X-ray Crystal Density Method to Determine the Avogadro and ... · Proposed New Definition of the...

X-ray Crystal Density Method to

Determine the Avogadro and Planck Constants

Horst BettinPhysikalisch-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

are defined in terms of c and ∆νCs.

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*) X represents one or more digits to be added at the time the new definition is finally adopted.

∞

=R

cMhN

2

)(e-2

A

α

NAh = 3.990 312 7110(18) × 10−10 Js/mol,

with relative uncertainty of 0.45 × 10-9Amedeo Avogadro

(1776-1856)

Max Planck

(1858-1947)

Avogadro Constant

Definition of Avogadro constant NA

• Number of molecules per mol

• 6.022... x 1023 mol-1

Current definition of molAmedeo Avogadro

(1776-1856)

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• Number “entities” like 12C atoms in 12 g

• i. e. 6.022... x 1023 12C atoms have a mass of 12 g

12 g/mol = NA m(12C )

Faraday constant F = NA e (e: elementary charge)

Molar gas constant R = NA k (k: Boltzmann constant)

Counting Atoms: XRCD Method

1. Volume a03 of the unit cell

2. Volume of an atom: a03 /8

3. Volume V of a sphere

4. Number N of the atoms

Use of a silicon crystal!

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30

8

aV

N =sphere

molA m

MaV

N ⋅=30

8

Lattice parameter measurement (INRIM)

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d220(2014)=192014711.98(34) am

d220(2011)=192014712.67(67) am

ur(2014) = 1.8 x 10-9

Lattice parameter set-up at PTB

S

M1

M M2

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AL

AA

O H

Sphere Interferometer of PTB

Camera 1 Camera 2

mK-temperature stabilisation

Collimator

Fizeau-Objective 1

Fizeau-Objective 2

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Diode laser

Diameter results (2014)

Sphere Lab. Mean apparent diameter/nm

Relative Uncertainty

in 10-9

AVO28-S5c PTB 93 710 811.38(83) 9

AVO28-S5c NMIJ 93 710 811.11(62) 7

AVO28-S5c

weighted mean

93 710 811.21(50) 5

AVO28-S8c PTB 93 701 526.24(66) 7

AVO28-S8c NMIJ 93 701 526.29(68) 7

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AVO28-S8c NMIJ 93 701 526.29(68) 7

AVO28-S8c

weighted mean

93 701 526.26(47) 5

u(volume) = 1.5 x 10-8 V

Molar Mass Determination

Modified IDMS: virtual element

approach

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- Three samples: 28Si-material („x“) 30Si-enriched („y“)

IDMS-blend („bx“)

- R(30Si/29Si) measured in x, y, bx

MSi

A. Pramann et al.

Metrologia 48 (2011) S20-S25

5.0E-08

5.0E-07

1 2 3 4 5(S

i))

2.7 x 10-7

2003 2011 2014 2015 2015

Si28-10Pr11„Avogadro“

PTB

Si28-23Pr10„kilogram-2“

PTB

Si28-24Pr7„kilogram-2“

PTB

WASOIRMM

2016

Molar Mass Uncertainty

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5.0E-10

5.0E-09

ure

l(M(S

i))

8.2 x 10-9

4.4 x 10-9

3 x 10-9

9.8 x 10-10

99.995 % 99.998 % 99.9995 %28Si: 92 %

Mass Determination (at BIPM)

Sartorius CCL 1007 mass comparatorwith its vacuum transfer system

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Surface artefacts∆S = 186 cm2

Mass

Mass measurements

• Extraordinary calibrations using the IPK

• Uncertainty of the weighted mean: 3.5 µg

0.999698445

0.999698450

NMIJ

0.999401325

0.999401330

5 µg

Sphere AVO28-S5 Sphere AVO28-S8

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PTB

BIPM

NMIJ

WM

0.999698420

0.999698425

0.999698430

0.999698435

0.999698440

m/ k

g

5 µgBIPM

NMIJ

PTB

WM

0.999401300

0.999401305

0.999401310

0.999401315

0.999401320

m/ k

g

5 µg

Surface Characterisation

XRR/XRF

Carbon

Water

Si oxide

XRF, XPS, IR15 µg

Gravimetric8 µg

XPS, XRF,XRR, SE

70 µg

MethodMassSurface

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Ellipsometry

XRR/XRF

Metals

Si crystal

XRF< 1 µg

Surface layer measurement: XPS/XRF

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Quantity Relative uncertainty/10-9 Contribution/%

Molar mass 5 6

Lattice parameter 5 6

Surface 10 23

Uncertainty budget (2014)

Only one sphere (AVO28-S5c):

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Surface 10 23

Sphere volume 16 59

Sphere mass 4 4

Point defects 3 2

Total 21 100

Values of the Planck Constant (2016)

6,6260696

6,6260698

6,6260700

6,6260702

6,6260704/ k

g m

2s-

1

NIST-32014

IAC

NRC 2014

IAC 2014 CODATA

2014

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6,6260688

6,6260690

6,6260692

6,6260694

6,6260696

1034

h/ k

g m

IAC2010 NIST-4

2016

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

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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

Existing Si-28 Single Crystals

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AVO28 kg-2.1 kg-2.2 ...99.995% 99.998% 99.9995%

Thank you very much for your attention!

Questions?

Comments?

Physikalisch-Technische BundesanstaltBraunschweig und BerlinBundesallee 10038116 BraunschweigGermany

Dr. Horst BettinWorking Groups “Solid State Density” and “Avogadro Constant”Telephone: +49 531 592-3330E-Mail: [email protected]

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X-ray Crystal Density Method to Determine the Avogadro and ... · Proposed New Definition of the...

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