Electron Spin and Proton Spin in the Hydrogen and Hydrogen-Like Atomic Systems

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1 Journal of Modrn Physics, 014, 5, Publishd Onlin Dcmbr 014 in SciRs. htt:// htt://dx.doi.org/10.436/jm Elctron Sin and Proton Sin in th Hydrogn and Hydrogn-Lik Atomic Systms Stanisław Olszwski Institut of Physical Chmistry, Polish Acadmy of Scincs, Warsaw, Poland Rcivd 16 Octobr 014; rvisd 1 Novmbr 014; acctd 5 Dcmbr 014 Coyright 014 by author and Scintific Rsarch Publishing Inc. This work is licnsd undr th Crativ Commons Attribution Intrnational Licns (CC BY). htt://crativcommons.org/licnss/by/4.0/ Abstract Th hanical angular momntum and magntic momnt of th lctron and roton sin hav bn calculatd smiclassically with th aid of th uncrtainty rincil for nrgy and tim. Th sin ffcts of both kinds of th lmntary articls can b xrssd in trms of similar formula. Th quantization of th sin motion has bn don on th basis of th old quantum thory. It givs a quantum numbr n = 1/ as th indx of th sin stat acctabl for both th lctron and roton articl. In ffct of th sin xistnc th lctron motion in th hydrogn atom can b rrsntd as a drift motion accomlishd in a combind lctric and magntic fild. Mor than 18,000 sin oscillations accomany on drift circulation rformd along th lowst orbit of th Bohr atom. Th smiclassical thory dvlod in th ar has bn alid to calculat th doublt saration of th xrimntally wll-xamind D lin ntring th sctrum of th sodium atom. This saration is found to b much similar to that obtaind according to th rlativistic old quantum thory. Kywords Sin Effct and Its Smiclassical Quantization, Elctron and Proton Elmntary Particls, Elctron Drift in th Hydrogn Atom, Saration of th Doublt Sctral Lins 1. Introduction In hysics w look usually for gnral ruls which govrn th rortis of a hysical objct, or a st of such objcts. For xaml th Bohr atomic modl givs a rathr rfct dscrition of svral quantum aramtrs charactrizing th hydrogn atom, but not th sin ffcts. Th main itms obtaind from th Bohr dscrition How to cit this ar: Olszwski, S. (014) Elctron Sin and Proton Sin in th Hydrogn and Hydrogn-Lik Atomic Systms. Journal of Modrn Physics, 5, htt://dx.doi.org/10.436/jm

2 S. Olszwski hav bn confirmd both on th xrimntal way, as wll as on th quantum-hanical footing which is considrd to b a mor flxibl formalism than th old quantum thory. Simultanously, howvr, quantum hanics smd to b nough comlicatd to giv no transarnt ida on th sin ffcts of th chargd articls ntring th atom. In consqunc a tratmnt of th sin ffcts of th lctron and roton was vidntly absnt in such siml modl as th smiclassical Bohr aroach to th hydrogn atom; s.g. [1]. Th aim of th rsnt ar is to bridg this ga. A gnral warning on th tratmnt of sin is that it should not b skd as a rsult of th circulation ffct of a articl about its own axis (s.g. []), and this viw is shard also in th rsnt aroach. But instad of th motion about an axis which crosss th articl body, a chargd articl may rform its sontanous circulation in th magntic fild about an axis locatd outsid th articl mass. Th sns of such bhaviour is as w shall s that in ffct of th articl intraction with th magntic fild cratd by th articl motion, th articl nrgy bcoms much lowrd blow th zro valu of nrgy which can b assumd to b associatd with th articl at rst. In dfining th osition of th axis of th articl circulation in th magntic fild, th uncrtainty rlation for nrgy and tim can b of us [3]-[5]. Byond of tim t and nrgy E, th rincil contains also a rfrnc to th articl mass m and th sd of light c : ( ) E t >. (1) Evidntly th rul (1) drivd for lctrons in [3]-[6] dos aly to th articls which oby th Frmi statistics. But, for xaml, instad of lctrons of th mass m= m considrd in [3]-[6], w can hav also th gas of th roton articls of th mass m = m distributd in th fild of a ngativ background which maks th gas lctrically nutral. A rasoning of [3]-[6] ratd in th cas of an nsmbl of th roton articls givs th rsult ( ) E t >. () This maks () diffrnt from (1) solly by a rlacmnt of m in (1) by m in (). Crtainly E and t in () rfr to th roton articl. A consqunc of th rincil in (1) and () is a rul that two Frmi articls of th sam kind cannot aroach togthr to an arbitrarily small distanc but thy should b saratd at last by th intrval which in viw of (1) is qual to [6] for lctrons, but bcoms qual to x = (3) x = (4) for th rotons cas; s.g. [7] for th roton mass, sin angular momntum and sin magntic momnt. Th minimal distancs (3) and (4) btwn articls rrsnt rsctivly th Comton lngth of th lctron and roton articl, on condition that th rationalizd Planck constant is rlacd by th original Planck constant h. Th kind of th formula givn in (3) and (4) has bn drivd bfor in [8]-[10]; s also [11]. In Sction w aly (3) and (4) to dfin th ositions of th axs of a sontanous articl circulation giving, rsctivly, th lctron and th roton sin. Bfor ths motions tak lac w assum that th articl nrgy of th lctron ( ) E and roton ( ). Sinning Procss of th Elctron and Proton E is at zro: E = E = 0. (5) A gnral law of hysics is that any articl tnds to assum a ossibly lowst lvl of nrgy. In cas of a chargd articl this can b attaind in ffct of th articl circulation about som axis along which th articl motion inducs th rsnc of th magntic fild. This situation imlis that th kintic nrgy of th orbital motion is associatd with a articl. Th axis of th motion can b locatd outsid th xtnsion ara of th 031

