The cross section for the capture in different excited s-states of an
electron by  an ion passing through gaseous mailer has been evaluated; the method followed is an extension to that of Brinkman and Kramers. It is found that the first term of the series obtained here for the Is-ns capture agrees with Brinkman and Kramer’s approximate result

Sil, N C

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ELECTRO N C A PT U R E BY IONS PA SSIN G  TH RO U G H  G A SE Sn y  N . C . S I Lr)KPAKTMKNT OF TnhORK‘nC\L PUYSTCS, INDIAN ASSOCIATION FOK THK CULTIVATIONofScikncf, Iadavpitr, Calcutta-33.{Received for publication, March 30, 1954)ABSTRACT—The cross section for the capture in different excited 5-states of an electron h\ an ion passinj;^  through gaseous mailer has been evaluated ; the method followed is an extension to that of Urinknian and Kramers. It is found that the first term of the scries obtained here for the is-ns capture agrees with Brinkman and Kramers* approximate result.28I N T R 0  I) U T I O NThe itiiportancc of the problem of the capture of an electron by an ion passing through gaseous matter has, of late, increased in view of its application to tlie explanation of some of the lines emitted by solar corona (Saha, 1942) and aurora borealis (Fan and Meinel, 1953). 'I he theoretical study in this Hue has been mainly done by Oppenheimer (1928), Brinkman and Kramers (1030), Massey and Smith (1933), Saha and Basu (1945), Jackson and Schiff (1953) and others. Brinkman and Kramers Iiave considered in detail the case in which an ion captures in its is  orbit an electron whicli was originally in the i.v orbit of the atom constituting the gaseous matter. The present work is an extension of the theoretical investigation of Brinkman and Kramers for the more general case of capture of the electron in excited orbits. The necessity of considering the cross section of capture of electrons to excited orbits arises from the suggestion of Saha (1942) to explain the origin of lie-lines in the spectrum of the chromosphere. In it we notice, in addition to the extremely strong neutral helium lines, the well-known line of ionised helium A 4686A whose appearence is unexpected in view of the fact that this line has an excitation potential of nearly 75-25 volts while the ordinary excitation of the chromosphere is 9 to 14 volts. The occurrence of this latter line has been ex|ilained by assuming that high velocity ^-particles generated in the solar body due to some nuclear process get slowed down by losing energy through ionisation. After being considerably slowed down they begin to capture electrons in excited orbits which jump back to lower orbits emitting the well-known lines of ionised helium. The capture problem is expected to play an important role in explaining the spectrum of aurora borealis. The aurora borealis is most likely to be excited by high-speed ions penetrating into the upper atmosphere. Meinel (1951) detected a violet Doppler w ing onHa line extending to an energy greater than 57 Kev thus proving the existence of protons as one of those causing the excitation. A preliininaiy experiment by ^leinel and Fan (1Q52) using accelerated protons with rarefied air sample, showed a striking similarity between the laboratory spectrum and that of the aurora.Electron Capture hy Ions Passing Through Gases 233D K R I V A T  I O N O I ' T  H  U  R K S U J ,  TThe type of collisoii considered in this paper comes under the general head of rearrangement collisions because here the electron initially attached to one hydrogen-like atom of charge /c  forms finally another hydrogen-like atom of charge Z'e hy simple rearrangement. The dilTerenlial cross section of a general binary collision, in which a system /I in state collides with a system B in state n to form systems C in state s and D in state f, has been derived (vSehiff, 1949) using the first Born approximation. In our problem, of the two initial