V. Determination of Vo by use of Capillary Tubes. 0 The results of this work show that I was unable to obtain pure radon; the relative amount of CO, varied in different experiments from 15 per cent. to 45 per cent. of the total quantity of gas. It will be remembered that in the method of compressing the purified radon into a capillary tube the initial volume undergoes generally a marked diminution ranging from 30 per cent. to 50 per cent. of its final value. It seems clear that in this method also pure radon was not obtained by use of chemical processes only. It is highly probable that the final purification takes place in the capillary, the impurities being absorbed by glass under the action of a-rays. In order to throw some light on this question, I have executed some experiments by the method of the capillary tube using an apparatus described in a previous paper*, and also the arrangement shown on fig 3. To the vertical tube V a sloped tube S, ending in a calibrated capillary C1, was sealed. Another capillary C2 of the same bore as C1 sealed to V, allowed the pressure of compressed gas to be determined by means of a cathetometer. The use of the bulb b will be explained later. The first experiment was performed with the apparatus described in my previous paper. The results were very similar to those obtained by Rutherford. 40.5 millicuries radon were introduced into a capillary. The contraction amounted to 30 per cent. of the final value, which corresponded to V。= 64.10 c.c. 66 a It would seem surprising that in presence of rarefied radon gas is evolved, while the concentrated radon acts as clean-up" agent. This apparent contradiction is easily explained if we consider that the first effect is due to collisions between a-particles and the walls of the apparatus, while the second one is a consequence of collisions between molecules and a-particles. It is probable that, in analogy with the clean-up effects observed in a triode valve, ionized or excited molecules are readily absorbed by the material of walls. If the pressure is very low the number of collisions. with free molecules is negligibly small, and therefore under these conditions the evolution of gas is more important than its absorption. In a concentrated state radon atoms as well as molecules of impurities are struck by a-particles, and it seemed there L. Wertenstein, loc. cit. fore probable that radon itself may be partly absorbed in the capillary tubes. In order to verify this hypothesis I have tried to repeat the experiment with radon as pure as possible; if radon was absorbed, the final volume ought to be smaller than its theoretical value. I have used for this experiment radon the degree of purity of which has been tested by the "damping" method. This radon was distilled into the tube S (wrapped for this purpose, as usual, with cotton wool, soaked in liquid air). This tube contained some fused KOH in order to diminish further the amount of CO2. Before compressing the radon into the capillary, it was submitted to another kind of purification based on laws of flow of highly rarefied gases. For that purpose the bulb b, connected to S through a 5 mm. wide capillary, was used. After the radon had been condensed, mercury was brought into such a position that the communication with b was shut off. When the radon was set free after removal of liquid air, the mercury was lowered for a few seconds below the opening of the capillary. During that short time a part of the radon passed into the bulb b, but, according to the laws of flow of gases, the relative amount of CO, which went to this bulb must have been much larger. The remaining gas was introduced into the capillary C1. The results of the experiment confirmed my hypothesis. The initial volume of the radon corresponded to Vo=5.9.10-c.c., the final volume (determined on the next day and corrected for radioactive decay) to V。=4·34.10-4 C.C. We see that nearly 30 per cent. of the radon was absorbed. In order to ascertain the nature of this absorption the radon was pumped out and the tube heated during exhaustion to 200°. Subsequent measurements of the y-ray activity showed that the tube retained about 20 per cent. of the initial quantity of radon (taking account of the radioactive decay). The amount of concentrated radon absorbed on glass seems to increase with the time during which the radon is kept in the capillary. In another experiment performed with less pure radon the initial volume corresponded to V1 = 7.91.10-4 c.c., the final value, measured on the next day, to 6.2. 