NEW INSTRUMENTS AND METHODS OF MEASUREMENT

ELECTROSTATIC ANALYZER OF ELECTRON VELOCITIES

V. I. Milotin and A. N. Kabanov

1. INTRODUCTION

The study of the distribution of electrons by velocity in an electron beam is highly important for solving a number of problems in electronics and in the structure of matter. Thus, in the development of electron-beam instruments with high resolving power, including the electron microscope, it is very important to know the energy composition of the electrons in the beams. In the electron microscope, for example, analysis of the velocities of electrons that have passed through an object is necessary for studying the mechanism of image formation and for determining the contrast of the image and the resolving power of the microscope.

Questions of elastic and inelastic scattering of electrons as they pass through matter, beginning with Lenard¹, have constantly been the subject of theoretical and experimental investigation. A large amount of material has already been accumulated concerning the energy losses of both slow and fast electrons as they pass through various substances. However, the methods that existed earlier did not make it possible to obtain a complete picture of the processes of electron scattering, especially of fast electrons, in matter.

Of particular interest is the study of elementary scattering processes caused by single collisions of electrons in interaction with matter. Experimental investigation of such processes requires the use of very thin layers of the substances under study and spectral instruments with high chromatic resolving power.

Ruthemann², and subsequently Lang³, using the magnetic method of semicircular focusing, obtained quantitative results on scattering processes in thin films for electrons with velocities of 5–8 kv. Discrete energy losses were found corresponding to the excitation energies of the K- or L-X-ray levels of carbon, nitrogen, and oxygen. Along with these large losses, associated with excitation of the inner shells of the atom, it proved possible to determine small losses of the order of 20 ev, characteristic of each substance.

Of especially great interest are the energy losses in the passage through matter of fast electrons with energies of the order of 50–100 kev, which are important, for example, in the study of electron-diffraction phenomena and in electron microscopy. In this case the most suitable proved to be, as Möllenstedt⁴ first showed, an electrostatic lens possessing high chromatic resolving power. As was shown⁵,

the chromatic resolving power of an electrostatic lens is 40–50 times higher than that of a magnetic lens.

In the course of developing a velocity analyzer using an electrostatic lens, it proved possible in some cases to obtain a resolution of the order of tenths of a volt at accelerating voltages of 30–50 kV. The most convenient for purposes of analysis is a cylindrical (slit) electrostatic lens, which has turned out to be a very sensitive dispersing element of the analyzer. If such a lens is introduced into the circuit of an electron microscope, then, after certain modifications, it can be used as an analyzer. Recently special analyzer instruments have also begun to appear. Their field of application has lately been expanding rapidly. Along with the study of discrete losses of electron velocities as they pass through matter, they are used for the analysis of electrons reflected from matter, for the analysis of electrons scattered through various angles in the process of electron diffraction, and also for the investigation of the chromatic composition of electrons in beams produced by various sources.

Below we shall briefly describe the principle of operation of the analyzer, various designs of the instrument, its fields of application, and some measurement results.

2. PRINCIPLE OF OPERATION OF THE ANALYZER

The principle of operation of the analyzer is based on the use of the strong chromatic action of the marginal zone (the region removed from the axis) of a single electrostatic lens. This region is very sensitive to changes in the velocity of the electrons passing through it.

Let us follow the trajectories of electrons entering the lens with the same velocity but at different distances from the lens axis. It turns out that the form of the trajectories and the deflection of the electrons from the axis as they pass through the lens depend on the distance \(y_0\) from the axis at which they enter the lens (Fig. 1). Electrons entering the lens strictly along the axis (\(y_0 = 0\)) are not deflected. As \(y_0\) increases, the form of the trajectory changes and, as a consequence, the deflection \(y_A\) of the electron beam on the screen changes. At first \(y_A\) increases (ray 1), and the trajectory crosses the lens axis once. Then the ray, having passed through the maxi-

Fig. 1. Paths of electron rays in an electrostatic lens and the dependence of their deflection \(y_A\) on the screen on the distance from the axis \(y_0\) at entry into the lens.

...of deflection, approaches the axis (ray 2), crosses it, and is deflected to the opposite side (ray 3). In this case the trajectory has two points of intersection with the axis. With a further increase, the deflection of the ray again reaches a maximum value and then again decreases (ray 4). Thus, the deflection of the ray on the screen, as the distance of entry into the lens changes, has an oscillatory character, the period of the oscillations decreasing with increasing \(v_0\) (see Fig. 1, right). For some \(v_0\), reflection of the electrons occurs, and the lens acts as a mirror. The operating regime of the lens in which the electron trajectories intersect the lens axis once is customarily called the first working region. With double or triple intersection of the axis, they are called, respectively, the second or third working regions.\(^6\)

The above-described character of the dependence of the deflection of rays on the screen on the distance from the axis at which they enter the lens is easy to obtain if, in front of an ordinary single electrostatic lens, a diaphragm with a very narrow straight slit is placed in the path of the electrons (Fig. 2). This slit cuts out from the beam incident on the lens a thin flat beam of electrons. Electrons of this beam, at different distances from the axis, are refracted differently by the lens; the image of the slit is then obtained in the form of a loop (see Fig. 3). The circle shown in Fig. 3 by a dashed line represents the edge of the aperture of the upper electrode. The loop-shaped curve on the screen corresponds to the region of variation of \(y_A\) between rays 1 and 3 in Fig. 1. As can be seen, the distances from the axis of the rays entering the lens in this case vary from \(y_{0\min}\) to \(y_{0\max}\). The corresponding deflections on the screen vary from \(y_{A\min}\) to \(y_{A\max}\). It is also seen that, for portions of the slit close to the edge of the lens aperture, the deflections on the screen are maximal. This shows that the region of the lens far from the axis is more sensitive to changes in the distance of the entering electrons from the lens axis. It is also more sensitive to changes in the electron velocity. This is explained by the fact that in the region far from the lens axis, i.e., close to the edge of the aperture of the negative electrode, strong retardation of the electrons occurs and, as a consequence, a large deflecting action of the lens field.

Fig. 2. Diagram of deflection of electron rays by a single electrostatic lens.

Fig. 3. Image of a slit located outside the axis of an electrostatic axially symmetric lens.

