JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Experimental Verification of the Relativistic Doppler Effect

HiRSCHI I. MANDELBERG* AND Louis WITTENRIAS, Baltimore, Maryland(Received August 9, 1961)

An experiment has been performed to measure the quadratic Doppler shift, -(1 -v'/c2)- 1, predicted bythe special theory of relativity. A moving beam of radiating hydrogen atoms with velocities ranging up to2.8X108 cm/sec has been viewed from the incoming and outgoing directions simultaneously. Averagingwavelength measurements of a particular spectral line for the two observations gives a measurement of thequadratic shift. The number () in the theoretically predicted exponent can be compared to the experimentalresults. The latter gives the value of this exponent to be 0.498-40.025. The predominant factor leadingto the experimental uncertainty is the width of the spectral lines involved in the measurement.

I. INTRODUCTION

COMPARATIVELY few nonnull experiments ofinterest with regard to the special theory of rela-

tivity have been made. The successful design of high-energy machines which accelerate particles over a rangeof velocities from zero to nearly the velocity of light isa striking confirmation of a nonnull prediction of thetheory, that involving the mass variation. A recentanalysis' has shown that the design and operation of themachines are such that they would work even if therewere a discrepancy of several percent between experi-ment and theory. Recent careful direct measurement"'of the mass-variation law for 385- and 660-Mev protonshave given agreement with theory with an accuracy of0.1 or 0.2%. The agreement between the experimentalmeasurements and the theory of the fine structure ofhydrogen shows that the relativistic mass variation lawwhich is assumed in the theoretical analysis is correctwith an accuracy of 0.05%.'

Another nonnull result of the theory which can bemeasured is the frequency or Doppler shift of spectrallines and its dependence on the square of the velocity(on ,2= v'/c2) together with the associated time-dilationprediction. A series of mi'easurements of the lifetimeof mu mesons moving over a range of velocities wellinto the relativistic region (v/c- 1), when comparedwith the rest-frame lifetime of mu mesons, gives agree-ment with theory within several percent. Atomic beammeasurements by Ives and Stilwell4 and by Otting' havebeen made on the Ha, and Ha9 lines emitted by a beam offast-moving hydrogen atoms. Their experimental ar-rangement is substantially the same as that used in theexperiment we have performed which will be describedin detail below. An analysis6 of the experiments of Ives

* Present address: Physical Sciences Division, 7338 BaltimoreBlvd., College Park, Maryland.

' P. S. Farago and L. Janossy, Nuovo cimento 5, 1411 (1957).2 D. J. Grove and I. C. Fox, Phys. Rev. 90, 378 (1953).3 V. P. Zrelov, A. A. Tiapkin, and P. S. Farago, J. Exptl. Theo-

ret. Phys. (U.S.S.R.) 34, 384 (1958).4 H. E. Ives and G. R. Stilwell, J. Opt. Soc. Am. 28, 215 (1938);

31, 369 (1941).5 G. Otting, Physik, Z. 40, 681 (1939).6H. I. Mandelberg, Ph.D. dissertation The Johns Hopkins

University, 1960 (unpublished). RIAS Technical Report 60-20is substantially the same as the dissertation and is available by

and Stilwell and of Otting indicates that although theirreported experimental points seem to fit the curve withan accuracy of about 2 to 3%, the experimental un-certainty is more nearly 10-15%. By use of the M6ss-bauer effect, a measurement of the quadratic term inthe Doppler shift has been made under conditions whenno first-order shift appears; the velocity involved in theexperiment was rather low ( < 10-6).

We wish to report an experiment whose purpose wasto measure the quadratic Doppler shift by techniquessimilar to those introduced by Ives and Stilwell and toestablish the limits of accuracy of the method. The ex-perimental result is that the exponent in the quadraticexpression for the Doppler shift, (1-1 2)A, is found tobe 0.49840.025.

II. PRINCIPLE OF EXPERIMENT

In this section we shall outline the ideas behind theexperiment. Consider a beam of radiating hydrogenatoms of velocity v observed at an angle 0, near zero.This beam is produced by accelerating hydrogen molecu-lar ions with voltages up to 76 kv or velocities up to2.8X108 cm/sec. These fast molecular ions are thenconverted to fast excited atoms through collisions withstationary hydrogen molecules.

