Non-Relativistic Limit of Neutron Beta-Decay Cross-Section in the Presence of Strong Magnetic Field

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1 Joual of Scics, Islaic Rublic of Ia 9(): (18) Uivsity of Tha, ISSN htt://jscics.ut.ac.i No-Rlativistic iit of Nuto Bta-Dcay Coss-Sctio i th Psc of Stog Magtic ild M. Sidi * Datt of hysics, aculty of Scics, Ila Uivsity, P.O.Box:16-691, Islaic Rublic of Ia Rcivd: Novb 17 / Rvisd: 7 Ju 18 / Acctd: 1 July 18 Abstact O of th ost iotat actios of th URCA that lad to th coolig of a uto sta, is uto bta-dcay ( ). I this sach, th gy scta ad wav fuctios of assiv fios takig ito accout th Aoalous Magtic Mot (AMM) i th sc of a stog chagd agtic fild a calculatd. o this uos, th Diac-Pauli quatio fo chagd ad utal fios is solvd by Ptubatio ad obius sis thod, sctivly. Th sults of th obius sis thod a i good agt with th sults of Nikifoov-Uvaov thod (NU). I cotiuous, usig th calculatd wav fuctios, th gal latio of uto dcay coss-sctio i th o-lativistic liit has b obtaid. This latio has b divd by th fou-fio agagia withi th fawok of th stadad odl of wak itactios. Ths calculatios fo th sctiv of ucla astohysics ca b iotat. Kywods: Diac-Pauli quatio; Aoalous Magtic Mot; ou-io agagia. Itoductio Study of wak ocsss i th sc of stog agtic filds i uto stas ad Radio-ulsas a vy iotat i viw of astohysical alicatios. Th uto stas a su-ds objcts with dsity 14 about of.8 1 g c [1]. Th stgth of th sufac agtic fild i ths objcts is i th ag of ~ 1 G [, ]. Also, th itsity of th agtic 14 1 filds ca b 1 ~ 1 G fo so agtas [4, ]. o o dtails about th otis of agtas, f to as by Gao t al. [6] ad Mghtti t al. [7]. Thy showd that agtas a uto stas i which a stog agtic fild is th ai gy souc. Th uto stas a th bst laboatoy fo studyig ds att hysics. Ths objcts a causd by suova xlosios with a ital tatu about 11 1 of1 1 K. Th ocsss of coolig i th uto stas a du to ittig utios. Duig ths 1 ocsss tatu dcass to T 1 K [8, 9]. Th wak itactios of uclos caus th bta-dcay ad th issio of utios fo uto stas (th URCA ocss) [1, 11]. Th stog agtic filds will stogly ifluc o th wak itactios. Th dtails of wak itactios latd to th URCA ocss a std i Rf. [1]. Ev lativly wak filds ca lay stog ol i vaious astohysical obls [1]. U to ow, it is widly cogizd that stog agtic filds ca b a sigificat facto latd to * Cosodig autho: Tl/ax: ; E-ail: asoudsidi@yahoo.co 81

2 Vol. 9 No. Su 18 M. Sidi. J. Sci. I. R. Ia divs astohysical ad cosological viots. Ud th ifluc of stog agtic filds i th uto stas th dict URCA ocsss a as follows [14, 1]:,, Th URCA ocsss hav b studid i difft lctoagtic filds. Th fist attts to cosid th URCA ocsss i stog costat lctoagtic fild hav b ivstigatd i Rfs [16, 17]. Th lativistic thoy of th uto bta-dcay i stog agtic fild has b dvlod i [18]. Also, th obability of olaizd uto bta-dcay i th sc of agtic fild was divd i Rfs [19, ]. Studiki ad Shikvich [1] dvlod a lativistic aoach fo calculatios of coss-sctios of th URCA ocsss i stog costat agtic filds. May iotat tchical dtails of th calculatios, also usful fo th lat studis, a xist i thi wok. Th dcay at fo th oto ivs dcay ocss accoutig th AMM of uclos i uifo costat stog agtic fild is