3 S. Olszwski articl mass. As a distanc of th axis from th articl location ( r for th lctron and r for roton) lt us assum that (3) and (4) hold rsctivly in th lctron and roton cas. Th magntic fild B causs th vlocity v along a circl normal to B, and th balanc of th forcs rquirs that v B = m (6) r whr m = m or m, v = v or v, and r = r or r. In ffct th forc in (6) rrsnts an quilibrium btwn th forc of th fild and th hanical forc du to th acclration of a articl toward th track cntr (s.g. [1]). W ostulat that r = x (7) in th cas of th lctron articl, and r = x (8) in th roton cas. Th hanical angular momnta of lctron and roton bcom rsctivly L = mrv = mrω (9) L = mrv = mrω. (10) For th sak of simlicity th sam siz of charg for th lctron and roton is assumd. Th ω and ω in (9) and (10) ar th lctron and roton circulation frquncis qual to Bc ω = (11) Bc ω =. (1) Th B c and B c ar th strngths of th magntic fild suitabl for th lctron and roton cas. For both kinds of articls w assum that th strngth of B is so larg that lctron or roton gyrat in th magntic fild with a sd clos to c. This rquirmnt for th articl vlocity is dictatd by xamination of th articl acclration xrssd in trms of th lctric fild E and magntic fild B [13]. In this cas and and dv v 1 1 = 1 E + [ v B] v ( ve ). (13) dt m c c c Evidntly th acclration (13) vanishs whn th articl vlocity bcoms a constant v = c. Thus w hav Bc v = rω = r (14) With th aid of (3) and (4) w obtain from (14), (15) and (16): Bc v = rω = r (15) v = v = c. (16) 3 B = B c r = (17) 3 B = B c r = (18) 03

4 S. Olszwski on condition th absolut valus of B ar takn into account. Th orbital radii r and r [s (7) and (8)] substitutd togthr with th vlocitis of (16) into th formula (9) and (10) for th angular momntum giv rsctivly for th lctron and for th roton articl. In ffct w hav L m c = (19) L m c = (0) L = L. (1) Evidntly th formula obtaind in (19)-(1) do not dnd on th articl mass. But a mass dndnt aramtr bcoms th magntic momnt M of a articl. For th lctron cas w obtain: (which is th Bohr magnton) and for roton M = L = = M M B = L = calld also th thortical nuclar magnton alid in considring th nuclar articls [7]. Th ratio btwn (3) and () is dfind by () (3) M M = m m (4) which is not vry far from th ratio obtaind from th xrimntal data for th magntic momnt of lctron and roton [7]. In many cass th xrimnts rformd on th nuclar magntic momnta M n giv th ratio Mn M not much diffrnt from m m n whr m n is th nuclar mass. Th nrgy of a sinning articl in th magntic fild is rsctivly rrsntd by 1 E = Bc M = (5) for an lctron, and by 1 E = Bc M = (6) for a roton. Thrfor th gain of nrgy in th magntic fild du to formation of th articl sin is larg. This gain of nrgy is xnsd to rovid th kintic nrgy to a sinning articl having its vlocity clos to c. 3. Magntic Flux of a Sinning Particl, Consrvation of Enrgy and Quantization of th Sin Motion A aramtr concrning sin which has its stablishd xrimntal countrart is th magntic flux. Lt us choos for an lmntary lanar ara of that flux th circl for lctrons, and th circl S = πr = π( x) = π S = πr = π( x) = π (7) (8) 033

