EFFECTIVE CROSS SECTION OF GAS MOLECULES WITH RESPECT TO SLOW ELECTRONS AND IONS*

K. Ramsauer and R. Kollath, Berlin—Reinickendorf

II. Ions**

§ 28. Preliminary remarks.

Until now, in Part I, we have considered interactions between molecules and electrons. We shall now give a survey of what is known about the behavior of molecules with respect to ions of various masses.

Speaking of the effective cross section (e.c.s.) of gas molecules with respect to ions, we mean the cross section obtained by constructing a circle with a radius equal to the sum of the effective radii of the two colliding particles, as was already indicated by us at the end of § 5 of this article. What is new, in comparison with the case of electrons considered above, arises in connection with the fact that between molecules and ions an entirely special kind of interaction proves possible: charge exchange.

This process consists in the ion giving up its charge to a neutral molecule, which as a result becomes an ion possessing the molecular velocity, while the former ion continues on its path as a fast neutral particle. Charge exchange plays an especially large role in the presence of ions with high energy.

As was already indicated in the historical survey (§ 2), among the very large number of ions only very few have been subjected to investigation. In view of this, it seemed advisable to us to consider the whole material not according to methods of measurement, experimental results, character of interactions, etc., but according to groups of ions, and thus to divide the material into three large parts. Within each of these parts, the corresponding works are arranged simply in chronological order. This method of subdivision seemed to us especially convenient because the methods of investigation in the various cases differed greatly from one another, as a result of which, for understanding

* Handb. d. Phys. 2nd ed., vol. XXII, part II, pp. 296–322, translated from the German by N. Khlebnikova.
** Continuation; see Uspekhi fizicheskikh nauk XIV, 1934, and XV, 1935.

in many works a special description of the method employed is necessary. The three groups of works mentioned above relate to slow: 1) hydrogen ions, 2) alkali-metal ions, 3) other ions.

It should be noted that, for the range of velocities considered by us, there are as yet no measurements relating to negative ions.

§ 29. Behavior of gas molecules with respect to slow hydrogen ions.

The first to investigate the mean free path of hydrogen ions in hydrogen was Ayx¹, whose experimental arrangement is shown in Fig. 1.

Fig. 1. Ayx’s apparatus.

The incandescent filament \(F\) is at the same potential as grid \(N_1\) and has a potential approximately 20 V lower than that of grid \(N_2\). Thus the electrons emitted from the filament enter the space between grids \(N_1\) and \(N_2\) with velocities of about 20 V, ionize and dissociate the hydrogen molecules located there (the whole apparatus is filled with hydrogen at a pressure of about \(0.04\) mm Hg). The positive ions thereby produced are drawn out by means of a weak accelerating field beyond grid \(N_3\), and between \(N_3\) and \(N_4\) are accelerated to 25 V. This accelerating voltage of 25 V is sufficient to repel back from \(N_4\) all electrons, while the ions continue to move farther in the field-free space between \(N_4\) and \(N_5\) and partly reach the collecting plate \(P\). This plate has, with respect to \(N_5\), so high a positive potential that only ions which have not undergone substantial changes in velocity and direction can reach it. \(P\) and \(N_5\) can be moved so that the distance between them and \(N_4\) changes. The measurements consist in determining the magnitude of the ion current at \(P\) for various distances between \(P\) and \(N_4\). On the basis of the change in the strength of this ion current with the change in distance, if, in addition, the gas pressure is also known, the mutual effective cross section of the hydrogen molecule and the ion can be calculated. The constancy of the electron emission from \(F\) is checked during the measurements.

For the effective cross section a value is obtained equal to \(20.6\ \mathrm{cm^2/cm^3}\), i.e. approximately equal to the cross section of the hydrogen molecule calculated according to the kinetic theory of gases. Ayx concluded from this that the ions with which he worked were predominantly ions of monatomic hydrogen. We shall see below that this conclusion is incorrect, since in the experiments described \( \mathrm{H_2^+}\) ions and \( \mathrm{H_3^+}\) ions could just as well have been taken into account (see below the work of Holtscher).

Whereas in Ayx’s method it was impossible to separate the different hydrogen ions from one another, Dempster² carried out

this separation by means of a magnetic field. Dempster’s apparatus is shown schematically in Fig. 2. The formation of ions is carried out by two different methods in the part of the instrument lying above diaphragm 1. One of the methods is the same as in Ayx’s apparatus; the second, developed by Dempster in connection with his investigation of isotopes, consists in the following: electrons from the filament \(F\), which is at a negative potential relative to the other parts of the apparatus, bombard a plate \(Li\) made of metallic lithium, thereby producing positive ions, which are then accelerated in the direction of diaphragm 1. The advantage of this second method is that the region of ion formation is more sharply bounded than in the case of gas ionization, and also that by it a larger percentage of protons can be obtained.

