excitation of dielectric spheres by external electron...

PHILOSOPHICALMAGAZINEB, 1987, VOL. 56, No. 3, 335-349

Excitation of dielectric spheres by external electron beams

By P. M. ECHENIQUEt, A. HOWIE and D. J. WHEATLEY

Cavendish Laboratory, Cambridge CB3 OHE, England

[Received 30 June 1986 and accepted 8 December 1986]

,~

ABSTRACT

Classical dielectric theory is developed to provide an analytical expression for theenergy losses experienced by a fasl charged particle passing close to a dieleclric sphere.The results are compared with energy loss spectra obtained from Ag catalysl particlesin a scanning transmission electron microscope. In practice a considerable number ofangular momentum components are required, particularly if the effects of thin oxideor other surface coatings are to be included. For close surface encounters at largespheres the results confirm the validity of a simple quasi-planar approximation.

éi', ........

§ 1. fNTROD\JCTION

Recent developments and applications of scanning transmission electron micro-scopy (STEM) have stimulated renewed interest in the interaction of high-energyelectron beams with surfaces and with small particles (see, for example, Batson 1982 a,b, Cowley 1982, Howie 1983, Colliex 1985). In STEM a well-focused approximately0.5 nm probe of about 100 keV electrons provides a high-resolution transmissionscanning image from thin samples with a complex structure such as catalysts orsemiconductor devices. It can also yield, from selected local regions of the structure,X-ray emission spectra and electro n energy loss spectr,", In the laller case, thecharacteristic losses associated with inner-shell excitations are now routinely used todetermine composition on the nanometre scale but interpretation of the valence lossspectra is far less advanced despite the wealth of information which these relativelyintense signals potentially convey about both bulk and surface electronic excitations.

The classical dielectric theory of stopping power airead y widely used in electronenergy loss spectroscopy of homogeneous systems (Raether 1980), is a convenientstarting point and has been employed by Krivanek el al. (1983)and Howie and Milne(1984,1985) to interpret observations obtained by STEM for planar interfaces in bothparallel and glancing-angle conditions of the incident beam. It has been shown(Ritchie 1981, Ritchie and Howie 1987) that the assumption employed here, oftreating the fast electron as a poinf classical particle, is generally valid provided thatthe spectrometer collects ,essentially all the scattered electrons and provided that theresults are convoluted over a range of impact parameters corresponding to the spatialdistribution of the current in the incident prob~.

The problem of extending the dielectric model to deal with the case of smallspheres has attracted considerable allention but the mathematical difficulties havelimited progress so far to situations where serious and sometimes unrealistic approxi-matións have been made. The free-electron model, originally used by Fujimoto and

t Also at Euskal Herriko Unibertsitalea, Quimicas, Donostia, Euskadi, Spain.

-

,Ir

\:

336 P. M. Echenique el al.

Komaki (1968) in broad-beam geometry, has been applied to the case of a localizedbeam by Schmeits (1981) and Kohl (1983) but only for the case of dipole 1= 1 orquadrupole 1= 2 excitations. In more recent work (Acheche 1985, Acheche, Colliex,Kohl, Nourtier and Trebbia 1986) the contribution of higher 1 modes has beenidentified. Using the more general dielectric formalism, Batson (1980, 1982 a, b, 1985)has determined the resonance frequencies of small spheres, including the effects ofsurface coatings and of contacts with neighbouring spheres. Valuable though it is, thismethod does not yield the intensity of the various spectral features. Here weinvestigate in detail a new form of the dielectric model (Ferrell and Echenique 1985),valid for electron trajectories outside a spherical particle and generalize it to includethe possible presence of a surface coating. Computations extend to high values of 1andare compared with experimental data obtained from Ag catalyst particles.

f:"" ".~

§2. EXPERIMENTAL DETAILS

The specimens used were Ag on y-A1203 catalysts used by ICI in the oxidation ofethylene. These contain a rather high 5 % loading of Ag, and previous work (Marksand Howie 1979, Cunningham 1982) has shown that the metal particles have a verywide size distribution with diameters ranging from about 2 nm to several hundrednanometres. A variety of structures including single crystals and singly and multiplytwinned particles are present but most of the particles are of quasi-spherical shape.Specimens suitable for transmission electron microscopy were obtained by crushingthe catalyst pellets in a mortar and dispersing samples on holey carbon films usingalcohol. This procedure usually provided a few examples of suitably sized particlesoverhanging the edge of a hole in the support film, thus providing a specimenconvenient for external, or so-called ALOOF beam, spectroscopy as shown schemati-cally in fig. 1.

