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Distribution function of magnetically confined electrons in a scattering atmosphere
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTI’RE
311
ISSGES
relations governing the transfer of energy in t,his system. Genera1 forms for these relations may be established by modifications of the theorem of t,he Poincarh invariant. These theorems enable one to associate a conservation relation with a closed (cyclic) family of dynamical systems; such a family may be formed by varying any one of the phase parameters which one may associate w-ith t,he fundamental frequencies. The relations are first set up in a form appropriate to systems with a finite number of degrees of freedom. When applied to a purely reactive electrical net,work, these relations reduce to those derived by Manley and Rowe. The relations are next est,ablished in a form appropriate to the study of nonlinear fields. The general relations are specialized for the following examples: nonlinear electromagnetic media; one-dimensional electron beam wit.h electrostatic field; and unrest.ricted electron flow wit,h electromagnetic field. These esamples reproduce and extend relations established by Haus and (Frau. in the ;1 = 4 System. ZZZ; 7’(p, n). J.~.\IEs IC. YOUNG :tnd PAI.L R. STEIN An analysis is presented of t’he charge exchange reaction T(p, n). The model used is basically that proposed by Selove, a direct interaction model. A cluster model (deuteron plus neutron) representation of the three-body ground state is assumed. Further, the relevant interactions, knock-on and core pickup, hetween proton and target are characterized through t,he Yamaguchi separable t-matrix, a non-local operator. The impulse approximation (t in the medium equals t for free scattering) is introduced. Certain other approsimations having t,o do with the nuclear form factors and the smallness of momenta components of the hound neutron are also employed. Calculat,ions are presented in which a romparison is made with t,he differential cross sections at 1.75, 3.0, and 5.5 Mev. The observed dependence of back to forward scatt,ering upon energy is well represented hy the theor>-. Reactions
A variational method is developed for calculating the thermodynamic potential of quan tun-mechanical many-body systems with pair-wise interactions. The method is based on Peierls’ theorem and yields an upper bound to the thermodynamic potential density in the limit of an infinite system; this hound can be minimized by variation of functions determining the properties of elementary escitations of the system. It is necessary to treat the diagonal part of the interaction between elementary escit,ations exactly in order to obtain an upper bound to the thermodynamic potential. This is done hy summing those BlorhdeDominicis graphs which contribute in the limit of an infinite system, thereby defining, by a linear integral equation, a temperature-dependent “screened” interartion between elementary excitations. The theory is illustrated by application to the BCS model of superconductivity. The results agree with those of BCS both at zero temperature and at, the transition, but. give small but nonvanishing corrections to t,he BCS results at. intermediat.e temperatures due to effects of interactions of element,ary excitations. In this connection, a weak point in the Bogolubov-Zubarev-Tserkovnikov“proof” of the exactness of the BCS thermodynamics is discussed. ZAstribution MACDONALD
Function
of Magnetically
Corlfined
Electrons
in
a Scattering
dlmosphe,x.
and 111. WALT A system of electrons confined in an inhomogeneous magnetic field in the nonuniform atmosphere of scattering particles is described at any instant by t,ion function appropriate to the situation in which no atmosphere is present. tion of the Fokker-Planck equation is then derived for the evolution in time of
W. M.
presence of a the distribu A generalizathe distribu-
Sl:!
ABSTHSCTS
OF
PAPERS
TO
tion function through these quasi-equilibrium this equation is given for the case of electrons solrltion yields the distribution of the electrons
rZPPEAlt
IN
FCTUltE
ISSUES
distributions. An approximate solution of trapped in the earth’s magnetic field. This in energy and angle as a function of time.
