Perpendicular, definition

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

To continue, we assume the following situation We concentrate on an x-y plane, which is chosen to be perpendicular to the seam. In this way, the pseudomagnetic field is guaranteed to be perpendicular to the plane and will have a nonzero component in the z direction only. In addition, we locate the origin at the point of the singularity, that is, at the crossing point between the plane and the seam. With these definitions the pseudomagnetic field is assumed to be of the form [113]. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

Anuther concept that is extremely powerful when considering lattice structures is the fi i i/imca/ lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar juoduct of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions ... 

PR = perpendicular recording LR = longitudinal recording HFRH = high frequency recording heads RH/HDTV = recording heads for high definition television and Pa = perpendicular perpendicular anisotropy. 

Now the effective conductivity ia the direction of the electric field is <7/(1 + /5 ), ie, the scalar conductivity reduced by a factor of (1 + /5 ) by the magnetic field. Also, the electric current no longer flows in the direction of the electric field a component j exists which is perpendicular to both the electric and magnetic fields. This is the Hall current. The conductivity in the direction of the Hall current is greater by a factor of P than the conductivity in the direction of the electric field. The calculation of the scalar conductivity starts from its definition ... 

If L is a charac teristic dimension of a ciystal of selected material and shape, the rate of growth of a ciystal face that is perpendicular to L is, by definition,... 

The first pseudo force, Fi, is called the Coriolis force, and its magnitude is directly proportional to the angular velocity of the rotating frame of reference and the linear velocity of the particle in this frame. By definition, this force is perpendicular to the plane where vectors Vi and o are located, Fig. 2.3a, and depends on the mutual position of these vectors. The second fictitious force, F2, is called the centrifugal force. Its magnitude is directly proportional to the square of the angular velocity and the distance from the particle to the center of rotation. It is directed outward from the center and this explains the name of the force. It is obvious that with an increase of the angular velocity the relative contribution of this force... 

Taking into account the fact that the vector of angular velocity is constant, the right hand side of this equation can be simplified and with this purpose in mind we make use of Fig. 2.3e, where the vector R is perpendicular to the axis of rotation. By definition,... 

Figure 32.8 shows the biplot constructed from the first two columns of the scores matrix S and from the loadings matrix L (Table 32.11). This biplot corresponds with the exponents a = 1 and p = 1 in the definition of scores and loadings (eq. (39.41)). It is meant to reconstruct distances between rows and between columns. The rows and columns are represented by circles and squares respectively. Circles are connected in the order of the consecutive time intervals. The horizontal and vertical axes of this biplot are in the direction of the first and second latent vectors which account respectively for 86 and 13% of the interaction between rows and columns. Only 1% of the interaction is in the direction perpendicular to the plane of the plot. The origin of the frame of coordinates is indicated... 

The rod forms also show considerable variation. A rod is usually considered to be a cylinder with the ends more or less rounded. Some rod forms are definitely ellipsoidal in shape. The ends of rods also show considerable variation. Some species are markedly rounded others exhibit flat ends perpendicular to the sides. Gradations between these two forms may be seen. 

As can be seen in Figure 8.23, mirror images of this structure (reflected through a vertical mirror perpendicular to the plane of the page in the figure) are nonsuperposable—classic chirality by the Lord Kelvin definition, extended... 

As indicated in Table 4.2, the eigenvalues of the Hessian matrix of fix) indicate the shape of a function. For a positive-definite symmetric matrix, the eigenvectors (refer to Appendix A) form an orthonormal set. For example, in two dimensions, if the eigenvectors are Vj and v2, v[v2 =0 (the eigenvectors are perpendicular to each other). The eigenvectors also correspond to the directions of the principal axes of the contours of fix). 

The spectra obtained from the chemisorption of methanol onto catalyst above 100°C indicated the progressive oxidation of methoxy species to formate via dioxymethylene/HCHO and finally to CO, CO2, and H2. Phenol adsorbed on the surface Lewis acid-base pair site and dissociated to phenolate anion and proton. The formation of phenolate anion and proton were discerned from the strong intense C-0 stretching vibration and the disappearence of phenolic 0-H stretching vibration, respectively. Importantly, there were series of definite low intensity bands between 2050 and 1780 cm" that were identified as the out-of-plane aromatic C-H bending vibrations [79, 84-85]. These bending vibrations are possible only if the phenyl ring of phenol is perpendicular to the catalyst surface. 

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