Experimentally-determined oscillator strengths for molecular hydrogen— III. Rotational variation of band strength

$ Quant Spectrosc. Radlat Transfer Vol 14, pp. 723-729 Pergamon Press 1974. Prmted m Great Bntam

EXPERIMENTALLY-DETERMINED OSCILLATOR STRENGTHS FOR MOLECULAR HYDROGEN-III. ROTATIONAL VARIATION OF BAND STRENGTH B. R. LEWlS* Department of Physics, The UmversRy of Adelmde, Adelmde, 5000, Austraha (Recewed 28 December 1973)

Abstraet--Thls paper presents the first unamblgUOUSabsorption measurements whlch verlfy the dependence of band strength on the rotauonal quantum number. A description is made of baghly accurate measurements on the Ro R~ doublet and the P3 line of the (3,0) Lyman band of molecular hydro8en, and it is found that the rotational variation of band strength mdicated by the results ts consistent with the theoretical calculations wRhin the expenmental uncertamty.

INTRODUCTION ALL TI~ oscillator strength results presented in the previous papers of this senes (1,2) were analyzed wlthin the framework o f the Born-Oppenhelmer approximation, no account hawng been taken of any rotation-vibration interaction due for example to centrifugal stretching. In this approximation the band strength p « « , depends only on the vibrational quantum numbers and the oscillator strength of a glven rotational line may be equivalently represented ~by a band oscillator strength, (1) the two bemg mterrelated by the appropriate H ö n l - L o n d o n factor. That these assumptions are good can be seen after an examinaUon of the previously¢l,2) presented osclllator strength results within given vlbratlonal bands. No definite rotational effects can be observed wlthin the experimental accuracy. I f t h e radial Schrödinger equation for a rotating diatomic molecule is solved it is found that the formalism of the Born-Oppenheimer separaUon may be maintamed provided that a shght dependence of the vibratmnal ware functlon on the rotational quantum number is allowed for. (3) As a result all quantities dependent on the vibrational ware function, such as the band strength, have a slmilar J dependence. As noted by VILLAREJOet al., (4) one of the principal effects of thls rotation-vlbratlon interaction is a shift in the potential curve minim u m to larger values of internuclear distance as a function of increasing rotation, and a corresponding shift in the radial positions of the nodes of a given vibratmnal eigenfunction. Thus if there is a relaüve displacement between the upper and lower states, appreciable changes in such overlap integrals as band strength, F r a n c k - C o n d o n factor and r centroid could occur when vibratlon-rotahon mteraction is considered. It has been shown by VILLAR~O et al. t») that the largest effect for molecular hydrogen is expected for the Lyman bands since the X and B potential curves have quite different characteristics. (6) They also note that the magnitude of the above mentJoned displacement is * Present address: School of Physlcal Sciences, The Fhnders UmversRy of South Austraha, Bedford Park, .5042, Australia. 723

724

B.R. LEWlS

inversely proportional to the molecular mass, and hence the largest vibratlon-rotation effeets are to be expected for hydrogen. JAMES(7) notes that these effects are amplified somewhat if the electronic transition moment has a cons~derable dependence on the internuclear distance r. These observations make it clear that, ff such effects are to be measured, the Lyman bands of molecular hydrogen are the obwous choiee. Vibration-rotation effects have been subject to considerable theoretical study over the years (3'a-~2) with molecular hydrogen commanding special attention. (s'13-~6) HALMANN a n d LAULICHT(la) and HALMANNO4) have used R.K.R. potential curves and the FRASER(17) approramation to calculate Franck-Condon factors and r centroids for the (0,0)--(3,0) Lyman bands. VILLARF.JOet al. (»'1 s) have calculated Franck-Condon factors for 450 Lyman bands uslng the adiabatic potential curves of KOLOS and WOLNIEWICZ(~S,l 9,2o) but neglecting the variation of the electronic transmon moment with internuclear distance. The most recent calculations are by WOLNIEWICZ(16) who glves the rotational variation of band strength for the (0,0)-(4,0) Lyman bands including the exphcit dependence on r of the electronic transmon moment. Although experimental intens~ty measurements exlst for some bands of the molecules studied theoretically, no conclusive evidence has been reported showing the aceuracy of the vibratlon-rotation calculatlons. Most of the experimental results deal with emission intensities from flame or discharge spectra, and here the accurate determlnation of number denslties is difficult. The electron energy loss data (21'22'23) dIscussed prewously °) either does not show the rotational structure or is too inaccurate to draw any definite conclusions. These comments have been supported by JAMES(7) who emphasizes the desirability of making accurate intensity measurements in absorption if verlfication of theoretical vabration-rotation calculations IS wanted. This paper presents the results of accurate absorption measurements on the R0 R~ doublet and the Pa hne of the (3,0) Lyman band of molecular hydrogen, the rotational variation of band strength indicated being conslstent with the theoretlcal calculations(~ «~ s,i ~) within the experimental uncertalnty EXPERIMENTAL PROCEDURE In vlew of the quite notlceable variation m band strength predlcted (16) for the low¢r Lyman bands of molecular hydrogen, it was decided to choose two suitable lines from a given band and take a series of accurate equlvalent width measurements for each, hoping to confirm the predicted &fference in band strength. Low lamp intensity, high operating pressure and poor instrument resolution ruled out studies of lines from the (0,0)-(2,0) Lyman bands which would otherwise have proved most suitable. It was decided to sc,an the Pa and the Ro or R 1 hnes of the (3,0) band. In this spectral region there was a high and weil behaved light intensity, the operating pressures were less than 2pro and large regions were available for accurate background determinatIons. A difference in band strength o f 7-8 per cent was expected (16) and this was large enough to detect with the present apparatus. Unfortunately it was not possible to resolve the R0 RI doublet and thus a weighted average band strength would be obtained for this palr. Since a difference In band strength of less than 2 per cent (~6) was expected for the Ro and R~ lines, no large uncertainty was lntroduced by the lack o f experimental resolution. The apparatus used for the eqmvalent wIdth scans has been fully des¢ribed prevlously(~) and IS shown schemaucally in Flg 1. The system conslsted essentially of a &spersed light

