Physical model for multi-point normalization of dual-inlet isotope ratio mass spectrometry data

Abstract

A simple model is presented for multi-point normalization of dual-inlet isotope ratio mass spectrometry (DI-IRMS) data. The model incorporates the scale contraction coefficient and the normalized working reference gas isotope delta value as its two physical parameters. The model allows the full use of isotope measurement data and outputs the normalized sample isotope delta value along with the mentioned parameters. The model reduces to the expected linear behavior on application to a natural range CO₂ isotopic composition sample, under typically observed scale contraction levels. Next, DI-IRMS measurements of the NIST CO₂ gas isotopic reference materials (RMs) 8562, 8563, and 8564 are used to construct a three-point linear calibration, spanning 40‰ for the δ⁴⁵CO₂ and 20‰ for the δ⁴⁶CO₂ raw data. Accuracy of the regression at the 0.009‰ level for δ¹³C and 0.01‰ for δ¹⁸O is observed for the three NIST RMs. The model derived scale contraction term is found to be a more accurate measure of cross-contamination in contrast to its end of day measurements by the enriched sample method. The constructed multi-point normalization model is next used to assign ${\delta }^{13}{\text{C}}_{\text{VPDB-}{\text{CO}}_2}$ and ${\delta }^{18}{\text{O}}_{\text{VPDB-}{\text{CO}}_2}$ isotope delta values on the Vienna PeeDee Belmnite-CO₂ (VPDB-CO₂) scale, for pure CO₂ gas samples in the natural isotopic range. A Monte Carlo analysis of the uncertainty, including estimates for the normalization step, is provided to assist future multi-point normalization with more than three reference points.

Introduction

Scale normalization [1] is routinely used for translating the raw isotope delta measurement scale to the reference scale. This calibration exercise is necessitated to correct the measurement bias present in differential (delta) methods on count of instrumental factors [2]. The measured “scale” is stretched or contracted to match the reference scale and made amenable to comparisons across measurement campaigns. For the case of CO₂ (carbon dioxide) dual-inlet isotope ratio mass spectrometry (DI-IRMS) has been the gold standard in relating ${\delta }^{45}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG}}$ and ${\delta }^{46}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG}}$ (isotope delta of 45/44 and 46/44 isotope ratios) of sample versus working reference gas (WRG) to sample versus VPDB-CO₂ scale, ${\delta }^{45}{\left({\text{CO}}_2\right)}_{\text{VPDB}}$ and ${\delta }^{46}{\left({\text{CO}}_2\right)}_{\text{VPDB}}$ [3]. Normalization schemes based on single- or multi-point are utilized for data treatment [1]. Each method comes with its advantages and limitations. While single-point requires a single reference material, it is strongly dependent on scale correction. The required correction, in the case of CO₂ DI-IRMS measurement, is dominated by scale contraction caused by the cross-contamination of the sample and reference gas during the differential measurement [4]. Multi-point scale normalization, in practice, does not require independent measurement of scale correction but depends on the availability of multiple RMs across a wide δ¹³C, δ¹⁸O range. In this regard, two carbonate-based RMs, NBS-19 (CaCO₃ with a δ¹³C_VPDB = + 1.95‰) and NIST RM 8545 (Li₂CO₃ referred to as LSVEC [5] with a δ¹³C_VPDB = − 46.6‰), have been used historically to achieve two-point normalization for ${\delta }^{13}{\text{C}}_{\text{VPDB-}{\text{CO}}_2}$ [6]. The NIST NBS-19, RM 8545 two-point normalization, was adopted in 2006 [6] to improve the consistency in δ¹³C measurements, and is referred to as the VPDB2006 scale realization. Limitations on availability (of NBS-19) and stability (of LSVEC) have resulted in efforts to find appropriate replacements. These include development of a high-purity CaCO₃-based RM, USGS44, with a high negative δ¹³C_VPDB2006 value of − 42.21‰ [7] for δ¹³C standardization work. Efforts at the International Atomic Energy Agency (IAEA) have resulted in the development of a metrologically traceable [8] primary reference material [9] IAEA-603 (δ¹³C = + 2.46‰, δ¹⁸Oδ¹⁸O = − 2.37‰) along with three [10] anchors in the form of IAEA-610, −611, and −612, respectively (covering δ¹³C from − 9.109 to − 36.722‰ and δ¹⁸O from − 4.224 to − 18.834‰). (This is referred to as the VPDB2020 [11] scale realization.) In an example of utilizing multiple stable RMs, the IAEA RMs have been recently used to value assign 25 isotope reference materials. Notably, a discontinuity of 0.18‰ at the negative end was seen between the VPDB2020 [11] and VPDB2006 (NBS-19, LSVEC) scale realizations.

