XXX. The Kinetic Energy of the Negative Electrons emitted by Hot Bodies. By O. W. Richardson, Professor of Physics, and F. C. Brown, Experimental Science Fellow, Princeton University [Communicated by the Authors.]. That the carriers of negative electricity emitted by hot metals are electrons was first proved by the experiments of J. J. Thomson [citation redacted] on their deflexion in a magnetic field. This result alone did not compel any definite view oE the origin of this ionization. Somewhat later one of the authors [citation redacted] showed that the phenomena then known were such as would arise if the electrons originate in the metals, from which they were able to escape when their velocity normal to the surface exceeded a certain value. This method of looking at the question was found to give a particularly satisfactory account of the thermal relations of the phenomena which were then accurately investigated for the first time. In Richardson's method of developing the subject (Joe. cit.) the assumption is justified by theoretical considerations, that the translational kinetic energy of the electrons inside the metal has the same value as that of the molecules of a gas at the same temperature as that of the metal. From the principles there laid down it also follows that the translational kinetic energy of the electrons outside the metal possesses the same value. This equality applies not only to the average value, [footer] 354 [header] but to the way in which the energy is distributed among the different electrons as well. Although a knowledge of the kinetic energy of the emitted electrons is of obvious importance, no attempt appears to have been made to determine it. The present paper embodies the results of an investigation of the portion of the kinetic energy which depends upon the component of the velocity normal to the emitting surface. What is determined is the value of [formula redacted], where m is the mass of an electron and u is its component of velocity perpendicular to the surface from which it is emitted. Both the average value of this portion of the energy and the law according to which it is distributed among the different particles given off are examined. The method employed gives no information about the part of the energy which depends on the component of velocity parallel to the surface of emission : the sideways energy will form the subject of another communication by one of the authors. Theory of the Method. Stated briefly the method used consisted in measuring the rate at which an insulated plate A charged up when a portion of another plate B parallel to A consisted of metal sufficiently hot to emit ions. It is clear that as the metal in B emits negative ions the plate A will become negatively charged, so that a difference of potential will be established tending to stop the flow of electricity. From the way in which the current between the two plates varies with the time or with the difference of potential between them the desired information can be obtained. Consider two parallel planes of indefinite extent perpendicular to the axis of x. The lower plane, determined by x = 0, has a small portion of its central region heated so that it is emitting ions. The potential of this plane is maintained at zero. The potential of the upper plane, determined by x = x0 has the value V at the instant considered. Consider an ion whose charge is e and mass m situated at a point between the planes whose coordinates are x, y, z. Its equation of motion will be [formula redacted] From these equations it follows [citation redacted] that the electron will only arrive at the upper plate provided [formula redacted] where u0 is the initial velocity perpendicular to the] plate. [header] 355 and V is the difference of potential between the two plates. There is no reason why all the ions should be emitted from the hot metal with the same velocity. If we consider a sufficiently great number the velocity or energy will be distributed, anions: them according to some regular law. Out of any large number of ions emitted by the plate let us denote the fraction having velocity components between n and v, those between v and [formula redacted] , and [formula redacted] and [formula redacted] and [formula redacted] respectively. Here r and ic are the initial components of velocity parallel to y and z respectively ; if the planes are of sufficient extent the condition that the ions should reach the upper plane will be independent of these components of velocity. If n is the total number of ions emitted by the lower plane per second the current i to the upper plane is clearly [formula redacted] where c is the capacity of the system. Circular Plate. In practice it is impossible to use planes of indefinite extent, so that it is important to determine the effect of the finite size of the two plates. In the experiments the plates were circular, the lower being somewhat larger than the upper one. The upper plate was surrounded by a guard-ring. The hot metal could be treated with sufficient approximation as a point at the centre of the lower plate. We shall now calculate the number of ions which reach the upper plate, everything being supposed to be symmetrical about the axis of the two circles. Consider a charged particle whose distance from this axis is p, and whose distance from the lower plate is x. Its equations of motion are [formula redacted] and [formula redacted] Integrating these subject to the conditions that when [formula redacted] and [formula redacted] [footer] 356 [header] (5) we get [formula redacted] where [formula redacted] is supposed to be constant. The equation to the path is thus [formula redacted] If xo is the distance between the two plates and po the radius of the upper one the condition that the particle should just reach the boundary of the upper plate is [formula redacted] If the value of W considered as a function of u , a , and p is < that given by equation (7) the particle will reach the upper plate, provided u also satisfies the inequality (2), otherwise it will either be returned to the lower plate or it will reach the guard-ring. Solving the quadratic equation we see that the particle will just reach the edge of the plate if [formula redacted] when [formula redacted] so that the positive sign is the one to take. Hence for the particle to reach the upper plate W must lie between the limits and [formula redacted], whilst u must lie between. 