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Title: Molar mass of silicon highly enriched in 28Si determined by IDMS
Author(s): Pramann, A.; Rienitz, O.; Schiel, D.; Schlote, J.; Guettler, B.; Valkiers, S.
Journal: Metrologia
Year: 2011, Volume: 48, Issue: 2
DOI: doi:10.1088/0026-1394/48/2/S03
Funding programme: iMERA-Plus: Call 2007 SI and Fundamental Metrology
Project title: T1.J1.2: NAH: Avogadro and molar Planck constants for the redefinition of the
kilogram
Copyright note: This is an author-created, un-copyedited version of an article accepted for
publication in Metrologia. IOP Publishing Ltd is not responsible for any errors or omissions in this
version of the manuscript or any version derived from it. The definitive publisher-authenticated
version is available online at doi:10.1088/0026-1394/48/2/S03
EURAMET
Secretariat
Bundesallee 100
38116 Braunschweig, Germany
Phone: +49 531 592-1960
Fax:
+49 531 592-1969
secretariat@euramet.org
www.euramet.org
Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
Molar mass of silicon highly enriched in 28Si determined by IDMS
Axel Pramann,*1 Olaf Rienitz,1 Detlef Schiel,1 Jan Schlote,1 Bernd Güttler,1
and Staf Valkiers2
1
Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Ger-
many
E-mail: axel.pramann@ptb.de
2
European Commission−Joint Research Centre−Institute for Reference Materials and Meas-
urement (EC-JRC-IRMM), Retieseweg 111, 2440 Geel, Belgium
Abstract
The molar mass of a new silicon crystal material highly enriched in
28
Si ("Si28", x(28Si) >
99.99 %) has been measured for the first time using a combination of a modified isotope dilution mass spectrometry (IDMS) technique and a high resolution multicollector−ICP−mass
spectrometer (MC−ICP−MS). This work is related to the redetermination of the Avogadro
constant NA with an intended relative measurement uncertainty urel(NA) ≤ 2·10-8. The corresponding experimental investigations of the International Avogadro Coordination (IAC) were
performed using this novel "Si28" material. One prerequisite of the redetermination of NA is
the determination of the isotopic composition and thus molar mass of "Si28" with
urel(M("Si28")) ≤ 1·10-8. At PTB, a molar mass M("Si28") = 27.97697027(23) g/mol has been
determined with an associated relative uncertainty urel(M("Si28")) = 8.2·10-9 opening the opportunity to reach the aimed uncertainty of NA.
-1-
Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
1. Introduction
A limiting factor for the redetermination of the Avogadro constant NA with an intended associated relative uncertainty urel(NA) ≤ 2·10-8 using natural silicon has been the uncertainty associated with the molar mass urel(M) ≤ 3·10-7 of this material [1−4]. This uncertainty constrained
that of NA drastically to urel(NA) ≈ 10-7. The production of an ultra pure silicon single crystal
material highly enriched in
28
Si (in the following called "Si28") had the intention to over-
come this very problem, amongst others, by applying the X-ray crystal density molar mass
(XRCDMM) method [4]. Both the advantage as well as the challenge and inherent difficulties
of the unprecedented high enrichment in 28Si with an amount−of−substance fraction x(28Si) >
99.99 % were approved. At PTB, a combination of several methodological and experimental
methods has been recently applied successfully to end up in the determination of the molar
mass of "Si28" with urel(M) ≤ 1·10-8. In general, a modified isotope dilution mass spectrometry (IDMS) technique was applied to determine the isotopic composition and finally the molar
mass. This has been performed with a high resolution multicollector inductively coupled
plasma mass spectrometer (MC−ICP−MS) for the first time. Additionally, a novel explicit
experimental route for the determination of calibration factors K used to correct the measured
ratios for mass bias effects has been applied. This paper outlines and summarizes these principles providing the results supported by uncertainty calculations with respect to the molar
mass of "Si28" and K factors according to the Guide to the Expression of Uncertainty in
Measurement (GUM) [5].
