Dimensionless spin

Here j = cJ/GM2 is a dimensionless spin parameter, where J is the stellar angular momentum. If in a particular case one believes that the observed frequency is in fact the orbital frequency at the ISCO, then the mass is equal to the upper limit given above. 

Figure 3. Calculated dimensionless wet film thickness profiles for Newtonian and non-Newtonian liquids at selected dimensionless spinning time T defined in ref. (12).

S dimensionless spin angular momentum quantum number (of a cation)... 

Physicists and chemists have preferred to discuss these concepts in terms of a dimensionless spin I (J = h, in which h is Planck s constant divided by 2 ir). Eq. Al-1 then becomes... 

We observe that at vanishing spin vecs/ x 0) the Muschelknautz-derived model has a value of 2, but then exponentially increases with increasing dimensionless spin velocity, vqcsI x The Barth model does not account for an entrance loss, and so the initial value of Eux is zero. The Barth and Muschelknautz models agree reasonably well from a VQcshx value of about... 

We introduce the dimensionless bending coordinates qr = t/XrPr anti qc = tAcPc ith Xt = (kT -r) = PrOir, Xc = sJ kcPc) = Pc nc. where cor and fOc are the harmonic frequencies for pure trans- and cis-bending vibrations, respectively. After integrating over 0, we obtain the effective Hamiltonian H = Ho + H, which is employed in the perturbative handling of the R-T effect and the spin-orbit coupling. Its zeroth-order pait is of the foim... 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

Figure 2.94 The lifting of the degeneracy of the Ms = - 1/2, a, and Ms = + 1/2, (l, spin states on the application of a magnetic field of flux density 8. g is the dimensionless g-factor,...

Since in the case of an isolated pair of spin nuclei the dipolar dephasing, and hence the REDOR evolution curve, is exclusively governed by the dipolar Hamiltonian, the data analysis proves to be straightforward employing a universal REDOR curve, in which the normalised difference intensity AS/Sq is plotted as a function of the dimensionless product NTRxd.2 ... 

Bohr magneton, N is the number of moles in the sample, and g is the (dimensionless) electron free-spin factor (g-factor), which has the value 2.0023. 

For configurations with one electron or hole above closed shells the moments of the spectrum are conditioned in the single-configuration approach by only one-electron spin-orbit interaction, and then the dimensionless quantities skewness jci and excess K2 (formulas (32.7)) as well as orbital quantum number and, therefore, do not depend on the accuracy of radial orbitals, namely,... 

Practical application of the one-dimensional theory developed to the calculation of the effects of losses on the detonation velocity is limited by the fact that even at the limit the reaction time is small and heat transfer and braking do not cover the entire cross-section of the tube. At the same time, in the vast majority of cases, long before the limit is reached one observes the so-called spin—a spiral-like or periodic propagation of detonation which is not described by our theory. Some thoughts are given concerning the dimensionless criteria on which the spin depends. 

The dimensionless criterion factor f = qv/a is of great importance in estimating liquid flow parameters the discussion of its values is beyond the frames of the present work (as outlined in the references cited). It may be noted, however, that the same expression appears in the equation evaluating the ability of polymers for thread and fiber spinning 15>. To the first approximation, the criterion f = qv/o may be treated as a measure of shape-keeping ability of liquid in flow. 

We have considered here the influence of dispersion asymmetry and Zee-man splitting on the Josephson current through a superconductor/quantum wire/superconductor junction. We showed that the violation of chiral symmetry in a quantum wire results in qualitatively new effects in a weak superconductivity. In particularly, the interplay of Zeeman and Rashba interactions induces a Josephson current through the hybrid ID structure even in the absence of any phase difference between the superconductors. At low temperatures (T critical Josephson current. For a transparent junction with small or moderate dispersion asymmetry (characterized by the dimensionless parameter Aa = (vif — v2f)/(vif + V2f)) it appears, as a function of the Zeeman splitting Az, abruptly at Az hvp/L. In a low transparency (D Josephson current at special (resonance) conditions is of the order of yfD. In zero magnetic field the anomalous supercurrent disappears (as it should) since the spin-orbit interaction itself respects T-symmetry. However, the influence of the spin-orbit interaction on the critical Josephson current through a quasi-ID structure is still anomalous. Contrary to what holds... 

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