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Konstantin Iakoubovskii, Kazutaka Mitsuishi, Yoshiko Nakayama, Kazuo Furuya

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[Mean free path of inelastic electron scattering in elemental solids and oxides using transmission electron microscopy: Atomic number dependent oscillatory behavior](https://mdr.nims.go.jp/datasets/62814966-cbb9-4262-a828-7187bfc83ac3)

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Mean free path of inelastic electron scattering in elemental solids and oxides using transmissionelectron microscopy: Atomic number dependent oscillatory behaviorKonstantin Iakoubovskii* and Kazutaka MitsuishiQuantum Dot Research Center, National Institute for Materials Science, 3-13 Sakura, Tsukuba 305-0005, JapanYoshiko Nakayama and Kazuo FuruyaHigh Voltage Microscopy Station, National Institute for Materials Science, 3-13 Sakura, Tsukuba 305-0005, Japan�Received 7 January 2008; revised manuscript received 6 February 2008; published 5 March 2008�Mean free path of inelastic electron scattering � has been measured with a 200 keV transmission electronmicroscope for the majority of stable elemental solids and their oxides. An oscillating behavior vs atomicnumber Z has been revealed, such that within one row of the Periodic Table, the minimum �maximum� of � isobserved for elements with completed �empty� outer d shells. A significantly weaker ��Z� dependence isobserved for the oxides. The ��Z� variation is ascribed to the three major factors: atomic density, number of“free” electrons per atom, and contribution of atomic core-loss transitions.DOI: 10.1103/PhysRevB.77.104102 PACS number�s�: 79.20.Uv, 68.37.Lp, 72.15.Lh, 61.85.�pI. INTRODUCTIONInelastic electron scattering has been a subject of numer-ous studies and reviews �see, e.g., Refs. 1–7�. Here, we shallmostly focus on scattering by single atoms and elementalsolids. The characteristic measures of the scattering are thecross section � or mean free path �. By definition, they arerelated through the number of atoms N per unit volume as1 /�=N� and thus will be used in this paper interchangeably.Those � �or �� values are important for a wide range ofsciences from radiation physics to transmission electron mi-croscopy �TEM� and are crucial for quantitative analysis ofresults obtained with various electron spectroscopies. Theydepend on several experimental parameters, such as electronenergy and geometry of the illumination and collection op-tics. Those dependences have been discussed previously,1,3,8and in this paper, we shall analyze variation of � with theatomic number Z.Two major ��Z� behaviors are known: periodic oscillationand smooth increase approximated as 1 /�=N��Za. Earlytheories of atomic scattering predicted the latter dependencewith a=1 /3 �Ref. 6� or a=1 /2.7 However, it was soon real-ized that this monotonic increase originates from oversimpli-fications and that more rigorous approach reveals periodicdependence.5 The latter can be described as follows:1 Withinone row of the Periodic Table, � decreases from alkali to-ward noble gases. In the alkali, the valence electrons arerather delocalized and thus produce low-energy scatteringspectra, while the noble gases have rather compact, filled sand p shells and therefore high-energy electron loss transi-tions. The scattering intensity rapidly decreases with energythat results in the above-mentioned � decrease.While the ��Z� dependence for atoms is well established,the situation in solids is rather uncertain. The crucial differ-ence in the inelastic electron scattering by atoms and solidsis that in atoms, electrons are localized and the scatteringoriginates from electronic transitions between differentatomic levels called “core-loss” transitions. The latter arealso observed in solids, however, the crucial difference isthat in solids, valence electrons are delocalized and producecollective excitations called plasmons. Compared to mostcore-loss transitions, plasmons have significantly