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Lanthanides in Luminescent Thermometry Chapter 281 361 Assuming that the error in the calibration temperature determination ðsTc Þ is negligible and the error in D is similar to its uncertainty, we get: sSa 1⁄4sD%dD (15) and, thus, Eq. (13) becomes: Sa D D sSr pffiffi dD S1⁄4 2D (16) r Finally, substituting Eq. (16) into Eq. (12), leads to: sdT pffiffi dD dI dT1⁄4 2D1⁄42I (17) That means that the relative error on the temperature uncertainty is essen- tially determined by dI/I. 3.2.3 Temperature Uncertainty and Thermometer Size A recent quantification of dT for distinct classes of nanothermometers (fluo- rescent polymers and proteins, CdSe QDs, and nanodiamonds) was recently done considering size and system-dependent properties and the spin-boson model (Alicki and Leitner, 2015): 4T dT1⁄4 pffiffi e 3 3TD 3TD=8T 1 pffiffiffiffiffiT (18) Na where Na is the number of atoms in the sample and TD its Debye temperature. For TD in the range 100–2000 K, the term in parenthesis changes between 0.9 and 1.3 and 1 dT % pffiffiffiffiffi T (19) Na For these authors, dT is termed “accuracy” in the sense that it represents how close a measured value is to the actual (true) value. Accuracy is defined in terms of systematic errors and should not be confused with precision that represents how close the measured values are to each other (random errors). Both for fluorescent organic- (with green fluorescent protein, GFP) and solid-state-based thermometers (with nanodiamonds or CdSe QDs) the model predicts an accuracy in accord with the experimental measurements, about 0.5K for GFP (Takei et al., 2014), 44103 K for nanodiamonds (Kucsko et al., 2013), and 0.6 K for CdSe QDs (Yang et al., 2011). Moreover, the sys- tem dimension determines the lowest temperature change that is possible to measure and in practice dT is controlled by the radius, r, of the thermometer, and varies as r3/2 (Alicki and Leitner, 2015). For instance, considering the criteria of temperature uncertainty referred earlier for portable (dT<0.5 K)PDF Image | HANDBOOK ON THE PHYSICS AND CHEMISTRY OF RARE EARTHS
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