Consider atoms in the presence of a -directed magnetic field of
strength . Suppose that all
atoms are identical spin- systems. It follows that either
(spin up) or (spin down), where is (twice) the -component
of the th atomic spin. The total energy of the system is
The physics of the Ising model is as follows. The first term on the right-hand side of Eq. (351) shows that the overall energy is lowered when neighbouring atomic spins are aligned. This effect is mostly due to the Pauli exclusion principle. Electrons cannot occupy the same quantum state, so two electrons on neighbouring atoms which have parallel spins (i.e., occupy the same orbital state) cannot come close together in space. No such restriction applies if the electrons have anti-parallel spins. Different spatial separations imply different electrostatic interaction energies, and the exchange energy, , measures this difference. Note that since the exchange energy is electrostatic in origin, it can be quite large: i.e., eV. This is far larger than the energy associated with the direct magnetic interaction between neighbouring atomic spins, which is only about eV. However, the exchange effect is very short-range; hence, the restriction to nearest neighbour interaction is quite realistic.
Our first attempt to analyze the Ising model will employ a simplification
known as the mean field approximation. The energy of the th atom is written
We can write
Consider a single atom in a magnetic field . Suppose that
the atom is in thermal equilibrium with a heat bath of temperature
. According to the well-known Boltzmann distribution, the mean spin
of the atom is
Let us assume that all atoms have identical spins: i.e., .
This assumption is known as the ``mean field approximation''.
We can write
It is helpful to define the net magnetization,
Figures 102, 101, and 103 show the net magnetization, net energy, and heat capacity calculated from the iteration formula (363) in the absence of an external magnetic field (i.e., with ). It can be seen that below the critical (or ``Curie'') temperature, , there is spontaneous magnetization: i.e., the exchange effect is sufficiently large to cause neighbouring atomic spins to spontaneously align. On the other hand, thermal fluctuations completely eliminate any alignment above the critical temperature. Moreover, at the critical temperature there is a discontinuity in the first derivative of the energy, , with respect to the temperature, . This discontinuity generates a downward jump in the heat capacity, , at . The sudden loss of spontaneous magnetization as the temperature exceeds the critical temperature is a type of phase transition.
Now, according to the conventional classification of phase transitions, a transition is first-order if the energy is discontinuous with respect to the order parameter (i.e., in this case, the temperature), and second-order if the energy is continuous, but its first derivative with respect to the order parameter is discontinuous, etc. We conclude that the loss of spontaneous magnetization in a ferromagnetic material as the temperature exceeds the critical temperature is a second-order phase transition.
In order to see an example of a first-order phase transition, let us examine the behaviour of the magnetization, , as the external field, , is varied at constant temperature, .
Figures 104 and 105 show the magnetization, , and energy, , versus external field-strength, , calculated from the iteration formula (363) at some constant temperature, , which is less than the critical temperature, . It can be seen that is discontinuous, indicating the presence of a first-order phase transition. Moreover, the system exhibits hysteresis--meta-stable states exist within a certain range of values, and the magnetization of the system at fixed and (within the aforementioned range) depends on its past history: i.e., on whether was increasing or decreasing when it entered the meta-stable range.
Figures 106 and 107 show the magnetization, , and energy, , versus external field-strength, , calculated from the iteration formula (363) at a constant temperature, , which is equal to the critical temperature, . It can be seen that is now continuous, and there are no meta-stable states. We conclude that first-order phase transitions and hysteresis only occur, as the external field-strength is varied, when the temperature lies below the critical temperature: i.e., when the ferromagnetic material in question is capable of spontaneous magnetization.
The above calculations, which are based on the mean field approximation, correctly predict the existence of first- and second-order phase transitions when and , respectively. However, these calculations get some of the details of the second-order phase transition wrong. In order to do a better job, we must abandon the mean field approximation and adopt a Monte-Carlo approach.
