Hysteresis butterfly

Molecules with small spin have also been discussed. For example, time-resolved magnetization measurements were performed on a spin 1/2 molecular complex, so-called V15 [81]. Despite the absence of a barrier, magnetic hysteresis is observed over a time scale of several seconds. A detailed analysis in terms of a dissipative two-level model has been given, in which fluctuations and splittings are of the same energy. Spin-phonon coupling leads to long relaxation times and to a particular butterfly hysteresis loop [58, 82],... 

Several sophisticated physical studies of the Vis cluster have been published, for example I. Chiorescu, W Werns-dorfer, A. Muller et al., Butterfly Hysteresis Loop and Dissipative Spin Reversal in the S = 1/2, Vis Molecular Complex, Phys. Rev. Lett. 2000, 84, 3454-3457 B. Barbara, I. Chiorescu, W Wernsdorfer et al. The Vis Molecule, a Multi-Spin Two-Level System Adiabatic LZS Transi-tion with or without Dissipation and Kramers Theorem, Progr. Theor. Phys. Suppl. 2002, 145, 357-369. 

Addition of galactomannan to agarose-water systems profoundly alters the optical rotation behavior. Thus, both in the absence and presence of locust-bean gum, there is, below 35°, a sharp shift in optical rotation. However, these shifts are opposite in sign, as shown in Fig. 6. The reheating curves also differ, in that, when locust-bean gum is present, a butterfly-shaped hysteresis curve is obtained.40... 

An example will help to clear this point up. Fixing the value of y at 15, we can compute the cusp of fifth-order (or butterfly) singular points originating from the sixth-order singularity in the feasible region. When these points are plotted in the v-B plane, the result is the upper graph in Fig. 13. Choosing v and B to have the values 0.7520 and 930,000, the hysteresis variety projected... 

The whole phenomenology of phase behavior and emulsion inversion was interpreted wifli a butterfly catastrophe model with amazing quahtative matching between theory and experiment. The phase behavior model used the Maxwell convention which allows the system to split into several states, i.e., phases at equilibrium. On the other hand, the emulsion-type model allows for only one state (emulsion type) at the time, with eventually catastrophic transition and hysteresis, according to the perfect delay convention. The fact that the same model potential permits the interpretation of the phase behavior and of the emulsion inver sion (204, 206) is a symptomatic hint that both phe-nomenologies are linked, probably through formulation and water/oil composition which are two of the four manipula-ble parameters in the butterfly catastrophe potential. 

The butterfly catastrophe model explains why the transitional inversion is not really an inversion but a surfactant transfer from one phase to the other, while the catastrophic inversion is a nonreversible hysteresis type instability. This approach, which is out of the scope of this chapter, is well documented elsewhere (197). 

Figure 6.12 Strain versus applied electric field loops (a) symmetrical butterfly loop for a ferroelectric with a symmetrical P-E hysteresis loop (b) experimentally determined loop for a fine-grained ceramic 0.95Bi Nag iO -0.05KNbO ...

Dipole reorientation under the influence of an electric field was investigated in oriented nylon 11 films with polarised IR spectroscopy for the amide A (N-H stretching) and amide I (carbonyl stretching) bands. Butterfly-shaped hysteresis loops were obtained from peak intensity versus applied electric field strength. Least-squares Gaussian... 

Soh, A.K., Song, Y.C., and Ni, Y. 2006. Phase field simulations of hysteresis and butterfly loops in ferroelectrics subjected to electro-mechanical coupled loading. Journal of the American Ceramic Society 89[2] 652-661. 

The buffer-fly curve of FMR fi-equencies vs. applied electric field was observed and shown in Fig. 2 (b), which resembled the widely observed piezoelectric strain vs. electric field butterfly curves for piezoelectric materials and matched the ferroelectric P-E hysteresis loop of PMN-PT single crystal as well. This once again confirmed that the change of FMR fi equency of the FeGaB film results from the ME coupling induced strain in the FeGaB film. 

Figure 2. Permeability spectra of the FeGaB/Si/PMN-PT heterostnicture under different electric fields (left) The butterfly sh d hysteresis of the FMR fiequency vs. electric field of the FeGaB/Si/PMN-PT heterostructure, and the ferroelectric P-E hysteresis loop of the PMN-PT single...

The FMR field of the Fe304/PZT and Fe304/PMN-PT muitiferroic composites exhibited the characteristic butterfly shape in their FMR field vs. electric field curves as shown in Fig. 7, which coincided with the ferroelectric hysteresis loops of the PZT and PMN-PT respectively and were similar... 

Figure 7. Butterfly curves of resonance fields vs. electric fields and ferroelectric hysteresis loops of muitiferroic composite Fe304/PZT (a) and FesOVPMN-PT (b).

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