady value in the p level, viz.

and introducing,

w

=

T

= Y =

!hl

llTPJ

[n(c)D(s) + A(s)] t

The general solution of equation

(2.6.7)

is

T

Y(T)

=

e-T

J

WeT) eTdT

o

subject to the initial condit~on Y(o)

=

0

(2.6.8)

(2.6.10)

If we againassume that the laser' s radiation has a step-like temporal behaviour i.e.

t

>

0

t

<

0

(2.6.11)

then equation

(2.6.10)

becomes

(2.6.12)

which gives the additional population density in the s level, N(s), ai aresult

of selective excitation

N(s)

=

n(p)B(p,s)[ Ii(Y)L(V)dV {l-exp[

-(n(c)~(s)+A(s)}t

lL

n(c)D(s) + A(s)

JJ

(2.6.13)

The intensified spontaneous emission between the s and p levels,

j*,

is given by

j*

~ ~

n*(s)A(s,p)

(2.6.14)

where we can put

and so we write the intensity of the scattered radiation (remember that the

scattered radiation was defined as being the proportion of the spontaneous emission that emanates from atoms that have been excited by absorption of the incident ,'- . beam of radiation)

J

=

~

N(s)A(s,p) (2.6.15) or from equation (2.6.13) _{"', ·f_,..}

J

=

hvnfp)AtS~PtB~P,S)Jlt(V)L(V)dV

{l _

~[n c D s +A(s)]

e~p[-(n(c)D(.s)::,=

A(S)}Jt_(4.6.16)

l

This is sChematically illustrated in Fig. 5 • . It is clear that for times,

t> [n(c)D(s) + A(s)]-l (2.6.17) The intensity of scattered radiation is given by

J (2.6.18)

2.7 Comparison Between Classical Scattering Theory and The Weak Beam Solution In order that a fair comparison be made between the weak beam

solution and the classical theory of scattering certain conditions must be applied. These are; the p and s levels correspond to the ground and first excited states respectively, collisional effects are completely negligible and the atoms are at

rest. Wi.th these assumptions the intensity of scattered radiation becomes • --'.!',

hV

t

J'

J

=

4;

n(1)B(1,2)JI (V!~(V)dV

and we see that it is proportional to the number density in the ground state. This concept has been used by Hoffman (Ref.

6)

to measure the localized ion density of Barium in a Q-machine plasma •. He monitored the light

scattered from an interrupted beam of radiation having a frequency equal to the first resonance line of the Barium ion.

The Milne absorption coefficient can be related to the absorption oscillator strength f(1,2) by the expression,

hv B(1,2)

=

4~ c ro f(1,2)

and under the conditions staUdabove L(V) is the natural line profile given by

L(V) 1

~

l~2

where Vo is the centre line frequency, and 1

=

~

2TITOV0

2

is the intensity halfwidth (classically termed the radiative decat

~onst~nt).

If we assume that on~y one mode of the laser falls within the absorption profile then from equation (2.2.22)

and we can write

J =

This should be compared with the classical resonance scattering relation, Jackson . (Ref. 1.)

(2.7.6)

which applies in the vicinity of the resonance frequency vOo In equation (2.7.5)

n(1)f(1,2) can be thought of as the equivalent number of classical oscillators. 2.8 Reabsorption Lengthof Scattered Radiation

In all of the above discussions the scattered radiation was assumed to experience no absorption within the plasma. Thät is to say if the observations are to be truly localized there should be no multiple scattering. The validity of this assumption will now be considered. The one dimensional radiative transfer equation for the spectral line of interest in the unperturbed region can be written in the form,

~ hv hv

dz

=

4; n(s) A(s,p)L(v) -

4;

[n(p)B(p,s)-n(s)B(s,p)] I(v)L(v)

The corresponding macroscopie equation is

where

and

dI(v)

=

j(v) _ K(v)I(v) dz

j(v) is the volume emission coefficient and

K(v) is the volume absorption coefficient

j(v)

=

~;

n(s) A(s,p)L(v) K(v)

=

~

[n(p)B(p,s) - n(s) B(s,p)] L(v) The solution of (2.8.2) is (2.8.1) (2.8.2) (2.8.3) (2.8.4)

I(v,t) = I(V, 0) exp(- T(V,t)j + eXP(-T(V,nu S(v,z)exp(T(v,Z)) dT(V,Z)

(2.8.5) where we introduce; the source function S(v,z) and the ·9ptical depth T(V,Z).

