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Next: Exercises Up: Applications of Statistical Thermodynamics Previous: Effusion


Consider a solid substance consisting of $ N$ identical atoms arranged in a regular lattice. Each atom has a net electronic angular momentum, $ \hbar {\bf S}$ , and an associated magnetic moment, $ \mu$ . [Here, we use $ {\bf S}$ , rather than $ {\bf J}$ , to denote atomic angular momentum (divided by $ \hbar$ ) to avoid confusion with the exchange energy, $ J$ , introduced in Equation (7.246).] The atomic magnetic moment is related to the corresponding total angular momentum via

$\displaystyle \mbox{\boldmath$\mu$}$$\displaystyle = g \mu_B {\bf S},$ (7.244)

where $ \mu_B$ is the Bohr magneton, and the $ g$ -factor, $ g$ , is a dimensionless number of order unity. (See Section 7.9.) In the presence of an externally-applied magnetic field, $ {\bf B}_0=B_z {\bf e}_z$ , directed parallel to the $ z$ -axis, the Hamiltonian, $ H_0$ , representing the interaction of the atoms with this field is written

$\displaystyle H_0=-g \mu_B\sum_{j=1,N}{\bf S}_j\cdot {\bf B}_0=-g \mu_B B_z\sum_{j=1,N}S_{j z}.$ (7.245)

Each atom is also assumed to interact with neighboring atoms. The interaction in question is not simply the magnetic dipole-dipole interaction due to the magnetic field produced by one atom at the position of another. Such an interaction is, in general, much too small to produce ferromagnetism. Instead, the predominant interaction is known as the exchange interaction. This interaction is a quantum-mechanical consequence of the Pauli exclusion principle. (See Section 8.2.) Because electrons cannot occupy the same state, two electrons on neighboring atoms that have parallel spins (here, to simplify the discussion of the origin of the exchange interaction, we are temporarily assuming that the electrons all have zero orbital angular momenta, so that their total angular momenta are solely a consequence of their spin angular momenta) cannot come too close to one another in space (else they would simultaneously occupy identical spin and orbital states, which is forbidden). On the other hand, if the electrons have antiparallel spins then they are in different spin states, so they are allowed to occupy the same orbital state. In other words, there is no exclusion-related restriction on how close they can come together. Because different spatial separations of the electrons give rise to different electrostatic interaction between them, the electrostatic interaction between neighboring atoms (which is much stronger than any magnetic interaction) depends on the relative orientations of their spins. This, then, is the origin of the exchange interaction, which for two atoms labelled $ j$ and $ k$ can be written in the form

$\displaystyle H_{jk}=-2 J {\bf S}_j\cdot{\bf S}_k.$ (7.246)

Here, the so-called exchange energy, $ J$ , is a parameter (with the dimensions of energy) that depends on the atomic separation, and which measures the strength of the exchange interaction. If $ J>0$ then the interaction energy is lower when spins on neighboring atoms are parallel, rather than antiparallel. This state of affairs favors parallel spin alignment of neighboring atoms. In other words, it tends to produce ferromagnetism. Note that, because the exchange interaction depends on the degree to which the electron orbitals of the two atoms can overlap, so as to occupy approximately the same region of space, the exchange energy, $ J$ , falls off very rapidly with increasing atomic separation. Hence, a given atom only interacts appreciably with its $ n$ (say) nearest neighbors.

To simplify the interaction problem, we shall replace the previous expression by the simpler functional form

$\displaystyle H_{jk}=-2 J S_{j z} S_{k z}.$ (7.247)

This approximate expression, which is known as the Ising model, leaves the essential physical situation intact, while avoiding the complications introduced by vector quantities. The Ising model is equivalent to assuming that all atomic magnetic moments are either directed parallel or anti-parallel to the $ z$ -axis.

The Hamiltonian, $ H'$ , representing the interaction energy between atoms is written in the form

$\displaystyle H'=\frac{1}{2}\left(2 J\sum_{j=1,N}\sum_{k=1,n}S_{j z} S_{k z}\right),$ (7.248)

where $ J$ is the exchange energy for neighboring atoms, and the index $ k$ refers to the nearest neighbor shell surrounding the atom $ j$ . The factor $ 1/2$ is introduced because the interaction between the same two atoms is counted twice in performing the sums.

The total Hamiltonian of the atoms is

$\displaystyle H = H_0 +H'.$ (7.249)

The task in hand is to calculate the mean magnetic moment of the system parallel to the applied magnetic field, $ \overline{M_z}$ , as a function of its absolute temperature, $ T$ , and the applied field, $ B_z$ . The presence of the interatomic interactions makes this task difficult, despite the extreme simplicity of expression (7.248). Although the problem has been solved exactly for a linear one-dimensional array of atoms, and for two-dimensional array of atoms when $ B_z=0$ , the three-dimensional problem is so complex that no exact solution has ever been found. Thus, in order to make further progress, we must introduce an additional approximation.