Experimentally-determmedoscillator strengths for molecular hydrogen--III monitor output

725

To Differentml Pumpir~

/

pre.omp

I monochromotor

Imo~tor I

•

ceu ldet~'tor

to pump

hr---~,~lv,

to '~ornD Jt ne~:lle •- t.-,..v. Tvalve ~gas

to pump

Fig. I. A schemat~cd]agram of the essentlal paxts of the experimental system. source and a differentially pumped absorption cell with channel electron mult~pher detectors fitted to each end. The light source was a capillary discharge tube operated in a thyratron modulated ¢ondensed &scharge mode to excite the argon resonance lines. The dispersing instrument was a 1-m monochromator operated at 0.38 A resolution with a stepping motor providing the wavelength drive. Admlssion of molecular hydrogen to the absorption cell was controlled by a solenoid operated gas admRtance valve and the pressure was measured with a calibrated ]on gauge. A few modifications were made to enable an improved accuracy to be obtained. The most important of these was the momtoring of the ion gauge pressure reading with a digital voltmeter, the pulsed output of wh]ch was recorded at each channel advance on a spare track of the digital data recordmg system. Thls techmque eliminated the gauge reading error and provided a complete pressure record over the period of a scan. The absorption cell temp¢rature was also measured several t]mes durmg a scan, the resultant average being accurate to withm +_½°K. Using the automatic scanning procedure discussed fully previously, (1J highly accurate transmission scans were done at room temperature on the P3 line and the Ro R1 doublet o f the (3,0) Lyman band, alternated to minimize the effects of any cumulaüve errors on the ratio of the corresponding band strengths. Each scan was composed of a large number o f channels so that very extensive regions were avallable for accurate background determinations. Pressures were used so that measured equivalent widths were below the Doppler saturation region of the curve of growth, tl) Equivalent w]dths were calculated from each transmission scan by a procedure slightly different from that discussed previously. °) In this case a linear least squares fit was made to the background points of every scan and all transm]sslon points were normalized accordingly before an equivalent width value was deduced. DISCUSSION

OF E R R O R S

Because of the small (7-8 per cent) theoretical magnitude of the difference in band strength for the above hnes, a thorough &scussion of the errors involved must be given. As noted m the introduction, m the presence of vibration-rotation interacüon the band oscillator strength has a dependence on the rotational quantum number and thus this term loses its significance as being representatwe of a given vibrational band. However, it is

