Ulf Jacobsson, Håkan Källgren,
Bengt Johansson, Leslie Pendrill
12th comparison between the
Swedish National Kilogram and
SP's principal kilogram mass
standards
SP Report 2004:32
Measurement Technology
Borås 2004
2
Abstract
The 12th comparison between the Swedish National Platinum-Iridium kilogram (Prototype number 40) and the principal kilogram standards for mass has been performed at SP
Swedish National Testing and Research Institute. The method used was a weighted least
square method with restraint developed by Dr Leslie Pendrill <1>. Weighings for the panEuropean key intercomparisons for 1 kg, EUROMET 509 and 510 were made together
with the regular kilogram comparison.
Key words: Sweden, SP, mass standards, kilogram, comparison, traceability
Cover: Swedish National Kilogram K40, photograph by Mats Johansson.
SP Sveriges Provnings- och
Forskningsinstitut
SP Rapport 2004:32
ISBN 91-85303-08-9
ISSN 0284-5172
Borås 2004
SP Swedish National Testing and
Research Institute
SP Report 2004:32
Postal address:
Box 857,
SE-501 15 BORÅS, Sweden
Telephone: +46 33 16 50 00
Telex: 36252 Testing S
Telefax: +46 33 13 55 02
E-mail: [email protected]
3
Contents
Abstract
2
Contents
3
Preface
5
1
1.1
1.2
Introduction
History
Present day
7
7
7
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Mass standards
K40
H
G1
MJV2
Me
Me2
Summary of weight properties
8
8
8
8
8
9
9
9
3
3.1
3.2
3.3
3.4
The balance and the weighing environment
Balance
Air tight chamber
Wobblestick and calibration weight
Atmospheric instruments
10
10
10
10
11
4
4.1
Procedure
Weighing dates
12
12
5
5.1
5.2
5.3
5.4
5.5
Calculations
Gravitational gradient
K40 Stability, drift model
Data acquisition and corrections
Model for a weighing process
Least squares fit
13
13
13
14
15
16
6
6.1
6.1.1
6.1.2
Results
Uncertainty estimation and calculation
Common contributions to the uncertainty
Individual contributions to the uncertainties
17
17
17
18
7
7.1
7.2
7.3
7.4
7.5
Tables and graphs showing mass change
National Kilogram K40
Gilded brass kilogram H
Stainless steel kilogram G1
Stainless steel kilogram MJV2
Stainless steel kilogram Me
19
19
19
20
21
22
8
8.1
Conclusions and Discussion
Summary of results
23
23
9
Acknowledgements
25
10
References
26
4
Annex A Traceability chain for mass in Sweden
Annex B Protocols from kg prototype handling
Annex C Certificates of equipment used
Annex D Control sheet for the comparisons
5
Preface
The following report describes the work performed in connection with the 12th comparison between the Swedish principal standards for one kilogram and the Swedish National
Prototype for one kilogram, no 40 (K40). It was manufactured from a rod of 90% Platinum and 10% Iridium in the 1880’s.
In general four mass standards for one kilogram made of stainless steel are used to maintain the Swedish kilogram. Every 6 to 10 years a comparison with the ultimate Swedish
reference K40 is performed.
In earlier times this interval was regulated by the Swedish act for weights and measures.
After the act’s latest revision in 1972 it is up to the National Measurement Institute to
decide when to do comparisons at the primary level.
The time chosen for this particular comparison fits nicely with the time schedule for the
key comparisons EUROMET projects 509 and 510, thereby confirming the Swedish Best
Measurement Capability as defined in the Mutual Recognition Agreement (MRA).
The Swedish kilogram K40 is regularly calibrated at the International Bureau for Weights
and Measures (BIPM) providing necessary traceability to the international kilogram prototype. The results for K40 from the latest, so called, 3rd verification was dated February
26th 1991. Since there is some time since the certificate was issued a drift model for mass
gain since calibration until the date of the comparison is used.
6
7
1
Introduction
1.1
History
The Metre Convention is a diplomatic treaty according to which every country signing it
should have the metre and the kilogram as the only legal units for length and mass.
Sweden was one among the 17 premier states signing this treaty in 1875.
Following the signing, the hard work to find a materialized artefact embodying these
measures began. Finally it was found that the metre and the kilogram could best and most
durably be materialized with a bar and a cylinder of an alloy of platinum-Iridium (90% Pt
10% Ir). Out of a batch of almost 50 prototypes the one with a mass most resembling the
mass of 1dm3 of pure water at 4 ºC was chosen as the international prototype for one
kilogram. The latter definition was proposed by the French chemist Lavoisier almost a
century earlier, in the 1790’s. The international prototype is accompanied by 6 other kilogram prototypes called “temoins” (witnesses), or official copies, that are used for more
routine measurements at the BIPM <2>.
Sweden was allotted copy no 40 (K40) of the kilograms and copy no 29 of the metres
manufactured by BIPM. Professor Robert Thalén brought the prototypes for the metre
and the kilogram from BIPM to Sweden in 1889 <3>.
1.2
Present day
Since 1960 the kilogram is the last man made artefact embodying a quantity. Work is in
progress to replace the kilogram with a definition relating the kilogram to “natural” quantities. In the future, the kilogram is expected to become a secondary unit either to electrical or atomic units. The technical challenges in redefining the kilogram to another physical quantity, while keeping the same accuracy as today, are enormous.
