The Avogadro Constant for the Definition and Realization of the Mole

Transcription

1 REVIEW The Mole The Avogadro Constant for the Definition and Realization of the Mole Bernd Güttler,* Olaf Rienitz, and Axel Pramann The International System of Units (SI) base unit of the quantity amount of substance is the mole (symbol: mol). After the revision of the SI to be implemented in 2019, when all SI units will be based solely on constants, the mole will be defined through a fixed value of the Avogadro constant N A.One mole contains exactly elementary entities, meaning the mole will no longer be linked to the kilogram. Currently, the mole is defined via the mass of exactly kg of the 12 C isotope which links it to the kilogram prototype. The history, changes, and implications of the revised definition of the mole are discussed here from the chemist s point of view. The ability to count entities such as atoms or molecules (precisely enough to enable a revision of the SI and preserve consistency of previous and future measurements) is crucial. This is achieved with the realization (Mise en Pratique) based on the X-ray-crystal density (XRCD) method (counting the atoms in a silicon sphere). The determination of N A, focusing on the measurement of the molar mass of silicon highly enriched in the 28 Si isotope, with the lowest uncertainty so far, is presented. 1. Introduction The scientific community is close to a paradigm change: the planned new definition of the International System of Units (SI) is solely based on exact and fixed fundamental constants without the need of keeping any artifacts. After the 24th meeting of the General Conference on Weights and Measures (CGPM) the need of a revised definition of the SI based on fixed fundamental constants was foreseeable and recommended by the 106th International Committee for Weights and Measures (CIPM) in [1,2] The revision is expected to enter into force as from May 20, In the present paper, we will focus on the SI unit mole (symbol: mol). With this background, a variety of (review) articles dealing with this subject have been published. However, the authors of this review do not intend to compile a complete list of the respective available literature and refer to the extensive references already cited in other review articles. [3 7] In 1971, the SI base unit mole had been introduced and defined by the CGPM as follows: [8] Dr.B.Güttler, Dr. O. Rienitz, Dr. A. Pramann Physikalisch-Technische Bundesanstalt Bundesallee 100, Braunschweig, Germany bernd.guettler@ptb.de The ORCID identification number(s) for the author(s) of this article can be found under DOI: /andp The mole is the amount of substance of a system that contains as many elementary entities as there are atoms in kilogram of carbon 12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, or other particles, or specified groups of such particles. This definition clearly demonstrates the relation of the mass of an entity and its amount of substance. The mole is one out of a set of seven base units of the SI. Although the mole and the kilogram are defined as independent base units, they are closely related to each other. In the 1971 definition, one mole of 12 C has a mass of exactly 12 g. This in turn means that the instrument to determine the mass of an object a balance is also used to determine the amount of substance. Every chemist can use a balance in a laboratory measuring the mass of a substance, dividing it by its respective molar mass M (unit g mol 1 ), finally yielding the amount of substance n expressed in mol. Here, the amount of substance is needed for applying the principles of quantitative chemistry and stoichiometry. If it is known that a certain amount of substance of an educt will react to a certain amount of substance of a product, the chemist can calculate the respective masses and can use balances or if the densities are known volumetric methods, to prepare the correct amounts for that reaction. This in turn means that the amount of substance differs from the mass in particular by the fact that identical amount of substances of different materials contain the same number of atoms, molecules, or other particles, but usually do not have the same mass. The revised definition of the mole is as follows: [9] The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 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. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles. In other words, one mole of a substance contains exactly specified entities. The above definition of the mole includes already a fixed value of the Avogadro constant (1 of 17)

2 BerndGüttler studied mineralogy at Leibniz University Hannover and received his Ph.D. in After a fellowship from 1987 to 1989 at the University of Cambridge, UK, he joined PTB in Starting as a senior researcher in solid-state chemistry, he became first head of the department Metrology in Chemistry in 2002 and in 2015 head of the division Chemical Physics and Explosion Protection. Since 2002, he has represented Germany at the CCQM, where he is head of the Ad-Hoc Working Group on the mole. In this capacity he paved the way for the revised definition of the SI unit mole. Figure 1. Photograph of silicon spheres used for the realization of the mole. Left: a mole -sphere with mass matched to the same number of atoms as one mole of silicon. Right: an Avogadro -sphere highly enriched in 28 Si with mass of 1 kg used for the determination of N A via the XRCD method. N A = mol 1, which has been released using a special least square adjustment (LSA) by the Task Group on Fundamental Constants (TGFC) of the Committee on Data for Science and Technology (CODATA) in 2017 after the deadline of June 31, 2017 for the input quantities for the revised definition. [10] N A will be fixed and no longer an uncertainty will be associated to it. One mole can be expressed as: ( ) 1mol= N A The proposed new definition will not affect or change relative atomic masses, but the atomic and molar mass constants will be provided with extremely small uncertainties (see Section 6.1), thus in practise almost no or a very small impact on the uncertainties of related measurements will be the consequence. For this purpose, the Avogadro constant N A used now since more than a century had to be redetermined with a hitherto unmatched accuracy. This was achieved by the X-ray crystal density method (XRCD), also known as Avogadro experiment or silicon route, initially started with the aim to redefine the kilogram via N A by counting the atoms in a single-crystalline silicon sphere with a mass of 1 kg(seefigure 1). Comprehensive reviews on N A with its historical development have been given by Becker and Cerruti. [11,12] The main principles of the XRCD method with special focus on the molar mass determination of enriched silicon are discussed in Section 6.1. This way also the link of the mole to the definition of the kilogram will be broken, underpinning the importance of the mole as a base unit in its own right. 