ELEMENTARY PROCESSES IN THE IONIZATION BY IMPACT OF MATERIAL PARTICLES*

G. Kallmann and B. Rosen, Berlin

g) Ionization processes in adsorbed gas layers.
h) Ionization of solids.
II. Probability of ionization.
Addenda.
I. An outline of the theory of ionization processes. 1. Classical theory.
2. Quantum-mechanical theory of collision.
II. Experimental method. 1. Objects of investigation and difficulties.
2. Method I. Directed vapor jet. 3. Method II. Differential pumping. 4. Method III. Low uniform pressure.
(The numbers next to the authors’ names refer to the bibliography at the end of the article.)

g) Ionization Processes in Adsorbed Gas Layers

Let us consider briefly also the ionization processes on solids. Here one must distinguish between processes occurring in the layer of gas adsorbed on the surface and processes taking place in the solid itself.

Processes on the surface have only a few times been the subject of detailed investigation. Here one should mention first of all the works of Wolfenden^122 and Kistiakowsky^123, ^124. Under electron bombardment of a metallic surface situated in a rarefied gas, the onset of ionization can often be observed already at potentials several volts lower than the ionization potential of the gas present in the volume. In this case these po—

* Continuation; see Uspekhi Fizicheskikh Nauk, vol. XII, issue 1.

potentials also depend on the type of surface. These effects can be explained by assuming that ionization occurs in layers of gas adsorbed on the surface. Apparently, the adsorbed molecules are in a different energy state than molecules in the bulk, and sometimes they are adsorbed not even in the form of molecules, but in the form of atoms. These circumstances apparently lower the ionization energy. We shall confine ourselves to presenting a summary of the processes of this kind that have been studied so far (Table VIII). It is possible that, in addition to the processes listed in the table, many cases in which the appearance of various kinds of ions was observed at such low potentials that, theoretically, they could not have arisen from normal molecules, are also explained by analogous processes.

TABLE VIII

Ionization of adsorbed gas layers
(according to Kistyakovsky)

Such, for example, is the formation of H+ from H₂ at 16.5 V[^125] and of O+ and CO+ from CO₂ at 17 and, respectively, 18 V.

3. Ionization of solids[^127–^148]

For the study of ionization processes in solids, one must first of all thoroughly degas the surface subjected to electron bombardment. After this, the secondary electrons knocked out by the primary electrons are studied, and from this an attempt is made to draw conclusions about the ionization process itself. The chief difficulty is—

...of this kind of research lies in the fact that, as a result of the appearance of X-rays, whose very origin is due to the primary ionization of atoms, it is often impossible to decide definitely which secondary electrons were ejected by X-rays and which by primary electrons.

There are, in the main, two methods by which one can investigate electrons ejected from a surface. The first method, consisting in the fact that the velocity of the ejected electrons is spectroscopically analyzed by a magnet or a retarding field, has recently been especially improved by Rudberg 130. The second method—the method of Richardson and his collaborators 132—138—consists essentially in studying the ratio of the intensity of the primary electron beam to the intensity of the secondary one as a function of the velocity of the primary electrons.

In spectroscopically analyzing the velocity of secondary electrons, it turns out that, on the one hand, a certain fraction of all electrons has the same velocity as the primary electrons (these are those secondary electrons which have undergone multiple scattering 128 without appreciable loss of velocity), while, on the other hand, the great bulk of the electrons have very small velocities 139—148. These are, evidently, electrons knocked out of the atom by the impact of primary electrons and thereby acquiring some additional kinetic energy.

The velocity distribution of these electrons, in a first approximation, does not depend on the velocity of the primary electrons 140, 141. At the same time, the number of secondary electrons formed per one primary electron increases with increasing velocity, and their yield at high primary velocities exceeds 100%. The fact that the yield may exceed 100% is explained by the fact that the primary electrons, along their path, can ionize several times and that the faster secondary electrons can also ionize.

In addition to these two groups of electrons, in the velocity spectrum of the ejected electrons there were observed electrons whose energy was smaller by a definite amount than the energy

primary electrons, and from both of the above-mentioned groups they were separated by a deep minimum in the velocity spectrum. These are, apparently, scattered primary electrons that have undergone an inelastic collision with the solid, since the same energy losses occur at different velocities of the primary electrons. The mechanism of these collisions is not yet fully explained. The energy losses depend for the most part on the substance of the solid. For some substances, well outgassed by heating in vacuum, the values of such energy losses are given in Table IX.

TABLE IX

Discrete losses (in V) of electrons reflected from solids (according to Rudberg¹³⁰)

Richardson and his collaborators determined the ratio of all electrons flying off from the surface (i.e. the sum of all velocity groups) to the number of primary electrons as a function of the primary velocity. They find that this ratio of intensities, at certain potentials, proves to be discontinuous, and they connect this with the velocity losses observed by Rudberg. Analysis of this discontinuity leads Richardson¹³² to think that what is involved here is the excitation of so-called structural electrons present in crystals, not bound to atoms, but also not identical with free electrons. The question of whether all the phenomena occurring when an electron strikes a solid can be explained by excitation or ionization of such electrons has not yet been resolved.