The 0.5 nm electron probe used in the experiments is generated in a VacuumGenerators scanning transmission electron microscope using a semi-angle y ofconvergence on the specimen of about 8 mrad. Energy loss spectra from thetransmitted electrons are obtained with a bending magnet spectrometer with acollection semi-angle pof about 5 mrad and usually, but not necessarily, operating onthe axis of the incident beam.

Fig.l

rq

Charged particle q moving at impact parameter b > a near a dielectric sphere of radius a.

</

I

~..,~

(

- -- ----

Excitotion of dielectric spheres 337

Other signals can be collected simultaneously, including the electron intensityreceivedin an annular detector at rather larger seattering angles Os~ 100mrad andarising mainly from electron-atom elastic seattering at fairly small impact parameters.This signal is invaluable, particularly for small-particle location in high-atomic-number elements (Crewe, Langmore and Isaacson 1975, Treacy, Howie andWilson 1978). Several procedures are available for collecting energy loss spectra fromvery precisely defined regions of the specimen such as near a small particle.

(o) The seanning probe can simply be stopped at a predetermined point on a high-magnification image: This stationary spot mode can be complieated, however,by problems of specimen drift, particularly when weak spectral featuresrequiring an acquisition time of several tens of seconds are of interest.

(b) The spectrum can be acquired continuously during a very-higb-magnifieationsean extending only I or 2 nm outside the particle. By displaying the annulardetector signal simultaneously throughout this procedure (Berger andPennycook 1982) the operator can maintain the particle in the centre of theseanned area. This method is very effective for microanalysis purposes butdoes not separate the rather different contribution to the valence losses fromfast-electron trajectories Iying partly inside the particle from that of theso-ealled ALOOF trajectories Iying wholly outside the partic1e.

(c) With the spectrometer adjusted to accept any selected .energy loss window, aline sean can be made across the particle. Detailed correlation with the annulardetector signal identifies the points where the beam just contacts the edges ofthe particle. Howie and Milne (1984) employed this method quite effectively inthe case of carefully aligned MgO smoke cubes whose well-defined geometry isa great advantage. The method is a liule tedious if the behaviour of a largenumber of spectral features is to be studied. Relative normalization of thedifferent line scans to changes in incident beam intensity is difficult since thezero-loss intensity is not continuously monitored.

(d) Wheatley, Howie and McMullan (1984) employed the annular detector signalto switch the spectrometer detection system on or off during a high-magnifica-tion image sean of the type described in (b) above. In its simplest form, thisprocedure was shown to separa te the contributions of inside and outsidetrajectories to the valence loss spectrum. With more sophistieated electronictiming and gating circuitry, arranged for example to switch on the spectro-meter detector and energy scan for a very short time at a predeterminedinterval after the instant when the seanned electron probe just leaves theparticle, complete spectra corresponding to a definite small range of impactparameters can be collected automatieally.

In our work, method (a) was used initially and then method (c) with a few runsusing method (d).

§ 3. FORMULA TION OF DIELECTRIC THEORY

Classieal dielectric theory yields an expression for the total energy loss Wexperienced by a particle of charge q moving on a straight-line trajectory through amedium each of whose composite parts j ean be individually described by anappropriate complex frequency-dependent dielectric constant tlaJ). W emerges fromthis theory as a sum of contributions at different frequencies aJ and, for spherieal

_ _.. _'__'_'h". ______

--


geometry, depends on the impact parameter b or the distance of the particle trackfrom the centre of the sphere.

