JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Calculation of the Raman Shift in Vacuum

ASCHER OPLERResearch Department, Western Division, The Dow Chemical Company, Pittsburg, California

(Received February 1, 1951)

A correction table is given to simplify conversion of the wavelengths of Raman lines observed as angstroms(in air) to wave numbers (in vacuum). Two more elaborate tables for direct conversion are described. Alltables were calculated by punched card machines using the Barrell and Sears dispersion formula.

INTRODUCTION

THE term wave number (waves/cm) is preferredT over frequency (waves/sec) and wavelength(cm/wave) in measuring the Raman shift. If measure-ment is made in vacuum, the wave number and wave-length values are reciprocal. However, when measure-ments are made in air, this simple relation no longerholds and is replaced by v= 1/nX, where n, the refractiveindex of air, is a function of wavelength, temperature,and composition.

It is relatively easy to calculate observed Ramanshifts in terms of wave numbers if one neglects to con-vert to the value in vacuum. In that case, it is necessaryonly to obtain the reciprocal of the wavelength (in cm)of the observed line and subtract from the wave numbervalue of the exciting line. If the vacuum values aredesired, recourse is usually had to a table giving thewave number value corresponding to each wavelength.The table of Kayser' is the standard work on thissubject. The usual procedure for converting to vacuumis succinctly given by Stamm,2 "The measurements ofX (A in air) are converted to v (cm-l in vacuum) bymeans of Kayser's Tabelle der Schwingungszahlen andthe zAv's then determined."

Punched card machines have been used to preparetables to simplify this conversion to vacuum. Threetables have been constructed and checked as describedbelow.

SELECTION OF CONSTANTS

Kayser used the dispersion formula for air given byMeggers and Peters.3 Herzberg's4 suggestion to use themore recent values of Barrell and Sears' was followed incalculating the tables below. After the refractive indiceshad been calculated, a recent paper by Meggers andKessler6 came to the writer's attention. In their paperthey indicate that a better value lies somewhere be-

'H. Kayser, Tabelle der Schiwingungzahlen (Hirzel, Leipzig,1925).

2 R. F. Stamm, Ind. Eng. Chem., Anal. Ed. 17, 318 (1945).3 W. F. Meggers and C. G. Peters, Bur. Stand. Sci. Papers 14,

724 (1918).4 G. Herzberg, Molecular Spectra and Molecular Structure,

(D. Van Nostrand Company, Inc., New York, 1950), secondedition, Vol. I.6 H. Barrell and J. E. Sears, Trans. Roy. Soc. (London) A238, 1

(1939).6 W. F. Meggers and K. G. Kessler, J. Opt. Soc. Am. 40, 737

(1950).

tween the values of Meggers and Peters3 and those ofBarrell and Sears.5 However, except for the most precisework, this will not affect the conversion to wavenumbers, as the maximum difference between the twodispersion formulas amounts to only 0.02 cm-l in theregion calculated. For the calculation of differences thisbecomes less than 0.01 cm-l.

The wavelength values of the mercury lines in Table Iare the most recent of Meggers and Kessler 6 and areadjusted to the new provisional primary wavelengthstandard, the green line of 80' Hg. The wavelengths inair are based on observations in dry, CO2-free air at15'C and 760-mm pressure. The wave number values inthis table were calculated by Meggers and Kessler6

using the Barrell and Sears dispersion formula.

CALCULATION OF THE TABLES

The tables were calculated using punched card equip-ment of the International Business Machines Corpora-tion. The steps in the calculation were as follows:

1. By the use of the Barrell and Sears5 formula, (-l)X 10-6was calculated at wavelength intervals of 5A.

2. The vacuum wave number values tz were obtained bycalculation 1/nXa.

3. The wave number values in air () were obtained by calcu-lating the reciprocals of the wavelengths in air (Xa).

4. Values of ,, were subtracted from the corresponding valuesof a at intervals of 20A to obtain a as listed in Table II.

5. Each calculated value of v, was subtracted from the wavenumber of the exciting line. This was done for each of the fiveprincipal visible mercury lines (4047, 4358, 5461, 5770, and 5791A).The calculation yielded Supplementary Table I.*

TABLE I. Wavelengths of some mercury lines in standard air andtheir corresponding wave number values according to the Barrelland Sears dispersion formula (reference 5).

Wavelength Wave numbersin air in vacuum

(angstroms) (cm-')

2536.5064 39412.5113650.1567 27388.2964046.5714 24705.3134077.8379 24515.8914358.3376 22938.0945460.7532 18307.4155769.5984 17327.4335790.6626 17264.403

* Supplementary Tables I and II may be obtained by orderingDocument 3230 from American Documentation Institute,1719 N Street N. W., Washington 6, D. C., remitting $1.00 formicrofilm (images one-inch high on standard 35-mm motion pic-

349

MAY, 1951VOLUME 4, NUMBER 5

ASCHER OPLER

TABLE II. Corrections () in cm'1 for converting Raman lines fromstandard air to vacuum in conjunction with equation.

