Coulomb scattering in intense, high-frequency laser fields

Volume 112A, number 3,4

PHYSICS LETTERS

21 October 1985

C O U L O M B S C A T F E R I N G IN I N T E N S E , H I G H - F R E Q U E N C Y L A S E R F I E L D S M.J. O F F E R H A U S a, J.Z. K A M I N S K I h and M. G A V R I L A a

FOM-Institute for Atornic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands h Institute of Theoretical Physics, Warsaw University, 00- 681 Warsaw, Poland Received 10 July 1985; accepted for publication 1 August 1985

A recently developed theory for free-free transitions in intense, high-frequency laser fields is applied to study the laser-modified elastic scattering in a Coulomb potential (case of linear polarization). Under the conditions of the theory the scattering can be viewed as happening in a time-independent dressed Coulomb potential. This is modelled here by its spherical average so that a phase-shift analysis can be applied. We study the change of the Rutherford cross section as a function of the electron energy, the laser parameters, and the nuclear charge Z. Large modifications are shown to appear for values of the parameters corresponding to presently existing high-frequency lasers. Characteristic Coulomb-interference oscillations are displayed at small angles.

Substantial effort is now being invested in the development of high-frequency lasers operated at high intensities. Atomic interactions are drastically modified in laser fields of this kind, and challenging theoretical problems arise. We have recently presented a theory for electron-atom collisions under such conditions [1,2], and in the following we shall discuss the case of elastic scattering from a Coulomb potential. The laser radiation was assumed to be purely single mode, and the electron-atom interaction was represented by a potential V(r). It was shown that for sufficiently high (time-averaged) intensities I and frequencies w, and for low incoming electron energies (the quantitative criteria were given in ref. [1 ] ), the elastic scattering in the radiation field is described nonperturbatively by a time-independent Schr6dinger equation H~b = E~k, with H containing the "dressed" potential: 1 1 du Vo(ato'r) =-~_fl V(r +OtoU) (1 - u 2 ) 1/2" (1) Here at 0 = _ s 0 e ,

s0 =i1/2 6 o - 2 ,

(2)

and e is the polarization vector o f the plane wave. We consider the case o f linear polarization, i.e. e is real. All our formulas are written in atomic units; the au of

0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(time averaged) intensity is I 0 = 3.51 × 1016 W/cm 2 . The dressed potential, eq. (1), can be looked upon as created by a linear distribution of "charges" extending from ---~0 to +a 0 along e, with density o(~) = (1/Trs0) [1 -- (~/s0)2]

-1/2,

(3)

the unit of "charge" generating the potential V(r). In the regime we are considering, 6o and I enter the scattering problem only through s 0. If the original V(r) is spherically symmetric, V0(0t0, r) is axially symmetric around at 0. This complicates the scattering problem, as the azimuthal quantum number l is no longer conserved. Here we shall adopt a simplified approach, in which V0(at0, r) is modelled by its spherical average V0(s0, r), which is the first term o f its Legendre expansion. The problem is thus reduced to a phase shift calculation. This should give the right order o f magnitude for the elastic scattering, particularly at low energies, when the electron wavelength is larger than the extension of the line of "charges", i.e. p s 0 £ 1 6o is the electron momentum). On the other hand, interesting features related to the dependence o f the cross section on the polarization vector e (or~to) will thus be lost. By inserting the Coulomb potential V(r) = - Z / r into eq. (1) and averaging over all directions of at0, one finds: 151

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-z (2 arcsin p

V0(a0, r) = ~

- p In 1 - (1 _ p 2 ) 1 / 2 ]" Z aop,

-

p1>l,

(4)

where p = r/a O. Thus, for r/> aO, VO coincides with the original Coulomb potential V. For r < ao, ~'0 can be expanded as

v0(%,r) = - ~

P +.=0 ~ An pz~

(5)

which displays a logarithmic singularity at the origin. This is much weaker than that of the Coulomb potential. As is easily seen, V0(a0, r) is the potential energy due to a spherical symmetric distribution of electric charges extending up to r = a0, of radial density: fir) = 2Zo(r),