calculatd i [1, ]. Sic th al agtic filds i uto stas a a o-uifo [], i th st w hav cosidd th gy scta, wav fuctios ad coss-sctio of th bta-dcay olaizd uto i a o-uifo stog agtic fild. Actually, studyig such a ocsss ca b attactiv fo astohysical ascts, scially i th coolig of whit dwafs [4]. I this wok, fistly, th gy scta ad th wav fuctios of chagd fios i a xtal costat agtic fild a calculatd i th o-lativistic liit, ad th th coctios of th gy scta ad wav fuctios a do i th sc of xtal chagd agtic fild usig th foal thoy of tubatio. o utal fio th gy sctu ad th wav fuctio has b obtaid usig obius sis thod. ially, usig th fou-fio wak itactio thoy [, 6], th coss-sctio of uto bta-dcay has b calculatd i a xtal o-uifo agtic fild, takig ito accout th AMM itactio of uclos i th o-lativistic liit. I ou calculatios th icoig utio is suosd to b lativistic ad th ffcts of o-zo ass of utio hav b glctd, ad w hav usd c 1. Th stuctu of this a is as follows. I sctio, w hav xssd foalis of th Diac-Pauli quatio (with AMM i xtal fild) ad th coss - sctio fo bta- dcay. Sctio icluds dtails of calculatios ad th obtaid sults. Matials ad Mthods o a dscitio of aticls with half si i lativistic liit, w us th Diac quatio. I th olativistic liit th Diac quatio covt to th Diac Pauli o Pauli quatio. I this sctio w giv a bif discussio of th Diac-Pauli quatio with AMM i xtal fild ad coss-sctio of bta-dcay. Diac-Pauli quatio with AMM i xtal fild I th lativistic quatu thoy, ovig of a chagd aticl with half si i th xtal lctoagtic fild dscibs by th Diac quatio [7, 8]. I th o-lativistic liit to obtai th gy sctu ad wav fuctio of ovig fios i a xtal agtic fild, it is cssay to covt th Diac quatio to th Pauli quatio. Th Diac quatio fo fios with AMM i xtal agtic fild ca b witt as followig [8]: i t, t P B, t, (1) Paats i this quatio a dfid as follows: I I, 1, P i A, =, = 1, I I () Wh - a Pauli atics, A - is th vcto ottial of agtic fild I ad - a th atixs, -is fio ass ad - is AMM of aticl. O ca asily obtai th Pauli quatio with cotibutio of AMM of fios, so w hav: A i t, t B (), t, as Th statioay stat of bisio, t, t x it 1 is dfid. (4) Wh - is th gy of fio i th olativistic liit. W a itstd i solvig th Eq. () fo utal ad chagd fios i th sc of a xtal agtic fild. W itoducd th agtic fild with cylidical syty: a B,, b () I th Eq. (), a ad b a costat aats. Th vcto ottial fo such agtic fild bcos: b A Az, A a. (6) 8

3 No-Rlativistic iit of Nuto Bta-Dcay Coss-Sctio i l Th xist ay gaugs which oduc th agtic fild, but fo covic w itoducd th gaug of tiod i th Eq. (6). Coss-sctio fo bta- dcay i a xtal fild I od to calculatig th bta-dcay of coss-sctio withi th fawok of th stadad odl of wak itactios w us th fou-fio agagia as follow [1, 6]: G I 1 1 (7), I I abov 1.6 G 44 is th i could costat ad is th atio of th axial ad vcto costats. Th total coss sctio of th ocss ca b witt as: M (8) T has sac I th Eq. (8) suatio is fod ov th has sac of th fial aticls ad M is th atix lt of th ocss. Th atix lt is latd to th fou-fio agagia ca b witt as G 4 M d x (9) l d x W accout fo th ifluc of th backgoud agtic fild i th Eq. (9). I this study th assiv fios hav b cosidd i th o-lativistic liit. Whas th utio is asslss, so