The Uncertainty Paradox: Molar Mass of Enriched Versus Natural Silicon Used in the XRCD Method

Abstract

The X-ray crystal density method uses silicon spheres highly enriched in ²⁸Si as a primary method for the dissemination of the SI base unit kilogram yielding smallest possible uncertainties associated with the mass m within a few parts in 10⁻⁸. This study compares different available and newly developed analytical methods and their results for the determination of the molar mass M of silicon highly enriched in ²⁸Si (M_e) and of silicon (M_x) with an almost natural isotopic distribution. While for M_e relative uncertainties u_rel(M_e) in the lower 10⁻⁹ range are obtained routinely, it was not possible to fall below a value of u_rel(M_x) < 4 × 10⁻⁶ in the case of natural silicon, which is approximately three orders of magnitude larger. The application of the state-of-the-art isotope ratio mass spectrometry accompanied with sophisticated thoroughly investigated methods suggests an intrinsic cause for the large uncertainty associated with the molar mass of natural silicon compared to the enriched material.

1 Introduction

After the revision of the International System of Units in 2019, the seven base units are defined via fixed fundamental constants [1,2,3,4]. The most far-reaching change was the new definition of the kilogram, the base unit of the mass, previously defined via a single artefact, a Pt-Ir cylinder, now defined via the Planck constant h. Since the revision, the dissemination of the SI units is still a challenge and an ongoing duty of the metrology community. The two most suitable methods—in metrological terms “primary methods”—for the realization and dissemination of the kilogram are the “Kibble balance” approach and the XRCD method [5, 6]. The first method directly determines the Planck constant h via a comparison of mechanical and electrical power; the XRCD method uses a combination of experiments for “counting” the number of silicon atoms in an Si sphere yielding the Avogadro constant N_A as a direct result. Since both N_A and h can be converted into each other [7, 8].

$$N_{\text{A}} = \frac{{c\alpha^{2} A_{\text{r}} (e)}}{{2R_{\infty } }}\frac{{M_{\text{u}} }}{h}$$

(1)

(with the Rydberg constant R_∞, the fine structure constant α, the speed of light in vacuum c, the relative atomic mass of an electron A_r(e), and the molar mass constant M_u), the two primary methods have a complementary character. Briefly, in the XRCD method a silicon sphere is characterized according to

$$N_{\text{A}} = \frac{8V}{{a^{3} }}\frac{M}{m} = \frac{8}{{a^{3} }}\frac{M}{\rho } = \frac{N}{n}$$

(2)

with the macroscopic volume V of a single crystalline sphere with eight atoms in the unit cell, a nominal mass m of approximately 1 kg, the lattice parameter a, the molar mass M, and the density ρ. After the revision of the SI, N_A is fixed, and the measurement of the parameters given in Eq. (2) yields the mass m of the respective sphere with a very low uncertainty. Once characterized, a respective sphere can be used as a primary mass standard with an associated uncertainty u_rel(m) < 2 × 10⁻⁸. Contributions to m of the surface layers and other impurities have to be determined from time to time [6]. Equation (2) indicates a proportional relation between N_A and M. Thus, during the efforts for the realization of N_A using the XRCD method (“Avogadro Project”), the relative uncertainty associated with M must be smaller than 2 × 10⁻⁸, because one of the accompanying preconditions of the revision was u_rel(N_A) ≤ 2 × 10⁻⁸ [8]. Initially, the XRCD method aimed at the determination of N_A only, finally leading to a redefinition of the kilogram [9, 10]. Prior to 2007, silicon spheres made of silicon with almost natural isotopic composition have been used. However, it was not possible to undercut a limit of u_rel(M_x) ≈ 3 × 10⁻⁷, which was at least one order of magnitude too large for the intended redefinition of the kilogram [11]. During the first years of the “Avogadro-Project”, the molar mass of the natural silicon crystals has been determined by pioneering gas-phase isotope ratio mass spectrometric measurements [12, 13]. At that time, the solid silicon samples had to be converted via several chemical steps into gaseous silicon tetrafluoride SiF₄ [12]. Additionally, the molar mass of natural silicon delivered the largest contribution to the uncertainty associated with N_A [13]. Therefore, it was of upmost importance to reduce the impact of the molar mass M to u(N_A) and to reduce u(M) by more than at least one order of magnitude. For this reason, both theoretical and experimental problems had to be solved. The availability of a silicon crystal highly enriched in ²⁸Si in 2007 was a milestone in the XRCD method: a new extremely pure and highly enriched silicon crystal with an amount-of-substance fraction x(²⁸Si) = 0.999 957 52(12) mol/mol [14]. The extremely low abundance of both ²⁹Si and ³⁰Si in these enriched crystals required new theoretical methods for the determination of the molar mass which led to the invention of the virtual element-isotope dilution mass spectrometry (VE-IDMS) method [15] which is based on the measurement of isotope ratios, $R_{j}^{\text{true}}$ in Eq. (7). The intensity ratios $R_{j}^{\text{meas}}$ actually measured by mass spectrometry deviate from the isotope ratios due to several unavoidable processes in the mass spectrometer starting from the ion source. The calibration or the so-called K factors convert the intensity ratios to yield the corrected isotope ratios (Eq. (7)). Usually, K factors can be determined by measuring an isotopic reference material. In case of the enriched silicon used for the XRCD method, no suitable isotopic reference material with a sufficiently low uncertainty associated with the isotope ratios exists. Therefore, an analytical closed-form approach has been developed for the determination of the K factors necessary for the correction of the measured intensity ratios [16]. The application of this primary method for the determination of the molar mass in combination with high-resolution multi-collector inductively coupled plasma mass spectrometry (MC-ICP-MS) yields also for the subsequent highly enriched silicon crystals relative uncertainties in the 10⁻⁹ range [17, 18]. The combination of these methods applied to the enriched silicon material reduced the relative uncertainty associated with the molar mass by two orders of magnitude within one decade. Additionally, the contribution of the molar mass to the uncertainty of N_A was reduced to < 10% [19].

In the meantime, a small pool of silicon spheres highly enriched in ²⁸Si and characterized by the XRCD method is available and guarantees the dissemination of the kilogram on the top level of traceability yielding u(m) < 2 × 10⁻⁸ [18, 20]. Usually, national metrology institutes (NMIs) can use the spheres for laboratory intercomparison programmes of the mass. However, for industrial and economic purposes, this high level of enrichment is not necessary. PTB initiated a concept for the use of silicon spheres of three different categories (and thus uncertainty ranges associated with m) for the dissemination of the mass [20]: (a) ²⁸Si-primary Si spheres (highly enriched in ²⁸Si as described), which are extremely expensive and need a time-consuming production, (b) ^natSi_qp-quasi-primary Si spheres, made of silicon with natural isotopic composition, and (c) ^natSi_sc-secondary Si spheres, the latter industrially manufactured. The uncertainties u(m) of the natural silicon spheres are in principal larger compared to the enriched spheres. One limiting factor of this increased uncertainty is the larger uncertainty associated with the molar mass of natural silicon M_x as outlined before. This was the motivation for the search and development of new analytical methods for the determination of the molar mass of silicon with the intention to further reduce the uncertainty using isotope ratio mass spectrometry. In this work, the current status of the different methods for the determination of M_x is described and compared with the aid of uncertainty calculations based on the “Guide to the Expression of Uncertainty in Measurement” (GUM) [21]. Figure 1 displays the isotopic composition in terms of x(ⁱSi) of the different available silicon crystals highly enriched in ²⁸Si in comparison with the distribution of natural silicon (according to IUPAC [22]).

Fig. 1

Amount-of-substance fractions x(²⁸Si), x(²⁹Si), and x(³⁰Si) of different silicon crystals used for the XRCD method [18] compared to the isotopic composition of natural silicon [22]. The data of Si28-31Pr11 are of preliminary character

Although the x(²⁸Si) of both enriched (> 0.999 9 mol/mol) and natural (≈ 0.92 mol/mol) silicon seem to be similar, the respective x(²⁹Si) and x(³⁰Si) of the enriched materials differ by several orders of magnitude from those of natural silicon.

2 Experiment and Materials

The determination of the molar mass M_x of crystalline silicon with natural isotopic composition is similar in sample preparation and isotope ratio mass spectrometry to the enriched material as described in detail elsewhere [14, 18, 23]. The different analytical methods are described in Sect. 3. Here, only a short summary of the main procedures is given.

2.1 Reagents and Materials

The different methods—if experimentally tested—are carried out using a sample (500 mg) from a silicon single crystal with the code V3182 (PTB, Germany) with natural-like isotopic composition. This crystal was also used as calibration material for neutron activation analysis (NAA) for the measurement of x(³⁰Si) in enriched silicon [24]. Calibration factors were always derived from the measurement of isotope ratios in samples of the WASO04 material with natural isotopic composition characterized in [14]. The characterization itself is extremely complex and time-consuming. It requires gravimetrically prepared solutions of highly enriched silicon materials of all three isotopes (so-called parent materials). From these, at least two binary blends need to be prepared. The intensity ratios ³⁰Si/²⁹Si and ²⁸Si/²⁹Si in all the blends and enriched parent materials have to be measured over several weeks to reach the necessary precision. These intensity ratios together with the accurate (buoyancy corrected) masses of the parent materials blended yielded the K factors needed to correct the above-mentioned ratios. This way, the isotope ratios in the WASO04 material were determined without any isotopic reference material, turning the WASO04 material itself into an isotopic reference material. The resulting uncertainties associated with the isotope ratios in WASO04 are low enough to determine the K factor needed for the molar mass measurement via the VE-IDMS method but too large for the direct measurement of the molar mass of any other natural silicon material. The WASO04 sample was cleaned and etched prior to weighing and dissolution in aqueous tetramethylammonium hydroxide (TMAH), yielding a final mass fraction of the silicon sample solution w(Si) = 4 µg/g in w(TMAH) = 0.0006 g/g.