The ions obtained acquire between 1 and 2 the required velocity and, after passing through 2, enter a magnetic field perpendicular to the plane of the drawing, which forces them to move in a circle. With a suitable choice of the field strength they pass through diaphragm 3 and fall on the collecting plate \(P\), connected to the electrometer \(E\)*. The gas under investigation continuously flows into the instrument near the middle of the semicircle and is pumped out through a tube located near the source of protons; in this way a certain mobile equilibrium is established. Dempster investigated helium at ion velocities from 14 to 1000 V.

Fig. 2. Dempster’s apparatus

Dempster used the method already described by us (§ 10) for taking velocity-distribution curves with the aid of a magnetic field. The difference between this case and the one described above is that here, in addition to the velocity, the ratio \(\frac{e}{m}\) is a variable quantity, since, generally speaking, ions of several kinds are formed. Keeping the ion velocity constant, Dempster varied the strength of the magnetic field and measured the charge on the plate corresponding to each value of the strength. Plotting the field strength along the axis of abscissae, and along the axis of ordinates the charge of the plate \(P\), he obtained, for a pressure of \(77 \cdot 10^{-4}\) mm Hg, the curve shown in Fig. 3a. The origin of the three principal maxima of this curve is very easily explained with the aid of the well-known relation

* In fact, in order to obtain greater accuracy of measurement, Dempster used a compensation method of measurement, on the description of which we shall not dwell.

between the magnetic-field strength and the square root of the mass, and a control experiment making it possible to establish that one of the maxima belongs to helium ions. This experiment may consist, for example, in taking a curve with helium entirely absent from the apparatus.

In order to investigate the influence of the gas on the ions, Dempster varied the pressure of helium in the apparatus, going over to higher and higher pressures (Fig. 3b—d). The following is observed here.* The He\(^+\) ions are partially absorbed already at low pressures; the H\(_2^+\) ions—at a somewhat greater increase of the pressure, whereas the maximum corresponding to the H\(^+\) ions is retained up to the highest pressures investigated. The broadening of the curve indicates that the action of the gas on the protons consists chiefly in scattering them through small angles. These experiments of Dempster have a qualitative character, but they establish beyond doubt that, for protons, helium atoms in the investigated range of velocities (800—900 V) possess a very small e.m.f.

Fig. 3. Transparency of helium for protons (after Dempster)

Fig. 4. E.m.f. of hydrogen with respect to H\(^+\), H\(_2^+\), H\(_3^+\) (after Golcher).

Golcher\(^3\) determined the mutual transverse cross section of hydrogen molecules with respect to H\(^+\)-, H\(_2^+\)- and H\(_3^+\)-ions. He used apparatus similar to Dempster’s apparatus and carried out measurements with one trap (Brodé’s method, see § 8). Of his results (Fig. 4) the following are especially noteworthy: a) the mutual transverse cross section of hydrogen molecules and H\(^+\)-ions is almost constant in the region from 80 to 800 V; b) the smallest cross section of the hydrogen molecule

* On the question of the first maximum, attributed by Dempster to additional dissociation of H\(_2^+\), see Dempster’s paper.

with respect to H₂ ions is considerably greater than with respect to H₃ ions.

In order to be able to judge the nature of the interactions, Holscher compared the velocity-distribution curves (obtained with the aid of a magnetic field) at various pressures and on this basis drew the following conclusions:

H₃⁺ ions—the transfer of charge is little probable;
  the principal phenomenon is scattering;
H₂⁺ ions—the transfer of charge plays an essential role; it may be that scattering at large angles exists;
H⁺ ions—the transfer of charge is little probable;
  the principal phenomenon is scattering.

Ramzauer, Kollath, and Lilienthal⁴ investigated the effective cross section for various gases with respect to protons in the range of velocities between 30 and 2500 V by a method substantially different from those described in the preceding works (Fig. 5). Protons were obtained by Dempster’s method by bombarding a lithium surface with electrons from the filament \(F\). They were then deflected through \(90^\circ\) by means of the magnetic field inside the magnet \(M\), whereby the beam was made homogeneous. On leaving the magnet, the protons, in the form of a rectilinear beam, entered the measuring apparatus, which consisted, as in the case of electrons, of two traps \(V\) and \(H\). The calculation

Fig. 5. Apparatus of Ramzauer, Kollath, and Lilienthal.

of the effective cross section was carried out according to equation (4), § 7, on the basis of measurements of the intensity with the traps connected together and with only the trap \(H\). No retarding fields were applied to the trap. Therefore protons that had undergone a change of speed without a simultaneous change of direction were recorded as if they had not been acted upon at all. He, Ne, Ar, H₂, and N₂ were investigated.