W(b) = So'"P",(b)hw d(hw),

where P",(b) is thus the 'probability' of an energy loss hw in a single-scatteringapproximation.

If we neglect retardation effects, the direct potential <I>darising at the poin!r = (r, e, q,) from a charge q at the point r' = b + VI, where r' > r, can be Fourieranalysed in the form

(1)

~~~

~ 1 = ~ " (2 _ c50) (l - m)! rl4neolr-r'l 4m;of::. m (l+m)!r'I+J

X Pi(cos e)Pi(cos e') cos [m(q, - q,')]

Ice rl

= t;. _ 00C1m(w)QiPi(cos e) cos [m(q, - q,')] exp (- ¡,Wl).

(2)

(3)

We can, without loss of generality, take q,' = O.In these summations, O~ 1~ 00 andO~ m ~ lo

Following Ashley and Ferrell (1987), Clm(w)is given by the expression

C ( ) = ~ (2 _ c5 )(l - m)! Ioo Pi(cos e') exp (iWI)

1mW 8 2 Om (l )" . o dtneo +mo -00 r

= ~(2 _.c50mW-m (IWla

)1 1

(~ )/-mKm (IW1b

), (4)4n2eov v (l + m)! ¡wl v

where Kmis a modified Bessel functionoOutside the sphere, i.eoin the region a ~ r ~ b, the total potential consists of this

direct part plus an induced potential and has Fourier components with respect to timegiven 'by

<I>(r,w) = t;.[ Clm(W)G)' + A1m(W>(~)'+]Pi(COS e) cos (mq,)o (5).,f.§'"~ Using the result

o(

pm(cos e»

) 1+ 1 - m pm (cos 8),1 = - 1+2 1+1- 1+1 uoz r (6)

we can evaluate the energy loss W due to the action of the induced electric field actingon the charged particle:

W = -qv I:00Ez<r = r', t) dt

I"" 1

(wa

)/+l

(w

)/+J-m

(IWlb

)=2q _""t.Alm(W)(l_m)!(-iy+l-m -;- Iwl Km ~ dwo(7)

The last line follows by once again using eqno(4).

.-~

-------

Excitation 01 dielecrric spheres 339

The-precise'Vatue.ofA'm -depends'on the dielectric response of the spherical regionr' :$ a, but this will be proportional to the direct field component. We therefore write

A'm(w) = Y'm(w)C'm(w). (8)

Substitution from eqns. (8) and (4) into eqn. (7) and comparison with eqn. (1) thenyields the result for the excitation probability

PO)(b)= LP~(b). ,

q1a 2 - O

(wa

)2'

(Wb

)= 'f (1- m)!(Io~m)! Im[-Y'm(w)] V- K;' V- .(9)

\...,

We have evaluated Y'm==¡./ (which is independent of m) for the situations described infigs. 1 and 2 by the standard procedure of matching scalar potentials and normalcomponents of the field at the surface boundary and find the following results.

(a) For a dielectric sphere e], embedded in a medium e2 (fig. 2), we find that

¡',(w) = l(e2- e])/[Ie] + (1+ l)e2Je2

which, when the outside medium is vacuum, reduces to

(10)

y, (w) = 1(1- e)/(Ie + 1+ 1). (11 )

and, when the electron moves in a dielectric outside a cavity, as in a mediumcontaining small voids, we have

" (w) = I(e- 1)/[1+ (1+ I)Ü (12)

(b) For an electroDmoving in vacuum outside a dielectric coated sphere asdescribed in fig.2, we obtain for y,(w)

1(1- e2) + 1,[1+ (1+ l)e2JR2/+]y,«(:) = le2 + (1 + 1) + 1,[(1 + 1)(1 - e2)JR2/+]'

(13)

where R = ada2 and

1, = (e2 - e1)1/[le1 + (1 + 1)e2J.

r We now look at some of these cases in greater detail.

(14)

Fig.2

(3)

Charged panicle moving at impact parameter b > Q2near a coated sphere.

---

- - -- --

340 P. M. Echenique et al.

~

§4. UNCOATED SPHERE

For an electron moving in a vacuum outside a dielectric sphere the energy lossprobabilit}' is given by eqns. (9) and (11) as first shown by Ferrell and Echenique(1985).

