by making use of the expression for Ko, formula (4). The expression for S can be obtained in the same manner from equation (29) by calculating the value of w, from which it is found that and This expression for S is of no direct use, because Q represents several types of resistance to motion, the chief of which are viscosity, air-damping, and the effect of the support of the bar. The resistance S also includes the effect of dielectric loss in the quartz, due to the electric fields produced by the alternating stress. The value of S is determined directly from the reaction of the resonator on an electric circuit by methods developed by W. G. Cady (4) D. W. Dye. The above analysis therefore leads to the conclusion that the piezo-electric quartz bar, in the neighbourhood of one of its resonant frequencies, has the same impedance as an electrical circuit made up of a resistance S, an inductance N, and a capacity K in series, shunted by a condenser K10, as in fig. 2, where the values of N, S, K, Kjo are given by formulæ (31), (33), (32), and (27). 10 The auxiliary quantities M, K, and K, are In the case of a disk cut perpendicular to the electric axis and vibrating along the electric axis, the product ot, wherever it occurs in the formulæ, should be replaced by πо2=πt', the sectional area. Similar formulæ hold for vibrations along the third axis t, if instead of H we write H, and if we use for K the value е given by formulæ (7), namely t 1 oe E The values of K and K10 are therefore the same as before, and N is altered in the ratio of t to e2 so that the resonant frequency is altered in the ratio of e to t. The above values hold only in the neighbourhood of resonance; by this we mean that they hold good for all его values of for which one can write without great error a Thus, whenever the value of w is such that is small compared with unity, the electrical circuit gives a good representation of the quartz resonator. Since the decrement of quartz is very small, and therefore the range of frequency over which it reacts on an electrical system only a small fraction of the fundamental frequencyat most one in a thousand, and generally not more than one in five thousand,-the above condition is always satisfied for the range of frequency with which we are concerned. The equivalence between mechanical vibrating systems and electrical oscillatory circuits is, of course, well known, and S. Butterworth (5) pointed out several years ago that many problems could be simplified by making use of the analogy. W. G. Cady (3) subsequently mentioned the equivalence in the particular case of quartz rods, and K. S. Van Dyke (6) gave expressions similar to the above for N, K, and K10 The equivalent capacity K can be measured experimentally, for example by D. W. Dye's (1) methods, and the average piezo-electric constant H can be calculated from formula (32). This requires that the quartz should be free from electrical twinning as well as from optical twinning, a condition very rarely met with even in the best specimens. An important figure, which is in some ways a measure of the suitability of the quartz for resonators and oscillators, is the ratio of K1 to K. From equations (3) and (32) this is found to be for the fundamental: Κι π H2 = -P, K 32 E which, with the values of H, E, and P, given at the beginning of this paper, is 140. Most pieces of quartz give a value well above this figure, but the nearer the value is to 140 the better is the quartz, provided its damping S is low. It is necessary to point out that the overtone frequencies are not given exactly by the formula (30): where k is an integer, because the formula was developed without taking into account the lateral motion of the vibrating bar. For a long thin rod the variations are small, although irregular, as shown by Giebe and Scheibe (8). An important result can be deduced from the above analysis, concerning the strain at any point of the vibrating rod. Since in all cases u is small, approximate expressions for 2A and 2B of formula (21), in the neighbourhood of In an approximate expression we can neglect A in comparison with B, and we obtain ZF cos wt + F, sin wt, substituting for F and F, by (20) gives Z=constant (sin vx cosh ux cos wt + cos vx sinh ux sin wt), and since a is never greater thane, sinh ux can be neglected in comparison with unity, and the simple expression for Z is Ꮓ = constant. sin vx. cos wt. The displacement is therefore sinusoidal, being zero at the centre and a maximum at the ends, where x=+e, and the strain is a maximum at the centre and decreases to zero at the