The Mobilities of Ions and Electrons in Pure Gases.

Transactions and Proceedings of the Royal Society of New Zealand, Volume 63, 1934, Page 285

The Mobilities of Ions and Electrons in Pure Gases. By C. M. Focken, B.Sc., B.M.E. (Melb.), D.Phil. (Oxon.), Otago University. [Read before the Otago Institute, 9th August, 1932; received by the Editor, 17th August, 1932; issued separately, June, 1933.] Purpose. The purpose of this paper is to compare the general mobility equations advanced by Langevin and by Compton, and to apply them to certain cases for the motion of electrons and positive ions in pure gases. Some aspects of the present state of our experimental and theoretical knowledge of the mobilities and nature of gaseous ions are discussed. Part I.—Mobilities of Ions in Pure Gases. 1.—Experimental Methods: In Conduction of Electricity Through Gases (1) Sir J. J. Thomson and G. P. Thomson describe eleven distinct methods for measuring the mobility of ions in gases. Some of these have proved much more satisfactory than others, and within the last few years a modification of one of them—Tyndall and Grindley's method (2)—has in the hands of Tyndall and his collaborators, and of Bradbury (3), yielded very valuable results. Tyndall and Grindley produce periodic flashes of ionization by a commutator which controls the electric field of rectangular wave-form between the measuring plates. The ionization is produced near one plate by a strip of polonium placed on a rotating wheel on the same shaft as the commutator. Laporte (4) has used a method similar in principle, but the flashes of ionization produced by the polonium are controlled by mechanical means, instead of by a commutator. He employed two similar circular discs with radial slits, rotating on a common axis. For mobility determinations in pure gases these methods (employing a polonium source) are unsuitable, for it has not been possible to devise such

an apparatus in metal and glass which can be thoroughly “baked out” and evacuated in order to eliminate gaseous impurities almost completely. Hence arises the importance of two methods developed from these periodic flash methods. They are identical in principle, and were independently devised by Van de Graaff (5), and by Tyndall, Starr, and Powell. (6). The ions are admitted to the constant field and from it to the collecting chamber by means of properly timed alternating potentials actuated by commutators, transformers, or valve oscillators. Some preliminary experiments with negative ions in moist air performed by Van de Graaff and the author (7) served to demonstrate the principle of the method, which has been used by Van de Graff (5) to measure the mobility of positive ions in hydrogen. Tyndall and his collaborators refer to their method as the “Four Gauze” method. The advantages they claim for it, which apply likewise to Van de Graaff's method, are that the apparatus can be made airtight, that the method has a higher resolving power than Tyndall and Grindley's method, that the time of measurement is shorter, and, in addition, the ions may be given any required age within wide limits. An attempt was made to prevent the interpenetration of the constant field and the field for introducing the ions. This method is not an absolute one, and compared with Van de Graaff's has the disadvantage that the ions are derived from polonium within the apparatus, so that there is grave risk of contamination when “baking out” the apparatus. Van de Graaff used a glow discharge from a needle point for the source of ions. Although this device may produce some metastable atoms, it has obvious advantages, and thus has been adopted by other experimenters, e.g., Tyndall and Powell (8). Tyndall and Powell (9) have modified the “Four Gauze” method to enable it to be used with pure gases at pressures far below atmospheric. In addition to the fairly high resolving power for ions of different mobilities which had already been attained, the new apparatus was constructed entirely of pyrex glass and metal so that it could be subjected to a rigorous heat treatment in accordance with the requirements of modern vacuum technique. The apparatus was still unsatisfactory in some ways, viz., only relative mobilities could be measured, and as a polonium source was used to produce the ionization the whole apparatus could not be heated to a sufficiently high temperature to bake it out effectively. Although gaseous impurities could not be completely eliminated, the mobility value observed for the fastest positive helium ions in helium was more than double that of any previous observer. In 1931 Tyndall and Powell (8) made further modifications to their apparatus in order that (1) absolute mobilities could be measured; (2) a glow discharge could be used for the ion source; and (3) low gas pressures could be used in order to reduce the effect of small traces of impurity. As this apparatus could be “baked out” as a whole at 550°C. while it was thoroughly evacuated, the chance of contamination of a very pure gas introduced into it was much reduced.

Other methods have been used recently which do not lend themselves so well to work with very pure gases. Loeb (10) has used an alternating current method by which he obtains absolute values of mobilities. He claims that with it there are no disturbing effects due to the use of gauzes, and that the ions definitely emerge from one plane of the parallel-plate electrode system. Zeleny (11) has used his blast method to study ions of moderate age in air with water vapour present. 2.—Recent Results from Mobility Experiments: In 1930 Tyndall and Powell (9) gave adequate reasons for their conclusion that in none of the experiments made before that time on the mobility of ions had the gas under examination been spectro-scopically pure. “In such conditions,” they pointed out, “in a given gas the ions may well consist of clusters of which the size and mass vary from one experiment to another. The reason why many observers have found the negative ions to be of molecular magnitude and not electrons in nitrogen, hydrogen, etc., becomes obvious, and it is not surprising that the actual mobilities found are smaller than those calculated theoretically from the standpoint of the classical dynamical theory of gases, assuming the ions to be monomolecular.” It is no exaggeration to say that this paper marked the commencement of a new era in the subject, and it is partly to draw attention to the importance of some aspects of the change that this paper has been prepared. These experimenters showed that the effects of impurities in the case of positive ions are even more critical than with negative ions, and in this they have been supported by Zeleny's work (12) on the ageing of ions in air and in nitrogen, and by recent experimental determinations of the mobilities of positive ions. The enormous difference in the mobility values found in the new experiments and those previously accepted can be illustrated by the case of positive ions in helium. In 1907 Franck and Pohl obtained the value 5.09 (cm./sec. per volt/cm.) at atmospheric pressure in this gas, and in 1928 J. S. Rogers obtained values ranging from 5.6 to 6.7. In 1930, however, Tyndall and Powell found evidence for the existence of positive ions in helium with mobilities as high as 17, and a year later they definitely obtained the value 21.4 at 20°C., which they consider to be accurate to 1 per cent. For the mobility of positive ions in hydrogen, Zeleny obtained the value 6.70 and Van de Graaff 5.8. It is probable that when experiments are conducted in the pure gas, values much in excess of these will be obtained. Bradbury (2) has already obtained evidence of positive ions with a mobility of 13.6. Loeb (10) in 1931 obtained values for positive ions, presumably of sodium, in hydrogen and in nitrogen. Provided their life did not exceed 10-4 sec, the mobilities at 0°C. were found to be 17.5 and 3.75 respectively. The possible error in this work was high on account of an uncertain temperature correction introduced by the temperature gradient from the Kunsman source through the gas. This error did not exceed 10 per cent. Bradbury (2) has confirmed some of Loeb's results by Tyndall and Grindley's method. In hydrogen Loeb found

for the normal positive ion a value 8.4, and for an intermediate ion of fairly short life a value 13.5, both at 0°C. Definite evidence for the existence of these two types of ion was obtained by Bradbury, but the detection of the highest mobility ion (17.5) was beyond the limits of his method. Bradbury used hydrogen ions generated by an intense beam of hard X-rays in his experiments instead of sodium ions, so there is little change in mobility of these two types of ion due to change in nature of the original ion. This is to be expected if the normal ions are clusters, and if the intermediate ion is formed by the attachment of an impurity molecule to the original ion, or by the exchange of an electron between an impurity molecule and the original positive ion. In their most recent paper Tyndall and Powell (13) have obtained accurate values for the mobilities of the alkali ions in argon, neon, and helium. The Kunsman sources were cooled by circulating water, so that the resulting temperature variation of the gas was of the order of the room temperature variation, and could be neglected, as the resultant error was probably less than 1 per cent. These experimenters (14) have also measured accurately the ratio of the mobility of the positive mercury ion to the positive helium ion in helium at 20 mm. pressure, for which they obtain the value 0.55. In the light of these recent experiments at least three of the six chief generalizations which had emerged from investigations made prior to 1928 (reference (1), page 156), will need revision. These three are:— (1) The mobility of the ions, especially in the lighter gases, is much less than it would be if the ion consisted of a single molecule, having the same free path as an uncharged ion. (2) The mobility of the ions depends only on the gas through which they move, so that though in a mixture of gas there may be ions of different kinds all the ions of the same sign have the same mobility. (3) In the lighter gases the mobility of the negative ions is greater than that of the positive. The considerations which led Tyndall and Powell to the conclusion that no significance can be attached to the values of the mobilities of positive ions determined before 1930 demand attention. The positive ions attain very small terminal energies as compared with electrons under similar conditions, and the monomolecular or atomic ions which are the primary products of the action of the ionizing agent may be replaced in the presence of extremely small concentrations of foreign molecules by others of an entirely different nature. Any or all of the following processes may be involved:— (1) A cluster round the positive ion may form as the result of its impact with a foreign molecule possessing a marked dipole. (2) A positive ion may capture an electron from any molecule or atom of lower ionization potential with which it collides. (3) Metastable atoms may be produced in the process of ionization.