Compared to carbonates [12], pure CO₂ gas [13] isotope reference materials do not require additional preparations, making them easier to use and adopt for VPDB scale realization. The availability of multiple pure CO₂ gas isotope reference materials is currently in the form of NIST RMs 8562, 8563, and 8564. These RMs originate from three distinct natural sources—limestone, petroleum, and biomass, and cover δ¹³C, δ¹⁸O in the (− 3.72 to − 41.59) ‰ and (− 33.52 to − 10.09) ‰ range, respectively, on the VPDB-CO₂ scale [13–15]. The NIST RMs, albeit in short supply, are particularly useful for VPDB scale realization of a range of pure CO₂ gases, for subsequent use in the development of CO₂ in air (CO₂-free) isotope reference mixtures [16]. Such mixtures are critical for monitoring [17] global trends in the CO₂ isotope delta value using field-based optical isotope ratio analyzers. In this regard, ongoing comparison studies (CCQM-P204) of CO₂ isotope ratio standards, coordinated jointly by BIPM (Bureau International des Poids et Mesures) and IAEA, are an important step [18].

Single-point normalization requires scale contraction correction. The correction can be minimized by a selection of reference material that lies close to the sample isotope delta value, and instrument parameters optimized to reduce the cross-contamination coefficient. For multi-point normalization, a measurement of the cross-contamination is not required but is inherently present in the linear statistical model [1], commonly employed for three or more points. However, a formulation of the multi-point normalization from first principles incorporating a non-linear cross-contamination coefficient term can lead to a fuller representation of the physico-chemical processes present in the DI-IRMS measurement. Such a treatment is surprisingly missing and easy to construct for routine use as a quality control and best practice. In this work, such a multi-point normalization model is built, for the case of CO₂ DI-IRMS, from the definition of isotope ratio and expressed in terms of a non-linear cross-contamination coefficient and working reference gas isotope delta value on the reference scale. The model is applied to NIST RMs 8562, 8563, and 8564 for a three-point normalization. Cross-contamination coefficients are obtained and compared to traditional approaches of determining them. The model is then applied to make value assignments of eight isotopically distinct samples on the VPDB-CO₂ scale, to assist future multi-point normalization implementation with more than three standards. Additionally, uncertainty evaluation, including its estimation for the normalization step, is provided using Monte Carlo simulation.

Experiment

The experimental details for DI-IRMS measurements are identical to Srivastava and Verkouteren [16] and only salient points are emphasized. NIST CO₂ isotope reference materials, RMs 8562, 8563, and 8564, were cryogenically transferred from respective glass ampoules to glass bulbs for experimental usage. The nominal isotope delta value for the CO₂ RMs ranged between (− 4 to − 42) ‰ for ${\delta }^{13}{\text{C}}_{\text{VPDB-}{\text{CO}}_2}$ and (− 10 to − 34) ‰ for ${\delta }^{18}{\text{O}}_{\text{VPDB-}{\text{CO}}_2}$, respectively (for reference values, see Table SAIV). (Note [16] ${\delta }^{13}{\text{C}}_{\text{VPDB-}{\text{CO}}_2}={\delta }^{13}{\text{C}}_{\text{VPDB}}$ and ${\delta }^{45}{\left({\text{CO}}_2\right)}_{\text{VPDB-}{\text{CO}}_2}={\delta }^{45}{\left({\text{CO}}_2\right)}_{\text{VPDB}}.)$ One ampoule was used per reference material. Two sets of four pure CO₂ gas samples each (all with isotope delta values in the natural isotope range), designated as N_i and B_i (i = 1 to 4), respectively, were prepared at NIST (procured from commercial sources) and BIPM. The nominal isotope delta value for the CO₂ samples ranged between (− 1 to − 50) ‰ for ${\delta }^{13}{\text{C}}_{\text{VPDB-}{\text{CO}}_2}$ and (− 9 to − 41) ‰ for ${\delta }^{18}{\text{O}}_{\text{VPDB-}{\text{CO}}_2}$, respectively. The samples were contained in separate single-ended 50 cm³ volume stainless steel sample cylinders. The CO₂ was cryogenically transferred from the sample cylinder to the sample glass bulb, attached to the dual-inlet system manifold. Adequate precaution was taken to maintain purity between sample handling steps, using helium pressure-vacuum purge cycles and tight vacuum levels. The experimental measurement configuration and daily measurement sequence are summarized in Table 1 and Table 2, respectively.

