Bilateral key comparison CCM.P-K3.1 for absolute pressure

measurements from 3 × 10-6 Pa to 9 × 10-4 Pa

J. A. Fedchak1, Th. Bock

2, and K. Jousten

2

Abstract. This report describes the bi-lateral key-comparison CCM.P-K3.1 between the

National Institute of Standards and Technology (NIST) and Physikalisch-Technische

Bundesanstalt (PTB) for absolute pressure in the range of 3 × 10-6

Pa to 9 × 10-4

Pa. This

comparison was a follower to the CCM.P-K3 comparison. Two ionization gauges and two

spinning rotor gauges (SRGs) were used as the transfer standards for the comparison. The

SRGs were used to compare the standards at a pressure of 9 × 10-4

Pa and to normalize the

ionization gauge readings. The two ionization gauges were used to compare the standards in

the pressure range of 3 × 10-6

Pa to 3 × 10-4

Pa. Both laboratories used dynamic expansion

chambers as standards in the comparison. The two labs showed excellent agreement to each

other and to the CCM.P-K3 key comparison reference value (KCRV) over the entire range.

1 NIST: National Institute of Standards and Technology, United States of America

2 PTB: Physikalisch-Technische Bundesanstalt, Germany

2

1. Introduction

The CCM.P-K3 was the first key comparison for absolute pressures in the range of 3 × 10-6

Pa to

9 × 10-3

Pa and was carried out from 1998 to 2002, with the final report being published in 2010 [1].

As discussed in the report, PTB discovered problems with their equipment that led to measurement

errors larger than their uncertainty budget. Consequently PTB did not show equivalence to the key-

comparison reference value (KCRV) over the pressure range of 9 × 10-6

Pa to 9 × 10-3

Pa. PTB fixed

the problems and, before the publication of the final report, it was decided to conduct a bi-lateral

comparison between NIST and PTB, with NIST as the pilot. It was decided that the protocol would

closely follow that of the CCM.P-K3, including using the same spinning rotor gauges (SRGs) and the

same Stabil-Ion1 gauge used in the P-K3 key comparison, and that the measurements would be carried

out from 2009 to 2011.

Unfortunately, in the first phase of the comparison both labs experienced technical problems with their

primary systems during the measurements in addition to problems with the transfer package, which

consisted of one Stabil-Ion gauge, two glass Bayard-Alpert gauges (BAGs), and two SRGs. The first

round of measurements was carried out by NIST during 2009 and the transfer standards were shipped

to PTB for the second round of measurements. During the second round of measurements at PTB, a

solenoid failure caused a roughing valve on the outlet of a turbo-molecular pump to close during their

bake-out procedure. The bake continued at an elevated pressure and caused a roughly 10% change in

the accommodation coefficient of both SRGs. Nevertheless, PTB completed their measurements and

the transfer package was shipped to NIST for the third round of measurements. During the third

round, NIST experience a power failure which caused their standard’s vacuum system to vent,

resulting in a filament failure in one of the glass-BAGs and a seal failure in the NIST standard, which

then required maintenance. It addition, the Stabil-Ion gauge controller was showing instability of the

scale factor and had physical damage from shipping.

Considering all the problems, both laboratories agreed to begin over and create a new and slightly

different protocol with a different transfer package. The new transfer package included the NIST

SI-404 Stabil-Ion gauge used in the CCM.P-K3, but with a new controller. PTB provided a second

Stabil-Ion gauge with controller and two spinning rotor gauges (SRGs) with rotors that could be

secured under vacuum during shipment by a spring attached to an all-metal vacuum valve. In

addition, it was decided to use N2 as the calibration gas instead of the Ar gas that was specified in the

original protocol. Originally NIST provided two heads for the two SRGs, but these where lost during

shipment between the second and final rounds of measurements. The measurements were performed

from January 2010 September 2011. Since the original protocol was not ultimately followed, there is

no need of further discussion of it here. In what follows, only the second protocol used for the

CCM.P-K3.1 will be discussed.

2. Primary Standards

2.1. NIST Dynamic Expansion Standard

In the dynamic expansion technique, a known flow of gas passes through an orifice of known

conductance into a region of lower pressure, and gas-dynamic calculations are used to determine a

standard pressure generated upstream of the orifice. The NIST high vacuum standard [2] with the

1Commercial equipment, instruments, or materials are identified in this paper in order to specify the

experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement

by the NIST or PTB, nor is it intended to imply that the materials or equipment identified are necessarily the best

available for the purpose.

3

associated constant-pressure flowmeter [3] was used to calibrate the gauges in this comparison. The

standard consists of two main elements: a low-range flowmeter, and a dynamic-expansion vacuum

chamber where the vacuum gauges are mounted for calibration. A partition containing the orifice

separates the chamber into two approximately equal volumes. The orifice is 1.1 cm in diameter and is

mounted in a plate that is attached to a lifting mechanism. When the orifice plate is in the raised

position, the upper and lower chambers are connected to a larger diameter hole that allows a larger

effective pumping speed for evacuating the upper chamber. When the orifice plate is lowered into

place, it seals against a liquid gallium-indium alloy that fills a groove in the partition. The vacuum

chamber has a volume of approximately 180 L and is evacuated by a turbo-molecular pump connected

to the lower chamber. The full calibration range of the standard is 10-7

Pa to 10-1

Pa.

2.2. PTB Dynamic Expansion Standard

The PTB primary standard used for this comparison is a dynamic expansion system, called CE3 [4],

with its flowmeter FM3 [5]. A known gas flow is injected into a flow divider chamber, which then

flows into either a UHV chamber through a larger conductance orifice, or into a XHV (extreme high

vacuum) chamber through a smaller conductance orifice with about 1/100 of the conductance to the

UHV chamber. The XHV and UHV chambers are each evacuated with cryo-pumps through pump

orifices of similar conductances; when both pumps are operating the generated pressure in the XHV

chamber is about a factor of 100 lower than that in the UHV chamber. For the present comparison,

only the pressure generated in the UHV chamber was used while the XHV chamber was pumped by a

turbo-molecular pump. This only slightly increased the uncertainty of the generated pressure.

2.3. PTB Static Expansion Standard

The static expansion technique is used to generate a low pressure by allowing gas to expand from a

small volume, where the gas pressure is high enough to be read by stable high-quality pressure gauge,

into a larger volume. The subsequent lower pressure in the larger volume is calculated from the

volume ratio of the two volumes, which is measured in a separate experiment, and the gas pressure in

the small volume. Several expansions may be required to obtain the desired lower pressure. The PTB

static expansion standard was not used to generate any of the target pressures in the comparison, but

their static expansion standard SE2 was used to make accommodation coefficient measurements that

were used to help establish the stability of the spinning rotor gauges. A more detailed description of

the SE2 standard can be found in Ref. [6].

Table 1. Transfer standards used in the CCM.P-K3.1

Identifier Pressure range for

calibration /Pa

Gauge

Description

Controller Comments

SI-404 3 × 10-6

to 9 × 10-4

Granville-Phillips

Stabil-Ion Gauge

Model GP370; s/n

370B1055, IG1, F1

Memory module

YEA053845 was

installed and loaded

SI-1004 3 × 10-6

to 9 × 10-4

Granville-Phillips

Stabil-Ion Gauge

Model GP360; s/n

97031709, IG1, F1

F11 9 × 10-4

Spinning Rotor

Gauge, chrome-

steel ball of

nominal diameter

4.75 mm

None Rotor was transported

in a thimble attached

to an all-metal high-

vacuum valve under

vacuum and secured

with a spring attached

to the bonnet seal.

4

3. Transfer Standards

The transfer package (Table 1) consisted of two Stabil-Ion gauges with controllers and cables, two

spinning rotor gauges with two heads, an electrical box for measuring the ion-gauge electrical

parameters, and a hand-held digital voltmeter. The SRG rotors where contained in a thimble

connected to an all-metal valve, to which was attached a spring that pushed against the rotors when the

valve was closed, thus securing the rotors under vacuum during shipment. The SRG heads, digital

voltmeter, and electrical box were lost during shipment between cycles one and two; therefore NIST

used a different voltmeter, SRG heads, and electrical box during the final round. These changes were

not considered critical for the comparison. Otherwise, no problems with the transfer standards were

reported during this comparison.

4. Organization of the key comparison

The chronology of the measurements reported in this key comparison is listed in Table 2. A total of

three cycles of data were taken: the first and third cycles were taken at NIST and are designated as

NIST1 and NIST2, and the second cycle was taken at PTB and is designated by PTB. The date range

given in the table only includes the time period during which the measurements were taken and does

not include, for example, the time required for bake-out or set-up. As discussed in the introduction,

the dates given in the Table 2 are those of the second protocol; none of the measurements taken during

the first protocol will be presented in this report.

