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Vacuum 81 (2006) 427–433

www.elsevier.com/locate/vacuum

The Reference Gauge technique for static expansion ratios—Applied toNPL medium vacuum standard SEA3

John Greenwood�

National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK

Received 23 March 2006; received in revised form 5 June 2006; accepted 5 June 2006

Abstract

Static expansion pressure standards are typically characterised in terms of the ratio between the density of an expanded sample of gas

and that of an initial sample. For standards that perform a series of expansions through several system vessels most of the existing

techniques to measure this expansion ratio involve disturbance to the system. They usually also involve operation with higher pressures

and with valve sequences that are not encountered under normal operational conditions, thereby introducing the risk of systematic

errors. A modification of the Reference Gauge technique for expansion ratios is described which has none of these disadvantages. The

technique is described with reference to the medium vacuum standard SEA3 at the UK National Physical Laboratory.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Vacuum standards; Expansion ratio; Reference Gauge

1. Introduction

The static expansion technique can be used to generatecalculable pressures as low as 10�6 Pa. In its simplest form,the technique requires a vacuum system consisting of twovessels separated by a valve and a means of measuring thepressure in the first of the vessels. Gas isolated in the firstvessel is then shared between the two vessels by opening theconnecting valve, thereby reducing the pressure in propor-tion to the change in volume of the gas. Pressure standards[1] based on the technique are maintained by a number ofNational Measurement Institutes [2].

Early workers [3] used gravimetric techniques to measurethe absolute volume of the individual vessels. Thesemethods are still useful [4], however they have the seriousdisadvantage that the system must be disassembled everytime re-measurement is required. Other techniques thatdetermine absolute volumes have also been reported, usinga calibrated gauge and a previously measured referencevolume [5–7]. As with the gravimetric techniques, these

ee front matter r 2006 Elsevier Ltd. All rights reserved.

cuum.2006.06.012

8943 7126.

ess: John.greenwood@npl.co.uk.

methods often have the disadvantage that the system mustbe disturbed for measurement to take place.Recognising that, under certain experimentally achievable

conditions, only the ratio of volumes is required rather thanabsolute values, a technique was developed at NPL toaccurately determine expansion ratios [8]. In this technique anadditional, high accuracy, calibrated pressure gauge isconnected to each expansion volume in turn and the pressureof the gas accumulated in the system is built up by multipleexpansions to the point at which it can be accuratelymeasured. This successive accumulation method has beendeveloped in recent years [4,9,10] and is widely used where theratio of the final to initial volume of the gas is below about200, though it has successfully been implemented at ratios upto 3000 [11]. The method has the disadvantages that eachexpansion volume must be disturbed to introduce the secondgauge and that the pressures are different to thoseexperienced in normal operation. Reported uncertainties forthe technique are better than 0.1%.For large volumetric ratios, a technique that involves

repeated single expansions has been developed [4,6,9].Unlike the accumulation technique, this method does notrequire a calibrated gauge to measure the initial pressure;however an additional, uncalibrated gauge that has a

ARTICLE IN PRESSJ. Greenwood / Vacuum 81 (2006) 427–433428

fundamentally linear response to pressure is also required.This technique has the same disadvantages as theaccumulation technique for multi-vessel expansion sys-tems. Careful comparisons of the results achieved by thistechnique and the accumulation technique show agreementto within experimental uncertainty [4,9].

Another technique, known as the Reference Gaugetechnique, has also been briefly described [4]. In thismethod, a pressure balance is used to establish a high initialpressure (up to 3� 105 Pa) that is subsequently expandedand measured by an uncalibrated gauge. The pressurebalance is then used to directly generate a low pressure thatreproduces the indication of the uncalibrated gauge. Theexpansion ratio is then calculated from the ratio of thepressures required to produce the same indication on theuncalibrated gauge.

