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RADIATION, THEORY OF
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See also:RADIATION, THEORY OF . The See also:physical activities that flourish on the See also:surface of the See also:earth derive their See also:energy, in a See also:form which is highly available thermodynamically, from the radiation of the See also:sun. This has been ascertained to be dynamic energy, transmitted in waves by the vibrations of a See also:medium occupying space, as the energy of See also:sound is transmitted by the vibrations of the See also:atmosphere. The See also:elasticity that transmits it may be assumed to be mathematically perfect: any slight loss in transit of the See also:light from the most distant stars, which See also:recent statistical comparisons of brightness with distance may possibly indicate, is to be explained far more suitably by the presence of nebulous See also:matter than by any imperfection of the See also:aether. The latter would thus be the one perfect frictionless medium known to us: it could not be such if it were constituted, like matter, of See also:independent molecules. It is thus on a higher See also:plane, and may even be considered to be a dynamical See also:specification of space itself. % See also:molecule of matter is a kinetic See also:system compounded of simpler elements; its energy may be classified into constitutive energy essential to its continued existence, and vibratory energy which it can receive from or radiate away into aether. A piece of matter isolated in See also:free aether would in See also:time lose ab. energy of the latter type by radiation; but the former will remain so See also:long as the matter persists, along with the energy of the See also:uniform translatory See also:motion to which it is ultimately reduced. Thus all matter is in continual See also:exchange of vibratory energy with the aether: it is with the See also:laws of this exchange of energy that the See also:general theory of Radiation deals, as distinguished from the mechanism of the aethereal vibrations, which is usually treated as the Theory of Light (see AETHER).
r. The See also:foundation of this subject is the principle, arrived
at independently by See also:Balfour See also: Assoc. See also:Report, 1871) that if the enclosure contains a radiating and absorbing See also:body which is put in motion, all being at the same temperature, the constituents of the radiation in front of it and behind it will differ in See also:period on See also:account of the Doppler-See also:Fizeau effect, so that there will be an opportunity of gaining See also:mechanical See also:work in its settling down to an See also:equilibrium; there must thus be some See also:kind of thermodynamic See also:compensation, which might arise either from aethereal See also:friction, or from work required to produce the motion of the body against pressure exerted on it by the surrounding radiation. The See also:hypothesis of friction is now excluded in ultimate molecular physics, while the thermodynamic bearing of a pressure exerted by radiation, such as is demanded by See also:Maxwell's electric theory, has been more recently See also:developed on other lines by See also:Bartoli and Boltzmann (1884), and combined with that of the Doppler effect by W. Wien (1893) in development of the ideas above expressed.
The See also:original reasoning of Stewart and Kirchhoff rests on the dynamical principle, that by no See also:process of See also:ordinary reflexion or transmission can the period, and therefore the See also:wave-length, of any See also:harmonic constituent of the radiation be changed; each constituent remains of the same wave-length from the time it is emitted until the time it is again absorbed. If we imagine a See also: In the See also:present See also:case of a field of radiation, this equalization cannot take See also:place directly between the various constituents of the radiation that occupy the same space, but only through the intervention of the emission and absorption of material bodies; the constituent radiations are virtually partitioned off adiabatically from See also:direct interchange. Thus in discussing the transformations of temperatures of the constituent elements of radiation, we are really reasoning about the activity of material bodies that are in thermal equilibrium with those constituents; and the theoretical basis of the See also:idea of temperature, as depending on the fortuitous See also:residue of the energy of molecular motions, is preserved. 2. Mechanical Pressure of Undulatory Motions.—Consider a wave-train of any kind, in which the displacement is = a See also:cos m(x+ct) so that it is propagated in the direction in which x decreases; let it be directly incident on a perfect reflector travelling towards it with velocity v, whose position is there-c—v from work done by the advancing reflector against pressure exerted by the radiation. That pressure, per unit surface, must therefore be equal to the fraction ? of the energy in a length c—v z c+v of the incident wave-train; thus it is the fraction ~z+u2 of the See also:total See also:density of energy in front of the reflector, belonging to both the incident and reflected trains. When v is small com- pared with c, this makes the pressure equal to the density of vibrational energy, in accordance with Maxwell's electrodynamic See also:formula (Elec. and Mag., 1871). The See also:argument may be illustrated by the transverse vibrations of a tense See also:cord, the reflector being then a lamina through a small See also:aperture in which the cord passes; the lamina can thus slide along the cord and sweep the vibratory motion in front of it. In this case the force acting on the lamina is the resultant of the tensions T of the cord on the two sides of the aperture, giving a lengthwise force 2Td(+S')2/dx2 when, as usual, See also:powers higher than the second of the ratio of See also:amplitude to wave-length are neglected; this, when v/c is small, is an oscillatory force of amount 2p(dt/dt)2, whose time-See also:average agrees with the value above obtained. If we consider a finite train of waves thus sent back from a moving reflector, the time integral of the pressure must represent force transmitted along the cord, or a gain of See also:longitudinal momentum in the reflected waves, or both together. When it is a case of transverse waves in an elastic medium, reflected by an advancing obstacle, the origin of. the working pressure is not so obvious, because we cannot easily formulate a mechanism for the advancing reflector like that of the lamina above employed. In the case of light-waves we can, however, imagine an ideal material body, constituted of very small molecules, that would sweep them in front of it with the same perfection as a metallic See also:mirror actually reflects the longer Hertzian waves. The pressure will then be identified physically, as in the case of the latter waves, with the mechanical forces acting on the screening oscillatory electric current-See also:sheet which is induced on the surface of the reflector. The displacement represented above by , which is annulled at the reflector, may then be taken to be either the tangential electric force or the normal component of the vector whose velocity is the magnetic force. The latter See also:interpretation is theoretically interesting, because that vector, which is the dynamical displacement in See also:electron-theory, usually occurs only through its velocity. The general case of oblique incidence can be treated on similar lines; each filament of radiation (See also:ray) in fact exerts its own longitudinal push equal to its energy per unit length, and it is only a matter of summation. The usual formula for the pressure of electric radiation is fore given at time t by x=vt. There will be a reflected train given by E'=a' cos m'(x-ct), the velocity of See also:propagation c being of course the same for both. The disturbance does not travel into the reflector, and must therefore be annulled at its surface; thus when x=vt we must have +'=o