Chowdhury, A Roy

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ta the £ditotThe Board o f  Editors does not hold itself responsible fo r  opinions expressed in  the letter published in this section. The notes containing short reports o f  original investigations commu­nicated to this section should not contain m any figures and should not exceed 500 words in length. The contributions reaching the Secretary by the 15th o f  any month may be expected to appear in the issue fo r  the i%ext month. N o  p roo f will be sent to the author.26A  N O T E  O N  A  R E L A T IV IS T IC  F O R M U L A E  F O R  IO N IZ A T IO N  B Y  P R O T O N  IM P A C TA . R O Y  CH O W D H U R YP h y s i o s  D e p a r t m e n t , J a d a v p u r  U n i v e r s i t y , I n d i a .(Received August 23, 1968)W e consider the collision of a proton with a hydrogen like atom, in which process the electron become ionized. (W e neglect the translatory motion of the nucleus).Let (}>, \[r denote the proton and electron field operators. Then the initial bound state wave function is defined as (Roy, 1960 a).where l^i> = f 2 d?hgt{k)a*-^ {k)\0>( 1 )(2)the bound atate vector for the sini^ le electron. a*\lc) is the usual notation for creation operator of an electron, and A  for pioton. Also we have=  f S  a*{p)Up*e*^ ‘’*d^ p iis the Dirac Spinor. Now (1), (2), (3) give :ilfp(x) =  S cPkg,{k)Up*{k)e*^ ‘'‘(.1(3)(4)Let Vp(k) — J ^p(x)er^*'*ihi. denote theFourier transform of bound state wave function; then (4) givesFp(i) =  S  u»g^k) (5)444Letters to the Editor 4 4 6which is the equation detarmining tho four-vector <7^ (i). It is interesting to note that we need not determine gr’s explicitly, but can use equation (5) to convert them in terms of F(fc), which are detorminod easily.The matrix element of the process is given b y :where I v'/,>  =  J dH-r,,(lc)a‘^k)A -nA) 10>I f f >  =  A+{i"f)nHgf) I <>>— Ij- [  J P\{ (^!r')y^A (^z')4>{T')}{ {^x)y^A„{x)\/r{x)]]dxdr' (b) with the help of ( 7) givesM „ _  D. j,,.(6)(7)( 8)with p =  Pf--Pi-\-gf, D contains some constant factors. In non-ndativistic limits (8) becomesM.fi _  <^lun(Pf-Pi+Q/) f f T. ‘ . e,i{P f-p t).R »?/•»• d m (9){P i-P f?  J Jwhich is the usual Born approximation (Mott and Massey 1949). For the Is state the solution of Dirac’s equation is (Akheiezer and Borottross Keii; 1953):^ ( r )  == N i ( \ + i k j y Q * f \  n =  rjrV(p) =  iV^ i( Ji+i^i7o?pJ^2MO) • • (fO)whcore J j =  — J\lv fcj =  J ctz, Fiom (8) we see S| ®propoitional toACP.-P,)whore A stands for:A [A^(P,)r'‘A+(A)r^]A[A+(?^)0A-(O)Q] and Q =  7^( J i+ » A  J.p J,)Now the differential oross-aection for the process is given by :dcr =  27rppL | Mft | ®/e.where (R oy, lOtiOb)( 1 1 )( 12)(13)(14)4 4 6 Letters to the Editorwhen (8), (11), (12) are ukihI in (13) the expression for the differential cross section boils down to:d(T 4a„»d^Pf(Pqf(2n)* P , (Pt -P f )*  ■ f  ’where:(15)(If.)A, =  [2(P,.P<)m4-4(P,.P/-M2)m+2(P/9^)(P,)„+2(P.-.g;)xX (P f)o -2 {P ( .P ; ) ig f )o -2 (q fU P i .  P f -  M^)]Az =  aV[2(P,)o(^/?/)+2(P/)o(Pj.(?'/)-2(5y)oX{Pt.P f-M ^)-2(Pi.P f)(q j)^^where (q) represents tho zorooth (iomponont o f the four-voetor q. The formula for cross-section contains two terms and A 2, of which tho latter is proportional to For tlie physically interesting situation wo may neglect 2nd part and tho formula (15) then gives., _  e< d^Pfd^qf S (ef+ E f—E)7T«a3 • mPi • (Pi -P f )*  • {z -^ + a ^ H P f-P t+ q fn *X /  2-1-\ ^  Jif* M m  M m  I . .  (17)(<?<> Pi) boiug energy momentum of tho incoming proton. Formula (17) is easily seen to bo reduced to that of tho usual Born approximation results in non-relati- vistic limit.Author is grateful to Dr. T. C. Ray of Physics Department for suggesting the problem and helping to solve it. Ho also wisher to thank the Government of India for a C.S.I.R. fellowship.R E F B B E N C E SAkheiezer A. I. and Berettross Keii., B., 1953, Quantvm Electrodynamics, Moscow. Mott. N. F. and Massey, H. S., 1949, Theory of Atomic Collision, Clarendon Press, Oxford.Roy, T., 1960a, Zeit, fur, Physik, 168, 142.Roy, T., 1960b, Indian J. Phys., 86, 29.

A Note on a Relativistic Formulae for Ionization by Proton Impact

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A Note on a Relativistic Formulae for Ionization by Proton Impact

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