2. Some Historical Remarks The quantitative treatment of chemical reactions dates back to 1792 when the chemist Jeremias Benjamin Richter introduced the principles of stoichiometry. [13] (1) Olaf Rienitz studied chemical engineering at the Technical University Berlin and received his Ph.D. in 2001 (analytical chemistry). He joined PTB in 1998 and works in the Inorganic Analysis Group. Since 2013, he has been head of this group. His current research topics cover inorganic clinical chemistry, highly accurate monoelemental standards, and isotope ratio measurements for the redefinition of the SI units mole and kilogram. Mass spectrometric methods are his main tools of trade. He is especially interested in the development and improvement of the necessary mathematical models. Axel Pramann studied chemistry at Technical University Braunschweig and received his Ph.D. at Humboldt- University Berlin in 2000 (physical chemistry). After postdoctoral fellowships ( Keio University, Japan; University of Osnabrück, Physics) he joined PTB in Since 2008, Pramann has worked in the department Metrology in Chemistry in the Inorganic Analysis Group as a senior researcher. His current main research topics include the redefinition of the SI units mole and kilogram ( Avogadro-Project ) with special focus on mass spectrometric methods. The interpretation of chemical reactions or processes as connections between atoms and molecules induced the concepts today known as mole and Avogadro constant with origins dating back to the eighteenth and early nineteenth century. In 1808, the British chemist John Dalton ( ) declared that atoms of an element do not differ from one another and that they must have a defined atomic mass and volume. Atoms cannot be further separated and in a chemical reaction the atoms of the educts will be rearranged in the products in certain relations (law of multiple proportions). Almost at the same time in (2 of 17)

3 Figure 2. Development of N A and its associated measurement uncertainty reported by various authors and institutions over the last one-and-a-half centuries. [11] Especially remarkable is the reduction of the relative uncertainty from 100% down to The dashed blue line shows the final value of N A, achieved by the CODATA LSA in 2017 used for the revised definition of the mole. [10] 1811, the Italian scientist Amedeo Avogadro ( ) postulated that same volumes of gases contain at the same temperature and the same pressure the same number of molecules. [12] This observation was almost forgotten until Stanislao Cannizarro ( ) published, in 1858, a consistent system of chemical formulas and atomic weights (relative atomic masses) of all known elements. [11] The concept of the mole can be mainly assigned to Wilhelm Ostwald ( ). [14] In his Manual and Auxiliary Book for the Performance of Physiochemical Measurements of 1893, he wrote: Let us generally refer to the weight in grams which is numerically identical to the molecular weight of a specified substance as one mole,.... [15] Using this definition, the unit mole was more closely related to a mass than to a number of entities and thus for a long time it was interpreted as the chemical mass unit. Although the atomic theory at that time had been linking the mole with a particle number and, therefore, would have required the introduction of a quantity like the amount of substance since Dalton s and Avogadro s pioneering works experimental results, which could confirm these models, were missing at that time. In the beginning of the twentieth century, similar concepts describing an amount of substance as an amount or mass have been used in parallel. One initial precursor of the mole, the grammolecule, the g-molecle or the g-mole with a comparable meaning were, however, at the same time also used in the scientific community. [16] Thus, the gram-molecule has been chosen in two ways: as a quantity and a unit. Also Einstein implicated the term gram-molecule in the context of his work in a way of an amount concerning the Avogadro constant in [17] The gram-molecule has also been used by Perrin in 1909 when examining the principles of Brownian motion and derived one of the first values of what is known today as the Avogadro number. In Perrin s work, the gram-molecule is used as a mass of the investigated substance. With the Avogadro number, it is thus possible to handle and to proof microscopic systems by transferring them to an artificial assembly. [18] After the experimental confirmation of the atomic theory and the determination of the Avogadro constant (particle number per mole), two different perceptions of the mole were formed. These were treated exactly by Stille ( ) for the first time using two concepts. [19] In the first concept, the mole was treated as a chemical mass unit associating the relative atomic masses A r (atomic weights) with the unit gram (1 mol = A r g). On the other hand, the mole is treated as number of moles, symbol l (in German Molzahl ), a particle number which is defined by in principle the Avogadro constant. Here, l is dimensionless. The concept amount of substance was derived from the German concept Stoffmenge, according to Stille and was later promoted by Guggenheim. [20] The unit mole was implemented into the SI system of units in 1971 at the 14th General Conference of the Metre Convention and this contradiction has been solved finally. Thus, a differentiation of the two concepts has become superfluous when the amount of substance" (symbol n) as the seventh base quantity and the mole as the respective unit have been officially established. As a consequence, the Avogadro constant N A had to be introduced. Figure 2 depicts the development of N A and the achieved improvements in terms of the associated measurement uncertainty over the last one-and-a-half centuries. The Avogadro constant connects microscopic with macroscopic scales as well as statistical mechanics with the principles of thermodynamics. First, there have been corresponding requests and recommendations of the International Union of Pure and Applied Physics (IUPAP), the International Union of Pure and Applied (3 of 17)

4 Chemistry (IUPAC), and the International Organization for Standardization (ISO), together with the note that the carbon isotope 12 C had to be selected as the reference and the mole adopted as an SI unit. [21] The isotope 12 C has been finally used in the 1971 definition of the SI unit mole on a scale based on the relative atomic mass A r ( 12 C) = 12. This was originating from a successive development starting with the invention of mass spectrometry and the discovery of isotopes in the beginning of the twentieth century. After the initial use of 16 O, experimental advances in spectroscopy led to a preferred use of 12 C by the end of the 1960s. [3,4] To further handle scientific and practical questions related to the base quantity amount of substance a committee of the Metre Convention (Comité Consultatif pour la Quantité de Matière CCQM) was founded in [22] 3. Relations of Chemical Quantities within the New SI Several chemical quantities are directly or indirectly related to the quantity amount of substance n, which is of utmost importance in the daily life of (analytical) chemists. A detailed and official description can be found in the latest draft of the brochure for the New SI. [23,24] A (pure) chemical substance X as an entity is best characterized by quantities which are related to an amount and to a mass, thus one is able to prepare samples gravimetrically using a balance according to the laws of stoichiometry. The most common quantities are: the particle number N or number of elementary entities of the substance X, the amount of substance n of the substance X, the mass m corresponding to the particle number N, the relative atomic or molecular mass A r (X) (also known as atomic weight ; dependent on the nature of X: for example, an element, molecule/compound), the molar mass constant M u, the atomic mass constant m u, the molar mass M(X), and the Avogadro constant N A. The respective relations which remain valid in the revised SI are (Da: Dalton): [25] n (X) = N(X) N A (2) n (X) = with m(x) = m(x) A r (X)M u M(X) M (X) = A r (X)M u (4) m a (X) = A r (X)m u (5) N A = M u m u (6) M u N A = m u = 1Da= 1u (7) In the special case of treating a single atom, with the atomic mass m a, Equation (4) can be written as M (X) = N A m a (X) (8) (3) Equation (8) can be best illustrated with the current definition of the SI unit mole via the mass of 12 g of 12 C. M ( 12 C ) = N A m a ( 12 C) (9) This equation immediately points out that a fixed N A with an associated uncertainty of zero requires uncertainties of the molar mass M(X) of a compound (in this case M( 12 C)). The relative atomic and molecular masses A r (X) will be kept unchanged in the new SI. In addition M u as well will in future have an uncertainty of u rel (M u ) < 10 9 (see Equation (13)). From Equations (4) and (5), it follows that in the new SI M u and m u have the same relative uncertainty, currently u rel (M u ) = u rel (m u ) = [5] Due to the comparably small value of this uncertainty, most uncertainties of processes in chemical operations are not influenced by this, illustrating that the impact of the revision of the SI can be almost neglected in most practical and scientific work. The evaluation of the amount of substance in practice is done via n (X) = N(X) N A = m(x) A r (X)N A m u = w(x)m = m(x) A r (X)M u M(X) (10) Here, the mass of the substance m(x) is determined by the total mass m of the sample and the mass fraction w(x)ofthesubstance in the sample. The same relation can be obtained for the number of entities N (which however is not commonly used in practice). N (X) = w(x)m m a (X) = m(x) A r (X)m u (11) A r (X) can be derived from the respective chemical formulae or tables containing the relative atomic masses A r. The tabulated atomic masses of the elements and isotopes are regularly updated in the AME or IUPAC tables. [26,27] Main uncertainty contributions of Equation (10) result from the mass determination of the sample, whereas the relative atomic masses are usually associated with relative uncertainties ranging from 10 4 to Before May 20, 2019, the date for the revised Si coming into force, the molar mass M(X) is equal to the relative atomic mass A r (X) with the unit g mol 1 resulting in M (X) = A r (X) g mol 1 (12) This implies that M u is exactly 1 g mol 1 in the current SI. Taylor gave a concise overview on the calculation of the molar mass and the respective changes after the introduction of the revised definition. [28] In that letter, it is shown, how to avoid a molar mass factor (1 + κ) where the numerical value of κ would have been zero with an associated uncertainty. For the sake of simplifying the SI and keeping the system of quantities transparent, the use of that factor has been abandoned. According to the relation M u = 2R N A h α 2 ca r (e) (13) (4 of 17)

5 with the Rydberg constant R, the fixed Avogadro constant N A, the fixed Planck constant h, the fine structure constant α, the speed of light in vacuum c, and the relative atomic mass of the electron A r (e), the main contribution to the uncertainty of M u is α. [5] Taylor pointed out that in the new SI M u will have an associated uncertainty. The value of M u will be mainly influenced from the new adjusted values of R, α, anda r (e) by approximately <10 9, which is too small to effect any usual chemical application. M u will therefore remain 1 g mol 1 for practical purposes. 4. The Mole as an SI Unit in Chemistry and Initial Concerns regarding a Revised Definition Using the principle of the mole and amount of substance, it is possible to trace back the results of measurements in chemistry to the SI units, a prerequisite for international comparability, shown in numerous pilot studies and key comparisons performed within the scope of the CCQM. To establish a traceability chain using the mole, an increasing number of national standards in chemistry were developed based on the mole and amount of substance. Chemical reactions usually are based on extremely large numbers of atoms and molecules. The unit mole therefore combines for both practical and historical reasons so many particles, that reference to other units (e.g., the kilogram) can be made using considerably small numbers. In practice, the traceability of the amount of substance n(x) of a substance X is often realized via its mass m(x) and the respective molar mass M(X) (see Equations (3) and (10)). Prior to the current revision of the SI, the molar mass M(E) of an element E was calculated from its average relative atomic mass (atomic weight) A r and the molar mass constant M u = 1g mol 1. The relative atomic masses use 12 C as the reference point: by definition: A r ( 12 C) = 12 and M( 12 C) = 12 g mol 1. The average value of A r of an element was calculated from the A r ( i E) of its isotopes i E and their respective amount of substance fractions x( i E). [27] M(E) = A r (E) M u (14) A r (E) = i [ x( i E) A r ( i E )] (15) In the previous SI, traceability to the mole ultimately required a reference to the mass. Once N A is fixed in the revised SI, traceability of an amount of substance to the mole may also be achieved by other means such as quantifying (elementary) entities based on other properties such as their charge or their optical or magnetic properties. These properties may be related to detectable transitions between energy levels (nuclear or electronic), specific for a particular molecular entity and also correlating with the number of particles. Concepts for measurements of the number of elementary entities have been available for a rather long time, although being limited to special cases. Particle measurements can, for example, be performed by determining the amount of substance of a crystalline solid with the aid of its microscopic, crystallographic lattice parameter, and its macroscopic volume. This is in principle what is done within the scope of the XRCD method, counting the silicon atoms in a high-purity 28 Si single crystal (see Section 6). Similar experiments although being much more simple have been carried out since the discovery of X-ray diffraction. Also experiments are known for the (direct or indirect) counting of elementary entities, for example, electrons (singleelectron-tunneling SET circuits) which is relevant for a revised quantum-metrological definition of the SI base unit ampere. [29 33] Prior to a fundamental change of a system as in the case of the planned revision of the SI, there are of course several pros and cons, at least about the correct time of such a critical endeavor. One central criticism is that the fundamental constants used may be interrelated. Moreover, the constancy in time may not be guaranteed and cannot be tested. [5] An often expressed doubt is the possibility of a fixed fundamental constant with a slightly wrong value and its impact. Actually, the acting organizations of the meter convention prepared recommendations in order to overcome respective problems. [5] In any case, unlike the other base units, the mole has to be applied, thus, in a twofold manner. The first is related to a number. The second requires the identification of the elementary entities and establishes the connection to chemistry. The complete description of the amount of substance of a material in the unit mole requires both the identification and quantification of the elementary entities (analyte). An amount of substance n = 1 mol always contains the same number of entities. This number is identical with the fixed numerical value of N A. 5. The Avogadro Constant and the Redefined SI Unit Mole The SI unit mole will be redefined using a fixed numerical value of the Avogadro constant N A one of the seven fundamental constants, each essential, fixed and a base for the realization of the SI units (Figure 3). When referring to the revised definition of the mole, it is mandatory to mention in the same breath the background of the revised definition of the kilogram. This is all the more understandable since before, the amount of substance was directly linked to the mass of 12 g carbon. A strong driving force initiating enormous scientific efforts enabling finally the revised definition of the SI is the fact that the SI base unit of the mass the kilogram was still defined by an artifact the international kilogram prototype (short IPK), which is stored at the BIPM the Bureau International des Poids et Mesures near Paris. Prior to the revision of the SI, the mass of the IPK is exactly one kilogram per definition. It is a platinumiridium cylinder which has been manufactured in 1889 together with six official copies. These and other prototypes located in national metrology institutes (NMIs) around the world have been compared with respect to their masses roughly all 40 years in the past. It is a well-known fact, that during the last 100 years the masses of the copies and other prototypes were biased to the mass of the IPK: in total by 50 μg per 100 years ( relatively). This was one practical motivation to exchange the kind of the definition of the kilogram by independent experiments using fundamental physical constants. The probably main motivation for a revision of the SI was the concern of the metrological community to standardize the definitions of the SI units (5 of 17)