0 and [formula redacted]. If the fraction of the ions emitted whose velocity parallel to the plates lies between W and W + dW is [formula redacted], the current received by the upper plate will be [formula redacted] Particular Case of Maxwell's Law of Distribution. If we assume tentatively that the law of distribution of velocity among the emitted ions is the same as that among [header] 357 the molecules of a gas which start from any surface bounding it or within it, then the above functions may be calculated by the methods of the kinetic theory of gases. They are [formula redacted] where 3/4k is the average energy of translation of a molecule at the temperature of the hot body. It is to be borne in mind that the above functions are expressed as fractions of the total number of ions leaving an element of area perpendicular to the axis x in unit time, and not in terms of the number in unit volume as is usually done. If we substitute these values of the functions in the preceding formulae and carry out the integrations, we shall obtain the current to the upper plate as a function of the potential-difference, provided the law of distribution of velocity among the emitted electrons is Maxwell's law. Under these 1 circumstances, in the case where the planes are of indefinite extent, the current to the upper plate becomes [formula redacted] if i0 is the value of the current at the initial instant when V = 0. Since [formula redacted]where R : is the constant in the gas equation [formula redacted], calculated for a single molecule, and is the absolute temperature, we have, taking logarithms [formula redacted] where v is the number of molecules in 1 c.c. of gas at 0° C. and 760 mms. pressure, and R is the constant in the equation [formula redacted] taken to refer to the quantity of gas occupying unit volume under these standard conditions. Assuming what is now fairly well established, that the charge e on an electron is equal to that carried by a monovalent ion during electrolysis, ve is equal to the quantity of electricity required to liberate half a cubic centimetre of hydrogen in a water voltameter under standard conditions of temperature and 358 [header] pressure, since hydrogen is a monovalent element having a diatomic molecule in the gaseous state. Thus on this view both ve and R are well known physical constants. The preceding relations may be used to determine the way in which the potential of the upper plate varies with the time t, during which the current from the hot body has been flowing. If C is the capacity of the upper plate and its connexions we have [formula redacted] so that [formula redacted] integrating this, [formula redacted], we have subject to the condition that V=0 when t=0, we have [formula redacted] or [formula redacted] and [formula redacted] The current, therefore, is always finite and vanishes when [formula redacted]. Nevertheless the potential is infinite when t is infinite. This approach to an infinite value of the potential is not observed in practice. This is probably due to the fact that the current falls off' with the time so rapidly that it soon becomes comparable with the small leaks inherent in the apparatus, and with the discharging current carried by ions of the opposite sign. For these reasons a limit is soon fcund to the potential to which the upper plate can be charged in this way. The two formula? (11) and (12), which give the current as a function of the potential-difference and the potential-difference as a function of the time respectively, are not independent of one another, since the former can be obtained from the latter by differentiation with respect to the time. To test the theory, therefore, it is only necessary to examine the truth of one of the two formulæ. This has been done for the formula [formula redacted] in a manner which will now be described. [header] 359 Experimental Arrangements. The general arrangements of the apparatus used to investigate the kinetic energy of the negative ions from hot platinum is shown diagrammatically in fig. 1. The central [figure redacted] portion of a narrow platinum strip H was bent upwards as shown through a square hole at the centre of the lower of two circular plates, so as to be flush with the upper surface of the latter. The upper plate U was somewhat smaller than the lower, and was surrounded by a guard-ring G. Both plates were perpendicular to the common axis passing through their centres. The upper plate was connected to one pair of the quadrants of a Dolazalek electrometer by means of the key kik2- The plug k 2 was connected with the guard-ring and with the second pair of quadrants. By means of the key b it could be connected either with the earth or any 360 [header] desired potential. In this way the rate o£ charging of the upper plate could be determined both with the two plates initially at the same potential and with any desired difference of potential between them. The charging quadrants could also be connected to a subdivided standard condenser not shown in the diagram. The platinum strip was heated by a current furnished by the battery B and regulated by two sliding rheostats in parallel (shown as one in the diagram) at r 2 . One of these had a much higher resistance than the other, and served as a fine adjustment. The temperature was determined