Effect of screening as an eikonal correction

Volume 53A, number 1

PHYSICS LETTERS

19 May 1975

E F F E C T O F S C R E E N I N G AS A N E I K O N A L C O R R E C T I O N ¢r E.A. BARTNIK, Z.R. IWllqSKI and J.M. NAMYSEOWSKI

Institute of TheoreticalPhysics, WarsawUniversity, Warsaw,Poland Received 20 March 1975 Imposing the known exact solution for the unscreeened Coulomb field we develop an eikonal formalism for a correction due to screening. This correction turns out to be a multiplieative phase factor, easy to calculate even in the case when the pptential is only given numerically. We avoid the partial wave expansion. Let us focus our attention on the high energy Sommerfeld-Maue wave functions, needed for the evaluation of the high energy bremstrahlung, pair production and elastic scattering in the presence of a given screened Coulomb field [1,2]. They are in ~kout = [exp (ip-r)] (1 - { i or

in

V/E) F°Utu ,

where = and u are Dirac matrices and spinor, respectively, E = (p2 + 1)1/2, and F ~°nutobey the equation (V2+ 2ip" V For

2EV)F= O.

(1)

V= VC = -Ze2r -1 , the exact solutions ofeq. (1) are known as

F~Ut= F(1 +- iZe2Ep -1) [exp ({,tZe2Ep-1)] 1FI(~ iZe2Ep -1 ; 1; ~ipr - ip" r ) .

(2)

The application of the Sommerfeld-Maue wave functions to atomic processes has been studied in detail by Bethe arid Maximon [1] and Olsen, Maximon and Wergeland [2]. We want to replace the second order differential equation (1) by a simply solvable first order differential equation. There are many ways of doing Uhis and our choice is specified by the following requirements: i) for V = VC the exact solution F C of eq. (1) has to be also the exact solution of our first order differential equation in the whole configuration space, ii) the first order differential operator 2 i p . V, present in eq. (1), has to appear in our first order differential equation. These requirements are met if we take the following first order differential equation

[2ip ~z - A(p, z) - 2EV? F = O ,

(3)

where the z-axis is chosen along p, p is the cylindrical coordinate perpendicular to p, and A(P, z) is chosen in such a way that for V = VO and F = F C, eq. (3) holds in the whole configuration space. An implicit definition of A(p, z) ts Z

Fc(P,z)=Bex p - i E p -1 f

[Vc(p, z') - E-1A(p,z')l dz'},

zo(p) where B is a constant, and zO(p) has to be determined from the boundary condition. The screened potential V we write as a sum VC + V1 and get the appropriate F from eqs. (3) and (4) as *~Supported by NSF Grant No 36217.

(4)

Volume 53A, number 1

PHYSICS LETTERS

19 May 1975

Z

F=Fcex p I - i E p - l f

Vl(P,z')dz'1 .

(5)

zo(o) Thus the multiplying phase, representing the effect of screening in eq. (5), has the form of an eikonal phase evaluated from the small and slowly varying potential V 1 = V - VC. To satisfy an appropriate boundary condition for the outgoing solution, for example we have to demand

zo(o) f v(P, z') dz' = Ze 2 In [pZp(zo + r0) -1 ]

(6)

where r 2 = p 2 + z 2, and p2 = (r - z) (r + z). Including both equations (5) and (6), we get F °ut = F~ ut exp ( i x ) ,

(7)

where Z

X(p,z)

= Ze2Ep-l ln [p(~/p2 + z2 -

z)]

-- Ep -1 f V( ov/~ z'2) dz ' .

(8)

The correcting phase X, which takes care of screening, satisfies a very simple equation ~z-zX=

E

V . ~" 1"

(9)

In a separate paper [3] we evaluated riumerically X, given by eq. (8), for Vtaken as the modified T h o m a s - F e r m i potential [4, 5]. We found there that X, thus also the effect of screening is most important in the p ~ z ~ p region, out of four regions O ~ 1, z ~ +p; p ~ p , z -~ -+p interesting for the high energy bremsstrahlung and pair production. The importance of X increases with the increasing charge of nucleus Z, while the variation of X with p is very mild and practically given by an additive term in p, present in eq. (8). It is possible to justify our method by inserting our solution given by eq. (7) into the exact second order differential equation (1). Since F c satisfies eq. (1) for V= VC , we get for X the following second order differential equation

LzP-I[iv2x-(VX) 2] +(p VFCFc p )" V X - ~ "

g 1 •

(10)

if we disregard the term i ( P F c ) -1 VF c in eq. (10), then eq. (9) is the eikonal approximation to eq. (10). Notice, that in both eqs. (9) and (10) in the inhomogeneous term we have the small and slowly varying potential V 1 . This is in contrast to the eikonal approximation applied to the starting eq. (1). In the last case the eikonal approximation can be justified only for large impact parameters p ~ p, where V is small. However, for small p ~ 1 in the usual treatment of the problem [2] one is forced to make WKB approximation in each partial wave. For p and z such, that a strip p < p - I and z ~> _ p - 1 is excluded, one can show [3] that it is legitimate to approximate i VFc p . ~

PFc

p_p__.

PP

Furthermore, for p i> 0.1 and z ~< 2p one can prove [3] that the ratio of the omitted terms in eq. (10) to the retained one

d=- ~ p -11i72x

-

(vx)ZI/(Ep-I V1) ,

with × givenby eq. (8), is smaller than 3%. This means that our × satisfies the exact equation (10) to accuracy better

Volume 53A, number 1

PHYSICS LETTERS

19 May 1975

than 3% practically in the whole space, namely, in the region p t> 0.1, z <~ 2 p , which expands with the increasingp. The initiation o f this work b y Professor R.H. Pratt and several stimulating discussions with him are gratefully acknowledged.

References [1] H.A. Bethe and L.C. Maximon, Phys. Rev. 93 (1954) 768. [2] H. Olsen, L.C. Maximon and H. Wergeland, Phys. Rev. 106 (1957) 27. [3] E.A. Bartnik, Z.R. lwi~iski and J.M. Namyslowski, Warsaw University preprint IFT/3/75, February 1975, submitted to the Physical Review A. [4] H.K. Tseng and R.H. Pratt, Phys. Rev. 3A O971) 100. [5] P. Csavinszky, Phys. Rev. 166 (1968) 53.