Characteristic Energy Losses of Electrons in a Solid*

G. Fridman

Introduction

The experiments of Franck and Hertz on collisions of electrons with mercury atoms showed that inelastic scattering of electrons is accompanied by discrete energy losses. In 1941 Ruthemann first discovered the presence of discrete energy losses, multiples of 14.7 eV, by electrons when they passed through aluminum foil. Whereas energy losses in gases can readily be correlated with the scheme of atomic terms by comparison with optical spectra, in the case of energy losses in a solid the corresponding spectra lie in the region of soft X-rays, and because of the great experimental difficulties such a comparison was impeded[^1]. Moreover, in a solid one should expect much more complicated relations for the interaction of the primary electron with the medium[^2]. At present nothing definite can be said about the mechanism of inelastic scattering of electrons in a solid. Along with attempts to interpret the observed phenomena in a manner quite analogous to the Franck–Hertz experiment (Bethe, Bloch, Slater), there is also possible, in particular, an explanation based on the idea of a collective interaction of the primary electron with the entire electron gas of the solid (the plasma theory of Pines and Bohm). In view of the fact that different theories often describe the quantitative results of observations almost equivalently, it is difficult to indicate an experiment that would make it possible to choose unambiguously one theory or another[^3]. At the same time, a correct interpretation is often made difficult by contradictory measurement results. The fact that the magnitudes of the energy losses in scattering are characteristic of a given scattering substance and that they are completely independent of the energy of the primary electrons has been established with complete certainty in a large number of works for all the substances investigated. A review of the energy losses measured up to the present time shows that there exist still some experimental parameters, not indicated in the papers, which affect the magnitude of these losses and, consequently, make it difficult to compare the data of different authors. The discrepancies in measurement results very often go far beyond the stated accuracy. Often the data of different authors on the intensity and half-width of the observed lines do not agree with one another. It is therefore necessary to control the experimental conditions more carefully.

Experimental Methods

In this section we shall dwell in more detail on the measurement methods used by various authors. In some respects the simplest apparatus is found in Rudberg[^4], who from 1929 to 1936 carried out

* Fortschritte der Physik 5, 51 (1957). Translated by V. A. Kiseleva.

the first accurate and systematic investigations. He already encountered two substantial experimental difficulties that arise in the study of characteristic losses in solids. These are, on the one hand, the difficulty of preparing the substance for investigation while meeting especially high requirements of purity and, on the other hand, the difficulty of obtaining high resolving power of the electron spectrograph. In Rudberg’s apparatus, which did not undergo essential changes in the subsequent works of Turnbull and Farnsworth, and also of Reichertz and Farnsworth\(^5\), electrons with energies from 40 to 300 eV (and in some cases up to 900 eV) were reflected from a thin layer deposited on a metal plate directly in the vacuum of the spectrograph. Immediately after reflection, through a narrow entrance slit, the electrons entered a homogeneous magnetic field and, describing a semicircle in it, were collected at the analyzer slit. Behind the slit there was placed a Faraday cylinder connected to a sensitive electrometer or to an electrometer tube. As is known, a beam of electrons emerging from one point and having equal velocities, after being deflected from its initial direction by \(180^\circ\) in a homogeneous magnetic field, is, to a first approximation, again gathered at a point. The velocity of the electrons was set by means of an accurately measured potential applied in addition to the primary voltage. The magnetic-field strength was then set so that, in the absence of the additional potential, only elastically scattered electrons were recorded. The displacements of the image points of beams having different velocities are proportional to the relative difference of velocities and to the radius of the trajectory. With a trajectory radius of 2.5 cm and an energy of the primary electrons of 200 eV, Rudberg used widths of the entrance and exit slits that corresponded to an energy difference of \(8 \cdot 10^{-3}\) eV, which was taken as the width of the instrumental function. The resolving power was limited to approximately 0.8 eV as a result of the thermal energy distribution of the primary electrons emitted by a heated tungsten filament. Many of the spectrograms reveal an asymmetry, characteristic of the Maxwellian distribution, in the shape of the curve for elastically scattered electrons. The half-width of the observed maxima increases by an amount equal to the half-width for elastically scattered electrons (in this case by 0.8 eV). The position of the maximum, however, is shifted, depending on the shape of the curve of elastic and inelastic scattering, by as much as several tenths of a volt. In the point-by-point method used here for constructing the scattering curve, fluctuations of the current, the primary energy, and the magnetic field, as well as the frequently occurring phenomena of charge accumulation, limit the accuracy in determining sharp characteristic maxima to a value approximately equal to 0.1 eV. In the case of flatter maxima, the sensitivity with which they can be detected is correspondingly reduced because of the limited accuracy of registration.