If the aperture in the middle electrode of the lens is made elongated, in the form of a slit, we obtain a cylindrical (slit) lens and an image of a narrow...

the slit on the screen assumes the form shown in Fig. 4. When the velocity of the electrons entering the lens is changed, the image of the slit on the screen will be displaced. The character of the change in the deflection of the electrons on the screen when their velocity is changed has the same form as in the case where their distance from the axis is changed (Fig. 5). Figure 6 shows images of a slit 5 μ wide at various accelerating voltages: 35 kV; 35 kV − 20 V; 35 kV − 40 V, etc. It is seen from the figure that electrons whose velocities differ by 20 V give images that are displaced sufficiently far relative to one another.

Fig. 4. Image of an obliquely positioned slit produced by a cylindrical electrostatic lens.

If a narrow slit is placed strictly parallel to the edge of the slit opening of the middle electrode of the lens (Fig. 7), then its image will be obtained in the form of a line (more precisely, a strip). When the electron velocity is changed, this image will move, forming a system of parallel lines.

The distance of the slit from the axis of the lens and the potential on its middle electrode should expediently be chosen so that a telescopic course of the rays takes place (see, for example, ray 2 in Fig. 1). In this case, high sensitivity to a change in velocity is achieved (the steep course of the curve in Fig. 1, right) and an almost linear dependence between the change in velocity and the magnitude of the displacement of the slit image.

Fig. 5. Course of electron rays in an electrostatic lens and the dependence of their deflection on the change in electron velocity.

Changes in electron velocity of 1 V, and even of several tenths of a volt, at a total accelerating voltage of 50 kV are easily detected.

Thus, with the aid of such a lens it is easy to detect small discrete losses of electron energy arising, for example, when they pass through a substance or are reflected from it. If a beam of electrons with a discrete distribution of velocities falls on the slit, then the cylindrical lens will give

on the screen (photographic plate) a series of images of the slit in the form of parallel lines, i.e., a velocity spectrum. By photometering the spectrum obtained, one can measure the distances between the spectral lines with sufficient accuracy. To estimate these distances in volts, the lens-analyzer is first calibrated; for this purpose, at

Fig. 6. Images of an obliquely positioned slit, obtained with a cylindrical lens at accelerating voltages: a) 35 kV; b) 35 kV; 35 kV—20 V; 35 kV—40 V; 35 kV—60 V; 35 kV—80 V.

Fig. 7. Transverse section (right) and top view of the analyzer lens with a slit positioned parallel to the edge of the aperture of the middle electrode of the lens.

specified positions of the slit and at the potential on the middle electrode of the lens, the accelerating voltage is reduced in steps, for example by 10 V, and each time the image of the slit is photographed. On the photographic plate a system of parallel lines is obtained, the distances between which are equal to 10 V. The resulting system of lines is usually called the calibration standard. The required jumps in the accelerating voltage can be produced by means of an auxiliary battery connected into the cathode circuit. In Fig. 8, by way of example, a calibration standard corresponding to a change in the electron velocity in steps of 20 V is shown, together with its photometric curve. The velocity spectrum of electrons that have passed through the substance under study is, as a rule, recorded on the same plate. In this case the potential on the middle electrode and the position of the slit remain the same as when obtaining the calibration standard, while the auxiliary battery is disconnected from the cathode circuit of the gun. Comparison of the velocity spectrum with the calibration standard makes it possible to estimate the magnitude of the energy losses of the electrons as they pass through various objects. In Fig. 9, as an example, a spectrum of 40-keV electrons that have passed through a nickel film 500 Å thick is presented, along with a calibration standard (below), the spacing between whose lines

Fig. 8. Photograph of the calibration standard and its photometric curve.

corresponds to 20 V. In the spectrum one sees the main, most intense line, corresponding to electrons that have passed through the film without energy loss, and, at some distance from it against the general diffuse background, several lines corresponding to discrete losses of electron energy.

Fig. 9. Photograph of the spectrum of 40-keV electrons that have passed through a nickel film and a calibration standard with a spacing between the lines of 20 V.

3. EXPERIMENTAL APPARATUS

For the analysis of electron velocities, modified arrangements of various electron microscopes have, as a rule, been used up to the present. Möllenstedt⁴ used for this purpose an electrostatic microscope of the AEG-Zeiss system. The schematic diagram of this microscope, adapted for velocity analysis, is shown in Fig. 10. The illumination system and the objective remained unchanged. Above the block of projection lenses an analyzer lens was additionally built in, with a slit lying close against the upper electrode. The slit could be removed in order to switch to the ordinary operating mode of the electron microscope. The analyzer lens is, as stated above, a cylindrical electrostatic lens consisting of three electrodes, the outer two of which are usually grounded, while the middle one, with the slit aperture, is at a high negative voltage. The final screen consists of two hinged halves, each of which covers one half of the photographic plate. Such a screen arrangement makes it possible to obtain on one plate both the electron spectrum and the calibration standard, which facilitates accurate measurement of the energy losses. The aperture angle of the beam illuminating the slit in this device is \(10^{-4}\) radians.

The electrical circuit differs from that of the electron microscope in that a battery is inserted in the anode circuit for obtaining calibration standards, and a high-voltage voltage divider is included for adjusting the potential of the middle electrode of the analyzer lens relative to the cathode (see Fig. 10).

It should be noted here that, for good operation of the analyzer, its circuit must provide sufficient smoothing of high-voltage pulsations. The presence of a high-resistance divider when the pulsations are insufficiently smoothed causes a phase shift between the voltages on the gun and on the analyzer lens, which leads to blurring of the image of the slit.

Soon after Möllenstedt, a similar analyzer began to be widely used by other investigators as well. Thus Goto⁷ used the design of an electrostatic microscope, replacing one of the projection lenses by a cylindrical one. The parameters of this lens in analyzer mode were calculated by Seto⁸. The slit was made of silver, which reduces the formation of carbon deposits. The slit width can be adjusted during operation. To increase the intensity of the electron beam incident on the analyzer lens, a cylindrical objective lens was used.

The electrical circuit of the instrument (Fig. 11) makes it possible to vary over wide limits the voltage on the Wehnelt cylinder by means of a potentiometer. The potential on the middle electrode of the analyzer lens is varied by means of a battery.

An analogous reconstruction of an electrostatic microscope for analyzing electrons by velocity was carried out by Marton and Leder[^9].

