Order parameter Bessel function

S 1 + 2 V2wdP/6rt [exp (—0r2/dP) ]/i (0r2/dr) in which J1 denotes the first-order modified Bessel function and X and 9 are adjustable parameters. X determines how much higher the velocity is near the wall than at the axis, and 6 determines how flat the profile is... 

Iq is the zero-order modified Bessel function of the first kind [24]. Equation 14.56 shows that a plot of x = C/Cq versus t depends only on the two dimensionless parameters Nrea and Req-... 

In Eq. (77), x = h(o/2kg T is the reduced internal frequency, q = EJhoi the reduced solvent reorganization energy, p = hElha> the reduced electronic energy gap and / (z) the modified Bessel function of order m. The quantity S is a coupling parameter which defines the contribution of the change in the internal normal mode ... 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

In Section 3 we derive that for the vacancy-mediated diffusion mechanism, one expects the shape of the jump length distribution to be that of a modified Bessel function of order zero. Both distributions can be fit very well with the modified Bessel function, again confirming the vacancy-mediated diffusion mechanism for both cases. The only free parameter used in the fits is the probability prec for vacancies to recombine at steps, between subsequent encounters with the same embedded atom [33]. This probability is directly related to the average terrace width and variations in this number can be ascribed to the proximity of steps. The effect of steps will be discussed in more detail in Section 4. 

The potential parameters for the free-electron case are interesting for several reasons. First of all they are easy to calculate analytically from standard expressions for the spherical Bessel functions [4.2] and therefore useful in order-of-magnitude estimates. Secondly, they may be used in empty-lattice tests of the LMTO method in order to indicate the accuracy of that method in various applications. Such tests are also useful for programme debugging purposes. Thirdly, muffin-tin orbitals with free-electron parameters are used in Sect.6.9 to derive a correction to the atomic-sphere approxir mation. 

In these equations, all variables and parameters are local except for NTU0, Cf, and e. Here / represents the modified Bessel function of the first kind and nth order. Shah [32] has tabulated the effectiveness of Eq. 17.46 for 0.5 < NTUe < 500 and 1 < C <... 

Ii (/) is the modified Bessel function of the first order. J is a dimensionless parameter defined as... 

In these equations 5 is the reciprocal of the X-radiation wave length and bo is the magnitude of the fibrillar axial period (ca. 640 A.). The voa and (Toa are mean values for the respective displacement parameters averaged over the fibrillar levels at which distortion occurs. The symbol Ji signifies the first-order Bessel function whose argument is included after it in parentheses. 

Here /o(x) and /i(x) are Bessel-functions of the first kind, with 0 and 1 order correspondingly. Transformation from electrical unit parameters to electrochemical ones is given for this case as ... 

Equations (17), (19) and (20) acquire considerable importance in the analysis of small angle neutron scattering from block copolymers and their nature is displayed in Fig. 3. In this semi logarithmic plot the characteristic maxima are observed with increasing Q, however, the higher order maxima are severely damped and only observable in such semi-logarithmic plots. The maxima in such plots appear at characteristic values of QD (where D is the dimension parameter used in the argument to the Bessel Function), Table 3 sets out these values for the three particle types. 

In the case of a very diluted polymer network or at high temperatures liquid crystal is ordered only in the vicinity of the polymer fibers. If the order parameter 5 is small the quadratic and cubic term in (12.53) can be neglected and an analytical solution can be expressed with the modified Bessel functions Ko r) and K r) [74] which asymptotically leads to s r) oc For a particular case, profiles are shown in Figure 12.27. 

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