Table 1.

Experimental measurement configuration

Parameter	Value
Instrument	MAT 253 (Thermo)
MS high voltage	9500 V
Filament emission current	1.5 mA
Configuration	Dual-inlet, CO2
Integration time	16 s
Idle time	15 s
Cycles per acquisition	8
Typical acquisition time	13 min
Bellow/bellow master	Sample
VISC* valve turns open	1.5 out of 6
Cup 1 (m/z = 44) resistor	3 ×10.8 Ω
Cup 2 (m/z = 45) resistor	3 × 10.10 Ω
Cup 3 (m/z = 46) resistor	1 × 10.11 Ω
Bellow pressure	30 mbar
Typical signal level, cup 2	8500 mV
Typical background	< 5 mV

^*VISC, variable ion source conductance

Table 2.

Daily measurement sequence of dual-inlet sample-reference bellow

Sequence run no	Sample bellow	Reference bellow	Acquisitions*
1	WRG§	WRG	2
2	Sample1	WRG	3
3	RM1 = RM8563	WRG	3
4	Sample2	WRG	3
5	RM2 = RM8564	WRG	3
6	Sample3	WRG	3
7	RM3 = RM8562	WRG	3
8	Sample4	WRG	3
9	ES†	WRG	2
10	WRG	WRG	2

^*Each acquisition has 8 cycle runs

^§WRG, working reference gas

^†ES, enriched sample

For each sample-RM sequence run, three continuous acquisitions were measured (each comprising 8 sample-reference cycles). To ensure maximum repeatability, peak center, background, and “pressure adjust” steps were included in each acquisition. The internal precision (standard deviation) over repeated cycles of a single acquisition is 0.008‰ for δ⁴⁵CO₂ and 0.012‰ for δ⁴⁶CO₂ and found to be consistent with shot-noise predictions. The typical run time for a complete sample-RM sequence run was close to 40 min.

The measured isotope delta values (${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{sam}/\text{WRG}}$ ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{RM}/\text{WRG}}$, sam = sample, RM = reference material versus WRG = working reference gas, p = 45, 46) of the individual cycles were pooled (across the constituent acquisitions) to obtain the mean and standard deviation for a given sample-RM sequence run. At the end of each sample-RM sequence run, an “end of day” enriched sample, ES (nominal δ¹³C = + 948‰, δ¹⁸O = + 1498‰) versus working reference gas run (sequence run no. 9), was conducted to obtain the cross-contamination coefficient, η_ES [16]. The VISC (variable ion source conductance) valve opening, MS high voltage, and filament emission current values (see Table 1) are optimized to maximize ion sensitivity (near 1000 molecules per ion for listed conditions). The measurement sequence for each sample set (N_i, B_i) was repeated over 3 days.

Results and discussion

Multi-point normalization model

To convert from the raw isotope delta scale (sample versus WRG) to the VPDB-CO₂ isotope delta scale (sample versus VPDB-CO₂), a multi-point normalization model is constructed from first principles. The stable isotopologue-amount ratios for CO2 can be expressed as

$$\frac{R_s^p}{R_{VPD\mathrm{B-}C{O}_2}^p}=\frac{R_s^p}{R_{WRG}^p}\frac{R_{WRG}^p}{R_{VPD\mathrm{B-}C{O}_2}^p}$$ (1)

where R, p, s, and WRG refer to stable isotopologue-amount ratio relative to mass number 44, mass numbers 45 and 46, sample, and working reference gas, respectively. In isotope delta notation, Eq. (1) becomes,