II. PROBABILITY OF IONIZATION

Let us now consider in somewhat greater detail the question of the probability of ionization. First of all, let us turn to ionization by slow electrons; here the question is only that of the ionization of the most weakly bound electrons. In measurements of the ion yield, generally speaking, only the total number of ions of one kind per electron is measured. It is not distinguished whether the formation of these ions was caused by different excitations; often even different kinds of ions are measured together. But since for the most part one kind of ion proves to be strongly predominant, the errors are not large and can be easily corrected. In such yield measurements two circumstances are of interest. First, the absolute value of the effective cross section with respect to ionization and, second, the dependence of the ion yield on velocity. These questions have recently been investigated in detail in many works 149–159a; the older works are discussed in the book by Franck and Jordan 1.

Fig. 8. Smith apparatus 157 for measuring the ion yield under electron impact.

Here we shall describe only the most recent method of measuring the yield in ionization. It consists in the use of Bleakney’s mass spectrograph 22, deprived of the part of the apparatus serving for the measurement of \(e/m\). The apparatus is shown in Fig. 8. It is located in a magnetic field \(H\), so that the electrons emitted by the incandescent filament are focused into a very narrow beam. Ions formed between \(P_2\) and \(P_3\) are attracted by a weak field to the plate \(P_2\). If the electron current passing through the diaphragm \(S_2\) is accurately known, then at a known (sufficiently small) pressure the ion yield is determined directly by the ratio of the measured ionic current to the ionizing electron current. In doing this, special care is taken to avoid any reflection of electrons from the walls. This is achieved by catching the electrons with a special Faraday cylinder \(F\), into which all electrons reflected from the walls enter (between \(P_1\) and \(F\) pri-

... a strong electric field was applied, attracting to the wall \(F\) all electrons aligned from \(P_1\)).

TABLE X

Effective cross section for ionization at electron velocities corresponding to maximum yields

The advantage of such an apparatus, proposed by Johnson \(^{154}\) and further improved somewhat by Smith \(^{157}\), is that a very well-defined electron beam is obtained and that, apparently, it is possible to avoid all secondary processes and reflections. With this apparatus only the total yield of ions of all kinds is measured.

If, at the same time, the relative yields of ions of different kinds are also measured by the Bleakney method \(^{155, 158}\), then the absolute yields of the individual ion kinds are obtained.

Measurements by this method in their general features confirmed the results of Compton and Van Voorhis \(^{151, 152}\) and of Jesse \(^{150}\), obtained with less perfect apparatus. The results of all these works are given in Figs. 9—13*.

* In a more recent work \(^{159a}\) Smith gives an ionization curve for Hg, somewhat different from Bleakney’s curves (see Fig. 9); \(N\) reaches a max-

They show the yields of various ions as a function of the energy of the impacting electrons, the measure of

Fig. 9. \(J\)-curves of singly and multiply charged mercury (after Bleakney \(^{158}\)).

Fig. 10. \(J\)-curves of singly and multiply charged argon (after Bleakney \(^{158}\)).

the ion yield being taken as the number \(N\) of ions formed by 1 electron over 1 cm of path at a pressure of 1 mm Hg and \(0^\circ\).

Fig. 11. \(J\)-curves of singly and multiply charged neon (after Bleakney \(^{158}\)).

Fig. 12. \(J\)-curves of noble gases according to Smyth \(^{157}\), according to the measurements of Compton and Van Voorhis \(^{151, 152}\), and according to the measurements of Yost and Klem \(^{149}\).

It is clearly seen that the yield of all ions increases with increasing electron energy, and that the maximum is reached only when this energy is several times greater than the energy

maximum at 85 V and equal to only 19.4. And the fall-off at high velocities is also not so steep.

ionization (and not by a factor of two, as should have been the case according to Thomson’s classical theory\(^5\)). After the maximum the yield decreases slowly.*

From these data it is easy to calculate the effective cross sections for various ionization processes. These values are given in Table X (third column).

Fig. 13. \(J\)-curves according to Compton and Van Voorhis\(^ {152}\).

In the first column the kind of ions appearing in the process just described is indicated; in the second—for comparison—the gas-kinetic effective cross sections of the corresponding neutral atoms or gas molecules.

As can be seen, the maximum effective cross sections in ionization are considerably (approximately by a factor of 10–20) smaller than the gas-kinetic effective cross sections (with the exception of Hg, for which the ionization effective cross section is about 60% of the gas-kinetic one). The effective cross sections for multiple ionization are, for all gases, much smaller than for simple ionization. On average they are equal to roughly 5–10% of the effective cross section for ionization. Only for Hg is the effective cross section still fairly large also for multiple ionization.

