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Broken symmetries and leptonic weak interactions
364 method. The half widths tions are greatly reduced mainly due to the Fermi
ABSTRACTS
of the yc (Lorentz factor of the center of mass system) and appear to be consistent with the assumption that momentum of the target nucleon in the target nuclei.
distributhey are
A Model of the Insulator-Metal Transition. II. Quasi-Chemical Equilibrium Theory. G. KEMENY, Ledgemont Laboratory, Kennecott Copper Corporation, Lexington 73, Massachusetts. A simple cubic hydrogenic lattice in the extreme tight binding approximation with only one energy band has been set up as a model for an insulator in Part I of this paper. Bound states of electrons and holes are combined using the quasi-chemical equilibrium theory to construct the electron structure of the crystal. At a particular value of a characteristic parameter all bound pairs dissociate simultaneously and a metal is obtained. The transition between the insulating and metallic ranges is sharp confirming a conjecture by Mott. 1Vuclear Alignment of Co60 in Cerium Magnesium Nitrate. FERNANDO CARBONI AND RICHARD C. SAPP, Department of Physics and Astronomy, The University of Kansas, Lawrence, Kansas. For a spherical single crystal of Ce-Mg nitrate containing Co60 cooled by adiabatic demagnetization from 24 kOe/” K, the gamma ray anisotropy E is quite low (7%) compared to that for isomorphous Ce-Zn nitrate under comparable conditions (26%). In order to understand this apparently paradoxical behavior we have carried through a calculation off in which cerium-cobalt spin-spin interactions are represented by a local magnetic field Hi A digital computer diagonalized the 22 X 22 matrix of the spin Hamiltonian >Sgl/3 Hi + AI,& + $@(Z+S+ I-S+) with magnetic resonance parameters appropriate for X-sites in the crystal lattice, giving the exact eingenvalues and eingenfunctions from which the angular distribution of gamma radiation has been calculated. An agreement within 2% can be obtained with $6 of the Co60 nuclei in X-sites and >$ in Y-sites if Hi = 165 Oe, as in Ce-Zn nitrate, and if l/T = 950 rather than 325, in accordance with recent work on the Ce-Mg nitrate temperature scale for a sphere. Possible sources of the remaining discrepancy are discussed. Unitarity and Partial Wave Amplitudes. PAUL B. KANTOR, Brookhaven National Laboratory, Upton, New York. In the last few years considerable effort has been invested in the single boson exchange approach to the nucleon-nucleon interaction. A crucial problem in this scheme is the calculation of amplitudes satisfying unitarity from nonunitary Born terms, and various methods have been applied. The present work is a comparison of these methods based on applying them to some simplified problems with no inelastic effects and numerically evaluating the resulting expressions for k cot 6. We consider S wave scattering from an exponential and a separable Yukawa potential and compare the following methods: (1) N/D; (2) Determinantal Approximation; (3) Schriidinger equation; (4) “Heitler damping” or “Kmatrix” method. We find that these methods do not, in general, agree at all well. We are also able to draw some conclusions about the dependence of solutions (1)) (2)) and (4) on remote singularities and find that it is not small. The significance of these results is discussed. Broken Symmetries and Leptonic Weak Interactions. stitute for Nuclear Studies and the Department Chicago, Illinois. The currents involved in lepton-lepton couplings
REINHARD OEHME, Enrico of Physics, University generate
the algebra
Fermi Inof Chicago,
of U(4).
