Amount of substance and the mole in the SI

The concepts 'mole' and 'Avogadro constant' are based on the understanding of chemical processes as interactions between atoms and molecules. In 1808, John Dalton proposed that atoms of an element do not differ from one another and that they have a defined atomic mass. At this time, there was no experimental evidence demonstrating the existence of atoms and Dalton was very much aware that he was proposing an ambitious and speculative idea, born nonetheless out of many empirical indications. Cautiously he titled his publication 'A New System of Chemical Philosophy' in order to express his concerns [6]. In 1811, Amedeo Avogadro published his hypothesis that the same volumes of all gases contain—at the same temperature and the same pressure—the same number of molecules, with reference to Dalton's publication [7]. Initially, Avogadro's observation was forgotten until Stanislao Cannizzaro revived it. In 1858 he published a consistent system of chemical formulas and 'atomic weights' (relative atomic masses, see below) of all elements. Subsequently, terms such as atomic and molecular weight as well as other terms based on atomic theory were developed and used in chemistry [8].

The origin of the concept 'mole' is attributed to Wilhelm Ostwald (see, e.g. [9]). In his 'Hand- und Hilfsbuch zur Ausführung Physiko-Chemischer Messungen' of 1893, he writes: 'Let us generally refer to the weight in grams of a substance that is numerically identical to the molecular weight of that substance, as one mole...' [10]. Similar terms such as, for example, 'g-Molekel' or 'g-Mol' with comparable meaning were also used contemporaneously by others such as Walther Nernst (see, e.g. [11]). According to this definition, the unit 'mole' was, therefore, closely connected with mass and for a long time it was interpreted as a 'chemical mass unit' although the atomic perception that links the mole with a number of particles had existed since Dalton and Avogadro. There was, however, a lack of experimental evidence, which could unequivocally confirm these models [11]. The experimental confirmation of the atomic theory finally came from the complete explanation of Brownian motion by Perrin and Einstein in 1909 [12] that also led to the definition of the Avogadro constant. It was followed by the discovery of x-ray crystal diffraction in 1912 by von Laue, Friedrich and Knipping [13] that was essential in fixing the numerical value of the Avogadro constant. This finally led to two different perceptions of the mole, which Stille differentiated by the concepts 'mole' (as a chemical mass unit) and 'mole number' as a unit related to a number of particles which is defined by the Avogadro constant and, therefore, suggesting the introduction of an additional base quantity, the 'amount of substance' [14, 15].

The integration of the unit 'mole' into the SI system of units resolved this ambiguity and made a differentiation of the concepts superfluous. It took place at a very much later date, in October 1971, after the 14th General Conference on Weights and Measures had decided to introduce the mole as a base unit of the SI. The English term 'amount of substance', the quantity of which the mole is the unit, was derived from the German term 'Stoffmenge', introduced by Stille [14, 15]. The decision had been preceded by a corresponding recommendation of the International Union of Pure and Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry (IUPAC) and the International Organization for Standardization (ISO), together with the note to select the carbon isotope ¹²C as the reference point [16]. The CIPM committee that deals with issues related to the base quantity 'amount of substance' (Consultative Committee for Amount of Substance—CCQM) was—again—established much later, in 1993 [17].

The mole establishes a connection between the SI and chemistry. With the aid of the mole, quantitative relations in chemistry can be made traceable to SI units and measurements internationally compared. Based on the mole, a large number of national standards for important measurands in chemistry have been introduced in recent years so that these measurements are now traceable to the SI. Generally these are for derived quantities such as the amount of substance concentration.

In usual practice (e.g. in the performance of chemical reactions in the laboratory or in the chemical industry), a very large number of atoms and molecules is involved in any reaction. The unit 'mole' therefore considers together many particles, such that reference to other units (the kilogram) can be made relatively easily. The mole is important in chemistry because it recognizes that atoms and molecules react together on an amount of substance basis and not on a mass basis—chemists call this stoichiometry.

