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

K. Ramsauer and R. Kollath
Berlin, Reinickendorf

§ 1. Introduction. The present article deals with the effective cross section (e. c. s.) of molecules with respect to slow electrons and ions. It is, of course, not possible to draw a sharp boundary between “slow” and “fast” electrons and ions. Nevertheless, the history of the development, the experimental technique, and the nature of the processes make it possible to single out a certain more or less definite region of “slow” electrons and ions.

If one takes the experimental technique as the criterion, i.e., the apparatus used in the investigation, then the upper limit of velocities for slow electrons should be regarded as the velocities corresponding to accelerating voltages of several hundred volts, and for ions of several thousand volts. From the point of view of the mechanism of the phenomena, however, this boundary proves to be too high. Therefore a large part of the material presented here relates to electrons having velocities not above 50 V; the limit of the accelerating voltage for ions lies at about 1000 V.

§ 2. Historical survey. Electrons. The cross section of gas molecules with respect to slow electrons was first measured by Lenard² in 1903. He established that, as the velocity of the electrons decreases, the size of the molecules increases, though not beyond a certain limiting value (Fig. 1). Repetition of these measurements by Robinson³ and extension of the range of investigated velocities, carried out by F. Mayer⁴, yielded nothing new. It should be noted here that the smallest voltage in Lenard’s experiments was 4 V, in Robinson’s 3.2 V, and that for both authors the number of individual measurements was insufficient to form an idea of the details—

* Handb. d. Phys., 2nd ed., Vol. XXII; translated from the German by N. Khlebnikov.
** Works that appeared between March 1 and mid-August 1931 are briefly reviewed in the addendum at the end of the present article.

stages of the course of the curve. These works were followed by Okesson’s qualitative experiments,^5 which revealed the presence of a maximum in the molecular cross section for certain noble gases at definite electron velocities. These experiments, however, were not sufficiently convincing.^6 Mayer,^7 for example, believed that, by increasing the number of points on the curves for the gases he investigated, he would be able to prove that, as the electron velocity decreased, the molecular cross section tended toward some definite value.

In 1920 Ramsauer^8 discovered that, at an electron velocity of 1 V, the argon atom has an unexpectedly small cross section, which decreases still further as the velocity is reduced. These results were soon confirmed

Fig. 1. Absorption cross sections (after Lenard, 1903).

by F. Mayer, who showed at the same time that the maximum of the cross section for argon, the presence of which could have been expected on the basis of the measurements of Lenard and Ramsauer, lies near 12 V. Almost at the same time, on the basis of his studies of electron diffusion in less rarefied gases, Townsend^9 concluded that there exists an “anomaly of the cross section” of the molecules of certain noble gases. Then, together with Bailey,^10 he found an extremely small value of the cross section of argon atoms at velocities of the order of 1 V, and somewhat later—the existence of a minimum of the cross section at still smaller velocities.^11 These works were followed by Ramsauer’s investigations of all the noble gases^12 and by the study of the noble gases by Townsend’s school.^13,14 Like Okesson, Townsend and his collaborators did not find any approach of the molecular cross section to a limiting value as the electron velocity decreased; nevertheless, these experiments also could not yet refute the results of F. Mayer’s direct measurements.

A significant step forward was provided by the measurements of Brode and Rusch. Brode^15 discovered the “argon-like” character of CH$_4$, as well as a sharp rise

curves for hydrogen and nitrogen in the region of low electron velocities. Ramsauer[^16] observed a drop in the curve for hydrogen at velocities below 1 V, as well as a minimum of the cross section for argon and krypton. Then Brode[^17] obtained complete cross-section curves for nitrogen and hydrogen molecules (two and one maxima, respectively), and thereby definitively refuted the above-mentioned results of Mayer, according to which, as the electron velocity decreases, the cross sections of the molecules of both gases should tend toward limiting values. In addition, Brode[^18] and Brode[^19] studied a considerable number of noble gases, which also exhibited the “cross-section anomaly.” Brode opened up an entirely new field, being the first to undertake a systematic study of the dependence between the effective cross section of molecules and their structure.[^20]

These works were followed by an expansion of the field of investigation toward lower electron velocities (below 1 V)—the work of Ramsauer and Kollath[^21], then the study of the “fine structure” of the cross section for various gases, carried out by Normand[^22], and, finally, the continuation of Brode’s work—a detailed investigation of complex organic compounds by Schmeider[^23] and, more recently, by Holst and Holtsmark[^24].

Simultaneously with the works whose aim was the investigation of the molecular cross section, studies of another kind were also carried out—directed toward elucidating the mechanism of the action of molecules on electrons. Already Lenard, in his fundamental work, attached great importance to a strict distinction between absorption, diffusion, and loss of electron velocity. In his measurements of absorption he used apparatus that made it possible deliberately to exclude the influence of diffusion and loss of velocity. It was clear to Lenard that the measured “absorbing cross section” represents two phenomena, which at that time had not yet been separated experimentally, namely: “true absorption,” when the velocity of the electron is brought in magnitude and direction to the velocity of the molecule as a result of a single collision, and “apparent absorption,” when, as a result of a single collision, the electron is deflected from the direction of the beam; moreover, unlike diffusion, all directions of deflection are equally probable. In contrast to Lenard’s concept of the “absorbing cross section,” Ramsauer introduced the concept of the “effective cross section” of a molecule, i.e., the “total cross section” in the sense that it characterizes a molecule acting on an electron in any manner: absorbing it, reducing its velocity, deflecting it, or reflecting it.[^25]

As a result of further investigations it became clear that, in the case of slow electrons, acts of “true absorption” must be regarded as an exception (Franck and Hertz[^26], Townsend and co-workers[^27], Loeb[^28], Wahlin[^29]); in exactly the same way, diffusion and loss of velocity recede into the background. The principal phenomenon proves to be the deflection of electrons (Lenard’s “apparent absorption”), pro-

emerging at velocities less than the excitation potential without a change in velocity, and at greater velocities—possibly accompanied (but not necessarily) by a change in it. Recently, what has mainly been studied is the distribution over directions of these deflected (scattered) electrons (Dymond ^30, Arnot ^31, Bullard and Massey ^32, Ramsauer and Kollath ^33, and others ^34).

Ions. Whereas for electrons there already exists a fairly complete body of experimental material, the investigation of the interaction between gas molecules and ions is still in its initial stage. The reason for this is that for many ions work with any single one is not of the general interest that investigation has in the case of electrons (what has been said, of course, does not apply to protons).