utio caot b o-lativistic. Th utios hav ot itactio with agtic fild, futho thy hav ot AMM, thfo th utio wav fuctio i th sc of costat agtic fild th sa as chagd agtic fild. Accodig to th att, w us th utio lativistic wav fuctio i ou calculatios which is dotd by f1 1 f x, t x i t. (1) f 1 f 1 1 i 1 1 cos, 1 cos, a th utio gy ad otu f f (11) sctivly. W glct ffcts of th utio ass, so. Th utio ola ad aziuthally agls a dotd by ad sctivly [1, 1]. Rsults To obtai th wav fuctios ad th gy scta of th assiv fios (lcto, oto ad uto), w will solv th Diac- Pauli quatio, th w will us th wav fuctios fo calculatig th coss-sctio of uto dcay. This sctio cotais th subsctios which a: utal fio, chagd fio ad th gal latio of coss-sctio i th o-lativistic liit. Nutal fio Substitutig th Eq. () ad Eq. (6) ito Eq. () fo utal fio w hav A a 1, b (1) 1,, Solutios of this quatio i cylidical coodiats a: 1 C x i l k z f, l=, 1,,. (1) 1 1, 4 C I this quatio, adial fuctio i cylidical coodiats, f, is calculatd usig th obius sis thod as follows: A l f 1, l 1; x, (14) A k b, x A, k ; 1,,,. Wh, th aats as,, a th uto ass, gy ad AMM ( g, N g 1.91 is ad facto ad 4 1 N.1 g Ga is ucla agto), sctivly. x y-gotic cofficits a 1, ; is Coflut fuctio [9]. Th uto si C 1 s, C 1 s. (1) 1 s s classifis th uto stats with sct to th si ojctio to z dictio ( s 1 cosods to th si oitatio Th uto si valu 1 aalll to th agtic fild B ). o Eq. (14) ad otis of y-gotic fuctio, th gy sctu fo uto is obtaid as follows: k a., l s b (16) l. This latio is i good agt with th NU 8

4 Vol. 9 No. Su 18 M. Sidi. J. Sci. I. R. Ia thod []. Chagd fios Th Pauli quatio fo chagd fio takig ito accout itactio of AMM aticls with xtal costat agtic fild B is as follow: 1 1, A B 1,. 17) Solvig th Eq. (17) lads to: 1 i y z C i t 1 1,, t U ; (18) C x, C1 1 s, C 1 s, s = 1. (19) Wh U is it fuctio. 1 d () U 1! d Th gy sctu of chagd fio (i.. lcto) is calculatd as: B s B 1 s, =. (1), w suos that aat a i Eq. () is sall. W calculat th cotibutio of th tubatio t ( a ). Thfo, usig th ti iddt tubatio thoy, th lcto wav fuctio 1 x, t ca b witt as follows: 1 C k 1 x t i t i y z U k k C, x x. () a 1 ab a k k. () o o 1 Wh x, t,,,, ad k a th lcto wav fuctio of o od coctio, gy, ass, otu coots ad gy coctio ud th ifluc agtic fild, 1 sctivly. Th oto wav fuctio x, t ca b xssd i a siila fo 1 C q 1 x, t x i tx i y zu. q q C (4) x, C1 1 s, C 1 s, S = 1. () Th oto aats a siila to lcto os. Th gy scta of goud stat ( o ) is: ab B 1 4, a,,. o (6) Also fo xcitd stat ( 1o 1), w hav: 1 47 ab 1 1,, a 4 o (7), ad a th ass of chagd fios, AMM ad chag fios, sctivly. Th gal fo of coss-sctio i th olativistic liit I ou calculatios w hav obtaid th xact solutio of th Diac-Pauli quatio fo uto i th sc of agtic fild. Also, th o-xact solutios of Diac-Pauli quatio fo lcto ad oto i th o-lativistic liit a obtaid usig tubatio thoy i th sc of o-uifo agtic fild with cylidical syty. Without loss of gality, th cylidical syty agtic fild B is tak i z-dictio. Substitutig ths wav fuctios i Eq. (9) ad aft so algba w hav: x x i t i y i z dydzdt x (8) Itgatig ov th coodiat x i th atix lt of bta-dcay, w gt: G M x, tc1 C1 C C C1 f1 C f C1 C C C1 C1 f1 C f C C1 C1 C C f C f C C C C C f C f dxdydzdt (9) This quatio ca b witt as: G M x, tc jdxdydzdt () Wh x, t is dfid as: q k q k 1 x, t x U U x i 1x 4 k q i txi y i z x x (1) Substitutig th Eq. (1) ito Eq. (), w obtai 84

5 No-Rlativistic iit of Nuto Bta-Dcay Coss-Sctio i k q j q k q k i 1 x G C M x U U dx () Usig th latios of followig [1]: T T T () i which T ad a th quatizatio lag ti ad distac i th y ad z dictios, sctivly, th squad o of th atix lt is obtaid as follow: TG q k M C 8 j 4, q, k k q A x 4 1 (4) i that w hav: C j 1 cos 1 1 cos 1 11 cos si C C S C C C C C C C C () Accodig to th Eq. ( 4), w a abl to calculat th total coss sctio of uto dcay ( Eq. (8)). Th has sac facto fo th lcto ad oto i th sc of a agtic fild i th goud stat is: d d d has sac S, S 4 d (6) Rgad to this dfiitio, w obtai 7 M M ddd d d d i, i s, s s s T 1 T s, s (7) d d d d d C j A x 4 i which G q k, q, k k q (8) Th itgatio ov th lcto otu coots, ad a fod usig th two ad dlta fuctios { }. Aft ths itgatios w gt th laws of cosvatio fo th two otu coots, that is,. ially, w hav obtaid th coss sctio of th bta dcay of olaizd uto i a cylidical syty xtal agtic fild i th olativistic liit as: 1 k q G ,, kq 8 a 1 x. 4 k q s B s B 1. (9) This latio is th gal fo of coss-sctio i th o-lativistic liit. w ca obtai coss-sctio of uto dcay i goud stat by istig k q 1 ito Eq. (9). Discussio I this suvy, w hav dvlod th bta-dcay of olaizd uto i th sc of stog xtal uifod agtic fild (vaiabl with cylidical syty). It is kow a stog agtic fild ca lad to bta-dcay i uto stas. I od to th dscitio of th gy sctu ad wav fuctios i th URCA ocss, w hav loyd th Diac-Pauli quatio. Th Diac-Pauli quatio has b solvd fo utal ad chagd fios with accout of AMM i th sc of a stog agtic fild with a cylidical syty., w hav usd th tubatio thod fo solvig th Diac-Pauli quatio. Th gy sctu fo assiv utal fios i th olativistic liit is calculatd ad coad with th Nikifoov-Uvaov thod (NU) []. Th sults a i good agt togth. This as that th wav fuctios fo th o-lativistic fios a valid. W kow that th utios ov with th sd of light. Thy hav ot itactio with agtic fild, so w hav usd th lativistic wav fuctios of utios i th sc of costat agtic fild [1] i ou calculatios. Ths wav fuctios hav b usd i calculatig th atix lt ( M ) of th ocss. Th alitud of th boud-stat dcay ocss ( M ) 8

6 Vol. 9 No. Su 18 M. Sidi. J. Sci. I. R. Ia has b foulatd by usig th fou-fio agagia withi th fawok of th stadad odl of wak itactios. Ths calculatios fo th sctiv of ucla astohysics ca b iotat i stiatig th uto dcay coss sctio. W hav obtaid th gal fo fo calculatig th coss sctio of bta dcay i th o-lativistic liit. This latio ca b iotat fo th sctiv of ucla astohysics. Sic, xital astohysics data's satisfy th xistc of stog cylidical sytic agtic fild o th sufac of focd agtizd uto stas, ou calculatios ad sults a valid, i oit of astohysics obsvatio. Rfcs 1. Kauts V.., Savochki A.M., ad Studiki A. I. Asyty of Nutio Eissio fo Nuto Bta Dcay i Suds Matt ad a Stog Magtic ild, Phys. Atoic Nucli, 69: (6).. Ghauga G. Magtic ilds of Dgat Stas, Au. Rv. Asto. Astohys, : (199).. Thoso C. ad Duca R. C. Th Nuto Stas. II. Quisct Nutio, X-Ray, ad Alfv Wav Eissio Soft Gaa Rats as Vy Stogly Magtizd, Astohys. J, 47: (1996). 