PHYSICAL REV JEW VOLUM E 79, NUM 8 ER 4 AUGUST ls, l9so High Enrgy Elastic Scattring of Elctrons on Protons M. N. RosKNBI.UTH Stanford Univrsity, Stanford, California (Rcivd March 28, 1950) Th thory of th lastic scattring of lctrons on protons at vry high nrgis is discussd in dtail. A formula is givn for th cross sction. This formula contains crtain paramtrs which dpnd on th action of th virtual photon and mson 6lds. In particular, curvs hav bn calculatd on th assumption of scalar and psudoscalar mson thory. awhil ths prturbation thory calculations ar not vry trustworthy, and th rsults dpnd on th choic of coupling constants, it is flt that qualitativ faturs can b chckd with xprimnt. It is concludd that at low rlativistic nrgis (E(50 Mv) th xprimnt provids a valuabl chck on quantum lctrodynamics. At highr nrgis it should yild data on th natur of th mson cloud of th proton. I. INTRODUCTION HE Stanford linar lctron acclrator program is xpctd to mak availabl larg currnts of rlativistic lctrons with various nrgis ranging from 6 to 1000 Mv. Among th xprimnts of considrabl intrst which may thn b prformd is th lastic scattring of lctrons on protons. This may b don on a hydrogn gas or liquid targt. Dspit th srnallnss of th cross sction at high nrgis, th xpctd larg intnsity of th bam should rndr th xprimnts possibl. It is th purpos of this papr to show that, at appropriat nrgis and angls, th xprimnt should giv considrabl information both about th validity of th "quantum lctrodynamical radiativ corrctions" to scattring, and about th structur of th mson cloud associatd with th proton. Procsss compting with lctron-proton lastic scattring can b groupd into two classs: (a) thos arising from lctron-lctron intractions; (b) othr lctron-proton procsss. Th lctron-lctron intractions hav a much largr cross sction at high nrgis than th lctron-proton intractions. Background from th lctron-lctron intractions may b liminatd by (1) angular coincidncs btwn th scattrd particls, (2) nrgy slction of th scattrd lctron at a givn angl [or a combination of (1) and (2)], or (3) dirct obsrvation of th rcoil protons by photographic plats. Th compting lctron-proton procsss ar brmsstrahlung and mson production. Thy will hav cross sctions comparabl to th corrctions to th lastic scattring which ar discussd blow. Mthods (1) and (2), discussd abov, would also liminat background from ths procsss. If th proton is obsrvd dirctly a dtrmination of its nrgy by grain counting and a corrlation of nrgy and angl could b usd to liminat ths procsss. At vry high nrgis it may prov xprimntally impossibl to sparat th digrnt lctron-proton procsss, in which cas th brmsstrahlung and mson production must b addd to th lastic scattring which is calculatd in this papr. II. ELASTIC SCATTERING OF AN ELECTRON AND PROTON Th lastic scattring of an lctron and a proton can b rprsntd schmatically on a Fynman' diagram as in Fig. 1. Figur 1 shows a proton of 4-momntum pl and an lctron of 4-momntum p2 xchanging a virtual quanturn of 4-momntum q= p3 pi= p2 y4 and bing scattrd to momnta p3 and p4, rspctivly. Hr M is th proton rst-mass', " is th ffctiv charg of th lctron, ' th ffctiv charg of th proton, and ~''/2M its fi'ctiv anomalous magntic momnt. Th Hctiv FlG. 1. Diagram for th lastic scattring of a physical proton and a physical Pp lctron. (Th lttr "q" with th bar through it in this Pl figur is th sam as rc'' %~pa &(A' th Grman lttr, q, usd in th txt. ) chargs and magntic momnts ar functions of q'= q4' g3' q2 qi'- as discussd blow. Th notation of a Grman lttr, q, mans q4p4 q3p3 q»2 q&p&, whr th y's ar givn in trms of th usual Dirac matrics by (v, v4) = (P~, 8) Th cross sction for this procss is computd by standard spur tchniqus to b '-' ('" q 8 8 1 +(E2/ )Msin'(8/2)+(8'-/M')[2(1+~')' tan'-(8/2) sin"-(8/2)+" sin'(8/2) j~ dn=1 ctn'- csc'- dq. (1) 1 &2E i 2 2 (2E/M) sin'(8/2) ]'- 1[1+ ' R. P. Fynman, Phys. Rv. 76, 749 and 769 (1949). Th mthods of calculation and th notation usd in this papr ar just thf. s of Fynman unlss othrwis indicatd. % also us natural units, h=c=1.