The wavelength of the light emitted from an oncom-ing atom of velocity v viewed at a small angle 0 to thebeam direction by an observer stationary in the labora-tory reference frame is given by

1-3 cosCXB = XO ( cos0-I-OS2 ),

(1-32)1(1)

where 0 is measured in the laboratory frame and Xois the wavelength observed in a reference frame at restwith respect to the radiating atom. The ,B cos0 termin the numerator is the first-order Doppler shift; thesubscript B indicates a Doppler shift to the blue. Thedenominator contains the quadratic shift; the approxi-mate expression is valid in this experiment because ofthe low velocity (<0.01) and the experimental un-certainty. Similarly, the wavelength of the light emitted

wricing to the RIAS librarian. This report contains many detailsof the experiment which are not reported here.

529

MAY, 1962VOLUME 52, NUMBER 5

H. I. MAN DELBERG AND L. WITTEN

in the backward direction, i.e., as seen by a laboratoryfixed observer looking at a receding atom, is given by

1+ cosoXR=XO ( 2 Xo(1+0 2cos0 ) (2)

the subscript R indicating a red Doppler shift. Theaverage of these two wavelengths,

XQ2(XB+XR) =X0 (1+ -! 2 ), (3)

equals the wavelength which would be observed at rightangles to the beam, corresponding to 0=7r/2 in Eqs.(1) and (2). There are many reasons why a perpendicularobservation of the beam is not feasible, the foremostamong these being the difficulty of achieving the ex-tremely accurate alignment necessary to reduce thefirst-order Doppler shift to a size sufficiently smallerthan the second-order effect to make accurate measure-ment of the second-order effect possible.

The experiment consisted in measuring XR and B,the wavelengths observed with and opposite to thebeam, and taking the average to determine XQ and hence

XkO2/2. By a subtraction,

2XDAXR-XB3 2XO cosO, (4)

j3 was determined by measuring XR, XB, Xo, and 0. Thevelocity was thus obtained by direct measurementwithout assuming a precise knowledge of the acceler-ating voltage and without making assumptions regard-ing the collision mechanism which produced the atom.To give an idea of the magnitudes of the parametersinvolved, a typical run was made with an acceleratingvoltage of 63.70 kv which produced a beam of excitedatoms whose measured velocity corresponded to#3=0.008176. For a wavelength Xo=6562.793, we meas-ured 2

D = 107.317 A and X-1

X0#32= 0.219 A.

III. DESCRIPTION OF EXPERIMENTAND APPARATUS

The apparatus for this experiment can be divided intotwo major categories, that for the production and thatfor the observation of the radiating atomic beam. Theapparatus for the production of the collimated beamconsists of an ion source and associated electrical

FIG. 1. Ion source and acceleration chamber: (3) oxide coated filament, (4) capillary, (6) beam exit aperture, (7) probe, (9) arcblock, (11) probe corona shield, (14) anode, (15) accelerating electrode. For all other components, see reference 6.

530 Vol. 52

RELATIVISTIC DOPPLER EFFECT

FIG. 2. Observation chamber: (20) inner observation chamber, (21) beam entrance tube, (22) mask, (23) "Red" mirror, (24)mask, (25) "Blue" mirror, (26) stopping plate, (27) hydrogen inlet tube, (38) inner chamber window, (46) main window. For othercomponents, see reference 6.

circuitry which produced the ion beam (i.e., a powersupply, electrostatic focusing system and circuitrywhich accelerate and collimate the beam from the low-velocity ions emitted by the source) and an observationchamber where the high-velocity ions are converted intohigh-velocity excited atoms which emit Ha radiation.The excitation process is a collision between a fastmolecular ion and a molecule stationary in the observa-tion chamber and can be described by the processes(5) and (6):

fast (H2+)+stationary H2 fast H*+other products, (5)

fast (H3+)+stationary H2 fast H*+other products. (6)

The fast excited atom has almost the same velocity asthe incident molecular ion. Since the excitation mecha-nism is a collision, the pressure in the observationchamber should be high enough to get as many singlecollisions as possible but not high enough to get a sub-stantial number of multiple collisions.

