Interference effects in the double ionization of helium by charged particles

PhysicsLettersA 159 (1991) 257—260 North-Holland

PHYSICS LETTE S R A

Interference effects in the double ionization of helium by charged particles V.A. Sidorovich institute of Nuclear Physics, Moscow State University, Moscow 119899, USSR Received 21 June 1991; accepted for publication 13 August 1991 Communicated by B. Fricke

The interference between different double-ionization mechanisms by collisions of charged particles with helium is studied in the second order of perturbation theory series in terms of the interaction potential of the projectile with the helium electrons and the correlation potential of atomic electrons, it is shown that the contribution to the interference term in the integrated doubleionization cross section of helium comes only from the interference between the process of independent electron removal and the process in which an ejected electron, as a result of the interaction with the projectile electron, scatters on another helium electron, resulting in double ionization.

The influence exerted by the interference between different double-ionization mechanisms by collisions of fast charged particles with helium upon the magnitude of the cross section is a point of wide discussions in the literature (see, e.g., 11—4]) in connection with the established dependence of the double-ionization cross section of helium upon the projectile sign [1]. A special interest in the interference effects stems from their promise to shed light upon the experimentally observed [2] difference in the double-ionization cross sections of helium for protons and antiprotons. As it follows from previous study 11,4 j, the interference term in the double-ionization cross section is proportional to the cubed projectile charge and has opposite signs for the scattering of positively and negatively charged particles on the helium target. It is this term that contains the dependence of the double-ionization cross section upon the projectile sign. However, a clear understanding of the interference, namely, which processes contribute to the double-ionization cross section, is still lacking. In the present paper the interference problem in the double ionization of helium by fast projectiles is considered rigorously. We shall proceed from the results of ref. [51,where an expression for the double-ionization amplitude of helium colliding with structureless charged particles was obtained within perturbation theory with allowance for the electron correlation potential. According to ref. [5], it is convenient to write down the double-ionization amplitude in the form a(b)=a~’(b)+a’~(b)

,

(1)

where the amplitude a~(b) describes the transition of the helium electrons to the continuum as a result of the interaction of each of the electrons with the projectile and the amplitude ac(b) describes the transition as a result of the interaction of the projectile with only one of the helium electrons with the subsequent (preceding) correlation interaction of the helium electrons. Up to the second order in the interaction potential of the projectile with the target electrons Vand the electron correlation potential VC the amplitudes a’~ and ac are determined by the expressions [5]

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257

Volume 159, number 4.5

aIEA(b) =

—

2flfJ

PHYSICS LETTERS A

~ (I +P,,,,)

J

14 October 1991

~-4exp( —iq~~b)~(a,;—a,,—q1~v) q

it~V

xJ~fexP(_irb)(JexP(ip.r)lii>s(~

(2)

11_a,~_~iv)

and ~

x(iEó(E~_E~) + E~E~ (ff2

+ E?—E~ <~ exP(Iq’.r)Ik,,,>)~a~(b)+a~(b).

J’~lk,,,ii>

(3)

Here Z and t’are charge and velocity of the projectile; />, 1k> and If> are the wavefunctions ofa single helium electron in the initial, intermediate and final state, respectively, in the self-consistent field approximation, which we shall define here approximately as the Coulomb wavefunctions in the field of some effective nuclear charge 7*. 1, K and F are the initial, intermediate and final states of helium electrons; E~and E~are the energies of the helium electrons in the L-state with and without the inclusion of the electron correlations: P,1 is the permutation operator of the indices / and j: the symbol P means that the integral over dEK is taken in the sense of the principal value: a,, and a1, are the energies of the jth helium electron in the initial and final state, respectively: b is the impact parameter; q(p) and q’ are the momenta transferred, respectively, to one electron and to the helium atom in the course of collision; q~1(p11),q~and q1 (p~j,q’ are their parallel and orthogonal components with respect to the velocity vector t’. The first term in (3) describes the process in which an ejected electron, as a result of interaction with the projectile electron, scatters on another helium electron with the subsequent transition of both electrons into the final states. The second and third term in (3) describe the transitions of the two helium electrons into the continuum as a result of two interactions J~’andV’ through the virtual states. As shown in ref. [5], the shakeoff mechanism does not contribute to the double ionization cross section of helium if only the first correction to the wavefunction in the correlation interaction is allowed for. The amplitudes a’1 and a~correspond, respectively, to the first and two last terms in (3). To make a further transformation of the amplitudes a~ and ac (or a~,a~), we shall use the series expansion of the exponent exp(itr) in the spherical harmonics [6]. cxp(it~r)z=42t ~ ijf(lr) Y~,,,(Q,)Y1,,,(Q,)

.

(4)

wherej1(/r) are the /th-order spherical Bessel functions; Y,,,(Q) are the spherical harmonics; Q~is the spherical angle of the vector t. The continuum wavefunctions /~>.I/i> and 1k,,,> will be represented as an expansion in terms of the Coulomb wavefunctions of stationary states with a certain energy a, the angular momentum / and its projection upon the quantization axis tn [6].

