Nuclear magnetic resonance (NMR) spectra of 23Na (I = 32) and 93Nb (I = 92) from a powdered sample of NaNbO3 have been studied. At room temperature the 23Na spectrum shows the presence of two distinct sites, one having axial symmetry with a coupling constant of 21.5 ± 0.2 MHz and the other having an asymmetry parameter lying between 0.80 and 1.0 with a coupling constant of 1.0 ± 0.1 MHz. The niobium spectrum arises from a single site with a coupling constant of 19.7 ± 0.5 MHz and an asymmetry parameter of 0.82 ± 0.02. The (12 ↔ − 12) transition in the 93Nb spectrum exhibits some unusual features which were confirmed by computer-simulated powder patterns. The temperature dependence of one 23Na site indicated a rather linear decrease in the coupling constant, from 2.6–1.6 MHz, and the asymmetry parameter was approximately zero as the temperature increased from − 170–280°C. The 93Nb quadrupole interaction showed a linear decrease not only in the coupling constant, from 19.7–10 MHz, but also in the asymmetry parameter, from 0.82–0.64, as the temperature rose from room temperature to 270°C. Electric-field-gradient calculations for 93Nb produced coupling constants lower than the measured value by about an order of magnitude while yielding values of the asymmetry parameter comparable to that observed. Similar calculations for 23Na gave relatively high values of the asymmetry parameter, and the coupling constants were of the same order of magnitude as those determined experimentally. Third-order corrections to the NMR frequencies perturbed by the quadrupole interactions for the case of the asymmetry parameter unequal to zero were calculated. These proved to be negligible except for the case of niobium at low frequencies.
ESR studies were performed on single crystals of LiNbO3 doped with a trace amount of iron. Fe3+(S=52) is in an axially symmetric site and is described by the spin Hamiltonian H=β (H· g· S)+B20O20+B40O40+B43O43. The crystal field parameters at room temperature were found to be B20=596 G, B40=−0.85 G, and B43=10 G.
ESR studies were performed on a single crystal of ferroelectric LiNbOa doped with chromium. For the main spectrum, attributed to Cr3+ at the Nb site, all transitions (including forbidden transitions) were identified. The experimental rotation pattern observed at X-band frequency was compared with that computed from the appropriate spin Hamiltonian {X=H.y.s+D[Sl-lS(S+lJI and the best fit ESR parameters obtained: D=0.411 cm-1 and g= 1.969. In examining these parameters as a function of temperature, the g value was found to be temperature independent, and the variation of D with temperature was attributed to anisotropic lattice expansion. On the main central transition, two additional peaks were observed. EPR parameters (D=0.21±0.02 cm-1 and g= 1.97) are given for the peak on the high-field side. It is proposed that this peak arises from Cr3+ at the Li site. A discussion of the peak on the low-field side is also given.
Fermat's principle asserts that light takes the path of minimum (actually extremal) time. Sailors often wish to find paths of minimum time. Thus by thinking of a sailboat as a light ray, a sailor can use Fermat's principle to describe the optimum sailing strategy. Huygens' principle and Hamiltonian optics follow from Fermat's principle, so a sailor can use Huygens' principle to visualize least-time paths, and Hamilton's optics provides a mathematical description of these paths. In especially simple cases, the optics-based formalism can be used to describe and quantify the basic tactics of sailboat racing.
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