e:-

Nuclear Instruments and Meth0ds in Physics Research B40/4] (1989) 333-339North-Holland. Amsterdam

333

ELECTRON DENSITY FLUCTUATIONS INDUCED BY ION CLUSTERSIN CONDENSED MATTER

1.M. PIT ARKE

Fisika Tearikaa Saila, Zientzi Fakultatea, Euskal Herrika Unibertsitatea. 644 Pasta Kurxatila, 48080 Bilba, Basque Calmtry. Spain

P.M. ECHENIQUEMaterialen Fisika Saila, Kimika Fakultatea, Euskal Herrika Ullibertsitatea. 1072 Pasta Kurxatila. 20080 Danastia.

Basque Cauntry, Spain

f~'*:~

R.H. RITCHIE

Oak Ridge Natianal Labaratory, P.O. Bax 2008, Oak Ridge, Tennessee, 37831.6113, USA

Valence electron density fIuctuations induced around a trailing ion by a leading ion of a swift dicluster moving throughcondensed matter depend on the orientation of the dicluster axis with respect to the beam axis. These density fIuctuations mayappear experimentally in the angle dependence of the knock.on collision electrons emitted under bombardment or molecular ions.We present linear-response theory caIculations of the density fIuctuations at the position of the trailing ion. as a funclÍon of theorientation. Second-order perturbation theory caIculations of the dependence on the angle of the scattering probabilities of targetvalence electrons by a two-center Coulomb field moving with different velocities are presented. The applicability of linear theories asthe charges and velocities of diclusters are changed is discussed.

1. Introduction

~

Since the pioneering work of Ritchie and coworkers [1), who first gave an explicit expression for thewake potential induced by coherent electron displacements due to the passage through condensed matterof swift charged partic1es, much theoretical [2,4) and experimental [5-9) work has been performed. Most ofthe experimental work is related to measurements of the electric field strength and the wake potential,quantities that are obtained from certain averages of the induced electron density fluctuations.

When molecular ions penetrate solid targets, valence electron density fluctuations induced around thetrailing ion by the leading ion depend on the orientation, 8, of the molecular axis with respect to the beamaxis. These fluctuations may appear experimental1y in the dependence on 8 of the knock-on collisionelectrons emitted under bombardment by molecular ions. In particular, measurements of the knock-oncollision electrons ernitted at Oo under N; molecular ion bombardment as a function of the orientation(8) of the molecular ions have been recently performed by Yamazaki [10].

In this paper we first present linear-response theory calculations, in the plasmon-pole approximation, ofthe density fluctuations induced by the.leading ion of the dic1uster at the position of the trailing ion, as afunction of 8. Next we go beyond first order and present the results of a second-order quantum-mechani-cal calculation of the 8 dependence of the scattering probabilities of target valence electrons by atwo-center Coulomb field.

J,

"I

2. Density fIuctuations

The electron density fluctuations 8n(p, z) in a homogeneous isotropic medium induced by a swift pointcharge Z having constant velocity u are related to the wake potential by the Poisson equation and given, in

0168-583X/89/$03.50 @ EIsevier Science Publishers B.V.(North-Holland Physics Publishing Division)

1. ATOMIC PHYSICS

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334 J. M. Pitarke et al. / Electron density fluclllatiom

T--~zo¡::«on::>01- .00::>...Ju..

o

T ~. .-: : ...

r t !................... ."... ..: : : ...¡ ¡ >.,.,. "'-

~-------. . '.. .. "-. - '--- -...

! ! ! ~~~ ------~.~--'.;; ~~~-;;-;. . .. . .. . .. .. .:: 1 1 L-

~Vizwo

(j"~ o 20 40 60 80

THETA

Fig. 1. Induced density fiuctuations at the position of the trailing ion versus the orientation of ion diclusters moving with velocilyv = 2.5 a.u. (solid curve), v = 4.0 a.u. (dashed curve) and v = 8.0 a.u. (dashed-doued curve). The medium is characterized by a

density parameter rs= 1.53ao.Alsoshownare the correspondingO=sin-I( ,B/v) angies(verticaldotted lines).