2. Principle of the molar mass determination using IDMS
The main invention on the way to the determination of the molar mass of the "Si28" material
is concentrated in the use − or better adoption − of IDMS [6, 7] in the problem of the molar
mass determination. The sum of 29Si and 30Si in the "Si28"−material was treated as a virtual
element (VE) in the matrix of all three silicon isotopes of the original element [8−10]. The
-2-
Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
mass fraction w of the VE is the sum of the mass fractions w(29Si) and w(30Si). The sample
material was spiked with another silicon single crystal material highly enriched in
30
Si
("Si30"). Then, it was mainly necessary to measure the isotope amount ratios R(30Si/29Si)
only, in order to obtain the complete isotopic composition x(28Si), x(29Si), x(30Si), and the
molar mass M. Typical isotope amount ratio determinations with respect to the isotope of
highest abundance (28Si) would produce values of R(30Si/28Si) and R(29Si/28Si) in the range of
1·10-6 and 4·10-5. That strong deviation of R from unity would exclude a sufficiently small
uncertainty in M as identified as necessary. Moreover, the detection of the extremely intense
28
Si+ signal with a commercial MC-ICP-MS of limited dynamic range as used in our study
was not possible. Applying the VE concept, essentially R(30Si/29Si) in the sample ("Si28"), in
the IDMS blend , and in the spike material "Si30" had to be measured. This provided isotope
amount ratios R(30Si/29Si) in the range of 3·10-2 (sample) and approximately 1 (IDMS blend)
with a correspondingly reduced measurement uncertainty. Figure 1 schematically displays the
relations between the amount−of−substance fractions x, mass fraction w of the VE and isotope
ratio R(30Si/29Si).
The molar mass M was derived from the amount−of−substance fractions x of the respective
isotopes. The molar masses M(28Si), M(29Si), and M(30Si) of the silicon isotopes were taken
from [11]
M =
∑ [x ( Si )⋅ M ( Si )]
30
i = 28
i
i
(1)
x
1 − wimp
xx
M ( Si )
( Si ) = 1 − w
w
+ (1 + R ) ⋅
M ( Si )
M ( Si ) + R
28
28
imp
28
(2)
imp
x
29
x
-3-
⋅M
( Si )
30
Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
wimp
xx
( Si ) = 1 −1x+ (R Si ) = 1 − w M ( Si )+ R ⋅ M ( wSi )
+ (1 + R ) ⋅
M ( Si )
M ( Si ) + R ⋅ M ( Si )
28
29
29
30
x
x
imp
x
(3)
imp
x
28
29
30
x
xx
Rx ⋅
( Si ) = R
x ⋅ xx
30
wimp
Si + Rx ⋅ M
)
( Si )
M(
( Si ) = 1 − w
w
+ (1 + R ) ⋅
M ( Si )
M ( Si ) + R ⋅ M ( Si )
29
29
imp
30
(4)
imp
x
28
29
30
x
wimp = wy ⋅
wy =
myx M
⋅
mx M
⋅M
y ⋅M
29
x
29
( Si ) ⋅ (R
( Si ) (R
30
y
− Rbx )
30
bx
( Si ) + R ⋅ M ( Si )
⋅ M ( Si ) + M ( Si ) + R ⋅ M ( Si )
M
Ry , 28
( Si ) + R
( Si ) + R
29
(5)
− Rx )
30
y
29
28
30
∧ Ry , 28 =
y
( Si ) .
x ( Si )
xy
28
(6)
29
y
A detailed derivation of the above relations is given in [8].
Measured isotope amount ratios R(30Si/29Si) suffer from mass discrimination effects. Therefore, these values were corrected by calibration factors K. In the course of the molar mass
determination of "Si28" a new analytical exact expression for the determination of K factors
was developed [12]. The K factors for the correction of the two isotope amount ratios of interest are given in equations (7) and (8)
Rj ≡ R
true
j
Ry , 28 ≡ R
= K 30 ⋅ R
true
y , 28
meas
j
= K 28 ⋅ R
meas
y , 28
with
R
meas
j
=
meas
y , 28
with R
I j (30 Si)
I j ( 29 Si)
=
I y ( 28 Si)
I y ( 29 Si)
and
j ∈ {x, y, bx}
.
(7)
(8)
These calibration factors K30 and K28 were experimentally obtained by measuring the isotope
ratios R(30Si/29Si) and R(28Si/29Si), respectively, in two blends, one made of natural silicon
(natural isotopic composition) and Si, highly enriched with respect to
other blend consisting of "Si29" and "Si30" [8, 10].