10-c.c. After four days the radon was pumped off as in the previous experiments. It was found that the glass retained strongly 40 per cent. of the initial quantity of radon absorbed. It is probable that the "clean-up" effect of a-rays is selective. It is obviously much smaller for radon than for other gases, and this explains probably why the method of the capillary tube gives in general results very close to truth. The work described in this paper was performed during the years 1925/1926 and 1926/1927 in the Cavendish Laboratory. It is a great pleasure for me to express my best thanks to Sir Ernest Rutherford for his kindness in receiving me in the Cavendish Laboratory, in placing at my disposal the large quantities of radon and other experimental resources, and in showing a permanent interest in the progress of my work. I am grateful to Dr. J. Chadwick for his help and interest. My stay in England was made possible by a fellowship granted by the International Education Board. I wish to express my thanks to this Board. III. The Classical Reasonableness of the Quantum Theory and Simple Operative Solutions of Schrödinger's Equation. By Prof. A. PRESS *. IN PART I. The Differential Equation of an Excitation Field. N a paper appearing in the Philosophical Magazine for Dec. 1927, the writer likened the action of a radiating harmonic oscillator to that of a reed or organ-pipe fed by a source of air under pressure. As the pressure was increased the organ-pipe responded but slightly to an increase in intensity of the note emitted, for a point was soon reached when the dominant note suddenly changed to a higher harmonic. It was later shown by strictly classical analysis that the amplitude of the radiation wave always bore a definite ratio to the amplitude of the normal non-radiating component. In the following, therefore, reference to the normal non-radiating component will be sufficient. For the pipe above referred to, the excitation field contemplated would be that due to the blowing stream causing it to emit a note. It will be on the basis of the above analogy with the radiating harmonic oscillator that the following analysis will be developed. In general there will be a * Communicated by the Author. Phil. Mag. S. 7. Vol. 6. No. 34. July 1928. D variable velocity potential P in such an exciting stream because of the action of the organ-pipe wall of the slit or the reed, which, due to the vibrations, affects the distribution of pressure head in the stream contiguous thereto. As the slit wall or reed vibrates outwardly and inwardly the stream becomes blocked and more open alternately. The wave-front of pressure intensity (and therefore of velocity potential) will travel backwards along the stream toward the source as long as the slit wall in its alternations gives freer egress to the excitation stream. With the return of the slit wall or reed and the blocking of the stream, the wave-front of pressure intensity (and therefore of velocity potential) returns in the direction of the stream-flow. There is thus an alternation of the wave-front position along the stream depending on the frequency of the note emitted. The equation of the velocity potential therefore becomes with u as the velocity with which a disturbance can travel along the stream for a given mean velocity w of the stream itself. Relationship between the various Velocities concerned. As the momentary velocity w of the stream is increased relative to a, the momentary velocity u at r, with which any disturbance in the velocity potential can proceed, will diminish in value, but correspondingly the momentum at any moment of the slit wall or reed (mv), and therefore the velocity v, will increase. That is to say, although there is phase correspondence between the v of the alternating slit wall or reed and u with which the phase of the potential disturbance travels with respect to the excitation stream velocity wat x, the one velocity increases, whereas the u at any a diminishes inversely with increasing w. The above complies with one of Schrödinger's conditions, in which u represents the velocity of his -function, and r the momentary velocity occurring in the kinetic energy expression of Hamilton's function Eo. Postulates of the Mechanical Constraints If, now, in the hypothetical case of a "Radiating Harmonic Oscillator" the maximum kinetic energy of the alternating member due to its extreme position (analogous to that of * See Schrödinger, "Undulatory Mechanics," Phys. Rev., Dec. 1926. the slit wall or reed above) is so limited that we have a connecting equation of the type then the function solving equation (1) must be of the form This will enable a considerable simplification to be performed in solving symbolically equation (1). It has already been stated that u, at any moment pertaining to the instantaneous velocity of the P wave-front, is inversely proportional to the mean stream velocity w. Similarly the v of the kinetic energy term, having reference to the radiator (the wall of the slit or reed, as it were), is for the same moment directly proportional to the mean stream velocity w. It would be therefore reasonable to write that and the Fourier law of dimensions will be satisfied. . Schrödinger's Differential Equation. Split Form of (4) By virtue of equation (2), the solution of equation (1) being of the form 2TTE P = P(x). sin t, ho (5) (9) ・・ Making use of the latter, (1) transforms to There then remains to express u in terms of Eo by virtue of (4). Introducing the latter, we have that The justification of this type of Quantum Formula will be gone into later. See end Part I. |