In some contradiction with these results are, as it were, the measurements of Tippel\(^ {153}\) and Funk\(^ {156}\), who determined the ion yield by an entirely different, very elegant

* Tate, on the basis of Smith’s measurements\(^ {157}\), gave an empirical formula representing the ionization function in He between \(V_a = 60\) and \(4500\ \mathrm{V}\) with sufficient accuracy. It reads:

\[ \varepsilon = 3.383\,(V_0/V_a)^{1/2}\left[1-e^{-54 V_0/V_a}\right]^{1/2} \times \left[1-e^{-(V_a - V_0)/2.28 V_0}\right], \]

where \(V_a\) is the ionization energy.

method. They ionized a beam of atoms by means of an electron beam perpendicular to it and measured the number of ions formed in the atomic beam. True, from such measurements one can obtain the absolute yield only by means of a rather complicated calculation. The authors find the maximum of ionization already at an electron energy equal to twice the ionization energy (agreement with Thomson’s theory). They measured only the yields in Na, K, and Hg. The results for Na and K are shown in Fig. 14. For Hg completely different values were found than by Bleakney’s method. It seems to us,

Fig. 14. Ionization functions of sodium and potassium according to Funku[^156]. The abscissae are the electron velocity. The ordinates on the left are the effective cross section in $\mathrm{cm}^2/\mathrm{cm}^3$, on the right—the yield as a percentage of the gas-kinetic e. c. On the left is the curve for potassium (ionization potential 4.3). On the right is that for sodium (ionization potential 5.1).

that Bleakney’s method, owing to its simplicity, gives more reliable results.

The curves representing the ionization function are, generally speaking, monotonic, except for their first part. No indications of ionization of inner shells were found. Only in one case, namely for potassium, were inflections found when the potential was increased, namely at 40.81 and 122 V, the second maximum being 18%, and the third 11%, higher than the first[^159]. From the brief communication that appeared on this it is not clear how these maxima can be explained. Inflections at the beginning of the ionization curve, i.e. at potentials only very slightly exceeding the ionization threshold, were discussed in detail on p. 113.

The ionization-yield curves considered so far correspond to ionization of the outer shells of the atom. Ionization of the inner shells of the atom was also investigated

in a whole series of works 160–172. To determine the ion yield, the yield of X-rays arising when solids are bombarded by fast electrons was quantitatively determined. In these measurements two kinds of difficulties arose. First, the experimentally obtained radiation intensity depends not only on the direct excitation of the radiation by electrons, but partly also on the photoelectric excitation by all the X-radiation arising in the solid (anticathode). Then, the velocity of the electrons inside the solid is nonuniform owing to braking, and therefore only the integral ion yield, produced by the impact of electrons of different velocities, is always measured. This latter difficulty can to some extent be avoided by making measurements in sufficiently thin layers (Webster 165–167 measured in a thin layer of silver 20–300 Å thick on a beryllium substrate; Lorentz 161, 162—in thin layers of aluminum). Another way is to measure the interfering effects and introduce the corresponding corrections into the values obtained (Wishak 168 for Cu, Cr, Mo, Ag).

The results obtained in this way show that the probability of ionization increases with increasing voltage. According to Webster 166 the maximum of the ionization probability for silver lies at an electron energy three times greater than the excitation energy.

Wishak finds values less than twice the ionization energy. Apparently, as the atomic number of the ionized atom decreases, the maximum approaches twice the ionization energy. For Al, according to Johnson’s measurement 172, the maximum occurs at an energy 2.6 times greater than the ionization energy.

However, up to now there have been no accurate quantitative data on the ionization probability at a definite electron velocity.

Appendix I

OUTLINE OF THE THEORY OF IONIZATION PROCESSES

1. Classical Theory

Application of the Laws of Conservation of Energy and Momentum

Let us denote the incident particle by \(S\), the atom or molecule being struck by \(A\), and the electron ejected from \(A\) by \(e\). In a collision of \(S\) with \(A\), the electron \(e\) may, as a result of the interaction energy between \(S\) and \(A\) (denote this energy by \(W\)), acquire sufficient energy to separate from \(A\). The minimum initial velocity \(v\) of the particle \(S\) (relative to \(A\)) at which ionization is still possible can be calculated by applying to the entire system \(SA\) and \(e\), before and after the collision, the laws of conservation of energy and momentum. Namely, only the energy

\[ E=\frac{Mv^{2}}{2}=\frac{m_A}{m_S+m_A}\,T_S, \tag{1} \]

can go into ionization, where

\[ M=\frac{m_S m_A}{m_S+m_A}. \]

(\(T_S\) is the kinetic energy of the incident particle in the system in which \(A\) is at rest. The mass of the atomic electron \(m_e\) may be neglected in comparison with \(m_A\).)

