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360 Handbook on the Physics and Chemistry of Rare Earths We should note, however, that recording a set of temperature readouts with statistical significance might not be always feasible. For instance, to record with high-resolution spectral region requested to define D, PMT detec- tors take typically one minute, which makes the evaluation of temporal fluc- tuations in temperature unpractical. Then, to overcome this limitation, dT can be computed using Eq. (7), with the relative uncertainty on D estimated through the integrated area of the transitions (dI/I). The typical dD/D values reported in Table 2 should not be used as they correspond to the lower uncer- tainty limit (maximum SNR operating conditions). In generally, the detector operates above that maximum and, thus, the dD/D values are always higher (eg, 0.5–2.5%) that those reported in the table. When the calibration curve is provided dD/D is assessed as the root mean square value of the deviations, otherwise we have: where dIi/Ii (i1⁄41, 2) is estimated using the SNR values. Typically, dividing the readout fluctuations of the baseline by the maximum intensity value allows one to estimate the order of magnitude of dIi/Ii. Assuming that the rel- ative uncertainties on I1 and I2 are similar, we have: dD pffiffi dI D1⁄4 2I (10) Thus, to evaluate the error in temperature determination we use the ther- mal dependence of Sr (Eq. 7) and the SNR of the spectra, resulting in: pffiffi dT1⁄4 2dI (11) Sr I The error in the determination of the temperature uncertainty (sdT) can be accessed by calculating the error propagation for Sr and dI/I: where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dD dI 2 dI 2 1⁄41+2 (9) D I1 I2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 sdT 1⁄4 sSr + sdI=I dT Sr dI=I Sr %sSr (12) considering that sdI/I is negligible relatively to sSr . The error in Sr is then given by: s1⁄4 rs + rs 1⁄4S Sa + D (13) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @S 2 @S 2 s 2 s 2 Sr @SSa @DDrSD aa sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s s2s2 Sa1⁄4 D + Tc (14) Sa D TcPDF Image | HANDBOOK ON THE PHYSICS AND CHEMISTRY OF RARE EARTHS
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