The author is indebted to Dr. A. W. Weiss for providing the wave functions, and to Dr. MI. Inokuti for many helpful discussions. 5. Pekeris, C. L. Phys. Rev. 126, 1470 (1962). 6. Pekeris, C. L. Phys. Rev. 115, 1216 (1959). 7. Pekeris, C. L. Phys. Rev. 126, 143 (1962). 8. Ilart, J. F. and Herzberg, G. Phys. Rev. 106, 79 (1957). 9, Rotenberg, M. and Stein, J. Phys. Rev. 182, 1 (1969). REFERENCES 1. Kim, Y.-K., and Inokuti, M. Phys. Rev. 166, 39 (1968). 2. Inokuti, M., and Kim, Y.-K. Phys. Rev. 178, 154 (1968). 3. Inokuti, M., Kim, Y.-K., and Platzman, RK. L. Phys. Rey. 164, 55 (1967). 4. Weiss, A. W. J. Res. Nall. Bur. Std. TEA, 163 (1967). 10. Hurst, R. P. Acta Cryst. 18, 634 (1960). 11. Freeman, A. J. Acta Crysf. 18, 190 (1960). 12. Fock, V. and Petrashen, M. J. Phys. Z. Sowjet. 8, 547 (1935). 13. Kim, Y.-K. and Invkuti, M. Inelastic-Seattering Cross Sections of Fast Charged Particles by Lit. This report. THE NUMBER OF BOUND STATES IN ION-ATOM SYSTEMS Smto Tant* and Mikio [nokutt The knowledge of the number of bound states and density of states of a molecular ion is very useful in the analysis of molecular spectra and also in the studyof ion-atom scattering at low incident energy. The WKB approximation in quantum mechanics has been used to estimate the total number of bound states and the density of states for the ion-atom complex represented by a simple parameter. The purpose of this work is to provide, in a quick esti- mate, parameters concerning boundstates of an ion with a neutral atom. Accordingly, we assume that the polari- zation (7*) potential plays the most essential role," and extend it to a distance which is a sum of effective radii of the ion and the atom. Inside we assumean infinitely high repulsive potential (hard core) for the sake of mathematical simplicity. The parameters we deal with are (1) the number of bound states for each rotational quantum number, (2) the density of bound states per unit energy interval, and (3) the radius of the largest classical orbit. In the above-mentioned simplified model, the polarizability a of the neutral atom,” the reduced mass » of the system, and the ionic and the atomic radii® @ion and Gneutr. Will appear only in a combination of the form z2=V ap/ (Bion + Gneutr.)5 where all quantities will be expressed in atomic units. Therefore, a handy tabulation of the results 1s possible byusing values of z as indices. The zero-energy resonances in this model can be determined from the zeros of a Bessel function.” We TT ooh: RTSUOEY 8 consider the rotational quantum number (/) as a con- tinuously varying parameter.t When the parameter z * Visiting Scientist for the period of July-August, 1969; permanent address: Physics Department, Marquette University, Milwaukee, Wisconsin, 53233. + A resonance considered as a function of angular momentum is called a Regge pole and is well studied in scattering theory. (See, for example, Reference 5.) introduced above is known, the upper I:mit of J will be determined for each vibrational quantum number (v). The results will be presented in the form of an ex- tensive table, and we can estimate the possible number of bound states straightforwardly. Besides being of direct help in the spectroscopy of molecular ions, the location of such resonances for variable J (Regge poles) is greatly relevant to the study of ion-atom scattering at very low energies. This is so because, as energy is raised above zero, these resonances continue to exist with a complex value of angular momenta, and some of them maybesignificant as a cause of a rainbowor a glory.* Both quantum numbers, J and »v, can take large values. Then, the situation is semiclassical. Namely, an estimate based on classical mechanics is quite close to the rigorous result, and the WKB approximation is valid, Therefore, we shall use this method in the evaluation of the density of the states. Since the problem is characterized by a single parameter z in the case of J = 0, we shall investigate this case in detail. It turns out that only a small correction is necessary for a nonvanishing value of /, unless J is very large. The orbital radius for a high lying level is large. If other atoms or ions are encountered along such a large orbit, the spee- tral line of a high lying level will be shifted. The density effect of the same kind in which an electron is orbiting instead. of an ion was first discussed by Fermi.It is planned to derive an effective value of the largest orbital radius, which will serve as a critical parameter in the shift of spectral lines of high lying levels. * The quantal effect of the Lit-He scattering was studied by Weber and Bernstein;® in the case of H-H, in which the potential behaves like r* at a large distance, an extensive tabulation and drawing of graphs was made by Waeeh and Bernstein;although there is a difference between the nature of their problem and ours, presentation of the final result has cer- tain common aspects in these two cases,