Evidence in support of the second process is now strong, and it is probably of paramount importance in many experiments, for it accounts satisfactorily for the marked effect on the mobility obtained in the presence of minute quantities of even non-polar impurity. The possibility that this process might alter to a large extent the nature of the ions being measured was ignored until a few years ago. 3.—General Mobility Equations: Langevin (15) has given the most satisfactory general theory for the mobility of ions in gases. This is based on rigorous mathematical investigations introduced into the kinetic theory of gases by Maxwell and Boltzmann. Hassé (16) pointed out the fundamental importance of Langevin's paper. He made a recalculation of the numerical part of it, and embodied his results in a more convenient and accurate form (Table III of Hassé's paper). Langevin considers the ions to be conducting spheres of mass M and charge e (the charge on an electron) moving under the influence of an external field in a gas whose molecules (mass m) have a high concentration (N per unit volume) compared with that of the ions. Let p be the density of the gas and D its dielectric constant. If a is the “electrical radius” of the molecule and r the distance between the centres of the molecule and ion, then the attractive force between the ion and the molecule due to the induced charge on the latter is equal to e2a2 − a2)/r2(r2 − a22, which reduces to 2e2a3/r5 when r is assumed to be large compared with a. (J. J. Thomson Elements of the Mathematical Theory of Electricity, 5th Edition, page 114.) It has been shown by J. H. Jeans (Mathematical Theory of Electricity and Magnetism, page 131) that D − 1 = 4 π Na3. Hence the force of attraction is given by F = (D − 1)/2 π Nr5 (1) and in the approximation we have underestimated this force in the ratio 1 to (1 + 3/2 a2/r2). Langevin considers two cases: (1) when the molecules are perfectly elastic spheres, and (2) when in addition the molecules exert attractive forces of the type given by Eq. (1). This consideration is only a first approximation to the rigorous and complicated mathematical treatment based on the kinetic theory of diffusion, but it happens to be a very good one in this case. Langevin arrives at the general mobility formula k = A/√ρ(D − 1) × (1 + m/M)1/2 (2) where A is a numerical coefficient, whose value depends on the comparative effect of the attractive forces and the elastic collisions. Write λ2 = 8(πpσ4/(D − 1) e2 (3) where p is the pressure of the gas and σ is the distance between the centres of the molecule and the ion at the instant of collision, i.e.,

the value of r during a collision. Then A was calculated as a function of λ by Langevin, and the result was confirmed by Hassé. There are two limiting cases: (1) When the attracting forces are negligible, in which case λ is large and the product λA approaches ¾. Substituting for (D − 1) from Eq. (3), we can write Eq. (2). k = λAe/σ2(8πρp)1/2(1+m/M)1/2 (4) In this limiting case we have k = ¾ e/σ2(8πρp1/2(1+m/M1/2 (5) (2) The attractive forces are of much greater importance than the collisions when λ is small, due to either σ being small or (D − 1) large. In this case A tends to the value 0.505 and λA to zero. Between these limits A is a continuous single-valued function of λ with a maximum value of about 0.590 for λ = 0.6. One of the best established generalisations emerging from experiments on ionic mobilities is that the mobility of the positive ion at constant temperature varies inversely as the pressure. This law holds also for negative ions over a more limited pressure range. Since according to the usual theory of dielectrics (D − 1) varies as p, we deduce from Eq. (3) that λ, and so A, is independent of p. Eq. (4) then shows that k is inversely proportional to p. According to this theory the mobility is independent of the electric field (X volt per cm.), unless this influences the dielectric constant, which effect would be expected to be small. Hassé has drawn attention to the rigorous methods which have been applied by Chapman and by Enskog to the kinetic theory of gases. In the case of elastic collisions the equation for the diffusion of gases, and hence Eq. (2) which has been derived from it, gives results about 13% too low, but the equation is exact in the other extreme case when we are dealing only with forces varying inversely as the fifth power of the distance. For intermediate cases, where both collisions and attractive forces are operative, the correcting factor will be between 1.00 and 1.13, but its value cannot be determined in the present state of theoretical knowledge. In most of the calculations, for which we use Langevin's equation, λ < 1, so that the correcting factor is not likely to exceed about 1.05. In their recent review—Electrical Discharges in Gases—K. T. Compton and I. Langmuir (17) deal with the question of the mobilities of ions and electrons in so far as they concern this subject. As it is clear that most of this section is due to Compton, I will ascribe it all to him for convenience. On the basis of the kinetic theory of gases, and assuming elastic collisions between ions and molecules, Compton, after allowing for the phenomenon of persistence of velocity, arrives at a general mobility equation which should apply to ions and electrons. Apart from some changes in notation and

the elimination of numerous typographical errors, his treatment is used in what follows. The terminal speeds of charged particles in the direction of the uniform electric field (X) are calculated by equating the rate of gain to the rate of loss of energy of the particle. In advancing a distance dx in the direction of the field each particle of charge e receives energy edU = eXdx, where U is the kinetic energy of a unit-charged particle in terms of equivalent potential drop. In the path dx the particle will lose energy vfeUdx, where v is the average number of collisions made while advancing unit distance and f is the average fraction of its energy lost at a collision as a result of momentum transfer. Hence the net rate of gain of energy by the particle is edU/dx = e (X − ufU. The terminal energy Ut is the value of U when dU/dx = 0 Therefore Ut = X/uf (6) If we write eUt = E1, and eΩ = E2 for the mean energies of charged particle (mass M) and gas molecule (mass m) respectively, Compton shows that u = 2Ut/λ02X (7) where λ0 is defined by 1/λ = φ N σ2. It is to be noted that λ0 is not in general the mean free path of the particles, but it is the free path the particle would have if its speed was very large compared with the speed of the molecules. The fraction of the energy lost at a collision between an ion and a molecule at rest, where both are treated as elastic spheres, is known to be f = 2mM/(m + M)2. When the molecules are also in motion the average fraction of the energy lost is less than this. In the general case of particles with Maxwellian velocity distributions Cravath (18) has shown f = 2.66 mM/(m + M)2 × (1 − E3/E1) (8) Substitute from Eqs. (7) and (8) for v and f in Eq. (6) Ut = Xvf = X2λ02(m + M)2/2Ut × 2.66 mM.E1/E1 − E2 But E1/E2 = Ut/Ω therefore Ut = X2λ02(m + M)2/5.32UtmM.Ut/Ut − Ω Hence Ut2 − Ω Ut − A = 0 where A = X2λ02(m + M)2/5.32mM Solve for Ut. Ut = ½ Ω + (1/4 Ω2 + A)1/2 (9)