$$\begin{gathered} {\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{VPDB-}{\text{CO}}_2}=[1+{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}]{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG}} \\ +{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2} \end{gathered}$$ (2)

to give a one parameter linear relationship between ${\delta }^p{\left({\text{CO}}_2\right)}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$ and ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG}}$. (Eq. (2) is also used for single-point referencing when working gas isotopic composition, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}$, is known [1].) To account for scale contraction in the sample to working reference gas isotope delta value, cross-contamination correction for DI-IRMS proposed by Meijer et al. [4] is used according to

$${\delta }_c=\frac{{\delta }_m}{1-2\eta -\eta {\delta }_m}.$$ (3)

Here δ_c and δ_m refer to the corrected and measured isotope delta value, while η is the cross-contamination coefficient. The ηδ_m term imparts non-linearity to the cross-contamination correction and is relevant only for significantly enriched samples. Inserting the correction, Eq. (2) becomes

$$\begin{gathered} {\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{VPDB-}{\text{CO}}_2}=[1+{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}]{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG-c}} \\ +{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2} \end{gathered}$$ (4)

$$\begin{gathered} {\delta }^{\text{p}}{({\text{CO}}_2)}_{\text{s}/\text{VPDB-}{\text{CO}}_2}=(1+{\delta }^{\text{p}}{({\text{CO}}_2)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2})\left(\frac{{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG-m}}}{1-2\eta -\eta \delta \text{P}{{\left({\text{CO}}_2\right)}_{\text{s}}}_{/\text{WRG-m}}}\right) \\ +{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2} \end{gathered}$$ (5)

where ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG-c}}$, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG-m}}$ are the corrected and measured values of the sample versus working reference gas isotope delta, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG}}$. It is to be noted that ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$ is non-linear with respect to the measured isotope delta value, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG-m}}$.

Equation (5) represents a complete model for multi-point scale normalization and includes two physical parameters, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}$ and η, which can be obtained by fitting the model to the “N” reference material ref_j data points, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{{\text{ref}}_{\text{j}}/\text{WRG-m}}=x_{\text{j}}=\text{measured}$; ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{{\text{ref}}_{\text{j}}/\text{VPDB-}{\text{CO}}_2}=y_{\text{j}}=\text{known}$, j =1 … N ). This treatment makes use of the available reference material measurement data and does not rely on separate measurements of the cross-contamination coefficient. A simulation of the model is presented in Fig. 1 for ${\delta }^p{\left({\text{CO}}_2\right)}_{{\text{ref}}_{\text{j}}/\text{WRG-m}}={\text{x}}_j$, ${\delta }^{\text{P}}{\left({\text{CO}}_2\right)}_{{\text{ref}}_{\text{j}}/\text{VPDB-}{\text{CO}}_2}=y_j$, η= 0.3, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}=-20\%$, and N = 101 points between ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{{\text{ref}}_{\text{j}}/\text{WRG-m}}-500$ to 500‰. Scale contraction is clearly visible on the raw measurement isotope scale when comparing (${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{{\text{ref}}_{\text{j}}/\text{WRG-m}}=x_j$; ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{{\text{ref}}_{\text{j}}/\text{VPDB-}{\text{CO}}_2}=y_j$) and (${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{{\text{ref}}_{\text{j}}/\text{WRG-c}}=x_j$; ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{{\text{ref}}_{\text{j}}/\text{VPDB-}{\text{CO}}_2}=y_j$) plots.

Fig. 1.

Simulation of multi-point normalization of reference versus woking gas, ${\delta }^{45}{\left({\text{CO}}_2\right)}_{\text{ref}/\text{WRG}}$, scale to the reference versus ${\delta }^{45}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}$, scale. Scale contraction and its correction are shown for an assumed cross-contamination coefficient, η = 0.3. The value of ${\delta }^{45}{\left({\text{CO}}_2\right)}_{{\text{WRG/VPDB-CO}}_{\text{2}}}$ is assumed to be − 20‰. The terms ${\delta }^{45}{\left({\text{CO}}_2\right)}_{\text{ref/WRG-m}}$, ${\delta }^{45}{{\left({\text{CO}}_2\right)}_{\text{ref}}}_{/\text{WRG-c}}$ represent the measured (bottom x-axis) and scale contraction corrected (top x-axis) isotope delta value of reference versus working reference gas. See text for details

