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On the dynamical determination of the regge pole parameters
436 extended theorems forbidden.
ABSTRACTS
OF
PAPERS
TO
to include the main cone at all higher which specify the nature of a boundary No numerical results are included.
APPEAR
IN
FUTURE
latitudes. The results of a class of trajectories,
ISSUES
are embodied in two none of which are
Scattering of High Energy Deuterons by Complex iVuclei. LARRY ZAMICK, Palmer Physical Laboratory, Princeton University, Princeton, New Jersey Making use of the high energy approximation developed for quantum mechanical scattering problems largely by Glauber, the theory of the scattering of high energy deuterons by complex nuclei has been considered. The optical potential describing this process was constructed in terms of the nucleon-nucleon scattering amplitude. This potential contains central, spin-orbit, and tensor terms. A calculation was made of the angular distribution, the vector and tensor polarizations for scattering of 420 Mev deuterons by carbon and compared with the experimental data of J. Button and R. Mermod. Collective OTTO
Aspects THEIMER,
of Bremsstrahlung New Mexico State
Emission University,
and Electrical Conductivity Research Center, University
in
a Plasma. Park, New
Mexico The Bremsstrahlung spectrum and the ac electrical conductivity of a plasma is calculated, by classical methods, from a model in which the electrons interact with the plasma as a whole, and not with individual particles. The Fourier expansion of the particle density is employed for calculating the time dependent electrical field at the position of a moving electron, and the Fourier coefficients of the electrical field are used for calculating the Bremsstrahlung emission and the conductivity. The method clearly exhibits the relation between these two quantities, and gives a fairly rigorous expression for the “cutoff” parameter pma, as a function of the Debye shielding length, the electron mean free path, and the frequency of the electrical field. On the Dynamical Determination of the Regge Pole Parameters. HGNG CHENG, California Institute of Technology, Pasadena, California, AND DAVID SHARP, California Institute of Technology, Pasadena, California and Lawrence Radiation Laboratory, Berkeley, California The purpose of this article is to outline a method, based on the principles of analyticity and unitarity in the t channel, which may help to det,ermine some dynamical properties of the Regge parameters cu(t) and P(t). In the introduction we list various applications of this method, and discuss the role of crossing symmetry and unitarity in all three channels in relation to the uniqueness of the solutions. In Section II we derive the unitarity condition satisfied by the partial wave amplitude A@, t), for 1 complex, for a relativistic, two-body scattering process. Upon neglecting intermediate states of more than two particles, the unitarity condition can be expressed in terms of Regge parameters. An approximate form for the unitarity condition, accurate at low t, is next derived. This form will be used for numerical work. In Section III we show, in the relativistic case, that the functions a(t) and p(t), describing a boson Regge pole, are analytic with only right-hand cuts, in addition to those arising from the crossing of Regge trajectories. Our proof is based on two assumptions: (I) validity of the Mandelstam representation; (2) analyticity of A (1, t) in the whole 1 plane, with at most poles and essential singularities. The consequence of the existence of essential singularities at 1 = -1, -2, -3, . . in relation to a(t) and B(t) is especially discussed. Finally, we note in this section how the preceding results are modified if the Regge pole being considered is a Fermion. In Section IV we write dispersion relations for a(t) and P(t). These, together with the unitarity condition of Section II, constitute a tentative method for the dynamical deter-
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE
ISSUES
437
minat,ion of the Regge parameters. We outline an extension of our method which is appropriate for discussions of Fermion Regge poles. The behavior of a(t) and p(t) at the elastic or inelastic thresholds is derived, and applied to perform subtractions in the dispersion relation for /3(t). Finally, in Section V, we turn specifically to a-a scattering and discuss an approximation which might possibly lead to a reasonably accurate description of this process. Estimates of the range of validity of the approximation are made. Spin Assignments in Even-Even Nuclei between 0’6 and Ca40. C. BROUDE AND H. E. GOVE, Atomic Energy of Canada, Ltd., Chalk River, Ontario, Canada The spins of fourteen levels in even-even nuclei from Ne20 to S32 have been measured together with the multipole mixing of transitions between the first excited state and higher states. The levels were excited by inelastic proton scattering using a proton beam from the Chalk River tandem accelerator. The method employed was to measure both gammagamma coincidence correlations and direct gamma ray angular correlations without observing the inelastically scattered protons. These correlations if done in a variety of different geometrical arrangements of the gamma counters permit unambiguous spin assignments to be made. A description of the computer program employed to make the assignment