lower ener-gies and higher scattering probabilities. As a result, experi-mental electron energy loss spectra �EELSs� from most ma-terials are dominated by plasmon excitations. Therefore, it isargued that the atomic calculations are inapplicable tosolids,3,4 and other theoretical approaches are considered.The most popular one is based on the free-electron �or“jellium”� model. When the width of the plasmon peak issmall compared to its energy Ep, this model predicts thefollowing relations:1/� = N� � Ep � n1/2 � ��z/A�1/2. �1�Here, n is the number of free electrons per unit volume, z isnumber of free electrons per atom, � is mass density, and A isatomic mass. Note that �1� we have focused only on Z de-pendence and thus omitted proportionality constants. �2� TheEp term should actually read Ep ln�B /Ep�, where B is a con-stant and the logarithmic term originates from angular de-pendence of the scattering. However, for typical collectionangles of �1 mrad, B /Ep�1. Therefore, the logarithm is aslow function of Ep, and, considering the experimental er-rors, it can well be assumed constant. �3� Equation �1�, dueto the specific ��Z�z�Z� dependence, actually does predictperiodic ��Z� behavior.Numerous modifications of the free-electron model exist.3The weakness of most of them is uncertain value of z, whichis usually taken as the most probable valence that a certainatom exhibits in chemical compounds.In a striking contrast to a vast number of theoretical stud-ies, systematical measurements of ��Z� are scarce. Egerton1summarized � values for 17 elemental solids and com-pounds. Despite significant data scatter, the ��Z� dependencewas approximated by a smooth 1 /��Z0.36 increase. Yet inanother EEL study,9 data for six solids were fitted with a1 /��Z0.57 function.Instead of using electron scattering for measuring �, op-tical absorption has also been employed, assuming theequivalence of those methods for small scattering angles.PHYSICAL REVIEW B 77, 104102 �2008�1098-0121/2008/77�10�/104102�7� ©2008 The American Physical Society104102-1http://dx.doi.org/10.1103/PhysRevB.77.104102Within this approach, � has been calculated for 34 elementalsolids.4 Again, despite a large scatter, results were approxi-mated by a 1 /��Z0.5 law.This brief discussion of the material dependence of inelas-tic scattering can be summarized as follows: Calculations onisolated atoms and solids predict periodic ��Z� or ��Z� be-havior, however, it has not been revealed by the experiment.This apparent contradiction is resolved in the present study.By improving the experimental accuracy and increasing therange of studied materials, we reveal a clear periodic behav-ior of the inelastic mean free path in elemental solids.II. EXPERIMENTAL DETAILSThe mean free path values were deduced from electronenergy loss �EEL� spectra measured with a Jeol 200 kV highvacuum 2500SES scanning transmission electron microscope�STEM� equipped with an Enfina EEL spectrometer. Samemicroscope was used for recording STEM images. The exci-tation and collection semiangles were set to �20 mrad.Previous studies1,4 indicated that variation of � withatomic number is rather small ��100% for the whole Zrange�, and therefore its reliable measurement requires im-proved accuracy. This was achieved in the following way.�1� All measurements were performed at nominally thesame microscope settings, without major microscope realign-ment in between.�2� Diffraction effects can modulate the EELS intensityand thus result in overestimation of � by up to 25%.10 Thoseeffects were minimized by selecting appropriate sample ori-entation.�3� Under electron irradiation, TEM samples often rapidlyaccumulate carbon-related contamination affecting the �measurements. Using high vacuum ��10−6 Pa� and short ac-quisition time ��50 ms� minimized this problem in thepresent study. Rapid measurement also allowed us to studymaterials unstable under focused electron beam such as sul-fur, phosphorus, iodine, and some oxides.