The first definition of the SI unit 'mole', which remains in place until 20 May 2019 is [16]:

The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is 'mol'.

When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

In this definition, it is understood that unbound atoms of carbon 12, at rest and in their ground state, are referred to [16]. Unlike Ostwald, whose definition related only to the mass, reference is now also made to a number of particles. The number is derived in this definition, however, from a mass measurement, namely the mass of 0.012 kg of ¹²C and, hence, was still directly dependent on the SI unit kilogram. Traceability to the mole still ultimately required a mass measurement at this stage.

In practice, traceability to the mole is (and will be) in most cases realized via a weighing process and reference to the relative atomic mass. The connection between the amount of substance n_A of an analyte A and its mass m (measured in kilogram) is performed via its molar mass M(A) (measured in kg mol⁻¹):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{n}_{{\rm A}}}={m}/{M\left(\rm A \right)} \nonumber \end{align} \tag{ 1 }$$

whereby the molar mass of ¹²C, M(¹²C)  =  0.012 kg mol⁻¹, serves as a reference basis for M(A). The molar mass of a substance can be calculated from the mean, relative atomic masses A_r of the elements involved, which are usually well known. These also have ¹²C as the reference point; A_r(¹²C)  =  12 (a dimensionless ratio) by definition. The mean relative atomic masses take into account the relative atomic masses of all isotopes of an element and their abundance on earth [16, 18]. The molar mass M(A) of the substance A is the sum of the mean relative atomic masses A_r of all elements of a molecule of the substance A in accordance with its stoichiometric composition A_r(A) multiplied by the molar mass constant M_u. In the definition of the mole from 1971 it follows that M_u is exactly 0.001 kg mol⁻¹:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle M\left(\rm A \right)={{A}_{{\rm r}}}\left(\rm A \right)\times {{M}_{{\rm u}}}.\nonumber \end{align} \tag{ 2 }$$

In the new definition the mole is defined by a fixed number of particles. This required the determination of the Avogadro constant with an accuracy that preserves continuity of measurement results related to the mole before and after the redefinition. This was made possible by the Avogadro experiment, described in more detail below. It allowed the numerical value of the Avogadro constant to be fixed and a definition of the mole based on a number of elementary entities without dependence on mass.

After redefinition, experiments for realizing the mole without reference to mass via determination of the number of elementary entities are conceivable, based on the fixed value of the Avogadro constant. Obviously, the Avogadro experiment itself is then best suited for this purpose as referred to in the mise en pratique of the mole [2] and explained in more detail in section 5. It then also allows for a realization of the kilogram based on counting a number of elementary entities [19].

Further to this, other properties of an elementary entity such as its charge might also be suitable for counting and the realization of the mole [20]. Concepts for measurements of the number of elementary entities have been available for a long time, although this had been limited to special cases. The principle utilized in the Avogadro experiment, namely the measurement of the amount of substance of a crystalline solid with the aid of its microscopic, crystallographic lattice parameter and its macroscopic volume, is not new. Similar—albeit much simpler—experiments have been carried out since the discovery of x-ray diffraction. Traceability to the SI was, however, not possible before the advent of x-ray interferometry [21]. For the redefinition of the SI much more sophisticated experiments were required in order to achieve improvements in uncertainty by several orders of magnitude. There are also experiments for the (direct or indirect) counting of elementary entities, e.g. of ions [22] and electrons (single-electron-tunneling—SET circuits). The latter is relevant to the realization of the revised SI base unit ampere [23]. Counting rates of several GHz have been achieved with this approach [24].

In addition, there are numerous procedures for the observation of single atoms (scanning probe microscopy) and also chemical-analytical procedures for the identification and detection of single molecules [25]. Except for the Avogadro experiment none of these procedures is, however, currently suited to quantify macroscopic sample quantities of the order of 1 mol. Some application fields such as nanotechnology do not require this though.