from the resistance, which was measured by the Wheatstone's bridge arrangement, of which H, R a , Rj, and R formed the four arms. This is the method previously used by Richardson. The essential conditions that R and R b should both be large compared with R a and H, and that R a should carry the current without heating, were satisfied. It is important that the middle point of the exposed portion of the hot strip should be at zero potential. If a fine wire was welded to the middle portion of H and soldered to the lower plate it was found that this gave rise to local variations in the heating, and also that any slight displacement of the strip during the experiments put it out of adjustment. These difficulties were overcome by shunting the whole Wheatstone's bridge circuit with a high resistance r1, any point of which could be connected to earth. By trial a point in r1 was found so that when it was connected to earth the initial rate of leak to the upper plate was unchanged on reversing the main heating current. The condition for this is evidently that the centre of the hot strip should be at zero potential. By simply reversing the main current from time to time this adjustment could be tested and a readjustment made, if it were required during the course of the experiments. The lower plate was permanently connected to earth. A section through the platinum strip, showing the plates and arrangement of apparatus in their immediate neighbourhood, is shown in fig. 2. The detailed construction of the plates will be described later. The lower plate consisted of two sections screwed together. The platinum strip was held between them and insulated from them by strips of mica. The thickness of the platinum strip was *0018 cm. Its other dimensions were '2 x *5 cm. Its ends were welded to heavier platinum leads which dipped into glass mercury-cups sealed into the heavy brass base-plate B. These served to introduce the heating current. The resistance of the portion [header] 361 of the platinum strip which emitted the ions was between .04 and '006 ohm. That of the leads about # 1 ohm. The heating current was usually in the neighbourhood of 2 amperes. The lower plate L was supported by a brass ring soldered to the brass plate B. The upper plate and guard-ring were rigidly connected together and insulated from each other by ebonite pieces. These are not shown in fig. 2. [figure redacted] Additional ebonite supports, also not shown, enabled it to rest on the lower plate, and also kept it insulated from the latter. These supports also ensured that the plates were always at a constant distance apart and parallel to each other. Connexion with the electrometer was made by means of a stout wire soldered to the top of the upper plate, which supported a mercury-cup M insulated from the shield S by ebonite. The wire connecting it to the electrometer system dipped into this mercury-cup. 362 [header] The whole of this portion of the apparatus was enclosed by a glass tube C, so that any desired degree of evacuation could be obtained. The bottom of this tube was ground flat and rested in a groove in the base-plate, to which it was joined with sealing-wax done over with soft wax. All sealing-wax joints were made air-tight in this way. The platinum wires E 1 and E 2 were of course fused into the glass. An outlet tube connected with the mercury-pump, McLeod gauge, drying-apparatus, and tap for letting in gases. The drying agent used was phosphorus pentoxide redistilled in oxygen. In order to make certain that effects were not being caused by charges accumulating on the glass the upper plate and the wire leading from it were shielded by the flanged brass tube S. This was connected to the guard-ring and also to earth by means of the extensible wire E 2 . It will be seen that the system was very thoroughly protected from any effects which might arise from charging up of insulation. The particular design of apparatus was adopted so that it might very readily be taken apart and a new platinum strip substituted whenever that became desirable. It was admirable from that point of view. The detailed construction of the two plates is shown in fig. 3. The three ebonite pieces which served to insulate the [figure redacted] upper plate from the guard-ring and the lower plate, and also to support both on the lower plate, are denoted by e. The plan of the lower plate shows the square hole where the hot strip came up flush with the surface. The two sections show the way in which the platinum strip was supported. The mica insulating strips are denoted by m. A sheet of [header] 363 thin platinum foil was soldered to the lower side of the upper plate so as to avoid any effects which might conceivably be due to anything of the nature of contact electromotive force. The diameter of the upper plate used in the experiments was 3*6 cms. and the distance between the two plates was two millimetres. In discussing the results the formulae for infinite planes will be used. Strictly speaking we ought to apply formula (8), substituting the values of F(u0) and F'(W) in (9). When this is done an integral is obtained which cannot be evaluated in finite terms. It is, however, easy to show from expression (8) that the fractional error introduced by neglecting the finiteness of the radius p of the upper plate will be comparable with a quantity lying between [formula redacted] and [formula redacted] depending on the value of V. A preliminary calculation showed that with the dimensions of the apparatus chosen this error would always be smaller than the expected error of observation. This conclusion was subsequently confirmed by experiments in which the current to the upper plate, to the guard-ring, and to the two together were measured and compared. Under all conditions the current to the guard-ring was small compared with that to the plate, the ratio of the two being smaller