XVII. On Lorentz's Theory of Long Wave Radiation. By G. H. Livens [Communicated by the Author.]. THE Lorentz form of the theory of radiation, which regards the radiation from a thin metallic plate as arising from the motion of the electrons inside the plate, will probably always remain as a deciding factor in the general theory of this subject, since it involves no principles which cannot certainly be regarded as well established by independent theory and experiment. The final formula to which this theory leads and the extent to which it depends on the assumptions made must therefore be matters of the first importance in the general theory. Basing mainly on the two assumptions that the period of the radiation considered is long compared with the interval of time between two consecutive collisions of an electron with an atom and that this latter interval is also long compared with the time of duration of a collision, Lorentz derives a formula, identical with the Rayleigh-Jeans formula, which is apparently correct in the long wave part of the spectrum but fails hopelessly for obvious reasons in the visible and ultraviolet regions. These two assumptions of course naturally restrict the analysis to long waves, but it is the expressed opinion of Prof. Lorentz f that the same Rayleigh-Jeans formula would be obtained as the general result for the other parts of the spectrum if only the difficulties of the analysis could be overcome. The main object of the present paper is the discussion of a partial generalization of Lorentz's analysis, in which one of the two above-mentioned restrictions is removed. A formula, which is concluded to be practically identical with the Kayleigh-Jeans formula, and applicable to all parts of the spectrum, is obtained on the single assumption that the duration of the impact of an electron with an atom is always negligibly small ; and the conclusion carries with it a partial confirmation and possible limitation of a certain well-known result in the optical theory of metals. The method to be followed is identical with that given by Lorentz with the single exception that it will not be found necessary to assume the relation between the period of oscillation and time interval between successive collisions which is implied in his theory. It is, however, necessary to retain the assumption he makes regarding the smallness of duration of a collision in order to avoid making arbitrary [header] 159 hypotheses regarding the dynamical character of the collisions. The case to be analysed is, therefore, virtually that in which the electrons and molecules are assumed to be perfectly rigid elastic spheres, the molecules being, however, of comparatively large mass so that their energy and motion may be neglected. I do not find it necessary to depart very widely from Lorentz's admirable exposition of his theory given in his book ; The Theory of Electrons' [citation redacted], and I shall take the liberty of quoting verbally in many cases from his work, to which I must here acknowledge my great indebtedness. We will, therefore, with Lorentz, consider a thin metallic plate in which a large number of free electrons are moving about in a perfectly irregular manner, consistent with the general laws of the conservation of their total energy and momentum. We know that an electron can be the centre of an emission of energy when its velocity is changing, thus, as a result principally of the numerous collisions of the electrons with the atoms, resulting in alterations of the directions and magnitudes of the velocities of the electrons, a part of the heat energy of the irregular motion of the electrons will be radiated away from the metal. This radiant energy, which is subsequently to be the subject of a detailed examination, is, however, presumed to be so small compared with the energy of motion of the electrons that it can be neglected in any dynamical considerations respecting those motions extended over any finite time. To this extent the analysis offered is only a first order approximation to the actual state of affairs. We know also, that as a result of the same collisions between the electrons and atoms, part at least of any regular or organized energy acquired by the electrons during their free motion between the atoms can be dissipated into heat energy of the irregular motion of the same electrons. In this way it is possible for a metal to absorb a portion of the energy from an incident beam of radiation, because the electric force in the electromagnetic field associated with the radiation will pull the electrons about during their otherwise free motion between collisions, imparting kinetic energy to them which will be dissipated by collision at the end of each path into irregular heat-motion. Now let tv and w' be two infinitely small parallel surface elements, to being on the plate itself and w' at a distance r outside it on the normal to the plate through the centre of iv. Then of the whole radiation emitted by the metal 160 [header] plate, a certain portion will travel outwards through w and so'. Suppose we decompose this radiation into rays of different wave-lengths and each ray again into its plane-polarized constituents in two planes at right angles through the chosen normal to the plate (these two planes and the plane of the plate being parallel to a system of properly chosen rectangular coordinate planes in which z = 0 is the plane of the plate). Now consider in particular those of the rays in this beam whose wave-length lies between the two infinitely near limits [lambda] and [citation redacted] and which are polarized in the plane [formula redacted]; the amount of energy