Comparison of the Isotopic Composition of Silicon Crystals Highly Enriched in 28Si

Article Comparison of the Isotopic Composition of Silicon Crystals Highly Enriched in 28Si

Olaf Rienitz and Axel Pramann * Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Germany; [email protected] * Correspondence: [email protected]

Received: 27 May 2020; Accepted: 9 June 2020; Published: 11 June 2020

Abstract: The isotopic composition and molar mass M of silicon in a new crystal (code: Si28-33Pr11) measured by isotope ratio mass spectrometry using a high-resolution multicollector-inductively coupled plasma mass spectrometer (MC-ICP-MS) is presented using the virtual-element isotope dilution mass spectrometry (VE-IDMS) method. For this new crystal, M = 27.976 950 48 (16) g/mol was determined with u (M) = 5.7 10 9. The “X-ray-crystal-density (XRCD) method”, one of the primary rel × − methods for realizing and disseminating the SI units kilogram and mole in the recently revised SI, is based on “counting” silicon atoms in silicon single crystal spheres. One of the key quantities is the isotopic composition—expressed by the molar mass M—of the three stable isotopes 28Si, 29Si, and 30Si in the material highly enriched in 28Si. M was determined with lowest possible uncertainty using latest improvements of the experimental techniques. All uncertainties were estimated according to the “Guide to the expression of uncertainty in measurement, GUM”. The results of the new crystal are discussed and compared with the four previously available crystals, establishing a worldwide limited pool of primary reference spheres of highest metrological quality.

Keywords: silicon single crystals; molar mass; isotope ratios; mass spectrometry; SI revision; mole dissemination; kilogram dissemination

1. Introduction The choice of silicon with the three stable natural isotopes 28Si, 29Si, and 30Si as an element which is ideally suitable for the realization and dissemination of the SI base units kilogram and mole dates back to its first comprehensive use for the X-ray-crystal-density (XRCD) method in the 1970s, first applied by the National Bureau of Standards (now National Institute of Standards and Technology, NIST) [1,2]. During the 1980s up to the beginning of the 2000s, first silicon with a natural isotopic abundance has been used for the production of single crystals in a macroscopic manner ready to produce perfectly round spheres with a mass of approximately one kilogram each [3]. Silicon has a long tradition in semiconductor research and applications [4], the underlying knowledge and improved techniques for the growth of macroscopic single crystals marks Si an ideal candidate for the XRCD method [5]. Moreover, the mechanical robustness of the macroscopic spheres including the stability of oxide surface layers turns it into a material of choice [6]. The broader background of the research—focused on the isotopic composition and molar mass M of the silicon material—was the revision of the Si units defined by fixed fundamental physical constants in 2019 and the scientific methods used therein [7,8]. The kilogram, now defined via the numerical value of the Planck constant h, and the mole, defined via the numerical value of the Avogadro constant NA, are best realized and disseminated by the Kibble balance (watt balance) and the XRCD method, respectively [5,9]. The exceptional advantage is that both units can be realized and disseminated by one of these two primary methods which are of comparative nature, yielding both smallest measurement uncertainties. From 2007, silicon highly enriched in 28Si is

Crystals 2020, 10, 500; doi:10.3390/cryst10060500 www.mdpi.com/journal/crystals Crystals 2020, 10, 500 2 of 11 used in the XRCD method, because when taking silicon with natural isotopic composition, the relative uncertainty associated with the molar mass M, a key quantity in this procedure, reaches a lower limit 7 in the 10− range, at least one order of magnitude too high for a respective small uncertainty of NA and thus a precondition of the revision of the SI at that time. Using the enriched silicon crystal material, 8 urel(M) is smaller than 10− applying the methods described later. The basic principle of the XRCD method is the “counting” of all silicon atoms of a macroscopic silicon sphere with lowest associated uncertainty yielding NA using the relation

8V M N N = = (1) A a3 m n here, V = V is the volume of the macroscopic silicon sphere ( 430 mL), 8 is the number of atoms sphere ≈ in the unit cell, M is the molar mass, m is the mass of the Si sphere ( 1 kg), a is the lattice parameter. ≈ This yields the total number N of silicon atoms and the amount of substance n. As stated, from 2018 N has a ﬁxed numerical value N = 6.022 140 76 1023 mol 1 without an associated uncertainty by A A × − deﬁnition. This in turn shows, that a silicon sphere characterized by the XRCD method (second term in Equation (1)) can serve as a primary mass standard which is expressed by

Ph i i i x( Si)Ar( Si) 8V 2R h i m = ∞ + m m (2) 3 2 sur def a cα Ar(e) − with the Rydberg constant R , the Planck constant h, the speed of light in vacuum c, the ﬁne structure ∞ i constant α, the relative atomic mass of the electron Ar(e), the relative atomic masses Ar( Si) of the silicon isotopes—given in respective updated tables [10], the mass of surface layers msur, and the mass of point defects mdef. In a similar way, the base quantity amount of substance n is realized and disseminated