When the term $\eta {\delta }_m^p$ in Eq. (3) becomes negligible, as in the case of natural samples, contraction correction becomes linear [4, 16] in δ (see SC, supporting information) and Eq. (5) reduces to

$$\begin{gathered} {\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{VPDB-}{\text{CO}}_2}=\left(1+{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}\right)(1+2\eta ){\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG-m}} \\ +{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2} \end{gathered}$$ (6)

This is similar in form to the linear relationship used for multi-point normalization,

$${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{VPDB-}{\text{CO}}_2}=m{\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{s}/\text{WRG}}+b$$ (7)

where m is the slope and b the intercept $\text{intercept}={\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}$. The slope is referred to as the “expansion factor” and the intercept as the “additive correction factor” [1]. For flow-based differential measurements, as in continuous-flow and elemental-analyzer isotope ratio mass spectrometry (CF-IRMS and EA-IRMS), m and b in Eq. (7) are treated as independent. The linear multi-point normalization serves as a statistical model. However, as Eq. (8) shows, in the case of DI-IRMS, the terms m and b are related as,

$$(1+b)(1+2\eta )=m$$ (8)

In this study, the constraint introduced in Eq. (8) is included to maximize the information from the DI-IRMS measurement data, lending a physical meaning to the calibration slope. The linear slope is influenced by both the isotope delta value (of the working gas) and the measurement of the cross-contamination coefficient. While the former, for typically used depleted (negative valued ${\delta }^{13}{\text{C}}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}$, ${\delta }^{18}{\text{O}}_{\text{WRG}/\text{VPDB-}{\text{CO}}_2}$) working reference gases, leads to shrinking, the second factor leads to stretching of the measurement scale when anchored to the VPDB scale.

Next, the residual error introduced by the linear approximation of the cross-contamination correction is simulated in Fig. 2, for CO₂ isotope delta values in the natural range as a function of three cross-contamination coefficient values (η = 2 × 10⁻², 2 × 10⁻³, 2 × 10⁻⁴, chosen to capture their observable range reported in literature [3, 7, 10, 19]). The simulated data is derived using the full non-linear model given in Eq. (5). Residual values for the predicted isotope delta values approach 0.01‰, 0.001‰, and 0.0001‰ for η = 2 × 10⁻², 2 × 10⁻³, and 2 × 10⁻⁴, respectively, at ${\delta }^P{\left({\text{CO}}_2\right)}_{\text{WRG-m}}=\pm 50\%$ for linear approximation (Eq. (6)) of the simulated data. At η levels < 2 × 10⁻³ linear approximation of the model, used in the rest of the paper, is adequate to achieve 0.001‰ accuracy for δ⁴⁵CO₂ and δ⁴⁶CO₂ in the natural isotopic range.

Fig. 2.

Residual dependence of non-linear and linear fit of the normalization model in the natural range of CO₂ isotope delta values as a function of cross-contamination coefficient. See text for details

Three-point scale normalization of NIST RMs: 8562, 8563, 8564

Six independent determinations of the three-point normalization of NIST RMs are made in terms of the described model. As illustrated in Fig. 3, the δ^45,46CO₂ values lie in the natural isotopic range and the model falls within the linear regime. The absolute value of the residuals for δ⁴⁵CO₂ and δ⁴⁶CO₂ are seen to be within 0.004‰ and 0.009‰, respectively, representing excellent fit. The fit residuals provide a quantitative measure of the normalization accuracy relative to the best estimate of the NIST RM true values (see Table SAIV, supporting information). The predicted δ¹³C and δ¹⁸O NIST RM values are within 0.009‰ and 0.01‰ of their true values, well within the reported standard uncertainties of the RMs. Consistent with calibration using Eq. (6), the cross-contamination coefficient, ${\eta }^p={\eta }_{\text{model}}^p$, and the isotope delta value of the working reference gas relative to the VPDB-CO₂, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{WRG/VPDB-}{\text{CO}}_2}$, are obtained for p = 45 and 46. (Complete regression parameters, its residuals are provided in the supporting information SA.) The cross-contamination coefficient values obtained in the fit are tabulated in Table 3 for further discussion.