systems, one is a hydrogen-like atom with charge /c  and mass AM  of the nucleus to which an elctron of charge and mass m is bound in the n,l,m 01 bit and the other is a free particle with charge //c and mass A'M  moving with the velocity Y with respect to the atom ; the final systems are respectively a hydrogen-like atom with charge //c and mass A^M of the nucleus, the electron being in the orbit, and a free ])article withcharge / e  and mass AM relative to which the atom moves with a velocity v'. P'igure I gives us the co-ordinate system for our collision process.Z c, AMFigure i. Co-ordinate syst<“iii. The points marked C  and C arc centies of mass of the initial and final atoms.Following the calculations given in L,. 1. Schiff’s ‘Quantum Mechanics', we get the differential cross section for capture of the electron in the final state from the initial n,l,m state as in Jackson and, Schiff’spaper U953) viz,(i;5—18521*—5234 N. C. Silwith T -  j cxpf (P') V exp W dr^drn, (2)where (**) the ncrmalised initial state wave-function for theelectron m aton /  and (v') is the normalised final state wave fniictioiifor the electron in the aitom Z \I (3)  [  (4)(h /27t)K ' =  /x/Y' ' ,  fi,f =  AM (A 'M  + m)l{A-\-A')M+ m ' ,and V is the perturbation  T Ianiiltonian .Using as the independent coordinates r, r' instead of r, R we may write(h i 27t) K ^  fii'v I1 =  jc xp {- i Cy ) '^ v\v ,m  y )  V{v t^')cxpii'B.T)xf r^t,i,m (v)where C =  K — and B  =  -  K -  K 'A M + ni AM +  VI5^)(6)The perturbation Hamiltonian consists of twn> parts ; (i) the Coulomb interaction between the electron and the incident particle and (2) the Coulomb interaction between the nucleus and the incident particle. Brinkman and Kramers (1930) considered only the first term neglecting the second one in their calculation of capture cross section for electrons from initial i.v orbit to final IS orbit. Saha and Basu (1945) extended the method of Brinkman and Kramers to calculate the capture of electron from i.v to 2p orbit. It lias been mentioned by Jackson and vSehiff (1953) that in an exact calculation, the second term in the perturbation Hamiltonian would have actually a negligible contribution ; howwer, they claim that better results are obtained by taking into consideration the contribution of the second term which is no longer negligible when the calculations are based on the approximate method of Born. Here we shall follow Brinkman and Kramers, and shall carry out the calculation for the capture of electron in the final ns state from initial ij? state by the first of the two alternative mothods given by them.As the perturbation potential wc may take either the initial perturbationy/ Ze^or the final perturbation F / =  — — . However, it has been proved r rby Ja ck so n  and SchilT (19 53) in the first Born approxim atio n  the a m b ig u ity  in the choice of the perturbation  potential does not lead to an y am b igu ity  in the resu lt fo r the transition  p ro b ab ility . 1 ‘hey show  that whenI -----I -------------- (15) IC),\ 2 m/ /and for V =  V //{hl2n)^B^2Wt+ e (B) <Pn',l-,m' (C),(7a)(7b)where nii and m/ are the reduced masses of the electron when attached to the atoms /  and Z' rcvspectivelyElectron Capture by Ions Passing Through Gases 23$AMni , A Mmm f =  , , , ,  ,A M +  m A M  ^ m(B) = J e x p  (iB.r) dr,p,/u' (C) =  J exp (/r 7') V'V./'.N/ (**',* dr A(8)(9)and <•:, are the binding cnergic"  ^ of the electron 111 atoms Z and Z ' respectively.Now from the conservation of eneigy requirement viz.  ^we get with the help of equations i.\\ (C>) and (8)ihl2irVC'^ ,—  ~r t  —  - —  ^ T" < (10)2 m f 2 in,Hence I =  l \The evaluation of <l>n,i,m (K) for a hydrogeii-like atom with charge Ze of the nucleus has been done by Podolsky aud Pauling (1029):— ( / ) !  I r ( "^.