This is of the right form, for it reduces to the equipartition value Ut = Ω for weak fields X = 0. The kinetic theory of diffusion leads to the mobility equation k = 0 921 eλ0/MC (1 + E2M/E1m)1/2 (10) where C is the root mean square velocity of the ions. This is comparable with the most familiar mobility equation, the simplified Langevin equation, k = 0.815eλ0/MC(1 + M/m)1/2 (11) The difference in the numerical factor arises from a different method of averaging used by Compton, and the absence of the ratio of the mean energies in Langevin's equation arises from an implicit assumption of equipartition of energy. Eqs. (9) and (10) are the basis of Compton's general mobility equation. Since eUt = E1 = ½ MC2 C = (2eU1/M)3 (12) Substitute from Eqs. (9) and (12) for C and Ut in the mobility Eq. (10) and we have k = 0.921eλ0(2eMUt)1/2 =0.921eλ0/(2eM1/2{1/2 Ω + (1/4Ω2 + λ02X2(m + M)2/5.32mM)1/2}−1/2 ×{1 + M/m.1/1/2 + λ02X2(m + M)2/5.32mMΩ2)1/2}1/2 (13) This is Compton's general mobility equation. In this equation the second bracket term replaces the factor (1 + M/m)1/2 in the usual mobility equations. The change allows for the influence of persistence of velocities. The first term in brackets reduces to the equipartition value of the energy Ω, when X is small. For the two cases of chief interest Eq. (13) takes particular forms. For monomolecular ions M = m, and λ0 = √2.λ1 where λ1 = molecular mean free path. Hence k = 0.921eλ 0/(2e M)1/2{1/2Ω + (1/4 Ω2 + λ02/1.33)1/2}−1/2 × {1 + 1/1/2 + (1/4 + λ02X2/1.33Ω2)1/2}1/2 (14) For electrons M « m, and λ = electron mean free path. Hence k = 0.921eλ0/(2e M)1/2{1/2Ω2 + λ02X2m/5.32M)1/2}−1/2 (15)

4.—Calculation of Mobilities and Comparison with Experimental Results: Eq. (14) can be simplified as follows:— λ0 = √2λ1 = √2λ2/p T/273 and Ω = E2/e = E0/e T273 where λ2 = molecular mean free path at 0°C. and 1 mm. pressure, E0 = mean molecular kinetic energy at 0°C., T is the absolute temperature and p is the gas pressure in mm. of mercury. Since E0 = 5.62 × 10−14 ergs and 1 electron-volt = 1.59 × 10−12 ergs, therefore Ω = 3.54.10−2 × T/273 electron-volts. Substituting for λ0 and Ω in Eq. (14), we obtain k = 0.873 × 10−5/M1/2 λ2/p (T/273)1/2 {1 + (1 + 4800 (λ2X/p)2))1/2}−1/2 × {1 + 2/1 + (1 + 4800 (λ2X/p)2)1/2}1/2 (16) Fig. 1. Helium ions in helium.

Values for helium: Table XV of Compton's paper (loc. cit. page 208) gives the kinetic theory values of the mean free paths of electrons and molecules in various gases. From it we find λ2 for helium = 0.02035 cm., and since for helium m = 6.59 × 10−24 gm., we have for helium ions in helium at 20°C. and normal pressure k = 94.4 [1 + (1 + 1.98X2/p2)1/2]−1/2[1 + 2/1 + (1 + 1.98X2/p2)1/2]1/2 The values of k calculated for values of X/p between 0 and 10 are shown by Graph I (Fig. I). It will be observed that the value of X/p has marked effect on the mobility. In this figure there are shown for comparison the values 26.5 calculated by Hassé (16), and 21.4 obtained experimentally by Tyndall and Powell for values of X/p between 0.3 and 4.2. It is of interest to calculate the kinetic theory value of σ2 corresponding to the value of λ2 used above. Since σ2 = 1/√2 π N λ1 = 760/√2 π N λ2 = 3.11 × 10−16 cm.2 at 0°C. therefore diameter of helium molecule σ = 1.76 × 10−8 cm. This value is much less than that quoted by J. H. Jeans (Dynamical Theory of Gases, 3rd edition, page 288) which is σ = 2.18 × 10−8 cm. This latter value is calculated from the coefficient of viscosity. From Langevin's theory it is easy to calculate the mobility at 20°C. in the following cases: (1) small attractive forces λA = ¾, and k = 54.3. (2) attractive forces more important than collisions, A = 0.505 and k = 22.8. (3) assuming σ = 2.18 × 10−8 cm., then λ = 0.622, and A = 0.590, and k = 26.5. For small electric fields Compton's equation gives k = 61.3. The ratio between this and Langevin's value 54.3 is 1.13. The ratio between the numerical constants 0.921/0.815 is also 1.13, which therefore completely accounts for the difference in values. The effect of neglecting the attractions is to increase the value of k by over 100%. Values for Hydrogen: From Eq. (16), using the value λ2 = 0 01322 cm. given by Compton, the values of k shown in Table I have been calculated. This kinetic theory value of λ2 corresponds to σ2 = 4 78 × 10−16 and σ = 2.19 × 10−8, whereas the value given by Jeans is σ = 2.71 × 10−8. Table I.—Mobilities of positive ions in hydrogen at 20°C. and 76 cms. pressure. X/p 0 0.1 1 4 10 k 86.4 86.3 76.5 47.0 32.8

In this gas the value calculated from Langevin's theory is 19.0, whereas if the attractions are neglected it becomes 49.9. Hence the attractions between ions and molecules account for a factor 2.62, which is greater than in the case of helium (2.05). Values for positive ions in various gases: Using Langevin's Eq. (2) values have been calculated for the mobilities of positive ions at 20°C. and normal pressure using the values for molecular diameters given by Jeans (loc. cit.). The following estimated values of σ which are not given by Jeans were employed. Table II. Nature of Particle. Hydrogen+ (atomic). Mercury+. Sodium+. Neon. Argon. σ × 108 in cms. 0 4.40 2.76 2.76 3.68 The quantities required in the mobility calculations are the density and dielectric constant of the gas; the masses of the ion and the molecule; and the radii of the ion and the molecule, the sum of whose values is assumed to equal σ. The statical values of the dielectric constants given by Hassé (loc. cit. Table IV) have been used. These are in fair agreement with the more recent values measured by Watson, Rao, and Ramaswamy (19). The mobilities for monomolecular ions moving in their own gas have been previously calculated by Hassé (16), and the mobilities for the positive clustered ions have been taken from his paper. These ions were assumed to have a layer of twelve gas molecules round the parent positive ion. In some cases it has not been possible to compare the calculated mobilities with recent experiments where a high degree of purity has been attained in the gases, in which case no experimental value is shown in Table III. Table III.—Mobilities of Positive Ions in Pure Gases. Mobility at 20°C. and normal pressure. Type of Ion. Gas. Calculated. Experimental. Observer. Reference Number. Helium Helium 26.5 21.4 Tyndall & Powell 8 Mercury " 14.7 11.8 " " 14, 8 12.2 " " Sodium " 20.1 23.1 " " 13 Sodium Neon 8.46 8.87 " " 13 Sodium Argon 3.46 3.21 " " 13 Hydrogen (molecular) Hydrogen 19.1 - Hydrogen (atomic) " 21.8 - Sodium " 14.1 18.1 Loeb 10 Sodium + water " 13.7 14.0 " 10 molecule (13.6) Bradbury 3 Clustered-ion " 8.5 8.7 Loeb 10 8.5 Bradbury 3 Nitrogen (molecular) Nitrogen 3.46 - Sodium " 3.64 3.78 Loeb 10 Sodium + water molecule " 2.96 3.10 " 10 Clustered-ion " 1.41 1.65 " 10

