Function Langevin

In the classical model the magnetic entities carry a localised magnetic moment m0 having an arbitrary direction. The magnetic energy in an applied field is 

The proportion of magnetic moments whose direction belongs to the solid angle 

The average value of the magnetic moment m along the field is 

For small values of the argument the Langevin function can be expanded to yield 

In such a case the magnetisation varies linearly with the applied field 

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On the basis of the classical theory it was shown by Debye2 that cos 9 is given by the Langevin function... 

The quantity in brackets is the Langevin function of rllkT, which may be written Jlffrl/kT). Hence... 

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)...

Lagrange multipliers 255-256 Lagrange s moan-value theorem 30-32 Lagperre polynomials 140, 360 Lambert s law 11 Langevin function 61n Laplace transforms 279—286 convolution 283-284 delta function 285 derivative of a function 281-282 differential equation solutions 282-283... 

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. 

If the molecular effects of the electric field are irrelevant to microwave heating of solutions, this assumption could be envisaged in the use of operating conditions very far from current conditions. On one hand, it will be necessary to use an electric field of higher amplitude, or to reduce the temperature according to the Langevin function. This last solution is obviously antinomic with conventional chemical kinetics, and the first solution is, currently, technologically impossible. It will, on the other hand, be necessary to avoid reaction media with dielectric loss. The molecular effects of the microwave electric field could, paradoxically, be observed for a medium which is not heated by the action of microwave irradiation. 

Here, I n is the reduced Bessel function, and its argument L (m) is the inverse Langevin function (see Callen and Callen 1963, 1965). Using the values given above for... 

Equations (28) and (29) are derived from the statistical theory based on the Gaussian statistics which describes the network behaviour if the network is not deformed beyond the limit of the applicability of the Gaussian approximation33). For long chains, this limit is close to 30 % of the maximum chain extension. For values of r, which are comparable with rmax, the force-strain dependence is usually expressed using the inverse Langevin function 33,34)... 

U(0) is the orientation dependent local potential energy and U (6) accounts for any anisotropy due to the local environment. For instance, if the sample were liquid crystalline, U (O) and U (rc) would represent the potential well associated with nematic or smectic director. In addition to local potentials it is clear that f(0) depends on the experimentally controllable quantities p, E, and T. The solution to equation 16 is the series expansion referred to as the third-order Langevin function L,3(p)... 

In the limit of the oriented gas model with a one-dimensional dipolar molecule and a two state model for the polarizability (30). the second order susceptibility X33(2) of a polymer film poled with field E is given by Equation 4 where N/V is the number density of dye molecules, the fs are the appropriate local field factors, i is the dipole moment, p is the molecular second order hyperpolarizability, and L3 is the third-order Langevin function describing the electric field induced polar order at poling temperature Tp - Tg. 

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)) ... 

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