A cm-' A cm-, A cm-,

2500 12.018 3500 8.167 4500 6.2302520 11.901 3520 8.116 4520 6.2012540 11.787 3540 8.065 4540 6.1722560 11.676 3560 8.017 4560 6.1432580 11.570 3580 7.967 4580 6.1152600 11.463 3600 7.919 4600 6.0882620 11.359 3620 7.871 4620 6.0592640 11.258 3640 7.824 4640 6.0312660 11.159 3660 7.777 4660 6.0052680 11.059 3680 7.732 4680 5.9772700 10.964 3700 7.685 4700 5.9492720 10.868 3720 7.640 4720 5.9242740 10.774 3740 7.596 4740 5.8972760 10.686 3760 7.553 4760 5.8722780 10.595 3780 7.508 4780 5.8462800 10.507 3800 7.466 4800 5.8192820 10.418 3820 7.424 4820 5.7962840 10.335 3840 7.380 4840 5.7692860 10.252 3860 7.339 4860 5.7452880 10.170 3880 7.299 4880 5.7202900 10.090 3900 7.259 4900 5.6952920 10.011 3920 7.221 4920 5.6712940 9.933 3940 7.180 4940 5.6472960 9.855 3960 7.142 4960 5.6232980 9.780 3980 7.101 4980 5.5993000 9.704 4000 7.063 5000 5.5773020 9.632 4020 7.025 5020 5.5523040 9.559 4040 6.988 5040 5.5303060 9.487 4060 6.951 5060 5.5063080 9.419 4080 6.914 5080 5.4853100 9.349 4100 6.878 5100 5.4623120 9.283 4120 6.842 5120 5.4403140 9.216 4140 6.808 5140 5.4163160 9.149 4160 6.772 5160 5.3963180 9.085 4180 6.738 5180 5.3733200 9.023 4200 6.705 5200 5.3533220 8.960 4220 6.671 5220 5.3303240 8.898 4240 6.637 5240 5.3103260 8.837 4260 6.604 5260 5.2873280 8.774 4280 6.571 5280 5.2673300 8.716 4300 6.538 5300 5.2463320 8.659 4320 6.506 5320 5.2263340 8.602 4340 6.472 5340 5.2063360 8.545 4360 6.443 5360 5.1853380 8.489 4380 6.411 5380 5.1663400 8.432 4400 6.380 5400 5.1453420 8.378 4420 6.349 5420 5.1263440 8.326 4440 6.318 5440 5.1073460 8.273 4460 6.290 5460 5.0873480 8.219 4480 6.259 5480 5.068

A cm-'

5500 5.0485520 5.0305540 5.0115560 4.9925580 4.9745600 4.9545620 4.9375640 4.9195660 4.9005680 4.8825700 4.8655720 4.8475740 4.8295760 4.8135780 4.7955800 4.7785820 4.7615840 4.7435860 4.7275880 4.7115900 4.6945920 4.6775940 4.6625960 4.6455980 4.6306000 4.6146020 4.5966040 4.5826060 4.5676080 4.5516100 4.5356120 4.5206140 4.5066160 4.4896180 4.4746200 4.4606220 4.4466240 4.4306260 4.4156280 4.4016300 4.3876320 4.3726340 4.3586360 4.3456380 4.3316400 4.3166420 4.3036440 4.2896460 4.2766480 4.261

A cm-,

6500 4.2486520 4.2356540 4.2226560 4.2086580 4.1956600 4.1836620 4.1706640 4.1576660 4.1446680 4.1306700 4.1196720 4.1066740 4.0936760 4.0806780 4.0686800 4.0566820 4.0446840 4.0326860 4.0196880 4.0076900 3.9966920 3.9856940 3.9716960 3.9616980 3.9497000 3.9387020 3.9267040 3.9157060 3.9027080 3.8917100 3.8817120 3.8697140 3.8597160 3.8477180 3.8367200 3.8257220 3.8157240 3.8047260 3.7937280 3.7827300 3.7727320 3.7617340 3.7517360 3.7417380 3.7317400 3.7217420 3.7097440 3.6997460 3.6897480 3.680

6. Cards were prepared containing wave number values 5, 10,15... 3495, 3500. These were interspersed successively with eachset of cards containing the wavelengths and corresponding wavenumber shifts. A linear interpolation was performed to solve forthe wavelengths corresponding to the constant wave numberintervals. This produced Supplementary Table II.*

7. At least seven figures were carried in all calculations in-volving wavelengths, wave number shifts, or (n-1)X10-6. In the

ture film readable in microfilm reader or under low power micro-scope) or $6.45 for photo-copies (6X8 inches readable withoutoptical aid.)

case of wave numbers, eight significant figures were carried. In thefinal operation, the wavelengths and wave number shifts wererounded to six significant figures.

8. All partial and final results were checked by finding first (andin some cases-second) differences. These were inspected forsmoothness.

9. Errors are assumed to result from rounding in multiplicationand division and from the use of linear interpolation in preparingSupplementary Table II. The maximum errors of the first kindrarely exceed 1 in the sixth place. In the region calculated, linearinterpolation introduces errors of 0.01A or less.

USE OF THE TABLES

Table II is to be used in conjunction with thefollowing formula:

AVv=JVO+-1/Xa,

where A'v, is the Raman shift (in wave numbers) cor-rected to vacuum, a is the correction read from Table II(with interpolation where necessary), and vo is the wavenumber value of the exciting line (given in Table I.)The reciprocal of ;\a may be found from a table or bymeans of a desk calculator.

Example: A Raman line excited by the 4358A mercury line isobserved at 4520A. What is the vacuum Raman shift?

Avv=22938.094+6.201-22123.894=820.401 cm-'.

Supplementary Table I may be used (with linearinterpolation) to find wave number shifts directly whenusing one of the five mercury lines for excitation.

Supplementary Table II may be used in a similarmanner. However, a second use of Table II is in thelocation of corresponding lines where more than oneRaman spectrum is recorded. Overlapping is en-countered in the case of the first two and the last threeof the five lines. As it is difficult to remove all traces ofthe interfering excitation, at least some of the linesobserved may be shifts from interfering lines. As thecorresponding positions of each shift from the fiveexciting lines are shown, it should be simple to confirmthe presence or absence of spurious lines. In fact, it maybe desirable, in conjunction with this table, to photo-graph as much of the five spectra as possible so thatfrequency checks may be obtained at several positions.

350 Vol. 41