�4� Moderately thin sample regions have been chosen�t /��1 or 70� t�180 nm� in order to reduce the surfaceplasmon contributions, which are non-negligible at smallthicknesses t and plural scattering gaining at largethicknesses.1�5� The � value for a certain material was typically ex-tracted from a single EEL spectrum. Note that usually anextra independent measurement�s� �of thickness, for ex-ample� is required. In those additional measurements, it isdifficult to precisely locate the sample area used for the EELspectra. This difficulty probably reduced the accuracy ofmany previous mean free path measurements.The following procedure has been applied to deduce the �values.8 The EEL spectrum J�E� was deconvoluted with thestandard routines1 to extract the integrated zero-loss peak I0and the single-scattering EEL spectrum S�E�. The � valueswere calculated as� =tln�I/I0�, �2�where I=�J�E�dE. This equation is derived1 using the onlyassumption of independence of electron scattering events�Poisson statistics� and therefore is rather reliable. Thesample thickness t was deduced from the same spectrumusing the “Kramers-Kronig sum rule,”1,8,10,11t =4a0FE0I0�1 − 1/n2� � S�E�dEE ln�1 + �2/ 2�. �3�Here, a0 is the Bohr radius, E0 is the electron energy, F is therelativistic factor, n is the optical refractive index, � is thecollection angle, and  �E / �2FE0�. This equation is deducedwith more approximations than Eq. �2�, but several experi-mental tests8,10,11 revealed that it is accurate to �10%.For many materials, a newly developed8 and even moreaccurate routine has been applied: Several layered structuresSi /SiO2 /Ta /CoO /Pt, Si /SiO2 /Ta /Pt /NiO /Al,Si /SiO2 /Cr /Pt /W, Si /Au, and Si/diamond were preparedusing conventional techniques of magnetron sputtering, ther-mal evaporation, and chemical vapor deposition. Cross-sectional TEM samples were prepared from those structuresby focused ion-beam cutting �Ga+, 5–30 keV�. Ion-beamcutting produced samples with uniform, almost constant,thickness profile, thus avoiding abrupt thickness variation atthe interfaces common for most other sample preparationtechniques.EEL spectra from those multilayered samples were auto-matically acquired at several hundred locations on a straightline running through all the layers. The thus obtained dataarrays were processed with Eq. �2� yielding a line profile ofrelative � values for several materials. It showed abruptjumps at the material interfaces superimposed on a slowbackground originating from minor thickness variations inthe studied sample. The latter contribution has been fitted bya second-order polynomial and removed resulting in a cor-rected line profile of relative � values. They were convertedinto absolute ones using the well-calibrated �=145 nm valuefor crystalline Si.8,12–15 It is important to note that similar�within 10%� � values were obtained for those layered struc-tures using Eq. �3� thus confirming the reliability of theKramers–Kronig sum method.III. EXPERIMENTAL RESULTSA. Material dependence of inelastic scatteringFigure 1 summarizes inverse � values measured at200 keV for the majority of stable �ambient conditions� el-emental solids and oxides. Preliminary results revealed nosignificant effect of crystalline structure, such that � valuesvaried within 10% among single crystal, polycrystalline, andamorphous forms. The largest variation was observed for ox-ides of light elements, such as SiO2, Al2O3, and B2O3: � foramorphous phase was �10% larger than for the crystalline.We attribute this effect to the smaller mass density commonfor amorphous phases of many materials. Therefore, singlecrystalline regions were selected for all the measurements.Figure 1 reveals a clear periodic dependence of � suchthat within one row of the Periodic Table, the minimum�maximum� of � is observed for elements with completed�empty� outer d shell. A smaller variation is observed for theoxides.IAKOUBOVSKII et al. PHYSICAL REVIEW B 77, 104102 �2008�104102-2B. Importance of �„Z… dependence for electron microscopyNote that the revealed above