8V 2R h 1 8V 1 n = ∞ = (3) 3 2 3 a cα Ar(e)Mu a NA in the third term of Equation (3), the molar mass constant Mu must be considered with Mu = 1.000 000 1 000 00 (45) g mol− [11]. The application of the XRCD method for the realization and dissemination of the kilogram and the mole thus constitutes one of the two best (on a metrological scale) respective experiments. In case of the XRCD method this means that not only one single sphere but a pool of several rather similar spheres of highest quality should be available. Due to the extreme enrichment of x(28Si) > 0.999 9 mol/mol (with the amount-of-substance fraction x), the production of a respective crystal is both time consuming and extremely expensive which explains the very limited number of spheres available on that level [12]. In this article, we report on the isotopic composition x(iSi) and molar mass M of another new silicon single crystal highly enriched in 28Si with the code Si28-33Pr11. From the new crystal ingot (Figure1), two spheres were cut as schematically shown in Figure2. The molar mass M has been derived from four samples ( 500 mg each) bracketing the outer ≈ limits of the spheres in the regions N and V. All data were calculated using an uncertainty analysis with respect to the “Guide to Expression of Uncertainty in Measurement” (GUM) [13]. We compared the results of the present crystal with the other available crystals highly enriched in 28Si: Si28-10Pr11 (“AVO28-crystal” (2007)), Si28-23Pr11 (2015), Si28-24Pr11 (2016), and Si28-31Pr11 (2018). From each crystal two spheres are available. CrystalsCrystals 20202020,, 1010,, x 500 FOR PEER REVIEW 33 of of 11 11 Crystals 2020, 10, x FOR PEER REVIEW 3 of 11

Figure 1. Single crystalline float-zone crystal ingot Si28-33Pr11 (length: 560 mm; mass: 6.216 kg). Figure 1. Single crystalline ﬂoat-zonefloat-zone crystal ingot Si28-33Pr11 (length: 560 mm; mass: 6.216 kg).