Fig. 6. Effective cross section of argon atoms with respect to protons (after Ramzauer, Kollath, and Lilienthal).

As an example of the results, Fig. 6 gives the curve for argon. Along the abscissa is plotted the velocity of the protons in \(\sqrt{V}\), along the ordinate—the reciprocal effective cross section in \(cm^2\,cm^3\) at \(1\) mm Hg and \(0^\circ\)C. On the right-hand side of the drawing, the line marked by the letter \(G\) indicates the magnitude of the gaskinetic cross section of the argon molecule,

As the proton velocity of the effective cross section increases, it first falls rapidly, reaching approximately half the gas-kinetic cross section, and then rises again to roughly three times the gas-kinetic value. Measurements corresponding to velocities above 40 V cannot be regarded as quantitative (shown by a dotted line), because this region lies at the limit of applicability of the apparatus.

Fig. 7 gives an idea of the results obtained for all five gases investigated. In all cases a rise of the curve after the minimum is observed (for the apparent exception for helium see below, Dempster’s work). There is apparently a connection between the positions of the minima and the values of the ionization potentials for all gases; the physical meaning of this is still unclear. Remarkable is the extremely small value of the effective cross section for helium, which

Fig. 7. Effective cross sections of various gases with respect to protons (according to Ramsauer, Kollath, and Lilienthal).

Fig. 8 and 9. Apparatus for investigating charge transfer (according to Goldman).

at velocities above 100 V is a full order of magnitude smaller than the gas-kinetic value. This fact very well confirms Dempster’s qualitative conclusions (see above) on the behavior of helium with respect to protons.

On the basis of an investigation of the character of the interaction between gas molecules and protons, the authors established that the course of the curve on either side of the minimum is due to two completely different processes. The rise toward low velocities is a consequence of the scattering that arises, while the rise toward high velocities is a consequence of the presence of a large number of slow particles. At the same time, the question remains open whether these particles arise as a result of charge transfer or as a consequence of large losses of proton velocity.

In Goldman’s work⁵ the interaction between hydrogen and argon molecules and protons was investigated. In Figs. 8 and 9 is shown

EFFECTIVE CROSS SECTION OF GAS MOLECULES

Scheme of the measuring apparatus. The protons were obtained by ionizing hydrogen with thermoelectrons (filament \(F\)); to increase the intensity, a special “nozzle” was used, which proved to be a very effective device. Ultimately the proton beam, through diaphragms 5 and 6, enters the measuring chamber, consisting of a system of plates, of which the measuring plate is \(P_M\) (see also Fig. 9). Protons that have not been acted upon fly past the plates and are caught by cylinder \(H\), connected, like \(P_M\), to an electrometer. Each of the plates can be charged, independently of the others, to the desired potential.

The most essential result of the work is the establishment of the fact that, in the investigated range of velocities (from 400 to 4000 V), charge transfer predominantly takes place, and not ionization of molecules by protons. This is illustrated by the direct results of the measurements shown in Fig. 10, where

Fig. 10. Dependence of the positive and negative saturation current on pressure (according to Goldmann).

Fig. 11. “Cross section of charge transfer” for argon and hydrogen with respect to protons (according to Thomson).

the gas pressure is plotted along the abscissa, while along the ordinate is plotted the charge \(i\) of plate \(P_M\), referred to unit intensity \(J\), both for the positive (\(i^{+}\)) and for the negative (\(i^{-}\)) current at \(P_M\). The potential of the plate was then \(\pm 10\ \mathrm{V}^{*}\).

As the pressure increases, only the number of positive particles in the space between the plates increases; the number of negative ones (electrons produced as a result of ionization) does not depend on the gas pressure. This indicates that the weak negative charge of the plate, already observed in vacuum, is not due to the presence of gas, and that the positive particles are formed only by charge transfer, not by ionization.

The apparatus also permits quantitative investigations. The measured charge-transfer cross sections for argon and hydrogen are presented in Fig. 11 as a function of the velocity of the proto-

* Special measurements established that, at a potential of 10 V, saturation is almost reached.

ones, expressed in volts. It is easy to see that the values obtained have the same order of magnitude as those found by Ramsauer, Kollath, and Lilienthal for the effective cross section. (A comparison of all measurements relating to protons is given in Fig. 14 at the end of this paragraph.)