In this paper, the Reference Gauge technique has beenadapted to determine the expansion ratios for seriesexpansion systems in which the initial pressure is itselfgenerated from an earlier expansion stage. Unlike thegravimetric and accumulation techniques, the ReferenceGauge technique does not require the system to bedisturbed—it uses the system’s own inlet pressure standardto initiate the generation of reference pressures in the finalcalibration volume. No additional equipment is requiredbesides high-quality sensors, such as capacitance dia-phragm gauges that are routinely calibrated on suchsystems. This is particularly useful for automated systemsas it offers the possibility for the expansion ratios of thesystem to be under continual and unattended evaluationwhenever the system is not in use for other purposes.

The paper describes the implementation of the modifiedReference Gauge technique with the National PhysicalLaboratory’s medium vacuum static expansion standard,SEA3. Following a short background description, applica-tion of the technique is described and some results andpoints of discussion are presented.

2. Static Expansion Apparatus SEA3

For the simplest isothermal system consisting of aninitial isolation vessel, of volume v, connected through avalve to an expansion vessel, of volume V, the expanded

inlet pressure

N2 supply

V1 V2 V3

A B C D

Fig. 1. Schematic of stat

pressure, P, of an ideal gas is related to the initial pressure,p, by the Boyle–Mariotte law, which gives

P ¼ pv

vþ Vð Þ. (1)

Still lower pressures can be generated when the initialpressure is established by a preceding stage (or series ofstages) of expansion. This could be achieved, for example,using a system such as the NPL medium vacuum standarddepicted schematically in Fig. 1.The system consists of alternate small and large vessels

{Vi: i ¼ (1, 2, 3, 4, 5, 6, 7)} connected by valves {j: j ¼ (A,B, C, D, E, F, G, H)}, where we shall denote the volume ofeach vessel Vi by vi. We shall consider that any volumeassociated with each end of a valve in its closed state isincorporated into the volumes of the connected vessels, andwe shall denote the change in volume when a valve isopened by dvj. The pressure of the initially isolated gas ismeasured using a high-quality transfer standard, such as aquartz bourdon gauge.In the NPL standard calculable reference pressures are

generated in the vessel V7. Expansion into V7 is performedfrom either V5 or V6 and can be:

Single expansion from V5—where initially valves F andH are closed and the other valves are all open so that anappropriate initial pressure can be established through-out the connected system V1–V5. Volume V5 is thenisolated by slowly closing valve E, before the pressure ofthe gas trapped in V5 is measured (by the inlet gauge),and then expanded into the connected volumes V5+V7.Alternatively, it can be:Single expansion from V6—where it is V6 that is isolatedby slowly closing valve G, before the pressure of the gastrapped in V6 is measured, and then expanded into theconnected volumes V6+V7.

Expansion can also be:

Double expansion—where the initial isolation takes placein V3, before expansion into V3+V4+V5+V6, followedby isolation in V5 (or V6) and then expansion into theconnected volumes V5+V7 (or V6+V7). Or expansioncan be:

V4

V6

V5

V7

E F

G H

ic expansion system.

ARTICLE IN PRESSJ. Greenwood / Vacuum 81 (2006) 427–433 429

Triple expansion—where the initial isolation takes placein V1, before expansion into V1+V2+V3, followed byisolation in V3 and subsequent completion of the processas for a double expansion.

To calculate the pressure in V7, we need to first calculatethe amount of gas in this vessel following single, double ortriple expansion. If the volumes, {vi} were known thecalculation could be made using a realistic gas distributionmodel [12]. Unfortunately, experimental difficulties meanthat the volumes cannot usually be determined withsufficiently low measurement uncertainty and a simplifiedapproach is usually taken for which so-called expansion

ratios are determined.Taking into account the non-ideal nature of a real gas,

the simplified measurement equation for static expansionbased on expansion ratios can be written as follows:

rf ¼ rin � Ex, (2)

where Ex is the overall expansion ratio, rf the final molardensity and rin the initial molar density. The density can becalculated using

r ¼1

2RTBT

R2T2 þ 4RTBT p� �1=2

� RT� �

, (3)

which is a solution of the second-order virial equation ofstate for pressure:

p ¼ rRT 1þ rBTð Þ, (4)

where R is the gas constant and BT is the second virialcoefficient for the particular gas species, evaluated attemperature T.