than the probable observational error. Since the guard-ring was constituted so as to intercept all the ions from the lower plate which were not received by the upper plate, it is clear that practically all the ions from the metal strip which were not returned to the lower plate by the field reached the upper plate. The fiducial points used in calibrating the temperature-resistance curve were the temperature of the room, the melting-point of potassium sulphate (1066° C.) and the melting-point of platinum (18:20° C.) The temperature varied perceptibly along the strip owing chiefly to the conduction of heat from the ends. In standardizing it, therefore, the potassium sulphate was placed on the hottest portion, as this, owing to the tremendous rapidity with which the emission of ions increases with the temperature, would be the region which gave rise to the greater proportion of the total. In determining the resistance at the melting-point of platinum, that at which the strip melted was observed. This would obviously give the temperature of the hottest portion. When the three temperatures were plotted against resistance they were found to lie as nearly as possible on a straight line. 364 [header] This is not in agreement with the known variation o£ the resistance of platinum with temperature. Presumably the discrepancy arises from a change in the relative distribution of the heat along the strip as the temperature changes. The experimental temperatures were obtained from the measured resistances by reference to a chart constructed in the above way. We believe that this method of getting the temperature is trustworthy though not very accurate. The electrometer and everything connected with it were insulated on blocks of paraffin or ebonite, so that the rate of leak to or from the upper plate could be measured when it was charged to any desired potential. Potentials as high as 400 volts were used in some of the experiments. Results of the Experiments. In the earlier experiments trouble was experienced on account of the occurrence of positive ionization as well as negative. With a new wire, as is well known, the positive ionization is large compared with the negative, but decays away with time, so that after long continued heating the positive ionization becomes small compared with the negative. The latter was found to remain practically constant under comparable conditions throughout the experiments. In order to get rid of the positive ionization it was found necessary to heat the platinum strip from ten to thirty hours before taking readings. In these experiments it was never found possible to get rid of the positive ionization completely ; a considerable leak was always obtained if the upper plate was charged to a high negative potential. We think that most of this positive ionization was caused by gas evolved from the apparatus when the metal strip was heated. Fortunately the positive leak was small when the negative potential applied to the upper plate was small. In the experiments it was found that the negative ionization emitted by the strip was never able to charge the upper plate to a potential greater than one volt. If, therefore, the positive leak when the upper plate was charged to a potential of —2 volts was inappreciable compared with the negative leaks against the potential which were measured, it was taken that the positive leak had been sufficiently got rid of. Great care was taken to ensure that this condition was always fulfilled in practice. One of the authors (Mr. F. C. Brown) observed that the positive leak could be got rid of more quickly by charging the upper plate to a high negative potential, for example 200 volts. This procedure was, however, found to entirely alter the nature of the subsequent discharge of the negative [header] 365 ionization, the emitted ions being capable of going against a much higher potential than formerly. A detailed investigation of this interesting effect has been carried out by Mr, Brown, and will shortly be published, so that it is not necessary to say much about it here. It is evident that after such treatment the hot metal is in a peculiar state. In order to ensure that the metal was in a normal state we were careful to always keep the potential of the upper plate in the neighbourhood of zero so as to make sure that the hot metal strip was never placed in a strong electrostatic field. Incidentally it may be mentioned that it was found that the abnormal state induced in this way could be got rid of by heating the strip for a short time to a very high temperature. This property was made use of in some of the experiments which follow. The pressure of the gas (air) in the different experiments varied considerably from '001 to as much as % 06 mm. So far as the authors were able to judge, the actual value of the pressure had no effect on the phenomena investigated, provided it was as low as the above limits indicate. The chief experimental problem in hand was the determination of the current to the upper plate, when this was allowed to charge up, as a function of the potential-difference between the two plates. This relation when obtained enables the applicability of the formulae (10) and (11) to be tested. The platinum strip and the lower plate were always maintained at zero potential. The potential of the upper plate at any instant was determined by the reading of the electrometer to which it was connected. The electrometer was arranged to give 115 divisions deflexion for a volt, as this degree of sensitiveness was found to be most convenient for these experiments. The deflexions could be estimated to one-tenth of a division. In measuring the currents a suitable capacity was connected to the quadrants of the electrometer. The readings of the latter, which was nearly dead beat, were recorded at definite intervals of time, and the current was obtained from