616 M. N. ROSENBLUTH Hr E is th nrgy of th incidnt lctron and 8 th angl through which it is scattrd, both as masurd in th systm whr th proton is initially at rst. Th rst-mass of th lctron has bn nglctd compard to its nrgy. K also introduc th usful paramtr 4E' sin'(/2) (2) 1+2(E/M) sin'(8/2) For lctron nrgis small compard to th proton rst-mass Eq. (1) rducs just to th usual Mott- Ruthrford formula for scattring of an lctron by a fixd lctrostatic potntial. Sinc ~', th ffctiv anomalous proton magntic momnt, may b largr than 1 (for q=o, K = K0=1.79) th magntic momnt (0) ( c) (d) FIG. 2. Diagram showing th ffct of a virtual chargd psudoscalar mson on lctron-proton scattring. will play an important rol in th vry high nrgy rgion. That th ffctiv lctron charg " diffrs from its "natural" valu is du to th so-calld radiativ corrction to scattring, i.., to th possibility that th lctron may mit and rabsorb a virtual quantum, or mit a low nrgy ral quantum, during th scattring procss. This modification has bn tratd xtnsivly by Schwingr. ' His formula is valid undr th assumption that th proton acts as a fixd lctrostatic potntial. This is th cas in th low nrgy rgion in which this is th most important corrction trm. At highr lctron nrgis, th mor xact xprssion could b drivd by a modification of th radiativ corrction to Mgllr scattring, which has bn calculatd at Cornll. ' Hr w rstrict ourslvs to th rmark that th Schwingr corrction is a slowly varying function of angl and nrgy and corrsponds to a dcras of th ordr of magnitud of fiv prcnt in th ffctiv lctron charg for th rgion of intrst. ' Strictly spaking, w should also giv th lctron an anomalous magntic momnt, but this is quit small and dcrass rapidly at high nrgy. 3 J. Shwingr, Phys. Rv. 76, 813 (1949). In our notation ("/)~= ~ whr 5 is givn by Schwingr in his Eq. (2.105). 4 R. P. Fynman (privat communication). IG. MESON FIELD COMkECTIONS TO ELECTRON-PROTON SCATTERING Th modification of th proton charg and anomalous magntic momnt is hr assumd to b causd by th action of a virtual mson fild. At lctron nrgis small compard with th mson rst-mass ths modifications will b small. Thus, at low nrgis th scattring will giv us information concrning chifly th radiativ corrctions to scattring; at highr nrgis w may xpct to larn somthing of th natur of th mson cloud which surrounds th proton. Th action of a scalar mson fild in modifying th fi'ctiv proton charg and magntic momnt can b undrstood qualitativly by assuming that during a fraction R of th tim th proton xists as a nutron and a positiv mson. Its charg and anomalous momnt thn will b sprad out lik ~/r' (th squar of th mson wav function) whr p is th mass of th mson. Thus a high nrgy lctron is abl to pntrat th mson cloud and hnc s a smallr ffctiv charg and magntic momnt. Undr ths assumptions Schiff' has givn th ffctiv charg and magntic momnt to b ('/) = [(1 R)+R2p/( q') & tan( q') l/2pj, (3) (z''/kp) = 2p/( q') & tan '( q') &/2 p, whr q' is givn by Eq. (2). % will hr calculat in th covariant mannr of Fynman th ffctiv charg and magntic momnt of th proton as givn by four thoris: Nutral and chargd scalar msons with scalar-coupling, nutral and chargd psudoscalar msons with psudoscalar coupling. Th rsults for symmtrical thoris may b obtaind simply by adding th rsults for chargd and nutral thoris. Othr mson thoris lad to divrgnt rsults. To illustrat th mthod, w will discuss brifly th cas of chargd psudoscalar thory. Th ffct of th virtual msons on th scattring is shown in Fig. 2. Figur 2(a) shows th usual lctromagntic intraction btwn two Dirac-typ particls of charg and ". Figur 2(b) shows th proton mitting a positiv mson which absorbs th virtual photon and is thn rabsorbd by th nutron. Figur 2(c) shows th mson bing mittd and rabsorbd bfor th scattring taks plac. Figur 2(d) shows th virtual mission and rabsorption taking plac aftr th scattring. Hr g is th mso-nuclar coupling constant; =ip&p2p3p4, th factor 2 is insrtd for simplicity in latr discussing symmtrical thory; and th 2k +q at th mson-quantum vrtx rflcts th fact that a Klin-Gordon particl intracts with th lctromagntic 6ld through th trms ib(a Q)/Bx +id BQ/Bx whr P is th mson wav function, and 3 th lctromagntic potntial. W ndavor to show that adding th diagrams 2(a) to 2(d) producs a situation lik that in Fig. 1, and to ~ I.. I. Schiff, Stanford Microwav Laboratory Rport. 'lo. 102, p. 8 (1949).
