ELECTRIC DISCHARGES IN GASES

K. Compton and I. Langmuir

(Conclusion)

II. MOTION OF ELECTRONS AND IONS

1. Classical kinetic theory of the motion of electrons and ions is understood by us as that treatment of particle kinetics which is successfully applied to the problem of molecular motions in the kinetic theory of gases. We shall first give an outline of the most important relations that can be obtained in this way, then consider the experimental tests of the classical theory, and finally mention some attempts to apply the new methods of quantum mechanics to the solution of our problem.

A. Maxwellian distribution of velocities²¹⁷, so well known from kinetic theory, describes the motion of electrons in most regions of the discharge very accurately. This result was obtained experimentally, but we shall see that there are theoretical grounds for expecting that the same Maxwellian law also describes the motion of ions, provided only that these ions are not under exceptional conditions. The distribution law may be expressed in various ways. If \(N\) is the number of particles per unit volume, then the formula*

\[ Nf(u)\,du = N\left(\frac{hm}{\pi}\right)^{\frac{1}{2}} e^{-hmu^2}\,du \tag{68} \]

* In this and the following formulas, \(e\) is the base of natural logarithms.

represents the number of molecules with velocity components between \(u\) and \(u+du\), moving in the given direction. The formula

\[ N \cdot f(uvw)\,du\,dv\,dw = N\left(\frac{hm}{\pi}\right)^{\frac{3}{2}} e^{-hm(u^2+v^2+w^2)}\,du\,dv\,dw \tag{70} \]

gives the number of particles with velocity components along the axes lying between \(u\) and \(u+du\), \(v\) and \(v+dv\), \(w\) and \(w+dw\). The formula

\[ NF(cdc) = 4\pi N\left(\frac{hm}{\pi}\right)^{\frac{3}{2}} e^{-hmc^2}c^2\,dc \tag{71} \]

gives the number of particles with speeds (independently of direction) lying between \(c\) and \(c+dc\). Integrating this equation between a given value of \(c\) and infinity, we find the number of particles with speeds greater than \(c\). Further,

\[ Nuf(u)\,du = N\left(\frac{hm}{\pi}\right)^{\frac{1}{2}} u e^{-hmu^2}\,du \tag{72} \]

is the number of particles which cross any plane with velocity components perpendicular to this plane and, in magnitude, lying between \(u\) and \(u+du\).

The constant \(h\) can be expressed through the mean kinetic energy \(\bar E\), or the temperature \(T\), or the mean-square molecular speed \(C^2\), by means of the relations

\[ \bar E=\frac{1}{2}mC^2=\frac{3}{2}kT=\frac{3}{4h}, \tag{73} \]

where \(k\) is Boltzmann’s constant.

By integrating equation (72) over all values of \(u\), we find the number of particles crossing, per second, a unit area in any direction,

\[ h = N\frac{1}{2(\pi hm)^{\frac{1}{2}}} = N\left(\frac{kT}{2\pi m}\right)^{\frac{1}{2}}. \tag{74} \]

The mean speed \(\bar c\) and the root-mean-square speed \(C\) are related by

\[ \bar c=\left(\frac{8}{3\pi}\right)^{\frac{1}{2}}C=0.921\,C. \tag{75} \]

Comparing equations (72) and (74), we see that

\[ u e^{-\frac{hmu^{2}}{2}}\,du \tag{76} \]

represents the fraction of particles crossing a surface with velocity components perpendicular to the surface lying between \(u\) and \(u+du\). Hence we find that the mean energy of those particles which cross the surface is \(2kT\), whereas the mean energy of the particles in the given volume is \(\frac{3}{2}kT\). The difference is due to the fact that, for particles crossing the surface, the mean energy associated with each of the two coordinates parallel to the surface has the usual value \(\frac{1}{2}kT\), but the energy associated with the coordinate perpendicular to the surface is \(kT\). This difference is caused by the fact that the surface is crossed by a relatively larger number of fast particles than slow ones.

If two regions are separated by a layer in which there is a force field, so that each particle must perform work \(W\) in passing from the first region into the second, then it can be shown by direct integration\(^{218}\) of Maxwell’s equations that, if in the first region the particles obey the Maxwellian distribution of velocities, then, after passing through the retarding layer, they enter the second region also with a Maxwellian distribution characteristic of the given temperature. Indeed, according to the Boltzmann equation\(^{219}\), the only effect of such a force field consists in a change of the concentration of particles in the two regions in the ratio

\[ \frac{n_{1}}{n_{2}}=e^{\frac{W}{kT}}, \tag{77} \]

where \(W\) is the difference of the potential energies of the particles in the two regions.

Mott-Smith and Langmuir\(^{220}\) proved an analogous theorem applicable to the more general case of particles with a Maxwellian distribution of velocities which pass

intersect a surface, passing through a retarding or accelerating field in the direction toward the second surface, with the problems of electrode-collectors being meant. It turned out that in this case the Maxwell–Boltzmann equation is applicable to particles reaching the second surface, but only to definite groups in the velocity distribution. Let us consider a conservative system consisting of a large number of particles moving in a closed space and continuously exchanging energy and momentum, so that a state of statistical equilibrium exists, to which the Maxwell–Boltzmann distribution law is applicable. In this space let us imagine the existence of a region in which no interaction between particles occurs, either by collisions or in any other way, and where there is a force field acting on the particles. Every inner bounding surface of the region \(A\) (for example, the surface of a collector) we shall at first regard as ideally reflecting. Particles penetrating into \(A\) from outside will in that case describe “orbits” (perhaps including reflections at the inner surfaces) and will randomly leave again, passing through the outer boundary. In addition, within there may be orbits which never intersect the outer bounding surface. This would be the case, for example, if there existed a force directing particles inward into \(A\), and if we introduced into \(A\) particles with kinetic energies too small to be able to leave the region \(A\). Now let us consider all possible orbits which we should have if we imagined that particles pass through any point inside \(A\) with any velocities between zero and infinity. Of this group of orbits some intersect the outer boundary. Others may exist (as in the case of a field accelerating particles inward) which lie wholly inside the space \(A\). These orbits, which never carry particles outside \(A\), we shall call “internal orbits.” The theorem mentioned states:

“If the field \(A\) has no internal orbits, then the distribution of particles and their velocities everywhere satisfies the Maxwell–Boltzmann distribution law. If, however, internal orbits exist, then the distribution also satisfies the Maxwell–Boltzmann law, with the exception of one class of particles, namely, with the exception of those particles which—if any such exist—describe internal orbits.”

The first case also includes the case of electrons or ions which cross a plane or convex surface, on reaching which they are detained and thrown back. The second class is illustrated by electrons or ions passing through a shell inside which they are accelerated in the direction toward the inner collecting electrode, or are repelled back to the boundary surface, as in a hollow collector.

If the internal surfaces are absorbing rather than ideally reflecting, then the distribution is modified in the sense that groups of particles having velocities directed away from this surface are absent. The simplest case is a collector surrounded by a layer of space charge parallel to its surface. Here the concentration of particles of those velocities which carry the particles in the direction toward the collector is exactly half of what it would be if the collector were ideally reflecting. For particles which are carried toward the collector, there is exactly one half of the Maxwellian distribution after exclusion of those particles which have internal orbits. For particles which penetrate, against the field, into the interior of the layer, there is the full Boltzmann distribution for those particles which cannot penetrate to the inner absorbing surface, and a half Boltzmann distribution for those particles which can penetrate. There are numerous possible applications of this theorem to ions located in the vicinity of electrodes of different forms and at different potentials with respect to the surrounding ionized gas.

b. Mean free path1 is the average distance traveled by a particle between two collisions. For real molecules this quantity, obviously, depends on our definition of a collision. For hard spherical particles, which are so often considered in kinetic theory as a convenient approximation, there can be no question here: two particles (1 and 2) collide whenever their centers approach one another to a distance \(\sigma_{12}=\sigma_1+\sigma_2\), equal to the sum of their radii. The mean free path \(\lambda_1\) of particles of type 1 colliding with particles of type 2 will in this case be:

\[ \lambda_1= \frac{1}{\pi N_2\sigma_{12}^{2}\left(1+\dfrac{h_1m_1}{h_2m_2}\right)^{\frac12}} = \frac{1}{\pi N_2\sigma_{12}^{2}\left(1+\dfrac{C_2^2}{C_1^2}\right)^{\frac12}} = \frac{1}{\pi N_2\sigma_{12}^{2}\left(1+\dfrac{\overline E_2m_1}{\overline E_1m_2}\right)^{\frac12}}, \tag{78} \]

assuming that both types of particles have Maxwellian velocity distributions corresponding to mean energies \(\overline E_1\) and \(\overline E_2\), or else to velocities \(C_1\) and \(C_2\). In problems of the kinetic theory of gases the mean energies of mixed gases are equal, and therefore the factors \(\dfrac{h_1}{h_2}\) and \(\dfrac{\overline E_2}{\overline E_1}\) drop out of equation (78), but this is not true for electrons or ions in gas discharges. Equation (78) reduces to a simpler form in the following important special cases:

1) when both particles are of one kind:

\[ \lambda_1=\frac{1}{2^{1/2}\pi N_2\sigma_{12}^{2}}, \tag{79} \]

2) when the velocity of particles of type 1 is many times greater than the velocity of particles of type 2, independently of the velocity distribution (this case obtains for electrons colliding with gas molecules):

\[ \lambda=\frac{1}{\pi N_2\sigma_{12}^{2}}, \tag{80} \]

3) when the collision radius \(\sigma_2\) of the particle being struck so greatly exceeds the collision radius of the incident particle that the former may be substituted for the sum of the radii of the colliding particles in equation (79):

\[ \lambda_1=\frac{1}{\frac{4}{2^{1/2}}\pi N_2\sigma_2^2}. \tag{81} \]

If \(\sigma_2\) is the radius of the molecule, then comparison of equations (80) and (81) shows that

\[ \lambda_{\text{electron}}=4\times 2^{1/2}\lambda_{\text{molecule}}, \tag{82} \]

since the share of the electron radius in the sum \(\sigma_{12}\) for the electron and the molecule is negligibly small. Equation (82) determines the so-called “gas-kinetic mean free path of the electron.” The values of this quantity for a number of gases, calculated from the accepted values of the molecular mean free paths, are compared in Table 15, where their reciprocals are also given, showing the average number of collisions per centimeter of path.

Table 15.

Gas-kinetic mean free paths \(\lambda\) and numbers of collisions per \(1\ \text{cm}\) of path in gases at pressure \(p=1\ \text{mm}\) and at \(25^\circ\text{C}\).