The entire vacuum apparatus could be heated to \(400^\circ\)C, and the layers under investigation to \(1500^\circ\)C. Rudberg achieved a vacuum below \(10^{-7}\) mm Hg, and Farnsworth below \(5 \cdot 10^{-8}\) mm Hg. The purity of the treated surfaces obtained by Rudberg was not achieved in any of the subsequent works.

IMPROVEMENT OF THE METHOD

Substantial progress in the observation of characteristic losses was achieved by Ruthemann in 1941–1948\(^6\), using electrons with a primary energy of the order of 6 keV. He was the first to discover unusually sharp, multiple characteristic losses of electrons passing through thin aluminum foil. Since the mean free path of characteristic—

...scattering of the scattered electrons decreases with a decrease in their initial energy, measurements of single scattering events of low-energy electrons are impossible even when the thinnest layers are used. Ruthemann achieved the necessary resolving power thanks to a radius of trajectory (17.5 cm) seven times larger than Rudberg’s, and also thanks, at least in part, to correction of second-order image errors.

Two coaxial current-carrying coils create, in the middle of the annular region lying between them, a highly homogeneous magnetic field (Fig. 1). The radial falloff of this field is altered by superposing an additional field from a correcting coil so that the image (hereafter everywhere called the line) of the entrance slit, rotated through \(180^\circ\), appearing on the screen or photographic plate is as sharp as possible. The exact position of the focal plane of the magnetic cylindrical lens is found experimentally by focusing the “shifted lines” obtained with a slightly changed primary voltage. The half-width of the energy distribution of electrons that did not undergo inelastic scattering on their path was, at a primary energy of 4 kev, 2.2 ev, and at 7.5 kev, 3 ev. If the thermal half-width of the electrons, 0.8 ev, is subtracted, the instrumental line width obtained is, respectively, 1.4 and 2.2 ev.

Fig. 1. Diagram of Ruthemann’s apparatus.

These quantities take into account, along with the slight imperfection of the electron optics, also fluctuations of the primary voltage, not exceeding, for example, in the case of 5-kev electrons, 0.03%. Low-frequency fluctuations (for exposure times up to 5 min) of the transformer supply circuit were eliminated by Ruthemann with the aid of ballast tubes; high-frequency fluctuations, by the use of an electronic stabilizer. Sufficient constancy of the magnetic field is easily ensured by the use of several accumulators. Behind the entrance slit there is placed a device that makes it possible to expose different specimens and to limit the angular aperture \(\beta\) of the scattered beam. The observed range of scattering angles \(\theta\) was less than \(4^\circ\); the diaphragm aperture \(\beta\) was \(1\text{–}3^\circ\), later \(0.5\text{–}2^\circ\). The image appearing on the photographic plate, after construction of a photometric curve with the aid of blackening marks, directly gave the energy distribution of the scattered radiation. Calibration of the spectrograph was carried out by means of “shifted lines” obtained by stepwise variation of the primary voltage in steps of 10 v. The sensitivity of detection of sharp maxima, determined experimentally from the scatter of calibration measurements of shifted lines, was approximately 0.2 ev. Thin layers were prepared either by depositing the substance on rock salt and subsequently dissolving it in water, or by deposition on thin collodion films. The changes thereby occurring in the layers, such as, for example, oxidation upon dissolution in water, can be detected only by comparing layers prepared by different methods. It is not known how the indicated layer thicknesses were determined.

Law’s measurements (1948) were carried out on similar apparatus; layers on a collodion substrate were used. The thickness of the layers was determined from their optical transparency[^7].

In 1951 Geschlössl used a 70-degree magnetic cylindrical lens and achieved the same resolution as in Ruthemann’s improved method. Since in this case the entrance slit and the image are sufficiently far removed from the lens, there is considerably greater freedom in setting up the thermocathode[^8]. Registration was carried out by the photographic method. The magnetic field (trajectory radius 20 cm) was produced by a very homogeneous magnet of soft iron. The scattering angle in this apparatus could be chosen practically arbitrarily. In subsequent work Friedman achieved still greater resolution by improving the correction of second-order errors, by correcting the distortions introduced by the fall-off of the field at the edge of the magnet, and also by even stronger diaphragming ($\beta \simeq 10^{-3}$) of the electron beam. The theoretical instrumental line width was, for 6-keV electrons, less than 0.1 eV. The experimentally obtained curve of the energy distribution of primary electrons typically had a Maxwellian form with a half-width of 0.8 eV. The sensitivity for detecting maxima, as determined from calibration measurements, was 0.1 eV. The layers under investigation were prepared by evaporation onto thin Formvar substrates. Both before the spectrogram was taken and after it, the layers were examined in an electron microscope. The structure of the layer was established from diffraction photographs. The thickness of the evaporated layers was determined by observing interference fringes of equal thickness (Tolansky method).