“” 1  (jj)(« 4- /} Iwhere are the polar co-ordinates of K,t,— with being the reduced mass of the electron itlY»,the atom, and d'J is llie Gegenbaur function which may be defined by the generating functionQ (12)w <C IHence from (i), (7) and (n ), the differential cross section is known in general for capture of the electron in any final state from any initial state. From (ii)  we have in particular236 N. C. Silwhere y i=  ^ withUo1 -M ^and V « =  —  with a „ — - —  |7ia o nifC j(1*1)Willi the Iielp of ( 7 )  and (10) we getmi + •15^To get the total cross section wc require the integral of | f I" over all directions of crieiitation of tlie solid angle IT Now (cos ©) is thecocfricieiit of the expansion of ( i --2w cos H + t r r ‘ when i< < i.( cf. eqii. (12) ).HenceIC i- ,  ( c » H , r  \ U . y ' ( - ) '  (. sin ei>( sill B  j } 2 ( r +  i)}!Putting/ •2 _ y  2” -  COS -  I — 2.V,we haveI — cos O(16)(17)(18)and from (61.v(i - x )  -/ ’3 / 2(19)dx =  - 2 \ K K ' , (ao)wliuie \ =  A 'M ! (A‘ M + m) ami d is the angle between K ami K'. Thus Utilising the above results we get from (15),'f:'T 2 •'* • o)2n\ 1 / 1'^ sill d  ^ ^  1 -8\ ’ (i “ A/d.v,  (21 ;. *4.1 c \ —r \r n ' ( w * - r * ;  arwuh 6 .(0 ,« )  =  ( - )  a ------- f ^ + l T y r ' '   ^ ■Under the usual condition/< = -  A K'  »  Yn» \ K - \ K ' \(aa)(23)Eleciron Capture by Ions Passing Through Gases 237i l;en from ^iS) and (6) ue havu(K'-i-AK'ran I x(f  ^=  o } = X o ^(/C~A/CT + Y1 >(Vi"+>'■ ) + -"o'" + i ,I I hirer- \ 'vide appeiuUx)withFor the evaluation of tlie iiitej^ral in the right hand bide of the c(]n.<21/ we may note that j  a * ' Mi x) 'dx  is the incomplete hcta-fuiictionF ho(5 ■<■ mi *+• r) the numerical value of which can he obtained easily.Now since m «  xV/ wc may write — m, A=s£j^  andah,—a„. Hence from 'i)  & (21) the total captiiie cross section from \s orbit to 7]s orbit is given by2  .Sh (0,7?) F x .>(5 + r,i + r). ... (25)=  ..a,"' /?/ 2.t) X’a/1) 1 s c r s s I () NThe e.Kpressioii given by B. K. for the i s —/os' capture cioss bcc<ion is approximate being only the first term of oui series in ecjii. (25). In fact the contribution of the other terms of Die scries is not at all negligible for low energy of the incident particle IJowevei, for very high energy of the incident particle the result on the basis of our expiession difTers very little from that of B. K ., the main contribution being due to the first term. Thus at high energies of the incident {/article vve have an inverse 77 ‘ law for the capture cross section given also by Oppenlieinicr.A P P Iv N I) I XTo prove'n^  + (/V -  A/\ 4. 4- -V - + l6n-‘Mr-wc have from (3) and (4)whereandS=*(A /V)M + m __ niv238 N- C. SilFrom (loj we may write with the help of (3), (4), (6) and (14),4^;=  Y. + V:r“ \ a M.A'I hM / ,v . mv\v — v ----h7i ^I 1 AM.A'M ,H i  '(•y — 7>') + -d'Mo-> —■ " Jupto the first order of approximation. We take {v — v' )lv  and m/M as small quantities of the hist order.Then2n] AM.A'M {v — v') + /I'Mcr V =- 7T - V  «47T(T— -I- 710-,S  ( h Hence follows the result.A C K  N 0  VV h H 1) G M H N TThe author is thankful to Dr. D. Basu for suggesting the problem and for his kind help and guidance during the progress of the work.R K F F R ]C N C E SBrinkman, H. C. and Kramers, H. A., 1930, Proi.  Awst., 33, 973.Fan, C. Y. and JMeinel, A. B., 1953, Asttophyi>. J o u t 118, 205.Jackson, J . I), and Sdiiff, II,, 1953, Pliy, Rev,,  89, 359.Massey, H. S. W. and Smith, R. A., 1933, Proc. Roy. Soc. {London)^ A 142, 14?.Meiuel, A. B., 1951, Astropliys. Jour., 113, 50Meinel* A. B.,aud Fan, C. Y., 1952, Astrophys. Jour., 118, 330.Oppenheimer, J . R , 192B, Phys. R ev  , 31, 349,Podolsky, B, and Pauling, L., 1929. Phys. Rev.,  3 4 , icg- Sahji, M. N., 1942, P w c .  Nat. Inst. Sc., 8 , 99.Saha, M. N. and Basu, D., 1945, lud. J. Phys., 19, 121.Schilf, E. T., 1949, Ouantum Mechanics, McOraw-Hill Book Co. P. .730.

Electron Capture by Ions Passing Through Gases

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Electron Capture by Ions Passing Through Gases

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