Thus there is a good chance that the Pt-Ir kilogram prototypes will be in use, and serve as
the principal standards for mass, several years from now.
This report presents the work that was carried out at the National laboratory for mass in
Sweden in autumn 2002. It is the 12th in an unbroken line of reports starting in 1895, each
report thoroughly describing the procedures used to determine the mass of the principal
mass standards in Sweden from the national prototype.
8
2
Mass standards
2.1
K40
This artefact is the Swedish National Prototype for one kilogram. It is made of a 90%
platinum 10% Iridium alloy in the shape of a cylinder 39 mm in diameter and 39 mm
high. It has been in the ownership of the Swedish Government since its delivery. K40 is
also at the top level of the traceability chain for mass in Sweden <Annex A>.
Over the years K40 has been calibrated at the BIPM several times, in 1889, 1948, 1991
following the international periodic verifications <4, 5, 6>. Upon request by the Swedish
National Laboratory for Mass, additional calibrations were performed in the years 1956
<7> and 1984 <8>.
Between the years 1890 – 1934 the kilogram was housed at the Swedish Royal Academy
of Sciences (KVA). In 1934 K40 was transferred from KVA in Frescati outside Stockholm to the Royal Mint (MJV) situated at Kungsholmen in central Stockholm <9>. After
the outbreak of World War II the kilogram was kept in a bombproof shelter in the basement of the Royal Mint. As a part of the Royal Mint’s centennial celebration in 1950,
K40 was exhibited to the public <10>. In 1973 the kilogram was transferred to SP. When
SP transferred to new premises in Borås 1976 K40 followed along. It has been located in
a vault at SP since then. Two excursions to BIPM have been performed though, one
between the years 1982 – 1985 <11>. Another was between the years 1988 – 1992 for
the 3rd periodic verification of the National Prototypes <12>.
2.2
H
The kilogram mass standard ”H” was manufactured of gilded brass in the 1890’s by instrument maker P.M. Sörensen in Stockholm <13>. The mass standard is of cylindrical
shape with rounded edges. This mass standard has “always” accompanied the National
Kilogram K40. However it is not entirely compatible with present day demands for precision mass standards. From the report 1890 by Ångström <14> the volume of the mass
standard at 15 ºC is 121.665 cm3 with a volume expansion coefficient of 58 10-6 K-1.
2.3
G1
G1 is a kilogram mass standard that was manufactured by Gragerts våg och viktservice
AB in Stockholm, in the year 1974 <15>. The mass standard’s shape is cylindrical with
the letters “G1” engraved at the top surface. A stainless steel alloy (DIN 4305, Uddeholm
AB) was used, with a composition of 18% Cr, 10.5% Ni 2% Mn, 1% Si and 0.15% C.
2.4
MJV2
MJV2 is an example of first generation stainless steel kilogram mass standards. The
Royal Mint purchased two copies in 1945. The copies were first used for the 6th comparison of mass standards against the national kilogram in 1945 –1949 <16>. The mass standard is in the form of a cylinder with the letters MJV2 engraved on the top face. The alloy
is an austenitic stainless steel composed of 18.6% Cr, 8.5% Ni and 0.08% C according to
an analysis performed by SP<17>.
Its sister weight, MJV1, was found to be unstable and was eventually scrapped in 1996
after the 11th comparison after considerable instability was detected<18>.
9
2.5
Me
As a reinforcement of the traceability chain among the most accurate stainless steel kilograms in Sweden the kilogram mass standard “Me” was received from Mettler-Toledo
Corporation in 1995. This mass standard is of the standardized OIML-shape and made
out of an austenitic stainless steel alloy of low magnetic susceptibility.
2.6
Me2
SP purchased the kilogram mass standard Me2 in 2001 from Mettler-Toledo Corporation.
This mass standard is a state of the art, high precision, OIML weight class E1. The austenitic stainless alloy used has virtually no magnetic susceptibility nor magnetization as
well as a density close to the desired value 8000 kg/m3. It underwent a laser marking
procedure by Svenska Maskinskyltfabriken AB, Linköping in spring 2002 when the letters “Me2” were put on the top surface. Laser marking will possibly not affect a mass
standard’s stability to the same extent as an engraving or a punch mark. But that remains
to be further investigated in comparisons that will follow.
2.7
Summary of weight properties
Volumes and densities
Table 1 Properties of SP’s principal one kilogram mass standards
Name Year of SP’s inVolume at
acquisi- ventory no 0 ºC /cm3
tion
(unc, k=1)
Volume expan- Density at
Ref., year,
sion coefficient 20 ºC /kg·m-3 Certificate
/ K-1
(unc, k=1)
No
K40
1890
46.411 5
25.869 · 10-5+
5.65 · 10-9 · ∆t
21435.4
BIPM, 1889
H
1890
121.563
58 · 10-6
8216.7
<19>, 1894
G1
600354
-6
1974
601364
124.578 (1)
43.5 · 10
8020.1 (1)
BIPM, 1984,
No 42
MJV2 1945
601354
126.657 (1)
48 · 10-6
7887.7 (1)
BIPM, 1956,
No 113
Me
1995
601380
125.317 (3)
48 · 10-6
7972.1 (2)
SP, 1996,
01-B96074
Me2
2001
602618
124.828 (3)
48 · 10-6
8011.0 (2)
SP, 2002,
P200140-50
10
3
The balance and the weighing environment
3.1
Balance
The balance used for all comparison weighings is a fully automatic commercial balance
of the type Sartorius C1000S working according to the principle of electromagnetic force
compensation. This particular balance has been in use since 1991 and served during the
11th comparison between the mass standards and the National kilogram <20>.
