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Unsteady solutions of kinetic models with velocity dependent collision frequency
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incoherent (quasi-elastic) scattering. Good agreement with all the data is obtained for a consistent set of parameters. The main result of this investigation is conclusive evidence for a sizable negative real part of the coherent scattering amplitude. In high-energy approximation this corresponds to a real part of the central proton-nucleus potential which is repulsive in the nuclear surface. The Normal Modes of Oscillation of a Uniformly Charged Drop about Its Saddle-Point Shape. JAMES RAYFORD NIX, Lawrence Radiation Laboratory, University of California, Berkeley, California. We consider the question of the normal modes of oscillation of an idealized uniformly charged axially symmetric liquid drop about its saddle-point shape and calculate the normal modes and their frequencies as functions of the fissility parameter 2. Both expansions to first order in 1 - z and formulas appropriate for numerical evaluations are derived. For the range 0.7 s z s 1 .O numerical results for the four lowest symmetric and the four lowest asymmetric modes are tabulated at intervals of 0.02 in z for the frequencies, normal-coordinate stiffness and inertia constants, and eigenvectors (with respect to the coefficients an in an expansion of the drop’s radius vector in Legendre polynomials). For 0.3 s z r 0.7 the results obtained are of limited accuracy, and in this range only graphs of the frequencies of the three lowest symmetric modes are included. Some applications of the results are discussed. These include the transition-state energy levels of collective oscillations, the probability distributions for t,he saddle-point states of motion, and the penetration of the fission barrier. The formula for the penetrability of a cubic barrier is derived by use of the WKB approximation. The calculated (purely imaginary) frequency for motion in the fission direction, which affects the penetrability of the fission barrier, is compared as a function of z with existing experimental data on fission widths, spontaneous-fission lifetimes and the variation of fission cross sections with excitation energy. The comparison, which is made without the use of any adjustable parameters, indicates that the calculations are capable of reproducing the correct order of magnitude of the fission-direction frequency. Most of the data are at present not sufficiently accurate to provide a sensitive test of the theory, but the points derived from spontaneous-fission lifetimes suggest that possibly 2.8 times as much mass is displaced in the fission mode as would correspond to irrotational motion. On the Mechanics of Radiation. L. INFELD AND R. MICHALSKA-TRAUTMAN, Southwest Center for Advanced Studies, Dallas, Texas. A general scheme is formed to find the relation between the energy per unit time lost by a radiating particle and that spreading through space. This scheme is applied to the gravitational radiation of an electric charge moving in a uniform magnetic field. Under certain restricting conditions this radiation turns out to be of the eighth order. Lrnsteady Solutions of Kinetic Models with Velocity Dependent Collision Frequency. CARLO CERCIGNANI. Applicazioni e Ricerche Scientifiche, Milano, Italy. The method of elementa,ry solutions recently extended to treat steady problems with kinetic models with velocity-dependent collision frequency is now extended to cover also time-dependent problems. The theory is somewhat different from the previous one holding for the Bhatnagar, Gross, and Krook model, since the continuous spectrum of space transients covers now a two-dimensional region of the complex plane. Consequently in order to solve explicitly
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half-space problems we are not faced with one-dimensional singular integral equations but two-dimensional equat,ions with integrable kernel. This circumstance does not make possible to use the Muskhelishvili techniques which were used in previous papers. However, a connection of t,he present theory with the theory of generalized analytic functions is shown, and, on this basis, an analytic method of solving the relevant equations is constructed. The present, theory would be easily applied to typical time-dependent problems, such as t,he Rayleigh problem, the oscillating wall, and the evolution of an initial discontinuity. Theory of the Condensation Point. J.IMES S. L.INGER, Department of Physics, Carnegie Institute of Technology, Pittsburgh, Pennsylvania. This paper is a report of some studies leading to a new mathemat,ical description of the condensation point for a simple class of models of first-order phase transitions. The paper consists of three main parts. In the first part it is pointed out that, althollgh the conventional droplet model of condensation predicts that the free energy has an essential singularit,y at the condensation point, this singularity is so weak as to be experimentally unobservable. Furt,hermore, the analytic continuation of the free energy beyond t,he singularit,y describes a met,astable phase according to the assumptions of the model. The second part of the paper is devoted to the study of a soluble functional integral which exhibits an essential singularity similar to that found in the droplet model. A met,hod is developed for completing the singular properties of such integrals in cases where it is not possihle to evaluate the integrals exactly. In the third part of the paper this method is applied to a simple model of a ferromagnet at temperatllres well below the Curie point. Most of t,he really characteristic features of the droplet model are recovered in this calculation. The detailed results have a bearing on problems of phase coexistence, surface energies, and possibly even condensation rates.