ELECTROSTATIC ANALYZER[^25]

The use of considerably higher energies of the primary electrons, up to 100 keV, while maintaining at least as good an absolute resolution as in the magnetic method, became possible thanks to Möllenstedt’s introduction, in 1949–1952, of the electrostatic velocity analyzer[^10]. In an electrostatic lens of the usual design, the opposing field of the intermediate electrodes sharply decelerates and deflects the electrons as the distance $r$ from the optical axis increases. Therefore, within the lens the electron crosses its optical axis several times before reaching the image plane, which leads to a displacement of the image of a small aperture with increasing $r$. Small relative differences in the velocities of the electrons, owing to retardation, increase sharply and lead to small changes in $r$, which can be very well resolved. To facilitate the interpretation of the spectrograms thus obtained, Möllenstedt replaced the round diaphragm by a narrow slit (5 μ), placed at the optimum distance $r$ from the optical axis of a large-diameter lens (Fig. 2). This distance $r$ is found experimentally. To avoid the influence of high-voltage fluctuations, which are difficult to eliminate, on the image of the slit, use is made of the same

Fig. 2a. Möllenstedt analyzer. Left—general view; right—section with depiction of the path of the electron beams. $Sp$—analyzer slit.

the fact that the refracting properties of the electrostatic lens depend only on the ratio of the primary voltage to the lens potential. If the voltage is supplied to the lens through a potentiometer fed by the primary voltage, then this ratio remains constant. Later, Möllenstedt used, as the entrance slit of the analyzer, an image of the optical slit reduced by means of an intermediate lens. Thanks to this he achieved an instrumental line width of the order of several hundredths of an eV. The measured half-width of the energy distribution of the oxidized cathode was 0.5 eV.

For calibration of the spectrograph, a battery voltage was still superimposed on the primary voltage, while the lens voltage remained unchanged. From the results of measurements on “shifted lines,” Möllenstedt estimated the sensitivity of his apparatus as 0.1 eV at 35 keV. Möllenstedt placed an electrostatic analyzer in an electron microscope of ordinary design and was able to study in greater detail the layer before taking the spectrogram; the 5 μ slit was thereby removed from the path of the beams. Möllenstedt discovered and studied the characteristic losses of many substances.

Fig. 26. Electron microscope with Möllenstedt’s analyzer.

Labels in the figure: electron gun; anode; object; objective; narrow slit; cylindrical lens of the spectrograph; projecting lenses; shutter for fogging marks; shutter for the velocity spectrogram; photographic plate; 55 kV; 0.26 μF; 100 V battery; resistance circuit 1000 MΩ.

Subsequently, many investigators used, with minor modifications, the measurement method developed by Möllenstedt. In 1954 Klein eliminated, by means of a weak cylindrical lens (also of large diameter) placed behind the analyzer, the longitudinal blurring of the slit image caused by the analyzer, and thereby achieved a noticeable increase in intensity. He observed electrons specularly reflected at an angle of 5° from layers that could be heated to 200°C. In attempting to study the reflection of electrons from a KBr single crystal, he found that the crystal was destroyed because of the high intensity of the beam. Klein does not describe the effect of the beams on the other substances he investigated.

Marton and Leder (1953) used, instead of a photographic plate, automatic recording of the spectrum by means of a fluorescent screen connected to a photomultiplier.^12 These authors avoided placing the objective behind the scattering layer between the 5 μ slit and the analyzer lens, so as to exclude distortions in the intensity distribution on the spectrogram caused by its chromatic aberration. In studying the structure of layers, the apparatus operated on the principle of a shadow microscope. For the investigation of alkali metals, the layers were prepared by evaporation directly in the vacuum of the spectrograph ($2 \cdot 10^{-5}$ mm Hg).

Watanabe (1954) placed the analyzer lens between the objective and the eyepiece of a magnetic electron microscope.^13

In order to use the electrostatic analyzer for investigating the angular distribution of inelastic scattering, Leonhard (1954) selected the diffraction image of the scattering layer by means of an additional

of a magnifying lens onto the plane of a \(5\,\mu\)-target (Fig. 3). By magnifying the diffraction image, the diffraction angles of order \(10^{-1}\) were reduced to the value of the aperture of the analyzer maximally permissible\({}^{14}\), amounting to \(2\cdot 10^{-3}\).

Fig. 3. Path of electron rays in Leonard’s apparatus for observing the angular distribution. \(B_3\) — slit of the analyzer. \(L_3\) — lens of the analyzer.

