JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

A Method of Determining Optical Cross Sections*t

EiNAR HiNNovtDepartment of Physics, Duke University, Durham, North Carolina

(Received August 2, 1956)

A method of evaluation of the optical cross sections, i.e., cross sections for broadening collisions betweenexcited atoms and foreign molecules, from experimental intensity-density data is described. This methodutilizes the analytical expressions of the asymptotes to the theoretical intensity-density curves at very low,and at very high densities of the emitting atoms. Only relative intensity measurements are necessary for theevaluation of the cross sections. An intensity-density graph for the Na5890 line is constructed from photo-electric measurements of the total line intensities, emitted from a Lundeg&rdh-type acetylene-air flame.The optical cross section for sodium corresponding to this line is found to be 64 square angstroms. A briefdiscussion of the possibilities of utilizing this method for the determination of the degrees of dissociation ofmetal salts in the flame, or the evaluation of absolute and relativef values of the spectral lines is includedat the end of the paper.

INTRODUCTION

THE optical cross section corresponding to a givenT spectral line is one of the important factorsdetermining the intensity contours of the line. Conse-quently, the self-absorption in a particular source andthe observed intensity of the line depend on the opticalcross sections of the emitting atoms. At the presenttime, very little is known about the exact nature of theinteractions between an excited atom and foreignmolecules which result in the broadening of the emittedline. The numerous attempts to achieve theoreticalinterpretations of these interactions have beenhampered by lack of experimental data. Only in a veryfew cases have the optical cross sections been measured.

This paper describes a method for determining theoptical cross sections based on the measurement of thetotal line intensity as a function of the density of theemitting atoms in a flame source. In this section will begiven, together with notation and definitions, a shortreview of the theoretical background, based largely onthe work of Ladenburg and Reiche,l SchUtz,2 van derHeld,3 and others. The experimental setup and theevaluation of experimental data in the case of theNa5890 line will be given in the following sections. Asubsequent paper will present the results for severalother elements obtained under the same experimentalconditions.

The notation follows as closely as possible that usedby Mitchell and Zemansky.4 The reader is referredto this book also for an excellent review of older workin this field, both experimental and theoretical.

* This work was supported by the Office of Naval Researchunder contracts N6ori-107, T.O.I. and Nonr-1181 (05) with DukeUniversity.

t Based on part of a thesis in partial fulfillment of the require-ment for the degree of Doctor of Philosophy in the GraduateSchool of Arts and Sciences of Duke University in 1956.

1 Present address: Department of Physics, University ofMaryland, College Park, Maryland.

l R. Ladenburg and F. Reiche, Ann. Physik 42, 181 (1913).2 W. Schlitz, Z. Astrophys. 1, 300 (1930).3 E. F. M. van der Held, Z. Physik 70, 507 (1931).4A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation

and Excited Atoms (Cambridge University Press, New York,1934).

The total intensity of a spectral line, emitted byatoms in thermal excitation in a homogeneous source,may be expressed according to Kirchhoff's law as

I=io(vo,T)A =io(vo,T) fA (v)dv

= io (PoT)f (-e-(^Wz)dv- (1)

In this expression io(vo,T) is the intensity of blackradiation at the temperature T of the source, and thefrequency vo of the center of the line, A (v) the absorp-tivity of the source at the frequency v, and k (v) theabsorption coefficient. The latter is so defined that afraction e-k( )I of incident radiation of frequency v istransmitted by the source of thickness 1. The integralis to be taken over the frequency range of the line, andmay be extended from zero to infinity in case of anisolated line.