Thus, owing to the law of conservation of the motion of the center of gravity of the whole system, only a part of the initial kinetic energy of the incident particle can be used for ionization.

If the incident particle is an electron, then ionization practically takes place only when

\[ T_S>J\,(M\sim m_S,\; m_S\ll m_A), \tag{1a} \]

where \(J\) is the ionization potential of \(A\). In the impact of ions and neutral particles (\(m_S\sim m_A\)), ionization is possible only at a large kinetic energy. If, for example, \(m_S=m_A\), then ionization can take place only when

\[ T_S>2J. \tag{1b} \]

For a more detailed analysis of the elementary process of ionization, it is necessary to examine more closely the mechanism of this process and to make several special hypotheses.

Thus, for example, Thomson \(^{5}\) indicates a method that makes it possible to calculate approximately the ion yield at least in the region of sufficiently large velocities (cf. also the literature 6–10). This device is based on the assumption that

one must take into account only the collision of the incident particle with the atomic electrons, i.e., the effect of the atomic nucleus on the incident particle may be neglected. In what follows the intrinsic velocity of the atom–electron system is also neglected. This is approximately possible in those cases when the velocity of the incident particle is sufficiently large and when sufficient energy is transferred to the electron. It is then assumed that ionization is caused by every collision in which the atom’s electron is given an energy exceeding the ionization energy. Thus the probability of ionization turns out to be equal to the probability of transfer by the particle \(S\) to the electron \(e\) of an energy greater than \(J\). These probabilities are formally described by an effective cross section. Formally, the effective cross section \(\Phi_W\) for collisions of two charged particles with charges \(Z_S\) and \(Z_e\) and masses \(m_S\) and \(m_e\), in which an energy exceeding \(W\) is transferred, is equal to

\[ \Phi_W = \frac{z_S^2 z_e^2 m_S}{4T_S m_e} \left[ \frac{1}{W} - \frac{1}{4T_S}\frac{(m_S+m_e)^2}{m_S m_e} \right], \tag{2} \]

assuming that the initial velocity of \(e\) is zero. Hence the effective cross section for ionization is

\[ \Phi_J = \frac{z_S^2 z_e^2 m_S}{4T_S m_e} \left[ \frac{1}{J} - \frac{1}{4T_S}\frac{(m_S+m_e)^2}{m_S m_e} \right]. \]

In order for ionization to be possible at all, the second term of the expression in brackets must be \(< 1/J\). From this one obtains the condition for the possibility of ionization, imposed on the kinetic energy \(T_S=\dfrac{m_S v_S^2}{2}\),

\[ T_S > \frac{J(m_S+m_e)^2}{4m_e m_S}. \tag{3} \]

For an electron impact this means

\[ T_S > J. \]

According to this theory, ionization by electron impact begins at \(T_S=J\), the yield increases from \(T_S=J\) to \(T_S=2J\), and then decreases again. At large velocities the yield decreases proportionally to \(\dfrac{1}{T_S}\). In the impact of ions, ionization according to this theory can begin only at \(T_S=\dfrac{m_S}{m_e}J\), i.e., at an energy exceeding the ionization energy by a factor of \(\dfrac{m_S}{m_e}\). Суще-

ELEMENTARY PROCESSES IN IONIZATION

there exist, however, numerous experiments showing that ionization begins considerably earlier. The disagreement with theory apparently arises from the fact that this theory is valid only for high velocities of the incident particle (high in relation to the velocity of motion of the electron in the Bohr orbit). From this it is already clear how dangerous it would be to extrapolate this theory too far; it is therefore also doubtful whether it can be applied to electron impact at electron energies close to the ionization energy. It is probably with this that the circumstance is connected that the maximum of the ionization probability does not always occur at twice the ionization energy, as this theory requires, but, generally speaking, lies considerably higher. It is true that at high velocities the formulas of Thomson’s theory make it possible correctly to estimate the order of magnitude of the energy loss of the incident particles. This classical theory was improved by Thomas ^10, who also took into account the velocity of the atomic electron in the initial state, i.e., in its Bohr orbit, as well as the influence of the atomic nucleus.

2. QUANTUM-MECHANICAL THEORY OF COLLISIONS

We now pass to the quantum-mechanical theory of collisions. Of course, this theory also introduces the same limiting conditions (1), (1a), and (1b) as the classical theory. The calculations here are based on the following assumption. It is assumed that there is given a certain plane monochromatic wave (the wave of the incident particle), falling upon an atom in which there is a moving electron. One investigates how, under the influence of this wave, the various states of the atom are excited. We are interested above all in the excitation of states corresponding to the continuous spectrum. In these states the electron, as is known, no longer revolves about the nucleus, but recedes to infinity. Thus, excitation of the continuous spectrum means ionization of the atom. The quantum-mechanical problem can be solved by means of Born’s approximate method ^11. Since this approximate calculation in practice reduces to an expansion in a series in powers of

\[ \left(\frac{v_0}{v}\right)^2 \]

(where \(v_0\) is the velocity of the electron in the Bohr orbit, and \(v\) is the velocity of the incident particle), and since in the practical realization of this method one can use only the first, or at best the second, approximation, this solution correctly conveys the results only for high particle velocities (\(T_s\) large in comparison with the ionization energy). This solution depends only on \(v^2\). Thus, for particles of different mass but of the same velocity, one obtains one and the same degree of approximation.