The observation system consists of mirrors and lenseswhich view the beam symmetrically from the beam and

antibeam directions, combines the light from these twodirections, and sends it through a Fabry-Perot etaloncrossed with a grating spectrograph which providesthe requisite spectral dispersion.

The vacuum system in which the atomic beam wasproduced and observed is shown in Fig. 1 (the accelera-tion chamber), Fig. 2 (the observation chamber), andFig. 3 (the electrical connections). The capillary-arc

FIG. 3. Ion beam circuitry.

May 1962 531

H. I. MANDELBERG AND L. WITTEN

FIG. 4. Optical system: (1) 1.5-m Wadsworth spectrograph,(2) 200-micron slit, (3) interferometer plates, (4) 1 mm spacer,(5) temperature controlled interferometer housing, (6) 400-,aslit, (7) monitor photomultiplier, (8) half reflecting mirror,(9) axis correction plate, (10) prism, (11) central masks, (12) mainvacuum system window, (13) "red" mirror, (14) "blue" mirror,(L,) 152 mm focal length f/3.5 lens, (L2 ) 135 mm focal lengthf/3.5 lens, (L,) 60 mm focal length f/5.6 lens, (L,) 90 mm focallength f/6.8 lens.

type of ion source (No. 1-14 of Fig. 1) was borrowedfrom Allison. He has described elsewhere7 the detailsof construction and operation of this ion source. Afterbeing produced by the ion source, the beam is acceler-ated to the full voltage of the high-voltage power supplyby electrode 15. A typical ion-beam current is 100 pa.The beam passes into the observation chamber (Fig. 2).It then passes through tube 21 into the inner observa-tion chamber, whose only opening to the outer chamberis through this tube. The ion beam then passes throughmask 22 which protects the mirror MR, 23, from thedirect impact of the beam. The beam goes through an0.093-in. diameter hole in MR, then through mask24, and through an 0.093-in. diameter hole in MB, 25.Hydrogen is admitted into the inner observation cham-ber through the tube 27. The pressure inside thischamber is thereby maintained at about 400 u of Hg.Collisions of the fast hydrogen ions with stationarymolecules excite the ion beam to a bright glow. Doppler-shifted lines are observed corresponding to hydrogenatoms produced from H2+ and H3+, respectively, andmoving with velocities approximately equal to(2eV/2mo)i and (2eV/3mo)1, where V is the acceleratingpotential and mo is the mass of a proton. Fast pumpsmaintain the acceleration chamber pressure at 2X10-4mm Hg.

The beam of radiating atoms is then observed bymeans of the optical system shown in Fig. 4. Lenses L1and L2 focus the light from each direction from a pointmidway between MR and MB, through a prism Pand half-reflecting mirror Ml onto a 400,gc slit in frontof the Fabry-Perot etalon. The axis-correcting platecorrects for the lateral displacement of the optic axisof the lens L2 caused by refraction through the 4-in.thickness of M-. The light is collimated by lens L, andpassed through the etalon which has a 1-mm spacer andwhose plates have a silver coating with a half-wavecoating at Ha of SiO. This gives each plate a reflectanceat Ha of 70%; and the interferometer, an interferometerefficiency [TI(T+A)] 2 of 75%; T being the transmit-tance and A the absorptance of each reflecting film.

7 S. K. Allison, Rev. Sci. Instr. 19, 291 (1948).

The interferometer fringe system is focused, togetherwith the image of the beam, on the 200-u slit of a 1.5-mWadsworth spectrograph. This system provides a lineardispersion on the photographic plate which is approxi-mately ten times that of the 1.5-m Littrow mountspectrograph which had been used by Ives and Stilwell.The optical system is such as to magnify the 2-mm beamdiameter to cover a 10-mm height on the spectrographslit which is sufficient to give five complete interferencefringes for measurement.