identically. This gives a'= -a, and m'(c-v)=m(c+v). The amplitude of the reflected disturbance is therefore equal to that of the incident one; while the wave-length is altered on the ratio c+U, which is approximately r-2~, where v/c is small, and is thus in agreement with the usual statement of the Doppler effect. The energy in the wave-train being See also:half potential and half kinetic, it is given by the integration of p(dE/dt)2 along the train, where p represents density. In the reflected train it is therefore augmented, when equal lengths are compared, in the but the length of the train is diminished by the reflexion in the ratio +U; hence on the whole the energy transmitted per unit time is increased by the reflexion in the ratio c+v This increase per unit time can arise only ratio (c±v) z c-U ' radiant energy. Then See also:contract the remaining radiant energy to its previous See also:volume, which requires' an See also:expenditure of less work on the walls of the enclosure than the expansion of the greater amount of radiation originally afforded; and, finally, gain still more work by again equalizing the temperatures of its constituents. The energy now remaining, being of smaller amount and under similar conditions, must have a temperature See also:lower .than the initial one. This process might be repeated indefinitely, and would constitute an See also:engine without an extraneous refrigerator, violating See also:Carnot's principle by deriving an unlimited See also:supply of. mechanical work from thermal See also:sources at a uniform temperature. Thus, independently of any knowledge of the intensity of the mechanical pressure of radiation, or indeed of whether such a pressure exists at all, it is established that the shrinkage of the enclosure must directly transform the contained radiation to the constitution which corresponds to some definite new temperature. Now we have seen that the wave-lengths of its constituents are all reduced in the same ratio by this process. If, then, we can prove that the intensities of these constituents are also all changed in a See also:common ratio by the reflexions at the shrinking envelope, it will follow that the distributions of the radiation among the various wave-lengths are, at these two temperatures, and therefore at any two temperatures, homologous, in the sense that the intensity curves, after the wave-lengths in one of them have been reduced in a ratio depending definitely on the two temperatures, differ only in the See also:absolute See also:scale of magnitude of the ordinates. This See also:procedure modifies Wien's argument by employing a uniformly shrinking spherical enclosure (cf. Brit. Assoc. Report, 1900). If the enclosure is not spherical, the angles of incidence at successive reflexions of the same ray will differ by finite amounts; we must then estimate the average effect of the shrinkage. In the form of enclosure here employed all rays are affected alike, and no averaging is required; while by the principle of Stewart and Kirchhoff what is established for any one form is of general validity. 4. Pressure of Natural Radiation.—The question reserved above has now to be settled. At first sight it might have appeared that the reflexion is simply total; but, as has been seen in § 2, the advancing perfect reflector does work against the pressure of the radiation, and this work must be changed into radiant energy and thus go to increase the intensity of the reflected ray. Considering electric radiation incident at See also:angle c, the tangential electric force is annulled at the reflector; hence the amplitude of the electric vibration is conserved on reflexion, though its phase is reversed. As already seen, the wave-length is shortened approximately by the fraction sv cos See also:tic in each reflexion; thus, just as in § 2, the energy transmitted per unit time per unit See also:area is increased in the same ratio; and allowing for the See also:factor cos L of foreshortening, there is therefore a radiant pressure equal to the total density of radiant energy in front of the reflector multiplied by cosh. This argument, being independent of the wave-length, applies to each constituent of the radiation in this direction separately; thus their energies are all increased in the same ratio by the reflexion, as was to be proved. When we are dealing with the natural radiation in an enclosure, which is distributed equally in all directions, this factor .cos2t must be averaged; and we thus attain Boltzmann's result that the radiant pressure is then one-third of the density of radiant energy in front of the reflector, this statement holding See also:good as regards each constituent of the natural radiation taken separately. 5. Adiabatic Relations.—Consider the enclosure filled with radiation of energy-density E at volume V, of any given constitution but devoid of special direction, and let it be shrunk to volume V - SV against its own pressure; if the density thereby become E- SE, the conservation of the energy requires EV+*ESV = (E-SE) (V-3V), so that IESV+VbE=o, or E varies as V. Again—but now with a restriction to radiation with its energy derived from a theory, namely, that of the ordinary electro- further gain of work can be obtained at the expense of the dynamic equations, which considers the velocity of the matter, or rather of the electrons associated with it, to be so small compared with that of radiation that the square of the ratio of these velocities can be neglected. The formula above obtained is of general application, and shows that for high values of v the pressure must fall off. It has been urged as an objection to the thermodynamic reversibility of a ray (§ 8) that the work of the radiant pressure exerted at its front is lost, as there is no obstacle to sustain it; but on an obstacle moving with the velocity of the wave-front the pressure would vanish, so that this objection does not now hold. In every such case of an advancing perfect reflector the aggregate amplitude of the superposed incident and reflected wave-trains, of different wave-lengths and periods, will be represented by { =2a See also:sin - (x - 1) sin—cu (x-vt); thus the See also:appearance presented will be that of a train of waves each of length (s-v/c)21r/m, and progressing with the velocity u of the reflector, which travels at one of the nodes of the train. This slowly travelling wave-train corresponds to the stationary train which would be produced by a stationary perfect reflector; but the amplitude is now a varying quantity which, once uniform vibration has been fully established along any path, may itself be described as See also:running on after the manner of a superposed wave-train of very See also:great wave-length (c/u-1)27r/m and of very great velocity c2/v. A somewhat similar state of things arises when a wave-train is incident on a stationary reflector very nearly normally, as may sometimes be seen with incoming rollers along a shelving See also:beach; the visible disturbance at a reflecting See also:ridge, arising from each single wave-See also:crest, then rushes along the ridge at a See also:speed which is at first sight surprising, as it is enormously in excess of the speed possible for any See also:simple train of waves travelling into quiescent aether. 3. Wien's See also:Law.