Thomson’s experiments\(^6\) concerned the scattering of fast (4000–26,000 V) protons and thus lie outside the region to which we have decided to confine ourselves. We nevertheless cite this work, as well as the following one—by Dotepl, who also worked with fast protons (2500–2000 V)—because they represent a transition to studies of fast canal rays,* and also because Thomson was the first to succeed in discovering an interesting analogy between the behavior of molecules with respect to electrons and protons.

Fig. 12. Scattering of protons (according to Thomson).

Thomson obtained protons in a canal-ray tube. By means of an electric field the homogeneity of the stream was established; it was then directed onto a diaphragm that cut out a narrow beam of protons. At some distance from the diaphragm a photographic plate was placed, on which the beam produced a black spot. The broadening of the spot when gas was admitted into the apparatus was measured photometrically. In this way the angular distribution of protons in vacuum and in gas was determined at various velocities. Choosing on the distribution curve obtained some abscissa and determining for it the difference in blackening in gas and in vacuum, one obtains the relative measure of scattering at the corresponding point—naturally at a very small angle. The relative scatterings found in this way for an unchanged pressure are plotted along the ordinate axis as a function of the proton velocity, plotted along the abscissa axis (Fig. 12). Whereas for hydrogen a continuous increase in scattering is observed as the proton velocity decreases, helium and especially argon reveal scattering maxima at certain definite velocities. Thomson pointed out that the position of these scattering maxima of protons by helium and argon almost coincides with the position of the maxima of effective cross section for electrons, if both curves are constructed by plotting on the abscissa axis the values of the linear velocity of electrons and protons.

According to a preliminary communication, Dotepl\(^7\) investigated the—

* See also the work of Bartels (H. Bartels) on charge transfer by protons in hydrogen at velocities between 4000 and 30,000 V.

...charge transfer by protons in helium at velocities from 2500 to 30,000 V. Q_H found a maximum in the number of transfers lying at a proton velocity of \(1\cdot 10^8\ \text{cm/sec}\) (Fig. 13), without indicating the absolute value of the maximum. His results constitute an interesting supplement to the measurements of Ramsauer, Kollath, and Lilienthal, who found a decrease of the e.c.s. within the range up to 50 V. Thus it turns out that helium, with respect to protons, is not an exception, but that charge transfer occurs in this gas only at considerably higher proton velocities.

Fig. 13. Charge transfer by protons in helium (after Dopel).

In conclusion of this paragraph, in Fig. 14 we give a comparison of the results of measurements of the mutual e.c.s. of hydrogen molecules and protons, because, with the exception of Dempster*, this case was investigated by all the authors mentioned above. In considering this figure one should remember that Goldmann’s curve \(^{5}\) refers only to charge transfer, while Thomson’s curve \(^{8}\) refers only to scattering at small angles. Holtsher’s curve \(^{3}\) is entirely eliminated, since it is in contradiction both with the data of Ramsauer, Kollath, and Lilienthal \(^{4}\), and with Goldmann’s results \(^{5}\), and the reasons for the discrepancy have not yet been clarified. Thomson’s curve \(^{8}\) should not be regarded as a continuation

Fig. 14. E.c.s. of hydrogen with respect to protons (according to literature data).

of the curve of Ramsauer, Kollath, and Lilienthal \(^{4}\), since the absolute height of its ordinates is completely arbitrary. This curve shows, nevertheless, that the e.c.s. curve, as the velocity increases, again begins to fall.**

* If one does not count certain qualitative measurements which showed the behavior of hydrogen with respect to protons analogous to the behavior of helium.

** For momentum transfer this was proved by Bartels (see Appendix).

§ 30. Behavior of gas molecules with respect to slow ions of alkali metals. The aim of the work of Ramsauer and Beesk[^9] was to measure the mutual effective cross sections of molecules of various gases and slow (velocities from 1 to 30 V) ions of alkali metals. The method of measurement was the same as that also used for electrons (the magnetic method with two traps; see § 9, Fig. 12). The source of ions was a platinum strip coated in a special way with an amalgam of one or another alkali metal.

Fig. 15. Mutual transverse cross section of argon and ions of alkali metals (after Ramsauer and Beesk).

All curves of the effective cross section for argon, regardless of the kind of ions, exhibit the same character (Fig. 15), falling, as the velocity increases, at first rapidly, then more and more slowly. The curves for all the other gases investigated are of the same form.

If one compares the effective cross sections of different gases for the same kind of ions (for example K), having a velocity corresponding (according to Fig. 15) to an approximately constant value of the effective cross section, and represents the obtained results in the form of a sum of radii, then very interesting relationships are obtained: the differences between the obtained values of the sums of radii and the theoretical value of the radius of the K-ion (according to Herzfeld and Grimm) differ from the gas-kinetic radii of the gas molecules the more, the larger (gas-kinetically) the molecule is (Fig. 16).

