Second-order Perturbation Correction Research Articles

Generalized Cumulant Expansions and Spin-Relaxation Theory

The generalized cumulant expansion method of Kubo is applied to an analysis of spin-relaxation theory appropriate for NMR and ESR studies of molecular systems. It leads to a general, formal solution of the equation of motion of a suitably averaged magnetization operator (and also the spin-density matrix). This solution permits a convenient perturbation expansion in the region of motional narrowing; i.e., when the random functional spin perturbation H1(t) obeys | H1(t) |τc<1, where τc is a correlation time for the random process. It is shown how this expansion valid for times t≫τc generates the time-independent R or relaxation matrix to all orders in | 1(t) |τc, and detailed expressions are given through fourth order. While the R matrix supplies the linewidth and dynamic frequency shift of the main Lorentzian resonance line, it is found that by formulating the cumulant method without the restriction t≫τc, weak subsidiary Lorentzian lines are predicted. These subsidiary lines appear as second-order perturbation corrections in |H1(t)|τc. They have widths given by τc−1 and, in general, large frequency shifts. The problem of (anisotropic) -rotational diffusion is discussed in detail, and it is shown that the spin-Hamiltonian approximation for liquids rests upon a Born–Oppenheimer-type approximation appropriate for random modulation of the nuclear coordinates, i.e., τc−1≪ωn,o, where ℏωn,o is the energy separation between the ground electronic state and the nth excited state coupled by the angular-momentum operator L. The cumulant method is then used to obtain higher-order corrections to the g-tensor line-broadening mechanism.

A study of the Hartree-Fock approximation as applied to finite nuclei

Hartree-Fock calculations have been performed for the closed-shell nuclei 16O, 40Ca and 80Zr, and, with certain simple approximations, for the open-shell nuclei 12C, 28Si and 60Zn. The velocity-dependent, two-body force acts in relative s-states only. The force saturates nuclear matter at the correct energy and density and yields small second-order perturbation corrections, but has too large a range to be in good agreement with the two-body data. However the choice of this simple force enables us to study the Hartree-Fock approximation in detail which is the main purpose of this paper. We calculate wave functions, energy levels, binding energies and density distributions for the six nuclei and obtain rough agreement with the experimental results. A discussion is given of the non-local Hartree-Fock potential.

Microwave Spectrum, Quadrupole Coupling Constants, and Heavy-Atom Geometry of 2-Bromopropane

The microwave spectra of (CH3)2CH79Br, (CH3)2CH81Br, (CH3)213CH79Br, and 13CH3CH79BrCH3 have been examined, and the coordinates of the Br and C atoms have been obtained by the substitution method. The resulting bond distances and bond angles are as follows: r(CBr) = 1.957 Å, r(CC) = 1.508 Å, ∠CCBr = 110.0°, and ∠CCC = 114.2°. The quadrupole coupling constants of (CH3)2CH79Br have been determined to be χaa = 479.5 Mc/sec, χbb = −264.9 Mc/sec, and χcc = −214.6 Mc/sec. Second-order perturbation corrections were included in the analysis of the hyperfine splittings. The quadrupole coupling constants are consistent with χzz = 529.8 Mc/sec, assuming a cylindrical charge distribution, and with χzz = 514.2 Mc/sec, ηbond = 0.0303, assuming that the z axis and the CBr internuclear line coincide.

Self-consistent perturbation treatment of impurities and imperfections in metals, II. Second-order perturbation corrections

The density matrix perturbation theory developed in an earlier paper has been used to make nu erical self-consistent calculations to second-order for electron densities and potentials round impurities and imperfections in metals. For attractive potentials, the second-order perturbation equations may be solved up to quite large values of the charge on the impurity centre and for a wide range of densities. However, low-order perturbation theory fails for bound states, and their formation may lead to some restrictions on the physical range of usefulness of our second-order solutions. Similar restrictions, to small values of the charge on the defects, and to high densities, are found to apply for repulsive potentials, but in this case the limitations arise because the second-order equations do not appear to have convergent solutions outside certain ranges of conduction electron density and charge Z . I t seems that to obtain convergent results therefore for larger Z or lower densities the second- and third-order term s should be included together for repulsive potentials, b u t this task is unfortunately beyond our present computational facilities. Finally, it is pointed out that the Fourier transform of the first-order self-consistent potential may be obtained, and this is used to give a brief discussion of residual resistances in alloys.