Standards and the Propagation of Standards

Chapter 2 Standards and the Propagation of Standards

2.1 Establishing the Standards of Units of Measurement international standards are established by convention, in the form of resolutions of the General Conference of Weights and Measures. The basis of the convention is the current state of knowledge, and the resulting conviction that a new settle­ ment is better than the previous one. To illustrate this, a historical outline of the most important settlements on the units of measurement of electric quantities will be presented. A number of the changes made in this field are quite significant. The units of measurement of electric quantities were initially defined by Gauss in the CGS system via mechanical quantities. To combine the electric and magnetic measurement units, it was necessary to introduce an electric base quantity. In 1881,1 ampere was accepted as the base unit of measurement of electric quantities. The ampere was defined in various ways: — through the volt and the ohm, — by Faraday's law, based on the mass of silver produced in the electrolysis of silver nitrate, — more recently, by the force of interaction between two conductors which carry flowing current. This refers to the MKS A system of units. The search for standard units of electric quantities continued for years, and different standards were in use. 1860—Standard resistance, constructed by Siemens. This was a mercury 2 column 1 metre long with a cross-sectional area of 1 m m at 0°C. This was used until 1911, and in Germany untill 1948. (In 1898, the column length was in­ creased to 106.3 cm). 1863—The BAU standard—a metal alloy, applied in Great Britain. 1873—The Clark cell, accepted in 1893 as the standard element. 1881 —The ohm and the volt accepted as the base units, with the ampere defined through the ohm and volt; empirical units (standards) of ohm and ampere were defined—a current of 1 A yields 1.11810 mg of silver during 1 second of silver nitrate electrolysis.

STANDARDS A N D THE PROPAGATION OF STANDARDS

38

Ch.2

1892—The Weston cell introduced as the new standard element. 1910—Application of the ampere definition based on electrolytic phenomena; the EMF of the standard cell was determined as 1.01830 V. 1 1927—An electrical laboratory created within the framework of BIPM; beginning of studies on standard cells and standard resistors. 1948—The General Conference of Weights and Measures introduced the MKSA system of units. 1960—Introduction of the SI system of units. 1931, 1969—Corrections of the value of EMF of cells in the bureaus of measu­ rement in different countries. 1962—The discovery of the Josephson effect, established in 1968 as the voltage standard. 1 2 1972—Constant Φ ο = h/2e accepted by the CIPM. 1977—Discovery of the quantum Hall effect [166]. 1980—Construction of silicon MOSFET elements as the prototype resistance standard, applying the quantum Hall effect [187]. 1990—Planned acceptance of the Josephson junction as the voltage standard. Current state: 6 — base unit—ampere, reproduced with an accuracy of 6 χ 10" using the current balance; — best reproduced standard—Weston standard cell (group standard), accuracy 6 better than 10" ; 7 — new standard being investigated—Josephson junction, accuracy 1 0 " ; the nominal value established arbitrarily by CIPM is [109]: φ

0

- ι *

β

2e

483597.9 '

M

μ ν

Β

HZ

Ι ΜΡ

* 7

8

9

The accuracy obtained initially was 10" to 10" , it has now reached 10" [194] 12 and is predicted to reach 10~ [162]. The forecasts of Kose and Wöger [162] are presented in Fig. 2.1. The new resistance standard based on the quantum Hall effect is the silicon MOSFET element or heterostructure of GaAs, AlGaAs, InGaAs, InP, whose resistance is jRm

—

ι

e

z

?

ζ=

l 5 2, 3, ...

For / = 1, a value of RK = 25 812.807 7

is called Klitzing constant. Its experimentally verified accuracy is 10~ -10" 1 1

BIPM—Bureau International des Poids et Mesures (in Sevres, Paris). CIPM—Comite International des Poids et Mesures.

8

2.1 ESTABLISHING THE STANDARDS OF UNITS OF MEASUREMENT

10-"I

1930

ι

l

1940

ι

Ι 1950

ι

I

1960 year

ι

I

1970

ι

I

1980

ι

39

I

1990

Fig. 2.1. The relative uncertainty of reproduction of 1 volt; black circle—Weston standard cell, white circle—Josephson junction, triangle—Josephson junction array at room temperature, square—junction array at helium temperature 4.2 Κ [162].

