Bare-mode approximation

Figure 4 shows the SCF results in spheroidal coordinates for the excited bending state (0,4) as a function of the coordinate parameter a. For each of the states, the result is compared with the energy given by SCF in polar (hyperspherical) coordinates. Also shown are the results of a bare-mode approximation, a crude model which assumes for the mode a potential = 0), and similarly postulates a separate potential F( cq, > ) for the q mode, without any self-consistency in the treatment of the two modes. It is evident from Fig. 4 that the physically motivated elliptical (spheroidal) SCF modes do better in this case than the hyperspherical coordinates. Also, the SCF correction gives an important improvement on the bare-mode results. Most important, coordinate optimization, that is, imposing condition, (25) yields a noticeably better result than the SCF energy in a spheroidal system that is not refined for the best a value. 

The increase in ILSS for the epoxy-sized fibers over the bare fibers is 12.4%, approximately 50% of the increase observed in the interfacial shear strength as measured by ITS testing. Changes in the failure mode at the fiber-matrix interface may account for the differences. The sized fibers produced large matrix cracks that grew quickly to catastrophic size under load. This would tend to limit the increase in composite shear properties if at every fiber break in the tensile surface of the coupon a matrix crack was created. The presence of these matrix cracks... 

The Markov approximation, in which we assume that the correlation time of the EM field is much shorter than the timescale of radiation processes of the bare systems. This approximation is equivalent to the white-noise (broadband) description of the EM field modes, and allows us to replace p(f - x) by p(f). 

The Raman spectra of thin films of transition metal molybdates are shown in Fig. 2. The spectrum of bare Si (100) wafer is also given as reference. The intense and sharp peak located at 520 cm and a broad band extending from approximately 930 to 1000 cm attributed to the vibrational mode of the silicon substrate [11], are observed in all studied systems. All thin films are free of microcrystalline M0O3 since no signal at 997 cm was observed. Moreover, for Ni-Mo-0 samples a weak band at 830 and a shoulder at 960 cm" were observed. These confirm the presence of a-NiMo04 phase, which has Mo-0 stretching modes in the Raman spectrum at 706, 830 (weak), 914 and 960 cm" ... 

Note that the mode index n = 0 has been omitted from the buckling Bi to simplify the notation. Thus, as a first approximation to the Eq. (8.337), we use the solution (space form) given by (8.340) and compute the eigenvalues ixf from (8.240), keeping in mind that the are given by the usual relations for the bare reactor (e.g., in the case of the sphere, Bi = B = t/5). 

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