periodic ��Z� behavior hasnot only academic but also practical value. Apart from thementioned significance for the various electron spec-troscopies and radiation physics, it is also important for theinterpretation of TEM images. For example, much of trans-mission electron microscopy is based on intuitive analysis ofthe so-called dark-field and bright-field images. Roughlyspeaking, dark-field images are constructed using electronsscattered at relatively large �30 mrad for STEM� anglesand are dominated by elastic and thermal diffuse scattering.Meanwhile, bright-field pictures are usually obtained atsmaller angles and contain a mixture of elastic, thermal dif-fuse, and inelastic scattering. When sample thickness ex-ceeds few inelastic mean free paths �few hundreds of nanom-eters�, the latter contribution dominates.Conventional intuitive analysis of dark-field and bright-field TEM images is based on the monotonic increase of��Z�, which is experimentally established for thermal diffuseand elastic1 and assumed for inelastic scattering �see Intro-duction�. Therefore, in the absence of diffraction effects,darker �lighter� areas in bright-field �dark-field� images, re-spectively, are associated with heavier elements, as shown inpanels �a� and �b� of Fig. 2 for a Pt /Cr interface. However,the oscillatory behavior of inelastic scattering revealed inFig. 1 suggests that caution should be exercised when usingthis interpretation. Indeed, a solid composed of element withlarger Z �i.e., stronger elastic and thermal diffuse scattering�can have smaller 1 /� value �i.e., weaker inelastic scattering�.An illustrative example is the bright- and dark-field picturesof a technologically important diamond/Si interface shown inFigs. 2�c� and 2�d�. A thick sample has been selected for thisdemonstration such that the �multiple� inelastic events domi-nated the scattering. Note that here diamond area appearsdarker than Si both in dark- and bright-field pictures. Thiscould cause misinterpretations of bright-field images ofmixed-phase samples where the spatial distribution of differ-ent materials is unknown.It is important to stress that the contrast of the bright-fieldimages can also be altered by diffraction effects and that theabove example merely demonstrates that if those effects areexperimentally avoided or negligible �e.g., in amorphous orfine-grain solids�, then the material dependence of inelasticscattering must be taken into account. As to Fig. 2, the Pt, Cr,and diamond areas have indeed fine-grain polycrystals. Sili-con is a single crystal, but it was carefully aligned to thelow-index zone axis.IV. ANALYSIS AND DISCUSSIONThe shape of the 1 /��Z� dependence of Fig. 1 resembles�cf. Figs. 1 and 5� that of the mass density ��Z�. Therefore, inour preliminary study,8 it was approximated as 1 /���0.3;however, no physical interpretation could be provided. Inthis section, we shall try to understand, at least qualitatively,the 1 /��Z� variations of Fig. 1. We shall focus our discussionon the elemental solids and will briefly discuss the oxides inthe end of the section.Let us first discuss the dominant cause of inelastic scat-tering. In Introduction, it was suggested to be plasmons,however, Fig. 3 reveals it is not always the case. This com-plex figure presents the plasmon energy Ep and the energiesof relevant core-loss transitions. The latter are conventionallylabeled as M23, M45, N23, N45, N67, O23, and O45 and corre-spond to the transitions from the 3p, 3d, 4p, 4d, 4f , 5p, and5d states to the excited state levels, respectively. Also shownFIG. 