These are defined in the following way

S(v,z)=

ji~~ll_

~ (2.8.6)

and

dT

(v,z)

= K(v,z)dz (2.8.7)

In this problem we are interested in knowing the absorption length for the initial photons that were generated by the selective excitation of a specific

transition. In which case from the first two terms of (2.8.5) i.e. I(v,t)

=

I(v,O) exp

(-T(V,t))

we can see that absoprtion can be neglected if

T(V,t)

:s

1 (2.8.8

Le. if

P-i

K( v, z) dz

:s

1

Now _{K(V,z) =}

~

_{n(p)B(p,s) [1 -}

~~;~~1;:~~J

_L(v,z) (2.8.4)

For low temperatures where n(s)/n(p)

«

1, certainly true in our ~ituation, we may write

hv (

K(v,z) ~

4i

n(p)B(p,s)L v,z)

and if we introduce n(p) for the mean number density in the p state along the line of observation and likewise L(vV the mean profile function, then there is no multiple scattering over a path t providing;

t

$ 4~

hv n(p) B(p,s)L(v) (2.8.10)

This g~v.es~the condition for ~medium to be optically thin for the transition considered.

3. A PRACTICAL DIAGNOSTIC SCHEME FOR POTASSIUM 3.1 Model for Potassium

The wavelength at which a ruby will lase can be varied over a range of 120A by adjusting the temperature at which it operates according to Abella and Cummins (Ref. 1). At a temperature of about -600

_c

_{the wavelength}

is 6939°A and coincides with the second member of the Sharp series in Potassium so that the ruby could be made to selectively excite the (42_P

3

/

2 - 6~lb)

transition. This, as it turns out, is a good combination as 1t reason~Ely satis-fies all of the requirements set forth in the second section 'of this report. This can be seen by inspection of the energy level diagram for Potassium, illustratedc,iliLJFig. 6.

For convenience let us review the main requirements. These are: (i) the frequency of the transition should coincide with that of some

strong laser line,

(ii) the energy separation of the two levels, comprising the transition, should be much larger than the temperature of interest.

(iv) the upper state should be as low as possible so that its collisional depopulation rate is minimal, (D(s) dominates the other processes in imposing the density limitation).

The low cost and low ionization potentialof potassium makes it a constituent of many plasmas that are of interest in the fields of magnetogas-dynamic power generation and alkali plasma propulsion engines. Consequently it is worth considering this example of the proposed concept in considerable detail.

A realistic model for potassium is sChematically shown in Fig.

7,

where the 4s, 4p, 5p, 6s, 6p states are represented simply by the numbers 1, 2, 3, 4, 5 respectively. There is no clear distinction made in this diagram be-tween the two 42 p levels. However, in the computation we are in fact referring to the upper level (the

42p3/2

term), except where carefully stated to the con-trary. It is seen from Fig.

7

that of all the levels neighboring on the upper

(6s) level of our(62S1/2 - 42p3/2) transition, only the 5p and 6p levels are con-sidered in determining the depopulation rate. This is justified by estimating the relative excitation rates using the collisional excitation cross-section given by Seaton's relation (Ref. 8) [which we assume may.be very approximately applied to these highly excited statesJand the relevant oscillator strengths as given by Anderson and Zilitis (Ref.

9).

It is appreciated that these estimates are likely to be very rough but at the present time there are no experimental values. A more accurate estimate of the cross-sections could possibly be attained using Gryzinski's semi-classical method (Ref.lD).

The rapid increase in the population density of level 4 (the 6s level) is partly transmitted to its neighbors as a result of collisions. Thus the loss term associated with the collisional excitation to level 5(the :6p level) increases the population density of level 5. Now part of this is communicated back to level 4 by a corresponding increase in the collisional de-excitation rate of level 5. However, since level 5 is even more strongly coupled (collision wise) to higher levels the propórtion of the population density directed back to level 4 is no more than about 10% .. In fact, we can say quite generally that the perturbation in the population densities of the levels neighboring level 4 reduces the net depopulation rate by no more than about 10% and effecti vely nelps compen-sate, for the fact that the use of Seaton's cross-sections might have been a slight underestimate.

