Intensity formulae are found for the rotational structure of bands arising from transitions between a singlet and a triplet state of a diatomic molecule. These intercombinations occur because the wave functions which diagonalize the orbit-spin interaction contain both singlet and triplet terms. In the transition 1 2-3 2 two cases arise, according as the two states have the same or opposite symmetry as regards reflection of the orbital motions in a plane containing the nuclei, a different set of branches appearing in either case. A comparison is made with measurements of intensities in the atmospheric absorption bands of oxygen; the agreement is satisfactory on the assumption that these bands are due to dipole transitions from the 3 S~ ground state to a 1 2~ state. Formulae are also found for the transitions 1 2-3 n, x 2-3 A when the triplet state comes under either of Hund's cases (a) or (b). T HEORETICAL investigations of intensities in electronic bands havehitherto dealt with transitions between states of the same multiplicity. Certain bands are however known from their structure to be due to "intersystem" combinations between terms of different multiplicities. In this paper only singlet-triplet combinations are considered, since all inter-system bands so far described are of this kind.The proof of the selection rule for molecular spectra that only terms of like multiplicity combine depends on the possibility of separating the wave function of any state of the stationary molecule into the product of two factors, one depending on the orbital motions, and the other on the electron spins. This separation can be made only when the orbit-spin interaction is negligibly small; the effect of the interaction will be to modify the wave functions in such a way as to permit inter-system transitions of small intensity. In the analogous case of intercombinations between singlet and triplet terms in an atom with two electrons 1 it is necessary to solve this perturbation problem completely before the intensities can be calculated. The molecular problem is simpler in that for our purpose the perturbation problem need not be completely solved, nor indeed would this be possible without definite knowledge of the complete molecular wave functions. We are here concerned with the way in which the intensities depend on the rotation quantum number J for a given electronic transition, and not, as in the atomic case, with the intensities of different electronic transitions. STATIONARY MOLECULEElectronic orbital wave functions. The position of an electron with respect to the molecule with nuclei held fixed is specified by Cartesian coordinates 1 W. V, Houston, Phys. Rev. 33, 297 (1929). 806
In his original paper on the theory of complex spectra Slater (1929) calculated, as one of his examples, the relative positions of the multiplets 3P, 1D, 1S, arising from a configuration of two equivalent p electrons p2. The intervals between these multiplets, in the order written, were found to be in the ratio of 2:3. The same result was shown to hold for p4. Slater's method depends on setting up wave-functions for the atom by combining suitably the wave-functions of single electrons in a central field. With these atomic wave-functions the mean values of the electrostatic energy are calculated for the various multiplets. These mean values involve integration over angular co-ordinates, which can be performed, as well as integrations containing the unknown radial functions, which appear as parameters in the final result. The same method was subsequently applied (Condon and Shortley, 1931) to the configurations p2s, p4s, and extended (Johnson, 1932) to include in addition to the electrostatic energy the energy of the magnetic interaction between the orbits and spins, with which Slater was not concerned.
One of the earliest successes of classical quantum dynamics in a field where ordinary methods had proved inadequate was the solution, by Schwarzschild and Epstein, of the problem of the hydrogen atom in an electric field. It was shown by them that under the influence of the electric field each of the energy levels in which the unperturbed atom can exist on Bohr’s original theory breaks up into a number of equidistant levels whose separation is proportional to the strength of the field. Consequently, each of the Balmer lines splits into a number of components with separations which are integral multiples of the smallest separation. The substitution of the dynamics of special relativity for classical dynamics in the problem of the unperturbed hydrogen atom led Sommerfeld to his well-known theory of the fine-structure of the levels; thus, in the absence of external fields, the state n = 1 ( n = 2 in the old notation) is found to consist of two levels very close together, and n = 2 of three, so that the line H α of the Balmer series, which arises from a transition between these states, has six fine-structure components, of which three, however, are found to have zero intensity. The theory of the Stark effect given by Schwarzschild and Epstein is adequate provided that the electric separation is so much larger than the fine-structure separation of the unperturbed levels that the latter may be regarded as single; but in weak fields, when this is no longer so, a supplementary investigation becomes necessary. This was carried out by Kramers, who showed, on the basis of Sommerfeld’s original fine-structure theory, that the first effect of a weak electric field is to split each fine-structure level into several, the separation being in all cases proportional to the square of the field so long as this is small. When the field is so large that the fine-structure is negligible in comparison with the electric separation, the latter becomes proportional to the first power of the field, in agreement with Schwarzschild and Epstein. The behaviour of a line arising from a transition between two quantum states will be similar; each of the fine-structure components will first be split into several, with a separation proportional to the square of the field; as the field increases the separations increase, and the components begin to perturb each other in a way which leads ultimately to the ordinary Stark effect.
