D S F Crothers

Ionisation of atoms by ion impact

Total cross sections are calculated for the ionisation of a hydrogen atom by multicharged fully-stripped ions in the 20-1000 keV amu-1 impact energy range. Distortion is accounted for in the entrance channel (via the eikonal approximation) and in the exit channel (via the continuum distorted-wave approximation). The transition amplitude is calculated in the post form so that the electronic nonorthogonal kinetic energy is treated as the perturbation. It is concluded that of the currently available models this theory is the most successful and versatile over a considerable range of energies and charges. Specifically for ionisation of a hydrogen atom by 50 keV protons the authors present doubly differential cross sections for electrons ejected in the forward direction and singly differential cross sections as a function of emission energy. The question of cusps and peaks in the differential cross sections is considered as is the question of charged scaling of the total cross section.

Exact solution for the extended Debye theory of dielectric relaxation of nematic liquid crystals

The exact solution for the transverse (i.e. in the direction perpendicular to the director axis) component α⊥(ω) of a nematic liquid crystal and the corresponding correlation time T⊥ is presented for the uniaxial potential of Martin et al. [Symp. Faraday Soc. 5 (1971) 119]. The corresponding longitudinal (i.e. parallel to the director axis) quantities α⊥(ω), T⊥ may be determined by simply replacing magnetic quantities by the corresponding electric ones in our previous study of the magnetic relaxation of single domain ferromagnetic particles Coffey et al. [Phys. Rev. E 49 (1994) 1869]. The calculation of α⊥(ω) is accomplished by expanding the spatial part of the distribution function of permanent dipole moment orientations on the unit sphere in the Fokker-Planck equation in normalised spherical harmonics. This leads to a three term recurrence relation for the Laplace transform of the transverse decay functions. The recurrence relation is solved exactly in terms of continued fractions. The zero frequency limit of the solution yields an analytic formula for the transverse correlation time T⊥ which is easily tabulated for all nematic potential barrier heights σ. A simple analytic expression for T∥ which consists of the well known Meier-Saupe formula [Mol. Cryst. 1 (1966) 515] with a substantial correction term which yields a close approximation to the exact solution for all σ, and the correct asymptotic behaviour, is also given. The effective eigenvalue method is shown to yield a simple formula for T⊥ which is valid for all σ. It appears that the low frequency relaxation process for both orientations of the applied field is accurately described in each case by a single Debye type mechanism with corresponding relaxation times (T∥, T⊥).

Thermally activated escape rates of uniaxial spin systems with transverse field: uniaxial crossovers.

Classical escape rates of uniaxial spin systems are characterized by a prefactor differing from and much smaller than that of the particle problem, since the maximum of the spin energy is attained everywhere on the line of constant latitude: theta =const, 0 < or= phi < or = 2 pi. If a transverse field is applied, a saddle point of the energy is formed, and high, moderate, and low damping regimes (similar to those for particles) appear. Here we present the first analytical and numerical study of crossovers between the uniaxial and other regimes for spin systems. It is shown that there is one HD-Uniaxial crossover, whereas at low damping the uniaxial and LD regimes are separated by two crossovers.

Field dependence of the temperature at the peak of the zero-field-cooled magnetization

The effect of an applied magnetic field on the temperature at the maximum of the zero-field-cooled (ZFC) magnetization, MZFC , is studied using the recently obtained analytic results of Coffey et al (Coffey W T et al 1998 Phys. Rev. Lett. 80 5655) for the prefactor of the Neel relaxation time which allow one to precisely calculate the prefactor in the Neel-Brown model and thus the blocking temperature as a function of the coefficients of the Taylor series expansion of the magnetocrystalline anisotropy. The present calculations indicate that even a precise determination of the prefactor in the Neel-Brown theory, which always predicts a monotonic decrease of the relaxation time with increasing field, is insufficient to explain the effect of an applied magnetic field on the temperature at the maximum of the ZFC magnetization. On the other hand, we find that the non-linear field dependence of the magnetization along with the magnetocrystalline anisotropy appears to be of crucial importance to the existence of this maximum.

An approximate formula for the relaxation time of a single domain ferromagnetic particle with uniaxial anisotropy and collinear field

Recently there have been renewed efforts to solve the Fokker–Planck equation resulting from Brown’s model [Phys. Rev. 130, 1677 (1963)] of single domain ferromagnetic particle relaxation for the case of uniaxial anisotropy and zero external field. In particular, the usefulness of a simple analytic formula for the relaxation time based on the reciprocal of the lowest non‐zero eigenvalue resulting from this model has been stressed. Here we suggest an improved analytic formula which we extend to the case of applied collinear field and compare it with the exact numerical solution and previous analytic expressions. The results are presented in terms of the pre‐factor dependence which is commonly taken as constant. The formula exhibits good agreement with the exact results throughout the full range of anisotropy and reduced external field values.