Thus the most interesting region—when the energy of the incident particle is equal to or greater than (but not much greater than) the ionization potential—is still inaccessible to a more exact theoretical

calculation. But the formula obtained for high velocities also gives remarkable results, and therefore we shall dwell briefly on it as well.

Bethe1, who developed this theory with the greatest completeness, gives the following expression for the production of ions. He calculates the frequency of the process in which the incident particle undergoes a change of momentum \(M(v-v')\), and the kinetic energy of the electron after the collision is equal to

\[ E_k=\frac{h^2}{8\pi^2 m_e}\,k^2, \]

where \(k\) is equal to the momentum of the electron after the collision multiplied by \(\dfrac{2\pi}{h}\), \(v\) and \(v'\) are the relative velocities of the particle and the atom before and after the collision, and

\[ M=\frac{m_S m_A}{m_S+m_A} \]

is the reduced mass. Hence, \(Mv\) and, correspondingly, \(Mv'\) are the momenta of the particles referred to a system at rest with respect to the center of gravity.

If by \(q\) one denotes the difference of the momenta before and after the collision, multiplied by \(\dfrac{2\pi}{h}\), i.e., if one puts \(q=\dfrac{2\pi}{h}M(v-v')\), then \(q\) is determined by the initial momentum, by the decrease in kinetic energy \(\Delta E=-E_0+E_k\) (\(-E_0\) is the ionization energy of the atom), and by the scattering angle \(\vartheta\) of the incident particle, i.e.,

\[ q^2=\frac{8\pi^2}{h^2}M\left\{(Mv^2-\Delta E)-\sqrt{Mv^2(Mv^2-2\Delta E)}\cos\vartheta\right\}. \tag{4} \]

The probability (and consequently also the effective cross section) for the elementary process in which \(q\) lies between \(q\) and \(q+dq\), the initial momentum is equal to \(Mv\), and an energy \(E\) is transferred to the electron, is equal to

\[ d\Phi_k(q)=\frac{2h^3}{\pi M^2v^2a^2}\left(\frac{M}{m_e}z\right)^2\frac{dq}{q^3}\,|\varepsilon_k|^2, \tag{5} \]

where \(z\) is the charge of the nucleus, \(a\) is the radius of the hydrogen atom, and \(|\varepsilon_k|^2\) is the sum of matrix elements of the form

\[ \left|\frac{\Sigma}{j}\int e^{i(qr_j)}\Psi_0\Psi_k(r_j)\,d\tau\right|^2, \]

where the sum over \(j\) extends over the different electrons in the atom, \(\Psi_0\) is the characteristic function of the atom in the normal state, and \(\Psi_k\) is the characteristic function of the continuous spectrum for which the electron possesses kinetic energy \(E_k\). The dependence on the angle through which the defle-

contained in \(q\), for, at given \(Mv\) and \(E_k\), \(q\) depends only on \(\vartheta\). Thus, in order to obtain the total yield of electrons of a definite energy, one must integrate (5) with respect to \(q\) from \(\vartheta=0\) to \(\vartheta=\pi\).

In this connection, in the first approximation, when \(\Delta E \ll \dfrac{Mv^2}{2}\), one may express \(q^2\) in the following way:

\[ q^2=\frac{8\pi^2}{h^2}M\left\{(Mv^2-\Delta E)(1-\cos\vartheta)+\frac{(\Delta E)^2}{2Mv^2}\cos\vartheta\right\}. \tag{6} \]

Owing to the presence in (5) of the factor \(\dfrac{1}{q^3}\), the effective cross section for impacts with small \(q\), i.e. for small deflections, is much larger than for large deflections. Practically, in general, the greater part of the ionization occurs as a result of these collisions with small deflections. For such collisions formula (5) is simplified and reads:

\[ d\Phi_k(q)=\frac{2h^2}{m_e^2v^2}\cdot\frac{z^2}{a^2}\,|x_{0k}|^2\,\frac{dq}{q}, \tag{7} \]

where \(|x_{0k}|^2\) is the sum of matrix elements of the form

\[ \int \sum_j x_j \Psi_0 \Psi_k(r_j)\,d\tau. \]

Since the probability of optical transitions is also equal to \(|x_{0k}|^2\), all ionization by fast electrons turns out to be approximately proportional to the probability of optical transitions. In particular, if ionization is also associated with excitation of another level (vibration of the nucleus or excitation of a second electron), then the probabilities of excitation of this kind by ionization should behave like the probabilities of optical excitations. All the results so far discussed are valid for any atoms.