—Let us consider a spherical enclosure filled with radiation, and having walls of ideal perfectly reflecting quality so that none of the radiation can See also:escape. If there is no material body inside it, any arbitrarily assigned constitution of this radiation will be permanent. Let us suppose that the See also:radius a of the enclosure is shrinking with extremely small velocity v. A ray inside it, incident at angle t, will always be incident on the walls in its successive reflexions at the same angle, except as regards a negligible See also:change due to the motion of the reflector (§ 2); and the length of its path between successive reflexions is 2a cos L. Each undulation on this ray will thus undergo reflexion at intervals of time equal to 2a cos t/c, where c is the velocity of light, and it is easily verified that on each reflexion it is shortened by the fraction 2v cos c/c of itself: thus in the very long time T required to See also:complete the shrinkage it is shortened by the fraction vTa, which is Sa/a where Sa is the total shrinkage in radius, and is independent of the value of L. The wave-length of each undulation in the radiation inside the enclosure is therefore reduced in the same ratio as the radius. Now suppose that the constitution of the enclosed radiation corresponded initially to a definite temperature. During the shrinkage thermal equilibrium must be maintained among its constituents; otherwise there would be a running down of their energies towards uniformity' of temperature, if material radiating bodies are present, which would be superposed on the mechanical operations belonging to the shrinkage, and the process could not be reversible. Such a state of affairs is not possible, for it would See also:land us in processes of the following type. Expand the enclosure, gaining the mechanical work of the radiant pressure against its walls, whatever that may be. Then equalize the intensities of the constituent radiations to those corresponding to a common temperature, by taking See also:advantage of the absorptions of material bodies at the actual temperatures of these radiations; when this is done, as it may actually be to some extent by aid of the sifting produced by partitions which transmit some kinds of radiation more rapidly than others, a distributed as regards wave-length so as to be of uniform temperature—the performance of this mechanical work iESV has changed the energy of radiation EV from the state that is in equilibrium of absorption and emission with a thermal source at temperature T to the state in equilibrium with an absorber of some other temperature T-ST, and that in a reversible manner; thus by Carnot's principle 3E5V/EV= -ST/T, so that T varies as V-i, or inversely as the linear dimensions when the enclosure is shrunk uniformly. Combining these results, it appears that E varies as T4; this is Stefan's empirical law for the complete radiation corresponding to the temperature, first established on these lines by Boltzmann. Starting from the principle that this radiation must be a function of the temperature alone, this adiabatic process has in fact given us the form of the function. These results cannot, how-ever, be extended without modification to each See also:separate constituent of the complete radiation, because the shrinkage of the enclosure alters its wave-length and so transforms it into a different constituent. 6. Law of See also:Distribution of Energy.—The effect of compressing the complete radiation is thus to change it to the constitution belonging to a certain higher temperature, by shortening all its wave-lengths by the proportion of one-third of the See also:compression by volume, the temperature being in fact raised by the same proportion; at the same time increasing in a uniform ratio the amounts corresponding to each See also:interval bA, so as to get the correct total amount of energy for the new temperature. In the compression each constituent alters so that TX remains See also:constant, and the energy ESA in the range 5X in other respects changes as a function of T alone. Hence generally EASX must be of form F(T)f(TA)5X. But for each temperature f E~SX is equal to E and so varies as T4, by Stefan's law; that is, T-'F(T)J f(TT)d(TX) ccT4, so that T"'F(T) ocT4. Thus, finally, EAbX is of form ATbf (TX)bX or AX 5 (TX)5X, which is Wien's general formula. 7. Transformation of a Single Constituent.—It is of See also:interest to follow out this adiabatic process for each separate constituent of the radiation, as a verification, and also in See also:order to ascertain whether anything new is thereby gained. To this end let now E(X,T)bX represent the intensity of the radiation between X and X+bX which corresponds to the temperature T. The pressure of this radiation, when it is without special direction, is in intensity one-third of this; thus the application of Carnot's principle shows, as before, that in adiabatic compression T ocV-A, so that' a small linear shrinkage in the ratio z-x raises T in the ratio r+x. We have still to See also:express the See also:equation of energy. The vibratory energy E(X,T)bX . V in volume V, together with the mechanical work *E(X,T)SX. 3xV, yields the vibratory energy E(T(r -x), T(r+x)}&X(r-x) . V(r -3x); thus, See also:writing E and See also:EA or E (X,T) we have, neglecting x2, E(r +x) = (E - xk-+xTdT) (r - 4x), so that 5E+XdE-T—=o dX dT energy unaltered, is already implicitly fulfilled; it would thus appear that any further advance must involve (§ r r) the See also:dynamics of the radiation and absorption of material bodies. 8. Temperature of an Isolated Ray.—The temperature of each independent constituent of a radiation has here been taken to be a function of the intensity EA, where EAbX is the energy per unit volume in the range between wave-lengths X and X+5A; the See also:condition is, however, imposed that this radiation is in-different as to direction. When a beam of radiation travels without loss in a definite direction across a medium, its form varies as it progresses; but it is reversible inasmuch as it can be turned back at any See also:stage, or concentrated without loss, by perfect reflectors. If the energy of the beam has a temperature, its value must therefore remain constant throughout the progress of the beam, by the principle of Carnot. Now by virtue of a relation in geometrical See also:optics, which on a corpuscular theory would be one aspect of the fundamental dynamical principle of See also:Action, the See also:cross-See also:section 5S at any place on the beam, and the conical angle bw within which the directions of its rays are there included, are such that the value of V-25Sbw is conserved along the beam, V being the velocity of propagation of the undulations. If we represent the amount of radiant energy transmitted per unit time across the section 5S of the beam by IbSSw, it will follow that in passing along the beam its intensity of See also:illumination I varies as V-2, or as the square of the See also:index of See also:refraction, provided there is no loss of energy in trans-See also:mission. This condition requires that changes of index shall be See also:gradual, otherwise there would be loss of energy by partial reflexions; in free aether I is itself constant along the beam. The volume-density of the energy in any part of the directed beam is V-'ISw; it is thus inversely as the solid angular concentration of the rays and directly as the See also:cube of the index of refraction. Now we may consider this beam, of aggregate intensity IbSSw, to form an elementary filament of the radiation issuing in the direction of the normal from a perfect radiator. As such a body absorbs completely and therefore radiates equally in all directions in front of it, the total intensity of radiation from its See also:element of surface Ss is Ss fI cos BSw, or Os. nI, while the volume-density of the total advancing and receding radiation in front of it is 2V-'f Idw, and therefore niV-'I. If we take here ISk to represent the intensity between wave-lengths X and X+bk, this density is the quantity EA of which the temperature of the radiator is a function. Thus the quantity I—which optically is a measure of the brightness of the beam, and is conserved along it to the extent that µ2I is the same from whichever of its cross-sections the beam is supposed to be emitted—also determines its temperature., the latter being that of an enclosure containing undirected radiation of the same range SA which is density EaAA given by EA=4irV-'I, where V is the velocity of radiation in the enclosure. When a beam of radiation travels without suffering absorption, its temperature thus continues to be that of its source multiplied by the coefficient of emission of the source for that kind of radiation, this coefficient being less"than unity except in the case of a perfect radiator; but when its intensity I falls by bI in any part of its path owing