emitted by the plate per unit time through both elements w and to r so far as it belongs to these rays, must be directly proportional to [citation redacted] and [citation redacted] and inversely proportional to r 2 , and it can therefore be represented by an expression of the form [formula redacted] The coefficient E is called the emissivity of the plate and is a function not only of the positions of w, v/, and [lambda] but also of the conditions and type of the metal composing the plate. Let us now consider the opposite process. Suppose that a plane-polarized beam such as that specified in the previous paragraph is incident, through the small surface w' , on the patch 10 of the metal plate : then we know that a certain portion of the energy of this beam will be absorbed in the metal and converted into heat-energy, instead of being re-emitted as a portion of the reflected or transmitted beams. The fraction expressing the proportion of the energy absorbed is called the coefficient of absorption of the plate under the conditions specified, and is denoted by A. Starting from the thermodynamic principle that in a system of bodies having all the same temperature, the equilibrium is not disturbed by their mutual radiation, Kirchhoff finds that the ratio [fomrula redacted] between the emissivity and absorbing powers under the same conditions is independent both of the direction of polarization and the position and peculiar properties of the metal plate. This ratio, a function merely of the temperature T and wave-length [lambda], is now the chief object of search in the general theory of radiation, determining as it does the complete circumstances of the steady thermal radiation from any body. [header] 161 But in the conceptions we have adopted, the calculation of both E and A under the assumptions specified can be directly accomplished. If we consider that the thickness [delta] of the metallic plate is so small that the absorption may be considered as proportional to it, we shall find by an obvious calculation, after Lorentz, that [formula redacted][citation redacted] e being the usual velocity constant and a the conductivity of the metal. Now the interpretation of a in terms of the electron constants of the metal, although a matter of some difficulty, is nevertheless fairly certain f. If N denote the number of free electrons per unit volume in the metal, each of mass m and with a charge e, moving with velocities the average square of which is [formula redacted] , then we know that in all applications involving steady or slowly varying currents the conductivity [sigma]o is given by [formula redacted] wherein l m is a constant, a certain mean length of path, which is determined by the formula [formula redacted] in which n is the number of atoms per unit volume in the metal and R the sum of the radii of an atom and an electron. However, in applications involving more rapid alternations in the current the above formula is found to be insufficient and requires modification along lines already laid down by various authors. According to Jeans J the correct form to be used for alternating currents with a frequency [formula redacted] is [formula redacted] [footer] 162 [header] or, on substitution of the value of [sigma]o, we get [formula redacted] a formula reducing to the Lorentz-Drude formula for large values of [lambda]. We have therefore for the coefficient of absorption under the conditions specified and for plane-polarized radiation of wave-length [lambda], [formula redacted] Now let us consider the radiation from the plate, still closely and often verbally following Lorentz. We need only consider the radiation normally from the small volume [formula redacted] of the plate, as this is the only part of all the radiation through w from the whole plate that gets through w' . Now according to a well-known formula of electrodynamics, a single electron moving with a velocity v (a vector with components [formula redacted] in the part of the plate under consideration, will produce at the position of w' an electromagnetic field in which the x-component of the electric force is given by [formula redacted] if we take the value of the differential coefficient at the proper instant. But on account of the assumption as to the thickness of the plate, this instant may be represented for all the electrons in the portion [formula redacted] by [formula redacted] , if t is the time for which we wish to determine the state of things at the distant surface w . We may therefore write for the x-component of the electric force in the total field at w' and then the flow of energy through id' per unit of time will be [formula redacted] as far as this one component is concerned. [header] 163 Since the motion of the electrons between the metallic atoms is highly irregular and of such a nature that it is impossible to follow it in detail, we must rather content ourselves with mean values of the variable quantities calculated for a sufficiently long interval of time. We shall, therefore, always consider only the mean values of our quantities taken over the large time between the instants t = and t = [theta]. For example, the flow of energy through 10' is, on the average, equal to [formula redacted] say. Now whatever be the way in which E x changes from one instant to the next, we can always expand it in a series by the formula [formula redacted] where s is a positive integer and [formula redacted] The frequency in the sth term of this series is [formula redacted] so that the wave-length of the vibration represented in it is [formula redacted] If is very large the part of the spectrum corresponding to the small interval of length