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Line Emission Sources for Concentration

Measurements and Photochemistry

476

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U.S. DEPARTMENT OF COMMERCE

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NBS TECHNICAL NOTE 476

ISSUED MARCH 1969

Line Emission Sources for Concentration

Measurements and Photochemistry

Walter Braun and Tucker Carrington

Physical Chemistry Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

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CONTENTS

Page

Abstract ^

Introduction 1

I. The Three layer Model 3

II. Integrated Absorption 5

III. Absorbed Power 5

IV. Multiplet Lines and Lamp Diagnosis 6

V. Discussion 8

VI. Appendix n

VII. References 12+

Figure Captions 16

II
Line Emission Sources for Concentration

Measurements and Photochemistry

4t Walter Braun and Tucker Carrington

Abstract

When a resonance line is absorbed in its own gas, measurements

of this absorption can be used to derive a value of the

product, nf, of concentration of the absorber and f value of

the transition. This line absorption process is also

important in the study of fluorescence and photochemistry,

where it is possible to produce atoms or molecules in a single quantum state without otherwise disturbing the system.

The interpretation and success of these applications of line

absorption depend critically on the shape of the line emitted

by the light source. This is strongly influenced by the

optical depth in the emitting region, and by the inevitable

presence of a reversing layer through which the light must

travel on its way out of the lamp. This paper presents a simple model which can be used to estimate the effects of

these properties of the light source on the threshold and

sensitivity of measurements of nf, and on the power which

can be delivered to absorbing atoms outside the lamp.

Emphasis is on the general principles of lamp design and

diagnosis, rather than on accurate description of a particular

lamp.

Key words: Spectroscopy; photochemistry; analysis; gas; vacuum ultraviolet; absorption.

INTRODUCTION

In spectroscopic observations designed to determine the concentration

or f value of atoms or molecules forming an absorption line, one can use either a light source which emits a continuumlj^ or one which emits

a discrete spectrum,,-^ In either case, we want to relate the observed

attenuation of the incident light beam to the product nf of the f value

and number of absorbing atoms traversed by the beam. When a continuum

is used, its intensity can be assumed to be independent of frequency over

the range where absorption is significant, so that the wavelength

dependence of the light source intensity is unimportant. The resolving

power of the spectrometer is however very important in determining the

sensitivity with which absorption by a narrow line can be detected and

measured. *-t^?5 If on the other hand, the light source is not a continuum,

^Present address: York University, Toronto, Canada.
but instead emits a relatively sharp line whose center coincides with that

of the absorption line, we are freed from the requirement of high

resolution of the spectrometer (assuming we are concerned with a single

line) but the intensity distribution in the emission line becomes of

critical importance. This distribution is strongly influenced by self

absorption and reversal in the light source.

This paper illustrates and discusses the effects of the intensity

distribution in the emission line on two somewhat different types of

absorption measurements,, In one case, one measures the fraction of

incident light which is absorbed, and relates this to the product, nf,

of absorber concentration and f value for the transition, using curves analogous to the curve of growth familiar in connection with continuum

light sources. In such measurements, the absolute intensity of the light source may be of little interest. Absolute measurement of the absorber

concentration is possible if the f value, or integrated absorption co- efficient of the transition is known. 5.?o if the f value is unknown,

empirical calibration curves can be constructed. ' Absorption measurements

can be used to derive f Qvalues if the concentration of absorbers can be

independently measured.

In a second type of absorption measurement, one is primarily interested in maximizing the absolute rate of absorption of energy in a gas

irradiated by the lamp in question. An example is the measurement of

absorber concentration by measurement of the intensity of re -emission

(fluorescence) of the absorbed light. In many cases this method is quite

sensitive.-^ Another application is to photochemical or photosensitized

processes, where, as in the fluorescence method, one is simply interested

in the rate of absorption of energy from a line emission source.

