arXiv:cond-mat/0504428v1 [cond-mat.mtrl-sci] 18 Apr 2005 Comparison of methods for estimating continuous distributions of relaxation times Enis Tuncer 1, ∗ and J. Ross Macdonald 2, † 1 Solid State Physics Division, ˚ Angstr¨ om Laboratory, Uppsala University, SE-75121 Uppsala, Sweden 2 Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA The nonparametric estimation of the distribution of relaxation times approach is not as frequently used in the analysis of dispersed response of dielectric or conductive materials as are other immit- tance data analysis methods based on parametric curve fitting techniques. Nevertheless, such distri- butions can yield important information about the physical processes present in measured material. In this letter, we apply two quite different numerical inversion methods to estimate the distribution of relaxation times for glassy Li0.5La0.5TiO3 dielectric frequency-response data at 225 K. Both methods yield unique distributions that agree very closely with the actual exact one accurately cal- culated from the corrected bulk-dispersion Kohlrausch model established independently by means of parametric data fit using the corrected modulus formalism method. The obtained distributions are also greatly superior to those estimated using approximate functions equations given in the literature. PACS numbers: 77.22.Gm 02.70.Rr 02.30.Zz 02.50.Ng 02.60.-x 02.30.Sa 66.30Dn 61.47.Fs 72.20.i 66.10.Ed Keywords: Distribution of relaxation times; dielectric relaxation; least squares approximations; Monte Carlo methods; inverse problems Broadband dielectric (also known as immittance or impedance) spectroscopy is widely used to character- ize materials and to help understand the mechanisms involved in such challenging areas of condensed-matter physics as conductivity, molecular relaxation, liquid-glass transition etc. [1]. In this experimental technique an electrical property of the material is recorded as a func- tion of probing field frequency ν . Data may be ex- pressed at one of the four specific immittance levels (i) the complex resistivity ρ(ω); (ii) the complex mod- ulus M (ω) ≡ ıωε 0 ρ(ω); (iii) the complex permittiv- ity ε ≡ [M (ω)] −1 ; and (iv) the complex conductivity σ(ω) ≡ ıωε 0 ε(ω) ≡ [ρ(ω)] −1 . Here, ω is the angular frequency ω =2πν ; ε 0 is the permittivity of free space; and ı = √ -1. Once a data set is acquired, it may be expressed at an appropriate immittance level and then analyzed to obtain valuable information about material pro- cesses. Often employed procedures that have been used to analyze frequency response data are (a) using the Kohlrausch-Williams-Watt (KWW) approached derived from stretched exponential behavior in the time do- main [2, 3]; (b) the Havriliak-Negami empirical expres- sion [4]; and (c) estimation of the distribution of relax- ation times (DRT) inherent in the data [3, 5, 6, 7, 8, 9], an approach not as commonly employed as the other two. Unlike the KWW analysis of (a), procedure (b) is a data fitting method that does not lead to added understanding of the physical processes presented in the experimental material. On the other hand, KWW analyses involve fitting models whose parameters are all of physical sig- nificance. Although they are useful for comparing fit pa- rameters for various materials at different state variable levels they are less appropriate for data involving several DRTs associated with different physical processes. The DRT approach of (c) is an elegant method for in- vestigating the contributions of relaxing units to the to- tal relaxation and for determining the influence of state variables on the relaxation. In the presence of differ- ent processes or broad relaxations, the DRT approach is superior to the parametric ones since (1) no a priori assumptions are needed, i.e., a sum of empirical expres- sions etc.; (2) the actual distributions in a given data set are initially unknown; (3) a DRT can be related to various physical parameters of the system; (4) and when there are two different overlapping relaxations present, their depencies on state variables would be easy to iden- tify and to observe the influence of the state variables on the distributions. As an example, the dynamic com- plexity of the relaxation system can be determined by estimating its DRT and thus establishing whether it is intrinsicly broadening (homogeneous) or a distribution of responses (heterogeneous) [10]. A distribution may be characterized as either discrete (composed of individual points) or continuous, and DRT analysis can unambigu- ously distinguish between these two possibilities [7, 8]. Recently, non-resonant spectral hole burning technique has been employed to resolve distinct continuous distri- butions experimentally in order to identify multiple re- laxing domains in materials [11]. In this letter, we compare the results of two different DRT inversion methods for analyzing a set of experimen- tal frequency-response data that involves a continuous distribution. We also compare the accuracy of two equa- tions for estimating appropriate distribution functions proposed by B¨ ottcher and Bordewijk [5]. Although esti- mation of discrete-point distributions is not an ill-posed mathematical problem [7], distribution estimation of con- tinuous distributions, the usual situation, is ill-posed. It is therefore particularly important to assess the utility and power of different DRT estimation procedures for a well-defined data situation. Experimental data, with M = 52 points, for the Li 0.5 La 0.5 TiO 3 (LLT) glass at 225 K [12], expressed as Typeset by REVT E X