Gas	\(\lambda\) (electron)	\(\lambda\) (molecule)	\(\nu\) (electron)	\(\nu\) (molecule)
\(Hg\)	0.0149	0.00263	67.0	380.0
\(A\)	0.0450	0.00795	22.2	125.9
\(Ne\)	0.0787	0.01390	12.7	72.0
\(He\)	0.1259	0.02221	7.95	45.0
\(H_2\)	0.0817	0.01444	12.2	69.1
\(N_2\)	0.0425	0.00751	23.5	133.0
\(O_2\)	0.0455	0.00805	22.0	124.2
\(HCl\)	0.0322	0.00570	31.0	175.5
\(CO\)	0.0420	0.00743	23.8	136.5

The formula is closely connected with these expressions:

\[ \nu_{12}=2\pi^{1/2}N_1N_2\sigma_{12}^{2} \left(\frac{1}{h_1m_1}+\frac{1}{h_2m_2}\right)^{1/2}. \tag{83} \]

for the total number of collisions in unit volume per unit time between particles of kind 1 and kind 2. When the particle whose mean free path we are considering is an ion, there exists an attraction between it and all neighboring uncharged particles, caused by the electric dipole moment induced in each of these uncharged particles. The magnitude of this force between an ion and a neutral molecule at a distance \(r\) will be:

\[ F=\frac{K-1}{4\pi N}\cdot \frac{e^{2}}{r^{5}}=Ar^{-5}, \tag{84} \]

where \(K\) is the dielectric constant, and \(N\) is the number of molecules per unit volume2. This attraction increases the number of approaches of the center of the ion to a distance \(\sigma_{12}\) from the center of the molecule, and also deflects the path of the ion when the ion and molecule pass near one another, but not near enough to collide. Each of these effects decreases the diffusion and mobility of the ion. Since the diffusion constant and the mobility constant are proportional to the mean free path in the absence of attraction, we may use their decrease in order to determine the mean free path in the presence of attraction, i.e.

\[ \frac{\lambda'}{\lambda}=\frac{\mu'}{\mu}=\frac{D'}{D}, \]

where the primed symbols represent the quantities in the presence of attraction, and the unprimed symbols the quantities which would be obtained if there were no attraction. Langevin3 calculated \(\mu\) and \(D\) for smooth elastic spherical ions moving among smooth elastic spherical molecules both in the presence of attraction and without it, when the ions and molecules are in thermal equilibrium and therefore have the same kinetic energies \(\overline{E}\). The influence of the attraction is determined by the ratio

\[ \frac{A}{4\sigma_{12}^{4}\overline{E}}, \]

where \(\dfrac{A}{4\sigma_{12}^{4}}\) is the work required to separate the ion and molecule from contact to infinity against the force expressed by equation (84). This quantity, thus

thus represents the dissociation energy of the elementary cloud of ions. The following table shows the magnitude of this effect. If we determined the mean free path through the loss of energy, and not through diffusion and mobility, then these values would be somewhat different, but the general conclusion would remain unchanged. This conclusion is that the free path of an ion is not appreciably decreased as a consequence of its charge, unless the mutual kinetic energy proves to be of the order of the dissociation energy of the cloud, or smaller than it. Therefore, in ionized gases, where accumulations of ions do not occur in appreciable quantity, we have no need to consider the influence of the ion’s charge on its free path. Apparently, this phenomenon does not play an appreciable role in ordinary vacuum discharge tubes, nor in any discharges in which considerable amounts of energy density are dissipated. It may be important in phenomena of weak electrical conduction through a gas at high pressures.

Table 16.

$\dfrac{A/4\sigma_{12}^{4}}{\bar E}$	$\dfrac{\lambda'}{\lambda}=\dfrac{\mu'}{\mu}=\dfrac{D'}{D}$
$\infty$	0
10	0,076
5	15
2,5	311
1,0	73?
0,5	93
0,3	97
0,0	1,00

C. Collisions and single scattering are very simple and, for many purposes, are treated quite satisfactorily by the kinetic theory of gases as collisions between elastic spheres.

The energy losses in collisions are easily calculated for the impact of elastic spheres. Suppose that a sphere of mass

\(M\) is at rest before it is struck by a moving sphere of mass \(m\) and initial velocity \(v\). Let \(\theta\) be the angle between the initial trajectory of \(m\) and the radius vector drawn to the point of impact.

Fig. 15. Collision of elastic spheres.

The results of the collision are computed from the equations of conservation of momentum and energy:

\[ \begin{gathered} mv - mv_1 \cos \varphi = Mw_1 \cos \theta,\\ mv_1 \sin \varphi = Mw_1 \sin \theta,\\ \frac{1}{2}mv^2 - \frac{1}{2}mv_1^2 = \frac{1}{2}Mw_1^2, \end{gathered} \tag{85} \]

whence

\[ f_\theta = \frac{\frac{1}{2}mv^2 - \frac{1}{2}mv_1^2}{\frac{1}{2}mv^2} = \frac{4Mm}{(M+m)^2}\cos^2\theta, \tag{86} \]

where \(f_\theta\) is the fraction of the initial energy lost by the moving particle in a collision at angle \(\theta\). The moving particle may strike the stationary one at any point on the surface of the latter, characterized by values of the angle \(\theta\) between zero and \(\frac{\pi}{2}\). Multiplying \(f_\theta\) by the probability of a collision at angles \(\theta\) and \(\theta + d\theta\), and integrating over all values of \(\theta\) between zero and \(\frac{\pi}{2}\), we obtain the average fraction of energy loss in a collision:

\[ f = \frac{2Mm}{(M+m)^2}. \tag{87} \]

From this theory we may predict that an electron striking a molecule loses on average a fraction \(f=\dfrac{2m}{M}\) of its energy (where \(m \ll M\)); on the other hand, ions colliding with molecules of the same mass should lose on average half their energy.

If the particles subjected to collisions are also in motion, then the average fraction of energy loss of the incident particle is smaller than that indicated by equation (87). If the incident particle moves very rapidly and has a small mass \(m\) in comparison with the mass \(M\), then, as Compton\(^{224}\) showed, the average fraction of energy loss of the incident particle in a collision will be:

\[ f=2\frac{m}{M}\left(1-\frac{E_m}{E_M}\right), \tag{88} \]

where \(E_m\) and \(E_M\) are the kinetic energies, respectively, of the incident particle and of the particle subjected to collision. This expression should be applicable to electrons colliding with molecules (regarded as elastic spheres), provided that the velocities of each of these two types of particles are homogeneous.

Recently Kravatt\(^{225}\) succeeded in solving the analogous problem for the general case of particles obeying a Maxwellian velocity distribution, and without restrictions concerning the masses or the mean energies of the two groups. He found the formula:

\[ f=2.66\,\frac{mM}{(m+M)^2}\left(1-\frac{E_m}{E_M}\right), \tag{89} \]

where \(E_m\) is the average fraction of the mean energy lost by particles \(m\) in a collision with one of the particles \(M\), whose mean energy is \(E_M\). Thus the application of equation (88) to problems of electron mobility must be corrected by replacing the factor 2 by the factor 2.66.

Single scattering is readily treated in the case when the colliding particles behave as elastic

sphere. From equation (85) we obtain the general relation between the angle of incidence and the angle of scattering \(\varphi\):

\[ m\left[\sin(\varphi-\theta)\sin(\varphi+\theta)+\sin^2\theta\right] = M\left[\sin^2(\varphi-\theta)-\sin^2\theta\right]. \tag{90} \]

The probability of a collision at an angle lying between \(\theta\) and \(\theta+d\theta\) will be \(2\sin\theta\cos\theta\,d\theta\), whence the probability of deflection through an angle between \(\varphi\) and \(\varphi+d\varphi\) is found by substituting into this expression \(\varphi\) instead of \(\theta\), with the aid of the relation between these angles given by formula (90). For the two most important cases, namely for the collision of an electron with a molecule \((m \ll M)\) and for the collision of an ion with a molecule of equal mass \((m=M)\), we thus find the following expressions for the relation between \(\theta\) and \(\varphi\) and for the probability \(P(\varphi)d\varphi\) that a particle undergoing a collision will be deflected through an angle between \(\varphi\) and \(\varphi+d\varphi\):

\[ \begin{gathered} \text{for an electron, } \varphi=2\theta \text{ and } P(\varphi)d\varphi=\frac{1}{2}\sin\varphi\,d\varphi,\\ \text{for an ion } \varphi=\frac{\pi}{2}-\theta \text{ and } P(\varphi)d\varphi=\sin 2\varphi\,d\varphi. \end{gathered} \tag{91} \]

We see, incidentally, that the most probable deflection of an electron occurs at an angle of \(90^\circ\), and for an ion—at an angle of \(45^\circ\); furthermore, that there are no ions which would be thrown backward, and that after the collision the paths of the ion and the molecule are at right angles. The distribution of scattering angles is shown graphically in Fig. 16. With the aid of equation (86) it is easy to find the energy transferred by the molecule for any angle of impact or deflection. In these expressions no distinction is made with respect to the azimuth of the deflection; all azimuths for a given angle \(\varphi\) enter into the probability \(P(\varphi)d\varphi\). Sometimes it is desirable to represent this distribution differently—through the relative probabilities of deflection within a given element of solid angle for different orientations of this element relative to the axis of the initial trajectory. In other words, if radius vectors are drawn from a common point for each collision, then what will be the density of these vectors in the given direc-

direction? This density is easily found from equation (91), by dividing each value \(P(\varphi)d\varphi\) by the element of solid angle between \(\varphi\) and \(\varphi + d\varphi\), i.e. by \(2\pi\sin\varphi\,d\varphi\). Denoting the result by \(F(\varphi)\), it is easy to see that this quantity characterizes the fraction of deflections per unit solid angle in the direction \(\varphi\). We have:

\[ \begin{aligned} &\text{for electrons}\quad F(\varphi)=\frac{1}{4\pi}\\ &\text{for ions}\quad F(\varphi)=\frac{\cos\varphi}{4}, \end{aligned} \tag{92} \]

Fig. 16. Distribution of scattering angles for elastic pairs: a) collision of electrons, b) collision of ions.

whence it is seen that the electrons are scattered uniformly in all directions in space, whereas the ions are scattered with greatest density in the direction of flight. Fig. 17 shows these distributions.