Unlike the other base units, the mole was, and still is, defined by two clauses. The first one defines a number of elementary entities. The second one requires the identification of the elementary entities and establishes—beyond the field of physics—the connection to analytical chemistry. The complete description of a measuring operation for the determination of the amount of substance of a measurand in the unit mole thus requires the identification and quantification of a specified entity (analyte). The amount of substance 1 mol always contains the same number of specified entities. This number is identical to the Avogadro number, the numerical value of the Avogadro constant.

As regards the identification of the entities, the additional notes in the SI Brochure also explain: 'It is important to give a precise definition of the entity involved (as emphasized in the second sentence of the definition of the mole); this should preferably be done by specifying the molecular chemical formula of the material involved' [1]. This implies that—in addition to the chemical formula—other information may be required for their complete description such as, for example, information about their structure.

Prior to SI revision the kilogram remained the only base unit defined and realized as a single material artefact to which all mass measurements across the world were ultimately traceable [16]. In this respect mass measurement had the benefits of precision and the drawbacks of lack of resilience that are usually associated with a non-SI measurement scale: but in this case a special measurement scale that defined the SI unit of mass [26]. The drawbacks of a unique material artefact defining a unit are well known and outweigh the benefits. For a long time metrologists had been keen to redefine the kilogram in terms of constants of nature. The development of the Kibble balance [27] brought this possibility into sharper focus and also prompted additional proposals to redefine with respect to fixed numerical values of constants three other base units of the SI: the ampere, the kelvin and the mole [28, 29]. (The metre was already defined with respect to the speed of light in vacuum and the second depends on the material property of an atom and until the advent of definitions in terms of Planck units it always will. The candela, related to the luminous efficacy technical constant related to a spectral response of the human eye, is a rather separate issue and was not part of these discussions.)

One can consider that the definition of the kilogram, the ampere, the kelvin and the mole in the SI prior to revision are statements of the fixed nature of the mass of the international prototype of the kilogram (IPK), the magnetic constant, the triple point of water and the molar mass of carbon-12. With the exception of the magnetic constant these are related to material properties, and in one case a unique material artefact. The proposals for revision of the SI considered a number of options for combinations of constants that were more universal, allowed realization across the quantity scale not just at one fixed point, and were not related to material properties [29, 30]. The number of constants with fixed values could not over-constrain the system, for instance there could not be two independent constants or groups of constants defining the same SI unit. Equally all proposals needed to ensure that the relationships between quantities was unaltered, regardless of the choice of unit definitions. In particular it is worth noting two such quantity relations relating to the mole:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle m\left(^{12}{\rm C} \right)=\frac{M\left(^{12}{\rm C} \right)}{{{N}_{{\rm A}}}}\nonumber \end{align} \tag{ 3 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{h}{{{m}_{{\rm u}}}}=\frac{{{N}_{{\rm A}}}h}{{{M}_{{\rm u}}}}=\frac{c{{\alpha }^{2}}}{2{{R}_{\infty }}}{{A}_{{\rm r}}}(\rm e)\nonumber \end{align} \tag{ 4 }$$

where m(¹²C) is the mass of one atom of ¹²C, M(¹²C) is the molar mass of ¹²C, N_A is the Avogadro constant, h is the Planck constant, m_u is the atomic mass constant, M_u is the molar mass constant, c is the speed of light in vacuum, α is the fine structure constant, R_∞ is the Rydberg constant and A_r(e) is the relative atomic mass of the electron. This highlights that the values of h and m_u cannot both be fixed, nor the three values of h, M_u and N_A. The Planck constant and the Avogadro constant are directly linked via the Kibble balance experiment that links h to macroscopic mass and the Avogadro experiment to count atoms in a near perfect silicon sphere that links N_A to macroscopic mass. It is possible to define mass in terms of either of these constants.

It soon became clear as these proposals developed that a fixed numerical value of N_A was a likely outcome in a revision of the SI, and could also be used to provide a new definition of the mole based on a fixed number of elementary entities. The main options for fixing the numerical values of constants and their effect on the relative uncertainties most relevant to chemistry are given in table 1.