SCAT'I ERING OF ELECTRONS ON PROTONS dduc th valus of ' and x'. For th cas of q, th photon momntum, qual to zro, diagrams (b), (c), and (d) ar found to add to zro as might b xpctd, sinc thr is thn no scattring procss. (Thr ar nonssntial mass rnormalization trms but thy do not concrn us. ) Morovr, th valu of q dos not affct th proton-mson parts of diagrams (c) and (d). Thrfor w can obtain th Anal proton-mson portion of th amplitud by adding th proton-mson parts of diagrams (a) and (b) and subtracting off th valu of diagram (b) for q=0. Th proton-mson amplitud matrix from diagram (a) is simply y, that from diagram (b) can b writtn ~a" i' ys(lii &+M)ys(2k. +q.) I.0 0 d4k 7ri & [(p,+it)' M'j[k' ii'1[(k+q)' ii'] Hr p, is th mson rst-mass; th intgration is to b prformd ovr all virtual msons; and w ar intrstd in th lmnt btwn initial and final proton stats of this matrix. Aftr th intgral ovr th virtual msons is prformd, and th amplituds from diagrams (a) and (b) addd, with th q=0 valu of diagram (b) subtractd, w obtain as final amplitud: g2 pi (i -1 y 1 ) ( - ( in( 1+ I dx uy 2x ~o ~o -2 ~ a] m[3)& (5 (a/2)) y'+ 2y'j (4) L(1 y)'+my'+ayj[(1 y)'+ayjqy. y, q g' I' r' ( y(1 y) dy( 2M 2 s ~, ", ~ (1 y)'+ay'+ay~ E sin s/s p (i + ae sirp ~/, Fio, 3. Ratio of Rctiv anomalous proton magntic momnt to its zro-nrgy valu for chargd mson thoris. "'S Fynman, rfrnc I, for a full discussion of th mthod notation, and calculation tchniqus. In particular, th Appndix, p, 785, givs a full discussion of th valuation of th radiativ corrction to scattring intgral which is vry analogous to our cas. '' fco I.O,7.5 0 2 F s)n~/2 2E p, l+ sin Frc. 4. Ratio of ffctiv anomalous proton magntic momnt to its zro-nrgy valu for nutral mson thoris. Hr x and y ar intgration paramtrs, a=@, '-//M'-'; u=q'(xs x)/ms. Th first trm hr rprsnts th R'ctiv charg of th proton, th scond its anomalous magntic momnt. For n=0 this trm givs just th valu for th anomalous momnt prviously drivd by Cas.' For th cas I/ 0 th intgrals ar vry complicatd. Th intgral on y can b prformd analytically. Th rmaining intgral on x thn dpnds on th paramtrs q'/ii' and q'/m'. If q' is of comparabl ordr of magnitud to p, ', but much smallr than M', a rgion of considrabl intrst, th intgral may b xpandd to first ordr in th paramtr q'/m-' and thn prformd analytically. For largr valus of q' it must b carrid out numrically. Th othr mson thoris rquir th sam typ of calculation. Figurs 3, 4, 5, and 6 giv th rsults of ths intgrations. Figurs 3 and 4 ar graphs of s''/kp, th ratio of th ffctiv anomalous magntic momnt to th zronrgy anomalous momnt. On Fig. 3 w hav also plottd th "classical" formula (3). W hav assumd in all calculations that p, th mson mass, is 276 lctron masss, consistnt with xprimntal valus for th x-mson. Ths ratios ar indpndnt of th coupling constant, which will dtrmin only th magnitud of th zro-nrgy momnt. It should b notd, howvr, that th scalar chargd and psudosc alar nutral thoris prdict th wrong sign for th proton momnt. Figurs 5 and 6 ar graphs of th ffctiv proton charg. For rasons discussd blow, w hav plottd '/=~, rathr than '/=1 5 as obtaind dirctly from (4). As can b sn from (4), 8 is dirctly proportional to g'. W hav plottd ' for thos valus of g' ncssary to prdict th corrct valu for th magnitud of th zro-nrgy proton momnt. Ths valus ar givn in Tabl I. To illustrat th us of th graphs, and to show how thy may b adaptd to symmtrical thory, lt us ' K. Cas, Phys. Rv. 76, 6 (1949).