A more general classical case is that in which the particles act upon one another with forces of the type \(F \sim r^{-n}\). In this case it is impossible to give a natural definition of collision. Nevertheless, relative scattering through various angles can be calculated, but it will now be a function of the number and concentration of the scattering particles and of the velocities of the scattered particles.

As an example, let us consider the scattering of electrons by centers of a force field inversely proportional to the squares of the distances. Assu-

with this, that an electron with kinetic energy \(\frac{1}{2}mC^{2}=eV\) passes through a region containing, per unit volume, \(N\) molecular ions each with unit charge. The attraction of the electrons to each ion will be \(\frac{e^{2}}{r^{2}}\). Along a path of length \(l\) through this ionized gas, the probability of deflection through an angle lying between \(\varphi\) and \(\varphi+d\varphi\), as a result of interaction with some positive ion, will be:

\[ P(\varphi)d\varphi=\frac{\pi eNl}{4V^{2}}\cotg\frac{\varphi}{2}\,\operatorname{cosec}^{2}\frac{\varphi}{2}\,d\varphi, \tag{93} \]

Fig. 17. Another graphical representation of the distribution of scattering angles:
a) impact of electrons, b) impact of ions.

and the fraction of deflections per unit solid angle in the direction \(\varphi\) will be:²²⁶

\[ F(\varphi)=\frac{e^{2}Nl}{16V^{2}}\operatorname{cosec}^{4}\frac{\varphi}{2}. \tag{94} \]

If the energy is expressed in volts, then the coefficients of the trigonometric expressions reduce to \(16.1\times10^{-5}\frac{Nl}{V^{2}}\) and \(1.28\times10^{-15}\frac{Nl}{V^{2}}\). These two distribution functions are illustrated by Figure 18.

Since collisions of elastic spheres are equivalent to deflections by force centers \(F\sim r^{-\infty}\), it is obvious that, for a law of force inversely proportional to a power of the distance higher than the second, the distribution functions

...will, in character, be intermediate between those illustrated by Fig. 18 and those illustrated by Figs. 16a and 17a.

Finally, if we investigate the single scattering of ions by \(N\) ions in unit volume with the same mass, we find \({}^{227}\):

\[ P(\varphi)d\varphi=\frac{2\pi e^2Nl}{V^2}\cotg\varphi\,\cosec^2\varphi\,d\varphi, \]
\[ F(\varphi)=\frac{e^2Nl}{V^2}\cotg\varphi\,\cosec^3\varphi. \tag{95} \]

Fig. 18. Single scattering of electrons by ions (inverse-square law).

These distributions are shown graphically in Fig. 19. If \(V\) is expressed in volts, then the coefficients will be: \(12.9\times10^{-15}\dfrac{Nl}{V^2}\) and \(20.5\times10^{-15}\dfrac{Nl}{V^2}\).

Comparison of the formulas shows that scattering by elastic spheres differs in one important respect from scattering by force centers: in the first case the concentration \(N\) of the scattering centers does not enter into the formulas. This is because in the first case the definition of a collision is quite precise, whereas in the latter there is no distinction between a collision and a non-collision. In fact, for comparing different types of scattering it is preferable to speak not of scattering in a collision, but of single scattering per unit path.

through a gas containing \(N\) molecules per unit volume. Then expressions (93), (94), and (95) remain unchanged, but for elastic spheres we shall have to modify equations (91) and (92) as follows: the total cross-sectional area for collision, which is presented by \(N\) molecules per unit volume, will be \(\pi\sigma_{12}^{2}N\). Along a path \(l\) the probability of collision will be \(\pi\sigma_{12}^{2}Nl\), and the probability of no collision will be \((1-\pi\sigma_{12}^{2}Nl)\). The first fraction of particles, after traversing the path \(l\), will be

Fig. 19. Single scattering of ions by ions (inverse-square law).

scattered according to equations (91) and (92). The last fraction will not be scattered at all. We thus have:

for electrons

\[ P(\varphi)d\varphi = \begin{cases} (1-\pi\sigma_{12}^{2}Nl), \ldots\ldots \varphi=0,\\ \pi\sigma_{12}^{2}Nl\,\dfrac{1}{2}\sin\varphi\,d\varphi \ldots 0<\varphi<\pi \end{cases} \tag{96} \]

\[ F(\varphi)= \begin{cases} \infty \ldots\ldots\ldots\ldots \varphi=0,\\ \dfrac{1}{4\pi}\sigma_{12}^{2}Nl \ldots\ldots\ldots 0<\varphi<\pi; \end{cases} \]

for ions

\[ P(\varphi)d\varphi = \begin{cases} 1-\pi\sigma_{12}^{2}Nl \ldots\ldots \varphi=0,\\ \pi\sigma_{12}^{2}Nl\sin 2\varphi\,d\varphi \ldots 0<\varphi<\dfrac{\pi}{2} \end{cases} \tag{97} \]

\[ F(\varphi)= \begin{cases} \infty \ldots\ldots\ldots\ldots \varphi=0,\\ \pi\sigma_{12}^{2}Nl\cdot\dfrac{1}{\pi}\cos\varphi \ldots\ldots 0<\varphi<\dfrac{\pi}{2}. \end{cases} \]

Fig. 20 illustrates the application of these expressions to the case of single scattering of electrons in helium. The total area bounded by each curve must obviously be equal to unity. In the case of scattering by elastic spheres the area \(OABC\) is equal to the fraction of particles that experienced collisions, while the line \(CD\) goes to infinity in such a way that the area lying under it has a finite value, with \(OBCD\) equal to unity. For

Fig. 20. Comparison of the scattering of elastic spheres and centers of force inversely proportional to the squares of the distances (electrons in helium). The ordinate is thousandths.

any inverse-power law with exponent higher than the second, the curves lie between the two types shown.

d) Diffusion is one of the two processes by means of which ions move during a discharge; the other-

that process is the action of a field. The fundamental law of diffusion is \(^{228}\):

\[ D_{12}=\frac{1}{3}\cdot\frac{N_1\lambda_2 c_2+N_2\lambda_1 c_1}{N_1+N_2} \tag{98} \]

for the mutual diffusion of gases 1 and 2, whose molecular concentrations, mean free paths, and mean speeds are denoted respectively by \(N\), \(\lambda\), and \(c\). The number of molecules of type 1 which cross a unit area in the \(z\) direction per unit time will be:

\[ n_1=-D_{12}\frac{dN_1}{dz}, \tag{99} \]

and the rate of increase of their concentration in a unit volume:

\[ \frac{dN_1}{dt}=D_{12}\nabla^2 N_1. \tag{100} \]

Substituting the values of \(\lambda\) from Eq. (78) and putting \(N=N_1+N_2\) for the total number of molecules per unit volume, we obtain:

\[ D_{12}=\frac{4}{(3\pi)^{\frac{3}{2}}N\sigma_{12}^{2}} \left(\frac{\overline{E}_1}{m_1}+\frac{\overline{E}_2}{m_2}\right) = \frac{6.921\,(C_1^2+C_2^2)^{\frac{1}{2}}}{3\pi N\sigma_{12}^{2}}, \tag{101} \]

where the mean speed \(c\) has been replaced by the root-mean-square speed \(C\) according to equation (75), and \(\overline{E}\) is the mean kinetic energy according to equation (73). We may note that, in substituting in equation (98) the values \(\lambda_1,\lambda_2\) given by equation (78), we assume that \(\lambda_1,\lambda_2\) refer to the free paths of molecules of one gas, which are limited by collisions with molecules of the other gas. In other words, we neglect collisions between molecules of one and the same gas. This, however, is permissible, since collisions of the latter type on the average have no effect on the rate of diffusion.

Let us introduce into these equations the quantity \(\lambda_0\), defined by the formula

\[ \lambda_0=(\pi N\sigma_{12}^{2})^{-1} \tag{102} \]

This is the free path which a particle would traverse if its speed were very large in comparison with the speeds of the particles undergoing collisions [equation (80)].

Let us confine ourselves to the special case of electrons diffusing through the gas \((m_1 \gg m_2\) and \(C_1 \gg C_2)\), and ions diffusing through the gas \((m_1 = m_2 = M)\). In this case equation (101) reduces to the following:

\[ D_{\text{electrons}} = 0.434 \lambda_0 \left(\frac{\overline{E}_1}{m_1}\right)^{\frac12} = 0.333 \lambda_0 c_1, \]

\[ D_{\text{ions}} = 0.434 \frac{\lambda_0}{M^{\frac12}} \left(\overline{E}_1^{\,2}+\overline{E}_2^{\,2}\right) = 0.333 \lambda_0 (c_1^2+c_2^2)^{\frac12}. \tag{103} \]

Here it should be noted that for electrons \(\lambda_0\) is the “gas-kinetic” mean free path; for ions the gas-kinetic value of \(\lambda_0\) is \(1/4\) of the corresponding value for electrons, owing to the negligibly small dimensions of the radii of the latter. However, \(\lambda_0\) for ions is \(2\frac12\) times greater than the molecular mean free path.

Equations (101) and (103) are practically identical with the equations derived from the direct consideration of the “conservation of velocities,” which in principle explains the deviation of the numerical values of the coefficients from the value \(1/3\) when the diffusing particles have finite mass[^229]. We have followed the method of Stefan–Maxwell, which avoids the direct consideration of conservation of velocities required by Meyer’s other theory.

e) The mobility \(\mu\) is defined through the electric-field strength \(E\) and the mean drift of the ion by the relation

\[ v = \mu E, \tag{104} \]

which is useful despite the fact that \(\mu\) is, generally speaking, not a constant but a function of \(E\). The quantity \(\mu\) has been derived by various methods[^230]; in this, the results obtained by different methods differ only by numerical coefficients, whose values vary by approximately a factor of two in extreme cases. A satisfactory method consists in deriving the constant \(\mu\) from the constant diffusion \(D\), expressed by equation (101), by means of the general relation:

\[ \mu = \frac{e}{kT}D, \tag{105} \]

K. Compton and I. Langmuir

where \(\dfrac{3}{2}kT\) is the mean kinetic energy of an ion \(E\). This relation was derived by Thomson by equating the mean drift velocity of the ions, caused by the electric field \(N\mu E\), to the mean drift that would be produced by a partial pressure gradient \(\dfrac{dp}{dz}\), sufficient to create the very same force for \(N\) ions per unit volume, i.e. \(NeF=-\dfrac{dp}{dz}\); consequently,

\[ N\mu E=-\frac{\mu}{e}\cdot\frac{dp}{dz} =-\frac{\mu kT}{e}\cdot\frac{dN}{dz}, \]

since \(p=NkT\). But this quantity must be identical with the drift due to diffusion, \(-D\cdot\dfrac{dN}{dz}\), whence equation (105) follows directly.