".

the plasmon-pole approximation, by eq. (26) of ref. [3]. p and z are the cylindrical coordinates relative tothe position of the point charge Z.

Fig. 1 shows plots of 8n(R sin O, - R cos 0)/2, that is, the electron density fluctuation induced at thetrailing ion position by the leading ion when a dicluster transverses the target, as a function of the diclusterpolar angle with respect to the beam direction, for different constant velocities. R is the distance betweenthe pair of ions of the dicluster, which is assumed to be constant and equal to 2.200 (00 = 0.529 Á). It canbe seen from this figure, that the density fluctuation of target valence electrons around the trailing ion isprominent when it is in the conical region behind the particle with half-angle of operung 0- sin-1(,B/v),which defines the envelope of the collective wake [3], and this fact may appear experimentally in theO-dependence of the knock-on collision electrons emitted under bombardment of molecular ions.

If we are interested in electrons associated with high momentum transfers, the scattering processes aremainly determined by the short strong range of the Coulomb potential; however, no detailed nonlineartheory exists at presento In the next section we present some theoretical work based on second-orderperturbation theory, discuss its validity comparing with the very scarce experimental data, and suggest newexperiments.

e

3. Quantum analysis

We consider the process of noninteracting electrons being scattered by a two-center Coulomb fieldmoving with velocity vo, each center having a charge equal to Z, in the coordinate system in which the twocenters, separated by a fixed distance R, are at rest.

If the electron is irutially in the state Ii), the probability per unit time of finding it in the state IJ)after the perturbation caused by the ion and represented by H' is given to second order, in atomic uruts,where le I = Ii = m = 1, by [11]

I

H "

1

2

, ¡nHn;y¡;=2-rr H¡¡+ L . 8{E¡-E¡),

n WJI1(1)

------ ----

l_M_ Pitarke et al_ / Electro" density J7l1clllotions 335

where H/t- H/" and H;¡ represent matrix dements of the perturbation.

Wj" = E¡ - En. (2)

and Ej. E, and E" are the eigenvalues of the unperturbed Hamiltonian corresponding to Ii). If) and theIn) interrnediate states, respectively_

The initial and final states are assumed to be plane waves. and the perturbation Hamiltonian

H' = V(r) + V(r+ R). (3)

where V(r) is a Yukawa potential, that is. we approximate the wake by a spherically symmetric form of thetotal potential that electrons in the medium feel when a point charge Z having a. constant velocity Vopasses through the medium:

Z e - ar

(

W

)V(r)=- a=~,

r Vo

.' which should be a good approximation for transitions to electronic states of high momentum_ Then weki:. .ain~::-

(4)

Hí¡= U IH' Ii) = ~ 1 + e-i(k,-k¡)-R2...2a2+lk.- k

", ¡

(5)

and

HínH;¡ = (1 IH' In)(n IH' Ii) = Z2 1 + e-i(k,-k¡)-R + e-i(k,-k.)-R + e-i(k.-k¡)-R4...4 l. ,\, . ."(6)

where k¡, k¡ and kn represent the wave vectors of ji), If) and In) states, respectively. Substituting eqs.(2), (5) and (6) in eq. (1) and integrating over final states, the probability per unit time and unit solid angleof the transition to one of a group of final states that have nearly the same k¡, is

2dy _ Z2k¡

1

1 + e-i(k,-k¡)-R Z

I

d n-~ 2 2+2(1\+12) ,~~ 2'Ir a + Ik¡ - k¡ I 'Ir

if k¡= k¡, where

d q 1 + e-i(k,-k¡)-R1\ = 1

k¡2_q2 (a2+ Iq-k¡12)(a2+ Ik¡-qI2)

(7)

(8)

Land

d q e-i(ki-q)-R + e-i(q-k¡)-R

12= 1 k;_q2 (a2+ Iq-k¡12)(a2+ Ik¡-qI2)'(9)