At present, some ten years after its acquisition, the balance is well characterized regarding
internal heat generation, short- and long-term stability etc. This makes it possible to fulfil
the special demands that this kind of measurements have.
3.2
Air tight chamber
To keep the environment as stable as possible, minimizing external influences from pressure change, an airtight chamber was used to enclose the balance and the mass standards.
The chamber is the same that was used during the 11th comparison and is described in
detail in that report <21>.
3.3
Wobblestick and calibration weight
In the previous comparison the need to dismantle the vacuum chamber for sensitivity
checks resulted in higher uncertainties as well as substantially more work <22>. To be
able to make sensitivity checks of the balance during the weighing process an item called
“wobblestick” was purchased from Nor-Cal Inc<23>. This makes it possible to manipulate small objects within the air tight chamber. Together with a specially “wrinkled” 10
mg weight it was possible to check the balance sensitivity during the weighings without
opening the chamber.
Figure 1. Pedestal and “wrinkled” 10 mg weight.
Figure 2. Wobblestick, an instrument used to manipulate small objects such as the sensitivity
weight shown above in the air tight chamber.
11
3.4
Atmospheric instruments
The displaced air volumes of a platinum- (46 cm3) and a stainless steel kilogram
(125 cm3) differ by a large amount. Since all objects are “floating” in air, air buoyancy,
this volume difference gives rise to systematic weighing errors. These errors are from the
difference in mass of the displaced air volume, which is about 90 mg. For this reason it is
crucial to know the air density as well as the mass standard density with good accuracy
and make a correction accordingly.
The air density was determined by using the BIPM formula for calculation of air density
from common air parameters <24, 25>.
The measured air parameters were pressure, temperature and relative humidity (dew
point). An auxiliary measurement was made to determine the CO2 content as well.
Table 2. Instruments used to check the atmospheric parameters
Quantity
Instrument used
s/n
Inventory no Calibration ref1
Temperature
Systemteknik 1228
6629
600 031
MTvP201555
Temp (switch)
Burster
1634
601 154
MTvP201555
Pressure
Druck DPI 141
775 / 00-03
601 797
P200140-61
Pressure (extra)
Texas 145-01
2908 / 6915
600 211
P200140-60
Humidity
Protimeter DP989M 315127
601 024
F2 09783 A
Dew point (extra) EG&G 660
0000736 / 903 600 068
F2 11160
Dew point (extra) Thommen HM30
1005578
601798
F2 11161B
CO2 content
50595
300 910
KMo300910
1
TSI 8551
Copies of all calibration certificates are collected in annex C
12
4
Procedure
The mass standards were set up in the C1000S balance in a way that minimized the number of times K40 had to be moved. Before each comparison a simple protocol was written
and signed by two persons thereby securing that all weight positions were thoroughly
checked. See annex D for an example.
Each run was performed during a prolonged time with a pre-run for a period of 12 – 20
hours before the actual comparison took place. The actual weighing scheme was to read
the balance indication for position 1 Æ 2 Æ 3 Æ 4 Æ 1 … etc. Every position was
weighed 30 times with time stamps so that the drift correlated with temperature increase
could be monitored.
The software used to govern the balance and weight handler is essentially the same as of
the 11th comparison. The weighing sequence is described in detail in <26>.
The influence of balance linearity can have a role when there are large differences in
display indication between standard and test weight. To minimize the effects from the
weighing differences between K40 and the stainless steel mass standards due to air buoyancy an additional 100 mg sheet weight was put on each of the stainless mass standards.
4.1
Weighing dates
Several weighings were performed during the dates in the tables below:
Table 3. Start and stop times for weighings where K40 was used as reference
Start date
200208-28
08-29
08-30
09-03
09-03
Start
time
23:42
22:13
23:46
08:34
23:15
End date
200208-29
08-30
08-31
09-03
09-04
End
time
06:10
04:41
06:13
15:03
05:43
Balance
pos “N”
K40
K40
K40
K40
K40
Balance
pos “1”
MJV2
MJV2
Me
Me
Me
Balance
pos “2”
K55
K651
Me2
Me2
Me2
Balance
pos “3”
Me
Me
MJV2
MJV2
MJV2
The kilograms K55 and K651 are kilogram prototypes owned by NPL, <27> and were
circulated during the intercomparison EUROMET 509.
Table 4. Start and stop times for weighings where MJV2 was used as reference
Start date
200208-20
08-23
08-24
09-11
Start
time
16:22
21:30
19:18
22:49
End date
200208-20
08-24
08-25
09-12
End
time
22:35
03:42
01:30
05:16
Balance
pos “N”
MJV2
MJV2
MJV2
MJV2
Balance
pos “1”
Me
Me
Me
G1
Balance
pos “2”
Me2
Me2
Me2
Me2
Balance
pos “3”
61
61d
61d
H
The kilograms are 61 and 61d are manufactured of stainless steel, owned by NPL, and
were circulated during the key intercomparison EUROMET 510.