In Section I we define the simple three layer model used to treat a lamp consisting of an optically thick emitter (1) with a separate

reversing layer (2) and an external absorbing layer (3). In Section II

this model is used to derive a relationship, analogous to a curve of

growth, relating total absorption to optical depth in the absorbing layer,

assuming Doppler shapes for the absorption coefficients, and different

temperatures in each layer. The effect of the reversing layer in the

light source on the sensitivity and linearity of the method are illustrated

and discussed. Section III discusses effects of parameters of the light source (optical depth of emitting and reversing layers) on the

sensitivity of the fluorescence method, i„

e

B on the rate of absorption

of energy in layer 3. A lamp which is optimum in this respect will not,

in general, be optically thin. After this discussion of the importance

of the optical depth of emitting and reversing layers in the light source,

a method of estimating these parameters for lamps emitting multiplet

lines is discussed in Section IV, There follows in Section V a brief

discussion of the application of these ideas to practical measurements,,
I. THE THREE LAYER MODEL

Figure 1 illustrates the model, an optically thick emitter whose

radiation is transmitted through a reversing layer and then through a separate absorbing layer, A more complicated model, including smooth

gradients of concentration rather than separation into layers, has been

treated by Hooymayers, for application to flames. Another continuum

model has been treated by Cowan and Dieke 5. In this reference the

general quantitative aspects of the mechanism of self -absorption

particularly in arc discharges is discussed in detail. The specific layer

model we adopt here is similar to that suggested by Bleeker. ° This

model has not been treated by Cowan and Dieke but falls between their two

extreme cases (a) where the emitting and absorbing atoms are uniformly

distributed in the discharge and (b) where the emitting and absorbing atoms

are completely separated. Since Cowan and Dieke found that the general

line intensity distribution depends very little on the particular spatial

distribution function describing the emitting and absorbing atoms the

simplified layer distribution model adopted here should very adequately

describe the end-on type discharges characterized in this paper.

The three layers shown in Figure 1 will in general have different

temperatures and optical densities of absorbers. The lamp is characterized by the number of emitting atoms (which we assume to be much smaller

than the number of absorbing atoms) and by the optical depth (at the

center of the line) and temperature in layers 1 and 2. To write an expression for the intensity emitted from layer 1 we need the path length

H} populations of absorber and emitter n and n', statistical weights g

and g', and absorption cross section c (v). Subscript numbers refer to the

layer in question and the prime denotes the upper level. The intensity

emitted from layer 1 is then-'-' LjW = B(v,Tex )J1

(v) J cm'

2

s"

1

sr"

1

(s"

1 )"1

(1)

where B(v) is the Planck function with temperature Tex defined for the

first layer by

^=exp - (E'-E)/kTex (2)

and J., is the emissivity,

J-^v) = 1 - exp(-a (vjn^). (3)

In the usual case, when n' « n and v is nearly constant over the line, we have the proportionality

n' _g_ hTM " ^ n

J

l<*' W
The intensities transmitted through layers 2 and 3 are then simply

obtained from this by multiplying by the factor exp(-0" (v)nJl) for one

or both of these layers.

Use of these expressions requires knowledge of the magnitude and

frequency dependence of o~ „ These problems are separated by defining

a (v) = s(v) \ ff (v)dv, \s(v)dv = l (5)

The integral of the cross section is then equivalent to the absorption

transition probability in its various forms,

s

a (v)dv v = ^ = ^. (6)

c mc x '

All integrals are extended over the range - 00 to +00

. The line shape

factor s(v), in its dependence on v and T for the three layers, has a profound effect on the absorption in layer 3. In the work presented

here we will assume pure Doppler broadening for which

s(v) - yffT- exP "C^pO

2

. (7)

This is appropriate to many low pressure systems with transitions in the

ultraviolet, and it is just such systems which are of interest in the

measurement of concentrations of ground states of atoms by the line

absorption method, and in many photochemical experiments. Lorentz or Voight prof iles-*-°^l9 could be used instead of the Doppler form, with

some increase in computing time.

