Materials Science and Engineering/Magnetic Materials/Saturation Magnetization Near Curie Temperature

Part 1
'''Discuss the physical implications of the use of the mean field theory in solving ferromagnetic problems. Where in the dynamical equations for magnetization does this approximation get used? What obvious problem in the solution of such equations does it obviate? When is it appropriate?'''

A material placed in a magnetic field can respond a number of different ways. The interaction of the field with the angular momenta of the atomic components is the origin of a response. In a paramagnetic material, there is a small interaction among magnetic moments. Interactions among atomic moments in ferromagnetic materials is so strong that there is magnetic ordering in the absence of applied fields, and this produces very large magnetizations.

There is complexity in the response of a ferromagnet to external magnetic fields. The ferromagnet consists of regions, called domains, whose net magnetic moments are in different directions. Mean field theory assumes only nearest neighbor interactions of spins in a ferromagnet exist. Magnetic energy is determined by interaction of spin with the magnetic field that arises from configurations of spins on nearest neighbors. In the mean field approach of Weiss, each configuration of neighbors is replaced by a statistical mechanical average. Each spin is subject to an effective internal magnetic field that is proportional to the overall magnetization of the crystal. The problem is reduced to one that is identical to the model of independent paramagnetic atoms. Each atom is treated as if it were an independent magnetic moment. Instead of using only the external field $$H$$, the internal field $$H_i$$ is added. A total field of $$(H + H_i)$$ is used in the Brillouin function.

The mean field approximation becomes unreliable in the case below. The term $$E_F$$ is the fermi energy, and $$B$$ is the band width.

$$U \ge E_F \approx B$$

Part 2
'''Show that in the mean field approximation the saturation magnetization just below the Curie temperature has the dominant temperature dependence ($$T_c - T)^{\frac{1}{2}}$$. How does this compare with experimental data?'''

The total angular momentum of an atom gives rise to a magnetic moment whose $$z$$ component is expressed below.

$$M_z = -g \mu_B M_s$$

$$\mu_B = \frac{e \hbar}{2 m_e}$$

Terms above are the Bohr magneton,$$\mu_B$$, electron mass, $$m_e$$, gyromagnetic ratio, $$g$$, and azimuthal quantum number of the total angular momentum.

Choose the z-axis to be in the direction of the field. The energy of the atomic moment in the field is expressed below with $$H$$, the magnitude of the field..

$$\mu_m = g M_s \mu_B H$$

The total magnetization is the atomic moment times the number of atoms per unit volume. Find the statistical mechanical average of $$M_s$$ since the energy and magnetization are proportional to the azimuthal quantum number.

$$\overline{M_s} = \frac{ \sum_s M_s e^{-g M_s \mu_B H / kT}}{\sum_s e^{-g M_s \mu_B H / kT}}$$

Note that an assumption is that the magnetic field does not affect the lattice vibrations. Write the partition function as below.

$$Z = \sum_{M_s = -S}^{M_s = S} e^{-M_s y}$$

$$Z = \sum_{M_s = -S}^{M_s = S} x^{M_s}$$

$$y = g \mu_B H / k T$$

$$x = e^{-y}$$

Sum by using the formula for the sum of a geometric series.

$$z = x^{-s} + x^{-s+1} + x^{-s+2}$$

$$z = x^{-s} (1 + x + x^2 + ... + x^{2s})$$

$$Z = \frac{x^{-s} - x^{s+1}}{1-x}$$

$$Z = \frac{e^{s y} - e^{-(s+1)y}}{1-e^{-y}}$$

$$Z = \frac{e^{(s+1/2) y} - e^{-(s+1/2)y}}{e^{y/2}-e^{-y/2}}$$

$$Z = \frac{\sinh [(s + 1/2)y]}{\sinh(y/2)}$$

Find the average $$\overline{M_s}$$. Differentiate the equation with respect to $$y$$. Divide by $$Z$$.

$$\frac{dZ}{dy} = \sum_{M_s = -S}^{M_s = S} M_s e^{-M_s y}$$

$$\overline{M_s} = \frac{-d\ln Z}{dy}$$

$$\overline{M_s} = \frac{1}{Z} \sum_{M_s = -S}^{M_s = S} M_s e^{-M_s y}$$

Define a variable $$z$$.

$$z = Sy$$

$$z = \frac{g S \mu_B H}{k T}$$

$$Z = \frac{\sinh [z(2S + 1) / 2S]}{\sinh(z / 2S)}$$

$$S \frac{d \ln Z}{dz} = \frac{d \ln Z}{dy}$$

$$\frac{d \ln Z}{dy} = S [ ( \frac{2 S + 1}{2S} ) \frac{ \cosh [ z (2S+1)/2S]}{\sinh [ z (2S+1)/2S]} - \frac{1}{2S} \frac{ \cosh(z/2S)}{ \sinh(z/2S)} ] $$

$$S \frac{d \ln Z}{d z} = \overline{M_s}$$

Define Brillouin function

$$B_s = ( \frac{2 S + 1}{2S} ) \coth [ z (2S+1)/2S] - \frac{1}{2S} \coth(z/2S)$$

$$\overline{M_s} = -S B_s (z)$$

Find the statistical average for the magnetization per atom by taking the average of the magnetic moment and the use the result of the azimuthal quantum number.

$$M_p = n \overline{M_s}$$

$$M_p = -g \mu_B \overline{m_z}$$

$$M_p = n S g \mu_B B_s(z)$$

Consider $$M_f$$ to be the magnetization arising from ferromagnetic interactions. Below is an expression of the internal field at each site.

$$H_i = \gamma M_f$$

Below is the expression of the magnetization arising from ferromagnetic interactions. There is a replacement of $$x$$.

$$M_f = n S g \mu_B B_s (x)$$

$$x = S g \mu_B (H + \gamma M_f) / k T$$

Retain the first three terms in the expansion below and approximate the Brillouin function with an expression below.

$$\coth x = \frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} - ...$$

$$B_s(x)=\frac{(S+1)}{3S} x - (\frac{1}{2s})^4 [(2s+1)^4 - 1] \frac{x^3}{45}$$

Put this result in the first expression below and use the definition of the Curie temperature.

$$M_f = n S f \mu_B B_s (x)$$

$$T_c = \frac{n \gamma}{3 k} S (S+1) (g \mu_B)^2$$

$$M_f = \frac{T_c}{T} (M_f + \frac{H}{\gamma} ) - K ( \frac{T_c}{T} )^3 (M_f + \frac{H}{\gamma} )^3$$

The term $$K$$ is a constant that can be readily evaluated from he definition of $$T_c$$. If the external field is zero, $$M_f$$ is the spontaneous magnetization $$M_f^s$$. Solve for $$M_f^s$$.

$$M_f^s = \frac{1}{K} ( \frac{T_c - T}{T} )^{1/2}$$

The experimental agreement is only approximate.

Part 3
Of what interest is this result?

The mean field theory correctly displays the Currie-Weiss law and the existence of a second-order ferromagnetic to paramagnetic transition

Response References

 * Girifalco, Louis A. Statistical Mehanics of Solids, Oxford University Press, New York, New York, 2000
 * Nagaosa, Naoto. Quantum Field Theory in Strongly Correlated Electronic Systems (Texts and Monographs in Physics), Iwanami Shoten, Publishers, Tokyo, 1998