Table 2. Chronology of the measurements

NMI Cycle Begin Date End Date

NIST1 July 6, 2011 July 13, 2011

PTB October 14, 2011 October 20, 2011

NIST2 June 20, 2012 June 29, 2012

5. General Calibration Procedure

Detailed procedures for performing the measurements were specified in the protocol. The calibration

gas used by both labs was N2 gas of at least 99.999% purity. All of the measurements at NIST were

performed using the high-vacuum standard [2], and at PTB using the CE3 standard [4]. Prior to

making measurements, PTB made accommodation coefficient measurements on the two SRGs using

their SE2 standard [6]. These measurements were used to help determine the stability of the SRGs,

discussed in Section 6.2, but are not otherwise used in the comparison.

5.1. Preparation for calibration

The two Stabil-Ion gauges were mounted with their filaments oriented vertically. The SRG gauges

were mounted with the thimble assembly horizontal, and the vacuum valve isolating the SRG closed.

After the transfer standards were installed, the vacuum chambers were evacuated to a pressure below

1 × 10-4

Pa and the SRG isolation valves were opened. The SRG gauge heads were mounted vertically

F14 9 × 10-4

Spinning Rotor

Gauge, chrome-

steel ball of

nominal diameter

4.75 mm

None See comment for F11

to the bonnet seal.

5

on the thimble, and both the SRGs and ionization gauges were turned on to verify their operation. As

described in section 5.3, the electrical parameters of the two ionization gauges were measured.

After the operating condition of the gauges was verified, the gauges and vacuum systems were

prepared for bake-out. Each lab followed its usual bake-out routine. Ionization gauges were operated

during bake-out using bake-able cables at NIST. PTB didn’t operate the ion gauges during the bake-

out. The SRGs were not operated during the bake-out and the heads were removed prior to bake-out.

NIST baked the vacuum chamber and gauges at 250 °C for approximately 3 days and PTB baked to a

temperature of 180 °C for approximately 10 days.

Following bake-out, at NIST the system was allowed to cool and the bake-able cables were replaced

with the cables provided in the transfer package. PTB started the operation of the ion gauges with the

provided cables direct after bake-out at a temperature of 100°C. The SRG rotors were re-suspended at

room temperature and operated for at least one day before measurements began in order to reach

equilibrium. Prior to calibration, the gauges were operated at an elevated pressure of 1 × 10-2

Pa for

approximately one hour to condition the gauges, and then the vacuum standards were re-evacuated to

base pressure. NIST used N2 gas for the high pressure conditioning whereas PTB used Ar gas. The

ionization gauges were then de-gassed for 10 minutes using the default conditions of the controllers.

At PTB the de-gas procedure was carried out at 100°C. Ionization gauge electrical parameters were

measured after the bake and de-gas, but before calibration measurements were made.

5.2. Calibration of the Gauges

The target pressure steps for this comparison were 3 × 10-x Pa and 9 × 10

-x Pa, with x = 6, 5, and 4,

with actual realized pressure specified to be within 5% of the target pressure. The entire calibration

sequence was completed in ascending order (from lowest pressure to highest) in a single day with each

pressure step generated twice and the calibration factor measured twice at each pressure. Thus a total

of twelve measurements were made in a single day. The entire calibration sequence was repeated on

three separate days for a total of 36 measurements, six at each pressure step.

Spinning rotor gauge measurements were only required at the pressure step 9 × 10-4

Pa. A total of six

effective accommodation coefficients, two for each day of the calibration sequence, were determined

during each round. To determine an effective accommodation coefficient, a nominal value of the rotor

diameter d = 4.762 mm and the density ρ = 7.715 g/cm3 were used. The SRGs were operated over a

frequency span of 405 to 415 Hz and with an integration time of 30 s for all measurements.

The base pressure readings of the ionization gauges were recorded before the calibration sequence was

begun on the each of the three days of measurements. For the SRGs, the residual drag or vacuum

decrement only needed to be recorded once and each of the labs could use their standard method of

determining the residual drag and its frequency dependence.

5.3. Electrical Parameter Measurements

The electrical parameters for the ion gauges were specified to be measured a minimum of three times

during the calibration: before bake-out, after bake-out, and after all calibration measurements were

complete. These were to be done using electrical break-out boxes included in the transfer package

which were inserted between the ionization gauge cables and controllers during the electrical

parameter measurement. Four voltage measurements were made on each gauge: the grid bias, Vg, the

high and low filament bias, VH and VL, and the voltage drop across a 1 kΩ resistor between the

controller and the grid, Ve.

6

NIST made electrical parameter measurements at the three times specified in the protocol, but failed to

record Ve before the bake during the first cycle. PTB made electrical parameter measurements twice

after the bake, and also made the electrical parameter measurements on each day calibration

measurements were made. The measurements are summarized in Table 3. The purpose of the

electrical parameter measurements is to assess whether the gauge and controller are working correctly

and consistently. The values in Table 3 show consistency between all three cycles of measurements.

6. Data Reduction and Analysis

The data reduction and analysis closely follows the procedures used in the CCM.P-K3, including the

nomenclature and symbols. Detailed references and explanations of the analysis methods can be

found in Ref. [1] and many details will not be repeated here. The goal is to obtain calibration ratios

for each of the transfer standards used in the comparison. These are adjusted to a consistent

temperature and target pressure. The range of the ionization gauges cover the entire pressure range of

the comparison, but are assumed to be less stable than the SRGs which were used as a transfer

standard at the highest pressure of the comparison, thus reducing the effects of pressure-independent

Table 3. Electrical Parameters determined for the ionization gauges

Gauge

Identifier

NMI Measurement

Date

Vg VH VL Ve Description

SI-404 NIST 22-6-2011 179.8 30.03 27.53 Before Bake

SI-404 NIST 30-6-2011 179.7 29.90 27.82 3.994 After Bake

SI-404 NIST 15-7-2011 179.7 29.90 27.76 3.990 End

SI-404 PTB 7-9-2011 178.9 29.9 27.15 3.998 Before Bake

SI-404 PTB 22-9-2011 179.1 29.88 27.29 3.996 After Bake

SI-404 PTB 6-10-2011 179.6 29.90 27.65 3.990 After Bake

SI-404 PTB 14-10-2011 179.6 29.90 27.65 3.990 Day 1

SI-404 PTB 18-10-2011 179.6 29.90 27.64 3.980 Day 2

SI-404 PTB 20-10-2011 179.6 29.90 27.64 3.992 Day 3, End

SI-404 NIST 29-5-2012 179.5 29.90 27.31 4.000 Before Bake

SI-404 NIST 13-6-2012 179.9 29.92 27.69 3.990 After Bake

SI-404 NIST 6-7-2012 179.7 29.92 27.77 4.000 End

SI-1004 NIST 22-6-2011 180.0 30.05 27.01 Before Bake

SI-1004 NIST 30-6-2011 180.2 29.78 26.98 4.000 After Bake

SI-1004 NIST 15-7-2011 180.3 29.80 27.25 4.001 End

SI-1004 PTB 7-9-2011 180.2 26.69 27.15 3.994 Before Bake

SI-1004 PTB 22-9-2011 180.3 29.75 26.89 4.003 After Bake

SI-1004 PTB 6-10-2011 180.2 29.73 27.23 3.999 After Bake

SI-1004 PTB 14-10-2011 180.3 29.70 27.18 3.999 Day 1

SI-1004 PTB 18-10-2011 180.3 29.66 27.14 4.000 Day 2

SI-1004 PTB 20-10-2011 180.3 29.71 27.17 4.000 Day 3, End

SI-1004 NIST 29-5-2012 180.6 29.78 26.88 4.000 Before Bake

SI-1004 NIST 13-6-2012 180.5 29.80 27.32 4.000 After Bake

SI-1004 NIST 6-7-2012 180.5 29.78 27.42 4.000 End

7

shifts in the ionization gauge readings. The SRGs calibration ratios are used to facilitate a comparison

at the target pressure PT = 9 × 10-4

Pa, and to normalize the ion gauge calibration ratios at 9 × 10-4

Pa.

The calibration ratios are then used to determine the indicated pressure achieved by the transfer

standards if the primary standards of both NMIs generated the exact same pressure.

Following the nomenclature used in the CCM.P-K3, the subscript “i” refers to the transfer standard

gauge; “j” refers to the NMI making the measurement; “m” refers to the calibration cycle for PTB and

NIST; and “k” is the individual reading of the gauge. For PTB, m = 1 in all cases and, for clarity, the

subscript “PTB” will often be used in place of the subscript j or the combined subscript jm where

appropriate. For NIST, m = 1 or 2, and the combined subscript jm will often be replaced by “NIST1”

or “NIST2” where appropriate. The symbol Nijm, is the total number of times the target pressure was

independently generated and, in this comparison, Nijm = 6 for all gauges, cycles, and NMIs.