13
5
Calculations
5.1
Gravitational gradient
A factor influencing high precision mass determination when masses have very different
shape and / or density is the difference in height of centre of mass for different artefacts.
This is not a large effect but comes from the property that the gravitational acceleration
reduces as one moves outward from the earth’s surface.
To calculate the magnitude of this effect the expression for the difference in gravitational
force ∆F = m·∆g associated with the gravitational gradient ∆g is used. Modelled with a
conceptual expression ∆F = ∆m·g it is assumed that ∆F depends on some apparent mass
difference ∆m instead of a difference in gravitational acceleration ∆g. According to work
made by NPL <28> the relative gradient of the gravitational acceleration can be set to
3,14·10-10 mm¯¹ near the sea level. The apparent mass difference can be calculated from
the relation below:
∆m ∆g
=
m
g
eq 5-1
where the value of the right hand expression is given by the NPL figure. A recalculation
factor 109 µg/kg leads to an apparent mass gradient of 0,314 µg/mm. There exist other
models as well <29> but in this work the NPL-figure is used.
The centre of gravity for a Pt-Ir kilogram is about 19 mm above the bottom whereas in
the case of a standard stainless steel kilogram the centre of gravity can vary depending on
its shape as seen in the following table.
Table 5. Centre of gravity for Sweden’s principal kilogram standards
Mass standard
5.2
Distance base-centre of Difference to K40
gravity / mm
/ mm
correction ∆m
/ µg )
K40
19.5
0.0
MJV2
27.2
7.7
2.4
G1
27.5
8.0
2.5
H
26.8
7.3
2.3
Me
40.2
20.7
6.5
Me2
40.2
20.7
6.5
K40 Stability, drift model
Even though all kilogram prototypes display excellent stability over time there must be
means to estimate the mass change since last calibration at the BIPM. In this work a
model described by Richard Davis, BIPM has been used <30>. From minute examinations of the BIPM official prototype No 25 the following conclusions were drawn after
having used the cleaning washing procedure in connection with calibration:
Mass increase first 3 months:
Annual increase thereafter:
Increase in uncertainty:
0.0032 mg
0.001 mg/yr
0.0004 mg/yr
14
K40 had a mass value of 1 kg - 0.035 mg with an uncertainty of 0.002 3 mg
(k=1, 12 degrees of freedom), by the time of the third verification of the National prototypes according to the calibration certificate dated May 18 1993 <31>.
The 12th comparison was performed an estimated 11.49 years after the 3rd verification
meaning that the corrected values used according to the above mentioned model were
1 kg - 0.0206 mg with an estimated uncertainty (k=1) of 0.0051 mg. These were the mass
and uncertainty values used for K40 throughout this comparison.
Simulated K40 mass change since 3rd verification
m - 1 kg (µg)
-10
-20
-30
-40
1990-09-04
1992-09-03
1994-09-03
1996-09-02
1998-09-02
2000-09-01
2002-09-01
date
Figure 3. Simulated mass change of K40 since 3rd verification. Note the bend 3 months after the
3rd verification. The other two points show the time for the 11th and 12th comparisons respectively.
5.3
Data acquisition and corrections
From the weighing sequence described in section 4 the input data consists of four time
series of balance indications
1. Measurement of time, balance indication and air density which are logged quantities
for each point.
2. Correction for heat expansion and each weight’s volume (density) according to:
V(t) = V(tref ) · α · ∆t
eq 5-2
where:
∆t is t - tref, where tref is 0 or 20 °C depending on original weight data
α is the coefficient of volume expansion for each weight (see section 2.7).
3. Calculation of each weight’s mean volume and standard deviation during one measurement series. The weight volume has an uncertainty of type A expressed as a standard deviation.
4. Adjustment of the balance readouts for air buoyancy and mass of sensitivity weight for
each measurement point, deflections.
5. Feed the deflection values into a weighted least squares fit to get the mass corrections
to the nominal mass of 1 kg.
15
5.4
Model for a weighing process
A fictitious counterweight cw can be used when modelling the weighing even though
there is a system of electromagnetic force compensation in a modern balance.
Figure 4 Symbolic view of a weighing model with a mass standard a counter weight and
a sensitivity weight.
It is easy to see that the balance indication is the difference in apparent mass between the
masses put on the load carrier to the left and the counterweight.
I = m + Tm – ρa · (Vm + VTm) – mcw + ρa · Vcw
eq 5-3
However the really interesting part is the deflection.
I’ = m – mcw
eq 5-4
that results in
I’ = I - Tm + ρa · (Vm + VTm) – ρa · Vcw
eq 5-5
If all deflections I’ are indexed,
I’N(ti) , I’1(tj) , I’2(tk) , I’3(tl)
eq 5-6
there are four curves described, one for each weight handler position, for each comparison.
A fit to a second-degree polynomial equation with respect to time
I’(t)= a + b ·t + c ·t2
eq 5-7
for the deflections I’ giving the corrected indication for each mass standard is created.
This can readily be done with the standard tools supplied with mathematical software
packages such as Excel, Mathlab or Mathcad.