Fig. 2. Diagram of the action of a molecule on electrons.

A considerable range of velocities is covered by the investigations on protons by Dempster ^35, Ramsauer—Kollath—Lilienthal ^36 and Goldmann ^37, by the work of Dempster and his collaborators ^38, and also by Ramsauer—Beck ^39 with ions of the alkali metals. Investigations with other ions, with the exception of the most recent work of Wolf ^40, such as, for example, the numerous measurements of Kallmann and Rosen ^41, relate only to definite values of the velocities.

With respect to ions, investigations are also being carried out of various kinds of interaction with molecules.

§ 3. Effective cross section. The meaning of the expression “effective cross section of a gas molecule with respect to electrons” is clarified by Fig. 2. In a beam, the individual electrons of which have the same direction and equal velocities, there is a gas molecule. In this case there will be electrons in the beam passing so far from the center of the molecule that neither the magnitude nor the direction of their velocity is changed under the influence of its force field. On the other hand, the velocities and directions of motion of some electrons which have come sufficiently close to the center of the molecule will be changed by the action of its force field. We shall call the “effective cross section of a molecule with respect to electrons” the area situated around the center of the molecule, the plane of which is perpendicular to the direction of the beam, through which an electron must pass in order that its velocity or direction of motion be changed. This definition was originally understood empirically, i.e., with the self-evident limitation concerning the limit of the changes measurable with the given apparatus.

Adopting this definition, one cannot assert that the cross-

cross section can have a definite value. According to the notions of classical theory, generally speaking, there is no such electron which would not experience the action of the field of the molecule, however far from it it might pass.

These notions proved to be incorrect, since, contradicting the results of experiments, they predicted a continuous strengthening of the action of the molecule on the electron as the velocity of the latter decreased. In contrast to the classical theory, quantum theory asserts that the number of electrons subjected to the action of a single molecule is finite. Hence follows the definition of an effective cross section that does not depend on the properties of the measuring apparatus.

Nevertheless, one may expect difficulties in measurements. It might be thought, for example, that some definite change in the direction of an electron could be taken into account by one device and remain undetected by a coarser one.

Fig. 3. (a and b). Toward the derivation of the absorption law.

As we shall see below, for the region of slow electrons these difficulties practically do not make themselves felt, and moreover, measurements with apparatuses having different resolving power lead, in general, to identical results.

§ 4. The fundamental experiment and the law of absorption. To calculate the effective cross section from experimental data we must establish certain quantitative relations. For this purpose let us imagine a homogeneous beam of definite intensity (with respect to the directions and velocities of the electrons forming it), propagating in a gas under some pressure (Fig. 3). The electrons of the beam collide with gas molecules, as a result of which changes occur in the velocities or directions of motion of the electrons. If we have an arrangement that allows us immediately to exclude from the beam every electron that has undergone a collision, the number of electrons will decrease from left to right (Fig. 3a). The law of decrease of the beam intensity can easily be obtained on the basis of the considerations on the interaction of molecules set forth by Clausius^44.

In our derivation we shall strive above all for clarity, and therefore carry out the calculations in a somewhat simplified way, taking the cross section of the molecule to be unchanged. In a more rigorous derivation it is necessary to use the formulas of kinetic theory, proceeding, for example, from consideration of the mean free path.

Let us imagine the part of space between the planes \(l_0\) and \(l_x\) divided into a large number of layers of equal thickness, perpendicular to the direction of the electron beam. Let the number of layers falling on each centimeter be \(n\). We imagine the layers, on the one hand, to be sufficiently thick for us to have the right to consider the number of molecules in them the same, and on the other hand—so thin that \(n\) is a large number. Both conditions can be satisfied simultaneously, since even at the highest rarefactions employed the number of molecules in \(1\ \mathrm{cm}^3\) is still very large. Thus, for example, at \(p = 10^{-5}\ \mathrm{mm}\ \mathrm{Hg}\) it is equal to \(10^{10}\).

Let us now consider one of these layers in the direction of propagation of the beam (Fig. 3b). Let the surface of the layer, equal to the cross section of the beam, be \(F\). The black dots in the figure represent the randomly distributed molecules present in the layer. Let the part of \(F\) opaque to the beam be \(f\). When incident on such a layer, the greater part of the electrons will fly past the molecules and thus pass through the layer without experiencing any effects. Some electrons will strike the molecules; their velocities and directions of motion will be changed, and they will consequently cease to belong to the beam. The number of these electrons knocked out of the beam will be to the total number of electrons in the beam as \(\frac{f}{F}\). If the intensity of the beam (i.e., the number of electrons falling on each square centimeter of the beam cross section) at the plane \(l_0\) was \(J_0\), then after passing through the first of the \(n\) layers the intensity will be diminished by \(J_0 \frac{f}{F}\) and will be equal to:

\[ J_1 = J_0 - J_0 \frac{f}{F} = J_0\left(1 - \frac{f}{F}\right). \]

Applying the same reasoning to the following layers, we obtain:

\[ J_2 = J_1 - J_1 \frac{f}{F} = J_1\left(1 - \frac{f}{F}\right) = J_0\left(1 - \frac{f}{F}\right)^2; \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

\[ J_n = \ldots\ldots\ldots = J_0\left(1 - \frac{f}{F}\right)^n; \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

\[ J_x = \ldots\ldots\ldots = J_0\left(1 - \frac{f}{F}\right)^{nx}. \]

If the subdivision is continued, increasing \(n\) without bound, the fraction \(\frac{f}{F}\) will tend to zero. On the contrary, the expression \(n\frac{f}{F}\), i.e. the “ne-

the “transparent” area of all molecules in \(1\ \mathrm{cm}^3\) will remain finite. Therefore we have:

\[ J_x=J_0\left\{\left(1-\frac{nf}{F}\frac{1}{n}\right)^n\right\}^{x} \]

and after passing to the limit:

\[ J_x=J_0 e^{-n\frac{f}{F}x}. \tag{1} \]

Consideration of dimensions shows the correctness of the meaning we assign to the expression \(n\frac{f}{F}\). Indeed:

\[ \left[n\frac{f}{F}\right]=\left[\frac{\mathrm{cm}^{-1}\cdot \mathrm{cm}^{-2}}{\mathrm{cm}^{2}}\right]=\left[\frac{\mathrm{cm}^{2}}{\mathrm{cm}^{3}}\right]. \]

The quantity \(n\frac{f}{F}\) can be determined experimentally, as we shall see in the following paragraphs.