It will be observed that if a correction factor of about 1.05 is applied to the calculated values to allow for the Chapman-Enskog correction the agreement with the experimental results will be improved in all cases except for sodium ions in argon, and for helium ions and mercury ions in helium. An explanation of the high calculated values for these two ions in helium is given in the discussion. 5.—Discussion: Immediate consequences from the calculations and comparisons made are:— (1) Mobilities calculated from Langevin's general equation are much closer to the experimental results than those calculated from Compton's equation. (2) In the gases considered the effect of neglecting the attractive forces between ions and molecules, as is done in Compton's equation and in Langevin's simplified equation, is to obtain mobilities much higher than the experimental values. (3) There is a sufficiently close agreement between the recent experimental results for mobilities of positive ions in pure gases and the predictions from Langevin's theory to quicken scientific interest in the mechanism of collisions between ions and molecules. (4) As there is practically no experimental evidence that the mobility of positive ions depends on X/p, the terms in Compton's equation expressing this dependence are unnecessary, except possibly when X/p is large. If we consider only small values of X/p which appertain in all mobility experiments, Compton's Eq. (13) becomes k = 0.921 e λ0/(2 e Ω M)1/2 × (1 + M/m)1/2 which is identical except for the numerical constant with Langevin's Eq. (11), in which the effect of the attractions has been ignored. Compton (17) has pointed out that his mobility equation, which has no arbitrarily adjustable constant, is probably more correct than Langevin's Eq. (11), and we will see in considering the motion of electrons that there is evidence in support of this contention. But, in the case of ions, it is clearly less correct than Langevin's general Eq. (2), which also contains no arbitrarily adjustable constant, and in addition leads to the well-established law connecting mobility and pressure, and to the fact that the mobility is independent of the electric field for small to moderate fields. Compton states his reasons for neglecting the attractive forces in the consideration of electrical discharges in gases, but these reasons are unsound when applied to mobility work. He writes for the law of force between the ion and neutral molecule (loc. cit., page 209) F = D − ¼πN × e2/σ5 = B σ−5, whereas the correct form (Eq. (1)) is double this. This attraction both increases the number of times that the centre of an ion comes

within the distance σ of the centre of a molecule, and also deflects the path of the ion when ion and molecule pass near each other, but not near enough to collide. Each of these effects reduces the diffusion and the mobility of the ion. Since the diffusion constant d and reduced mobility K (i.e., the mobility at 0°C. and normal pressure) are proportional to the mean free path in the absence of attraction, Compton uses their reduction to define the mean free path in the presence of attraction. So λ1′/λ1 = K′/K = d′/d where the primed symbols represent the values in the presence of attraction. Langevin has calculated K and d for elastic spheres both with and without attraction when the ions and molecules are in thermal equilibrium, and hence have the same average kinetic energy E. The effect of attraction depends on the ratio ¼ B/(E σ4) where ¼ B/σ4 = work to separate an ion and molecule from centres distant σ to infinity against the attractive force, so that it equals the energy of dissociation of an elementary cluster. Compton gives values for this ratio and for σ1′/σ1 which presumably are taken from Langevin and Hassé's papers to which references are made. These values are shown in Column I (Table IV). In making his estimates a numerical error of considerable size has been made which may have influenced Compton in his judgment of the relative unimportance of the attractive forces. The correct values obtained by Langevin and Hassé are shown in Column II. If the mean free paths are defined by energy loss the effect of the attractions is found by Sir J. J. Thomson (Conduction of Electricity Through Gases 3rd edition, page 51) to be rather greater (see Column III). Table IV. Values of λ1/λ1 = K′/K = d′/d ¼ B/(E σ4) I (Compton). II (Langevin & Hassé). III (Thomson). ∞ 0 0 0 10 .076 .197 .07 5 .15 .28 .10 2.5 .31 .40 .14 1 .73 .63 - 0.5 .93 .79 - 0.3 .97 .88 - 0 1.00 1.00 - We are mainly concerned with the value of ¼ B/(E σ4) for which there is an appreciable change in the mobility due to the attractions. To produce a 20% reduction in the mobility ¼ B/(E σ4) must equal about 0.9 according to Compton, and 0.5 according to Langevin and Hassé, so Compton has definitely underestimated the effect of the attractive forces. From his figures Compton concludes that “the ion free path is not appreciably diminished by the fact of its charge unless the mutual kinetic energy is of the order of the cluster dissociation energy, or less.” Since this conclusion is sufficiently vague it cannot be called wrong, but the corrected figures show that if the mutual kinetic energy equals the cluster dissociation energy the mobility is reduced by 37%, which is a considerable effect. Further, Thomson (loc. cit.) draws a different conclusion to

Compton from his approximate investigation, viz., “for the mobility of the charged ion to be much reduced, the clustered dissociation energy must be considerable compared with the average kinetic energy of a molecule.” This is the condition that aggregates of charged and uncharged molecules should be formed, so that it follows that if the mobilities of the ions are much reduced by their charges then clusters will form, and when no clusters form the mobility will not be greatly reduced. Compton concludes that this phenomenon will play no appreciable role in ordinary vacuum discharge tubes or any discharges which dissipate large amounts of energy. These are not the conditions under which mobilities are experimentally determined, so this phenomenon may be important in such cases of feeble electrical conduction through a gas at considerable pressure. It is interesting to note that Tyndall and Powell (8) found no evidence of ionic clusters in their work on pure helium. From the shape of their curves they are able to say with confidence that they dealt with positive ions of helium, unaffected by impurity. Yet even in this case of complete absence of clusters the effect of the attractive forces reduces the mobility by about 50%. The method adopted for allowing for the attractive forces is only a first approximation, but it is not possible to make a much closer approximation until more knowledge is obtained about the nature of the forces between ions and molecules when they are in close proximity. Debye (21) has shown that an inverse fifth-power law of attraction would hold whether the polarization of the molecule were due to a dielectric displacement or the orientation of a permanently dipolar molecule in the field of the ion. Loeb's (10) contention—that the law of force has been deduced from ordinary forces of dielectric polarization in weak homogeneous fields, and it is improbable that this would prove accurate for the huge inhomogeneous fields existing within a molecular diameter or two of an ion—appears reasonable, but the inaccuracy caused is probably small. One of the greatest sources of inaccuracy in the use of Langevin's general equation is in the value of σ Eq. (3) shows that σ2 is required to obtain λ, from which A is derived. It is true that the values of A (and so of the mobility) are not very sensitive to changes in λ: nevertheless such crude approximations are made in arriving at σ and such large differences exist in the values of molecular and ionic diameters obtained by various methods (see Loeb Kinetic Theory of Gases, Appendix I) that this is an important source of error. The value of σ is assumed to be the sum of the radii of the molecule and the ion. The method of Hassé (16), which has been adopted in this paper, is to take the effect of the attractive and repulsive forces of the neutral molecule as equivalent to that given by an elastic sphere whose radius can be determined from measurements of viscosity. Hassé states that this assumption is very unsatisfactory, and so is the second assumption that the effect of the attractive force between the ion and the neutral molecule calculated from Eq. (1) can be superposed on that due to the attractive and repulsive forces of the molecule itself. These assumptions cannot be very far from the truth,

