XXI. A Criticism of Wien's Distribution Law. 1. Introduction. THE purpose of this article is to criticise Wien's distribution law from a mathematical point of view :-To show (1) that although the derivation of Wien's distribution law is generally made by steps which are not mathematically justified, or for which no rigorous justification is given, still, by using Wien's assumptions †, a rigorous derivation is possible; (2) that the law obtained by Wien is inconsistent with other results which follow from the same assumptions; and (3) that this inconsistency is eliminated and a new law ‡ obtained if Wien's implicit assumption be replaced by a simpler and more probable one. Incidentally several theorems, new so far as the author knows, in the kinetic theory of gases will be obtained from the Maxwell law for the distribution, with respect to their velocities, of the molecules in a gas; a simple proof of the Wien displacement law will be obtained from these theorems and the Wien assumptions, and two interesting relations regarding the dependence of the radiation of a molecule upon its velocity will be given. Also there will be found some criticisms of the treatment of the distribution law and of the displacement law as given in the standard treatises. Mendenhall and Saunders §, Waidner and Burgess ||, Rayleigh and others have criticised from a physical point of view the assumptions used by Wien to prove his distribution law, while Lummer and Pringsheim**, Paschen ††, and others have considered the agreement of this law with experimental results. So far as the author knows, no criticism from a mathematical point of view has been published. *Communicated by the Author. † It will be necessary to include under Wien's assumptions one which he has made implicitly, but not explicitly; or else to assume that he has made a fundamental error. I am indebted to Professor Lunn, of the University of Chicago, for this new formula, and for the observation that Wien either made a mistake or an unstated assumption. Astrophysical Journal, xiii. p. 25 (1901). Bull. Bur. of Standards, i. p. 189 (1904). ¶ Phil. Mag. xlix. p. 539 (1900). Verh. d. Deutsch. Phys. Ges. i. p. 1 (1900). + Astrophysical Journal, x. p. 40 (1899); xi. p. 288 (1900). where (x, 0)dλ represents the intensity of radiation of a black body at a temperature produced by waves whose lengths lie between X and X+dλ, and where C and care constants*. M. Planck † obtained this same formula (and also another ) but from considerations entirely different from those used by Wien. The criticism of this article does not apply to Planck's work; however, it has been suggested that a similar criticism might apply to Planck's derivation of this same law. Other formulas for p(x, 0) have been obtained by Callendar § and Rayleigh ||; although these formulas may be in closer accord with experimental results, still the Wien formula has considerable importance due to its use by Drude and many other investigators. 2. A derivation of the Distribution Law along the lines proposed by Wien. Wien takes a gas as the black body, and uses Maxwell's law that the number of molecules whose velocities lie between v and v+dv is proportional to where a2, and is the root-mean-square velocity; a is proportional to 0, the absolute temperature of the gas. Wien makes the hypotheses: (a) That the length of the wave sent out by a molecule depends only upon the velocity of that molecule :-then v is a function of only. (b) That the intensity of the radiation for wave-lengths between λ and λ+d is proportional to the number of molecules, as given by Maxwell's law (2), which send out waves with lengths between A and λ + dx. Wien states that it follows from these two hypotheses that where F(X) and f(x) are two unknown functions. *Wied. Ann. lviii. p. 662 (1896). † Wied. Ann. i. pp. 69, 719 (1900). Verh. d. Deutsch. Phys. Ges. ii. p. 202 (1900). Phil. Mag. xxvi. pp. 787 (1913); xxvii. p. 870 (1914). Phil. Mag. xlix. p. 539 (1900). (3) In our development of Wien's distribution law, it is assumed that F(X) and f(x) are continuous functions. Just what physical significance these assumptions have can be seen from § 6 of this article. He then states: "Now the variation of the radiation with the temperature according to the law given by Boltzmann and myself consists of an increase of the total energy in proportion to the fourth power of the absolute temperature, and a variation of the length of the waves associated with an energy quantum and lying between λ and λ+dλ in such a way that the corresponding wave-lengths are inversely proportional to the temperature. So if one plots for any one temperature the energy as a function of the wave-length, then for any other temperature this curve will be the same if the scale units of the graph are so varied that the ordinates are made smaller in the ratio and the abscissæ are made 1 larger in the ratio 0. This latter is possible for our value of (, ) only when λ appear in the exponential as a and product 0. Then Since I have found no simple proof of (4) from the above statements, I propose the following derivation of the form of F(x) and f(x) based entirely upon Wien's statements. Let C be the curve obtained by plotting λ as abscissa and y=(x, 0) as ordinate for an arbitrary temperature 1; then the equation of C1 will be _f(A) y= F(X)e. Let C2 be a corresponding curve for a temperature 01⁄2 ; then the equation of C2 will be y= F(x) e ̄ 2. Now Wien's statement, as corrected, is that the transformation This trans will transform C, into a curve congruent to C1. formation gives as the equation of the transform of C 1893); Lorentz, 'The Theory of Electrons,' p. 74. Now if we make the transformation y', ', this curve with equation (6) coincides with C1, and so This is an identity for all values of 01, 0, and X. (7) (8) If 0 and 0, be replaced by k0, and ke,, where k is an arbitrary constant different from zero, the value of the lefthand member of (8) is unchanged, and therefore the value of the right member is also unchanged. This gives {1s (1944) - 1,~~) } _ }{4,ƒ(011) – }; ^»)} which is an identity for all values of k#0. Therefore In order to obtain the form of F(x) and f(x) we will prove the Lemma: The most general solution of the functional equation C xa, kap(kx) is p(x) = where k (k#0), C, and a are constants. and then A(x) is a continuous function of x. (13) it follows that A(kx)=A(x). Suppose first <1; then the A function has the same value at the points a, ka, ..... ka,... This set of points converges to the point a =0, and therefore A(x) = lim A(x); but since A(x) is continuous lim A(x) = A(0), and A(x) = A(0). x=0 x=0 This is true for every value of x, and therefore A(a) is a constant. Now suppose k>1; make the transformation kay in (14), then (14) becomes A(y)=A(), and therefore the y function has the same value at the set of points y,,.... y which converges to the point x=0, and therefore, by kn the same reasoning as for k<1, A(x) is a constant. C Therefore A(x)=C in every case, and by (13) p(x) = which proves the lemma. = Equations (10) and (11) are of the form (12), where These values, substituted in (3), give Wien's distribution law. 3. A criticism of Drude's proof of the Wien Displacement * and Distribution Laws. Equation (5) is the Wien displacement law and is generally derived from the Stefan-Boltzmann law, which states that or that the total radiation varies directly as the fourth power The relation λ=a const. is sometimes known as the Wien displacement law, and sometimes as a part of that law, but in this paper will be regarded as a distinct law. The equation λ=a const. of itself means nothing, since A and are independent. The equation is a shorthand way of stating that the radiation for λ=λ,, 0=0, will become the radiation for λ=λ, when 0=02, where λ ̧Ø1=λ202. Wien obtains this relation by actually determining the change of wave-length, the temperature being increased by an adiabatic compression of the gas. No criticism of the derivation of this relation is intended in this article. |