d[lambda] between wave-lengths [lambda] and [formula redacted] will contain the large number [formula redacted] of spectral lines represented by terms of this series. If now we substitute the Fourier series for E, into the expression for the mean energy flux through [formula redacted], we shall And in the usual manner that it is equal to [formula redacted] To obtain the portion of this flux corresponding to wave-lengths between [lambda] and [formula redacted] we have only to observe that the [formula redacted] spectral lines, lying within that interval, may be considered to have equal intensities. Tn other words, the value a s may be regarded as equal for each of them, so that they contribute to the sum 2 in (2) an amount [formula redacted] 164 [header] Consequently the energy flux through w' belonging to the interval o£ wave-lengths d\ is given by [formula redacted] and we now want to find a s . From the value of E z given by equation (1) we see that [formula redacted] where the square bracket round the Vx serves to indicate the value of this quantity at the time [formula redacted] . The sign [sigma] now refers again to a sum taken over all the electrons in the part [formula redacted] of the plate. On integration by parts we find [formula redacted] or what is the same thing [formula redacted] Now each of the integrals on the left is made up of two parts,, arising respectively from the intervals between the consecutive impacts of the electrons and from the intervals during these impacts. If, as mentioned above, we can suppose the duration of an encounter of an electron with an atom to be very much smaller than the time between two successive encounters of the same electron, we may neglect altogether the part that corresponds to the collisions and confine ourselves entirely to the part corresponding to the free paths between the collisions. But while an electron travels over one of these free paths, its velocity v x is constant. Thus the part of the integrals in a s which, corresponds to one electron and to the time during which it traverses one of its free paths is therefore [formula redacted] where t is now the instant at which this free path is commenced and t the duration of the journey along it ; but this is equal to [formula redacted] [header] 165 We now fix our attention on all the paths described by all the electrons under consideration during the time [theta], and we use the symbol S to denote a sum relating to all these paths. We have then [formula redacted] We now want to determine the square of the sum S. This may be done rather easily because the product of two terms of the sum whether they correspond to different free paths of one and the same electron, or to two paths described by different electrons, will give if all taken together. Indeed the velocities of two electrons are wholly independent of one another, and the same may be said of the velocities of one definite electron at two instants separated by at least one encounter. Therefore positive and negative values of v x being distributed quite indiscriminately between the terms of the series S, positive and negative signs will be equally probable for the products of two terms. We have therefore only to calculate the sum of the squares of the terms in S or simply [formula redacted] Now since the irregular motion of the electrons takes place with the same intensity in all directions, we may replace [formula redacted] by [formula redacted] . Also in the immense number of terms included in the sum (3) the quantities r and v are very different, and in order to effect the summation we may begin by considering only those terms for which the product [formula redacted] has a certain value. In these terms which are [formula redacted] still very numerous, the angle [formula redacted] has values that are distributed at random over an interval ranging from to sir. The square of the cosine may therefore be replaced by its mean value 1/2, so that [formula redacted] or if we introduce, after Lorentz, the length of the path l instead of the time in it, this may be written [formula redacted] 166 [header] The metallic atoms being considered as practically immovable, the velocity of an electron will not be altered by a collision. Let us, therefore, now fix our attention on a certain group of electrons moving along their zigzag lines with the definite velocity u. Consider one of these electrons and let us calculate the chance of its colliding with an atom at rest in a unit of time. This chance is obviously equal to the number of atoms in a cylinder of base [formula redacted] and height u, R being as before the sum of the radii of an atom and an electron ; it is therefore equal to [formula redacted] n being the number of atoms per cubic centimetre in the metal. But in unit time the electron under consideration travels a distance in hence the chance of a collision of the electron with an atom per unit length of its path is [formula redacted] and thus the mean free path of an electron is, as before, [formula redacted] [Lorentz does not make it clear that the l m introduced here is, in fact, identical with that l m used in the formula for the conductivity ; the expanded argument here given, however, proves directly what was probably already known to him.] It is important to notice for future reference that l m is independent of u. This is a consequence of the assumed rigidity of the atoms. Now during the time 6 one of the electrons moving with a velocity u describes a large number of paths, this number being given by [formula redacted] and we now want to know how many of these paths are of given length l. For this, let f(l) be the probability