OutlineOutline

液中ひょう量法による固体密度測定と液中ひょう量法による固体密度測定と不確かさの解析不確かさの解析

Kenichi FUJIIFluid Properties Section

Material Properties and Metrological Statistics Division

National Metrology Institute of Japan

• 測定原理

• 装置

• 測定例

• 不確かさの解析例

Basic principle of the hydrostatic weighingBasic principle of the hydrostatic weighing

g(z): acceleration due to gravity

zFluid with a density ρ(z)

Solid materialvolume Vmass m zzgzppp

z

z∫=−=Δ2

1

d)()(12 ρ

Buoyancy force: ρVg

Gravity force: mgHydrostatic pressure

p1: hydrostatic pressure at z = z1

p2: hydrostatic pressure at z = z2

National Metrology Institute of Japan (NMIJ)National Institute of Advanced Industrial Science and Technology (AIST)

固体密度標準物質固体密度標準物質

シリコン単結晶

• 高純度、無転位、大寸法の単結晶が容易に入手可能

• 完全に近い結晶性を有するので密度が極めて安定

• 使用中の化学的純度低下による密度変化がない

• 安定同位体28Si、29Si、30Siの同位体組成のばらつきによるシリコン単結晶の密度の相対偏差：1 × 10−5

• 20 ℃、101.325 kPaにおいて約2329 kg/m3

Solid density comparison by hydrostatic weighingSolid density comparison by hydrostatic weighing

from H. A. Bowman, R. M. Schoonover, and C. L. Carroll: A Density Scale Based on Solid Objects, J. Res. Nat. Bur. Stand., Vol. 78A, No. 1, 1974, pp. 13-40.

First attempt at the NBS in 1970s

Reference standard: steel spheres

Working liquid: fluorocarbon

Polishing of 1 kg silicon spheres at the CSIRO

from Applied Optics, Vol. 26, No. 4, pp. 600-601, 1987

Solid density comparison by hydrostatic weighingSolid density comparison by hydrostatic weighing

from A. M. Peuto, A. Sacconi, M. Mosca, K. Fujii, M. Tanala, and Y. Nezu: Comparison of Silicon Density Standards at NRLM and IMGC, IEEE Trans. Instrum. Meas., Vol. 42, No. 2, 1993, pp. 242-246

Single-pan knife-edge balance

single-pan knife-edge balance

Reference standard: silicon spheres

Hydrostatic weighing apparatus developed at the IMGC

Working liquid: water

Optical interferometer at the NMIJ to measure Optical interferometer at the NMIJ to measure the diameter of the silicon spherethe diameter of the silicon sphere

Flat etalon with optical-frequency tuning system

National Metrology Institute of Japan (NMIJ)National Institute of Advanced Industrial Science and Technology (AIST)

Density of Silicon SpheresDensity of Silicon Spheres

Diameter, volume, mass, density, and their uncertainties⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯Quantity State Unit Sphere S4 Sphere S5 Combined Relative

standard combineduncertainty standarduc uncertainty

uc,r /10−6

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯Diameter 22.5 °C, 0 Pa mm 93.617 8242 93.618 8591 0.000 0022 0.023

Volume 22.5 °C, 0 Pa cm3 429.609 872 429.624 120 0.000 032 0.07420.0 °C, 101 kPa cm3 429.601 149 429.615 397 0.000 046 0.107

Mass 0 Pa g 1000.578 606 1000.612 025 0.000 015 0.015101 kPa g 1000.578 610 1000.612 029 0.000 016 0.016

Density 22.5 °C, 0 Pa kg m−3 2329.040 07 2329.040 62 0.000 18 0.07620.0 °C, 101 kPa kg m−3 2329.087 37 2329.087 92 0.000 25 0.108

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯at 22.5 °C and 0 Pa, correlation coefficient for the volumes: r(VS4, VS5) = 0.777

correlation coefficient for the masses: r(mS4, mS5) = 0.971

Hydrostatic weighing apparatus of the NMIJHydrostatic weighing apparatus of the NMIJ

Weights

Automatic handler

Single-pan flexure-hinge electronic balance

Glass vessel

Tridecane

Temperature controlledwater bath

Counter weight

Silicon sphere S4

Silicon sphere S5

Solid sample

from K. Fujii, A. Waseda, and N. Kuramoto: Development of a silicon density standard and precision density measurements of solid materials by hydrostatic weighing, Meas. Sci. Technol., Vol. 12, 2001, pp. 2031-2038