is included. The results are examined for evidence of any systematic pattern and, for Mgz4, a detailed discussion of the results is given in terms of the collective model. Studies of Polaron Motion Part III: The Hall Mobility of the Small Polaron. L. FRIEDMAN AND T. HOLSTEIN, Department of Physics, University of Pittsburgh, Pittsburgh, Pennsyl vania The one-dimensional molecular crystal model of polaron motion, developed in parts I and II, is suitably generalized to consider the existence of a Hall effect. As in II, the treatment is confined to the case for which the electronic-overlap term of the total Hamiltonian is a small perturbation. In zeroth order-i.e., in the absence of this term-the electron is localized at a given site. The vibrational state of the system is specified by a set of quantum numbers, h’h , giving the degree of excitation of each vibrational mode. The existence of a nonvanishing electronic bandwidth then gives rise t,o transitions to neighboring sites. Of principal interest in the present paper is the high temperature regime (T > Tt , as defined in II) where polaron motion is predominantly by means of random jumps between neighboring sites. Although the lowest order jump rate is adequate in considering the polaron drift mobility, higher order processes, involving the occupation of (at least) three sites, must be taken into account in treating the Hall effect. In particular, it is demonstrated the relative probability of the electron, initially located on one of the three sites, hopping to one or the other of the remaining two sites, is modified by a contribution which, both in sign and magnitude, is linearly proportional to the applied magnetic field. This effect is shown to arise from the interference between the amplitude for the direct transition from the initial to the final site, and the amplitude for an indirect, second order transition, involving intermediate occupancy of the third site. The (magnetic) field induced component.s of the jump rates, corresponding to the above processes, are first calculated by a classical occurrence-probability approach which treats the lattice vibrational coordinates as given functions of time. The second approach presents a full quantum mechanical calculation of the jump rates. The results of this t,reatment agree with those of the occurrence-probability approach in the classical limit (T >> &bye). The order of magnitude of the calculated Hall Coefficients are found to be greater than or comparable to the “normal” result (R = -l/net) depending on whether or not the three sites involved in the transition are mutually nearest neighbors. A final note corrects an error in a previous paper by one of the authors (T. H.) concerning the sign of the Hall effect in impurity conduction.
ABSTRACTS
OF
PAPERS
TO
to include the main cone at all higher which specify the nature of a boundary No numerical results are included.
APPEAR
IN
FUTURE
latitudes. The results of a class of trajectories,
ISSUES
are embodied in two none of which are
Scattering of High Energy Deuterons by Complex iVuclei. LARRY ZAMICK, Palmer Physical Laboratory, Princeton University, Princeton, New Jersey Making use of the high energy approximation developed for quantum mechanical scattering problems largely by Glauber, the theory of the scattering of high energy deuterons by complex nuclei has been considered. The optical potential describing this process was constructed in terms of the nucleon-nucleon scattering amplitude. This potential contains central, spin-orbit, and tensor terms. A calculation was made of the angular distribution, the vector and tensor polarizations for scattering of 420 Mev deuterons by carbon and compared with the experimental data of J. Button and R. Mermod. Collective OTTO
Aspects THEIMER,
of Bremsstrahlung New Mexico State
Emission University,
and Electrical Conductivity Research Center, University
in
a Plasma. Park, New
Mexico The Bremsstrahlung spectrum and the ac electrical conductivity of a plasma is calculated, by classical methods, from a model in which the electrons interact with the plasma as a whole, and not with individual particles. The Fourier expansion of the particle density is employed for calculating the time dependent electrical field at the position of a moving electron, and the Fourier coefficients of the electrical field are used for calculating the Bremsstrahlung emission and the conductivity. The method clearly exhibits the relation between these two quantities, and gives a fairly rigorous expression for the “cutoff” parameter pma, as a function of the Debye shielding length, the electron mean free path, and the frequency of the electrical field. On the Dynamical Determination of the Regge Pole Parameters. HGNG CHENG, California Institute of Technology, Pasadena, California, AND DAVID SHARP, California Institute of Technology, Pasadena, California and Lawrence Radiation Laboratory, Berkeley, California The purpose of this article is to outline a method, based on the principles of analyticity and unitarity in the t channel, which may help to det,ermine some dynamical properties of the Regge parameters cu(t) and P(t). In the introduction we list various applications of this method, and discuss the role of crossing symmetry and unitarity in all three channels in relation to the uniqueness of the solutions. In Section II we derive the unitarity condition satisfied