In this way equations (73), (101), and (105) lead to the formula

\[ \mu_{12}= \frac{2e}{(3\pi)^{\frac12}\pi N\sigma_{12}^{2}\overline{E}_{1}} \left( \frac{\overline{E}_{1}}{m_{1}}+ \frac{\overline{E}_{2}}{m_{2}} \right)^{\frac12} = \frac{0.921\,e}{\pi N\sigma_{12}^{2}m_{1}C_{1}} \left( 1+\frac{\overline{E}_{2}m_{1}}{\overline{E}_{1}m_{2}} \right) \tag{106} \]

for the mobility of charged particles 1 moving through gas 2. If the mean velocity \(c_{1}\) is used instead of the root-mean-square velocity \(C_{1}\), then the numerical coefficient will be 0.85 instead of 0.921.

Introducing now \(\lambda_{0}\) from equation (102), we have in general

\[ \mu_{12}= \frac{0.921\,e\lambda_{0}}{m_{1}C_{1}} \left( 1+\frac{\overline{E}_{2}m_{1}}{\overline{E}_{1}m_{2}} \right)^{\frac12}; \]

for electrons,

\[ \mu= \frac{0.921\,e\lambda_{0}}{mC} = \frac{0.85\,e\lambda_{0}}{mc}; \]

for ions,

\[ m_{1}=m_{2}=M;\quad \mu= \frac{0.921\,e\lambda_{0}}{MC} \left( 1+\frac{\overline{E}_{2}}{\overline{E}_{1}} \right)^{\frac12}. \tag{107} \]

Comparing the value \(\mu_{12}\) found by us with Langevin’s most widespread mobility equation\(^{232}\)

\[ \mu_{12}=\frac{0.815\, e\lambda_0}{m_1 C_1}\left(1+\frac{m_1}{m_2}\right)^{\frac12}, \tag{108} \]

we observe a twofold difference: 1) the difference in the numerical factors is due to the difference in the method of averaging and is typical for a number of constants lying between 0.5 and 1, which are obtained depending on one or another degree of rigor in the derivations. We believe that the value given here is just as acceptable as any other; 2) the absence of the ratio of the mean energies in the mobility equations of Langevin (and of all others) is due to the implicit assumption of a uniform distribution of energy when carrying out the integration. This circumstance is probably explained by the fact that when mobility equations were first derived, interest was concentrated on cases in which the ions were practically in equilibrium with the gas molecules.

In equations (107) and (108) the following circumstance remains entirely unclear: \(\lambda_0\) in these equations is not necessarily the mean free path, but is determined by equation (102) and is \(2\frac12\) times greater than the mean free path when the velocities of the ions and molecules are of the same order of magnitude\(^{233}\).

It should be noted that Langevin derived a more general equation for mobility, based on the assumption of attractive forces between ions and molecules inversely proportional to some power of the distance\(^{234}\). But we have already seen that attractive forces do not have a noticeable influence on free paths when the energy of the ion is too great to give it the possibility of forming a “cloud”; therefore such a refinement should not be considered essential for the theory of electrical discharges in gases.

To the extent that collisions can be treated as collisions of elastic spheres, equation (107) should

may permit an estimate of the mobility within a small uncertainty in the numerical coefficient, provided only that the mean velocities and energies are known. In strongly ionized gases these quantities can be measured directly for electrons and estimated with considerable accuracy for ions by means of special probe methods,^235 which will be described in Part II. Where these direct data cannot be obtained, equations (107) can be used only in conjunction with some additional theory for estimating the velocities and energies. An attempt to do this is made below on the basis of the work of Hertz^236 and Compton.

The steady-state velocity can be calculated by equating to one another the rates of energy gain and energy loss in the motion of a charged particle through a gas in a uniform field \(E\). Moving a distance \(dx\) in the direction of the field, each particle with charge \(e\) acquires energy \(edU=eE\,dx\), where \(U\) is the kinetic energy of the charged particle, expressed through the equivalent potential drop. Along this same path \(dx\) the particle loses energy \(\nu' fU\,dx\), where \(\nu'\) is the mean number of collisions occurring in a displacement per unit distance, and \(f\) is the mean fraction of energy lost in a collision as a result of momentum transfer. Thus the quantity

\[ e\frac{dU}{dx}=e\left(E-\nu'fU\right) \tag{109} \]

represents the final rate at which the particle gains additional energy. The state of the steady-state velocity is determined by the condition \(\dfrac{dU}{dx}=0\), whence for the limiting energy \(U_t\) we obtain:

\[ U_t=\frac{E}{\nu'f}. \tag{110} \]

Hence the mean number of collisions \(\nu'\) experienced by the charged particle in moving a unit distance can be calculated as:

In equation (107) let \(eU=E_1\) and \(eQ=E_2\) for the mean energies of the charged particle and of the gas molecule. Then the mean velocity of translational displacement in the direction of the field will be expressed as:

\[ v=\mu E=\frac{0.921\, e\lambda_0}{mC}\,E\left(1+\frac{Qm}{UM}\right)^{\frac12}. \tag{111} \]

But the mean number of collisions undergone by the charged particle per unit time will be \(\dfrac{c}{\lambda}\). Hence the mean number of collisions undergone in displacement over a unit distance will be:

\[ \nu'=\frac{\dfrac{c}{\lambda}}{\mu E} =\frac{0.921\,C}{\lambda E} =\frac{mC}{0.921\,e\lambda_0\left(1+\dfrac{Qm}{UM}\right)^{\frac12}} = \]

\[ =\nu'\,\frac{2U}{\lambda\lambda_0 E\left(1+\dfrac{Qm}{UM}\right)^{\frac12}} =\frac{2U}{\lambda_0^2 E} \tag{112} \]

according to equation (78). Taking into account that \(\dfrac{1}{\lambda_0}\) is of the order of magnitude of the number of collisions per unit path, we see that the number of collisions undergone in translational displacement over a unit distance is of the order of magnitude of the square of the number of collisions undergone per unit path along the trajectory.

Returning to equation (110) and substituting for \(f\) and \(\nu'\) their values from equations (89) and (112), we obtain for the final energy of a charged particle of mass \(m\) and with mean free path \(\lambda\) in a medium whose molecules have mass \(M\) and mean energy \(Q\):

\[ U_t=\frac{Q}{2}+\left[\frac{Q^2}{4}+\frac{\lambda_0^2 E^2(M+m)^2}{5.32\,Mm}\right]^{\frac12} \tag{113} \]

This quantity reduces to the value for a uniform distribution \(U_t=Q\) for very weak fields \(E=0\), then

as in strong fields the limiting energy becomes almost proportional to the field according to the equation:

\[ U_t=\frac{\lambda_0 E(M+m)}{2.31\, d M}. \]

Since the mean square velocity is expressed by the formula:

\[ C_1=\left(\frac{2eU_t}{m}\right)^{\frac12}, \tag{114} \]

then, substituting the value of \(U_t\) from equation (113) into equation (114), and the value of \(C_1\) thus found into equation (107), we obtain the general equation for the mobility:

\[ \mu= \frac{0.921\, e\lambda_0}{(2em)^{\frac12}} \left[ \frac{Q}{2} + \left( \frac{Q^2}{4} + \frac{\lambda_0^2 E^2(M+m)^2}{5.32\,Mm} \right)^{\frac12} \right]^{-\frac12} \tag{115} \]

\[ \times \left[ 1+\frac{M}{m}\cdot \frac{1}{ \frac12+ \left( \frac14+ \frac{\lambda_0^2 E^2(M+m)^2}{5.32\,Mm} \right)^{\frac12} } \right]^{\frac12}. \]

In this formula the first term in brackets is a generalization of an analogous expression previously derived by one of us \(^{237}\). The second term in brackets replaces the factor

\[ \left(1+\frac{m}{M}\right)^{\frac12}, \]

which appeared in the first mobility equations (compare Langevin \(^{232}\)). Obviously, in the original equations the effect on the mean free path of the fact that the mean energy of the ions exceeds the mean energy of the molecules [cf. the discussion of equation (78)] was neglected in this term, which by its nature is a correction term. Attention should be drawn to the fact that \(\lambda_0\) is determined by the formula \(\lambda_0=(\pi\sigma_{12}^{2}N)^{-1}\) and therefore does not depend on the velocities.

ELECTRICAL DISCHARGES IN GASES

Equation (115) is simplified in two cases that are of greatest interest to us:

α) For electrons \(m \ll M\), \(\lambda_0=\lambda=\) the mean free path of the electron,

\[ \mu_{\text{electrons}} = \frac{0.921\,e\lambda_0}{(2em)^{\frac12}} \left[ \frac{\Omega}{2} - \left( \frac{\Omega^2}{4} + \frac{\lambda_0^2 E^2 M^{\frac12}}{5.32\,m} \right)^{\frac12} \right]^{-\frac12} \tag{116} \]

β) For ions \(m=M\), \(\lambda_0=\sqrt{2}\lambda\), where \(\lambda\) is the mean free path of a molecule,

\[ \mu_{\text{ions}} = \frac{0.921\,e\lambda_0}{(2eM)^{\frac12}} \left[ \frac{\Omega}{2} + \left( \frac{\Omega^2}{4} + \frac{\lambda_0^2 E^2}{1.33} \right)^{\frac12} \right]^{-\frac12} \times \]

\[ \times \left[ 1+ \frac{1}{ \frac12+ \left( \frac14+ \frac{\lambda_0^2 E^2}{1.33\Omega} \right)^{\frac12} } \right]^{\frac12} \tag{117} \]

Both equations reduce to simpler forms in the limiting cases of very small fields and very strong fields.

The effect of inelastic collisions consists in an increase of mobility; this result will not appear paradoxical if we recall that inelastic collisions diminish the velocity of the translational motion of the particles and that this velocity enters into the denominator of the mobility equation (107). If all collisions experienced by electrons were completely inelastic, their mobility would be expressed by the formula \(^{237}\):

\[ \mu_i=\left(\frac{\pi i e}{2mE}\right), \tag{118} \]

which is at least in qualitative agreement with observations \(^{238}\).