An implicit assumption of this simplified approach is thatthe molar density is the same throughout a system ofconnected vessels. This assumption holds under ‘‘isother-mal’’ conditions where the temperature of the gas is constantthroughout. Under these particular circumstances, forexpansion from V5 into V5+V7 we have the expansion ratio,

E5;7 ¼v5

v5 þ dvF þ v7, (5)

and for expansion from V6 into V6+V7 we have

E6;7 ¼v6

v6 þ dvH þ v7. (6)

For the expansion from V3 into V3+V4+V5+V6 withsubsequent isolation of V5 we have

E3;5 ¼v3

v3 þ dvD þ v4 þ v5 þ dvG þ v6(7)

and similarly, for the expansion from V3 intoV3+V4+V5+V6 with subsequent isolation of V6 we have

E3;6 ¼v3

v3 þ dvD þ v4 þ v5 þ dvE þ v6. (8)

Finally, for the expansion from V1 into V1+V2+V3 withsubsequent isolation of V3 we have

E1;3 ¼v1

v1 þ dvB þ v2 þ v3. (9)

In practice, isothermal conditions are difficult to maintainwithout special measures [4] and the actual expansion ratioachieved in practice will not be equivalent to the ratio of thevolumes as in (5)–(9) [18]. For the purposes of this paper thedistinction is not critical.Applying the simplified measurement equation for the

expansions performed by SEA3, we can write

r7 ¼ rin � Ex, (10)

where for example, for a triple expansion via V5

Ex ¼ E1;3 � E3;5 � E5;7. (11)

3. Reference Gauge technique applied to NPL vacuum

standard SEA3

The Reference Gauge technique essentially involvesgenerating the same gas density by two different means.To measure the generated pressure, a sensor with anappropriate full-scale range is connected to a test port ofV7. Capacitance diaphragm gauges are ideally suited forthis purpose because of their excellent (short-term) stability[13], though for expanded pressures below about 0.1 Pa aspinning rotor gauge may be more suitable. For optimumsensitivity, the best sensor for measuring the pressureestablished in V7 would have a full-scale range only slightlyhigher than the highest generated pressure. However, suchspecific devices are usually not available. For the measure-ments reported here capacitance diaphragm gauges withfull-scale ranges of 1000Torr (1.33� 105 Pa), 10 Torr(1.33� 103 Pa) and 1Torr (1.33 Pa) were used to registerthe generated pressures.

3.1. Expansion ratio E5,7—from V5 into V5+V7

From previous experience and consideration of thesystem volumes, the expansion ratio E5,7 is estimated tobe in the region of 0.02. For a maximum inlet pressure ofpmax ¼ 105 Pa, the maximum pressure that can be gener-ated is therefore about 2000 Pa. This expanded pressurecan also be directly measured with an acceptable level ofuncertainty using the inlet gauge—a Ruska 7050i with anoperating range of 105 Pa.To determine the expansion ratio E5,7 the technique is

therefore implemented as follows: first, an inlet pressure,pki (ppmax) is established and the initial temperature, Tki, ismeasured. The single expansion is then performed and thesensor response, bko and the temperature of the expandedgas, Tko is measured. The vessel V7 is then evacuated andthe expansion process is repeated a number of times withnominally the same inlet pressure. From the data collectedthe mean values of inlet density, rk, and normalised sensoroutput, bk, are calculated:

rki ¼1

2R Tki BTki

R2T2ki þ 4R Tki BTki

pki

� �1=2� R Tki

� �,

(12)