by the partial wave amplitude A@, t), for 1 complex, for a relativistic, two-body scattering process. Upon neglecting intermediate states of more than two particles, the unitarity condition can be expressed in terms of Regge parameters. An approximate form for the unitarity condition, accurate at low t, is next derived. This form will be used for numerical work. In Section III we show, in the relativistic case, that the functions a(t) and p(t), describing a boson Regge pole, are analytic with only right-hand cuts, in addition to those arising from the crossing of Regge trajectories. Our proof is based on two assumptions: (I) validity of the Mandelstam representation; (2) analyticity of A (1, t) in the whole 1 plane, with at most poles and essential singularities. The consequence of the existence of essential singularities at 1 = -1, -2, -3, . . in relation to a(t) and B(t) is especially discussed. Finally, we note in this section how the preceding results are modified if the Regge pole being considered is a Fermion. In Section IV we write dispersion relations for a(t) and P(t). These, together with the unitarity condition of Section II, constitute a tentative method for the dynamical deter-
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE
ISSUES
437
minat,ion of the Regge parameters. We outline an extension of our method which is appropriate for discussions of Fermion Regge poles. The behavior of a(t) and p(t) at the elastic or inelastic thresholds is derived, and applied to perform subtractions in the dispersion relation for /3(t). Finally, in Section V, we turn specifically to a-a scattering and discuss an approximation which might possibly lead to a reasonably accurate description of this process. Estimates of the range of validity of the approximation are made. Spin Assignments in Even-Even Nuclei between 0’6 and Ca40. C. BROUDE AND H. E. GOVE, Atomic Energy of Canada, Ltd., Chalk River, Ontario, Canada The spins of fourteen levels in even-even nuclei from Ne20 to S32 have been measured together with the multipole mixing of transitions between the first excited state and higher states. The levels were excited by inelastic proton scattering using a proton beam from the Chalk River tandem accelerator. The method employed was to measure both gammagamma coincidence correlations and direct gamma ray angular correlations without observing the inelastically scattered protons. These correlations if done in a variety of different geometrical arrangements of the gamma counters permit unambiguous spin assignments to be made. A description of the computer program employed to make the assignment is included. The results are examined for evidence of any systematic pattern and, for Mgz4, a detailed discussion of the results is given in terms of the collective model. Studies of Polaron Motion Part III: The Hall Mobility of the Small Polaron. L. FRIEDMAN AND T. HOLSTEIN, Department of Physics, University of Pittsburgh, Pittsburgh, Pennsyl vania The one-dimensional molecular crystal model of polaron motion, developed in parts I and II, is suitably generalized to consider the existence of a Hall effect. As in II, the treatment is confined to the case for which the electronic-overlap term of the total Hamiltonian is a small perturbation. In zeroth order-i.e., in the absence of this term-the electron is localized at a given site. The vibrational state of the system is specified by a set of quantum numbers, h’h , giving the degree of excitation of each vibrational mode. The existence of a nonvanishing electronic bandwidth then gives rise t,o transitions to neighboring sites. Of principal interest in the present paper is the high temperature regime (T > Tt , as defined in II) where polaron motion is predominantly by means of random jumps between neighboring sites. Although the lowest order jump rate is adequate in considering the polaron drift mobility, higher order processes, involving the occupation of (at least) three sites, must be taken into account in treating the Hall effect. In particular, it is demonstrated the relative probability of the electron, initially located on one of the three sites, hopping to one or the other of the remaining two sites, is modified by a contribution which, both in sign and magnitude, is linearly proportional to the applied magnetic field. This effect is shown to arise from the interference between the amplitude for the direct transition from the initial to the final site, and the amplitude for an indirect, second order transition, involving intermediate occupancy of the third site. The (magnetic) field induced component.s of the jump rates, corresponding to the above processes, are first calculated by a classical occurrence-probability approach which treats the lattice vibrational coordinates as given functions of time. The second approach presents a full quantum mechanical calculation of the jump rates. The results of this t,reatment agree with those of the occurrence-probability approach in the classical limit (T >> &bye). The order of magnitude of the calculated Hall Coefficients are found to be greater than or comparable to the “normal” result (R = -l/net) depending on whether or not the three sites involved in the transition are mutually nearest neighbors. A final note corrects an error in a previous paper by one of the authors (T. H.) concerning the sign of the Hall effect in impurity conduction.