Let us focus attention on a particular atom, $ j$ , which we shall refer to as the ``central atom.'' The interactions of this atom are described by the Hamiltonian

$\displaystyle H_j =-g \mu_B B_z S_{j z}-2 J S_{j z}\sum_{k=1,n}S_{k z}.$ (7.250)

The final term represents the interaction of the central atom with its nearest neighbors. As a simplifying approximation, we shall replace the sum over these neighbors by its mean value. In other words, we shall write

$\displaystyle 2 J\sum_{k=1,n}S_{k z}\simeq 2 J\overline{\sum_{k=1,n}S_{k z}}=g \mu_B B_m,$ (7.251)

where $ B_m$ is a parameter with the dimensions of magnetic field-strength, which is called the molecular field, and is to be determined in such a way that it leads to a self-consistent solution of the statistical problem. In terms of this parameter, expression (7.250) becomes

$\displaystyle H_j=-g \mu_B (B_z+B_m) S_{j z}.$ (7.252)

Thus, the influence of neighboring atoms has been replaced by an effective magnetic field, $ B_m$ . The problem presented by expression (7.252) reduces to the elementary one of a single atom, of normalized angular momentum, $ S$ , and $ g$ -factor, $ g$ , placed in an external $ z$ -directed magnetic field, $ B_z+B_m$ . Recall, however, that we already solved this problem in Section 7.9.

According to the analysis of Section 7.9,

$\displaystyle \overline{S_{j z}}=S {\cal B}_S(\eta),$ (7.253)


$\displaystyle \eta= \beta g \mu_B (B_z+B_m),$ (7.254)

and $ \beta = 1/(k T)$ . Here, $ {\cal B}_S(\eta)$ is the Brillouin function, for normalized atomic angular momentum $ S$ , that was introduced in Equation (7.113).

Expression (7.253) involves the unknown parameter $ B_m$ . To determine the value of this parameter in a self-consistent manner, we note that there is nothing that distinguishes the central $ j$ th atom from any of its neighboring atoms. Hence, any one of these atoms might equally well have been considered the central atom. Thus, the corresponding mean value $ \overline{S_z}$ is also given by Equation (7.253). It follows from Equations (7.251) and (7.253) that

$\displaystyle 2 J n S {\cal B}_S(\eta)=g \mu_B B_m.$ (7.255)

The previous two equations can be combined to give

$\displaystyle {\cal B}_S(\eta)=\frac{k T}{2 J n S}\left(\eta-\frac{g \mu_B B_z}{k T}\right),$ (7.256)

which determines $ \eta$ (and, hence, $ B_m$ ). Once $ \eta$ has been found, the mean magnetic moment of the system follows from Equation (7.253):

$\displaystyle \overline{M_z} = g \mu_B\sum_j\overline{S_{j z}}= N g \mu_B S {\cal B}_S(\eta).$ (7.257)

Figure: Graphical solution of Equation (7.256) that determines the molecular field, $ B_m$ , corresponding to the intersection of the curves at $ \eta =\eta '$ . The dash-dotted straight-line corresponds to the case where the external magnetic field, $ B_z$ , is zero.
\epsfysize =4in

The solution of Equation (7.256) can be found by drawing, on the same graph, both the Brillouin function $ y={\cal B}_S(\eta)$ and the straight-line

$\displaystyle y =\frac{k T}{2 J n S}\left(\eta-\frac{g \mu_B B_z}{k T}\right),$ (7.258)

and then finding the point of intersection of the two curves. See Figure 7.9.

Consider the case where the external field, $ B_z$ , is zero. Equation (7.256) simplifies to give

$\displaystyle {\cal B}_S(\eta)=\left(\frac{k T}{2 J n S}\right)\eta.$ (7.259)

Given that $ {\cal B}_S(0)=0$ , it follows that $ \eta=0$ is a solution of the previous equation. This solution is characterized by zero magnetization: that is, $ \overline{M_z}=0$ . However, there is also the possibility of another solution with $ \eta\neq 0$ , and, hence, $ \overline{M_z}\neq 0$ . (See Figure 7.9.) The presence of such spontaneous magnetization in the absence of an external magnetic field is, of course, the distinguishing feature of ferromagnetic materials. In order to have a solution with $ \eta\neq 0$ , it is necessary that the curves shown in Figure 7.9 intersect at a point $ \eta\neq 0$ when both curves pass through the origin. The condition for the existence of an intersection with $ \eta\neq 0$ is that the slope of the curve $ y={\cal B}_J(\eta)$ at $ \eta=0$ should exceed that of the straight-line $ y=(k T/2 J n S) \eta$ . In other words,

$\displaystyle \left(\frac{d{\cal B}_S}{d\eta}\right)_{\eta=0}> \frac{k T}{2 J n S}.$ (7.260)

However, when $ \eta\ll 1$ , the Brillouin function takes the simple form specified in Equation (7.119):

$\displaystyle {\cal B}_S(\eta)\simeq \frac{1}{3} (S+1) \eta.$ (7.261)

Hence, Equation (7.260) becomes

$\displaystyle \frac{1}{3} (S+1)> \frac{k T}{2 J n S},$ (7.262)


$\displaystyle T< T_c,$ (7.263)


$\displaystyle k T_c=\frac{2 J n S (S+1)}{3}.$ (7.264)

Thus, we deduce that spontaneous magnetization of a ferromagnetic material is only possible below a certain critical temperature, $ T_c$ , known as the Curie temperature. The magnetized state, in which all atomic magnetic moments can exploit their mutual exchange energy by being preferentially aligned parallel to one another, has a lower magnetic energy than the unmagnetized state in which $ \eta=0$ . Thus, at temperatures below the Curie temperature, the magnetized state is stable, whereas the unmagnetized state is unstable.

