- Email: [email protected]
On a formal theory of resonance reactions
ABSTRACTS
OF
PAPERS
179
where the function J(kb) varies from zero at kb = 0 to unity as kb -+ 0~. Hall’s well-known experiments are consistent with this purely hydrodynamic analysis and give a core radius 0 of -4*x, compared with the earlier estimates of -30 -7A. The studies of vortex motions in an atomistic framework are also discussed. Stzcdies of Many-Channel Scattering. II. Connection between Total and Partial Widths. HANS A. WEIIIENM~~LLER, Institut fur Theoretische Physik, Universitat Heidelberg, Heidelberg, Germany. Results that were derived in a previous paper on many-channel scattering are used to investigate the formula which expresses the total width of a resonance as a sum of the partial widths of the open channels. If a resonance is defined by a pole of the S matrix, it is shown that this formula is valid only in the limit of arbitrarily sharp resonances and only if one considers the sum of the absolute values of the partial widths of the open channels (these widths are in general complex). On a Formal Theory of Resonance Reactions. LUCIANO FONDA, Istituto di Fisica dell’lmiversita, Trieste, Italy; and Istituto Nazionale di Fisica Nucleare, Sottosezione di Trieste, Trieste, Italy. A recent theory of resonance reactions is reconsidered for the purpose of deriving from it a complex potential model and the decay law of the compound nuclear states. Second-Order Z-Dependent Theory of Many-Electron Atoms. DAVID LAYZER, ZDENEK HO&K, MARGARET N. LEWIS, AND DANIEL P. THOMPSON, Harvard College Observatory, Cambridge, Massachusetts. A quantitative description of complex term spectra requires accurate knowledge of the three leading coefficients in the Z expansions E = EoZ* + RZ + Ez + E&l + . . of term energies. The screening approximation (described in a previous paper) yields exact values for Eo and El but only estimates of the remaining expansion coefficients. The present paper is concerned with the coefficient Ee , which may be split into two physically and mathematically distinct components, E?’ and E2”. The intermediate states that figure in the perturbation formulas for E2’ and Ez” differ from the initial state in one and two principal quantum numbers, respectively. The Hartree-Fock and extended Hartree-Fock approximations to Ee prove to be successive approximations to the component E?‘. EFF , EFHF and Eel can all be readily evaluated directly from their perturbation formulas. Examples of such calculations are given and the results are compared with the results of conventional calculations of the same quantities. The component E?” can be expressed as a linear combination of values of Ez” for two-electron states in a manner first described, schematically, by Bather and Goudsmit in 1934. For two-electron states, the coefficient ET can be evaluated by existing techniques, and from Ez one easily obtains Ez”. By using published variational calculations of Eg for the two-electron states ls2 18, ls2.G and ls2s3S, we evaluate Ee” for the three-electron state 1.~~1~ %‘. The properties of the coefficient E2 , together with considerations based on the screening approximation, enable one to understand why, in low-lying configurations, termenergy patterns appropriate to high degrees of ionization usually persist down to quite low degrees of ionization. First-order calculations of the matrix elements of one-electron operators are shown to correspond in complexity, not to complete second-order energy calculations as one might expect on general perturbation-theoretic grounds, but to calculations of EFHF. They can either be based on perturbation expansions which involve single generalized sums of products of hydrogenic radial integrals, or they can be made to depend on the solution of ordinary, inhomogeneous, second-order differential equations.
OF
PAPERS
179
where the function J(kb) varies from zero at kb = 0 to unity as kb -+ 0~. Hall’s well-known experiments are consistent with this purely hydrodynamic analysis and give a core radius 0 of -4*x, compared with the earlier estimates of -30 -7A. The studies of vortex motions in an atomistic framework are also discussed. Stzcdies of Many-Channel Scattering. II. Connection between Total and Partial Widths. HANS A. WEIIIENM~~LLER, Institut fur Theoretische Physik, Universitat Heidelberg, Heidelberg, Germany. Results that were derived in a previous paper on many-channel scattering are used to investigate the formula which expresses the total width of a resonance as a sum of the partial widths of the open channels. If a resonance is defined by a pole of the S matrix, it is shown that this formula is valid only in the limit of arbitrarily sharp resonances and only if one considers the sum of the absolute values of the partial widths of the open channels (these widths are in general complex). On a Formal Theory of Resonance Reactions. LUCIANO FONDA, Istituto di Fisica dell’lmiversita, Trieste, Italy; and Istituto Nazionale di Fisica Nucleare, Sottosezione di Trieste, Trieste, Italy. A recent theory of resonance reactions is reconsidered for the purpose of deriving from it a complex potential model and the decay law of the compound nuclear states. Second-Order Z-Dependent Theory of Many-Electron Atoms. DAVID LAYZER, ZDENEK HO&K, MARGARET N. LEWIS, AND DANIEL P. THOMPSON, Harvard College Observatory, Cambridge, Massachusetts. A quantitative description of complex term spectra requires accurate knowledge of the three leading coefficients in the Z expansions E = EoZ* + RZ + Ez + E&l + . . of term energies. The screening approximation (described in a previous paper) yields exact values for Eo and El but only estimates of the remaining expansion coefficients. The present paper is concerned with the coefficient Ee , which may be split into two physically and mathematically distinct components, E?’ and E2”. The intermediate states that figure in the perturbation formulas for E2’ and Ez” differ from the initial state in one and two principal quantum numbers, respectively. The Hartree-Fock and extended Hartree-Fock approximations to Ee prove to be successive approximations to the component E?‘. EFF , EFHF and Eel can all be readily evaluated directly from their perturbation formulas. Examples of such calculations are given and the results are compared with the results of conventional calculations of the same quantities. The component E?” can be expressed as a linear combination of values of Ez” for two-electron states in a manner first described, schematically, by Bather and Goudsmit in 1934. For two-electron states, the coefficient ET can be evaluated by existing techniques, and from Ez one easily obtains Ez”. By using published variational calculations of Eg for the two-electron states ls2 18, ls2.G and ls2s3S, we evaluate Ee” for the three-electron state 1.~~1~ %‘. The properties of the coefficient E2 , together with considerations based on the screening approximation, enable one to understand why, in low-lying configurations, termenergy patterns appropriate to high degrees of ionization usually persist down to quite low degrees of ionization. First-order calculations of the matrix elements of one-electron operators are shown to correspond in complexity, not to complete second-order energy calculations as one might expect on general perturbation-theoretic grounds, but to calculations of EFHF. They can either be based on perturbation expansions which involve single generalized sums of products of hydrogenic radial integrals, or they can be made to depend on the solution of ordinary, inhomogeneous, second-order differential equations.