Relaxation times spherical harmonics

Westlund developed also a theory for PRE in the ZFS-dominated limit for S = 1, which included a stringent Redfield-limit approach to the electron spin relaxation in this regime (118). Equations (35) and (38) were used as the starting point also in this case. Again, the correlation function in the integrand of Eq. (38) was expressed as a product of a rotational part and the spin part. However, since it is in this case appropriate to work in the principal frame of the static ZFS, the rotational part becomes proportional to exp(—t/3tb) (if Tfl is the correlation time for reorientation of rank two spherical harmonics, then 3t is the correlation time for rank one spherical... 

This leads to an infinite set of differential-difference equations, on substitution into Brown s equation. The calculation of the relaxation times is then achieved by selecting the relevant indices n and m, from which we obtain a differential-difference equation describing the time evolution of the average value of the desired spherical harmonic. Those spherical harmonics which are of interest to us here are P,(cos ), with n = l and m = 0, which describes the evolution of the alignment with the 2 axis and Pj(cos i )cos , with n = 1 and m = l which describes alignment perpendicular to the z axis. 

To enable us to calculate the relaxation time of the magnetization in a direction perpendicular to the applied field it is necessary to expand IV in spherical harmonics involving the two space coordinates i and . For this we use the form given in the introduction to this section which is... 

We first outline the approach of the following sections. Starting with Brown s equation we expand the distribution W(r, t) of orientations of M in spherical harmonics as in the previous chapter whence we obtain an infinite set of differential-difference equations. We then select the spherical harmonic of interest P for the longitudinal relaxation and P e" for the transverse relaxation. The relaxation times can then be expressed as in the previous section in terms of a continued fraction. Since we consider only the response with respect to a small applied field H, such that 1, it is only necessary to evaluate quantities linear in Furthermore we assume that equilibrium has been attained for the ratio R s) in the continued fraction and that this ratio is expressible in terms of the equilibrium values of the relevant spherical harmonics. This expression is achieved as follows. The average value of any spherical harmonic is... 

We have derived closed form expressions for the relaxation times Ty and Tj by assuming that spherical harmonics (in the differential-difference equations) of order two and higher reach their equilibrium values consid-... 

We reiterate that the availability of the closed form expressions for the relaxation behavior depends on the assumption that the ratio of the Laplace transform of the first and second order spherical harmonic averages has reached its final value appreciably faster than the ratio of the first and zero order. This means that for step-on and step-off solutions, we are considering processes that take place in times of order of magnitude... 

The spherical harmonic analysis so far presented for uniaxial anisotropy is mainly concerned with the relaxation in a direction parallel to the easy axis of the uniaxial anisotropy. We have not considered in detail the behavior resulting from the transverse application of an external field and the relaxation in that direction for uniaxial anisotropy. Thus we have only considered potentials of the form V(r, t) = V(i, t) where the azimuthal or dependence in Brown s equation is irrelevant to the calculation of the relaxation times. This has simplified the reduction of that equation to a set of differential-difference equations. In this section we consider the reduction when the azimuthal dependence is included. This is of importance in the transition of the system from magnetic relaxation to ferromagnetic resonance. The original study [17] was made using the method of separation of variables on Brown s equation which reduced the solution to an eigenvalue problem. We reconsider the solution by casting... 

This multiexponential approximation (one for each spherical harmonic) is more accurate than a single exponential description, yet still only depends on Dx and equilibrium averages. Equations (35) thereby provide a reasonable way of estimating from simulations D of lipids or other probes in membranes (assuming they can be modeled as rigid cylinders on the time-scale of rotational diffusion). Figure 6 plots these functions for parameters that model the long-axis rotation of a lipid in a bilayer Dx = 10 s, P ) = 0.8836, (P2) = 0.7, and Pi,) = 0.3467 (these Pi) are consistent with a Maier-Saupe potential with a field value of 3.8646 bT). Included (in dashed lines) are Ci(r) and C2(0 for unrestricted rotational diffusion. Note that the two relaxation times are somewhat shorter and closer to each other when diffusion is restricted. 

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