Fig. 10. Diagram of an electron microscope adapted for analyzing electrons by velocity.

Fig. 11. Electrical circuit of the analyzer. \(P\) — potentiometers: \(P_1\) — for bias, \(P_2\) — for adjusting the potential on the objective, \(P_3\) — for obtaining a calibration standard.

Watanabe[^10] used a magnetic microscope for the same purposes, introducing an electrostatic analyzer lens between the objective and the projector.

Fig. 12. Diagram and cross section of an electron spectrograph:

1 — electron source; 2 — condenser; 3 — diaphragm; 4 — object; 5 — target; 6 — analyzer lens; 7 — screen; 8 — high-voltage input; 9 — insulator; 10 — protective screen; 11 — filament and Wehnelt cylinder; 12 — focusing device; 13 — observation window with mirror; 14 — magnetic condenser; 15 — object holder on object table; 16 — observation window; 17 — intermediate screen; 18 — target; 19 — two-part screen; 20 — magnifier; 21 — pumping branch pipe; 22 — screen shutter; 23 — 12-plate camera; 24 — flange for a photomultiplier.

Recently, a report appeared11 on the construction in Germany of a special instrument—an electron spectrograph with high chromatic resolution. The layout and section of the instrument are shown in Fig. 12. The main element of the instrument is an electrostatic analyzer lens. The accelerating voltage is 50 kV.

The illuminating system consists of an electron gun and a magnetic condenser focusing the electron beam on the object. The design of the object holder provides for horizontal displacement, tilting relative to the axis, and azimuthal rotation of six objects, which are necessary for obtaining both velocity spectra and diffraction spectra in transmission and reflection. The slit is movable and is displaced parallel to the edge of the slit opening of the middle electrode of the analyzer lens. The instrument uses a camera for 12 photographic plates of size \(5 \times 5\ \mathrm{cm}\), and provision is also made for connecting a self-recording device with a photomultiplier to obtain photometric curves.

In the Soviet Union an analyzer instrument, in principle similar to that described above, has been developed by the authors of the present article.

4. DISPERSION AND RESOLVING POWER OF THE ANALYZER

The main characteristics of a velocity analyzer, as of any spectrograph, are dispersion and chromatic resolving power. Here we shall give definitions of these concepts as applied to an electron velocity analyzer and indicate the factors on which they depend.

Dispersion determines the dependence of the magnitude of the displacement of a spectral line (the image of the slit) on the change in electron velocity. Numerically, the dispersion is equal to the ratio of the distance between two spectral lines in millimeters to the corresponding change in the accelerating voltage of the beam in volts (linear dispersion)

\[ \delta = \frac{\Delta y_A}{\Delta U}\left[\frac{\mathrm{mm}}{\mathrm{V}}\right]. \tag{1} \]

Below we shall call the linear dispersion simply the dispersion. The magnitude of the dispersion is determined by the electron-optical properties of the analyzer lens, by the position of the slit relative to the lens axis, and by the electron velocity.

As was already mentioned above, the high sensitivity of the electrostatic analyzer is due to the use of the edge zone of the analyzer lens, where the coefficient of chromatic aberration increases strongly. Ultimately this coefficient determines the magnitude of the dispersion. It is known that the displacement of the slit image on the screen \(\Delta y_A\) upon a change of the accelerating voltage by \(\Delta U\) is determined by the following expression:

\[ \Delta y_A = M C_{\mathrm{chr}} y_0 \frac{\Delta U}{U}, \tag{2} \]

where \(M\) is the magnification of the lens, \(C_{\mathrm{chr}}\) is the coefficient of chromatic aberration, and \(U\) is the accelerating voltage.

The linear dispersion is then determined by

\[ \delta = \frac{\Delta y_A}{\Delta U} = \frac{M C_{\mathrm{chr}} y_0}{\Delta U}. \tag{3} \]

Vatanabe10 gives the following estimate of this coefficient for the lens he used. Taking \(M=1\), \(y_0=0.5\) mm, he obtained the value \(MC_{x_p}y_0=3.4\cdot 10^3\) mm. Then, for \(U=22000\) V, the value \(\delta=0.154\) mm/volt is obtained.

As \(y_0\) (the displacement of the slit relative to the axis) is increased, the dispersion also increases. However, a large increase in the value of \(y_0\) is undesirable, since the lens passes into the regime of the 2nd or 3rd operating regions, where the image of the slit has a large width and is unstable. As we shall see below, an increase in the width of the slit image decreases the resolving power of the analyzer. It is therefore necessary to find the optimum position of the slit relative to the axis, at which the linear dispersion is sufficiently large without a noticeable deterioration of the resolving power.

As for the accelerating voltage, with an increase in the latter the dispersion decreases.

The resolving power of an analyzer is conventionally defined as the ratio of the accelerating voltage (electron velocity) to the smallest distance between the lines of the slit image, in volts, that are still visible separately:

\[ A=\frac{U}{\Delta U_{\min}} . \tag{4} \]

It is clear that the smallest distance between the lines is equal to the width of the line itself, i.e., to the width of the slit image. An increase in the width of the slit image leads to a decrease in resolving power.

Fig. 13. Path of the electron beam in an analyzer with a collimating lens.

The width of the slit image is determined by: the width of the slit itself, the spread of the initial velocities of the electrons, the aperture angle in the plane of the slit, and the aberrations of the lens (chiefly spherical). The production of very narrow slits is connected with difficulties of a technical nature, as well as with a strong decrease in the intensity of the beam.

Some decrease in the width of the slit image can be achieved by using an additional collimating cylindrical lens12 (Fig. 13). However, the overwhelming majority of analyzers operate without a collimating lens.

The resolving power of the analyzer also depends on the distance of the slit from the axis of the lens. As this distance is increased, the dispersion and the width of the slit image increase. But the change in resolving power due to dispersion, as a rule, exceeds its change due to the increase in the width of the slit image. Therefore one may say that with an increase in dispersion the resolving power of the analyzer also increases.

As reported in the literature11, in an electron spectrograph a resolving power of the order of 50,000 was obtained \((\Delta U=0.75\ \text{V},\ U=35\ \text{kV})\). The use of a point oxide cathode made it possible to obtain a resolving power of the order of 70,000 \((\Delta U=0.5\ \text{V},\ U=35\ \text{kV})\)12.

