Legendre polynomial, second order

Here Pj is the Legendre polynomial of order i, the angular brackets refer to the ensemble average over all conformational transitions undergone within the time interval x by the chain segment to which m is rigidly aflbced. The subscript i assumes the values 1 and 2 in the respective cases of first and second OACFs. 

Here 0 is the recoil angle relative to the electric vector of the dissociating light and P2 is the second-order Legendre polynomial. 

Fig. 2 The values of second-order P2 and fourth-order P4 Legendre polynomials as functions of [iRL. The two polynomials do not have a common root, i.e., the anisotropic terms in the second-order quadrupolar Hamiltonian cannot both be canceled at once, regardless of the orientation of the rotor axis...

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... 

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... 

In elastomer samples with macroscopic segmental orientation, the residual dipolar couplings are oriented as well, so that also the transverse relaxation decay depends on orientation. Therefore, the relaxation rate 1/T2 of a strained rubber band exhibits an orientation dependence, which is characteristic of the orientational distribution function of the residual dipolar interactions in the network. For perfect order the orientation dependence is determined by the square of the second Legendre polynomial [14]. Nearly perfect molecular order has been observed in porcine tendon by the orientation dependence of 1/T2 [77]. It can be concluded, that the NMR-MOUSE appears suitable to discriminate effects of macroscopic molecular order from effects of temperature and cross-link density by the orientation dependence of T2. 

The orientational order parameter is the second Legendre polynomial of cos9 and, where the chemical shift dispersion is reduced by rotational averaging, is given by... 

---