Exponential analysis in physical phenomena

Abstract

Many physical phenomena are described by first-order differential equations whose solution is an exponential decay. Determining the time constants and amplitudes of exponential decays from the experimental data is a common task in semiconductor physics (deep level transient spectroscopy), biophysics (fluorescence decay analysis), nuclear physics and chemistry (radioactive decays, nuclear magnetic resonance), chemistry and electrochemistry (reaction kinetics) and medical imaging. This review article discusses the fundamental mathematical limitations of exponential analysis, outlines the critical aspects of acquisition of exponential transients for subsequent analysis, and gives a comprehensive overview of numerical algorithms used in exponential analysis. In the first part of the article the resolution of exponential analysis as a function of noise in input decays is discussed. It is shown that two exponential decays can be resolved in a transient only if the ratio of their time constants is greater than the resolution limit, which can be explicitly calculated from the signal-to-noise ratio in the transient. Although the signal-to-noise ratio is generally limited by the sensitivity of the equipment, it is shown that digitalization of the decays may be a major source of noise. The requirements for type of analog-to-digital converter, number of digitized data points and duration of digitized transients, which must be met to obtain the theoretical resolution limit and to improve stability of the exponential analysis, are formulated. The second part of the review article gives an overview and comparison of major numerical techniques of exponential analysis, such as the nonlinear least squares fit, the Prony method, the method of modulating functions, the method of moments, the Laplace–Padé approximation, the Tikhonov regularization method, the Gardner transformation, the method of maximum entropy and others.

Temporal phase correction of multiple echo T2 magnetic resonance images

“…The resolution limit that defines the minimum ratio of T 2 times that can be resolved is noise dependent such that the higher the SNR, the better the resolution limit [33]. The SNR after TPC is slightly improved because the real-valued decay data are retained and the imaginary-valued data, which should contain mainly noise, is discarded.…”

On the importance of grain size in luminescence dating using quartz

“…It is also important to note that the two fitted D 0 values for each curve differ by an order of magnitude and so it is unlikely that these components are fitting artifacts (Istratov and Vyvenko, 1999). It is also interesting to note that the dose response curves of the two grain sizes overlap up to doses of about 100 Gy.…”

Processo para produção e caracterização de rochas sintéticas com porosidade controlada para aplicações em Petrofísica por RMN de alto e baixo campo.

“…Da mesma forma, sob o ponto de vista matemático, o processo de ajuste de dados de relaxação nãoé algo trivial, e uma série de fatores associadosà aquisição dos dados, assim comoàs abordagens matemáticas do problema, podem induzir a erros na interpretação dos parâmetros obtidos nos ajustes [20]. Essa mesma ressalva há de ser feita também para a interpretação dos dados de relaxação transversal (T 2 ) que serão apresentados a seguir.…”

“…Se considerarmos materiais geológicos com suscetibilidade magnética desprezível, que possuam grandes concentrações deíons paramagnéticos (altos valores de ρ) [14], a taxa de relaxação dos spins nesses materiais será dominada pelo termo de relaxação superficial, de modo que a equação (20) pode ser simplificada para a forma:…”