(6)

with the function o defined by eq. (3). Our scattering problem resembles thus that of an electron probing an extended charge nucleus of density r(r) and radius a0, see ref. [3] ; in contrast to this, however, we are dealing with a nonrelativistic case. The determination of the continuum solutions of the potential F"0 of eqs. (4) and (5) raises some interesting mathematical points on account of its logarithmic singularity at the origin. A number of results have been obtained lately for the bound state problem of the purely logarithmic case (all A n = 0 in (5)), e.g. see refs. [ 4 - 6 ] . By expanding the lth reduced radial partial wave as follows oo

oo

PI(P) = pl+l ~ (p2 in p)m ~ rn=0

a(lm)p2n,

(7)

n=0

we have uniquely determined the coefficients 4 ) n by

recurrence relations. [Eq. (7) turns out to be a generalisation to our problem of the expansion of Gesztesy and Pittner (ref. [4], eq. (4.16)), and Mtiller-Kirsten and Bose (ref. [5], eq. (62)).] We have checked numerically the convergence of the series for small p (uniform convergence could be proven by Gesztesy and Pittner for their problem). On account of its behaviour for p ~ 0, it represents the physically acceptable solution. It can be used to start the numerical so152

21 October 1985

lution in the vicinity of the origin. Beyond/9 = 1 the solution goes over into a linear combination of the regular and irregular Coulomb functions, and the extra phase shifts 6/, due to the deviation of V0 from a Coulomb potential, can be determined. For an attractive potential (Z > O) 5 l is negative, and for given l and E at small values a 0 , its dominant behaviour reads: 5 / = 7 22/e-'WlP(l + 1 +i7)12 r ( l +-~) [(2l + 1) !] 2 (l + 1)(2l + 3)x/-ffX (pa0)2/+2 [1 + O(a0)] ,

(8)

in which 7 = - Z i p . Eq. (8) was obtained from the well-known expression for sin 5 l given in ref. [7], eq. (X.72), by setting iz equal to our ~'0 of eq. (4), putting V = - Z / r , and replacing the Yt and Yl by their common dominant behaviour at the origin. For values a 0 ~< 10 -2 , eq. (8) agrees quite well with the numerical computation. For large values of a0, on the other hand, we find analytically (at given l and E) the behaviour: 5l = 7 In ao + XI(P) + O(1/a0),

(9)

where ×I(P) is independent of a 0. Eq. (9) can be derived from an extension of the usual JWKB phase shift expression to the case of a modified Coulomb potential like that of eq. (4), e.g. see ref. [8], § 9.2.3. It is limited by the condition: P~0 >> l + I- Note that, although p is fixed and can be small in our case, the JWKB approximation still applies, because at large a 0 the magnitude of the potential V0 (and its derivatives) becomes small, while its range increases as c~0 . The logarithmic build-up of 5 l with a 0 displayed by eq. (9), is peculiar to the Coulomb long-range behaviour of F"0. For a0 ~> 102, this was quite well checked numerically. Because of the long range behaviour of the potential, eq. (4), the scattering amplitude is given by the theory for modified Coulomb scattering (e.g. see ref. [8], § 6.3.):

f(O ) = fe(O ) + f'(O ) , fc(o)

-

(1 O)

-v

2p sin2(0/2)

X e x p [ - i 7 In sin2(0/2) + 2io0] ,

(11)

Volume 112A, number 3,4

PHYSICS LETTERS I

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Fig. 1. Ratio R of the elastic scattering cross section for the averaged dressed Coulomb potential, eq. (4), to the Rutherford cross section. Nuclear charge Z = 1. Values of the electron energy E (in Ry) and of s 0 , as indicated (for the definition of s 0 see eq. (2)).

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f'(O) = ~

PHYSICS LETTERS

tion, eq. (13), to the Rutherford cross section at Z = 1 for a number of electron energies E and values of s 0. The energies E chosen satisfy the condition of validity of our theory: E N co [ 1 ], for co attainable with some existing high-frequency lasers [10,11 ]. The values considered for s 0 have also been achieved experimentally. (When the laser of refs. [10,11 ] is operated at I = 1015 W/cm 2, we have s 0 = 3.1 and at I = 1016 W/cm 2 , % = 9.8). In all cases R ~ 1 as 0 -> 0. This is due to the fact that fc(0 ) and d o c / d ~ become infinite as0 ~ 0,whereas f'(O) stays finite. Moreover, for 0 -~ 0, R has infinitely many oscillations of increasing frequency and decreasing amplitude. These are due to the presence of the Coulomb phase factor, see eq. (11), in the interference term ofeq. (13) (but not in doc/df2 ). Indeed, by setting 7 = 0 in the phase factor, the oscillations disappear and R either becomes a monotonically decreasing function of 0 (for small % ) , or has a broad maximum and then decreases (for larger s0). The amplitude of these Coulomb-interference oscillations decreases for 0 -+ 0 because the relative importance of the interference term in eq. (13) diminishes compared to doc/d~2. Note that, as seen in fig. 1, the Coulomb-interference oscillations occur in an accessible range of experimental parameters. (They were not de-

~ (21 + 1) exp(2io/) l

X [ e x p ( 2 i 6 l ) - 1] Pl(cOsO).