Precision Measurement oJ L S-Ray Wavelengths and Line IVidths ji)foT 74 5 Z 5 95 and Their InferpretatiorL in Terms on Nuclear Perturbations. JOHN Jay MERRILL and J. W. M. k-MO,uD Techniques for the precision measurement of S-ray lines using the two-crystal spect,rometer are discussed, and then applied t,o measurement of the L S-ray spectra of the transuranic elements uranium, neptunium, plutonium and americium. In all, 52 emission lines and 1 LI~I absorption edges were measured, all of them with higher precision than has heretofore been obtained and many of them for the first time. Using these dat,a, the binding energies of the electrons for t)hese atoms were computed. The LII-/,I II level splitting was romputed and compared with t,heory and it was found that current theory is not sufficient, to acrount quantitatively for the observed data. Certain features of the emission line widths are discussed and given qualit,ative explanations in terms of the Coster-Kronig transition LI~-LII~!Mv and hyperfine structure. The observed hyperfine structure is due widths of certain of the to the large magnetic moment, of Npza7 which leads to increased Np lines. It is helieved that this is the first experimental ohservat,ion of hyperfine structure in S-ray spectra. The Scattering of Electrons by Atomic Hydrogen. MARVIN H. MITTLE~MAS A method is presented which reduces the problem to the scattering of a single electron in an equivalent potential. The complexities of the electron-electron interaction are embodied in this potential. An exact expression is given for the potential, and various approximation methods are discussed. The adiabatic approximat,ion is investigated and the imaginary part of the potent,ial is examined. Comparison of Two Models for the Many Fermion System. E. M. HEKLEY and L. WILETS A critical comparison is made between the Overhauser and Bardeen-Cooper-Schrieffer t,rial wave functions for attract,ive fermion systems employing variational techniques. Firstly, a two-particle separahle int,eraction, constant over a thin shell near the Fermi surface is considered in one, two or three dimensions. The exact Overhauser ground states energies always lie higher than those of BCS. Furthermore, as pointed out by Kohn and Nettel, a finite interaction strength is required in two or more dimensions before the Overhauser wave function gives any improvement over plane waves, hecause of t,he necessit,y of distorting t,he Fermi surface. We find the critical value of t)he strength paramet,er t,o he greater than unity. Secondly, for a nonseparable one-dimensional Gaussian potential, t,he BCS equations are solved exactly in strong and weak coupling and approximately for intermedint,e values. The BCS ground state energy is always lower. Finally, for the one dimensional shell interaction, a trial wave function is constructed which pairs time-reversed Overhauser states in the BCS spirit. Such a trial function is inferior to one composed of plane wave BCR pairs. Thus the BCS correlations appear to destroy those of Overhauser, and we conclude t,hat the Overhauser representation is not necessarily a better starting point, than plane waves for improved calculations in the many fermion system. The Guiding (‘enter Approximation to Charged Particle Motion. THEODORE G. NORTHROP The equations governing the guiding center motion of a charged particle in an electromagnetic field are obtained simultaneously and deductively, without considering indi-
OF
PAPERS
TO
APPEAR
IN
FUTI’RE
311
ISSGES
relations governing the transfer of energy in t,his system. Genera1 forms for these relations may be established by modifications of the theorem of t,he Poincarh invariant. These theorems enable one to associate a conservation relation with a closed (cyclic) family of dynamical systems; such a family may be formed by varying any one of the phase parameters which one may associate w-ith t,he fundamental frequencies. The relations are first set up in a form appropriate to systems with a finite number of degrees of freedom. When applied to a purely reactive electrical net,work, these relations reduce to those derived by Manley and Rowe. The relations are next est,ablished in a form appropriate to the study of nonlinear fields. The general relations are specialized for the following examples: nonlinear electromagnetic media; one-dimensional electron beam wit.h electrostatic field; and unrest.ricted electron flow wit,h electromagnetic field. These esamples reproduce and extend relations established by Haus and (Frau. in the ;1 = 4 System. ZZZ; 7’(p, n). J.~.\IEs IC. YOUNG :tnd PAI.L R. STEIN An analysis is presented of t’he charge exchange reaction T(p, n). The model used is basically that proposed by Selove, a direct interaction model. A cluster model (deuteron plus neutron) representation of the three-body ground state is assumed. Further, the relevant interactions, knock-on and core pickup, hetween proton and target are characterized through t,he Yamaguchi separable t-matrix, a non-local operator. The impulse approximation (t in the medium equals t for free scattering) is introduced. Certain other approsimations having t,o do with the nuclear form factors and the smallness of momenta components of the hound neutron are also employed. Calculat,ions are presented in which a romparison is made with t,he differential cross sections at 1.75, 3.0, and 5.5 Mev. The observed dependence of back to forward scatt,ering upon energy is well represented hy the theor>-. Reactions
A variational method is developed for calculating the thermodynamic potential of quan tun-mechanical many-body systems with pair-wise interactions. The method is based on Peierls’ theorem and yields an upper bound to the thermodynamic potential density in the limit of an infinite system; this hound can be minimized by variation of functions determining the properties of elementary escitations of the system. It is necessary to treat the diagonal part of the interaction between elementary escit,ations exactly in order to obtain an upper bound to the thermodynamic potential. This is done hy summing those BlorhdeDominicis graphs which contribute in the limit of an infinite system, thereby defining, by a linear integral equation, a temperature-dependent “screened” interartion between elementary excitations. The theory is illustrated by application to the BCS model of superconductivity. The results agree with those of BCS both at zero temperature and at, the transition, but. give small but nonvanishing corrections to t,he BCS results at. intermediat.e temperatures due to effects of interactions of element,ary excitations. In this connection, a weak point in the Bogolubov-Zubarev-Tserkovnikov“proof” of the exactness of the BCS thermodynamics is discussed. ZAstribution MACDONALD
Function
of Magnetically
Corlfined
Electrons
in
a Scattering
dlmosphe,x.