We are now in a position to calculate l(v) and I, its integral over

frequency, for light emitted from each of the three layers. In the first

group of calculations we have chosen the Doppler temperatures T, = 700,

T = 350, T = 300 K, though in most ofpthe following only the ratios

are important,, With these fixed, Ji = \ Ji(v)dv depends only on the

optical depth at the center of the line, Di = 0"i(0)niii. Jg, the

integral of Ja(v), depends on D2 as well as Di,

J2 = \ [l - exp(-ai(v)niXi)] exp(-a2(v)n2 ^2)dv (8)

and similarly J3 depends on all the optical depths,
J3

= \[l - exp(-ai(v)tii^i)] exp(-a2 (v)n2 £2 ) exp(-a3 (v)n3 £3 )dv. (9)

Here the a (v) functions are expressed in terms of (6) and (7) with the

Doppler widths 6 appropriate to each temperature,, As in Eq. (k) , we

have the proportionalities

V-^-S-*.*--*—5-*

(10)

The calculations reported in this paper were done with the OMNITAB general

purpose computer program.

^

II. INTEGRATED ABSORPTION

The integrated absorption in layer 3 > {T-z'^-s) 1^-2, depends strongly

on the characteristics of the lamp, that is on the optical depths in

layers one and two. The curves in Fig. 2 show this dependence on optical depth in the reversing layer. The curves have two properties of

interest. The smallest D3 which can be detected (corresponding for

example to 1$ absorption) depends strongly on D2. The sensitivity, in

this sense of detection limit, is greatly decreased by absorption in the

reversing layer. The second important property of these curves is the

slope, which is related to sensitivity in the sense of distinguishing

one value of D3 from another. As is to be expected, all the curves have

unit slope at sufficiently small D3 , but for the curves corresponding to

relatively large D2 , this region of unit slope extends to higher D3 than

for the other curves. This is because the lamp emits relatively little

light at the center of the absorption line, so that the effective optical

depth in layer 3 is much less than D3 , the optical depth at the center of

the line. Since it is only in this region of unit slope that the

integrated absorption is simply proportional to D3 , the presence of a reversing layer in the lamp does give a certain advantage in the measurement of large D3 values. Another way of saying this is that although all

the curves in Fig. 2 level off toward unity at sufficiently high D3 , the curves for large D2 do so more slowly, and thus retain a useful dependence

on D3 up to higher D3 values.

III. ABSORBED POWER

For fluorescence, and for photochemical applications, the quantity of

5
interest is the absolute rate of absorption of energy in layer 3> I2 -I3 , rather than the normalized absorption (I2 _ l3)/l2 just discussed. The

absolute rate of absorption also depends strongly on lamp parameters,

as is illustrated in Fig. 3 > showing curves of I2-I3 for various ratios

D2 /Di. For all cases where D2 4 0, the curve goes through a maximum, and

the height and width of this maximum decrease strongly as D2 /D;l increases.

Though the presence of a reversing layer is frequently unavoidable in

practical light sources, Fig. 3 emphasizes its profoundly detrimental

effect on absorbed power, and hence on the intensity of fluorescence or the rate of a photochemical process. The assumption of D2 /Di = const

with Di as independent variable in Fig. 3 corresponds to a type A lamp in

which increasing the power input increases the density of absorbing atoms

(for example H in a He discharge containing a trace of H2 ) in the emitting

layer and hence in the reversing layer also, but does not change the

excitation temperature. Alternatively, the plot in FigQ 3 also corresponds

to the change of I2 -l3 on increasing addition of an active species, i.e.,

H2 to an inert gas discharge, since in this case too, D2 will be

approximately proportional to Di, in this case I2 -l3 is proportional, for

example, to the rate of absorption of Ly a in layer 3«

In a lamp for which I2 -l3 is a maximum it will generally be

impossible to make D2 = 0, hence it is necessary to be able to predict

lamp performance when Di and D2 are not negligibly small,,

If one is interested in absorption by a continuum rather than a line,

in layer 3> there is no need to worry about matching of emission and

absorption line shapes, and one must simply maximize I2. This criterion

can lead to lamps with very large optical depths in the emitter. In the

case of Kr, resonance lamps which have been described as optimum21j22

have optical depths at the center of the line of the order of 103 per cm

of length in the emitting layer.