6.1. SRG Calibration Ratios and pressure comparison at 9 × 10-4

Pa

Both labs recorded the SRG decrement DCRijmk and rotor frequency ω at the target pressure

PT = 9 × 10-4

Pa. The decrement is directly read from the SRG controller and is defined by:

ijmk

ijmk

DCR . (1)

Both labs took many SRG readings at the target pressure, and DCRijmk represents the average of all

readings at the target pressure. Each lab determined the frequency-dependent residual drag RDijmk(ω)

at a different time according to their usual methodology, and the recorded ω reading allowed the

determination of the residual drag at the time the SRG reading was recorded . The SRG pressure is

given by:

8

20

jmk i iijmk ijmk ijmk

RT dp DCR RD

M. (2)

Tjmk is the gas temperature of the standard, di and ρi are the diameter and density of the SRG rotor, R is

the universal gas constant, and M is the molar mass of N2. Since NIST and PTB used exactly the same

values of R, M, di and ρi ; the true values of these quantities is not relevant for the comparison and the

determination of the calibration ratio. The SRG calibration ratio aijm is defined by:

1 1

1 1

ijm ijmN N

ijmk

ijm ijmk

k kijm ijm ijmk

pa a

N N P. (3)

This definition is exactly the same as what is commonly known as the accommodation coefficient,

often given by σ; here we use aijm instead of σ to be consistent with the nomenclature used in the

CCM.P-K3 analysis and with the ionization gauge calibration ratio, and also to not confuse the

calibration ratio with the standard deviation. Pjmk is the generated pressure of the NMI primary

standard and Nijm, is the total number of times the target pressure was independently generated and, as

previously stated, Nijm = 6 for both labs in all three cycles. For NIST, the subscript i could be dropped

in Pijmk since the generate pressure is the same for both SRGs but, for PTB, there is a slight difference

in Pijmk between the two SRGs because the pressure in their standard decreased with time and the two

SRG readings were taken at slightly different times.

8

Table 4. SRG calibration ratios at PT = 9 × 10-4

Pa defined by eq.(3).

aijm

F11 F14

NIST1 1.0852 1.0951

PTB 1.0835 1.0930

NIST2 1.0850 1.0913

The measured aijm is given in Table 4 and is shown in Figure 1 as a function of cycle. Rotor F14

shows a 0.24% difference between the aijm measurements made by NIST in cycles NIST1 and NIST2,

and rotor F11 shows only a 0.02% difference between measurements made during cycle NIST1 and

NIST2. The SRG calibration ratio measurements made by PTB are within 0.2% of all NIST

measurements. Considering that the uncertainty of the standard pressure is approximately 0.2% for

both laboratories, it is evident from Figure 1 that there is excellent agreement between the two labs for

SRG measurements made at a target pressure of 9 × 10-4

Pa.

Figure 1. The spinning rotor gauge (SRG) calibration ratio according to eq. (3) as a function of measurement

cycle for both SRG transfer standards F11 and F14.

The calibration ratio is used to calculate a predicted gauge pressure reading on SRG i, where i = F11

or F14, when the primary standard of each NMI (at calibration cycle m) is set to target pressure PT:

ijm ijm Tp a P . (4)

Note that the predicted gauge reading of eq. (4) is not the same as the SRG pressure given by eq. (2);

pijmk is the actual reading SRG pressure determined from eq. (2) during the calibration, whereas pijm is

the predicted reading of the SRG when the primary standard is set to the target pressure PT, given the

determined calibration ratio aijm. The predicted gauge reading allows the measurements at NIST and

1.075

1.080

1.085

1.090

1.095

1.100

NIST1 PTB NIST2

aij

m

F11

F14

9

PTB to be compared at a common target pressure. The predicted gauge pressure readings are

presented Table 5.

Table 5. SRG predicted gauge pressure readings pijm, defined by eq.(4), and mean

cycle gauge pressure readings pjm , defined by eq. (5), determined at PT = 9 × 10-4

Pa.

pijm pjm

F11 F14

NIST1 9.767E-04 9.856E-04 9.812E-04

PTB 9.752E-04 9.837E-04 9.794E-04

NIST2 9.765E-04 9.821E-04 9.793E-04

To compare the measured pressures of both NMIs, a single mean cycle gauge pressure reading pjm

was calculated as the simple arithmetic mean of the predicted gauge readings of the two SRGs:

11 14

2

F jm F jm

jm

p pp . (5)

These values are given in Table 5. Finally, to compare the two NMIs it is desired to calculate a single

mean gauge pressure reading, pj, for each NMI. For PTB, the subscript m can be dropped in eq. (5) to

define as the mean gauge pressure reading as

11, 14,

2

F PTB F PTB

PTB

p pp . (6)

For the pilot laboratory (NIST), a single value of pj was calculated as the arithmetic mean of the two

mean cycle gauge pressure readings defined in eq. (5):

1 2

1

2 NIST NIST NISTp p p . (7)

For PTB, the mean gauge pressure reading pj is pPTB = 9.794 × 10-4

Pa, and for NIST

pNIST = 9.802 × 10-4

Pa.

6.2. Estimates of uncertainty in the predicted gauge pressure readings and mean gauge pressure

readings at 9 × 10-4

Pa based on the SRGs

Since the mean gauge pressure readings are constructed of averages of the predicted gauge pressure

readings, we will first estimate the uncertainty of the predicted gauge pressure readings. The

combined standard uncertainty in the predicted gauge pressure readings, pijm, for each SRG at each

NMI and each cycle, is estimated from the root-sum-square of the component uncertainties [7]:

2 2 2( ) ( ) ( ) ( )c ijm B ijm A ijm LTS ijmu p u p u p u p , (8)

Where uB(pijm) and uA(pijm) are the Type B and Type A standard uncertainties in pijm, and uLTS(pijm) is

the standard uncertainty associated with long-term shifts in the SRG calibration ratio. The values of

the relative component uncertainties for the SRGs are given in Table 6 and will be discussed below.

10

Table 6. Relative component standard uncertainties used to estimate uc(pijm) using the SRGs

at a target pressure of 9 × 10-4

Pa. All uncertainties are k = 1.

Component NIST1 PTB NIST2

F11 F14 F11 F14 F11 F14

( )std ijm

ijm

u p

p

0.0023 0.0023 0.0022 0.0022 0.0022 0.0022

( )T ijm

ijm

u p

p

0.0001 0.0001 0.0013 0.0013 0.0003 0.0003

( )RD ijm

ijm

u p

p

0.0004 0.0007 0.0048 0.0008 0.0008 0.0013

( )B ijm

ijm

u p

p

0.0023 0.0024 0.0055 0.0027 0.0024 0.0026

( )A ijm

ijm

u p

p

0.0001 0.0001 0.0006 0.0006 0.0001 0.0002

( )LTS ijm

ijm

u p

p

0.0008 0.0013 0.0008 0.0013 0.0008 0.0013

( )c ijm

ijm

u p

p

0.0025 0.0027 0.0056 0.0031 0.0025 0.0029

As seen from eqs. (2) – (4), there are three uncertainty components comprising uB(pijm): the standard

Type B uncertainty of the primary standard ustd(pijm), the standard uncertainty in the gas temperature

uT(pijm), and the standard uncertainty due the residual drag uRD(pijm). There is no Type B uncertainty

associated with the decrement reading DCR. The Type B uncertainty can be expressed as

2 2 2( ) ( ) ( ) ( ) B ijm std ijm T ijm RD ijmu p u p u p u p . (9)

The relative standard uncertainty of ustd(pijm) may be expressed as

( ) ( )

std ijm std ijm

ijm ijm

u p u P

p P. (10)

Pijm is the average of the 6 SRG readings pjmk taken during the cycle m by NMI j. Similarly, the

relative standard uncertainty of uT(pijm) may be expressed as

( ) ( )

2

T ijm jm

ijm jm

u p u T

p T, (11)

where Tjm is the average of 6 temperature readings (which is the same for F11 and F14, hence we drop

the subscript i) and, as was done in the CCM.P-K3 analysis, we use one-half the maximum difference

11

in recorded gas temperatures drift as an estimate of u(Tjm) for each cycle at each NMI. This

component is the same for both SRGs and for all readings made during a cycle. The relative

uncertainty due the residual drag uRD(pijm) is given by

( ) ( )

RD ijm ijm

ijm ijm ijm

u p u RD

p DCR RD. (12)

In this case, ijm ijmDCR RD is a typical value for each SRG for each cycle of each NMI. For NIST,

u(RDijm) is estimated from the standard deviation of five separate measurements of the residual drag at

410 Hz made over the course of the cycle. For PTB, a rectangular distribution between the lowest and

highest rotor frequency was assumed; consequently u(RDijm) was estimated as the half of the change in

RD divided by 3 .