An electrostatic velocity analyzer of a particularly convenient design was used by Gabor in 1954\({}^{15}\). Here the electrostatic field of a cylindrical capacitor focuses electrons with a definite tangential component of velocity onto the surface of a coaxial cylinder, in which the entrance and exit slits lie. The electrons fall at an angle of \(45^\circ\) relative to the axis of the cylinder. Electrons with a somewhat different tangential velocity are deflected to a cylindrical surface of a somewhat different radius (Fig. 4). Gabor and Joll studied, in transmission, gold films deposited on a carbon substrate. They studied the structure of the film in an electron microscope; the thickness of the film was determined by the Tolansky method.

Blajstock, Birkhoff, and Slater (1954) studied the energy distribution of electrons scattered in a layer as a function of the primary energy. In studying the dependence of the mean free path of characteristically scattered electrons on their energy, these authors reduced the velocity of the scattered electrons from \(20\)—\(100\,\mathrm{keV}\) to a value of the order of \(100\,\mathrm{eV}\) by means of a retarding field applied beyond the scattering substance, and analyzed these relatively slow electrons in a simple electrostatic analyzer\({}^{17}\). The resolutions achieved by them at \(100\,\mathrm{keV}\) corresponded to an instrumental line width of \(1.2\,\mathrm{eV}\).

Fig. 4. Gabor electrostatic analyzer. a) Path of the rays with radial focusing when observed along the axis of the cylinder. b) Path of the rays when observed perpendicular to the axis of the cylinder. \(K\) — cathode, \(F\) — scattering layer, \(A\) — entrance diaphragm, \(E\) — receiver.

To study the energy distribution of fast electrons, many authors also used the classical retarding-field method (Lenz, Börsch^18, Haberstroh^19). The beam of scattered electrons was limited by a diaphragm and was retarded by a strong retarding field applied between the diaphragm and the receiver. The retarding-field voltage differed from the cathode potential by a precisely measured value of the battery voltage. The current of all particles with energies above a specified value was measured. To achieve high sensitivity it is necessary to diaphragm the beam of scattered electrons. By heating the receiver above \(200^\circ\mathrm{C}\), it is possible to avoid the formation on it of interfering carbon layers. A substantial drawback of the retarding-field method is that it gives only integral curves of the energy distribution. Steinmann^20 eliminated this drawback by applying a differentiating pulse circuit. His apparatus made it possible to record the Maxwellian energy distribution of electrons at a primary voltage of \(10\ \mathrm{kev}\).

RESULTS OF MEASUREMENTS

In this section we shall first become acquainted with the spectra of characteristic losses observed for a number of elements and compounds. We shall then dwell on the question of the influence on these spectra of individual experimental conditions, such as, for example, the thickness and structure of the layer, the “scattering angle,” and the energy of the primary electrons.

In conclusion we shall consider the angular distribution of characteristically scattered electrons, as well as the dependence of the mean free path between two acts of inelastic scattering on the energy of the primary electrons.

Spectra of characteristic energy losses. At present the spectra of characteristic losses are known for many substances. These experimental data were obtained mainly by Möllenstedt, Marton, Klein, and Watanabe on an electrostatic spectrograph (see the detailed review by Marton^2). In the present review we shall become acquainted with the spectra of characteristic losses of those substances which were investigated independently by several authors.

In such a case, one should first of all turn to aluminum. Experimental data on its characteristic losses are given in Table 1. As is seen from this table, all experimenters found

Table 1

Characteristic energy losses in aluminum (in ev)

a sharp and intense maximum at 14.7 eV; the majority also indicate subsequent discrete losses, multiples of 14.7 eV, at 30, 44, and 59 eV, as well as a broad and weak maximum near 7 eV. In some cases multiple 7 eV losses were found at 22 eV and 36 eV (losses that are even multiples of 7 eV are imperceptible in the presence of the series of strong multiples of the 14.7 eV losses). The existence of the maximum near 7 eV cannot be explained by partial oxidation of the aluminum layers, since in studying aluminum oxide, Al₂O₃, no energy losses by electrons at 7 eV were observed. The mean deviation, according to data from different authors, from the averaged value of the characteristic loss at 14.7 eV is ±0.2 eV, and here the result of Marton (13.9 eV), which differs sharply from the others, is also taken into account. The mean deviation, according to the data on the characteristic loss at 7 eV, is about 0.6 eV. The indicated deviations still do not exceed the expected experimental accuracy of determining the magnitude of these characteristic losses.

Fig. 5. Characteristic energy losses in an Mg layer 600 Å thick; Watanabe’s photograph for the case of 25 keV electrons.