The main power supply, which generates the highelectrostatic potential with which the beam is acceler-ated, was designed and built on special order by the NJECorporation, Kenilworth, New Jersey (their model No.46H59). The output voltage can be varied continuouslyfrom 10 to 80 kv and is stable to =0.007% over a 24-hrperiod with a negligibly small hum and noise level andwith an output current up to 10 ma. A tap on themain voltage divider is provided that produces a smallvoltage (3 v max) which can be used for monitoring thepotential. This voltage was monitored during a run bya John Fluke differential dc voltmeter having a accuracyof 0.002%. The monitoring tap and differential dc volt-meter permitted the high-voltage output to be set beforethe run to within 0.1% of the desired value and alsopermitted checking the stability of the power supplywhich remained within the specified tolerance of4+ 0.007% over a 24-hr period.

IV. APPARATUS DESIGN CONSIDERATIONS

Some of the special points of interest in the designof the experiment are briefly outlined.

A. Velocity or Voltage Stability

A variation of the velocity of the atoms in the beamwould produce a variation in the first-order Dopplershift XD, which would shift the spectral line considerably.Obviously XD-q3 or XD- V1, where V is the acceleratingpotential. Hence

5XD/XD= 25V/V. (7)

For the power supply used 3V/V=7X10-5; hence18X1 <0.002A for XD=6 0A (the maximum attainedin the experiment). Therefore, 0.002 A was arbitrarilyset as the limit within which other errors should be kept.

B. Alignment Errors

As can be seen from Fig. 4, the observation angle a

varies from almost zero to the angle subtended by theoutermost portions of the focusing lenses L, and L2.Because of the cylindrical symmetry around the beamaxis, all the light emitted by the beam in the angularrange of interest is intercepted by the optical system.To determine the precision required in the alignmentof the optics, it is sufficient, because of this symmetryabout the beam, to consider the effect of a misalignmentof a zero-aperture optical system, which ideally is

532 Vol. 52

RELATIVISTIC DOPPLER EFFECT

aligned along the beam. For this model, XD-XD(0=0)=Xo#3. If now the system is misaligned by a small angle0, XD=XO/ COS02XD(1-02/2), and the error caused bythe misalignment is (XD)02. If this error is to be keptbelow 0.001 A, the resultant condition on the align-ment error is that 0<0.005 rad.

C. Collimation Effects

The atomic beam is not a perfectly parallel beam buthas some angular divergence. However, because of thecylindrical symmetry of the beam and viewing mirrors,MR and MB, and because of the symmetry of the opticalsystem about the common focal point of the lenses, L1and L2, the only effect of the divergence is to increasethe effective viewing angle or aperture of the opticalsystem. No asymmetry of aperture between the beamand antibeam observations is introduced. Measure-ments of the beam divergence were made with a cathe-tometer as the beam traveled down the observationchamber. The measured divergence was about 0.25 degwhich agrees with estimates that can be made fromexperiments using counting techniques.8

D. Stark Effect

Another factor to consider is the shift in the wave-length of the emitted line caused by the Stark effectarising from the fact that the radiating atoms are in anelectric field caused by the ion beam. The fields en-countered can be estimated from a knowledge of thebeam current, radius, and velocity and turn out tobe of the order of 10 v/cm. The shift in the wavelengthof the Ha lines due to quadratic Stark effect is much lessthan the 0.002 A error tolerance and can be ignored.

E. Aperture Effects

Most of the effort in the design of the optical systemused in the experiment went into ensuring that apertureeffects would introduce no significant errors. The im-portant point here is that the major contribution tothe linewidth is that due to the variation in B and XR

caused by the variation in 0 due to the finite size of theaperture. The variation in observed wavelength goesas cos0, which is an even function of 0; i.e., a largeraperture will give a broader line, and its width will in-crease only toward smaller values of R and largervalues of B, thereby causing a shift in the apparentcenter of the lines. For this reason the angular aperturesused in viewing the light from the beam and antibeamdirections must be closely matched. If this is not done,the asymmetry of first-order Doppler shifts from aper-ture asymmetries could cause large errors in themeasurement of the quadratic effect. Now let us con-sider the required symmetry. Since XD(minimum)= X0XcosOmax, d(XD min) ~Xo#fOmaxdmax for small max

8 D. R. Sweetman, Proc. Roy. Soc. (London) A256, 416 (1960).

(maximum viewing angle). An error of 0.004 A is al-lowable here for the shift of wavelength at the edge ofthe line; this corresponds to a shift of the center of0.002 A.