Differences at pre determined times t1 t2 t3 ... tM which are mean times between weighings
for each pair of weights according to the design are made:
d1 N =
1 ⎛M '
⎞
⎜ ∑ I1 (t i ) − I N' (t i ) ⎟ + ∆m ⋅ ∆h1N
M ⎝ i =1
⎠
d 21 =
1
M
d 32 =
1 ⎛M '
⎞
⎜ ∑ I 3 (t k ) − I 2' (t k ) ⎟ + ∆m ⋅ ∆h32
M ⎝ k =1
⎠
eq 5-10
dN3 =
1 ⎛M '
⎞
⎜ ∑ I N (t l ) − I 3' (t l ) ⎟ + ∆m ⋅ ∆hN 3
M ⎝ l =1
⎠
eq 5-11
(
⎛M '
⎜ ∑ I 2 (t j ) − I1' (t j )
⎜
⎝ j =1
(
(
(
)
)⎞⎟⎟ + ∆m ⋅ ∆h
⎠
21
)
)
eq 5-8
eq 5-9
The term ∆m · ∆h gives a mass correction based on the vertical gravitational gradient as
described in section 5.1. The four different dis make up the vector d together.
16
5.5
Least squares fit
Written in matrix form the design for the differences mentioned above can be described
in matrix form as
1
1
A= 0
0
−1
0
0
0
−1 0 0
1 −1 0
0 1 −1
0
0
1
where each column symbolizes a mass standard and each row symbolizes a comparison.
Row 0 symbolizes the reference standard and is used to create the restraint together with
the weighting element W00.
The difference vector d is fed into the weighted least squares fit:
eq 5-12
c = (AT · W · A)-1 · AT · W · d
Where the c-vector contains the mass corrections to the nominal mass of 1 kg for all four
weights.
W is the weighting matrix constructed with the diagonal elements:
Wii = s (d j ) 2 ⋅
1
i≠0
1
∑ s(d ) 2
j
eq 5-13
and all off-diagonal elements = 0
The normalization condition is
∑W
ii
= 1 for i ≠ 0
The restraint for this fitting is given by the element W00, which is the mathematical
weighting assigned to the reference. The figure used in this work is 106. It is chosen by
the experimenter from experience and thus tells something about the experimenter’s confidence in the reference.
This design A with its associated least squares fit is based on the same technique as used
when performing subdivision of mass standards. <32, 33>]
17
6
Results
6.1
Uncertainty estimation and calculation
One of the most extensive tasks to do in a comparison of this type is the uncertainty calculation. Several standard publications have been issued to give guidance how to determine measurement uncertainty in calibration <34, 35>.
The uncertainty for each weight consists of a number of components. Some components
are common for all mass standards, such as balance parameters and air density. Other
components are individual for each mass standard like, for example, the result of a
weighing process or the result of density determination. In subsection 6.1.1 through 6.1.2
the method for presenting uncertainties according to EA-4/02 is used <36>.
6.1.1
Common contributions to the uncertainty
Table 6. The reference (K40) and balance uncertainty components uref and ubal.
Estimated1
unc. (k=1) Divisor
Quantity
Ref. Mass
0.0054 mg
1
Ref. Drift
1
Bal. scale div
0.0006 mg
1
Bal. sensitivity
0.0030 mg
1
Standard un- Sensitivity Contribution
certainty
coefficient
/ mg
Symbol
0.005400 mg
1 0.0054
MS
0.000000 mg
1 0.0000
δmD
0.000577 mg
1 0.0006
δmC
0.003000 mg
1 0.0030
δmS
total (k=1) 0.006 2 mg
or
6.2 µg
1
The estimated uncertainty is based on the assumption that the drift is taken care of and
the balance scale division has a rectangular distribution. Uncertainty from balance sensitivity and reference mass are taken to have normal distributions.
u ref = MS 2 + δmD 2
eq 6-1
ubal = δmC 2 + δmS 2
eq 6-2
Table 7. Air density uncertainty uair.
Quantity
Pressure
Temp
Dew pt
Estimated Relative senuncertainty
sitivity1
ci/ρa
(k=1)
4 Pa
1.00·10-5
0.035 °C
-4.00·10-3
0.27 °C
-3.00·10-4
Relative un- Uncertainty in
certainty
air density
in air density
/ kg·m-3
3.775·10-5
0.00005
-4
1.386·10
0.00017
-5
8.07·10
0.00010
Unc based on volume difference2
/ mg
0.0036
0.0131
0.0077
1.385·10-5
0.00002
0.0013
-5
6.00·10
0.00007
0.0057
Total (k=1) 0.000 21
0.0167 mg
Or
0.21 µg/cm³
16.7 µg
1
The relative sensitivity transforms an absolute uncertainty into a relative uncertainty in
air density. Multiplied with the reference density used (1.200 kg·m-3) gives the absolute
uncertainty in air density.
2
The following volumes were used: Pt-Ir 46 cm3 stainless steel 125 cm3.
CO2
Formula
35 ppm
0.4
6.00·10-5
18
Common uncertainty components
contribution / µg
20
15
10
5
0
Ref.
Ref.
drift
scale
div.
sens
press
temp
dew pt.
CO2
CIPMformula
total
Figure 5. Uncertainty components shown graphically.