3.2 Duration of the Intensified Emission Pulse

In the second section of this report it was shown that the inten-sified spontaneous emission will decay to about l/e of its peak value in a time

g(s) + g(p)

n(c)[g(p)D(p) + g(s)D(s)J+ g(p)A(p)+g(s)A(s) For the situation outlined above this can be re-written in the form

(3.2.1) where for the highly excited states we use the approximate relation for the excitation rate given. by Allen (Ref. HJ).

This is based on Seaton's cross-sections and can be put in the form

X ( 4 , 5 ) = 7 10 -3 [ . f ( 4

4

5 ) ]

1. x . Ti E ( , 5 ) exp

{

_11600E(4,5)}nf~1600E(4,5J

T

l

T

where, . f( 4',5) is the absorption oscillator strength for the transit ion between levels 4 and 5.

E(4,5) is the energy separation between levels 4 and 5

P {1l600 E ( 4,5 )} _{T - - - lS a quan urn mec anlca correc lon ac or,} ° t h O l t O f t t b _{a u a e} 1 t d b _y Allen (Ref. 1(1) T is in oK, E(4,5) is in eV Likewise; -3 [

f~4,3~

'

1.

7

x 10 . ~jE

3,

4

]

K(4,3)

It should be noted that for the higher levels, 3 and 5, the statistical weights g(3) and g(5) refer to the multiplet degeneracy, whereas g(2) refers only to the degeneracy of the 42p3/2 term. The numerical values of the radiative transition probabilities A(2) and A(4) have been calculated by Heavens (Ref. ~) using the Bates and Wamgaard method.

The value of tf has been calculated ,and plotted. in Fig.

8

as a function of electron number density 'for tho'se elec,tron d>emperatures of interest. It is clearly seen that as the electron' number density increases the decay time, tf' of the intensified emission pulse decreases. At an electron number density greater than 1013 cm- 3 the decay time would be less than 10 nanosecs and would be difficult to resolve. For very low densities it is seen that a limit on the decay time is set by the radiative lifetime of the levels involved. The.values

of tf in general and it's radiative limit, of 160 nanosecs, is independent of

the laser intensity providing the criteriEm given by expression (2.3.1) is satisfied.

3.3 Amplitude of the Intensified Emission Pulse

The upper limit on the electron density imposed by the short decay time of the pulse of intensified emission serves to indicate that in evaluating the amplitude of the intensified emission LTE cannot be assumed • . The exception to this is the thermal plasma, such as producedin an oven, where the electron temperature is the same as the atom' ; temperature. Consequently,in this case the excitation temperature is the same as the atom's translational temperature. Under these conditions a calibration experiment toevaluate the depopulation rate coèfficients is possible.

The ratio of the peak intensifiedvolume emission coefficient to the value prior to the laser excitation is given by

0*

J_max j n p n s + 1 = + + -P g s + 1

This is seen by integrating equation (2.3.7) over the frequency range that constitutes the line. For the (42p3/2 - 62S_{1/2) transition in}

potassium we may write

0*

J_max j

~

nfl

+1 2 + 1 g

For electron densities of below 1013cm-3 the ratio n(p)/n(s), as required by equation (2.3.7), is given by a coronal equation,

n(4) _ n(c)X

t

2,4) ,

IiT2J -

A()

,

(3.3.2)

This neglects the fact that level 4 mayalso be populated by collisions involving atoms excited to level 3. However, this is true only if,

n(2)X(2,4»> n(3)X(3,4)

but the small value of X(1,3) makes n(3) so small that this is reasonable. The condition of obtaining a considerable intensification of the spontaneous emission was that the energy separation between levels 2 and 4 should be large compared to the temperatures of interest, viz E(2,4) »kT. This require-ment excludes the use of Allen's relation for the excitation rate coefficient, but enables us to approximate the excitation collision cross-section by a linear

expression of the form,

rr(2,4)

=

c(2,4) (E(c) - E(2,4)}

where E(c) is the free electron energy prior to impact. c(2,4) can be estimated from the experimental work of Volkova (Ref. 13). In which case the exci tation rate coefficient x(2,4) is found by averaging the cross-section given by

equation (3.3.3) over the Maxwellian velocity distribution corresponding to the relevant electron temperature. The appropriate expression for x(2,4) is found to be,

[1 + E(2,4)] exp {_ E(2,4)} _{2kT kT ( 3 . 3 .4)} or using the values for c(2,4) and E(2,4) this becomes,

x(2,4) = 1.69 x lÖ12 93/2[1 +

10~5]

exp

{_20~9}

where 9

=

T/103.

