Power-exponential relaxation

The actually observed relaxation function in the fast limit does not alw s follow strictly an exponential decay. In practice a power exponential relaxation is often used... 

Fig. 133. Temperature dependenee of the power in the power-exponential relaxation fit to the ZF speetra of single-erystalline CePt2Snj. From Luke et al. (1997a).

As indicated above, exponential relaxation does not generally provide a satisfactory description of experimental data. The response of dielectric materials is instead generally found to be characterized by certain power laws, both in the time and frequency domains (95,96). As observed by Curie and von Schweidler, a long time ago (97,98), the polarization current following the application of a constant electric field typically decays as a power function of time. The same is true for the dielectric response function, since it is proportional to the current oc l/t , where a is a positive constant. This response... 

Fig. 102. Left Temperature dependence of the near ZF muon spin relaxation rate in i-Tb Mg,2Zn5(i on double logarithmic scales. The vertical dashed line indicates the onset of loss of asymmetry due to instnunental dead time. The inset shows the temperature dependence of power used in the power exponential fit to the relaxation spectra. The spin-glass temperature is near 8K. Right pSR asymmetry spectra of i-Gd Mg42Zn5o at 1.5 K, in the applied longitudinal fields shown. The solid lines are a least-squares fit of a mildly inhomogeneous freezing...

In the paramagnetic regime (T > Tm), the spectra in a weak LF (needed to suppress the depolarization by Cu nuclear dipoles) for x > 0.08 were most easily fitted to a power exponential (exp[—(At) ]) relaxation. Hence the summary label relaxation rate in fig. 113 (left) refers to the static width Aeff (see eq. 74) for T dynamic rate A for r > Tu- The variation of power p was studied in some detail for the 10% sample. A decrease fromp w 1 at high temperatures top w 0.6 close to Tm was found. This is another indication that a disordered spin-glass-like state is approached and 7m might best be considered a spin freezing temperature. This spin-glass-like state, however,... 

In CePd2Sn2 an AFM transition with 7 w 0.5 K has been established by Beyermann et al. (1991a,b) for polycrystalline material. The pSR data by Lidstrom et al. (1996a) are fully compatible with this view. Since there is no significant nuclear depolarization in the Pd compound, the temperature variation of (exponential) relaxation rate A directly reflects the dynamics of Ce moments. For T > one observes a critical power... 

In the nineteenth century, the Maxwell equations appeared, based upon electromagnetic theory and juxtaposed with light. In the twentieth century came the Debye tradition of regarding biomolecules as polar materials with exponential relaxation. However, experimental findings showed that most dielectrics do not show exponential relaxation, but fractional power law relaxation. Fractional power law was the universal property. Then came the Cole models based upon a new component, the constant phase element (CPE). 

Figure 10.10 shows measurements on the 187, 1030, and 39200 kDa systems, and fits to the temporal scaling ansatz. The ansatz fits well at most frequencies. In some cases a second high-frequency stretched-exponential relaxation, additive to the power-law relaxation, is demanded by the data. The G"(co) values in Figure 10.10c make clear that the high-frequency relaxation is not a second, additive power-law decay. For the storage modulus, while some measurements on the 187 and 1030 kDa material extend into some low-frequency regime in which a power-law description was inapplicable, measurements did not reach sufficiently... 

Graessley, et al. report t]o, r](K), 7 , and Ni for linear, four-arm star, and six-arm star polyisoprenes in tetradecane(35). Figure 13.23 shows y(/c), which is uniformly consistent with a simple exponential relaxation exp(—a/c) at lower shear rates, and a power-law form k at higher shear rates, seen most clearly in Figure 13.23a for the 1611 kDa linear chains and the 1950 kDa four-arm stars. For some concentrations and polymer molecular weights, the observed frequency range includes only an exponential or only a power-law k dependence. Graessley,... 

Ito and Shishido examined shear thinning of 170 kDa polydimethylsiloxane in the siloxane pentamer(37), leading to Figure 13.25. In the more concentrated solutions, ri(K) varies sufficiently to reveal a stretched-exponential c-dependence at lower k and a power-law relaxation and large-shear-rate constant at larger k. 

---