Garnwell carried out some investigations of a qualitative character concerning the motion of Cs+- and K+-ions in helium, neon, argon, hydrogen, and nitrogen. His apparatus was of the same construction as the Eijk apparatus described in § 28. Garnwell found that the mean free path has a value of the same order as that calculated according to the kinetic theory of gases, whereas the losses of velocity observed experimentally prove to be much smaller,

than might have been expected on the basis of the usual ideas about the collision mechanism.

The works cited below belong to researchers of Dempster’s school. They must therefore be regarded as parts of a certain whole.

With the aid of Dempster’s apparatus, described in § 28 (Fig. 2), Durbin¹¹ investigated potassium ions with velocities from 8 to 350 V in He, Ar, H₂, N₂, O₂, and air at pressures from 0 to \(150 \cdot 10^{-4}\) mm Hg. Durbin measured the charge of the collecting plate \(P\) as a function of the gas pressure.

Fig. 16. Discrepancies between experimental and theoretical values of sums of radii (after Ramsauer and Beeck).
Labels in the figure: molecular radius (gas-kinetic); sum of radii (experimental); ion radius (according to Herzfeld and Grimm); He; Ar; N₂; K⁺.

In order to make his results comparable with the results of other works, we shall express them in terms of e.c.s. As is seen from Fig. 17, Durbin obtained curves similar to those of Ramsauer and Beeck. In doing so he assumed that, by extrapolating the found e.c.s. values to an ion velocity equal to zero, for the e.c.s. of molecules of various gases one obtains quantities equal to the gas-kinetic ones, if the latter are calculated using, instead of the cross section of the K-ion, the cross section of the argon atom. This conclusion, which seemed plausible when Durbin’s results were plotted (with volts on the abscissa axis and the ratio of the experimental value of the free path length to its value calculated by the indicated method on the ordinate axis), ceases to appear correct with the method of plotting adopted here (abscissae—\(\sqrt{V}\), ordinates—the mutual e.c.s.). Moreover, it proves to be at variance with the results of Ramsauer and Beeck, which lead to considerably smaller velocity values, and it no longer appears in the later works of Dempster’s school.

Fig. 17. Mutual cross section of K⁺ ions and molecules of various gases (after Durbin).
Labels in the figure: mutual cross section; cm²; velocity of ions in \(\sqrt{V}\); O₂; air; N₂; Ar; H₂; He.

Using an entirely identical apparatus, Kennard¹² investigated Na⁺, Rb⁺, and Cs⁺ ions in He, Ar, and H₂. He studied the free path length and, chiefly, the character of the action of molecules on ions. For this purpose he used the method already described in its application to electrons (§ 10), based on measu-

in the curves of distribution by velocities caused by the presence of gas. To clarify the conclusions drawn by the author, we must consider the influence on the form of the curve of a new kind of interaction—charge transfer.

Kennard distinguishes three limiting cases of change in the form of the curve, shown schematically in Fig. 18:

a) A decrease in the area enclosed between the distribution curve and the abscissa axis. Only charge transfer is present.

Fig. 18. Change in the form of distribution curves.

b) A shift of the curve without change in its form or area toward lower magnetic-field strengths. Only loss of velocity is present.

c) Broadening of the curve without change in area. Only scattering through small angles is present.

Kennard’s results, illustrated by Fig. 19, are as follows:

Fig. 19. Loss of velocity (left) and charge transfer (according to Kennard).

1) Cs\(^+\) ions in H\(_2\) and He. The interaction is expressed mainly in loss of velocity.

2) Cs\(^+\) ions in Ar (35 and 90 V). The process consists mainly in charge transfer. Scattering through large angles possibly exists.

3) Na\(^+\) ions in H\(_2\) (445 V; not shown in Fig. 19). All three kinds of interaction are observed—charge transfer, loss of velocity, and scattering through small angles.

Thus heavy ions in light gases behave quite differently from heavy ions in heavy gases.

EFFECTIVE CROSS SECTION OF GAS MOLECULES

Kennard’s work is supplemented by Cox’s investigations13, concerning light ions in heavy gases; moreover, an extreme case was chosen where possible—lithium ions in mercury vapor. Cox’s apparatus differed somewhat from those described so far with respect to the device for trapping ions. When the cylinders \(H\) and \(V\) (Fig. 20) were connected, it was possible to record velocity-distribution curves, as Kennard did. In addition, it was possible to measure the e.m.f. by the straight-beam method. Both kinds of measurements were made for ions with velocities from 25 to 250 V.

