The quantum theoretical treatment has been considered in a previous note: Stueckelberg, Comptes rendus 207, 387 (1938).
The continuous absorption of 0 2 has lately been measured by Ladenburg, Van Voorhis and Boyce. Their suggested explanation in the term diagram of the molecule is compared to the corresponding matrix elements. PART I. THEORY OF CONTINUOUS ABSORPTION IN DIATOMIC MOLECULES/CONTINUOUS absorption and emission in diatomic molecules have been ^^ explained as transitions between electronic states with discrete vibrational energy and electronic states with continuous energy.Condon 1 gave in his paper a qualitative way of obtaining the relative intensities of the observed continua. This has been applied to the continuous emission of H 2 by Winans and Stueckelberg 2 and by Finkelnburg and Weizel. 3 Other applications have been made by Kuhn. 4 The present paper gives a quantitative explanation of the continuous absorption of 0 2 as measured by Ladenburg, Van Voorhis and Boyce. 5 The probability (number of transitions per second) of a transition from the discrete state k to the energy continuum W is:Huw i s the matrix element of the perturbing energy H, and dQ w /dW is the number of states per region dW. In our case we have H=ME where M is the electric moment of the system, and E= E 0 cos lirvt the electric field strength of the incoming light wave. One obtains | H k w\ 2 = | M k w\ 2 E 0 2 /3. The factor J is due to the averaging over the angle between E 0 and M. Let N k be the number of molecules in the state k per cm 3 , and let n be the number of incident light quanta per cm 2 and sec. The number of quanta absorbed in the thin layer AZ per sec. is
The electronic energies W~(n", ny, n, ) of the hydrogen molecular ion are calculated by means of the wave mechanics as functions of the nuclear separation c = 2p, for several values of the quantum numbers n", n~and n, . The wave function is separable in the elliptical coordinates y = (ri+r, )/2p, @ and x=(r& -r&)/2p. A qualitative idea of the behavior of these energies as p changes from infinity to zero is gotten by an investigation of the behavior of the nodal surfaces. The number of these surfaces in any coordinate equals the quantum number in that coordinate. When p= oo the resulting system is that of a hydrogen atom and a separated nucleus, the nodes are paraboloids and planes with quantum numbers n", ny and n~, and the electronic energy is W"=R/(n"+n~+ng+1)' where R is the lowest energy of the hydrogen atom. When p =0 the system is that of a helium ion, the nodes are spherically symmetric with quantum numbers n", n@ and n|i, and the electronic energy is W0=4R/ (n, +ny+np+1)'. As p changes from zero to infinity it is shown that the quantum numbers are related in the manner n"~n"~n"; n~~n@~n~, ' nylon, ~2ng or 2ng+1.Therefore W0=4R/n"+ny+2n~+1)' or =4R/(n"+n~+2n~+2)'. By this rule it is possible to check the following quantitative calculations. The first order perturbations of the various electronic energies of the first three degenerate levels of the helium ion resulting when p=0 were calculated; the perturbation being the slight separation of the nuclei (p )0). The first order perturbations of the various electronic energies of the first two degenerate levels of the hydrogen atom resulting when p = were calculated when the perturbation was the diminution of the separation (p & oo).The first method is not valid for p) a/2, where a is the radius of the first Bohr orbit of the hydrogen atom, and the second is not valid for p&3a/2. The gap between was extrapolated by means of the nodal reasoning above. These electronic energies plus the energy of nuclear repulsion give the molecular potential energies. A calculation of these shows that of the eight curves obtained only three, the 1so, 3do and 4P states show minima, and therefore are stable configurations to this order of approximation (the Hund molecular notation is used for the states). The numerical results check with previous calculations and with the data available. 2l+1 @0(N, I, m) =Se(mlm) e' &. sin 0. P~(cos8) e "I"" (2r/lao)'L"qg(2r/mao)(3) These quantum numbers bear relation to n~n"and n the quantum numbers for each coordinate, for n& = m, ng =lm, n =n"+n~+ng+1. The relation between the spherical coordinates used above and the general ellip-3 F. Hund, Zeits. f. Physik 40' 742 (1927).
E. C. G. STUECKEL HER Gholes are immobile, being bound to the impurity atoms.Under these circumstances the polarization is large and temperature independent. Absorption of radiation in the long wavelength region of the fundamental absorption band is expected to result in the production of excitons. If these are thermally dissociated at room temperature, free electrons and holes are produced, and the polarization is small. At low temperatures, where the excitons are not thermally dissociated, they wander to impurity atoms where dissociation does occur, the electron becoming free but the hole remaining bound to the impurity atom. Under thele conditions of excitation, as the temperature is reduced, the polarization increases.The dissociation energy of an exciton can be estimated from the relation" E. -pr'me4/u4k'n' by taking n= i. The index of refraction, p, of diamond "F. Seitz, Phys. Rev. 76, 1376Rev. 76, (1949.is 2.42 and EI~.2 ev. The lifetime of the exciton is given by r = rp exp(+Eg/kT).Estimating rp~10 " sec, one obtains r(100'K)~10 ' sec and r(300'K)~10 ' sec. If the cross section for collision with an impurity atom is taken to be j.0 " cm', the concentration of impurity atoms of the order of 10" cm ', and the velocity of the exciton as 10' cm/sec, then at 100'K an exciton will make a million collisions with impurity atoms during its thermal lifetime. At room temperature, the corresponding number of collisions is less than unity. The behavior of the exciton is therefore in agreement with the preceeding interpretation.One of us (C.C.K.
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