For the optical system used in this experiment, as-suming that the limiting stop of the entire optical systemis at the spectrograph collimating mirror, it can beshown' that the maximum viewing angle fOmax, for eitherlens L1 or L2 is given by

Omax= (A/2)(f/fo)m.

A is the spectrograph aperture, f and f are the focallengths of the interferometer projection and collimationlenses, respectively, and m is the lateral-magnificationfactor of L1 or L2 for projecting an image of the beamon the interferometer slit. Therefore, even though L1and L2 must have different focal lengths because of themechanical arrangement, it is only necessary that theirrespective lateral-magnification factors be matched.For the tolerances allowed here, it is necessary tochoose focal lengths and distances such that the magni-fication factors of L1 and L2 (m; 3) will be matched to1%.

Another important point in the design of the opticalsystem is the requirement that every point on the imageof the beam upon the spectrograph slit be illuminatedby light from the same angular range with respect tothe beam axis. Because of the cosO dependence of theDoppler shift, any variation in angular range wouldcause a variation in wavelength along the spectrographslit, thereby rendering the interferometer fringes mean-ingless. An analysis 6 will show that this requirement canbe satisfied by the proper choice of the distance betweenthe interferometer collimation and projection lenses,and is critically dependent upon this distance. For thisreason the collimation lens was mounted in a movablecollar, a light source was set up inside the spectrographand the collimation lens was adjusted until the effectiveapertures of the focusing lenses were independent ofslit height. The change in the magnification factors ofL1 and L2 because of this motion is a second-order effectand has no significant effect on the magnification-match-ing condition.

Masks were inserted in the focusing lenses to elimi-nate the light originating from the central 5-mmdiameter of the internal mirrors because the beam de-stroyed this portion of the surface of the blue mirror(25, Fig. 2) after a short time. This portion of the bluemirror and the identical portion of the red mirror hadto be masked off to preserve the angular symmetry.Finally, the depth of field of this optical system, dueto the high longitudinal-magnification factor of 20,was less than 1 mm; therefore no variations in 0 maxccould arise from a large depth of field.

V. EXPERIMENTAL PROCEDURE

The internal mirrors (MR and MB) were alignedwithin 0.002 in. and the external optics were aligned

May 1962 533

H. I. MANDELBERG AND L. WITTEN

to the smaller of 4-0.003 in. or 0.001 rad. The data runswere taken over a period of 6 months. Immediatelyprior to each run the alignment of the entire system waschecked by using a light source in the spectrograph.Referring to Fig. 4, it is obvious that in this configura-tion L, and L2 have mutual focal points. Therefore theimage of a light source in the focal plane of L0, i.e., atthe collimator slit, will be focused back onto itself,provided the system is in perfect alignment. In checkingthe alignment, such a light source was provided byimaging a light source inside the spectrograph ontothe collimator slit. If the system is misaligned in somefashion, the image as focused back in the collimatorfocal plane will not be in register with its object andthis lack of register can be visually detected. It can bedemonstrated, for the optical system in use, that thismethod can detect an error of 0.001 rad in the centralviewing angle. This provides an adequate verificationof the alignment of the optical system.

Data runs were taken for operating voltages between24.50 and 76.00 kv. The voltages were chosen so as toallow 4+0.4 A clearance between the "beam" lines(Doppler-shifted Ha lines) and the lines of the hydrogenmolecular spectrum. Some of the resultant "allowed"voltage ranges, where none of the four (red and blueshifted mass 2 and red and blue shifted mass 3) beamlines were within 0.4 A of a molecular line, were rathernarrow. This necessitated that the accelerating voltagebe known fairly accurately. As has already been re-marked, by calibrating the output voltage against thevoltage from a low voltage (1-3 v) tap on the power-supply divider chain, it was possible to set the acceler-ating voltage to +0.1%, by the use of a precision(0.001%) voltmeter.