ARTICLE IN PRESS

�ko

�o

�/m

ole.

m-3

p /P

a

5.500E-35.495E-3

�/volts.K-1

5.490E-3

b /volts

1.61.41.21.00.8

42.12

42.11

42.10

42.09

42.08

42.07

42.06

2200

2000

1800

1600

1400

1200

1000

(a)

(b)

Fig. 2. (a) e5,7 data for determination of E5,7. Standard uncertainty for

each point is u(rki)E0.02molm�3, u(bko)E3� 10�6 VK�1. (b) ‘‘Direct

measurement’’ data for determination of E5,7. Standard uncertainty for

each point is u(p)E0.45 Pa, u(b)E8� 10�4 V.

J. Greenwood / Vacuum 81 (2006) 427–433430

rk ¼ rki

� �, (13)

bko ¼bko

Tko

, (14)

bk ¼ bko

� �. (15)

The mean expanded gas temperature and a correspondingsensor response is then calculated as

To ¼ Tkoh i, (16)

bo ¼ bk � To. (17)

The reference pressures in V7 are then established, anddirectly measured, while all of the inter-volume valves areopen. This is achieved by setting a small range of inletpressures, p, and recording the corresponding output, b, forvalues spanning bo. A generalised least-squares linearregression fit [14] is made to the direct measurement datato yield the coefficients (a,c) where

p ¼ a� bþ c, (18)

which is then used to calculate po, the pressure correspond-ing to bo:

po ¼ a� bo þ c. (19)

The corresponding molar density in the system can thenbe calculated as

ro ¼1

2R To BTo

R2 T2o þ 4R To BTo

po

� �1=2� RTo

� �. (20)

The single expansion data indicates that this molar densityro was produced by single expansion from initial molardensity rk, i.e.

ro ¼ e5;7 � rk, (21)

where e5,7 is the measured expansion ratio; hence,combining the results of Eqs. (13) and (21) we calculate

e5;7 ¼ro

rk

. (22)

Some typical results for this measurement are shown inFig. 2.

The measured expansion ratio incorporates the addi-tional volume, vA, of the capacitance diaphragm gaugeconnected to V7. The basic, zero added-volume expansionratio is therefore

E5;7 ¼1

e5;7�

vA

v5

� ��1. (23)

Volumes vA and v5 can be determined with sufficientaccuracy by expansion or dimensional techniques.

3.2. Expansion ratio E6,7—from V6 into V6+V7

The expansion ratio E6,7 is estimated to be in the regionof 0.0055, therefore the maximum pressure that can begenerated by single expansion from V6 is about 550 Pa.Unlike the case for determination of E5,7, this pressure is

too low to be measured directly with sufficient accuracywhen using the inlet pressure gauge; however, this pressurecan also be generated by a single expansion from V5 withan initial pressure of about 27,500 Pa.To determine the expansion ratio E6,7 the technique is

therefore implemented as follows. For the expansions fromV6, a small range of inlet pressures, {pui:pmax4pui40.95pmax} is established and at each of these the initialtemperature, Tui, is measured and an expansion isperformed. The sensor response, buo and the temperatureof the expanded gas, Tuo is recorded and the vessel V7 isthen re-evacuated ready for the next inlet pressure.The recorded data are used to compute the inlet density

rui and the corresponding normalised sensor output buo

and a generalised least-squares linear regression fit is thenmade to the data to determine coefficients (a,c) where

rui ¼ a� buo þ c. (24)

Next, pressures are generated by expansion from V5 withinlet pressure, pki, such that the sensor output is bko, a valuewithin the range of the sensor indications produced by theexpansions from V6. The initial and expanded gastemperatures Tki and Tko are also recorded. From the

ARTICLE IN PRESS

1.4628E-2

�ko

�k

38

39

40

41

42

43

44

0.0150

�uo

�/volts.K-1

�/m

ole.

m-3

�/m

ole.

m-3

�/volts.K-1

11.281

11.280

11.279

11.278

11.277

11.276

11.275

1.4624E-2 1.4626E-2 1.4630E-2 1.4632E-2

0.01450.01400.01350.0130

(a)

(b)

Fig. 3. (a) e5,7 data for determination of E6,7. Standard uncertainty for

each point is u(rki)E4� 10�3molm�3, u(bko)E9� 10�6 VK�1. (b) e6,7data for determination of E6,7. Standard uncertainty for each point is

u(rui)E0.015molm�3, u(buo)E9� 10�6 VK�1.