1. �Color online� Variation of inverse mean free path ofinelastic electron �200 keV� scattering with atomic number Z mea-sured in stable elemental solids and oxides. For oxides, Z corre-sponds to the main element, e.g., Z=14 for SiO2.FIG. 2. ��a� and �c�� Dark-field and ��b� and �d�� bright-fieldSTEM images from ��a� and �b�� Pt /Cr and ��c� and �d�� diamond/Siinterfaces. Note a usual contrast reversal between dark-field andbright-field images for Pt /Cr. This, however, does not occur in thediamond/Si sample because of “abnormal” � behavior.FIG. 3. �Color online� Energies of the core-loss �open symbols�and plasmon �EP, solid squares� transitions and a characteristic EELenergy EC �black solid line� �see text for details�.MEAN FREE PATH OF INELASTIC ELECTRON… PHYSICAL REVIEW B 77, 104102 �2008�104102-3in Fig. 3 are the characteristic EEL energies EC, which in thiswork were calculated from the experimental spectra asEC2 =� S�E�EdE� S�E�EdE. �4�Note that the function EC is special by that it can be easilyevaluated both experimentally and analytically.1 For ex-ample, most other combinations of scattering functions dodepend on the �plasmon� linewidth that brings extra compli-cation to the analysis. On the contrary, within the jelliummodel, EC is simply equal to Ep. Note, however, a noticeabledifference between EC and Ep in Fig. 3 which can be inter-preted as follows.In the light-element solids �Z�20�, the core-loss energiesare relatively large �hundreds of eV� and thus do not signifi-cantly affect the total EELS intensity. The scattering is domi-nated by plasmons, EC is close to Ep, and their difference isdue to the asymmetry of the plasmon peak, which is ignoredin the free-electron model. For heavier elements, the contri-bution of the low-energy core-loss transitions becomes sig-nificant thus increasing the EC−Ep difference. Note, how-ever, that the largest difference is observed at Z�29, 48, and79 for elements with completed outer d shells, but thoseatoms do not show low-energy core-loss edges. This appar-ent inconsistency is discussed below �see Fig. 6�, and it isargued that for those elements, no single plasmon energy canbe defined, at least in terms of high-energy electron scatter-ing.The above discussion reveals that the ratio of the core-loss and plasmon contributions changes significantly with Zthereby complicating the analysis. In order to discuss theplasmon part only, we have removed the core-loss fractionand replotted the 1 /��Z� data of Fig. 1 in Fig. 4. The removalprocedure is discussed below �see Fig. 6�. In order to provideclear reference data, we have also copied the � values ofFigs. 1 and 4 into Table I.Figure 4 reveals that removing the core-loss componentdoes not alter the shape of the 1 /��Z� oscillations, but in-creases their amplitude. In attempt to fit their shape with thefree-electron model �Eq. �1��, we have also plotted in Fig. 4a ��z /A�1/2 function. The fitting parameter was the number zof free electrons per atom, which was restricted to be equalto the number of either s or s+ p or s+ p+d electrons in thecomposite outer atomic shell. The latter includes the nd, �n+1�s, and �n+1�p levels, n=3,4 ,5. They usually have closeenergies and therefore should be considered together. A rea-sonable agreement between the plasmon contribution and the��z /A�1/2 function is observed suggesting that the free-electron model can, at least qualitatively, explain the materialdependence of the plasmon scattering.Let us discuss z numbers, which were used for fitting thedata of Fig. 4 and are summarized in Fig. 5. Values z�6correspond to the s+ p electrons. Larger z �up to 14� are,however, observed in Fig. 5 and originate from s+ p+d elec-trons in the outer shells. In order to understand such unusu-ally large “valence” numbers, let us look back at the proce-dure, which we used to separate plasmon and core-losscontributions.Figure 6 shows two representative cases: yttrium �Z=39,free-atom outer shell configuration of 5s25p1, z=3� and pal-ladium �Z=46, 4d10, z=10�. The former example is represen-tative of z�6 cases. Here, plasmon and core-loss peaks canwell be separated, for which we have used the followingprocedure: The high-energy tails of the EEL spectra werefitted with a polynomial function in materials with negligiblecore-loss contribution such as B, Be, P, S, and diamond. Acharacteristic function was found and applied to other mate-rials. In most cases, however, only a small part of the plas-mon spectrum could be used to adjust the fitting curve �seered line in the top part of Fig. 6� that brought extra uncer-tainty to the thus deduced plasmon contributions presented inFig. 