Considering table 1, in the SI prior to revision the mass of the IPK had no uncertainty and the molar mass constant is 0.001 kg mol⁻¹ exactly: values of the other quantities are experimentally determined [32]; from (4), u_r(N_A)  =  u_r(m_u). After the revision, the numerical values of h and N_A are fixed and thus the molar mass constant and the atomic mass constant have equal relative uncertainties, again inferred from (4). Another viable option, not chosen, would have been to fix the numerical value of h and leave the definition of the mole unchanged. This would have resulted in reduced uncertainties for u_r(N_A) and u_r(m_u), although would still have required the 1980 clarification appended to the definition of the mole concerning unbound atoms in their ground state. A further option would have been to fix the values of m_u, M_u and N_A. This proposition, which results in the kilogram being defined by m_u, is very appealing to chemists but the value of u_r(h) under these circumstances is not considered negligible by physicists. Relative atomic masses and relative molecular masses are ratios, not dependent on the definition of the kilogram or the mole, and are unaffected by the new definitions of the kilogram and the mole.

When considering a change of definition (for any SI unit, not just the mole) it is essential that at the point of re-definition the unit system remains coherent and consistent without any step changes in unit size. (The re-definition of the ampere resulted in a change of about 1 part in 10⁷ but this is because under the previous definition the most precise electrical measurements are defined by non-SI conventions. The new definitions of the ampere and kilogram will bring electrical metrology within the SI once again). In particular, in the SI prior to revision M_u is exactly 0.001 kg mol⁻¹ and M(¹²C) is exactly 0.012 kg mol⁻¹. The continuity conditions agreed for the SI must ensure that these quantities be consistent with their historic values to within their newly acquired uncertainties, at the time of redefinition.

Equations (3) and (4) can be combined to yield (m(e) denotes the electron mass):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle M\left({}^{{\rm 12}}{\rm C} \right)=\frac{2{{N}_{{\rm A}}}h{{R}_{\infty }}m\left({}^{{\rm 12}}{\rm C} \right)/m\left(\rm e \right)}{c{{\alpha }^{2}}}=\frac{24{{N}_{{\rm A}}}h{{R}_{\infty }}}{c{{\alpha }^{2}}{{A}_{{\rm r}}}\left(\rm e \right)}.\nonumber \end{align} \tag{ 5 }$$

As may be deduced from equation (5), the relative uncertainty of the fine structure constant will dominate the experimental uncertainty of M(¹²C)—which will essentially be equal to 2u_r(α)  ≈  0.45  ×  10⁻⁹ since the relative uncertainties of the relative atomic mass of the electron and the Rydberg constant are significantly smaller. (As in the past, the value, and uncertainty, of α will continue to be updated as new data become available, the next occasion being the adjustment of the Committee on Data of the International Council for Science (CODATA-2018) in May 2019. The value of α is a pure number which is identical in all unit systems.) This consideration in turn informs the debate on the number of significant digits preferable for N_A [33]. Nine significant figures were decided, such that N_A is exactly equal to 6.022 140 76  ×  10²³ mol⁻¹ in the SI after the revision [32].

There are a number of benefits to the presence of the mole, and therefore chemical measurement, within the SI. These are as relevant now as they were when the mole was introduced into the SI in 1971, and remain so after revision of the SI [34]. In particular:

• It formally described a unit which was directly proportional to the number of elementary entities in a sample of a substance. This was an important requirement for chemists in order to describe chemical relationships in stoichiometric terms, without recourse to mass.
• It introduced amount of substance as having its own dimension [16], thereby distinguishing 'number of entities' from a pure number and extending the power of quantity calculus to chemistry.

