Mass bias correction models

Calibration of the measured isotope amount ratio, 787/86, is achieved by admixing known amounts of strontium that is enriched in Sr and Sr [24-27]. The isotopic composition of the double-spike strontium has to be known. Matrix matching does not have to be achieved per se. If the amount ratio of the non-radiogenic strontium isotopes, say N( Sr)/N( Sr), is well known and does not vary significantly in Nature, it can be used to calibrate the measured ratio 787/86 without any admixing of the double spike [28]. In both of its variations, this procedure involves the selection of an appropriate mass bias correction model, such as the exponential law (see below). 

Albeit among the simplest mass bias correction models, this is the least favorable of all calibration strategies as it combines the shortcomings of all other techniques -the calibration is sequential and there is a need for a mass bias correction model and for matrix matching. 

The need for a formal mass bias correction model arises when a specific isotope amount ratio is used to calibrate the amount ratio of another pair of isotopes, either of the same or of a different element, as for example when N( Sr)/N( Sr) is used to caUbrate N( Sr)/N( Sr) or when is used to caUbrate... 

Mass bias in MC-ICP-MS varies with both time and nuclide mass. Since calibration is typically performed sequentially in a calibrator-measurand-cahbrator bracketing mode, the effect of time has to be separated from that of mass in the mass bias correction model. Hence the logic of creating a mass bias discrimination model is to express the calibration factor, Ki/j, as a product of two functions one that varies with time only and the other that varies with the nuclide masses only [16, 17, 19]. For example,... 

Russell s law [40], on the other hand, is not equivalent to the exponential mass bias correction model, although these two names are often used interchangeably. However, the difference in the correction factors for the two models is determined solely by the masses of all nuclides involved. Combining Eqs. (5.5) and (5.6) leads to the following ... 

To date, interrogation of the efficacy of mass bias correction models has largely resorted to attempts to determine the value of the discrimination exponent. In such experiments, the slope of the log-linear two-isotope ratio regression is used, which, in turn, leads to the discrimination exponent by solving the following expression [43-46] ... 

The ability to measure simultaneously the isotopes of two different elements with high precision has never been a particular challenge in mass spectrometry. The challenge has always been to ensure the accuracy of the mass bias transfer. Hence a fundamental limitation of the traditional mass bias correction models is their... 

Note that the cahbrator element in this method serves as the mass bias correction proxy. Hence, for example, even though the value obtained for the Zr isotope ratio may itself be biased due to the limitations of the mass bias correction models employed (e.g., assumptionysr= ), this bias is largdy negated in the second step of the calibration (Zr Sr). Owing to error cancellation with this method, it is akin to the use of isotope dilution methods wherein reverse and direct isotope dilution protocols are performed in tandem. The error cancellation, however, occurs only when matrix separation and concentration matching are fully attained. 

Yang, L. and Sturgeon, R.E. (2003) Comparison of mass bias correction models for the examination of isotopic composition of mercury using sector field ICP-MS. J. Anal. At. Spectrom., 18, 1452-1457. 

Depending on the isotopes monitored and the nature of the sample, several types of mass bias correction models can be used when coupling chromatography to MC-ICP-MS. Some studies report the use of internal mass bias correction with an exponential law when an invariant isotope ratio is available (as is the case, for example, for Sr or Nd) [27, 29-31]. It is the simplest model, as merely another isotope ratio of the element of interest is used for the correction. 

This is the model equation for the calibration of isotope amount ratios based on the log-linear temporal isotope amount ratio regression. Note that a and b are perfectly correlated (p = +1) if Rtp < 1 (inRup < 0) and perfectly anti-correlated (p = —1) if Rk/i > 1 (InRfe/ > 0). It is important to stress that this calibration method is fundamentally different from the conventional mass bias correction la vs. Since the regression model does not invoke the principle of time-mass separation, it does not need either the discrimination exponent or the equality of the discrimination functions [17]. 

What is more important, this calibration model is not derived from either the exponential or Russell s laws as is commonly perceived (and originally presented) [15, 32], Rather, it only requires the mass spectrometer response be linear [Eqs. (5.27) and (5.28)]. It is the interpretation of the slope and the intercept that can lead to the reliance on the exponential mass bias correction or even erroneous results. Consider, for example, the substitution of Eq. (5.34) in Eq. (5.32) ... 

Although this mass bias correction method is superior to all other currently known models, it demands large investments in measurement time and sample size. This cannot always be fulfilled in practice hence there is a need for alternative calibration models, even at the cost of performance. Moreover, theoretical limitations of this calibration model are yet to be comprehended. 

Like the traditional mass bias correction approaches, the double-spike method also relies on the choice of the mass bias model. The original formalism of the double spikes employed the linear mass bias law and, although double-spike calibration equations adapted for the exponential mass bias discrimination are available, linear models are still often used owing to their simplicity (see, for example, [50-52]). The caveat here is that erroneous results can be obtained when a linear correction is applied to data that do not follow such behavior. This is illustrated below. 

Baxter, D.C., Rodushkin, 1., Engstrom, E., and Malinovsky, D. (2006) Revised exponential model for mass bias correction using an internal standard for isotope abundance ratio measurements by multi-collector inductively coupled plasma mass spectrometry. J. Anal. At. Spectrom., 21, 427-430. 

As research into absolute isotope ratio measurements by MC-ICP-MS continues, some new limitations have been identified recently. Although in MC-ICP-MS, mass bias is generally considered to be mass-dependent fractionation (MDF) and is corrected by various mass-dependent correction models, mass-independent fractionation (MIF) in MC-ICP-MS was described by Lu Yang et al., of the National Research Council Canada, at the 2013 European Winter Conference on Plasma Spectrochemistry. Their study looked at numerous elements and concluded that this appears to be a common phenomenon, and consequently has serious implications on the absolute isotope amount ratio measurements. 

In practice, the analyst monitors the bias correction through analysis of a reference standard on a routine, often daily, basis. This value comes to be known very well and makes insignificant contributions to overall precision. Even though it may not be truly applicable to the sample being analyzed, using it is far better than applying no correction it is the best that can be done in an imperfect world. A model of thermal fractionation on mass spectrometer filaments has been developed by Habfast [58]. 

---