XVIII. Calculation of Scattered Radiation from a Plate exposed to a beam of X-rays. By Oskar Klein, Lic. phil. [Communicated by the Author.] In a paper treating of the absorption and scattering of [gamma]-rays, Ishino [citation redacted] has undertaken a few theoretical calculations of scattered radiation issuing from the layer of a substance. Such calculations combined with measurements of the radiation scattered within known solid angles would lead to a determination of the scattering coefficient. Some time ago T. E. Aurén, in connexion with his researches on the absorption of X-rays, performed some experiments in this direction. He measured the absorption coefficient in different positions of the plate exposed to the primary radiation. If the scattered radiation had been of any considerable amount, he would have found different values for the apparent absorption coefficient in the different positions. Now the differences were very small. Therefore it was of interest to Aurén to know how far this was in agreement with the theory. For this purpose I have undertaken the calculation below. As mathematical formulae are obtained which may be used in numerical calculation, the results may perhaps have a certain interest for those who treat these questions experimentally. In this connexion it must be mentioned that Ishino in his paper just cited, has made an error in reckoning, which has caused his final value to take the Since this paper was finished and sent to the Phil. Mag. a quite similar calculation has been published by Glocher, Phys. Zeitschr. xix. p. 251 (1918). 208 [header] form of a divergent integral. Moreover, he never tried to evaluate this integral, a fact which accounts for his not observing the mistake. At the outset of the calculation I have been guided completely by Ishino. There is nothing new in what follows as regards the mathematics. Suppose a plate of thickness l is struck by a radiation, whose intensity is I. The absorption coefficient may be [kappa], the scattering coefficient [sigma]. When the radiation has penetrated the plate to the depth x, its intensity has been diminished to [formula redacted]. The quantity of scattered radiation issuing from the distance between x and x+dx is evidently [formula redacted]. But it is not distributed alike at all angles. J. J. Thomson has theoretically arrived at the following expression for that part which is radiated from the solid angle dw, whose direction makes the angle [theta] with the incident radiation: [formula redacted] If this expression be integrated over all directions we ought to have [sigma]. Consequently [formula redacted] This expression, according to experimental investigations, does not seem to be quite correct. For what we have in view, however, the error will scarcely be of any importance. I therefore adopt the following expression for the radiation that is scattered from the path dx between two cones, whose generatrices form the angles [theta] and [theta] + d[theta] with the incident radiation when leaving the plate: [formula redacted] Owing to absorption and scattering, the intensity of this radiation when leaving the plate will be [formula redacted] for the radiation must travel the distance [formula redacted]. The radiation issuing from the whole plate and falling within a [header] 209 cone whose generatrices form the angle [theta] with the incident radiation will then be [formula redacted] In the first place, I effect the integration in regard to x and obtain [formula redacted] The next thing is to find an expression for this integral adapted to numerical calculation. For this purpose I put [formula redacted] and obtain [formula redacted] For the sake of brevity, I write [formula redacted] I now pass to an examination of the integral K. First of all we may write [formula redacted] By using this relation we obtain, after some reductions: [formula redacted] 210 [header] I commence by examining the integral [formula redacted] We may write [formula redacted] and [formula redacted] But [formula redacted] is equal to the constant of Euler: [formula redacted]. Besides, this may be proved without difficulty by starting from the ordinary integral form of C, which in the manner known may be directly derived from the above-mentioned definition, i. e. [formula redacted] If we take the difference between the two integrals and exchange the limits and 0 and [formula redacted] for t and T, we obtain by effecting the integration: [formula redacted] The limit value of this expression when t goes to and T to [formula redacted] will evidently be 0. The other part of U may be written as follows : [formula redacted] Consequently [formula redacted] [header] 211 To reduce the remaining integrals I consider the following functions : [formula redacted] (M an entire number). It is readily shown that the subsequent expressions hold : [formula redacted] These will be employed below. [formula redacted] may by partial integration be reduced to [formula redacted]. By simple calculation we get [formula redacted] I am now going to show r that the integrals found in K of the form [formula redacted] may be reduced to expressions which only contain the function [phi] together with rational functions of z and k. I therefore examine the integral [formula redacted] 212 [header] Consequently [formula redacted] So we have reduced all the integrals of this form. There remains only [formula redacted] This one is immediately reduced by the relation (10), which gives [formula redacted] By effecting all these reductions of K, we finally get: [formula redacted] For z equal to [formula redacted], i.e. [formula redacted], this expression is transformed to [formula redacted] This value must be used in order to calculate the scattered radiation entering the ionization-chamber, when the plate is quite close to its opening. It now remains to show how [formula redacted] is connected with known tabulated functions. This is extremely simple and follows from (10). Then the function [formula redacted] is found in the tables of Jancke and Emde under the designation [formula redacted]. (10) gives [formula redacted] [header] 213 Besides [phi] is in a close connexion with the well-known function [formula redacted] which under the denomination of integral logarithm has been the object of special study. Indeed, we get by exchange of variables [formula redacted] The expression obtained allows of calculating, without any difficulty, the scattered radiation, which under a given visual angle penetrates into the ionization-chamber, as soon as we know the intensity, absorption and scattering coefficient of the incident radiation. It is evident that we may by the aid of this expression be able to calculate the said coefficients from the experimental determinations. I leave to the experimental physicists the consideration of how this may be beat executed in practice. In conclusion I wish to make a numerical application of the above results to some data taken from the experiments of Aurén. He placed the plate in two different positions, and measured the radiation entering the ionization-chamber in both cases. The distance to the centre of the plate was in the former position 2.1 cm., in the latter 12.3 cm. The distance between the ionization-chamber and the focus of the bulb was 42 cm. In one case Aurén examined a graphite plate 1 cm. thick. The mass-scattering coefficient for C, according to Aurén, is 0.142. Thus the scattering coefficient [sigma] = 1.7. 0142, because the density is about 1.7. For one of the wave-lengths used by him (0.34 . 10^-8 cm.) k =0.094 . 1.7. Thus [formula redacted] By the aid of these facts, I get the following values of K in both positions : [formula redacted] For the sake of comparison, I calculate [formula redacted], and get [formula redacted] The scattering leaving the plate in the different positions is moreover inversely proportional to the square of the distance of the plate from the focus. With this in view, I calculate the ratio of scattered radiation entering the chamber [footer] 214 [header] in the respective positions to that which would enter it if the plate were in the immediate proximity of the opening, i.e. if r1, r2, and [formula redacted] are the distances in the respective cases, [formula redacted] I calculate [formula redacted] and [formula redacted]. For the two expressions we obtain the values 0.0827 and 0.00602 respectively. The difference between the two expressions is 0.077. If I assume that about equal scattered radiation is propagated backwards, this difference, which comes out as a difference in the apparent absorption, will amount to ca. 3.9 per cent. of the absorption caused by scattering. According to the figures given above it only amounts to [formula redacted] of the total absorption. The difference will then be 1.5 per cent. of the total absorption. This is in fair agreement with the Aurén experiments. In addition to the above calculation I wish to mention that, for small values of z, K is more readily denoted by developing in the formula (4), the integrand of a power series and integrating term by term. So the following expression is arrived at: [formula redacted] Nobel Institution of Physical Chemistry, Stockholm, June 1918.