to absorption or other irreversible process, this involves a further fall of temperature of the energy of the beam and a rise of entropy which can be completely determined when the relation connecting ,u-2EA with T and k is known. Any directed quality in radiant energy increases its effective temperature. Splitting a beam into two at s reflecting and refracting surface diminishes the temperature of each part; it is true that if the reflecting surface were non-molecular the operation could be reversed, but actually the reversed rays would encounter the reflecting molecules in different collocations, and could not (§ IT) recombine into the same detailed phase-relations as before. The direct See also:solar radiation falling on the Earth is almost completely convertible into mechanical effect on account of its very high temperature; there seems ground for believing that certain constituents of it can actually be almost wholly turned to account by the a partial See also:differential equation of which the integral is E = AX5'b(TX), the same formula as was before obtained. This method, treating each constituent of the radiation separately, has in one respect some advantage, in that it is necessary only to postulate an enclosure which totally reflects that constituent, this being a more restricted hypothesis than an absolutely complete reflector. To determine theoretically the form of the function 4 we must have some means of transforming one type of radiation into another, different in essence from the adiabatic compression already utilized. The condition that the entropy of the in-dependent radiations in an enclosure is a minimum when they are all transformed to the same temperature with total See also:green leaves of See also:plants. But the same solar radiation, when broken up into diffused See also:sky light, which has no definite direction, has fallen into equilibrium with a much lower temperature, through loss of its reversibility. It has been remarked that the temperatures of the See also:planets can be roughly compared by means of this principle, if their coefficients of absorption of the solar radiation are assumed; that of See also:Neptune comes out below — zoo° C., if we suppose that it is not kept higher by a supply of See also:internal See also:heat. To ,obtain dynamical precision in this discussion an exact See also:definition of the narrow beam such as is usually called a ray is essential. It can be specified as a narrow filament of radiation, such as may be isolated within an infinitely thin, impermeable, bounding See also:tube without thereby producing any disturbance of the motion. If either the tube or the surrounding radiation were not present to keep the beam in shape, it would spread sideways, as in See also:optical diffraction. But the function of the tube is one of pure constraint; thus the change of energy-content of a given length of the tube is represented by energy flowing into it at the end where the radiation enters, and leaving it at the other end, but with no leakage at the sides. The total radiation may be considered as made up of such filaments. g. Temperature of the Sun.—The mean temperature of the radiating layers of the Sun may be estimated from Stefan's law, by computing the intensity of the radiation at his surface from that terrestrially observed, on the basis of the law of inverse squares; the result is about 6500° C. The application of Wien's law, which makes the wave-length of maximum energy vary inversely as the temperature, for the case of a perfectly radiating source, gives a result 5500° C. These See also:numbers will naturally differ because (i) the Sun is not a perfect radiator, the constitution of his radiation in fact not following the law of that of a See also:black body, (ii) the various radiating layers have different temperatures, (iii) the radiation may be in part due to chemical and See also:electrical causes, and in so far would not be determined by the temperature alone. The See also:fair agreement of these two estimates indicates, however, that the radiation is largely regulated by the temperature, that the layers from which the See also:main part of it comes are at temperatures not very different, and that not very much of the complete radiation established in these layers and emitted from them is absorbed by the overlying layers. ro. See also:Fluorescence.—When radiation of certain wave-lengths falls on a fluorescent body, it is largely absorbed, but in such manner as directly to excite other radiation of different type which is emitted in addition to the true temperature-radiation of the body. The distinction involved is that the latter radiation is spontaneously convertible with the heat of the absorbing body at its own temperature, without any See also:external stimulus or compensation; it is, in fact, on the basis of this convertibility that the thermodynamic relations of the temperature-radiation have been established. According to the experimental law of See also:Stokes, the wave-lengths of the fluorescent radiation are longer than those of the radiation which excites it. If the latter were directly transformed, in undiminished amount, into the fluorescent kind, this is what would be expected. For such a spontaneous change must involve loss of availability; and, beyond the wave-length of maximum energy in the spectrum, the temperature of a given density of radiation is greater the shorter its wave-length, as it is a function of that density and the wave-length alone such that greater radiation always corresponds to higher temperature. But it would appear that the opposite should be the case for radiation of long wave-lengths, lying on the other See also:side of the maximum, in which the tendency would thus be for spontaneous change into shorter waves; this may perhaps be related to the fact that the lines of longer wave-lengths in spectra often come out brighter at lower temperatures, for they are then thrown on the other side of the maximum and cannot be thus degraded. The principle does not, however, have free See also:play in the present case, even when the incident radiation is diffused and so has not the abnormally high temperature associated with a directed beam(§ 8), since part of it might be degraded into See also:low-temperature heat, or there might be other compensation of chemical type for any abnormally high availability that might exist in the fluorescent radiation. It has been found that fluorescent radiation, showing a continuous or banded spectrum, can be excited in many gases and vapours; milky See also:phosphorescence of considerable duration, and thus doubtless associated with chemical change, is produced in vacuum tubes, containing See also:oxygen or other complexly constituted gases, by the electric See also:discharge. rr. Entropy of a Ray.—If each definitely constituted beam of radiation has its own temperature and everything is reversible as above, a question arises as to the location of the process of averaging which enters into the idea of temperature. The See also:answer can depend only on the fact, that although the beam is definite as to wave-length and intensity, yet it is far from being a simple wave-train, in that it is constituted of trains of limited lengths and various phases and polarizations, coming from the independent radiating molecules. When such a beam has once emerged, it travels without change, and can be reflected back intact to its source, and is in so far reversible; but when it has arrived there, the molecules of the source will have changed their positions, and it cannot be wholly reabsorbed in the same manner as it was emitted. There must thus be some feature in the ultimate averaged constitution of the beam, emitted from a body in the definite steady state of internal motion determined by its temperature, which adapts it for spontaneous uncompensated reabsorption into a body at its own (or a lower) temperature, but not at a higher one. The question of the determination of the form of the function 4i in § 6 would thus' appear to be closely connected with the other problems, hitherto imperfectly fathomed, See