2. Comparison of these results of the classical kinetic theory with experiment can be made on several points.

a) The Maxwellian distribution of velocities was found in electrons emitted by heated bodies (cf. B 1). Likewise, in various parts of discharge tubes it was found\(^{239}\) (by the methods described in Part II) that the electrons possess a Maxwellian distribution with a mean energy determined by the ionization potential of the gas and by the discharge conditions, this energy rarely exceeding the minimum ionization potential by more than one third. Often, in regions of intense ionization near the cathode, two Maxwellian distributions superposed on one another are found: one—with a relatively high energy—evidently consisting of primary electrons from the cathode which have undergone scattering, and the other—with a considerably lower energy—consisting of electrons produced by ionization of the gas. The mechanism by which these Maxwellian distributions are established so rapidly has not yet been elucidated (cf., however, section D 4). It is well known that electrons carried through a gas under the influence of a uniform field and obeying the law of conservation of momentum are transported with an established translational speed, at which they have a Maxwellian distribution of velocities with respect to a coordinate system moving with the velocity of their uniform drift\(^{240}\). However, an approximate calculation of the number of collisions required to attain such conditions shows that, in an ionized gas, the scattering of velocities must be due chiefly to some other and, moreover, more effective agent. Generally speaking, electrons exhibit a velocity distribution very close to Maxwellian in all parts of the discharge where appreciable luminosity is observed, as well as in the Faraday dark space. Large departures from a complete Maxwellian distribution of velocities are sometimes found in the dark spaces between striations and always in the regions of the cathode and anode potential drops.

Fig. 21 shows two typical velocity distributions. The criterion for a Maxwellian distribution of velocities

consists in the fact that the graph of \(\log i_{-}\) (the electron current to the collector) as a function of \(V\) (the collector potential) must be a straight line in the region where \(V\) is negative with respect to the space potential. The slope of this straight line must be \(\dfrac{3}{2\overline V}\), where \(\overline V\) is the mean energy of the electrons in the surrounding space, expressed in equivalent volts. Fig. 21a was obtained for an arc in argon, and Fig. 21b—for an arc in helium, heated cathodes being used in both cases.

Fig. 21. a) One Maxwellian distribution; b) two Maxwellian distributions superposed on one another.

The Boltzmann distribution of concentrations also proves to be valid for electrons in discharge tubes. An excellent example of this can be found in the work of Killian \(^{24}\).

b) In the case of mean free paths, the experimental results indicate a much more complicated mechanism of collisions than that which is assumed in the calc—

considered above, “classical” kinetic theory. Nevertheless, the results are—in general—of the same order of magnitude, so that the values given by classical kinetic theory are useful for rough estimates or for use in cases where more accurate data are not available. There are three direct methods for measuring the free paths of charged particles, due to Lenard, Ramsauer and Langmuir, and Jones.

In Lenard’s method²⁴² the charged particles were accelerated by a field to the desired velocity; the sharply defined beam thus obtained passed through a system of narrow diaphragms into a chamber where it could collide with gas molecules. Those particles which passed through the gas without being deflected could then be caught by a corresponding collecting electrode. The number of charged particles \(N\) which have traveled a distance \(x\) without deflection will be

\[ N_0 e^{-\frac{x}{\lambda}} \]

or

\[ N_0 e^{-\frac{px}{\lambda_1}}, \]

where \(\lambda_1\) is the mean path length at unit pressure. The ratio of the numbers \(N'\) and \(N''\), corresponding to two pressures \(p'\) and \(p''\), or to two distances \(x'\) and \(x''\), will be

\[ e^{-\frac{x(p'-p'')}{\lambda_1}} \]

or, respectively,

\[ e^{-\frac{p(x'-x'')}{\lambda_1}}, \]

from which the mean path length \(\lambda_1\) at unit pressure can readily be calculated.

It is necessary to investigate what real meaning is possessed by the mean path lengths determined in a given experiment, for in each case the meaning of the mean free path is determined by the apparatus used for finding it. In Lenard’s apparatus the free path is a path for which the angular scattering is very small—small enough not to make it possible for the particle to reach the electrode. Energy losses here are not taken into account, or are taken into account only to the extent to which they accompany deflection.

In Ramsauer’s method (Fig. 22) the charged particles are liberated photoelectrically or thermionically, for example from an incandescent filament \(F\), and are accelerated to the desired velocity by the electric field between \(F\) and electrode \(C\). The beam emerging from the slit \(S_c\) is bent by a magnetic field

so that it passes through the system of slits \(S_1, S_2, S_3, S_4, S_5\) arranged in a circle and finally enters the Faraday cylinder \(B\).

Thus this apparatus selects a very narrow group of velocities, determined by the magnetic field and the system of slits. The current to the Faraday cylinder \(B\) is measured for a series of different gas pressures; as before, the ratio of the currents for two given pressures is expressed as

\[ e^{-x(p'-p'')/\lambda_1}, \]

where \(x\) is the path length from \(S_c\) to \(S_3\). Brode \(^{243}\) indicated certain necessary precautions and corrections of second-order magnitude.

In this method, as is evident, not only angular deflections, but also energy losses, even those not accompanied by deflection, remove particles from the beam. For this reason the absorption coefficient \(\frac{1}{\lambda_1}\), measured by this method, is often called the total absorption coefficient.

Fig. 22. Ramsauer’s method for measuring mean free paths.

Fig. 23. Langmuir and Jones’s method for studying electron collisions with gas molecules.

In the method of Langmuir and Jones \(^{244}\) (Fig. 23) the ionization of the gas is maintained between an emitting heated equipotential filament \(F\) and the anode plates \(A\) and \(B\), while the surrounding coaxial cylinder \(C\) is used as the collector electrode, whose current is measured at various potentials relative to the filament. As will be shown in Part II, the principal mass of the enclosed ionized gas is located approximately—

considerably at the same potential, approximately coinciding with the anode potential, and the potential differences between this mass of gas and the electrodes are concentrated “in the space-charge layers” near the electrodes, the thickness of these layers being relatively small if the ionization is very intense. Thus the electrons emitted by the filament are ejected radially into the gas with velocities corresponding to the cathode fall of potential in the layer. If the collector \(C\) is charged—

Fig. 24. Results of Langmuir and Jones with nitrogen; \(\Delta\) is a measure of the number of electrons that do not lose translational momentum.

negatively with respect to the filament \(F\) (and therefore also with respect to all parts of the discharge), then positive ions pass into it through the surrounding space-charge layer, but all electrons entering the layer are repelled back into the main part of the discharge and thus do not reach the collector. But if the collector \(C\) is charged exactly to the potential of the filament, then those electrons from the filament which have not suffered a loss of translational momentum prove able to penetrate through this layer and reach the collector. For more positive potentials

electrons of the collector \(C\) which have lost some part of their translational momentum are nevertheless still capable of reaching the collector. In this way the current to the collector at different potentials of it can be used to indicate the fraction of electrons emitted by the filament which collide (lose translational momentum) while passing through the gas between the filament and the collector. This experiment also gives a large amount of information concerning the nature of collisions, as we shall see in the next section. Fig. 24 shows a typical curve for nitrogen. In the region \(AB\) only positive ions reach the collector \(C\). In the region \(BC\) the cylinder is no longer negative with respect to \(F_2\), and the length \(BC=\Delta\) represents the relative change of the current to the collector in passing from \(B\) to \(C\). In this case the fraction of primary electrons from the filament which reaches the collector \(C\) without collisions will be:

\[ \frac{\Delta}{1-\gamma+\beta_c}=e^{-\frac{ap}{\lambda_1}}, \]

where \(\gamma\) and \(\beta_c\) are small corrections due to the reflection of the fraction \(\gamma\) of electrons incident on \(C\), and to the change of the current of positive ions to \(C\), owing to the loss of this group of electrons as ionizing agents in the discharge; \(a\) is the radius of the collector \(C\). All these quantities can be measured or estimated with considerable certainty, and therefore \(\lambda_1\) can be found. Further results obtained by this method concerning the nature of the collisions themselves are discussed in the following section.

Data concerning the mean free paths of electrons found by these methods are given in Figs. 25 and 26. In accordance with established custom, the ordinates represent the “absorption coefficients” of the gas at a pressure of \(1\) mm for electrons, or the total “effective” cross-section of all molecules in a cubic centimeter at \(1\) mm pressure, or the mean number of collisions made by an electron over \(1\) cm of path through the gas at \(1\) mm pressure. All these expressions are mutually equivalent and equal

reciprocal quantity \(\frac{1}{\lambda_1}\), the mean path length at a pressure of \(1\ \mathrm{mm}\).

Data concerning the mean free paths of positive ions are very scanty and are completely absent

Fig. 25. Absorption coefficients \(a=\frac{1}{\lambda_1}\) for electrons in gases at \(1\ \mathrm{mm}\) and \(25^\circ\mathrm{C}\) as functions of the velocity of the electrons. K. T. denotes kinetic values calculated from equation (82). (1) and (2) indicate the corresponding ordinate scales.

in most important cases, when the positive ions move among molecules of the very same gas, with the exception of the case of protons moving in hydrogen \(^{246}\). Here Demster found that protons with velocities between 50 and 2 thousand V pass through a multitude of hydrogen molecules,

without appreciably changing its velocity, whereas Thomson found an angular distribution of scattering of such a type as would be expected if, in the collisions here, the inverse-cube law of distance were operative. Similarly, for protons with velocities of 900 V, moving—

Fig. 26. Absorption coefficients \(\alpha\) (Fig. 25).

—in helium, Dempster\(^{246}\) found anomalously long free paths (or else unexpectedly small collision effects), whereas for protons of various velocities in helium and argon Thomson found maximum scattering (minimum free path) at velocities respectively \(0.4 \times 10^8\) cm/sec and \(0.7 \times 10^8\) cm/sec, by analogy with the “Ramsauer effect” for electrons.

Recently the mean free paths of alkali ions in various gases have been studied. Kennard\(^{247}\) and Thomson\(^{248}\) found that ions of lithium, sodium, potassium, or cesium,

moving as a beam through hydrogen, helium, or argon, are scattered considerably less than could have been expected on the basis of kinetic radii, and also that the scattering is less at high velocities than at low ones. Ramsauer and Beeck\(^ {249}\) likewise found a rapid decrease of scattering at high velocities, but discovered that the scattering—

Fig. 27. Absorption coefficients \(a=\frac{1}{\lambda_1}\) of alkali ions in argon.

approaches a limiting value at velocities of the order of 50 V or more, and that this limiting value agrees very well with the predictions of the kinetic theory. This means that the mean free path

\[ \lambda_1=\frac{1}{a} \]

is such as could have been expected if the effective collision radius were equal to the sum of the radii of the gas molecule and the ion. (The data concerning the magnitude

radii of ions differ very considerably among themselves, as is shown in the table on p. 22 in the article cited under No. 249. Agreement is reached if one takes the “true dimensions” of the ions according to Fajans and Herzfeld. Figs. 27 and 28 give some of the results of Ramsauer and Beeck. In Fig. 28 the lines to the right of the curves, marked

Fig. 28. Comparison of experimental and gas-kinetic mean free paths of positive ions.

