Lambda transition

Simple kinds of phase change, such as melting and vaporization, are characterized by considerable changes of volume, and also of entropy and enthalpy, at the point of transition. Thus, whereas the chemical potentials of the two phases are equal when they are at equilibrium together, their volumes, entropies and enthalpies are far from equal. 

In the ordinary type of phase change, if fiy s, v, etc., are plotted as functions of temperature and pressure, we obtain curves as shown diagrammatically in Fig. 21. The chemical potential itself shows a change of gradient at the point of phase change, but no discofUinuUy. The latter is manifested in the entropy, volume and all other higher derivatives of the chemical potential. X 

Apart from the discontinuous changes of entropy and volume, an important feature of these normal types of phase change is that the values of Cp and k do not usually change at all rapidly as the transition 

X With regard to the heat capacity, it may be noted that its value may either increase (as in the ice-water transition) or decrease (as in the water-steam transition) in passing from the lower to the higher temperature phase. 

Other types of phase change have been discovered which show quite a different character. In these there seems to be no difference of volume between the two forms of the substance and also little or no difference in entropy or enthalpy, i.e. zero or almost zero latent heat. The transition is manifested simply by a sharp change in the heat capacity and compressibility. These properties also vary rather rapidly as the transition point is approached. 

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The type (e) transition for 4He is very useful as reference point at 2.18 K (normal-to-superfluid lambda transition). Such second orders are represented in Fig. 8.1. 

We now distinguish solid state transformations as first-order transitions or lambda transitions. The latter class groups all high-order solid state transformations (second-, third-, and fourth-order transformations see Denbigh, 1971 for exhaustive treatment). We define first-order transitions as all solid state transformations that involve discontinuities in enthalpy, entropy, volume, heat capacity, compressibility, and thermal expansion at the transition point. These transitions require substantial modifications in atomic bonding. An example of first-order transition is the solid state transformation (see also figure 2.6)... 

In lambda transitions, no discontinuity in enthalpy or entropy as a function of T and/or P at the transition zone is observed. However, heat capacity, thermal expansion, and compressibility show typical perturbations in the lambda zone, and T (or P) dependencies before and after transition are very different. 

Table 3.2 lists the optimal values of the interpolation coefficients estimated by Berman and Brown (1987) for the most common oxide constituents of rock-forming minerals. These coefficients, through equations 3.78.1, 3.78.2, and 3.78.3, allow the formulation of polynomials of the same type as equation 3.54, whose precision is within 2% of experimental Cp values in the T range of applicability. However, the tabulated coefficients cannot be applied to phases with lambda transitions (see section 2.8). 

It may of course be unnecessary to consider all these terms and the equation is much simplified in the absence of magnetism and multiple electronic states. In the case of Ti, it is possible to deduce values of the Debye temperature and the electronic specific heat for each structure the pressure term is also available and lambda transitions do not seem to be present. Kaufman and Bernstein (1970) therefore used Eq. (6.2), which yields the results shown in Fig. 6.1(c). 

Other phases are then characterised relative to this ground state, using the best approximation to Eq. (6.1) that is appropriate to the available data. For instance, if die electronic specific heats are reasonably similar, there are no lambda transitions and T 6o, then the entropy difference between two phases can be expressed just as a function of the difference in their Debye temperatures (Domb 1958) ... 

Very few examples of heat capacity or compressibility behavior of the type shown in the second column have been observed experimentally, however. Instead, these two properties most often are observed to diverge to some very large number at Tt as shown in the third column of Figure 13.1.1 The shapes of these curves bear some resemblance to the Greek letter, A, and transitions that exhibit such behavior have historically been referred to as lambda transitions. 

Figure 13.1 Changes in fi, S, V, Cp and k for a first-order, second-order, and lambda transition.

Figure 13.12 Heat capacity of liquid 4He. The lambda transition temperature is 2.172 K.

Another type of phase transition is called a lambda transition, because a graph of heat capacity versus temperature for this type of transition resembles the Greek letter X, as shown in Fig. 4. This type of transition is usually associated with a change from an ordered state to a state with some disorder (order-disorder... 

The heat capacity is taken from the drop calorimetry of Dworkln and Bredig (384 to 1260 K) (6). Between 298 and 820 K, the observed enthalpy differences and the constraint of passing through zero at 298.15 K are fit by a linear least squares technique Cp = 15.99 + 6.22 X 10 T cal K" mol" (298-820 K). The heat capacity in the observed diffuse lambda transition region, 820-1100 K, was adjusted to properly reproduce the observed enthalpies. The sharp heat capacity maximum occurs at 1050 K (6) and... 

The diffuse lambda transition has been discussed above. 

The heat capacity is based on the drop calorimetry of May (6) (400-1500 K), The pre-melt S-shaped enthalpy curve is reinterpreted as incorporating a lambda transition in view of the enthalpy measurements on K2S by Dworkin and Bredig (7 ) and the occurrence of lambda transitions in other materials having the fluorite or anti-flourite type of structure ( ). The adopted heat capacity shows the maximum of the lambda transition at 50.65 cal... 

The peak temperature for the specihc heat, marking the lambda transition, is [155, 193]... 

A surprising result emerging from the quantum simulations [65, 66] of small ( He)jv clusters and the analyses in Sections II.B and ll.C is the manifestation of a well-characterized, broadened, high-order phase transition for small (" He) y clusters (i.e., N = 8) for the superfluid density [155], and N = 32 for the appearance of the lambda transition [65]. An open question pertains to the threshold size of these equations What is the system s smallest size for the exhibition of superfluidity and what are the corresponding phase transitions ... 

Tt o- Taking the short correlation length 2 A, we roughly estimate that 77 6 A, so that the smallest " He cluster will consist of a central atom and its first coordination layer. Thus the threshold size domain for the realization of the lambda transition is TVmin " 5-13. Such a low value of Amin is consistent with the value Amin < 8 for the exhibition of the superfluid density in finite systems [155]. Finally, the threshold size for the appearance of rotons in the elementary excitation spectra of ( He) y clusters [128] is realized for 20 < Amin < 70 (Section l.D). 

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