Coefficient coupling

E, E2, Eg = Young s (extension) moduli in the 1-, 2-, and 3-directions V j = Poisson s ratio (extension-extension coupling coefficient), i.e., the negative of the transverse strain in the j-direction over the strain in the i-direction when stress is applied in the i-direction, I.e.,... 

Lekhnitskii defines the coefficients of mutual influence and the Poisson s ratios with subscripts that are reversed from the present notation. The coefficients of mutual influence are not named very effectively because the Poisson s ratios could also be called coefficients of mutual influence. Instead, the rijjj and ri y are more appropriately called by the functional name shear-exitension coupling coefficients. 

Note that the Chentsov coefficients are more effectively called the functional name of shear-shear coupling coefficients. 

Compare the transformed orthotropic compliances in Equation (2.88) with the anisotropic compliances in terms of engineering constants in Equation (2.91). Obviously an apparenf shear-extension coupling coefficient results when an orthotropic lamina is stressed in non-principal material coordinates. Redesignate the coordinates 1 and 2 in Equation (2.90) as X and y because, by definition, an anisotropic material has no principal material directions. Then, substitute the redesignated Sy from Equation (2.91) in Equation (2.88) along with the orthotropic compliances in Equation (2.62). Finally, the apparent engineering constants for an orthotropic iamina that is stressed in non-principal x-y coordinates are... 

An important implication of the presence of the shear-extension coupling coefficient is that off-axis (non-principal material direction) tensile loadings for composite materials result in shear deformation in addition to the usual axial extension. This subject is investigated further in Section 2.8. At this point, recognize that Equation (2.97) is a quantification of the foregoing implication for tensile tests and of the qualitative observations made in Section 1.2. 

Extension-extension coupling coefficients (Poisson s ratios) Shear-extension coupling coefficients (coefficients of mutual influence) Shear-shear coupling coefficients (Chentsov coefficients)... 

Boys, S. F., and Sahni, R. C., Trans. Roy. Soc. [London) A246, 463, (ii) Electronic wave functions. XII. The evaluation of the general vector-coupling coefficients by automatic computation. ... 

In general, Q is a nonsymmetrical matrix whose components off the main diagonal are the coupling coefficients among the various fluxes involved. 

The mixing coefficients a and b in (4.10) depend upon the efficiency of the spin-orbit coupling process, parameterized by the so-called spin-orbit coupling coefficient A (or for a single electron). As A O, so also do a or b. Spin-orbit coupling effects, especially for the first period transition elements, are rather small compared with either Coulomb or crystal-field effects, so the mixing coefficients a ox b are small. However, insofar that they are non-zero, we might write a transition moment as in Eq. (4.11). 

The first two terms in the expansion are strictly zero because of the spin selection rule, while the last two are non-zero, at least so far as the spin-selection rule is concerned. So a spin-forbidden transition like this, X VT , can be observed because the descriptions X and are only approximate that is why we enclose them in quotation marks. To emphasize the spin-orbit coupling coefficients for the first row transition elements are small, the mixing coefficients a and b are small, and hence the intensities of these spin-forbidden transitions are very weak. 

Reminder The one-electron spin-orbit coupling coefficient, is intrinsically positive. The many-electron spin-orbit parameter X is defined by... 

Some useful spin-orbit coupling coefficients ... 

Harnung SE, Schaffer CE (1972) Phase-fixed 3-G Symbols and Coupling Coefficients for the Point Groups. 12 201-255... 

The QUICK scheme has a truncation error of order h. However, similarly as in the case of the central differencing scheme, at high flow velocities some of the coupling coefficients of Eq. (37) become negative. 

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,... 

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