Magnetic heat capacity order-disorder

The Inden model [20] is frequently used to describe second-order magnetic order-disorder transitions. Inden assumed that the heat capacity varied as a logarithmic function of temperature and used separate expressions above and below the magnetic order-disorder transition temperature (TtIS) in order to treat the effects of both long- and short-range order. Thus for z = (T/TtIS) < 1 ... 

Figure 8.24 Heat capacity of C03O4 [23-25]. The insert shows the magnetic order-disorder transition at around 30 K [24] in detail.

Not only do the thermodynamic properties follow similar power laws near the critical temperatures, but the exponents measured for a given property, such as heat capacity or the order parameter, are found to be the same within experimental error in a wide variety of substances. This can be seen in Table 13.3. It has been shown that the same set of exponents (a, (3, 7, v, etc.) are obtained for phase transitions that have the same spatial (d) and order parameter (n) dimensionalities. For example, (order + disorder) transitions, magnetic transitions with a single axis about which the magnetization orients, and the (liquid + gas) critical point have d= 3 and n — 1, and all have the same values for the critical exponents. Superconductors and the superfluid transition in 4He have d= 3 and n = 2, and they show different values for the set of exponents. Phase transitions are said to belong to different universality classes when their critical exponents belong to different sets. 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

However, just as is the case for many other kinds of critical phenomena (e.g. the one-component fluid, magnetism, order-disorder transitions in solids, etc.) such predictions do not agree either with the results of careful experimental measurements or with simple theoretical models that can be treated nearly exactly. The coexistence curve is more nearly cubic than parabolic, the critical isotherm is of distinctly higher order than cubic, and the heat capacity Cp,x,m diverges at the critical point. 

There are some phase transitions that do not fall either into the first-order or second-order category. These include paramagnetic-to-ferromagnetic transitions in some magnetic materials and a type of transition that occurs in certain solid metal alloys that is called an order-disorder transition. Beta brass, which is a nearly equimolar mixture of copper and zinc, has a low-temperature equilibrium state in which every copper atom in the crystal lattice is located at the center of a cubic unit cell, surrounded by eight zinc atoms at the corners of the cell. At 742 K, an order-disorder transition occurs from the ordered low-temperature state to a disordered high-temperature state in which the atoms are randomly mixed in a single crystal lattice. The phase transition between normal liquid helium and liquid helium II was once said to be a second-order transition. Later experiments indicated that the heat capacity of liquid helium appears to approach infinity at the transition, so that the transition is not second... 

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