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Stability of the thermal Hartree-Fock approximation
ANNALS
OF PHYSICS:
Abstracts
26, 479-480
(1962)
of Papers
Lore&z’s Pendulum Problem. Cambridge, England.
to Appear
J. E. LITTLEWOOD,
Trinity
in Future College,
University
Issues of Cambridge,
On Resonances in a System of Conpled Two-Particle Channels. SATL BARSHAY, Physics Department, Brandeis University, Waltham, Massachusetts. ,4 simple model of a system of coupled two-particle channels is solved exactly. It, is shown that a narrow resonance can occur at a relatively low energy, especially if one of the twoparticle systems has orbital angular momentum greater than zero. The resonance manifests itself at a given energy in all of the T-matrix elements, although it may be considerably weaker in some element,s than in others. Within the framework of the model, the essential dynamical origin of the resonance is the coupling between the channels, plus, in addition, an interaction peculiar to one of the channels. An attempt is made t,o see whether this t,ype of resonance might describe the isotopic spin zero, negative strangeness, meson-baryon resonances observed at 1405 Mev and at 1525 Mev. Both of these resonances satisfy a requirement of t,he model that the cent)er-of-mass moment,a in all of the coupled two-particle channels he relatively low. Within t,he framework of this model, the 1525 Mev resonance might be viewed as a “virtual-bound” state of the h hyperon and an isoscular, scalar meson (or x--?T resonant stat,e) with total energy of about 400 Mev. The 1405 Mev resonance might similarly be viewed as a “virtual-bound” state of the A hyperon and an iaosoalar, scalar system wit,h t.otal energy about twice the pion mass. Stability of the Thermal Hartree-Fock Approximation. S. DAVID MERMIN, Department of Mathematical Physics, University of Birmingham, Birmingham, England. The thermal Hartree-Fock approximation of Green’s fun&ion theory is shown to be equivalent,, in its thermod,ynamic consequences, to the use of that densit,y marris which makes the free energy stationary over a simply restricted class of trial density matrices. The variational formulation provides a st,ability condition requiring one t,o reject those solutions of the t,hermal Hartree-Fock equat,ions which do not, correspond to local minima of the free energy. If only stable Hartree-Fock propogators are used, then the spectrum of densit,?- oscillations as calculated in the thermal random phase approximation can be shown to have only real frequencies. Reasons for the thermal RPA becoming unst,able may therefore be found from studying the significance of unstable Hartree-Pock solutions. It is shown that isothermals with positive slope can occur in the Hartree-Fock equation of state only if an unstable solution has been used. This suggests that one type of instability is associated with a first order phase transition. This is confirmed when the approximation is applied to a lattice gas with attractive interact,ions. The equation of st,ate is of the van der Waals type if the stability condition is ignored. If only solutions which give global minima to the free energy are used, t,he corrected isothermnls are those which result from applying the Maxwell equal area construction to the naive ones. Unstable solutions correspond to the parts of the van drr Waals isot,hermals with positive slope. The physically metastable parts correspond t)o solutions which are local but not, global minima. A lattice gas w-ith repulsive interactions illustrates another type of instability, characterizing a second order phase transition. In this case the naive solut,ion becomes unstahle even though no pathological hehaviour 479
480 appears density
ABSTRACTS
OF
m its equation of state, of part,icles are required
PAPERS
TO
APPEAR
IX
and solutions corresponding to restore stability.