The total combined common standard uncertainty
2
2
2
u common = u ref
+ ubal
+ u air
eq 6-3
3
is calculated to be 17.8 µg for all mass standards with a density near 8000 kg/m or a
volume of 125 cm3. For the mass standard H made of gilded brass with slightly higher
density, this figure amounts to 18.3 µg. Uncertainties in temperature and dew point are
the dominant uncertainty components.
6.1.2
Individual contributions to the uncertainties
Apart from the common contribution there is an individual contribution for each mass
standard, which might differ slightly depending on the conditions during the weighing
sequence.
2
2
u = u individual
+ u common
eq 6-4
Table 8. Total uncertainty (k=1) calculated from the individual and common components respectively
Mass
Standard destandard viation of the
mean / mg
Common
contribution
/ mg
Uncertainty
(k=1),
/ mg (±)
MJV2
Me
Me2
G1
H
0.0178
0.0178
0.0178
0.0178
0.0183
0.018
0.019
0.021
0.019
0.019
0.0031
0.0035
0.0098
0.0024
0.0035
19
7
Tables and graphs showing mass change
In this section both the total drift over time as well as drift during this comparison where
applicable for each mass standard are shown. The weights in section 2.4 MJV2, 2.5 Me
and 2.6 Me2 used K40 as reference. Therefore a correction from the difference in height
of centre of mass was applied after calculation of the mean mass value. The weights in
section 2.2, H and section, 2.3 G1 used MJV2 as reference. No correction for difference
in centre of mass was applied since it is almost at the same level.
7.1
National Kilogram K40
Mass for the Swedish National Kilogram K40 1894-1991
0,000
m - 1 kg / mg
1. verification
2. verification
-0,020
1984
-0,040
1991
1948
1889
1956 3. verification
-0,060
1885
1905
1925
1945
1965
1985
2005
Figure 6. K40 mass values and uncertainties (k=1), from all verifications and auxiliary weighings.
7.2
Gilded brass kilogram H
Mass for the gilded brass kilogram H 1894-2002
6,90
1988
m - 1 kg (mg)
1945
2002
1965
6,80
1980
1924
1996
1935
1914
6,70
1894
1915
1904
6,60
1885
1905
1925
1945
1965
1985
2005
Figure 7. H mass values. The value from 1955 <37> has been excluded from the graph due to its
high deviation (m – 1 kg = 7.00 mg) from the other values making the graph less readable.
20
7.3
Stainless steel kilogram G1
Below is the graph from the mass development of G1from purchase and onwards
Mass for the stainless kilogram G1 1984-2002
m - 1 kg (mg)
2,50
2,40
BIPM value
2,30
2,20
1980
1985
Figure 8. G1 mass values
1990
1995
2000
2005
2010
21
7.4
Stainless steel kilogram MJV2
Mass for the stainless kilogram MJV2 1945-2002
m - 1 kg (mg)
0,60
1955
1956
1988
0,50
1956
1945
1965
BIPM value
0,40
1940
1950
1960
1996
1970
1980
1990
2002
2000
Figure 9. MJV2 mass values. The values in 1956 were given without uncertainty <38>.
Table 9. Mass drift of MJV2 during 12th comparison.
Date
2002-08-29
2002-08-30
2002-08-31
2002-09-03
2002-09-04
m -1 kg
/ mg
0.444
0.444
0.446
0.431
0.433
unc k=1
/ mg
0.018
0.018
0.018
0.018
0.018
MJV2 mass 12th comp
0,460
0,450
0,440
0,430
0,420
0,410
2002-08-28
2002-09-01
2002-09-05
Figure. 10 MJV2 mass change during 12th comparison
The mean value from above (m – 1 kg = 0.440 mg) was then corrected for the height of
centre of mass for this particular mass standard (0.0024 mg) to become
m – 1 kg = 0.442 mg.
22
7.5
Stainless steel kilogram Me
Mass for the stainless kilogram Me 1996-2002
m - 1 kg (mg)
0,70
0,60
0,50
1994
1996
1998
2000
2002
2004
Figure 11. Me mass values. In 1998 a comparison was made with the mass standard MJV2 as
reference <39>.
Table 10. Mass drift of Me during 12th comparison.
Date
2002-08-29
2002-08-30
2002-08-31
2002-09-03
2002-09-04
m - 1 kg
/ mg
0.639
0.639
0.642
0.624
0.628
unc k=1
/ mg
0.018
0.018
0.018
0.018
0.018
Me mass 12th comp
0,660
0,650
0,640
0,630
0,620
0,610
2002-08-28
2002-09-01
2002-09-05
Figure 12. Me mass change during 12th comparison
The mean value from above (m – 1 kg = 0.634 mg) was then corrected with the height of
centre of mass for this particular mass standard (0.0065 mg) to become
m – 1 kg = 0.641 mg.
23
8
Conclusions and Discussion
In August – September 2002 the 12th comparison of the Swedish principal kilograms
against the Swedish National prototype K40 was performed. The equipment used was
essentially the same as in the 11th comparison in 1994 – 1996 except that the refractometer was omitted. The BIPM formula for air density was used for air density determination
in this work. A drift model for the National kilogram’s mass was used based on work by
BIPM on kilogram number 25.
To check the balance sensitivity during the comparison a “wobblestick” was used together with a specially (de)formed 10 mg wire weight. During the course of the comparison the weighing results showed excellent reproducibility, indicating that knowledge
about the present mass for the Swedish principal mass standards has been gained with
good confidence.