Corre-
365
ABSTRACTS
sponding U(4)algebras can be constructed from the hadron currents; they may be considered as subalgebras of higher symmetries like U(6) X U(6), etc. The U(4)-algebras are used for the construction of two models of a universal weak coupling. One model is based upon the U(4)-algebra generated by the total hadron current involved in semileptonic interactions, and a second one is obtained by considering strangeness-changing and nonchanging currents separately. The first model involves the Cabibbo angle as a parameter and it requires this angle to be the same for all currents. The second model has no such parameter; it depends upon a possible larger manifestation of the symmetry breaking interactions to account for the damping of AS = 1 transitions. In both cases the neutral generators of U(4) are identified with the (1 + 76).projections of electromagnetic currents and of lepton number or triplet-number currents. For triplets with charges z + 1, z, z, the first model gives preference to z = 0 and the second one to z = -1. The two models are discussed briefly in connection with the phenomenological situation and with the assumption of broken SU(3)-symmetry for the hadrons. Renormalization effects are considered in detail, especially in view of the damping factor required in the second model mentioned above. The commutation relations of the hadron currents, together with the hypothesis of partially conserved axial-vector currents, give a reasonable universal damping factor XA - m,/mK for axial vector matrix elements. For vector currents one obtains an analogous result only under rather restrictive assumptions. Otherwise, commutation relations, partially conserved currents, and XV - rn*/rnK are not consistent as long as other manifestations of the symmetry breaker are considered to be small. Calculation of Inelastic Scattering in Terms of Elastic Scattering. N. AUSTERN, University of Pittsburgh, Pittsburgh, Pennsylvania, AND J. S. BLAIR, University of Washington, Seattle, Washington. It is shown that the adiabatic distorted-waves theory of inelastic scattering, under suitable approximations, can be re-expressed in terms of derivatives of elastic scattering amplitudes. The methods are very general, and have the interesting aspect that they display relations which are directly among scattering amplitudes, so that the role of the optical potential is minimized. Results are obtained for all orders of the inelastic coupling constant, hence for single excitation (normal inelastic scattering), for double excitation, and for the reaction of inelastic coupling on the elastic scattering. The theory easily takes the form of a generalized “diffraction model,” and yields information about the reliability of such models. The older, Fraunhofer model of Blair is obtained as a special case. Scattering of Hf by H. MARGARET J. FULTON AND MARVIN H. MITTLEMAN, University of California, Lawrence Radiation Laboratory, Livermore, California. An attempt is made to describe the H+ on H collision in the intermediate energy range where neither the high energy nor the low energy theories are expected to work. A wave function that gives good results at both high and low energies is used in a variational expression to obtain intermediate energy results. This problem is considered as a prototype of ion-parent atom collisions where in general molecular wave functions are not available. Hence only atomic wave functions are used throughout. Results are presented for exchange to the ground state, total exchange, total ionization, and total non-exchange excitation. An attempt is also made to describe Everhart’s experiment. A “damping” of the resonances is obtained with some improvement over previous results. Minimum Size Department,
of Dense Source Distributions Northeastern University,
in General Relativity. Boston, Massachusetts,
R. ARNOWITT, C. W. MISNER,
Physics Depart-
ABSTRACTS
of the yc (Lorentz factor of the center of mass system) and appear to be consistent with the assumption that momentum of the target nucleon in the target nuclei.
distributhey are
A Model of the Insulator-Metal Transition. II. Quasi-Chemical Equilibrium Theory. G. KEMENY, Ledgemont Laboratory, Kennecott Copper Corporation, Lexington 73, Massachusetts. A simple cubic hydrogenic lattice in the extreme tight binding approximation with only one energy band has been set up as a model for an insulator in Part I of this paper. Bound states of electrons and holes are combined using the quasi-chemical equilibrium theory to construct the electron structure of the crystal. At a particular value of a characteristic parameter all bound pairs dissociate simultaneously and a metal is obtained. The transition between the insulating and metallic ranges is sharp confirming a conjecture by Mott. 1Vuclear Alignment of Co60 in Cerium Magnesium Nitrate. FERNANDO CARBONI AND RICHARD C. SAPP, Department of Physics and Astronomy, The University of Kansas, Lawrence, Kansas. For a spherical single crystal of Ce-Mg nitrate containing Co60 cooled by adiabatic demagnetization from 24 kOe/” K, the gamma ray anisotropy E is quite low (7%) compared to that for isomorphous Ce-Zn nitrate under comparable conditions (26%). In order to understand this apparently paradoxical behavior we have carried through a calculation off in which cerium-cobalt spin-spin interactions are represented by a local magnetic field Hi A digital computer diagonalized the 22 X 22 matrix of the spin Hamiltonian >Sgl/3 Hi + AI,& + $@(Z+S+ I-S+) with magnetic resonance parameters appropriate for X-sites in the crystal lattice, giving the exact eingenvalues and eingenfunctions from which the angular distribution of gamma radiation has been calculated. An agreement within 2% can be obtained with $6 of the Co60 nuclei in X-sites and >$ in Y-sites if Hi = 165 Oe, as in Ce-Zn nitrate, and if l/T = 950 rather than 325, in accordance with recent work on the Ce-Mg nitrate temperature scale for a sphere. Possible sources of the remaining discrepancy are discussed. Unitarity and Partial Wave Amplitudes. PAUL B. KANTOR, Brookhaven National Laboratory, Upton, New York. In the last few years considerable effort has been invested in the single boson exchange approach to the nucleon-nucleon interaction. A crucial problem in this scheme is the calculation of amplitudes satisfying unitarity from nonunitary Born terms, and various methods have been applied. The present work is a comparison of these methods based on applying them to some simplified problems with no inelastic effects and numerically evaluating the resulting expressions for k cot 6. We consider S wave scattering from an exponential and a separable Yukawa potential and compare the following methods: (1) N/D; (2) Determinantal Approximation; (3) Schriidinger equation; (4) “Heitler damping” or “Kmatrix” method. We find that these methods do not, in general, agree at all well. We are also able to draw some conclusions about the dependence of solutions (1)) (2)) and (4) on remote singularities and find that it is not small. The significance of these results is discussed. Broken Symmetries and Leptonic Weak Interactions. stitute for Nuclear Studies and the Department Chicago, Illinois. The currents involved in lepton-lepton couplings
REINHARD OEHME, Enrico of Physics, University generate
the algebra
Fermi Inof Chicago,
of U(4).