6 As described above, the SI unit mole will be defined via the Avogadro constant N A. N A,andh are related via Equation (13). After rearrangement follows N A = cα2 A r (e) 2R M u h (16) Before the revision of the SI, Equation (16) was a kind of consistency check for h or N A. Both constants can be derived from each other via the molar Planck constant hn A and prior to the revision hn A was associated with a relative uncertainty in the lower range. 6. The XRCD method: Primary Realization of the SI Unit Mole Figure 3. SI logo: Schematic of the seven SI base units and their relation to the fixed fundamental constants (by courtesy of BIPM, using stable and most likely invariant physical constants which are theoretically accessible to the whole metrological community: fundamental constants (see Figure 3). [34 37] Basically, definitions of this kind enable a unit to be realized at any location and at any given time. Therefore, for more than 30 years the attempt has been made to define the kilogram via an atomic constant or a fundamental constant of physics. Two experimental most advanced routes dominated the search for a revised definition of the kilogram: The watt-balance (today referred to as Kibble-balance) experiment using a moving-coil for the determination of the Planck constant h, and the XRCD method (also referred to as the silicon route or Avogadro-experiment) counting the atoms in a silicon crystal yielding the Avogadro constant N A. With both experiments, it is possible to realize the kilogram with a relative uncertainty in the range of at least the lower 10 8 level, which was one of the prerequisites for a revised definition. [38 45] Finally, it has been agreed to use the fixed Planck constant h for the revision of the definition of the kilogram in the new SI and a fixed N A for the revision of the definition of the mole. A possible motivation of the decision to redefine the kilogram using h and the mole using N A is the perception that nowadays extremely precise electrical measurements can be conducted to generate h. [5] This in turn enables the use of N A as an additional free variable to define the mole. At the same moment also other constants related to N A like the universal gas constant R, and the Faraday constant F will be fixed. The SI unit of the mass the kilogram will be defined as follows [46] : The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be when expressed in the unit J s, which is equal to kg m 2 s 1, where the metre and the second are defined in terms of c and ν Cs. Parallel to the first report of the XRCD method by Deslattes et al. in 1974, PTB started experiments to determine the lattice parameter of silicon with a natural isotopic composition by applying X-ray-interferometry. [42,43,47] These efforts led to the so-called Avogadro-Project by setting up experiments to count the atoms in silicon with natural isotopic composition. [48] In the XRCD method, the lattice parameter of the single crystalline silicon is measured as well as the volume of the silicon sphere used with amassof 1 kg, the molar mass of the silicon, the density and mass of the sphere, its surface properties, and impurities of the crystal (chemical impurities and vacancies). A polishing technique for silicon spheres with a mass of 1 kg has been developed and the respective experiments for the determination of N A with further reduced uncertainty have been improved. [49,50] In 2003, N A has been determined with an associated uncertainty of u rel (N A ) = which is about one order of magnitude too large to be applicable for a revised definition of the kilogram and mole. At that time, the quantity which limits a further reduction of the uncertainty and serves as a kind of bottleneck of that project was the molar mass M or isotopic composition of the natural silicon material with u rel (M) = [51] The contribution of u rel (M) to the uncertainty associated with N A averaged out at around 60% compared to the other quantities. To overcome this problem, the idea came up to produce silicon crystals with an amount of substance fraction of the 28 Si isotope, x( 28 Si) almost equal to 100% by isotope enrichment. In the following years, the International Avogadro Coordination (IAC) was established, aiming at the reduction of the uncertainty associated with N A by the production of a single silicon crystal highly enriched in 28 Si. This crystal should be of sufficient size and mass to produce spheres with a mass of 1 kg, appropriate to be used for the XRCD experiment. [52] In 2008, the first measurements started with a new silicon crystal called AVO28 (exact notation: Si28-10Pr11, Figure 4) highly enriched in 28 Si with x( 28 Si) > mol mol 1. At the same time, a novel method for the determination of the molar mass of this material with reduced uncertainty has been developed by Rienitz et al. [53,54] This paved the way for the IAC to finally determine N A with u rel (N A ) < , a prerequisite for the planned revision of the SI. At present, the uncertainty associated with the molar mass of enriched silicon contributes to less than 6% to the overall uncertainty of N A. Table 1 gives an overview of the latest values (6 of 17)

7 In the XRCD method, the number N of silicon atoms in the macroscopic sphere and the respective amount of substance n is given by N = 8V a 3 = nn A (17) with V denoting the volume of the macroscopic sphere, 8 as the number of atoms in the unit cell of silicon with the edge length a, also known as the lattice parameter. Expressing N using Equations (2) and (3) the Avogadro constant N A is given by the following equation N A = 8V a 3 M m = 8 M a 3 ρ = N n (18) Figure 4. First single crystal ( AVO28, Si28-10Pr11) consisting of silicon highly enriched in 28 Si with a mass of 5 kg (by courtesy of Leibniz- Institute for Crystal Growth (IKZ), Berlin, Germany). Table 1. Published values of N A by the IAC. Ref. Year N A (10 23 mol) u rel (N A ) (10 8 ), k = 1 [55] (18) 3.0 [56] (12) 2.0 [57] (70) 1.2 [58] (15) 2.4 Prior to the deadline of the CODATA LSA in 2017, the smallest relative uncertainty so far associated with N A is only , one precondition for the revision of the SI. of u rel (N A ) obtained by the IAC with various enriched silicon crystals. A frequently asked question is why silicon is the material used for counting atoms in the XRCD method. Since large crystals with a mass of up to 6 kg are needed to cut at least two spheres from one crystal, well-established and reliable crystal growth procedures had to be available. This is the case for silicon due to the still increasing need and advanced research in the semiconductor area. Silicon single crystals can be produced