-4-
29
Si ("Si29") and an-
Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
3. Experiment
The experimental details were reported in [10, 13]. Ion currents for the determination of isotope amount ratios were measured using a commerical MC−ICP−MS (Neptune, Thermo Scientific, Bremen, Germany) [14, 15]. The main parameters were: RF power = 1200 W; Ar flow
rates: 16.0 L·min-1 (coolant), 0.8 L·min-1 (auxiliary), 1.2 L·min-1 (sample), 0.02 L·min-1 (additional: spray chamber); nebulizer: 50 µL·min-1 (self-aspirating). Medium and/or high resolution modes (M/ΔM = 6000 (MR), 8000 (HR)) were used. The sensitivity of natural silicon for
the 28Si+ signal is (0.50 ± 0.05) V/(µg/g) in the high resolution mode. In a first sequence the
"Si28" samples and IDMS blends, bracketed with the respective blanks (aqueous NaOH,
w(NaOH) = 0.001 g/g) were measured using the Faraday cups for 29Si and 30Si only. In a following sequence, the K factors were measured: natural silicon, "Si29", "Si30", blends b1 and
b2, all bracketed with respective blanks (aqueous NaOH, w(NaOH) = 0.0001 g/g).
A single experiment takes about 13 hours. The Neptune sample introduction was modified
using a sapphire torch system and a PFA/PEEK spray chamber (AHF analysentechnik, Tübingen, Germany). This reduces contamination by natural silicon effectively. After a measurement, blank data were subtracted from sample data in order to correct for contamination due
to natural or enriched silicon.
The chemicals used were of highest commercial purity [10]. All solutions and respective dilutions were prepared gravimetrically. Silicon crystals were cut from original float-zone ingots
of the respective material. IR spectroscopic investigations revealed no evidence for significant
impurities. The crystal samples were etched, weighed, and eventually dissolved in aqueous
sodium hydroxide followed by dilution steps (w(NaOH) = 0.0001 g/g: for natural Si) and
w(NaOH) = 0.001 g/g: for "Si28") to form the respective silicate solutions. During all steps
for preparation and cleaning, ultrapure water was used (ρ ≥ 18 Ω·cm). Solutions of natural
silicon, "Si29", "Si30", blends b1 and b2 had concentrations of w(Si) = 4 µg/g with w(NaOH)
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Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
= 0.0001 g/g. The "Si28" sample solutions used had high concentrations w(Si) = 4000 µg/g
with w(NaOH) = 0.001 g/g, because the amount-of-substance fraction x(30Si) in "Si28" is in
the range of 1·10-6. This small amount generates only very weak
30
Si+ signals, usually < 1
mV.
4. Results and discussion
"Si28" crystal samples of parts 4, 5, 8, and 9 corresponding to the notation of the areas of the
original float zone crystal were investigated with respect to the molar mass. Table 1 displays
the numerical values of M and the amount−of−substance fractions x(iSi) from seven samples
with their associated relative uncertainties. The last rows contain the mean values of parts 5
and 8, and their average, used as the actual data of M and x(iSi) .
Each value is the average of an experimental series of usually 5 single experiments (one per
day). The actual value of the molar mass of "Si28" is M = 27.97697027(23) g/mol with an
associated relative uncertainty urel = 8.2·10-9 (k = 1). A relative standard deviation over all
single data with respect to the mean of srel = 3.5·10-9 was obtained. All values of table 1 are
located inside a band of the combined standard uncertainty (k = 1), which indicates a high
homogeneity of the crystal with respect to M. Coincidently, this meets the requirements of the
Avogadro project for a redetermination of NA with the aim of urel(NA) ≤ 2·10-8.
