Curling mode

The Stoner-Wohlfarth approach works fairly well for very small particles, where VM = 0 is a good approximation. However, it has been known for decades that neither the Stoner-Wohlfarth theory nor the additional consideration of the curling mode account for the coercivity of real materials. For example, the coercivity of optimized permanent magnets is only 20-40% of the anisotropy field 2Kx nMs, and only a part of the discrepancy can be ascribed to the curling terms in Eq. (11). The reason is that real-structure imperfections make it impossible to consider the magnets as perfect ellipsoids of revolution. 

Goolaup S, Singh N, Adeyeye AO, Ng V, Jalil MBA (2005) Transition from coherent rotation to curling mode reversal process in ferromagnetic nanowires. Eur Phys J B 44 259-264... 

Figure 10. Nucleation modes in homogeneous magnets (a) coherent rotation in a sphere, (b) curling in a sphere, and (c) curling in a cylinder. The arrows show the local magnetization M = Mz ez + m, where ez is parallel to the axis of revolution of the ellipsoid (cylinder).

Figure 13. Spin-wave modes in nanowires (a) is the coherent mode (kL = 0) and (b) is of the curling type. When k l/R, then the perpendicular spin waves are essentially superpositions of plane waves exp(iA x) and t p( ikyy). Since the diameter of typical nanowires is much larger than interatomic distance, there are many excited perpendicular modes, and the finite-temperature magnetization MS(T) is reminiscent of bulk magnets.

Figure 13 shows various types of spin-wave modes in long nanowires. In very thin nanowires, where R < Rcoh, curling-type modes can be ignored [55, 114], and the perpendicular magnetization components obey Mx =... 

Here V n and V x n are the divergence and the curl of n. The three contributions to Wd are associated with the three independent modes of distortion splay, twist, and bend, depicted in Fig, 10-6. Terms of higher order than quadratic in Vn are only required if spatial distortions become severe. The Frank constants K, K2, and are of the order u /a,... 

Figure Ic illustrates an example of GSAMS in shrink mode using reduced stepwidths of El ew curl produces a higher probability of locating the...

Further shrinkage of the search space to Ll ew curl excludes the possibility of a direct move from C to E in one step while the probability for a move to E in a second step remains the same as before, i.e., 2 4 x 1/4 = 0.125. Note that a shrink process is suppose to be preceded by an expand mode but this aspect was not demonstrated in this brief discussion. See Appendix B for the pseudo-code listing of the GSAMS algorithm. 

Here e1 is the fundamental transverse microscopic electric held operator and b is the corresponding magnetic held operator. The superscript on the electric held operator designate its transverse character with respect to the direction of propagation, redundant in the case of the magnetic held as it is intrinsically transverse, namely, divergence-free, since it arises from the curl of a vector potential held a(r). Since the electric held also derives from a(r), we concentrate first on the second-quantized form of this vector potential, which is cast in terms of a summation over radiation modes as follows ... 

Mode of Preparation.—The trees are generally cut down, and the bark, after removal om the stem and branches, is carefully dried, so that it may retain its bright colour the larger and thicker portions are dried, so as to form flat pieces, while the smaller are allowed to curl into quills. Unless very coarse or injured, the epidermis, with the lichens which grow on it, is carefully preserved on the bark. 

Simon U, Miller CE, Bradley CC, Hulet RG, Curl RE, Tittel FK. 1993. Difference frequency generation in AgGaS2 using single-mode diode laser pump sources . Opt. Lett. 18(22) 1062-1064. 

Extensive laser-microwave investigations have also been performed in the spectrum of NH2. Hills and Curl, Jr. observed strong electric dipole microwave transitions between a previously unobserved rovibronic level and the J = 1/2 and J = 3/2 spin-rotational levels of ho, A ir(0,10,0), respectively. The NH2 molecules passed through a resonant half-wave microwave cavity with a 60-mW single-mode cw dye laser beam along the axis. The microwave field was amplitude modulated, and the laser-excited fiuorescence signal was detected with a phase sensitive amplifier. Microwave transitions in that part of the NH2 spectrum were used by Hills to assign numerous optical transitions. 

Considering first the magnetic modes s = 1, it is convenient to shift the curl operator appearing in the vector potential dA r) to the orbitals... 

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