Equation (3.3.1) may be approximated by

.*

J max '"

.!

Il(21 _

A(

4)

j - 3

nr+Y -

3~(c)X(2,4)

(3.3.6)

which on inserting equation (3.3.5) and the calculated value of A(4), see Heavens (Ref. 12), becomes

.*

J_max

j

2.16 x 1021 exp (

~)

n(c)

9

3/

2[1+

1~.5]

The variation of jfuax/j with temperat~e is shown in Fig. 9, for electron number densities of 1011 , 1012 and 1015 cm-jo

It is clear that the ratio j*max/j is an extremely sensitive function of the electron temperature in this coronal situation and that the intensification is at least eight orders of magnitude. This means that the intensified signal should be easily identified against the background radiation.

3.4 The Number of Photons Created by Selective Excitation

Although, as indicated above, the ratio of j*max/j is favourab1e for a wide range of conditions we should a1so determine the number of photons

emitted in the pulse of.i.:Lntensified emission to establish whether there are

sufficient to be measured against other forms of noise, e.g., photomu1tiplier noise.

We have introduced, j*

=J

j*(v)dv , and we now write ID* as the

number of photons emitted in the pulse of intensified emissionper unit volume

and per unitsolid angle. In fact we define N.* as the integrated intens i ty, j*,

over a time 2tf' viz, .

2tf N*=

Ll

j*dt

=

hv 0 2tf n(s)A(s,p)

r

S(t)dt

47r

-b

(3.4.1)

This is of course an underestimate of the actua1 number of photons created by

selective excitation since at t

=

2tf, S has on1y fallen to 1/e2 of its peak

value. If we assume that

gE + 1» ~

g+l g(:;+T) (3.4.2)

then since (:;« 1 and T]« 1,

~» gEC

g+l g+l (3.4.3)

and so if we use the definitions of tr and tf as given by equations (2.3.4) and

(2.3.5) we can write for times t ~ 2 tf,

S(t)

=

{:!~l}

exp (-

~~)

-

{:!î

g} exp (-

~r)

(3.4.4)

Inserting this into the integral of equation (3.4.1) yields

assuming

Now the basic requirement for very strong laser coupling of the two levels given by condition (2.3.1) can be restated as

In which case we can write,

N* - (0.87) n(p)A(s,P)tf

47r(g

+1) which can be put in the form

N* ~ (0.87) j*

max

(3.4.6)

where J0* _{max :: 41T(g+1)} n(p)A(s,p)hV from equatlon 2.5.1. ° ( )

Let N be the equivalent number of photons that would have been emitted by the same element of plasma in the same time interval if there had been no excitation by the laser, then; .

assuming N*

~

(0.87) N - \: 2 2t_f

N

=

~r

° dt

= 2j t f

hV~

J

-nv-jfuax j

that is to say the plasma emission (with no laser excitation) can be regarded as constant over the time interval of 2tf. Equation (3.4.7) gives the number of photons emitted, in the 6939°A line of potassium, as

N* :::::. (3.4.10)

The population density in level 2 can be related to the ground state population by the expres sion

~=

nnJ

n(c) X(l 2)

n(c)K(2,1~+A(2,1)

(3.4.11)

Where because of the large energy separation between levels 2 and 1 the collisional excitation coefficient X(1,2) is giv.en by an expression similar to that for x(2,4j viz. X(1,2) 2C(1,2)(2kT)3/ 2 [1 E(1,2)] { E(1,2)} I + exp -(1Tffi e) 2 2kT kt (3.4.12)

As before the collision cross-section is assumed to be a linear function of the electron energy so we may write;

CT (1, 2) = C ( 1 , 2 ) {E ( c ) - E ( 1 , 2) } (3.4.13) using the experimental value for C(1,2) as given in.the work of Zapesochnyi and Simon (Ref. l!t) and writing T = 103

e

the collisional excitation rate is

X(1,2)

=

1.2 x 10-

8

e

3/ 2 [1 +

9~35]

exp (_

1~.7)

Statistical mechanics leads to the relation

( g{1l ( ) {E(1,2)}

K 2,1)

=

gJ2)

X 1,2 exp kT

(3.4.14)

(3.4.15 ) and if the possible values of this collisional de-excitation coefficient are com-pared with the value for A(2,1) it is reasonably clear that a coronal._ type of expression relates n(2) to n(l), i.e.