Table 2 gives an overview of a representative uncertainty budget of the molar mass of "Si28",
showing the relevant quantities included. The uncertainty budget was calculated according to
the Guide to the Expression of Uncertainty in Measurement (GUM) [5] using the GUMWorkbench software [16]. xx(iSi) displays the amount-of-substance fractions of the respective
silicon isotopes; M(iSi) corresponds to the molar masses of each Si isotope; wimp is the sum of
the mass fractions w(29Si) and w(30Si) in the sample x ("Si28"), whereas wy is the sum of the
mass fractions w(29Si) and w(30Si) in the spike y ("Si30"); Rx, Ry, and Rbx, denote the
-6-
Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
Table 1. Molar mass and amount-of-substance fractions x of "Si28" samples. Combined
measurement uncertainties (coverage factor k = 1) are given in brackets (last digits). In row i
= 8 the average of samples originating from the crystal part 5 are displayed (row i = 9: crystal
part 8, respectively). The last row contains the final averaged molar mass and
amount−of−substance fractions of the "Si28" crystal spheres (parts 5 and 8).
M
x(28Si)
x(29Si)
x(30Si)
i
g/mol
mol/mol
10-5 mol/mol
10-6 mol/mol
1 4.4
27.97697030(28)
0.99995748(24)
4.122(23)
1.30(3)
2 5.B2.1.4
27.97697009(20)
0.99995764(14)
4.111(12)
1.25(4)
3 5.B3.1.1.3
27.97697039(23)
0.99995738(16)
4.132(14)
1.30(5)
4 5.B4.1.1.4
27.97697035(23)
0.99995741(18)
4.132(16)
1.27(4)
5 8.A2.1.4
27.97697006(24)
0.99995768(17)
4.104(14)
1.28(5)
6 8.B4.1.1.3
27.97697047(24)
0.99995733(19)
4.135(17)
1.32(4)
7 9.8
27.97697033(21)
0.99995740(16)
4.134(14)
1.26(4)
sample
8
1 4
∑ Mi
3 i=2
27.97697028(22)
0.99995750(16)
4.123(14)
1.27(4)
9
1 6
∑ Mi
2 i =5
27.97697027(24)
0.99995751(18)
4.119(16)
1.30(4)
10
1 9
∑ Mi
2 i =8
27.97697027(23)
0.99995750(17)
4.121(15)
1.29(4)
respective isotope amount ratios R(30Si/29Si) of the sample, spike and IDMS blend; Ry,28 is
R(28Si/29Si) in the spike; K30 and K28 are the calibration factors to correct the measured isotope
amount ratios R(30Si/29Si) and R(28Si/29Si), respectively. The mass of the spike solution used
for the preparation of the IDMS blend bx is given by myx with its total silicon mass fraction
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Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
wy,total; mx is the mass of the solid sample material ("Si28") in the IDMS blend. The unit of the
sensitivity coefficients ci in table 2 is [ci] = g/mol/[Xi], “Index” represents the relative uncertainty contribution (in %) of the respective quantity Xi to the molar mass M(Si). Blank cells
originate from intermediate results representing no real input quantities. Displayed isotope
amount ratios R were already corrected with respect to background and contamination by
blank subtraction. The main uncertainty contribution of 46 % was caused by the isotope
amount ratio R(30Si/29Si) = Rx of the "Si28" sample. This very ratio needed to be determined
with a relative standard uncertainty of at least 5 %.
The isotope amount ratio Rbx (IDMS blend prepared from "Si28" and the spike material
"Si30") accounted for the uncertainty with 20 %. A third contribution originated from the molar mass M(28Si) of the 28Si isotope. This value was taken from [11], but has been reinvestigated in [18], which lead to a smaller measurement uncertainty. We have calculated the molar
mass of "Si28" using the data from [18], too, yielding a final expanded measurement uncertainty reduced by 13 % compared to the result when using the data from [11]. The numerical
value does not change significantly. Due to this and the fact that the data used in [18] are not
yet commonly used in metrology in chemistry as reference values (not yet adopted in the according IUPAC table) and to maintain a more conservative and reliable uncertainty budget,
we used the values given in [11].
Table 2. Uncertainty budget of the molar mass M of the "Si28" crystal material (representa-
tive budget from a crystal sample of part 8, calculated according to [16]). Quantities are explained in the text and in [8] with the notation according to [17]. Standard uncertainties as
well as the combined measurement uncertainty are given for a coverage factor k = 1.