From the quantity \(n\frac{f}{F}\) one can calculate the “opaque” cross-section of a single molecule (by division by the number of molecules in \(1\ \mathrm{cm}^3\)). One may also proceed otherwise, by calculating the sum of the cross-sections of the molecules contained in \(1\ \mathrm{cm}^3\), for example at a pressure of \(1\ \mathrm{mm}\) Hg and \(0^\circ\mathrm{C}\). The first method has the advantage that it leads to a constant characterizing an individual molecule. The second method is practically applicable only in the case where the cross-section for any pressure can be calculated directly. Denoting the total e.c.s. of all molecules contained in \(1\ \mathrm{cm}^3\) at a pressure of \(1\ \mathrm{mm}\) Hg and a temperature of \(0^\circ\mathrm{C}\) by \(Q_{\mathrm{eff}}\), and the pressure under certain specified conditions by \(p\), we can give expression (1) the form:

\[ J_x=J_0 e^{-Q_{\mathrm{eff}}px} \tag{2} \]

The dimension of \(Q_{\mathrm{eff}}\) is:

\[ \left[\frac{\mathrm{cm}^{2}}{\mathrm{cm}^{3}}\frac{1}{\mathrm{mm\ Hg}}\right] \quad (\text{at }0^\circ\mathrm{C}). \]

The sum of the cross-sections of all molecules in \(1\ \mathrm{cm}^3\) at \(1\ \mathrm{mm}\) Hg and \(0^\circ\mathrm{C}\), determined in this way, will always be understood below when we speak simply of the e.c.s. of gas molecules.

In principle, other laws of decrease of the beam intensity are conceivable. However, it has been established experimentally that precisely the expression given above describes the phenomenon correctly. This has been shown both by varying the distance \(x\) and by varying the pressure \(p\) [formula (2)].

§ 5. INTERPRETATION OF THE RESULTS. In the preceding discussion we tacitly assumed that electrons have an infinitely small cross-section. This assumption, very convenient because of the simplifications it introduces, is in itself incorrect, since we are dealing with interactions between molecules and

electrons, in which both partners must be regarded as equal. The inaccuracy arising under the indicated assumption becomes especially obvious if one turns to the consideration of collisions not between molecules and electrons, but between molecules and ions. In this case one can speak only of the “mutual” cross section of a neutral molecule and an ion, and, moreover, the separation of the “mutual” cross section into separate ones is in principle impossible. But since the concept of a mutual e.c.s. is extremely unsatisfactory from the formal point of view, they prefer to use the concept of the sum of radii*. In the present article we shall nevertheless use the concept of e.c.s., because this concept is the only acceptable one when considering the interaction between molecules and electrons, which is foremost for us.

It should be pointed out here that, in addition to the cross section, one can use the concept of the mean free path. Historically this concept was borrowed from the kinetic theory of gases, and from a purely formal point of view it is equivalent to the concept of e.c.s. Nevertheless, its application to the region under consideration encounters certain difficulties, which will be discussed below.

The mean free path of an electron is that distance which the electron flies between two successive collisions with molecules. Hence it is clear that this concept acquires a physical meaning only under the condition of considering two successive collisions. In contrast to this, one can speak of the e.c.s. as applied to each individual collision, which in fact is the object of our investigation. In all other respects (i.e. formally) there are no difficulties in passing from the e.c.s. \(Q_{\mathrm{eff}}\) to the effective mean free path \(\lambda_{\mathrm{eff}}\). At a pressure of \(1\ \mathrm{mm}\ \mathrm{Hg}\) and a temperature of \(0^\circ \mathrm{C}\), the two quantities are related by:

\[ Q_{\mathrm{eff}}\lambda_{\mathrm{eff}} = 1. \tag{3} \]

Having settled on the concept of cross section, it would be consistent in all cases to speak of the cross section of individual molecules, as was done in the first works \(^{43}\). However, in what follows, guided by purely practical considerations, we shall use another of the quantities indicated above: by e.c.s. we shall mean the sum of the cross sections of all molecules contained in \(1\ \mathrm{cm}^3\) of gas at a pressure of \(1\ \mathrm{mm}\ \mathrm{Hg}\) and a temperature of \(0^\circ \mathrm{C}\).

§ 6. Quantitative relations. In concluding this introductory chapter we shall give relations by means of which the reader will easily be able to pass from one mode of expression to another.

* The “mutual” cross section defined in this way in the case of collision of particles of equal diameter is equal to four times the cross section of each of them.

The electron velocity is directly proportional to the square root of the accelerating potential difference. The proportionality factor is found from the equation:

\[ \frac{mv^2}{2}=eV, \]

where \(e\), \(m\), and \(v\) are the charge, mass, and velocity of the electron, and \(V\) is the accelerating potential difference.

All quantities are expressed in CGSE units. Thus the electron velocity at an accelerating potential difference of \(1\ V\) is equal to

\[ 1\sqrt{V}=0.59_5\cdot 10^8\ \frac{\text{cm}}{\text{sec}}. \]

The relation between the electron velocity and the de Broglie wavelength is given by

\[ mv=\frac{h}{\lambda}, \]

where \(m\), \(v\), and \(\lambda\) are the mass, velocity, and de Broglie wavelength of the electron, and \(h\) is Planck’s constant.

From this relation it follows that

\[ \lambda=\frac{12.2}{v}\left(\lambda\text{ in }\text{\AA},\ v\text{ in }\sqrt{V}\right). \]

From the effective cross section \(Q_{\mathrm{eff}}\) of all molecules contained in \(1\ \text{cm}^3\), we can pass to the effective cross section \(q\) and radius \(\rho\) of an individual molecule by means of the equalities:

\[ q=0.28\cdot 10^{-16}Q_{\mathrm{eff}};\quad \rho=0.30\cdot 10^8 Q_{\mathrm{eff}}, \]

where \(\rho\) is expressed in centimeters, \(q\) in square centimeters, and \(Q_{\mathrm{eff}}\) in

\[ \left(\frac{\text{cm}^2}{\text{cm}^3}\right) \]

(at \(p=1\ \text{mm Hg}\) and \(t=0^\circ\text{C}\)).

Electrons

A. Methods for the Experimental Determination of the Effective Cross Section

§ 7. Preliminary Remarks

In what follows we shall assume that the experimental arrangement is such that the measurement of electron velocities, the determination of the direction of the electron beam, the determination of the number of electrons, etc., are carried out by methods that leave no doubt as to the reliability of the results. How this is achieved in practice will be explained in § 12.