also:relating to the See also:statistics of kinetic molecular theory. A very interesting attack on the problem from this point of view has recently been made in various forms by See also:Planck. It of course suffices to examine some simple type of radiating system, and the results will be of general validity. He considers an enclosure filled with radiation involving an entirely arbitrary See also:succession of phases and polarizations along each ray, and also containing a system of fixed linear electric oscillators of the Hertzian type, which are taken to represent the transforming action of radiating and absorbing matter. The radiation contained in the enclosure will be passed through these oscillators over and over again, now absorbed, now radiated, and each constituent will thus See also:settle down in a unilateral or irreversible manner towards some definite intensity and See also:composition. But it does not appear that a system of vibrators of this kind, each with its own period, can perform one of the main functions of a material absorber, namely, the transformation of the relative intensities of the various types of radiation in the enclosure to those corresponding to a common temperature. There would be equilibrium established only between the r_•i,tn internal vibratory energy in the vibrators of each period ans the density of radiation of that period; there is needed also some means of interchanging energy between vibrators cf. different periods, which probably involves doing away with their fixity, or else employing more complex vibrators and assuming a law of distribution of their internal energy. In the See also:absence of any method of introducing this temperature equilibrium directly, Planck originally sought, in the case of each independent constituent, for a function of its intensity of energy and its wave-length, restricted as to form by a certain assumed molecular relation, which has the See also:property of continually increasing after the, manner of entropy, during the progress of that constituent of the radiation in such a system towarde its steady state. If the actual entropy S per unit volume could be thus determined, the relation of See also:Clausius 8S =SE/T would supply the connexion between the temperature and the density of radiant energy E. This procedure led him, in an indirect and tentative manner, to a relation d2S/dE2=—a/E, so that S =— a E See also:log/9 E, where a, tit are functions of X an expression which conducts through Clausius's relation to E= (eg)—'e 11°T. The previous argument then gives E(X,T)SX=c,X-5e °/'TSX, 1.4435 in c.g.s. measure, but not so well when the range is farther a type of formula which was originally suggested by Wien on extended: it appeared that a larger value of c was needed to the basis of the See also:analogy that it assigns the same distribution represent the radiation for high values of TX, that is, for high for the radiant energy, among the various frequencies of vibra- temperature or for very long wave-lengths. Thiesen proposed tion, as for the energy of the molecules in a See also:gas among their the somewhat more general form c,(TX)''e c/Th, and suggested various velocities of See also:translation. But the experimental in- that the value k = z agrees better with the experimental numbers adequacy of this formula afterwards suggested a new See also:pro= than Wien's value k=o. See also:Lord See also:Rayleigh was led (Phil. Mag., cedure, as infra. See also:June 1900) towards this form with k equal to unity from entirely Processes may be theoretically assigned for the direct continuous different theoretical considerations, on the See also:assumption of the transformation of radiant into mechanical energy. Thus we can Maxwell-Boltzmann distribution of the energy of a system, imagine a radiating body at the centre of a See also:wheel, carrying oblique vanes along its circumference, which reflect the radiation on to a consisting of an isolated See also:block of aether, among its free periods See also:ring of parallel fixed vanes, which finally See also:reverse its path and return of vibration, See also:infinite in number; in some cases this form appeared it to the centre. The pressure of the radiation will drive the wheel, to give as good results as Wien's own. and in case its motion is not resisted, a very great velocity may be Acting on a See also:suggestion advanced by Lord Rayleigh, See also:Rubens theoretically obtained. The thermodynamic compensation in such and Kurlbaum soon afterwards widely extended the test of the cases lies in the reduction of the effective temperature of the portion of the radiation not thus used up. We might even do away with formulae by means of the so-called Reststrahlen. A substance the radiating body at the centre of the wheel, and consider a beam such as an See also:aniline dye, which exhibits selective absorption of of definite radiation reflected backwards and forwards across a any See also:group of rays, also powerfully reflects those rays; and dia that diam tthemwork donelbyit yn See also:driving thelwheel willlberconc m to tewi h Rubens has been able thus to isolate in considerable purity the increase of the wave-length, and therefore with expansion of the rays belonging to absorption bands very far down in the invisible length occupied by the beam. The thermodynamic features are ultra-red, having wave-length of order ro 3 cm., which are thus analogous to those of the more See also:familiar case of an envelope intensely absorbed by substances such as sylvine, by means of filled with gas, which can change its thermal energy into mechanical five or six successive reflexions of the beam of radiation. By energy by expansion of the envelope against mechanical resistances. experiments ranging between temperatures -200° C. and In the case of the expanding gas pv E,, where Es is the total trans- latory energy of the molecules, while in adiabatic expansion p=kv-v. + soo° C. of the source of radiation, it has been found that the Thus the work gained in unlimited expansion, fpdv, is 1E0/(y-r). intensity of this definite radiation tends to vary simply as T, The final temperature being absolute zero, this should by Carnot's with See also:close approximation, thus increasing indefinitely with principle be equal to the total initial energy of the gas that is in the temperature, whereas Wien's formula would make it tend connexion with temperature, constitutive energy of the molecules being excluded; when y-i is less than i there is thus internal to a definite limit. The only existing formula (except the thermal energy in the molecules in addition to the translatory one suggested by Lord Rayleigh) that proved to be in See also:accord energy. In the case of the beam of radiation, of length 1, between with this result was a new one advanced shortly before and n and n+Sn reflexions, where Sn is an integer, its total energy E is supported on theoretical grounds by Planck, namely, EaSX= by § 2 reduced according to the law EE = -.-4cv6n 51 =2v3n g (c+v)2 Also 1 c+v ; CX 5SX/(e`/' T- r), which for small values of XT agrees with Wien's EE _ 2c sl original form, known to be there satisfactory, while for larger thus E = c+v l' When v is small compared with c, this gives values it tends towards C/c.X-4T; the new formula is, in fact, the E =K1-2; and p is then 2E/1, so that fpdl = E, the temperature simplest and most likely form that satisfies these two conditions. of the beam being ultimately reduced to absolute zero by the The point of Lord Rayleigh's argument was that, at any See also:rate at unlimited expansion. This is in accord with Carnot's principle, in that the whole energy of the beam travelling in a vacuum is mechani- low frequencies, the law of distribution would suggest an equable cally available when reduction to absolute zero of temperature is See also:partition of the energy between temperature heat and radiant in our See also:power. vibrations, and that therefore the energy of the latter should 12. Experimental Knowledge.