F. Spieweck, A. Kozdon, H. Wagenbreth, H. Toth, and D. Hoburg: A Computer–controlled Solid-density Measuring Apparatus, PTB-Mitteilungen, Vol. 100, 1990, pp. 169-173

Details of the hydrostatic weighing apparatusDetails of the hydrostatic weighing apparatus

Weight exchange mechanism

To electronic balance

Temperature controlledwater bath

Counter weight

Silicon sphere S4

Silicon sphere S5

Solid sample

Details of the hydrostatic weighing apparatusDetails of the hydrostatic weighing apparatus

single-pan knife-edge balance

Silicon sphere S4

Counter weight

Solid sample

Silicon sphere S5

to electronic balance

to vertical translation stages

ABCD

ABCD

Triangular cage

Loading device A

Guide

Wind insulator

Sample support

Loading device B

Loading device C

Loading device D

Adapter

Counter weight

Silicon sphere S4

Silicon sphere S5

Solid sample A

Mettler AT-1005

Handler

Loading of 1 kg weight

Mass: m1000

Volume: V1000

Tridecane

Gravity force: m1000gbal

Air buoyancy: ρairV1000gbal

Balance reading: B0 < 1 gDifference from 1 kg

Balance force: KB0gbal + Fconst

Cage

Step 1

Mettler AT-1005

Handler

Counter weight

Loading of silicon sphere S4

Silicon sphere S5

Solid sample A

Loading of 325 g weightMass: m325

Volume: V325

Tridecane

Gravity force: m325gbal

Air buoyancy: ρairV325gbal

Balance reading: BS4 < 1 gDifference from 1 kg

Balance force:KBS4gbal + Fconst

Cage

Step 2

Gravity force: mS4gS4

Liquid buoyancy: ρliq,S4VS4gS4

Liquid density near S4 at temperature tliq

from Steps 1 and 2

mS4 gS4 − ρliq,S4(t)VS4(t)gS4 + m325 gbal − ρair(tair)V325(tair)gbal − KBS4 gbal

= m1000 gbal − ρair(tair)V1000(tair) gbal − KB0 gbal

( ) ( ) [ ] ( )[ ]( )[ ] 4SliqSiS4

airSSbal3251000airairbalS40bal1000325S4S4liqS4liq, C2031)C20(

C201)C20()C20()()(gtV

tgVVtgBBKgmmgmt°−+°

°−+°−°+−+−+=

αβρρ

Liquid density near S5 at temperature tliq( ) ( ) [ ] ( )[ ]

( )[ ] S5liqSiS5

airSSbal3251000airairbalS50bal1000325S5S5liqS5liq, C2031)C20(

C201)C20()C20()()(gtV

tgVVtgBBKgmmgmt°−+°

°−+°−°+−+−+=

αβρρ

Step 3: loading of S5 and 325 g weight

Liquid density near solid sample A at temperature tliq

( ) ( ) ( )2

liqS5liq,liqS4liq,liqAliq,

ttt

ρρρ

+=

Step 4: loading of solid sample A (1 kg silicon sample) and 325 g weight

Volume of solid sample A at temperature tliq

( ) ( ) [ ] ( )[ ]AliqAliq,

airSSbal3251000airairbalA0bal1000325AAliqA )(

C201)C20()C20()()(gt

tgVVtgBBKgmmgmtVρ

βρ °−+°−°+−+−+=

Density of solid sample A at temperature tliq

ρA(tliq) = mA/VA(tliq)

Step 5: calibration of the balance sensitivity K

Steps 1 to 5: a single density-determination of the solid sample, taking approximately 20 minutes

Δρ = ρA − (ρS4 + ρS5)/2

Example of the density measurementExample of the density measurementSolid sample: 1 kg silicon crystalSolid sample: 1 kg silicon crystal

2329.080

2329.081

2329.082

2329.083

2329.084

2329.085

2329.086

2329.087

2329.088

2000/5/2612:00

2000/5/270:00

2000/5/2712:00

2000/5/280:00

2000/5/2812:00

2000/5/290:00

2000/5/2912:00

Date & time

Den

sity

at 2

0 ℃

and

101

kP

a /(k

g m

−3)