Corre-
365
ABSTRACTS
sponding U(4)algebras can be constructed from the hadron currents; they may be considered as subalgebras of higher symmetries like U(6) X U(6), etc. The U(4)-algebras are used for the construction of two models of a universal weak coupling. One model is based upon the U(4)-algebra generated by the total hadron current involved in semileptonic interactions, and a second one is obtained by considering strangeness-changing and nonchanging currents separately. The first model involves the Cabibbo angle as a parameter and it requires this angle to be the same for all currents. The second model has no such parameter; it depends upon a possible larger manifestation of the symmetry breaking interactions to account for the damping of AS = 1 transitions. In both cases the neutral generators of U(4) are identified with the (1 + 76).projections of electromagnetic currents and of lepton number or triplet-number currents. For triplets with charges z + 1, z, z, the first model gives preference to z = 0 and the second one to z = -1. The two models are discussed briefly in connection with the phenomenological situation and with the assumption of broken SU(3)-symmetry for the hadrons. Renormalization effects are considered in detail, especially in view of the damping factor required in the second model mentioned above. The commutation relations of the hadron currents, together with the hypothesis of partially conserved axial-vector currents, give a reasonable universal damping factor XA - m,/mK for axial vector matrix elements. For vector currents one obtains an analogous result only under rather restrictive assumptions. Otherwise, commutation relations, partially conserved currents, and XV - rn*/rnK are not consistent as long as other manifestations of the symmetry breaker are considered to be small. Calculation of Inelastic Scattering in Terms of Elastic Scattering. N. AUSTERN, University of Pittsburgh, Pittsburgh, Pennsylvania, AND J. S. BLAIR, University of Washington, Seattle, Washington. It is shown that the adiabatic distorted-waves theory of inelastic scattering, under suitable approximations, can be re-expressed in terms of derivatives of elastic scattering amplitudes. The methods are very general, and have the interesting aspect that they display relations which are directly among scattering amplitudes, so that the role of the optical potential is minimized. Results are obtained for all orders of the inelastic coupling constant, hence for single excitation (normal inelastic scattering), for double excitation, and for the reaction of inelastic coupling on the elastic scattering. The theory easily takes the form of a generalized “diffraction model,” and yields information about the reliability of such models. The older, Fraunhofer model of Blair is obtained as a special case. Scattering of Hf by H. MARGARET J. FULTON AND MARVIN H. MITTLEMAN, University of California, Lawrence Radiation Laboratory, Livermore, California. An attempt is made to describe the H+ on H collision in the intermediate energy range where neither the high energy nor the low energy theories are expected to work. A wave function that gives good results at both high and low energies is used in a variational expression to obtain intermediate energy results. This problem is considered as a prototype of ion-parent atom collisions where in general molecular wave functions are not available. Hence only atomic wave functions are used throughout. Results are presented for exchange to the ground state, total exchange, total ionization, and total non-exchange excitation. An attempt is also made to describe Everhart’s experiment. A “damping” of the resonances is obtained with some improvement over previous results. Minimum Size Department,
of Dense Source Distributions Northeastern University,
in General Relativity. Boston, Massachusetts,
R. ARNOWITT, C. W. MISNER,
Physics Depart-