J. Greenwood / Vacuum 81 (2006) 427–433 431

measured V5 expansion data the mean inlet density and themean normalised sensor output are calculated using(12)–(15).

The fit to the V6 expansion data is then used todetermine the density, ru corresponding to buo ¼ bk:

ru ¼ a� bk þ c. (25)

Since the expansion from V6 generates the density ro

corresponding to the sensor indication bk by expansionfrom ru we have

ro ¼ e6;7 � ru, (26)

whereas single expansions from V5 generated the same

density by expansion from rk:

ro ¼ e5;7 � rk. (27)

Hence, equating (26) and (27), and expressing in termsof the basic, zero added-volume ratios E5,7 and E6,7

we have

1

E5;7�

vA

v5

� �� rk ¼

1

E6;7�

vA

v6

� �� ru. (28)

The ratio E6,7 is therefore given by

E6;7 ¼rk

ru

1

E5;7�

vA

v5

� ��

vA

v6

� ��1. (29)

Typical results for this measurement are shown in Fig. 3.

3.3. Expansion ratio E3,5—from V3 into (V3+V4+V5+V6)

followed by isolation of V5

The procedure for determination of E3,5 is similar toE6,7. In this case, the expansion ratio E3,5 is estimated to bein the region of 0.011 so the maximum pressure that can begenerated in V7 by double expansion is therefore about22 Pa. This expanded pressure can also be generated by asingle expansion from V5 with an initial pressure of about1100 Pa.

The double expansions are performed with a range ofinitial pressures and the recorded data are used to computethe inlet density rui and the corresponding normalisedsensor output buo and a fit is then made to the data.

Next, pressures are generated by single expansion fromV5. From the measured single expansion data, the meaninlet density and the mean normalised sensor output arecalculated.

A fit to the double expansion data is then used todetermine the density, ru corresponding to buo ¼ bk. Sincethe double expansion generates the density ro correspond-ing to the sensor indication bk by expansion from ru wehave

ro ¼ e3;5 � e5;7 � ru, (30)

whereas single expansions generate the same density byexpansion from rk:

ro ¼ e5;7 � rk. (31)

The measured expansion ratio e3,5 is therefore given by

e3;5 ¼rk

ru

. (32)

Since there is no additional volume associated with thisparticular expansion ratio, the basic, zero added-volumeratio E3,5 is therefore

E3;5 ¼ e3;5. (33)

Typical results for this particular measurement are shownin Fig. 4.

3.4. Expansion ratio E1,3—from V1 into (V1+V2+V3 )

followed by isolation of V3

The procedure to determine E1,3 is similar to theprocedure for E3,5 except that triple and double expansionsare performed in place of double and single expansions,respectively. The expansion ratio E1,3 is estimated to be inthe region of 0.02, therefore the maximum pressure thatcan be generated in V7 by triple expansion is about 0.44 Pa.This expanded pressure can also be generated by a doubleexpansion from an initial pressure of about 2000 Pa.