Indeed, the definition of the mole prior to revision makes it clear that a number of elementary entities are specified: 'The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kg of carbon 12...'. The problem is that the quantitative statement is one relating to mass, not to what this number of elementary entities might be. There is evidence that this is bypassed by most educators who use a proxy definition that one mole is equal to an 'Avogadro number' of elementary entities [31]. Apart from not explicitly mentioning the Avogadro constant in its definition (or indeed the molar mass of carbon 12⁶) we might consider that the definition prior to revision has other drawbacks. First, its reliance on mass therefore incurring all the drawbacks of traceability to the IPK. This still has the potential for some to confuse amount of substance with mass. Second the definition was closely linked to the realization, but in practice the mole cannot easily be realized in the way described. This is a problem with other SI units which are being redefined—although we should recognize that the definition of a unit in no way implies its realization. It is common practice to make small correction between real experiments and idealized definitions. This is necessary for the definition of the second, which specifies that the frequency measured must be 'unpertubed'—an impossibility. More important the former definition of the mole specifies a specific material property at a particular point on the amount of substance scale.

The new definition relates the mole to a fixed number of elementary entities, explicitly based on a fixed numerical value of the Avogadro constant: 'The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.022 140 76  ×  10²³ elementary entities. This number is the fixed numerical value of the Avogadro constant, N_A, when expressed in the unit mol⁻¹ and is called the Avogadro number' [1]. A diagrammatic representation of the relationships between relevant chemical quantities and how their uncertainties change as a result of the redefinition of the mole is given in figure 1.

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The uncertainty in N_A prior to redefinition moved to M_u and M(¹²C), albeit much reduced in relative terms thanks to h now having a fixed value, and following redefinition the numerical value of N_A has no uncertainty. This has the effect of linking the microscopic and macroscopic worlds with zero uncertainty: this more clearly displays the nature of the Avogadro constant as a concept synthesizer [36] relating amount of substance to a number of defined elementary entities. A further consequence of an exact numerical value of the Avogadro constant is that atomic mass scales and the molar mass scales have equal relative uncertainty: a pleasing symmetry for chemical metrologists. There is also the benefit that the definition of the mole is no longer tied to the definition of the kilogram, although for the foreseeable future the vast majority of practical realizations of the mole and its derived units will still involve weighing via n  =  m/M where n is the amount of substance, m is the mass of a pure sample and M is the molar mass (mass per amount of substance). It is noteworthy that this equation can be used to determine the amount of substance without knowledge of the value of the Avogadro constant. This is useful because relative atomic masses were known accurately long before similarly accurate measurements of the Avogadro constant were made, and it will still be useful after redefinition [37].

The major benefit of the new definition is that it explicitly defines the mole in the way that most chemists think about it [31]. It also provides a definition not dependent on a material artefact that is more universal in its accessibility and applicability across the amount of substance scale. It conforms to a format similar to the other unit redefinitions, but was adapted to make it more understandable for the chemistry community, as requested by the International Union of Pure and Applied Chemistry [38]. The new definition says nothing about its realization, thereby making the new definitions a better fit to the technologies of the 21^st century which will be used to realize them. This is especially true at very low amount of substance levels, where techniques such as single molecule counting may be employed [34].

A drawback of the new definition is the diminished importance of the molar mass constant and the molar mass of ¹²C, both becoming experimentally determined quantities. It has been argued that this change is one that could be seen as risking a lack of coherence in the system [30]. Although the situations where the newly acquired uncertainty in the molar mass constant might need to be considered, for instance for isotopes with very precisely known relative atomic masses, are currently only hypothetical because purity considerations dominate by many orders of magnitude. It has also been argued that an inexact molar mass constant might be difficult to accommodate in teaching. However, the uncertainty in M_u is still an order of magnitude lower that the most accurate realization of the mole—via the silicon sphere experiment—and several orders of magnitude smaller than the relative uncertainties in more common realizations of the mole, for instance in the mise en pratique for the mole. A more fundamental way to consider the newly acquired uncertainty of the molar mass constant in the new SI is that this is the level at which the assumption about the conservation of mass in chemical reactions begins to breakdown. This is insignificant for normal analytical chemistry but not for other areas of science. The final option in table 1, where m_u, M_u and N_A are all fixed, does not confer enough benefit to chemistry to warrant imposing it on the non-chemical community [31].