M. N. ROSEN BLUTH I I.O.8 E sin ~/p )+ sifp' Fro. 5. Effctiv proton charg for chargd mson thoris with coupling constants chosn to fit th magnitud of th obsrvd proton anomalous magntic momnt. calculat th ffctiv charg and magntic momnt for a 500-Mv lctron scattrd through 90' on th basis of symmtrical psudoscalar thory with coupling constant 57.2. Th abscissa 0~ f 2E, 8~' E, sin ( [ p( 1+ sin'-' =( ) q')&/2p 2& & M 2) in this cas is qual to 2.03. Sinc th nutral and chargd thoris giv opposit signs for th magntic momnt: K'',= K'', K''. Using Tabl I and Figs. 3 and 4, (~''/xo), = 0.81(57.2/16. 1) 0.94(57.2/22. 4) = 0.48. To obtain th Rctiv charg: 8,= 8,+5 =0.22(57.2/16. 1)+0.15(57.2/22. 4) = 1.16, ('/), = xp( 8, ) = 0.31. To obtain th final cross sction ths valus for ' and K and Schwingr's' valu for " ar substitutd in (1). It will b notd that symmtrical thory prdicts a rapid dropping or of magntic momnt and charg du to th larg coupling constant. IV. CONCLUSIONS It will b notd at onc that th valus of g'-' listd in Tabl I ar so larg as to throw grav doubts on th us of scond-ordr prturbation thory. This is spcially tru for th psudoscalar cas whr w xpct scondordr trms to b small compard to highr ordr trms. In this connction it may b notd that th valus of g' listd in Tabl I do not giv th corrct nutron momnt. Som justification for th prturbation thory procdur may b found in th fact that xprimntal rsults on photo-mson production do sm to agr with th qualitativ prdictions of scond-ordr psudoscalar prturbation thory. ' (Thr has bn no ffort to masur th absolut cross sction so that no xprimntal valu of g' is obtaind. ) It is bcaus of doubt of th adquacy of th scond-ordr thory that w hav plottd '/=xp( 5), thus considring at last som of th highr ordr trms. It will b notd that vn though th mson clouds ar mor tightly bound than a naiv pictur would prdict (s Fig. 3), thr is nonthlss a vry sizabl dcras in proton charg and magntic momnt to b xpctd at high nrgis. This is spcially tru if w assum th larg valus of coupling constant ncssary to prdict th propr proton momnt. Evn if only th qualitativ faturs of ths curvs ar dpndabl th xprimntal rsults should at last indicat (1) if th proton magntic momnt is rally du to th x-mson ' -5 = ~ 8.5 0 2 E sin Fio. 6. Effctiv proton charg for nutral mson thoris with coupling constants chosn to fit th magnitud of th obsrvd proton anomalous magntic momnt. TABLE I. Coupling constants ncssary for corrct magnitud of proton magntic momnt. Thory chargd nutral symmtrical Psudoscalar chargd Psudoscalar nutral Psudoscalar symmtrical 2.76* 9.67 3.86* 16.1 22.4* 57.2 * Indicats wrong sign for magntic momnt. J. Stinbrgr, xprimnts prformd at Brkly and not yt publishd.