Hydrogen atom

For hydrogen atoms we have:

\[ |x_{0k}|^2=\frac{2^8}{3}\cdot\frac{\alpha^6 k}{(\alpha^2+k^2)^5}\cdot \frac{e^{-4\frac{\alpha}{k}\operatorname{arc\,tg}\frac{k}{\alpha}}} {1-e^{-2\pi\frac{\alpha}{k}}},\qquad \left(\alpha=\frac{1}{a}\right). \tag{8} \]

It follows from this formula that the probability of ionization is much greater for those processes in which the electron of the atom acquires very little kinetic energy. Here \(k\) is determined from the equation:

\[ E_k=\frac{h^2}{8\pi^2 m_e}\,k^2. \]

The probability that, in ionization by electron impact, an electron of the ionized atom will receive, in excess of the ionization energy, a portion of kinetic energy four times larger is already a thousand times smaller than the probability of ionization with kinetic energy equal to zero. If the incident particle undergoes a strong deflection during ionization, then the atomic electron, generally speaking, will receive a considerable amount of kinetic energy. But such processes are in general comparatively rare.

The total number of primarily formed ions can be obtained from (5) by integration over \(q\), or, correspondingly, over \(\theta\) and over \(k\). Thus we obtain

\[ \Phi_i=0.285\,\frac{2\pi e^2 z^2}{R h m_e v^2}\ln\frac{2mv^2}{0.048\,Rh}. \tag{9} \]

This result is valid for incident particles of any mass, provided only that the velocity \(v\) is considerably greater than the velocity of the electron in its Bohr orbit. Here \(R\) is the Rydberg constant; hence \(Rh\) is equal to the ionization energy of the H atom. If \(m_e v^2\) is expressed through \(Rh=\dfrac{e^2}{2a}\), then we obtain

\[ \Phi_i=1.14\,\frac{z^2}{x}\,\pi a^2\ln\frac{2m_e v^2}{0.048\,Rh}. \]

Thus, the effective cross section for ionization is smaller than the gas-kinetic cross section of the H atom (equal to \(\pi a^2\)); namely, it is equal to this latter quantity multiplied by the logarithmic factor and by the factor \(\dfrac{z^2}{x}\).

Application to Atoms with Many Electrons

For more complex atoms with many electron shells, one can approximately determine [calculate from (7)] the effective cross section for ionization of an \(n\)-, \(l\)-shell, using hydrogen-like characteristic functions,

\[ \Phi_i^{nl}=\frac{2\pi e^2 z^2}{m_e v^2}\cdot\frac{c_{nl}Z_{nl}}{-E_{nl}}\ln\frac{2mv^2}{c_{nl}}. \tag{10} \]

Here \(Z_{nl}\) is the number of electrons in the \((nl)\)-shell. \(E_{nl}\) is the work of removing an electron from the \((nl)\)-shell, approximately equal to \(-E_{nl}\); \(c_{nl}\) is a constant, which for different shells lies approximately between 0.3 and 0.05; its values for different shells are:

Thus it is always somewhat more difficult to excite a shell with a larger azimuthal quantum number than a shell with a smaller azimuthal quantum number. In all other respects, the ionization of different shells is inversely proportional to

work function and to the square of the velocity of the incident particle. The classical theory set forth on pp. 301–302 would give, for the corresponding effective cross section, the value

\[ \frac{2\pi e^4 z^2}{m_e v^2}\cdot \frac{z_{nl}}{-E_{nl}} . \]

This value is similar to that obtained from the quantum-mechanical theory, only without the logarithmic term and without the factor \(C_{nl}\). A test of these formulas against Wisshak’s experiments \(^{168}\) on the excitation of the \(k_\alpha\) line as a function of velocity showed that the experiments agree rather with the classical formula than with the quantum-mechanical one.

At the same time, however, one must not forget that these experiments were carried out in a region lying somewhat above the ionization energy, i.e., where these formulas are certainly not yet strictly valid.

We have dwelt in relatively great detail on the theory of ionization itself because this is the only theoretical quantitative result so far obtained in this field. Thus, the probability of ionization for fast particles is approximately proportional to the probability of optical transitions and inversely proportional to \(v^2\), and the predominant processes are those in which the atomic electrons acquire only small velocities. This theory is applicable only for large velocities and for charged particles. If the incident particle is an ion, then the influence of the electron shell of the ion must also be taken into account.