In the introductory chapter an exponential law was derived, according to which the intensity of an electron beam passing through a gas decreases. There we did not consider the question of how, in practice, the intensity is measured. Now, before turning to the description of individual experimental

methods, we shall consider the relation between the relations derived above and direct experiment.

In Fig. 4 the plate $Z$ is a source of electron flux, to which, on the path from $Z$ to diaphragm 1, the desired velocity is imparted and which, with this velocity, partially penetrates through the aperture of the diaphragm into the space $S$. Thus the aperture 1 turns out to be the source of the electron beam. Of the electrons that have entered the space $S$, some part $J$ passes through aperture 2 into the “trap” $K$ and is measured there. The number of electrons entering $K$ in vacuum characterizes the beam intensity and is a quantity essential for measuring e.p.s.

Fig. 4. Schematic of an apparatus for measuring the effective cross section.

Suppose that the entire space containing the described apparatus is filled with a gas at a certain pressure. Some portion of the electrons will undergo collisions with gas molecules and will be knocked out of the beam. The number of electrons entering $K$ when gas is present in the apparatus, equal to $J'$, will be less than $J$. If the numbers of electrons emerging from 1 in vacuum and in the presence of gas were equal, then from the ratio of $J'$ to $J$ we could calculate $Q_{\mathrm{eff}}$. But since, for a whole series of reasons, $J'$ and $J$ may be unequal, it becomes necessary to measure each time the numbers of electrons $J_0'$ and $J_0$, emerging from 1 in the presence and in the absence of gas. Thus, for each separate measurement of $Q_{\mathrm{eff}}$, it is necessary to know four quantities:

1) the number of electrons entering $K$ in vacuum, — $J$;
2) the number of electrons entering $K$ at gas pressure $p'$, — $J'$;
3) the number of electrons emerging from 1 in vacuum, — $J_0$;
4) the number of electrons emerging from 1 at gas pressure $p'$, — $J_0'$.

These four numbers, in one form or another, appear in every measurement of $Q_{\mathrm{eff}}$.*

By dividing $J$ by $J_0$, and $J'$ by $J_0'$, we obtain relative numbers of electrons, comparable with one another, entering $K$ in the presence of gas and in vacuum. We substitute the ratios $J/J_0$ and $J'/J_0'$ into formula (3) in place of $J_x$ and $J_0$ and, by simple calculations, obtain $Q_{\mathrm{eff}}$:

\[ Q_{\mathrm{eff}}=\frac{1}{l p'}\left\{\ln\left(\frac{J}{J_0}\right)-\ln\left(\frac{J'}{J_0'}\right)\right\}. \tag{4} \]

* Analogous reasoning shows the necessity of knowing four values of the beam intensity if, instead of changing the pressure, the path length of the electrons is changed.

Moving on to other pressures \(p'', p'''\), etc., we substitute into equation (4), instead of \(\ln \dfrac{J'}{J'_0}\), the expressions \(\dfrac{J''}{J''_0}\), \(\dfrac{J'''}{J'''_0}\), etc. It is not difficult to see that, plotting along the abscissa axis the differences \(p' - p\), \(p'' - p'\), \(p''' - p'\ldots\), and along the ordinate axis the logarithms of the corresponding intensity ratios, we should obtain a straight line. Such “pressure straight lines” serve as a good check in determining the effective cross section.

It is necessary to distinguish two groups of methods for determining the effective cross section. The first group is that in which the principal process is a single collision of an electron with a molecule; the second comprises those methods in which the electrons undergo multiple collisions with molecules. The methods of both groups may be subdivided into “quantitative” and “qualitative.” By quantitative methods we shall mean those in which, in order to find one value of the effective cross section, four beam intensities are measured. Qualitative methods will be those in which one value of \(Q_{\mathrm{eff}}\) is obtained on the basis of only two measurements.

Of the quantitative methods, we shall describe first of all those in which a rectilinear electron beam is used, and then those in which the electron beam is guided along a circle by means of a magnetic field. To make the discussion of the following paragraphs easier for the reader, we set out the above subdivisions in a visual form:

\[ \begin{array}{l} \text{Methods of single collision} \left\{ \begin{array}{l} \text{Quantitative} \left\{ \begin{array}{l} \text{Rectilinear beam (I a, b)}\\ \text{Magnetic deflection (I c, d)} \end{array} \right.\\ \text{Qualitative (II a, b, c)} \end{array} \right.\\ \text{Methods of multiple collisions (III, a, b, c, d)} \end{array} \]

In parentheses are indicated the sections in which the corresponding methods are described.

With the great variety of apparatus to be described, we shall not be able to adhere to the letter designations used in the individual original papers: that would make the consideration of the drawings too complicated. We introduce the following standard designations:

\(Z\) — photoelectric source of electrons (for example, a zinc plate);

\(F\) — thermionic source of electrons (incandescent filament);

\(1, 2\) — diaphragms through which the beam passes;

\(N_1, N_2\) — grids;

\(P\) — plate of the electron trap;

\(A, K, V, H\) — Faraday cylinder of the electron trap;

\(R\) — annular electron trap;

\(S\) — space in which the interaction under investigation between the electrons and the gas molecules takes place;

\(M\) — magnet (the direction of the field is perpendicular to the plane of the drawing);

\(W\) — metallic screen (electrostatic shielding);

$E$—electrometer;
$G$—galvanometer;
$H$—magnetic field with lines of force lying in the plane of the drawing;
$E$—electric field with lines of force lying in the plane of the drawing.

§ 8. Quantitative method; rectilinear beam, single collision (1 a, b).

Ia. With the apparatus shown in Fig. 5, Lenard² carried out the first measurements of the cross section of molecules for slow electrons. Apparatuses essentially the same in their main features were used by Robinson³ and F. Mayer⁴.

Through a quartz window, ultraviolet light falls on the plate $Z$. The electrons released by it from $Z$ acquire the desired velocity in the field between $Z$ and the grid $N_1$ and fly in the space free from electric forces between the grids $N_1$ and $N_2$. Some of these electrons pass through the diaphragm $l$ into the Faraday cylinder $K$. The charge of the cylinder is measured in vacuum and in the presence of gas (two intensities); to make the method quantitative, the total emission from $Z$ is measured in vacuum and in gas (a second pair of intensities). After this the gas pressure is measured. The distance over which absorption of electrons occurs is a constant of the instrument. The cross section is calculated on the basis of the data obtained by means of formula (4). The cross section of a molecule measured with such an apparatus Lenard called the “absorbing” cross section.

