X-ray Crystal Density Method to Determine the Avogadro and Planck Constants
Horst Bettin Physikalisch-Technische Bundesanstalt Braunschweig, Germany Proposed New Definition of the Kilogram
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 6.626 070 040 x 10 -34 when expressed in the unit J s, which is equal to kg m2 s-1, where the metre and the second c ∆ν are defined in terms of and Cs .
*) X represents one or more digits to be added at the time the new definition is finally adopted.
α 2 - = M (e )c NAh 2 R∞
N h −10 A = 3.990 312 7110(18) × 10 Js/mol, Amedeo Avogadro Max Planck with relative uncertainty of 0.45 × 10 -9 (1776-1856) (1858-1947)
Page 2 of 19 Avogadro Constant
Definition of Avogadro constant NA • Number of molecules per mol • 6.022... x 10 23 mol -1
Amedeo Avogadro Current definition of mol (1776-1856) • Number “entities” like 12 C atoms in 12 g • i. e. 6.022... x 10 23 12 C atoms have a mass of 12 g 12 12 g/mol = NA m( C )
Faraday constant F = NA e (e: elementary charge)
Molar gas constant R = NA k (k: Boltzmann constant)
Page 3 of 19 Counting Atoms: XRCD Method
3 Use of a silicon crystal! 1. Volume a0 of the unit cell 3 2. Volume of an atom: a0 /8 3. Volume V of a sphere 4. Number N of the atoms
8 V8 VM N = = ⋅ mol A N 3 3 a0 a0msphere
Page 4 of 19 Lattice parameter measurement (INRIM)
d220 (2011)=192014712.67(67) am
d220 (2014)=192014711.98(34) am -9 ur(2014) = 1.8 x 10
Page 5 of 19 Lattice parameter set-up at PTB
S M1
M M2
AL
AA
OH
Page 6 of 19 Sphere Interferometer of PTB
mK-temperature stabilisation Camera 1 Camera 2
Fizeau- Fizeau- Collimator Objective 1 Objective 2
Diode laser
Page 7 of 19 Diameter results (2014)
Relative Mean apparent Sphere Lab. Uncertainty diameter/nm in 10 -9 AVO28-S5c PTB 93 710 811.38(83) 9 AVO28-S5c NMIJ 93 710 811.11(62) 7 AVO28-S5c weighted 93 710 811.21(50) 5 mean AVO28-S8c PTB 93 701 526.24(66) 7 AVO 28 -S8 c NMIJ 93 701 526.29(6 8) 7 AVO28-S8c weighted 93 701 526.26(47) 5 mean
u(volume) = 1.5 x 10 -8 V
Page 8 of 19 Molar Mass Determination Modified IDMS: virtual element approach
- Three samples: 28 Si-material („ x“) MSi 30 Si-enriched („ y“) IDMS-blend („ bx “) A. Pramann et al. - R(30 Si/ 29 Si) measured in x, y, bx Metrologia 48 (2011) S20-S25
Page 9 of 19 Molar Mass Uncertainty
1 2 3 4 5 2003 2011 2014 2015 20152016 5.0E-07
2.7 x 10 -7
WASO Si28-10Pr11 Si28-23Pr10 Si28-24Pr7 5.0E-08 IRMM „Avogadro“ „kilogram-2“ „kilogram-2“ PTB PTB PTB (Si)) (Si)) M ( 28 Si: 92 % 99.995 % 99.998 % 99.9995 % rel u
5.0E-09 8.2 x 10 -9
4.4 x 10 -9 3 x 10 -9
9.8 x 10 -10 5.0E-10
Page 10 of 19 Mass Determination (at BIPM)
Sartorius CCL 1007 mass comparator with its vacuum transfer system
Surface artefacts ∆S = 186 cm 2
Page 11 of 19 Mass Mass measurements • Extraordinary calibrations using the IPK • Uncertainty of the weighted mean: 3.5 µg
0.999698450 0.999401330 Sphere AVO28-S5 Sphere AVO28-S8 0.999698445 0.999401325 5 µg NMIJ 0.999698440 0.999401320 PTB NMIJ PTB / kg / / kg / 0.999698435 0.999401315 m m WM WM BIPM 0.999698430 0.999401310 BIPM 5 µg 0.999698425 0.999401305
0.999698420 0.999401300
Page 12 of 19 Surface Characterisation
Surface Mass Method Carbon 15 µg XRF, XPS, IR Water 8 µg Gravimetric
XPS, XRF, Si oxide 70 µg XRR, SE XRR/XRF
Metals < 1 µg XRF
Si crystal
Ellipsometry
Page 13 of 19 Surface layer measurement: XPS/XRF
Page 14 of 19 Uncertainty budget (2014) Only one sphere (AVO28-S5c):
Quantity Relative uncertainty/10 -9 Contribution/% Molar mass 5 6 Lattice parameter 5 6 Surface 10 23 Sphere volume 16 59 Sphere mass 4 4 Point defects 3 2 Total 21 100
Page 15 of 19 Values of the Planck Constant (2016)
6,6260704
6,6260702
6,6260700 NRC IAC NIST-3 2014 CODATA
-1 2014
s 6,6260698 2014 2014 2
6,6260696 IAC
/ kg kg kg //m m 2010 NIST-4 h
34 6,6260694 2016 10
6,6260692
6,6260690
6,6260688
Page 16 of 19 Improvements until June 2017
Aim: Avogadro value with relative uncertainty below 1.5 x 10 -8
Improvements: a) New XPS/XRF apparatus for spheres at PTB b) New XPS apparatus for spheres at NMIJ c) Spheres with better roundness (smaller wavefront abberration) d) New lattice parameter measurement at PTB e) Avogadro constant determined using Si-28 with higher enrichment
Page 17 of 19 Existing Si-28 Single Crystals
AVO28 kg-2.1 kg-2.2 ... 99.995% 99.998% 99.9995%
Page 18 of 19 Thank you very much for your attention!
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Physikalisch-Technische Bundesanstalt Braunschweig und Berlin Bundesallee 100 38116 Braunschweig Germany
Dr. Horst Bettin Working Groups “Solid State Density” and “Avogadro Constant” Telephone: +49 531 592-3330 E-Mail: [email protected]