To obtain the scattering probability per unit time and unit solid angle of any target valence electron tobe emitted after the perturbation with a wave vector k¡, it is necessary to perform the average of eq. (7)over all allowed initial states. In particular, if electrons emitted in a velocity region of AIVO:$;v¡:$;A2VO'atOo with respect to the beam direction of bombarding diclusters,are considered, the only contributions toeq. (7) come from initial states whose wave vector k, in the dicluster rest system, satisfies

AlkO:$; Ik - ko I :$; A2kO' (10)

where ko is the wave vector of the bombarding dicluster. Then, by taking k¡= k¡ko/ko, we obtain thefollowing average scattering probability per unit time and unit solid angle:

(dy )3Z2

1>'2ko 3 (l

12... 2

dQ = :\) dk¡k¡J.,dp.¡ d4>d/(k¡,p.¡,I/>¡)I,av 8'1r k F >'.ko k o(11)

1. ATOMIC PHYSICS

336 l./v/. Pitarke el al. / Eleclron density flllCllIaliolls

with k' = (k,2 + k~ - k})/(2k¡ko) and where

2f(k¡. Jl¡, e!>¡)= f¡(k¡, Jl¡, </>¡)+ -df21(k¡, Jl¡, e!>¡)+ f22(k¡, Jl¡, </>;)),'Ir

(12)

with

1 + e-ir

fl(k¡, Jl¡, e!>¡)=a2+2k,2(1+Jl¡)'

- -ir1

"'" q2 fl

12" ~

f21(k¡,Jl¡,</>¡)-(1+e ) dq 2 2 dJlde!>Ao k¡-q -10

(13)

(14)

and

f (k Jl </» =1 "'" d q2 f

l

12" e-ir, + e-ir2

22 " " I q 2 2 dJl de!> . .o k¡-q -10

(15)

ffZ~ In these expressions,

A = (a2 + q2 + k,2)2 - 2k¡q( a2 + q2 + k,2 )(Jl + xk) + 4k,2q2Jlxk,

xk = - JlJl;J(l - ¡.t;)(1 - ¡.t2)(cos e!> cos </>¡ + sin e!> sin e!>¡),

t = k¡R[Jl- ¡.t; cos </>¡sin 8 - (1 + ¡.t¡) cos 8],

tI = R [( k¡Jl- ¡.t; cos e!>¡- q/1 - ¡.t2COS</»sin 8 - (q¡.t+ k¡Jl¡) cos 8]

and

t2 = R [qV1 - ¡.t2 COS e!> sin 8 - (k¡ - q¡.t)cos 8] .

kF is the Fermi momentum (kF = 1.92/,s), and 's is the mean distance between electrons, Le. n = 3/(4'1r's3).If the target valence electrons are considered to have zero velocity in the laboratory system, we have

(~ ) = 322k~[

(k~-k~)(A22-A~) 2kHA32-A~)_ kHXi-A~)]

( )2

dQ 8 3k 3 2 + 3 4 If ko, 1, </>¡ I ,av 'Ir F

(16)

".

L where f(ko, 1, </>¡} is given by eq. (12).We first considered ion dic1usters moving with various velocities, each ion having a charge equal to

2 = 1, and calculated the scattering probabilities per unit time and unit solid angle for electrons to beemitted in a velocity region of 1.95vo::;;vf::;;2.15vo at 0° with respect to the beam. Figs. 2a-c exhibit plotsof the scattering probabilities computed from eq. (16), as a function of 8, when only the first-order term isconsidered and when both first- and second-order terms are incIuded in our calculations, for the velocitiesgiven in the figure caption. Notice that as the velocity increases, the second-order term becomescomparatively smaller, a~ we expected, since the perturbation also becomes smaller, and more fluctuationsoccur in the dependence of the scattering probabilities on the orientation of bombarding molecular ions, 8.

