Background information

This section serves to further explain details of the databases and notations in a scientific concept. Here, background information is given that can help the user to better understand the various elements of the package and the logic behind it. Usage of the module is not discussed here.

Databases

Currently, the default database that is loaded is “lodders09”.

“lodders09”

The database named “lodders09” is based on the work by Lodders et al. (2009). This database is loaded by default, unless a different database is specified.

The measurements loaded when “lodders09” is selected are the data in Table 10 for elements and isotopes. The elemental abundances are simply gained by adding up the isotopic abundances. Note that this introduces a total abundance of Si that is 999700, which is within uncertainties equal to 10⁶.

The solar abundances loaded with Lodders et al. (2009) are nuclide abundances 4.56 Ga ago.

Note

The nuclide abundance of ¹³⁸La in Table 10 of Lodders et al. (2009) is given as 0.000. This seems to be an error originating too few significant figures. Using the atom percentages and the nuclide abundance for ¹³⁹La, we calculated a nuclide abundance of 0.0004 for ¹³⁸La and used this calculated abundance for our database.

“asplund09”

The database named “asplund09” is based on the work by Asplund et al. (2009).

The loaded elemental abundances are taken from Table 1 in Asplund et al. (2009) and represent the present-day solar photosphere (column “Photosphere”). The isotope abundances are taken from Table 3 in Asplund et al. (2009) and are the representative isotopic abundance fractions in the Solar System. According to the authors, most isotopic values are taken from Rosman & Taylor (1998) with some updates discussed in Section 3.10 of Asplund et al. (2009).

“nist”

The database named “nist” is based on the online-available abundance table of the National Institute of Standards and Technology. The database can be found here.

To directly quote the database: “In the opinion of the Subcommittee for Isotopic Abundance Measurements (SIAM), these values represent the isotopic composition of the chemicals and/or materials most commonly encountered in the laboratory. They may not, therefore, correspond to the most abundant natural material. The uncertainties listed in parenthesis cover the range of probable variations of the materials as well as experimental errors. These values are consistent with the values published in Isotopic Compositions of the Elements 2009.”

More details can be found here.

Notations

δ-values

The δ-value of a given isotope ratio, generally used in cosmo- and geochemistry, is defined as:

\[\delta \left( \frac{^{i}X}{^{j}X} \right) = \left(\frac{\left(\frac{^{i}X}{^{j}X}\right)_{\mathrm{measured}}} {\left(\frac{^{i}X}{^{j}X}\right)_{\mathrm{solar}}} - 1\right) \times f\]

Here, the measured isotope ratio of element X and isotopes \(i\) and \(j\) represents the ratio as measured in a stardust grain or as modeled in a stellar model. The solar isotope ratio for the same isotope ratio is the one chosen from the database.

Subtracting unity form the ratio of ratios determines the deviation of the measurement from the solar abundance.

Note

The part of the equation in parenthesis should correctly be referred to as the δ-value, i.e., the δ-value is defined when setting \(f=1\).

This is important to remember. However, many measurements, especially of stardust, are expressed in parts per thousand or per mil. This means that the δ-value must be multiplied by a factor \(f=1000\).

On the other hand, bulk measurements of meteorites generally detect smaller deviations from solar. Thus, such measurements are often expressed in so-called ε- or µ-values. These generally only differ from the δ-value by using a different factor \(f\). The table below gives an overview of different notations and the respective \(f\)-values:

Notation	\(f\)-value
absolute deviation	1
%, percent	100
‰, per mil	1,000
ε, parts per ten thousand	10,000
µ, parts per one-hundred thousand	100,000
ppm, parts per million	1,000,000
ppb, parts per billion	1,000,000,000
ppt, parts per trillion	1,000,000,000,000

Internal normalization

In cosmo- and geochemistry, measured isotope ratios are often internally normalized. This is especially true for measurements that suffer from mass-dependent fractionation.

Above figure shows an example of the two normalization scenarios. On the left side is the regular δ-value notation as described above. As the normalization isotope, ⁵⁸Ni is chosen. The red, dashed line shows the internal, mass-dependent fractionation that was introduced into the system artificially. Clearly, ⁶⁰Ni shows some positive deviation from this line. After internal normalization, a clear excess in ⁶⁰Ni can be seen in the figure.

Internal normalization (right side) normalizes the same dataset to a second isotope. Here, ⁶²Ni is chosen. Assuming that any anomaly in ⁶²Ni is due to mass-dependent fractionation, all isotope ratios can be corrected for this mass-dependent fractionation. To do so, a mass-dependent fractionation law must be applied. These, internally normalized values, if expressed in permil, are often described with a capital delta (Δ).

Note

The same pre-factors as discussed above are applied for internal normalization. Often, measurements obtained using inductively-coupled plasma mass spectrometry (ICP-MS) are internally normalized and results are expressed in ε- (parts per 10,000) or µ-values (parts per 100,000). Note that the same notation is frequently used for both normalizations.

A detailed description on mass fractionation laws can be found in Dauphas and Schauble (2016).

In the iniabu package, corrections using an exponential (default) and linear law can be applied.

The exponential law, which is applied by default, assumes an exponential relation for the mass dependent mass fractionation. Let us assume the example from the above figure. The major normalization isotope ^jNi here is ⁵⁸Ni, the minor normalization isotope ⁱNi is ⁶²Ni. For a given sample, an exponential factor β can be calculated as:

\[\beta = \frac{\log(^{i}\mathrm{Ni}/^{j}\mathrm{Ni})_{\mathrm{sample}}/ \log(^{i}\mathrm{Ni}/^{j}\mathrm{Ni})_{\mathrm{solar}}} {\log(m_{i} / m_{j})}\]

Using this exponential factor, the mass-dependent fractionation corrected value of an isotope ratio of interest, e.g., ^xNi/^jNi can be calculated as:

\[\left(\frac{^{x}\mathrm{Ni}}{^{j}\mathrm{Ni}}\right)_{\mathrm{sample}}^{*} = \frac{(^{x}\mathrm{Ni}/^{j}\mathrm{Ni})_{\mathrm{sample}}} {(m_{x} / m_{j})^{\beta}}\]

Using this corrected ratio, the Δ-value can be calculated as:

\[\Delta^{x}\mathrm{Ni}_{i/j} = \left(\frac{(^{x}\mathrm{Ni} / ^{j}\mathrm{Ni})_{\mathrm{sample}}^{*}} {(^{x}\mathrm{Ni} / ^{j}\mathrm{Ni})_{\mathrm{solar}}} - 1 \right) \times k\]

Here \(k\) is the delta factor and defines the unit as described above for δ-values.

The linear law to correct for mass-dependent fractionation can be calculated as following:

\[\Delta^{x}\mathrm{Ni}_{i/j} = \delta^{x}\mathrm{Ni}_{j} - \frac{m_{j} - m_{x}}{m_{j} - m_{i}} \times \delta^{i}\mathrm{Ni}_{j}\]

Here, \(^{x}\mathrm{Ni}_{j}\) is short for the ratio \(^{x}\mathrm{Ni}/^{j}\mathrm{Ni}\).

The delta factor \(k\) is part of the δ-value calculation. With the linear law, values smaller than -(delta factor) are theoretically possible, however, such values are unphysical. The iso_int_norm routine automatically detects such values and sets them to the minimal possible value of -(delta factor).