In most flame sources the frequency dependence ofthe absorption coefficient k(v) is determined by twoimportant processes: the collision broadening, due toperturbation of the emitting atoms by foreign molecules,and the Doppler broadening, due to thermal motionof the atoms. The Lorentz collision theory yields theequation,u f exp(-2)dy

k ()1= 7r'rd2 f a2+ (y)2' (2)

for the absorption coefficient, resulting from thecombined effect of these two broadening processes.The quantities appearing in Eq. (2) are defined as

2e2 Nflr=-

Mc VR

AVR

a= (ln2)1,AVD

2(v-vo)w= (ln2)1,

AVD

(3a)

(3b)

(3c)

151

VOLUME 47, NUMBER 2 FEBRUARY, 1957

EINAR HINNOV

log r

FIG. 1. Theoretical intensity-density curves.

e,m=charge and mass of the electron, respectively,c=velocity of light, N=number of atoms per cm3

capable of absorbing the line in question, M=atomicweight of these atoms, f=oscillator strength for theline, R= gas constant, N'= number of perturbingparticles per cm3 , =mean speed of the perturbingparticles relative to the emitting atom, and oA=crosssection for the broadening collisions. The Dopplerwidth of the line, AVD, is given as

v0 f8RTAVD=-( -ln2 * (4)

C MI

If the line were broadened by the Doppler effect alone,AVD would represent the total width of the functionk(P) at a point where k(v)= k(vo). Similarly, thecollision width APR would determine the width ofk(v) if the line were broadened by collisions alone, i.e.,in absence of the Doppler broadening. In terms of thecross section for the broadening collisions, the collisionwidth is given by

AvR=N'OoT2. (5)

Various approximation methods have been used toevaluate the integral in (2) and to calculate the totalabsorption A (r,a) as a function of r for different valuesof the absorption parameter a. For further references,derivations of (2), and discussions of the methods ofevaluation, the reader is referred to Mitchell andZemansky4 and Penner and Kavanagh.5

Figure 1 shows curves of log(A/AvR) vs log r forseveral values of a. These curves show several importantfeatures. For all values of a they approach asymp-totically the straight line t of slope 1 as becomes small,and the straight line 't' of slope as r becomes large.In the intermediate range of r, the shape of the curvesdepends on the a value, but all curves for a> 1 practi-cally coincide.

The physical interpretation of these curves is quite

6 S. S. Penner and R. W. Kavanagh, J. Opt. Soc. Am. 43, 385(1953).

straightforward. Since A is proportional to the totalintensity of the line and r is proportional to the"optical density" Nfl, each of these curves describesthe actual variation of the emitted intensity as afunction of ATfl for given conditions of the source,i.e., for given a and APR. At small optical densities theself-absorption is negligible, and the emitted intensityis proportional to Nfl independent of the broadeningprocesses. As the optical density becomes large, theradiation at the center of the line will become that of ablackbody at the temperature of the source, andfurther increase in intensity will occur only in theradiation emitted in the wings of the line. In case ofnegligible Doppler broadening, corresponding to a= o,the emitted intensity becomes proportional to thesquare root of NYfl at large Nfl. Even if the Dopplerbroadening is not negligible, the extreme wings of theline will have the "dispersion" shape (6) because theinfluence of the Doppler broadening decreases ex-ponentially with v- PoI. Consequently, at very largeATfl the influence of the Doppler broadening of theline on the self-absorption is negligible for any valueof a>O. From Fig. 1, it is evident that the qualitativebehavior of the intensity-density curve is influencedby the Doppler broadening only if the Doppler widthis larger than the collision width of the line, i.e., if a< 1.

When the Doppler broadening is negligible, theabsorption coefficient has the dispersion form

[2 (v-vO)/APR] 2+ 1

which is a special case of (2) for a= oo. In this case, thetotal absorption can be evaluated analytically (Laden-burg and Reichel). Equation (6) is substituted into (1)and the variable changed to P-Po. An integration byparts and the substitution

2 (v- vo)/APR= tanO/2yield

A =A vRT-e-rI2[Jo( -t-i(-)], (7)

where the functions in the brackets are the Besselfunctions.

i- rJn(Z) = _ feiz cosocos(O)dO.

From (7) can be obtained the equations for the low- andhigh-density asymptotes in Fig. 1 as

A A llog-=log(7rr/2) and log-=- log(7rr),

AVR APR 2(8a,b)

respectively. For the latter result, the asymptotic

152 Vol. 47

DETERMINING OPTICAL CROSS SECTIONS

expression for Bessel functions (Jahnke and Emde6 )

fir \ ne r/2Jnj > - as roo

2 (err)

has been used. The asymptotes intersect at r=4/7r,A/AvR=2.