Concerns about teaching are still most acute when it comes to distinguishing counting quantities from amount of substance. In this respect the new definition will not help, and may cause even more confusion—this area needs attention from the chemical measurement community in future. The new definition also will not help to resolve the awkward historical relationship the mole has had with the quantity of which it is the base unit, amount of substance. This has mainly arisen because, unlike in physical metrology where we first conceive of a quantity and then of its unit, this happened in reverse for chemistry.

Overall the change is one of clear net benefit for chemistry, even if in the short term there are no practical implications and any improvements may take some time to realize. It is also a necessary change to keep metrology in chemistry aligned with the rest of metrology.

The new definition of the mole will remove any link between mass and amount of substance: the mole will not rely on any other units or defining constants for its definition. The mole will also be clearly linked to a count, in such a way that the Avogadro constant acts as the constant of proportionality linking counting of elementary entities with amount of substance. It has also been highlighted that the new definition of the mole may also find use in the emerging areas of ultra-low chemical and biological quantification [34]. However, extra vigilance will need to be used to avoid confusion on whether measurement results are being reported in terms of the number of entities or amount of substance.

The XRCD method exploits the existence of large silicon single crystals where the atoms are perfectly ordered over macroscopic ranges. The principle set-up for measuring the distance of lattice planes is shown in figure 2 [43]. It consists of three lamellas, cut out of the Si crystal, oriented in direction of the {2 2 0} lattice planes. The first, the 'splitter' lamella ('S') splits the x-ray into two beams. In the second lamella, the 'mirror' ('M') the two beams are reflected and finally recombined in the third lamella ('A'). This analyzer lamella can be moved transverse to the beam yielding in a periodic intensity variation of the transmitted x-rays with the period of the diffracting-plane spacing. The movement of the analyzer is measured by an optical interferometer, so that the period of the x-ray fringes (i.e. the lattice spacing) is measured in units of optical fringes and thus traced back to a wavelength standard.

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As the analyzer lamella is separated from the crystal with the other two interferometer lamellas, the adjustment is very difficult and requires exact positioning [42]. An optical interferometer with polarization encoding and phase modulation is used to monitor the displacement with picometre accuracy and the rotation in pitch and yaw with nanoradian resolution.

The x-ray interferometer (XINT) is cut out of the crystal in an axial position near to the spheres. Except for carbon, oxygen and nitrogen, the contents of which are measured (see section 5.4), the contaminant concentrations are significantly less than one atom in 10⁹ Si atoms (compare table 3). Since the contaminants in the silicon slightly change the mean atomic distances, the measured lattice parameter a(XINT) has to be corrected to calculate the mean lattice parameter of the sphere

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle a\left({\rm sphere} \right)=a\left({\rm XINT} \right)\left[ 1+\sum\limits_{i}{\left({{\beta }_{i}}\,\Delta {{C}_{i}} \right)} \right]\nonumber \end{align} \tag{ 10 }$$

using the strain coefficient β_i of the point defect i and the concentration difference ΔC_i between the x-ray interferometer (XINT) and the sphere [42].

Apart from the strain due to contaminants and vacancies, it must be ensured that the lattice parameter is determined for the same temperature and pressure as the volume (20 °C in the International Temperature scale of 1990 (ITS-90) and 0 Pa).

For the determination of the sample volume in equation (6) spheres are used, since they are less susceptible to damage than cubes or cylinders, and only one parameter—the mean diameter—has to be determined. The diameter of the silicon sphere is measured by optical interferometry from many different directions [40]. If the deviation from a perfect spherical shape is very small, the volume can be calculated from the measured mean diameter D with high accuracy: V  =  (π/6)D³.