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE
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incoherent (quasi-elastic) scattering. Good agreement with all the data is obtained for a consistent set of parameters. The main result of this investigation is conclusive evidence for a sizable negative real part of the coherent scattering amplitude. In high-energy approximation this corresponds to a real part of the central proton-nucleus potential which is repulsive in the nuclear surface. The Normal Modes of Oscillation of a Uniformly Charged Drop about Its Saddle-Point Shape. JAMES RAYFORD NIX, Lawrence Radiation Laboratory, University of California, Berkeley, California. We consider the question of the normal modes of oscillation of an idealized uniformly charged axially symmetric liquid drop about its saddle-point shape and calculate the normal modes and their frequencies as functions of the fissility parameter 2. Both expansions to first order in 1 - z and formulas appropriate for numerical evaluations are derived. For the range 0.7 s z s 1 .O numerical results for the four lowest symmetric and the four lowest asymmetric modes are tabulated at intervals of 0.02 in z for the frequencies, normal-coordinate stiffness and inertia constants, and eigenvectors (with respect to the coefficients an in an expansion of the drop’s radius vector in Legendre polynomials). For 0.3 s z r 0.7 the results obtained are of limited accuracy, and in this range only graphs of the frequencies of the three lowest symmetric modes are included. Some applications of the results are discussed. These include the transition-state energy levels of collective oscillations, the probability distributions for t,he saddle-point states of motion, and the penetration of the fission barrier. The formula for the penetrability of a cubic barrier is derived by use of the WKB approximation. The calculated (purely imaginary) frequency for motion in the fission direction, which affects the penetrability of the fission barrier, is compared as a function of z with existing experimental data on fission widths, spontaneous-fission lifetimes and the variation of fission cross sections with excitation energy. The comparison, which is made without the use of any adjustable parameters, indicates that the calculations are capable of reproducing the correct order of magnitude of the fission-direction frequency. Most of the data are at present not sufficiently accurate to provide a sensitive test of the theory, but the points derived from spontaneous-fission lifetimes suggest that possibly 2.8 times as much mass is displaced in the fission mode as would correspond to irrotational motion. On the Mechanics of Radiation. L. INFELD AND R. MICHALSKA-TRAUTMAN, Southwest Center for Advanced Studies, Dallas, Texas. A general scheme is formed to find the relation between the energy per unit time lost by a radiating particle and that spreading through space. This scheme is applied to the gravitational radiation of an electric charge moving in a uniform magnetic field. Under certain restricting conditions this radiation turns out to be of the eighth order. Lrnsteady Solutions of Kinetic Models with Velocity Dependent Collision Frequency. CARLO CERCIGNANI. Applicazioni e Ricerche Scientifiche, Milano, Italy. The method of elementa,ry solutions recently extended to treat steady problems with kinetic models with velocity-dependent collision frequency is now extended to cover also time-dependent problems. The theory is somewhat different from the previous one holding for the Bhatnagar, Gross, and Krook model, since the continuous spectrum of space transients covers now a two-dimensional region of the complex plane. Consequently in order to solve explicitly
ABSTRACTS
OF PAPERS
TO APPEAR
IN
FUTURE
ISSUES
379
half-space problems we are not faced with one-dimensional singular integral equations but two-dimensional equat,ions with integrable kernel. This circumstance does not make possible to use the Muskhelishvili techniques which were used in previous papers. However, a connection of t,he present theory with the theory of generalized analytic functions is shown, and, on this basis, an analytic method of solving the relevant equations is constructed. The present, theory would be easily applied to typical time-dependent problems, such as t,he Rayleigh problem, the oscillating wall, and the evolution of an initial discontinuity. Theory of the Condensation Point. J.IMES S. L.INGER, Department of Physics, Carnegie Institute of Technology, Pittsburgh, Pennsylvania. This paper is a report of some studies leading to a new mathemat,ical description of the condensation point for a simple class of models of first-order phase transitions. The paper consists of three main parts. In the first part it is pointed out that, althollgh the conventional droplet model of condensation predicts that the free energy has an essential singularit,y at the condensation point, this singularity is so weak as to be experimentally unobservable. Furt,hermore, the analytic continuation of the free energy beyond t,he singularit,y describes a met,astable phase according to the assumptions of the model. The second part of the paper is devoted to the study of a soluble functional integral which exhibits an essential singularity similar to that found in the droplet model. A met,hod is developed for completing the singular properties of such integrals in cases where it is not possihle to evaluate the integrals exactly. In the third part of the paper this method is applied to a simple model of a ferromagnet at temperatllres well below the Curie point. Most of t,he really characteristic features of the droplet model are recovered in this calculation. The detailed results have a bearing on problems of phase coexistence, surface energies, and possibly even condensation rates.