Fig. 5. Apparatus for measuring the absorbing cross section (after Lenard).

Owing to the large cross section of the beam (compared with the aperture $l$ of the diaphragm), equal to the area of the light spot on $Z$, compensation occurs for electrons lost by the beam as a result of their deflection through a small angle. In place of electrons from the middle part of the beam that have been deflected through a small angle and for this reason have not entered $l$, the same number of electrons, belonging to the periphery of the beam and enabled by small deflections to enter $l$, enter $l$. We have already indicated above that this circumstance, essential when working with fast electrons, has practically no effect in the case of slow ones. Moreover, in Lenard’s original method, electrons which have undergone a change in the magnitude of their velocity without a change in its direction are not removed from the beam. Their removal, however (see below), is possible and осу-

...is carried out by applying to \(K\) a negative potential whose magnitude is only slightly less than the accelerating voltage.

A somewhat modified form of the method described was proposed by Mayer\(^{7}\), who used a cylinder \(K\) movable in the direction of the beam. Fig. 6 shows the construction belonging to this author. Electrons emitted from the heated cathode \(F\) are accelerated between \(F\) and \(1\), pass through diaphragms \(2\) and \(3\), and enter the field-free space \(S\) in the form of an electron beam*. Diaphragm \(3\) is, in the sense indicated above, an electron source.

When \(K\) is moved close up to diaphragm \(3\), it captures all electrons passing through it. If \(K\) is moved away from \(3\) by a known distance \(x\), the decrease in the number of electrons will serve as a measure of the number of electrons that have undergone the action of molecules over the segment \(x\). Of course, here too, in order to obtain comparable results, two pairs of measurements are necessary, which is also needed to take into account changes in the source intensity that occur even in “vacuum” owing to the influence of residual gas or to geometrical features of the apparatus.

Fig. 6. Lenard’s method as implemented by Mayer.
Fig. 7. Lenard’s method as implemented by Brose and Jones.

More convenient is the “two-cylinder” construction proposed by Ramsauer\(^{44}\) and used for rectilinear electron beams by Brose\(^{45}\) and Jones\(^{46}\) (Fig. 7). Electrons emitted from \(Z\) are accelerated between \(Z\) and \(1\) and traverse the field-free space between \(1\) and \(2\) (\(2\) is the “electron source”). The beam thus diaphragmed then passes through the first cylinder \(V\) and through diaphragm \(3\), and enters the second cylinder \(H\). \(V\) and \(H\) are connected together or separately to the electrometer \(E\). When both cylinders are connected simultaneously \((V+H)\), the total number of electrons that have passed through \(2\) is measured. When \(V\) is grounded and the charge of only \(H\) is measured, the beam intensity in the plane \(3\) is measured. The distance between \(2\) and \(3\) is the path \(x\) over which absorption occurs. From what has been set forth above it is clear that here, too, four intensity measurements are necessary: two (\(V+H\) and \(H\)) in vacuum and two in the presence of gas.

In his measurements with the apparatus described, Brose also took into account the loss of velocity by electrons, not accompanied—

* Thus the influence of electron deflections through small angles is excluded not by means of a narrow diaphragm and a broad beam, but by means of a narrow beam and a comparatively broad diaphragm.

undergoing considerable changes of direction. For this purpose he charged, to the corresponding potential relative to the other parts of the apparatus, either both cylinders or only cylinder \(H\).

In what follows, Lenard’s method, both in its original form and in the form improved by Mayer (Fig. 6) and by Broche and Johnson (Fig. 7), will be denoted by us as method 1a.

1b. Figs. 8 and 9 show diagrams of the devices used in work with rectilinear beams by Brode\(^{47}\) and Rusch\(^{48}\). In comparison with those described above, these designs represent something new only in the respect that they make use of all directions of electron emission, which provides better conditions as regards the magnitude of the beam intensity. The design shown in Fig. 8 is symmetric with respect to the axis \(x\text{—}\cdot\text{—}x\), and that shown in Fig. 9 with respect to the axis \(y\text{—}\cdot\text{—}y\). The electron trap in Fig. 8 is the cylinder \(R\); in Fig. 9, the sphere \(R\). The electrons which are knocked out of the beam are those which, being directed along the

Fig. 8. Brode’s rectilinear-beam method.

Fig. 9. Rusch’s rectilinear-beam method (the number of holes in the sphere reaches 400).

radius, have undergone along their path a change in the direction of motion. Just as in Lenard’s original apparatus, in addition to the intensity at \(R\), the total emission of the electron source—the filament \(F\) (Fig. 8) or the zinc sphere \(Z\) (Fig. 9)—can be measured. In Rusch’s method, measurements of this kind are not made. We shall return to it in describing the qualitative methods.

The two methods just described for working with rectilinear beams will hereafter be denoted as methods 1b.

§ 9. Quantitative method: single collision; direction of the beam by a magnetic field (1c). The direction of the beam by means of a magnetic field was introduced by Ramsauer\(^{43}\). The advantages of this method are as follows.

1. The electron beam is quite homogeneous both with respect to direction and in respect of the velocities of the individual electrons. In the case of a rectilinear beam it is possible to achieve homogeneity with respect to the direction of the electrons, but as for the distribution of electrons by velocities, it depends on the emission properties of the cathode and cannot be altered.

1. In the magnetic method all electrons that have undergone one or another action are excluded from the beam without any special devices, i.e. by the action of the magnetic field itself*. In the rectilinear method, electrons that have experienced a change in velocity without a change in direction must be stopped by a retarding field. Retarding fields may introduce distortions. When working with a magnetic field, electrons with changed velocities behave in the same way as those that have undergone a deflection, and do not enter the opening of the trap. Owing to this, the superposition of retarding fields becomes unnecessary.

Ic. For convenience in clarifying the essential features, we shall consider the action of instruments with a magnetic field not on the example of Ramsauer’s original construction, but on the simplest model, used by Brode and shown in Fig. 10. Elec-

Fig. 10. Magnetic method as implemented by Brode.

