The Analysis of Matter

THE atomicity of matter is a hypothesis as old as the Greeks, and in no way repugnant to our mental habits. The theory that matter is composed of electrons and protons is beautiful through its successful simplicity, but is not difficult to imagine or believe. It is otherwise with the form of atomicity introduced by the theory of quanta. This might possibly not have surprised Pythagoras, but it would most certainly have astonished every later man of science, as it has astonished those of our own day. It is necessary to understand the general principles of the theory before attempting a modern philosophy of matter; but unfortunately there are still unsolved physical problems connected with it, which make it improbable that a satisfactory philosophy of the subject can yet be constructed. Nevertheless, we must do what we can. As everyone knows, the quantum was first introduced by Planck in 1900 in his study of black-body radiation. Planck showed that, when we consider the vibrations which constitute the heat in a body, these are not distributed among all possible values according to the usual law of frequency which governs chance distributions, but on the contrary are tied down by a certain law. If is the energy of a vibration, and its frequency, then there is a certain constant ,[7] known as Planck's constant, such that is or , or , or some other small integral multiple of h. Vibrations with other amounts of energy do not occur. No reason is known for their non-occurrence, which remains so far of the nature of a brute[Pg 31] fact. At first, it was an isolated fact. But now Planck's constant has been found to be involved in various other kinds of phenomena; in fact, wherever observation is sufficiently minute to make it possible to discover whether it is involved or not. A second field for the quantum theory was found in the photo-electric effect. This effect is described as follows by Jeans:[8] "The general features of the phenomenon are well known. For some time it has been known that the incidence of high-frequency light on to the surface of a negatively charged conductor tended to precipitate a discharge, while Hertz showed that the incidence of the light on an uncharged conductor resulted in its acquiring a positive charge. These phenomena have been shown quite conclusively to depend on the emission of electrons from the surface of the metal, the electrons being set free in some way by the incidence of the light. "In any particular experiment, the velocities with which individual electrons leave the metal have all values from zero up to a certain maximum velocity , which depends on the conditions of the particular experiment. No electron is found to leave the metal with a velocity greater than this maximum . It seems probable that in any one experiment all the electrons are initially shot off with the same velocity , but that those which come from a small distance below the surface lose part of their velocity in fighting their way out to the surface. "Leaving out of account such disturbing influences as films of impurities on the metallic surface, it appears to be a general law that the maximum velocity depends only on the nature of the metal and on the frequency of the incident light. It does not depend on the intensity of the light, and within the range of temperature within which experiments are possible it does not depend on the temperature of the metal.... For a given metal this maximum velocity increases regularly as the frequency of the light is increased, but there is a certain frequency below which no emission takes place at all." [Pg 32] The explanation of this phenomenon in terms of the quantum was first given by Einstein[9] in 1905. When light of frequency falls on the conductor, it is found that the amount of energy absorbed by an electron which the light separates from its atom is about five-sixths of , where is Planck's constant. It may be supposed that the other one-sixth is absorbed by the atom, so that atom and electron together absorb exactly one quantum . When the light is of such low frequency that is not enough to liberate an electron, the photo-electric effect does not take place. Explanations not involving the quantum have been attempted, but none seem able to account for the data. Another field in which the quantum hypothesis has been found necessary is the specific heat of solids at low temperatures. According to previous theories, the specific heat (at constant volume) multiplied by the atomic weight ought to have the constant value 5·95. In fact, this is found to be very approximately correct for high temperatures, but for low temperatures there is a falling off which increases as the temperature falls. The explanation of this fact offered by Debye is closely analogous to Planck's explanation of the facts of black-body radiation; and as in that case, it seems definitely impossible to obtain a satisfactory theory without invoking the quantum.[10] The most interesting application of quantum theory is Bohr's explanation of the line spectra of elements. It had been found empirically that the lines in the hydrogen spectrum which were known had frequencies obtained from the difference of two "terms," according to the formula: where is the frequency, is "Rydberg's constant," [Pg 33] and are small integers, and are what are called "terms." After the formula had been discovered, new lines agreeing with it were sought and found. Certain lines formerly attributed to hydrogen, and not agreeing with the above formula, were attributed by Bohr to ionized helium; they are given by the formula: Bohr's theoretical grounds for attributing these lines to helium were afterwards confirmed experimentally by Fowler. It will be seen that they fit into the formula (1) when is substituted for , a fact which Bohr's theory explains, as well as the more delicate fact that, to make the formula exact, we have to substitute, not