Effective Cross Section of Gas Molecules with Respect to Slow Electrons and Ions*

K. Ramsauer and R. Kollath, Berlin—Reinickendorf

I. Electrons

B. Results of Measurements of the Effective Cross Section**

In what follows, when the results of measurements of the effective cross section are represented graphically, it will be plotted along the ordinate axis, expressed in \(\frac{\mathrm{cm}^2}{\mathrm{cm}^3}\) at \(1\ \mathrm{mm}\) Hg and \(0^\circ\mathrm{C}\); along the abscissa axis the electron velocity will be plotted in \(\sqrt{V}\).

First of all we shall give several examples of agreement and disagreement among the results of measurements by different methods. Then we shall give a summary of the most important measurements of the effective cross section (e.c.s.) carried out up to the present time. Next we shall set forth details on the most interesting e.c.s. curves, taken from individual original works. Finally, at the conclusion of the section, a table will be given in which, in alphabetical order by authors, all measurements of e.c.s. carried out up to now are summarized.

§ 13. Most probable course of the e.c.s. curve. Measurements with various degrees of homogeneity of the beam were carried out by Grin\(^1\) and Palmer\(^2\). In both cases the measurements were made with an apparatus similar to that shown in Fig. 7,*** and only one trap \(H\) was used, while diaphragm 3 had a variable aperture. The results of the measurements are presented in the form of the dependence of the magnitude of the e.c.s. on the size of the diaphragm aperture. In contrast to Grin, who did not find for He, Ar, H\(_2\), and mercury vapor any noticeable dependence of the e.c.s. on the size of the diaphragm, Palmer found for He and Hg vapor a more or less strong dependence between the e.c.s. and the size of the diaphragm. On the basis of considerations regarding the angular distribution of scattered electrons, it should be assumed that Palmer’s results are closer to the truth. It should be noted here that when working with two traps, with equal sizes of the diaphragm apertures of both traps—

* Handb. d. Phys., 2nd ed., vol. XXII, part 2, pp. 263–296; trans. with it by N. Khlebnikov.

** Continuation (see Uspekhi fizich. nauk XIV, issue 8, 1934).

*** See U. F. N. XIV 969, 1934.

the dependence on the dimensions of the diaphragms should not be detectable in the range of velocities used, since the presence of two traps acts like a strong narrowing of the diaphragm apertures (§ 27).

The agreement of the results of different authors is in general quite good, as is seen, for example, from the curves in Fig. 1 for argon. It should be noted that—as in this drawing—the values of the effective cross section obtained by Ramsauer generally prove to be the highest, Brode’s numbers the lowest, and the repetition of Brode’s measurements by Normand, as a rule, gives higher values. The numbers obtained by Broch usually occupy an intermediate position. Townsend’s data at small velocity values approximately coincide with the results of other authors, but as the electron velocities increase, noticeable deviations toward smaller values are found. An explanation of these discrepancies can be obtained only as a result of very detailed special investigations. Until this is done, one must beware of taking anyone’s results as “correct.”

Fig. 1. Comparison of the results of measurements of the effective cross section for argon according to the data of various authors.

Normand\(^3\) found a “fine structure” in a number of sections of the effective-cross-section curves, i.e., the presence of deviations from a smooth course—the appearance of small maxima and minima. For helium and carbon monoxide at velocities of about 1 V, Normand’s results agree well with the older measurements of Ramsauer and Kollath.\(^4\) For some other gases, especially neon, Normand’s measurements are not confirmed by the earlier measurements of Broch.\(^5\) In such cases Normand’s curves are given alongside the older curves. In such regions both curves are drawn with dashed lines.

At the very smallest electron velocities (less than 1 V), the construction of quantitative curves still encounters difficulties, since the measurements of Ramsauer and Kollath\(^4\) and Normand\(^3\) (the only ones made for this region by a directly quantitative method) give, in a small part, results that differ from one another. In this region, the data obtained by Townsend’s method (III a) and Rush’s qualitative measurements (II c)\(^6\) speak in favor of the results of Ramsauer and Kollath.

A repetition of the measurements for argon and hydrogen by means of a completely different method (I), carried out by Gaertner,\(^7\) gave for hydrogen the same course of the curves as was obtained by Ramsauer and Kollath. In the case of argon, Gaertner qualitatively confirmed the proposition—

...the minimum. Therefore we do not regard the new rise of the curve found only by Normand for H₂, N₂, and CO in the region of very low velocities as reliable, especially since Townsend’s measurements for the region between 0.2 and 0.4 V and the separate points obtained by Wahlin

Fig. 2. E.p.s. as a function of electron velocity in various gases and vapors.

for a velocity of 0.17 V, precisely for these three gases, give a sharp fall of the curves toward very low velocities.

Figure 2 shows the most important e.p.s. curves for gases and vapors. The range of electron velocities extends from 0 to 6 V and is sufficiently wide to make it possible to see the most essential features in the course of the variation of the e.p.s. It should, however, be noted that for a number of gases and vapors the e.p.s.

was investigated over considerably broader ranges; for example, in Brode’s\(^9\) and Normand’s\(^3\) work the velocities reached 20 V, and in individual cases in Brode’s work they went up to 50 V. The curves do not include data relating to complex chemical compounds, which were investigated by Brox\(^ {10}\) and especially by Schmieder\(^ {11}\), Holst and Holtsmark\(^ {12}\). We shall return to these works in the next section in connection with the question of the dependence of the effective cross section on the structure of molecules.

The rendering of parts of some curves by a dashed line indicates that these parts still cannot be regarded as quantitatively reliable. The presence of two dashed sections of one and the same curve indicates that it was impossible to give preference to either of the two measurements. As regards the absolute height of the curves, for simplicity we follow the comparison made by Kollath,\(^ {13}\) giving below explanations for each of the curves concerning its origin.

§ 14. Explanations of the curves. Helium (Fig. 1). Above \(1.5\sqrt{V}\)—the mean between the curves of Ramsauer\(^ {14}\) and Normand\(^3\), and coincides with the results of Brox\(^5\). In the region below \(1.5\sqrt{V}\)—the fine structure discovered by Ramsauer and Kollath\(^ {15}\), and Normand’s\(^3\) course of the curve according to Ramsauer and Kollath. The end of the curve at \(0.17\sqrt{V}\) according to Wahlin\(^8\).

Neon (Fig. 1). Course of the curve according to Ramsauer\(^ {14}\), Brox\(^5\), and Ramsauer and Kollath\(^ {15}\). Fine structure according to Normand\(^3\).

Argon, krypton, xenon (Fig. 2). Course of the curve according to Ramsauer\(^4\) and Ramsauer and Kollath\(^ {15}\). Fine structure for argon according to Normand\(^ {13}\). The final point of the curve for argon at \(0.17\sqrt{V}\) according to Wahlin\(^8\).

Hydrogen (Fig. 3). Above \(1\sqrt{V}\)—the mean between the curves of Brox\(^ {16}\) and Normand\(^3\). Fine structure according to Normand\(^3\). Below \(1\sqrt{V}\)—according to Ramsauer and Kollath\(^ {17}\). Final point at \(0.17\sqrt{V}\) according to Wahlin\(^8\).

Nitrogen (Fig. 4). Above \(1\sqrt{V}\)—the mean between the curves of Brox\(^ {16}\) and Normand\(^3\). Fine structure at \(2.5\sqrt{V}\). Below \(1\sqrt{V}\)—Ramsauer and Kollath’s curve, lowered by 10% (for better agreement).\(^ {17}\)

Oxygen (Fig. 4). Above \(1\sqrt{V}\)—according to Brox\(^ {18}\), below—according to Ramsauer and Kollath\(^ {17}\).

Alkali metals (Fig. 5). According to Brode\(^ {19}\)—the only one who investigated the alkali metals.

Zinc, cadmium, mercury (Fig. 6). According to Brode\(^ {20}\), taking into account the comparability of all three curves, of which the curves for zinc and cadmium are found only in Brode’s work.

Thallium (Fig. 7). According to Brode\(^ {21}\).