ANOMALOUS MOLF. CULAR ROTATION 619 fild, (2) whthr a loos-bound scalar typ thory or a tight-bound psudoscalax-typ thory is prfrabl, and (3) how much faith can b placd in th "quantum lctrodynamical radiativ corrctions" to scattring and in th far mor dubious scond-ordr mson corrctions to scattring. I would lik to thank Profssor L. I. Schiff for many hlpful discussions and suggstions. I would lik to xprss my gratitud also to Dr. Ross Thompson of Cornll Univrsity who has prformd most of ths calculations indpndntly, and who vry kindly chckd, and improvd upon on of, my rsults. P EE V S I C A L R E V I F. KV VOLUM1. 79, NUM HER 4 AUGUST 15, 1950 Anomalous Molcular Rotation and th Tmpratur of th Uppr Atmosphr* Lzms M. BRANscoMa** Lyman Laboratory of Physics, Harvard Univrsity, Cambridg, Massachlstts (Rcivd May 12, 1950) This xprimnt vrifis th prdiction of Oldnbrg that th spctroscopically masurd rotational tmpratur of a diatomic gas will b lowr than th translational tmpratur whn (1) th prssur is low, (2) th gas is xcitd by lctron impact, and (3}th xcitd lctronic stat from which th masurd bands ar radiatd has an quilibrium nuclar sparation gratr than th intrnuclar distanc in th ground stat. For gas tmpraturs from 400 to 670'K rotational tmpraturs from th scond ngativ bands of 02 v r found in qualitativ agrmnt with th prdictd rlation T t=tt,b'/b". Uppr atmosphr tmpraturs drivd from band profils in night sky spctra ar consistntly lowr than tmpraturs stimatd from othr data. Th possibl occurrnc of anomalous rotation of th night sky molculs casts som doubt on th maningfulnss of th night sky tmpratur masurmnts. A partial rotational analysis in th cours of this xprimnt suggsts rvisions of th Bo and a-valus for th O. + molcul in th 'll - and 'II,-stats. I. INTRODUCTION A ERO%LEDGE of th molcular dnsity and tmpratur at high altituds is fundamntal to an undrstanding of th procsss occurring in th arth's uppr atmosphr. For th rgion abov 80 km th atmosphric tmpratur is dducd from indirct vidnc from various sourcs. Unfortunatly most of th data is only qualitativ and, to mak mattrs wors, much of it is contradictory. Evidnc for a stadily incrasing tmpratur abov 80 km (a gradint of prhaps 4'/km) is found in th rlativ widths of th ionosphr layrs, th apparnt scarcity of hlium at high altituds, and th slow dcras in dnsity at vry grat hights (as is indicatd by high altitud auroral rays). ' On th othr hand, th mor dirct spctroscopic masurmnts on bands in th spctra of th aurora and th night sky luminscnc ar intrprtd by som authors as conclusiv vidnc for a constant tmpratur of about 250'K abov 90 km, a rsult which sms quit incompatibl with th othr tmpratur stimats. Ths spctroscopic tmpratur masurmnts ar basd on th rlationship btwn th quilibrium tmpratur of a radiating diatomic gas and th rlativ intnsitis of th lins in th rotational fin struc- * This work is dscribd in dtail in th author's Ph.o. thsis, Harvard Univrsity (1949). ~* Now a mmbr of th Socity of Fllows, Harvard Univrsity. 'S. K. Mitra, Th Uppr Atmosphr (th Royal Socity of Bngal, Calcutta, 194'7}; G. P. Kuipr, Th Atmosphr of th Earth and P/ants (Univrsity of Chicago Prss, Chicago, 1947). S spcially chaptrs by P. Swings and L. Spitzr, Jr. tur of an mission spctrum. Th rlativ intnsitis of th rotational lins dpnd on two quantitis: th rlativ transition probabilitis of th lins, and th population of th initial rotational lvls. If th tmpratur dpndnc of th populations of th initial rotational stats is known, on can calculat th transition probabilitis from quantum thory and can thn calculat th gas tmpratur from th xprimntal rlativ intnsitis of th lins in a band. In th masurmnts on uppr atmosphric spctra it was assumd that th xcitd molculs in th aurora and night sky ar in a Boltzmann distribution with rspct to rotational nrgy, at an quilibrium tmpratur T. Th significanc of th spctroscopic tmpratur masurmnts dpnds on th validity of this assumption. Laboratory invstigations of bands xcitd by lctron impact in a glow discharg hav affordd many xampls of cass in which th xcitd radiating molculs ar indd in a Boltzmann rotational nrgy distribution, for th spctroscopically dtrmind tmpraturs wr quit clos to th dirctly masurd gas tmpraturs. In 1934 Oldnbrg' pointd out that although th molculs in a low prssur glow discharg ar initially in thrmal quilibrium, th assumption of thrmal quilibrium in th xcd rotational stats is justifid only for molculs whos nuclar sparation is th sam in th xcitd and ground stats. Although this is usually th cas, thr ar crtain molculs in which th xcitd stat has a much largr nuclar sparation ' O. Oldnbrg, Phys. Rv. 46, 210 (1934).