[148, 195]. The advantage of the quantum Hall effect is that the value of the measure of the standard is determined by fundamental physical constants. The value of voltage in the Josephson junction is frequency-dependent. Progress in determination of fundamental physical constants is presented bv

x 1950

ι

ι 1960

ι

1 1970

-J

1— 1980

year

Fig. 2.2. The uncertainty of the physical constant hjlr

[162].

STANDARDS A N D THE PROPAGATION OF STANDARDS

40

Ch.2

Kose and Wöger [162]. Figure 2.2 shows the improvement in accuracy of the constant h/2e. A turning point can be shown to have occurred in the field of standards of measurement of electric quantities. Let us review its history. The value of Φ0 obtained from the EMF of a standard cell with an accuracy 6 5 of 6 x 10~ differed from the previous data by 3 χ 10~ [61]. The standard com­ monly applied for 90 years in the domain of electric quantities is the Weston cell. 3 The life-time of the cell is estimated at about 20 years. The EMF of a cell may be derived from a model of the chemical reactions occurring in the cell. The physicochemical model of the cell is empirically proven, but knowledge of the physicochemical properties is not profound enough to determine the value of EMF theoretically. It is also impossible to determine the EMF through the standard current, because of the inferior accuracy of the latter. In the field of electric quantities, a capacitance standard with an accuracy of 7 about 1 χ 10" may be applied. This is a so-called calculable standard, the measure of which is derived from a mathematical model by finding the dimensions of the standard. The construction of such a standard is shown in Fig. 2.3 [128].

Fig. 2.3. Cross-section of the calculable Thompson-Lampard capacitance standard (cross-capa­ citor). Capacitance value per length unit is C — (ε 0/π)1η2 F/m [128].

Means are also available for the construction of a standard current by applying the effect of paramagnetic resonance. The accuracy of this standard would be 6 7 10" [49] to 4 x 1 0 " [189]. Application of the Weston standard cell as the standard has been restricted by the difficulty of assuring the continuity of the reference state, because of the limited durability of the element. The method applied was later called the group standard. The group standard consists of a number of selected standards, and is 3

As Hamer [31] states, in 1951 the group standard of NBS included the cells from 1906, 1933 and 1948. Thus the durability is even greater.

2.1 ESTABLISHING THE STANDARDS OF UNITS OF MEASUREMENT

41

provided with new ones at the proper time. The standards whose measures depart from average are eliminated. Selection of the elements to be included in the group standard is carried out on the basis of the metrological properties expressed as the class 1 characteristic (see Chapter 4), investigated for 1-2 years. At present, the EMF of a cell is meas­ ured with an uncertainty of single hundredths of a microvolt. The level of noise, in the form of oscillations and fluctuations, is thus very low for these elements. For example, the temperature is kept constant within 0.001 °C. In Fig. 2.4, the

(a)

+ France ο ° • • x

(b)

Groat Britain USSR West Germany USA

^

20

I

L

15

Japan

* -~X2 H

10

[

Ξ ^ 5 .3

\" 10.4/jV

t

11.69

9- I

o-!l 2 5

-5. 1953

1958 1. 2. 3. 4 5. 6

1910

1920

1930

1940

1963

... 1 1 1 1 1968 1973

PTB West Germany BIMP ETZ Japan IMM USSR NBS USA Ν PL Great Britain

1950

Fig. 2.4. The results of comparison of the national group voltage standards: (a) data according to Hamer [31], (b) data according to Froehlich [25] referred to the PTB standard.

variation of the nominal value of national group standards is shown. The correc­ tions of the nominal values of the standards are indicated, as introduced in 1929-1936 and in 1969. According to M. Froehlich [25], the stability of the mean 8 value of EMF of a group standard consisting of 38 cells was 1 χ 10" V/year, as compared with the Josephson standard. Such good stability allows comparison over intervals of a few months [25]. The repeatability of the results is shown in Fig. 2.5. The problem of maintaining the measure of a standard constant despite the limited durability of the standard has been solved by the application of a gradually restored group standard. The elements of the group standard are selected on the basis of their metrological properties. The criterion for the selection is the insta­ bility error not being too great.