Mean density: 2329.08377 kg m−3

Standard deviation: 0.00055 kg m−3

Degrees of freedom: 193

y = f(x1, x2, . . . , xN)

Evaluation of UncertaintyEvaluation of UncertaintyGuide to the Expression of Uncertainty in MeasurementGuide to the Expression of Uncertainty in Measurement

∑ ∑∑

∑∑−

= +==

= =

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=

∂∂

∂∂

=

1

1 11

22

1 1

2c

),(2)(

),()(

N

i

N

ijji

ji

N

ii

i

N

i

N

jji

ji

xxuxf

xfxu

xf

xxuxf

xfyu

2],),(,,,[],),(,,,[ 2121 NiiNii

ixxuxxxfxxuxxxfZ ⋅⋅⋅−⋅⋅⋅−⋅⋅⋅+⋅⋅⋅

=

∑∑= =

=N

i

N

jjiji xxrZZyu

1 1

2c ),()(

Uncertainty evaluation of the density difference, Δρ = ρA − (ρS4 + ρS5)/2, under the presence of covariance. Correlation coefficients: r(mS4, mS5) = 0.971, r(mA, mS4) = 0.964, r(mA, mS5) = 0.936, and r(VS4, VS5) = 0.777.⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯Influence quantity Symbol u(xi) Zi

2/(kg/m3)2 or2ZiZjr(xi, xj)/(kg/m3)2

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯Mass

S4 at 0 Pa mS4 15 μg 1.32 × 10−9

S5 at 0 Pa mS5 15 μg 1.32 × 10−9

1 kg weight m1000 28 μg 0.00 × 10−9

325 g weight m325 51 μg 0.00 × 10−9

1 kg silicon sample mA 16 μg 6.00 × 10−9

VolumeS4 at 22.5 ºC and 0 Pa VS4 0.000 031 cm3 0.00 × 10−9

S5 at 22.5 ºC and 0 Pa VS5 0.000 032 cm3 0.00 × 10−9

Standard deviation of the mean 0.000 041 kg/m3 1.71 × 10−9

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯without evaluation of covariances uc, r(Δρ) = 4.4 × 10−8

Covariance between mS4 and mS5 r(mS4, mS5) 2.56 × 10−9

Covariance between mA and mS4 r(mA, mS4) −5.42 × 10−9

Covariance between mA and mS5 r(mA, mS5) −5.26 × 10−9

Covariance between VS4 and VS5 r(VS4, VS5) 0.00 × 10−9

with evaluation of covariances uc, r(Δρ) = 2.0 × 10−8

National Metrology Institute of Japan (NMIJ)National Institute of Advanced Industrial Science and Technology (AIST)

共分散の評価共分散の評価

Example: density measurement of 1 kg Si crystalwithout covariance with covariance

uc,r(ρA) 7.2 × 10−8 7.4 × 10−8

uc,r(VA) 8.1 × 10−8 7.3 × 10−8

uc,r(Δρ) 4.4 × 10−8 2.0 × 10−8

Reduction of the uncertainty in the density difference measurement: uc,r(Δρ) = 2.0 × 10−8

when the effect of covariance is evaluated.

質量 1 kg のシリコン単結晶の密度校正における不確かさ 不確かさの要因 記号 値又は平均

値

単位 標準不確かさ 単位 感度係数 密度校正への影響

(kg/m3)

自由度

単結晶シリコン球体S4の質量 mS4 1000.578610 g 0.000026 g 4.213E+00 0.00011 63.3

単結晶シリコン球体S5の質量 mS5 1000.612029 g 0.000026 g 4.213E+00 0.00011 63.3

1 kgステンレス鋼製分銅の質量 m1000 1000.001085 g 0.000075 g 2.435E-03 0.00000 50.0

325 g分銅群の質量 m325 325.000209 g 0.000029 g -2.435E-03 0.00000 50.0

校正器物の質量 mS 1000.249152 g 0.000027 g -4.841E+00 -0.00013 17.1

単結晶シリコン球体S4の体積 VS4(20 °C) 429.601149 cm3 0.000058 cm3 -3.361E+00 -0.00019 163.1

単結晶シリコン球体S5の体積 VS5(20 °C) 429.615397 cm3 0.000059 cm3 -3.361E+00 -0.00020 166.0

1 kgステンレス鋼製分銅の体積 V1000(20 °C) 125.421000 cm3 0.037500 cm3 -2.889E-07 0.00000 50.0