Q S R T Vol 14 N o 8 - - E

726

B.R. Lewm

convenient to retain the terminology and define the ' band oscillator strength' in the presence of vlbratlon-rotation interactlon as «j, WT(2J" + 1) F',j,, = 11"70 at' S(ko)Pl2oZS~r;,Ä,, (1) This relatlon is obtained from equaUons (8) and (9) of Ref. (1) with the normal oscillator strength symbol being changed to distinguish this from the rotationless case. Here W mA represents the expenmentally measured eqmvalent wldth, T°K the temperature of absorption, etj,, the fractional dlstribution of molecules in the rotational state J" at T°K, S(ko) the dimenslonless parameter discussed previously, t~) P #m the absorbmg gas pressure, l cm the absorption path length, 20 A the wavelength of the line centre and SJ,,A,, cA' the HönlLondon factor appropnate to the rotational line of mterest. When applying this equation to measurements on different lines zt is important to remember that any absolute errors due to pressure calibratlon, McLeod gauge reading and cell length calibratlon will be common to both measurements. In other words the probable absolute error of 5 per cent dlscussed prevlously °) may be lgnored for the purposes of comparmg relative band oscillator strengths measured using the same pressure gange. It must also be assumed that the effective teil length I does not change wlth the pressure of absorbing gas (about 1 /~m for the P3 line, and about 0.2 # for the Ro R1 doublet). This is a good assumption. (24) The linearity of the pressure calibration and the correspondence between the various scales of the 1on gauge were estabhshed accurately by experiment, and, since the digital voltmeter readout of pressure ehminated meter reading errors for a properly calibrated and zeroed gauge, the relative pressure values for the two types of scan were assumed a¢curate to better than 1 per cent. True drffts of pressure dld not cause a problem because of the channel by channel recordmg of this parameter. The cahbraUon of the ion gauge head was not observed to change measurably over the period of the experiment. The HÖnl-London factors are exact quantitles while the errors in wavelength and relative populatlon «j,, are trivial. Assuming a possible temperature reading error of _+0"5°K, it can be shown using equatlon (1) that a neghgible error in band osclllator strength is expected for an Ro or R~ line while an error of only -T-0"5per cent would be expected for a P3 line. Experimentally, the ion gauge readings were corrected for ambient temperature variation.(2,24) The above arguments indicate that the error in the rauo of band oscillator strengths for two lines is completely dominated by the random errors in W/S(ko) for each of the lines. Thus by performing a large number o f sufliciently accurate equivalent width scans the random error may be reduced to the point where a very small difference in band oscillator strength would be detectable. RESULTS Twelve scans of the Pa hne at a pressure close to 1 gm and eleven scans of the Ro R1 doublet at about 0-2 #m were performed. The ambient temperature was close to 21°C for all seans. The resoluüon was 0.38 A and the wavelength increment between channels 3/400 A In the most accurate scan a total o f 515 data channels was reeorded. The band oscdlator strength values calculated via equaüon (1) are given in Table 1 together with the ¢orresponding standard deviations. It should be stresseä that the Ro Rx doublet eqmvalent width values were split between the two lines according to the methods

Expefimentally-determined oscillator strengths for molecular hydrogen--III

727

Table 1. Expertmental band oscillator strengths for the Ps fine and the Re R~ doublet of the (3,0) Lyman band. Relative standard deviations are also shown

Sc.an

F(P3) x 102

1 2 3 4 5 6 7 8 9 10 11 12

2"074 1 763 1974 2 092 1.605 2 018 1 699 1"723 2"016 1 689 1.687 2 019

a(F(Ps)) F(Pa)

~

Scan

6"1 7"8 77 87 10"8 11 1 9"5 10 0 16"3 12 5 21"7 16"6

F(RoR~)x

1 2 3 4 5 6 7 8 9 10 11

102

o(F(Ro RD)

1"559 1"775 1677 1 730 1 623 1 821 1 449 1.991 1"736 1"437 1"519

F(RoR:) 11.1 10"2 97 8-7 11-3 17 4 19 3 17"3 15"4 19-7 14"7

deseribcd previouslytl) which of coursc ignorc rotation-vibration effccts.In other words the band oscillatorstrcngth valuc for the Re RI doublct will rcprcscnt somc weighted average of the true individual Re and R I band oscillator strengths. It is clear from Table I that the band osciIlatorstrength for the Pa line is dcfinitclygrcater than the avcrage for the Re R~ doublct, and when an overall value is calculated using a statisticalwcighting tcchniquc on the mcasurcd valucs of equivalcnt width,t24) it is found that

F(P»)~ =

1-879 x 10 -2,

o = 0-053 x 10 -2,

F(RoR1)oo =

1"672 x 10 -2,

o = 0"063 x 10 -2,

where o represents one s t a n d a r d deviation. I n a d d i ü o n there is a c o m m o n p r o b a b l e absolute error o f 5 per °ent. F r o m these results, u s i n g the quadratic s u m m a t i o n o f errors, it follows that

F(P»)«dF(RoR~)oo =

1.12 _ 0.05.

T h e results o f WOLNIEWICZ (16) are given in terms o f b a n d strengthsp~'ss~, a n d b a n d oscillator strengths m a y be calculated from these b y using the relation t*)

ca'

303.7p~:Ss:. 2:,:.