Nitric oxide (Fig. 8). Above \(1\sqrt{V}\)—according to Brox\(^ {18}\), below \(1\sqrt{V}\)—dashed according to Skinker and White\(^ {22}\).

Nitrogen dioxide (Fig. 8). According to Brox\(^ {18}\) and Ramsauer and Kollath\(^ {23}\).

*

Carbon monoxide (Fig. 9). Above \(1\sqrt{\overline V}\)—the mean between the curves of Brose\(^{24}\) and Normand\(^{3}\). Below \(1\sqrt{\overline V}\)—the mean between the curves of Ramsauer and Kollath\(^{17}\) and Normand\(^{3}\); the rise in the region of the very lowest velocities, which according to Normand exists, has not been taken into account.

Carbon dioxide (Fig. 9). According to Brose\(^{24}\) and Ramsauer and Kollath\(^{17}\).

Paraffins (Fig. 10). According to Brose\(^{24}\), taking into account the comparability of all four curves. For methane below \(1\sqrt{\overline V}\)—according to Ramsauer and Kollath\(^{17}\). At small velocities, according to Brose\(^{25}\), the curve for butane splits into two, corresponding to its two isomers—normal butane \((N)\) and isobutane \((J)\).

§ 15. On certain curves of the e.m.f. We shall now consider some especially noteworthy curves from Fig. 2, turning to the corresponding original works.

Fig. 3. E.m.f. of heavy noble gases (according to Ramsauer and Kollath).

Fig. 4. Velocity distribution in the region of the e.m.f. minimum of xenon.

Curves of plate 2, Fig. 2. These curves may be seen in more detail in Fig. 3, taken from the works of Ramsauer\(^{26}\) and Kollath\(^{15}\). How clearly the existence of the e.m.f. minimum manifests itself may be judged on the basis of Fig. 4, which shows the shape of the electron velocity-distribution curve, obtained by the Ps method for vacuum and after admitting xenon into the apparatus. The electron beam, whose mean velocity corresponds precisely to the minimum of the cross section of the argon molecule, proves, after passing through the gas \((\times—\times)\), to be more homogeneous in velocity than before passing through it \((\circ—\circ)\).

Curves of plate 3, Fig. 2. For hydrogen, in the region of velocities below \(2\sqrt{\overline V}\), different authors have obtained highly contradictory results, as can be seen from Fig. 5, taken from the work of Gerthner\(^{7}\). In discussing these contradictions it must first of all be borne in mind that the rise of the curve in the region of very

small velocities, observed by Normand[^3], contradicts the results of all other investigations and therefore can hardly be regarded as actually existing. The curves of Townsend[^27] and Rush[^6] can be compared with the others only with their qualitative character in mind. They agree with the rest in the sense that they give a maximum at velocities greater than \(1\sqrt{V}\). There remains a contradiction between Bruche’s first[^28] and second[^29] curves. But the second curve of this author agrees well with the curve of Ramsauer and Kollath[^17], including the separate points found by Ramsauer[^26]. The peculiarities of Bruche’s first curve may partly be explained by his method, as will be seen from § 27. Hertner’s curve has the same character as Bruche’s second curve; moreover, it may be assumed that the considerable differences in the magnitudes of the ordinates can satisfactorily be explained by the peculiarities of the method used by Hertner (a longitudinal magnetic field). On the whole, the contradictions among the measurements of different authors, even in this exceptional case, prove to be not as great as they appear at first sight.

Fig. 5. E.c.s. for hydrogen according to data of various authors

Curves in chart 4 of Fig. 2. The sudden rise of the e.c.s. curve for oxygen in the region of the very smallest velocities is so unexpected that this result should be supported by data on the measurements. In Fig. 6, borrowed from the work of Ramsauer and Kollath[^29], the “pressure lines” are given for the velocities indicated in the table in the same figure. It is easy to discern a gradual decrease of the e.c.s. within the limits from \(0.5\) to \(0.8\sqrt{V}\), followed then by a sharp rise continuing right up to the boundary of the measurements, which lies at \(0.4\sqrt{V}\).

Curves in chart 5 of Fig. 2. In Fig. 7 are shown the e.c.s. curves for potassium obtained by Brode[^19]. On the same scale the curve for argon is plotted (as a dashed line).

Curves in chart 8 of Fig. 2. In Fig. 8 we give a whole series of e.c.s. measurements for mercury vapor, studied by many authors because of the interest it presents for technology. The position of two points found by Bäte[^30] for small velocities is apparently erroneous, whereas his other measurements agree well with the results of other investigators and very clearly show a jump in the curve at \(4.9\ \mathrm{V}\) (the excitation potential).

Curves in chart 9 of Fig. 2. Taking as an example the curves for carbon monoxide—

…of carbon (Fig. 9) the fine structure of the e.c.s. curve, discovered by Normand^3 at \(1\sqrt{V}\) and between \(2\) and \(3\sqrt{V}\), is clearly visible. The details of the course of the curve at \(1\sqrt{V}\), according to Brüche^18, are confirmed by a change—

Fig. 6. Pressure straight lines for oxygen at various electron velocities (after Ramsauer and Kollath).

Fig. 7. E.c.s. for potassium (after Brode) and argon on the same scale.

Fig. 8. E.c.s. of mercury-vapor molecules according to data of various authors.

Fig. 9. Fine structure of the e.c.s. of carbon monoxide molecules (after Normand).

Fig. 10. Fine structure of the e.c.s. of carbon monoxide molecules (bending of the curve at \(1.2\sqrt{V}\) according to Brüche).

Fig. 11. Pressure straight lines for normal butane and isobutane (after Brüche).

—of the shape of the distribution curve (Fig. 10), which is shown in Fig. 10; at \(1.2\sqrt{V}\) the curve for the gas suddenly departs from the vacuum curve, and as the velocity decreases this divergence increases more and more.

Just as clearly, this was found by Ramsauer and Kollath[^17], the discontinuity being at \(1.15\sqrt{V}\).

Curves of Fig. 10, drawing 2. The existence of a discrepancy between the curves for normal butane and isobutane at equal electron velocities follows quite clearly from the curves of Fig. 11, borrowed from Brüche’s work[^24]. The discrepancy between the “pressure straight lines” lies far beyond the limits of possible experimental errors.

§ 16. Summary table of measurements of e. p. c. The table in which, in alphabetical order (by authors), all measurements of e. p. c. made up to now are listed is on p. 136.

C. Applications of the Experimental Material

§ 17. Discharges in gases. The most important field of phenomena in which the studies briefly considered by us in the preceding section find application is discharges in gases, where we encounter the passage of electrons through a space filled with gas. Discharge phenomena in gases and vapors play, as is well known, an ever larger and greater role in modern electrical engineering, and the processes occurring in them have by no means yet been fully investigated. In general terms we know only what factors determine the course of a gas discharge. We shall be able to attain exact knowledge of the quantitative laws of all the phenomena taking place in a discharge only by studying all the elementary processes and their joint course.

The basis of all elementary processes is the interaction between gas molecules and electrons. The first essential question is the question of the probability of the occurrence of some interaction or other. It can be answered if one has at one’s disposal the e. p. c. curves. Below we shall show, by several examples, how on the basis of such modest data certain interesting conclusions of a qualitative character concerning discharges in gases may be obtained. The next question is the question of the nature of the interactions between molecules and electrons. This question is still not sufficiently clear. Something known about it will be set forth in the following section.

Before the “anomalies” of the e. p. c. were discovered, the number of collisions between molecules and electrons was calculated on the basis of the data of the kinetic theory of gases. To what errors this could lead is easy to show by several examples.