with highest chemical purity and are long-term stable evolving a constant thin oxide layer. Moreover, the knowledge of crystal parameters as well as the physical and chemical parameters of silicon compared to other elements is highly developed for a long time. [59] Meanwhile, also the handling of the silicon spheres for practical use as, for example, a primary standard for dissemination purposes has been further developed. In Equation (18) the main quantities to be determined in the XRCD method are summarized: the volume V of the macroscopic sphere (determined via an optical sphere interferometer), the lattice parameter a of the unit cell (determined with an X-ray interferometer), the molar mass M (determined by mass spectrometry), the mass m of the macroscopic silicon sphere (measured with a balance), or instead of m and V, the density ρ of the sphere. Additional experiments like surface characterizations and the measurement of chemical impurities, as well as the determination of the oxide layer are used to correct the respective quantities of Equation (18). This in turn means that the mass of a macroscopic single crystalline sphere is related to N A via Equation (18). In the revised SI, the amount of substance n and its unit mole is realized via Equation (18) expressing a fixed N A. The exact and primary realization of the mole using a fixed N A and a fixed h can be deduced as follows: N A and h are related by Equation (16). [34] Using the rearranged Equation (18) m = 8V a 3 M N A (18a) and the basic definition of the molar mass M (in this case the molar mass of silicon, M = M(Si)) M = i [ x( i Si)M( i Si) ] = i [ x( i Si)A r ( i Si)M u ] (19) with the amount of substance fractions x( i Si) ( i Si = 28 Si, 29 Si, and 30 Si) and the respective molar masses M( i Si) which are accessible via the products of the relative atomic masses A r ( i Si) (often referred to as atomic weights) and the molar mass constant M u, Equation (18a) can be expressed as m = 8V i x(i Si)A r ( i Si)M u (20) a 3 N A Inserting Equation (16) into Equation (20), the equation for the primary realization of the mass is obtained m = 8V 2R h i x(i Si)A r ( i Si) a 3 cα 2 A r (e) (21) (7 of 17)

8 With this equation, the mass can be realized and disseminated using a fixed h and, for example, silicon spheres (see Section 7) as primary standards of which the quantities of the right hand side of Equation (21) have to be measured beforehand via the XRCD method. With the basic relation given by Equation (3) for the amount of substance n follows m = 8V 2R h M (22) a 3 cα 2 A r (e)m u n = 8V 2R h 1 = 8V 1 (23) a 3 cα 2 A r (e)m u a 3 N A Equation (23) gives the definition of the amount of substance n via the Avogadro constant accessible with the XRCD method using, for example, a sphere of single crystalline silicon in the new SI. The validity and applicability of a fixed h for the revision of the kilogram and vice versa using a fixed N A for the revision of the mole as well as their relation expressed by Equations (21) (23) are based on several very general specifications: Current physical relations between relevant quantities must still be valid after the revision (especially the relation N A m a ( 12 C) = M( 12 C) with the atomic mass m a ( 12 C) of 12 C and the respective molar mass). It must be further guaranteed that the set of fixed fundamental constants defining the SI unit is as simple as possible. The validity of Equation (16) requires that M u cannot be fixed when fixing h and N A : obviously, an uncertainty M u will be associated with M u in the new SI. Note, that Equation (23) is a proof of the initial requirement expressing a particle number N (equal to the term 8 V/a 3 )by counting atoms (compare Equation (17)). The second factor of Equations (22) and (23) is equal to the mass of an electron m e to give a comprehensible idea of this term. Thus, Equation (23) serves as a pendant to Equation (22) which can be used in practice for the determination of a mass m using a silicon sphere. The quantities needed to perform the XRCD method (given in Equation (18)), must be measured traceable to the SI units. An additional requirement is that each quantity should be measured by different institutes in order to cross-check the validity and the accuracy of the numerical value. The associated uncertainties must also be checked independently. Currently, all quantities necessary for the realization of the mole, except the determination of the lattice parameter, via the most accurate method the XRCD method are available at PTB and at other NMIs, in part members of the IAC. Detailed descriptions of the respective experiments used to measure the quantities necessary to realize the mole (and the kilogram) using a fixed Avogadro constant N A and Planck constant h via the XRCD method are given in numerous publications and a special issue of Metrologia. [48,55 57,60] If a new crystal is used, all quantities of Equation (18) have to be measured, whereas using an already characterized silicon sphere according to Equations (22) and (23), mainly the surface conditions, the mass, and the volume of the sphere have to be determined again. Briefly, the volume V of the silicon sphere is determined by interferometric measurements of the sphere diameter providing a complete surface topography of the sphere. The deviation from an almost perfect spherical shape ( roundness ) is only a few 10 nm, necessary to achieve the required uncertainty associated with V. One of the main advantages of the XRCD method the use of a stable device: a silicon sphere to realize first a constant and later on disseminate a unit without the need of a multitude of sophisticated and delicate experiments, is simply based on the fact that the silicon spheres are covered by a stable and almost constant oxide layer. The mass and geometry of this layer must be determined regularly to correct for the main quantities given in Equation (18). Additional layers on top of the oxide coating must be carefully characterized, for example, adsorbed layers of water and hydrocarbons. The mass of the silicon sphere must be determined by weighing under vacuum conditions because of the large density differences between silicon