Fig. 11. Magnetic method, Ramsauer’s first construction.

trons fly out from the heated cathode (filament) \(F\) and are accelerated between \(F\) and \(C\). Some of them pass through diaphragm \(1\). The magnetic field, whose direction is perpendicular to the plane of the drawing, carries a part of these electrons through diaphragms \(2, 3, 4, 5\) and \(6\) into the trap \(K\). The magnetic-field strength \(H\), the electron velocity \(v\), and the radius \(r\) of the circumference along which the electrons move are connected by the known relation:

\[ H \cdot \frac{e}{m}\, r = v. \tag{5} \]

If \(H\) is constant, each value of the electron velocity corresponds to a definite radius of the circle. If, in a collision, the electron velocity changes, the radius of the corresponding circle also changes. The electron will be ejected from the circle \(2, 3, 4, 5\) and \(6\) along which it was moving.

* This is strictly true only for “ideal” apparatus having diaphragms with infinitely small openings. The complications caused by the finite sizes of the diaphragms will be considered in § 27.

Of the four necessary intensity values, two are determined from the magnitude of the charge of trap \(K\) in vacuum and in the presence of gas, and the other two by measuring the total emission of filament \(F\) (the first Brode design) or the number of electrons passing through diaphragm \(1\) (the second Brode design) in vacuum and in gas.

In Ramsauer’s method the method described is combined with the above-mentioned two-trap method, and in two different ways. The first design, shown in Fig. 11, contains two traps (\(V\) and \(H\)) placed side by side and at different distances from the zinc plate emitting the electrons. The four intensities are obtained by measuring the charges \(H\) and \(V\) in vacuum and in the presence of gas. Since this device permits variation of the electron velocity only by changing the wavelength of the incident light, it can be used for investigating the cross section of molecules only at velocities of about \(1\) V.

In the second design (Fig. 12) the traps \(V\) and \(H\) are placed one behind the other. This makes it possible to vary the electron velocity over wide limits, and also affords a number of conveniences in manipulation. The magnitudes of the four intensities are obtained by measuring the total charge of both cylinders and of the second one alone, in vacuum and in gas. In this latter form (Fig. 12) the magnetic method was used in most of the investigations described below.

The three methods just described will hereafter be denoted as methods 1c.

1d. In the methods described above, in order to make the electron beam homogeneous, a magnetic field perpendicular to the direction of propagation of the beam was used. According to Rusch, the same results can be achieved by using, instead of a “transverse” field, a longitudinal magnetic field, i.e., one whose lines of force are directed parallel to the axis of a weakly conical electron beam.

Rusch \(^{49}\) used in his method the well-known results of Busch \(^{50}\), applying them to slow electrons, and in this way made this method of controlling the electron beam suitable for measuring the E.P.C. Replacing the transverse magnetic field by a longitudinal one may help to decide the question whether the magnetic field as such affects the magnitude of the E.P.C.

The scheme of Rusch’s method is shown in Fig. 13. Electrons leave \(F\), pass through diaphragm \(1\) into space \(C\), moving along paths slightly inclined to the axis of the beam, and are guided by magnetic field H along helical trajectories. The strength of the magnetic field is chosen so that between \(1\) and \(2\) the electrons make exactly one turn of the helix. Electrons of “higher orders” are held back by ring \(B\) and plate \(A\). When the lengths of the “monochromator” \(C\) and the first trap \(V\) are equal, diaphragms \(1\) and \(2\) are the “foci” of the electron beam. The four intensity values are obtained in the usual way, with the aid of traps \(V\) and \(H\).

Referring to this method, we shall designate it as method 1d.

§ 10. Qualitative method; a single collision (IIa).

IIa. Okesson⁵ used an apparatus essentially similar to Lenard’s apparatus (Fig. 5). He determined the dependence of the electron current to \(K\) on the accelerating voltage between \(Z\) and \(N_1\) in vacuum and in a gas. In the case of vacuum this current increases uniformly with increasing voltage. In the presence of a gas, a uniform increase can occur only when the effective cross section does not change with the change in electron velocity, or else changes in the same direction as this velocity. If, however, there are any anomalies in the change of the effective cross section—for example, if there is a maximum at some value of the electron velocity—then, at voltages corresponding to this velocity, especially many electrons will be knocked out of the beam, and the intensity curve will show a bending toward smaller values. As the gas pressure is increased, this curvature will become more and more noticeable (see, for example, Fig. 14). In the case where a minimum of the cross section exists, there will be observed

Fig. 12. Magnetic method, Ramsauer’s second construction.

Fig. 13. Longitudinal magnetic-field method (after Rusch).

a deviation of the curve from a uniform course in the opposite direction, i.e., toward larger intensity values. On the basis of such deviations from a uniform course Okesson concluded that there exists a maximum of the effective cross section in nitrogen molecules at an electron velocity of about \(1.5\sqrt{V}\), and two maxima for carbon dioxide gas at \(2\sqrt{V}\) and \(5\sqrt{V}\). Later, Glocker⁵¹ used the same method, with slight improvements, for qualitative investigations of the effective cross section. He applied low negative voltages to the trap \(K\), whereby electrons with excessively small velocities were retained. Below, Okesson’s method is designated as method IIa.

IIb. Brose⁵² applied Ramsauer’s construction (Fig. 12), using only one trap. The number of electrons reaching the traps connected together was measured at a constant beam velocity in vacuum and in a gas. The path along which absorption occurs was an arc of a circle from \(Z\) to diaphragm 5.

Determination of the effective cross section by this method gives in itself only

qualitative relations, because the constancy of electron emission with \(Z\), and consequently the comparability of the results of measurements for vacuum and gas, have not been proved. According to Brose, quantitative relations can be obtained by comparing the qualitative results for certain definite velocities with the results of measurements at the same velocities with two traps. If agreement is established between the results of measurements by both methods in different velocity intervals, the method described offers certain advantages. The point is that, because the measurements take considerably less time, it permits a very dense plotting of points on the curve. Moreover, these numerous points turn out to be considerably less scattered

Fig. 14. Qualitative measurements (according to Okesson)

Fig. 15. (a and b). Change of the distribution curves under the action of a gas.

than in the method with two traps. This occurs because the differences of ratios of intensities are considerably more sensitive to random deviations than the deviations themselves. With the aid of this method Brose succeeded in constructing, with great accuracy, a considerable number of e.v.d. curves.

We shall hereafter denote Brose’s method as method IIb.

IIc. As was shown earlier by Ramsauer\(^{43}\) in his first paper, in the case of the magnetic method a qualitative investigation of the e.v.d. can be carried out by comparing the curves of the electron velocity distribution for vacuum and gas. This method was further developed by other authors and brought to a form convenient for application.