The characteristic spectrum of magnesium contains a series of discrete, multiple 10.3 eV losses (see Table II). In Watanabe’s photographs (Fig. 5), even quadruple losses at 41.5 eV are quite clearly visible. The half-width of the first two “lines,” according to this author’s data, is 2 eV and only slightly exceeds the half-width of the primary “line.” In the spectrograms of Marton and Klein the maximum at 20 eV is much broader than the 10 eV maximum. Subsequent multiple losses were not found by them. The appearance in Klein’s spectrograms of a weak maximum near 4.7 eV is apparently due to partial oxidation of the magnesium layer, since in studying magnesium oxide Watanabe observed two weak maxima at 4.5 and 5.5 eV.

Table II

Characteristic energy losses in magnesium (in eV)

Table III

Characteristic energy losses in beryllium (in eV)

Beryllium (Table III). Both Watanabe and Ruthemann observed a sharp characteristic loss at 19 eV, as well as the next three losses that are multiples of it. The weak maximum at 6.3 eV, appearing in the photographs of Marton and Leder, may apparently be attributed to beryllium oxide,

since, in studying BeO, Watanabe found a maximum near 6.5 eV. Let us note that magnesium and beryllium oxidize in air much more rapidly than aluminum. According to Rudeman and Watanabe, the half-width of the first characteristic loss of beryllium is twice as large as the half-width of the first characteristic loss of aluminum; according to Marton, it is five times as large. The loss values obtained by Klein do not agree with the results of other authors. Whereas for the other authors the deviation from the mean value is less than 0.1 eV, the value of the first characteristic loss according to Klein differs from this mean value by 1.7 eV.

Table IV

Characteristic energy losses in copper (in eV)

Copper (Table IV). Multiple losses were not detected. No sharp and intense maxima were observed. In investigating the characteristic losses in copper, experimenters encountered exceptionally great difficulties. Hardly any one of them found agreement between his results and the results of other investigators. Let us take as an example the carefully performed work of Rudberg. He studied several massive copper specimens at high temperature (600–700° C) and in a high vacuum. Measurements of the positions of the maxima, carried out by him many times on each of the specimens, did not fall within the limits of the experimental accuracy. (Thus, the relative error in determining the position of the 25 eV maximum reached 20%.) The four spectrograms obtained by Rudberg did not give concordant results. For example, the maxima on the first spectrogram lay near 3.5, 6.9, and 25 eV; on the second, the only maximum was near 8.1 eV, etc. In subsequent experiments with cold cathodes sputtered with copper layers, Rudberg found three maxima (near 4.2, 7.3, and 23.6 eV).

The relative intensity of these lines underwent large changes from experiment to experiment.

The data of different authors, as is seen from Table IV, also agree poorly with one another. At small primary energies weak discrete maxima sometimes appear. For high primary energies a broad and intense maximum at 20 eV is characteristic (Fig. 6).

Fig. 6. Photometric curve of Lang’s Cu spectrum.

Similar difficulties are encountered in the study of nickel and silver, although these metals are apparently considerably less subject to

oxidation than copper or aluminum. Since characteristic losses in oxides of heavy metals have not been studied, it is impossible to isolate the characteristic spectrum of pure metals.

Table V

Characteristic energy losses in gold (in eV)

Gold (Table V). Because of the exceptional chemical stability of this element, the foils studied were apparently completely free from oxides and impurities. Nevertheless, despite this, the measurements had a clearly pronounced irregular character, which is generally typical of substances possessing multiple and usually broad characteristic lines. Almost all experimenters found characteristic losses at 23 and 32 eV, with the mean deviation from the values averaged over the data of different authors being about 2.5 eV. The maxima sometimes appearing on the spectrograms and lying in the region from 3 to 15 eV are weak but rather sharp. Experimental errors do not at all explain the observed shifts and distortions in the shape of these maxima. An example of such a contradiction is provided by two series of Rudberg’s data, obtained with unchanged experimental accuracy.

Table VI

Characteristic energy losses in aluminum oxide (in eV)

Al₂O₃ (Table VI). Films of aluminum oxide can be obtained by various methods, in particular by anodic oxidation of a deposited layer of aluminum. During the preparation process various kinds of contamination are quite possible. Nevertheless, despite this, the results of different works for aluminum oxide, in contrast to the above-mentioned studies of copper, nickel, silver, and gold, agree surprisingly well with one another. Thus, the first characteristic maximum, with a half-width of the order of 15 eV and lying near 22.5 eV, is indicated by various authors with a mean deviation from this value of only 0.3 eV. The double weak and broad maximum at 46 eV is absent only in Möllenstedt’s spectrum. Other maxima, in particular at energies below 20 eV, were not observed.