and, for example, platinum-iridium, in order to render an air buoyancy correction obsolete. The volume and mass determinations are corrected with respect to the coating layers. The lattice parameter a, basically the edge length of the unit cell of the silicon crystal is determined from the lattice distance of the crystal using an X-ray interferometer. Chemical impurities as well as vacancies (crystal defects in general) in the crystal lattice must be determined accurately, currently using infrared (IR) spectroscopy and instrumental neutron activation analysis (INAA). [61,62] Recently, at PTB complementary experimental methods applicable to silicon for the purity characterization like glow discharge mass spectrometry (GD MS) and laser ablation (LA) ICP mass spectrometry are under development to complete and improve the lack of the ability of determining as many impurities as possible using primary methods in the near future. The molar mass M of silicon is determined by mass spectrometry. As long as natural silicon was in the focus of the XRCD method, M has been determined by gas phase isotope ratio mass spectrometry (IRMS) at the former Institute for Reference Materials and Measurements (EC-JRC-IRMM) today renamed as Joint Research Centre Geel (JRC Geel), Belgium. [63] The determination of the isotopic composition of silicon via IRMS requires a stable and predominant sample gas in this case silicon tetrafluoride (SiF 4 ). The initial solid silicon crystals had to be transferred into gaseous SiF 4 via several delicate chemical steps. The solid silicon was preconditioned using sodium hydroxide. The resulting silicate was transformed into a stable oxide (SiO 2 ) which was converted with aqueous HF into hexafluorosilicate H 2 SiF 6. Using an aqueous solution of an alkaline earth metal, for example, BaCl 2, a precipitate of BaSiF 4 has been obtained. After careful pyrolysis of the latter, SiF 4 was formed. It is evident, that this method although an indispensable pioneering work at that time is by no means suitable in the case of silicon highly enriched in the 28 Si isotope due to the risk of contamination with natural silicon and other chemical impurities during the preparation steps. As stated above, in 2008 PTB started successfully the determination of M of enriched silicon using a novel modified isotope dilution mass spectrometry (IDMS) method in combination with a novel closed-form method for the absolute determination of calibration (K) factors needed to correct for the bias of measured isotope ratios. [53,54,64] An introduction to the currently applied mass spectrometric method used to determine M with lowest associated uncertainty is given in Section 6.1 As pointed out, the first silicon crystal ( AVO28, Si28-10Pr11) and the respective two spheres (AVO28 S5 and AVO28 S8) have been characterized (8 of 17)

9 with a mass of 6 kg. This polycrystal was converted to a chemically extremely pure single crystal (>5 kg) at the Leibniz-Institute of Crystal Growth (IKZ) in Berlin, Germany, using a float zone technique. The spheres as well as additional parts were manufactured at PTB ready for use. Preliminary simulations have demonstrated that an increase of x( 28 Si) coincides with a decrease of u rel (M). Measurements of M of the first new crystal materials from Russia (with x( 28 Si) ranging from to mol mol 1 ) showed the predicted reduction of u rel (M) with increasing enrichment, giving u rel (M) 10 9 as a routine result. Due to the promising results of the kg-2 project, a follow-up project called kilogram-3 started in 2015 aiming at the production of another three ingots of single crystalline silicon highly enriched in 28 Si to produce finally six additional spheres. In total, after the completion of the kilogram- 3 -project, 12 spheres highly enriched in 28 Si will be in circulation to apply the XRCD method with lowest associated uncertainty. Figure Si-single-crystal production steps. intensively in the forefront of the revision of the SI units: In 2011 and 2015, u rel (N A ) = and , respectively, have been obtained. [55,56] However, one precondition for the revised SI was a relative uncertainty of < Simulations have shown that an appropriate reduction of u rel (M) and therefore of u rel (N A ) is possible with a further increased enrichment of silicon with x( 28 Si) > mol mol 1. The availability of a pool of additional silicon spheres highly enriched in 28 Si will guarantee a cross-check of the realization of N A by a number of physically different spheres, which can be used for key comparisons, lent to other NMIs or even sold to interested user facilities. This is also necessary to further establish and maintain the basic idea of the XRCD method via enriched silicon as one of the two complementary state-of-the art methods to realize the mole and the kilogram. In 2012, a project called kilogram-2 ( kg-2 ) was initiated by PTB with the intention to produce two additional silicon single crystal ingots, both even higher enriched in 28 Si as in the previous AVO28 crystal in order to get four more silicon spheres for the realization of N A and the future dissemination of the mass and amount of substance. The detailed description of the project and the technical and experimental areas as well as the improvements in chemical analysis related with the production of the new crystals are described elsewhere (Figure 5). [65,66] Briefly, several 10 kg of silicon with natural isotopic composition (x( 28 Si) = mol mol 1, x( 29 Si) = mol mol 1, x( 30 Si) = mol mol 1 ) were shipped from Germany to an isotope enrichment facility, the company JSC PA Electrochemical Plant (ECP) located in Zelenogorsk, Russia. At ECP, the silicon was converted with purified fluorine gas to silicon tetrafluoride gas (SiF 4 ). Using numerous newly developed cascades of gas centrifuges, the SiF 4 was finally highly enriched in 28 Si which was subsequently converted to extremely pure silane (SiH 4 )atthe Institute of Chemistry of High-Purity Substances (IChHPS RAS) of the Russian Academy of Sciences in Nizhny Novgorod, Russia. Here, the enriched SiH 4 was chemically deposited on a 28 Si single crystalline precursor rod (slim rod) using chemical vapor deposition (CVD) in