IV. MULTIPLET LINES AND

LAMP DIAGNOSIS

Up to now we have discussed only single, isolated lines. We may

be concerned instead with an atomic multiplet or with isotope or nuclear

spin hyperfine structure. The situation is the same as for a single

isolated line only in the limiting case in which the line separation is

much less than the width of the shape function s(v) for a single line*

In this case the shape function is the same as that for a single line,

but the integral of the absorption cross section is correspondingly

larger. In the opposite limiting case, when the lines are so well

separated that they do not significantly overlap, but all are nevertheless

transmitted by the spectrometer or filter used with the detector, the

results for single lines must be appropriately averaged. In the

quantity I2 -I3 /l2 , the weighting factor for each line is just the fraction

of the total observed intensity of the multiplet due to that line. If

the lines are neither coincident nor well separated, their overlap
produces a complex line shape which is the sum of the shape factors for

the component lines, suitably displaced and weighted according to the

relative intensities,, The integral of the composite absorption

coefficient is the integral for the weakest line (assigned a relative

strength of unity) times the sum of the relative strengths, nf, of all

the lineSo For multiplets obeying L-S coupling, with populations

proportional to degeneracy, these relative strengths, gf, are conveniently tabulated„ ^3

The discussion in Sections II and III has emphasized the importance

of the parameters Di and especially D2 of the light source. In many

cases of practical interest these parameters can be estimated directly

in light sources emitting three or more well resolved lines in the same multiplet. If Di and D2 are large enough so that the behavior of the

light source differs significantly from that characteristic of the small

optical depth limit, the relative intensities of the multiplet lines in

emission will differ from the theoretical values (or the limiting values

at small optical depth). Di and D2 may be estimated from the observed

intensity ratios. For simplicity we will first discuss the case of a triplet with common upper level in which each line is resolved by the

spectrometer. We then have two independent observable intensity ratios

for use in estimating Di and D2 .

In Figure h, calculations for a two layer model are compared with

experimental relative intensities 21,25 f tke components of the oxygen

I triplet ( 3P2 1 o3

S) near 1302A. In this particular plot the optical

depth in the weakest line in the reversing layer is assumed to be one- half its value in the emitting layer. A family of such plots can be

constructed for various ratios, N, of this optical depth in the two

layers. One procedure in lamp diagnosis is to choose from this family

a plot which gives both of the observed intensity ratios at the same value of the abscissa. This value of the abscissa gives Di, and the N

value associated with the particular set of curves gives D2 , both for the

weakest line. In calculating such curves, it is sometimes inappropriate

to assume that the relative population in the multiplet levels corresponds

to equilibrium at the kinetic temperature of the gas, or that the populations are in the ratio of the statistical weights (infinite temperature).

In some cases values of the population ratios intermediate between these

extremes must be chosen, and adjusted to get agreement with the observed

relative intensities. This assumption of relative populations, plus

experimental or theoretical gf values for the transitions in the multiplet,

is sufficient to determine curves such as those in Fig. K, using Eqs. (8)

and (10)„ At small values of Di, the ratios of intensities approach

the ratios of the theoretical gf values, 5:3:1 for this oxygen multiplet.

At higher optical depths the curves bend over in such a way that the

intensity ratios are strongly altered, and even inverted, due to the

reversing layer

„

An interesting feature of these curves is the existence of a maxi- mum at moderate optical depths when T2 is equal to or only slightly less

than Ti. This is due to the fact that Ii cannot exceed the black body

7
intensity corresponding to the assumed excitation temperature, while the

fraction of light transmitted by layer 2 can decrease to values arbitrarily

near zero as Dg is increased. If T2 is sufficiently smaller than Tx> there

will be no maximum, since the reversing layer cannot absorb light in the

wings of the emission line. At intermediate values of T2, the maximum

in the I2 vs Di curve will be followed by a minimum,, All these results

are consequences of the assumption that the excitation temperature is at

least approximately independent of changes in discharge power, which

affect only the densities of emitter and absorber, but not their ratic

If the energy splitting in the multiplet is small compared to kTi,

the levels will be populated nearly in the ratio of their statistical

weights whether or not they are strictly in thermal equilibrium. If the

splitting is large, their relative population will be unknown, and values

must be assumed which, in combination with a chosen N value, give results

consistent with observed intensity ratios of the multiplet.