Figure 2. Float-zone crystal ingot Si28-33Pr11: schematic cross section. The measured samples (cubic shape with approx. 500 mg each) bracketing the sphere areas (P and U) are taken from parts N and V. Figure 2. Float-zone crystal ingot Si28-33Pr11: schematic cross section. The measured samples (cubic 2. Theoretical Background shapeFigure with 2. Float-zone approx. 500 crystal mg each)ingot Si28-33Pbracketingr11: the schematic sphere areas cross (P section. and U) The are measured taken from samples parts N (cubic and V.Theshape silicon with approx. material 500 described mg each) inbracketing this article therequires sphere areas a special (P and methodology U) are taken from for the parts measurement N and of theV. molar mass and isotopic composition, because of the extreme enrichment in 28Si and thus the ultra-lowThe molar abundances mass M of has29Si been and derived30Si. For from this four reason, samples typical (≈ 500 mass mg spectrometric each) bracketing methods the outer used limitsfor theThe of determination themolar spheres mass in M ofthe hasM regions andbeen/or derived Nx (andiSi) V. mightfrom All fourdata fail orsampleswere will calculated lead (≈ 500 to largemg using each) uncertainties. an bracketinguncertainty Detailsthe analysis outer of withtheselimits respect proceduresof the tospheres the have “Guide in beenthe to regions publishedExpression N and elsewhere of V.Unce Allrtainty data [14– were16 in]. Measurement” Brieﬂy, calculated a modiﬁed using (GUM) an isotope-dilution uncertainty [13]. We compared analysis mass 28 thespectrometrywith results respect of tothe method the present “Guide (virtual-element crystal to Expression with the isotope ofot herUnce dilution availablertainty mass in crystals Measurement” spectrometry: highly enriched (GUM) VE-IDMS) in[13]. Si: wasWe Si28-10Pr11 compared applied in 28 (“AVO28-crystal”combinationthe results of withthe present isotope(2007)), crystal ratioSi28-23Pr11 mass with spectrometry the (2015), other Si28-24Pr11 available (high resolution crystals (2016), highlyand multicollector Si28-31Pr11 enriched inductively-coupled in(2018). Si: Si28-10Pr11 From each crystalplasma(“AVO28-crystal” two mass spheres spectrometer (2007)), are available. Si28-23Pr11 MC-ICP-MS). (2015), Mass Si28-24Pr11 bias correction (2016), of and the Si28-31Pr11 measuredisotope (2018). From ratios each was performedcrystal two usingspheres a gravimetricare available. mixtures method yielding the necessary calibration (K) factors [15]. 2.The Theoretical measurement Background of the isotope ratios will give accesses to the respective x(iSi). The extremely 2. Theoretical Background highThe enrichment silicon material will generate described too largein this uncertainties article requires if ratios a special e.g., methodology30Si/28Si are measured.for the measurement Therefore, ofthe the siliconThe molar silicon material mass material and is treatedisotopic described as compositio consisting in this articlen, of because (theoretically) requires of the a special extreme29Si methodology and enrichment30Si only for (thein the28 virtualSi measurement and thus element the ultra-lowwhichof the molar can abundances be mass regarded and of isotopic as 29 anSi and impurity) compositio 30Si. For in this then, because matrixreason, of oftypical 28theSi (anextreme mass excess spectrometric enrichment amount). Thein methods 28Si measurements and usedthus forthe therequireultra-low determination mainly abundances the of determination M of and/or 29Si and x( ofi30Si)Si. the mightFor isotope this fail reason, ratios or willR typical( 30leadSi/29 to Si)mass la=rgex (spectrometric30 uncertainties.Si)/(x(29Si) and methods Details the knowledge usedof these for i 28 proceduresofthe the determination masses have (myx beenandof Mm publishedxand/or) of components x( Si)elsewhere might in thefail [14– blendor 16].will bx Briefly,lead of the to samplelaa rgemodified uncertainties. material isotope-dilution x (Si Details enriched of in masstheseSi) spectrometryandprocedures a “spike” have method material been (virtual-element y (Sipublished enriched elsewhere in isotope30Si). Finally, dilu[14–tion16]. the mass Briefly, amount-of-substance spectrometry: a modified VE-IDMS) isotope-dilution fractions wasx(i Si)applied and mass the in 9 combinationmolarspectrometry mass M withmethodwill isotope be (virtual-element derived, ratio the mass latter isotope spectrometry with relativedilution uncertainties (highmass spectrometry:resolution in the multicollector 10 VE-IDMS)− range. was The inductively- applied VE-IDMS in coupledmethodcombination hasplasma been with mass applied isotope spectrometer and ratio validated mass MC-ICP-MS). byspectrometry several Mass national (highbias metrologycorrection resolution institutesof themulticollector measured (NMIs) isotope andinductively- serves ratios as wasacoupled primary performed plasma method usingmass [17 –spectrometera20 gravimetric]. MC-ICP-MS). mixtures me thodMass yielding bias correction the necessary of the measured calibration isotope (K) factors ratios [15].was Theperformed measurement using a of gravimetric the isotope mixtures ratios will me givethod accesses yielding to the the necessary respective calibration x(iSi). The (extremelyK) factors high[15]. enrichmentThe measurement will generate of the isotopetoo large ratios uncertainties will give accessesif ratios e.g.,to the 30Si/ respective28Si are measured. x(iSi). The Therefore, extremely thehigh silicon enrichment material will is generatetreated as too consisting large uncertainties of (theoretically) if ratios 29 e.g.,Si and 30Si/ 3028SiSi onlyare measured. (the virtual Therefore, element whichthe silicon can bematerial regarded is treated as an impurity) as consisting in the of matrix (theoretically) of 28Si (an 29 Siexcess and 30amount).Si only (the The virtualmeasurements element which can be regarded as an impurity) in the matrix of 28Si (an excess amount). The measurements

Crystals 2020, 10, 500 4 of 11

3. Materials and Experimental Methods The silicon samples (each with a mass of approximately 500 mg) were cut from two axial positions (parts N and V) of the original ingot (compare Figure2). From each part, two samples (adjacent to the axis and with a small distance) were used, bracketing the intended area of the two spheres. The crystal has been produced in Russia (isotopic enrichment at the Stock Company Production Association Electrochemical Plant (Sc “PA ECP”) in Zelenogorsk and the production of the polycrystal in the G.G. Devyatykh Institute of Chemistry of High-Purity Substances of the Russian Academy of Sciences (IChHPS RAS) in Nizhny Novgorod, and finally at the Leibniz-Institute for Crystal Growth (IKZ) in Berlin in the framework of a joint project aiming at manufacturing several silicon crystal-spheres highly enriched in 28Si [12,21,22]. The sample notation is: Si28-33Pr11 N.2.1, Si28-33Pr11 N.2.3, Si28-33Pr11 V.2.1, and Si28-33Pr11 V.2.3. Prior to the preparation of the sample solutions used for the mass spectrometric measurements, the solid samples were carefully cleaned and etched to remove possible surface layers of e.g., oxides [23]. The rather large size of the individual samples was necessary, because of the extremely high enrichment in the 28Si isotope and thus the respectively very small abundances in 29Si and 30Si with the latter being the analytes of interest. The cleaned and etched samples were weighed (with buoyancy correction) dissolved in aqueous tetramethylammonium hydroxide (TMAH)(Electronic Grade, 99.9 999 %, Alfa Aesar, Thermo Fisher GmbH, Kandel, Germany) with a mass fraction of the final sample solution w(TMAH) = 0.0006 g/g. This yields the following mass fractions of the solutions to be measured with the MC-ICP-MS: sample wx(Si) 4000 µg/g, blend ≈ w (Si) 3000 µg/g, and natural silicon for mass bias correction ww(Si) = 4 µg/g. All mass spectrometric bx ≈ measurements were performed with a Neptune XT™ MC-ICP-MS (Thermo Fisher Scientific GmbH, Bremen, Germany) using the operation parameters listed in Table1.