The relative Type A standard uncertainty of each SRG is given by the standard deviation of the mean

of all Nijm = 6 measurements, corrected for limited sample size [8, 9]:

( ) ( ) ( ) 1

3

A ijm A ijm ijm ijm

ijm ijm ijm ijm

u p u a s a N

p a a N. (13)

It is clear from Figure 1 that both F11 and F14 demonstrated excellent stability. The measurements

for F11 made at NIST1 and NIST2 differed by only 0.02%, which is less than typical long term

stability (LTS) estimates for SRGs and is comparable to the Type A uncertainty. For F14, the

difference between the NIST1 and NIST2 is 0.35%, which is smaller than the estimated LTS for the

SRGs used in the CCM.P-K3, but is within reason for a rotor that has been shipped, baked, and re-

suspended [10]. In addition to the SRG calibration ratios made during the two cycles at NIST, PTB

carried out accommodation coefficient measurements using their SE2 standard before and after the

measurements were made on their CE3 standard. This required removing the SRGs from one standard

and installing them on another; therefore these measurements may also be used to estimate the LTS

and are presented in Table 7. For F11 the difference between the mean accommodation coefficient

measured before and after the PTB cycle is 0.22%, and is 0.04% for F14. There is not enough

information to adequately determine if the two SRGs have a linear drift over time and therefore we

assume a constant model for the LTS where the SRG accommodation coefficients are assumed to vary

randomly. For the LTS we simply use the root-mean-squared average of the LTS determined by NIST

and PTB:

2 2( ) ( ) ( ) ( )1

2

LTS ijm LTS ijm LTS NIST LTS PTB

ijm ijm NIST PTB

u p u a u a u a

p a a a. (14)

Given the excellent stability of the SRGs observed by both PTB and NIST, a more sophisticated

approach to determine the LTS is not warranted here. For uLTS(aNIST), aNIST is the arithmetic mean of

the accommodation coefficients measured at NIST1 and NIST2 and uLTS(aNIST) is estimated from one

half the difference between the calibration ratios measured at NIST1 and NIST2. Similarly, for

uLTS(aPTB), aPTB is the arithmetic mean of the accommodation coefficients measured by PTB using SE2,

and uLTS(aPTB) is estimated from one half the difference between the accommodation coefficients

measured using SE2 before and after the measurements made using CE3. The LTS results are given

in Table 6.

12

Table 7. Accommodation coefficients made by PTB using the SE2

standard.

Date F11 F14

25-8-2011 1.0869 1.0926

26-8-2011 1.0851 1.0915

Mean 1.0860 1.0921

7-11-2011 1.0837 1.0923

9-11-2011 1.0836 1.0909

Mean 1.08365 1.0916

Now that all the relevant component uncertainties have been discussed, the standard uncertainty in the

mean gauge pressure readings, uc(pj), is found by applying the methods of [7] to eqs. (6) and (7). The

generated pressure of the primary standard is correlated for both SRGs and for the two cycles at NIST.

The gas temperature is correlated for both SRGs but not between the two NIST cycles. For NIST,

with 2 SRGs and 2 cycles, pj is the mean of 4 values of pijm and uc(pj) is given by

1/22

2 22

1 1

2 22 2 2

1 1

( ) 1( )

2 4( )

1( ) ( ) ( )

16

std jm

T jm

m m

c j

RD ijm A ijm LTS ijm

m i

u pu p

u p

u p u p u p

. (15)

The uncertainty in the generated pressure was averaged between the two NIST cycles since these are

correlated but of different magnitude. For PTB, with 2 SRGs but only 1 cycle, pj is the mean of only 2

values of pijm and we can drop the subscript m and write:

1/2

22 2 2 2 2

1

1( ) ( ) ( ) ( ) ( ) ( )

4

c j std j T j RD ij A ij LTS ij

i

u p u p u p u p u p u p . (16)

The relative uncertainty of the mean gauge pressure readings, uc(pj)/pj, for the SRGs at a target

pressure of 9 × 10-4

Pa are 0.0025 for NIST and 0.0039 for PTB.

6.3. Calibration ratio and pressure comparison from 3 × 10-6

Pa to 3 × 10-4

Pa based on the

ionization gauge

Comparisons for the pressure range 3 × 10-6

Pa to 3 × 10-4

Pa are based on measurements of two

ionization gauges: SI-404 and SI-1004. Measurements were also made with the ion gauges at the

target pressure PT = 9 × 10-4

Pa, and these were used to correct for pressure-independent shifts in the

ion gauge calibration ratios by normalizing to the SRG measurements at the same target pressure.

First, the gauge readings were corrected for their “zero” reading with the vacuum chamber evacuated

and at the base pressure:

0 ijmk Gijmk G ijmkp p p , (17)

were pGijmk is the uncorrected gauge reading, pG0ijmk is the zero-pressure gauge reading, and pijmk is the

gauge reading corrected for zero-pressure offsets. For each ion gauge i, each NMI j, and each

calibration cycle m, an average ion gauge inverse correction factor , Sijm(Pijm), was calculated from the

Njm = 6 readings of the generated pressure Pijmk:

1 1

1 1( ) ( )

6 6

jm jmN N

ijmk

ijm ijm ijmk ijmk

k k ijmk

pS P S P

P. (18)

13

Ionization gauges are sensitive to gas density [11, 12]; therefore equivalent inverse correction factors

were determined for the common reference temperature of 23 ºC. This was done by multiplying the

individual inverse correction factors by the ratio Tjmk / 296.15 K.

In eq. (18), Pjm is the arithmetic mean of the 6 generated pressures. Since the measured pressures

typically differed from the target pressure, the average ion gauge inverse correction factors were

corrected to values at the target pressures. This was done by assuming that Sijm varied linearly

between the value at Pijm and the target pressure PT. Values of Sijm bounding the pressure interval are

used in the linear interpolation. In fact, this correction is negligible, less than 0.1% in all cases, and

therefore the uncertainty introduced by this correction is also negligible. The result is a set of Sijm(PT)

values, which is presented in Tables 8 and 9 and Figures 2 and 3.

Table 8. Average inverse ion gauge inverse correction factors Sijm(Pijm) for SI-404 as defined in

eq. (18). Pijm is the pressure generated by the standard. Sijm(Pijm) is corrected to a common

temperature of 23 ºC and interpolated to the target pressure PT. Shown are the uncorrected and

corrected values of Sijm(Pijm).

NIST1 PTB NIST2

Sijm(Pijm) Sijm(Pijm) Sijm(Pijm)

PT

/Pa

Pijm

/Pa

Un-

corr.

Corr. Pijm

/Pa

Un-

corr.

Corr. Pijm

/Pa

Un-

corr.

Corr.

3.E-06 3.006E-06 1.109 1.112 3.041E-08 1.112 1.119 3.019E-06 1.096 1.098

9.E-06 8.925E-06 1.112 1.115 9.566E-08 1.109 1.115 8.979E-06 1.095 1.097

3.E-05 2.929E-05 1.106 1.109 3.171E-07 1.111 1.118 2.992E-05 1.092 1.094

9.E-05 8.830E-05 1.106 1.108 8.798E-07 1.106 1.112 8.974E-05 1.089 1.091

3.E-04 2.954E-04 1.107 1.110 2.933E-06 1.110 1.116 2.973E-04 1.093 1.095

9.E-04 8.971E-04 1.113 1.115 8.855E-06 1.115 1.121 8.917E-04 1.096 1.098

Table 9. Average inverse ion gauge inverse correction factors Sijm(Pijm)for SI-1004 as defined in

eq. (18). Pijm is the pressure generated by the standard. Sijm(Pijm) is corrected to a common

temperature of 23 ºC and interpolated to the target pressure PT. Shown are the uncorrected and

corrected values of Sijm(Pijm).

NIST1 PTB NIST2

Sijm(Pijm) Sijm(Pijm) Sijm(Pijm)

PT

/Pa

Pijm

/Pa

Un-

corr.

Corr. Pijm

/Pa

Un-

corr.

Corr. Pijm

/Pa

Un-

corr.

Corr.

3.E-06 3.006E-06 0.990 0.992 3.036E-08 1.002 1.008 3.019E-06 1.010 1.012

9.E-06 8.925E-06 0.992 0.995 9.553E-08 1.001 1.007 8.979E-06 1.009 1.011

3.E-05 2.929E-05 0.985 0.987 3.166E-07 1.001 1.007 2.992E-05 1.003 1.004

9.E-05 8.830E-05 0.985 0.987 8.786E-07 0.998 1.003 8.974E-05 1.000 1.002

3.E-04 2.954E-04 0.985 0.987 2.929E-06 1.001 1.006 2.973E-04 1.003 1.005

9.E-04 8.971E-04 0.984 0.986 8.843E-06 0.997 1.002 8.917E-04 1.002 1.003

14

Figure 2. Inverse ion gauge correction factor for SI-404. The Sijm are corrected to 23 °C.

Figure 3. Inverse ion gauge correction factor for SI-1004. The Sijm are corrected to 23 °C.