325 g分銅群の体積 V325(20 °C) 40.800600 cm3 0.024750 cm3 2.889E-07 0.00000 50.0

液体の温度（平均値） tliq 19.984 ℃ 0.003 ℃ -1.516E-10 0.00000 50.0

空気温度（平均値） tair 20.083 ℃ 0.200 ℃ -1.100E-08 0.00000 50.0

空気密度（平均値） ρ’air 1.1867 kg/m3 0.0024 kg/m3 -2.060E-04 0.00000 50.0

シリコン単結晶の体膨張係数 βSi 0.00000767 /K 0.00000003 /K -1.516E-05 0.00000 50.0

ステンレス鋼製分銅の体膨張係数 βSS 0.00004500 /K 0.00000300 /K -2.046E-05 0.00000 50.0

電子天びんの感度 K 0.99973572 0.00000162 -1.689E+00 0.00000 50.0

重力勾配の影響 C 0.99999969 0.00000002 -4.590E-02 0.00000 50.0

液体中の密度勾配の非線形性 0.00000004 2.329E+03 0.00009 50.0

平均値の標準偏差 0.00006 kg/m3 1.000E+00 0.00006 34.0

密度の合成標準不確かさ 0.00036

密度の有効自由度 νeff 412.2

体積の有効自由度 νeff 215.2

t分布表から求められるt95(νeff) t95(νeff) 1.966

t分布表から求められるt95(νeff) t95(νeff) 1.971

密度（合成標準不確かさ） ρS(20 °C) 2329.07274 kg/m3 0.00036 kg/m3

体積（合成標準不確かさ） VS(20 °C) 429.462392 cm3 0.000072 cm3

密度の拡張不確かさ U95(ρS) 0.00071 kg/m3

体積の拡張不確かさ UU9955((VVSS)) 0.000142 cm3

National Metrology Institute of Japan (NMIJ)National Institute of Advanced Industrial Science and Technology (AIST)

Examples of density measurements for Examples of density measurements for solid materials by hydrostatic weighingsolid materials by hydrostatic weighing

Material Mass Density Volumem/g ρ/(kg/m3) U95(ρ)/(kg/m3) V/cm3 U95(V)/cm3

Silicon crystal 1000.25 2329.072 74 0.000 71 429.462 39 0.000 14

Silicon crystal 500.57 2329.1580 0.0029 214.912 87 0.000 29Silicon crystal 202.21 2329.0795 0.0089 86.817 64 0.000 36Silicon crystal 78.09 2329.110 0.023 33.529 65 0.000 35Silicon crystal 30.81 2329.169 0.060 13.228 29 0.000 35

Stainless steel 1000.00 7965.966 0.021 125.534 07 0.000 33Stainless steel 500.00 7995.164 0.037 62.537 78 0.000 29Stainless steel 200.00 7994.93 0.10 25.015 84 0.000 33Stainless steel 100.00 7995.26 0.20 12.507 41 0.000 32Stainless steel 50.00 7995.13 0.43 6.253 81 0.000 34Quartz 170.61 2648.944 0.014 64.405 69 0.000 36Polystyrene 1.43 1048.14 0.19 1.364 82 0.000 33Gold 101.21 19279.9 1.3 5.249 72 0.000 39

National Metrology Institute of Japan (NMIJ)National Institute of Advanced Industrial Science and Technology (AIST)

Certified Reference Materials for Density MeasurementCertified Reference Materials for Density MeasurementDensityDensity--calibrated Si crystalscalibrated Si crystals

Cylinder (1 kg)Diameter: 90 mm

Sphere (1 kg)Diameter: 94 mm

Ring (200 g)Diameter: 90 mm

National Metrology Institute of Japan (NMIJ)National Institute of Advanced Industrial Science and Technology (AIST)

TraceabilityTraceabilityOptical frequency standard Prototype kilogram

SI base unit Length standard (m) Mass standard (kg)