In order to compare our results with the experimental data [lO), next we have considered 4.5 MeV N;molecular ions (v = 2.5 a.u. and R = 2.2 a.u.) bombarding a carbon foil target ('s = 1.53ao) and calculatedthe scattering probabilities per unit time and unit solid angle for electrons to be emitted in a velocityregion of 1.95vo::;;vI::;;2.15vo at Oo with respect to the beam. Fig. 3 shows the results obtained from eqs.(11) and (16), plotted as a function of 8, assuming that the effective charge, 2, of the ions of the dic1usteris given by the following expression [12):

2 = 20 [1 - exp( - vZo 2/3 )], (17)

"

,ox r-~---'--""'---~--'---'-'-'-N l

l;

¡:: l:J

~ JO tg:b~

1- f~ ,v> L_

[ --'"r ,,

J.M. Pitarke et al. / Electron density jlllctuations

.Io

X r-'~."-.-r.--'~--~-~--'-._~-,, ,N,r C-¡ .i

S l- I~ ,o '

g:6~().- :Z !

~'"

,a~

oo 40 60

lHETA

\..

fIfI

lJ

o-=--

20

337

To

Ef~---r---' '-~ '

f

tli' r

, I

T

b-

oo ea

lHETA

.l

...:::=:::..40

~ea

~60

lHETA

Fig. 2. The solid and dashed curves show second- and first-order perturbation theory ealeulations, respectively, of the seatteringprobabilities of target valence electrons versus O,obtained froro eq. (16) for 2 = 1 and ror (a) /J= 2.5, (b) /J= 4.0 a.u. and (e) /J= 8.0

a.u.

, r---r r ,

~,17 , \i ~ '-"_'~_,

~. ~~ .--'-'- """ '«'.:.~~ 'A

J~ ~.::.::.::.::.~.::.::.::.::...,.~.~.;;¿~...~:.::;:0_. r-.'

-~.' L.. 80M"-.T.".".".;: ;.:~- . 60o I . 20O

(-~.

i

~

THETA

Fig. 3. The solid and dashed eurves show second- and first-order perturbation theory caleulations, respeetively, of the scatteringprobabilities versus O, obtained froro eq. (16) for 20 = 7, /J = 2.5 a.u. and the effeetive eharge given by eq. (17). The dashed-dottedand dotted curves show second- and first-order perturbation theory caleulations, respectively, obtained froro eq. (11) for 20 = 7 and

/J= 2.5 a.u.

1. ATOMIC PHYSICS

---

338 1. JI. Pirarke el l/l. / Eleclr/m densily jlllCIlll/liOllS

,oa b

~::J

!:i1to.. o<.>.~~~'"

5!o~""0..0<.>.Z~

~'"

j

oo 20

oo 20

lHETA

~ 5!o~""o..o<.>.z ~

~'"

40

lHETA

Fig. 4. Perturbation theory calculations of the scattering probabilities of target valence electrons versus 9, obtained from eq. (16) forv = 2.5 a.u. and effective charges given by eq. (17), with (a) 20 = 1, (b) 20 = 3, (c) 20 = 5 aI!d (d) 20 = 7. Second- and first-order

calculations are represented by solid and dashed curves, respectively.

".

where Zo is the bare ion charge. Notice from this figure that the specific contributions from all the initialstates to the average scattering probability do not change qualitatively its angular structure. It can beconcluded, also, that the second-order term is quite large compared with the first-order one. And, on theother hand, agreement with the experimental data is not very good [lO). It may be due to the fact that forsuch effective charges and velocities second-order perturbation theory is not valid any more. Indeed, thesmaller the effective charge the smaller the second-order terms, as it is obvious from figs. 4a-d which showthe way the results change as the effective charge increases, for a fixed velocity.L

4. Conclusions

It has been shown that second-order perturbation theory is more accurate for small charges and highvelocities, as we expected. On the other hand, as the ion velocity increases, the angular structure of thescattering probability chánges appreciably. However, the available experimental data have been obtainedfor rather large effective charges, so new experiments performed for Z = 1 and at different velocities wouldbe very useful in order to investigate the applicability of linear theories and capture and loss theories, andto detect the effects of electron density fluctuations. We hope our work will provoke such experiments.