2.2 Instrumentation

The molar mass measurements were performed using a high-resolution MC-ICP-MS (Neptune™, Thermo Fisher Scientific GmbH, Bremen, Germany) with typical operating conditions given in [14, 17]. The samples are aqueous solutions which are introduced into the ion source as aerosols. An argon plasma serves as the ion source running at approximately 10 000 K. The necessary power of around 1200 W is inductively coupled into the plasma through a radio frequency generator running at 27 MHz. The ions are accelerated into the mass spectrometer and separated according to their mass over charge ratio by an electrostatic and subsequently a static magnetic field. The ion currents of the different masses are then simultaneously measured using an array of up to nine detectors. Prior to each sample measurement, a blank solution (w(TMAH) = 0.0006 g/g) was measured to correct for carryover effects. In a respective sequence, the sample (x) was measured four times, then the blend(s) (bx) is/are measured four times, and finally the WASO04 sample for the K factor determination was measured four times.

3 Analytical Methods for Molar Mass Determination

3.1 Classical Approach (MM A)—Direct Measurement of all three isotopes

The isotopic distribution in natural silicon is roughly: x(²⁸Si) = 0.92 mol/mol, x(²⁹Si) = 0.05 mol/mol, and x(²⁹Si) = 0.03 mol/mol. Therefore, a direct measurement of all three isotopes should yield reasonable signals and sensitivity. The molar mass is defined via

$$M_{\text{x}} = x_{\text{x}} ({}^{28}{\text{Si}})M\left( {{}^{28}{\text{Si}}} \right) + x_{\text{x}} ({}^{29}{\text{Si}})M\left( {{}^{29}{\text{Si}}} \right) + x_{\text{x}} ({}^{30}{\text{Si}})M\left( {{}^{30}{\text{Si}}} \right)$$

(3)

The index x denotes the respective sample (here: natural silicon V3182); the M(ⁱSi) are the molar masses of the respective isotope taken from [25]. By measuring the intensity ratios and correcting them using the K factors measured within the same sequence, the following isotope ratios were determined

$$R_{\text{x}} = \frac{{x_{\text{x}} \left( {{}^{30}{\text{Si}}} \right)}}{{x_{\text{x}} \left( {{}^{29}{\text{Si}}} \right)}};R_{\text{x},28} = \frac{{x_{\text{x}} \left( {{}^{28}{\text{Si}}} \right)}}{{x_{\text{x}} \left( {{}^{29}{\text{Si}}} \right)}}$$

(4)

yielding

$$x({}^{i}{\text{Si}}) = \frac{{R_{i} }}{{\sum\limits_{j = 28}^{30} {R_{j} } }}$$

(5)

3.2 Virtual Element VE-IDMS (MM B)—Measurement of ²⁹Si and ³⁰Si, One Blend with ³⁰Si-Enriched Silicon

This method is adopted from the molar mass determination developed for silicon highly enriched in ²⁸Si, where only the isotopes ²⁹Si and ³⁰Si (the virtual element) are measured [15]:

$$M_{x} = \frac{{M\left( {{}^{28}{\text{Si}}} \right)}}{{1 + \frac{{m_{\text{y}} }}{{m_{\text{x}} }} \times \frac{{M\left( {{}^{28}{\text{Si}}} \right) \times \left( {1 + R_{\text{x}} } \right) - M\left( {{}^{29}{\text{Si}}} \right) - R_{\text{x}} M\left( {{}^{30}{\text{Si}}} \right)}}{{R_{\text{y},28} M\left( {{}^{28}{\text{Si}}} \right) + M\left( {{}^{29}{\text{Si}}} \right) + R_{\text{y}} M\left( {{}^{30}{\text{Si}}} \right)}} \times \frac{{R_{\text{y}} - R_{\text{bx}} }}{{R_{\text{bx}} - R_{\text{x}} }}}}$$

(6)

Here, m_y and m_x are the respective masses of the solid spike material y and sample material x (natural silicon) in the blend bx; R_j = x_j(³⁰Si)/x_j(²⁹Si) and R_j,28 = x_j(²⁸Si)/x_j(²⁹Si).

The measured intensity ratios R ^meas_j of the materials x (natural silicon), y (silicon enriched in ³⁰Si), and bx (a blend of x and y) were corrected by the K factors K₃₀ and K₂₈ yielding the correct or “true” isotope ratios R ^true_j (Fig. 2)

Fig. 2

Mixing scheme in the case of the original VE-IDMS approach (MM B). For details refer to [15]

$$R_{j}^{\text{true}} = K_{30} \times R_{j}^{\text{meas}} \quad {\text{with}}\quad R_{j}^{\text{meas}} = \frac{{I_{j} ({}^{30}{\text{Si}})}}{{I_{j} ({}^{29}{\text{Si}})}}\quad {\text{and}}\quad j \in \{ \text{x,y,bx}\}$$