A discharge tube filled with mercury vapor (Fig. 2, drawing 6). Mercury molecules act on electrons at the greater distance, the smaller the velocity of the latter. Thus at first, until the electrons have as yet acquired a considerable velocity, we have a process resembling diffusion in a comparatively dense gas. The greater the velocity of the electrons becomes under the action of the accelerating voltage, the less resistance the mercury vapor offers to their motion. This continues up to approximately

TABLE 1

Investigations of e.p.s. (in alphabetical order by authors)

Continuation of Table 1

4.9 V, when the electrons begin to excite the luminescence of mercury atoms, thereby completely or partially losing their velocity and therefore again beginning to experience great resistance to motion. The whole process then repeats from the beginning. On the whole the phenomenon proceeds approximately as one might have expected without knowing anything about the course of the e.m.f. curve, namely: the influence of the molecules proves weaker the greater the velocity of the electrons.

A discharge tube filled with argon (Fig. 2, curve 2). At the very lowest velocities the e.m.f. has a certain mean value, which decreases as the velocity increases and reaches, at about 0.4 V, the value which it should have according to the kinetic theory of gases, in order then, at 13 V, to prove more than three times greater than the gas-kinetic value. In other words, at 13 V argon offers 150 times greater resistance to the motion of electrons than at 0.4 V. This is diametrically opposite to what should have been expected on the basis of the prevailing ideas, and is substantially at variance with the assumption of the constancy of the cross section adopted by kinetic theory.

A discharge tube filled with vapors of an alkali metal, for example potassium (Fig. 2, curve 5) (or the addition of such a vapor to an already existing filling, for example argon or mercury; additions of this kind, owing to the small magnitude of the ionization potential of alkali-metal vapors, may be advantageous). At an electron velocity of 2 V, potassium atoms have an e.m.f. 8 times greater than mercury atoms, and 160 times greater than argon atoms at the same velocity. Therefore a small addition of potassium vapor can exert a completely disproportionate influence on the discharge. If the cross sections of the atoms obtained on the basis of other measurements were taken as the basis of the calculations, then at 2 V we would have values approximately 50 times smaller than the actual ones and, on this basis, would strive to obtain an entirely unsuitable quantity of vapor.

§ 18. E.m.f. and the structure of molecules. On the relation between the magnitude of the e.m.f. of molecules and their structure, a review article by Brose[^31] was placed in Ergebnisse der Exakten Naturwissenschaften, to which we refer readers interested in the details of the question. We shall dwell only on a few characteristic examples, especially from the works of Schmieder[^11] and Holst and Holtsmark[^12], which had not yet been published at the time Brose’s review appeared.

Fig. 2 shows, on the one hand, the great variety of the types of curves encountered, and on the other—evident even at a cursory glance—the similarity of certain curves to one another. It is possible here to distinguish clearly separate groups of e.m.f. curves having the same character. Such are, for example, the groups of curves for the heavy noble gases (curve 2), for alkali-metal vapors (curve 5), for the vapors of zinc, cadmium, and mercury (curve 6), and also for paraffins (curve 10). Some

the curves reveal a similarity to one another that is close to complete coincidence. Such, for example, are the cases of nitrogen and carbon monoxide, methane and krypton (Fig. 12). As Broch showed in his investigations, the reason for such similarity lies in the structure of the outer electron shells of the molecules of the corresponding gases.

Fig. 12. Curves for molecules of identical structure (according to Broch).

In addition to such general questions, the study of the e. c. s. makes it possible to pose and solve other, much more particular questions concerning details, such as, for example: in what manner does the combination of a hydrogen atom with some other atom—for example, a carbon or nitrogen atom—take place? To what extent are the e. c. s. curves of molecules that, according to Grimm’s law, ought to have identical properties similar? In what way are the dipole properties of molecules manifested in the course of the e. c. s. curves? Do the e. c. s. curves differ for two molecules having the same chemical composition but different structure (isomers)? etc.

In its present form the method has the disadvantage that the chemical meaning of each separate curve cannot be indicated directly, and this becomes possible only by collecting extensive experimental material and comparing the individual curves with one another.

Without being able to go into this question here, we shall confine ourselves to only a few examples. In Fig. 13 are shown the results of measurements made by Schmieder^11 on the compounds \(CH_3\). When \(O\) is replaced by \(NH\) or \(CH_2\), the curve changes only quantitatively, while its character remains unchanged. In Fig. 14 are shown the results obtained by Holst and Holtsmark^12.

Fig. 13. Replacement of \(O_2\) by \(NH_3\), or \(CH_2\) (according to Schmieder)

Fig. 14. E. c. s. of molecules \(CCl_4\), \(CHCl_3\), \(CH_2Cl_2\), and \(CH_3Cl\) (according to Holst and Holtsmark).

The authors believe, on the basis of their results and comparison with Brode’s measurements (hydrogen chloride),^29 that the maximum lying between \(2.6\)—\(2.7\sqrt{V}\) is characteristic of the chlorine atom. As was already indicated above, such conclusions cannot be considered fully justified so long as the matter concerns only individual cases.

§ 19. Theoretical explanation of the effective-cross-section curves. Here we must first of all say that we do not intend to give an exhaustive theory of the dependence of the effective cross section on velocity. Such a theory, which would predict the course of the curves for various gases in agreement with experiment, does not yet exist today, although with the aid of wave mechanics we have already advanced considerably in this direction. Thus, for example, the transparency of argon atoms for slow electrons is amenable without particular difficulty to qualitative explanation, whereas only a few years ago, in attempting to do this with the aid of classical theory, one had to resort to extremely artificial assumptions. Below we shall cite works essential for the development of the theory, limiting ourselves, with respect to each of them, to a statement of the basic ideas and to a comparison of some results of the calculations with experiments.

F. Hund^32 was the first to consider theoretically the question of the transparency of argon atoms for slow electrons. In his work he investigated under what assumptions an explanation of this effect by means of the classical theory would be possible. These assumptions proved, however, to be so artificial that in the second part of his work he attempted to conduct the investigation with the aid of quantum theory.

F. Zwicky^33 attempted to explain the course of the effective-cross-section curve for argon on the basis of the classical theory, assuming that at a certain value of the velocity of the electrons in the beam resonance phenomena arise between them and the atomic electrons. A quantitative treatment proved impossible.

M. Born^34 was the first to consider the problem of collisions between molecules and electrons from the point of view of quantum mechanics. He constructed a general theory of collisions and gave a solution of the equations for the case of electrons possessing large velocities.

L. Menzing^35 calculated, with the aid of the equations of wave mechanics, the process of collision between atoms and electrons, regarding the atoms as charged hard spheres. The results differ considerably from those obtained in the analogous optical problem (scattering of light by small particles). Quantitative data were not obtained, since such a model of the atom proved to be too crude an approximation to reality.

W. Elsasser^36 likewise attempted to consider the phenomenon as an analogue of the scattering of light by small particles, the electrons being treated (as also in Born’s work) as waves with a wavelength determined by the de Broglie relation.

H. Holtsmark^37, in a whole series of papers, investigated the effective cross section from the theoretical side in very great detail. He calculated—analogously to the scattering of light by small spheres—the scattering of electron waves in a force field possessing spherical symmetry, with the magnitude of the potential (the refractive index) inside such a sphere being a function of the coordinates. It is in this nonconstancy of the refractive index that the complication of the problem, as compared with the optical one, lies. Although the calculations proved to be exceptionally complicated, Holtsmark succeeded in obtaining exact solutions of the equations. This method, however, turned out to be practically applicable only to the heavy noble gases, whose atomic fields were known thanks to the investigations of Hartree^38.

Fig. 15. Calculated effective cross section for krypton atoms (after Holtsmark).

The effective-cross-section curves obtained by calculation agree well with the experimental ones. In order to obtain quantitative agreement, it is necessary to assume a definite dependence of the polarization of the atom on the distance between it and the electron of the beam.

In Fig. 15a are shown the effective-cross-section curves for krypton: the experimental curve, (——), obtained by Ramsauer^39 and by Ramsauer and Kollath^15, and that calculated by Holtsmark^40 (+++). Fig. 15b depicts the field of the krypton atom calculated by Hartree (——), and the field with polarization taken into account (— · — · —).