Many calculations ofthe interaction offast particles with dielectric spheres restrictthemselves in practice to the dipolar / = 1 termo Equation (9) shows that thisapproximation will always be valid for wa/v ~ 1 but in practice this can be a ratherrestrictive condition typically implying a < 1nm for 100 keV electrons. Some physicalinsight into the nature ofthe dipole approximation, corresponding to the / = 1 term inthe sum of eqn. (9) can be achieved by realizing that the dipole form can be easilyobtained (R. H. Ritchie, 1985,private communicatión) by assuming that the externaldirect field is constant over the sphere, leading to a uniform induced polarization andinstantaneous induced current J in the sphere and thus an energy dissipation given by

W = I x def E. J dt =I.

x P ",(b)hw d(hw).- ex. spbere o

(15)

One easily obtains

P",(b) = (a3/4n2eoh2) 1m (8 -

21»

)IE",12. (16)(8 + )

The Fourier components of the field E", can be taken from any textbook onelectrodynamics (see. for example, Jackson 1975),leading in the non-retardation limitto P~(b), the first / = 1term ofthe sum in eqn. (9). Even for large values of a, therefore,the dipolar approximation can be valid at impact parameter b ~ a sufficiently greatfor the external field to be taken as uniform over the whole sphere. Such distanttrajectories, however. are mainly associated with very low excitation frequencies

Fig.3

0.7

0.6

.,~ 0.5.D

-3o.:~- 0.";§

0..3 0.3

(

0.2

----------.",...,.,.,--...0.1

oO

;

2 3 " 5 6 7 8 9 10

Relative weigbt of the 1mode in the probability function P",(b) for hw = 3.6 eV for an 80 keVelectron passing at impact parameter b from the centre of an Ag sphere of radius a:

,., a = 450 A, b = 455 Á; - __, a = 2000 Á, b = 2010 Á.

<~

Excilalion of dieleclric spheres 34]

~

w ::Sv/b, as is confirmed by the asymptotic behaviour of the Km functions in eqn. (9).The essential dipole condition wa/v ~ 1 therefore still applies provided that the spherehas sufficient dielectric 1055at these frequencies.

In many experimental situations b ~ a and wa/v ~ 1 and one therefore needs toinclude many 1 values in eqn. (9). The most important characteristic frequenciesinvolved in an energy 1055process will vary both with 1and with the dielectric functione(w). Since the excitation of the various modes also depends on wa/u for a givenelectron energy, spheres of different materials can show substantial differences inenergy loss spectra as well as in the number and nature of the modes excited.

To illustrate these points, we compare the results of our calculations with theexperimental data for Ag, obtained as described in§2. In fig. 3 we show, as afunctionof 1,the relative weight P~/ P '" oflhe 1mode for hw = 3.6 eV and for a 80 keV electronpassing at impact parameter b of the centre of an Ag sphere of radius a for twodifferent values of the radius. For a large radius a of about 2000 A we see that thecontribution of small 1 values is negligible. Even for a radius of 450 A the dipolecontribution will, at small impact parameters, be less than 10% of the total. In fig.4 we

Fig.4

lO

eL~W2!-......

eL2L-J2!

." \",,0,,~, - ...,...-------

a. ~ :::;::;.".,.....----",'"

",'"",'"

~-

.a=4S0A 1=1-

OID W ~ ~ ~ ro ro w ~

SCA)Contribution to the los5 probability P",(b)for hw = 3.6 eVfor an 80 keV electro n incident on an

Ag sphere ofradiu5 a as a function ofreduced impact parameter B = b - a (lmu= 100):-, representthe contributionof the 1= 1,1 =2 and tbe first ten 1values to P",(b)for aspbere of radiu5 450 Á; - - -, the contribution of the first ten modes to P",(b) fora = 2000 Á.


~

show the dependence of the relative weight of.the first two modes and ~.1he.sum DIthe first ten I values (i.e. 120 distinct modes) as a function of the reduced impactparameter B = b - a. Atlarge impact parameters the relative weight of smalIl valuesincreases, the more so, the smalIer the radius, as discussed above. AII the computa-tions for Ag reported here have employed the dielectric function 8(W)published byHagemann, Gudat and Kunz (1974).

