Vectors critical, angle

Under this condition, the vectors are co-aligned (with critical angle 0X2c = 0), so the Schwarz inequality (11.4) necessarily becomes an equality... 

Equation (11.126) shows that the critical angle coc relating base vectors Rx, R2 to eigenvectors Ui, u2 at (TC,PC) is determined by the ratio of geometrical and arithmetic means of the diagonal metric elements. Inserting the specific metric expressions from Table 11.1, we obtain finally... 

Figure 7. Scattering geometry vertical scattering (reflectivity) and in-plane scattering (diffraction). The x-rays (or neutrons) with incident wave vector k strike the surface at a small angle of incidence (/), which is less than the critical angle for total external reflection, . In reflectivity measurements, the vertical incidence and exit angles are equal so the difference between k and the reflected wave vector k, is perpendicular to the scattering plane. In diffraction measurements, the difference k — k,i lies essentially in the plane.

Fig. la. Schematic showing the optical field (magnetic component) at an interface which supports surface plasmons. The dielectric function in the dielectric medium is the diectric function in the metal can be approximated hy the Drude-Lorentz expression given in the upper right hand corner. Notice that the field extends much farther into the dielectric than the metal, b. The reflectivity in an ATR configuration. The 0 is the critical angle and 0gp is the angle at which the surface plasmon is excited. Reflectivity extends from zero to one. Notice that the reflectivity from s waves, i.e., those waves with their electric vector perpendicular to the plane of incidence do not excite a surface mode.. 

In internal reflection, at angles of incidence larger than the critical angle, electromagnetic radiation is totally reflected (attenuated total reflectance, ATR. see Section 16.2.2.4 and Fig. 5). This special ca.se is very important in analysis for two approaches. First, simple transportation of radiation within the fiber (or a waveguide). Second, in total reflection, an evane.scent field appears in which the electrical field vector decays exponentially in the optically less dense medium. Every change within the medium with lower refractive index influences the field vector coupled to the field in the optically denser medium. Therefore, the totally reflected radiation contains information about effects on the other side of the phase boundary (the medium with lower refractive index) [20], [144]. Various principles to interogate this effect are known and used in evanescent field sensors. 

Fig. 7.4 A-C. A model of control of morphogenesis by vectors of auxin wave propagation in the final phase (autolysis) of axial tracheid maturation in secondary xylem of the stem. A Trajectories of vectors of auxin wave propagation (a) in the cambial region and in differentiating secondary tissues as seen in radial (b) and transverse (c) sections. The breakdown of the cytoplasm is initiated when some critical angle (a) between the vector of auxin wave propagation and the cell axis is attained. B and C Vector trajectories associated with a model of regulation of earlywood and latewood differentiation. Ph conducting phloem Ph dividing phloem Cj cambial initial dividing xylem ...

Fig. 13—Normalized o>/b as function of tglb, cta/b is the critical shear stress to move a dislocation from the B layer into the A layer, Q=(G -Gb)/(G +Gg), G and Gg are the shear moduli of A and B, b is the Burgers vector, fg is the thickness of one single B layer, and e is the angle between the A/B interfaces and the dislocation glide plane.

Let us have a closer look at the differences between the minimal and the extended set of equations and follow these differences along some paths in the parameter space. As mentioned in Sect. 2.3, we can omit some of the physical parameters by using dimensionless parameters. In Figs. 5-9 we show the dependence of the critical values of the tilt angle and wave vector on the dimensionless parameters (as defined... 

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