Allis and Morse^41 calculated the effective-cross-section curves by the same method as Holtsmark, considering, only for the sake of simplification, the atom to consist of a nucleus with a single electron shell. Thanks to this considerable simplification the authors were able to find numerical values of the effective cross section for many atoms (Fig. 16). The form of the calculated curves agrees qualitatively well with the experimental data. Of particular interest is the sharp difference in the absolute height of the curves and the clearly manifested dependence of the height on the position of the corresponding atom in the periodic system.

Fig. 16. Effective-cross-section curves for various elements. Dashed lines—experimental; solid line—theoretical, according to Allis and Morse.

Oppenheimer^42 improved Born’s method, taking into account the possib-

EFFECTIVE CROSS SECTION OF GAS MOLECULES

the so-called electron exchange. In a wave-mechanical treatment of the collision problem, owing to the indistinguishability between the electrons belonging to the beam and to the atom, and on the basis of the Pauli principle, an additional term (the exchange term) appears in the solution. Oppenheimer gave an approximate solution for such a conception, but did not make applications of it to concrete cases.

Massey and Mohr^43 applied Oppenheimer’s theory to slow electrons and obtained good agreement with experimental data. The chief value of this work lies in the determination of the excitation function of definite spectral lines and in the study of the angular distribution of elastically reflected electrons. Here it was found (as will also be discussed at the end of the section) that only by means of “electron exchange” can agreement between theory and experiment be obtained in some cases.

In conclusion we shall attempt, without going into details, to give briefly the general train of thought underlying the wave-mechanical treatment of the problem.

The basic process in the scattering of electron waves may, according to Mott^44, be represented in the following way:

1. The atom is a force field with a definite distribution of potential. Each volume element of the scattering atom is a source of a spherical wave, whose amplitude is proportional to the potential of the corresponding volume element and to the amplitude of the oscillations existing there, which arise on account of the spherical waves sent out by the other volume elements. Thus the opposite influence of the different volume elements of the atom on the scattering is taken into account here (Faxén and Holtsmark, Allis and Morse).

2. At high velocities of the beam electrons the scattering amplitude is inversely proportional to their velocity. Therefore, in the case under consideration, the amplitude of the scattered oscillations is small in comparison with the amplitude of the incident ones. Owing to this, at a definite point of the atom the oscillation consists, chiefly, of the incident wave. Thus each volume element of the atom sends out a wave whose amplitude is proportional to the potential at this point and to the amplitude of the incident wave. It follows from this that at high electron velocities the opposite influence of the various elements of the atom on the scattering may be neglected (Born).

When the problem is treated according to point 2, good agreement between theory and experiment is obtained for fast electrons. However, the corresponding point 1, the general case, does not correctly represent the phenomena at small electron velocities. This may have the following grounds:

1. The approaching electron electrostatically influences the distribution of charges and, consequently, also the distribution of potential in the scattering atom. These influences can be taken into account by introducing into the calculation the polarization energy (Holtsmark).

2. For many cases, for example for strong back scattering,

for the scattering of slow electrons in helium and hydrogen, taking polarization alone into account proves insufficient. In this case the assumption of electron exchange has proved very fruitful (Oppenheimer, Massey, and Mohr).

D. Experimental Investigation of Elementary Processes

§ 20. Components of the e.p.s.

In the preceding section we were interested only in the question of the number of electrons undergoing action in collisions with gas molecules under specified experimental conditions. We shall now consider what the action of molecules on electrons consists in.

In general, the action of a molecule on an electron may be divided into two types:

1) a change in the direction of motion of the electron, and
2) a change (decrease) in velocity.

Both types of action may be combined with one another in an arbitrary manner.

Let us see what these combinations may be and how the distinction between them can be established experimentally. On the basis of experiment we must acknowledge, first, that a change in the direction of motion of electrons can occur without a change in their velocity, and second, that the loss of velocity by an electron occurs by jumps. A special position is occupied by cases in which the velocity is reduced to zero. The mechanism of the phenomenon in this case differs from that corresponding to partial losses of velocity.

Thus we distinguish three kinds of action on an electron:

a) “deflection” (without change of velocity),
b) “loss of velocity” (with or without change of direction),
c) “capture” (bringing the velocity of the electron, in magnitude and direction, to the velocity of the corresponding molecule).

The phenomenon placed by us under heading “a” is called by Lenard “fictive absorption”; by “true absorption” he meant what we call “capture.” From heading “b” it is clear that losses of velocity not accompanied by a change in direction are a special case of loss of velocity in general.

Losses of velocity (heading “b”) substantially affect the form of the e.p.s. curves usually only in the region of velocities greater than those corresponding to the excitation and ionization potential of molecules. This phenomenon is analyzed in detail in the article by Grotrian and Penning (Handb. d. Phys. XXIII, first half).

Capture (heading “c”) plays a very secondary role in the case considered by us, because for some of the gases of interest to us it is altogether absent; in those cases where it is present, this phenomenon does not have a substantial effect on the magnitude of the e.p.s. According to Loeb’s data^45, for example, for the case

oxygen out of 5,000 collisions leads to capture only once. Thus the “capturing” cross section amounts to only \(0.02\%\) of the entire effective cross section. The phenomenon in itself is, of course, of great interest, since it reveals forces of attraction between the electron and the neutral molecule. However, its consideration lies beyond the scope of the present review.

Thus, in the region of small electron velocities, only the phenomenon that we have placed under heading “a”—deflection without loss of velocity—plays a role. We shall outline the train of thought according to which we shall examine it in detail in the following paragraph.

Deflections without loss of velocity (heading “a”) can be detected already at comparatively high electron velocities. As the velocity decreases, the number of these deflections approaches ever more closely the number of electrons that have undergone any interaction at all, and finally, at velocities below a certain limit characteristic for each gas, it becomes the sole kind of interaction between electrons and molecules. Thus, at small velocities, the number of electrons that have undergone interaction is equal to the number of electrons that have experienced deflection without loss of velocity. Since, as a result of measurements of the effective cross section, this number has been determined for many gases and vapors, it remains only for us to consider the question of what happens to each individual electron, i.e. in what direction it is deflected. We shall return to this in the paragraph devoted to the consideration of the angular distribution of electrons.

We shall now give a survey of the results of several works devoted to the deflection of electrons in general and, in their development, leading to the investigation of the angular distribution of electrons deflected and scattered by molecules.* Then we shall describe the methods by which the investigation of the angular distribution of electrons is carried out, give examples of results obtained by various authors, compare the results of certain theoretical works with this experimental material, and finally discuss the influence of the angular distribution of electrons on the value of the effective cross section when it is measured by various methods.

§ 21. Works on the question of the existence of deflections without loss of velocity. The first experiments proving the existence of the reflection of electrons without loss of velocity (elastic reflection) were carried out by Franck and Hertz\(^{46}\). They used an apparatus whose scheme is shown in Fig. 17. The electrons, emerging from the incandescent filament \(F\), acquire the desired velocity in the electric field between \(F\) and the grid \(N_1\), and then enter the field-free space \(S\), where they collide with gas molecules (the plate \(P\) should be imagined as being removed

* Below, by the word “deflection” we shall denote the process of deflection of a single electron; “scattering,” on the other hand, will refer to the deflection of many electrons.

from the position it occupies in the figure). Some of the electrons entering \(S\) are reflected by the molecules at large angles and, after passing through the grid \(N_2\), strike the collecting electrode—the ring-shaped plate \(R\), connected to the galvanometer \(G_2\). The velocity distribution of these electrons can be determined from the curves of the retarding potential (applied between \(N_2\) and \(R\)) and compared with the distribution for electrons emitted from the filament. The latter distribution is determined by placing the movable plate \(P\), connected to the galvanometer \(G_1\), opposite the grid and measuring its charge as a function of the applied retarding potential. It turns out that both distribution curves extend to one and the same value of the maximum velocity (in

Fig. 17. Proof of the existence of elastic reflection in gases (after Franck and Hertz).

Fig. 18. Velocity distribution of primary and reflected electrons.