1k> =

~

~ i’ exp(i~/)R~/(r) Y7,,,(Qk) Y1,,,(Q,),

(5)

where ~, is the phase shift; k is the value of the electron momentum: Rk/ are the Coulomb wavefunctions defined in the field of the effective charge Z’~. Substituting expressions (4) and (5) into (2) and (3) for a~, a~and a’~we get, respectively: 258

Volume 159, number 4,5

a~A(b)=

Z

—

PHYSICS LETTERSA

2

exp[ —i(ö~1~ +~2))]

(—1 ~

~

K1 K2t)

14 October 1991

l,.,n,/z,’nz

Y,,,,, (QK,) Y,2,~2(Q~2)

xD(K,K7v;blI,17,rn,rn2), a~(b)=

~

—

(6)

I )/

(—

K1 K2 V i,,,, /~.~flI,/2,~fl2

exp[ —i(~,’~ +ô~) ]C~,20C~’~,12,,,2

2~/1±/2+~fl

>< Y~,,,,,(Q,(, ) Y/2,,,2(Qk2)DI (ic, K2 U;

a~(b)=

bI 1, 1,

/2,

in),

(7)

2~) I

—i— K

~

(—I

)II+I2~/~+/2+m+I

exp[

—i(ô~+ö~

1 K2 V 1.,n 1,..,,,. 12 ~

X Y1,,,,, (Q,~,) Y12,,,2(Q,~2)D2(ic,,

ic2

v; bj 1, 1,

/2,

in)

(8)

,

where ic and ~ are the momenta of the helium electrons scaled to the continuum; ~ are the Clebsch— Gordan coefficients. The real functions D, D1 and D2 are used to denote the integral expressions from the product of the Coulomb functions Rk,, the Bessel functions, the spherical Bessel functions, the associated Legendre polynomials and the power functions. The associated Legendre polynomials andYirn(Qq) the Bessel functions appear 0g and çoq in the spherical functions Y~n(6q, ~) and the as a result of the respect separation ofazimuthal the variables integration with to the angle in the integral over the vector of the momentum transfer q with the use of the expression

J

d~qexp(—iqb+irn~q)=2~(—i)’V,,,(bqj).

(9)

The arguments, which the functions D, D 1 and B2 depend on, are given in brackets. The interference term in the double-ionization cross section of helium in accordance with (1 )—(3) and (6)— (8) originates from the quantity 2Re(W)=2Re(W1)+2Re(W2)

(10)

,

where K

W1 =a~(ai~A)~=

1~ +~2) Z 1K2 V

Ii, ,I~

~

I’i.,I’2

exp[

~

~l)

~ô~1)]

—i(~

‘fl,’flI,’fl2

x CRO!2oC/,,,/2,,,, Y1,~,(Q~) Y,2,fl2(.QK2 ) Y~,,,,(.Q,~) Y7’2,fls(QK2) XD(K~, ic2

v, bIt’, “2, rn’s, rn~)D1(ic1,K2 v; bIt, tel,

/2.

rn)

(11)

and 2~

_~,‘)

K W2 =a~(a~)~= 1 K2 V

~ 1,1,12

exp[

ih±l2±~n/’Il~~n~m’2+1

~,2>)J

—i(d)~~ +ô~

1’~.1’2

tfl,~flIjfl2 Pfl~,~fl2

x~

C~’1’1,,IZ,,I2Yi,~,(QKI)

XD(K~,ic2

Y12,~2(Q~2)~

v; bII’1, 1’~,rn~,m’2)D2(ic1,

K2

(Q,~)~ v; bli, 1, 12, rn)

.

(12)

If we take care of the differential with respect to the emission angles of one or two electrons of the doubleionization cross section of the helium atom, then, as one can easily see from (11) and (12), the contribution to the interference comes from both processes (described by the amplitudes a~and a~)entering the correlation 259

Volume 159, number 4,5

PHYSICS LETTERS A

14 October 1991

mechanism. Integrating expressions (11) and (12) with respect to the emission angles of the electrons w~get =

J

dQ~,dQ~,W1

xD(K,K2

i’;

=

~

KiK2L

1.f,,I2

i”

-- “i

‘~~c’~1211 ~

D (K~,

K7 V,

hi

12.

rn~ rn7) ,

hi!, 1,12, in)

(13)

and =

J

dQ~,dQ~2U~2=

~ /.1,./2

KiK2V

i”~”n~~n~+

‘C~o,2oC~,, 1,,,,,D(K1,

K~,~‘; b~l1,12. rn1,

in2)

XD2(ic1 , K2: v; hi!, ‘‘/2, in) (14) It follows from the properties of the Clebsch—Gordan coefficients C~0~20 and ~ that l+/~+ ‘2 = 2n (n=0, 1,2,...) and ,n=rn~+rn2.Then from relations (13) and (14) it follows that l~’~ is real and W2 is imaginary. Thus, the contribution to the interference term of the integrated double-ionization cross section of helium comes from the interference between the process of independent electron removal and the process in which an ejected electron, as a result of interaction with the projectile electron, scatters on another helium electron, resulting in double ionization (a part of the correlation mechanism determined by the amplitude .

a’1).

References [1] J.H. McGuire, Phys. Rev. Lett. 49 (1982)1153. [2] L.H. Andersen. P. Hvelplund. H. Knudsen, S.P. Møller, K. Elsener. K.-G. Rensfelt and E. liggerhoj, Phys. Rev. Lett. 57 (1986) 2147. [31i.E. Reading and A.L. Ford, J. Phys. B 20 (1987)3747. [4] L.H. Andersen, P. Hvelplund, H. Knudsen. S.P. Møller, A.H. Sørensen, K. Elsener, K.-G. Rensfelt and E. Uggerhøj, Phys. Rev. A 36 (1987) 3612. [5] V.A. Sidorovich, Phys. Lett. A 152 (1991)53. [6] L.D. Landau and E.M. Lifshitz, Quantum mechanics. Nonrelativistic theory (Nauka. Moscow, 1989).

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