Fig. 3.

Multi-point normalization (shown as linear fit) of the measured, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{\text{ref/WRG-m}}$ (x-axis) to the true (known best estimate) VPDB-CO₂ isotope scale, ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{{\text{ref/VPDB-CO}}_{\text{2}}}$ ref/VPDB-CO₂ (y-axis) for NIST RMs 8562, 8563, and 8564 (p = 45, 46). Residuals, representing deviation of the true (known best estimates) from their fit predicted values, have a maximum absolute value of 0.004‰ and 0.009‰ for ${\delta }^{\text{p}}{\left({\text{CO}}_2\right)}_{{\text{ref/VPDB-CO}}_{\text{2}}}$, p = 45 and 46

Table 3.

Cross-contamination values for δ⁴⁵CO₂, δ⁴⁶CO₂ using three approaches, with uncertainties in brackets for the model method

η45/10−3			η46/10−3
Model	PW*	ES§	Model	PW	ES
2.12(0.02)	2.13	0.64	0.34(0.15)	0.36	2.21
2.19(0.06)	2.22	0.67	0.65(0.27)	0.60	2.21
2.10(0.00)	2.10	0.69	0.85(0.25)	0.89	2.32
2.12(0.10)	2.16	0.69	1.97(0.24)	2.01	2.28
2.04(0.04)	2.05	0.68	1.95(0.35)	2.01	2.25
2.13(0.04)	2.15	0.69	2.51(0.08)	2.52	2.26

^*PW, pairwise is a calculated value for RM pair with the largest isotope delta difference.

^§ES, enriched sample values are based on measurement of enriched sample versus working reference gas

Two other determinations of the cross-contamination coefficient are provided. One is based on measurements of the enriched sample versus working reference gas method $\left({\eta }^p={\eta }_{ES}^p\right)$ at the end of the day. The second value is the calculated pairwise value for RM pair with the largest isotope delta value difference. This value is computed to mimic the pairwise measurement method of cross-contamination determination between two isotopically different CO₂ gases with large isotope delta value difference for p = 45, 46. The model derived cross-contaminations are observed to be a close match to the pairwise calculated values, within the fitting errors of the ${\eta }_{\text{model}}^p$. In the limit, of the fit residuals approaching zero, the two would become identical. In contrast, the model and pairwise determined cross-contamination coefficients are 3.1(0.1) and 0.6(0.4) times relative to the enriched sample method, expressed as mean(uncertainty) for η₄₅ and η₄₆, respectively. Additionally, the model scale-contraction parameters show larger variability for δ⁴⁶CO₂ compared to δ⁴⁵CO₂ with mean (standard deviation) values of ${\eta }_{\text{model}}^{46}=1.38(0.80)\times {10}^{-3}$ and ${\eta }_{\text{model}}^{45}=2.12(0.05)\times {10}^{-3}$ across the six measurement sequences. This suggests scale contraction during sample and RM measurement is not entirely represented by the end of day enriched sample cross-contamination correction and would require consideration of additional sampling, memory effects during the measurement sequence. Rather, the model derived value is a more realistic capture of the contraction during the sample-RM measurement sequence.

Sample VPDB-CO₂ scale realization

The model regression parameters are used to obtain VPDB-CO₂ scale isotope delta values in the analyzed samples (N_i, B_i, i = 1 to 4). Sample ${\delta }^{13}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$, ${\delta }^{18}{\text{O}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$ values are obtained using appropriate ¹⁷O interference correction parameters (${}^{13}{R}_{VPDB}=0.011180(28)$, ${}^{17}{R}_{VPD\mathrm{B-}C{O}_2}=0.0003931(9)$, λ=0.528, K=0.01022461, ${}^{18}{R}_{VPDB-C{O}_2}=0.00208835$, uncertainties reported at 95% confidence interval) and ${\delta }^{18}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$, ${\delta }^{46}{\left({\text{CO}}_2\right)}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$ to ${\delta }^{13}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$, ${\delta }^{18}{\text{O}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$ linearization approximation provided by Brand, Assonov, and Coplen [20]. The uncertainty due to 17-O correction parameter selection was negligible, ≤ ± 0.001‰ across the sample range. As shown in Fig. 4, the derived values span a wide range, (− 50 to − 1) ‰, (− 41 to − 9) ‰ for ${\delta }^{13}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$ and ${\delta }^{18}{\text{O}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$, respectively. The ${\delta }^{13}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$, ${\delta }^{18}{\text{O}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$ values obtained involve extrapolation of (N₁, N₄, B₁, B₂), (N₄, B₁, B₂) for δ⁴⁵CO₂, δ⁴⁶CO₂ normalization. The uncertainty of each sample comprises type A and type B and is tabulated along with the mean isotope delta values in Table 4.