and 111. WALT A system of electrons confined in an inhomogeneous magnetic field in the nonuniform atmosphere of scattering particles is described at any instant by t,ion function appropriate to the situation in which no atmosphere is present. tion of the Fokker-Planck equation is then derived for the evolution in time of
W. M.
presence of a the distribu A generalizathe distribu-
Sl:!
ABSTHSCTS
OF
PAPERS
TO
tion function through these quasi-equilibrium this equation is given for the case of electrons solrltion yields the distribution of the electrons
rZPPEAlt
IN
FCTUltE
ISSUES
distributions. An approximate solution of trapped in the earth’s magnetic field. This in energy and angle as a function of time.
Precision Measurement oJ L S-Ray Wavelengths and Line IVidths ji)foT 74 5 Z 5 95 and Their InferpretatiorL in Terms on Nuclear Perturbations. JOHN Jay MERRILL and J. W. M. k-MO,uD Techniques for the precision measurement of S-ray lines using the two-crystal spect,rometer are discussed, and then applied t,o measurement of the L S-ray spectra of the transuranic elements uranium, neptunium, plutonium and americium. In all, 52 emission lines and 1 LI~I absorption edges were measured, all of them with higher precision than has heretofore been obtained and many of them for the first time. Using these dat,a, the binding energies of the electrons for t)hese atoms were computed. The LII-/,I II level splitting was romputed and compared with t,heory and it was found that current theory is not sufficient, to acrount quantitatively for the observed data. Certain features of the emission line widths are discussed and given qualit,ative explanations in terms of the Coster-Kronig transition LI~-LII~!Mv and hyperfine structure. The observed hyperfine structure is due widths of certain of the to the large magnetic moment, of Npza7 which leads to increased Np lines. It is helieved that this is the first experimental ohservat,ion of hyperfine structure in S-ray spectra. The Scattering of Electrons by Atomic Hydrogen. MARVIN H. MITTLE~MAS A method is presented which reduces the problem to the scattering of a single electron in an equivalent potential. The complexities of the electron-electron interaction are embodied in this potential. An exact expression is given for the potential, and various approximation methods are discussed. The adiabatic approximat,ion is investigated and the imaginary part of the potent,ial is examined. Comparison of Two Models for the Many Fermion System. E. M. HEKLEY and L. WILETS A critical comparison is made between the Overhauser and Bardeen-Cooper-Schrieffer t,rial wave functions for attract,ive fermion systems employing variational techniques. Firstly, a two-particle separahle int,eraction, constant over a thin shell near the Fermi surface is considered in one, two or three dimensions. The exact Overhauser ground states energies always lie higher than those of BCS. Furthermore, as pointed out by Kohn and Nettel, a finite interaction strength is required in two or more dimensions before the Overhauser wave function gives any improvement over plane waves, hecause of t,he necessit,y of distorting t,he Fermi surface. We find the critical value of t)he strength paramet,er t,o he greater than unity. Secondly, for a nonseparable one-dimensional Gaussian potential, t,he BCS equations are solved exactly in strong and weak coupling and approximately for intermedint,e values. The BCS ground state energy is always lower. Finally, for the one dimensional shell interaction, a trial wave function is constructed which pairs time-reversed Overhauser states in the BCS spirit. Such a trial function is inferior to one composed of plane wave BCR pairs. Thus the BCS correlations appear to destroy those of Overhauser, and we conclude t,hat the Overhauser representation is not necessarily a better starting point, than plane waves for improved calculations in the many fermion system. The Guiding (‘enter Approximation to Charged Particle Motion. THEODORE G. NORTHROP The equations governing the guiding center motion of a charged particle in an electromagnetic field are obtained simultaneously and deductively, without considering indi-