Of the other substances, only a few have been studied simultaneously by several experimenters, but even in these rare cases there is a certain agreement of the results. Of undoubted interest is the comparison, carried out by Marton and other investigators, of the characteristic spectra of metals with the spectra of characteristic losses of some of their compounds². For a number of compounds and the corresponding metals (SiO₂ and Si, TeO₂ and Te, PbS and Pb, as well as SbS₃ and Sb), Marton found a similarity of their characteristic spectra. The maxima of the compounds are only considerably broader and shifted by a large amount relative to one another. In view of the noticeable difference among the results of numerous investigations, the question of the influence of various experimental conditions on characteristic losses becomes of great importance. It is difficult to say whether the data obtained up to the present and presented below on the influence of some experimental parameters are sufficient to explain the observed differences in the measurements, since a careful determination of the various experimental parameters was not carried out. Apparently, further extensive measurements will be required here.

Influence of layer thickness and scattering angle. An increase in layer thickness causes a noticeable shift of both sharp (Al)⁶ and comparatively broad (Se)⁹ characteristic maxima (Fig. 7). Moreover, the first characteristic maximum broadens as the layer thickness is increased. Evidently, fluctuations of the layer thickness lead to still greater broadening of the characteristic lines. The ratio of the intensities of the characteristic losses, if the multiple character of their sequence is interpreted as the result of multiple inelastic scattering of electrons, is directly related to the layer thickness \(d\). Moreover, for the intensities of the characteristic maxima \(J_{nE}\), one should naturally expect in this case a Poisson distribution:

Fig. 7. Shift \(\delta E_1\) of the characteristic energy losses in Al₂O₃⁶ and Se⁹ with increasing layer thickness \(d\).

Fig. 8. Dependence of the ratio of the intensities of multiple maxima on the layer thickness for Al (according to the data of work ⁷) and Se (according to work ⁹). (The scattering background has been subtracted.)

\[ J_{nE}=\frac{1}{n!}\left(\sigma_{\mathrm{diff}}\cdot dN\right)^n e^{-Nd\sigma_{\mathrm{tot}}}, \tag{1} \]

where \(N\) is the particle density, \(\sigma_{\mathrm{tot}}\) is the total effective scattering cross section, \(\sigma_{\mathrm{diff}}=\dfrac{d\sigma_n}{d\Omega}\Delta\Omega\); \(\dfrac{d\sigma_n}{d\Omega}\) is the differential effective cross section for inelastic scattering. This assumption finds its experimental confirmation in Lang’s work on Al and Friedman’s work on Se. The multiple characteristic losses of aluminum are the most suitable material for testing relation (1) and the thicknesses of the investigated foils measured by Lang. In Fig. 8, with the aid of relation (1), the results of measurements of the intensity of the characteristic maxima and the layer thicknesses carried out by Lang and Friedman are compared.

Ruthemann\(^6\) had already discovered the increase in characteristic losses that occurs with increasing scattering angle. In studies with strongly diaphragmed beams of electrons (Se\(^9\), Mg, and others\(^ {13}\)) this displacement of the characteristic maxima becomes noticeable even at very small scattering angles \(\theta \sim 10^{-3}\) (Fig. 9). Along with the displacement of the characteristic line, with increasing \(\theta\) a characteristic change in the shape of the lines is observed, especially evident for broad characteristic maxima. Namely, the contour of the line becomes steeper on the side of smaller energy losses and, conversely, more gently sloping on the side of larger energy losses. At the same time the width of the maximum increases. From what has been said it is clear that, at sufficiently high absolute resolution, the aperture of the analyzed electron beam affects the width of the characteristic maximum, since its shape is a superposition of line contours, each of which corresponds to a definite value of \(\theta\). In this connection it becomes clear that measurements in reflection cannot be directly compared with measurements in transmitted radiation. We shall dwell on this point in more detail in discussing the angular distribution for inelastic scattering.

Fig. 9. Displacement \(\delta E_2\) of the first characteristic loss with increasing scattering angle \(\theta\), for Al (according to \(^ {13}\)) and Se (according to \(^9\)).

Dependence of characteristic losses on the primary energy of the electrons. The very concept of a characteristic loss already emphasizes the fact that the position of the maxima does not depend directly on the energy of the primary electrons. This was established by Rudberg and Ruthemann in their first investigations and was definitively confirmed by subsequent work. As an example, one may point to the very good agreement of the losses measured in aluminum over the range of primary energies from 6 to 35 keV. At the same time, there was not a single case in which differences between spectrograms obtained by different authors for one and the same substance were explained by a difference in the energies of the primary electrons. It follows from Rudberg’s work that measurements carried out with electrons of different energy on the same specimen of the substance under study agree with one another much better than measurements carried out on different specimens at unchanged energy of the primary electrons. However, the ratio of the intensities of the characteristic lines depends on the energy of the primary electrons\(^9\). Thus, when comparing the results of Rudberg and of Klemperer (both for the case of reflection), it is immediately apparent that losses below 10 eV, which are intense at a primary energy of the order of 200 eV, at an energy of the primary electrons of 30 keV are considerably inferior in intensity to the higher characteristic losses.