If, in the apparatus shown in Fig. 12, the magnetic-field strength is varied (by changing the current in the coils), starting from zero, and the charge thereby received by the traps \(V\) and \(H\), connected together, is measured, it turns out that the magnitude of the charge-

...range increases from zero to a certain maximum and then again falls to zero. This occurs because the linear velocity of the electrons moving along a circle of a definite radius (circle 1, 2, 3, 4, 5) is proportional to the magnetic-field strength. Thus the curve obtained by the method described represents nothing other than the velocity distribution of the electrons emitted from plate \(Z\).

Having recorded similar distribution curves for vacuum and in the presence of a gas, it is not difficult to see that they differ from one another. The reason for this is the action of the gas. One may distinguish an “indirect” action, consisting in a change of the emission properties of the cathode, and a “direct” action, consisting in the fact that electrons of different velocities are absorbed differently by the gas on their path from plate \(Z\) to the trap. These actions can readily be separated from one another by comparing the results obtained with quantitative measurements of the effective cross section.

It turns out that, when working with photoelectrons, the “indirect action” is practically equal to zero for most gases. In these cases the change in the shape of the distribution curve makes it possible directly to draw a conclusion concerning the course of the curve expressing the dependence of the effective cross section on the electron velocity, as will be shown in several examples.

If, in the velocity interval under investigation, the effective cross section is constant (Fig. 15a, upper drawing), then in the presence of gas fewer electrons will enter the trap than in vacuum. However, in percentage terms this decrease in intensity will be the same for all parts of the distribution curve. Therefore the distribution curve for the gas (— · — · —), when reduced to the same height as the vacuum curve (——), coincides with it (······), as shown in Fig. 15a (lower drawing).

If the effective cross section increases with increasing electron velocity (Fig. 15b, upper drawing), then the slower of the electrons emerging from \(Z\) (the left side of the distribution curve) will be absorbed less strongly than the faster ones (the right side of the curve). Reducing the distribution curves (Fig. 15, lower drawing) for vacuum (——) and gas (— · — · —) to the same maximum height, we shall see that the maximum of the curve for the gas will be shifted relative to the maximum of the vacuum curve toward lower velocities.

From entirely analogous reasoning it follows that if, when the gas and vacuum distribution curves are reduced to the same maximum height, the gas curve proves to be shifted toward higher velocities, the effective cross section decreases with increasing electron velocity. A broadening of the curve for the gas on both sides indicates the presence of a maximum of the effective cross section; a narrowing indicates the presence of a minimum, and so forth.

We have dwelt in such detail on this qualitative method because, in those cases where it is applicable, it is very sensitive and gives a very clear picture of the variation of the effective cross section. Below we refer to this method as method IIc.

Rusch[^1] applied his method for studying the e.m.f. (transparent sphere) to entirely analogous qualitative investigations. The difference was that, in order to obtain velocity distribution curves, he used not a magnetic field but a retarding electric field,* and obtained distribution curves by differentiating the trap charge–voltage curves thus recorded. By comparing the distribution curves for gas and vacuum, in a manner quite analogous to that described above, he drew conclusions about the course of the curves expressing the dependence of the e.m.f. on the electron velocity. In doing so he had no possibility of quantitative control. Therefore he artificially shifted the distribution curve along the abscissa axis (by increasing the accelerating voltage) and observed whether the effect expected on the basis of the distribution curve was in fact found.

Fig. 16. Diffusion method (after Townsend).

§ 11. Method of multiple collisions. The methods described below, in contrast to those considered up to now, are characterized by the fact that a conclusion about the behavior of an electron in a single collision with a gas molecule is drawn on the basis of its behavior after multiple collisions. The most important of these methods is the diffusion method, due to Townsend[^9], which is so well grounded and developed from the mathematical side that the results obtained by it in most cases agree with the results of the methods described above.** If, despite this, Townsend’s results for a long time did not receive sufficient attention, the reason lies in the complexity of the method. The conclusions were obtained in so complicated a way that they did not seem sufficiently convincing for the substantiation of extremely unexpected facts. When, however, the correctness of such a method had been proved by more direct measurements, it could in many cases be applied to the solution of particular questions. The range of measurements by Townsend’s method extends from 0.2 to 2.5 \(\sqrt{\mathrm{V}}\).

Townsend’s apparatus is shown schematically in plan and in profile in Fig. 16. Electrons are torn from \(Z\) by ultraviolet light (in some works thermionic emission was also used) and, being accelerated under the action of the field \(E\) in the direction toward the plate \(R\), diffuse through the gas, undergoing multiple collisions with molecules. Some part of them penetrates through

* For details of the experiment see § 12.
** On the discrepancies see § 27.

orifice \(I\) into the space between \(R\) and \(R_4\), where the same conditions hold with respect to the gas pressure and the electric field as between \(Z\) and \(R\). Plate \(R_4\) is divided into three parts, as is shown in plan in the same Fig. 16. The rings \(R_1\), \(R_2\), and \(R_3\) have potentials corresponding to their distances from \(R\) and \(R_4\), and thus serve to make the field uniform. The mean velocity of motion of the electrons between \(R\) and \(R_4\) is approximately constant, since in this region an almost stationary state is established. This occurs for the following reasons.

1. Upon collision with a molecule an electron loses its velocity, the loss being the greater the greater its velocity was before the collision.

2. In the interval between two collisions the electron acquires a velocity in the direction of the field. When the velocity of the electron has such a magnitude that the mean loss of velocity (1) becomes equal to the mean increment of velocity (2), a stationary state sets in. Now, for the conical beam diffusing from \(I\) to the plates \(P_1\), \(P_2\), \(P_3\), two independent measurements are made:

a) the ratio is measured of the number of electrons \(n_2\) falling on plate \(P_2\) to the total number of electrons \(n_1 + n_2 + n_3\) falling on the plates \(P_1\), \(P_2\), \(P_3\);

b) the intensity of the magnetic field (directed perpendicular to the plane of the drawing) is measured that is required in order that \(n_1 + n_2 = n_3\), i.e. such a field as displaces the plane of symmetry of the beam by one half the width of \(P_2\).