and they give a simple method of treating the problem, so it is justifiable to adopt them until there is a definite advance in the state of our knowledge. A minor difficulty is that the temperature variation of the molecular radii is uncertain, but fortunately it is small over the range 0° to 20°C. Attention should be drawn to the fact that the values of the molecular radii given by Jeans (loc. cit.) have been calculated from viscosity determinations by a formula due to Chapman, and that this formula yields values considerably greater than some other methods. Chapman (22) has corrected the usual kinetic theory expression for the coefficient of viscosity (⅓ m C′)/(√2 π σ2), where C′ is the mean velocity of the molecules, and obtains the expression 0.499 m C′/(√2 σ2), by using for the molecular mean free path of molecules an expression 1.5/(√2 π N σ2), instead of the usual kinetic theory value 1/(√2 π N σ2). Chapman also calculates the molecular diameters from the constant b of Van der Waals' law, and obtains values in close agreement with those calculated from viscosity data, using the rigid methods he applies to the kinetic theory. It is probably due to a considerable difference in the values he adopts for σ, that Loeb's calculations (10) from Langevin's theory for the mobilities of positive sodium ions in hydrogen and nitrogen are much lower than those we have calculated (Table III). Kallmann and Rosen (23) discovered that the probability of charge transfer is very great for gas ions moving in a gas of their own molecular or atomic species. This experimental fact is well established, and has been utilised by Beeck (24) to produce beams of neutral argon atoms, and to explain the results found in the ionization of inert gases by slow alkali molecules. We have seen that there is strong experimental evidence (Tyndall and Powell (8), (9), and (14); and Zeleny (12)) that the phenomenon of charge transfer plays an important role in some ionic mobility determinations both on account of the probability of an ion capturing an electron from one of its own atoms, and also the possibility that it will transfer its charge similarly in an encounter with an impurity molecule. If the ionization potential of the impurity molecule is less than that of the gas atom, the impurity ion so formed will not lose its charge in collisions with other gas atoms, and therefore a very small concentration of impurity is sufficient to change completely the average mobility rate. This phenomenon of electron capture is likely to prove of great importance. In the case of a gas like helium with a high ionization potential it affords a method of determining the mobility of positive ions of other gases present in minute quantities, and if the gases are present in known concentrations it may be possible to determine the target area for electron capture presented by their molecules to helium ions at very slow speed (see Tyndall and Powell (14)). It was because the ionization potentials of the alkali metals are far below those of any common gaseous molecules or organic impurities that Loeb (10) suggested using them for mobility determinations. Hassé and Cook (25) have calculated the effect on the mobility of positive ions of the transference of charge from the ions to their own atoms. In all cases this results in a decrease of mobility, and the decrease

depends on the importance of the attractive force between ion and atom, being negligible for a small attractive force. In the case of helium ions in pure helium where the probability of transference is high, it is reasonable to assume that about half the encounters result in a transfer. On this basis Hassé and Cook calculate from the Langevin theory that the mobility is 21.2, instead of 26.5 for no transference. The most accurate experimental value is 21.4. Since calculation shows that the ratio of the mobility of the mercury positive ion to the helium ion is 0.555, the mobility of the mercury ion becomes 11.8 on the basis helium 21.2. This is in close agreement with recent experimental values (see Table III). The only other large discrepancy between the calculated and experimental mobilities in Table III is the case of sodium ions in hydrogen where the values are 14.1 and 18.1 respectively. In this case it is just possible on account of the high temperature of the Kunsman source that the short-lived fast ions which Loeb measured were hydrogen ions and not sodium ions, so that their velocity would be expected to be somewhat less than 19.1 on account electron transfers. The best experimental results for ions point to the following relations for the mobility k:— (1) k is independent of X. (2) k is inversely proportional to p. (3) k proportional to T if the pressure is constant. These three relations hold over large ranges. (4) k depends on the mass of the ions and of the molecules through which they move. (5) k depends on the nature of the charge in the case of normal ions. The bearing of the mobility equations on (1) and (2) have been discussed. The temperature effect on k is still uncertain according to Loeb (20). Langevin's theory shows that k (at constant pressure) is a function of the temperature, and the form of the function depends on the assumptions made regarding the actions between ions and molecules. (1) If they are regarded as elastic spheres whose size and mass are independent of the temperature, then k ∝ T. Compton's equation leads to the same result. (2) If account is taken of the effect of attractions in increasing the number of collisions, then Langevin's theory leads to the relation k′ (at constant density) ∝ T½/(C + T). This is the relation suggested by Sutherland (27), and is in fair agreement with the experiments of Phillips. (3) If the actions are like those between centres of force varying inversely as the (n + 1)th power of the distance, then Langevin's theory leads to the relation k′ ∝ T(2/n − ½). This will be in best agreement with experiment if 2/n − ½ = 0, i.e., n = 4. So an inverse fifth power law of attraction, which has been assumed throughout, gives good agreement

for the temperature effect. Compton's equation, which makes no allowance for the attractions, does not give the correct temperature variation. It was believed until a few years ago that the mobility of ions was independent of their mass, but recent experiments have disproved this. Langevin's, Compton's, and other mobility equations include a mass factor (1 + m/M)½ which can assume the limiting values 1.41 for monomolecular ions and 1 for very heavy ions. As this difference was inconsistent with the experimental evidence then available, some physicists concluded that the ions if monomolecular cannot remain unaltered during their path through the gas, but go through various phases in which they have different mobilities, and that the mobility measured by the usual methods is an average, and not its value in any particular phase. In view of later experiments the reason for this conclusion is removed, but it should be noted that the influence of different phases is clearly shown by the carriers of negative electricity—the negative ion and electron. The most accurate measurements illustrating the effect of ionic mass are those of Tyndall and Powell (13), (14). These experimenters found that the ratio of the mobility of the mercury to the helium positive ion in pure helium was 0.55. If we adopt the value 2.20 × 10-8 cm. for the radius of the mercury ion, the calculated ratio between these mobilities is, on Langevin's theory, A1 (1 + 4/200.6)½/A2(1 + 1)½ = 0.458/0.590 × (51)½ = 0.555, which is in very close agreement with the experimental value. These experimenters also showed that in the case of argon the fall in mobility of the positive alkali ions from sodium to caesium closely follows the (1 + m/M)½ relation, but that in the case of neon, and still more so in the case of helium, the fall in mobility with increase in mass is greater than that given by this expression. This can be accounted for by the decreasing polarizability (∝ (D − 1)) of argon, neon, and helium, which are related as 8:2:1. With a gas of low polarizability the size of the elastic spheres cannot be neglected, and the increase in size of the ions from sodium to caesium produce a decrease in A, which reduces the mobility, in addition to the reduction due to the increased mass of the ion. It is clear that relative values of the mobilities can be estimated from the theory with a higher degree of accuracy than absolute values. In the lighter gases the mobility of the normal negative ions is greater than that of the positive. Hassé (16) explains the difference on the assumption that the negative ion has a layer of molecules about an electron, whereas the positive ion has a single layer of twelve molecules about a charged molecule. The values he obtained are of the right order, but the electron attachment results of V. A. Bailey and of Loeb speak against it, besides which Loeb (26) and Luhr (28) have found experimental evidence to show that the normal ion has only one or two molecular additions which are the result of electrochemical forces depending on the molecular constitution and the sign of the charge. Loeb tries to account qualitatively for the mobility difference by ascribing it to the specific electrochemical combinations

which result by reaction of positive and negative ions with various molecular impurities or ionization products present in some gases. Reliable experiments indicate that there is a spectrum or range of mobility among the normal positive and negative ions in air of ages from about 0.5 to 2.0 seconds (see Loeb and Bradbury (26)). This is probably due to the difference in mass and diameter caused by the one or two electrochemically bound addition molecules. 6.—Conclusions: Though the mobility of gaseous ions has been the subject of numerous investigations during this century, there are still many aspects of the subject which require explanation. The recent experiments on positive ions in pure gases should give a marked impetus to this study. The experimental results obtained are much higher than previous ones, and are in good general agreement with mobilities calculated from Langevin's theory. For ions, Langevin's general equation is definitely superior to Compton's equation, which is inaccurate, as no allowance is made for the effect of the attractive forces. Until the new quantum mechanics yields a more detailed knowledge of the mechanism of ionic encounters, it is likely that little progress will be made beyond the theory of ionic mobility advanced by Langevin and somewhat modified by Hassé. Part II.—Mobilities of Electrons in Pure Gases. 1.—Mobility Equations for Electrons: The mean velocity (W) of an electron in the direction of a uniform electric field (X) is under some conditions not proportional to X, so that the term mobility has, under these circumstances, no definite meaning [vide V. A. Bailey, Phil. Mag. 46, 213, 1923]. Although the extension of the concept of mobility to the motion of electrons really serves no useful purpose, K. T. Compton has found it convenient to evaluate the velocity of an electron in the direction of the electric field as if it had a mobility [= W/X], and to study the deviations from this in terms of an interpretation from the kinetic theory. In order to avoid confusion, the terms used by Compton have not been altered in the discussion of his work in this section, except in the case of the so-called “mobility constant” for which the more rational term—reduced mobility—has been used. The application of the kinetic theory to the motion of electrons in gases was first suggested by Townsend (20), and has been considerably amplified by K. T. Compton. Owing to their exceedingly small mass compared with ions, electrons will have mobilities thousands of times greater than ions. For ions the mobility varies inversely as the pressure over a large pressure range, i.e., the product pk is a constant. Lattey (30) found an abnormal increase in this product in the case of negative ions in carefully dried air at pressures less than atmospheric and considerably higher than those for which the increase normally began. This may be explained by supposing that at low pressures the negative ion is an electron for part of its life. At these pressures the mobility depends on the electric field. Further, the mobility of electrons will depend on the velocity they possess before