Next a pressure-independent correction is applied to the entire set of Sijm(PT) values at each NMI j and

each calibration cycle m. It is assumed that the generated pressure at 9 × 10-4

Pa at each NMI was the

same whether it was being measured with an ion gauge or an SRG. The ion gauge calibration ratio,

Kijm(PT), is defined such that the predicted gauge pressure reading using the ion gauge is the same as

the mean gauge pressure reading from the SRGs at 9×10-4

Pa :

1.085

1.090

1.095

1.100

1.105

1.110

1.115

1.120

1.125

-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0

Sij

m

Log Pstd /Pa

NIST1

PTB

NIST2

0.980

0.985

0.990

0.995

1.000

1.005

1.010

1.015

-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0

Sij

m

Log Pstd /Pa

NIST1

PTB

NIST2

15

4

4

ˆ ˆ( )ˆ( ) ( )

ˆ ˆ(9 10 Pa)

( )ˆwith ( )

(9 10 Pa)

ijm T j j

ijm T ijm T

ijm R R

ijm T

ijm T

ijm

S P p pK P S P

S p p

S PS P

S

, (19)

ˆjp is the mean value of pj determined from the SRGs at 9 × 10

-4 Pa, and ˆ

Rp = 9 × 10-4

Pa. The

parameter Ŝijm(PT) is the ion gauge inverse correction factor normalized to the value at 9 × 10-4

Pa,

such that Ŝjm(9 × 10-4

Pa) = 1.0. For NIST, we are assuming that ˆjp is representative of the behavior

of the standard at 9 × 10-4

Pa for both calibration cycles. The Kijm are given in Figs. 3 and 4.

As with the SRG analysis, the ion gauge calibration ratios are used to calculate a predicted gauge

pressure reading when the primary standard of NMI j (at calibration cycle m) is set to target pressure,

PT:

( ) ijm ijm T Tp K P P . (20)

The calibration ratios and predicted gauge pressure readings are presented in Tables 10 and 11. For

PTB, the subscript m can be dropped in eq. (20).

Table 10. Calibration ratios Kijm for ion gauge SI-404 as defined in eq. (19) and the predicted gauge

pressure reading pijm for ion gauge SI-404 defined in eq. (20). Ŝijm is also defined in eq. (19) and is the

ion gauge inverse correction factor normalized to the value at 9 × 10-4

Pa. The value of pijm at 9 ×10-4

Pa in this table is not used to compute the reference value; that value is taken from the SRG

measurements.

NIST1 PTB NIST2

PT

/Pa Ŝijm Kijm

pijm

/Pa Ŝijm Kijm

pijm

/Pa Ŝijm Kijm

pijm

/Pa

3.E-6 0.997 1.086 3.26E-6 0.997 1.085 3.26E-6 0.999 1.088 3.27E-6

9.E-6 0.999 1.089 9.80E-6 0.994 1.082 9.74E-6 0.999 1.088 9.79E-6

3.E-5 0.994 1.083 3.25E-5 0.996 1.084 3.25E-5 0.996 1.085 3.25E-5

9.E-5 0.994 1.082 9.74E-5 0.992 1.079 9.71E-5 0.993 1.082 9.74E-5

3.E-4 0.995 1.084 3.25E-4 0.995 1.083 3.25E-4 0.997 1.086 3.26E-4

9.E-4 1.000 1.089 9.80E-4 1.000 1.088 9.79E-4 1.000 1.089 9.80E-4

Table 11. Calibration ratios Kijm for ion gauge SI-1004 as defined in eq. (19) and the predicted gauge

pressure reading pijm for ion gauge SI-1004 defined in eq. (20). Ŝijm is also defined in eq. (19) and is the

ion gauge inverse correction factor normalized to the value at 9 × 10-4

Pa. The value of pijm at 9 × 10-4

Pa in this table is not used to compute the reference value; that value is taken from the SRG

measurements.

NIST1 PTB NIST2

PT

/Pa Ŝijm Kijm

pijm

/Pa Ŝijm Kijm

pijm

/Pa Ŝijm Kijm

pijm

/Pa

3.E-6 1.006 1.096 3.29E-6 1.006 1.094 3.28E-6 1.008 1.098 3.29E-6

9.E-6 1.009 1.099 9.89E-6 1.004 1.093 9.83E-6 1.007 1.097 9.88E-6

3.E-5 1.001 1.091 3.27E-5 1.005 1.093 3.28E-5 1.001 1.090 3.27E-5

9.E-5 1.001 1.090 9.81E-5 1.002 1.090 9.81E-5 0.998 1.087 9.79E-5

3.E-4 1.001 1.091 3.27E-4 1.004 1.093 3.28E-4 1.001 1.090 3.27E-4

9.E-4 1.000 1.089 9.80E-4 1.000 1.088 9.79E-4 1.000 1.089 9.80E-4

16

Figure 4. Calibration ratios according to eq. (19) for ion gauge SI-404.

Figure 5. Calibration ratios according to eq. (19) for ion gauge SI-1004.

Next, a single mean cycle gauge pressure reading pjm was calculated as the simple arithmetic mean of

the predicted gauge readings of the two ionization gauges:

1.075

1.080

1.085

1.090

-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0

Kij

m

Log PT /Pa

NIST1

PTB

NIST2

1.085

1.090

1.095

1.100

-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0

Kij

m

Log PT /Pa

NIST1

PTB

NIST2

17

SI-404 SI-1004

2

jm jm

jm

p pp . (21)

Again, similar to the SRG analysis, a single mean gauge pressure reading, pj, is calculated for each

NMI. For PTB, the subscript m can be dropped in eq. (21) to define the mean gauge pressure reading

as

SI-404, SI-1004,

2

PTB PTB

PTB

p pp . (22)

For NIST, a single value of pj was calculated as the arithmetic mean of the two mean cycle gauge

pressure readings defined in eq. (21):

1 2

1

2 NIST NIST NISTp p p . (23)

The mean gauge pressure readings pj are presented in Table 12.

Table 12. Mean ion gauge pressure readings pj, from eqs. (22) and

(23), and their associated relative standard uncertainties (k = 1). The

value at PT = 9 × 10-4

Pa is not used to compare the two labs; that

value is taken from the SRG measurements.

NIST PTB

PT

/Pa

pj

/Pa

( )c j

j

u p

p

pj

/Pa

( )c j

j

u p

p

3.E-6 3.276E-06 0.0039 3.270E-06 0.0065

9.E-6 9.838E-06 0.0032 9.785E-06 0.0055

3.E-5 3.261E-05 0.0029 3.266E-05 0.0059

9.E-5 9.769E-05 0.0025 9.762E-05 0.0040

3.E-4 3.263E-04 0.0024 3.264E-04 0.0037

9.E-4 9.802E-04 0.0024 9.794E-04 0.0040

6.4. Estimates of uncertainty in the predicted gauge pressure readings and mean gauge pressure

readings from 3 × 10-6

Pa to 3 × 10-4

Pa based on the ionization gauge

Similar to the SRG uncertainty analysis, the combined standard uncertainty in the predicted gauge

pressures using the ion gauges is:

2 2 2( ) ( ) ( ) ( ) c ijm B ijm A ijm LTS ijmu p u p u p u p . (24)

The relative uncertainty in the predicted ion gauge readings uc(pijm)/pijm and the component

uncertainties given in eq. (24) are listed in Tables 13 -15. The individual components will now be

discussed.

The Type B components, uB(pjm) are derived from eqs. (17 - 20) and are given by

18

22ˆ( ) ( )( )

ˆ

B jm SRG jstd T

jm T j

u p u pu P

p P p

. (25)

The subscript “i” has been dropped since the Type B is the same for either ion gauge. The correction

for zero offset is included in ustd(PT) and ˆ( )SRG ju p is the standard uncertainty in the mean gauge

pressure from the SRGs at 9 × 10-4

Pa. It does not include the uncertainty of the standard, since it is

contained in both ˆjp and Sijm(9 × 10

-4 Pa)and therefore cancels. ˆ( )SRG ju p is given by:

2 2ˆ ˆ ˆ( ) ( ) ( ) SRG j c j std ju p u p u p . (26)

The Type A standard uncertainties, uA(pijm), are evaluated both at PT and at 9×10-4

Pa, and the Type A

standard uncertainty at 9 × 10-4

Pa must be included for all pressures due to the pressure-independent

correction:

2 24

4

( ) ( ( )) ( (9 10 ))

( ) (9 10 )

A ijm A ijm T A ijm

ijm ijm T ijm

u p u S P u S

p S P S. (27)

As was done for the SRGs, the standard deviation of the mean, ( ( ))jm Ts S P , corrected for limited

sample size, is used to calculate the components in eq. (27):

( ( )) ( ( )) 6 1

( ) ( ) 6 3

A ijm T ijm T

ijm T ijm T

u S P s S P

S P S P. (28)

uLTS(pijm) is the standard uncertainty arising from long-term shifts in the normalized ion gauge inverse

correction factor, ˆ ( )ijm TS P . Only the NIST data are used to evaluate uLTS(pjm). We assume that long-

term shifts in ˆ ( )ijm TS P vary randomly in time, and these are evaluated by taking one-half the

difference between the NIST1 and NIST2 measurements at each target pressure. As can be seen from

examining Figures 4 and 5, ˆ ( )ijm TS P is not a strong function of pressure. This observation is

consistent with other reported LTS values for Stabil-Ion gauges [13,14]. Therefore, we obtain a

constant value of uLTS(pijm)= uLTS(pi) by taking the root-mean-square average:

23E 4

1

3E 6 2

ˆ ( )1 1( ) 1

ˆ5 4 ( )T

iNIST TLTS i

P iNIST T

S Pu p

S P

. (29)

From eq. (29) we get uLTS(pSI-404)/pSI-404 = uLTS(pSI-1004) /pSI-1004 = 0.08%. This value is small but

consistent with the data presented in Figures 4 and 5. It is also interesting to perform a similar analysis

to the un-normalized correction factors Sijm(PT) shown in Figures 2 and 3, since that would be a better

measurement of the gauge stability even though it will not be used in the present analysis. In that case,

we get 0.73% for SI-404 and 0.86% for SI-1004, which is lower than what is typical for this gauge

(see discussion in Ref. [13]), but it is not unreasonably small. As a comparison, in the CCM-PK.3the

LTS of SI-404 ranged from 0.3% at PT =3 × 10-4

Pa to 1.9% at PT =3 × 10-6

Pa. As with the SRG

measurements, the ion gauge calibration ratios compare very well between NIST and PTB and a more

19

sophisticated evaluation of the LTS is not likely to change the conclusions of the comparison

presented in Sections 7 and 8.

The standard uncertainty in the mean gauge pressure readings, uc(pj), is derived from eqs. (21), (22),

and (23). The Type B uncertainty is correlated for both ion gauges and between the two cycles at

NIST. For PTB, the standard uncertainty in the mean gauge pressure readings is given by

1/2

22 2 2

1

1( ) ( ) ( ) ( )

4c j B j A ij LTS ij

i

u p u p u p u p

. (30)

For NIST, pj is the average of two gauges over two cycles, and the standard uncertainty in the mean

gauge pressure reading is given by:

1/2

22 2 2

2 2

1 1 1

( ) 1( ) ( ) ( )

2 16

B jm

c j A ijm LTS ijm

m m i

u pu p u p u p

. (31)

Table 13. Estimates of the relative standard uncertainty (k = 1) in the predicted ion gauge pressure readings

uc(pijm)and its components for NIST1 (m = 1). uB(pjm) is the same for both ion gauges, therefore the

subscript “i” is dropped. The subscripts “j” and “m” are dropped for the LTS component since this value is

the same for all cycles and both NMIs.

SI-404 SI-1004

PT

/Pa

( )B jm

jm

u p

p

( )LTS i

i

u p

p

( )A ijm

ijm

u p

p

( )c ijm

ijm

u p

p

( )LTS i

i

u p

p

( )A ijm

ijm

u p

p

( )c ijm

ijm

u p

p

3.E-6 0.0036 0.0008 0.0012 0.0039 0.0008 0.0018 0.0095

9.E-6 0.0026 0.0008 0.0007 0.0028 0.0008 0.0012 0.0091

3.E-5 0.0026 0.0008 0.0010 0.0029 0.0008 0.0011 0.0091

9.E-5 0.0023 0.0008 0.0007 0.0025 0.0008 0.0011 0.0090

3.E-4 0.0023 0.0008 0.0008 0.0026 0.0008 0.0012 0.0090

20

Table 14. Estimates of the relative standard uncertainty (k = 1) in the predicted ion gauge pressure readings

uc(pij)and its components for PTB. The subscript “m” is dropped since PTB had only 1 measurement cycle.

uB(pj) is the same for both ion gauges, therefore the subscript “i” is dropped. The subscripts “j” and “m” are

dropped for the LTS component since this value is the same for all cycles and both NMIs.

SI-404 SI-1004

PT

/Pa

( )B j

j

u p

p

( )LTS i

i

u p

p

( )A ij

ij

u p

p

( )c ij

ij

u p

p

( )LTS i

i

u p

p

( )A ij

ij

u p

p

( )c ij

ij

u p

p

3.E-6 0.0053 0.0008 0.0057 0.0079 0.0008 0.0055 0.0077

9.E-6 0.0052 0.0008 0.0037 0.0065 0.0008 0.0024 0.0058

3.E-5 0.0052 0.0008 0.0041 0.0067 0.0008 0.0044 0.0069

9.E-5 0.0037 0.0008 0.0029 0.0047 0.0008 0.0026 0.0046

3.E-4 0.0036 0.0008 0.0023 0.0044 0.0008 0.0021 0.0042

Table 15. Estimates of the relative standard uncertainty (k = 1) in the predicted ion gauge pressure readings

uc(pijm)and its components for NIST2 (m = 2). uB(pjm) is the same for both ion gauges, therefore the

subscript “i” is dropped. The subscripts “j” and “m” are dropped for the LTS component since this value is

the same for all cycles and both NMIs.

SI-404 SI-1004

PT ( )B jm

jm

u p

p

( )LTS i

i

u p

p

( )A ijm

ijm

u p

p

( )c ijm

ijm

u p

p

( )LTS i

i

u p

p

( )A ijm

ijm

u p

p

( )c ijm

ijm

u p

p

3.E-6 0.0041 0.0008 0.0014 0.0044 0.0008 0.0012 0.0044

9.E-6 0.0038 0.0008 0.0010 0.0040 0.0008 0.0010 0.0040

3.E-5 0.0030 0.0008 0.0022 0.0038 0.0008 0.0021 0.0038

9.E-5 0.0028 0.0008 0.0013 0.0032 0.0008 0.0012 0.0032

3.E-4 0.0025 0.0008 0.0014 0.0030 0.0008 0.0015 0.0031

7. Results of the comparison

7.1. Pair-wise difference between NIST and PTB and the reference pressure

The mean gauge pressure readings pj are used to calculate the pair-wise differences between NIST and

PTB. For PT ranging from 3 × 10-6

Pa to 3 × 10-4

Pa the pj are determined from the mean ion gauge

pressure readings defined in eq. (22) and (23). For PT = 9 × 10-4

Pa, the pj are determined from the

mean SRG pressure readings defined in eq. (6) and (7). We use pNIST and pPTB to denote the complete

set of mean gauge pressure readings for NIST and PTB. The pair-wise difference is defined as:

ij NIST PTBd p p . (32)

The standard uncertainty associated with dij is given by combining the standard uncertainties in the

mean gauge pressure readings, uc(pj):

21

2 2( ) ( ) ( )ij c NIST c PTBu d u p u p . (33)

For PTB, the values of uc(pPTB) are determined by eqs. (16) and (30), and the uc(pNIST) are determined

by eqs. (15) and (31). The expanded uncertainty (k = 2) is given as U(dij) = 2u(dij). The results of the

two laboratories are considered to be equivalent if dij is less than the expanded uncertainty:

1( )

ij

ij

d

U d . (34)

The results of the pair-wise differences are summarized in Table 16. The largest absolute value of

eq. (34) is 0.424. This demonstrates a high degree of equivalence between PTB and NIST. The

fractional pair-wise difference is shown in Figure 6. The difference between NIST and PTB is well

with the uncertainty of the measurements.

Table 16. The pair-wise difference between NIST and PTB, dij, and the associated standard

uncertainty u(dij).

PT dij

/Pa dij/PR

uc(dij)

/Pa

Uc(dij)

/Pa ( )

ij

ij

d

U d

3.E-6 6.02E-09 1.84E-03 2.480E-08 4.960E-08 0.121

9.E-6 5.31E-08 5.41E-03 6.260E-08 1.252E-07 0.424

3.E-5 -4.45E-08 -1.36E-03 2.145E-07 4.290E-07 0.104

9.E-5 7.02E-08 7.19E-04 4.601E-07 9.202E-07 0.076

3.E-4 -1.33E-07 -4.08E-04 1.457E-06 2.914E-06 0.046

9.E-4 7.97E-07 8.14E-04 4.284E-06 8.569E-06 0.093

22

Figure 6. The fractional pair-wise difference between NIST and PTB. The uncertainty bars are

Uc(dij)/PR (k = 2).

We can define a reference pressure PR as the average of the mean gauge pressure readings from both

NMIs:

1

2R NIST PTBP p p . (35)

This reference pressure PR is the reference value for the CCM.P-K3.1. The standard uncertainty in the

reference pressure is given by:

2 21

( ) ( ) ( )2

R c NIST c PTBu P u p u p . (36)

The difference between the NMI j results and the reference value is given by

j j RD p P , (37)

and its standard uncertainty is given by

2 2( ) ( ) ( )j c j Ru D u p u P . (38)

The expanded uncertainty is U(Dj) = 2u(Dj) (k = 2). The degree of equivalence (DOE) is defined in

the usual way as:

( )

j

n

j

DE

U D . (39)

-0.02

-0.01

0.00

0.01

0.02

1.E-06 1.E-05 1.E-04 1.E-03

dij

/

PR

PT / Pa

23

The DOE is summarized in Table 17. In general, labs are considered equivalent to the reference value

if En < 1. For PT = 9.0 × 10-6

, both labs have En ≤ 0.2, and both labs have En ≤ 0.05 for all other

target pressures. In reality, the En does not have a strong meaning in a bilateral comparison since

Dj = 1/2 dij. The equivalence between the two laboratories is demonstrated in Table 16 in the quantity

ijd /U(dij). Nevertheless the quantities defined in eqs. (35) through (39) are convenient for linking to

other key comparisons, as in section 7.2.