(7)

$$R_{\text{y},28}^{\text{true}} = K_{28} \times R_{\text{y},28}^{\text{meas}} \quad {\text{with}}\quad R_{\text{y},28}^{\text{meas}} = \frac{{I_{\text{y}} ({}^{28}{\text{Si}})}}{{I_{\text{y}} ({}^{29}{\text{Si}})}}.$$

(8)

3.3 Double VE-IDMS (MM C)–Measurement of ²⁹Si and ³⁰Si, Two Blends with ³⁰Si-Enriched Silicon

This method is a further development of the MM B approach (3.2). When applying the VE-IDMS method using enriched silicon samples, the measurement of the isotope ratios in the sample x is difficult due to the extremely low abundance of ²⁹Si and especially ³⁰Si. Therefore, a second blend bx2 was prepared with isotope ratios x(³⁰Si)/x(²⁹Si) slightly different from those in bx1 and according to Eq. (9), the measurement of R_x is now obsolete.

$$M_{x} = \frac{{M\left( {{}^{28}{\text{Si}}} \right)}}{{1 + \frac{{\left[ \begin{aligned} \left[ {M\left( {{}^{28}{\text{Si}}} \right) - M\left( {{}^{29}{\text{Si}}} \right)} \right] \times \left[ {\frac{{m_{\text{y}1} }}{{m_{\text{x}1} }}\left[ {R_{\text{bx}1} - R_{\text{y}} } \right] + \frac{{m_{\text{y}2} }}{{m_{\text{x}2} }}\left[ {R_{\text{y}} - R_{\text{bx}2} } \right]} \right] \hfill \\ + \left[ {M\left( {{}^{28}{\text{Si}}} \right) - M\left( {{}^{30}{\text{Si}}} \right)} \right] \times \left[ {\frac{{m_{\text{y}1} }}{{m_{\text{x}1} }}R_{\text{bx}2} \left[ {R_{\text{bx}1} - R_{\text{y}} } \right] + \frac{{m_{\text{y}2} }}{{m_{\text{x}2} }}R_{\text{bx}1} \left[ {R_{\text{y}} - R_{\text{bx}2} } \right]} \right] \hfill \\ \end{aligned} \right]}}{{\left[ {R_{\text{y},28} M\left( {{}^{28}{\text{Si}}} \right) + M\left( {{}^{29}{\text{Si}}} \right) + R_{\text{y}} M\left( {{}^{30}{\text{Si}}} \right)} \right] \times \left[ {R_{\text{bx}2} - R_{\text{bx}1} } \right]}}}}$$

(9)

In contrast to approach MM B (Sect. 3.2), here two additional masses, m_y2 and m_x2, have to be determined gravimetrically (the respective masses of the solid spike material y and sample material x in the second blend bx2) (Fig. 3).

Fig. 3

Mixing scheme in the case of the double VE-IDMS approach (MM C). The preparation of a second blend bx2 renders the measurement of R_x obsolete

3.4 VE-IDMS (MM D)—Two-Step Blending of the Natural Silicon with ²⁸Si- and ³⁰Si-Enriched Silicon, Retracing M _x

After the application of the VE-IDMS method to several silicon crystals with different enrichment in ²⁸Si and additional simulations (calculating u_rel(M) with increasing x(²⁸Si)), it became an apparent postulate that (theoretically) the higher the x(²⁸Si) the smaller the u_rel(M). The quintessence of this idea was to prepare a pre-mixture of the natural silicon sample (x) with silicon enriched in ²⁸Si (e) with an already known molar mass M_e. This blend was then handled as the “sample” (ex) in a subsequent VE-IDMS measurement. The “sample” (ex) consists of known amounts (masses) of natural silicon (m_x) and silicon enriched in ²⁸Si (m_e). Subsequently, a known mass m_ex of this “sample” was then mixed with the spike material (y), silicon highly enriched in ³⁰Si, to form the blend bx. The molar mass M_ex of the pre-blend was determined according to Eq. (10) analogue to Eq. (6).

$$M_{\text{ex}} = \frac{{M\left( {{}^{28}{\text{Si}}} \right)}}{{1 + \frac{{m_{\text{y}} }}{{m_{\text{ex}} }} \times \frac{{M\left( {{}^{28}{\text{Si}}} \right) \times \left( {1 + R_{\text{ex}} } \right) - M\left( {{}^{29}{\text{Si}}} \right) - R_{\text{ex}} M\left( {{}^{30}{\text{Si}}} \right)}}{{R_{\text{y},28} M\left( {{}^{28}{\text{Si}}} \right) + M\left( {{}^{29}{\text{Si}}} \right) + R_{\text{y}} M\left( {{}^{30}{\text{Si}}} \right)}} \times \frac{{R_{\text{y}} - R_{\text{bx}} }}{{R_{\text{bx}} - R_{\text{ex}} }}}}$$

(10)

Since the masses of natural m_x and enriched silicon m_e in this blend (ex) and the molar mass M_e of the enriched silicon component are known, the molar mass of the natural silicon M_x can be calculated using the following relation (Fig. 4):