the field is applied. If this velocity is not that corresponding to the energy due to thermal agitation—and in the case of electrons the attainment of this value is a slow process, while the chance of attachment to a molecule is in some gases appreciable—then the ordinary expression for its mobility will not apply. The mean free path of an electron, due to its much higher velocity, does not appear to be reduced by the presence of its charge. These considerations and others of a similar nature require that the mobilities of electrons should be treated independently to a large extent from those of ions. From Eq. (11), which was deduced by Langevin from the elastic collision theory when the influence of the charges was neglected, we deduce for electrons, since M « m, and λ0 = electron mean free path, k = 0.815e λ0/MC. Since E1 ≥ E2, Eq. (10) similarly gives k = [0.921e λ0]/MC (17) Now according to Pidduck's calculations (31), the numerical constant in the case of electrons moving under a uniform electric force is about 0.92, which is in much closer agreement with Compton's constant (0.921) than with Langevin's. These simple equations, which include no term depending on the field strength, can only be expected to give the order of magnitude of electronic mobilities. Compton's Eq. (15) would be expected to give a greater approach to accuracy. If X is small it simplifies to k = (0.921e λ0)/(2e Ω M)½, and this is identical with Eq. (17) only when E1 = E2, which is not in general true. If K represents the reduced mobility, i.e., the mobility under standard conditions, then K = IV/X × p/760 × 273/T = k × p/760 × 273/T (18) where k is the mobility at pressure p mm. of mercury and absolute temperature T. K is often called the mobility constant, but this term is misleading. By Eq. (18), Compton's Eq. (15) can be reduced to a more convenient form for application to particular cases. Let λ = electron mean free path at 0°C. and 1 mm. pressure, then λ0 = λ/p × T/273 (19) If E0 = kinetic energy of a gas molecule at 0°C., then E2 = E0T/273. Let M = molecular mass on the basis mass of the hydrogen atom = 1, then we have on substituting from Eqs. (15) and (19) in Eq. (18) and then inserting the values of known constants K = 271000 λ (273/T)½ {1 + (1 + 1 106.106 M λ2 (X/p)2)½}−½ (20) This is the equation Compton uses (loc. cit., page 234) in comparing his theory with the experimental results. 2.—The Transference of Energy in Collisions between Electrons and Molecules: It is not proposed in this paper to attempt a detailed discussion of this complicated and at present unsolved problem, but merely to point out some salient features which will be of assistance in

attempting to interpret the experimental facts. In the initial stages of the motion of electrons which start with a small velocity, the kinetic energy of the electrons increases as they move in the direction of the electric force; and since the direction of motion is changed in the collision with molecules of gas, the kinetic energy of the electrons appears as energy of agitation. After moving some distance in the direction of the force, a steady state of motion is attained where the electrons lose energy in collisions with molecules at the rate at which they acquire energy by moving in the direction of the force. The root mean square velocity of agitation (C) is then constant, and Townsend first pointed out that, in this steady state of motion, the energy of agitation of an electron may be much greater than that of a molecule of the gas. This property of the electrons has been proved experimentally (32). Let E1 = ½ MC2 = e U represent the mean energy of agitation of an electron, and E2 = ½mC12 = eΩ represent the energy of agitation of a gas molecule at 15°C.; then the former may be expressed in terms of the latter by the relation MC2 = k1mC12, where C1 = root mean square velocity of a gas molecule at 15°C., and k1 ≥ 1. When the electron gains its terminal velocity in the field, its terminal energy is often much higher than the energy of the gas molecules, and if the field is removed it loses this gradually, eventually attaining equipartition. Physicists have adopted different viewpoints in considering the average fraction (f) of an electron's energy which is lost at an impact with a molecule and the influence that these losses have on the electrical properties of the gas. The method adopted by Town-send and his school, which has proved adequate to account for the large array of experimental results they have now collected, is described by Townsend in Motions of Electrons in Gases (Clarendon Press, Oxford, 1925). A summary of the experimental results obtained at the Electrical Laboratory, Oxford, is also given in this pamphlet. In a paper on The Transference of Energy in Collisions between Electrons and Molecules (33) Townsend and the writer criticised certain hypotheses relating to the transference of energy from electrons to molecules which some physicists had said were required by the (old) quantum theory. The conditions in mobility experiments are somewhat simplified, since the values of X/p are usually small, so that the electrons only lose a small fraction of their energy in collisions with molecules. Townsend defines the mean free path of an electron as the average distance an electron moves in a given direction with the velocity C before all directions of motion become equally probable, when the electric force is zero. He assumes that the average energy lost in a collision is equal to the average increase of energy of an electron in traversing a free path in the direction of the field. Then in the simple case where all the free paths are taken equal to the mean free path and the velocity of agitation is constant, the proportion of its energy an electron loses in a collision (f) is 4W2/C2. When the distribution of the free paths about their mean, and the distribution of the velocities of agitation about their mean value are taken into consideration a different value is obtained for the numerical factor in the expression for f. Langevin assumed

that the velocities of agitation were distributed according to Maxwell's law, and arrived at the numerical constant 0.815 in the mobility formula. This gives f = 2.46 W2/C2 (21) Pidduck (loc. cit.) found that the distribution of the velocities of electrons is represented more accurately by a law given by Lorentz than by Maxwell's law. This led to a constant 0.92 in the mobility equation, which is very close to Compton's value and leads to f = 2.17 W2/C2 (22) Eq. (21) has been criticised by Compton, Loeb, and others, but it has been useful in explaining the experimental results obtained by Townsend and his collaborators, who have found no adequate reason for discarding it, although it is clearly realised that the exact numerical coefficient is uncertain. In his discussion of this subject Loeb (Kinetic Theory of Gases, page 518) says that the reason why Townsend's simple relations so nearly fit the experimental facts is still a mystery. He admits that they are at least good approximations which will form the first and most important terms in more accurate and complicated equations based on a more detailed consideration of the factors concerned. For a perfectly elastic collision between an electron and a stationary molecule f = 2M/m × (1 − Ω/U) = 2M/m × (1 − 1/k1) (23) Eq. (23) is used by Compton in his earlier papers (34), but in his recent review (17) he uses the corrected form due to Cravath f = 2.66M/m × (1 − Ω/U) = 2.66M/m × (1 − 1/k1 (24) in obtaining his mobility equation for electrons (Eq. 15)). If, in the derivation of Eq. (15), we do not substitute for f from Eq. (24), we arrive at an expression for electron mobility K = k × p/760 × 273/T = 4.16.104 (p λ/X)½ × f1/4 Substitute for λ0 from Eq. (19) and we get for the reduced mobility K = k × p/760 × 273/T = 4.16.104 (p λ/X)½ × f1/4 (25) where it has been assumed that k is measured at 15°C. and pressure p mm. Eq. (25) can be used to calculate reduced mobilities of electrons for comparison with experimental results provided the appropriate value of f is substituted. Eq. (25) is identical with Eq. (20) when the value of f adopted by Compton is used. According to Compton, for electron speeds below a certain critical value determinable for each gas, the collisions with molecules are elastic, so Eq. (24) applies accurately. Townsend considers that Eq. (21) is the best expression to use for f, and the coefficient 2.46 is the most probable one. The values obtained experimentally for the losses of energy of electrons in collisions with atoms of monatomic gases at small values of X/p are nearly the same as calculated values for smooth elastic spheres. For larger values of X/p the losses are much greater,