Table 1. Summary of the main operation parameters and instrument settings of the MC-ICP-MS.

1 Argon gas (cooling, auxiliary, sample)/L min− 16, 0.8, 0.9 ... 1.2 power (radio frequency)/W 1175 torch, bonnet sapphire 1 1 nebulizer (PFA ; flow rate/µL min− ) 50 spray chamber (PFA 1, PEEK 2) cyclonic/Scott sampler, skimmer (orifice/mm) nickel (1.1), nickel (0.8; XT type) mass resolution M/∆M 8000 (high resolution) slit size/µm 25 autosampler CETAC ASX 110 FR Faraday detectors C, H3 operation mode static virtual amplifier™ left rotation integration time/s 4 number of integrations (cycles) per block 1 (3) number of blocks 6 1 PFA: perfluoroalkoxy alkane, 2 PEEK: polyether ether ketone.

The isotope ratio measurements were carried out in exactly the same way for the samples x, the blends bx, the natural material w, and the blank solutions. Prior to each silicon solution, a blank solution was measured to enable the signal correction for blank and carry-over eﬀects. Each silicon solution was measured four times in the order sample x, blend bx, and natural silicon w yielding the uncorrected intensity ratios (in V/V) of 30Si/29Si. The natural silicon solution w was measured at the end of each sequence in order to avoid a contamination of the isotopic composition of the sample and blend solution. Moreover, the measured 30 29 intensity ratio Rmeas( Si/ Si) in w was used to calculate the respective K factor applying the “true” 30 29 values of Rw given elsewhere [24]. With that K factor, each measured ratio Rmeas( Si/ Si) was converted to a corrected isotope ratio R(30Si/29Si) of that sequence. Crystals 2020, 10, x FOR PEER REVIEW 5 of 11

solution was measured four times in the order sample x, blend bx, and natural silicon w yielding the uncorrected intensity ratios (in V/V) of 30Si/29Si. The natural silicon solution w was measured at the end of each sequence in order to avoid a contamination of the isotopic composition of the sample and blend solution. Moreover, the measured intensity ratio Rmeas(30Si/29Si) in w was used to calculate the respective K factor applying the “true” values of Rw given elsewhere [24]. With that K factor, each measured ratio Rmeas(30Si/29Si) was converted to a corrected isotope ratio R(30Si/29Si) of that sequence.

Crystals 2020, 10, 500 5 of 11 4. Results and Discussion

4. Results4.1. Molar and DiscussionMass of Si28-33Pr11 In this study, four samples were measured aiming at a first glance overview of the 4.1. Molar Mass of Si28-33Pr11 behavior/distribution of the molar mass (isotopic composition) along the crystal axis acting as a first Indecision this study,guidance four for samplesthe cutting were regions measured of the two aiming spheres. at It a is first the first glance study overview on the molar of the mass of behaviorenriched/distribution silicon in of our the group molar using mass a (isotopic new further composition) development along of thethe crystalNeptune™ axis MC-ICP-MS, acting as a the first decisionnew Neptune guidance XT™. for Each the cuttingsample regionswas measured of the twoin exactly spheres. the It same is the way first using study the on same the molar sequences massand of enriched operating silicon conditions. in our group Four usingto six asequences new further per development samples were of measured the Neptune (in™ totalMC-ICP-MS, 21 sequences, the newsee Neptunesupplementary XT™. Each material). sample The was av measurederage molar in exactlymass values the same of the way respective using the samples same sequences are shown in and operatingFigure 3 (with conditions. associated Four uncertainties, to six sequences k = 1). per The samples respective were data measured as well as (in the total amount-of-substance 21 sequences, see Supplementaryfractions x are Materialslisted in ).Table The 2. average In contrast molar to mass comparable values of studies the respective performed samples with are other shown crystals in of Figureenriched3 (with associatedsilicon, the uncertainties, average uncertaintyk = 1). Theof the respective average data molar as mass well asof the all amount-of-substancefour crystals measured in fractionsthis xstudyare listed is slightly in Table increased2. In contrast yielding to comparableM(Si) = 27.976 studies 950 performed48 (16) g/mol with with other urel crystals(M) = 5.7 of × 10−9 enriched(including silicon, data the averagescattering) uncertainty [21,22,24]. of the average molar mass of all four crystals measured in this study is slightly increased yielding M(Si) = 27.976 950 48 (16) g/mol with u (M) = 5.7 10 9 rel × − (including data scattering) [21,22,24].