Fig. 4.

Relative position of the sample (N_i, B_i) and NIST RMs ${\delta }^{13}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$, ${\delta }^{18}{\text{O}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$. See text for details

Table 4.

Mean and uncertainty values of the sample ${\delta }^{13}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$ and ${\delta }^{18}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$

Sample	${\delta }^{13}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}\%$ (Mean of 3 days)	${\delta }^{18}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}\%$	$u_A^p\%$		$u_B^p\%$		$u_c^p\%$
			p = 13	p = 18	p = 13	p = 18	p = 13	p = 18
N1	−2.85	−16.51	0.012	0.014	0.03	0.13	0.03	0.13
N2	−10.98	−12.27	0.012	0.019	0.02	0.16	0.03	0.16
N3	−39.89	−33.73	0.013	0.024	0.04	0.23	0.04	0.23
N4	−50.42	−41.33	0.015	0.027	0.06	0.32	0.06	0.32
B1	−1.48	−9.43	0.011	0.021	0.02	0.15	0.03	0.15
B2	−43.11	−36.00	0.021	0.030	0.04	0.24	0.05	0.24
B3	−8.84	−14.34	0.011	0.020	0.03	0.19	0.04	0.19
B4	−34.30	−30.18	0.017	0.033	0.03	0.18	0.04	0.18

$u_{A(B)}^p=\text{type}\text{A}(\text{B})\text{standard}\text{uncertainty}$, $u_c^p=\sqrt{{\left(u_A^p\right)}^2+{\left(u_B^p\right)}^2}=\text{combined}\text{standard}\text{uncertainty}$, p = 13, 18

The type A standard uncertainty contribution forms the minor component of the overall budget and lies in the (0.011 to 0.021) ‰ and (0.014 to 0.030) ‰ range for ${\delta }^{13}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$ and ${\delta }^{18}{\text{C}}_{\text{s}/\text{VPDB-}{\text{CO}}_2}$, respectively. The reported uncertainty estimates are obtained by the Monte Carlo simulation method and include its evaluation for the normalization step (see supporting information, SB).

Conclusion

A simple two parameter physical measurement model is presented for multi-point normalization of DI-IRMS isotope data. The contribution of scale contraction, in terms of cross-contamination coefficient, is explicitly included to derive its value from the model regression under experimental conditions. This approach represents the full use of the measurement data and gives the working gas isotope delta on the VPDB-CO₂ scale as its second parameter, for the case of CO₂ DI-IRMS. The commonly used linear model for continuous-flow isotope ratio measurement calibration is refined, in terms of cross-contamination correction, to obtain a physical interpretation of the model parameters for the case of DI-IRMS. In contrast to the end of day measurement of cross-contamination coefficient by enriched sample method, the model derived values mimic experimental scale contraction conditions better. Consequently, a pairwise method or a multi-point normalization model-based method, as established in this work, is potentially a better alternative to the end of day enriched sample method for correcting the cross-contamination introduced bias in DI-IRMS measurements.

The model is applied to the NIST RMs 8562, 8563, and 8564 to construct scale normalization with accuracy at the 0.009‰ level for δ¹³C and 0.01‰ for δ¹⁸C. The three-point normalization is used to make VPDB-CO₂ isotope delta value assignments of eight isotopically distinct CO₂ samples in the natural range. A full uncertainty analysis using the Monte Carlo method is also provided. Future studies will allow usage of these CO₂ samples as standards for multipoint normalization with more than three points.