4 and Table I.The palladium example is characteristic of z6 situation.Here, the plasmon peak ��8 eV for Pd� is accompanied by anumber of other features; their separation can hardly beachieved unambiguously and therefore it has not been at-tempted. Those features probably correspond to different ex-citations of the composite outer shell consisting of different4d+5s+5p electronic configurations. This interpretation sug-gests that the number of valence electrons per atom involvedin plasmon scattering is larger than number of electrons tak-ing part in chemical reactions.In summary, the above analysis suggests that the periodicshape of the 1 /��Z� dependence can well be explained by thecombination of three major factors: mass density �, numberof electrons z at the outer shell, and the core-loss transitions.In order to analyze their relative weights, we have plotted thecorresponding values in Fig. 5 and found that those contri-butions to 1 /� are comparable: for example, elements fromK �Z=19� to Ni �Z=28� show tenfold increase both in z�from 1 to 10� and � �0.89–8.9 g /cm3� values. The core-lossfraction varies, but it can be �50% or larger for many ma-terials.Finally, as we have understood the 1 /��Z� dependence inelemental solids, let us briefly discuss the oxides. Figure 1FIG. 4. �Color online� Solid circles show plasmon contributionto the inverse mean free path deduced for elemental solids from thedata of Fig. 1. Open squares represent the fitting function ��z /A�1/2suggested by the jellium model.IAKOUBOVSKII et al. PHYSICAL REVIEW B 77, 104102 �2008�104102-4TABLE I. Values of total mean free path �200 keV� of inelastic scattering � and of plasmon contribution�P copied from Figs. 1 and 4. Accuracies are �5% –10% for � and �10% –30% for �P.Z Material��nm��P�nm� Oxide��nm�4 Be 160 1695 B 123 126 B2O3 1206 Diamond 112 11612 Mg 150 214 MgO 13313 Al 134 160 Al2O3 14014 Si 145 168 SiO2 15515 P 160 16016 S 200 20020 CaO 13021 Sc2O3 12522 Ti 120 202 TiO 12023 V 109 158 V2O5 11624 Cr 104 149 CrO3 11825 Mn 106 14626 Fe 102 121 Fe2O3 11627 Co 98 108 CoO 11528 Ni 98 103 NiO 11529 Cu 100 10030 Zn 106 106 ZnO 11731 Ga 11032 Ge 120 126 GeO2 13034 Se 130 205 SeO2 13538 SrO 12639 Y 124 354 Y2O3 12240 Zr 113 268 ZrO2 11541 Nb 105 19442 Mo 98 163 MoO3 11144 Ru 90 13446 Pd 94 118 PdO 11047 Ag 100 125 Ag2O 11248 Cd 107 13049 In 110 12950 Sn 115 273 SnO2 11551 Sb 120 23452 Te 130 216 TeO2 12853 I 140 23356 BaO 12557 La2O3 13058 Ce2O3 12559 Pr2O3 12260 Nd2O3 12062 Sm 112 280 Sm2O3 12063 Eu2O3 11864 Gd 110 275 Gd2O3 12565 Tb2O3 12566 Dy 118 310 Dy2O3 12667 Ho2O3 120MEAN FREE PATH OF INELASTIC ELECTRON… PHYSICAL REVIEW B 77, 104102 �2008�104102-5reveals much weaker Z dependence for the oxides that can beunderstood as follows. Introducing a fixed element oxygeninto elemental solids results in averaging of the mean densi-ties of atoms and valence electrons. For example, those den-sities decrease for transition metals and increase for alkali.This results in flattening of the 1 /��Z� dependence in oxides,and we can anticipate a similar effect for other compounds�e.g., sulfides, nitrides, chlorides, etc.�. Our experimentaldata support this intuitive suggestion, however, due to theirmajor incompleteness, they are not presented here.In the above paragraph, we have attempted to qualita-tively explain the 1 /��Z� variation in the oxides. It is pos-sible to apply the quantitative analysis of the elemental sol-ids, performed in this section, to the oxides. However, itwould involve additional speculative quantities, such as av-erage atomic and electronic densities, and therefore has notbeen attempted.V. SUMMARY AND CONCLUSIONSIn this paper, we have reported the values of the mean freepath � of inelastic scattering for 200 keV electrons, system-atically measured from most of the stable �ambient