5 S. Olszwski for rotons. From (7), (8), as wll as for th magntic fild strngth takn rsctivly from (17) and (19), w obtain and Φ = 1 hc c Bc S = = 3 π Φ = 1 hc c Bc S = = 3 π rsctivly in th lctron and roton cas. An vidnt rsult is that (9) (30) 1 hc Φ c =Φ c =. (31) Thrfor th flux xtndd ovr th lmntary aras in (7) and (8) dos not dnd on th articl kind rrsntd by th articl mass. Morovr, th flux calculatd in (9) and (30) is qual to a constant quantum trm obsrvd xrimntally sinc a long tim in surconductors [14]. Th tim drivativ of th flux trm is zro, so w hav th fundamntal rlation of lctrodynamics d d hc E d l = d 0 L dt B S = = S dt. (3) Physically this mans that a linar intgral ovr E rrsnting th lctric fild along a circular ath of th lctron is qual to zro, thrfor th nrgy of th circular motion in th magntic fild of B is consrvd. Having th magntic flux hc πrb c c = = Φ c (33) th sin motion can b quantizd according to a rul of th old quantum thory [1] [15]. It originats from a gnral rul givn by Sommrfld that momntum intgratd ovr a closd ath dr of th articl motion should b a multil of th Planck constant h : r d = nh (34) hr n is usually considrd as an intgr numbr. But according to [1] Equation (34) can b transformd into πr c B c = nh. (35) c By taking into account th first quation in (33) w obtain for (35) th rlation h = nh (36) from which th sin quantum numbr bcoms: n = 1. (37) This is a wll-known rsult confirmd xrimntally by th masurmnts on th gyromagntic ratio in frromagnts [16] rformd a tim bfor th sin discovry [17]. 4. Drift Vlocity of a Sinning Elctron in th Elctric Fild of th Proton Nuclus Till th rsnt tim no othr fild than B c sontanously cratd by a sinning articl has bn considrd. Now lt us assum that th sinning lctron mts th lctrostatic fild of th roton nuclus. A minimal distanc which can aar btwn th lctron moving articl and th roton bing at rst is dfind in (3) bcaus (4) is too small to hav a dcisiv influnc. In this cas 034

6 S. Olszwski E c = = = ( x ) ( m c) min whr E c is th absolut valu of th lctric fild acting on th lctron. Anothr forc acting on th lctron is B c whr B c is th magntic fild intnsity of th lctron sin; s (17). Assuming that E c is normal to B, th driving lctron vlocity obtaind as a rsult of th joind action of both filds is [18] c But it is asy to chck from (17) and (38) that is th fin-atomic-structur constant [] [19], so c c c c c vd c c c B c B c B c (38) E B E B E = = =. (39) E v B d c c 1 = = α c 137 (40) = c = c. (41) Th rsult in (41) is rcisly th lctron vlocity on th lowst orbit of th Bohr atom [1]. Thrfor a combind action of th sin magntic forc of th lctron and lctrostatic forc acting btwn lctron and th roton nuclus, givs th sd of lctron qual to that ossssd on th lowst quantum stat in th hydrogn atom. Th sin action of th roton on th lctron sin momnt rsnt on th orbit has bn nglctd. In ffct th vlocity along th lowst orbit of th Bohr s hydrogn atom can b considrd as a consqunc of a drift motion bing a rsult of surosition of many sinning rotations along vry small orbits having thir radii qual to (7) and travlld with a sd qual to c. Th tim ncssary to travl along th Bohr orbit having th wll-known radius ab = (4) m is 3 πab π π T1 = = = (43) 4 v m m d whras th travl tim along th sin orbit calculatd from (7) and (3) is qual to πr T = = π. (44) c In consqunc th numbr of sinning circular motions which tak lac in cours of th lctron drift along th first Bohr orbit is qual to 3 T1 π c 1 = = = 4 4 T m π α (45) This is a numbr indndnt of th mass m. A diagram rsnting schmatically th motion of a sinning lctron along th lowst Bohr orbit in th hydrogn atom is givn in Figur 1. Th circular frquncy of a sinning lctron is π 1 1 = = sc. (46) T Th mass m has to b rlacd by m in cas of a sinning roton frquncy. 5. Smiclassical Aroach to th Doublt Saration in th Sodium Atom Exrimntally th doublt sarations in th sctra of atoms ascribd to th rsnc of th lctron sin ar 035