Influence of the structure of the layer\(*\). Careful control of the structure of the layers studied by means of electron diffraction and electron-optical devices is absent in most works. Therefore, when comparing the results of different authors it is impossible to take into account the influence of the layer structure. However, in two works there is an indication that in some cases this influence is insignificant. Turbul and Farnsworth established that the spectrum of characteristic losses in single crystals of se-

\(*\) On the connection with the angular distribution of elastic scattering, see below.

edge does not differ from the spectrum of characteristic losses in polycrystalline silver. In Friedman’s work, selenium layers were investigated, brought by annealing to a definite crystalline state. The structure of the layer was studied by electron diffraction both before the spectrograms were taken and after they were taken. It was established that, for a sufficiently short irradiation time and a weak electron flux, the irradiation does not produce changes in the layer (changes of structure, formation of carbon layers). In this work it was also established, by comparing characteristic losses in amorphous layers with losses in crystalline layers, that the structure of the layer has no appreciable influence on the position of the characteristic maxima. The accuracy of determining the maximum was equal to \(0.1\) eV. Gabor and Joule investigated gold films deposited on carbon layers. The layer under study consisted of several gold films about \(20\ \text{Å}\) thick, separated by thin carbon layers. Whether such an alternation of layers causes any specific changes in the spectrum of characteristic losses is still not clear. In the main, the characteristic spectra of pure and laminated gold foils agree with one another. But the intensity of the first characteristic maximum for the case of a pure gold layer exceeds by more than a factor of 10 the intensity of this maximum for the case of a laminated gold foil having the same total layer thickness \(^{16,2,3}\). Moreover, in the investigation of pure foils a number of weak maxima are observed which are absent in the case of laminated gold foils. Gabor and Joule interpret them as weakly excited characteristic losses. The ratio of the intensities of these maxima has not yet been sufficiently studied, and further investigations in this direction are necessary.

In many works, layers deposited on substrates were used. The influence of the latter on the scattering pattern was not eliminated \(^{9}\). However, owing to the fact that the characteristic maxima of the substrates used were very broad, in the main one should have expected only an increase of the background. The possibility was also not excluded of some displacement of the maxima as a consequence of the nonuniformity of this background \(^{6}\).

To clarify the influence of other changes—for example, chemical changes occurring in the layers under investigation in the course of their preparation—further studies are necessary. It would also be desirable to determine whether the scattering pattern is affected by the absorption of vapors by certain substances (as was found in the case of the formation of carbon layers).

Angular distribution of characteristically scattered electrons. The study of the angular distribution of characteristically scattered electrons became possible only after Leonard \(^{14}\) improved Möllenstedt’s electrostatic apparatus. In investigating Be (Al and Au), Leonard found that the maxima of the angular distribution of electrons inelastically scattered with an energy loss equal to the first characteristic loss correspond to the same angles as the interference maxima of elastically scattered electrons. The ratio of the intensity of elastic scattering to the intensity of inelastic scattering remains constant for all scattering angles. Since incoherently scattered electrons cannot give an interference pattern, it follows from this that either before or after inelastic scattering they are elastically scattered within the investigated range of interference angles \((\theta_1 = 5 \cdot 10^{-3}\) at \(50\ \text{keV})\). From the slight broadening of the interference maxima of the inelastically scattered electrons in comparison with the maxima of the angular distribution of elastically scattered electrons, one may conclude that the half-width of the angular distribution of the inelastically scattered electrons does not exceed the aperture of the analyzed

beam of \(2\cdot 10^{-3}\). From photographs obtained for Be layers \(300\,\text{\AA}\) thick it is seen that approximately 20% of the elastically scattered electrons additionally undergo an inelastic collision with an energy loss of \(20\,\text{eV}\) (the first characteristic loss). The probability of direct inelastic scattering into the region of significant interference angles is very small. Unfortunately, at scattering angles \(\theta<\theta_1\) it is impossible to separate elastic and inelastic scattering completely. Determination of the intensity of only elastic scattering is possible up to the aperture angle \(\beta\). In studying the angular distribution of inelastically scattered electrons the aperture of the beam must be small in comparison with the half-width of this distribution. A study of the angular distribution using strongly diaphragmmed electron beams \((6\,\text{keV},\ \theta_1=2\cdot 10^{-2},\)