In the interferometer the sphere is placed between two reference surfaces (the etalon) and the distances d₁ and d₂ between the sphere surface and the reference surfaces are measured. The fractional orders of interference are measured by phase-shifting interferometry using optical frequency tuning. With an additional measurement of the length L of the empty etalon, the (apparent) diameter is given by D  =  L  −  d₁  −  d₂.

Two different types of optical configuration are used (figure 3): At the Physikalisch-Technische Bundesanstalt (PTB, Germany) the reference surfaces are spherical, which enables diameter measurements in numerous directions without rotating the sphere [40]. This interferometer can measure about 10 000 diameters of the sphere simultaneously in an aperture angle of 60°. In the set-up of the National Metrology Institute of Japan (NMIJ) an etalon with flat reference surfaces is used (figure 3(b)) [40]. By rotating the sphere, the diameter topography of the entire sphere surface is available, and a topography can be determined (figure 4).

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As in the lattice parameter measurements, the volume is measured in vacuum and at 20 °C. Standard platinum resistance thermometers are used calibrated traceable to the ITS-90 at the triple point of water (0.01 °C) and at the melting point of gallium (29.7646 °C) [40].

The sphere is covered by surface layers, which cause a small phase retardation in the reflected light beam. The diameter measured by the interferometer therefore provides information only on the 'apparent diameter'. To deduce the diameter of the silicon core or the diameter of the whole sphere, the phase retardation on reflection from the sphere surface is evaluated from surface layer measurements (section 5.4) [40].

The determination of the Avogadro constant using silicon with natural isotopic composition failed to reach uncertainties of 10⁻⁷ or below. To reach the highest accuracy with the XRCD method, the enrichment of the ²⁸Si isotope should be higher than 99.99%. Gaseous silicon tetrafluoride SiF₄ is used to enrich the ²⁸Si isotope by centrifuges, since fluorine consists of only one isotope [45]. Then the enriched SiF₄ is transformed into silane (²⁸SiH₄) using calcium hydride (CaH₂):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {}^{28}{\rm Si}{{{\rm F}}_{4}}+2\,{\rm Ca}{{{\rm H}}_{2}}\to {}^{28}{\rm Si}{{{\rm H}}_{4}}+2\,{\rm Ca}{{{\rm F}}_{2}}.\nonumber \end{align} \tag{ 11 }$$

Since the silicon material must be also chemically extremely pure, the silane is cleaned in a very sophisticated procedure by cryofiltration with subcooled boiling and rectification [45]. Eventually, the enriched silicon is deposited by pyrolytic vapour deposition on a slim, electrically heated Si-28 rod. The produced polycrystalline silicon rod is purified by the float zone technique, first in vacuum to remove oxygen and later in argon [45].

The measurement of the isotopic composition and molar mass by 'virtual element isotope dilution mass spectrometry' is described in detail in section 6.

The surface of a silicon sphere is typically covered with a 1 nm oxide layer and additional carbonaceous and water layers which have a total thickness of about 0.5 nm to 1 nm (see table 2). Measurements by spectroscopic ellipsometry (SE), x-ray reflectometry (XRR), x-ray photoelectron spectroscopy (XPS), and x-ray fluorescence (XRF) spectrometry can be performed to characterize the surface layers [40]. XPS is most suitable for the thin layers on silicon, since it cannot only measure the amount of the chemical elements on the surface but also the chemical binding state of the elements. This allows the stoichiometric composition of the layers to be determined. Table 2 lists the amounts of substance on the Si-28 sphere Si28kg01a [46]. The amount of the element X can be calculated from the measured mass deposition m_d(X) by n_S(X)  =  m_d(X)A/M(X), where A  =  πD² is the surface area of the sphere with the diameter D and M(X) the molar mass of the element X. The number of silicon atoms in the oxide

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{N}_{{\rm s}}}\left({\rm Si} \right)={{N}_{{\rm A}}}{{n}_{{\rm s}}}\left({\rm Si} \right)={{N}_{{\rm A}}}m_{\rm d}\left({\rm Si} \right)\,{A}/{M\left({\rm Si} \right)}.\nonumber \end{align} \tag{ 12 }$$

has to be considered for the calculation of the total amount of silicon in a silicon sphere, see section 5.5.