7 S. Olszwski Figur 1. A schm rrsnting th motion of a sinning lctron along th shortst (lowst) circular Bohr orbit of th hydrogn atom. Th orbit circl is rrsntd by a dashd lin, th saration distanc btwn two circls nclosing th motion is twic th radius r givn by th Formula (3) and (7). For th numbr of th sin oscillations along th orbit s Formula (45). wll known sinc a long tim; s.g. [0]. Th roblm is with a thortical aroach to ths valus. In th author s oinion no satisfactory agrmnt btwn xrimnt and thory has bn rortd in this domain. Our aim is to calculat a doublt saration for th sodium atom in th cas of th lctron transition btwn n = 3 but having diffrnt angular momnta: two lvls bing on th sam atomic shll ( ) 3 S 3 P (47) [0]. Th lvl nrgis ar aroachd by th quantum-dfct mthod. W follow first th ida dvlod by th old quantum thory, nxt th formalism of th rsnt ar is alid. Th considrd lctron of th sodium atom is th valnc lctron moving outsid th atomic cor. Th lctron nrgy is givn by th formula Z W =. (48) 0 an B 0 Hr a B is th first Bohr orbit radius, Z 0 = 1 and n 0 is an ffctiv quantum numbr associatd with th lctron lvl n = 3 by th quantum-dfct formula n n 0 = µ. (49) W aly µ = 1.37 (50) for trm S ( l = 0) and for trm P ( 1) l = ; s Tabl 7. in [0]. A diffrnc of nrgy (48) µ = 0.88 (51) Z W = rg = a B n0 n01 calculatd rsctivly from (50) and (51) givs th lngth of th sctroscoic lin qual to 7 10 hc λ = = = cm = 5517 Å (53) 1 W which is not far from th xrimntal lngth λ = 5685 Å (54) x (5) 036

8 S. Olszwski masurd for th xamind doublt [0]. A roosal of calculating th doublt saration basd on th rlativistic old quantum thory alis th following formula for th chang of nrgy connctd with that saration [1] [] Hr ( ) 4 4 hr Z s m U = h ν = α = n l l 1 α. (55) hr = ( ) 1 m 4 ( ) For th ffctiv nuclar charg qual to that alid bfor in (5), i.. morovr w obtain (56) 5 α = = (57) Z0 = Z s = = 1 (58) n = 3 (58a) l = ( ) 7 ( ) 4 4 m ( ) (58b) U = α = = rg. (59) A smiclassical aroach of th rsnt ar is basd on th intraction nrgy of two magntic diols. On of thm is rovidd by th angular momntum of th lctron circulating about th atomic cor, anothr diol is du to th lctron sin. For th sak of simlicity w assum that th magntic momnta of th orbital motion and th sin motion ar ithr aralll, or antiaralll, in thir mutual arrangmnt. For both cass th absolut valu of th couling nrgy is th sam. On th lvl of ty s th lctron has its angular momntum qual to, on th lvl of ty lt this momntum b. This lads to two orbital magntic momnta on s and qual rsctivly to M lvl s = (60) and M lvl =. (61) W assum that momnta (60) and (61) ar locatd at th nuclus. Th absolut valu of th sin magntic momnt (locatd at th lctron osition) is th sam in both cass bing qual to th Bohr magnton M B givn in (). Th lctron in cours of an xcitation dos not chang its sin, but a saration distanc btwn th magntic momnta of th orbital motion and th sin momntum is changd. For stat s w hav and for stat rs = n01ab = 1.63 ab (6) r = n0a =.1 a. (63) B B Thrfor in cas of a aralll arrangmnt of th orbital momntum and sin momntum th nrgy chang of th momnta intraction du to th lctron xcitation bcoms: 3 4 M lvl lvl MB s B α rs r ab M M 1 m 1 1 U = = 4 3 (64) 037