27.9769510

27.9769508

27.9769506 -9 urel(M) = 5.7 x 10

/ (g/mol) 27.9769504 M

27.9769502

27.9769500 .3 .2.1 .2 N V.2.3 N 1 V.2.1 1 Pr11 Pr1 3 3Pr1 -33 -3 8-3 8 2 28 Si28-33Pr11 Si Si2 Si

FigureFigure 3. Molar 3. Molar masses massesM Mof of the the four four crystal samples samples (Si28-33Pr11) (Si28-33Pr11) investigated investigated in this in this study. study. For each For eachaverage average value, value, the combined the combined uncertainty uncertainty (k = 1) (k is= given1) is given by error by errorbars. bars.The average The average value value(arithmetic M (arithmeticmean) mean) of all ofmeasurements all measurements is M(Si) is =(Si) 27.976= 27.976 950 95048 (16) 48 (16)g/mol g/mol (solid (solid line). line). The The average average relative 9 relative uncertainty is urel(M) = 5.7 −910 − including the contributions of data scattering. uncertainty is urel(M) = 5.7 × 10× including the contributions of data scattering. Table 2. Average molar masses and amount-of-substance fractions (isotopic composition) of diﬀerent samplesTable (Si28-33Pr11). 2. Average Numbersmolar masses in brackets and amount-of-substance denote uncertainties fractions in the last (isotopic digits. Uncertaintiescomposition) (ofk = different1) do notsamples cover data (Si28-33Pr11). scattering. Numbers in brackets denote uncertainties in the last digits. Uncertainties (k = 1) do not cover data scattering. 28 29 30 Sample M x( Si) x( Si) x( Si) 1 g mol− mol/mol mol/mol mol/mol 10 5 10 7 × − × − N.2.1 27.976 950 371 (71) 0.999 976 461 (64) 2.3232 (60) 3.07 (17) N.2.3 27.976 950 390 (86) 0.999 976 266 (85) 2.3602 (85) 1.312 (47) V.2.1 27.976 950 655 (99) 0.999 976 061 (98) 2.3747 (96) 1.926 (86) V.2.3 27.976 950 519 (76) 0.999 976 161 (74) 2.3683 (74) 1.562 (74) average 27.976 950 484 (84) 0.999 976 237 (81) 2.3566 (80) 1.97 (10) Crystals 2020, 10, x FOR PEER REVIEW 6 of 11

Sample M x(28Si) x(29Si) x(30Si) g mol−1 mol/mol mol/mol mol/mol × 10−5 × 10−7 N.2.1 27.976 950 371 (71) 0.999 976 461 (64) 2.3232 (60) 3.07 (17) N.2.3 27.976 950 390 (86) 0.999 976 266 (85) 2.3602 (85) 1.312 (47) V.2.1 27.976 950 655 (99) 0.999 976 061 (98) 2.3747 (96) 1.926 (86) V.2.3 27.976 950 519 (76) 0.999 976 161 (74) 2.3683 (74) 1.562 (74) average 27.976 950 484 (84) 0.999 976 237 (81) 2.3566 (80) 1.97 (10)

Crystals 2020, 10, 500 6 of 11 The reason for this increase is suggested to originate from data scattering between the different measurements as will be shown below. At a first glance, the molar masses of the measured samples Theagree reason within for the this limits increase of uncertainty is suggested and to thus originate the crystal from datacan be scattering treated betweenas homogeneous the different along the measurementsmain axis aswhich will beis supported shown below. by the At general a first glance, manufacturing the molar process masses of the measuredfloat-zone samplescrystal as has agreebeen within described the limits previously of uncertainty [12]. andThe thus relative the crystal uncertainties can be treatedassociated as homogeneous with M of alongthe single the mainmeasurements axis which range is supported in the lower by the10−9general region not manufacturing including data process scattering of the as float-zonereported in crystal [22]. Figure as has4 beendisplays described the detailed previously measurement [12]. The results relative of uncertaintiesM of the crystal associated sample withV.2.3.M Here,of the the single error bars 9 measurementsindicate the range respective in the lower uncertainties 10− region without not including includin datag data scattering scattering. as reported It is evident in [22 ].that Figure some4 data displayspoints the are detailed not included measurement in the kresults = 1 range. of M Sinceof the this crystal behavior sample is observed V.2.3. Here,when themeasuring error bars a single indicatesample, the respective it is clear that uncertainties this kind of without fluctuation including cannot data be attributed scattering. to It inhomogeneities is evident that some of the data material points(e.g., are notdisordering included or in impurities the k = 1 in range. the crystal Since latti thisce). behavior The scattering is observed is a when combined measuring result aof the singleexperimental sample, it is clearconditions that this (plasma kind of stability, fluctuation decrease cannot of be sensitivity attributed due to inhomogeneities to strong depositions of the of Si materialcomponents (e.g., disordering on the sampler or impurities and skimmer in the crystal cones latticeduring). the The measurement scattering is aof combined the highly result concentrated of the experimentalSi solutions, conditionsgeneral low (plasma signal abundance). stability, decrease of sensitivity due to strong depositions of Si components on the sampler and skimmer cones during the measurement of the highly concentrated Si solutions, general low signal abundance).