After checking the optical-system alignment, andstabilizing the beam operation, reference exposureswere made with a neon discharge tube. The exposurewas then made on Kodak 103aE plates, varying in timefrom 5 to 25 hr, depending on the light intensity.Another reference exposure was made following therun, after which the system alignment was againchecked. During the exposure the interferometer-hous-ing temperature was maintained at a constant valuewithin 4t0.01C; the housing was sealed against at-mospheric pressure changes. The limitation on theexposure time arose from the fact that stray atoms orions struck the active part of the blue mirror. Therewas not enough intensity to destroy the mirror surfacerapidly, as was the case in the central part of the beam,but these atoms did cause a polymerization of a sur-face layer of hydrocarbons from the vacuum grease andbackstreaming pump oil. Consequently, a carbon layerbuilt up on the mirror which reduced its reflectivityto about 5% in 24 hr of running time. The carbon layerwas assumed to be uniform over the very small regionat the center of the mirror whose reflected light enteredthe interferometer; hence the angular distribution of thelight was assumed to be unchanged by the layer. In

FIG. 5. Sample of photographic plate. Reference is attop, beam spectrum at bottom.

about 75% of the runs a usable density was present onthe photographic plates after this exposure time. Theblue mirror was then replaced for another run.

A typical plate is shown in Fig. 5. This shows thesection around Ha, which is the region of interest. Theupper spectrum is the neon reference and the importantlines are indicated. The lower spectrum is that of thebeam. The broad central line, its width comparable tothe interferometer free spectral range, is seen. TheDoppler-shifted lines, which were the ones measured,are marked in pairs with the mass of the molecularion to which their velocity corresponds indicated. Notethe presence of mass 2, 3, and 19, the 19 belonging toH3 0+. Velocities corresponding to proton accelerationwere observed very weakly on some plates; these were,however, too weak to measure accurately. The lowintensity of these lines can be attributed to two factors;a low percentage of protons in the beam, and a lowcross section for electron capture as compared to thecross sections for the molecular collisions. The othersharp lines are molecular hydrogen lines. By directcomparison of two plates taken at different voltages, itwas determined that there was no shift in these molecu-lar lines. Since the cross section for the production of fastneutral molecules from the molecular ions of H2+ isstill one-fourth that for the production of fast atoms,8

the conclusion to be drawn is that the fast moleculeswhich result from the collision are not in excited states,while many of the atoms are. We have found no othermeasurements that can verify this point.

VI. MEASUREMENTS AND CALCULATIONS

The plates were measured on the photoelectric settingcomparator (of the type described by Dieke, Dimock,and Crosswhite9 ) at the Johns Hopkins Universityspectroscopy laboratory. It has an accuracy of +0.25 u.The neon reference lines measured were those at

I G. H. Dieke, D. Dimock, and H. M. Crosswhite, J. Opt. Soc.Am. 46, 456 (1956).

Vol. 52534

RELATIVISTIC DOPPLER EFFECT

6678.276, 6598.953, 6532.882, 6506.528, 6402.246,6382.991, and 6266.495 A. The fringe diameters of theselines were treated by the least-squares method describedby Meissner"' to get the fractional order of interferencee of each pattern. The method of exact fractions'was then used to find the exact order of interference foreach fringe pattern. For the closest match between thecalculated order of interference and the measured valueof e (averaged over 4 measurements of the fringepattern for each wavelength: 2 on the pre-exposurereference plate and 2 on the post-exposure referenceplate), there was a difference which increased withdecreasing wavelength. This can be attributed to aphase-shift dispersion with wavelength caused by theSiO protective layer over the silver. The difference be-tween these two numbers, or the phase correction factor,was plotted for all the data runs which were usableand a phase-correction curve obtained. A phase correc-tion from this average curve was then applied in thecalculation of the wavelengths of the beam lines.

The wavelengths of the Doppler-shifted Ha lines werecalculated by the method of Crosswhite," which utilizeseach fringe diameter to get a wavelength determination.This method weights no individual reading more thanany other as contrasted with other calculation methods.Hence, a more reasonable idea of the uncertainty inthe wavelength measurement can be derived. Themethod is based on the following considerations. It caneasily be shown from the fundamental interferometerequation that

X= (2t-CD 2 )/n. (6)

The interferometer spacer thickness t and the constantC can be derived from the reference line data, and theorder of interference at the fringe center n can be ob-served from an approximate knowledge of , the wave-length at the fringe center. D is the fringe diameter.The approximate value of X is taken from the calculatedvalue by assuming the relativistic shift to be correct.