Table 17. The difference between the NMI j and the reference pressure Dj and the associated standard

uncertainty u(Dj), and the DOE, En. At the target pressure of 9 × 10-4

Pa the analysis only includes the

SRG readings.

NIST PTB

PT PR /Pa u(PR) /Pa Dj /Pa u (Dj) /Pa En Dj /Pa u (Dj) /Pa En

3.E-6 3.273E-06 2.48E-08 3.01E-09 2.79E-08 0.05 -3.01E-09 3.27E-08 0.05

9.E-6 9.812E-06 6.27E-08 2.66E-08 7.00E-08 0.19 -2.66E-08 8.29E-08 0.16

3.E-5 3.264E-05 2.14E-07 -2.22E-08 2.34E-07 0.05 2.22E-08 2.89E-07 0.04

9.E-5 9.765E-05 4.60E-07 3.51E-08 5.23E-07 0.03 -3.51E-08 6.02E-07 0.03

3.E-4 3.264E-04 1.46E-06 -6.66E-08 1.66E-06 0.02 6.66E-08 1.90E-06 0.02

9.E-4 9.798E-04 4.29E-06 3.99E-07 4.87E-06 0.04 -3.99E-07 5.60E-06 0.04

7.2. Linking the PTB results to the CCM.P-K3

With the results of the previous section, we can link the present PTB CCM.P-K3.1 results to the

CCM.P-K3 and transfer the DOE. First, we can calculate the difference between the PTB results and

CCM.PK-3 reference pressure:

(P K3) ( 3) ( 3.1) ( 3.1)PTB NIST P K NIST P K PTB P KD D D D . (40)

We have used a prime to indicate that D′PTB(P-K3) is derived from the present comparison, and is not the

PTB results reported in Ref. [1], whereas DNIST(P-K3) are the NIST results from Ref. [1]. By examining

eqs. (36) and (32), we see that the last two terms in eq.(40) is just the pair-wise difference between

NIST and PTB in the present comparison.

There are four contributions to the uncertainty in D′PTB(P-K3): The uncertainty in the CCM.P_K3

reference pressure u(PR(P-K3)), the uncertainty in the PTB mean gauge readings for the present

comparison u(pPTB), the uncertainty in the NIST mean gauge pressure readings for the present

comparison u(pNIST), and the uncertainty in the mean gauge pressure readings for the CCM.P-K3,

u(pNIST(P-K3)). The latter two are not identical, but are highly correlated. A reasonable estimate of the

contribution of the NIST mean gauge pressure reading to u(D′PTB(P-K3)) is the simple arithmetic mean of

the two:

( 3)

1( ) ( ) ( )

2NIST NIST NIST P Ku p u p u p . (41)

The expanded uncertainty is U(D′PTB(P-K3)) = 2u(D′PTB(P-K3)) and is expressed as

2 2 2

( 3) ( 3) NIST PTB( ) 2 ( ) ( ) ( )PTB P K R P KU D u P u p u p . (42)

24

With eq. (40) and eq. (42), the CCM.P-K3 DOE for PTB is determined from

( 3)

( 3)( )

PTB P K

n

PTB P K

DE

U D

. (43)

The transferred CCM.P-K3 DOE results are presented in Table 18. As usual a lab is considered

equivalent to the KCRV if 1nE . PTB shows 0.4nE for all target pressures of the CCM.P-K3

and can therefore be considered equivalent to the CCM.P-K3 KCRV.

In a similar way, the pairwise difference between PTB and the all of the NMIj participating in the

CCM.P-K3 can be transferred from the present results to the CCM.P-K3. The pairwise difference

between NMI j and j′ can be written as:

jj j jd D D . (44)

In particular, for the CCM.P-K3, we use the following notation:

( 3) ( 3) ( 3)jj P K j P K j P Kd D D . (45)

To calculate the pairwise difference for PTB to the labs in the CCM.P-K3 based upon the present

results, we use:

( 3) PTB( 3) ( 3)

( 3) ( 3) ( 3)

PTBj P K P K j P K

jPTB P K j P K PTB P K

d D D

d D D

. (46)

As before, the prime represents results derived from the present work and not the PTB results reported

in Ref. [1]. The pairwise differences are given in Table 19. For all labs except PTB, the pairwise

differences given in Table 19 are taken from Ref. [1]. For PTB, the results given in Table 19 are

derived from the present work using eq. (46).

The standard uncertainties for the pairwise differences are given by:

2 2

3( ) ( ) ( )P K jj c j c ju d u p u p . (47)

The uP-K3(djj′) are given in Table 20. For all labs except PTB, uc2(pj) is taken from Ref. [1] and the

uP-K3(djj′) results presented in Table 20 are the same as those reported in Ref. [1]. For PTB,

uc2(pj) = uc

2(pPTB) from the present work and is given in Table 18; the uP-K3(djj′) given in Table 20 are

derived from the present work using eq. (47).

The pair-wise differences between two labs are considered equivalent if

(P K3)

3

1( )

jj

P K jj

d

U d

. (48)

25

Here UP-K3(djj′) = 2 uP-K3(djj′). The results for djj′(P-K3)/ UP-K3(djj′) are given in Table 21 and are easily

derived from the values in Tables 19 and 20. Again, the PTB results are taken from the present work

whereas the results for the other labs are taken from Ref. [1].

8. Conclusions and Recommendations

It is clear from the pairwise difference results presented in Table 16 and Fig. 6 that PTB and NIST

show equivalence. The DOE for the pairwise difference between the two NMIs is less than 0.424 over

the range of target pressures from 3 × 10-6

Pa to 9 × 10-4

Pa, and for all target pressure except

9 × 10-6

Pa, the DOE is less than 0.121. Based on the differences between NIST and PTB in the

present work, we have derived the difference between PTB and the KCRV for the CCM.P-K3 as well

as the pairwise difference to all the NMIs that participated in CCM.P-K3. From Table 18, we see that

En < 0.5 for all target pressures, and from Table 21 we see that PTB demonstrates equivalence to all

the NMIs that participated in the CCM.P-K3. We therefore strongly recommend that these present

results be used to support a CMC for PTB in the pressure range of 3 × 10-6

Pa to 9 × 10-4

Pa.

26

Table 18. The DOE for PTB transferred to the CCM.P-K3 key comparison based upon the present results and the NIST results for the CCM.P-K3. The NIST data

for the CCM.P-K3 is taken from Ref. [1].

PT DNIST(P-K3)

/ Pa

DNIST(P-K3,1)

/ Pa

DPTB(P-K3.1)

/ Pa

D′PTB(P-K3)

/ Pa

u(PR(P-K3))

/Pa

u(pNIST(P-K3))

/ Pa

u(pNIST)

/ Pa

u(p′NIST)

/ Pa

u(pPTB)

/ Pa

U(D′PTB(P-K3))

/ Pa

DOE

E′n

3.E-6 -3.30E-08 3.01E-09 -3.01E-09 -3.90E-08 2.90E-08 3.43E-08 1.27E-08 2.35E-08 2.13E-08 8.60E-08 0.45

9.E-6 -2.58E-08 2.66E-08 -2.66E-08 -7.89E-08 6.37E-08 7.11E-08 3.12E-08 5.11E-08 5.43E-08 1.96E-07 0.40

3.E-5 3.27E-08 -2.22E-08 2.22E-08 7.72E-08 1.92E-07 1.98E-07 9.34E-08 1.46E-07 1.93E-07 6.18E-07 0.12

9.E-5 -1.26E-07 3.51E-08 -3.51E-08 -1.96E-07 4.49E-07 4.20E-07 2.48E-07 3.34E-07 3.88E-07 1.36E-06 0.14

3.E-4 6.07E-07 -6.66E-08 6.66E-08 7.40E-07 1.29E-06 1.17E-06 7.99E-07 9.84E-07 1.22E-06 4.06E-06 0.18

9.E-4 -6.16E-07 3.99E-07 -3.99E-07 -1.41E-06 3.31E-06 3.12E-06 2.32E-06 2.72E-06 3.60E-06 1.12E-05 0.13

27

Table 19. The pair-wise difference for the CCM.P-K3. For all labs except PTB, the pair-

wise differences are taken from Ref. [1]. The results for PTB are derived from this work

and represent the pair-wise difference transferred to the CCM.P-K3 key comparison based

upon the present results.