—Under the stimulus of Wien's ultimately vary as T; and this prediction, which has thus been investigation and of improvements in the construction of linear verified, may be grafted on to any formula that is in other thermopiles and bolometers for the refined measurement of the respects appropriate. distribution of energy along a spectrum, the general See also:character Recognizing that his previous hypothesis, restricting the nature of the See also:curve connecting energy and wave-length in the complete of the entropy in addition to its property of continually in-radiation at a given temperature has been experimentally creasing, had thus to be abandoned, Planck had in fact made ascertained over a wide range. At each temperature there is a a fresh start on the basis of a train of ideas which was introduced wave-length X,,, of maximum radiation, which is displaced by Boltzmann in 1877, in order to obtain a precise physical towards the ultra-See also:violet as the temperature rises, and Wien's conception of entropy. According to the latter, for an indefinitely law of homology (§ 6) shows that X„,T should be constant. This numerous system of molecules, with known properties and in See also:deduction, and the law of homology itself, as also the law of given circumstances, there is a definite See also:probability of the Stefan and Boltzmann that the total radiation varies as T4, occurrence of each statistical distribution of velocities, or say have been closely verified by the experiments of Rubens and each " complexion " of the system, that is formally possible Kurlbaum, Lummer and See also:Pringsheim, Paschen and others. when all velocities consistent with given total energy are See also:con-They established a steady field of radiation inside a material sidered to be equally likely as regards each molecule; the distrienclosure by raising the walls to a definite temperature, and bution of greatest possible probability is the state of thermal measured the radiant intensity emitted from it through an equilibrium of the system, and the probability of any other opening or slit in the walls, by means of a bolometer or thermos state is a function of the entropy of that state. This conception See also:pile, this being the radiation of the so-called perfectly black body. can be developed only in very simple cases; the application to The principle here involved formed one of the See also:foundations of an ideal monatomic gas-system led Boltzmann to take the Balfour Stewart's See also:early treatment of the theory, and had already entropy proportional to the See also:logarithm of the probability. This been employed by him and Stokes (1860) in experiments on the logarithmic law is in fact demanded in advance by the principle polarized emission from See also:tourmaline: cf. Stokes, Math. and that the entropy of a system should be the sum of the entropies Phys. Papers, iv. 136. It has been remarked by Planck and of its parts. By means of a priori considerations of this nature, by Thiesen that the coefficient of T4 in Stefan's law, and the referring to the distribution of internal vibratory energy among value of X,mT, are two absolute physical constants independent a system of linear electric vibrators of given period, and its of any particular kind of matter, which in See also:conjunction with the equilibrium of exchanges with the surrounding radiant energy, constant of See also:gravitation would determine an entirely absolute Planck has been guided to an expression for the law of depend-system of physical See also:units. The form of the function ¢(TX) ence of the entropy of that system on the temperature, which adopted by Wien and in Planck's earlier discussions, namely, corresponds to the form of the law of radiation above stated. c,e-`'TA, was found to agree fairly with experiment over the The result gains support from the fact that the expressions for range from ro0° C.. to 1300° C., when c, = I.24 X Io 5, and c the coefficients to which he is led give determinations of the absolute physical constants of molecular theory, such as the radiation) should be equal. This proposition is a general constant of See also:Avogadro, which are in close accord with other recent determinations. But on the other See also:hand these determinations are already involved in the earlier formula of Rayleigh, which expresses the distribution for long waves, based merely on the Maxwell-Boltzmann principle of the equable partition of the energy among the high free periods belonging to the enclosure which contains it. It is maintained by Jeans that the reason why this principle is of avail only for very long wave-lengths is that a steady state is never reached for the shorter ones, a See also:doctrine which as he admits would entirely remove the foundations of the application of thermodynamic principles to this subject. By an argument based on the theory of dimensions, Lorentz has been led to the conclusion that consistency between temperatures, as measured molecularly, and as measured by the laws of radiation, requires that the ultimate indivisible electric charges or electrons must be the same in all kinds of matter. The abstract statistical theory of entropy, which is here invoked, admits of generalization in a way which is a modification of that of Planck, itself essentially different from the earlier idea of Boltzmann. The molecules of matter, whose interactions See also:control physical phenomena, including radiation, are too numerous to be attended to separately in our knowledge. They, and the phenomena in which they interact, must thus be sorted out into differential See also:groups or classes. Elements of energy of specified types might at first sight constitute such classes: but the identity of a portion of energy cannot be traced during its transformations, while an element of physical disturbance can be definitely followed, though its energy changes by interaction with other elements as it proceeds. The whole disturbance may thus be divided into classes, or groups of similar elements, each with permanent existence: and these may be considered as distributed in See also:series of cells, all See also:equivalent in extent, which constitute and See also:map out the material system or other domain of the phenomena. The test of this equivalence of extent is superposition, in the sense that the same element of disturbance always occupies during its wanderings the same number of cells. This framework being granted, the probability of any assigned statistical distribution of the elements of disturbance now admits of calculation; and it represents, as above, the logarithm of the entropy of that distribution, multiplied however by a coefficient which must depend on the minuteness of scale of the statistics. But in the calculation, all the physical laws which impose restrictions on the migrations of the elements of disturbance must be taken into account; it is only after this is done that the See also:rest of the circumstances can be treated as fortuitous. All these physical laws are, however, required and used up in determining the complex of equivalent cells into which the system which forms the seat of the energy is mapped out. On this basis thermodynamics can be constructed in a priori abstract See also:fashion, and with deeper and more complete implications than the formal Carnot principle of negation of perpetual motions can by itself attain to. But the ratio of the magnitude of the See also:standard element of disturbance to the extent of the standard See also:cell remains inherent in the results, appearing as an absolute physical constant whose value is determined somehow by the other fundamental physical constants of nature. A prescribed ratio of this kind is, however, a different thing from the hypothesis that energy is constituted atomically, which underlies, as Lorentz pointed out, Planck's form of the theory. It has indeed already been remarked that the See also:mere fact of the existence of a wave-length X. of maximum radiation, whether obeying Wien's law X,,,T=constant or not, implies by itself some, prescribed absolute physical quantity of this kind, whose existence thus cannot be evaded, though we may be at a loss to specify its nature.