27.9769508

27.9769507

27.9769506

27.9769505 / (g/mol) M

27.9769504

27.9769503

123456 sequence

Figure 4. Single measurements of molar masses M of the sample Si28-33Pr11 part V.2.3 (error bars for k = 1 without scattering contribution). The solid line represents the average molar mass M of Figure 4. Single measurements of molar masses M of the sample Si28-33Pr11 part V.2.3 (error bars for all sequences (including the measurements of all the other samples). The dashed lines denote the k = 1 without scattering contribution). The solid line represents the average molar mass M of all respective upper and lower average uncertainties (k = 1) without the contribution of scattering. sequences (including the measurements of all the other samples). The dashed lines denote the respective upper and lower average uncertainties (k = 1) without the contribution of scattering. A consistency analysis calculating the degrees of equivalence (di) has been performed for the molar mass results of the four samples (Figure5) with di = Mi M (diﬀerences of the single samples A consistency analysis calculating the degrees of equivalence− (di) has been performed for the values and average value of the molar mass). Within the limits of uncertainty, all di enclose the zero molar mass results of the four samples (Figure 5) with di = Mi − M (differences of the single samples line, clearly showing the consistency of the molar masses of the crystal samples. A representative values and average value of the molar mass). Within the limits of uncertainty, all di enclose the zero uncertainty analysis (single measurement of sample N.2.1) is given in Table3. The uncertainty analysis was performed using the GUM Workbench Pro™ software (version 2.4.1 392, Metrodata GmbH, Braunschweig, Germany) according to the model equation of the molar mass of the VE-IDMS principle e.g., [21]. As in previous studies on other crystals highly enriched in 28Si, one of the main contributions to the uncertainty associated with M is the measured intensity ratio (index 2 denotes the ratio 30Si/29Si) meas 30 Rbx,2 in the blend of the sample and the “spike” material (Si enriched in Si); here with 61%. The second main contribution (24.5%) is the corrected isotope ratio Rw,2 of the natural silicon material w measured meas for the K factor determination. The measured intensity ratio Rx,2 in the sample contributes with Crystals 2020, 10, x FOR PEER REVIEW 7 of 11

line, clearly showing the consistency of the molar masses of the crystal samples. A representative uncertainty analysis (single measurement of sample N.2.1) is given in Table 3. The uncertainty analysis was performed using the GUM Workbench Pro™ software (version 2.4.1 392, Metrodata GmbH, Braunschweig, Germany) according to the model equation of the molar mass of the VE-IDMS principle e.g., [21]. As in previous studies on other crystals highly enriched in 28Si, one of the main contributions to the uncertainty associated with M is the measured intensity ratio (index 2 denotes meas 30 29 30 the ratio Si/ Si) Rbx,2 in the blend of the sample and the “spike” material (Si enriched in Si); here

with 61%. The second main contribution (24.5%) is the corrected isotope ratio Rw,2 of the natural meas Crystalssilicon2020 material, 10, 500 w measured for the K factor determination. The measured intensity ratio Rx7,2 of 11 in the sample contributes with another 13%. The masses of the components in the blend bx as well as the molar masses of the individual silicon isotopes do not contribute to the uncertainty at all. another 13%. The masses of the components in the blend bx as well as the molar masses of the individual silicon isotopes do not contribute to the uncertainty at all.

) 2 -7

0 / (g/mol x 10 i d

-2

-4

3 .2.1 N V.2.1 V.2. r11 r11 r11

8-33Pr11 N.2.3 2 Si28-33P Si Si28-33P Si28-33P

Figure 5. Degrees of equivalence di of the four samples of the crystal Si28-33Pr11 (average molar mass Figure 5. Degrees of equivalence di of the four samples of the crystal Si28-33Pr11 (average molar mass without scattering). Uncertainties for k = 2. All data are consistent due to the enclosure of the zero line. without scattering). Uncertainties for k = 2. All data are consistent due to the enclosure of the zero line. Table 3. Representative uncertainty budget of sample N.2.1 (single measurement). meas The comparable small contribution of R to u(M) is unusual, because of the isotopic Best Estimate x,2 Standard Sensitivity Quantity Unit Index composition of this very crystal. Although(Value) the enrichmentUncertainty in 28Si Coewithﬃ cientx(28Si) = 0.999 976 24 (19)