FUTURE
ISSCES
to a spatially
nonurliform
Probability
Distribution of iVeutrons and Precursors in a Multiplying ;lssembly. GEORC:E I. Division of Engineering and Applied Physics, Harvard University, Cnml)ridgc, Massachusetts. In this paper we consider the probabilit,y distribution of the number of neutrons nn(l delayed neutron precursors in a multiplying assembly. Particular emphasis is placed OII the probability distribution for a system which is brought t,o a supercritical stat,e in the presence of a neukon source which is so weak that deviations from the average population may be large. A space independent model is used with one group of neutrons. The problem is formulated in t,erms of the probability distribution generat,ing frmctiou which satisfies a partial differential equation, derived in Section I. Its solution is attempted by t,he method of characteristics and general properties of the characteristic curves and the solution are discussed. In Section II, the method is applied to a system without delayed neukons, and, for a constant source, the generating function and probability distributiou are found. We nest consider simplifications that are possible when the neutron lifetime is very short compared t,o precursor lifetimes. In Section III the multiplication of precursors is treated in a short neutron lifetime model for a system which is below prompt critical. For a single group of precursors and constant reactivity, the generating function is derived. The short neutron lifetime model is extended, in Section IV, to a system above prompt critical, and again we find the generating function in a simple case. Finally, in Section V the resulk of our calculations are compared with some experiments on Godiva and good qualitative agreement is shown. BELL,
On the Levinson Theorem in the Multichannel Case. G. C. GHIRARDI, M. P’UURI, ANLI LI1. RIMINI, Istituto di Fisica dell’ Universita, Parma, Italy. With the aid of simple multichannel models, the possibility is investigated of obtaining :L connection between the number of bound states of the Hamiltonian operat,or and only :I part of the scattering phase shifts at zero and infinit,e energy. Regge Poles ccs Consequences of dnalyticit:y und lfnitarity. S. MANDELSTAM, I)epartment. of Mat,hematical Physics, Universit,y of Birmingham, Birmingham, England. It is shown that a scattering amplit,ude which satisfies a double-dispersion relation and elastic unit,arity is moromorphic in the complex I-plane for ReZ greater than some negative quantity, and that, it,s asymptotic behavior in the momentum transfer is therefore given by the Regge formula. In the proof it is necessary to assume that the scattering amplitude is bounded by ts-’ (for pion-pion kinematics) as both s and t become infinite, but it is argued that, if this limit is not satisfied, the dispersion integrals in s will be dominated by the highenergy region and, in all probabilit,y, those in t will require an infinite number of suhtmctions. In contrast to the SchrGdinger or Bethe-Salpeter cases, the positions of t,he Reggo pole at high energy will depend on the strength of the coupling. This high-energy Regge pole appears to come int,o the right half-plane for repulsive potentials and to correspond to a “ghost” rather than bound state. If and only if it were to reach the value 1 = 1, the boundctlness condition referred to above would no longer be satisfied.
OF PHYSICS:
Abstracts
26, 479-480
(1962)
of Papers
Lore&z’s Pendulum Problem. Cambridge, England.
to Appear
J. E. LITTLEWOOD,
Trinity
in Future College,
University
Issues of Cambridge,
On Resonances in a System of Conpled Two-Particle Channels. SATL BARSHAY, Physics Department, Brandeis University, Waltham, Massachusetts. ,4 simple model of a system of coupled two-particle channels is solved exactly. It, is shown that a narrow resonance can occur at a relatively low energy, especially if one of the twoparticle systems has orbital angular momentum greater than zero. The resonance manifests itself at a given energy in all of the T-matrix elements, although it may be considerably weaker in some element,s than in others. Within the framework of the model, the essential dynamical origin of the resonance is the coupling between the channels, plus, in addition, an interaction peculiar to one of the channels. An attempt is made t,o see whether this t,ype of resonance might describe the isotopic spin zero, negative strangeness, meson-baryon resonances observed at 1405 Mev and at 1525 Mev. Both of these resonances satisfy a requirement of t,he model that the cent)er-of-mass moment,a in all of the coupled two-particle channels he relatively low. Within t,he framework of this model, the 1525 Mev resonance might be viewed as a “virtual-bound” state of the h hyperon and an isoscular, scalar meson (or x--?T resonant stat,e) with total energy of about 400 Mev. The 1405 Mev resonance might similarly be viewed as a “virtual-bound” state of the A hyperon and an iaosoalar, scalar system wit,h t.otal energy about twice the pion mass. Stability of the Thermal Hartree-Fock Approximation. S. DAVID MERMIN, Department of Mathematical Physics, University of Birmingham, Birmingham, England. The thermal Hartree-Fock approximation of Green’s fun&ion theory is shown to be equivalent,, in its thermod,ynamic consequences, to the use of that densit,y marris which makes the free energy stationary over a simply restricted class of trial density matrices. The variational formulation provides a st,ability condition requiring one t,o reject those solutions of the t,hermal Hartree-Fock equat,ions which do not, correspond to local minima of the free energy. If only stable Hartree-Fock propogators are used, then the spectrum of densit,?