Fig. 18 the curves are shown for helium at a maximum electron velocity of 4 V). From the equality of the maximum velocities for both distributions it follows that at least some of the reflected electrons undergo reflection without loss of velocity. In the case of hydrogen analogous results were obtained, but there it was already possible to observe a loss of velocity, which was especially clearly revealed for oxygen.

These experiments were then repeated for hydrogen by Berwald\(^ {47}\), who, using a somewhat modified apparatus, in addition to confirming the results described, obtained data on the dependence of the number of reflected electrons on their velocity and on the gas pressure.

Since, on the basis of the experiments of Franck and Hertz described above, it was unclear precisely what part of the deflections occurs without loss of velocity by the electrons, they undertook a new investigation\(^ {48}\) using the apparatus shown in Fig. 19. Electrons from the filament \(F\) are accelerated between \(F\) and \(N\) and enter the retarding field between \(N\) and the collecting electrode \(P\), connected with a galv-

manometer \(G\). The distance \(x\) between \(F\) and \(N\) can be changed by moving the plate with the filament \(F\) attached to it. At constant pressure, retarding-potential curves were taken for different distances \(x\) (curves \(I\) and \(II\), Fig. 20). Despite the considerably increased number of collisions of electrons with molecules as \(x\) was increased (curve \(II\)), the distribution of the first ones by

Fig. 19. Proof of the existence of elastic reflection in gases (Frank and Hertz apparatus).

Fig. 20. Retarding-potential curves (after Frank and Hertz). Curve \(I\)—distance between the filament and the plate \(4\) mm. Curve \(II\)—\(18\) mm.

velocities did not change. It follows from this that all collisions occur without loss of velocity.

In later works, which appeared much later (after 10 years and more), the distribution of electrons by velocities for definite angles of deflection was investigated. With very small angles (diffusion) Baumann,\(^{49}\) Shahman,\(^{50}\) and Goldman\(^{51}\) worked. Deflections through angles of the order of \(90^\circ\) were investigated by Kollath\(^{52}\) and Werner.\(^{53}\)

In the works of the first group, by measuring the distribution of the intensity of the electron beam on both sides of its middle, its broadening at various pressures and in various gases was investigated. The apparatus used for this purpose is shown schematically in Fig. 21. Electrons fly out from the incandescent filament \(F\) and, after passing through diaphragms 1 and 2, enter the space \(S\), free of field. The electron projector can rotate about an axis, the angle of rotation being read with the aid of a pointer on a circular scale. Opposite the projector there is located (stationarily) a device for catching the electrons, consisting of a narrow slit diaphragm 3, perpendicular to the plane of the drawing, and a Faraday cylinder \(K\) situated behind the slit. The grids \(N_1\) and \(N_2\) serve to retard electrons possessing velocities less than the initial one. Figure 22 shows Shahman’s measurement results for argon at various electron velocities. In that

Fig. 21. Measurement of scattering at small angles (after Shahman).

whereas at an electron velocity of 6.5 V no appreciable broadening of the beam is observed upon admission of the gas, at 11 V the broadening proves very noticeable. If, following Ramsauer, one plots the dependence of the broadening of the curve (measured at half its height) on the electron velocity at constant gas pressure, then for hydrogen there is obtained a continuous rise of the curve in the interval of velocities from 5.5 to \(1.5\sqrt{V}\). For argon, on the contrary, a maximum is observed at \(3.5\sqrt{V}\), with a sharp fall toward lower velocities.

Fig. 22. Distribution by directions for argon at various pressures and velocities (after Ramsauer).

In the second group of works, deflections at angles lying within definite narrow limits were investigated, owing to which it was possible to obtain quantitative data on the dependence of the angle of deflection on the gas pressure and the electron velocity. The method used by Kollath\(^{52}\) was as follows: electrons emitted by the incandescent filament \(F\) (Fig. 23) form, passing through diaphragms 1 and 2, a beam, which then passes through the scattering space \(S\) and is caught by the Faraday cylinder \(A\). The arrangement (the setup of Fig. 23 has cylindrical symmetry with respect to the axis \(x\text{—}x\)) serves to ensure that only those of the electrons scattered in the space \(S\) whose angles of deflection are close to \(90^\circ\) (more precisely, lie between \(87^\circ\) and \(93^\circ\)) enter the annular trap \(R\). With these electrons entering \(R\), two independent measurements can be made. First, negative potentials of various magnitude may be applied to \(R\), and in this way the velocities of the electrons entering \(R\) may be determined. Measurements of this kind showed that in all the investi-

Fig. 23. Measurement of scattering at a right angle (after Kollath).

Fig. 24. Effective cross section (dashed) and deflection at a right angle (solid line).

in the gases investigated (He, Ne, Ar, Kr, H₂, N₂, CO, CO₂, N₂O, CH₄) there occur deflections not associated with loss of velocity.* Secondly, one can quantitatively compare the number of electrons that enter \(R\) without losing velocity (a high negative potential on \(R\)) with the number of beam electrons that enter \(S\).

Plotting the numbers of electrons measured in this way as a function of their velocities, we obtain curves (Fig. 24) which, for example in the case of carbon monoxide and carbon dioxide, have maxima lying at \(1.5\sqrt{V}\) and \(2\sqrt{V}\), respectively, i.e. precisely where the first maxima of the e.c.s. curves lie. As for the second maxima of the e.c.s. curves, they evidently have no relation whatever to elastic collisions.

Werner \(^{53}\), in order to verify Mott’s scattering formula, investigated in his work the number of electrons deflected through an angle of \(90^\circ\) at higher velocities (from 40 to 300 V). He found, up to the highest of the velocities he investigated, measurable numbers of electrons deflected through \(90^\circ\) without loss of velocity.

The next step in studying the distribution of electrons over directions is to find the distribution in all directions. Here two groups of methods may be distinguished. In the first of them movable traps are used; in the second, many rigidly fixed traps are employed.

Fig. 25. Scattering of electrons in gases. Apparatus with a movable trap.

§ 22. Movable-trap method. This method, first proposed by Dymond \(^{54}\), was subsequently used, with minor changes, by many authors. To clarify the essential features of the method we shall consider a simple scheme (Fig. 25), and return later to the peculiarities of the original works.

Electrons are emitted by the heated cathode \(F\) and, after passing through diaphragms 1 and 2, enter the scattering space \(S\) in the form of a beam. Electrons that have not undergone collision fly into the Faraday cylinder \(A\). In order that a definite point of the beam (the center \(S\) in Fig. 25) could be observed at all times, the detecting device, consisting of diaphragms 3 and 4 and the trap (Faraday cylinder) \(K\), is made movable, in such a way that in any position the continuation of the axis of the cylinder and diaphragms passes through the point under investigation. With such an arrangement, obviously, the investi-

* Data on the loss of velocity obtained by this method are, of course, not distinguished by such definiteness as in the experiments of Franck and Hertz, because here the electron velocity is determined after a single collision, whereas in the experiments of Franck and Hertz the electron velocity is measured after numerous collisions.

scattering of electrons occurring in the volume \(\Delta S\), determined by the cross section of the beam and by the configuration, and also by the position of the diaphragms of the collecting device. This highly elegant method of isolating the scattering region requires great accuracy of measurement.

By varying the angle \(\vartheta\) between the directions \(F \to A\) (the primary beam) and \(\Delta S \to K\) (the scattered beam), which we shall call the scattering angle, we shall measure the charge of the trap \(K\) corresponding to each value of \(\vartheta\). In vacuum no charge will be observed until the directions of the primary and scattered beams almost coincide. Thereupon a rapid increase of the charge will begin, reaching a maximum value and then again rapidly falling to zero. On repeating the same measurements in the presence of gas in the apparatus, we shall obtain values of the charge determined by the number of electrons scattered in \(\Delta S\) in the given direction.

Fig. 26. Scattering in mercury vapor at 379 V (after Arnot).

Fig. 27. Apparatus for studying the distribution according to directions (after Dymond).