mol/mol Xisi not that large[Xi] compared toxi other highly enrichedu(xi) silicon crystalsci (see Section 3.2), the 29 30 characteristic28 of the crystal Si28-33Pr11 is the large ratio of12 x( Si) and x( Si) with two orders of M( Si) g/mol 27.976926534940 540 10− 1.0 0.0% 6 × 12 29 magnitude.myx This is an additionalg 5.611000 experimental10− challenge:704 10 the− small abundance4.2 of Si is 0.0% accompanied 30 × × 9 6 by a muchmx smaller signalg of Si leading 0.062474995 to a difficult569 data10 collection.− 380 10− 0.0% 29 × 12 − × 6 M( Si) g/mol 28.976494669090 610 10− 23 10− 0.0% 30 × 9 × 9 M( Si)Table 3. gRepresentative/mol 29.9737701360 uncertainty budget27.0 of sample10− N.2.1 (single560 measurement).10− 0.0% × − × 9 Ry,3 mol/mol 1.5855 0.0222 81 10− 0.1% − × 9 Ry,2 mol/mol 269.04Best Estimate 5.65 Standard2.0 10Sensitivity1.4% Quantity Unit × − Index Rmeas V/V 0.010380(value) 686 10 Uncertainty6 50 10Coefficient6 13.0% x,2 × − × − Rmeas 6 bx,2Xi V/V[Xi] 3.7540xi 0.0116 u(xi) 6.4 10− ci 60.9% 3 − × 6 MRw,2(28Si) mol/mol g/mol 0.66230 27.9769265349401.32 10− 540 × 10-1236 10− 1.0 24.5% 0.0% meas × 6 − × 6 Rw,2 V/V 0.7028500 73.8 10− 33 10− 0.0% myx g 5.611000 × 10-6 × 704 × 10-12 × 4.2 0.0% Ymx [Y] g y 0.062474995 uc(y) 569 × 10-9 -380 × 10-6 0.0% MM(29Si) g/mol g/mol 27.9769502183 28.97649466909095.1 10 9 610 × 10-12 23 × 10-6 0.0% × −

meas The comparable small contribution of Rx,2 to u(M) is unusual, because of the isotopic composition of this very crystal. Although the enrichment in 28Si with x(28Si) = 0.999 976 24 (19) mol/mol is not that large compared to other highly enriched silicon crystals (see Section3), the characteristic of the crystal Si28-33Pr11 is the large ratio of x(29Si) and x(30Si) with two orders of magnitude. This is an additional experimental challenge: the small abundance of 29Si is accompanied by a much smaller signal of 30Si leading to a diﬃcult data collection. Crystals 2020, 10, 500 8 of 11

4.2. Comparison of the Available Highly Enriched Silicon Crystals The realization and dissemination of the kilogram and mole via the XRCD method is based on silicon spheres highly enriched in 28Si with extremely high chemical purity and an almost perfect crystal lattice [5]. Again, only an enrichment in 28Si of this order of magnitude is capable to yield an uncertainty of u (M) < 1.0 10 8 which is essential to obtain u (N ) < 2 10 8, which was a rel × − rel A × − prerquisite for the realization of NA prior to the revision of the SI [5]. A silicon sphere matching this condition is called a “primary” sphere according the notation given in [6]. A sphere of this kind must be well characterized by the XRCD method. Due to the large amount of material (1 kg per sphere), the manufacturing is extremely expensive, but also time consuming with the need of extraordinary logistic eﬀorts [12]. This is the reason why only a small pool of spheres has been manufactured yet, each a kind of an individual material artefact with properties of its own. Currently, the production of ten spheres has been completed/is under way according to the original crystals listed in Table4. Each of the spheres can be used as a primary mass standard with lowest associated uncertainty u(m) on the metrological top level suitable for the calibration/comparison of other weights on the next lower metrological level. Therefore, the spheres are interchanged in part for the reason of laboratory intercomparisons on the highest level. Additionally, they are object of basic metrological research (XRCD properties and time-dependent changes of surface characteristics). The spheres with the highest enrichment in 28Si are originating from the crystal Si24-Pr11 (the second crystal manufactured during the “kilogram-2 project” [12]) with an amount-of-substance fraction x(28Si) > 0.999 99 mol/mol, followed by the spheres from Si28-31Pr11 (obtained during the “kilogram-3 project”), the latter not yet fully characterized. All available crystals show an enrichment of at least x(28Si) > 0.999 9 mol/mol. In Figure6, both the average molar masses and x(28Si) of the respective crystals are compared. It is evident that there is a strong correlation between M and x(28Si).

Table 4. Comparison of M and x(iSi) of the available silicon spheres highly enriched in 28Si used for the XRCD experiment (uncertainties for k = 1).

Crystal M x(28Si) x(29Si) x(30Si) Year Ref 1 Si28- g mol− mol/mol mol/mol mol/mol 10 5 10 7 × − × − 10Pr11 27.976 970 12 (12) 0.999 957 52 (12) 4.136 (11) 11.21 (14) 2015 [24] 23Pr11 27.976 942 666 (40) 0.999 984 470 (39) 1.492 1 (38) 6.095 (48) 2017 [21] 24Pr11 27.976 933 787 (77) 0.999 993 104 (66) 0.653 6 (51) 3.60 (17) 2019 [22] 31Pr11 27.976 941 260 (42) 0.999 985 501 (40) 1.426 4 (39) 2.338 (75) 2018 1 33Pr11 27.976 950 48 (16) 0.999 976 24 (19) 2.357 (24) 1.97 (78) 2020 2 1 Preliminary results; 2 this work.

