Crystal field heat capacity

Fig. 6. Experimental and calculated crystal field heat capacity of PrAl3 as a function of temperature according splitting scheme shown in Fig. 7.

Spin-state transitions have been studied by the application of numerous physical techniques such as the measurement of magnetic susceptibility, optical and vibrational spectroscopy, the Fe-Mbssbauer effect, EPR, NMR, and EXAFS spectroscopy, the measurement of heat capacity, and others. Most of these studies have been adequately reviewed. The somewhat older surveys [3, 19] cover the complete field of spin-state transitions. Several more recent review articles [20, 21, 22, 23, 24, 25] have been devoted exclusively to spin-state transitions in compounds of iron(II). Two reviews [26, 27] have considered inter alia the available theoretical models of spin-state transitions. Of particular interest is the determination of the X-ray crystal structures of spin transition compounds at two or more temperatures thus approaching the structures of the pure HS and LS electronic isomers. A recent survey [6] concentrates particularly on these studies. 

The state of polarization, and hence the electrical properties, responds to changes in temperature in several ways. Within the Bom-Oppenheimer approximation, the motion of electrons and atoms can be decoupled, and the atomic motions in the crystalline solid treated as thermally activated vibrations. These atomic vibrations give rise to the thermal expansion of the lattice itself, which can be measured independendy. The electronic motions are assumed to be rapidly equilibrated in the state defined by the temperature and electric field. At lower temperatures, the quantization of vibrational states can be significant, as manifested in such properties as thermal expansion and heat capacity. In polymer crystals quantum mechanical effects can be important even at room temperature. For example, the magnitude of the negative axial thermal expansion coefficient in polyethylene is a direct result of the quantum mechanical nature of the heat capacity at room temperature." At still higher temperatures, near a phase transition, e.g., the assumption of stricdy vibrational dynamics of atoms is no... 

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. 

ZT Y r A A A A A AC dimensionless thermoelectric figure of merit electronic coefficient of heat capacity (1+ZT)F2 crystal field singlet non-Kramers doublet (crystal field state) crystal field triplet crystal field triplet hybridization gap jump in heat capacity at Tc K KL -min P 6>d X JCO total thermal conductivity of solid thermal conductivity of electrons or holes thermal conductivity of lattice minimum lattice thermal conductivity electrical resistivity Debye temperature magnetic susceptibility magnetic susceptibility at T = 0... 

Fig. 12. Total heat capacity of a single crystal of PrFe4Pi2 vs. temperature in various applied magnetic fields (a) low fields and (b) high fields. The dashed fines in (b) correspond to the best fit of the heavy fermion state to die resonant level model (Crlm)-Cph is the estimate of die phonon contribution to the heat capacity (Aoki et al., 2002).

NdRu4Sbi2 is metallic and undergoes some type of magnetic transition near 1.3 K. The magnetic susceptibility follows a Curie-Weiss law above 50 K with an effective moment of 3.45/u.b and a Weiss temperature of -28 K. Crystal fields likely effect the susceptibility and magnetic interactions for temperatures below 50 K. Low temperature heat capacity data confirm the bulk nature of the magnetic transition (Takeda and Ishikawa, 2000b). 

Fig. 5. The excess heat capacity of PrFj, NdF3, DyFj and ErF3 as calculated from the crystal field energies.

The experimental excess heat capacity thus obtained as the difference between measured Cp and Qat can then be compared to the values calculated from the crystal field levels. As an example, fig. 7 shows the good agreement of the experimental and calculated excess heat capacity of DyF3. 

The data shown in fig. 10 are not the values reported by Gorbunov et al. (1986) and Tolmach et al. (1987, 1990a, 1990b, 1990c), because they did not extrapolate their measurements to 0 K in all cases. To derive S° (298.15 K) we have assumed that the heat capacity of L11CI3 represents the lattice component, and Am at the lower temperature limit is derived from the results for this compound. The excess contribution at this temperature is calculated from the crystal field energies (see table 5) derived from spectroscopic studies of the ions in transparent host crystals (Dieke et al., 1968 Morrison and Leavitt, 1982 ... 

The heat capacities for the other compounds were derived using the estimation procedure described for the trichlorides, i.e., from the lattice and excess contributions. The former was derived from the enthalpy measurements, the latter from the crystal field energies. As the crystal energies of the tribromides and triiodides are poorly known, we have used the values for the trichlorides to approximate Cexs- The results thus obtained are listed in tables 10 and 11. The calculated data for TmG agree within 2% with the DSC results of Gardner and Preston (1991). 

The variation in the heat capacity and entropy of the solid lanthanide trihalides can be described by a lattice contribution that linearly varies with atomic number within each crystallographic class of compounds, and an excess contribution that depends on the electronic configuration (crystal field) of the lanthanide ions. A distinct difference is observed between... 

The lack of auxiliary data such as crystal field energies for the tribromides and triiodides further limits the value of the semi-empirical approach used here to estimate high temperature heat capacities. 

The crystal field interaction has pronounced effects on the heat capacity behavior of the system. At very low temperatures, ions occupy the lowest crystal field states. With increasing temperature, excitation within the crystal field spectrum takes place resulting in a significant contribution to the heat capacity (38). This contribution is given by the expression... 

Deenadas et al. (69) reported the heat capacity of NdAl2. This shows superimposition of a Schottky-type heat capacity excess over the normal X-type anomaly. Magnetization measurements (6S) on single crystals of NdAl2 in three crystallographic directions in fields up to 350kG and neutron inelastic scattering experiments (70) yielded consistent crystal field parameters (A (r4) = 40K and A (r6> = - 12K). 

CeAl3 is a very interesting case where the superposition of the Kondo effect on crystal field interaction gives rise to anomalous heat capacity (7(5) and resistivity behavior (46, 77, 78) at low temperatures. The heat capacity measurements of Mahoney (75) showed that the full entropy of R In 6 was recovered under the excess... 

The heat capacity behavior of PrNi2 confirms that it becomes a Van Vleck paramagnet at low temperatures (65). It exhibits no X-type thermal anomaly expected if there is magnetic ordering only a Schottky-type heat capacity excess is observed. This is a consequence of the thermal population of the higher crystal field states. 

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