Fig. 4

Mixing scheme in the case of the VE-IDMS approach (MM D) applied to a pre-blend (ex) of the natural sample (x) and ²⁸Si-enriched silicon (e)

$$M_{\text{x}} = \left[ {\frac{1}{{M_{\text{ex}} }} \times \left( {\frac{{m_{\text{e}} }}{{m_{\text{x}} }} + 1} \right) - \frac{1}{{M_{\text{e}} }} \times \frac{{m_{\text{e}} }}{{m_{\text{x}} }}} \right]^{ - 1}$$

(11)

3.5 Double VE-IDMS (MM E)—Two-Step Blending of the Natural Silicon with ²⁸Si- and ³⁰Si-Enriched Silicon, Retracing M _x, No Knowledge of M _e and R _e Necessary

This procedure is a combination of 3.4 and 3.3 and thus a further development of 3.4. Method MM D requires the a priori knowledge of the molar mass M_e of the enriched silicon material. In the method MM E, the molar mass M_e is expressed by the molar mass of a second blend M_ex2 so that the initial knowledge of M_e and R_e becomes obsolete. Thus, two pre-blends ex1 and ex2 have to be prepared consisting of the natural silicon sample (M_x) and a silicon material highly enriched in ²⁸Si(e) with an unknown molar mass M_e. From each of these two pre-blends, subsequently two VE-IDMS blends bx1 and bx2 have to be prepared by mixing known amounts m_ex1 and m_ex2 of the pre-blends with silicon enriched in ³⁰Si (with known masses m_y1 and m_y2). It is recommended to prepare blends with slightly different ratios (R_bx1< 1 <R_bx2 or vice versa). By applying the VE-IDMS principle to these blends, the molar masses M_ex1 and M_ex2 in the pre-blends are determined according to Eqs. (12) and (13):

$$M_{\text{ex}1} = \frac{{M\left( {{}^{28}{\text{Si}}} \right)}}{{1 + \frac{{m_{\text{y}1} }}{{m_{\text{ex}1} }} \times \frac{{M\left( {{}^{28}{\text{Si}}} \right) \times \left( {1 + R_{\text{ex}1} } \right) - M\left( {{}^{29}{\text{Si}}} \right) - R_{\text{ex}1} \times M\left( {{}^{30}{\text{Si}}} \right)}}{{R_{\text{y},28} \times M\left( {{}^{28}{\text{Si}}} \right) + M\left( {{}^{29}{\text{Si}}} \right) + R_{\text{y}} \times M\left( {{}^{30}{\text{Si}}} \right)}} \times \frac{{R_{\text{y}} - R_{\text{bx}1} }}{{R_{\text{bx}1} - R_{\text{ex}1} }}}}$$

(12)

$$M_{\text{ex}2} = \frac{{M\left( {{}^{28}{\text{Si}}} \right)}}{{1 + \frac{{m_{\text{y}2} }}{{m_{\text{ex}2} }} \times \frac{{M\left( {{}^{28}{\text{Si}}} \right) \times \left( {1 + R_{\text{ex}2} } \right) - M\left( {{}^{29}{\text{Si}}} \right) - R_{\text{ex}2} \times M\left( {{}^{30}{\text{Si}}} \right)}}{{R_{\text{y},28} \times M\left( {{}^{28}{\text{Si}}} \right) + M\left( {{}^{29}{\text{Si}}} \right) + R_{\text{y}} \times M\left( {{}^{30}{\text{Si}}} \right)}} \times \frac{{R_{\text{y}} - R_{\text{bx2}} }}{{R_{\text{bx}2} - R_{\text{ex}2} }}}}$$

(13)

Additionally, the masses of the components of natural silicon (m_x1, m_x2) and silicon enriched in

²⁸Si (m_e1, m_e2) in the pre-blends have to be known. The molar mass M_x of the natural silicon sample is calculated according to eq. (14) using M_ex1 and M_ex2 from Eqs. (12) and (13) (Fig. 5):

Fig. 5

Mixing scheme in the case of the double VE-IDMS approach (MM E) applied to two pre-blends (ex1 and ex2) of the natural sample (x) and ²⁸Si-enriched silicon (e)

$$M_{\text{x}} = \frac{{\frac{{m_{\text{e}2} }}{{m_{\text{x}2} }} - \frac{{m_{\text{e}1} }}{{m_{\text{x}1} }}}}{{\frac{1}{{M_{\text{ex}1} }}\frac{{m_{\text{e}2} }}{{m_{\text{x}2} }}\left( {\frac{{m_{\text{e}1} }}{{m_{\text{x}1} }} + 1} \right) - \frac{1}{{M_{\text{ex}2} }}\frac{{m_{\text{e}1} }}{{m_{\text{x}1} }}\left( {\frac{{m_{\text{e}2} }}{{m_{\text{x}2} }} + 1} \right)}}$$

(14)

4 Results and Discussion

The results of the different methods developed and applied for the determination of the molar mass of silicon with natural isotopic composition of the crystal V3182 are displayed in Fig. 6 and listed in Table 1. As a guideline for the absolute order of magnitude, the range of molar mass values of natural silicon reported by IUPAC-CIAAW is plotted [26].