-8-
Revised manuscript for publication in Metrologia (Special Issue 2011)
Quantity
Unit
28.1.2011
Best estimate
Standard
Sensitivity
(value)
uncertainty
coefficient
ci
Index
Xi
[Xi]
xi
u(xi)
xx(28Si)
mol/mol
0.99995753
1.6·10-7
xx(29Si)
mol/mol
4.119·10-5
1.3·10-7
xx(30Si)
mol/mol
1.28·10-6
6·10-8
M(28Si)
g/mol
27.97692649
1.10·10-7
1.0
23.1
M(29Si)
g/mol
28.97649468
1.10·10-7
4.1·10-5
0.0
M(30Si)
g/mol
29.97377018
1.10·10-7
0.0
0.0
wimp
g/g
4.40·10-5
1.6·10-7
wy
g/g
9.978·10-7
7·10-10
Rx
mol/mol
0.03273
1.4·10-3
1.1·10-4
46.3
Ry
mol/mol
219.34
4.62
3.1·10-9
0.4
Rbx
mol/mol
1.3663
3.12·10-3
−3.3·10-5
20.3
Ry,28
mol/mol
1.021
0.115
−200·10-9
1.0
K30
1
0.94991
1.48·10-3
−4.3·10-5
7.7
K28
1
1.0438
9.8·10-3
−200·10-9
0.0
myx
g
7.24744
9.1·10-4
6.0·10-6
0.1
mx
g
0.12798
1.2·10-5
−3.4·10-4
0.0
wy,total
g/g
1.00263·10-6
5.5·10-10
44
1.1
Y
[Y]
y
uc(y)
urel(y)
M(Si)
g/mol
27.97697022
2.29·10-7
8.2·10-9
-9-
Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
Uncertainties of masses and mass fractions were not contributing significantly. The last noticeable uncertainty contribution of 7.7 % is related to the calibration factor K30 which corrects each measured isotope amount ratio R(30Si/29Si) of a single experiment. Uncertainty
budgets of the respective calibration factors K30 and K28 are given in tables 3 and 4. There,
mz1, mz2, mw2, and my1 are the masses of the silicon materials "Si29" (z), "Si30" (y), and natural silicon (w) contained in the K factor blends b1 (index 1) and b2 (index 2). Rb1 and Rb2 are
the isotope amount ratios R(30Si/29Si) measured in these two blends. Main uncertainty contributions of K30 are Rz,28 with 23 %, the mass of mz1 (19 %) and Rb1 (16 %). In the case of K28
(table 4) the uncertainty contributions are mainly limited to Rb2 (44 %), Rz (33 %), and Rw,28
(15 %). Actually, when regarding the uncertainty budget of the molar mass (table 2), both
uncertainties of K30 and K28 have only a minor influence (7.7 % and 0 %). Compared to the
uncertainty of the molar mass of a natural silicon sample measured under equal conditions,
the uncertainty associated with M("Si28") is reduced by two orders of magnitude below the
relative 10-8 level [10]. The reason for this special behaviour is connected with the isotopic
abundances of 29Si and 30Si in each material: In the case of "Si28" xx(29Si) is by a factor of
30 larger than xx(30Si), which benefits the impact of spiking with the "Si30" material. In the
case of natural silicon the ratio of xx(29Si) vs. xx(30Si) is only 1.5. This very fact reduces the
uncertainty associated with M("Si28") by applying the modified IDMS method using a "virtual element" in the case of the silicon material highly enriched in 28Si.
Table 3. Uncertainty budget of K30 of the measurement corresponding to table 2. Standard
uncertainties as well as the combined measurement uncertainty are given for a coverage factor
k = 1.
- 10 -
Revised manuscript for publication in Metrologia (Special Issue 2011)
Quantity
Unit
28.1.2011
Best estimate
Standard
Sensitivity
(value)
uncertainty
coefficient
Index
Xi
[Xi]
xi
u(xi)
ci
mz1
g
0.00020752
1.35·10-7
−4800
19.4
mz2
g
0.00020934
2.20·10-7
−190
0.1
mw2
g
0.00019368
1.36·10-7
200
0.0
my1
g
0.00018905
0.95·10-7
5300
11.5
Rz
mol/mol
0.04924
1.4·10-4
3.3
9.3
Rz,28
mol/mol
0.04265
7.4·10-4
0.96
23.2
Ry
mol/mol
219.34
4.62
6.5·10-5
4.1
Ry,28
mol/mol
1.021
0.115
−0.0047
12.9
Rw
mol/mol
0.6984
0.00115
0.062
0.2
Rw,28
mol/mol
18.9025
0.0664
−0.0019
0.7
Rb1
mol/mol
1.04684
5.9·10-4
−1.0
16.4
Rb2
mol/mol
0.07938
1.6·10-4
−1.4
2.2
Y
[Y]
y
uc(y)
urel(y)
K30
1
0.94991
1.48·10-3
1.55·10-3
Table 4. Uncertainty budget of K28 of the measurement corresponding to table 2. Standard
uncertainties as well as the combined measurement uncertainty are given for a coverage factor
k = 1.