Constructing curves expressing the charge of the trap as a function of the angle \(\vartheta\) for vacuum and for gas, we obtain a picture like that shown in Fig. 26, borrowed from the work of Arnot[^55]. The number of scattered electrons we shall call the difference between the ordinates of the curves for vacuum and for gas. The curve expressing the dependence of this difference on the scattering angle will be the desired distribution curve. It should be noted here that measurements between \(\vartheta = +15^\circ\) and \(\vartheta = -15^\circ\) are omitted because of the finite dimensions of the primary beam and of the diaphragms of the collecting device.

The study of the distribution of electrons by velocities can in our case be carried out by creating retarding fields between \(K\) and diaphragm 4 (Fig. 25). By applying a sufficiently high retarding potential, we can directly determine the number of electrons scattered without loss of velocity.

With the aid of charging \(A\) it becomes possible to avoid disturbances from secondary electrons and positive ions. Owing to this, the method described is quite suitable for work with fast electrons. At very small velocities some difficulties arise, due to the fact that the number of collected—

...of the \(K\) electrons constitutes, owing to the geometrical features of the apparatus, only a negligible part of the primary beam.

As has already been indicated, the first to apply an apparatus of this kind to the study of the angular distribution of electrons was Dymond \(^{54}\). In his original method, in contrast to the one just described, diaphragms 2 and 3 of the collecting device were stationary (Fig. 27), while the electron source was moved. The source of the scattered electrons was the point of intersection of the primary beam (direction \(F \to 1\)) and the scattered beam (direction \(2 \to 3\)). Diaphragms 2 and 3 and trap \(K\) in Dymond’s apparatus were separated in order to make it possible to investigate the velocity distribution of the electrons with the aid of a magnetic field.

The scattered electrons pass from \(S\) through diaphragms 2 and 3 into \(D\) and move there under the action of the magnetic field along a semicircle, finally passing through diaphragm 4 into \(K\).* The magnetic field made it possible, on the one hand, to study the distribution of electrons with respect to velocities (by the method described in § 10), and, on the other, to determine the number of electrons scattered without loss of velocity.

Fig. 28. Apparatus for studying the angular distribution (according to Bullard and Massey).

The apparatus used by Arnot \(^{56}\) differs hardly at all from Dymond’s. The difference consisted in the method of investigating the velocity distribution of the electrons. Arnot used a radial electric field deflecting the electrons through \(90^\circ\). It should be noted that a radial electric field produces very good focusing when the electrons are deflected through an angle of \(127^\circ\). This was shown by Hughes and McMillen \(^{57}\), who worked with an apparatus identical with Arnot’s, but with the electrons deflected not through \(90^\circ\) but through \(127^\circ\).

In conclusion let us dwell on the design of the apparatus belonging to Bullard and Massey \(^{58}\) (Fig. 28), which is of interest because these authors obtained very valuable results precisely for the region of very slow electrons. The electron source \(F\), together with diaphragms 1 and 2, is fixed immovably. The device for collecting the electrons, consisting of two guiding diaphragms 3 and 4 and trap \(K\), can be rotated by means of the ground joint \(J\). Thus the scheme of Fig. 28 is very close to that shown in Fig. 25.

It should not be thought that in all the methods described the scattering direction obtained was exactly the same, as can be

* The magnetic field was produced by means of a large coil placed near \(D\), and was compensated for in the other parts of the apparatus by a coil connected in the opposite direction.

conclude from the schematic drawing in Fig. 25. In reality the question is never one of a single direction, but of an entire group lying within the limits of a certain angle, determined, on the one hand, by the divergence of the primary electron beam and, on the other, by the geometrical relations in the collecting device. Thus, for example, in the experiments of Bullard and Massey this range of angles was \(11^\circ\).

§ 23. The method of fixed traps. A characteristic feature of the methods described up to now has been the mobility of the electron source and of the collecting device relative to one another. In contrast to this it is also possible to use arrangements all parts of which are fixed, but which instead contain several collecting devices, separate for each direction of scattering.

Fig. 29. Diagram of an arrangement with fixed traps (after Ramsauer and Kollath).

Such an apparatus, shown in Fig. 29, was used, for example, by Ramsauer and Kollath[^59]. The source of electrons was the incandescent filament \(F\); the electron beam was formed with the aid of diaphragms 1, 2, 3, and 4. The beam passed through the chamber \(S\) and was collected by the Faraday cylinder \(A\). Around the point \(S\), with a radius of 3 cm, a sphere was described, consisting of 11 zones (\(R_1\)—\(R_{11}\)) of equal width, serving as the collecting devices (the arrangement in Fig. 29 is symmetrical with respect to the axis \(X—\cdot—X\)). By these zones the electrons scattered by the gas in the region \(S\) in various directions are collected. The angle of scattering is given, in first approximation, by the angle between the axis \(X—\cdot—X\) and the line joining the center \(S\) with the middle of the corresponding zone. Each of the zones has an independent lead to an electrometer, which can be connected for measuring the charge to any of the zones by means of a switch. At the same time the remaining zones and the cylinder \(A\) are connected to a second electrometer. In this way, by successively connecting the electrometer to the 11 zones, 11 intensities are obtained corresponding to 11 definite scattering angles (more precisely, to 11 regions of scattering), whose magnitude is determined geometrically from the length of the path of scattering of the beam and the position of the zone. Each of the values obtained corresponds to a certain mean scattering angle \(\vartheta_m\). Experiment showed that, with the aid of these 11 zones, the scattering curve can be constructed with sufficient accuracy. The dimensions of the scattering region are determined in this method not by its essence, but by the geometrical properties of the apparatus. There are no reasons that would prevent bringing the degree of accuracy of this method at least up to that inherent in the movable-trap method, especially since

in terms of the intensity of the scattered beams, the method described has advantages.

For the region of fast electrons this method is applicable only up to a certain limit. Namely, for velocities lying above the excitation potential of the molecules, it does not make it possible to distinguish between elastically and inelastically reflected electrons; while at velocities greater than the ionization potential, the results are distorted owing to the presence of positive ions and secondary electrons, which here cannot be eliminated, as is done in methods with a movable trap. On the other hand, this method is extremely convenient for the region of very low velocities, since, owing to the large surface area of the zones capturing the scattered electrons, one can work with very low intensities of the primary beam. Thus, for example, it proved possible, without any difficulties in the sense of insufficient intensity or space-charge phenomena, to carry out investigations at velocities of 1 V and below.

For completeness it should be noted that the device described had a predecessor. Ramsauer and Kollath initially used an apparatus with three zones for comparing scattering in the forward (the first zone from 5 to 75°) and backward (the third zone from 175 to 110°) directions. The idea of a stationary collecting arrangement recording electrons of a definite direction is still older and goes back to Kollath’s apparatus described in § 21, where electrons scattered at an angle of 90° were captured by a stationary ring-shaped trap.

§ 24. PRESENTATION OF THE EXPERIMENTAL RESULTS. Before speaking of the results of the experiments, it is necessary to say a few words about the methods of representing them, of which there are two. Both have their advantages and their shortcomings.

The number of scattered electrons may be referred: a) to a unit solid angle; b) to a unit spherical zone. In the first case, regarding the scattering molecule as lying at the center of a sphere, the number of electrons scattered at an angle \(\vartheta\) is understood as the number of electrons falling in the direction \(\vartheta\) inside a sufficiently small solid angle (equal, for example, to \(1/100\) of a unit one). In the second case, the molecule is regarded as lying at the center of a sphere having an axis coinciding with the direction of the primary beam. Electrons scattered at an angle \(\vartheta\) are considered to be electrons falling in the direction \(\vartheta\) onto a spherical zone corresponding to an angle of \(1^\circ\). The two methods are related by a conversion factor which is the product of a constant by the sine of the scattering angle; the constant can easily be calculated. It does not, however, play an essential role in what follows, since in most cases only the relative height of the different parts of the distribution curve is determined.