In future work, we shall investigate the nonlinearity appearing for high values of Z. Indeed, the factthat in order to calculate the average scattering probability it is sufficient to compute the transitionprobability of the electron having zero velocity in the laboratory system enables to perform third-orderperturbation theory calculations. Furthermore, the wave functions of the continuum for the two-center

"

- --. -- -40

o'20

lHETA

o

.d

..

5!o"''oo.._<.>.z

"L808040o 20o

lHETA

--

1.114. Pilllrke el al. / EleClrDn densily jlllclllalions 339

problem, exact solutions of the Schródinger equation, may be computed, so the exact scattering probabili-ties of electrons on two Coulomb centers could be obtained, though this would be a cumbersome task.

One of the authors (P.M.E.) gratefully thanks Iberduero S.A. and the March Foundation for their helpand support.

References

\:"..

[1] J. Neufeld and R.H. Ritchie, Phys. Rey. 98 (1955) 1632;see also R.H. Ritchie, Ph.D. Thesis, Uniyersity of Tennessee (1959), unpublished;V.N. Neelayathi, R.H. Ritchie and W. Brandt, Phys. Rey. Let!. 33 (1974) 302; Phys. Rey. Let!. E33 (1974) 670; E34 (1975) 560.

[2] M.H. Day, Phys. Rey. B12 (1975) 514;Z. Vager and D.S. Gernrnell, Phys. Rey. Let!. 37 (1976) 1352;V.N. Neelayathi and R.H. Kehr, Phys. Rey. B14 (1976) 4229;R.H. Ritchie, W. Brandt and P.M. Echenique, Phys. Rey. B14 (1976) 4808;M.H. Day and M. Ebel, Phys. Rey. B19 (1979) 3434;P.M. Echenique and R.H. Ritchie, Phys. Rey. B21 (1980) 5854;R.H. Ritchie and P.M. Echenique, Philos. Mag. 45 (1982) 347.

[3] P.M. Echenique, R.H. Ritchie and W. Brandt, Phys. Rey. B20 (1979) 2567.[4] P.M. Echenique and R.H. Ritchie, Elhuyar 7 (1979) 1;

F. Guinea, F. Flores and P.M. Echenique, Phys. Rev. Let!. 47 (1981) 604;Phys. Rey. B25 (1982) 6109;A. Mazarro, P.M. Echenique and R.H. Ritchie. Phys. Rey. B27 (1983) 4117.

[5] W. Brandt, A. Ratowski and R.H. Ritchie, Phys. Rev. Let!. 33 (1974) 1325; Phys. Rey. Lett. E25 (1975) 130.[6] D.S. Gemrnell, J. Rernillieux, M.J. Gaillard, R.E. Holland and F. Vager, Phys. Rey. Lett. 34 (1975) 1420.[7] S. Datz, C.D. Moak, O.H. Crawford, H.F. Krause, P.F. Dittner, J. Gómez del Campo, J.A. Biggerstaff, P.D. Miller, P.

Hyelplund and H. Knudsen, Phys. Rey. Let!. 40 (1978) 843.[8] O.H. Crawford and R.H. Ritchie, Phys. Rey. A20 (1979) 1848;

J. Tejada. P.M. Echenique, O.H. Crawford and R.H. Ritchie, Nucl. Instr. and Meth. 170 (1980) 249.[9] F. Bell, H.D. Betz, H. Pank and W. Stehling, J. Phys. B9 (1976) 3017;

D.H. Jakubassa, J. Phys. C10 (1977) 4491.[10] Y. Yamazaki, to be published.[11] L.\. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955).[12] H.D. Betz, Rey. Mod. Phys. 44 (1972) 465.

v

l. ATOMIC PHYSICS