due to large losses of energy in some collisions, when ionization by collision becomes appreciable. With diatomic or polyatomic molecules the losses even with fairly small values of X/p are considerably greater than those calculated on the assumption of smooth elastic spheres. Townsend (35) has pointed out clearly that, when the collisions are taken as perfectly elastic and the spheres as smooth, his equation for f leads to approximately the same result as Compton's value. For this case Pidduck (31) found k1 − 1 = 1 2 W2/C12 approximately, which gives W2/C2 = (1/1.2) (1 −1/k1)M/m Hence f = 2.46W2/C2 = 2 M/m × (1 −1/k1) approximately, in agreement with Eq. (23). So under these conditions Compton's and Townsend's methods will be in close agreement, but Townsend's value for f has a much wider range of application. 3.—Calculation of Electron Mobilities and Comparison with Experimental Results: In 1923 Compton (34), using his mobility equation with constants slightly less accurate than used in Eq. (20), showed that the agreement with experiment was good in the case of hydrogen, and within a small factor in the case of helium, nitrogen, and argon for values of X/p less than the critical value which he specified as 20 for hydrogen, 1.3 for nitrogen, 0.5 for argon, and > 0.4 for helium. In this computation he used a fixed value for the electron mean free path which was calculated for each gas from the kinetic theory, and the experimental results were those of Townsend and Bailey, and of Loeb. In the case of oxygen the agreement was not good, indicating a very low critical value of X/p. Compton concluded that when electrons collide at speeds greater than that corresponding to the critical values of X/p, the collisions are no longer elastic as assumed in his theory. Deviations from the theory could possibly be accounted for in two ways:— (1) The electronic mean free path may not be equal to that predicted from the kinetic theory. (2) The electron impacts may not be perfectly elastic. Compton showed that an increase in the value assumed for the free path increases the mobility more for small values of X/p than for large ones, thus leading to a steeper curve. On the other hand, an increase in the fractional energy loss at impacts tends to flatten the curve by increasing the mobility more for large values of X/p. In a recent review (17) Compton uses his Eq. (20) to compare his theory with the best experimental results for hydrogen and for nitrogen. The same experimental results as were used in his earlier comparison were used for hydrogen, but in the case of nitrogen he used Wahlin's results (36) in place of Loeb's. For values of X/p < 0 1 in nitrogen the theoretical curve lies well below the experimental, and as the hypothesis of inelastic collisions would not increase the

mobility at X/p = 0, Compton can only suggest that the fact that scattered electrons are much more concentrated in the forward direction than would be true of elastic spheres would introduce a correction in the right direction. In hydrogen and helium, on the other hand, he finds that the theoretical curves are above the experimental ones. He suggests that this might be accounted for by using the experimentally determined mean free paths which are much smaller for these two gases than the kinetic theory values. This would over-correct the discrepancy, so, in addition, he suggests it is again necessary to take account of the excess of scattering in the forward direction, which the quantum theory of scattering predicts is less pronounced at small, than at large, velocities. His general conclusion is that his theory gives results of the right order of magnitude and about the right type of variation of reduced mobility with X/p. There appear to be simpler and more convincing methods of explaining the considerable variation between experimental results and those predicted by Compton's theory. Since 1923 more accurate experiments from which electron mobilities are obtainable have been performed on some gases. For these reasons the subject has been re-examined by considering the gases oxygen, nitrogen, and helium. Of the six gases he studied, Compton found that oxygen departed furthest from his theory. The values obtained by Brose (37) with this gas have been used in place of Townsend and Bailey's earlier results as being probably more accurate, but the difference between them is small. Brose states that the number of ions in the streams of electrons he used was quite inappreciable even when comparatively large gas pressures were used. From the values of W listed by Brose, the reduced mobility K has been estimated at values of X/p up to 7 using Eq. (18). In making calculations from Compton's theory Eq. (25) has been used, after substituting for f from Eq. (24). This leads to the same result as using Eq. (20), but the calculation is much shorter. Except when k1 < 5, the error is negligible for our purpose in writing f = 2 66 M/m, since it is f which is used to calculate K. The maximum percentage error involved in this approximation is 5. We have also calculated the reduced mobilities from Eq. (25), using Townsend's value of f given by Eq. (21). In this case f is calculated from the experimentally determined values of W and C. It would be more consistent to use Eq. (22) for f in place of Eq. (21), since the same numerical coefficient (0.815) has been used in deducing Eqs. (22) and (25). This procedure would merely make the calculated values of K about 3% higher, so it is unimportant. We shall assume that λ remains constant over the range of X/p considered. λ is calculated from the kinetic theory value = 4 √2 × molecular mean free path. The molecular mean free paths are obtained from Compton's Table XV (loc. cit., page 208), which are the commonly accepted values. For oxygen λ = 0.0417 cm., and 2.66 M/m = 0 452 × 10−4. Since for X/p = 0.4, k1 = 6, we can with sufficient accuracy write 1 − 1/k1 = 1 for all the values of X/p considered. The comparison is set out in Table V, and the curves showing the reduced mobility in oxygen as a function of X/p are

shown in Fig. 2. The experimental mobilities are labelled Brose, and the calculated ones Compton and Townsend, depending on which expression for f was used in their calculation. Fig. 2. Oxygen Table V.—Reduced Electron Mobilities in Oxygen. f × 104 Value of K X/p Compton Townsend Brose Townsend Compton Difference % Townsend/Brose 7 .452 83.0 884 935 262 + 6 5 .452 73.0 1070 1110 312 + 4 2 .452 69.8 1870 1740 494 − 7 1 .452 69.1 2740 2450 697 − 11 0.8 .452 62.6 2960 2670 779 − 10 0.6 .452 56.6 3320 3010 899 − 6 0.4 .452 27.5 3120 3090 1100 − 1

It is clear that in the case of oxygen the use of Townsend's expression for f is justified, and that for values of X/p > 0.4 the collisions are not of the perfectly elastic kind. In the case of nitrogen we employ the experimental results of Wahlin (36) for values of X/p < 0.1 and those of Townsend and his collaborators for values of X/p from 0.25 to 2. These results lie well on a smooth curve although obtained by entirely different methods. It was not possible to calculate the values of f in the ordinary way from Wahlin's data. For comparison with Wahlin's results, therefore, mobilities were calculated by Compton's theory using Eq. (20). Then from these calculated values k1, and hence f, could be calculated indirectly. The value of f obtained by Townsend's method was fairly constant between X/p = 1 and X/p = 0.25, so this value for f was assumed to hold down to the lowest value of X/p (0.002). There is little justification for this assumption, so the figures dependent on it have been shown in brackets. For nitrogen λ = 0.0389 cm., and 2.66 M/m = 0.516 × 10-4. Since for X/p = 0.25, k1 = 7.5 we can use the approximation 1 − 1/k1 = 1 down to this value. The results of the comparison are shown in Table VI, and the corresponding curves are drawn in Fig. 3. Fig. 3. Nitrogen

Table VI.—Reduced Electron Mobilities in Nitrogen. f × 104 Value of K X/p Townsend Compton Townsend & Wahlin Townsend Compton 2 .516 10.3 817 1035 493 1 516 6.5 1080 1310 696 0.5 .516 5.5 1550 1770 976 0.25 .516 6.5 2570 2630 1390 0.08 .46 (6.5) 5000 (4640) 2390 0.06 .45 (6.5) 6000 (5350) 2740 0.04 .41 (6.5) 8050 (6550) 3280 0.02 .32 (6.5) 12000 (9250) 4380 0.01 .21 (6.5) 16000 (13100) 5570 0.002 .027 (6.5) 18000 (26800) 7080 According to Compton the critical value of X/p for this gas is about 1.3, so below this the collisions should be perfectly elastic. The curves show that this is only approximately true. Fig. 4. Helium The experimental data referring to helium are those quoted in Townsend's pamphlet Motion of Electrons in Gases (page 590), and were determined by Townsend and Bailey. These are in close agreement with the results obtained by Loeb, which were used by Compton in his comparison. For helium λ = 0.115 cm., and 2.66 M/m

