2.5　Thermal properties of HTS bulk

It is well known that the critical temperature Tc ，critical current density Jc ，and upper critical field Hc2 of HTS bulks are the most important parameters for high field applications of bulk superconductors, which have been discussed above. HTS bulks have a series of unique properties, which are very important for the applica tions.These problems should be mainly for the physicists and materials scientists to investigate.In addition to the three critical parameters Tc ，Jc ，and Hc2 ，some thermal and mechanical properties of HTS bulk materials associated with the ap plications will be briefly discussed in the section.Details of the HTS bulk properties can be found elsewhere125.

2.5.1　Specific heat

The specific heat capacity, often simply called specific heat, is the heat capacity per unit mass of a material. An object's heat capacity C is defined as the ratio of the amount of heat energy transferred to an object and the resulting increase in temperature of the object，

where Q is the amount of heat, C is the specific heat capacity, m is the mass, andΔT is the temperature change after absorption or release of heat.In the Interna-tional System of Units, heat capacity has the unit J/（kg·K）.

The specific heat C in a normal conductor consists of two contributions：from electrons in the conduction band Cel and from the lattice or phonons Cph . The elec tronic specific heat Cel is defined as the ratio of that portion of the heat used by the electrons to the rise in temperature of the system.The free electron contribution to the specific heat is typically less than 1% of the phonon specific heat at room temperature.When a small amount of heat is put into a superconductor, some of the heat is used to increase the lattice vibrations, and the rest is used to increase the energy of the conduction electrons.The specific heat of the electrons in a su perconductor varies with the temperature in the normal and the superconducting state.The electronic specific heat in the superconducting state Cels is smaller than in the normal state Celn at enough low temperatures, and the precise measurements indicate that, at temperatures considerably below the transition temperature, the logarithm of the electronic specific heat is inversely proportional to the tempera ture.However, Cels becomes much larger than Celn as the transition temperature Tc is approached.The transition to the superconducting state is accompanied by quite drastic thermodynamic changes in the superconductor.The specific heat at the transition from the normal to the superconducting state in zero magnetic field appears as a jump at the critical temperature Tc .

Specific heat experiments on bulk superconductors are insensitive to the phase of the order parameter. However, they can provide valuable information on the density of states near the Fermi level because the electronic specific heat Ce is proportional to the density of states.

Due to the low specific heat of LTS materials in superconducting state, prac-tical LTS wires are produced as multifilamentary composites in order to prevent quenching. In contrast to LHS materials, there are some peculiar features in the specific heat of YBCO compared with those of conventional BCS superconductors, namely bulk HTS are thermally stable even in large sample sizes due to their rel atively large specific heat in the superconducting state126.However, when a HTS bulk magnet is activated another new thermal instability is brought in by the flux motion.For thermal stability, the cooling power of the system must be larger than the local heat generation.

Myers et al.126reported the specific heats of four superconducting materials at temperatures from 0 K to 300 K and in magnetic fields from 0 T to 14 T（Fig. 2.5.1）.The specific heats of the Bi2212 samples were relatively independent of applied field, the zero field specific heat of a two-dimensional random oriented single stack sample increased from 0.155 J/（kg·K）at 4 K to 254 J/（kg·K）at 250 K.At 2 K, the specific heat of a Nb3Sn rod-in-tube strand increased from 0.0257 J/（kg·K）at 0 T to 0.0716 J/（kg·K）at 14 T.In zero field the specific heat of a MgB2 sample increased from 0.623 J/（kg·K）at 4 K to 382 J/（kg·K）at 250 K.

Naitoa et al.127have measured the temperature dependence of specific heat C（T）for a Ag（10-20μm thick）deposited YBCO coated conductor（YCC）film（about 1.5μm thick），YCC reinforced by a thin Cu tape（300μm thick），and a Hastelloy substrate with a buffer layer.Fig.2.5.2 shows the temperature dependence of the specific heat of YCC-Ag20（20μm），YCC-Ag10（10μm）Cu300（300μm）and Hastelloy+buffer.The C（T）of all the samples decreases monotonically with decreasing temperature.Absolute values of C（T）at 300 K are about 370 J/（kg·K），390 J/（kg·K）and 400 J/（kg·K）for YCC-Ag20，YCC-Ag10Cu300 and Hastelloy+buffer, respectively.

Gahtori et al.128reported the temperature dependence of the specific heat C（T）of GdBa2 （Cu1−x Mnx ）3 O7−δ for different x（see Fig 2.5.3），and gave the theoretical calculation results129.The model based on separate electron and phonon contribu-tions was used to interpret the specific heat data.The substitution of Mn in the system has been found to effectively suppress the specific heat jump observed in the pristine compound.In the lower right corner of Fig 2.5.1，it can be seen clearly that the specific heat of HTS Bi2212 is higher than LTS Nb3Sn and MgB2 in the same temperature range.

