Are super-exponential luminescence decays possible? Tiago Palmeira, Mário N. Berberan-Santos ⇑ CQFM – Centro de Química-Física Molecular and IN – Institute of Nanoscience and Nanotechnology, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal article info Article history: Received 29 August 2014 In final form 7 October 2014 Available online 22 October 2014 Keywords: Luminescence decay Stretched exponential Relaxation kinetics Fluorescence Phosphorescence Triplet–triplet absorption abstract Luminescence decay functions describe the time dependence of radiation emitted by a sample after excitation. An overview of the mathematical aspects and systematics of luminescence decays is presented. In particular, super-exponential (faster-than-exponential) decays are defined and the possibility of their observation in single species physicochemical systems discussed. It is shown that this type of behavior can be both spontaneous and induced (by acting upon the system in real time). Spontaneous super-exponentiality is identified for the first time in experimental decays, these being phosphorescence decays affected by triplet–triplet absorption. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction A luminescence decay function, I(t), is the function describing the time dependence of the intensity of radiation spontaneously emit- ted by a previously excited sample at a given wavelength. This emission may result from a single species (containing a luminophore) or from several species. A strict decay function is a monotonically decaying function dI dt < 0 for all t   with a finite  light sum  ( R 1 0 IðtÞdt, a time constant proportional to the total number of light quanta emitted) and results from a single species. For convenience, and without loss of generality, the decay function is normalized at t= 0, I(0) = 1. The decay function can in principle be related to a model describing the luminescence mechanism and respective dynamics, but remains a valuable tool if this model is not available. The luminescence decay function can be written in the following form, see Ref. [1, pp. 308–312]: IðtÞ¼ Z 1 0 gðkÞe kt dk: ð1Þ This relation is generally valid as I(t) always has an inverse Laplace transform, g(k). The function g(k), also called the eigenvalue spec- trum, is normalized, as I(0) = 1 implies R 1 0 gðkÞdk ¼ 1. In many situ- ations the function g(k) is nonnegative for all k>0, and g(k) can be regarded as a distribution of rate constants - strictly, a probability density function (pdf) [1-3]. It was previously shown [2] that this is always the case for completely monotonic functions [4], i.e., functions for which the nth-order derivatives I (n) (t) obey ð1Þ n I ðnÞ ðtÞ > 0 ðn ¼ 0; 1; 2; ...Þ: ð2Þ However, in a few cases the decay function does not comply with this definition and the respective g(k) also takes negative values [2,3]. For the discussion of the time behavior of decay functions a sec- ond formal approach is useful [2,3,5]. Let us consider the following defining equation wðtÞ¼ d ln IðtÞ dt ; ð3Þ where w(t) is the decay rate coefficient, with a possible time depen- dence. For a monotonic decay function, w(t) > 0 for all t. Eq. (3) implies that the decay can be written as IðtÞ¼ exp  Z t 0 wðuÞdu   : ð4Þ Using Eq. (1), the time-dependent rate coefficient becomes wðtÞ¼ R 1 0 kgðkÞe kt dk R 1 0 gðkÞe kt dk ¼ Z 1 0 khðk; tÞdk; ð5Þ where h(k, t) is a new time-dependent distribution function that remains normalized for all times [2,3], hðk; tÞ¼ gðkÞe kt R 1 0 gðkÞe kt dk : ð6Þ http://dx.doi.org/10.1016/j.chemphys.2014.10.011 0301-0104/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author. E-mail address: berberan@tecnico.ulisboa.pt (M.N. Berberan-Santos). Chemical Physics 445 (2014) 14–20 Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys