SCALING OF SYNCHROTRON RADIATION WITH MULTIPOLE ORDER J. C. Sprott PLP 821 Novmbr 1979 Plasma Studis Univrsity of Wisconsin Ths PLP Rports ar informal and prliminary and as such may contain rrors not yt liminatd. Thy ar for privat circulation only and ar not to b furthr transmittd without consnt of th authors and major profssor.
SCALING OF SYNCHROTRON RADIATION WITH MULTIPOLE ORDER J.C. Sprott Krst (PLP 819) has stimatd th synchrotron radiation loss for a low bta mu1tipo1 of arbitrary ordr. This not addrsss th sam problm using a simplr and mor gnral flux-spac argumnt. Also includd ar som usful, numrically-drivd, scaling laws for maximum fild and stabl flux as a function of mu1tipo1 ordr. Th synchrotron radiatd powr from a volum dv containing plasma with dnsity n, non-rlativistic lctron tmpratur T, and magntic fild B is givn by (Ros and Clark, pag 251): dp = 6.2 x 10-20 B 2 nt dv All units ar mks xcpt T which is in V. Avraging ovr a flux tub containing magntic flux d givs whr dl is a unit lngth paralll to B. If n and T ar constant along a fild lin, th radiatd powr can b writtn as dp = ' 6.2 x 10-20 nt 1 Bdl From Ampr's law, g, Bdl is just o tims th total hoop currnt I T ' providd S is ngligibl. (For high S, on would also includ th diamagntic currnt
-2- nclosd by th flux surfac.) Hnc th total synchrotron radiatd powr is which can b writtn in a convnint, machin-indpndnt form: p whr th brackts < > dnot a flux spac avrag, and T is th total flux in th dvic. This rlation has bn vrifid by numrical intgration of th Larmor radiation formula for a particl trappd nar th sparatrix of various ordr multipols. In ordr to procd furthr with a dtrmination of th influnc of multipol ordr on radiation losss, it is ncssary to hav rlations btwn I T ' T ' and maximum fild. A numrical calculation was don for an idal, linar multipol in which N currnt filamnts with qual currnts wr placd uniformly around a circl of radius a. Th maximum fild on th sparatrix and th flux pr unit lngth L btwn th sparatrix and -critical wr thn calculatd for various N. Th total flux T was assumd qual to twic th flux btwn th sparatrix and -critical, in ordr to account for th privat flux. Th rsults, normalizd to th sam maximum fild on th sparatrix, ar listd in th tabl on pag 3, assuming a constant -spac distribution of lctron nrgy dnsity.
-3- N I T (MA) IBa I/! T (wbrs)/bal 2 2 P(watts)/nT B a L 2 (quadrupol) 3.55 0.127 3.52 x 10-20 3 (hxapo1) 3.17 0.176 4.35 x 10-20 4 (octupo1) 3.00 0.177 4.14 x 10-20 5 2.92 0.168 3.82 x 10-20 6 (dodcapo1) 2.85 0.157 3.48 x 10-20 8 2.79 0.137 2.98 x 10-20 10 2.75 0.126 2.69 x 10-20 20 2.70 0.080 1.68 x 10-20 50 2.62 0.041 0. 84 x 10-20 (tokamak) 19.48 x 10-20
-4- Th tokamak cas was calculatd simply from and so it is rprsntativ of any circular cross-sction dvic in which th magntic fild is ssntially uniform. Low ordr multipols hav typically 5 to 10 tims lss synchrotron radiation than th corrsponding tokamak, and th advantag incrass linarly with ordr at vry high ordr. An additional improvmnt of <nt > /<nt > V could b xpctd if th -spac avragd lctron dnsity in a multipol is smallr than th volum-avragd lctron nrgy dnsity in a tokamak. A first ordr rlativistic corrction can b includd by rplacing T with T (1 + T /204000) vrywhr. Finit S and rabsorption of th radiation will, of cours, altr ths conclusions.