If loga is added to the ordinates and abscissas ofeach curve, the position of the low-density asymptoteremains unchanged, while the high-density asymptotesare shifted vertically by loga from the asymptote ofthe curve for a= 1. Figure 2 shows some of the resultingcurves, log(A (ln2)V AVD) vs log (ra). This representationis more appropriate for the evaluation of experimentaldata, since AVD can be calculated if the temperatureof the source is known. The equations for the trans-formed low- and high-density asymptotes are evidently

A (ln2)llog = log (7rra/2)

AvDand

A (ln2)1log = 2 log (7rra2 ),

AVD

respectively. The ordinate of the point of intersectionof the asymptotes is

A (ln2)1log = log2a. (9)

AVD

Thus, if the value of A for this point is determinedexperimentally and the temperature of the source isknown, the absorption parameter a, and consequently,AvR and o-2 can be obtained immediately. Once thea value has been found, the absolute values of r areautomatically determined for the experimental in-tensity-density curve.

EXPERIMENTAL

The source of the spectral lines used in these experi-ments is an acetylene-air flame of the LundegArdhtype,' 8 with two auxiliary burners attached in orderto keep the temperature constant along the line ofsight over the colored part of the flame. The measure-ments are taken about 6 mm above the inner reactionzone, where the flame gases can be expected to be inapproximate thermodynamic equilibrium. Along theline of sight the thickness of the colored part of theflame is about 25 mm. The element under investigationis introduced into the flame in the form of a saltsolution, which is converted into a fine spray by theair supply of the burner. Changing the concentration

6 E. Jahnke and F. Emde, Tables of Functions (Dover Publi-cations, New York, 1945), 4th edition, pp. 137-138.

7 H. Lundeg~rdh, Lantbruks-H15gskol. Ann. 3, 49 (1936).8 L. Huldt, Eine spekroskopische Untersuchung des elektrischen

Lichtbogens und der Azetylen-Luftfiamme (Uppsala, 1948).

I:i'

log o

FIG. 2. Theoretical intensity-density curves.

of the salt solution provides the means of varying thedensity of the emitting atoms in the flame by knownrelative amounts. The proportionality between theconcentrations of the salt solution and the densitiesof the emitting atoms in the flame has been checkedby measuring the intensity-density relationship forsome weak line of the same element, over a large rangeof concentrations.

The temperature of the flame is measured by theline reversal method, using many lines in the wave-length range from the red (K7665) to the near ultra-violet (Cu3247). For different lines, the deviationsfrom the mean value of 2480'K do not exceed 20° andno correlation between the deviations and the wave-lengths has been observed. This temperature, 2480'K,is also in good agreement with the theoretical tempera-ture of the flame, calculated from the known initialcomposition of the flame gases.

The partial pressures of the free element underconsideration range from about 10-9 to 10-5 atmos.Since the total pressure in the flame is 1 atmos, thecollisions of the emitting atoms with atoms of the samekind can be neglected in comparison with the collisionswith foreign molecules. This condition is importantsince the collision cross section and, consequently, thea value must remain constant over the entire intensity-density curve.

Behind the flame in the optical setup stands a5.5-v/30-amp G. E. tungsten ribbon lamp. The bright-ness temperature of the tungsten filament, at about6650 A, is measured directly by means of a Leeds andNorthrup Optical Pyrometer. A quartz lens forms animage of the lamp filament at the center of the flame.This image, together with the radiation from the flame,is focused on the entrance slit of a monochromator byanother quartz lens. The lens combination is so chosenthat the second lens and the optical system of themonochromator are both completely filled by the lightfrom the background source. During measurements ofthe flame intensity alone, an opaque screen is insertedbetween the background source and the flame.

The dispersing instrument is an Ebert-Fastie.type

153February 1957

EINAR HINNOV

ground intensity can then be expressed as

B = i (AT)D ()S (I),

log ra

FIG. 3. Intensity-density graph for the Na5890 line in experi-mental and absolute coordinates. The point of intersection of theasymptotes has the ordinate 0.236 corresponding to a=0.86.

grating monochromator,9 which has a linear dispersionof about 11 A/mm in the first order. Selective filtershave been used in front of the entrance slit in order tominimize the effects of stray light.