A new mass standard (OIML class E1) was brought into the traceability chain with this
comparison. A new feature with this mass standard is the laser marking “Me2” on the top
face. Whether this marking method influences the mass standard’s long time stability
remains to be seen.
Looking at a larger view one could compare the consistency between the mass standards
MJV2, G1 and Me. This has been done in an auxiliary measurement with MJV2 as standard in 1998 when a high precision mass standard from Estonia was calibrated at SP.
Again this shows a good agreement as can be seen in Figure 8 and Figure 11.
Simultaneously with this comparison two international intercomparisons were made.
EUROMET 509 deals with the mass determination of the Platinum Iridium kilograms
K55 and K651 provided by NPL (UK). The project EUROMET 510, also piloted by NPL
(UK) used the stainless steel kilograms 61 and 61d. EUROMET 510 is also registered as
a key intercomparison aimed to tie all National Measurement Institutes results together,
thereby stating the Best Measurement Capability for each laboratory <40>.
From the results in the preliminary report draft A <41>, there is an indication that the
drift model used for K40 might exaggerate the mass change slightly, however within the
uncertainty for the comparison. It is hard to quantify the result on this level due to the
uncertainties involved.
8.1
Summary of results
When performing the measurements the atmospheric parameters were logged. The values
in the following table are not taken from the automatic logs obtained during the measurements, but excerpts from notes taken during each part of the comparison to give an
estimate of the environmental conditions during the comparison measurements and its
contribution to the uncertainty.
24
Table 11. Summary of results from measurements of atmospheric parameters.
Quantity
Pressure
Temperature
Dew point
Min value
99280 Pa
20.285 ºC
10.0 ºC
Max value
101015 Pa
20.374 ºC
12.0 ºC
406 ppm
409 ppm
CO2
Contribution to uncerEstimated1 tainty in air density
unc (k=1)
/ kg·m-3
4 Pa
0.00005
0.035 °C
-0.00017
0.3 °C
-0.00010
35 ppm
BIPM-formula
0.00002
0.00007
Air Density 1.17251 kg/m³ 1.19097 kg/m³
0.000 21
The estimations are based on values in calibration certificates and repeatability of instruments.
1
As can be inferred from the tables in section 4.1, and the graphs in section 7 each mass
standard obtains several mass values with aid of the calculations described in detail in
section 5. The mean values from these results corrected for centre of mass were taken as
each mass standard’s actual mass value.
Table 12. Masses for the Swedish principal mass standards for one kilogram, 2002
Mass
Real mass,
standard m - 1 kg
/ mg
Uncertainty
(k=1),
/ mg (±)
Density at
20 °C
/ kg·m-3
MJV2
Me
Me2
G1
H
0.018
0.019
0.021
0.019
0.019
7887.6
7972.1
8011.0
8020.1
8216.7
0.442
0.641
-0.120
2.396
6.848
Uncertainty in
density, (k=1)
/ kg·m-3
0.2
0.2
25
9
Acknowledgements
The authors want to thank Mr Rauno Pykkö, SP, KM for help with CO2 measurements.
Svenska Maskinskyltfabriken AB for help with laser marking of the mass standard
“Me2”.
26
10
References
1 ”11th Comparison Between the Swedish National Kilogram and SP Principal Standards for One
Kilogram”, Johansson B., Källgren H., Pendrill L., SP-Report 1996:50, 1996
2 La Troisème Vérification périodique des prototypes Nationaux du Kilogramme, Extrait des
Procès-verbaux du Comité international des poids et mesures 82e session, BIPM, 1993
3 ”Jämförelse mellan Svenska Riksprototypen för Kilogrammet och några Staten tillhöriga
Hufvudlikare och Normalvigter”, Ekstrand Å. G., Ångström K.,
KVA handlingar, 27, 5, 1895, p 3
4 Comité Consultatif pour la masse et les grandeurs apparentées, Rapport de la 5e session, BIPM,
1993, ISBN 92-822-2132-6
5 Reference 2
6 ”The Third Periodic Verification of National Prototypes of the kilogram 1988-1992”, Girard G.,
Metrologia, 31, 1994, 317-336
7 ”Sjunde jämförelsen mellan svenska riksprototyperna för metern och kilogrammet och myntoch justeringsverkets huvudlikare”, Swensson T., Glansholm D., Walldow E., KVA
handlingar fjärde serien, 7, 3, 1958
8 ”10th Comparison of Swedish National Kilogram with National Testing Institute principal
kilogram standards” Pendrill L., Källgren H., SP-Report 1988:38, 1988
9 ”Femte jämförelsen mellan Svenska riksprototyperna för metern och kilogrammet och mynt och
justeringsverkets huvudlikare”, Grabe A., Swensson T., Walldow E., KVA handlingar tredje
serien, 15, 5, 1935, p 3