9 S. Olszwski (th dot roducts of th vctor joining th sin and orbital momnta with ths momnta can b nglctd bcaus th vctor is assumd to b normal to th momnta). Th rsults obtaind in (59) and (64) diffr solly by a factor ( ) ( ) : 1. (65) A substitution of th valus W ± U, whr U is obtaind in (59), in lac of W into a formula similar to (53) rovids us with two wav lngths which diffr by th intrval qual to about Å 5 λ (66) in W th wav lngth calculatd in (53) has bn takn into account. Th rsult (66) is smallr by th factor of about Å 10 (67) than th xrimntal doublt saration qual to 6 angstroms [0]. A substitution of (59) givs a similar saration to that obtaind in (66); s (65). U from (64) instad of 6. A Look on th Dirac Thory and th Prsnt Thory of th Elctron Sin A diffrnc of both thortical tratmnts of sin is vidnt. Dirac s thory is ssntially a rlativistic quantumhanical aroach to th lctron motion; s.g. [] [19] [3] [4]. Aftr th Hamiltonian of th roblm is linarizd, th four-dimnsional matrics ar alid as substitutions of th Hamiltonian orator. In th rsnc of an xtrnal lctromagntic fild a simlification of th roblm can b obtaind by sarating larg and small comonnts of th Dirac quation. In this way th sin-dndnt intraction nrgy with th fild can b calculatd. Th sin magntic momnt is could with th sin angular momntum by a constant trm which is twic as larg as in th classical lctrodynamics. This imlis that th sin quantum numbr should hav th siz of 1/. Dirac s lctron articl considrd in th fild of th Coulomb otntial givs rathr comlicatd formula for th lctron wav functions which hav no countrart in th rsnt smiclassical thory. An advantag of th Dirac thory is that it givs an insight into antiarticls lik ositron, and rsnts an. On th othr sid, no intrval in th nrgy sctrum of articls and antiarticls of th siz qual to aroach to th sin and magntic momnt of such articls lik rotons has bn xlicitly outlind by Dirac. Th thory of th rsnt ar is much diffrnt than th Dirac aroach. First th mthod is ssntially of a smiclassical natur sinc no wav functions ar considrd. A basic rfrnc to th quantum thory is th uncrtainty rincil alid to th changs of nrgy and tim; s (1) and (). Th trm (68) includd in th formalism is obtaind in ffct of th drivation rocdur of th rincil; s [3]-[5]. A furthr analysis of th chang E of a fr-articl nrgy ntring th rincil givs a minimal distanc for th gomtrical saration btwn th articls; s [6]. This saration allowd us to mak a roosal of th sin as a rsult of a sontanous circulation of th lctron, or roton, rformd about an axis locatd outsid th articl mass; s Sction. Anothr advantag of th rsnt thory is that both Frmi articls lctron and roton can b considrd on an qual footing bcaus of th frmion charactr of ths articls; s (1) and () which diffr solly in thir mass symbol. This allowd us to obtain an insight into th sin and magntic momnt of rotons togthr with similar lctron rortis. Th thortical rsults obtaind for both kinds of th articls ar confirmd by th xrimntal data to a larg dgr. Morovr, th Dirac thory assums that crtain magntic fild should b rsnt in ordr to obtain a sinning lctron articl, but th siz of such fild is not dfind. In th rsnt aroach th siz and sourc of th magntic fild acting on th articls ar th rsults of th thory. 7. Summary A smiclassical modl of two sinning chargd articls (lctron and roton) has bn roosd on th basis of 038