exactly , but a slightly smaller quantity. The form of the equation (1) suggested to Bohr that a line of the hydrogen spectrum is not to be regarded as something which the atom emits when it is in a state of periodic vibration, but as produced by a change from a state connected with one integer to a state connected with another. This would be explained if the orbit of the electron were not just any orbit possible on Newtonian principles, but only an orbit connected with an integral "quantum number"—i.e. with a multiple of . The way in which Bohr achieved a theory on these lines is as follows. He supposed that the electron can only revolve round the nucleus in certain circles, these being such that, if is the moment of momentum in any orbit, we shall have: where is, as always, Planck's constant, and is a small whole number. (In theory might be any whole number, but in[Pg 34] practice it is never found to be much larger than 30, and that only in certain very tenuous nebulæ.) The reason why the quantum principle assumes just this form will be explained presently. Now if is the mass of the electron, the radius of its orbit, and its angular velocity, we have: But, on grounds of the usual theory, since the radial acceleration of the electron is and the force attracting it to the nucleus is we have: From equations (3) and (4) we obtain: The possible orbits for the electron are obtained by putting = 1, 2, 3, 4, ... in the above formulæ for . Thus the smallest possible orbit is: and the other possible orbits are , , , etc. For the energy in an orbit of radius we have, since the potential energy is double the kinetic energy with its sign changed:[11] in virtue of (5). Thus when the electron falls from an orbit whose radius is to one whose radius is , there is a loss of energy: [Pg 35] It is assumed that this energy is radiated out in a light-wave whose energy is one quantum of energy , where is its frequency. Hence we obtain the frequency of the emitted light by the equation: This agrees exactly with the observed lines if [see equation (1)]: where is Rydberg's constant. On inserting numerical values, it is found that this equation is verified. This striking success was, from the first, a powerful argument in favour of Bohr's theory. [Pg 36] Bohr's theory has been generalized by Wilson[12] and Sommerfeld so as to allow also elliptic orbits: these have two quantum numbers, one corresponding, as before, to angular momentum or the moment of momentum (which is constant, by Kepler's second law), the other depending upon the eccentricity. Only certain eccentricities are possible; in fact, the ratio of the minor to the major axis is always rational, and has as its denominator the quantum number corresponding to the moment of momentum. In order to explain the Zeeman effect (which arises in a magnetic field) we used a third quantum number, corresponding to the angle between the plane of the magnetic field and the plane of the electron's orbit. In all cases, however, there is a general principle, which must now be explained. This will show, also, why, in Bohr's theory, the quantum equation (2) takes the form it does.[13] The first thing to observe is that the quantum principle is really concerned with atoms of action, not of energy: action is energy multiplied by time. Suppose now that we have a system depending upon several co-ordinates, and periodic in respect of each. It is not necessary to suppose that each co-ordinate has the same period: it is only necessary to suppose that the system is "conditionally periodic"—i.e. that each co-ordinate separately is periodic. We must further assume that our co-ordinates are so chosen as to allow "separation of variables" (as to which, see Sommerfeld, op. cit., pp. 559-60). We then define the "momentum" (in a generalized sense) associated with the co-ordinate as the partial differential of the kinetic energy with respect to —i.e. calling the generalized momentum , we put: where is the kinetic energy. The quantum condition is to apply to the integral of over a complete period of —i.e. we are to have: where the integration is taken through one complete period of . Here will be the quantum number associated with the co-ordinate . The above is a general formula of which all known cases of quantum phenomena are special cases. This is its sole justification. The above principle is exceedingly complicated—more so, even, than it appears in our summary account, which has omitted various difficulties. It is possible that its complication may be due to the fact that quantum dynamics has had to force its way through the obstacles which the classical system put in its way; it is possible also that quantum phenomena may turn out to be deducible from classical principles. But before pursuing this line of thought, it may be well to say[Pg 37] a few words about the developments of Bohr's theory by Sommerfeld and others. In its original form, in which circular orbits were assumed, Bohr's theory accounted for the main facts concerning the line spectra of hydrogen and ionized helium. But there were a number of more delicate facts which required the hypothesis of elliptic orbits: with this hypothesis, together with some niceties derived from relativity, the most minute agreement has been obtained between theory and observation. But perhaps this great success has made people think that more was proved than really was proved. The great advantage obtained from admitting elliptic orbits is that they provide a second quantum number. In the emission of light by atoms, what we have is essentially as follows. The atom is capable of various states, characterized by whole numbers (the quantum numbers). There may be more or fewer quantum numbers, according to the degrees of freedom of the system. The loss or gain of energy when an atom passes from a state characterized by one set of values of the quantum numbers to a state characterized by another set is known.