As already noted above, the narrowest image of the slit is obtained in the regime of the telescopic passage of rays. Deviation from this regime—for example, a change in the accelerating voltage, with an unchanged potential on the middle electrode of the analyzer lens, which usually occurs when obtaining a calibration standard—broadens the slit image, and the greater the broadening, the larger the slit width. In Fig. 14 this broadening is illustrated by photo-

...graphs of calibration standards for slits of different widths. To preserve the same image width of the slits over the entire spectrum, a device was proposed1 with a second slit located behind the analyzer lens (Fig. 15). Its distance from the axis is chosen so that only telescopic rays can pass through it. When the accelerating voltage is changed, the voltage on the analyzer lens is changed by such an amount that the telescopic path of the rays is restored. In Fig. 15 the voltage on the analyzer lens is denoted as \(U+\Delta U(t)\), where \(U\) is the voltage on the lens corresponding to the telescopic path of the rays at the given accelerating voltage, and \(\Delta U(t)\) is the change in the voltage on the lens, proportional to the change in the accelerating voltage. Thus, in the device the telescopic path of the rays is constantly maintained and, consequently, the image of the slit has a constant width and is not displaced. If the photographic plate is moved through distances proportional to the change \(\Delta U(t)\), then a spectrum or calibration standard with constant resolution is obtained. The use of a photomultiplier with an automatic recording device makes it possible to obtain spectrograms and calibration standards rather rapidly. Figure 16 shows a spectrum and calibration standard obtained with such a device.

Fig. 14. Dependence of the resolving power of the analyzer on the width of the collimator slit at a voltage of 35 kV on the slit. Slit width: \(a\)—\(3\mu\); \(b\)—\(10\mu\); \(v\)—\(25\mu\).

Fig. 15. Diagram of an analyzer with constant resolving power over the entire spectral region.

Fig. 16. Photometric curve of the spectrum and calibration standard of 35 keV electrons that have passed through an Al film. Obtained with the analyzer whose diagram is shown in Fig. 15.

5. ANALYZER LENS

A cylindrical electrostatic lens is the principal element of the analyzer instrument. Its electron-optical properties mainly determine the dispersion and resolving power of the instrument. However, the literature devoted to velocity analyzers does not contain sufficiently detailed information on the characteristics of the lenses used. Only in work \(^{11}\) are the principal parameters of the analyzer lens used given. Figure 17 shows the geometrical dimensions of the lens electrodes and the dependence of the refracting power for the paraxial region on \(\dfrac{U_k-U_l}{U_k}\) (in %), where \(U_k\) is the voltage at the cathode and \(U_l\) is the voltage on the lens. The paper does not indicate how the

Fig. 17. Dependence of the refracting power of the analyzer lens on \(\dfrac{U_k-U_l}{U_k}\) for positive rays.

optimal geometry of the lens was found. Most often, apparently, it was found experimentally.

A rigorous calculation of such lenses is a difficult problem. There are, however, attempts at approximate calculation which provide some guidance for the practical construction of lenses. We shall point here to the works of Sethe \(^{8}\) and Lippert \(^{14}\), which take as the basis of the calculation an approximation of the potential distribution in the plane of symmetry of the lens by parabolas.

Sethe considers the lens as a combination consisting of three diaphragms with an infinitely long slit. The potential distribution is approximated in the regions of the end slits (electron entrance and exit) by the parabola

\[ \Phi_1(x)=1-n(x\pm x_0)^2, \tag{5} \]

and in the region of the middle electrode by the parabola

\[ \Phi_2(x)=\Phi_0+px^2, \tag{6} \]

where \(n\), \(p\), and \(\Phi_0\) are constants and \(x\) is a coordinate measured along the axis of the system. For \(n=0.0234\), \(p=0.0387\), \(\Phi_0=0.067\), the following non-Gaussian (non-paraxial) trajectory equations are obtained:

in the entrance region

\[ y_1=y_0\,\operatorname{ch}\left[\arcsin(x+x_0)\sqrt{\frac{\beta^2}{1+y_0^2\beta^3}}\right]\qquad(\beta=\sqrt{n}), \tag{7} \]

in the region of the middle electrode:

\[ y_2=A\cos\left[\operatorname{arcsh}\left(x\sqrt{\frac{a^2}{1-A^2a^2}}\right)-\varphi\right] \qquad \left(a=\sqrt{\frac{p}{\Phi_0}}\right), \tag{8} \]

in the exit region:

\[ y_3=B\,\operatorname{sh}\left[\operatorname{arcsh}(x-x_0)\sqrt{\frac{\beta^2}{1+B^2\beta^2}}-\psi\right]. \tag{9} \]

The constants \(A,\ \varphi,\ B,\ \psi\) are determined from the conditions for matching the trajectories at the boundaries of the regions; \(y_0\) is the ordinate at entry into the lens. Trajectories were calculated for several values of \(\varepsilon\), and, in particular, the trajectory possessing telescopic passage, the most advantageous one, as shown above, for the analyzer, was found; it is obtained for \(y_0=0.87\ \mathrm{mm}\). The deviation of the beam on the screen for the corresponding change of \(y_0\) was found. For a screen located at a distance of \(500\ \mathrm{mm}\) from the lens, this deviation is \(y_A=47.5\ \mathrm{mm}\) for \(\Delta y_0=0.15\ \mathrm{mm}\). If a slit \(1\ \mu\) wide is placed in front of the lens, then its image on the screen will be \(0.3\ \mathrm{mm}\) (without allowance for the spread of electron velocities) and \(0.45\ \mathrm{mm}\) with allowance for the thermal spread of velocities of the heated cathode (approximately \(0.3\ \mathrm{V}\) for tungsten). A change in electron velocity of \(10\ \mathrm{V}\) corresponds to a displacement on the screen \(\Delta y_A=5\ \mathrm{mm}\), i.e. the dispersion coefficient is equal to \(0.5\ \mathrm{mm/V}\). The voltage on the lens was assumed to be \(30\ \mathrm{kV}\). These calculations were checked experimentally in the analyzer described above and gave satisfactory agreement with experiment.

