Bose–Einstein Statistics

The properties of the ideal Bose gas are calculated from the integral equations for the energy and the number of particles as a function of the temperature and chemical potential. It is shown that the integral equations break down below the Einstein temperature that corresponds to the transition to the low-temperature state. The lowest single-particle energy level must be treated explicitly to get the proper equations. With the inclusion of the lowest single-particle energy level, the low-temperature behavior is calculated. The occupation of the lowest level becomes comparable to the total number of particles in the system below the Einstein temperature, and equal to the total number of particles at zero temperature. A numerical solution to the properties of the Bose gas is discussed, and the detailed calculations are assigned to the problems at the end of the chapter.

Study of elastic moduli and thermal properties of RMnO3 (R = La, Nd) compounds

In this paper, we develop a theoretical model for quantitative analysis of temperature-dependent heat capacity calculation of the magnetoresistance compounds RMnO 3 ( R = La , Nd ). The results on heat capacity obtained by us are in good agreement with the measured values. An effective interionic interaction potential (EIoIP) with the long-range Coulomb, van der Waals (vdW) interaction and short-range repulsive interaction up to second neighbor ions within the Hafemeister and Flygare approach was formulated to estimate the Debye and Einstein temperature and was found to be consistent with the available experimental data. In addition, the properties studied are the cohesive energy, molecular force constant, Restrahlen frequency and Gruneisen parameter. After characterizing thermal properties, a systematic investigation of elastic behavior has been undertaken and it has been found that the elastic moduli are decreasing continuously with increasing temperature.

Temperature Dependent Photoluminescence of ZnO Nanorods Synthesized by Hydrothermal Method

Temperature dependent photoluminescence of ZnO nanorods synthesized by hydrothermal method is studied. According to fifteen photoluminescent curves which were measured from 78 K to 288 K with an interval of 15 K, peak energy of exciton emissions, integral intensity of exciton emission peaks and integral intensity of deep-level emission peaks as a function of temperature were studied. The experimental data were fitted by Bose-Einstein relation and thermal activation function. By fitting, some important parameters were obtained and compared, such as the Einstein temperature for the excitons, the thermal activation energy of excitons or deep-level defects etc.

Variation in the Einstein Temperature of Guest Atoms in Ba-Ge-X type-III Clathrate Compounds

Einstein temperatures of guest atoms in Ba-Ge-(Al, In) type-III clathrate compounds have been estimated from the temperature dependence of the atomic displacement parameters determined by synchrotron X-ray powder diffractions. The lowest temperature is obtained for the vibration of Ba(2) atoms along the x-direction, which corresponds to the “rattling motion” of the guest atoms in the compounds. The temperature estimated is significantly low of about 50 K, which agrees with the fact that the compounds have small lattice thermal conductivities of about 0.6 W/mK. Though the lattice thermal conductivity of Ba24Ge88Al12 is larger than that of Ba24Ge88In12, the Einstein temperature of Ba24Ge88Al12 is slightly smaller than that of Ba24Ge88In12. This discrepancy can be explained by the consideration of higher Debye temperature of Ba24Ge88Al12 than that of Ba24Ge88In12, that is, lattice thermal conductivity without “rattling motion” is larger for Ba24Ge88Al12 than that for Ba24Ge88In12.