Knowledge of specific heat and thermal conductivity of HTS is essential to un-derstand the response of the superconductor to heat released due to variations of the applied magnetic field, for instance, the heat response of the superconduc-tor during local heating when HTS bulks are activated by pulsed fields and the phenomenon of flux jumps at low temperature.

2.5.2　Thermal conductivity

According to Fourier's law, the temperature gradientΔT imposed across an isotro-pic sample of a crosssectional area A results in a heat flow Q given by

the minus sign indicating that heat always flows from a warmer to a colder region of a substance. The units of thermal conductivity are W/（m·K）.The thermal con-ductivity k consists of both the thermal conductivity of the electrons kel and the lattice or phonon kph .

In pure metals, the electronic component accounts for nearly all the heat con ducted, while the lattice component is almost negligible. In superconductors at tem peratures well below Tc , electrons condense into Cooper pairs that cannot transport entropy.Thus, they do not contribute to the thermal conductivity.The phenomenon can also be understood in terms of the superconducting state being a perfectly or dered state, i.e.one of zero entropy, thus vanishing thermal conductivity due to the electrons.At sufficiently low temperatures, the thermal conductivity is attributable entirely to lattice waves and is similar to the form of the thermal conductivity of an insulating material.The thermal conductivity in the normal state（kn）approaches that in the superconducting state（ks）as the temperature approaches the transi tion temperature for all materials, either pure or impure.In normal metals like copper, large electrical conductivity is accompanied by large thermal conductivity.The ratio between the two is approximately constant.The thermal conductivity in the superconducting state is smaller than that in the normal state and almost van ishes at very low temperatures.Even in the superconducting state, the existence of normal electrons cannot be completely avoided, due to the so-called quasipar ticles in the superconducting state.The amounts of these quasiparticles in YBCO will affect the thermal conductivity, and their effect has been observed to decrease rapidly with temperature.It is now widely accepted that the rapid increase of k below Tc and the peak of k are mostly due to the contribution of quaispartides located in the CuO2 planes.However, further study is needed to clarify the effects of phonons and quasipartides to the specific contribution.

Marchal et al. 130 reported the dependence of thermal conductivity, thermoelec tric power and electrical resistivity on temperature for a bulk, large grain melt processed YBCO HTS containing two grains separated by a well-defined grain boundary（see Fig.2.5.4）.Transport measurements at temperatures between 10 K and 300 K were carried out both within one single grain（intra-granular properties）and across the grain boundary（inter-granular properties）.The influence of an applied external magnetic field of up to 8 T on the measured sample properties was also investigated.The presence of the grain boundary is found to strongly affect the electrical resistivity of the melt-processed bulk samples, however it has almost no effect on its thermoelectric power and thermal conductivity, within experimen tal error.The results of this study provide direct evidence that the heat flow in multi-granular melt-processed YBCO bulk samples should be virtually unaffected by the presence of grain boundaries in the material.

The coefficient of thermal conductivity of HTS bulk is about 2-10 W/mK at 77 K and very small below 10 K.Generally, there is a peak at around 50 K[see Fig.2.5.4（a），（b），and（c）].The planar structure of HTSCs makes the thermal con ductivity anisotropic.The thermal conductivity of bulk YBCO is anisotropic：3.5 W/mK along the c-axis, and 14 W/mK along the a-b plane110.Other anisotropic re sults are 4 W/mK and 20 W/mK along the c-axis and the a-b plane, respectively131.

2.5.3　Thermal expansion

The thermal expansion coefficients，α，are defined as the change of length or volume per degree of temperature under constant pressure.The linear thermal expansion coefficient is defined byαl=dl/（ldT），dl/l is the change rate of length, dT is tem perature change.The unit of linear thermal expansion coefficient is K−1.Thermal expansion of superconducting materials, the change of the superconductor dimen sion between room and low operating temperature, are important for both trapped flux and engineering applications.The stresses resulting from the differential ther mal expansion should be as small as possible.This is especially important for highly brittle HTS bulk materials, where these stresses may actually damage the super conducting bulks.The stresses also directly affect the maximum trapped flux in YBCO bulks.The thermal expansion is closely related to the specific heat.Thermal expansion is also very useful for predicting how solid state properties respond to volume changes in the limit of zero applied pressure.

The thermal expansion of sintered HTS bulk is similar to that of non-supercon ducting ceramics. The HTS single crystals are necessary for study of structural anisotropy.Since HTS bulk has a layered orthorhombic structure, one needs to measure linear coefficients of thermal expansion along all three crystallographic axes, i.e.the a-axis and b-axis in a-b plane, and in the c-direction perpendicular to the CuO2 planes.The linear thermal expansion coefficient of HTS bulk is in the range 1×10-6−1×10-5 K−1 in the temperatures of 40 K-300 K in a-b plane.The coefficient is in the range 3×10−6−1.5×10−5K−1in the temperatures of 40 K-300 K along the c-axis, and rises approximately linearly up to 150 K, then more slowly rising132.Some measurement methods and data of thermal expansion can be found elsewhere133.The thermal expansion associated with trapped flux will be discussed in Section 2.7.