Heat capacity Schottky

Figure 10-16 The Schottky heat capacity curve for a single excited state with energies of 30, 300 and 1000 cm-1 and a degeneracy of one.

Figure 10.17 The effect of degeneracy on a Schottky heat capacity from a single excited state 30 cm-1 above the ground state, (a), o= i = l (b). o = 3. gi = 1 and (c), go — Ug =3.

This study [13] calculated the Schottky heat capacity for a hypothetical spin of s = 10, with an axial anisotropy, — D, of —0.5K. Across three different fields, Figure 9.9 shows that the larger the field, the smaller the contribution of this heat capacity. More relevant here is that the derived —ASM values showed the maximum —ASM shifts and decreases with increasing anisotropy from D = —0.5 to —1.5K and — 3.0K, as shown in Figure 9.9, for AH = 0 — 7T up to 200 K. 

The excess contribution is due to the distribution of the valence electrons over the energy levels, and includes the splitting of the ground term by the crystalline electric field (Stark effect) and is called the Schottky heat capacity or Schottky anomaly. It can be calculated from... 

The excess (magnetic) heat capacity may be represented to a good approximation as the sum of (a) the electronic transitional (or Schottky) heat capacity, (b) the effects of interaction between the electrons and the nuclear spin of the paramagnetic ion, (c) the dipolar interaction between these ions, and (d) interactions for other types of interionic coupling. The last three terms are often small above 2°K. and in some instances can be obtained from paramagnetic relaxation data. In principle, the second and third can also be obtained from paramagnetic resonance data. 

In this way Inoue et al. (54) obtained parameters x and W which best fit the experimental ACp. The calculated Schottky heat capacity is compared with the experimental results in Figs. 3 and 4. Good agreement with experiment was obtained for two sets of parameters x and CFOAS (1) x = 0.5, CFOAS = 80.3 K and (2)x = — 0.15, CFOAS = 109K. The Schottky heat capacity curves resulting from the calculations utilizing the above sets of parameters labeled Scheme A and Scheme B, respectively, are shown in Figs. 3 and 4. 

This behaviour is characteristic of any two-state system, and the maximum in the heat capacity is called a Schottky anomaly. 

The electronic contribution is generally only a relatively small part of the total heat capacity in solids. In a few compounds like PrfOHE with excited electronic states just a few wavenumbers above the ground state, the Schottky anomaly occurs at such a low temperature that other contributions to the total heat capacity are still small, and hence, the Schottky anomaly shows up. Even in compounds like Eu(OH)i where the excited electronic states are only several hundred wavenumbers above the ground state, the Schottky maximum occurs at temperatures where the total heat capacity curve is dominated by the vibrational modes of the solid, and a peak is not apparent in the measured heat capacity. In compounds where the electronic and lattice heat capacity contributions can be separated, calorimetric measurements of the heat capacity can provide a useful check on the accuracy of spectroscopic measurements of electronic energy levels. 

Einstein heat capacity equation 569-72 Schottky effect 580—5 solid + solid phase transitions 399-404 first-order 402-4 solutes 6... 

Debye heat capacity equation 572-80 Einstein heat capacity equation 569-72 heat capacity from low-lying electronic levels 580-5 Schottky effect 580-5 statistical weight factors in energy levels of ideal gas molecule 513 Stirling s approximation 514, 615-16 Streett, W. B. 412... 

Let us examine the data of the third measurement on the metallized wafer. There are two contributions to the heat capacity, a linear contribution and a spurious one. The spurious contribution may be interpreted as the high temperature side of a Schottky anomaly. In this hypothesis, the heat capacity per unit volume of the metallized wafer may be... 

As such, nuclear contributions to the heat capacity due to the interaction between germanium crystalline electric field gradients and the quadrupole moments of boron nuclei could account for the observed onset of the Schottky anomaly. 

Up to now, in the formulation of a bolometer model, only the heat capacity of itinerant carriers was considered [57], However, our measurements show that, even at 24 mK, the presence of a spurious heat capacity in the thermometer increases the expected value of the pulse rise time. We expect that the spurious contribution in Fig. 12.17 increases down to the temperature of the Schottky peak at T = k.E/khT about 10 mK. Since gc decreases at low temperatures, the total effect on pulse rise time and pulse amplitude can be dramatic at lowest temperatures. In reality, the measured rise time of CUORICINO pulses is about three times longer than that obtained from a model which neglects the spurious heat capacity of the thermistor. For the same reason, also the pulse amplitude is by a factor two smaller than the expected value (see Section 15.3.2). 

Figure 8.25 The Schottky-type heat capacity of Nd2S3 [28]. The insert shows the total heat capacity of ErFeC>3 [29].

Also, the heat capacity is affected by the axial ZFS parameter and, in excess of the lattice contribution, it shows a Schottky anomaly as modeled in Fig. 2. In the zero magnetic field the isofield heat capacity Ch collapses to the usual Cp and stays isotropic. 

The low temperature heat capacities, 52.67-296.29 K were obtained from Weller (6). The high temperature heat capacities were measured by Schottky (7, 282 C), Ewald (8, 275-373 K), Krestovnlkov and Peigina (9, 288-873 K), and Chiznikov and Khirik (IJO,... 

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). 

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. 

---