4. Chalo N. ad asl P., Physics of Nuto Sta Custs, ivig Rv. Rl., 11: 1 (8).. Pg Q.. ad Tog., Th hysics of stog agtic filds i uto stas, Mo. Not. R. Asto. Soc., 1: (7). 6. Gao Z.., Wag N., Xu Y. ad i X. D., Th ffcts of suhigh agtic filds o th quatios of stat of uto stas, Asto. Nach., 6: 866 (1). 7. Mghtti S., Pos J. ad Mlatos A., Magtas: Potis, Oigi ad Evolutio, Sac Sci. Rv., 191: 1-8 (1). 8. Ya X., Guag-Zhou., Chg-Zhi., Cu-Bo., og- Ya W., Mig-g Z., E-Guag Z. Th Nuclo Dict URCA Pocsss i a Coolig Nuto Sta, Chi. Phys. tt., : (1). 9. Yakovlv D. G., Pthick C. Nuto Sta Coolig, A. Rv. Asto. Astohys. 4: (4). 1. Calso J., Cat M. P., Cast R., Elst C., allo P., Gad A., Goss C., ag G., ays A. C., igibotha D. W., owll C. R., oowitz C. J., Jos K.., Kodv. G., ai S., Macchiavlli A., McCutch E. A., Natowitz J., Nazawicz W., Pabock T., Rddy S., Rily M. A., Savag M. J., Savad G., Shill B. M., Sobotka. G., Stoy M. A., Tsag M. B., Vtt K., Widhov I. ad Wuosaa A.., Yllo S., Whit a o ucla astohysics ad low-gy ucla hysics, Pat : ow-gy ucla hysics, Pog Pat Nucl Phys., 94: (17). 11. Pag, D. ad Algat, J.., Th coolig of uto stas by th dict URCA ocss, Astohys. J, 94: 17- (199). 1. iu, J. J., Gu W.M.: A w isight ito utio gy loss by lcto catu of io gou ucli i agtas sufac, Astohys. J. Sul. S., 4: 9 (16). 1. iu J. J. ad iu D. M., Ifluc of su-stog agtic filds o bta dcay of uclid 9- Co i agta sufac, Astohys Sac Sci., 61:46 (16). 14. ujii., Muto T., Tatsui T., Taagaki R., Effcts of wak itactio o kao codsatio ad coolig of uto stas, Nucl. Phys. A, 71: (1994). 1. atti J. M., Pthick C. J., Pakash M., asl P., Dict URCA Pocss i Nuto Stas, Phys. Rv. tt, 66: 71-4 (1991). 16. Mats J., O Coll R., Nuto bta-dcay i a uifo agtic fild, Phys. Rv., 18: (1969). 17. assio-cauto., Nuto bta-dcay i a stog agtic fild, Phys. Rv., 187: (1969). 18. Studiki A., Poto coil ffcts i bta dcay of olaizd utos i a stog agtic fild, Sov. J. Nucl. Phys. 49: 11-4 (1989). 19. Tov I., Rodioov V., Zhulgo V. ad Studiki A., β- dcay of Polaizd Nutos i Extal Elctoagtic ilds, Sov. J. Nucl. Phys. 8: (1978).. Baaov I., β-dcay of a u-olaizd uto i a its lctoagtic fild, Sov. Phys. J., 17: -8 (1974). 1. Shikvich S., Studiki A., Rlativistic thoy of ivs bta-dcay of olaizd uto i stog agtic fild, Paaa J. Phys., 6: 1-44 ().. Bad M., Rubisti. R., Poto bta dcay i lag agtic filds, Phys. tt. B, 11: (199).. Skoblv V. V., O th Pssu of a Nuto Gas Itactig with th No-Uifo Magtic ild of a Nuto Sta, Russ Phys J, 6: 7 (18). 4. Schwab J., Bildst. ad Quatat E., Th iotac of Uca-ocss coolig i acctig o whit dwafs, Mothly Notics of th Royal Astooical Socity, 47: 9-46 (17).. Wibg S., Quatu thoy of filds, Cabidg Uivsity ss, (). 6. Oku., Quaks ad tos, Noth-ollad (198). 7. adau. M., ifshitz E. M., Quatu Elctodyaics, Pgaa Pss, (1979). 8. Bjok J. B., ad Dll S. D., Rlativistic Quatu Mchaics, McGaw-ill, (1964). 9. Abaovitz M., Stgu I. A., hadbook of Mathatical uctio with oula, Gahs ad Mathatical Tabls, Dv, Nwyok (197).. Sidi M., Calculatig of uto gy sctu with accout of AMM i th sc of a stog agtic fild usig th Nikifoov-Uvaov thod, 4 th Iaia Nucla Cofc, Ia, Uivsity of Isfaha (18). 86