Djj′(P-K3)

j Dj(P-K3)

/ Pa NIST NPL NPLI KRISS PTB

P T = 3.0E-6 Pa

NIST -3.30E-08 7.09E-08 1.97E-08 4.13E-08 -6.02E-09

NPL 3.79E-08 -7.09E-08 -5.12E-08 -2.96E-08 -7.69E-08

NPLI -1.33E-08 -1.97E-08 5.12E-08 2.16E-08 -2.57E-08

KRISS 8.29E-09 -4.13E-08 2.96E-08 -2.16E-08 -4.73E-08

PTB -3.90E-08 6.02E-09 7.69E-08 2.57E-08 4.73E-08

P T = 9.0E-6 Pa

NIST -2.58E-08 1.40E-07 -9.00E-10 -3.62E-08 -5.31E-08

NPL 1.14E-07 -1.40E-07 -1.41E-07 -1.76E-07 -1.93E-07

NPLI -2.67E-08 9.00E-10 1.41E-07 -3.53E-08 -5.22E-08

KRISS -6.20E-08 3.62E-08 1.76E-07 3.53E-08 -1.69E-08

PTB -7.89E-08 5.31E-08 1.93E-07 5.22E-08 1.69E-08

P T = 3.0E-5 Pa

NIST 3.27E-08 2.24E-07 -5.87E-08 -2.97E-07 4.45E-08

NPL 2.57E-07 -2.24E-07 -2.83E-07 -5.21E-07 -1.80E-07

NPLI -2.60E-08 5.87E-08 2.83E-07 -2.38E-07 1.03E-07

KRISS -2.64E-07 2.97E-07 5.21E-07 2.38E-07 3.41E-07

PTB 7.72E-08 -4.45E-08 1.80E-07 -1.03E-07 -3.41E-07

P T = 9.0E-5 Pa

NIST -1.26E-07 8.29E-07 4.92E-07 -8.17E-07 -7.02E-08

NPL 7.03E-07 -8.29E-07 -3.37E-07 -1.65E-06 -8.99E-07

NPLI 3.66E-07 -4.92E-07 3.37E-07 -1.31E-06 -5.62E-07

KRISS -9.43E-07 8.17E-07 1.65E-06 1.31E-06 7.47E-07

PTB -1.96E-07 7.02E-08 8.99E-07 5.62E-07 -7.47E-07

P T = 3.0E-4 Pa

NIST 6.07E-07 -1.26E-06 2.03E-06 -3.20E-06 1.33E-07

NPL -6.57E-07 1.26E-06 3.30E-06 -1.93E-06 1.40E-06

NPLI 2.64E-06 -2.03E-06 -3.30E-06 -5.23E-06 -1.90E-06

KRISS -2.59E-06 3.20E-06 1.93E-06 5.23E-06 3.33E-06

PTB 7.40E-07 -1.33E-07 -1.40E-06 1.90E-06 -3.33E-06

PT = 9.0E-4 Pa

NIST -6.16E-07 -3.51E-06 7.11E-07 5.27E-06 -7.97E-07

NPL -4.13E-06 3.51E-06 4.23E-06 8.78E-06 2.72E-06

NPLI 9.54E-08 -7.11E-07 -4.23E-06 4.55E-06 -1.51E-06

KRISS 4.65E-06 -5.27E-06 -8.78E-06 -4.55E-06 -6.06E-06

PTB -1.41E-06 7.97E-07 -2.72E-06 1.51E-06 6.06E-06

28

Table 20. The uncertainty in the pair-wise difference for the CCM.P-K3. For all labs

except PTB, the uncertainties are taken from Ref. [1]. The results for PTB are derived from

the present work.

uP-K3(djj′)

j uj(P-K3)

/ Pa NIST NPL NPLI KRISS PTB

PT= 3.0E-6 Pa

NIST 3.43E-08 7.05E-08 7.20E-08 7.02E-08 4.45E-08

NPL 6.16E-08 7.05E-08 8.83E-08 8.69E-08 6.78E-08

NPLI 6.33E-08 7.20E-08 8.83E-08 8.81E-08 6.94E-08

KRISS 6.13E-08 7.02E-08 8.69E-08 8.81E-08 6.76E-08

PTB 2.13E-08 4.04E-08 6.52E-08 6.68E-08 6.49E-08

PT= 9.0E-6 Pa

NIST 7.11E-08 1.53E-07 1.60E-07 1.49E-07 1.06E-07

NPL 1.36E-07 1.53E-07 1.97E-07 1.89E-07 1.57E-07

NPLI 1.43E-07 1.60E-07 1.97E-07 1.94E-07 1.63E-07

KRISS 1.31E-07 1.49E-07 1.89E-07 1.94E-07 1.53E-07

PTB 5.43E-08 8.95E-08 1.46E-07 1.53E-07 1.42E-07

PT= 3.0E-5 Pa

NIST 1.98E-07 4.54E-07 4.84E-07 4.49E-07 3.34E-07

NPL 4.08E-07 4.54E-07 6.02E-07 5.73E-07 4.89E-07

NPLI 4.42E-07 4.84E-07 6.02E-07 5.98E-07 5.18E-07

KRISS 4.03E-07 4.49E-07 5.73E-07 5.98E-07 4.85E-07

PTB 1.93E-07 2.77E-07 4.51E-07 4.82E-07 4.47E-07

PT= 9.0E-5 Pa

NIST 4.20E-07 9.85E-07 1.18E-06 1.05E-06 8.00E-07

NPL 8.91E-07 9.85E-07 1.42E-06 1.31E-06 1.12E-06

NPLI 1.10E-06 1.18E-06 1.42E-06 1.46E-06 1.29E-06

KRISS 9.59E-07 1.05E-06 1.31E-06 1.46E-06 1.18E-06

PTB 3.88E-07 5.72E-07 9.72E-06 1.17E-06 1.03E-06

PT= 3.0E-4 Pa

NIST 1.17E-06 2.79E-06 3.62E-06 2.85E-06 2.52E-06

NPL 2.53E-06 2.79E-06 4.26E-06 3.63E-06 3.38E-06

NPLI 3.43E-06 3.62E-06 4.26E-06 4.30E-06 4.09E-06

KRISS 2.60E-06 2.85E-06 3.63E-06 4.30E-06 3.43E-06

PTB 1.22E-06 1.69E-06 2.81E-06 3.64E-06 2.87E-06

PT = 9.0E-4 Pa

NIST 3.12E-06 6.99E-06 9.30E-06 7.80E-06 7.51E-06

NPL 6.25E-06 6.99E-06 1.08E-05 9.50E-06 9.26E-06

NPLI 8.76E-06 9.30E-06 1.08E-05 1.13E-05 1.11E-05

KRISS 7.15E-06 7.80E-06 9.50E-06 1.13E-05 9.89E-06

PTB 3.60E-06 4.77E-06 7.21E-06 9.47E-06 8.01E-06

29

Table 21. The DOE for pairwise differences for the CCM.P-K3. The DOE is

calculated from the results presented in Tables 19 and 20. For all labs except

PTB, the results are identical to those given in Ref. [1]. The PTB results are

derived from the NIST and PTB measurements of the present work.

( 3)

3( )

jj P K

P K jj

d

U d

j NIST NPL NPLI KRISS PTB

PT= 3.0E-6 Pa

NIST 0.50 0.14 0.29 -0.07

NPL -0.50 -0.29 -0.17 -0.57

NPLI -0.14 0.29 0.12 -0.19

KRISS -0.29 0.17 -0.12 -0.35

PTB 0.07 0.59 0.19 0.36

PT= 9.0E-6 Pa

NIST 0.46 0.00 -0.12 -0.30

NPL -0.46 -0.36 -0.47 -0.66

NPLI 0.00 0.36 -0.09 -0.17

KRISS 0.12 0.47 0.09 -0.06

PTB 0.30 0.66 0.17 0.06

PT= 3.0E-5 Pa

NIST 0.25 -0.06 -0.33 0.08

NPL -0.25 -0.24 -0.45 -0.20

NPLI 0.06 0.24 -0.20 0.11

KRISS 0.33 0.45 0.20 0.38

PTB -0.08 0.20 -0.11 -0.38

PT= 9.0E-5 Pa

NIST 0.42 0.21 -0.39 -0.06

NPL -0.42 -0.12 -0.63 -0.46

NPLI -0.21 0.12 -0.45 -0.24

KRISS 0.39 0.63 0.45 0.36

PTB 0.06 0.46 0.24 -0.36

PT= 3.0E-4 Pa

NIST -0.23 0.28 -0.56 0.04

NPL 0.23 0.39 -0.27 0.25

NPLI -0.28 -0.39 -0.61 -0.26

KRISS 0.56 0.27 0.61 0.58

PTB -0.04 -0.25 0.26 -0.58

PT = 9.0E-4 Pa

NIST -0.25 0.04 0.34 -0.08

NPL 0.25 0.20 0.46 0.19

NPLI -0.04 -0.20 0.20 -0.08

KRISS -0.34 -0.46 -0.20 -0.38

PTB 0.08 -0.19 0.08 0.38

30

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