Cs\(^+\), Rb\(^+\), etc., indicate the sum of the radius of the argon atom and of each of these ions.

In order to reconcile the results of Thomson and of Ramsauer and Beeck with one another, it appears necessary to assume that the mean free paths do indeed have the order of magnitude indicated by the kinetic theory, especially at high velocities, but that the collision is manifested in a loss of velocity without a noticeable change of direction.

c) The nature of collisions and the single scattering of electrons by molecules were investigated by two methods, one of which was developed by Dymond, and the other by Langmuir and Jones. The advantage of the first is that it is more direct and has greater resolving power, whereas the second is characterized by greater experimental simplicity. As we shall see, both these methods lead to results that are in general agreement—since the matter concerns the basic features of scattering in collisions—but differ somewhat in certain details.

Fig. 29. Dymond’s method for investigating electron scattering.

In Dymond’s method^250 electrons from an incandescent filament (Fig. 29) are accelerated to any desired velocity and are brought into a sharply defined beam by a system of slits \(S_1\). Those electrons which collide with molecules in the small region \(A\) and are deflected through an angle \(\varphi\) pass through an evacuated system of slits into the evacuated chamber \(D\), where electrons of any desired velocity \(v_z\) are selected and deflected into a Faraday cylinder by means of a magnetic field (Dymond) or an electrostatic field (Garnwell). The scattering angle in observations can be varied by rotating the electron “gun” \(FS_1\) about the axis \(A\). In this way one can find the probability of deflection through any angle and the probability of any loss of energy for electrons of all initial velocities colliding with molecules of any gas. Typical results of this method are given in Fig. 30.

From a number of analogous experiments shown in Fig. 30, angular scattering in helium was determined. The results are given in Fig. 31, where, for comparison, some theoretical curves are also shown. As can be seen, the observations agree satisfactorily with the new quantum theory of scattering^251, but do not agree with the classical kinetic theory...

ries. Further, the area bounded by each separate curve gives the absorption coefficient \(a\) (the reciprocal of the mean free path) for the corresponding type of scattering. The most remarkable feature of these results is the relatively high probability of scattering through small angles, except in the cases of the very lowest velocities.

Fig. 30. Distribution of the velocities of electrons scattered through \(10^\circ\) in single collisions in helium for initial velocities 102, 226, and 386 V (Dymond and Watson \(^{250}\)). The three upper peaks represent elastic scattering; the pair of lower peaks represent scattering with sufficient loss of energy to excite one or another of the lower levels of the excited state of the helium atom.

Harnwell’s results \(^{250}\) show that the scattering curves for helium, neon, atomic hydrogen, and nitrogen have, in their essential features, the same character. The mean angle of elastic collision increases somewhat with increasing molecular weight.

In the method of Langmuir and Jones \(^{244}\) the apparatus shown in Fig. 23 is used, and results of the type shown in Fig. 24 are obtained. It should be recalled that the segment \(AB\) of the curve in Fig. 24 represents the current of positive ions to the collector, whose potential is \(V_c\) with respect to a homogeneous source of electrons, which are accelerated with any desired velocity directly

to the collector. When the collector potential is varied from slightly negative to slightly positive with respect to the source, those electrons which have not lost translational momentum on their way prove capable of reaching the collector. The conclusion as to the nature

Fig. 31. Scattering of electrons in helium. \(F(\Phi)\) is the fraction scattered per unit solid angle through the angle \(\Phi\). Observations of the elastic scattering of electrons of 210 V compared with the predictions of quantum theory. In addition to the theoretical scattering for other voltages, observations of the inelastic scattering of electrons at 210 V are also given. The observations are plotted on an arbitrary scale, which is the same for elastic and inelastic scattering. The curve \(\cdots\) represents the graph of equation (119).

of the collisions can be obtained by studying the character of the current–voltage curve for positive values of \(V_c\). The basis for the interpretation of these curves is the following.

If all electrons which collide lose all their energy, then it is obvious that the curve \(CDEF\) must be

completely plane and correspond precisely to those electrons which have passed from the source to the collector without collisions. If, however, some electrons prove to be slightly and elastically deflected, they will not be able to penetrate through the space-charge layer surrounding the collector when \(V=0\), but they will be able to penetrate through it when \(V\) has a sufficiently positive value, so that the potential drop in this layer will be less by an amount corresponding to the loss of energy associated with the translational component of the momentum. Thus the form of the rising curve \(CD\) gives information about the angular distribution of elastic scattering.

The break in the curve at the point \(D\) occurs approximately at 13 volts. At this voltage a \(V_c\)-electron, which has lost energy in just such an amount as to excite nitrogen molecules and continue its motion further forward, turns out to be just capable of penetrating through the layer and reaching the collector. At larger values of \(V_c\), electrons which have lost additional energy or were deflected in the process of exciting atoms can be collected in the collector.

In carrying out the calculations, described in detail in the original article, it is necessary to take into account certain secondary effects, such as, for example, reflection of an electron from the collector and the change in the total ionization when the electrons have already been collected. The experiment does not make it possible to distinguish sharply between loss of energy and angular deflection, since the loss of translational momentum is precisely the quantity which can be measured in the given case. But certain additional experiments confirm the conclusions drawn in Table 17.

The results agree with a probability law of angular scattering having the form:

\[ F(\varphi)=F(\varphi)_0 e^{-\frac{\varphi^2}{\varphi_0^2}}, \tag{119} \]

where \(F(\varphi)\) is the fraction of electrons which pass unit distance and, for each unit solid angle of scattering,

Table 17.

Summary of data concerning electron impacts in gases (Langmuir and Jones). \(P_k, P_e, P_r, P_i\) are the probabilities that an electron, in \(1\ \mathrm{cm}\) of path through a gas at \(1\ \mathrm{mm}\) pressure and \(20^\circ\mathrm{C}\), undergoes a collision: inelastic, elastic, one producing the first excited state, ionizing. \(\Phi_e\) and \(\Phi_r\) are the root-mean-square angles of deflection for elastic scattering and for scattering accompanying the first excitation. \(E_e = v\Phi_e^2\) (volt-radian\(^2\)) proves to be “approximately” constant for each gas and may be useful for preliminary calculation of \(\Phi_e\) for other electron energies \(v\) (volts). Bear in mind that \(\Phi_e\) must be expressed in radians. \(P_k + P_e\) is the total probability of collision.

Gas	Volts	\(P_k\)	\(P_e\)	\(P_r\)	\(P_i\)	\(\Phi_e\)	\(\Phi_r\)	\(E_e\)
\(\mathrm{He}\)	50	7,7	2,1	2,8	0,9	25°	(25)°	9,5
\(\mathrm{He}\)	100	6,0	1,5	0,7	1,6	19	16	11,0
\(\mathrm{Ne}\)	75	9,3	1,1	1,5	1,8	21	(21)	9,8
\(\mathrm{Ne}\)	100	8,2	0,9	1,3	2,4	19	(19)	11,0
\(\mathrm{A}\)	30	19,1	24,1	5,9	4,7	24	(24)	5,3
\(\mathrm{A}\)	50	18,4	20,8	1,9	9,5	18	(18)	5,2
\(\mathrm{A}\)	100	18,7	14,5	1,7	11,4	12	(12)	4,7
\(\mathrm{A}\)	250	18,4	12,8	2,7	11,4	10	(10)	4,5
\(\mathrm{Hg}\)	30	48,0	33,3	17,3	13,6	17	(17)	2,5
\(\mathrm{Hg}\)	50	49,1	29,5	14,7	19,9	11	13	3,1
\(\mathrm{Hg}\)	100	50,3	25,9	16,7	21,7	10	12	3,3
\(\mathrm{Hg}\)	250	32,0	21,3	6,7	20,4	6	(6)	8,2
\(\mathrm{H}_2\)	100	9,2	5,3	3,5	3,9	5	16	7,3
\(\mathrm{H}_2\)	250	7,2	6,3	2,1	3,6	9	5	6,7
\(\mathrm{N}_2\)	75	20,2	14,9	5,3	9,3	16	(16)	5,9
\(\mathrm{N}_2\)	100	16,2	10,4	4,8	10,3	14	8	5,8

are scattered through an angle \(\varphi\), \(F(\varphi)_0\) is the fraction at angle \(\varphi = 0\), and \(\varphi_0\) may be called the root-mean-square angle of scattering. Fig. 31 shows that this equation is also in approximate agreement with the experiments of Dymond and Watson in helium, the value of \(\varphi_0\) here being equal to \(22.7^\circ\) for elastic scattering of electrons at 210 volts. Although this equation may also be inaccurate, it is in satisfactory agreement with the facts, and its form is such that it is convenient to use in calculations. It may therefore be assumed that this law should be very useful in interpreting the effect of angular scattering in re-

— in sealed discharge tubes. The corresponding constants that should be used in this equation for a number of gases may be borrowed from Table 17, although more direct measurements are also available, as, for example, in the case of helium.

The total probability of scattering as a result of a collision, for displacement through a unit distance, is obtained from equation (119), namely:

\[ P=\int_{0}^{\pi}2\pi\sin\varphi\cdot F(\varphi)_0\cdot e^{\frac{\varphi^2}{\varphi_0^2}}\,d\varphi =\pi\varphi_0^2 F(\varphi), \tag{120} \]

whence, approximately,

\[ F(\varphi)=\frac{P}{\pi\varphi_0^2}\,e^{-\frac{\varphi}{\varphi_0}} . \tag{121} \]

The quantities \(P\) and \(\varphi_0\), obtained by substituting the experimental results into this equation, are given in Table 17. From these, the values of \(F(\varphi)\) for any angle can be calculated.

In some cases the curve above the first excitation potential proves unsuitable for an accurate estimate of \(\varphi_r\). Since \(\varphi_r\), generally speaking, proves approximately equal to \(\varphi_e\), it is assumed that in these unreliable cases both quantities are in fact equal to one another. These values are given in parentheses.

When using equation (121), \(\varphi_0\) is taken equal to \(\varphi_r\) or \(\varphi_e\), depending on the circumstances, and the corresponding value of \(\rho\) is used.