Let us consider a two-dimensional square array of atoms. Let be the size of the array, and the number of atoms in the array, as shown in Fig. 108. The Monte-Carlo approach to the Ising model, which completely avoids the use of the mean field approximation, is based on the following algorithm:
In order to demonstrate that the above algorithm is correct, let us consider
flipping the spin of the th atom. Suppose that this operation causes the
system to make a transition from state (energy, ) to state (energy, ).
Suppose, further, that . According to the above algorithm, the probability
of a transition from state to state is
Now, each atom in our array has four nearest neighbours, except for atoms on the edge of the array, which have less than four neighbours. We can eliminate this annoying special behaviour by adopting periodic boundary conditions: i.e., by identifying opposite edges of the array. Indeed, we can think of the array as existing on the surface of a torus.
It is helpful to define
Figures 109-116 show magnetization and heat capacity versus temperature curves for , 10, 20, and 40 in the absence of an external magnetic field. In all cases, the Monte-Carlo simulation is iterated 5000 times, and the first 1000 iterations are discarded when evaluating (in order to allow the system to attain thermal equilibrium). The two-dimensional array of atoms is initialized in a fully aligned state for each different value of the temperature. Since there is no external magnetic field, it is irrelevant whether the magnetization, M, is positive or negative. Hence, is replaced by in all plots.
Note that the versus curves generated by the Monte-Carlo simulations look very much like those predicted by the mean field model. The resemblance increases as the size, , of the atomic array increases. The major difference is the presence of a magnetization ``tail'' for in the Monte-Carlo simulations: i.e., in the Monte-Carlo simulations the spontaneous magnetization does not collapse to zero once the critical temperature is exceeded--there is a small lingering magnetization for . The versus curves show the heat capacity calculated directly (i.e., ), and via the identity . The latter method of calculation is clearly far superior, since it generates significantly less statistical noise. Note that the heat capacity peaks at the critical temperature: i.e., unlike the mean field model, is not zero for . This effect is due to the residual magnetization present when .
Our best estimate for is obtained from the location of the peak in the versus
curve in Fig. 116. We obtain . Recall that the mean field model
yields . The exact answer for a two-dimensional array of ferromagnetic atoms
Note, from Figs. 110, 112, 114, and 116, that the height of the peak in the heat capacity curve at increases with increasing array
size, . Indeed, a close examination of these figures yields
for , and
Figure 117 shows
plotted against for , 10, 20, and 40.
It can be seen that the points lie on a very convincing straight-line, which strongly suggests that
Of course, for physical systems, , where is Avogadro's number. Hence, is effectively singular at the critical temperature (since ), as sketched in Fig. 118. This observation leads us to revise our definition of a second-order phase transition. It turns out that actual discontinuities in the heat capacity almost never occur. Instead, second-order phase transitions are characterized by a local quasi-singularity in the heat capacity.
Recall, from Eq. (374), that the typical amplitude of energy fluctuations is proportional to the square-root of the heat capacity (i.e., ). It follows that the amplitude of energy fluctuations becomes extremely large in the vicinity of a second-order phase transition.
Now, the main difference between our mean field and Monte-Carlo calculations is the existence of residual magnetization for in the latter case. Figures 119-123 show the magnetization pattern of a array of ferromagnetic atoms, in thermal equilibrium and in the absence of an external magnetic field, calculated at various temperatures. It can be seen that for the pattern is essentially random. However, for , small clumps appear in the pattern. For , the clumps are somewhat bigger. For , which is just above the critical temperature, the clumps are global in extent. Finally, for , which is a little below the critical temperature, there is almost complete alignment of the atomic spins.
The problem with the mean field model is that it assumes that all atoms are situated in identical environments. Hence, if the exchange effect is not sufficiently large to cause global alignment of the atomic spins then there is no alignment at all. What actually happens when the temperature exceeds the critical temperature is that global alignment disappears, but local alignment (i.e., clumping) remains. Clumps are only eliminated by thermal fluctuations once the temperature is significantly greater than the critical temperature. Atoms in the middle of the clumps are situated in a different environment than atoms on the clump boundaries. Hence, clumps cannot occur in the mean field model.