Comparison with experimental results for the same two sphere sizes is made infigs. 5 and 6. We show the probability oflosing energy hw = 3.6 eV (surface plasmonfor silver) as a function of the reduced impact parameter B = b - a. In order toinelude most of the observed loss peak, both theory and experiment have beenintegrated over tbe range 2.3-4-9 eVo Evidently the elassical tbeory of energy loss,ineluding alI the necessary I contributions, is capable of describing tbe experimentalfacts reasonably well.

To cover tbis important case of medium to large radius a but smalI values ofB = b - a, where many 1values are required, Wheat]ey el al. (1984) have suggested asimplified calculation wbich can lead to useful results. They modified the die]ectrictheory for a p]ane surface (see, for examp]e, Howie and Milne 1984, 1985)by using ateach point oftbe track (fig. 7) tbe instantaneous impact parameter B', and introducinga further variable factor cos e required ir the s]owing-down force exerted by themedium is resolved in the track direction. In this model the probabi]ity of ]osingenergy hw at impact parameter b is therefore given by

b

(e - 1

)fa>

(2w 2 1/2 )dy

P,.,(b)= 2 h2 21m _ 1 Ko - [b(l + y) - a] (1 hll?' (17)7t eo v e + o ¡; + y

Fig.5

3

oo~x

>- 2 o- I. ~~ ]

~Q.

e::18

60 .SeA)

Probability ol excitiog energy ha>(over tbe raoge 2.3-4-9 eV) as a luoctioo ol the reducedimpact parameter B = b - a lor an 80 ke V electroo incident 00 ao Ag sphere ol radius450 Á: x, calculanoo; O, experimental points.

o 10 20 30 40 50

,

Probability of exciting energy hro (over the range 2.3-4.9 eV) as a function of the reducedimpact parameter B = b - a for an 80 keV electron incident on an Ag sphere of radius2000 Á: x, theoretical calculation; O, experimental points.

One would expect this approximation to apply at large values of aw/v and smalJvalues of (b - a)/a, when many 1values are involved. It does indeed emerge after somealgebra (see Appendix) as the correct limit to eqn. (9) in this situation.

A possible shortcoming ofthe approximation, which may show up at insufficientlylarge values of wa/v, is its simplified dependence on w which prevents fitting the wholew spectrum with the same accuracy. In the table, we compare the results ofcaIculations for Ag with E = 80 keV, a = 2000 A and b = 2010 A using eqn. (9) andthe approximate result given by eqn. (17). For close surface encounters at such largevalues of a, the approximation ofeqn. (17) does indeed seem to be useful, giving resultswhich agree with the results obtained from the exact expression to an accuracy ofabou\ 1O/~.

e Fig.7

Planar approximation to the energy loss in terms of the local impact parameter B'.

Excilalion of dieleclric spheres 343

Fig.6

I31-

I "'-O52x I EJ-::: 215O

.DoLo.el.

e:Jo I !:]u

,.- -,..

o 10 20 30 /,() 50 60 70 80 90 100 110 120

BeA)

--------


Comparison ofthe results of calculations for Ag with E = 80 keV.a = 2000A and b = 2010Ausing eqn. (9) and the approximate result from eqn. (17).

hw (eV)

3.253.63.774.2

Pw(b) sphere (eqn. (9»

Pw(b) plane (eqn. (17»

}.1}.110-890-98

§ 5. COATED SPHERE

For an electron moving outside a coated sphere the probability of loss is given byeqns. (9) and (13). y,(w) has the right limits since, if al ~ a2, R ~ 1, thene y,(w) -+ 1(1 - 1:2)/(11:2+ 1+ 1), (18)

and this is especiaIly so for large 1values. However, ir (a2 - al)/a2 ~ (21+ 1)-1, then

