Langevin magnetization

External magnetic field Volume-average Langevin magnetization... 

Fig. 6.15 Induced magnetic hyperfine fields, estimated from the spectra in Fig. 6.14, as a function of the reciprocal applied magnetic field. The full lines are linear fits in accordance with (6.20). The dotted line is a fit to the Langevin function. (Reprinted with permission from [58] copyright 1985 by the American Chemical Society)...

The function (a) is known as the Langevin function, after Paul Langevin, French physicist (1872-1946). The magnetic susceptibility of a paramagnetic substance can be expressed as (jim(B/kT). where fim is the magnetic moment, (S the magnetic flux, k the B and T die absolute temperature. 

We will present the equation of motion for a classical spin (the magnetic moment of a ferromagnetic single-domain particle) in the context of the theory of stochastic processes. The basic Langevin equation is the stochastic Landau-Lifshitz(-Gilbert) equation [5,45]. More details on this subject and various techniques to solve this equation can be found in the reviews by Coffey et al. [46] and Garcia-Palacios [8]. 

Weber in 1854 had attributed paramagnetism to the orientation of little permanent magnets in the substance (and diamagnetism to induced currents, as discussed above). A quantitative treatment was developed by Paul Langevin in 1895, by application of the Boltzmann principle. The theory is the same as for the orientation of electric dipoles (see App. IX). It leads to the equation... 

The conference was opened with a speech by Lorentz on the theory of electrons he had developed about 20 years before, followed by papers by Joffe on the electrical conductivity of crystals, Kamerlingh Onnes on superconductivity, and Hall on the metallic conduction and the transversal effects of the magnetic field. This last speech was followed by a discussion in which Langevin and Bridgman injected a few interesting remarks. 

For a system of nickel particles covering a certain size range, the magnetization curve will be composed of the Langevin functions of the individual particles (see also Becker (lie)) ... 

Figure 2. Magnetization curves of the tCo = 0.2 ran sample fit by averaging a Langevin curve over a Gaussian distribution of sizes with a =0.26. Inset, dotted line, fit to the paramagnetic contribution of non-aggregated Co above 2500 Oe/K, thick line, contribution due to the Co clusters.

The temperature dependence of the individual linewidth, eq. (21), is rather complicated. At very low temperatures the Langevin function in this equation becomes saturated for the majority of the nanoparticles, so that the main mechanism of this dependence is the thermal modulation of the magnetic anisotropy energy. Indeed, one can see from figure 8 (bottom) that the Ar temperature dependence provides a good estimate of the experimental low-temperature linewidth. As the damping factor linearly depends on the linewidth, it follows the same temperature dependence [11],... 

Further applications to new fields can be found in the work of P. Langevin On the Recombination of Electrically Dissociated Gases (ThSse, Paris 1902) and On the Magnetic Permeability of Gases (/. d. Phye., 4 [1905], 678). 

Magnetic and electric double refraction in liquids. Its explanation by the suspension of unobservable little crystallites whose complete alignment is impeded by the heat motion. A. Cotton and H. Mouton, Bull. aoc. de phya., 1910, p. 189 P. Langevin, Le Radium, 7 (1910), 249 O. M. Corbino, Phya. Zeitachr., 11 (1910), 756. 

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