A more complete investigation of the analyzer lens was later carried out by Lippert[^14]. Approximating, as above, the potential distribution along the axis of the cylindrical lens by a parabolic law and assuming that the electrons enter the lens parallel to its axis, he obtained the following expressions for the refracting power of the lens (the reciprocal focal length) and the position of the focus as a function of the distance \(y_0\) of the electron from the axis at entry into the lens:

\[ \frac{h}{f}=A\cos\left(\ln\frac{t-Fy_0^2}{t_m}\right), \tag{10} \]

\[ \frac{z_f}{h}=B+\frac{1}{A}\operatorname{tg}\left(\ln\frac{t-Fy_0^2}{t_m}\right), \tag{11} \]

where \(A,\ B,\ F\), and \(t_m\) are experimentally determined constants depending on the geometry of the lens electrodes; \(h\) is the distance between the end electrodes; \(t\) is determined by the voltage on the middle electrode \(U\) (relative to the end electrodes) from the following relation:

\[ t=\frac{U_s-U}{U_s}. \tag{12} \]

\(U_s\) is the voltage on the middle electrode at which the electrons in the lens are completely retarded; \(t_m\) is that value of \(t\) at which \(h/f\) is maximal.

The formulas given above were obtained under the assumption \(t\ll 1\). These formulas were experimentally checked for a lens whose electrodes are formed by planes parallel and perpendicular to the axis (Fig. 18), and for sufficiently small \(t\) and \(y_0\). The experimental investigation gives the dependence of \(h/f\) on \(t\), and at the same time the values of \(A\) and \(t_m\) for different values of the thickness \(d\) of the middle electrode and the diameter \(b\) of the aperture in it (see Fig. 18). An exact determination of the constant \(F\) was difficult, since distortion appeared when \(y_0\) was increased. The measured value of \(h^2F\) has, for different dimensions of the middle electrode, an approximately constant value between 50 and 100. The constant \(B\) has no substantial significance.

Next, if we assume that the distance \(s\) from the middle of the lens to the image (screen) is large in comparison with the distance to the focus, which is actually the case, we can find the dependence \(y_A=f(y_0)\). Assuming

Fig. 18. Dependence of the constants \(|A|\) and \(t_m\) on the geometry of the middle electrode of the analyzer lens and the diagram of the analyzer.

\[ \frac{y_A}{y_0}=\frac{s}{f} \tag{13} \]

and substituting this expression into (10), we obtain:

\[ y_A=y_0\frac{s}{h}\,A\cos\ln\frac{t-Fy_0}{t_m}. \tag{14} \]

This expression can be simplified if we denote:

\[ \eta_0=y_0\sqrt{\frac{F}{t_m}};\qquad \eta_A=y_A\frac{h}{s\cdot A}\sqrt{\frac{F}{t_m}};\qquad \tau=\frac{t}{t_m}. \tag{15} \]

Then we obtain

\[ \eta_A=\eta_0\cos\ln(\tau-\eta_0^2). \tag{16} \]

Figures 19 and 20 show the relationships among the three quantities: \(\eta_A\), \(\eta_0\), and \(\tau\). From Fig. 19 it is evident that, for small \(\eta_0\), the quantity \(\eta_A\) increases proportionally to \(\eta_0\), and the more rapidly, the larger \(\tau\) is. For large \(\eta_0\), a deviation appears

Fig. 19. Relationship between \(\eta_0\) and \(\eta_A\) for various \(\tau\).

Fig. 20. Relationship between \(\tau\) and \(\eta_A\) for various \(\eta_0\).

from this proportionality. From expression (16) one can find the main characteristics of the analyzer considered above by means of the following differential relations:

\[ \delta=\frac{\partial \eta_A}{\partial \tau}; \qquad R=\frac{\partial \eta_0}{\partial \tau}; \qquad M=\frac{\partial \eta_A}{\partial \eta_0}. \]

These relations mean: \(\delta\)-dispersion, i.e., the change of \(\eta_A\) with a change of \(\tau\) at a constant electron entrance height; \(R\)—the change of the electron velocity interval with a change of the width of the entrance slit; \(M\)—differential magnification.

6. SOME RESULTS OF THE ANALYSIS OF ELECTRONS THAT HAVE PASSED THROUGH MATTER; COMPARISON WITH THEORY

With the aid of the instruments described above, a fairly large amount of material has already been obtained on the energy losses of fast electrons that have passed through layers of various substances with thicknesses of the order of 200–500 Å. The results of investigations by various authors are summarized in Table I, which gives the discrete velocity losses at primary-beam voltages of \(25\)—\(40\) kv\({}^{15}\). In the same table, for comparison, data obtained by Ruthemann\({}^{2}\) with the aid of a magnetic analyzer (the magnetic method of semicircular focusing) at voltages of 5–8 kv are also given. Thin metal layers, as a rule, were obtained by evaporating the metal in vacuum onto a substrate (for example collodion, salt crystals, etc.) and subsequently dissolving the substrate in the corresponding solvents.

It is evident from the table that, when an electron beam passes through thin layers, several discrete energy losses are found for each substance, corresponding to peaks on the photometric curves. Moreover, the intensities and widths of these peaks (see Table I) are different. It is also seen that, to a first approximation, the magnitude of the discrete losses for the layers investigated does not depend on the accelerating voltage.

For most of the substances listed in the table, the discrete losses determined by different authors agree within the accuracy of the measurements. The exceptions are Au and Sb, which is explained by differences in the methods of preparing the films and by possible contamination. Leder and Marton\({}^{16}\) investigated the influence of the chemical composition of a number of substances on the discrete energy losses of 30-kev electrons. Their results are summarized in Table II. As can be seen from the table, the values of the discrete energy losses in compounds are shifted toward larger values in comparison with the pure elements.

As for the influence of the thickness of the layers on the discrete energy losses, complete investigations of this question are not yet known. Some information for aluminum is given in a recent work by Watanabe\({}^{15}\). He obtained energy spectra of 25-kev electrons for the following layer thicknesses: 150, 250, 300, 450, 500, and 650 Å (Fig. 21). The values of the layer thicknesses were measured interferometrically.

Fig. 21. Changes in the appearance of the spectrum with increasing thickness of an Al film. Layer thickness: \(a\)—150 Å; \(b\)—250 Å; \(v\)—300 Å; \(g\)—450 Å; \(d\)—500 Å; \(e\)—650 Å.

Table I

...rometer. It is seen that, with increasing layer thickness, the number of discrete losses and the intensity of the diffuse background increase. For thick layers (500 and 650 Å) the intensity of the lines corresponding to electrons without energy loss is less than that of the lines corresponding to the first discrete loss.