y,(w) -+ 1(1- 1:1)/(11:1+ 1+ 1). (19)

,;v...,

These results imply that one has to go to very large values of 1to detect a thin coating.In fact, Howie and Milne (1984) have pointed out that small-angle energy loss fromhigh-energy electrons is insensitive to surface reconstruction or surface states. Similarinsensitivity.at high energies was reported by Krivanek el al. (1983), in contrast withthe high sensitivity reported in low-energy electron diffraction (Ichinokawa, Ishikawa,Awaya and Onoguchi 1981).Thé physical reasonfor this i:'1sensitivityis the same asthat behind the need to go to large values of 1to detect the effect of a thin coating andcan be easily understood, as argued by Howie and Milne (1984), by noting that asurface layer of thickness d will affect only those surface modes of wavenumber q forwhich qd > 1. If d :::::0.5 nm, such modes will not belong to the dominant part q :::::w/vof the usualloss spectrum unless the incident energy is less than 200 eVoAt the highincident beam energies used in electron microscopy, the sensitivity of the energy lossprocesses to near-surface effects will be poor. However, surface sensitivity could beincreased by studying the energy loss spectrum at larger scattering angles.

§ 6. FREE-ELECTRON GAS MODEL

Many caIculations (Schmeits 1981, Kohl1983) of the interaction of fast electronswith spherical particulates assume that the response of the system can be formulatedin a normal mode type of Hamiltonian, with a set of well-defined modes existing,usually free electron like, with eigenfrequencies given by w~ = w: 1/(21+ 1)for spheresor w¡ = w:(l + 1)/(21+ 1) for voids. The excitation probabilities are then calculated,usually for the dipole mode or at the most the first few 1vaIues. The same remarksabout the dependence of Pw(b) on the number of modes quoted above apply, ofcourse, to this special case. Now, however, because of the restricted nature of theresponse function, each 1mode contributes only to its own characteristic sharp loss atw = w" in contrast with the general case where alll values, in principIe, contribute toany w loss. This means that in such undamped free-electron cases, some informationabout the modes actually excited at different impact parameters can be inferreddirectly from the position of the surface loss peak in the observed loss spectrum(assuming that adequate energy resolution is available).

;;

-- - - - -

Excitation of dielectric spheres 345

The probability of an energy loss hw, at impaet parameter b for a free-eleetronsphere with low damping is thus given by

p' (b) = q2a I (2 - ÓOm) (w,a

)2'K2 (w/b

) (20)'" 2n¡;ohv2 m (1- m)!(I + m)! w, v m V '

where the total energy loss is given by

w = I P~(b)hw,.,

(21)

The mean energy loss to the surface modes is then defined as

_ I,w,P~(b)w.(b) = I,P~(b) .

..{~;,- As pointed out by Aeheehe et al. (1986), w.(b) is mainly governed by the relativeweight of the low or high I modes and, from the foregoing diseussion, it is clear thatthe relative eontribution of high 1values beeomes greater the smaIler the distance fromthe surface, partieularIy for large values of w,a/v. One eould expeet therefore that, inthe lightly damped free-eleetron model, w.(b)/wp will deerease Crom 1/.)2 (high 1values dominate eompletely for small b) towards the value 1/.)3 charaeteristie of thedipole model.