TABLF. I. Calculation of quadratic Doppler shift fromdata run 15, 68.75 kv, mass 2.

Measured quantities:XR=6618.808±40.0075 AXB = 6507.25340.0041

cosO=cosO.075 rad =0.9972

Predicted quadratic Doppler shift:2

XD=2

Xo COSO=XR-XB= 111.55540.0122Xo,6= 111.86840.012#measured = 0.008 5 2 29 O0.000000101XoI3)predicted = 0.23840.0004

Measurement of (Q ofl2):

XQ= 2(XBX+R) = 6563.031i0.006X)=6562.793 (Taken from literature)(2Xof2)measured =XQ-Xo=0. 2 3 8 O.OO6

10 K. W. Meissner, J. Opt. Soc. Am. 31, 410 (1941).11 H. Crosswhite, The Johns Hopkins University Spectroscopic

Report No. 13 (1958) (unpublished).

A.250

.200

8)X RELATIVISTIC2X.R .

.150

.100i

.050

- MASS PRESENT RESULTS

* IVES AND STILWELL+ OTTING °

rMOSSBAUER EFFECT| MEASUREMENTS: 1

-I=7 X10' v /. o

I oil

AX DOPPLER=2X.8 0 2O

8 0 .001

V(MASS 2)KILOVOLTS 0

4,0 160 8,0.093.0041 .0061

10 20 40

ooI 21 140 A.008 T .010

60 80

FIG. 6. Results-Relativistic shift as function of first-orderDoppler shift and of fl. Results of this experiment are shown to-gether with previous results.

This introduces no circularity of argument in thecomputation, as the assumed value need be accurateto only 1 A. From this, an approximate n can be deter-mined and an unambiguous assignment of the exactorder of a given fringe easily made. To this n the phase-correction factor derived from the appropriately pre-pared curve is added to obtain the true order ofinterference.

The fringe system for each beam line was measured4 times to achieve good statistics. Since 4 or 5 fringeswere measurable for each line, a total of 16 or 20 wave-lengths were calculated. The average of these was de-termined together with the corresponding uncertainty.The corresponding pair of wavelengths were then treatedby Eqs. (3) and (4) to arrive at the measured valuesof the relativistic shift. The cosO term correspondingto the mean viewing angle of the lenses L and L2 wastaken to be 0.9972. A sample of this calculation is shownin Table I.

The major uncertainty in the beam wavelengths isdue to the width of the beam lines themselves. As canbe seen from Fig. 5, the width of the lines is about3 of the free spectral range, which is 2A. The lines aretherefore approximately 0.6 A wide, which precludedresolving the doublet nature of Ha. It is possible tolocate the center of these to within about 1% of theirwidth, i.e., an uncertainty of about hE0.006 A is to beexpected. This is the approximate value of the uncer-tainty observed. The uncertainties introduced by syste-matic errors in alignment, and other causes have beendiscussed earlier and are small compared to the un-certainty caused by the linewidth. The uncertaintyin the spacer thickness t is estimated to give a wave-length uncertainty of 4-0.002 A. The net uncertaintyin this measurement is therefore 0.007 A.

The results of this experiment are shown in Fig. 6.The solid line represents the theoretical prediction, thedata obtained are indicated as shown. All the platesmeasured are shown here. None of the deviations issignificantly greater than the measured uncertainty of

May 1962 535

H. I. MANDELBERG AND L. WITTEN

the individual measurements. The standard deviationof the entire group from the theoretically predictedresults is 0.009 A, which is in agreement with the wave-length uncertainties. Since the precision of the measure-ments is constant in angstroms for all the shifts, thepercentage uncertainty is smallest for the higher veloci-ties. It is approximately 3% for the highest velocitiesmeasured, up to 16% for the lowest. In the region de-scribed approximately by 0.0045 <# <0.0065, lie eightconsecutive points we have measured, each of whichlies below the theoretical line. However, in this region,most of the points measured by Ives and Stilwell lieabove the line; taking the experiments together givesa reasonable spread of measured points about the theo-retical line in this velocity range.