Despite the fact that at present there is already much experimental material on the values of the discrete energy losses of electrons that have passed through thin films of various substances, there is still no complete theory explaining the nature of these losses. Möllenstedt[^4], for example, suggested that discrete losses are the result of multiple collisions of electrons with the atoms of the substance. However, this assumption cannot be verified experimentally because of the difficulties connected with obtaining the dependence of the distribution of the electron intensity on the film thickness.

Table II

Comparison of the observed discrete energy losses with the fine structure of the Al \(K\)-absorption edge[^17] and the good agreement between the measured quantities led to the supposition that the losses are caused by transitions of electrons from occupied levels to unoccupied ones. However, the sharpness of the lines observed in the spectrograms (approximately \(3\text{--}4\) eV) cannot be explained from this point of view, since the width of the unoccupied levels in a metal is considerably greater.

Pines and Bohm[^18],[^19] interpreted the energy losses from the standpoint of the theory of the vibrational properties of the electron gas. (For the theory of the vibrational properties of the electron gas see [^20].) They proposed that such energy losses are caused by excitation of collective oscillations of the electron-gas plasma in metals with frequency \(\omega_p\). It was also noted that the calculated values \(\hbar\omega_p\) are in good agreement with the observed values. In the calculation, however, it was assumed that all valence electrons are free, which is not always actually the case. This, apparently, explains the fact that the calculated values of the basic quantum of energy loss agree only for some metals.

Recently papers have appeared in which experimental evidence is given in favor of this theory. Gabor[^21], on the basis of analysis of the velocity spectra of 10 keV electrons that had passed through gold films of two types (continuous and with carbon interlayers), comes to the conclusion that the discrete energy losses are due to the vibrational properties of the electron gas. Watanabe[^22] classifies all substances into two groups. The first group includes the elements Be, Mg, Al, Ta, and Mo, for which the calculated values agree well with those measured by the analyzer. The second group comprises all noble and transition metals. With the aid of the analyzer he obtained velocity spectra of diffraction images of films of metals of the first group. The author found that the energy losses increase with increasing scattering angle. This increase is well explained by the dispersion relation of Pines and Bohm (see equation (67) in [^19]). This circumstance and the agreement of the energy-loss values with the calculated values of \(\hbar\omega_p\) are sufficient, in the author’s opinion, to conclude that the sharp lines observed in the velocity spectra are due to oscillations of the electron-gas plasma. The diffuse lines and the energy-loss lines that do not depend on the scattering angle agree with observations of the fine structure of the \(L\)-absorption edge.

7. ENERGY ANALYSIS OF ELECTRON DIFFRACTION IMAGES

In studying the angular distribution of scattered electrons that have passed through thin films of substances, a number of difficulties arise, both experimental and theoretical in character. Whereas investigations of the scattering of $\alpha$-particles, carried out by Geiger and Marsden, gave good agreement with Rutherford’s formula (single scattering of $\alpha$-particles), investigations of electron scattering gave only very approximate agreement. The discrepancies are explained mainly by the presence of multiple and inelastic scattering of electrons. As was already stated above, the discrete energy losses of scattered electrons still do not have a satisfactory explanation. For further theoretical study of this problem it was necessary to carry out an analysis, according to velocities, of electrons scattered through large angles ($\sim 5^\circ$), which usually occur in electron diffraction phenomena. Of special interest in the study of diffraction spectra is the analysis of the so-called diffuse background. Attempts to obtain the spectral composition of the diffuse background were unsuccessful for a long time because of the low resolving power of the spectrographs, and the results of theoretical investigations still could not be verified experimentally.

The appearance of the electrostatic velocity analyzer made it possible for Leonhard^23, and then for Watanabe^24, to carry out quantitative investigations of the intensity and spectral composition of diffraction images of a number of substances, and especially of the spectral composition of the diffuse background.

Fig. 22. Scheme of energy analysis of electron diffraction images.

The scheme of the electron-optical system used for this purpose is shown in Fig. 22. An almost parallel beam of electrons passes through the specimen under investigation and undergoes diffraction. The objective projects the diffraction image, obtained in its back focal plane, onto the plane in front of the intermediate lens. Then, with the aid of the intermediate lens at a magnification of 1:1, it is transferred to the plane in which the slit is located. The diaphragm placed above the intermediate lens, as is seen from the figure, makes it possible to vary the aperture $\beta$ of the diffracted beam passing through the slit. The slit cuts out from the diffraction image a narrow strip that includes the central beam and portions of all the diffraction rings. The narrow beam cut out by the slit passes through the analyzer lens and forms a spectral pattern on the screen. In Fig. 23a the pattern formed in the plane of the slit is shown schematically, and in Fig. 23b the formation of the spectrum is shown, where the solid lines correspond to electrons that have passed through the object without loss of velocity, and the dashed lines to inelastically scattered electrons. The spectrum obtained in this way makes it possible to determine energy losses both for the principal spot and for each ring, and to reveal the dependence of the energy losses on the scattering angle.

The electron-optical scheme described above for obtaining diffraction images and analyzing them can be realized in any electron microscope with certain modifications in the column and in the electrical circuit. Leonhard, for example, used for this purpose an AEG-Zeiss electrostatic microscope. The illumination system was used without modification. The objective, with a focal length of 5 mm, had a lower electrode with an enlarged aperture so that the entire diffracted beam could pass through it. A long-focus lens was used as the projector. The analyzer lens was an ordinary slit lens. The focal lengths of these two lenses were adjusted by means of a potentiometer. The distance between the lenses was set, by means of special intermediate tubes, in accordance with the desired magnifications of the diffraction pattern on the intermediate and final screens. These distances were respectively: between the objective and the projector 35 cm, between the projector and the analyzer lens 45 cm, and from the latter to the final screen 40 cm. The aperture of the diaphragm before the projector had a diameter of 0.1 mm. The slit in front of the analyzer lens was formed by two plates. The slit width was adjusted by changing the distance between the plates. In addition, the slit could be moved parallel to the slit aperture of the analyzer lens. The smallest slit width was 1–2 μ, and the resolving power of the analyzer was then of the order of 50,000.

Fig. 23. a) Image of the diffraction spectrum in the plane of the slit; b) image of the diffraction spectrum after passage through the analyzer lens.