- 11 -
Revised manuscript for publication in Metrologia (Special Issue 2011)
Quantity
Unit
28.1.2011
Best estimate
Standard
Sensitivity
(value)
uncertainty
coefficient
Index
Xi
[Xi]
xi
u(xi)
ci
mz1
g
0.00020752
1.35·10-7
−68
0.0
mz2
g
0.00020934
2.20·10-7
−5700
1.6
mw2
g
0.00019368
1.36·10-7
6100
0.7
my1
g
0.00018905
0.95·10-7
75
0.0
Rz
mol/mol
0.04924
1.4·10-4
40
32.8
Rz,28
mol/mol
0.04265
7.4·10-4
1.1
0.7
Ry
mol/mol
219.34
4.62
9.1·10-7
0.0
Ry,28
mol/mol
1.021
0.115
−6.5·10-5
0.0
Rw
mol/mol
0.6984
0.00115
1.9
4.7
Rw,28
mol/mol
18.9025
0.0664
−0.058
15.2
Rb1
mol/mol
1.04684
5.9·10-4
−0.014
0.0
Rb2
mol/mol
0.07938
1.6·10-4
−41
44.3
Y
[Y]
y
uc(y)
urel(y)
K28
1
1.0438
9.8·10-3
9.4·10-3
5. Conclusion
The molar mass of the "Si28" material highly enriched in
28
Si has been determined for the
first time, combining the benefits of the powerful IDMS technique and a high resolution MCICP-MS. Isotope amount ratios of samples were corrected for contaminations by measured
blank data. A novel and simple preparation route transferring the silicon crystal in a single
step into the aqueous silicate solutions avoids cumulative contamination with natural silicon.
Compared to previous investigations using natural silicon for a redetermination of the
Avogadro constant [19], the measurement uncertainty associated with the molar mass of sili- 12 -
Revised manuscript for publication in Metrologia (Special Issue 2011)
28.1.2011
con was reduced by more than one order of magnitude when using the "Si28" material instead
as predicted. This, indeed was one key step to proceed in the redetermination of NA. With the
reported measurement uncertainty of the molar mass of "Si28", it has become possible to reassess NA with a prevailing associated relative measurement uncertainty urel(NA) < 3·10-8 [20].
Finally, this paper summarizes and concentrates the current results of the molar mass data of
"Si28" used to determine NA [20], whereas the associated detailed investigations and validation experiments (e. g. influence of interferences, resolution of the MS, dependence of K factors on different variables) concerning this crystal material were described comprehensively
in [13].
Acknowledgements
The authors benefitted from valuable discussions with Giovanni Mana (Istituto Nazionale di
Ricerca Metrologica (INRIM), Italy) and Rüdiger Kessel (National Institute of Standards and
Technology (NIST), USA). Technical discussions with Johannes B. Schwieters and Michael
Deerberg (Thermo Fisher Scientific) are gratefully acknowledged. The research within this
EURAMET joint research project receives funding from the European Community’s Seventh
Framework Programme, ERA-NET Plus, under Grant Agreement No. 217257.
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Figure 1. The molar mass Mx of the "Si28" material was derived by determining the
amount−of−substance fractions x of the respective silicon isotopes. These were accessible by
measuring the isotope amount ratios R(30Si/29Si) in the sample x, the spike material y, and the
IDMS blend bx. The definition of the virtual element VE (consisting of 29Si and 30Si) of x is
visualized by a dotted frame in the arrangement of the isotopic composition of "Si28" (x).
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