As the temperature, $ T$ , is decreased below the Curie temperature, $ T_c$ , the slope of the dash-dotted straight-line in Figure 7.9 decreases so that it intersects the curve $ y={\cal B}_S(\eta)$ at increasingly large values of $ \eta$ . As $ T\rightarrow 0$ , the intersection occurs at $ \eta\rightarrow\infty$ . Because, $ {\cal B}_S(\infty)=1$ [see Equation (7.117)], it follows from Equation (7.257) that $ \overline{M_z}\rightarrow N g \mu_B S$ as $ T\rightarrow 0$ . This corresponds to a state in which all atomic magnetic moments are aligned completely parallel to one another. For temperatures, $ T$ , less than the Curie temperature, $ T_c$ , we can use Equations (7.257) and (7.259) to compute $ \overline{M_z}(T)$ in the absence of an external magnetic field. We then obtain a magnetization curve of the general form shown in Figure 7.10.

Figure: Spontaneous magnetization of a ferromagnet as a function of the temperature in the presence of zero external magnetic field. The solid curve is the prediction of molecular field theory for $ J=1/2$ and $ g=2$ . The points are experimental data for nickel [from R. Kaul, and E.D. Thompson, J. appl. Phys. 40, 1383 (1969)].
\epsfysize =3in

Let us, finally, investigate the magnetic susceptibility of a ferromagnetic substance, in the presence of a small external magnetic field, at temperatures above the Curie temperature. In this case, the crossing point of the two curves in Figure 7.9 lies at small $ \eta$ . Hence, we can use the approximation (7.261) to write Equation (7.256) in the form

$\displaystyle \frac{1}{3} (S+1) \eta = \frac{k T}{2 J n S}\left(\eta -\frac{g \mu_B B_z}{k T}\right).$ (7.265)

Solving the previous equation for $ \eta$ gives

$\displaystyle \eta=\frac{g \mu_B B_z}{k (T-T_c)},$ (7.266)

where use has been made of Equation (7.264). Now, making use of the approximation (7.261), Equation (7.257) reduces to

$\displaystyle \overline{M_z} = \frac{1}{3} N g \mu_B S (S+1) \eta.$ (7.267)

The previous two equations can be combined to give

$\displaystyle \chi \simeq \frac{\mu_0 \overline{M_z}}{V B_z} =\frac{N \mu_0 \mu_B^{ 2} ,g^{ 2} S (S+1)}{3 V k (T-T_c)},$ (7.268)

where $ \chi$ is the dimensionless magnetic susceptibility of the substance, and $ V$ is its volume. This result is known as the Curie-Weiss law. It differs from Curie's law, $ \chi\propto 1/T$ (see Section 7.3), because of the presence of the parameter $ T_c$ in the denominator. According to the Curie-Weiss law, $ \chi$ becomes infinite when $ T\rightarrow T_c$ : that is, at the Curie temperature, when the substance becomes ferromagnetic.

Figure: Plot of $ \rho /\chi $ (where $ \rho $ is the mass density) versus temperature for gadolinium above its Curie temperature. The curve is (apart from some slight departures at high temperatures) a straight-line, in accordance with the Curie-Weiss law. The intercept of the line with the temperature axis gives $ T_c= 310$ K. The metal becomes ferromagnetic below $ 289$ K. Data from S. Arajs, and R.V. Colvin, J. appl. Phys. 32, 336S (1961).
\epsfysize =3.5in

Experimentally, the Curie-Weiss law is well obeyed at temperatures significantly above the Curie temperature. See Figure 7.11. It is, however, not true that the temperature, $ T_c$ , that occurs in this law is exactly the same as the Curie temperature at which the substance becomes ferromagnetic.

The so-called Weiss molecular-field model (which was first put forward by Pierre Weiss in 1907) that we have just described is remarkably successful at explaining the major features of ferromagnetism. Nevertheless, there are some serious discrepancies between the predictions of this simple model and experimental data. In particular, the magnetic contribution to the specific heat is observed to have a very sharp discontinuity at the Curie temperature (in the absence of an external field), whereas the molecular field model predicts a much less abrupt change. Needless to say, more refined models have been devised that considerably improve the agreement with experimental data.

next up previous
Next: Exercises Up: Applications of Statistical Thermodynamics Previous: Effusion
Richard Fitzpatrick 2016-01-25