(2)

where the band strength is in atomic units and the rotational linc wavelength 2:,sù, takcn from Dreie, <2s) is in A. Band oscillatorstrengths for several lines of the (3,0) Lyman band calculated by this tcchnique are givcn in Table 2. The unrcsolvcd doublct thcory introduced previously,cI)and discussed fullyelsewherc,~24) can easily be modified to include the effects of rotation-vibraüon intcraction mcrcly by alt°ring the relative strength factors, and on this basis a relation can be found betwcen the average doublet band oscillator strength and the band oscillator strength of one of the components. This varics with the expcrimental conditions, but in the case of the Re Rx doublet of the (3,0)Lyman band under the average experimental condifions of the resultsgiven above (a total equivalent width of about 14 mA), it can be shown that F(Ro RI)/F(RI) = 1.010, and thus a theoreücal value for F(Ro RI) °an be given in Table 2. The results of

728

B.R. Lewts

Table 2. A compartson of band osofllator strengths for several hnes of the (3,0) Lyrnan band. Stattstlcal standard deviations and probable absolute errors are givea for the present work VILLAREJO

F(Ro) × 102 F(RI) × 102 F(Pa) × 10z F(RoRI) × 102 F(Pa)/F(RoRx)

WOLNIEWICZ( x õ )

HALMANN etc)

e t at,. t~5)

1 719 1-681 1"825 1 698 1"075

----1"073

----1 075

Thls w o r k

1"879, « = 0"053 [ -4- 5 ~ 1"672, ~r=0.063 JABS 1"12, ~ = 0"05

WOLNIEWICZ(16) in fact i m p l y t h a t F(P3)/F(Ro R1) = 1-075, in quite g o o d a g r e e m e n t with the result o f this work. T a b l e 2 also shows s i m d a r b a n d oscillator str,ength r a t l o s d e d u c e d f r o m the F r a n c k C o n d o n factors o f HALMANNO4) a n d VILLAgmO et aL, tls) a s s u m i n g a c o n s t a n t electronic t r a n s i t i o n m o m e n t over the (3,0) b a n d . A l l o f the theoretlcal ratios agree with the results o f this w o r k within the e x p e r i m e n t a l error, wlule the a b s o l u t e a g r e e m e n t between the b a n d oscillator strengths F ( P s ) , F(Ro RI) o f WOLNIEWICZt16) a n d those o f thls w o r k is excellent c o n s i d e r i n g the p r o b a b l e a b s o l u t e e x p e r i m e n t a l e r r o r o f 5 p e r cent. U s i n g the e x p e r i m e n t a l results for F(P3) a n d F(Ro R t ) together wlth a simple i n t e r p o l a ü o n p r o c e d u r e b a s e d on the theoretlcal v a r i a t i o n o f b a n d strength, tl 6) a p r e d i c t e d best value o f b a n d oscillator strength for the P1 hne o f the (3,0) L y m a n b a n d was f o u n d to be

F(PI) = 1.811 x 10 -2,

tr = 0.041 x 10 - 2

this value being necessary for calculations o f the t h e r m a l t r a n s p i r a t i o n r a t l o &scussed m the p r e v i o u s p a p e r in this senes. (2) CONCLUSION T h e results o b t a i n e d in this w o r k , as well as d e m o n s t r a t i n g t h a t v i b r a t i o n - r o t a t i o n intera c t i o n effects a r e quite noticeable in this L y m a n b a n d , also serve to d l u s t r a t e t h a t very high relative a c c u r a c y is available f r o m the a p p a r a t u s e m p l o y e d , p r o v i d e d t h a t sufficient time is s p e n t o n d a t a a c c u m u l a t i o n . W h i l e it is believed t h a t these a r e the first a b s o r p U o n m e a s u r e m e n t s a c c u r a t e e n o u g h to u n a m b i g u o u s l y detect v i b r a t i o n - r o t a t i o n interaction, it is r e c o g m z e d t h a t rauch m o r e w o r k needs to be d o n e in this fiel& It is a n t i c l p a t e d t h a t m o r e extensive results wdl b e c o m e avaIlable o n c o m p l e t i o n o f the 6-m m o n o c h r o m a t o r c u r r e n t l y being designed a n d b u d t in this d e p a r t m e n t .

Acknowledgements--The author is grateful to Prof. J. H. CARVeXand Drs. L TOROP and W. FAmANfor helpful diseusSions during the course of thls work. Fmanclal support for the projeet was provided by the Austrahan Research Grants Committe¢ REFERENCES 1. W. FAmANand B. R. LEwIs, JQSRT 14 (1974). 2. B. R. L~vm, JQSRT 14 (1974). 3. G. I-IvJt~ERO,MolecularSpectraandMolecularStructure. I. SpectraofDiatomie Molecules. Van Nostrand, Princeton, New Jersey (1950) 4. D Vn,LAR~O, R. STOCKSAUERand M G. IsorIRm~, Chera. Phys. Letters 2, 11 (1968). 5. D. Vmi.Agmo, R. STOCICnAUeRand M G. INGX-Igm~,J. chem. Phys. 50, 1754 (1969). 6. T. E. SRARP,Atamic Data 2, 119 (1971). 7. T. C. JAMes,J. chem. Phys 32, 1770(1960).

Expenmentally-detenmnedoscfllator strengths for molecular hydrogen--III 8. 9. 10. 11. 12. 13. 14. 15.

729

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