In the graphical representation of the results by each of the methods, one may use both rectangular and polar

coordinates, and thus there turn out to be four methods of representation in all. Polar coordinates are preferable from the standpoint of clarity, while Cartesian coordinates are more convenient for calculations. To explain all four ways of presenting the results, let us consider a simple example—uniform scattering in all directions, which occurs, for example, in elastic scattering of a beam of small spheres by a large sphere. Fig. 30 presents this case in all four possible forms. The constancy of the scattered quantity, clearly visible in diagrams \(a_1\) and \(a_2\), changes in \(b_1\) and \(b_2\) into an ever stronger increase as the angle \(\vartheta\) approaches \(90^\circ\), which occurs because the surface area of the zones is proportional to the sine of this angle. Representation by means of zones has the disadvantage that the details of the course of the curve near \(0\) and \(180^\circ\) appear very weakly, since, owing to multiplication by \(\sin \vartheta\), the ends of the curve are smeared out. On the other hand, this method in the form \(b_1\) (Fig. 30) has one essential advantage: the area between the abscissa axis, the curve, and two ordinates (shaded in Fig. 30, \(b_1\)) is a direct measure of the total number of scattered electrons. Many authors have used this to establish the connection between scattering measurements and measurements of the mean free path.

Fig. 30. Various methods of representing the number of scattered electrons as a function of the scattering angle.

In order not to make it difficult to compare the results of individual experiments with one another, we everywhere adhere to only one of the four indicated methods, namely, method \(a_1\). Thus all subsequent curves are plotted in rectangular coordinates, with angles from \(0\) to \(180^\circ\) laid off along the abscissa axis, and along the ordinate axis—the numbers of electrons referred to unit solid angle.

§ 25. Results of measurements of the angular distribution. With regard to the angular distribution of electrons that have undergone scattering, there is at present very abundant experimental material. The course of the scattering curves for Ne, Ar, Kr, Xe, \(\mathrm{H_2}\), and \(\mathrm{CO_2}\) is known in the interval from 800 to 1 V; for mercury vapor from 800 to 8 V; for \(\mathrm{N_2}\) and \(\mathrm{CH_4}\) from 800 to 4 V; and for \(\mathrm{CO_2}\) from 10 to 1 V. It should be noted that the results of different authors not only do not contradict one another, but, on the contrary, agree well among themselves.

All measurements made up to the present are listed in alphabetical order by author in Table 2. The most important results belong to Arnot, who worked predominantly with high velocities (from 800 to 30 V), to Bullard and Massey, who investigated the region of intermediate velocities (30–4 V), and, finally,

Ramsauer and Kollath, who studied scattering at velocities below the excitation potential and went down to 0.5 V. All the data presented here have been taken by us from the works of these authors, since their results completely cover the results of all the others both with respect to the range of velocities investigated and with respect to angles. All authors, with the exception of Ramsauer and Kollath, who used for the investigation the apparatus described in § 23, employed the method of the movable trap.

We shall begin with Arnot’s excellent curves,^55 which relate to mercury vapor at electron velocities from 800 to 8 V (Figs. 31 and 32). The abscissas are the values of the angle \(\vartheta\), varying from 0 to

Figs. 31 and 32. Scattering in mercury vapor at various electron velocities (after Arnot).

180°, and the ordinates are the numbers of scattered electrons in relative units. In doing so, following Arnot, we use a two-sided representation, in which the scattering angle runs through values from 0 to 180° on both sides of the direction of the beam. Thus on each of the drawings in Figs. 31 and 32 there are two curves, if they are understood in the sense of drawing \(a_1\) in Fig. 30.

Arnot himself already pointed out the great similarity of the electron-scattering curves to the curves for the scattering of light by randomly distributed small spheres. The latter phenomenon consists, as is well known, in the fact that in the direction of propagation of the beam there is observed the principal maximum of intensity, around which, at equal distances, secondary maxima are located, whose height decreases with distance from the principal maximum. These secondary maxima, having in the figure an order up to the fourth inclusive, in full agreement with what is observed in the scattering of light, as the wavelength decreases, i.e. as the electron velocity increases, come closer and closer to the principal maximum.

TABLE 2

Studies of the angular distribution (in alphabetical order by author). (The last three works appeared after the article was written and are briefly reviewed in the addendum)

and, in the end, become inaccessible to observation. This exact analogy between electron scattering and light scattering is preserved, however (in its essential features), only for high electron velocities and does not extend to the region where the individual peculiarities of the e.c.s. curves are found. The deviations from it observed in mercury vapor for slow electrons are explained by the fact that, with increasing velocity, the outer maximum at first moves outward and only then, following the analogy mentioned, begins to move inward.

For further experimental material the analogy with light also has a certain value, at least until the electron velocity becomes too small.

Fig. 33. Scattering in argon (from the curves of Bullard and Massey).

Fig. 34. Scattering in helium (after Bullard and Massey).

The curves of Fig. 33, borrowed from the work of Bullard and Massey\(^{60}\), have the same character as the curves for mercury vapor; here, however, the first secondary maximum moves outward as the velocity increases. This similarity to mercury vapor is also shown by the noble gases Ne, Ar, Kr, Xe, as was shown for fast electrons by Arnot. The curve for 50 V in Fig. 34 (borrowed from the work of Bullard and Massey\(^{61}\)) can also be brought under the analogy with light, if one assumes that the first minimum has moved so far outward that the first secondary maximum has become unobservable, as happens in optics for very small particles (similar phenomena were observed by Arnot in CO, H\(_2\), N\(_2\), and CH\(_4\) for velocities from 800 to 30 V\(^{62}\)). From the remaining curves of Fig. 34 one can see how the minimum gradually becomes more and more clearly expressed as the electron velocity decreases and moves inward. It should especially be noted that there is an even stronger decrease in scattering at small angles as the electron velocity decreases. For example, in helium at 4 V there is observed

almost uniform distribution of the scattered electrons over directions.

Scattering phenomena become exceptionally varied upon a further lowering of the electron velocity—in the range of 1 V and below. To show the variety of the curves for this interval of velocities, Figs. 35–37 schematically show “forward” (Fig. 35),

Figs. 35–37. Different types of scattering curves at the very lowest electron velocities.

“backward” (Fig. 36), and “side” (Fig. 37) scattering. The data concerning this region are borrowed from the works of Ramsauer and Kollath \(^{63}\), the only ones devoted to electron scattering at velocities below 4 V.

“Backward” scattering is observed in hydrogen at velocities below 2 V, in CO—below 1.5 V, and in helium—around 4–5 V. As a result

Fig. 38. Transition from forward to backward scattering in hydrogen (after Ramsauer and Kollath).

Fig. 39. Transition to side scattering in argon (after Ramsauer and Kollath).

of this, these phenomena were noted by Bullard and Massey only for helium (at 4 V). In Fig. 38 one can see how, as the electron velocity decreases, “forward” scattering in hydrogen turns into “backward” scattering. It is very clearly seen (on the right-hand side of Fig. 38) how, as the number of volts decreases, the number of electrons scattered through small angles decreases, and the number of electrons scattered through large angles increases. This “backward” scattering was discovered by Ramsauer and Kollath \(^{59}\) with the aid of their instrument-

of the apparatus with three zones. The “scattering ratio” for a velocity of 1 V in hydrogen and carbon monoxide was equal to \(1/2\), i.e., approximately twice fewer electrons fell on the front half of the sphere than on the rear half.

“Lateral” scattering in pure form is found for argon at velocities of about 2 V, and for krypton—between 2 and 1.5 V. Fig. 39 shows how the scattering for argon at 4 V changes, as the velocity decreases, into “lateral” scattering.

In conclusion, let us show with several examples how good the agreement is between the measurements of various authors, despite the fact that the investigations were carried out by different methods. In Fig. 40 are shown the curves obtained by Arnot⁵⁵ and by Pearson and Arnquist⁶⁴ for mercury vapor at electron velocities of 200 V; in

Fig. 40—42. Directional distribution (according to various authors).

Fig. 41—the curves of Hughes and McMillen and of Arnot for argon at 50 V, and finally, in Fig. 42—the curves of Bullard and Massey⁶⁰ and of Ramsauer and Kollath⁶⁵ for argon at 6 V. In all these cases, as in many others, the results of different authors agree fully with one another and in no case is any contradiction observed.