that the electron shall describe a path at least equal to I, then [formula redacted] is the probability that the electron has described a path I and shall describe a further distance dl, and this will necessarily be the product of f(l) and another factor, this second factor expressing the probability of no collision occurring within [header] 167 the length dl. This factor is known from the above to be [formula redacted] so that [formula redacted] or what is the same thing so that [formula redacted] the arbitrary constant of the integration being determined by the condition that [formula redacted]. Thus the probability of the electron describing a free path between l and [formula redacted] is expressed as the product of the probability that it has described a free path I, and that it will collide in the next small distance dl, and is therefore [formula redacted] Thus of the total number of paths described by the electron in the time [theta] the number whose length lies between I and [formula redacted] is [formula redacted] so that the part of the sum in (4) contributed by these paths is [formula redacted] On investigation of this expression from Z=0 to l=co , we find the part of the sum in (4) due to one electron, which is therefore [formula redacted] Now the total number of electrons in the part of the metallic plate under consideration is [formula redacted] and by Maxwell's 168 [header] law, among these [formula redacted] have velocities between u and [formula redacted] ; the constant q is related to the velocity u m already introduced above by the formula [formula redacted] Thus the total value of the sum in (4) is given by [formula redacted] or, using [formula redacted] , by [formula redacted] This integral cannot be evaluated in definite terms, being of the integral-logarithmic type, but we can obtain various good approximations to its value. Iu fact a direct use of the first theorem of mean values in the integral calculus soon shows that we have [formula redacted] zo denoting some mean value of z, which is ultimately, however, a function of one constant [formula redacted] in the integral ; I find on trial that z is such a function of this constant that its value lies between 1 and 2, the values it assumes for small and large values respectively of the constant. If, therefore, we define u by the relation [formula redacted] we shall know that [formula redacted], ultimately a function of [formula redacted] must, however, lie between the limits [formula redacted] and [formula redacted] [header] 169 and then we shall have [formula redacted] and the expression for the partial energy flux through the element w l thus takes the form [formula redacted] But in virtue of the relation [formula redacted] this becomes [formula redacted] We therefore conclude that the emissivity of the plate is given by [formula redacted] On combining the two expressions for E and A we find that [formula redacted] Which is exactly Lorentz's result if [formula redacted] a value which certainly lies within the above limits possible for [formula redacted] , but which can only be said to be satisfied exactly for one particular value of [formula redacted]. If, therefore, the formula adopted for [sigma] is exact, our analysis verifies that Kirchhoff's law does not apply exactly in the case under investigation. Of course the discrepancy is small except perhaps for extremely short waves, but it is worth noticing. 170 [header] It would, however, appear more probable that it is the formula for a that is at fault [This probability is fully borne out by a more detailed investigation of the question as to the proper expression for [sigma] ] and not Kirchhoff's more general law, the truth of which can hardly be doubted. It is, however, in any case interesting to notice that although the formula for a adopted above may not be exact, the formula necessitated by Kirchhoff's law in combination with the above analysis provides an interesting verification of its general form. In any case, however, we may conclude that for all practical purposes the complete radiation formula applicable all along the spectrum is given by the usual Rayleigh- Jeans formula [formula redacted] as Lorentz predicts, a formula which is, however, only physically applicable in the extreme ultra-red part of the spectrum. But this general conclusion is utterly absurd both from a mathematical and a physical point of view, and it therefore appears that some fundamental error has been committed either in the physical assumptions made or in the mathematical analysis based on these assumptions. It is very difficult, if not quite impossible, to indicate any steps in the above analysis about whose mathematical rigour any doubts can be raised, but it is worth noticing that the final result obtained is not consistent with the preliminary assumptions, inasmuch as the Fourier series initially assumed, which can have no meaning unless it is convergent, ultimately turns out to be divergent, so that the theory would appear to lead to a result which is ultimately a contradiction in terms, or, at least, apparently so. Some light is, however, thrown on this question by an examination of the physical basis of the theory. The one advantage possessed by the present form of theory over Lorentz's original form is that the number of physical assumptions on which it is based is reduced from two to one, so that it is now possible to determine the actual extent to which the physical basis of the theory is responsible for the result obtained. We have merely assumed that the duration of every collision of an electron with an atom is vanishingly small compared with the other periods