Thus

N* :: (0.87)n(c)n(1)A(4,2)X(1,2)tf

4~ A(2,1)[g~

+ lJ

(3.4.17)

An estimate for the possible values of N* are plotted in Fig. 10 as a function of the temperature,

e,

for the electron number densities, 1011 , l012 and 1013cm-3. The ground state population density n(l) was related to the free electron density n(c) by modifying the results of Bates et al (Ref. 15.) to account for the difference in the ionizationpotential between their pseudo-alkali and potassium.

Within the uncertainties of the approximations made in the above discussion it is clear from Fig. 10 that even for volume elements of about 0.01 cm- 3 the number of photons emitted, during the pulse of intensified emission, should be adequate to be measured. It is worth pointing out that from the pre-dicted extreme intensification of the emission, see Fig. 9, a very large solid angle can be accepted by the detector as the background radiation should be very small. This will help ingiving a good signal to noise ratio where the noise in this context is photomultiplier noise. Furthermore, because the radiation moni-tored is effectively that scattered from a small volume no loss of resollition re-sults from having the detector accept a large solid angle. It is again worth mentioning that by monitoring an alternative radiative decay mode, such as the

69110_{A line, much of the problem associated with picking up laser light scattered}

from walls etc. is avoided. j!ax/j will be no different for this line but N* will be reduced in the ratio of the radiative transition probabilities (i.e., a factor of about two).

3.5 Laser Power Bequirement and Optical Depth for Resonance and Scattered Radiation As indicated in Section 2.3 the condition for the laser radiation to closely couple the population densities in the pand s levels of an atom, by selective excitation,is

B(p,s)J

I~(V)L(V)dV

»

n(c)D(s) + A(s) }

and B(p,s)J

I~(V)L(V)dV

»

n(c)D(p) + A(p)

U~.~.l)

Both of these inequalities are satisfied if

B(p,s)J

I

~(V)L(V)dV

»

n(c)[g(p)D(p)+g(s)D(s)] +g(p)A(p)+g(s)A(s) _{ges) + g(p)} (3 5 _{. •} 1)

that is to say

If we have a single mode laser with a frequency centre of the absorption line, v_o' then

JI~(V)L(V)dV

corresponding to that at the

=

QL( v )

o

where Q is the power density radiated into the relevant mode. In this case we find the condition on Q to be

Q»

B(p,s)L(v )tf o

1

Now for the 6939°A line in potassium, B(2,4)

~

1.74 x 1016 cm3 jOule-l and

L( v o)~ 0

.9i;!..j

_{6v D as the line is likely to be Doppler broadened for the low densi ty} regime. ,6.vD~,being the half intensity width of the line.

This then gives,

Q

»

102 watts/cm2

under the most unfavourable conditions (electron number density

=

1013cm-3 and T

=

50000K). This is a power level easily achieved with a ruby laser.

From equation (2.8.10) the plasmawill be optically thin~.o _tlie. ILine radiation between levels 2 and 4 providing

41T

$

~

_{h vn(2)B(2,4)L(v o)}

For the 6939°A line the plasma may be regarded as being optically thin ($ ~ 5 cm) for n(2) ~ 1013 cm-3. This means that except for the dense low temperature plasma multiple scattering can be neglected. However, when this relation is applied to the resonance line, 7664°A, linking levels 2 and 1 self absorption is likely to be significant. This was taken into account to some extent in deter-mining tf as A(2.1) was reduced by about an order of magnitude.

3.6 High Density Limit for a Potassium Plasma

The steady value of the intensified emission coefficient in the high density limit is described by equation (2.5.5) which in the case of

potassium yields

thus

~=

j n(1)~(4)X(1 2~

where the normal emission coefficient

and for potassium, .I. ,

j:;:-

~

n(4)A(4,2) , . .1. , n(2) n(4) K(2,1) x(4,5) + K(4,3) (3.6.1) (3.6.2) (3.6.4) Moreover if LTE prevails as is likely in the high density regime, then n(l) and n(4) are related by a Boltzmann distribution, i.e.