§ 26. Comparison of experimental and theoretical data on the distribution of electrons by direction. The theoretical calculation of the directional-distribution curves is in a definite connection with the theoretical investigation of the total effective cross section. (Let us recall the possibility of calculating the effective cross section directly from the directional-distribution curves (§ 24), under the assumption that the interactions between electrons and molecules reduce exclusively to the deflection of the electrons.) Therefore we shall now give a brief survey of the attempts made thus far to calculate the directional-distribution curves in connection with the theories proposed to explain the effective cross section.

In Fig. 43 are given the curves obtained by calculations on the basis of Born’s theory⁶⁶. They reproduce well the experimental results for fast electrons. This theory, however, could not ex-

explain the rise of the curves for large values of the scattering angles. Goldsmark’s theory^67, whose applicability is not limited to the region of high electron velocities, reproduces well, in qualitative terms, the complex angular distribution observed for heavy noble gases with slow electrons (Fig. 44). (It should be borne in mind that in Figs. 43 and 44, for the various curves the abscissa axis has been shifted for greater clarity; its true position relative to the corresponding curves is indicated by horizontal dashes in the right-hand part of the figures.) Calculations using simplified ideas about the force fields of atoms by the method of Allis and Morse^41 give satisfactory agreement with experiment for electron velocities

Fig. 43. Scattering at higher electron velocities (circles—experimental data of Arnot; straight lines—results of Arnot’s calculations according to Born’s theory).

Fig. 44. Scattering curves in argon (solid line—experimental data of Bullard and Massey; dashed line—results calculated by Bullard and Massey according to Goldsmark’s theory).

between 40 and 20 V. For velocities below 20 V these simplifications are already no longer permissible. Oppenheimer’s theory^42, which takes into account “electron exchange,” introduces entirely new features into the investigation. Massey and Mohr^43, who carried out calculations by Oppenheimer’s method, showed that this method can give a qualitative reproduction of the experimental results with respect to the remarkable fact of the appearance of “inverse” scattering for hydrogen and helium at small velocities (Fig. 45). A unification of the theories of Goldsmark and Oppenheimer, i.e. a simultaneous allowance for polarization of atoms and exchange of electrons, has not yet been carried out.

§ 27. Consequences from the angular distribution.
Below we shall consider the significance that angular-distribution curves have for measurements of the effective cross section, and especially the influence of diaphragm sizes when using direct methods. In this connection we shall briefly dwell on the distinction between measurements of the “diffusion” (Townsend) and effective (Ramsauer)

cross section. For clarity, a few examples will be given.

Measurements with one trap. a) Variation of the sizes of the diaphragms as applied to the one-trap method was carried out by Green^1 for helium, argon, hydrogen, and mercury vapor at electron velocities from 11 to 196 V, and by Palmer^22 for helium and mercury vapor at velocities between 20 and 135 V. As an example let us consider mercury vapor at 82 V, i.e., a case which, in the sense of the relations observed in the usual region of investigation of the effective cross section, should be regarded as an extreme one.

Fig. 45. Explanation of back scattering by electron exchange.
I—without taking exchange into account (Born), II—with scattering taken into account (Oppenheimer). Curves according to Massey and Mohr.

In order to clarify the action of the apparatus with one trap, let us turn to Fig. 31, from which it is seen that for mercury vapor at 82 V the scattering in the angular interval from 0 to 60° is very sharply pronounced, and the smaller maximum lying at large scattering angles is very small in comparison with it. For further reasoning let us idealize this curve, assuming that scattering occurs only in some mean direction, for example at an angle of 30°. In such a case, for the apparatus shown in Fig. 46a, for which, with a narrow diaphragm 3 (not shown in the drawing), the absorbing segment would extend from \(F\) to diaphragm 3, in the presence of a wide diaphragm this segment will turn out to extend only from \(F\) to \(A\), i.e., it will be shorter than it should be. This will occur because electrons scattered at point \(A\) through an angle of 30° will also enter the trap as “unscattered” electrons. Since the true length of the absorption path turns out—

Fig. 46. Diagrams of apparatus for investigations of the effective cross section.

would be smaller than is assumed in calculating \(Q_{\mathrm{eff}}\) by formula (4), § 7; the quantity \(l\) in the denominator will prove to be greater than the true one, and the computed \(Q_{\mathrm{eff}}\) will be smaller than the true value. The numerical magnitude of this diminution of \(Q_{\mathrm{eff}}\) was determined by Palmer for mercury vapor at 82 V with the aid of the idealized relations introduced above. He showed that the experimental dependence of \(Q_{\mathrm{eff}}\) on the dimensions of the diaphragm found by him approximately corresponds to that predicted by the calculations. In contrast to this, Green found no noticeable dependence of the value of \(Q_{\mathrm{eff}}\) on the dimensions of the diaphragm.

b) From considerations entirely similar to those given in item “a,” it follows that, when there is pronounced scattering through large angles, the value of \(Q_{\mathrm{eff}}\), measured with large diaphragms, will prove too large. The validity of these results, which at first sight seem unexpected, can easily be verified by considering Fig. 46b, taking for clarity again the limiting case, namely assuming that all the scattered electrons are scattered through an angle of \(150^\circ\). With a narrow diaphragm 3 (not shown in the figure), the absorption segment would have a length from \(F\) to 3. With a wide diaphragm it becomes longer—equal to the distance between \(F\) and \(A'\), owing to the fact that electrons scattered at point \(A'\) through an angle of \(150^\circ\) are not yet captured by the trap.

From all this it is clear that, when working with a single trap, no absolute significance can be ascribed to the results obtained, especially with regard to the height of the curves, until an exact investigation of the experimental conditions has been carried out.

Measurements with two traps. As the examples given above show, in order to avoid errors in measurements with one trap, it is necessary to have diaphragms of infinitely small dimensions. The same can, however, also be achieved in another way—by using two traps. In Fig. 46 the case of the use of two traps is considered under the same conditions as were considered in “a” (scattering only through an angle of \(30^\circ\)). Diaphragms 2 and 3 must be of the same size.* The numbers of electrons entering both traps \((V + H)\) and only the second \((H)\) are measured. These measurements give us, however, the beam intensities not at points 2 and 3, but at points \(A\) and \(B\), which is quite clear from the figure. The scattering absorption is therefore equal not to the segment \(2 \to 3\), but to the segment \(B \to A\). This distance, however, is exactly equal to the trap length \(2 \to 3\), \(V\), entering into equation (4). Thus, when working with two traps with identical diaphragms, the measured values of the effective cross section are in principle independent of the sizes of the diaphragms. This, of course, is true only insofar as

* When working with two traps and strongly differing diaphragm sizes, the same result is obtained as with one trap. Thus, for example, the strong discrepancy of Brose’s results for hydrogen and nitrogen is explained by the fact that in the first method (1a) he used diaphragms of very different size, whereas in the second (1c) the diaphragm size was the same.

*

since the geometrical ratios for both diaphragms can be kept exactly the same. Difficulties in this direction may be expected in those cases where the number of electrons actually* scattered through very small (or very large) angles exceeds the number scattered through medium angles by an order of magnitude. In the region of velocities below 50 V, such a situation is not encountered.

“Diffusion” and “effective” cross sections. With the aid of considerations similar to those set forth above, one can clarify certain discrepancies between measurements of the effective cross section made by the Townsend method and by the Ramsauer method. Discrepancies are to be observed owing to the essential difference between the principles underlying the two methods. Because of the existence of electron velocity distributions that change rapidly with the electron velocity, not only differences in the height of the curves must be observed, but also shifts of the maxima of the curves along the abscissa axis.

In this way, taking as a basis the effective-cross-section curves obtained by Brüche^16, the distribution of electrons over directions found by Ramsauer and Kollath^65, it is easy to explain the discrepancy between the results of the study of hydrogen by one method and by the other.^68 It should be pointed out, however, that what has been set forth is insufficient for explaining all the discrepancies revealed between the results of investigations by the Townsend method and by the Ramsauer method.

(To be continued)

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