13. Modification by a .Aagnetic Field.—The theory of ex-changes of radiation, which makes the equilibrium of radiating bodies depend on temperature alone, requires that, when an element of surface of one body is radiating to an element of surface of another body at the same temperature, the amounts of energy interchanged (when reflexion is counted in along with
dynamical consequence—on the basis of the laws of See also:reciprocity developed in this connexion (after W. Rowan See also: In order to make the system self-contained, reflectors must be added to it, so as to send back into the sources the polarized constituents that are turned aside out of the direct See also:line by the nicols. Then, as Brillouin has pointed out, and as in fact Rayleigh had explained some years before, the radiation from B does ultimately get across to A after passage backward and forward to the reflectors and between the nicols: this, it is true, increases the length of its path, and therefore diminishes the concentration of a single narrow beam, but any large change of path would make the beam too wide for the nicols, and thus require other corrections which may be supposed to compensate. The explanation of the slight difference that is to be anticipated on theoretical grounds might conceivably be that in such a case the magnetic influence, being operative on the phases, alters the statistical constitution of the radiation of given wave-length from the special type that is in equilibrium with a definite temperature, so that after passage through the magnetic medium it is not in a condition to be entirely absorbed at that temperature; there would then be some other element, in addition to temperature, involved in equilibrium in a magnetic field. If this is not so, there must be some thermodynamic compensation involving reaction, extremely small, however, on the magnetizing system. 14. Origin of Spectra.—In addition to the thermal radiations of material substances, those, namely, which establish temperature-equilibrium of the enclosure in which they are confined, there are the fluorescent and other radiations excited by extraneous causes, radiant or electric or chemical. Such radiations are an indication, by the presence of higher wave-lengths than belong in any sensible degree to the temperature, that the steady state has not arrived; they thus fade away, either immediately on the cessation of the exciting cause, or after an interval. The radiations, consisting of definite narrow See also:bright bands in the spectrum, that are characteristic of the gaseous state in which each molecule can vibrate freely by itself, are usually excited by electric or chemical agency; thus there is no ground for assuming that they always constitute true temperature radiation. The absorption of these radiations by strata of the same gases at low temperatures seems to prove that the unaltered molecules themselves possess these free periods, which do not, therefore, belong specially to dissociated ions. Although very difficult to excite directly, these free vibrations are then excited and absorb the energy of the incident waves, under the influence of resonance, which naturally becomes extremely powerful when the tuning is exact; this indicates, moreover, that the true absorption bands in a gas of sufficiently low density must be extremely narrow. There is direct See also:evidence that many of the more permanent gases do not sensibly emit light on being subjected to high temperature alone, when chemical action is excluded, while others give in these circumstances feeble continuous spectra; in fact, looking at the matter from the other side, the more permanent gases are very transparent to most kinds of radiation, and Therefore must be very See also:bad radiators as regards those kinds. The dark radiation of flames has been identified with that belonging to the specific radiation of their gaseous products of See also:combustion. There is thus ground for the view that the impacts of the colliding molecules in a gas, or rather their mutual actions as they See also:swing sharply See also:round each other in their orbits during an encounter, may not be sufficiently violent to excite sensibly the free vibrations of the definite periods belonging to the molecules. But they may produce radiation in other ways. While the velocity of an electron or other electric See also:charge is being altered, it necessarily sends out a stream of radiation. Now the orbital motions of the electrons in an actual molecule must be so adjusted, as appears to be theoretically possible, that it does not emit radiation when in a steady state and moving with constant velocity. But in the violent changes of velocity that occur during an encounter this equipoise will be disturbed, and a stream of radiation, without definite periods, but such as might constitute its See also:share of the equilibrium thermal radiation of the substance, may be expected while the encounter lasts. At very high temperatures the energy of this thermal radiation in an enclosure entirely overpowers the kinetic energy of the molecules present, for the former varies as T 4, while the latter See also:measures T itself when the number of molecules remains the same. The radiation which can be excited in gases, confined as it is to extremely narrow bands in the spectrum, may indeed be expected to possess such intensity as to be thermally in equilibrium with extremely high temperatures. That the same gases absorb such radiations when comparatively See also:cold and dark does not, of course, affect the case, because emissive and absorptive powers are proportional only for incident radiations of the intensity and type corresponding to the temperature of the body. Thus if our adiabatic enclosure of § 3 is prolonged into a tube of unlimited length which is filled with the gas, then when the temperature has become uniform that gas must send back out of the tube as much radiation as has passed down the tube and been absorbed by it; but 'if the tube is maintained at a lower temperature, it may return much less. The fact that it is now possible by great optical See also:dispersion to make the line-spectra of prominences in the See also:middle of the Sun's disk stand out bright against the background of the continuous solar spectrum, shows that the intensities of the radiations of these prominences correspond to a much higher temperature than that of the general radiating layer underneath them; their luminosity would thus seem to be due to some cause (electric or chemical) other than mere temperature. On the other hand, the general See also:reversing gaseous layer which originates the dark See also:Fraunhofer lines is at a lower temperature than the radiating layer; it is only when the light from the lower layers is eclipsed that its own direct bright-line spectrum flashes out. It is not necessary to attribute this selective flash-spectrum to temperature radiation; it can very well be ascribed to fluorescence stimulated by the intense illumination from beneath. When the radiation in a spectrum is constituted of wide bands it may on these principles be expected to be in equilibrium with a lower temperature than when it is constituted of narrow lines, if the total intensity is the same in the cases compared; this is in keeping with the easier excitation of See also:band spectra (cf. the banded absorption spectra), and with the fact that various gases and vapours do appear to emit band spectra more or less related to the temperature. 15. Constitution of Spectra.—In the problem of the unravelling of the constitutions of the very complex systems of spectral 'ines belonging to the various kinds of matter, considerableprogress has been made in recent years. The beginning of definite knowledge was the See also:discovery of Balmer in 1885, that the frequencies of vibration (n) of the See also:hydrogen lines could be represented, very closely and within the limits of See also:error of observation, by the formula n oc I - 4m-2, when for m is substituted the series of natural numbers 3, 4, 5, . . . 