Legendre polynomial times

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

The quantity (3 cos2 w(t) — l)/2 is the orientation autocorrelation Junction it represents the probability that a molecule having a certain orientation at time zero is oriented at co with respect to its initial orientation. The quantity (3x — 1)/2 is the Legendre polynomial of order 2, Pi(x), and Eq. (5.32) is sometimes written in the following form... 

The values of these autocorrelation functions at times t = 0 and t = 00 are related to the two order parameters orientational averages of the second- and fourth-rank Legendre polynomial P2(cos/ ) and P4 (cos p). respectively, relative to the orientation p of the probe axis with respect to the normal to the local bilayer surface or with respect to the liquid crystal direction. The order parameters are defined as... 

Orthogonal collocation on two finite elements is used in the radial direction, as in the steady-state model (1), with Jacobi and shifted Legendre polynomials as the approximating functions on the inner and outer elements, respectively. Exponential collocation is used in the infinite time domain (4, 5). The approximating functions in time have the form... 

All the correlation functions above are normalized, therefore equations (4 and 5) are identical to correlation functions over linear momentum p = mv and angular momentum J — lu, respectively. Note that, in this context I is the moment of inertia tensor The correlation function in equation (6) is calculated over the spherical harmonics. If m = 0, this reduces to time correlation function over Legendre polynomials ... 

Orientational dynamics is also affected with T/ and T2, correlation times relevant to first and second Legendre polynomial for a vector defining the... 

As already mentioned in Sects. 1.1 and 1.2, a characteristic feature of intramolecular mobility in polymers is the existence of relaxation time spectra. In this case, the time dependence of the mean value of the Legendre polynomials of the 2nd order = (3/2)[(cos2e>- (1/3)] is given by Eqs. (1.2.9) and(1.2.10). 

For all Azo-PURs, the quantum yields of the forth, i.e., trans—>cis, are small compared to those of the back, i.e., cis—>trans, isomerization—a feature that shows that the azo-chromophore is often in the trans form during trans<->cis cycling. For PUR-1, trans isomerizes to cis about 4 times for every 1000 photons absorbed, and once in the cis, it isomerizes back to the trans for about 2 absorbed photons. In addition, the rate of cis—>trans thermal isomerization is quite high 0.45 s Q 1 shows that upon isomerization, the azo-chromophore rotates in a manner that maximizes molecular nonpolar orientation during isomerization in other words, it maximizes the second-order Legendre polynomial, i.e., the second moment, of the distribution of the isomeric reorientation. Q 1 also shows that the chromophore retains full memory of its orientation before isomerization and does not shake indiscriminately before it relaxes otherwise, it would be Q 0. The fact that the azo-chromophore moves, i.e., rotates, and retains full orientational memory after isomerization dictates that it reorients only by a well-defined, discrete angle upon isomerization. Next, I discuss photo-orientation processes in chromophores that isomerize by cyclization, a process that differs from the isomeric shape change of azobenzene derivatives. 

The second-order quadrupole interaction depends on the Wigner rotation matrices, which become time-dependent when the sample is rotated about an angle 6 with respect to B. The average second-order quadrupolar shift then depends on the Legendre polynomials... 

P (z) are the Legendre polynomials [51] which now constitute the appropriate basis set), Eq. (132) may be solved to yield the corresponding results for rotation in space, namely, the aftereffect function [Eq. (123)] and the complex susceptibility [Eq. (11)], with x and Xo from Eqs. (81) and (84), respectively. Apparently as in normal diffusion, the results differ from the corresponding two-dimensional analogs only by a factor 2/3 in Xo and the appropriate definition of the Debye relaxation time. 

In such cases, Y is a spherical harmonic and w is a product of a power of r, an exponential function, aud a Legendre polynomial in r. We want to take a snapshot so time is fixed and we only care about the spatial coordinates. If we want to visualize these solutions, it is useful to think about where the functions are zero and what sorts of symmetry they have. 

The factor—1/2 is obtained from the second Legendre polynomial Fa = (3cos 0 — l) for the angle 0 = 90°, which the quantization axis of the spins forms wiA the Bq held during spin locking. As a result of self-compensation of the dipolar interaction in the different evolution intervals, the initially excited coherences are completely refocused under the magic echo. Another echo arises after half the echo time during the spin-lock... 

In this expression, P2 is the second Legendre polynomial and i(t) is a unit vector with the same orientation as the transition dipole at time t. The brackets indicate an ensemble average over all transition dipoles in the sample. The correlation function has a value of one at very short times when the orientation of y(t) has not changed from its initial orientation. At long times, the correlation function decays to zero because all memory of the initial orientation is lost. At intermediate times, the shape of the correlation function provides detailed information about the types of motions taking place. Table I shows the three theoretical models for the correlation function which we have compared with our experimental results. 

The function (151) can be simplified in the long-time limit t> 1, when the average number of created photons, /L n h fa (V +U)/2 exceeds 1. Then the mean-square fluctuation of the photon number has the same order of magnitude as the mean photon number itself, s/2 Jf, and the most significant part of the spectmm corresponds to the values n > 1. Using the Laplace-Heine asymptotical formula for the Legendre polynomial [283]... 

---