ends. This was illustrated in a beautiful manner by Giebe and Scheibe (8) with their luminous resonators. This work has been carried out under the auspices of the Radio Research Board, to whom thanks are due for permission to publish. List of References. (1) D. W. Dye, Proc. Phys. Soc. xxxviii. (5) pp. 399–458. (2) H. Lamb, 'Dynamical Theory of Sound.' (3) W. G. Cady, Phys. Rev. xix. p. 1 (1922). (4) W. G. Cady, Proc. Inst. Radio Eng. x. p. 83 (1922). (5) S. Butterworth, Proc. Phys. Soc. xxvii. p. 410 (1915). (6) K. S. Van Dyke, Abstract 52, Phys. Rev., June 1925. (7) And more recently Proc. Inst. Radio Engrs., June 1928. (8) Giebe and Scheibe, Elektrotechnische Zeitschrift, pp. 380-385 (1926) Phil. Mag. S. 7. Vol. 6. No. 40. Dec. 1928. 4 F CXIV. The Power Relation of the Intensities of the Lines in Na former paper by Wood and Gaviola † we have seen that the prediction of Wood ‡, that the intensity of some of the mercury lines appearing in the fluorescence of the optically-excited vapour (3650 for example) should be proportional to the cube of the exciting intensity, while others, such as 3654, 4358, etc., are proportional to the square, and others (2537) to the primary intensity itself, has been proved experimentally in a large range of variation. It has been found, for instance, that a 10-5-fold increase of the intensity of the exciting light was able to increase the intensity of the fluorescence line 3654 about 120 times, and of the line 3650 no less than 1200 times. The opticallyexcited vapour represents then a light-source which radiated energy, regarded in the light of the line 3650, apparently increases with the third power of the absorbed energy, or at least with the third power of the exciting intensity. This seems to be in contradiction with the principle of conservation of energy, and one must ask oneself, if it were possible to increase a hundred-fold the intensity of the mercury arc, would 3650 increase a million times in its brightness? And if so, where would the energy come from? To answer those questions it is necessary to make a quantitative study of the relations between the intensities of the primary exciting lines of the arc and of the secondary fluorescence lines in the vapour. We shall see that this theory will enable us at the same time to explain many of the results of Wood's experimental observations that could not be understood otherwise, for instance, the differences in the falling off of the intensity of the diverse lines along the cross-section of the tube when viewed "end on," and the changes in the intensity curves if gases are admitted to it. The Intensity of 3650. If we consider the diagram of energy levels of fig. 1 we see that the line 3650 is emitted by the 3D, level. Its intensity is then proportional to the number of atoms that * Communicated by Prof. R. W. Wood. † Phil. Mag., Aug. 1928, p. 352. R. W. Wood, Phil. Mag., Sept. 1927, p. 471. happen to be excited in that level. This level is reached in our case of mercury vapour at room temperature, where the time between two collisions is large in regard to the lifetime of an excited atom, so that collisions of the first or second kind do not play a significant role, only by the absorption of the line 3650 of the arc by the atoms on the level 23P2. We will call Na, Ns, Np., N, etc., the number of atoms in the different levels, as illustrated by fig. 1. Na, the number of atoms in the level 33D3, is then proportional to Np, and to the intensity of the line 3650 of the arc, that we will call 13650 or simply I3. It is then where c is a constant factor that does not interest us for the moment. The level 23P, is supplied with electrons chiefly by the emission of the line 5461 by atoms with electrons on the level 23S,. The emission of 3341, 3663, 3654, and 3023 by the fluorescent vapour also brings electrons to that level (fig. 2), but the sum of the intensities of these lines in fluorescence is about 0.08 of the intensity of the green line 5461, so that one can neglect them in a first approximation. This is the reason why Wood found that only electrons coming down from 23S, seemed to be effective in producing the absorption and re-emission of 3650. The absorption of 4358 in the exciting light by means of a filter placed between arc and tube made 3650 practically disappear, in spite of the fact that 3131 and 3650 were not reduced by the filter. The number of atoms on the level of 23P, is then proportional to the intensity of the green line in fluorescence that we will call J5461. In general we will denote *Phil. Mag., Oct. 1925, p. 786. |