Applying Maxwell’s diffusion equation and the continuity equation, it proves possible to establish relations between the mean free path of the electron \((\lambda = 1/Q_{\mathrm{eff}})\), the velocity of the electrons, and the measurement data. The process is considered as the diffusion of one gas (electrons) through another having a considerably greater density. In order to obtain a solution, convenient for investigation, of the complicated system of equations, it is necessary to make certain assumptions. Without dwelling on questions of averaging the velocities and the mean free path, we shall turn to one assumption that will be of importance later. Namely, we shall assume that, upon reflection of an electron from a gas molecule, all directions of reflection may be regarded as equally probable. In a gas-kinetic treatment of the problem this assumption is quite natural (the case of collision between spheres of small and infinitely large mass). However, as direct measurements show, for the electron-velocity interval considered it does not correspond to reality (§ 25).

In what follows we shall refer to the method just described as method IIIa.

On the basis of the same theoretical considerations, Loeb\(^{28}\) and Wahlin\(^{53}\) determined the mean free paths of electrons in some-

gases. In their mobility measurements, carried out by them according to the well-known Rutherford variable-field method, they determined the mean free path of the electrons, using Townsend’s diffusion theory. The special significance of their measurements of the e.c.s. lies in the fact that they worked with electron velocities considerably lower than those which other authors had been able to attain.

IIIb. Another diffusion method was applied by Minkowski and Sponer[^54]. They caused electrons to diffuse through a gas at a pressure of the order of several millimeters Hg and measured the strength of the electron current for various values of the initial electron velocities. In the resulting intensity curves one can discern characteristic parts of the cross-section curves, especially if the apparatus is calibrated using already known cases. By comparing the curves for argon, krypton, and xenon, with the apparatus calibrated from the known curves for argon, Minkowski and Sponer were able to conclude that krypton and xenon behave qualitatively in the same way as argon. Their other conclusions[^55] were confirmed somewhat less well.

We shall designate this method as method IIIb.

IIIc. The method described below, developed by Ornstein and his collaborators[^56] and called the “optical” method for determining the e.c.s., differs substantially from all the other methods by the manner of measuring the intensity of the electron beam. Its basic idea is as follows. An electron beam of a definite velocity moves in a field-free space containing gas and, along its path, excites the luminescence of the molecules of the latter. The intensity of the luminescence is measured at different points along the path of the beam. From the attenuation of the luminescence one calculates the decrease in the beam density; from this, knowing the gas pressure and the path length of the beam, it is easy to determine, by the usual method, the e.c.s. of the gas molecules. The obtained values of the e.c.s. agree, within the limits of experimental error, with the data of measurements by other methods.

This method will henceforth be designated as method IIIc.

§ 12. Details of the experiment. Electron sources may be either photoelectric or thermionic devices. The advantage of the first method is the negligible influence of gases on the emission capacity of the cathode.* Its disadvantages are the weakness of the emission and the complexity of the construction of the apparatus. The advantages of the thermionic method are: the ease of obtaining considerable electron currents (especially when using oxide cathodes) and simplicity of construction. Its disadvantage is the strong influence of the gas on the number of emitted electrons and especially on their distribution with respect to velocities. To this is also added the magnetic

* In this connection it is necessary to distinguish between the influence of the gas on the number of emitted electrons (for example, CO₂, NO₂, CH₄; established by Brüche) and the influence of the gas on the distribution of electrons with respect to velocities (for CH₄ and O₂; established by Ramsauer and Kollath).

field of the heating current, which—especially in magnetic methods—may have a considerable disturbing effect.

In creating a homogeneous electron beam one should distinguish between its homogeneity with respect to the directions of the electrons, easily achieved by means of diaphragms, and homogeneity in the sense of electron velocities. When working with rectilinear beams, the distribution of electrons by velocities is determined by the properties of the cathode, and the velocities lie within the range from 0.6 to 1 V.

When the beam is directed by a magnetic field, a considerably narrower one is selected out of this “natural” interval of electron velocities. Fig. 17 shows the velocity distribution for a rectilinear beam on the left and for one directed by a magnetic field in the center. On the right is shown the distribution calculated by Druyvesteyn[^57] for conditions close to those existing in Townsend’s diffusion method.*

Fig. 17. Velocity distribution for different methods.

Measurement of the number of electrons at low velocities is carried out with electrometers. At appreciable current strengths, galvanometers are also used. In constructing collecting devices it must be kept in mind that electrons are very readily reflected from a smooth metallic surface (the number of reflected electrons may reach 50% of the number incident). Therefore, closed and deep traps are used whenever possible; and if flat plates have to be used as the collector, they are sooted, whereby the number of reflected electrons is greatly reduced (to a few percent of the number incident).

Measurement of the velocity of electrons when working with rectilinear beams is possible only with the aid of retarding fields. In the apparatus shown in Fig. 7, the number of electrons is measured which, at various negative potentials (applied to \(V\) and \(H\)), still reaches \(V\) and \(H\). The magnitude of the current then indicates the number of electrons in the beam whose velocities are greater than those corresponding to the applied voltage. The results obtained in the meas—

* Townsend indicates a distribution which, in the descending part beyond the maximum, agrees well with Druyvesteyn’s data (Phil. Mag. 9, 1145, 1930). Recently similar results have also been obtained by M. Didlauchs (Z. Physik 74, 624, 1932).

curves of this kind are, therefore, integral curves; by differentiating them one obtains the distribution curves of the electrons of the beam with respect to velocities. The results of measurements of this kind can easily be distorted by contact potential differences, which are usually difficult to eliminate and difficult to introduce into the calculations. In addition, one must keep in mind the geometrical properties of the field.

In contrast to this, in magnetic methods the velocity of the electrons can easily be determined from the radius of the circle \(r\) and the intensity of the magnetic field \(H\) by means of the equality

\[ v=\frac{rH}{3.36}. \]

Here \(v\) is expressed in \(\sqrt{V}\), \(r\) in centimeters, and \(H\) in gauss.

Sources of error. We have already spoken above about the reflection of electrons from metallic parts. It should also be noted that these phenomena may also occur at the diaphragms limiting the beam. It is therefore expedient to make the diaphragms with sharp edges and to terminate them with a capillary.

Space-charge phenomena may be neglected as long as the strength of the electron current at diaphragms of ordinary dimensions does not exceed \(10^{-6}\) A. When working with very slow electrons it is necessary to compensate the Earth’s magnetic field.

Gas impurities affect different methods in different ways. In methods with a single collision, the impurity introduces into the measurements an error proportional to its amount. Conversely, in the case of multiple collisions, even insignificant contaminations (about 1%) make the results completely unreliable. Residual vapors (of mercury and stopcock grease) act in the same way.

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