condi-tions� elemental solids and some oxides. The recently devel-oped routine allowed us to increase the accuracy ofmeasurements and consequently reveal a clear periodic de-pendence of � as a function of atomic number Z, which wasmissed in the previous attempts1,4,9 and which is the mainachievement of this study. This ��Z� variation has not onlyacademic but also practical value. In particular, our TEMobservations �see Fig. 2� reveal its importance to the inter-pretation of bright-field TEM images.In attempt to qualitatively explain the ��Z� dependence,we have split it up into the contributions of low-energy core-TABLE I. �Continued.�Z Material��nm��P�nm� Oxide��nm�68 Er2O3 11570 Yb 110 275 Yb2O3 11572 Hf 95 23773 Ta 88 183 TaO 11074 W 82 151 WO3 11075 Re 78 14177 Ir 78 121 IrO 11078 Pt 82 12079 Au 84 12080 HgO 11681 Tl 95 13582 Pb 99 141 PbO 12283 Bi 105 147 Bi2O3 125FIG. 5. �Color online� Relative contributions to the inversemean free path data of Fig. 1 for elemental solids: number of va-lence electrons per atom z �solid circles�, volume density of atoms� /A �solid triangles�, and the core-loss fraction �open squares�. Allthose contributions appear significant.0 20 40 60 80 100 120 140plasmoncore loss (N23)ZLP x0.02PalladiumYttriumZLP x0.1Electronenergylosssignal(arb.units)Energy loss (eV)FIG. 6. �Color online� Two representative examples of separat-ing the plasmon and core-loss contributions in the EEL spectra.ZLP marks the zero-loss electron scattering peak. The separation israther straightforward for low-valence elements such as yttrium�upper spectrum�, but not for elements with completed d shells,such as Pd �lower spectrum�.IAKOUBOVSKII et al. PHYSICAL REVIEW B 77, 104102 �2008�104102-6loss excitations and plasmons and analyzed the latter withthe free-electron model. The ��Z� variation was explained bya product of three comparable factors, namely, atomic den-sity �, number of “free” electrons per atom z, and the core-loss contribution.An ambiguity of the EELS analysis has been revealed,such that the separation between the plasmon and low-energycore-loss transitions is hardly possible for transition metalswith �nearly� filled outer d shell. This ambiguity probablyoriginates not from imperfect mathematical procedures butfrom physical reasons—it may be argued that those elementsnot have a single plasmon peak, but a plasmon spectrumoriginating from various d+s+ p configurations of the outershell.ACKNOWLEDGMENTSThe authors are grateful to H. Shima and H. Akinaga�AIST, Tsukuba, Japan� for provision of theSi /SiO2 /Ta /CoO /Pt and Si /SiO2 /Ta /Pt /NiO /Al materials.This research was partially supported by Ministry of Educa-tion, Science, Sports, and Culture �MEXT� through theGrant-in-Aid for Young Scientists �B� 2005 17710120-6816and the Nuclear Research Project.*iakoubovskii.konstantin@nims.go.jp1 R. F. Egerton, Electron Energy Loss Spectroscopy in the ElectronMicroscope �Plenum, New York, 1986�.2 H. Bethe, Ann. Phys. 5, 325 �1930�.3 C. J. Powell and A. Jablonski, J. Phys. Chem. Ref. Data 28, 19�1999�.4 S. Tanuma, C. J. Powell, and D. R. Penn, Surf. Interface Anal. 25,25 �1997�.5 M. Inokuti, J. L. Dehmer, T. Baer, and J. D. Hanson, Phys. Rev.A 23, 95 �1981�.6 F. Lenz, Z. Naturforsch. A 9A, 185 �1954�.7 A. Crewe, J. P. Langmore, and M. S. Isaacson, in Physical As-pects of Electron Microscopy and Microbeam Analysis �Wiley,New York, 1975�.8 K. Iakoubovskii, K. Mitsuishi, Y. Nakayama, and K. Furuya,Microsc. Res. Tech. �to be published�.9 P. A. Crozier, Philos. Mag. B 61, 311 �1990�.10 Y. Y. Yang and R. F. Egerton, Micron 26, 1 �1995�.11 R. F. Egerton and S. C. Cheng, Ultramicroscopy 21, 231 �1987�.12 Q. Jin, Microsc. Microanal. 10, 882 �2004�.13 O. L. Krivanek, C. C. Ahn, and G. J. Wood, Ultramicroscopy 33,177 �1990�.14 R. Uemichi, Y. Ikematsu, and D. Shindo, J. Jpn. Inst. Met. 65,427 �2001�.15 C.-W. Lee, Y. Ikematsu, and D. Shindo, J. Electron Microsc. 51,143 �2002�.MEAN FREE PATH OF INELASTIC ELECTRON… PHYSICAL REVIEW B 77, 104102 �2008�104102-7