The error bars (uncertainties) of both M and x(28Si) are not shown in Figure6, because of their comparable small range which is covered by the data points itself. On an absolute scale, the “abundance” of 29Si and 30Si is “tiny” yielding ranges of approximately 6 10 6 < x(29Si) < 4 10 5 mol/mol. The × − × − range of x(30Si) is lowered by up to two orders of magnitude: 2 10 7 < x(30Si) < 1 10 6 mol/mol. In × − × − fact, the measurement of the 30Si isotopes is challenging, and the more diﬃcult, the smaller x(30Si). This experimental problem is not inﬂuencing the uncertainty associated with M because of the absence of any input values of amount-of-substances in the conditional equation of M given in [24]. Crystals 2020, 10, x FOR PEER REVIEW 9 of 11

Crystals 2020, 10, 500 9 of 11

27.976980 0.999995

0.999990 27.976970

0.999985

27.976960 0.999980

0.999975 / (g/mol)

27.976950 Si) / (mol/mol) M 0.999970 28 ( x 0.999965 27.976940 0.999960

0.999955 27.976930

1 r1 r11 r11 r11 P P 3P -10P 8 8-2 8-24 8-31 2 2 Si2 Si2 Si Si Si28-33Pr11

Figure 6. LeftLeft scale scale (filled (filled circles): Average Average molar molar masses masses MM ofof the available silicon crystals highly 28 28 enriched in 28Si.Si. Right Right scale scale (filled (filled squares): Amount-of-substance fractions x(28Si)Si) displaying the 28 “enrichment” ofof thethe respective crystal.crystal. Uncertainties forfor both M and x((28Si)Si) are are not not displayed, displayed, because because of their extremely smallsmall valuesvalues being being hidden hidden by by the the data data points. points. Data Data for for the the crystal crystal Si28-31Pr11 Si28-31Pr11 are are yet yetpreliminary preliminary due due to the to the limited limited number number of samples. of samples. 5. Conclusions The error bars (uncertainties) of both M and x(28Si) are not shown in Figure 6, because of their comparableThis work small presents range latest which results is covered of the molar by the mass data and points amount-of-substance itself. On an absolute fractions scale, of a newthe 28 “abundance”chemically ultrapure of 29Si and silicon 30Si crystal is “tiny” (Si28-33Pr11), yielding ranges highly enrichedof approximately in Si, to 6 be × used10−6 for< x( the29Si) preparation < 4 × 10−5 mol/mol.of two additional The range spheres of x(30 inSi) the is verylowered limited by up pool to oftwo primary orders reference of magnitude: standards 2 × 10 for−7 the< x( dissemination30Si) < 1 × 10−6 mol/mol.of the kilogram In fact, and the the measurement mole. For the correctof the estimation30Si isotopes of Siis atoms,challenging, it is necessary and the to determinemore difficult, chemical the 15 3 13 3 14 15 3 smallerimpurities x(30 likeSi). C,This N, andexperimental O (in the lower problem ranges is ofnot 10 influencingcm , 10 cmthe ,uncertainty and 10 –10 associatedcm , respectively) with M becausevia IR spectroscopy of the absence most of any exactly. input Thevalues reported of amount-of-substances crystal produced duringin the conditional the “kilogram-3-project” equation of M givenserves in as [24]. the origin of two new unique spheres. The isotopic composition expressed by the molar mass M has been determined at several crystal positions yielding an average M = 27.976 950 48 (16) g/mol with u (M) = 5.7 10 9. Now, crystals highly enriched in 28Si enable the use of 10 respective rel × − spheres, characterized by the XRCD method, which can be compared worldwide acting as the highest standards for the mass and amount-of-substance. However, the silicon crystals differ in their absolute compositions of 28Si, 29Si, and 30Si aiming first at the high enrichment in 28Si. The uncertainty associated with M is correlated at a first glance with the enrichment in 28Si. In parallel, the measurability of the isotopes 29Si and 30Si—which are the main target measurands using the VE-IDMS principle—is a limiting factor. Due to the extremely tiny abundance of both isotopes, urel(M) might increase although x(28Si) increases. Thus, care must be taken that for future productions of silicon spheres highly enriched in 28Si (x(28Si) > 0.999 9 mol/mol), there should be a proper balance between high enrichment and a reasonable ratio 29Si/30Si on a measurable scale.

Supplementary Materials: The following are available online at http://www.mdpi.com/2073-4352/10/6/500/s1. Author Contributions: Conceptualization; methodology; validation; formal analysis; investigation; resources; data curation; writing—original draft preparation; writing—review and editing; visualization, O.R. and A.P. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding.

Crystals 2020, 10, 500 10 of 11

Acknowledgments: The authors gratefully acknowledge the provisioning of silicon samples from the ingot crystal by Daniela Eppers (PTB). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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