the formula [formula redacted], where c is the capacity of the electrometer and its connexions, including the condenser, At is the interval of time between two readings, and AY is the corresponding increment of potential. Strictly, of course, this formula is only true for infinitesimal intervals, but by inserting a sufficiently large capacity in the system it was found in practice that the rate of increase of potential with the intervals used did not diminish very much during any one interval. The error thus introduced was also averaged 366 [header] out to some extent by taking the current thus obtained to correspond to the potential at the middle of the interval. In some o£ the experiments a different method of reducing the observations was made use of. The observations of potential and time were used to plot a curve connecting these two variables. The values of [formula redacted] were then obtained from this curve by geometrical differentiation. This method was not found to give results which were either more consistent or more accurate than those given by the other, so that as it was much more laborious it was discarded. The maximum potential of the upper plate against which a measurable current would go was always about *6 of a volt. With potentials of this magnitude the current was small, and the rate of change of voltage was only measurable when either a very small capacity or no capacity at all was added to that of the electrometer. Generally speaking it was found that the insertion of two capacities which changed the total capacity in a ratio of about 50 to 1 enabled the currents to be measured conveniently throughout the whole range. Two series of readings were usually taken with the heating current in opposite directions. By taking the mean of two deflexions corresponding to any given potential any error arising from the central point of the hot strip not being connected to earth was eliminated. As the potential-difference between the two plates was always small, the fall of potential along the hot strip itself is of considerable importance. It was not possible to determine what this amounted to, but it was estimated to be between *08 and *012 volt, and was probably nearer the lower than the higher of these two limits. Assuming that the middle of the strip was at zero potential the greatest potential at any point of the hot metal would lie between the limits fOi volt and ±'006 volt. It is difficult to be quite certain how the results would be affected by the existence of this external field along the strip. A large number of series of observations were taken in the manner indicated with slightly varied conditions. The numbers recorded in one of them are shown in the accompanying table. It will be noticed that with the same mean potential-difference ('11 volt) between the plates the currents measured were independent of the capacity used. The actual determinations were 21*5 x 10~ 12 ampere with *001 microfarad and 22xl0~ 12 ampere with -1 microfarad. The temperature in this experiment was 1283° C. [header] 367 Negative Ionization Electrometer Readings Capacity microfarads Pressure Time Heating current Direct revised mean Increment of potential, scale divisions Interval of time, seconds Current amperes Mean potential, volts [table redacted] Resistance 7570, temperature 1556 absolute scale. Gas constant [formula redacted] A series of observations similar to those in the table have been plotted in fig. 4, so as to exhibit the potential to which [figure redacted] the upper plate charges as a function of the time. 1 [ The general form of this curve is in agreement with the equation (12) which was deduced theoretically. Fig. 5 exhibits the current to the upper plate as a function of the potential-difference tending to stop it. The points 368 [header] with circles round them denote the values of the current recorded in the preceding table at the potentials given by the abscissae. These points are seen to be distributed fairly evenly about the smooth curve shown. In order to test the theoretical formulae (10) and (11) a series of points on the [figure redacted] smooth curve were taken, and the logarithms of the corresponding values of the current were obtained. These are plotted against the potential-difference in the figure. The scale of the logarithmic curve is shown on the right-hand side of the diagram. [header] 369 If we use logarithms to the base 10 formula (11) becomes [formula redacted] As [formula redacted], and R are constant in any one experiment, the curve obtained by plotting [formula redacted] against V should be a straight line. The accuracy with which the linear relation was fulfilled is shown by the diagram. From the slope of this line the value of the coefficient [formula redacted] could be deduced. Substituting the known value of ve, the quantity of electricity required to liberate half a cubic centimetre of hydrogen at 0° C. and 760 mms. by electrolysis, and the value of [theta], the absolute temperature, found experimentally, a value of R the gas constant could be determined by these experiments and compared with the well-known value of this constant. This particular experiment gave for R the value 4*1 x 10 3 C.G.s. units compared with the standard value 3*7 x 10 5 . The agreement is fairly good when all the difficulties of the investigation are taken into account. A large number of experiments were made with platinum under varied conditions, and this linear relation was always verified provided the general method of treatment of the platinum which has been described was adhered to. In most of the experiments the surrounding gas was air at a low pressure ; but the effect of replacing the air by hydrogen at a similar pressure was examined. In another set of experiments the platinum was covered with calcium oxide by heating a drop of calcium nitrate solution placed on it. In both