As a second example of lamp diagnosis, we have carried out this

procedure ^with measurements of intensity ratios in the CI I multiplet

near I35OA, emitted from a microwave discharge in 1 torr (133 Nra"^) He

containing ifo QA. ^ . The results are quite sensitive to values of the

optical depth and to the population ratios in the lower levels of the

multiplet. For example, photoelectric measurements of the intensity ratio

136^A/13^-7A °f lines from the common upper level ^3/2. give the value

'+/1, whereas the corresponding gf ratio for these lines is I/5. The

other pair of lines in this multiplet, from the upper level 2

Pi, has an intensity ratio of 2.5 and the corresponding gf ratio is 2. These

observations can only be explained by assuming large optical depths, and

that the ratio Cl2Pi/2P3 / 2 of population of the lower levels of the two

transitions is much larger than the equilibrium value at any temperature

which could reasonably exist in the lamp. Although the paparaeters

characterising the lamp (optical depths, multiplet population ratios,

Doppler temperature) are more numerous than the observations (two

independent intensity ratios) other physical constraints on the parameters

make it possible to estimate them within reasonably narrow bounds for a given lamp,,

The results for CI are however somewhat uncertain due to our lack of

knowledge of isotope and hyper fine splittings in the multiplet involved.

The nuclear spin splitting observed in the ground state is too small to be

significant. 2" Splittings in the upper state have never been observed

and are likely to be small compared to the Doppler width of 0.2 cm" .

' If hyperfine splitting comparable to the Doppler width is present, the

qualitative result that the population in the ground multiplet levels is

out of equilibrium would not be affected.

V. DISCUSSION

In recent years the line absorption method has been applied both to

the measurement of atom concentrations, and to the determination of f

8
values. The question of line shape and optical depth is of great

importance in both cases. The work of Parkes, Keyser, and Kaufman on f

values for resonance lines of represents the ideal case in which the

light source was designed to reduce Di and D2 to negligible magnitudes,

thus avoiding the complications discussed in this paper. In other work

aimed at the determination of f values or concentrations, the light source was not optically thin, and accounting for this considerably com- plicated interpretation of the results. °>10 It has been a common practice to assume that Ii(v), the frequency distribution of intensity from

the emitting layer, has a Doppler shape, and to describe effects of

optical depth in this layer by using an unnaturally large width, or "temperature" in this shape function„ In Fig„ 5 we compare this approximation with the correct treatment outlined in the preceding sections,

using curves like those in Fig. 2. Even though the pure Doppler approximation will give a good fit to the exact curve in many cases, the

"temperature" which must be used has no physical significance, and will

depend strongly on the optical depth in the emitting layer. On the other

hand, when optical depth in the emitting layer is treated correctly, a measured value of Ti, obtained for example by the pressure increase

method , can be used directly.

Most of the results we have given apply to the problem of allowing

for self reversal in the interpretation of line absorption measurements.

Since self -reversal is undesirable from almost every point of view, it

is clearly better to reduce it than to correct for it with extensive

calculations. It is then appropriate to ask to what extent self -reversal

can be reduced to insignificant values without greatly decreasing the

total effective intensity of the lamp. 21 Here it is necessary to distinguish two different types of lamps. Let us call type A lamps those

in which absorbing atoms are formed only by the discharge itself, along

with the excited atoms which radiate. Absorbing atoms will then be

removed by diffusion to the wall, and perhaps by chemical processes in

the volume. A typical example of type A lamps is a discharge in He with

a trace of H2 emitting the Lyman a line of H. In type B lamps, on the

other hand, the absorbing atoms are unreactive and are present more or less uniformly through the gas volume. A typical example here is a Kr resonance lamp. In either of these cases, it is convenient to think

of a trace additive, H2 or Kr, being added to a much larger amount of

He or other carrier gas in which the discharge is maintained.

