Magnetization, spontaneous temperature-dependence

The magnitude of the spontaneous magnetization M in ordered magnets is temperature dependent due to spin wave excitations. Spin wave frequencies are so fast (THz) that they are fully motional-narrowed in jxSR. All one will observe is the expectation value of the internal field which is coupled (but not necessarily directly proportional) to the expectation value of M T). For this reason one calls the of an ordered magnet a quasistatic field. The spontaneous precession vanishes at a second-order magnetic transition point and reaches a saturation value for T —> 0. An example is shown in fig. 27. 

Fig. 2.19. Temperature dependence of the amplitudes of coherent phonons of Gd(0001) and Tb(0001). On the right axis, and Ap show the square of calculated spontaneous magnetization given by the Brillouin function with Jq =7/2 and =6/2 representing the magnetic moment of 4f electrons. From [59]...

The collected data allow us a rough estimation of the hypothetical Curie temperature of the martensitic phase. Figure 10 shows temperature dependencies of the reduced spontaneous magnetization m = Ms(T)/Ms(0) of the alloys as a function of reduced temperature t = T/Tc. It is seen that the magnetization of the austenitic phase and... 

For one-sublattice magnets, such as Fe and Co, the Akulov or Callen and Callen theory [81] relates the temperature dependence of the anisotropy to the spontaneous magnetization and yields hi and Af° power laws for uniaxial and cubic magnets, respectively. This theory has become popular far beyond its range of applicability [82] but is unable to describe structures such as rare-earth transition-metal magnets [16, 60], actinide magnets [83], and L10 type compounds [44, 84]. 

Figure 15.4 (a) Temperature dependence of x ag for a ferromagnetic solid that undergoes a transition at the Curie temperature Tc- (h) Spontaneous magnetization H = 0) of a ferromagnetic crystal as a function of temperature. 

The temperature dependence of the spontaneous magnetization of the ferromagnetic clusters is shown in Table II. Figure 5 shows the present data fitted to a... 

Fig. 27. Temperature dependence of the spontaneous muon spin precession frequency in single-crystalline TbNij with a Curie temperature of 23 K. The dashed curve is the appropriate fiee-ion Brillouin function. Inclusion of CEF interaction in the ground-state multiplet gives the solid curve which fits the data well. The behavior around the critical temperature is typical for a second-order magnetic phase transition. After Dalmas de Reotier et al. (1992).

Thus far, we have neglected the existence of magnetic domains. An ordered magnet usually consists of a multitude of imiformly magnetized sub-volumes, the domains. They all have the same magnitude of magnetization (namely the temperature-dependent spontaneous magnetization) but Mdom points in different spatial directions for different... 

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