10 S. Olszwski a quantum uncrtainty rincil for nrgy and tim and th classical lctromagntic thory. Th main rason of a sontanous formation of a sinning articl is a strong lowring of th articl nrgy in th magntic fild associatd with th xistnc of th sin circulation. Th hanical angular momntum connctd with th sin is found to b th sam for lctron and roton, and th mass diffrnc btwn th articls bcoms sound only for th magntic sin momnt. This vry fact is confirmd by xrimnt (s.g. [7]) which rovids us with th ratio of th magntic momnts similar to that obtaind by th rsnt thory. It could b notd that th hanical momnt of a roton qual to that of a sinning lctron smd to surris many hysicists sinc a long tim; s.g. [5]. This kind of fling is stimulatd by th fact that th 3 10 tims smallr than that of lctron. Th indndnc of th magntic momnt of roton is about hanical sin momnta of both articls on thir mass can b xlaind by a rfrnc to th fact that th articls oby th sam (Frmi) statistics and hav th sam absolut valu of th lctric charg. Thrfor th uncrtainty rincil for nrgy and tim alid to lctrons and rotons is diffrnt just in th mass valu; s (1) and (). But th orbit radius of ach of ths sinning articls is invrsly roortional to thir mass. Sinc th angular momntum is by dfinition roortional to th mass, th both mass xrssions cancl togthr in th angular momntum formula which bcoms indndnt of th mass siz. Whn a sinning lctron mts th lctrostatic fild of a roton, it can b dmonstratd that th rsultd drift vlocity of th lctron bcoms qual to th vlocity of that articl on th lowst quantum lvl of th Bohr modl of th hydrogn atom. Th ffct of th sctral doublt saration has bn also xamind for th atomic sodium takn as an xaml. A smiclassical calculation of th rsnt ar givs almost th sam rsult as it is rovidd by th rlativistic old quantum thory. Rfrncs [1] Bohr, N. (19) Th Thory of Sctra and th Atomic Constitution. Cambridg Univrsity Prss, Cambridg. [] Landau, L.D. and Lifshitz, E.M. (197) Quantum Mchanics (in Russian). Izd. Nauka, Moscow. [3] Olszwski, S. (011) Journal of Modrn Physics,, htt://dx.doi.org/10.436/jm [4] Olszwski, S. (01) Journal of Modrn Physics, 3, 17. htt://dx.doi.org/10.436/jm [5] Olszwski, S. (01) Quantum Mattr, 1, 17. htt://dx.doi.org/ /qm [6] Olszwski, S. (014) Journal of Modrn Physics, 5, 164. htt://dx.doi.org/10.436/jm [7] Tolansky, S. (1948) Hyrfin Structur in Lin Sctra and Nuclar Sin. nd Edition, Mthun, London. [8] Ruark, A.E. (198) Procdings of th National Acadmy of Scincs of th Unitd Stats of Amrica, 14, 3. htt://dx.doi.org/ /nas [9] Flint, H.E. (198) Procdings of th Royal Socity A, London, 117, 630. htt://dx.doi.org/ /rsa [10] Flint, H.E. and Richardson, O.W. (198) Procdings of th Royal Socity A, London, 117, 637. htt://dx.doi.org/ /rsa [11] Jammr, M. (1966) Th Conctual Dvlomnt of Quantum Mchanics. McGraw-Hill, Nw York. [1] Slatr, J.C. (1967) Quantum Thory of Molculs and Solids. Vol. 3, McGraw-Hill, Nw York. [13] Landau, L.D. and Lifshitz, E.M. (1969) Mchanics. Elctrodynamics (in Russian). Izd. Nauka, Moscow. [14] Kittl, C. (1987) Quantum Thory of Solids. nd Edition, Wily, Nw York. [15] Onsagr, L. (195) Th London, Edinburgh, and Dublin Philosohical Magazin and Journal of Scinc, 43, htt://dx.doi.org/ / [16] Bck, E. (1919) Annaln dr Physik, 305, htt://dx.doi.org/10.100/and [17] Uhlnbck, G.E. and Goudsmit, S.A. (195) Di Naturwissnschaftn, 13, htt://dx.doi.org/ /bf [18] Matvv, A.N. (1964) Elctrodynamics and th Thory of Rlativity (in Russian). Izd. Wyzszaja Szkola, Moscow. [19] Schiff, L.I. (1968) Quantum Mchanics. 3rd Edition, McGraw-Hill, Nw York. [0] Whit, H.E. (1934) Introduction to Atomic Sctra. McGraw-Hill, Nw York. [1] Millikan, R.A. and Bown, I. (194) Physical Rviw, 3, 1. [] Rubinowicz, A. (1933) Handbuch dr Physik. In: Gigr, H. and Schl, K., Eds., Vol. 4, Part 1, Sringr, Brlin. 039

11 S. Olszwski [3] Ros, M.E. (1961) Rlativistic Elctron Thory. Wily, Nw York. [4] Avry, J. (1976) Cration and Annihilation Orators. McGraw-Hill, Nw York. [5] Kobos, A.M. (013) Postĕy Fizyki, 64,

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