According to the theory,

5Xrelativistic = KX 012; K= 2 .

A least-squares calculation of the coefficient K was madefrom the experimental data. The result was K = 0.498

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

+:0.025. This implies an over-all precision in this experi-ment of 5%, the limit on the accuracy being imposedby the width of the beam lines. It is not possible tolocate the center of these lines significantly more ac-curately than has been done here. The other errors,resulting from faulty alignment, aperture effects, voltagevariations, etc., are small with respect to the uncertain-ties arising from the line width; they add only slightlyto the experimental uncertainty.

ACKNOWLEDGMENTS

The authors are grateful to Professor S. K. Allisonfor lending them the ion source and for the valuableadvice he gave in the design of the beam accelerationand observation chambers and to Professor G. H.Dieke for his advice and guidance in the design andoperation of the optical measurement system. They alsowish to thank Professor H. M. Crosswhite, ProfessorD. E. Kerr, and W. Fastie for helpful discussions andsuggestions.

VOLUME 52, NUMBER 5 MAY, 1962

Secondary Standards of Ar I and Hgl98 I in the Near Infrared*

EDSON R. PECK, BAIj NATH KHANNA, AND N. C. ANDERHOLMNorthwestern University, Evanston, Illinois

(Received September 28, 1961)

A final report is given on wavelength measurements in air of eleven lines of Ar I and three of Hg198 i in thenear infrared, for use as secondary standards. Measurements of a preliminary character on three other lines

are also included. These measurements have been made in the course of several years' time by means of a

corner-reflector interferometer of the Michelson type with reversible fringe counter and static fringeinterpolator.

COMMISSION 14 of the International AstronomicalUnion stresses the need of numerous secondary

standards of wavelength in the near infrared.' Itdefines class B secondary standards as those known to0.001 A or better. Concordant measurements by atleast two laboratories are required for formal adoptionof these standards. The spectra of Ar i and Hg'98 i havereceived considerable attention for this purpose be-cause of their ease of excitation, sharpness, and freedomfrom structure. Photographic measurements of Ar iwere made in 1934 by Meggers and Humphreys. 2 Afew lines of Ar i were measured in 1953 by Burns andAdams.'

Extensive work in both spectra has been reported

* This work has been supported by a research grant from theNational Science Foundation. It was begun with the help of threegrants from the Research Corporation of New York.

' Draft Report of Commission 14 of IAU for meeting of August,1961.

2 W. F. Meggers and C. J. Humphreys, J. Research NatI. Bur.Standards 13, 293 (1934).

3 K. Burns and K. B. Adams, J. Opt. Soc. Am. 43, 1020 (1953).

by Humphreys and Paul.4 5 They used a Fabry-Perotinterferometer whose pattern is scanned by a slit.Littlefield and Rowley6 have measured the Ar I spec-trum relative to the Kr-86 primary standard, by usinga reflecting echelon. Rank, Bennett, and Bennett7 useda Fabry-Perot interferometer with variable air pressurein measuring some infrared lines of Hg'9 8.

The present paper is the report of work which hasbeen done at Northwestern University over a periodof several years by means of a corner-reflector inter-ferometer of the Michelson type. This instrument andits manner of use have been described in severalpapers.8 Preliminary results were quoted in a paper

4 C. J. Humphreys and E. Paul, Naval Ordnance LaboratoryReports 390, 429, 443 (1958); 464 (1959).

6 C. J. Humphreys and E. Paul, J. phys. radium 19, 424 (1958).6 T. A. Littlefield and W. R. C. Rowley, draft report of Com-

mission 14 for meeting of August, 1961.7 D. H. Rank, J. M. Behnett, and H. E. Bennett, J. Opt. Soo.

Am. 46, 477 (1956).8 E. R. Peck, J. Opt. Soc. Am. 38, 66, 1015 (1948); 45, 795,

931, (1955); E. R. Peck and S. W. Obetz, ibid. 43, 505 (1953).

536 Vol. 52