Fig. 6

Comparison of the molar mass results of the sample V3182 (with natural isotopic composition) determined with the several analytical methods (MM A–MM E) compared to the range of the IUPAC average molar mass of silicon (dashed black lines) [26]. The different methods are described in Sects. 3.1–3.5. Error bars denote combined uncertainties with k = 1

Table 1 Molar mass results of the sample V3182 (with natural isotopic composition) determined with the several analytical methods (MM A–MM E) compared to the IUPAC range of values [26]

Within the limits of uncertainties, all results of M agree, although the respective uncertainties associated with M cover a range of more than one order of magnitude depending on the analytical method applied. The range of the molar mass reported by IUPAC was taken from the 2016 publication [26]. The IUPAC range dates back to 2009 recommended by the CIAAW (Commission on Isotopic Abundances and Atomic Weights) including long-term fractionation of silicon in natural environment which aids as an appropriate guideline for a molar mass value of an artificial silicon crystal material with almost natural composition as in the case of the V3182 crystal. The direct determination of all three amount-of-substance fractions x(ⁱSi) via the measurement of the isotope ratios (MM A, classical approach, Sect. 3.1) according to Eq. (4) yields u_rel(M_x) = 1.4 × 10⁻⁵, which is at least three orders of magnitude too large to be suitable for the XRCD method. Table 2 contains an uncertainty budget for the V3182 sample according to MM A using the GUM Workbench Pro™ software (version 2.4.1 392, Metrodata GmbH, Germany) according to Eqs. (3)–(5).

Table 2 Uncertainty budget of a single selected molar mass determination of the sample V3182 (with natural isotopic composition) determined with the classical method MM A. Molar masses of Si isotopes are taken from [25]

R_w,28 and R_w (the corrected isotope ratios x(²⁸Si)/x(²⁹Si) and x(³⁰Si)/x(²⁹Si)), in the “calibration” material (natural silicon, WASO04), have the largest impact on u_c(M) with 85.5% and 8.5%, respectively. No direct measured quantity of the sample has a significant impact on u_c(M_x). The relative standard uncertainties u_rel($R_{\text{x}}^{\text{meas}}$) and u_rel($R_{{{\text{x}},28}}^{\text{meas}}$) range in the order of 0.0003 …0.0004 as a consequence of the high repeatability of the MC-ICP-MS measurements.

The smallest possible relative uncertainty associated with M_x was u_rel(M_x) = 4.2 × 10⁻⁶when using the method MM B—the VE-IDMS method—originally developed for enriched silicon directly applied on the natural silicon sample (Sect. 3.2). This method was already validated in the long term during the determination of molar masses of enriched silicon [14, 15, 17, 18]. A representative uncertainty budget of the molar mass of the V3182 crystal determined with the VE-IDMS method is given in the left-hand side of Table 3. The main contributions originate from R_w and R_y with 69% and 27%. In this budget, u_rel(M_x) is 4.2 × 10⁻⁶—approximately three orders of magnitude larger than in the case of Si enriched in ²⁸Si. On the right-hand side of Table 3, a representative budget of the crystal Si28-23Pr11 (x(²⁸Si) > 0.999 99 mol/mol) shows the sensitivity coefficients and indices of the respective input quantities with u_rel(M_x) = 1.4 × 10⁻⁹ [17]. In the case of the enriched silicon, the main uncertainty contributions stem from the same or similar input quantities as in the case of the natural silicon. However, the sensitivity coefficients (partial derivatives) differ significantly by several orders of magnitude which might finally explain the strongly reduced uncertainty in the case of enriched silicon when using the VE-IDMS method (MM B).

Table 3 Uncertainty budget of a single selected molar mass determination of the sample V3182 (with natural isotopic composition) determined with the VE-IDMS method (MM B, left-hand side)

In contrast to the original VE-IDMS method (MM B), the further developed method (MM C) did not yield a further reduced u_rel(M_x). According to Table 1, u_rel(M_x) = 4.6 × 10⁻⁶ is slightly increased, although the measurement of R_x in the sample is now obsolete. This elevated uncertainty is suggested to be induced by an increasing number of the measurands (MM B: 5 measurands; MM C: 7 measurands).