Fig. 21 shows the distribution curves at different gas pressures. Plotting the dependence of the areas bounded by the curves on the pressure, we obtain a decrease according to an exponential law, the exponent at \(e\) being the reciprocal e.m.f. at the corresponding ion velocity. Fig. 22 shows the values of the e.m.f. obtained by the two-trap method with a straight beam (\(\bullet\bullet\)); for comparison, the values found by the above-mentioned method are given (\(\circ\circ\)).

On the basis of the results obtained, Cox concluded that Li\(^+\)-ions, when passing through mercury vapor, do not undergo a loss of velocity (absence of a shift of the curves). From the strong discrepancy between the e.m.f. values obtained by the two methods, the conclusion is drawn that the principal process is scattering at small angles.

Fig. 20. Measurement of the e.m.f. according to Cox.

Fig. 21. Velocity-distribution curves for Li\(^+\)-ions in mercury vapor (according to Cox).

Fig. 22. Cross section of mercury-vapor molecules with respect to lithium ions for different methods of measurement (according to Cox).

With an apparatus analogous to the one just described, Thomson14 carried out the following measurements:

1) Recording of velocity-distribution curves by the magnetic method (as above).

2) Measurement of the e.m.f. with the aid of two traps (as above).

3) Recording of curves with the application of a retarding potential in traps 1 and 2 connected together.

The investigations were carried out with He and H\(_2\) (light gases) and Li\(^+\) (a light ion)

and Cs$^+$ (heavy ion) at ion velocities from 5 to 500 V. The results obtained confirm Kennard’s results: Cs$^+$ ions in He undergo predominantly only loss of velocity. The decrease in the velocity of the beams was found by Thomson not only in the form of a shift of the distribution curves, but also as a change in the shape of the retarding-potential curves (Fig. 23).

Fig. 23. Retarding-potential curves for Cs$^+$ ions in helium (after Thomson).

Helium behaves quite differently with respect to lithium ions. On the basis of the retarding-potential curves, as well as from the change in the shape of the velocity-distribution curve, one may conclude that in this case only scattering without loss of velocity occurs. The results of measurements of the effective cross section relating to Li$^+$ ions in He are shown in Fig. 24. Curve I represents the results obtained with a rectilinear beam with two traps; curve II represents the change in the area of the velocity-distribution curves. A comparison of both curves in Fig. 24 reveals a strong influence of the measurement method on the value of the effective cross section obtained, which is regarded by the author as evidence that Li$^+$ ions in He undergo only scattering through small angles.

When the points obtained by Ramsauer and Beeck (··) for Li$^+$ in He are plotted in Fig. 24, they form, as it were, a continuation of curve I. Thomson explains this remarkable agreement, as well as the low position of curve I, by the use of wide diaphragms. This does not seem correct, since, as was found in § 27, the two-trap method with diaphragms of equal width gives the same results as the one-trap method with a very narrow diaphragm. On this basis one should expect agreement of the results of Ramsauer and Beeck with curve II.

Fig. 24. Mutual cross section of helium atoms and Li$^+$ ions according to Thomson and Ramsauer–Beeck.

In conclusion, we shall give a brief survey of the results of measurements with

ions of the alkali metals. All investigators found that the effective-cross-section curves decrease as the ion velocity increases—at first rapidly, then more slowly. Between the absolute heights of the effective-cross-section curves obtained by different authors there are considerable discrepancies, which can partly be explained by scattering at small angles.

The character of the interaction depends strongly on the atomic weight of both colliding particles: light ions exhibit, both in light and in heavy gases, only scattering without loss of velocity; heavy ions in heavy gases undergo charge exchange, and in light gases—loss of velocity.

§ 31. Other ions. The information available on the behavior of gas molecules with respect to other ions, apart from protons and ions of the alkali metals, was obtained chiefly by Kallmann and Rosen¹⁵, who worked with very diverse ions, including ions carrying several (2) charges, for a rather narrow range of velocities. The following table indicates the ions and gases investigated.

Initially Kallmann and Rosen used apparatus essentially identical with the Dempster apparatus described in § 29 (Fig. 2). The ions were obtained by ionization of the corresponding gas. In most cases the ion velocity was equal to 400 V. The decrease in the intensity of the ion beam was measured as a function of the gas pressure. The principal result of these investigations was as follows: the increase in the decrease of intensity (absorption) with increasing pressure is the greater, the closer the neutralization energy of the ion lies to the ionization potential of the gas under investigation.