~

=

KUl

J

E(1,4) }

and so we can write for the ratio of intensified emission to the normal emission, (3.6.6)

This can be understood by the following argument. When a steady state has,> been., attained, a balance exists between the rate of collisional de-population of the coupled s and p states and the rate of collisional de-population of the lower state. That is to say for potassium,

n*(s)K(2,1) + n*(4) (x(4,5)+K(4,3)} == n(1)X(1,2)

But if the laser intensity is sufficient~to strongly couple the populations of levels 2 and 4 then

Thus we can write

or n*(4) .sË:. ==

n* (

4 )

j n( 4)

n*t~~

==

~==

g i l l g 2

n*

'B[2,'4)

gr+)

(3.6.8) g(1)X(1,2)exp{E(1,4)jkT} (3.6.6) [ g(2)K(2,1)+g(4){x(4,5)+K(4,3)} ]

Using the same approximate excitation and de-excitation coefficients, as was used in the previous work, the value of j*/j is calculated and plotted as a function of the electron temperature in Fig. 11. These calculations lead to two important conclusions; firstly the ratio j*/j, in this high intensity LTE limit, is independent of the density, but is astrong function of temperature, secondly because of the relatively small values of j*/j the ratio of the scattered radiation to the background radiation from the plasma will be small.

This implies that although monitoring the strength of the intensi-fied emission (i.e. the scattered radiation) is a sensitive technique for deter-mining the localized values of the electron te~perature, phase lockin~ techniques will probably be required to identify the signal from the background radiation. This should be possible using the ruby laser ,in its regular spiking mode, see Davis and Keller (Ref. 16). The Q-switching of a CW ruby laser using saturable dyes, Roe~ (Ref. l~), is a better technique from the phase locki~ point of view but is only applicable for steady plasmas.

At these high densities self absorption of the scattered radiation could be a problem so as a check the mean value of n(2) is calculated from

Saha 1 s equation

(3.6.10 ) where Z+ is the ion partition function and Te is the electron temperature.

It turns out that for n(c) ~ 1015 cm- 3 and Te ~ 20000_{K the plasma can be}

re-garded as being optically thin to the scattered radiation at either 6939°A or 691loA.

The above discussion of the high density regime assumes that ground state excitation dominates downward cascading as a means of populating level 2. Furthermore, temperature measurements would only be possible if either nel) or

4.

SlJMMARY

In this report is has been suggested that "Selective Excitation Spectroscopy" might enable localized (and in certain instances temporal) esti-mates to be made on one or more of the following parameters;

(i)

(ii) (iii) (iv) (v)

the excitation temperature, the electron number density

the translational temperature of the radiating~spect~s, ~ _ J' " .

the magnetic field the macroscopic mot ion

In the present context selective excitation spectroscopy refers to the technique of using radiation, of a frequency that coincides with that of aspectral emission line of one of the species comprising the plasma, to

selectively excite a specific transition. With a sufficiently intense beam of radiation the population density in the upper level of the excited transition is enhanced to the point that the spontaneous emission emanating from this optically pumped region can be identified against the general background radiation at the same frequency. The general requirement on the intensity of the radiation (usually a laser beam) is characterised by the statement

tf » tr

where "tr" is the rise time of the enhanced radiation, j*, t

r

g(s)

t

[g(p)+g(s)JB(p,s)JI

(v)L(v)dv

due to selective excitation of the transition between the pand s levels,

and "tf" is the decay time of j* due to collisional and radiative depopulation of the coupled levels,

t

=

f n(c)

Under these circumstances the approximate maximum value of j*/j is given by

"*

Q(cl

+ 1

Jmax

nrsJ

j -:ili207-

p

+--+

-1-g(s)

where the volume emission rate j

=

n(s)A(s,p) ~. It is worth noting that in the intense beam limit the amplitude of the int~nsified emission (the scattered radiation) is in fact independent of the laser intensity. Clearly

n(p)

»

n(s) if a large intensification of the spontaneous emission is to be attained. This necessitates that hv

»

kT ~nd this imposes a limit on the temperature, for which this technique is applicable, in terms of the frequency of the laser radiation.

It was found that collisional depopulation of the upper level, of the excited transition, controls the duration of this intensified emission and consequently limits the range of densities for which this technique could be used. With the 6939°A line in potassium the peak in the enhanced emission can probably only be resolved for n(c) ~ 1013 cm- 3 • At the higher densities, where LTE may apply, the steady value of j*/j was found to be a sensitive function of the excitation temperature and independent of the density. However, in this regime