Uncertainty charge collection

Thorium. Multiple-collector measurement protocols by TIMS for thorium isotopic analysis typically involve the simultaneous measurement of Th and °Th (for silicate rocks), or Th and °Th, then Th and Th (for low- Th samples), using an axial ion counter and off-axis Faraday collector (Table 1). Various methods are used to correct for the relative gain between the low-level and Faraday detectors and 2a-uncertainties of l-5%o are typically obtained (Palacz et al. 1992 Cohen et al. 1992 McDermott et al. 1993 Rubin 2001). Charge-collection TIMS protocols enable Th, °Th and Th to be monitored simultaneously on a multiple-Faraday array and can achieve measurement uncertainties at the sub-permil level (Esat et al. 1995 Stirling et al. 1995). 

In the past three years, MC-ICPMS has emerged as an alternative to TIMS for precise measurement of the U-series isotopes with comparable or better precision. U-Th isotopes can now be routinely measured at the sub-permil level. Previously, this had only been demonstrated using charge-collection TIMS applied to thorium isotope measurement. Data collection efficiency, sample size requirements, and detection limits can also be greatly improved over TIMS. For the U- U- Th system applied to carbonate samples, this has extended the dating range beyond 600,000 years, and °Th-age uncertainties of 2000 years are now attainable on 300,000 year-old samples (e g., Stirling et al. 2001). 

Of these terms the electronic noise over which we have some control, will be our main concern in this chapter although we shall have to consider uncertainty of charge collection, Wq, later. The remaining term, Wp, the charge production uncertainty, will be covered in Chapter 6. 

An alternative, and simpler, estimate of the electronic noise can be made from the FWHM vs. energy calibration (Chapter 6, Section 6.5). Equation (4.1) relates the FWHM of a peak to the various uncertainty contributions. Common sense tells us that if the gamma-ray energy is zero, the uncertainty on the charge production and on the charge collection must also be zero. Hence, if we extrapolate the width calibration to zero energy we will have an estimate of the electronic noise. 

In the first edition of this book, I discussed at various points the best function to fit to FWHM calibration data. I compared a linear fit with the square root function recommended by Debertin and Helmer (see below). Ultimately, I suggested that a linear fit to FWHM calibration data was more satisfactory than any alternative and that this could mathematically be justified if one assumed that the charge collection uncertainty was linear with respect to energy. A little more thought on the matter revealed some inconsistency in the justification. I have looked at the matter again. 

We can equate B with our and A with our This function implies that either the charge collection uncertainty is negligible or that it has the same square root dependence on energy as the charge production, in which case A = p + [Pg.139]

Clearly, the square root quadratic function provides the best fit. (Hurtado et al. (2006) come to the same conclusion in a similar comparison.) The closeness of the fit gives some confidence to the speculative proposal that the charge collection uncertainty can be modelled by assuming it is hnear with energy. We are no nearer to explaining why that should be so, but from a modelling point of view it is satisfactory. The estimated parameters for these particular data were ... 

From that, I estimate the Fano factor to be 0.108, somewhat higher than the value of 0.058 generally used, but well within the range of reported values. The value from the Debertin and Helmer equation is somewhat higher, but that does, in effect, include an allowance for the uncertainty of charge collection. 

The resolution of peaks in a spectrum is much worse than any natural spread in the gamma energy. The extra uncertainty is added in the processes of charge production, charge collection and electronic conversion. We can only influence the latter two. 

Both the theoretical and experimental data for the classic 2S i/2 — 2Pi/2 Lamb shift are collected in Table 12.2. Theoretical results for the energy shifts in this Table contain errors in the parenthesis where the first error is determined by the yet uncalculated contributions to the Lamb shift, discussed above, and the second reflects the experimental uncertainty in the measurement of the proton rms charge radius. We see that the uncertainty of the proton rms radius is the largest source of error in the theoretical prediction of the classical Lamb shift. An immediate conclusion from the data in Table 12.2 is that the value of the proton radius [27] recently derived form the analysis of the world data on the electron-proton scattering seems compatible with the experimental data on the Lamb shift, while the values of the rms proton radius popular earlier [28, 29] are clearly too small to accommodate the experimental data on the Lamb shift. Unfortunately, these experimental results are rather widely scattered and have rather large experimental errors. Their internal consistency leaves much to be desired. 

Fig. 16. Schematic picture of the region of delocalization of energy losses of a fast charged particle. Ax is the uncertainty in the coordinate of the point of energy transfer, bpl is the radius of collective excitations, and 9 and ft are the strengths of the electric and magnetic fields.

The paired samples which were collected at each site were compared to the results of the 10 sampler experiment to detect samples of poor quality. This procedure was utilized to insure than experimental uncertainties did not contribute to the observed variation in composition. The variance for each species listed in Table 1 from the control site was statistically compared to the variance for the paired samples at all sites with an F-test. One of the paired measurements at a site was rejected if their variance was significantly (p <0.005) greater than measured at the control site and one of the pair had a significantly (p <0.005) poorer charge balance (sum of... 

The limitations of these methods should be understood. Their application requires the determination of the empirical parameters in eqns (5.2) and (5.3), as well as the partial atomic charges in the coulombic term. The former are usually parameterized from experimental solid state data such as vibrational frequencies or sublimation enthalpies, which in themselves contain some experimental uncertainties and variability from system to system. The partial atomic charges can and do vary with the choice of basis set for the calculations from which they are derived. The function chosen and the complete set of parameters are often collectively termed a force field . Ideally, one would like to develop a universal force field, but given the diversity... 

A linear fit of I versus cos 0 or I versus sin 0 will reveal whether orientation is consistent for the collected spectra. If no systematic deviation from linearity is observed, there are probably no gross experimental artifacts. The sample charging artifact described in Section 4.2.2, however, can sometimes result in a linear fit because the illuminated spot size increases trigonometrically with incident angle. The standard deviations of the fit parameters provide some statistical uncertainty of the orientation measurement. With a sufficient number of points, a confidence interval can be determined. Small differences in orientation can then be judged on a statistical confidence basis. The parameters A and a can also be fit directly using nonlinear least-squares methods. 

A variety of techniques to identify and quantify acid and base components in rainwater are applied to data for southern California. Charge balance calculations using major cation and anion concentration data indicate southern California probably had alkaline rain in the 1950 s and the 1960 s with the exception of the Los Angeles area which probably had acidic precipitation. Measurements of the chemical composition of precipitation collected in Pasadena, California, from February 1976 to September 1977 are compared with the charge balance and conductivity balance constraints. A chemical balance is used to determine the relative importances of different sources. The pH is found to be controlled by the interaction of bases and strong acids with nitric acid being 32% more important on an equivalent basis than sulfuric acid. The uncertainties in the various calculations are discussed. 

Each of these terms can be replaced by the mathematical representations introduced above. In Section 6.2,1 showed that the width due to charge production is proportional to the square root of the gamma-ray energy. In Section 6.3, I suggested a linear relationship between collection uncertainty and energy, and in Section 6.4, I stated that the electronic contribution to peak width is independent of energy. Hence ... 

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