- oscillations as calculated in the thermal random phase approximation can be shown to have only real frequencies. Reasons for the thermal RPA becoming unst,able may therefore be found from studying the significance of unstable Hartree-Pock solutions. It is shown that isothermals with positive slope can occur in the Hartree-Fock equation of state only if an unstable solution has been used. This suggests that one type of instability is associated with a first order phase transition. This is confirmed when the approximation is applied to a lattice gas with attractive interact,ions. The equation of st,ate is of the van der Waals type if the stability condition is ignored. If only solutions which give global minima to the free energy are used, t,he corrected isothermnls are those which result from applying the Maxwell equal area construction to the naive ones. Unstable solutions correspond to the parts of the van drr Waals isot,hermals with positive slope. The physically metastable parts correspond t)o solutions which are local but not, global minima. A lattice gas w-ith repulsive interactions illustrates another type of instability, characterizing a second order phase transition. In this case the naive solut,ion becomes unstahle even though no pathological hehaviour 479
480 appears density
ABSTRACTS
OF
m its equation of state, of part,icles are required
PAPERS
TO
APPEAR
IX
and solutions corresponding to restore stability.
FUTURE
ISSCES
to a spatially
nonurliform
Probability
Distribution of iVeutrons and Precursors in a Multiplying ;lssembly. GEORC:E I. Division of Engineering and Applied Physics, Harvard University, Cnml)ridgc, Massachusetts. In this paper we consider the probabilit,y distribution of the number of neutrons nn(l delayed neutron precursors in a multiplying assembly. Particular emphasis is placed OII the probability distribution for a system which is brought t,o a supercritical stat,e in the presence of a neukon source which is so weak that deviations from the average population may be large. A space independent model is used with one group of neutrons. The problem is formulated in t,erms of the probability distribution generat,ing frmctiou which satisfies a partial differential equation, derived in Section I. Its solution is attempted by t,he method of characteristics and general properties of the characteristic curves and the solution are discussed. In Section II, the method is applied to a system without delayed neukons, and, for a constant source, the generating function and probability distributiou are found. We nest consider simplifications that are possible when the neutron lifetime is very short compared t,o precursor lifetimes. In Section III the multiplication of precursors is treated in a short neutron lifetime model for a system which is below prompt critical. For a single group of precursors and constant reactivity, the generating function is derived. The short neutron lifetime model is extended, in Section IV, to a system above prompt critical, and again we find the generating function in a simple case. Finally, in Section V the resulk of our calculations are compared with some experiments on Godiva and good qualitative agreement is shown. BELL,
On the Levinson Theorem in the Multichannel Case. G. C. GHIRARDI, M. P’UURI, ANLI LI1. RIMINI, Istituto di Fisica dell’ Universita, Parma, Italy. With the aid of simple multichannel models, the possibility is investigated of obtaining :L connection between the number of bound states of the Hamiltonian operat,or and only :I part of the scattering phase shifts at zero and infinit,e energy. Regge Poles ccs Consequences of dnalyticit:y und lfnitarity. S. MANDELSTAM, I)epartment. of Mat,hematical Physics, Universit,y of Birmingham, Birmingham, England. It is shown that a scattering amplit,ude which satisfies a double-dispersion relation and elastic unit,arity is moromorphic in the complex I-plane for ReZ greater than some negative quantity, and that, it,s asymptotic behavior in the momentum transfer is therefore given by the Regge formula. In the proof it is necessary to assume that the scattering amplitude is bounded by ts-’ (for pion-pion kinematics) as both s and t become infinite, but it is argued that, if this limit is not satisfied, the dispersion integrals in s will be dominated by the highenergy region and, in all probabilit,y, those in t will require an infinite number of suhtmctions. In contrast to the SchrGdinger or Bethe-Salpeter cases, the positions of t,he Reggo pole at high energy will depend on the strength of the coupling. This high-energy Regge pole appears to come int,o the right half-plane for repulsive potentials and to correspond to a “ghost” rather than bound state. If and only if it were to reach the value 1 = 1, the boundctlness condition referred to above would no longer be satisfied.