ARTICLE IN PRESS

�ko

�k

�/m

ole.

m-3

�/m

ole.

m-3

38

39

40

41

42

43

44

�uo

�/volts.K-1

�/volts.K-1

6.260E-36.258E-36.256E-3

0.4792

0.4790

0.4788

0.4786

0.4784

0.4782

6.254E-3

0.0056 0.0060 0.0062 0.00640.0058

(a)

(b)

Fig. 4. (a) e5,7 data for determination of E3,5. Standard uncertainty for

each point is u(rki)E2.5� 10�4molm�3, u(bko)E4� 10�6 VK�1. (b) e3,5data for determination of E3,5. Standard uncertainty for each point is

u(rui)E0.015molm�3, u(buo)E4� 10�6 VK�1.

�ko

�k

38

39

40

41

42

43

44

�uo

�/volts.K-1

�/m

ole.

m-3

�/m

ole.

m-3

1.0091

1.0089

1.0087

1.0085

1.0083

1.0081

1.526E-2 1.527E-2 1.528E-2

0.0137 0.0142 0.0147 0.0152 0.0157

�/volts.K-1

(a)

(b)

Fig. 5. (a) e3,5 data for determination of E1,3. Standard uncertainty for

each point is u(rki)E4� 10�4molm�3, u(bko)E1� 10�5 VK�1. (b) e1,3data for determination of E1,3. Standard uncertainty for each point is

u(rui)E0.015molm�3, u(buo)E1� 10�5 VK�1.

J. Greenwood / Vacuum 81 (2006) 427–433432

Since the triple expansion generates the density ro

corresponding to the sensor indication buo ¼ bk by expan-sion from ru we have

ro ¼ e1;3 � e3;5 � e5;7 � ru. (34)

Double expansions generate the same density by expan-sion from rk so

ro ¼ e3;5 � e5;7 � rk. (35)

The measured expansion ratio e1,3 is therefore given by

e1;3 ¼rk

ru

. (36)

Again, since there is no additional volume associatedwith this particular expansion, the basic, zero added-volume ratio E1,3 is therefore

E1;3 ¼ e1;3. (37)

Typical results are shown in Fig. 5.

4. Results

During determination of each particular ratio, themeasurement processes was repeated five times to deter-mine a single measured value. Each complete data setshown in Figs. 2–5 was collected, unattended, overapproximately 12 h.A brief summary of the results is presented in Table 1,

where the values of the ratios are computed from betweentwo and four complete data sets.The standard ISO-GUM [15] approach has been

followed to estimate the uncertainty of measurement forthe expansion ratios. In forming this estimate, theuncertainty of the individual measurement points has beenevaluated from the brought-in measurement uncertainty ofthe respective measuring instruments in combination withvarious operational influences, such as zero drift andlocation of temperature probes. This is summarised in thefigure captions for the typical data shown in Figs. 2–5. Theuncertainty of the corresponding mean values is evaluatedby combining the (Type B) uncertainty for individual

ARTICLE IN PRESS

Table 1

Summary of expansion ratios measured by Reference Gauge technique

Ratio Value Standard uncertainty

E5,7 0.020805 2.3� 10�5

E6,7 0.005525 7.5� 10�6

E3,5 0.011253 9.7� 10�6

E1,3 0.023626 1.9� 10�5

Table 2

Typical standard uncertainty in intermediate values

Ratio u(rk)

(molm�3)

u(bk)

(VK�1)

u(Tk)

(K)

u(pu)

(Pa)

u(ru)

(molm�3)

E5,7 0.017 3.1� 10�6 0.18 1.7 8.7� 10�4

E6,7 4.5� 10�3 9.7� 10�6 — — 0.030

E3,5 2.5� 10�4 4.0� 10�6 — — 0.029

E1,3 4.1 � 10�4 1.0� 10�5 — — 0.030

J. Greenwood / Vacuum 81 (2006) 427–433 433

measurements and the (Type A) variation in the calculatedmean. For those parameters calculated from linear regres-sion fits to experimental data, the uncertainty in thepredicted value is evaluated by a generalised least-squaresfitting program [16]. The full uncertainty evaluation isdescribed in detail elsewhere [17]; however, some typicalresults for intermediate values are summarised in Table 2.