Simple approximate formulae for the magnetic relaxation time of single domain ferromagnetic particles with uniaxial anisotropy

Abstract Approximate simple formulae valid for all barrier heights for the longitudinal relaxation time of a single domain ferromagnetic particle with uniaxial anisotropy are compared with the exact solution. It is concluded that a formula based on the application of a variational principle to the Sturn-Liouville equation associated with Browns Fokker-Planck equation yields the closest approximation to the exact solution.

Semi-classical treatment of atomic collisions

The semi-classical treatment of atom-atom collisions involving electronic transitions is discussed. As is well know n difficulties occur if the classical trajectories associated with the various states of importance in a collision process differ significantly. A method designed to overcome these is described. It will be referred to as the forced-common-turning-point method. The four coupled first-order differential equations which describe the new version of the semi-classical two-state treatment for an atom-atom collision may be reduced to a pair of generalized impact parameter equations. The first Born approximation to the cross-section obtained from the straightforward semiclassical treatment differs from the corresponding cross-section obtained from the full quantal treatment mainly in that it contains an anomalous multiplying factor equal to the ratio of the initial to the final velocity of relative motion. This anomaly does not arise with the forced-common-turning-point method. A model collision process which provides a very searching test is considered. Only two states are included. The initial interaction is zero, the final interaction is Coulombic and the transition matrix element is exponential. Curve-crossing may occur. The distorted wave approximation to the excitation cross-section may be found exactly and may also be computed using the forced-common-turning-point method. There is remarkable accord between the results. Thus in a case w here the reduced m ass of the colliding systems is 2 on the chemical scale, w here the excitation energy is 3.4 eV and where the incident kinetic energy of relative motion is only 0.85 eV above this the excitation cross-sections obtained differ by as little as 0.01 %; and, moreover, the patterns of the contributions to the cross-sections from the separate partial waves are similar.

Continuum Distorted Wave Methods in Ion—Atom Collisions

Publisher Summary This chapter discusses the continuum distorted wave method in both time-dependent and time-independent form and presents the clarification of its physical interpretation. An extension of the continuum distorted wave (CDW) Ansatz has been developed to a multistate variational close coupling theory that removes the normalization difficulties encountered in the standard CDW approximation. In addition, it shows the new formalism to possess the desirable properties of Galilean invariance, gauge invariance, flux conservation and, above all, the absence of divergences. This last point has been seen to emerge in a natural and essential way from the use of wave functions that satisfy the correct Coulomb asymptotic boundary conditions. The CDW theory is understood most easily within the impact parameter treatment, which indeed was the format originally used by Cheshire (1964) in its inaugural presentation. By contrast, both Oppenheimer–Brinkman–Kramers (OBK) and strong-potential Born (SPB) wave theories are obliged to resort to off shell effects in a futile attempt to compensate for failure to satisfy the correct asymptotic boundary conditions.

Quantal threshold ionisation

A semiclassical quantal theory of threshold electron impact ionisation is developed. Singular wavefunctions, classical differential cross sections and matching procedures are avoided. The Kohn variational principle is applied perturbatively to evaluate the scattering amplitude. The absolute electron-helium singlet ionisation total Wannier cross section is estimated to be 2.37 (Ea0/e2)1.127a02, in good agreement with the experiment of Pichou et al. (1978). Both helium electrons and the first four partial waves are included. Doubly differential cross sections p0.8(Er,1/2 pi ) at E=0.8 eV are found at 2.58*10-19 cm2 sr-1 eV-1, in good agreement with the experiment of Pichou et al. (1978). Triple differential cross sections are compared favourably with the recent experiment of Fournier-Lagarde et al. (1985).

Transfer and ionization processes during the collision of fast H+, He2+ nuclei with helium

Total cross sections for single electron capture, single ionization, resonant double capture and transfer ionization processes during the collision of H+, He2+ nuclei with helium are calculated within the continuum distorted wave approximation for the energy range 100-1600 keV amu-1. The semiclassical independent-event model of the collision is adopted, while the effect of electron correlation in the helium atom is explicitly accounted for by the use of a two-electron model of the helium ground state developed by Pluvinage (1950). For all but the lowest energies considered, results for single capture, single ionization and resonant double capture are typically one-third to two-thirds those of experiment. The discrepancies may be attributed to contributions from capture into excited states and from simultaneous capture/ionization and excitation processes, which have not been included in the calculations.