A comparison of these results with the results of Diamond and Yarnveld, described earlier, shows agreement, since the question concerns the predominance of scattering through small angles and the magnitude of the mean scattering angle. Langmuir and Jones found inelastic collisions to be relatively more probable than did Diamond and Yarnveld. In both cases the observed distribution \(F(\varphi)\) for small velocities tends toward the distribution of spherical particles.

For the important case of scattering in mercury vapor, Table 18 gives the figures found by both methods.

Table 18.

Single scattering of electrons in mercury vapor. \(F_e(\Phi)\) is the number of electrons elastically scattered through an angle \(\Phi\) per 1 unit solid angle, per primary electron, per 1 cm of path, at \(0.001\) mm pressure and at \(20^\circ\) C. \(F_k(\Phi)\) is the corresponding number for inelastic collisions. Primary electrons at 82 volts.

Diamond’s method (Arnot\({}^{250}\)) Langmuir and Jones	Diamond’s method (Arnot\({}^{250}\)) Langmuir and Jones	Diamond’s method (Arnot\({}^{250}\)) Langmuir and Jones	Diamond’s method (Arnot\({}^{250}\)) Langmuir and Jones	Diamond’s method (Arnot\({}^{250}\)) Langmuir and Jones
\(\Phi\)	\(F_e(\Phi)\)	\(F_k(\Phi)\)	\(F_e(\Phi)\)	Ratio
\(0^\circ\)	—	—	0.226	—
5	0.127	0.167	184	1.45
10	051	063	102	2.00
15	024	025	038	1.58
20	0144	0111	0089	0.69
25	0089	0061	0013	0.15
30	0056	0 36	0003	0.05
40	0028	0019	—	—
50	0015	0011	—	—
60	0005	0004	—	—

In Table 18 the first values were calculated from Arnot’s data, the last values were calculated from the data of Table 17 and equation (121), assuming \(P_e = 0.0204\), \(\varphi_e = 11.3^\circ = 0.198\) radians.

d) Diffusion of electrons has not yet been studied experimentally with sufficient accuracy to make it possible to use the results for checking theories similar to that developed in section D(1d). Experimentally it is easier to measure mobilities and from them to calculate diffusion constants by equation (105).

e) Mobilities of positive ions have not yet been studied under conditions in which the nature of the ion would be established sufficiently well for the results to be usable for checking equations of the type of equation (117). This quantity is more suitable for combining theory and experiment in order to investigate the nature of ions than for checking the theory itself.

The electron mobilities are known for many gases over a wide range of fields and pressures, and these data make it possible to test the theory. In this respect the only theory that lays claim to a rigorous derivation and that contains no arbitrary constants is the theory expressed by equation (116). Following custom, this equation may be represented by means of the “mobility constant” \(K\) (which in reality is not a constant, since it depends on temperature), defined as the mobility under standard conditions of temperature and pressure. \(K\) may be expressed by the formula:

\[ K=\frac{P}{760}\cdot \frac{273}{T}\mu, \tag{122} \]

if \(\mu\) is substituted from equation (116) and the values of the constants are introduced, then this equation takes the form:

\[ K= \frac{271\,000\,\lambda_1 \dfrac{273}{T}} {\left[1+\left(1+1\,106\,000M\lambda_1^2\left(\frac{E}{p}\right)^2\right)^{-\frac{1}{2}}\right]^{\frac{1}{2}}}, \tag{123} \]

where \(\lambda_1\) is the mean free path of the electron at \(p=1\) mm and \(273^\circ K\), \(M\) is the molecular weight expressed with respect to \(M_H=1\), and \(E\) is the field in volts per centimeter. The numerical constants differ slightly from those published earlier \(^{252}\), and, as we believe, are better suited for representing the theory.

The experimental results agree unexpectedly closely with the predictions of this theory, if one takes into account the considerable deviations of the conditions of actual collisions from the simple kinetic theory, as well as the fact that the constants of the equation are not arbitrarily assigned constants. Figs. 32 and 33 give a comparison of the theory with experiment for two cases; other cases have been considered by Compton \(^{252}\).

These and analogous results show that the theory gives results of the correct order of magnitude and correctly

indicates the character of the change of \(K\) as a function of \(\dfrac{E}{p}\) even with respect to the peculiar intersection of the curve with the axis at \(\dfrac{E}{p}=0\). Thus the general idea underlying the theory is probably correct, and the inaccuracy is due to the fact

Fig. 32. The “constant” of electron mobility; curve (1)—theoretical according to equation (123); curve (II)—experimental results of Bailey \(^{258}\).

that the theory uses an excessively simplified representation of the kinetic theory of collisions. In precisely what way the theoretical results should be modified by introducing experimental values of the mean free paths is not yet clear. For example, Fig. 25 shows that experimental determinations of mean path lengths give values considerably smaller than those to which the kinetic theory leads for the case of such small velocities as occur in the mobility experiments. However, the introduction of such small free paths should rather emphasize,

could eliminate the discrepancies shown in Fig. 32. Taking into account a possible nonideal elasticity of the collision does not give agreement, since this cause does not change the value of \(K\) for \(\frac{E}{p}=0\), but only changes the subsequent course of the curve. Nevertheless, in the case of nitrogen the fact

Fig. 33. “Constant” of electron mobility; solid curve drawn from theoretical equation (123); experimental results according to Loeb[^254], Townsend and Bailey.

that the scattered electrons are concentrated in the direction of the initial flight (small scattering angles) to a greater degree than would be the case for an elastic impact introduces a correction that increases the mobility and hence goes in the direction of smoothing out the difference between theory and experiment.

On the other hand, in hydrogen and helium the experimental values of the mobility constants are somewhat smaller than those predicted by theory. Consequently, in these cases the experimental free paths at small velocities are significantly smaller than the quantities obtained from the kinetic theory. This in turn gives a correction in the necessary direc-

influence that smooths out the discrepancy shown in Fig. 33 between theory and experiment.

However, this correction is somewhat too large, and in order to obtain agreement again it is necessary once more to take into account the excess scattering in the initial direction. In the case of hydrogen and helium, the quantum theory of scattering—which, as we have seen, is well confirmed by experiment—predicts that the excess scattering in the initial direction is less pronounced at low velocities than at high ones, and that as the velocity approaches zero the scattering tends to become uniform in all directions, as is required by the simple kinetic theory. It may be said that these considerations are quite sufficient to modify equation (123) so that it agrees with experiment; however, no test of the theory has yet been undertaken in this direction.

Clearly, a theory of mobilities based on the true laws of electron scattering at low velocities must be extremely complicated. It may be that, by adopting approximate analytic functions to represent the variation of the mean free paths and scattering angles as functions of the electron velocities, one will be able to find an approximate solution more satisfactory than equation (123).

There is another way of testing the mobility equations. This way avoids all considerations of a definite energy on which equations (117) and (123) are based, and proceeds from the more general equation (107), constructed only on the assumption of a Maxwellian distribution of velocities and the assumption that all directions of motion after a collision are equally probable. The method of Langmuir and Mott-Smith, already mentioned earlier285, makes it possible to test the character of the velocity distribution and to find the mean electron velocity in the case of a Maxwellian distribution, provided that the ionization of the gas is sufficiently intense. In these cases equation (107) makes it possible to calculate directly the mobility \(u\), if the mean free path is known, or conversely. Experi-

mentally easy to determine \(\psi\), for the total current carried by the electrons is equal to \(\int NeE\mu\,dS\), the integration extending over the entire transverse section of the current, while \(N\) and \(E\) are given by measurements with probes. Hence, in practice, the mean free paths of electrons \(\lambda\) are calculated by introducing the experimental results into equation (107). For mercury vapor it was found \(^{256}\) that the mean free paths calculated in this way are somewhat smaller than the values found from the kinetic theory. This, in turn, is in agreement with the experimental values shown in Fig. 26 for small velocities, with which we are concerned here (for most parts of the discharge—of the order of 2 volts or less). Further checks, carried out by this method for various conditions and other gases, are very desirable.

1. Quantum theories of impact and scattering require a special survey, and therefore are considered here only insofar as is necessary for completeness in discussing scattering phenomena, which are so important for gas discharges. There are three cases that are treated with great success by quantum theory: a) mean free paths for very small deflections—such as occur in measurements of mean free paths; b) the distribution of angular scattering as a function of velocity, which leads to a theory of the type of scattering studied by Dymond and Tarnwell; c) the anomalous transparency of certain molecules for very slow electrons—the Ramsauer effect, apparently observed for all atoms or molecules with a closed outer shell of 8 electrons. All these theories apply only to electrons, but the theory for case a) can also be extended to the case of deflections of positive ions. We shall consider these three cases very briefly.

a) The mean free paths of electrons were investigated by Wick \(^{257}\) by means of the perturbation method and through the apparatus of the old quantum theory. The problem may be formulated as follows: “how far from the center of the molecule must the initial trajec-

the trajectory of the electron, so that it experiences a deflection \(\varphi'\). Since the theory applies only to small values of \(\varphi\), the method of perturbations is used, in the following way: first it is assumed that the electron moves without deflection, in the direction of its initial trajectory. On its path it exerts a perturbing force on the molecule. Then the force with which this perturbed molecule acts on the electron is used to compute its deflection. This force also includes the polarization force equal to \(\dfrac{(K-1)e^2}{2\pi N r^5}\), where \(K\) is the dielectric constant, \(N\) is the number of molecules per unit volume, and \(r\) is the distance between the electron and the molecule. This force may also include the action of the permanent field of the molecule, if the latter possesses an invariable dipole moment and if the electron in the orbit of the molecule has a period \(\tau_1\) comparable with the time during which the electron passes through the molecule. Here a peculiar resonance effect occurs, due to the appearance in the molecule of a dipole moment on account of the orbital electron, even in those cases when this molecule has no dipole moment in phenomena with a considerably larger natural period. The deflection \(\varphi\), calculated in this way, proves to be proportional to:

\[ \varphi \sim \frac{U(a)}{\frac{mv^2}{2}}\cdot \Phi\!\left(\frac{\tau_1}{\tau_2}\right), \tag{12a} \]

where \(U(a)\) is the potential energy of the electron at the distance \(a\), \(a\) is the closest distance of the molecule from the initial trajectory of the electron, \(\tau_2\) is determined by the relation \(\tau_2=\dfrac{a}{v}\), and \(\Phi\) is a kind of resonance function.