5. Discussion

Besides uncertainty of measurement, the most likelysources of error in calculated ratios are uncorrected,systematic operational effects such as: temperature gradi-ents between vessels [12], pressure dependent valvediaphragm deflection [11], volume changes after valveclosing (changes of about 0.01 cm3 have been measured atNPL), or differences between nominally identical isolationand expansion valves [8]. These effects can all account forsystematic deviations, particularly when using expansionratios determined by techniques that work with higherpressures and different valve operating sequences to thoseexperienced in normal static expansions.

The consequences of these systematic effects are greatlyreduced by using the Reference Gauge technique; which isperformed at similar pressures, and uses identical valveoperating sequences to those encountered in normaloperation. This is reflected in better reproducibility of theReference Gauge data when compared with accumulationdata. The overall uncertainty associated with the vacuumstandard is therefore improved.

The Reference Gauge technique works well here because,for each stage, the expanded pressure can be generated bytwo different means, each requiring an initial pressure thatis measurable with a low uncertainty using the standard’s

own inlet pressure gauge. In situations where this is notpossible, the measurement range could be extended by useof an additional gauge to establish inlet pressures outsidethe normal range of operation.

6. Conclusion

A modified version of the Reference Gauge techniquehas been applied to the medium vacuum standard SEA3,where it is shown to be a reliable technique capable ofachieving low uncertainties.The technique is easy to implement and offers several

advantages over the more commonly applied successiveaccumulation technique. It is convenient in that it does notdisturb the system or require additional calibrated gauges,and it is more realistic, since it measures expansion ratiosusing the exact sequences of valve operations and pressuresexperienced in practice.

Acknowledgement

The author is pleased to acknowledge the support of theUK National Measurement System. http://www.dti.gov.uk/nms/.

References

[1] Jousten K. In: Lafferty JM, editor. Foundations of vacuum science

and technology. New York: Wiley; 1998 (Chapter 12).

[2] Messer G, et al. Metrologia 1989;26:183–95.

[3] Schumann S. Transactions of the ninth national vacuum symposium

of the American Vacuum Society. The Macmillan Company; 1962.

p. 463–7.

[4] Jitschin W, Migwi JK, Grosse G. Vacuum 1990;40:293–304.

[5] Elliott KWT, Woodman DM, Dadson RS. Vacuum 1967;17:439–44.

[6] Berman A, Fremery JK. J Vac Sci Technol A 1987;5:2436–9.

[7] Bergoglio M, et al. Vacuum 1988;38:887–91.

[8] Elliott KWT, Clapham PB. The accurate measurement of the volume

ratios of vacuum vessels. NPL report MOM 28, January 1978.

[9] Jousten K, Rohl P, Aranda Contreras V. Vacuum 1999;52:491–9.

[10] Redgrave FJ, Forbes AB, Harris PM. Vacuum 1999;53:159–62.

[11] Sharma JKN, Mohan P. J Vac Sci Technol A 1988;6:3148–53.

[12] Greenwood JC. Vacuum 2006;80:548–53.

[13] Miiller AP. Metrologia 1999;36:617–21.

[14] Barker RM, Cox MG, Forbes AB and Harris PM. Software support

for metrology best practice guide no. 4—discrete modelling and

experimental data analysis. London: HMSO; April 2004, ISSN 1471-

4124, version 2.

[15] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML. Guide to the

expression of uncertainty in measurement, 1st ed. Geneva, Switzer-

land: International Organisation for Standardization; 1993. ISBN 92-

67-10188-9.

[16] Cox MG, Forbes AB, Harris PM, Smith IM. The classification and

solution of regression problems for calibration. NPL report CMSC

24/03, August 2004.

[17] Greenwood JC. NPL report DEPC-EM 005, London: HMSO;

March 2006 ISSN 1744-0254.

[18] Jitschin W. Metrologia 2002;39:249–61.