Fig. 10. Angular distribution (in arbitrary intensity units) of the electrons scattered with the first characteristic loss in an Se layer, with primary energy \(6\,\text{keV}\). For comparison: the corresponding half-width of the angular distribution of elastically scattered electrons is \(\theta_H \approx 2\cdot 10^{-2}\). \(I_E/\beta t\) is the normalized intensity.

\[ \frac{\beta}{\theta_1}=5\cdot 10^{-2} \]

was carried out on selenium layers.^9 Multiple scattering greatly complicates the measurement of the intensity of inelastic scattering. In order to eliminate this drawback, very thin layers were used. The experiment showed that the angular-distribution curve of electrons inelastically scattered with an energy loss equal to the first characteristic loss falls with increasing scattering angle \(\theta\) (in the interval \(10^{-3}<\theta<2\cdot 10^{-2}\)) three times more rapidly than the theoretically calculated distribution of elastically scattered electrons (Fig. 10). In all cases in which the angle of the scattered electrons \(\theta>\theta_1\), i.e., primarily in spectrograms obtained in reflection, averaging of the inelastic scattering over small angles \(\theta<\theta_1\) occurs. The observed electrons are elastically scattered into the observation region.

Fig. 11. Dependence of the mean free path between two acts of characteristic scattering^17 on the primary energy.

Legend in Fig. 11:

• Paine and Bohm theory
• Film No. 1, first measurement
• Film No. 1, second measurement
• Film No. 2

Dependence of the intensity of characteristic losses on the primary energy. Studies of the dependenc...

...the mean free path length \(\lambda\) on energy for inelastic scattering were carried out with apparatus especially convenient for this purpose by Blackstock, Birkhoff, and Ritchie. They determined values of \(\lambda\) in the energy range from 20 to 100 keV for aluminum, magnesium, and copper layers deposited on thin zapon films. It was established that \(\lambda\) is proportional to the magnitude of the primary energy \(E_p\). The probability of characteristic scattering of electrons is inversely proportional to \(E_p\). In determining the intensity of characteristic losses, the intensities of multiple energy losses must be taken into account. The results for Al (Fig. 11) are in very good agreement with theoretical predictions.

COMPARISON OF THE RESULTS WITH VARIOUS THEORIES

In our review we have gathered the principal results of the experimental investigation of characteristic energy losses and, in conclusion, shall briefly mention those theoretical ideas that are used to explain them. The first model was developed by Bethe, Bloch, and Slater \(^{21}\). According to this theory, in an inelastic collision the primary electron transfers to an individual electron of the solid one of the possible excitation energies. But if, in the case of gases, the quantum-mechanical calculation of excitation energies and excitation probabilities is possible in the Born approximation and gives good results, then quantitative calculations for a solid, because of their complexity, encounter great difficulties. In a qualitative treatment \(^{9}\) it is assumed that characteristic energy losses arise because of scattering by electrons of ionic shells. The distance to the nearest free level corresponds to the characteristic loss. Free electrons have a substantially smaller excitation energy and therefore a greater probability of excitation (multiple scattering). Since the width of the conduction band is very large compared with the excitation energy, scattering by free electrons leads only to the appearance of a background on the scattering curve.

Another frequently discussed model was developed by Pines and Bohm within the framework of the theory of metals. They consider the excitation, by primary electrons, of collective oscillations of the density of free (or valence) electrons (plasma oscillations) \(^{22}\). The characteristic energy loss then corresponds to the natural frequency of the plasma oscillations. Making a reasonable estimate of the density of free electrons (from considerations of chemical valence), one obtains in many cases—at least in order of magnitude—correct values of the energy losses (for Al, Be, Mg, and some other substances even good agreement is obtained). Both theories give identical expressions for the scattering cross section \(^{23}\). The theoretical angular distribution of inelastic scattering and the dependence of inelastic scattering on the primary energy are in agreement with experiment. Until energy losses have been studied experimentally with sufficient thoroughness, the question remains open as to the nature of the excitation of each substance investigated, except for those quite definite cases when, for example, the absence of dependence on density can be demonstrated directly \(^{9}\). In most cases, unambiguous conclusions still cannot be drawn from comparing the energy levels of the solid, determined by measurements of the fine structure of X-ray \(K\)-quanta, with characteristic energy losses \(^{1}\). In any case, the good coincidence for Al of the values of the terms, deter-

…determined by the X-ray method and by the method of characteristic losses, is an essential argument in favor of the fact that, for this substance, excitation of a separate electron takes place.

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