order to produce a polycrystalline silicon crystal 6.1. Principle of the Molar Mass Determination via the VE-IDMS Method As mentioned, in 2008 a new ultrapure silicon crystal ( AVO28 ) highly enriched in 28 Si was available to reduce the measurement uncertainty associated with N A by applying the XRCD method. The enrichment of the 28 Si isotope was almost 100%: x( 28 Si) = (12). [67] However, also the composition of the remaining two isotopes 29 Si and 30 Si had to be measured to obtain a value of the molar mass M with an associated uncertainty of u rel (M) This was the intended upper limit of u rel (M), because one of the prerequisites for a revised definition of the kilogram and the mole was an uncertainty u rel (N A ) , implying that all relevant quantities necessary to determine N A via the XRCD method exhibit associated uncertainties of the same or even smaller size. Although it was well known that the application of isotope ratio mass spectrometry would be the best technique to determine the molar mass, for example, by the determination of the amount of substance fractions, the extreme enrichment in 28 Si provoked several technical and methodological problems. The amount of substance fractions of the two minor isotopes in the first enriched silicon crystal were x( 29 Si) = 4.136(11) 10 5 mol mol 1 and x( 30 Si) = 1.121(14) 10 6 mol mol 1. [67] When measuring the respective isotope ratios related to the isotope with the highest abundance 28 Si, extremely small isotope ratios far from unity will result: R = x( 29 Si)/x( 28 Si) = mol mol 1 and R = x( 30 Si)/x( 28 Si) = mol mol 1. The uncertainties associated with these isotope ratios would be far too large to ensure the required uncertainty of M smaller than Moreover, it was not possible to measure isotope ratios within a range of six or more orders of magnitude with the required small uncertainty using the available state-of-the art mass spectrometers. The problem was solved by the idea of transferring the principle of isotope dilution mass spectrometry (IDMS), a common primary method in analytical chemistry to determine the content of elements in a matrix, to the silicon problem. [68 72] At PTB the so-called Virtual-Element-IDMS -principle (VE- IDMS) has been developed. [53] In this principle, the enriched silicon was treated as containing an impurity the so-called (9 of 17)

10 VE that consists theoretically of only the isotopes 29 Si and 30 Si in the matrix of the 28 Si, apparent in extreme excess. The sum of the mass fractions w of 29 Si and 30 Si can be considered as a sort of impurity in the nearly pure 28 Si crystal. In the VE-IDMS approach, a blend (bx) has to be prepared, consisting of the sample material x (the enriched silicon) and a spike y (silicon highly enriched in 30 Si). One advantage of this method is the fact that the knowledge of the exact isotopic composition of the spike material y prior to the gravimetric preparation of the blend bx is not necessary and an approximate assumption is sufficient. In order to avoid the contamination of the blend solution with natural silicon, the respective components (x, y, and blank solution) were transferred to the blend bottle by simply pouring a precalculated respective amount instead of using a pipette. Thus, only the exact transferred mass of the respective component must be known. The isotope ratio measurement is divided into two sequences: In a first sequence (K-factor sequence), the isotope ratios of the materials w (natural silicon) and y (spike) are measured and the respective true ratios R j,2 = x j ( 30 Si)/x j ( 29 Si) and R j,3 = x j ( 28 Si)/x j ( 29 Si) are determined. Once they are known, in a second sequence (the VE-IDMS sequence, which can be performed separate in time) only R j,2 = x j ( 30 Si)/x j ( 29 Si) of the sample x, the blend bx, and the material w are measured. This provides shorter measurements and saving of sample material. In the blend bx, the ratio R j,2 = x j ( 30 Si)/x j ( 29 Si) could be adjusted around unity, enabling smallest associated uncertainties. The respective values of R = x( 30 Si)/x( 29 Si) then fell in the range (x) and 1 8(bx) and could be determined with uncertainties (k = 1) smaller than 1%, relatively, sufficiently precise to obtain a respective molar mass fulfilling the required small uncertainty. Generally, the molar mass M and the respective x( 28 Si), x( 29 Si), and x( 30 Si) are connected by the relation 30 [ ( M = x i Si ) M (i Si )] (24) i=28 with the respective molar masses of the three silicon isotopes M( i Si). The amount of substance fractions x( i Si) are accessible via the respective isotope ratios of the ith silicon isotope. x (i Si ) = R i 30 j =28 R j (25) Applying the VE-IDMS method, the molar mass M of the highly enriched silicon material can be obtained via Figure 6. Schematic of the principle of the VE-IDMS method. In the mass spectrometric measurement, the 28 Si signal is completely ignored. The 28 Si-enriched silicon sample x with a 30 Si-enriched material y is blended by adjusting a 30 Si/ 29 Si ratio of 1 in the blend bx. From the measured isotope ratios, the mass fraction w(ve) is calculated yielding the amount of substance fractions x( i Si) of all three isotopes. From this, the molar mass M of silicon in the enriched sample is calculated. blends are R j,2 = x j ( 30 Si)/x j ( 29 Si) and R j,3 = x j ( 28 Si)/x j ( 29 Si). [53] The principle of the VE-IDMS method is displayed in Figure 6. It was a fortunate coincidence that the same result has been independently developed from the scratch at the Istituto Nazionale di Ricerca Metrologia (INRIM, Italy), thus validating the VE-IDMS principle. [54] M = 1 + m yx m x M ( 28 Si ) M ( 28 Si ) (1 + R x,2 ) M ( 29 Si ) R x,2 M ( 30 Si ) ( ) Ry,2 R bx,2 R y,3 M ( 28 Si) + M ( 29 Si) + R y,2 M ( 30 Si) (R bx,2 R x,2 ) (26) (m yx and m x denote the masses of the solid spike material y and the solid sample material x, respectively in the IDMS blend bx; the isotope ratios in the respective endmember materials and Isotope ratios measured using an inductively coupled plasma (ICP) source always suffer from mass bias. [73] Therefore, the measured isotope ratios must be corrected by calibration factors (10 of 17)