(12)

Here 0 is the scattering angle,f c is the Coulomb amplit u d e , f ' is the extra contribution due to F"0 - V, and a! are the Coulomb phases. The modified cross section is do/d~2 = doc/d~2 + 2 Re fcf'+ if'(0)l 2 ,

(13)

where the first term at the right is the Rutherford cross section and the second representsthe interference of the Coulomb and short-range amplitudes. Eqs. (10) - ( 1 3 ) are similar to those used in the analysis of the classical p r o t o n - p r o t o n collision experiments at low (nuclear) energies with the short range nuclear force taken into account; see ref. [9], ch. 10, § § 5 and 9 (exchange effects are absent here). For smalla0, eq. (8) shows that only the l = 0 phase should be considered; to lowest order, 8 0 = O(~2). The cross section, eq. (13), can then be cast into the form used for p r o t o n - p r o t o n scattering (with exchange neglected), see ref. [9], ch. 10, eq. (5.1). In fig. 1 we present the angular dependence of the ratio R = (do/d~2)/(doc/dfZ) of the modified cross sec1.6

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21 October 1985

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tected in the p r o t o n - p r o t o n collision experiments, since there the energy had to be high enough, 7 ~ 0, to overcome the Coulomb repulsion so that the particles could approach within the range of nuclear forces). For large scattering angles fig. 1 shows that the modified cross section is considerably reduced with respect to the origical value, because o f destructive interference in eq. (13). The scattering amplitude for nuclear charge Z is related to that for nuclear charge 1 by the scaling law:

f(Z, ao,E,O)=Z-lf(1,oLoZ, E/Z2,0 ) .

(14)

In fig. 2 we display the effect on R o f the increase o f Z, at given E and a 0. It is apparent that the Coulomb oscillations will extend to higher angles, and become quite regular. F o r Z = - 1 (positron-proton scattering) and the parameters of fig. 2, R = 1 for all angles (not drawn). This happens because in this case the positron wave function does not penetrate into the region r ~< a 0 where the potential is modified by the laser field, and the scattering remains purely coulombic. We conclude that in the high-frequency, high-intensity laser regime studied here large effects on the elastic cross section may be expected in a range o f parameters now becoming accessible to experiment.

21 October 1985

This work was supported by the foundation FOM, which is a division of ZWO (the Netherlands Organization for the Advancement of Pure Research), and by the Polish Ministry of Higher Education, Science and Technology, under the project MR-I-7.

References [1] M. Gavrila and J.Z. Kaminski, Phys. Rev. Lett. 52 (1984) 613. [2] M. Gavrila, in: Atomic physics, Vol. 9, eds, R. van Dyck and E.N. Fortson (World Scientific, Singapore, 1985) p. 523. [3] L.R. Elton, Nuclear sizes (Oxford Univ. Press, London, 1961). [4] F. Gesztesy and L. Pittner, J. Phys. A11 (1978) 679. [5 ] H.J. Miiller-Kirsten and S.K. Bose, J. Math Phys. 20 (1979) 2471. [6] C. Quigg and J. Rosner, Phys. Rep. 56 (1979) 167. [7] A. Messiah, Quantum mechanics, Vol. 1 (North-Holland, Amsterdam, 1961). [8] Ch. Joachain, Quantum collision theory (North-Holland, Amsterdam, 1975). [9] R. Evans, The atomic nucleus (McGraw-Hill, New York, 1955). [10] T.S. Luk, H. Pummer, K. Boyer, M. Shahidi, H. Egger and C.K. Rhodes, Phys. Rev. Lett. 51 (1983) 110. [11] K. Boyer, H. Egger, T.S. Luk, H. Pummer and C.K. Rhodes, J. Opt. Soc. Am. B1 (1984) 3.

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