Observation on Probability Function by Measuring Time Series of Electric Property in Gadolinium Nitrate Crystal at Low Temperature

Abstract

The observation of the dynamics of fluctuation in a nonlinear nonequilibrium material system in a low-temperature region gives valuable information for developments in the physics of both nonlinear quantum dynamics and nonequilibrium thermodynamics. Rare-earth nitrate crystals have the same crystal habit and triclinic space group. The crystal structure is independent of temperature above 80K. Nonlinear and nonequilibrium phenomena have been observed in rare-earth nitrate crystals. Distinct intermittent nonperiodic oscillations (bursts) were observed in the measurement of the time series data for conductivity. The phenomena could be classified into a macroscopic chaos in a dissipative dynamical system. The effect of rare-earth elements on the fluctuation was found in the electric properties of the crystals. The characteristic properties of rare-earth nitrate crystals were observed in the electric measurement of the crystals in the lower-temperature region below 200K. We have measured the time series of ac conductivity, and clarified unknown properties of rare-earth nitrate crystals in the low-temperature region below 200K. A singularity in the probability distribution function was observed by measuring the time series of ac conductivity in the direction perpendicular to the c-axis of the gadolinium nitrate crystal at 11K. The result was given in the present note. A single crystal was grown in a gadolinium nitrate aqueous solution by evaporation at temperatures from 312 to 309K for 1994 h. The sample was 0:040 0:005 cm in thickness and 0:276 0:005 cm in area. The time series data of ac conductivity at 2 kHz were measured using a twophase digital lock-in amplifier (EG&G Princeton Applied Research 5210) and an analog memory recorder (Yokogawa AR1100). The experimental apparatus was controlled by a computer with a general-purpose interface bus (GPIB). The time series data of ac conductivity perpendicular to the c-axis (c2-axis) were measured at a sensitivity within 10 11 1 cm . The measurement was performed under isothermal condition controlled within 0:1K using a closed-cycle refrigerator system (Janis Research CCS-300). At 11K, with a sampling time of 50ms, 5000 data points were collected for the time series data. Figure 1 shows the time series of the fluctuation 0 for the real part 0 of the complex conductivity at 2 kHz for the electric field E applied in the direction perpendicular to the c-axis at 11K in a cooling cycle. The fluctuation 0 is defined by the relation 0 1⁄4 0 0, where 0 is the average derived from the data of the conductivity 0, given by 0 2:0298 10 6 1 cm 1 at 11K. Bursts occur randomly as nonperiodic unstable oscillations in the time series of 0 at the temperature. The probability distribution function of the time series data { 0} on the semilogarithm scale is given in Fig. 2. The probability function could not be classified into both standard Gaussian and Lorentzian line shapes. The functional form of the probability function is symmetrical and has a sharp peak as seen from Fig. 2. Figure 3 shows a log– log plot of the probability distribution function versus the fluctuation 0. The result indicates that the singularity of the probability function is proportional to ð 0Þ , where the exponent s of the power law is given by s 1⁄4 1:43 0:2. The temperature variation in the probability functions was observed in other rare-earth nitrate crystals, which were classified into the Gaussian one. 0 50 100 150 200 250 -10 -5 0 5 10