10 Reference 7 p 9
11 Reference 8 p 15
12 Reference 4 – 6
13 Reference 3 p 12
14 Reference 3 p 14
15 Reference 8 p 18
16 ”Sjätte jämförelsen mellan svenska prototyperna för metern och kilogrammet och mynt- och
justeringsverkets huvudlikare”, Grabe A., Swensson T., Walldow E., KVA handlingar fjärde
serien, 1, 7, 1950
17 Reference 8 p 17
18 Reference 1 p 30
19 Reference 3 p 14
20 Reference 1 p 9
21 Reference 1 p 8
22 Reference 1 section 2.3
23 Nor-Cal Products, Inc., P.O. Box 518, 1967 So. Oregon, Yreka, CA 96097,
http://www.n-c.com
24 "Equation for determination of the density of moist air",' Giacomo P.,
Metrologia, 18, 1982, p33-40
25 "Equation for the Determination of the Density of Moist Air 1981/91", Davis R. S.,
Metrologia, 29, 1992, 67-70
26 Reference 1 p 17-19
27 National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, United
Kingdom, www.npl.co.uk
28 ”Initial Stages in Determining the UK Mass Scale: From the National Prototype Kilogram to
the Stainless Steel Reference Kilograms”, Havard D. C., Lewis S. L., NPL report MOM 98,
1995 p 25
29 "Absolute determination of the vertical gradient of gravity", Hipkin R. G.,
Metrologia, 36, 1999, 47-52
30 BIPM Calibration of 1 kg mass standards in Platinum-Iridium since 3rd periodic verification
Davis R.S, Coarasa, CCM 2002-09
31 BIPM certificate No 22 dated May 18 1993
32 Reference 1 p 23
33 ”Neddelning av Kilogrammet”, Pendrill L., SP-Rapport 1989:22, SP, 1989
34 ISO guide to uncertainty in measurement, (GUM)
35 ”EA-4/02 Expression of the Uncertainty of Measurement in Calibration”, EA 1999
27
36 Reference 35
37 Reference 7 p 27
38 Reference 7 p 22-24
39 SP Reference No 98V12652
40 BIPM, MRA www.bipm.fr
41 After a comparison is completed the first draft (draft A) of the report is confidential. Only the
part taking laboratories may review it. Draft B i a more public draft followed by the final report.
28
1
Annex A
Traceability chain for mass in Sweden
Spårbarhetskedjan för massa
K40
1 kg
Vid behov, vanligen 6 år
MJV2
Me
Me2
1 kg
1 kg
1 kg
Vid behov, vanligen 6 år
G
3 år
1 kg
1 år
E1
1 mg-10 kg
E1kk
E1k
E11
1 mg-2 kg
1 mg-10 kg
1 mg-2 kg
1 år
1 år
1 år
E25
5 - 50 kg
E22
E2
E23
1 mg-2 kg
20 kg
1 mg-2 kg
2 år
E26
S
5 - 50 kg
1 - 20 kg
1 år
2 år
B
50 kg
F1
500 kg
Cirkel betecknar enstaka vikt,
kvadrat betecknar viktsats
1 år
LS
50 kg
10 vikter
K
500 kg
10 vikter
2 år
2 år
F14
1 mg-2 kg
H
1 kg
1
Protocols from kg prototype handling
Annex B
2
40
Annex B
3
Annex B
4
40
Annex B
1
Certificates of equipment used
Annex C
2
Annex C
3
Annex C
4
Annex C
5
Annex C
6
Annex C
7
Annex C
8
Annex C
9
Annex C
10
Annex C
11
Annex C
12
Annex C
13
Annex C
14
Annex C
1
Annex C
1
Annex D
Control sheet for the comparisons
Kontrollblankett för komparationsvägning
Förinställd starttidpunkt för vägningarna
datum:_______________
klockslag:______________
Position
kilogramvikt
Tilläggsvikt
Notering
N
______________
_____________________
____________________
1
______________
_____________________
____________________
2
______________
_____________________
____________________
3
______________
_____________________
____________________
Före komparationen Känslighetskontroll, våg 6 enligt metod 32
Visad massa känslighetsvikt
Osäkerhet i visad massa
_________________ mg
________________ mg
Före komparationen med öppna kranar på burken daggpunktsbestämning och tryck
Operatör:___________
datum:_______________
klockslag:______________
EG&G 660 visar (okorrigerat)
________________ °C
inv. nr: 600068
Protimeter DP989 visar (okorr)
________________ °C
inv. nr: 601024
Thommen HM 30 visar (okorr)
________________ %
inv. nr: 601798, 601799
Druck DPI 141 visar okorr.
________________ hPa
inv. nr: 601797
Thommen HM 30 visar okorr.
________________ hPa
inv. nr: 601798, 601799
Texas 145 visar (heltal)
________________
inv. nr: 600211
Efter komparationen med öppna kranar på burken daggpunktsbestämning och tryck
Operatör:___________
datum:_______________
klockslag:______________
EG&G 660 visar (okorrigerat)
________________ °C
Protimeter DP989 visar (okorr)
________________ °C
Thommen HM 30 visar (okorr)
________________ %
Druck DPI 141 visar okorr.
________________ hPa
Thommen HM 30 visar okorr.
________________ hPa
Texas 145 visar (heltal)
________________
2
Annex D
Example of a sheet showing weight placement on the handler for each comparison.
2
E 510 61
3
Me2
Me
1
Datum
____________
Placerat
____________
Kontrollerat ____________
MJV2
N
2
E 510 61d
3
Me2
Me
1
Datum
____________
Placerat
____________
Kontrollerat ____________
MJV2
N