Neutron Stars 1: Equation of State and Structure

There has recently been growing evidence for the existence of neutron stars possessing magnetic fields with strengths that exceed the quantum critical field strength of 4.4 × 1013 G, at which the cyclotron energy equals the electron rest mass. Such evidence has been provided by new discoveries of radio pulsars having very high spin-down rates and by observations of bursting gamma-ray sources termed magnetars. This paper will discuss the exotic physics of this high-field regime, where a new array of processes becomes possible and even dominant and where familiar processes acquire unusual properties. We review the physical processes that are important in neutron star interiors and magnetospheres, including the behaviour of free particles, atoms, molecules, plasma and condensed matter in strong magnetic fields, photon propagation in magnetized plasmas, free-particle radiative processes, the physics of neutron star interiors and field evolution and decay mechanisms. Application of such processes in astrophysical source models, including rotation-powered pulsars, soft gamma-ray repeaters, anomalous x-ray pulsars and accreting x-ray pulsars will also be discussed. Throughout this review, we will highlight the observational signatures of high magnetic field processes, as well as the theoretical issues that remain to be understood.

Physics of strongly magnetized neutron stars

Recent publications have suggested that the satisfaction of a principle we call kinetic balance can provide variational safety in Dirac calculations. The theoretical foundation for this proposal is examined, first in simple one‐electron problems, and then (less rigorously) in SCF calculations. The conclusion is that finite basis calculations using kinetic balance are safe from catastrophic variational collapse, but that the ‘‘bounds’’ provided by these calculations can be in error by an amount of order 1/c4. The bounds are applicable to total SCF energies, but not to SCF orbital energies. The theory is illustrated by a series of one‐electron calculations.

Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations

The geometrical structures which regulate transformations in dynamical systems with three or more degrees of freedom (DOFs) form the subject of this paper. Our treatment focuses on the (2n − 3)-dimensional normally hyperbolic invariant manifold (NHIM) in nDOF systems associated with a centre × · ·· ×centre × saddle in the phase space flow in the (2n − 1)dimensional energy surface. The NHIM bounds a (2n − 2)-dimensional surface, called a ‘transition state’ (TS) in chemical reaction dynamics, which partitions the energy surface into volumes characterized as ‘before’ and ‘after’ the transformation. This surface is the long-sought momentum-dependent TS beyond two DOFs. The (2n − 2)-dimensional stable and unstable manifolds associated with the (2n − 3)-dimensional NHIM are impenetrable barriers with the topology of multidimensional spherical cylinders. The phase flow interior to these spherical cylinders passes through the TS as the system undergoes its transformation. The phase flow exterior to these spherical cylinders is directed away from the TS and, consequently, will never undergo the transition. The explicit forms of these phase space barriers can be evaluated using normal form theory. Our treatment has the advantage of supplying a practical algorithm, and we demonstrate its use on a strongly coupled nonlinear Hamiltonian, the hydrogen atom in crossed electric and magnetic fields.

https://iopscience.iop.org/article/10.1088/0951-7715/15/4/301/pdf

The geometry of reaction dynamics

The energy values of many low-lying states of the one-electron problem in the presence of a homogeneous magnetic field of arbitrary strength (0<B<or=4.7*108 T) are calculated with high numerical accuracy for a sufficiently dense mesh of B. The wavefunctions are expanded either in terms of spherical harmonics (weak and moderate fields) or in terms of Landau states (strong and very strong fields), with r- or z-dependent expansion functions that are determined with the use of an adopted version of the MCHF code of Froese Fischer (1978). At intermediate field strengths up to 24 expansion terms are included. The structural change of the wavefunctions with magnetic field is discussed quantitatively for a few representative states. As an application, the splittings of the components of the Lyman- alpha , beta , and the Balmer- alpha lines of the hydrogen atom are presented (including the effects of the finite proton mass) as continuous functions of the field strength over the whole range of B considered.

http://iopscience.iop.org/article/10.1088/0022-3700/17/1/010/pdf

Hydrogen atoms in arbitrary magnetic fields. I. Energy levels and wavefunctions

A thorough unified treatment is given of the quantum-mechanical operators and wave functions for a molecular system (composed of N moving charged particles) in static uniform electric and magnetic fields E and B. The treatment is rigorous within the nonrelativistic approximation. The system may either be neutral or charged. The fields may have arbitrary intensities and orientations. Close correspondence is maintained between the classical and quantum-mechanical treatments. The wave functions are expressed both in time-independent energy representation and in time-dependent wave packets. Three types of momentum play important roles. For single-particle systems they are the canonical momentum P; the mechanical momentum Pi = P(e/c) A = MR; and the pseudomomentum K = Pi-(e/c)R x B-eEt.B. For neutral molecules, the Schroedinger equation is ''pseudoseparated'' and the internal degrees of freedom are coupled to the center of mass motion by only the ''motional Stark Effect,'' which involves the constant of motion scrK. For ionic systems, only one component of the center of mass is coupled to the internal motion.