Due to the sophisticated cleaning procedures, the crystals contain only very few impurities. Most important are carbon and oxygen which can be quantified by Fourier transform infrared absorption spectroscopy (FTIR) measurements [40]. The point defects in silicon influence the amount of silicon in the sphere. For example, carbon is incorporated substituting silicon atoms, thus reducing the amount of silicon. On the other hand, oxygen atoms are present on interstitial sites and do not change the number of lattice places as calculated by equation (6). All impurities and vacancies affect the lattice parameter and the density of the silicon, but these effects cancel each other. Vacancies in the lattice can be interpreted as substitutional atoms with zero mass. Table 3 lists the concentrations of the main point defects (impurities and vacancies) in the sphere Si28kg01a together with the relative amounts of substance C(X)/C(Si) with the 'concentration' of the silicon atoms: C(Si)  =  5.0  ×  10²² cm⁻³. The impurity atoms which substitute lattice places in the crystal have to be subtracted from the number of lattice places calculated by equation (6). The same is true for empty lattice places, i.e. for vacancies. This yields the correction N_corr for substitutional point defects:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{N}_{{\rm corr}}}={{V}_{{\rm core}}}\sum\limits_{X}{C\left(X \right)}\nonumber \end{align} \tag{ 13 }$$

where ${{V}_{{\rm core}}}=\left(\pi /{\rm 6} \right)D_{{\rm core}}^{3}$ is the volume of the sphere without surface layers and the sum has to be taken over all substitutional impurities and the vacancies. The point defects change the amount of silicon atoms usually by less than 50 nmol mol⁻¹ (compare table 3).

The number of silicon atoms in a silicon sphere can roughly be calculated by equation (5). Subtracting the substitutional impurities and adding the silicon atoms of the oxide layer yields:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle N\left({\rm Si} \right)=8\,{{{V}_{{\rm core}}}}/{{{a}^{3}}} - {{N}_{{\rm corr}}}+{{N}_{{\rm s}}}\left({\rm Si} \right)\nonumber \end{align} \tag{ 14 }$$

with N_corr from equation (13) and N_s(Si) from equation (12). Thus, the amount of silicon in the sphere reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle n\left({\rm Si} \right)=\frac{N\left({\rm Si} \right)}{{{N}_{{\rm A}}}}=\frac{8\,{{V}_{{\rm core}}}}{{{a}^{3}}{{N}_{{\rm A}}}}-\frac{{{V}_{{\rm core}}}\sum\nolimits_{X}{C\left(X \right)}}{{{N}_{{\rm A}}}}+\frac{m_{\rm d}\left({\rm Si} \right)A}{M\left({\rm Si} \right)}.\nonumber \end{align} \tag{ 15 }$$

Table 4 shows that the realization of the mole in the SI after the revision with the sphere Si28kg01a can reach a relative uncertainty of about 1  ×  10⁻⁸. Thus, the XRCD method yields the lowest relative uncertainty for the realization of the mole, although we still have to wait until this finds practical applications.

The mass fraction of silicon in the sample can be calculated by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle w\left({\rm Si} \right)=1-{\left[ {{V}_{{\rm core}}}\sum\limits_{X}{\left[ C\left(X \right){{m}_{{\rm A}}}\left(X \right) \right]}-{{m}_{{\rm SL}}}+m_{\rm d}\left({\rm Si} \right)A \right]}/{{{m}_{{\rm sphere}}}} \nonumber \end{align} \tag{ 16 }$$

where the sum has to be taken over all impurities X, m_a(X) is mass of the atom X and m_SL is the total mass of the surface layers. 1  −  w(Si) for the sphere Si28kg01a is about 5  ×  10⁻⁸ proving the high purity of the sphere.