these cases it was noticed that for some time after the change in the conditions had been made the above law connecting the current with the potential-difference was obeyed just as with platinum alone surrounded by air. It was also noticed that on first heating the strip after letting in hydrogen or after coating it with lime, the value of the current at any given temperature was practically unchanged from that which had obtained for pure platinum at the same temperature in air. The subsequent increase in the emission of ions, whether caused by hydrogen or by lime, appeared to take some time to establish itself, and as soon as a marked increase in the initial absolute value of the current occurred, it was found that the law was no longer obeyed. Thus the character of the phenomena appears to change when the metal is heated for a long time either in a hydrogen vacuum or when coated with lime. The results which we obtained after heating for some time in hydrogen were too irregular to draw very definite [footer] 370 [header] conclusions from ; but in the case of the lime-covered strip more consistency was observed. The main feature of the change consisted in a reduction of the curvature of the curve connecting the current and the potential. This reduction was greater the higher the temperature of the lime-coated strip, and also therefore the greater the absolute value of the current. At higher temperatures the current appeared within the limits of experimental error to be a linear function of the potential instead of an exponential function, In one experiment the current was 3*4 x 10 -8 amp. for V = 0 and diminished as a linear function of the potential-difference to the value zero for Y = l*22 volts. Usually the current reached zero for a somewhat smaller voltage than this. The general effect of both hydrogen and lime on the hot platinum appears to be to change the law of distribution of energy among the emitted electrons entirely, and also to change the average value of the kinetic energy to some extent. In every case the average value of the kinetic energy was found to be greater than what it would have been for pure platinum at the same temperature in air. This change is, comparatively speaking, not very great. The greatest increase recorded in our experiments amounted to about twice the value for platinum alone. It has been pointed out that the normal behaviour of the strip was also deranged if the hot metal was subjected to the action of a strong electric field at any time. In this case both the law of distribution of energy and the average value of the latter appear to be entirely changed. This peculiar state of the metal can be got rid of very rapidly by heating it for a short time to a high temperature. After this treatment the current against the potential again obeys the formula [formula redacted] with the same value of the coefficient [formula redacted] as in the normal case. The value of II calculated from eight different series of experiments which have been selected as illustrating all the different conditions under which the linear relation between log i and V was found to be satisfied are given in the accompanying table. It will be observed that other conditions besides the previous treatment of the hot metal were varied during the experiments. The pressure of the gas varied from *006 to '06 mm., the temperature from 1473 to 1840 ab- lute, and the maximum current from 3 x 10 -12 to 4*7 X 10 -12 ampere. The last series in the table was obtained after the hot metal had been put into the peculiar state already described by charging the upper plate with a high negative [header] 371 potential, and this peculiar state had afterwards been destroyed by strongly heating the strip. The last observation but one refers to similar conditions, except that the peculiar state was induced by charging the upper plate negatively. It is possible that some change took place in the strip which made the recorded temperatures too high in these two experiments. The values of U calculated from the different experiments are shown in the last column of the table. These range from 2-9 x 10' 3 to 4-2 x 10 3 with a mean value of 3*5 x 10 3 . The disagreement of these numbers among each other is probably greater than could arise from errors in the measurements of any of the physical quantities involved, such as the temperature for example. When we consider the number of things which appear to affect this phenomenon in a way which is not yet understood, the agreement is probably as satisfactory as could be expected. The agreement of these numbers with the theoretical value [formula redacted] is very striking, and seems scarcely likely to be a chance coincidence. Date Treatment of Platinum before Observations Pressure Absolute Temperature Maximum Current, amperes [table redacted] In order to test the theory further the obvious thing to try was whether the coefficient [formula redacted] deduced from the log i, V diagram really was universally proportional to the absolute temperature. A glance at the preceding table will show that in the case of platinum this is a difficult if not impossible task. The disagreement between the different determinations of R shows a greater ratio of variation than the fractional change in the absolute temperature over the whole range of temperature during which the effect could conveniently be measured. For this reason it seemed likely that [footer] 372 [header] it would prove impossible to disentangle any change in the coefficient due to 6 being changed from changes due to unknown causes. We have therefore not attempted to test this part of the theory by experiments on platinum itself. The negative ionization from the liquid alloy of sodium and potassium suggested itself as a more likely way of attacking this question. With this substance the current can conveniently be measured at temperatures