Radial parameter, definition

The radial parameter about any point is defined by r = rtfiN) since this function is constrained to be monotonic, its inverse exists so that, by definition, N=f l r/rf). Suppose that we now introduce the scale constant mo then Nmo = mo/ 1(r/ro) = M r) can be interpreted as quantifying the total amount of material inside a sphere of radius r centered on the assumed origin. Although r = rof(N) and M r) = Nm ) are equivalent, the development that follows is based on using M r) as a description of the mass distribution given as a function of an invariant radial distance parameter, r, of undefined calibration. 

Figure 10.3 Definition of geometrical parameters for a CMA shown for the case of off-axis focusing of a point source (Q). Rt and Ra are the radii of the inner and outer field cylinders, respectively 4, is the radial image distance to the inner cylinder (in analogy, one can introduce a radial source distance ds, in the present case one has ds = R,) z is the total distance between source and image measured along the symmetry axis of the analyser zf is the corresponding distance for the field region is the entrance angle into the analyser (due to symmetry properties this is equal to the exit angle).

Note the appearance of the AB equilibrium geometry (Rm, which can be obtained by a self-consistent procedure136) and Le Roy s parameter142 [R0, which represents the smallest value of the internuclear distance for which the asymptotic series of the dispersion energy is still a good representation of the damped series (49)] in the definition of the reduced coordinate x represents the expectation value of the square of the radial coordinate for the outermost valence electrons, which is tabulated in the literature143 for atoms with 1 120. Other important parameters in the dispersion damping... 

The experimentally determined values of parameters Ki and K2 allow the Pigford model to be used to examine the extent of the spin-up zone on the disk surface. Before this can be done, however, an unambiguous mathematical definition for the spin-up zone is required. This was provided by Burns et al. as the location at which the radial velocity stops accelerating and starts following the decelerating profile described by the synchronized flow model as shown in Eq. (3). ... 

Since the ss and the soo interactions are represented by two-electron operators, they comprise, in analogy with the electrostatic interaction, both a direct and an exchange part. For the spin-spin interaction, these parts are described in terms of the radial integrals A/ (n/, n / ), M ln l, nl)-and N nl, n l ) respectively. The definitions of these parameters are given below (Marvin, 1947). 

Fdyn and Fstat are the dynamic and static relaxation matrices, respectively, is the direction cosine of the intemuclear vector r with respect to the Cartesian coordinate or, and AaA represents the thermal fluctuations of the spin pair v = (i, j). For a static structure, y is equal to the order parameter S, which is by definition less than or equal to 1. For a dynamic structure, y can be greater than 1 if radial fluctuations exceed angular ones. 

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