It should also be noted that this theory does not take into account the fact that, in electron impact, the incident particle and the atomic electrons are particles of the same kind. When this circumstance is taken into account, it would turn out that the yields of ionization processes in which very fast atomic electrons appear would be somewhat smaller. For the region immediately adjacent to the ionization energy, a satisfactory theory still does not exist.*

In this chapter, whose aim is to describe known elementary processes, we shall also encounter ionization processes of a somewhat different character, such as, for example, charge exchange, etc. But since in this connection only separate groups of experiments will be discussed, the theory of these phenomena will be given in greater detail later, when the processes themselves are discussed.

Addendum II

EXPERIMENTAL METHOD*

1. OBJECTS OF INVESTIGATION AND DIFFICULTIES

Every apparatus, naturally, represents a certain compromise among all the various requirements imposed on it. Let us enumerate these requirements:

* Borrowed from the article by Smyth, Rev. of Mod. Phys. 3, 347, 1931.

a) the possibility of controlling the energy of the electrons,
b) high intensity, or sensitivity,
c) absence of collisions in the analyzer,
d) high resolving power of the analyzer,
e) absence of thermal dissociation.

Fulfilling condition (a) presents the same difficulties as arise in ordinary experiments for finding ionization potentials. They can best be avoided by calibrating the apparatus with the aid of a gas whose ionization potential is known. This method is not especially difficult, but it is not beyond reproach. The chief difficulties, however, are presented by fulfilling the last four conditions, which interfere with one another. Thus, for example, in order that the ionization intensity be large, there must be many collisions, i.e., a high gas density in the ionization chamber, and the slits of the mass spectrograph must be wide. But these conditions hinder the fulfillment of requirements (c) and (d). If, on the other hand, one tries to increase the number of collisions by increasing the power of the electron emitter, then it becomes necessary to use a hot filament with a large surface, which contradicts condition (e).

How far it has been possible to circumvent all these difficulties will be seen from a detailed consideration of the various experimental methods used in this field. These methods naturally fall into three classes, differing from one another in the way they meet requirements (b) and (c). Let us call them methods I, II, and III. We now proceed to their description.

2. Method I. Directed jet of vapor

It is clear that one way of satisfying conditions (b) and (c) is to make the gas density in the region of ionization higher than in the analyzer. This was done in the author’s first experiments by means of a directed jet of vapor crossing the ionization tube perpendicular to the stream of electrons, as shown in Fig. 15.

Fig. 15. Method I. Directed jet of vapor.

Mercury vapor from a heated reservoir enters tube \(A\) and condenses on the trap with liquid air \(K\). Electrons from filament \(F\) are accelerated by the field \(V_1\) and collide with Hg atoms in the space \(I\). The weak retarding field \(V_2\) extracts all the ions formed toward \(G_2\), whence the stronger accelerating field \(V_3\) accelerates them to the pla-

the wall \(D\). Slits \(S_1\) and \(S_2\) cut out a beam passing through the transverse magnetic field in the space \(M\). For certain values of \(\frac{e}{m}\), \(V_3\), and \(H\), the ions are deflected by just such an amount that they fall on slit \(S_3\) and charge plate \(P\), connected to a highly sensitive quadrant electrometer. The current through the electrometer is regarded as a function of \(H\) or \(V_3\) at various values of \(V_1\). In this way the relative number of ions with different \(\frac{e}{m}\), produced by the impact of electrons of various velocities, is determined. This method was developed by the author* for the study of mercury vapor and, somewhat later, by Kondratieff and Semenoff** for the study of vapors of a number of salts.

For several years this method received no further development, although it is undoubtedly the best for the study of vapors. Recently Nielsen*** used it to study negative ions in mercury vapor, and Ditchborn and Arnott**** used it in studying the ionization of potassium vapor. This latter work is of particular interest. In studying photoionization and ionization by positive ions from a Kunz-Makov source*****, Ditchborn and Arnott made the first step toward applying the general method of analysis of positive ions to ionization processes occurring not by electron impact.

Fig. 16. Method II. Differential pumping (Smyth).

3. Method II. Differential pumping

In many earlier works with positive rays it was customary to keep the pressure in the discharge tube higher than in the analyzer; to accomplish this, the “channel” in the cathode was made extremely narrow. Such a procedure is very desirable for our purposes as well, for it satisfies condition (c). But requirements (a) and (b) make its application difficult.

* Smyth, Proc. Roy. Soc., 102 A, 283 (1922).
** Kondratieff u. Semenoff, Z. Physik, 22, I (1924).
*** Nielsen, Proc. Nat. Acad., 16, 721 (1930).
**** Ditchborn a. Arnott, Proc. Roy. Soc., 123 A, 516 (1929).
***** \(Fe_2O_3\) with a small admixture of a salt of the corresponding alkali metal. Translator’s note.

Fortunately, work on this problem began soon after diffusion pumps, which make it possible to exercise greater control over pressure, had come into general use; this also made it possible to construct an apparatus of the type shown in Fig. 16.