If the molecule possesses high symmetry, then the potential energy is due solely to the above-mentioned polarization force, and \(\varphi\) will be proportional to

\[ \varphi \sim \frac{(K-1)e^2}{4\pi N a^4 m v^2}. \tag{12b} \]

Experimentally we find the fraction of electrons deflect-

…at an angle greater than a certain given angle \(\varphi_0\), determined by the construction of the apparatus. In other words, we find the fraction of electrons whose trajectories pass within a certain distance from the molecules. This fraction will be:

\[ Nq = N\pi a^2 = \alpha = \frac{1}{\lambda}, \]

where \(q\) is the effective cross-section of the molecule, and \(\alpha\) (Figs. 22 and 26) is the total effective cross-section of the molecules per unit volume. Thus, from equation (125) we find

\[ \alpha v = \frac{v}{\lambda} = \operatorname{const.}(K-1)^{\frac{1}{2}} . \tag{126} \]

It is evident from Figs. 25 and 26 that the curve for \(\mathrm{H}_2\) resembles a hyperbola and that the curve for Hg is more or less similar to it. In fact, it turns out that a quantitative estimate of the constant in equation (126) leads to calculated values of \(\alpha\) which reproduce the real values, in the cases of \(\mathrm{H}_2\), Hg, Zn, Cd, with an accuracy of about 10%.

If, on the other hand, the molecule possesses a dipole moment \(\mu\), oscillating or rotating as a consequence of the orbital motion of the electron with period \(\tau_1 = \frac{1}{\omega}\), then equation (124) takes the form

\[ \varphi \sim \frac{\mu v e}{a^2 m v}\,\varphi\!\left(\frac{v}{\omega a}\right) \tag{127} \]

or

\[ \alpha v^2 = \operatorname{const.}\left[\varphi\!\left(\frac{v}{\omega a}\right)\right]^{\frac{1}{2}} . \]

Here \(\alpha\) must decrease with increasing \(v\) more rapidly than in the case of merely polarizable molecules; it must be greater than in the case in which polarization alone acted; it must increase with atomic volume for atoms lying in one and the same column of the periodic system; and finally it must attain a maximum value near \(v = \omega a\).

All these characteristic features are found in a sharply pronounced form for the noble gases and, to a lesser degree, for Hg and N₂.

From what has been said it is clear that the theory of Dvicke set forth above is based on precisely those physical phenomena which in fact play a role in the scattering of electrons in collisions. The very same considerations can be used in the theory of the scattering of positive ions through small angles, with the special feature that in this latter case it is necessary to take mutual polarization into account.

b) The distribution of the scattering angles of electrons as a function of velocity was treated by means of wave-mechanical methods for atomic hydrogen and other one-electron systems—by Born²⁵⁸ and Somerfeld²⁵⁹—and by Mott²⁶⁰ for the case of helium. Some results of these investigations have already been discussed in connection with Fig. 31. We would go too far beyond the subject under discussion if we were to develop here the wave-mechanical theory of scattering, but the basic ideas and results must be formulated.

An electron with mass \(m\) and velocity \(v\) is treated as a plane wave with wavelength \(\dfrac{h}{mv}\) and amplitude \(\psi\); the product of \(\psi\) by the complex conjugate quantity \(\psi\), for each given point, measures the mean charge density or the probability of finding the electron at that point. The wave function satisfies Schrödinger’s equation:

\[ \Delta\psi+\frac{8\pi^{2}m}{h}(E-V)\psi=0, \tag{128} \]

where \(E\) is the total energy, and \(V\) is the potential energy of the electron. If the corresponding values of \(E\) and \(V\) are substituted into this equation and it is solved for \(\psi\) as a function of the coordinates, then the distribution of \(\psi\) gives the scattering distribution. To apply this method, the quantities \(V\) characteristic of the fields surrounding the various types of atoms must be known.

The Ramsauer effect, or the great transparency of certain gases for electrons of low velocities, is in turn well explained by wave mechanics[^261]. This phenomenon is observed for such gases as Ne, A, etc., whose molecules are so symmetrical that their electric fields decrease very rapidly with distance. Thus the function \(V\) in equation (128) has appreciable values only at very small distances from each molecule. Therefore, as far as \(V\) is concerned, the molecule is an object of very small linear dimensions. On the other hand, an electron of very small velocity \(v\) has a relatively large equivalent wavelength \(\frac{h}{mv}\). Thus this combination of a molecule possessing a very small external field and a slow electron with a large equivalent wavelength represents a case analogous to a small object in the path of a stream of waves of large wavelength. In this case the object has a relatively small capacity for scattering waves, which pass through the object unchanged.

4. Oscillations of the “plasma” of electrons and ions. These constitute a quite distinctive type of oscillations that can be obtained in discharge devices and are characterized by the fact that they are entirely independent of the network constants \(L, C, R\). These oscillations were first discovered by Penning[^262], and it was assumed that they were the cause of the unexpectedly rapid scattering of electrons in strongly ionized gases, discovered by Langmuir[^263] and later studied by Dittmer[^264]. Langmuir and Tonks[^265] partially explained this phenomenon in the following way.

“Plasma” is defined as a region of ionized gas in which the concentrations of electrons and positive ions are approximately equal to one another. If in such a region a group of electrons at \(x\) is displaced in the direction \(x\) by an amount \(\xi(x)\), the function \(\xi(x)\) satisfying the condition that \(\xi(x)\) on two parallel boundary planes, then an electric force is created, caused by the nonequilibri—

shifted by the space charge, which tends to displace these electrons back toward the equilibrium state of zero space charge. The change in the electron concentration and the resulting field may be calculated from Poisson’s equation and are equal to:

\[ \delta n=-n\frac{d\xi}{dx} \quad \text{and} \quad \frac{dE}{dx}=4\pi e\delta n, \]

where

\[ \frac{dE}{dx}=4\pi ne\frac{d\xi}{dx} \]

the integral of which will be

\[ E=4\pi ne\xi. \]

Since the restoring force, as is seen from the last equation, is proportional to the displacement, it is clear that the motion of each electron is a simple harmonic oscillation occurring according to the equation

\[ m\ddot{\xi}+4\pi ne^2\xi=0. \]

The solution of this equation gives the natural frequency

\[ \nu_0=\left(\frac{ne^2}{\pi m}\right)^{\frac{1}{2}}=8980n^{\frac{1}{2}}. \tag{129} \]

Thus we obtain a natural frequency proportional to the square root of the electron concentration \(n\). The usual values of \(n\) in low-pressure discharge tubes are \(10^{10}\ \text{cm}^{-3}\), which leads, according to equation (129), to the value \(\nu_0=9\times10^8\) cycles, corresponding to a radio-wave length of approximately \(33\ \text{cm}\).

The velocity of propagation of these waves will be:

\[ v=\lambda\left(\frac{ne^2}{\pi m}\right)^{\frac{1}{2}}. \tag{130} \]

Since the velocity is proportional to the wavelength, the waves exhibit great dispersion, and the group velocity of the waves tends to zero. Thus, although these waves can propagate in space, they do not carry energy.

The mean amplitude of these oscillations can be estimated if each volume element is regarded as an independent harmonic oscillator in equilibrium with the electrons themselves, assuming that we know the lower limit of the magnitude of this volume element. As a plausible guess we may take it to be equal to the cube of the “Debye” distance \(^{266}\), expressed by the formula:

\[ \lambda_D=\left(\frac{kT_e}{8\pi n e^2}\right)^{\frac12} =4.90\left(\frac{T_e}{n}\right)^{\frac12}\ \text{cm}, \tag{131} \]

where \(T_e\) is the “temperature” of the electrons (\(\lambda_D\) is the distance at which the mean potential near a charged plane in an ionized gas is \(1/e\) part of the potential of the plane itself). If this assumption is adopted, then the total energy density of the electric field of the oscillations will be:

\[ \frac{E^2}{8\pi} =\frac{3}{2}\left(\frac{1}{\lambda_D}\right)^3 kT_e, \tag{132} \]

where

\[ \begin{aligned} E&=96^{\frac12}\pi^{\frac54}e^{\frac32}n^{\frac34}(kT_e)^{-\frac14}\\ &=1.17(10)^{-6}\, n^{\frac34}T_e^{-\frac14}\ \text{volts/cm}. \end{aligned} \]

This relation probably gives values for \(E\) that are too large, since it is likely that the minimal volume element should be larger than \(\lambda_D\), although of the same order of magnitude.

It is evident that any homogeneous beam of electrons, passing through an ionized gas, tends to acquire a random distribution of velocities about the mean velocity, at the expense of the fields of these oscillations of the electron plasma. These oscillations therefore provide the possibility of exchange between ions and electrons on a par with individual collisions.

The described oscillations of the space charge of electrons relative to the more inert space charge of positive ions constitute the “oscillations” of the electron plasma. There also exist proper oscillations

positive ions, which Langmuir calls ion-plasma oscillations. For this case, arguments analogous to the preceding ones give the frequency

\[ \nu_p=\left(\frac{ne^2}{\pi M+\dfrac{ne^2MK^2}{kT_i}}\right)^{\frac12}. \tag{188} \]

If the wavelength is small, then this equation in form reduces to equation (129), with the ion mass \(M\) instead of the electron mass. On the other hand, for long waves the frequency approaches \(\left(\frac{kT_i}{M}\right)^{\frac12}\lambda\), which gives waves propagating with velocity \(\left(\frac{kT_i}{M}\right)^{\frac12}\). These latter waves are analogous to sound waves passing through an ionized gas, and their frequencies are usually less than \(5\times 10^5\) per second. The boundary between these two types of ion-plasma oscillations is given roughly by the Debye wavelength (equation 131).

In Part II of this article the authors will consider real types of discharges—the arc, spark, glow discharge, and corona—and will attempt to interpret their characteristics on the basis of the data on the fundamental processes considered in the present Part I.

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17. Faxen and Holtsmark. Ztschr. f. Physik 45, 307, 1927; Holtsmark. Naturwiss. 16, 814, 1928; ibid. 17, 365, 1929; Ztschr. f. Physik 48, 231, 1928; ibid. 55, 437, 1929.

18. Penning. Nature August 28, 1926; Physika 6, 241, 1926.

19. Langmuir. Phys. Rev. 26, 585, 1925.

20. Dittmer. Phys. Rev. 28, 507, 1906.

21. Langmuir. Proc. Nat. Acad. Sc. 14, 627, 1928; Tonks and Langmuir. Phys. Rev. 33, 195, 1928.

22. Debye and Huckel. Phys. Ztschr. 24, 185, 305, 1923.

1. Mean free path. ↩