Interaction of atoms, molecules, and ions with constant electric and magnetic fields

The classical spin of a particle is essentially its angular momentum of rotation about an axis passing through its center of mass. The classical Poisson brackets are generalized to include spin for any system for which the rate of change of spin is given by $\frac{d\stackrel{\ensuremath{\rightarrow}}{s}}{\mathrm{dt}}=\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{s}\ifmmode\times\else\texttimes\fi{}(\frac{\ensuremath{\partial}H}{\ensuremath{\partial}\stackrel{\ensuremath{\rightarrow}}{s}})$ where $H$ is the Hamiltonian. In a previous paper, we showed that this equation is satisfied by the Breit-Pauli Hamiltonian for a molecular system of electrons and nuclei interacting with an electromagnetic field. This Hamiltonian contains all of the fine-structure terms with the exception of the Lamb shift. Thus, our generalized Poisson brackets should apply to almost all molecular problems. Our generalized Poisson brackets are the same as those which Sudarshan and Mukunda derived by group-theoretic arguments for the direct product of a pure spin group with the group of transformations of generalized coordinates and their conjugate momenta.

Generalizations of classical Poisson brackets to include spin

Classical models of electrons and nuclei are developed which behave surprisingly like their quantum mechanical counterparts. The particles are finite in size and have both electrical charge density and current density. Classical spin results from the angular momentum of the particles rotating about their ’’center of mass’’. Internal electromagnetic fields are produced by the motion of the charged particles. These internal fields have dynamical properties of their own and produce the interaction between the particles. In this paper, the internal fields are derived through the second order of retardation or terms in 1/c2. Since higher order terms in the internal fields are not included, no difficulties are encountered with respect to the classical particles radiating. This same result could also be obtained by using equal parts of the retarded and advanced potentials. The Lagrangian, the (canonical, mechanical, and angular) momenta, the energy, and the Hamiltonian are derived in a gauge invariant manner for...

Interaction of molecules with electromagnetic fields. I. Classical particles and fields

Semirelativistic force and energy operators for the interaction of molecules (with moving nuclei) are derived for intermediate range separations. These operators are accurate through all of the (α4mc2 or 1/c2) fine‐structural terms (but not the Lamb shift). The molecules interact by means of the internal electromagnetic fields which result from their charges, multipoles, and motions. Thus, the force on molecule a due to b is the generalized Lorentz force: QaE(i)(ra;b)+ (1/c)Ia×B(i)(ra;b), where Qa and Ia are the effective charge and current on a, while E(1)(ra;b) and B(i) (ra;b) are the internal fields of b at ra. It is shown that the interaction Hamiltonian Hab is minus the Lagrangian of the cross internal fields of a and b and can be expressed as Hab = QaQb[1/rab]− [1/c2]{Ia ⋅ Ib[1/rab]+ IaIb:[rabrab/rab3]}, where rab = ra−rb is the distance between the centers‐of‐mass. Both Qa and Ia are expressed in terms of the electric and magnetic multipole moments. To obtain full semirelavistic precision, the elec...

Intermolecular force and energy operators. III

This paper is a summary of a detailed treatment, given elsewhere, in which (gauge invariant), equations of change (at points in configuration space) of arbitrary dynamical properties of an N-particle quantum-mechanical system in the presence (or absence) of an arbitrary (time- and space-varying) classical electromagnetic field are derived for an arbitrary Hamiltonian by making use of both the Maxwell and the Schrodinger time-dependent equations. The properties at a point in configuration space are defined so as to be closely related to observables and the forces are related to their classical counterparts. The rates of change of linear momentum and angular momentum are closely related to the classical Lorentz force and the Lorentz torque, respectively. The rate of change of energy is the work done on the system by E(t), the time-dependent part of the external electromagnetic field. The reason for E(t) and its definition of the energy for a system in a time-varying electromagnetic field is explained. The Breit-Pauli Hamiltonian is required to treat “state-to-state” molecular problems and the detailed equations of change using this Hamiltonian are used as examples.

Kuo-Ho Yang

Papers

Interaction of atoms, molecules, and ions with constant electric and magnetic fields

Generalizations of classical Poisson brackets to include spin

Interaction of molecules with electromagnetic fields. I. Classical particles and fields

Intermolecular force and energy operators. III

Equations of change in configuration space