as low as 500 p absolute; so that if the theory were obeyed by this substance the value ve of [formula redacted] ought to be about three times as great as in the experiments on platinum. In making experiments with this substance we have had considerable experimental difficulties to contend with, and so far have only been able to obtain results of a qualitative character. So far as our experiments go, they indicate that the nature of the negative ionization from the substance is not at all what was expected. Instead of the current falling off more rapidly with the potential than with hot platinum, it falls off less rapidly, indicating that the emitted electrons have a much greater quantity of kinetic energy than those emitted from platinum at a much higher temperature. The experiments that we have been able to make with the alloy so far are not sufficiently accurate to decide whether the formula [formula redacted] is obeyed with a different value of k or not. We hope to be able to resume this part of the investigation in the autumn. Discussion of Results. In presenting the results of this investigation the method has been adopted of making certain hypotheses as to the distribution of energy among the emitted electrons. From these hypotheses formulas have been deduced which were then compared with the experimental observations. This method is justifiable on the ground that these theoretical considerations first suggested the investigation, and also because in the case of platinum the phenomena are in accordance with the theory. We have seen, however, that a number of cases have arisen where the law of distribution of energy among the particles does not coincide with Maxwell's law, so that to analyse these cases it is important to have a more general method of deducing the mode of distribution of the energy than that which has been made use of. This may always be done in the following way : — First of all construct the curve giving the current-densities as ordinates in terms of the opposing potentials as abscissas. Our interpretation [header] 373 of this curve is that the current Cv corresponding to any potential V is equal to e the charge on an ion multiplied by the number of ions shot off in unit time for which [formula redacted] is greater than eV. Calling this number N ( v) we have then [formula redacted]. But if the number of ions emitted per second for which the normal component of the energy lies between eV and [formula redacted], (i. e. between [formula redacted] and [formula redacted] is denoted by [formula redacted] we shall have [formula redacted] So that [formula redacted] To obtain the number which have velocity components perpendicular to the emitting surface lying between u and [formula redacted] we have simply to replace eV by its kinetic equivalent [formula redacted]. We thus get [formula redacted]. The number of particles whose normal velocity or energy lies between given limits can thus always be calculated from the tangent to the CV curve. If we apply this method to the experimental numbers obtained for platinum in what we have called the normal condition, the function giving the number having energy between assigned limits is that required by Maxwell's law. This is sufficiently obvious, since otherwise the equations obtained previously would not have been satisfied. In the case where the electrons were emitted from platinum covered with lime the CY curve lost its exponential character and became a straight line at high temperatures. In this case [formula redacted] is constant, so that the number of particles whose energy lies between x and x + dx is proportional to dx and independent of x, or, in other words, the number of particles having an amount of [formula redacted] lying within a given range is independent of the amount itself. This is only true within certain limits ; in the case referred to the number of particles which had an amount of energy greater than that which corresponds to 1*22 volts was too small to be detected. The measurements that we have made in the cases in which the distribution of energy is abnormal are too meagre 374 [header] as yet to enable much to be said positively as to the causes which make the distribution of the energy differ from the Maxwell type. There are, however, certain obvious causes which might change the distribution of energy. If a double layer formed outside the metal, and the direction of the electric force in the double layer was such that it tended to drive the ions away from the metal, the distribution of energy would be altered. Admitting that the free electrons inside the metal have the distribution of velocity given by Maxwell's law, those which escaped into the double layer would also have this distribution provided they were enabled to get out by virtue of their kinetic energy overcoming the surface forces. All the ions which reached the double layer would escape into the region outside, but the value of [formula redacted] for each of them would be increased by the work done in passing through the layer. Thus, in this simple case, the distribution of energy among the emitted particles would be that given by Maxwell's law + a constant. If the electrons escaping were deflected by collisions with atoms inside the double layer itself this simple law of distribution would be altered and would become very complex, but in any case we should expect that any variation from the normal would result in an increase in the average energy of the particles. This explanation is supported by the facts, so far as this particular conclusion is concerned ; for in all the cases of deviation from the type which we have examined the mean kinetic energy appeared to be greater than that required by Maxwell's law. The most reasonable way of interpreting the results which have been obtained so far appears to us to be to suppose that generally speaking