Interferometetry Mass measurement

HydrometerCalibration for users Standard liquid Solid materials

CalibrationComparison

Hydrometer Vibrating-tube densimeter

Stainless steel weight, glass, Si crystal, metal

Silicon cylinderSecondary standardWeighing Hydrostatic

weighingHydrostatic weighing

Silicon ring

Primary standard Silicon sphere (kg/m3)

Hydrostatic weighing

National Metrology Institute of Japan (NMIJ)National Institute of Advanced Industrial Science and Technology (AIST)

International ComparisonInternational Comparison韓国 KRISSとの２国間比較 （ステンレス分銅） 2001

KRISS: ρ = 8006.017 kg/m3 U95(ρ) = 0.030 kg/m3

NMIJ: ρ = 8006.046 kg/m3 U95(ρ) = 0.020 kg/m3

KRISS/MO-2001-058: Final results of bilateral density comparison between NMIJ and KRISS for 1 kg weight

ドイツPTBとの2国間比較 (シリコン結晶) 2005ρNMIJ (hydrostatic weighing) - ρPTB (pressure-of-flotation) = -0.000 13 kg/m3

ρNMIJ (hydrostatic weighing) - ρPTB (Absolute measurement) = 0.000 08 kg/m3

H. Bettin, H. Toth, A. Waseda and K. Fujii: Comparison of density difference measurements at PTB and NMIJ, IEEE Trans. Instrum. Meas., Vol. 54, No. 2, 2005, pp. 877-881

国際度量衡委員会の密度基幹比較CCM.D-K1（シリコン結晶) 20068 NMIs（NMIJ, PTB, INRIM, CEM, CENAM, METAS, KRISS, NRC）

ρNMIJ - ρRef = -0.000 10 kg/m3 U95(ρNMIJ - ρRef ) = 0.000 24 kg/m3

Comparison of the relative mass differences from the reference Comparison of the relative mass differences from the reference value. The bars express the expanded uncertainties.value. The bars express the expanded uncertainties.

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

NM

IJ

PTB

IMG

C

KR

ISS

ME

TAS

NR

C

CE

M

CE

NA

M

Ref

eren

ce v

alue

Rel

ativ

e m

ass

diffe

renc

e in

10

−6

CCM.DCCM.D--K1K1

Comparison of the relative volume differences from the referenceComparison of the relative volume differences from the referencevalue. The bars express the expanded uncertainties.value. The bars express the expanded uncertainties.

CCM.DCCM.D--K1K1

-6

-4

-2

0

2

4

6

NM

IJ

PTB

IMG

C

KR

ISS

ME

TAS

NR

C

CE

M

CE

NA

M

Ref

eren

ce v

alue

Rel

ativ

e vo

lum

e di

ffere

nce

in 1

0−6

Comparison of the relative density differences from the referencComparison of the relative density differences from the reference e value. The bars express the expanded uncertainties.value. The bars express the expanded uncertainties.

CCM.DCCM.D--K1K1

-6

-4

-2

0

2

4

6

NM

IJ

PTB

IMG

C

KR

ISS

ME

TAS

NR

C

CE

M

CE

NA

M

Ref

eren

ce v

alue

Rel

ativ

e de

nsity

diff

eren

ce in

10

−6

National Metrology Institute of Japan (NMIJ)National Institute of Advanced Industrial Science and Technology (AIST)

CMCsCMCs for Solid Densityfor Solid Density

Calibration item

Calibration range CMC(k = 2)

Silicon single

crystals

2320 kg/m3

to2340 kg/m3

Solid materials

800 kg/m3

to20 000 kg/m3

20 °C30 g to less than 1000 g

0.29 mm3 to 0.36 mm3

(Uncertainty in density depends on the mass of silicon single crystals)

20 °C1000 g to 1010 g

0.000 71 kg/m3

Solid density

20 °C1 g to 1010 g

0.29 mm3 to 0.39 mm3

(Uncertainty in density depends on the mass of solid materials)

National Metrology Institute of Japan (NMIJ)National Institute of Advanced Industrial Science and Technology (AIST)

まとめまとめ

Principle of hydrostatic weighingArchimedes’ principle

Hydrostatic weighingVery precise solid-density comparator

Evaluation of covarianceAchieving uc,r(Δρ) = 2.0 × 10−8

Uncertainty in the volume calibrationfor any solid samples: 0.29 mm3 to 0.39 mm3 (k = 2)

International and bilateral comparisonsShowing consistency and equivalence