This note exhibits some old results in what is possibly a new form.The simple harmonic oscillator, whose kinetic and potential energies are given in terms of a single coordinate x bywhere 6 is a positive constant, has the equation of motionThe solution, appropriate to the initial conditions x = x 0) x =x 0 at t = 0 is conveniently writtenThe problem of the small oscillations of a conservative dynamical system of n degrees of freedom about a position of stable equilibrium is a generalisation of the problem of the oscillator; in seeking a generalisation of the solution (1) a matrix notation at once suggests itself. Let the dynamical system be defined by the two quadratic formswhere B is a symmetric square matrix, x a column matrix and x' its transpose. The form T is of course positive definite, and F is assumed to be non-negative. The equations of motion areThe standard method of solution is to find the latent roots p\, pi p\ of the matrix B, which will for the moment be assumed positive and distinct, and to construct the matrix L whose columns are the normalised latent vectors. Then L'BL is a diagonal
637former to the latter, denoted by R, was determined when the factors enumerated above were varied.Experiment shows that this ratio remains constant for pressures varying from 5 to 70 atmospheres, when air, oxygen or carbon dioxide is used in the chamber. This is in accord with the preliminary results 1 obtained with pressures varying from 10 to 50 atmospheres.R is independent of the volume of the chamber within the limits of the volume available in the chamber used. The volume was varied by a factor of four. It is also of interest to note that the shape of the ionization chamber seems to have no effect upon the ionization current per unit volume.The effect of temperature upon the ionization by gamma-rays has been reported. 1 We were interested in finding whether temperature was a factor in the case of ionization produced by cosmic rays. The chamber was subjected to various temperatures, while the batteries and other parts, by means of which temperature might cause a change in the sensitivity of the electrometer, were kept at a constant temperature. It was found that, within experimental error, R remained constant. Separate tests showed temperature to be a factor in each case. Hence, the temperature coefficient is the same for cosmic as for gammarays.It is hard to interpret the results when dif-Following a suggestion of Professor Van Vleck, we have investigated the influence of electric fields of different symmetries 1 on the paramagnetic susceptibilities of atoms in crystals. Experimentally, the susceptibility can often be represented by the Curie-Weiss law X -C/(r+A), provided the temperature is not too low. We find that if the energy levels are all either high or low compared with kT then the rigorous expansion for the susceptibility averaged over all directions is of the form x = G/r+CVr 3 + • • • , the term in 1/r 2 being absent. This means that if we measure the susceptibility of a crystal powder and plot 1/x against T, the curve tends to an asymptote passing through the origin and not through the point (0, -A). At lower temperatures it is necessary to use exact expressions and it is found that the susceptibility may simulate the law x -c/(T+A) over a wide range of temperature. This is the case, for example, with ferent liners are used. This is complicated by several factors, of which local radiation due to contaminations of the liners is difficult to determine. There is a variation of R after corrections are made for absorption. Further work is in progress on this point.At present the value of R is being determined for applied chamber voltages varying from 6 to 450 volts over the same range of pressures used when the effect of pressure was being studied. It is hoped that this may help to determine whether saturation with pressure is due to the rate of recombination of ions becoming equal to that of diffusion 1 or to the complete absorption 2 by the gas of the betaparticles ejected from the walls of the ionization chamber.These results indicate the process of ionization is the same for cosmic as for gamma...
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