The relative intensities have been measured bymeans of a photomultiplier tube. An RCA P21 tubehas been used in the red, and an RCA 1P28 in the shorterwavelength region. The photocurrents are measureddirectly with an RCA WV-84A Ultrasensitive dcMicroammeter. Both phototubes have good linearity ofresponse over a wide range of intensities, as checked bymeans of a rotating sector with variable openings.

For the line intensity measurements, a series ofconcentrations of a salt solution of the element inquestion is successively introduced into the burner,proceeding from weaker to stronger concentrations inorder to avoid cleaning the burner between each twomeasurements. The corresponding relative total lineintensities are determined by measuring the radiationintensity at the wavelength of the line and correctingfor any background radiation from the band systemsemitted by the flame. At higher concentrations, wherethe absorption in the flame is sufficiently large, twoadditional measurements are taken at the wavelengthof the line: the intensity of the radiation from the lampfilament at some known brightness temperature, withand without the flame in the path of the rays.

EVALUATION OF THE EXPERIMENTAL DATA

Let the photocurrents, corresponding to the correctedline intensities, the background source intensity, andthe combined line and background intensity be desig-nated by R B, and RB, respectively. Let D(X) be theeffective wavelength range (in cm) of the radiation fromthe background source, given as the product of theentrance slit width and linear dispersion of the mono-chromator at the wavelength X. The measured back-

9 W. G. Fastie, J. Opt. Soc. Am. 42, 641 (1952).

(10)

where i(X,TB) is the black radiation intensity at the3 brightness temperature TB of the source. S(X,Q) is a

a, proportionality factor, which depends on the geometryof the optical setup, and on the wavelength through theselective reflectances and transmittances in the opticalsystem and the selective response of the photomultipliertube. In a similar manner, the measured line intensityis given by

(11)

where T is the flame temperature and A the totalabsorption of the line, defined in Eq. (1). It is apparentfrom the definition of A that i(X,TB) and io(X,T),besides corresponding to different temperatures, differby a factor c- 2 .

Let it be defined as the ratio of the black radiationintensities io(X,T) to i(X,TB), corresponding to theflame temperature and the brightness temperature ofthe lamp filament, respectively. It can be shown thatn is given in terms of the experimentally determinedquantities as

n=R/(B-RB+R). (12)

Eliminating S(X,Q) from (10) and (11) and solving forA, finally yields

G D(X)RA=-

X2 B(13)

The absorption parameter a for the line underconsideration can now be evaluated. Figure 3 showsthe experimental intensity-density graph for theNa5890 line, logR vs logC, where C stands for theconcentrations of the NaCl solutions, in M/l. The twoasymptotes of the curve are drawn, and the absolutevalue of the ordinate for any one of the experimentalpoints is calculated from (13) and (4). The absolutevalue of the ordinate of the intersection of the asymp-totes is thereby automatically determined, and the avalue of the line can be obtained from (9).

The choice of the location of the asymptotes to theexperimental graph is somewhat arbitrary, but onlywithin quite narrow limits. The error limit of theexperimentally determined a value is estimated to benot more than -l 10%. This estimate is based on trialswith possible locations of the asymptotes, and includeserrors in the measurements of the intensities, and theentrance slit width. The absolute value of the flametemperature is used explicitly only in the calculationof the Doppler width of the line, which is proportionalto the square root of the temperature. Thus, eventhough a depends implicitly on the temperature, anyuncertainty in the temperature measurements is of nogreat importance in the determination of the a value.

At the highest concentrations used, the experimentalgraph in Fig. 3 is seen to deviate slightly from the

154 Vol1. 47

R=Aio(XT)S(XQ),

DETERMINING OPTICAL CROSS SECTIONS

asymptote. The reason for this deviation is twofold:at these high concentrations, (1) the proportionalitybetween the concentrations of the solutions and theoptical densities in the flame does not hold exactly, and(2) the line becomes so broad that the entrance slit isnot quite wide enough for the measurement of thetotal intensity of the line. In case of sodium, theentrance slit width had to be limited to 0.40 mm due tothe proximity of the other component of the doublet.

