Orthogonal components

As we are particularly interested in surface reactions and catalysis, we will calculate the rate of collisions between a gas and a surface. For a surface of area A (see Fig. 3.8) the molecules that will be able to hit this surface must have a velocity component orthogonal to the surface v. All molecules with velocity Vx, given by the Max-well-Boltzmann distribution f(v ) in Cartesian coordinates, at a distance v At orthogonal to the surface will collide with the surface. The product VxAtA = V defines a volume and the number of molecules therein with velocity Vx is J vx) V Vx)p where p is the density of molecules. By integrating over all Vx from 0 to infinity we obtain the total number of collisions in time interval At on the area A. Since we are interested in the collision number per time and per area, we calculate... 

The vector, pnoT, is computed, which is the part of the spectrum of the nth component orthogonal to the spectra of all other components in the original matrix. 

Trim pulses are short spin-locking periods without compensation for rf inhomogeneity. Magnetization components orthogonal to the spin-lock axis are dephased. The minimum duration of a trim pulse for elimination of the unwanted magnetization components depends on the rf amplitude and can be estimated by... 

FlP(E — e— jPaijPa) Density of states in momentum space that is a convolution of the components orthogonal and parallel to the direction of pa. E e—jPa is the kinetic energy allowed in the orthogonal directions. 

If there is still systematic variation left after fitting the two first principal components, it is possible to determine a third component, orthogonal to the first two. This can then be repeated until all systematic variation has been exhausted. Principal components modelling will thus constitute a projection of the data in the K dimensional descriptor space down to a lower dimensional (hyper)plane spanned by the principal component vectors. The projections are made so that the systematic variations of the original data are described by the coordinates along the principal component vectors. 

In the common case of ft consisting only of a z-component orthogonal to the vector f, we define ... 

Figure 10.40. Loadings for the two-component orthogonal PARAFAC model.

It is known from defects in the C3v symmetry group, that an externally applied field with components orthogonal to the three-fold symmetry axis of the defects, which is defined to be along z, lifts the degeneracy of the excited state 3E level. A.Kaplyanski et al., Soviet Physics Solid State (USSR) 12, 1193(1970). 

Furthermore, the field components orthogonal to the x-directed elements, namely the y- and z-directed components as expressed for example by (3.55) and (3.60), respectively, has so far not been possible to integrate into a simpler form. 

The shear stress may be further resolved into two in-plane components (orthogonal to each other). 

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