Brillouin functions

The projection of the magnetic moment along the quantum axis (z) can adopt only discrete values and the corresponding Zeeman Hamiltonian term is 

This is a finite geometric series which can be summed exactly to yield 

Notice some expressions for the hyperbolic functions sinh(x) = exp exp(—-v) 

The magnetisation obeys the relation of statistical thermodynamics according to which 

This function is plotted in Fig. 6.1 for some representative values of spins. The Brillouin function can be met in literature also in the form 

T) can be obtained by replacing the bulk J and with the size-dependent J K) and EexcW that are given as follows  

Differentiating Eq. (19.3) against exc( leads to the size- and temperature-dependent Hs(.K, T)  

---

NiCr204 Study of magnetic hyperfine and quadrupole splitting, H(T)IH(To) = fit to Brillouin function with 5=1... 

Figure 9.4 Magnetization versus field plots for Gd(lll)s for a range of temperatures. The green line is the Brillouin function for five uncoupled spins, each of s = 35/2 and g = 2.0. (Taken from Ref. [23] with permission from Royal Society of Chemistry.)...

This tells us that the reduced magnetic entropy change will be lower in an antiferromagnetic system than in an uncoupled one, as can be seen below, as the spins are not fully saturated below the conditions of lowest temperature and strongest field. The antiferromagnetic coupling in Gd(III)7 is also a hindrance, with behaviour similar to Gd(III)5, when compared to the relevant Brillouin functions. 

Fig. 11 Left experimental XmT vs. T for 10 at Hic of 0.5, 1, 5, 10, and 25 KOe. Right field dependence of the magnetization of 10 at 2 K (open circle) compared with the Brillouin function for seven uncoupled Gd3+ ions (blue line) and the Brillouin function for an S = 42/2 state plus an 5 = 7/2 state red line)...

Fig. 17 Magnetization isothermals of [Fe(Cp )2][Ni(dmit)2] (squares) and [Fe(Cp )2][Pt(dmit)2] 0circles) at 1.8 K. The solid line corresponds to calculated values from the Brillouin function, while the dashed line corresponds only the contribution from the cation molecules. From [53]... 

In the case of [Fe(Cp )2]2[Cu(dcdmp)2], yj is nearly temperature independent, decreasing slightly at low temperatures (T < 30 K). This suggests that the magnetic behavior is due to weak AF D+A interactions. The isothermal obtained at 1.7 K, presents magnetization values slightly below those predicted by the Brillouin function, confirming the presence of very weak AF interactions [63]. 

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]...

One can measure H as a function of temperature to see where Hx vanishes. If one assumes Hx is proportional to the magnetization, one can fit a Brillouin function through the data and extrapolate to zero Hx to get the magnetic transition temperature. As an example, the Neel temperature of -Fe203 has been determined by this technique [see Figure 1 of Ref. 12]. 

Since in this limit the magnetization process of the Ising model is just given by the Brillouin function (h = H/kgT remaining finite)... 

The average magnetization I is given by (15), where Bj a) is called a Brillouin function, and a by (16). When a 1, Bj(a) may be expanded, and, if we take only the first term, then (17) results. The paramagnetic susceptibility X is inversely proportional to the absolute temperature T. This relation is called the Curie law, and the proportionality constant C is the Curie constant. 

Fig. 333 Ground-state splitting of Fe in Au determined by Mossbauer effect (Violet and Borg 1966) (a) observed (b) Brillouin function for s= 1.

Fig. 10. Magnetization for six samples of Gai tMnr As/GaAs with Mn compositions x ranging from 0.015 to 0.071 at 2 K. The magnetic field is perpendicular to the sample surface for x = 0.035-0.071 and parallel to it for x =0.015 and 0.022. The dashed lines show fit to the mean-field Brillouin function and the solid line for x = 0.015 (paramagnetic sample) a fit to the Brillouin function (Oiwa et al. 1998a).

Fig. 19. Magnetic field dependence of the diagonal resistivity p (open circles) and magnetization Afnaii (close circles) determined from the ratio of the Hall and diagonal resistivities, Afnall = PHM/CP< where c = 6.3, for a 1.3-rrm thick film of lni tMnr As with x = 0.013. The solid line is a fit by the modified Brillouin function B (y), where S = 5/2 and y = SgpgB/(T + T0) with T0 = 1.5 K. The inset shows the hysteresis observed in the Hall resistivity at 3.5 K (Ohno et al. 1992).

Here, h(M0) denotes the inverse function to M0(h), where M0 is the macroscopic magnetization of the spins in the absence of carriers at a field h and temperature T. It is usually possible to parameterize M0(h) by the Brillouin function Bs according to... 

For a state showing only the first-order Zeeman effect the magnetization as a function of field strength is given by the Brillouin function (equation 57). 

Brillouin function at higher Curie law at low field temperature... 

Derivation van Vleck equation (linear magnetics) or expansion of Brillouin function Restrictions n Bg /kT < 1 - low fields and higher temperatures Formula Curie law Mean magnetic susceptibility... 

Fig. 9 Magnetization curves for various ZFS systems modeled as a powder average at T = 4.2 K top panel D/k = -5 (solid), - 10 (long dashed), - 15 (medium dashed),- 20 K (short dashed) bottom panel D/k = + 5 (solid), + 10 (long dashed), + 15 (medium dashed), + 20 K (short dashed) Brillouin function for a Curie paramagnet - dot-dashed...

Fig. 119 Experimental (open circles) and fitted (full points) data on [Ni(imidazole)4(ace-tate)2] complex Brillouin function - dashed...

For the size determination of spin polarons from the field-dependent magnetization data (SQUID results in Fig. 6) we first employed a fitting based on the Brillouin function /3(./), which is justified since the respective temperatures around T]T are sufficiently above the ferromagnetic transition. The paramagnetic magnetization as a function of field is given by... 

When all microscopic magnetic moments are aligned with the external field vector, we speak of saturation. To describe the saturation effect, observable at higher fields, the so-called Brillouin functions are used ... 

---