In addition to the measurements indicated, the same authors investigated charge transfer, using for this purpose the apparatus shown schematically in Fig. 25, whose special feature was a ring \(R\) arranged around trap \(A\). Trap \(A\) served for measuring the intensity of the ion beam, while ring \(R\) served to determine the number of slow ions formed by the beam during its passage between diaphragm 4 and the trap. These slow ions carry a positive charge and are formed as a result of transfer

charge by ions belonging to the beam. That the slow ions are not scattered ions of the beam was established by special control experiments using a somewhat modified apparatus. The charge-transfer cross section calculated on the basis of these measurements has a value approximately equal to the gas-kinetic one. Since in these experiments scattering plays almost no role, the total effective cross section is the effective cross section for charge transfer.

Penning and Venemans[^16] compared the passage through argon of \(\mathrm{Ar}^{+}\) and \(\mathrm{K}^{+}\) ions. In their apparatus, shown schematically in Fig. 26 and very similar to Eich’s apparatus (§ 29, Fig. 1), argon ions were produced by means of a gas discharge, and potassium ions by means of a Kunsman anode, which could be placed in the position \(F—N_{1}\). The ions were produced by electron impacts in the space between \(N_{1}\) and \(N_{2}\); between \(N_{2}\) and \(N_{3}\) they were accelerated (160–200 V) and then entered the retarding field between \(N_{3}\) and \(P\).

Fig. 25. Investigation of charge transfer by ions (after Kalman and Rosen).

Fig. 26. Measurements of charge transfer according to Penning and Venemans.

From the retarding-potential curves it may be concluded that \(\mathrm{Ar}^{+}\) ions lose considerably more energy in argon than do \(\mathrm{K}^{+}\) ions. The authors consider the cause of this to be charge transfer, which is stronger for \(\mathrm{Ar}^{+}\) in Ar. The measurements performed make it possible to calculate the cross section for charge transfer for \(\mathrm{Ar}^{+}\) in Ar; according to the authors cited, it is equal to 0.8 of the gas-kinetic cross section, i.e. approximately \(74\ \mathrm{cm}^{2}/\mathrm{cm}^{3}\). (For a comparison of this value with the results of other measurements, see the following paper.)

In his first paper Wolf[^17] measured, by means of an apparatus with a rectilinear beam and two traps, the mutual cross section of argon and \(\mathrm{Ar}^{+}\) at ion velocities from 25 to 900 V. Without dwelling on this, we shall pass to Wolf’s second paper, whose aim was to investigate charge transfer and whose results include those of the first and, in addition, give quantitative information concerning the relation between various kinds of mutual-

actions of ions and molecules. Wolf’s apparatus is shown in Fig. 27. The \(Ar^+\) ions are obtained by means of a gas discharge and are freed from all other ions by the magnetic field \(M\). Having passed through diaphragms 1, 2, and 3, the beam enters the measuring device. At a certain constant gas pressure one measures, on the one hand, the total number of ions that have passed through 3, and, on the other, the number of them that have reached the ring \(R\). In this case a negative potential of such magnitude is applied to \(R\) that all ions (saturation curves) formed as a result of charge transfer in the space \(S\) between \(N_4\) and \(N_5\) reach the ring. After this the same measurements are repeated for another pressure. From two values of the intensity, the pressure, and the distance between \(N_4\) and \(N_5\), the cross section for charge transfer is calculated.

Fig. 27. Measurement of charge transfer and ionization according to Wolf.

With the same apparatus it is possible to measure the ionization cross section. For this purpose a positive potential of such magnitude is applied to the ring \(R\) that the ions formed as a result of charge transfer can no longer reach it, and all electrons produced owing to ionization of molecules in \(S\) are attracted to the ring.

Fig. 28. Cross section for charge transfer and ionization cross section of argon atoms with respect to \(Ar^+\) ions (according to Wolf).

Wolf’s measurement results are shown in Fig. 28. The charge-transfer cross section \((\times — \times — \times)\) has a maximum at 6 V and a minimum at 21 V, and then slowly increases as the velocity rises. The magnitude of this cross section is at all times less than twice the cross section of the neutral argon atom, calculated according to kinetic theory; the latter value is shown in Fig. 28 by the line on the right. From the course of the curve of the ionization cross section \((\circ — \circ — \circ)\) it is easy to see that ionization of argon by \(Ar^+\) ions begins only at velocities of the latter greater than 300 V, and slowly

increases with increasing ion velocity. At the end of the investigated velocity interval (about 1000 V), the ionization cross section is still only about 10% of the charge-transfer cross section.

For comparison of Wolf’s results\(^{18}\) with the results of other authors, in the same Fig. 28 separate points from the measurements of Kalman and Rosen\(^{19}\) for a velocity of about 400 V (◎) and of Penning and Venemans\(^{16}\) (◎) for a velocity of \(\sim 200\) V are plotted. In making this comparison it must be borne in mind that both Kalman and Rosen, and Penning and Venemans, are not very certain of the quantitative reliability of their results. This comparison is intended only to show that some authors obtain values of the charge-transfer cross section fairly close to the gas-kinetic cross section.

