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Coupled Schroedinger equations for diatomic molecules internal motion solved using best adiabatic approximation to obtain energy correction to Born-Oppenheimer method
Adiabatic approximation for nonreactive molecular collisions obtained from variational principle
Ground state energy of molecule in adiabatic approximation
A simple formula for the transition probability for electron exchange between unlike ions and atoms is established within the adiabatic approximation by employing the Linear Combination of Atomic Orbitals (LCAO) method. The formula also involves an adiabatic parameter, introduced by Massey, and thus the difficulties arising from the internal energy defect and the adiabatic approximation are avoided. Specific reactions Li(+++) + H to Li(++) + H(+) and Be(4+) + H to Be(3+) + H(+) are considered as examples. The calculated capture cross section results of the present work are compared with the experimental data and with the calculation of other authors over the velocity range of 10(7) cm/sec to 10(8) cm/sec.
A counterexample to the adiabatic approximation theorem is given when degeneracies are present. A formulation of an alternative version is proposed. A complete asymptotic decomposition for n dimensional self-adjoint Hamiltonian systems is restated and used.
Nonrelativistic energy of excited vibrational state of molecule calculated in adiabatic approximation shown to be upper bound to exact nonrelativistic energy
A theory for the pitch-angle scattering of cosmic rays in a turbulent magnetic field can be derived from first principles by applying both the quasi-linear and adiabatic approximations to the master equation for the ensemble averaged distribution function. A proof of the failure of these approximations, taken together, is given. Some predictions of the quasi-linear theory, are given. New wave-like propagation modes have been discovered.
An adiabatic approximation is applied to the calculation of turbulent MHD dynamo magnetic fields in thin disks. The adiabatic method is employed to investigate conditions under which magnetic fields generated by disk dynamos permeate the entire disk or are localized to restricted regions of a disk. Two specific cases of Keplerian disks are considered. In the first, magnetic field diffusion is assumed to be dominated by turbulent mixing leading to a dynamo number independent of distance from the center of the disk. In the second, the dynamo number is allowed to vary with distance from the disk's center. Localization of dynamo magnetic field structures is found to be a general feature of disk dynamos, except in the special case of stationary modes in dynamos with constant dynamo number. The implications for the dynamical behavior of dynamo magnetized accretion disks are discussed and the results of these exploratory calculations are examined in the context of the protosolar nebula and accretion disks around compact objects.
By integrating the exact equations of motion, particle orbits have been followed in a good model magnetospheric field consisting of a planetary dipole, forward magnetosphere, and magnetotail current system. Proton energies from 2 eV to 20 keV were used for the full range of equatorial pitch angles and phase. Despite considerable pitch angle scattering in the equatorial plane crossings, it is found, first, that the bounce-averaged cross-tail drift velocity is approximately independent of pitch angle. Second, it is found that, averaged over initial gyrophase, the drift velocity (due to field curvature and gradient) is proportional to proton energy and is given to good approximation by adiabatic approximations, even up to 20 keV, despite the extreme lack of meeting the adiabatic criteria.
A new transport equation for the cosmic-ray omnidirectional intensity is obtained. This equation follows exactly from the coupled pair of differential moment equations we presented earlier. It can be characterized as a nonlocal convection-diffusion equation in which the usual transport coefficients are replaced by time integral operators. The nonlocal equation is shown to reduce to the standard convection-diffusion form if the adiabatic approximation can be applied. In general, the adiabatic approximation does not apply; however, by going to the limits of infinite and zero gyroradius and, in addition, applying the adiabatic approximation, the large- and small-gyroradius transport theories due originally to Jokipii are regained. The validity of these theories as asymptotic limits and as approximate theories in the interplanetary magnetic field is discussed.
Adiabatic exchange equation in electron scattering from multi-electron atoms - elastic scattering cross sections for sodium and lithium
The equation which governs the quasi-linear approximation to the ensemble and gyro-phase averaged one-body probability distribution function is constructed from first principles. This derived equation is subjected to a thorough investigation in order to calculate the possible limitations of the quasi-linear approximation. It is shown that the reduction of this equation to a standard diffusion equation in the Markovian limit can be accomplished through the application of the adiabatic approximation. A numerical solution of the standard diffusion equation in the Markovian limit is obtained for the narrow parallel beam injection. Comparison of the diabatic and adiabatic results explicitly demonstrates the failure of the Markovian description of the probability distribution function. Through the use of a linear time-scale extension the failure of the adiabatic approximation, which leads to the Markovian limit, is shown to be due to mixing of the relaxation and interaction time scales in the presence of the strong mean field.
To study the problem of electron heating in quasi-perpendicular shocks, under the combined effects of 'reversible' motion, in the shock electric potential and magnetic field, and wave-particle interactions a diffusion equation was derived, in the drift (adiabatic) approximation and it was solved by using a Monte Carlo method. The results show that most of the observations can be explained within this framework. The simulation has also definitively shown that the electron parallel temperature is determined by the dc electromagnetic field and not by any wave particle induced heating. Wave-particle interactions are effective in smoothing out the large gradients in phase space produced by the 'reversible' motion of the electrons, thus producing a 'cooling' of the electrons. Some constraints on the wave-particle interaction process may be obtained from a detailed comparison between the simulation and observations. In particular, it appears that the adiabatic approximation must be violated in order to explain the observed evolution of the perpendicular temperature.
Particle energization in Earth's and Jupiter's magnetospheres is discussed. Understanding of the large scale magnetic and electric fields in which charged particles move is reviewed. Orbit theory in the adiabatic approximation is sketched. General conditions for adiabatic breakdown at each of three levels of periodicity are presented. High energy losses and lower energy sources argue for the existence of magnetospheric accelerations. Nonadiabatic acceleration processes are mentioned. Slow diffusive energization by particle interactions with electromagnetic fluctuations is outlined. This mechanism seems adequate at Earth but, operating alone, is unconvincing for Jupiter. Adding spatial diffusion in the radially distended Jovian magnetodisk may resolve the difficulty.
This paper presents some exact solutions of problems in random function theory for the purpose of testing the validity of an approximate method known variously in the many different fields of its application as first order smoothing theory, first order cumulant discard, quasilinear theory, or the adiabatic approximation. The hydromagnetic dynamo equations are used here, as particularly appropriate for such an investigation. The calculations show that in one case the exact and approximate solutions agree. In the other case the approximate solution is wrong. Hence, in the absence of a general criterion for validity, a result based on first order smoothing theory is a conjecture rather than a fact. This impacts strongly on much of the recent work on hydromagnetic dynamos.
Subexcitation molecular collisions resonances demonstrated by numerical solution of integral equations for rotational excitation, exhibiting adiabatic approximation limitations
Moment analysis of atomic spectral lines of Cs-Ar and Cs-He systems, using adiabatic approximation
As a result of the Apollo 13 incident, increased emphasis is being placed on materials compatibility in a high pressure GOX environment. It is known that in addition to impact sensitivity of materials, approximately adiabatic compression conditions can contrive to induce materials reactivity. Test runs at high pressure using the ABMA tester indicate the following: (1) The materials used in the tests showed an inverse relationship between thickness and impact sensitivity. (2) Several materials tested exhibited greater impact sensitivity in GOX than in LOX. (3) The impact sensitivity of the materials tested in GOX, at the pressures tested, showed enhanced impact sensitivity with higher pressure. (4) The rank ordering of the materials tested in LOX up to 1000 psia is the same as the rank ordering resulting from tests in LOX at 14.7 psia.