In many cases in the vacuum ultraviolet it is impractical to run the

discharge directly against the window, since this may decrease its

transparency, or contamination may develop from the window sealing

material. If the discharge cannot be operated directly against the

window, there will certainly be a reversing layer in either type of lamp,

and in type B discharges the optical depth of this layer will be

proportional to the distance between the discharge and the window.

If the discharge can be operated against the window, the structure

of the reversing layer is determined by gradients of absorber and emitter
density between the uniform discharge (if it has a uniform region) and the

wall or window. In constricted discharges these gradients will be large,

particularly for the emitting atoms, and in general the gradient for the

emitters will be steeper than that for the absorbers,, This situation is

equivalent to the presence of an absorbing layer,, In type B lamps the

gradient of absorbers will be small, and their concentration may even be

higher near the wall than at the center of the discharge if there is an appreciable radical temperature gradient near the wall,, In type A lamps

the gradient of absorber concentration toward the wall will be largely

determined by the diffusion rate and the efficiency of removal of absorbers

at the wall. If this removal is inefficient, the ratio n'/n will become

small toward the wall and this will constitute a reversing layer. Note

that a two-layer light source (Fig. l) can produce a self reversed line

shape even if the two layers are at the same Doppler temperature, provided that n'/n in the second layer is sufficiently small and that layer

1 is not optically thin.

The preceding discussion indicates that, due to wall effects, an absorbing layer will be present in any lamp. The optical depth of this

layer can be reduced arbitrarily by simply reducing to sufficiently low

values the concentration of the species from which the emitter is produced

(H2, Kr). Figure 3 indicates that this will also reduce the effective

output of the lamp. The lamp with maximum efficiency, that is, which puts

the maximum power into layer 3 > will not be optically thin, and this fact

must be allowed for in interpreting the results. The absorption cross section at the center of a strong atomic line Doppler broadened at room

temperature can easily be as high as 10

* cm

2

. In this case a pressure

of 1 m torr (~ .1 Nm'2) of absorbing atoms (H, Kr ) in a 1 cm path length

gives an optical depth of 3» In such cases, careful control of the

additive concentration is required to avoid a greater than optimum

optical depth.

Interpretation of observations on the oxygen and chlorine multiplets

in terms of the two layer model leads to several generalizations applicable

to microwave discharges. These may be helpful in more refined calculations or calculations for other line multiplets. The results

can be summarized as follows:

a) The reversing layer 2 is best considered as a continuation of

the emitting layer 1 with the Doppler temperature T2 only slightly less

than Ti.

b) The length of layer 2 is less than layer 1 as should be typical

of the type A discharge defined above.

c) The optical density of layer 2 depends not only on the physical

length of this layer, but is particularly sensitive to the relative population of atoms in the ground state multiplet levels. These populations

10
are in general not fixed by the Doppler temperature of the layer since the

system is often far from being at thermal equilibrium,,

d) The discharge temperature is determined primarily by the carrier

gas so that the temperature of the discharge and the ratio T3./T2 for one line source may apply rather well to that of a different line source.

VI. APPENDIX

The model discussed in I assumes that there is no emission from

layers 2 and 3- Since light is being absorbed in these layers, it is

producing excited atoms and these can radiate" if they are not otherwise

deactivated. In the preceding discussion we have not allowed for this

radiation. It is the purpose of this appendix to do so in a very crude

and qualitative way.

Taking layer 3 as an example, the problem has two parts D One is to

estimate the steady state concentration of emitters as a function of depth

in the layer, and the other is to estimate the total intensity radiated

from the layer by these emitters. This second part is similar to the

calculation made for layer 1, except that in that case the emitter

concentration was assumed constant within the layer.