the distribution of velocity among the free electrons inside the metal is that given by Maxwell's law for a molecule of gas at the same temperature as that of the metal. That when the electrons which escape simply have to do a certain amount of work against surface forces this law also holds for the distribution of energy among the electrons which have escaped. It seems probable, however, that there are a large number of cases of the escape of electrons from hot metals when the mechanism is not so simple as this. There may for instance be a double layer like the one already alluded to ; there is some evidence that the very large change produced by absorbed hydrogen on the leak from hot platinum may be due to an effect of this kind. It is possible also that the expulsion of these electrons in some cases is due to a more indirect process. It might for [header] 375 instance be an effect of the radiant energy o£ the metal analogous to the photoelectric effect. In this case the kinetic energy would probably be much greater than the thermal value. We hope that further research will throw light on this point. It seems to us an important point to have established that in one case at any rate, that of platinum heated in air at low pressure, the distribution of the square of the velocity component normal to the surface among the electrons emitted is identical with Maxwell's law of distribution of the same quantity for a gas at the temperature of the metal. It has been suggested to us that experiments of this kind do not necessarily enable us to deduce the law of distribution of velocity among the emitted particles, for the reason that formulae similar to those on which our conclusion is based might be deduced by a purely hydrodynamical kind of treatment assuming that the particles exerted a pressure which was related to the temperature according to the law [formula redacted]. Since this kind of treatment supposes the matter concerned to behave as though it were continuous it would follow that the experiments would not warrant any conclusion as to the distribution of velocity among the particles. It appears to us, however, that this is an unfair view of the question to take. It is now well established that the electric currents under investigation are carried by charged particles whose charge and mass are known. Admitting for the moment our interpretation of the experiments, it follows that at 1650° absolute the mean value of u, the component of the velocity, perpendicular to the plate, of the ions emitted is about 1*5 x 10 7 cm. per sec. The distance between the plates being 2 millims., the average time occupied by the ions in crossing under zero field would be 1*3 X 10 -s sec. The maximum current in any of the experiments was 4' 7 x 10 -11 ampere, which corresponds to an emission of 3*6 x 10 8 ions per sec. The number of ions present at any instant between the two plates would therefore be comparable with 5. The average distance between them would be so great that their mutual forces would be entirely negligible. On these grounds it appears to us that the only reasonable view to take is that the current is carried by discrete charged particles whose motion after they have left the plate is determined solely by the magnitude of the electric field and their initial velocity. Unless we are prepared to deny the atomic theory of electricity there appears to be no escape from the conclusion that the distribution of velocity among; 376 [header] the emitted particles is that which has been deduced from these experiments. This method does not enable us to determine by experiment the distribution of velocity among the electrons in a closed space including a piece of hot metal when the final state of statistical equilibrium has been reached. All that we are able to do is to examine the distribution of velocity among the particles emitted from the hot metal at any instant, and to show that in the case of platinum at least the results are consistent with what would be required if in the state of statistical equilibrium the distribution of velocity among the electrons outside the metal were determined by Maxwell's law. This leads to a strong presumption that the distribution of velocity among the external electrons in the steady state would be given by Maxwell's law, with the mean translational kinetic energy identical with that of the molecules of a gas at the temperature of the metal. This involves the further conclusion that the distribution of velocity among the free electrons inside the metal is also determined by Maxwell's law. For if the free electrons inside the metal are free in the sense of the kinetic theory of gases, the only difference between those inside and those outside the metal will be due to the difference of their potential energy. There is a well-known theorem in the kinetic theory of gases which proves that when two regions of the same gas at the same temperature are compared, the regions being such that the mean potential energy of the molecules is different in the two regions, the mean translational kinetic energy is the same in both, and is distributed according to the same law. The only effect of the difference of potential energy is to make the concentration of the molecules different in the two regions. Applying this theorem to the case of the electrons inside and outside a piece of hot metal, it follows that the mean translational energy and the way in which it is distributed among the electrons will be the same both inside and outside the metal. The conclusion that the average translational kinetic energy and the law of distribution of velocity of the electrons inside a metal are identical with those among the molecules of a gas at the same temperature as that of the metal is of great importance in the electron theory of metallic conduction and thermal radiation. Princeton, N. J., June 5, 1908.