The process of obtaining spectra of electrons by velocities is no different from obtaining the spectra described in Section 6. A certain difference arises when obtaining spectra of electrons scattered at large angles. In this case the accuracy of the measurements is impaired by the astigmatic image of the diffraction spectrum produced by the cylindrical analyzer lens. However, this astigmatism can be corrected by an appropriate inclination of the intermediate lens.

Fig. 24. Spectrum of the diffraction image of gold.

Figure 24 presents a photograph of the spectrum of a diffraction image of gold. At the top of the photograph a scale in volts has been applied, obtained from measurements of a calibration standard. It is evident from the figure that the intensity of the inelastically scattered electrons is greatest in the central spot and smaller in the rings.

Photometry of the spectrum along lines of equal velocities makes it possible to obtain dependences of the intensity of electron scattering on the scattering angle in each line (Fig. 25). Analysis of the spectra of diffraction images of thin films of gold, beryllium, and aluminum allowed Leonhard^23 to obtain the ratios of both elastically and inelastically scattered electrons to the total number of electrons. Conducted

also a comparison of these ratios with the calculated values following from the formulas of Debye and Koppe. Experimentally, the presence is established of elastic and inelastically scattered electrons in diffraction images, of a general diffuse background and a background with a discrete distribution for inelastically scattered electrons, and of a general background and a background within each ring for elastically scattered electrons. In a later work by Watanabe^22 it is shown experimentally that the magnitude of the losses in electron velocity depends on the scattering angle. Figure 26 gives a photograph of the diffraction image of aluminum resolved into a velocity spectrum. From the photograph it is seen that the magnitude of the discrete energy losses increases somewhat with increasing scattering angle.

Fig. 25. Photometric curves of the diffraction image of gold, resolved in the velocity spectrum. Each curve corresponds to a certain electron velocity: 1 — unretarded electrons; 2 — electrons retarded by 20 V; 3 — by 40 V; 4 — by 60 V; 5 — by 80 V; 6 — by 100 V; 7 — by 120 V.

Fig. 26. Spectrum of the diffraction image of aluminum

8. ANALYSIS OF ELECTRONS REFLECTED FROM MATTER

Using an electrostatic analyzer, Klein^25 obtained energy spectra of 35-keV electrons reflected from various substances. For this purpose he used an electrostatic microscope. The layout of the instrument is shown in Fig. 27. To increase the intensity of the electron beam, the microscope objective was used as a condenser lens. The change in the inclination of the beam was carried out by means of deflecting plates. The specimen holder allowed displacement in two mutually perpendicular directions and rotation about the horizontal and vertical axes, and the presence of an air lock made it possible to change the specimen without disturbing the vacuum in the column. Next followed an intermediate screen and the analyzer lens.

Electrons reflected from the specimen enter through a slit into the analyzer lens and form on the screen (photographic plate) a velocity spectrum. To increase the brightness of the spectrum, after the analyzer lens there was a second cylindrical lens, which compressed the spectrum in the direction perpendicular to the direction of dispersion.

The position of the specimen and the path of the beam are shown in Fig. 28. The angle between the plane of the specimen and the incident beam was \(\varepsilon = 2.5^\circ\), and the angle between

Electrostatic Electron Velocity Analyzer

Cathode
Anode
Condenser
Object and air lock
Screen
Slot
Lens-analyzer
Cylindrical lens
Magnifier
Screen
Photographic plate

a)

b)

Fig. 27. Diagram (a) and general view (b) of the instrument for the analysis of reflected electrons.

a)

Deflecting condenser
Object
Deflecting screen
Slot

Fig. 28. Position of the object and path of the electron rays in the analysis of reflected electrons.

by the plane of the object and by the reflected ray \(\alpha = 1^\circ\). The aperture of the beam entering the analyzer lens is \(2 \cdot 10^{-4}\) radian. Since the electron beam passes through the slit somewhat inclined to the axis, the dispersion of the analyzer lens with the slit located to the right of the lens axis will differ from the dispersion when the slit is located to the left of the axis. Usually the slit is placed on that side of the lens axis where the dispersion is greater.

For a successful investigation, the specimens must be specially prepared. Polished and not previously heated specimens of various substances gave identical spectra of reflected electrons. A small difference in them appears when the object is heated, in the course of obtaining the spectrum, to \(150\)—\(300^\circ\). If the surface of the object is made rough and the object is simultaneously heated to \(300^\circ\) C, then a good difference is obtained in the spectra of the substances. In Klein’s opinion, heating the object removes films of carbon-containing contaminants, whose spectra remain constant.

The determination of discrete energy losses is carried out in the usual manner. As an example, Fig. 29 shows the spectra of aluminum, beryllium, and molybdenum and their photometric curves. The main maximum of the photometric curve corresponds to elastically reflected electrons, and the maxima located to the left correspond to inelastically reflected electrons. The results of measurements of the discrete energy losses of reflected electrons are collected in Table III.

Table III

Discrete energy losses of electrons reflected from various substances

Comparing these discrete losses with the losses upon passage through the same substances (Table I), one may note that some of them coincide and, in addition, there are additional losses.

Klein\({}^{25}\) compared the experimentally obtained discrete energy losses with the calculated values \(\hbar\omega_p\). It was shown that there is approximate agreement of the results for Al, Be, Mo, Ta, C, Mg, Pb, and In. For the metals Pt, Cu, Ag, Zn, Cd there is no agreement. Determining from the photometric curve the intensity distribution for elastically and inelastically reflected electrons, Klein found the ratio of the number of elastically reflected electrons to the number of in-

elastically reflected electrons. For polished aluminum, for example, a value was obtained approximately equal to \(\frac{1}{14}\).

Fig. 29. Velocity spectra of electrons reflected from aluminum (a), beryllium (б), and molybdenum (в), and their photometric curves.

The work of Klein presented here is, in essence, the first attempt to apply an electrostatic velocity analyzer to the study of reflected electrons. Therefore its results, naturally, cannot be regarded as exhaustive.

9. ANALYSIS OF ELECTRONS BY VELOCITY IN CATHODE RAYS

In conclusion we shall point out one more field of application of the electrostatic velocity analyzer—the investigation of the energy composition of electrons produced by various sources.