15. Soon afterwards series of related lines were picked out from the spectra of other elements by Liveing and See also:Dewar. See also:Rydberg conducted a systematic investigation on the basis of a modification of Balmer's law for hydrogen, namely, n=no-N/(m+,u)2. He found that in the group of alkaline metals three series of lines exist, the so-called See also:principal and two subordinate series, whose frequencies See also:fit approximately into this formula, and that similar statements apply to other natural groups of elements; that the constant N is sensibly the same for all series and all substances, while no and µ have different values for each; and that other approximate numerical relations exist. In each series the lines of high frequency See also:crowd together to--wards a definite limit on the more refrangible side; near this limit they would, if visible, constitute a band. The principal or strongest series of lines shows reversal very readily. The lines of the first subordinate series are usually nebular, while those of the second subordinate or weakest series are See also:sharp; but with a tendency to broaden towards the less refrangible side. In most series there are, however, not more than six lines visible: See also:helium and hydrogen are exceptions, no fewer than See also:thirty lines of the principal series of the latter having been identified, the higher ones in stellar spectra only. But very remarkable progress has recently been made by R. W. See also:Wood, by exciting fluorescent spectra in a metallic vapour, and also by applying a magnetic field to restore the lines sensitive to the Zeeman effect after the spectrum has been cut off by crossed nicols. The large aggregates of lines thus definitely revealed are also resolved by him into systems in other ways; when the stimulating light is confined to one period, say a single bright line of another substance, the, spectrum excited consists of a limited number of lines equidistant in frequency, the interval common to all being presumably the frequency of some See also:intrinsic orbital motion of the molecule. In this way the series belonging to some of the See also:alkali metals have been obtained nearly complete. Simultaneously with Rydberg, the problem of series was attacked by See also:Kayser and Runge, who, in reducing their extensive standard observations, used the formula n=A+Bm-2+Cm-4, higher terms in this descending series being presumed to be negligible. This cannot be reconciled with Rydberg's form, which gives on expansion terms involving m-3; but for the higher values of m the discrepancies rapidly diminish, and do not prevent the picking out of the lines, the frequency-See also:differences between successive lines then varying roughly as the inverse squares of the series of natural numbers. For low values of m neither mode of expression is applicable, as was to be expected; and it remains a problem for the future to ascertain if possible the rational formula to which they are approximations. More complex formulas have been suggested by Ritz and others, partly on theoretical grounds. Considered dynamically, the question is that of the determination of the formula for the disturbed motions of the system which constitutes the molecule. Although we are still far from any definite line of attack, there are various indications that the quest is a practicable one. The lines cf each series, sorted out by aid of the formulae above given, have properties in common: they are usually multiple lines, either all doublets in the case of See also:monad elements, or generally triplets in the case of those of higher chemical See also:valency; in very few cases are the series constituted of single lines. It is found also that the components of all the See also:double or triple lines of a subordinate series are equidistant as regards frequency. In the case of a related group of elements, for example the alkaline metals, it appears that corresponding series are displaced continually towards the less refrangible end as the atomic See also:weight rises; it is found also that the interval in frequency between the double lines of a series diminishes with the atomic weight, and is proportional to its square. These relations suggest that the atomic weight might here See also:act in part after the manner of a load attached to a fundamental vibrating system, which might conceivably be formed on the same See also:plan for all the metals of the group; such a load would depress all the periods, and at the same time it would split them up in the manner above described, if it introduced dissymmetry into the vibrator. The discovery of Zeeman that a magnetic field triples each spectral line, and produces definite polarizations of the three components, in many cases further subdividing each component into lines placed usually all at equal intervals of frequency, is explained, and was in part predicted, by Lorentz on the basis of the electron theory, which finds the origin of radiation in a system of unitary electric charges describing orbits or executing vibrations in the molecule. Although ifiese facts form substantial sign-posts, it has not yet been found possible to assign any likely structure to a vibrating system which would lead to a frequency formula for its free periods of the types given above. Indeed, the view is open that the group of lines constituting a series form a harmonic See also:analysis of a single fundamental vibration not itself harmonic. If that be so, the intensities and other properties of the lines of a series ought all to vary together; it has in fact been found by See also:Preston, and more fully verified by Runge and others, that the lines are multiplied into the same number of constituents in a magnetic field, with intervals in frequency that are the same for all of them. When the series consists of double or triple lines the separate components of the same See also:compound line are not affected similarly, which shows that they are differently constituted. The view has also found support that the different behaviours of the various groups of lines in a spectrum show that they belong to independent vibrators. The form of the vibration sent out from a molecule into the aether depends'on the form of the aggregate See also:hodograph of the electronic' orbits, which is in keeping with Rayleigh's remark that the series-laws suggest the kinematic relations of revolving bodies rather than the vibrations of steady dynamical systems. According to Rydberg, there is ground for the view that a natural group of chemical elements have all the same type of series spectrum, and that the various constants associated with this spectrum change rapidly in the same directions in passing from the elements of one group to the corresponding ones of the following groups, after the manner illustrated in graphical representations of Mendeleeff's law by means of a continuous wavy curve in which each group of elements lies along this same ascending or descending See also:branch; the chemical elements thus being built up in a series of types or groups, so that the individuals in successive groups correspond one to one in a See also:regular progression, which may be put in evidence by connecting them by transverse curves. Illustrations have been worked out mathematically by J. J. See also:Thomson of the effect of-adding successive See also:outer rings of electrons to See also:stable vibrating collocations. The frequencies of the series of very close lines which constitute a single band in a banded spectrum are connected by a law of quite different type, namely, in the simpler cases n2--A-B1;12. It may be remarked that this is the kind of relation that would apply to a See also:row of independent similar vibrators in which the neighbours exert slight mutual influence of elastic type. If denote displacement and x distance along the row, the equation d+k2E=–g -would represent the general See also:fea- tures of their vibration, the right-hand side arising from the mutual elastic influences. If the ends of the line of vibrators, of length 1, are fixed, or if the vibrators form a ring, the appropriate type of See also:solution is oc sin ,ux sin p, where µl = ma and m is integral; further – p2+k2 = gµ2, hence p2 = k2– j _ m2, which is of the type above stated. Dynamical systems of this kind are illustrated by the Lagrangean linear system of connected bodies, such as, for example, a row of masses fixed along a tense cord, and each subject to a restoring elastic force of its own in addition to the tension of the cord. A single spectral line might thus be transformed into a band of this type, as the effect of disturbance arising from slight elastic connexions established in the molecule between a system of similar vibrators. But the series in line-spectra are of entirely different constitution; thus for ,the series expressed by the formula p2 = p2 – Bm-2 the corresponding period-equation might be expressed in some such form as sin k(p2 – pot)–z = constant, which belongs to no type of vibrator hitherto analysed. Additional information and CommentsThere are no comments yet for this article.
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