involved in our analysis, and as long as this assumption is justified our result must be correct. But, as a matter of fact, in actual [header] 171 practice this assumption is justified only to a comparatively rough extent and only when all the other times involved in the analysis are large compared with the usual intra-molecular periods, so that the results obtained can only be applicable in the extreme ultra-red region of the spectrum, where it is of course known to apply. As soon as the encounters between the electrons and atoms are sufficiently long compared with the period of the light discussed, the effect of the collision will make itself felt in modifying the radiation formula, a conclusion drawn some time previously by J. J. Thomson. It is just the assumption concerning the shortness of the collisions which is the predominating factor in restricting the general application of the Rayleigh-Jeans formula. The fact that the general radiation formula which is to be applicable all along the spectrum must ultimately contain some general account of the actual collisions of the electrons with the atoms can be illustrated in various ways. Let us confine our attention to one of the free electrons in the metal considered in the above analysis ; the average flux of energy through iv' arising as a result of its motion is [formula redacted] Now if the collisions are all of short duration and [formula redacted] denotes the total change [formula redacted] during a typical one of these collisions of total duration [formula redacted], then the energy radiated through w' during this collision is of total amount [formula redacted] which is inversely as At. Thus if, as in the above, we assume the duration of all the collisions to be infinitely short, the total amount of energy radiated away will be infinitely large. This merely means of course that as soon as the time of a collision becomes appreciable a closer investigation will be necessary, involving necessarily some account of the nature of the collision. It is now no longer surprising that a divergent series is obtained in the expression for the total energy : in fact the shorter the collisions the farther up the spectrum does the agreement between theory and practice hold, but then the bigger is the total energy.' These results are still further illustrated by the result of a general theory developed by Thomson [citation redacted] on molecular 172 [header] kinetic principles. The method followed by Thomson is analogous to that of Lorentz, but it avoids the probability considerations involved in that author's theory. He views, with Lorentz, the radiation as a result of the changes of velocity produced in the collisions of the electrons against the molecules, and he concludes that the manner in which these changes take place must, as stated above, ultimately be of influence on the final formula for the radiation. Assuming, then, as possible arbitrary types of acceleration of an electron during a collision functions of the time of the form [formula redacted] in which A and a are constants, he arrives at forms of [formula redacted] of the following types respectively, [formula redacted] which give for the total energy radiated respectively [formula redacted] and [formula redacted] wherein each integral on the right we have written [formula redacted]. Now in each of the two cases here illustrated the constant a turns out to be approximately equal to the time of duration of an encounter of an electron with an atom, so that if we assume this time to be infinitely small both forms of Thomson's theory agree in giving [formula redacted] as the complete radiation formula all along the spectrum, but in both cases the total energy is infinite of the order [formula redacted]. It would thus appear, both from these two examples and also from the more general case discussed above, that any general theory which leads to the Rayleigh-Jeans formula as the formula generally applicable all along the spectrum must involve some assumption which essentially implies that the total amount of energy radiated is infinite, so that it [header] 173 cannot represent any real physical example of a radiating body of any known type. Physically this implies that the general restrictions limiting in actual practice the validity of the physical hypotheses on which the theory is based, must also limit the applicability of the formula obtained from the theory, an obvious remark which it appears, however, necessary to insist upon, because it is the expressed opinion of certain mathematical physicists that, for example, the Rayleigh- Jeans formula obtained by Lorentz on certain obviously restricted assumptions is of a general validity in no way limited by the restrictions naturally imposed on these assumptions. The results obtained by Thomson are almost conclusive evidence that this contention is in no wise justifiable, and the results of the above form of Lorentz's theory are also against such an opinion. I hope to discuss, in further detail, in a future communication some of the points raised in the latter part of this paper and not fully disposed of. The University, Sheffield, 1914. Note added Dec. 2nd, 1914. — Since the above paper was sent to press I have discovered that Prof. H. A. Wilson has anticipated the main point of the above analysis although he apparently failed to appreciate its bearing [formula redacted]. He. however, unfortunately includes it as a small part of a paper, all the other results of which are either incomplete or inaccurate, and I think it deserves better and more elaborate treatment. Some advantage may therefore be gained by amplifying the point as above.