Addendum

The works that appeared in the interval of time between the writing of the article and the receipt of the proofs we divide into three groups: experimental works on electron scattering, theoretical works on the same question, and works on ion scattering, and we briefly report on them in alphabetical order by author.

1. Experimental works on electron scattering. Hughes and McMillen\(^{20}\), in two papers, investigated the scattering of electrons by hydrogen and helium molecules at velocities from 25 to 700 V in the angular interval from 0 to 170°, using their previous method (see § 22). The angular-distribution curves obtained by them show, in the case of helium—up to 100 V, and in the case of hydrogen—up to 200 V, a minimum of scattering for angles close to 90°. In hydrogen, moreover, for velocities between 35 and 50 V a scattering maximum is observed at an angle of 155°. In addition, the same authors studied, as in their earlier work, the angular distribution of electrons scattered with loss of velocity, as well as of electrons knocked out of atoms in the process of ionization.

Tate and Palmer\(^{21}\) studied the scattering of electrons with velocities from 80 to 700 V by mercury-vapor molecules in the angular interval from 10 to 130°. Their apparatus was similar to the first of Harnwell’s arrangements\(^{22}\): a movable incandescent filament—a fixed trap. The velocity distribution was determined by the retarding-potential method. With the application of strong retarding fields it is possible to investigate the velocity distribution only of elastically scattered electrons. The results in this part agree well with Arnot’s data. In addition, as in Hughes and McMillen, the velocity distribution of electrons scattered with loss of velocity and of ionization electrons was investigated. On the basis of their data the authors made quantitative determinations, i.e., they calculated the total e. p. c., the excitation cross section, and the ionization cross section. The results of the calculations agree satisfactorily with the data of Brode (e. p. c.) and of Smith (ionization cross section).

2. Theoretical work on electron scattering.

Feenberg²³ showed that the wave equation hitherto used for calculating the effective cross section is an approximate equation for the case of many electrons. Further, in the case of scattering, exchange of electrons plays a very small role, since allowance for this phenomenon affects only the first approximation. Feenberg gave an approximate formula for the probability of scattering which, at high electron velocities, coincides with the first approximation of Born’s formula. In a note added in proof Feenberg reports that the results of numerical calculations give good qualitative agreement with experimental data also for the slowest electrons.

Henneberg²⁴, using the method of Faxén and Holtsmark²⁵, calculated, on the basis of the Fermi potential distribution, the scattering of electrons with velocities from 135 to 800 V by mercury-vapor molecules. The agreement with experiment (Arnot, Pearson, and Arkvist), especially for high velocities, is good.

Massey and Mohr²⁶, in their first paper, applied the Born and Oppenheimer theory to collisions between electrons and molecules. The scattering curves in hydrogen and nitrogen coincide with the experimental data only for high electron velocities (Born’s theory was applied only to this case). For velocities \(< 50\ \mathrm{V}\) in hydrogen and \(< 30\ \mathrm{V}\) in nitrogen, discrepancies begin. In a second paper by the same authors²⁷, in calculating by Born’s method, electron exchange is taken into account in the first approximation. The results obtained are the more reliable, the smaller the role played by exchange. The authors draw attention to the fact that deviations of the scattering function from the plane-wave equation and the exchange effect act, generally speaking, in opposite directions, which is the reason for the apparent applicability of Born’s theory to low electron velocities. In this work they succeeded not only in explaining from the qualitative side (as had been done earlier) the “backward” scattering in hydrogen, but also in indicating the value of the velocity at which “forward” scattering changes into “backward” scattering.

Stier²⁸ applied Holtsmark’s scattering theory to the case of a symmetrical diatomic molecule, taking, instead of a finite field with spherical symmetry, a finite ellipsoidal one. With the aid of these representations he was able to calculate the position of the sharp maximum of the effective cross section in nitrogen and to construct scattering curves for nitric oxide in the velocity interval from 0.5 to 2 V, in good agreement with the experimental curves.

3. Work on ion scattering.

The only work in this field belongs to Bartels²⁹, who investigated charge transfer by protons in hydrogen in the interval of potential differences from 4 to 30 kV, thus throwing a bridge from studies of this phenomenon with fast protons to studies for the case of low velocities (in particular, Goldmann’s work). He found (with good agreement of his results with the results—

with the data of other authors in regions already investigated), there is a maximum of charge transfer, lying at about 7000 V, where the cross section for charge transfer in hydrogen at a pressure of 1 mm Hg proved to be equal to 50 cm\(^2\)/cm\(^3\).

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