One semi-quantitative, but central result of the molar mass determinations of enriched silicon was the inverse proportionality of enrichment x(²⁸Si) and u(M) when using the VE-IDMS method (MM B) [17]. Therefore, it would be necessary to investigate the relation between x(²⁸Si)—the “enrichment”—and the respective u_rel(M) more quantitatively. A first glance simulation was carried out using Excel™ and the GUM Workbench Pro™ software. In this simulation, the x(²⁸Si) were changed in the range 0.922079 mol/mol ≤ x(²⁸Si) ≤ 0.999990 mol/mol. $R_{\text{w}}^{\text{meas}}$, $R_{\text{x}}^{\text{meas}}$, $R_{\text{bx}}^{\text{meas}}$, and their associated standard uncertainties were kept constant. The respective molar mass and its associated uncertainty was calculated for three isotope ratios x(²⁹Si)/x(³⁰Si) = 1.5, 5, and 10. The starting values x(²⁸Si) = 0.922079 mol/mol with x(²⁹Si)/x(³⁰Si) = 1.5 were chosen, because these are the respective values of natural silicon. The intention was to find a range of x(²⁸Si) where the respective u_rel(M) falls below a value of 1 × 10⁻⁷. Figure 7 shows the rough course of u_rel(M) vs x(²⁸Si) for the ratios x(²⁹Si)/x(³⁰Si) = 1.5, 5, and 10.

Fig. 7

Comparison of u_rel(M) vs x(²⁸Si) for three different ratios x(²⁹Si)/x(³⁰Si) = 1.5, 5, and 10 (for details see text)

An almost linear decrease in u_rel(M) with x(²⁸Si) can be observed. However, the ratios x(²⁹Si)/x(³⁰Si) do not play that important role as assumed in the past. Recent experimental investigations of enriched silicon crystals with different ratios support the finding that the enrichment and not the ratios x(²⁹Si)/x(³⁰Si) are mainly responsible for the small u_rel(M) [18]. Therefore, we concentrated on the changes of u_rel(M) as a function of x(²⁸Si) only, using a natural-like ratio of 1.5. Figure 8 shows the development of u_rel(M) with a stepwise increase in x(²⁸Si). It is obvious that it is necessary to have at least an enrichment of x(²⁸Si) > 0.98 mol/mol to fall below u_rel(M) = 1 × 10⁻⁷. A clearer result is shown in Fig. 9: between the two last data points (x(²⁸Si) = 0.991333 mol/mol and x(²⁸Si) = 0.999 990 mol/mol), u_rel(M) drops down drastically from 4.6 × 10⁻⁷ to 5.3 × 10⁻¹⁰: three orders of magnitude.

Fig. 8

Comparison of u_rel(M) vs x(²⁸Si) for x(²⁹Si)/x(³⁰Si) = 1.5 (for details see text)

Fig. 9

Same as Fig. 8 with u_rel(M) on a logarithmic scale

As a central result of this simulation, it can be concluded that simply an enlarged x(²⁸Si) (> 0.99 mol/mol) should be a key for u_rel(M) < 10⁻⁷ or less.

With this knowledge in mind, the idea came up to prepare a pre-blend (ex) of the sample (natural silicon) and silicon material enriched in ²⁸Si. The molar mass and its associated uncertainty of this blend determined using the VE-IDMS method should be smaller than that of the pure natural silicon due to the higher enrichment of the blend. Knowing the molar mass of the enriched component (M_ein eq. (11)) from previous investigations and the masses m_e and m_x of the components in the pre-blend, it should be possible to recalculate the molar mass M_x of the natural silicon sample—hopefully with a reduced u_rel(M_x). This is the idea behind method MM D (Sect. 3.4).Unfortunately, the application of this method (MM D, Sect. 3.4) did not succeed in a reduction of u_rel(M_x). Instead, u_rel(M_x) increased to 6.3 × 10⁻⁶, approximately 50% larger than applying the original VE-IDMS method (MM B). Table 4 shows a representative uncertainty budget of the molar mass of a natural silicon sample (V3182) using method MM D. The main contributions are $R_{\text{ex}}^{\text{meas}}$, the measured intensity ratio of x(³⁰Si)/x(²⁹Si) with 41% in the blend of natural silicon (x) and enriched silicon (e), followed by the contributions of R_w (29%) for the K factor calculation, and $R_{\text{bx}}^{\text{meas}}$ with 26% which is measured in the blend bx of natural (x) and enriched silicon (e) spiked with silicon (y) enriched in ³⁰Si.

Table 4 Uncertainty budget of a single selected molar mass determination of the sample V3182 (with natural isotopic composition) determined with method (MM D, Sect. 3.4) according to model Eqs. (10) and (11). Molar masses of Si isotopes are taken from [25]

Method MM E [Sect. 3.5, Eqs. (12)–(14)] yields an even worse result: u_rel(M_x) = 5.4 × 10⁻⁵, which is at least one order of magnitude larger than after applying the original VE-IDMS method (MM B). Again, this degradation can be explained by an increased number of measured input quantities. At this point, the application of the VE-IDMS method (MM B) yields the smallest uncertainty associated with M_x. However, u_rel(M_x)—in the lower 10⁻⁶ range—is still too large to use natural silicon as a sphere material in the XRCD method on a primary dissemination level.