In this apparatus the gas is continuously fed into the ionization chamber \(J\) and is pumped out of the accelerating space \(A\), the focusing space \(E\), and the magnetic space \(M\). In this way a pressure difference between \(J\) and \(M\) is maintained in the ratio from \(10/1\) to \(1000/1\), depending on the apparatus and the pressure region.

The arrangement of the electric and magnetic fields is practically identical with method I, namely—there is a field accelerating electrons from \(F\) to \(E_1\), \(V_2\), a weak retarding field attracting ions to \(E_2\), \(V_3\)—

Fig. 17. Method II. Differential pumping (Hogness and Lunn).

a strong field accelerating ions between \(S_1\) and \(E_2\), and \(H\)—a transverse magnetic field deflecting the ions toward \(V\) and toward the Faraday cylinder.

In the space \(C\) there is no electric field, and it is shielded from the magnetic field. The deflection in the magnetic field is \(180^\circ\), owing to which this apparatus for ion analysis is practically identical with Dempster’s apparatus* for the study of isotopes.

The method just described, with a number of small modifications, was used by the author, Hogness, Kalman, and their collaborators; moreover, this method enabled us to obtain a large part of the results we achieved. We shall not discuss in detail the apparatuses of different authors. The apparatus shown in Fig. 17 belongs to Hogness and Lunn**. It differs—

* Dempster, Phys. Rev., 11, 316 (1918).

* Hogness a. Lunn, Phys. Rev., 26*, 44 (1925).

differs from that shown in Fig. 3 in two essential respects, namely: the ionization chamber \(I\) is much longer, and the focusing space \(C\) has almost completely disappeared. Details may be found in the original paper.

We mention here only one apparatus, since in it condition (e)—the absence of thermal dissociation—was specially provided for. Smith and Stueckelberg*, working with \(N_2O\) and \(NO_2\), made special efforts to reduce thermal dissociation.

For this purpose they 1) used an oxide cathode, which operates at low temperature, and 2) designed the apparatus in such a way that, in addition to the pumping through \(S_1\), the gas current flowed all the time from the ionization chamber \(I\) past the filament to the outside. Their apparatus is shown in Fig. 18. In a very recent work Stewart and Olson**, working with propane and butane, did almost the same thing, placing a narrow slit between the space where the filament was located and the ionization chamber. They were able to maintain, in the space with the filament, a pressure of \(10^{-5}\) mm, whereas in the ionization chamber it was \(10^{-2}\) mm. They give no numerical data, perhaps because their intensities were too small for the critical potentials to be determined.

Fig. 18. Method II. Differential pumping. Special precautions to avoid thermal dissociation (Smith and Stueckelberg).

4. Method III. Low uniform pressure

In this method, used by Dempster and recently by Bleakney***, the pressure throughout the apparatus was so low that the mean free path of the ions was greater than their path in the analyzer. Bleakney satisfied requirement (b) by making the ionization region very long and parallel to the slits, also long. His analyzer also differs from the analyzers used in methods I and II. Drawings of his apparatus are given in Figs. 19 and 20.

The whole apparatus is inside a solenoid focusing the electrons from the electron beam \(FS\) into a narrow ribbon from \(S\) to \(P\). The ions are drawn out by the field \(V_2\) to the slit \(B\). In the space between the plates of the condenser \(C\) and \(V\), a magnetic field \(H\) and an electric field \(E\) act on them, com-

* Smiths a. Stueckelberg, Phys. Rev., 36, 472 (1930).

** Stewart a. Olson, J. A. C. S., 53, 1236 (1931).

*** Bleakny, Ph. Rev., 34, 157, (1929).

compensating one another when

\[ \frac{e}{m}=\frac{E^{2}c^{2}}{VH^{2}}, \]

where \(V\) is the part of \(V_{2}\) accelerating the ions.

Therefore ions with this value of \(\frac{e}{m}\) reach the collector \(K\), and the experiment consists in studying the current at \(K\) as a function of \(E\).

It is clear that this compensation is not an essential attribute of the method; the solenoid, however, is such a part, since a very long homogeneous magnetic field is required. This imposes restrictions on the magnitude of the magnetic field. This fact, and also the fact that there are so few ionizing collisions that even with a great length of the slits only small intensities are obtained, constitute a serious shortcoming of this method—namely, its low resolving power. It seems to the author that any method in which a uniform pressure is employed is, to some extent, subject to this shortcoming. On the other hand, it should be noted,

Fig. 19. Method III. Low homogeneous pressure. Bleakney’s scheme; side view.
Fig. 20. The same. Perpendicular section.

that the use of a solenoid and long slits can be combined with differential pumping. Nevertheless, the limitation on the magnitude of the magnetic field would still remain, and the method could hardly give great resolving power except in the case of very light ions or specially constructed powerful solenoids.

These three described methods may be varied in application to different problems, but these modifications are not so substantial that it would be worthwhile here to enter into a discussion of the various typical installations and the results obtained with them.

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1. Bethe. ↩