Up to now our treatment has described plane parallel layers,

although we have had in mind a cylindrical discharge tube viewed end -on. Now we must include the effects of cylindrical geometry, particularly the

escape of photons perpendicular to the tube axis through a gas that may

not be optically thin. The concentration of emitters at a given cross section of the cylinder will be determined, roughly speaking, by the

balance between their rate of formation by absorption of light in that

volume element, and the rate loss by radiation (or diffusion) of photons

radially toward the tube walls. If we take the effective diffusion

constant to be3 D = [ (2"o n) 2 T ] "1 then the mean time required to diffuse

the average distance R/2 is t = T Df: . Here T is the radiative lifetime

of the transition and D" is the average (over frequency) optical depth

in the radial direction. This form for the diffusion time suggests that

a reasonable qualitative form for the rate constant for escape of photons

would be

Escape

[T^ + D?)]"1 (Hi

11
Balancing this rate of escape (the net destruction rate for n') against

the rate of formation by absorption of light gives as the estimate of the

emitter concentration

,i_

n'U) = aU)n T (1 + D?) I2exp(-n\ a(i)di) (12)

The average optical depths or cross sections appearing here_and below

hide a good deal of complexity, but, with the exception of D , they can

all be defined rigorously in terms of the sorts of integrals we have

previously evaluated.

When Eq„ (12) is used to calculate the intensity due to fluorescence,

and this is combined with I3

* ', the directly transmitted intensity of

I2 through layer 3, the result for the total intensity per unit solid

angle leaving layer 3 is

= Ia

(tr) [l + ^ + y (l _ e

-(D--D)

) D-D

(13)

Since this result is only intended as a rough estimate, it is sufficient

to say that D is an average optical depth appropriate to light with the

frequency distribution Is(v), while D is appropriate for emission with a pure Doppler distribution at the same temperature as the absorber, i.e , the scattered light in layer 3° If Ia(v) is strongly broadened due to

large Di, or reversed due to large D2, we will have 5 « D3 . We may

expect on the other hand that D ^ D3 except for very high D3 . In Eq. (13

)

the term

n -(D-D)

D" - D

is unity only in the limit D -> D and decreases as D - D increases. In well

designed lamps D will seldom be greater than 1, so the correction to

I^ tr J due to fluorescent emission will be of the_order of D, perhaps

reduced by a factor of two or so due to the D-D term. Calculations

indicate that D will be small even when D3 is not small, for many

practical lamps. When D3 itself is small, D is smaller still. Whenever

the fluorescent emission is a small fraction of the transmitted intensity,

it can be roughly estimated using Eq. (13 ).

In all the preceding we have been considering light intensity per unit

solid angle, avoiding the unnatural constructs of parallel light and

point source. We have not explicitly introduced the aperture of the

12
spectrometer which isolates the emission line, but in fact the geometry

of light collection is quite important when fluorescent emission is

significant," If the spectrometer collects fluorescent light from a larger volume of system than that from which it collects transmitted

light, the effects of fluorescence will be exaggerated beyond the

predictions of Eq„ (13 )o

13
VII. REFERENCES

1. G. M. Lawrence, D„ L„ Mickey, and K. Dressier, J. Chem. Phys. 1+8, I989 (1968).

2. T Carrington, JQ Chem. Phys. 3_1, 121+3 (1959).

3. A„ C. Go Mitchell and M. Wc Zemansky, Resonance Radiation and Excited

Atoms, Cambridge Univ. Press (193*+).

k. H. J. Kostkowski and A. M, Bass, J. Opt. Soc. Am. 1+6, 1060 (1956).

5o T Gc Kyle and J. 0. Green, J Opt. Soc. Am. 5%, 895 (I965).

60 J. V Michael and R. E Weston, Jr., J. Chem. Phys. hj_, 3632 (I966).

7„ A. Walsh, in Advances in Spectroscopy 2_, 1-22 (I96I), edited by

W„ H„ Thompson (Interscience )„

80 D. A, Parkes, L. F. Keyser and F. Kaufman, Astrophys J» ll+9. 217

(1967).