Measurements of w.(b)/wp have been reported by Acheehe (1985) for 100 keVeleetrons incident near an Sn sphere of radius of 400 Á; These are eompared in fig. 8with the predietions of our theory based on a free.:electron dieleetrie eonstant¡;(w) = 1 - w~iw(w + ia) with hwp = 14eVoIn the limit ofsmall damping a, shown ineurve A, agreement with experiment is poor, partieularIy at large values of b. Thefailure of curve A to deseend to the value 1/.)3 can be traced, in the light of thediseussion following eqn. (16) above, to the faet that aw1/v = aWp/v.)3 ~ 3.5 so thatthe dipole model never applies. The situation ehanges radically, however, as shown byeurve B where a damping eonstant of ha.= 4 eV has been employed, chosen to fit thedieleetric data of Sn (MacRae, Arakawa and Williams 1967).The various surfaee losspeaks are now substantially broadened and include a low-energy tail extending to

. below 3 eV which dominates the results (and more nearIy satisfies the dipole" approximation) at large values of b. Clearly the agreement with the experimental data

stillleaves much to be desired but it should be borne in mind that, Corlarger values ofb/a, w.(b)/wp must be eomputed from very weak signals. The small spheres studiedmay alSOhave a modified damping constant and their dielectric response may havebeen further changed by the presenceof an oxide coating. .

A detailed study of surface plasmon excitation on oxide-coated spheres has beenmade by Munnix and Schmeits (1985) using for ¡;1(W)in fig. 2 the lightly dampedfree-electron moderand ufor the oxide dielectric constant ¡;2(W)a fixed value ¡;.According to eq~. (13), the response fUDctionof the system is then found to be peakedat values of W, given by

(22)

w, = coJ[l + (1 + 1)¡;O,/lJ1I2. (23)

Here

o, = [(le + 1+ I)R -(2'+ 1)- 1(&- 1)]/[(/¡; + 1+ I)R -(2/+ 1)+ (1+ lXt: - 1)]. (24)

- - - --

- --

346 P. M. Echenique et al.

Fig.8

3D.::::.

ñ Wp =14eV

.D

13 (1'1./2)0.7

l~

A a =400 A

C'{3)

0.51.0 1.1 1.2 1.3 1./. 1.5 1.6 1.7 1.8

¡.:-~

b/a

Mean energy loss to surface modes in an Sn particle of radius 400 A as a function of the impactparameter b: curve A, theoretical curve for zero damping: curve B, includes more realisticdamping.

This expression agrees with the formulae derived by Batson (1982 a) and byMunnix and Schmeits (1985). In fig. 9 we compare the results of ca1culations for a100 keV electron incident on an oxide-coated Al sphere for two models of the oxidedielectricresponsefunction,usingfirstlythe experimentale(oo)(Hagemannet al. 1974)and secondly a COnstantt value of3.73 as employed by Munnix and Schmeits (1985).The results are quite different,not only at high 00where tbe approximation of COnstantt is inapplicable but also in regions where supposedly it should be appropriate.Differences of a factor of 2 arise in the probability of excitation and the peakmaximum is shifted by about 1eVo It appears likely therefore that for detailedcomparison with experimental data, particularly over a wide range of 00, the fulldielectric functions should be used with realistic damping inc1uded.

~

-.-------

Excitation of dielectric spheres 347

Fig.9

:co:

1D

10 20 w(eV)

0.5' - - - -.,,-'- --......

~ ', ,, ," ",,,

L

o

Probability of excita tion P",(b)for a coated Al sphere calculated with the experimental aluminat(w) (---) and taking a constant value t = 3.73 (-) (a2 = 150 Á; al = 105 Á; b = 160Á;E = 100 keV).

ACKNOWLEDGMENTS

This research was sponsored by the SERC and by a grant from the U.S.A.-SpainJoint Committee for Scientific and Technological Co-operation and by the SpanishCAICT. P. M. Echenique gratefully acknowledges help and support from lberdueroSA and Gipuzkoako Foru Aldundia. D. J. Wheatley thanks the SERC and ICI for aCASE award. The authors wish to thank Professor T. Ferrell and Professor R. H.Ritchie for several profitable discussions and Dr C. Colliex for communicating resultsin advance of publication.