Aside from the preparation of the solutions, theentire measurement of an experimental intensity-density graph and associated data for the evaluationof the a value can be performed in 15-20 minutes.The only quantities for which absolute values must bemeasured are the width of the entrance slit of themonochromator and the brightness temperature of thebackground source.

For the Na5890 line the foregoing method yieldsa=0.86. From (3b) and (5), using T=2480'K, theoptical cross section is obtained as 64 square angstroms.Since nitrogen constitutes more than seventy percentof the flame gases, this cross section is essentially due tocollisions between the emitting sodium atoms and thenormal nitrogen molecules.

In Table I this result is compared with previousmeasurements of the optical cross sections of sodium byvarious investigators under different experimentalconditions. The result of James and Sugden followsfrom measurements of line intensities emitted by ahydrogen-air flame and the assumption that a= 1 forsodium. This a value is probably quite accurate, sincethe hydrogen-air flame was at a somewhat lowertemperature than that of the present work, and thevalue of a increases with decreasing temperature.Sobolev and co-workers have determined the contoursof the line from an acetylene-air flame by an inter-ferometric method. The remaining three values havebeen obtained from absorption measurements ofpressure broadening of the sodium D lines, performedat comparatively low temperatures and high pressuresof nitrogen.

The intersection of the asymptotes of an intensity-density curve has been shown to occur at r= 4/7r. Ifthe concentration of the solution at this point is Co,the proportionality constant Q between the abscissasof the absolute and the experimental intensity-densitycurves is given by

Q = 4a/irC,. (14)

The value of logCo is obtained directly from the graph.At any concentration C of the solution, the concen-tration of the atoms in the flame capable of absorbingthe line under consideration is, therefore, accordingto (3a,b)

mcA -DQN = ~ C. (15)

2e2fl(1n2)1

TABLE I. Comparison of measured optical cross sectionscorresponding to the Na5890 line. The cross sections are given insquare angstroms.

Author 2

James and Sugdena 68Sobolev et al.b 85Margenau and Watson" 42Minkowskid 59SchUtze 69Present work 64

a C. G. James and T. M. Sugden, Nature 171, 428 (1953).b Sobolev, Mezhericher, and Rodin, Zhur. Eksptl. i Teort. Fiz. 21, 350

(1951).o H. Margenau and W. W. Watson, Phys. Rev. 44, 92 (1933).d R. Minkowski, Z. Physik 36, 839 (1926).o W. Schfitz, Z. Physik 45, 30 (1927).

In case of the Na5890 line, N is the total concen-tration of the free atoms in the flame. With Q= 3.89 -103

from (14), l=2.5 cm, f=0.67, a=0.86 , and AvD(ln2)-= 7.26- 109 sec', this concentration is found to be

N=6.2 -1014 C atoms/cm3 . (16)

This value in itself is of little importance because itdepends on the particular experimental conditions.However, sodium is probably completely dissociated inthe flame. James and Sugden' 0 have found this to betrue even in flames of considerably lower temperature,ranging from 1650'K to 2250'K. It seems reasonableto assume that the total concentration, free or bound,of some other element M in the flame is the same as thatof sodium, at the same concentration of the salt solution.Therefore, if the numerical factor analogous to thatin (16) is found for the free element M under the sameexperimental conditions, then (16) provides the meansof obtaining the degree of dissociation of M in theflame.

On the other hand, if the f value for the line of theelement M is not known, but the element is completelydissociated in the flame, Eq. (15) may be employedto obtain the absolute f value for the line. If the elementis not completely dissociated, this method yields onlya lower limit for the f value. Finally, if the intensity-density curves are measured for two or more lines ofthe same element, the relative f values for these linescan be evaluated regardless of the degree of dissociationof the element.

ACKNOWLEDGMENTS

The topic of this paper has been suggested by Dr.Hedwig Kohn, to whom I am indebted for valuablecooperation and advice throughout this work.

Most of the intensity-density curves shown in Figs. 1and 2 have been constructed from unpublished data.calculated under the direction of Professor F. Reiche(then at the University of Breslau).

15 C. G. James. and T. M. Sugden, Proc. Roy. Soc. (London)227, 312 (1955).

155February 1957