What are the reasons—or is the reason—for this comparable elevated uncertainty associated with M in the case of silicon with natural isotopic composition compared to silicon highly enriched in ²⁸Si? Since the VE-IDMS method (MM B) yields the smallest u_rel(M_x), it was analysed in more detail. Table 3 shows that the sensitivity coefficients

$$u_{\text{c}}^{2} (y) = \sum\limits_{i = 1}^{N} {\left( {\frac{{\partial M_{\text{x}} }}{{\partial x_{i} }}} \right)}^{2} \times u^{2} (x_{i} )$$

(15)

(partial derivatives, second term in Eq. (15)) are significantly different when setting up an uncertainty budget for the molar mass [according to Eq. (6)] of enriched or natural silicon. Figure 10 displays the evolution of the respective partial derivatives: (a) ${{\partial M} \mathord{\left/ {\vphantom {{\partial M} {\partial R_{\text{x}}^{\text{meas}} }}} \right. \kern-0pt} {\partial R_{\text{x}}^{\text{meas}} }}$, (b) $-{{\partial M} \mathord{\left/ {\vphantom {{\partial M} {\partial R_{\text{bx}}^{\text{meas}} }}} \right. \kern-0pt} {\partial R_{\text{bx}}^{\text{meas}} }}$, and (c) ${{\partial M} \mathord{\left/ {\vphantom {{\partial M} {\partial R_{\text{w}}^{\text{meas}} }}} \right. \kern-0pt} {\partial R_{\text{w}}^{\text{meas}} }}$ with increasing enrichment x(²⁸Si).

Fig. 10

Evolution of sensitivity coefficients: a) ${{\partial M} \mathord{\left/ {\vphantom {{\partial M} {\partial R_{x}^{\text{meas}} }}} \right. \kern-0pt} {\partial R_{x}^{\text{meas}} }}$, b) $-{{\partial M} \mathord{\left/ {\vphantom {{\partial M} {\partial R_{bx}^{\text{meas}} }}} \right. \kern-0pt} {\partial R_{bx}^{\text{meas}} }}$, and c) ${{\partial M} \mathord{\left/ {\vphantom {{\partial M} {\partial R_{w}^{\text{meas}} }}} \right. \kern-0pt} {\partial R_{w}^{\text{meas}} }}$, as a function of the enrichment x(²⁸Si) applied for the VE-IDMS method (MM B, Sect. 3.2, eq. (6))

The three intensity ratios (bias corrected: isotope ratios) are the only quantities being measured (except the masses of the blend components) in the VE-IDMS method and should thus probably be influenced by a controlled improvement in the experiment. The respective standard uncertainties u($R_{\text{x}}^{\text{meas}}$), u($R_{\text{bx}}^{\text{meas}}$), and u($R_{\text{w}}^{\text{meas}}$) are kept constant for each x(²⁸Si) in this simulation. They do not influence u(M) in a comparable way and can be neglected in this comparison. When analysing the evolution of the three sensitivity coefficients, the largest change can be observed for $R_{\text{x}}^{\text{meas}}$ ranging over five orders of magnitude (Fig. 10). This is one of the key factors for the strongly decreased uncertainty associated with M when using enriched instead of natural silicon. As stated above, from the experimental point of view, the measurement of isotope ratios in natural silicon is much easier due to the significantly higher abundance of ²⁹Si and ³⁰Si which directly improves the respective repeatability and standard uncertainties of the measured intensity ratios. Nevertheless, from the evolution of the sensitivity coefficients of the main measurands with increasing enrichment, it is suggested at this stage that the isotopic composition itself—either in the natural or in the enriched material—influences the sensitivity coefficients most efficiently which will finally determine the combined uncertainty associated with the molar mass.

5 Conclusion

One primary method of the realization and dissemination of the kilogram and mole is the XRCD method using silicon spheres highly enriched in ²⁸Si. To promote a broader application of this method, the use of silicon with a natural isotopic composition would cut the costs of the spheres dramatically at the disadvantage of an elevated uncertainty. We have developed several new analytical methods based on the VE-IDMS method which is applied for enriched silicon in combination with K factor determination to reduce the uncertainty associated with the molar mass. The methods are compared by analysing the uncertainty budgets with respect to u(M_x). Currently, the best method is still the original VE-IDMS method yielding u_rel(M_x) in the lower 10⁻⁶ range, which is about three orders of magnitude larger than in the case of Si enriched in ²⁸Si. Simulations were carried out to investigate the evolution of u_rel(M) as a function of x(²⁸Si). For the time being, it seems that an enrichment of x(²⁸Si) > 0.99 mol/mol is an unconditional requirement for obtaining an uncertainty associated with the molar mass of u_rel(M) ≤ 10⁻⁷. The improvement in experimental designs to optimize the repeatability to reduce the standard uncertainties did not reduce u_rel(M_x). A central observation of this study is the strong influence of the sensitivity coefficients especially of the main measurands in the VE-IDMS method when comparing natural and enriched silicon. It is suggested at this stage that the impact on the sensitivity coefficients and thus the final uncertainty u_rel(M) is based on intrinsic properties of the crystal material, e.g. the amount-of-substance fractions impacting the uncertainty modelling in a serious manner. Further theoretical and experimental efforts are under way.