6"»'\f'. .

APPENDIX

To derive eqn. (17) from eqn. (9) in the limit of close encounters with largerspheres, we employ the following standard Bessel function relations:

""

I,,(z) = I 1 (z)

"+2m

m=O m!(m + n)! 2 '(A 1)

cosh [z cosh (~) ]= ,,=~ co12,,(z) cosh (n<f»,(A 2)

K,,(z) = L"" cosh (nt) exp (-x cosh t) dt.(A 3)

----

- ---

í

\;.-

f""


On the assumption that, for sufficiently large value~ of awjv, the dominant contribu-tions to Pw come from large valuesof /, 1m(- y,)can be replaced by 1m(-)' "J =1m [ - 2/( I + e)] for a sphere in vacuo. We can thus write

Pw(b) = 2C I (2 - c5om)K;,(B) f A2' 1m=O I=m

00

= 2C "L K;,(B)! 2m(2A), (A 4)m=-co

where

C = (q2a/1t2eoh2v2) 1m {-1/[1 + e(w)]},

A = wa/v, B = wb/v.

Now, usingeqn. (A 3) foreach Kmfunction,followedby someminor algebraand useof eqn. (A2), we obtain

P OJ(b)= !C f: eX) du f: eX) dv cosh [2A cosh ()2)]

x exp [-2B cosh (;J cosh(;2)]

~ ~C f: eX)du f: eX) dv exp {{ A - B cosh (;2) ]cosh (;2) }

= C f:oo Ko[2(B cosh ~ - A)] de;

J.

oo d

= 2C o Ko{2[B{1 + y2)1/2 - A]} y~. ...

This is identical with eqn. (17) apart from a factor a/b which is close to unity under theconditions of the approximation. In sorne cases, it may be advantageous to compute acorrection to (A 5) by replacing ')1,in eqn. (17) with ')11- ')100 and summing the first fewterms of a now more rapidly convergent series.

(A 5)

REFERENCES

ACHECHE,M., 1985, These de Troisieme Cycle, Orsay.ACHECHE,M., COLLIEX,C., KOHL, H., NOURTIER,A., and TREBBIA,P., 1986, Ultramicroscopy,20,

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Microscopy (New York: Wiley), p. 47.CUNNINGHAM,T. G., 1982, Ph.D. Thesis, University of Cambridge.FERRELL,T. L. and EcHENIQUE, P. M., 1985, Phys. Rev. Lect., 55, 14, 1526.FUJIMOTO, A. A., and KOMAKI,K., 1968, J. phys. Soco Japan, 25, 1679.

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lil'p¡!,f1:

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--

"

Excitalion of dieleclric spheres 349

HAGEMANI\,H. J., GUDAT, W., and KUNZ, c., 1974, DESY Report No. SR-74/7.HowlE, A., 1983, Ultramicroscopy, 11, 141HOWIE, A., and MILNE, R. H., 1984, J. Microsc., 136, 279; 1985, Ultramicroscopy, 18,427.IcHINOKAWA, T., ISHIKAWA,Y., AWAYA,N., and ONOGUCHI, A., 1981, Scanning Electron

Microscopy, Part I (Chicago: SEM Ine.), p. 271.JACKSOI\.J. D., 1975, Classical Electrodynamics, seeond edition (New York: Wiley), p. 396.KOHL, H.. 1983, Ultramicroscopy, 11, 53.KRIVANEK.O. L., TANISHIRO,Y., TAKAYANAGI,K., and YAGI,K., 1983, Ultramicroscopy, 11,215.MACRAE,R. A., ARAKAwA,E. T., and WILLlAMS,M. W., 1967, Phys. Re¡;., 162, 615.MARKS, L. D., and HOWIE, A., 1979, NaLUre, Lond., 282, 196.MUNNIX, S.. and SCHMEITS,'M. 1985, Phys. Re¡;. B, 32, 4192.RAETHER.H.. 1980, Excitations of Plasmons and lnterband Transitions by Electrons, Springer

Traets in Modero Physies, Vol. 88 (Heidelberg: Springer).RITCHIE, R. H., 1981, Phil Mag. A, 44,931.RITCHIE. R. H.. and HOWIE, A., 1987. (lo be published).ScHMEITS, M., 1981, J. Phys. C, 14, 1203.TREACY,M. M. J., HOWJE, A.. and WILSON, C. J., 1978, Phil. Mag. A, 38,569.WHEATLEY.D. 1., HOWJE, A.. and MeMuLLAN, D., 1984, Electron Microscopy and Analysis.

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excitation of dielectric spheres by external electron...

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