9. F. A. Morse and F. Kaufman, J. Chem. Phys. 42_, I785 (1965).

10. A, B. Prag, C. E. Fairchild, and K. C. Clark, Phys. Rev» 13 7A, I358

(1965).

11. S. E. Frish and 0. P. Bochkova, Vestnik Leningrad Univ„ 16, Ser.

Fizo and Khim. 16, 1+0 (I96I); Vestnik Leningrad Univ. k, Ser. Fiz«

and Khim. 1, I73 (I962) (Russian). U. S. Dept. Commerce, Office

of Technical Services, Washington, D. C. 20230, AD 1+22 5I5.

(English).

12. W. Braun and M. Lenzi, Dis. Faraday Soc. Kk, 252 (I967).

13. Ho E. Gunning and 0. P. Strausz, in Advances in Photochemistry, Vol. 1,

edited by W. A. Noyes, Jr. , G. S. Hammond, and J. N„ Pitts, Jr.

(interscience, I963).

Ik. H. P. Hooymayers and C. Th. J. Alkemade, J. Quanto Spec try, and

Radiative Transfer 6, 501 (I966).

15. R. D. Cowan and G. H. Dieke, Rev. Mod. Phys., 20_, 1+18 (191+8)0

16. W. Bleeker, Zeit. f. Physik, 5_2_, 808 (I929).

11+
17. S. S. Penner, Quantitative Molecular Spectroscopy and Gas

Emissivities, Addis on -Wesley, Reading, Mass. (1959 )•

18. D. Go Hummer, Mem. Roy. Astron. Soc. JO, 1 (1965).

19. H„ Prugger, Optik 21, 320 (1964).

20. J. Hilsenrath, G. G. Ziegler, C. G Messina, P. J. Walsh, and

R. J. Herbold, Natl. Bur. Std. Handbook 101 (March I966) .

21. H. Okabe, J Opt. Soc. Am. 5k, iVjQ (1964).

22 L. J. Stief and R. J. Mataloni, Appl. Opt. k, iG^k (1965),

23» C„ Wo Allen, Astrophysical Quantities, Univ. London (1963)0

2.K J. Vanier, Appl. Opt. 6. I67 (I967).

25. Do Davis and W. Braun, Appl. Optics ]_, 2071 (I968).

26. L. Davis, Jr., B T. Feld, C. W. Zabel, and J. R. Zacharias,

Phys. Revo 23_, 525 (19^8).

27. L. Jo Radziemski and V. Kaufman, private communcation, to be

publishedo

15
FIGURE CAPTIONS

Figure 1: Three layer model for light source and absorber,, T, D,

and I are Doppler temperature, optical depth at center

of line, and outgoing intensity.

Figure 2: "Curves of Growth" showing effect of optical depth of

reversing layer on absorption. For each curve the temperature ratios are Ti/T2/T3 = 1/0. 5/0. 39 and Di = 3.

From top to bottom, D2 = 0, 3 > 5 , and 7°

Figure 3: Power absorbed in layer 3 as a function of Di for various

values of the ratio D2/Di. For each curve D3 = 0.1 and

T1/T2/T3 = 1/0,5/0.39.

Figure k; Relative intensities of the oxygen resonance triplet

3p2 1,0

3

S as a function of optical depth in the line

with smallest gf. S, M, and W refer to the lines with

relative gf values 5; 3> and 1. Curves are calculated,

points are intensities observed25 with addition of small

amounts (< .01$) of 2 to a 2V?0 MHz discharge in 1 torr

He. The calculations were done with the value D2 /Di = 1/2 for the line with smallest gf. The relative

populations in the three levels of the ground multiplet

were assumed to correspond to Ti = 600 K, T2 = 500 K.

Figure 5: Comparison of "curve of growth" for optically thick emitter,

D2 = 3 from Fig. 2, with approximation corresponding to

optically thin emitter with unnaturally large Doppler

temperature, Ti/T2 o

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