= 3.62 × 10−4. For X/p = 0.1, k1 = 6.2, so we can use the approximation 1 − 1/k1 = 1 down to this value. The results of the comparison are set out in Table VII, and the corresponding curves are drawn in Fig. 4. According to Compton the critical value of X/p for this gas is > 0.4, so below this the collisions should be perfectly elastic. We notice that for this gas Townsend's values of f are less than the value calculated on the assumption of perfectly elastic spheres. This is not unreasonable, for in some collisions the electrons gain energy from the molecules, so the mean loss per collision may be lower than the value calculated for a perfectly elastic collision. Table VII.—Reduced Electron Mobilities in Helium. f × 104 Value of K X/p Compton Townsend Townsend & Bailey Townsend Compton 1 3.62 2.4 1030 1755 1960 0.5 3.62 2.3 1430 2450 2760 0.2 3.62 2.5 2450 3990 4380 0.1 3.62 2.6 3690 5700 6190 0.05 2.64 2.3 5350 7770 8050 0.02 1.91 1.56 8300 11200 11700 0.013 1.58 1 30 16500 13300 14000 4.—Discussion: In all the gases considered, even when the values of X/p are less than the critical values specified by Compton, the mobilities calculated by the use of Townsend's expression for the average fractional energy loss at a collision are in better agreement with experiment than those calculated on the assumption of perfectly elastic collisions. In the case of helium, for values of X/p < 1, there is not much difference between mobilities calculated in these two ways, and in this gas the collisions are very nearly elastic for the small values of X/p. There is adequate evidence to show that the collisions between electrons and molecules are not perfectly elastic under the conditions which some physicists have assumed them to be. Throughout our calculations the electronic mean free path has been assumed a constant independent of X/p, and equal to the kinetic theory value. Even so the general agreement between theory and experiment has been good, especially when it is remembered that on account of experimental errors and various approximations which have been made an agreement of closer than 10 to 15% could not be expected. The agreement would be much better if we used other values for the mean free paths—a smaller value for helium, and higher values for oxygen and nitrogen. There is good experimental evidence to show that the mean free path of an electron is not always equal to the kinetic theory value and that it is a function of X/p. Townsend determines W and C by experiment for various values of X/p. He then calculates f from Eq. (21) and λ from the mobility equation W = 0.815 × X/p × e/M × λ/C

In this λ is the mean free path of an electron in the gas at 1 mm. pressure when the electrons are moving with the velocity C. The value of λ required to satisfy this equation is not the same for all values of C. This was the first suggestion that the mean free path of an electron depends on the velocity, and it was for many years viewed with suspicion, but has now been established by other experimental methods devised by Lenard (38), Ramsauer (39), and Langmuir and Jones (40). A comprehensive historical and critical account of this important work has been written by Brose and Saayman (41). These investigations point to a collision mechanism much more complicated than that assumed in the classical kinetic theory, and an allowance for the variation in λ from the kinetic theory value would improve the agreement in the case of those gases we have considered. Wahlin (42) from his experimental investigations finds the ratio between his observed values for the mean free path and the kinetic theory value for various gases. For nitrogen his figure is 2.53, and for helium 0.565. The adoption of these figures in our calculations would give a closer approximation to the experimental mobility values. In the case of helium, for the range of X/p considered, Townsend's mean value of λ agrees with Wahlin's value (0.066 cm.) and the adoption of this value in place of the one previously used (0.115 cm.) gives the curve labelled “Wahlin” in Fig. 4 for values calculated from Townsend's f. Loeb questions the measurements in oxygen on account of the relatively high probability of electron attachment, but this effect was not appreciable in Brose's experiments. In a recent paper Massey (43) uses Born's method of applying the new quantum theory to the problem of the collision of electrons with rotating dipoles. He points out that the solution of the general case of collision at slow to moderate velocities of electrons with molecules is practically impossible, for the complete solution of the motion of an electron in the static field of the molecule would be required. He shows that if the rotator is initially in the ground state, then the probability of exciting the first rotational state is much greater than that of an elastic collision. If the rotator is not initially in the ground state, then the first rotational level will not, in general, be most strongly excited, but the probability of an inelastic collision will still be greater than that of an elastic one. The magnitude of these excitation probabilities is quite appreciable at low velocities, indicating that beams of slow electrons will exchange energy readily with dipolar molecules. The result of Massey's calculations is in general qualitative agreement with the experimental results of Townsend, Bailey, and others (44) on the loss of energy in collisions between slow electrons and molecules of polyatomic gases. These workers have shown that the losses in all the gases they have used, except helium, argon, and neon, are greater than can be explained by elastic collisions. The losses are greater in polyatomic gases than in monatomic ones. The support given to these experiments by Massey's calculations can only apply to those gases whose molecules have an electric moment, e.g., triatomic molecules and diatomic molecules having a slight dissymmetry.

In an interesting paper Bailey (45) has pointed out that the new quantum theory is in closer agreement with the experimental results obtained by Townsend and himself than was the old quantum theory, which some physicists believed conflicted with these experiments. For this purpose Bailey considered the absorption of light in the infra-red region and also the Raman effect in the scattering of light by the gases used. It was shown to be probable that with polyatomic molecules, the energy transferred from a slow electron to a molecule has a relation to the energy transferred from radiation to a molecule, whether in the process of absorption or of scattering. 5.—Conclusions: Probably the best general mobility equation which has been proposed for electrons is that due to K. T. Compton. A modification of Compton's equation which improves its agreement with the experimental results is suggested. This is to adopt the expression for the average fractional energy loss at an encounter between an electron and a molecule proposed by Townsend, which implies that the collisions are not in general like those between perfectly elastic spheres, although in monatomic gases they are very nearly elastic for small values of X/p. No mobility equation which does not make allowance for the variation in electronic mean free path with variation in the mean velocity of agitation of the electron can hope closely to approach accuracy over a large range of conditions. There are indications that the new quantum theory will help to explain in greater detail the nature of the collisions between electrons and molecules at slow speeds, and so lead to advances in this subject. References. 1. Sir J. J. Thomson and G. P. Thomson. Conduction of Electricity through Gases, 3rd edition, Ch. III (1928). 2. A. M. Tyndall and G. C. Grindley. Proc. Roy. Soc., A 110, p. 341 (1926). 3. N. Bradbury. Phys. Rev., 38, p. 1905 (1931). 4. M. Laporte. Comptes Rendus, 183, pp. 620 and 781 (1926); Ann. de Phys. 8, p. 466 (1927). 5. R. J. Van de Graaff. Phil. Mag., (7) 6, p. 210 (1928). 6. A. M. Tyndall, L. H. Starr, and C. F. Powell. Proc. Roy. Soc., A 121, p. 172 (1928). 7. C. M. Focken. Australasian Association for the Advancement of Science, Section A. January (1927). 8. Tyndall and Powell. Proc. Roy. Soc., A 134, p. 125 (1931). 9. Tyndall and Powell. Proc. Roy. Soc., A 129, p. 162 (1930). 10. L. B. Loeb. Phys. Rev., 38, p. 549 (1931). 11. J. Zeleny. Phys. Rev., 34, p. 310 (1929). 12. J. Zeleny. Phys. Rev., 38, p. 969 (1931). 13. Tyndall and Powell. Proc. Roy. Soc., A 136, p. 145 (1932). 14. Tyndall and Powell. Nature, 127, p. 592 (1931). 15. P. Langevin. Ann. de Chimie et de Physique 5, p. 245 (1905); 8, p. 238 (1905) and also 28, pp. 317 and 495 (1903). 16. H. R. Hasse. Phil. Mag., (7) 1, p. 139 (1926). 17. K. T. Compton and I. Langmuir. Reviews of Modern Physics, 2, p. 123 (1930). 18. A. M. Cravath Phys. Rev., 36, p. 248 (1930).

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