Alongside thermionic cathodes, gas-discharge sources of electrons are also widely used. Möllenstedt and Düker\(^ {26}\) carried out an investigation of beams of electrons produced by certain gas-discharge sources. Of particular interest were the studies of the Induni discharge gun\(^ {27}\), which is used in the Swiss electron microscope, possessing comparatively high resolution.

The arrangement used for investigating the Induni gun is shown in Fig. 30. In the upper part of the arrangement is located the gun under investigation. It consists of an aluminum circular rod inserted inside a steel tube situated perpendicular to the axis of the system. The aluminum rod is the cathode, at a potential of minus \(40\ \text{kv}\) relative to ground. The grounded steel tube is the anode. The distance between cathode and anode must be smaller than the mean free path of the molecules of the gas present in this space. On the side surface of the anode tube, at the point of intersection with the optical axis of the instrument, there is an aperture whose diameter is greater than the mean free path of the gas molecules (\(\sim 5\ \text{mm}\)). When the high voltage is switched on, a gas discharge is ignited near this aperture. The necessary gas pressure (\(\sim 10^{-2}\ \text{mm Hg}\)) is maintained by means of a special valve. The beam of electrons, form-

…ing in the discharge, passes through a throttling diaphragm (whose diameter is 0.1 mm) into the high-vacuum space.

Below the gun described there is an electrostatic objective with an aperture diaphragm of 0.05 mm, and then an analyzer lens with a slit whose width is 1–2 μ. The aperture angle in the plane of the slit is \(10^{-4}\) radian. Adjustment of the potentials on the middle electrodes of the objective and of the analyzer lens is carried out by means of a potentiometer with a resistance of 1200 megohms.

Fig. 30. Diagram of an apparatus for analyzing cathode rays.

Figure 31 shows a photometric curve corresponding to the energy spectrum of the electrons in the beam produced by Induni’s gun. It is characteristic that this spectrum contains an asymmetric principal maximum and an additional maximum on the left, corresponding to slowed electrons. The energy loss corresponding to this maximum is 13 eV. This energy loss is explained by the interaction of the electrons with gas atoms on the path from the cathode to the anode of the gun. Investigation of various gases (air, nitrogen, oxygen) and various cathode materials (aluminum, platinum) showed that the energy distribution corresponding to the principal maximum, when the accelerating voltage is varied from 25 to 40 kV, depends neither on the kind of gas nor on the cathode material.

In addition to Induni’s gun, a Hailer gas-discharge gun\(^{28}\) with a tubular cathode, shown in Fig. 32, was investigated by an analogous method. In this case too the energy spectrum of the electrons in the beam was the same as for Induni’s gun.

To eliminate the influence of the gas on the energy distribution of the electrons in the beam, a beam of electrons was obtained by bombarding a cathode located…

ELECTROSTATIC ELECTRON-VELOCITY ANALYZER

in vacuum \((10^{-4}\ \mathrm{mm}\ \mathrm{Hg})\), with ions from a special source. In this case an analogous energy distribution is obtained in the beam of secondary electrons released from the cathode, except that there is no additional maximum of slowed electrons.

In Fig. 33, for comparison, photometric curves are shown for all

Fig. 31. Photometric curve of the spectrum of electrons emitted from Induni’s gas-discharge tube.

Fig. 32. Section of Haidner’s gas-discharge tube.

Labels in Fig. 32: retarding electrode; cathode; anode; to pump; gas.

Fig. 33. Photometric curves for various tubes: ○ — Induni tube, + — Haidner tube, ● — cathode in vacuum, irradiated with ions.

these cases. A comparison of the energy spectra of an electron beam obtained from a gas-discharge source and a beam obtained with the aid of a heated cathode, at identical voltages, shows that in the first case a broader interval of velocities is obtained (Fig. 34).

Fig. 34. Spectral distribution of electrons in electron beams: A—40-keV electrons obtained by bombardment with 40-keV ions; B—40-keV electrons emitted by a tungsten cathode \((T = 2900^\circ \mathrm{C})\).

REFERENCES

1. P. Lenard, Ann. Phys. 56, 256 (1895).
2. G. Rutheman, Ann. Phys. (6) 2, 113 (1948).
3. W. Lang, Optik 4, 233 (1948).
4. G. Möllenstedt, Optik 5, No. 8/9, 499 (1949).
5. F. Lenz, Naturwiss. 22, 524 (1951); Optik 10, 439 (1953).
6. F. Heise, O. Rang, Optik 5, 201 (1949).
7. B. Gaute, Compt. Rend. 239, No. 5, 399 (1954).
8. A. Septier, Compt. Rend. 239, No. 5, 402 (1954).
9. L. Marton, L. Leder, Phys. Rev. 94, 203 (1954).
10. H. Watanabe, Journ. Phys. Soc. Japan 9, 920 (1954).
11. W. Dietrich, H. Düker, G. Möllenstedt, Zeits. Metallkunde 47, 240 (1956).
12. G. Möllenstedt, Optik 9, 473 (1952).
13. G. Möllenstedt, W. Dietrich, Optik 12, 246 (1955).
14. W. Lippert, Optik 12, 467 (1955).
15. H. Watanabe, Journ. Electronmicroscopy 4, 24 (1956).
16. L. Leder, L. Marton, Phys. Rev. 95, 1345 (1954).
17. J. Gauchois, Acta Crystallogr. 5, 351 (1952).
18. D. Pines, Phys. Rev. 85, 338 (1952).
19. D. Bohm, D. Pines, Phys. Rev. 92, 609 (1953).
20. A. A. Vlasov, Theory of the Vibrational Properties of an Electron Gas and Its Application, Scientific Notes of Moscow State University, issue 75, book 2, part 1, Moscow, 1945.
21. D. Gabor, G. Jull, Nature 175, No. 4460, 718 (1955).
22. H. Watanabe, Journ. Phys. Soc. Japan 11, 112 (1956).
23. F. Leonhard, Zeits. Naturforsch. 9a, 727 and 1019 (1954).
24. H. Watanabe, Journ. Phys. Soc. Japan 10, 321 (1955).
25. W. Klein, Optik 11, 226 (1954).
26. G. Möllenstedt, H. Düker, Zeits. Naturforsch. 8a, 79 (1953).
27. Induni, Helv. Phys. Acta 20, 463 (1947).
28. C. Hailer, Veröff Siemens-Werken 17, 115 (1938).

1. visible reference marker in the source. ↩