42

STANDARDS A N D THE PROPAGATION OF STANDARDS

ι January 1971

ι January 1972

1 January 1973

J January 1974

Ch.2

1 »» January 1975

Fig. 2.5. The results of comparison of the NML group voltage standard (Australia) consisting of 7 cells, with Josephson effect standard [33].

The criterion is difficult to formulate for a small number of elements of the group standard, as well as for a small number of results of comparison with other standards. Thus, arbitrary decisions can be made here. The state of the standards and the conditions of their reproduction are different for different physical quantities. For example, the mass standard was established a long time ago and, as may be supposed, the limits of its reproduction accuracy have already been reached. A fundamental change in accuracy and feasibility of reproduction occurred in length measurements when the prototype of metre was substituted by the atom effect standard. Changes occur as progress in science takes place. The prospects have been presented by Kose and Wöger [162]. 2.2 Propagation of Standards When two standards Wt and W2 are compared, it is stated a priori that one of them is the primary standard and the other is the secondary standard. The primary standard measure is transferred onto the secondary standard, and the unavoid­ able error is attributed to the secondary standard. For this reason, the standards of the same unit of measure, with the same value of measure, are classified at a proper level in the hierarchy of standards. The primary standard is placed at a higher level and the secondary standard at a lower one. The number of levels in the hierarchy is limited, for practical reasons. The lowest level includes utility

2.2 PROPAGATION OF STANDARDS

43

tools, following inspection tools. The assignment of a measure to the standards to be accepted for use is called calibration, although in this case the measure of the standard is propagated and the term the propagation of standards is frequently used. The primary standard measure is transferred onto the secondary standard, and the uncertainty of the measure is determined from the course of the compa­ rison process. Let us consider the case of two compared standards having approximately the same measure. The primary standard W1 has a nominal measure vv10 established at the instant T01 of the service time of the primary standard. The uncertainty of measure of the primary standard Λ = w'i-Wio

(2.1)

is described by a non-stationary random proces {A x}: {A1(T)} = gi(T){s1}+hi(T),

(2.2)

where gi(T), hx{T) are time functions characterizing the random and systematic errors, respectively, {ε χ } is a stationary random process with mean value equal to 0, unitary variance and the correlation function Ke(r)

= E[{et}{et+T}]

=

\ρ(τ)\ < 1,

ρ(τ),

(2.3a) (2.3b)

Γ is the service time of the standard, t is the time, t e T. For obvious reasons, the nominal value of the standard was established at the instant T0l in order to avoid introducing a systematic error in the standard. The nominal value of the standard may vary during its life period by the value hl0(T) (by a known rule) which may be predicted for a time Τ > Γ 0 1 , based on checks done over the time interval Τ < T01. Therefore, *>io(T) = w10(T0l)-hl0(T),

(2.4)

where h10(T) is the prediction of the systematic error. To simplify the notation, 4 equations (2.1), (2.4) are further interpreted a s

Λσ)= ^(D-HUT).

(2.5)

2. The systematic error h^T) in equation (2.2) does not include the component with the known value of hi0(T)- The error hx{T) is systematic in origin, but its value is unknown. The sources of this error are: the systematic error of the higher order standard (by which the measure w 1 0 has been established), the comparison errors occurring in the process of establishing the measure of the standard, and 4

This means that the measure corrected as in (2.4) is considered the nominal measure of the standard.

44

STANDARDS A N D THE PROPAGATION OF STANDARDS

Ch.2

the estimation error of the component A (prediction error). The process of establishing the measure of the secondary standard presented below will allow a glance at the process by which this error arises. The properties of the systematic error h (T) are as follows: — its value 10

x

/ (D = Α (Γ+ΔΓ) ?1

(2.6a)

Χ

is constant during the whole period of comparison with the secondary standard; — the probability of realization of the value h is expressed by the probability density ft(A);

(2.6b)

— the value of h is x

h = 5%(A)dA;

(2.6c)

x

— the value of this error depends on the time Τ because of the ageing process of the standard, also because of the dependence on Τ of the confidence interval of prediction, etc.; — the value of h is unknown and therefore may not be introduced as a correc­ tion to the measure of the standard; — the error A must be considered random, thus increasing the uncertainty of the measure of the standard. The error of the standard which is random in origin, expressed by the compo­ nent g! {ε }, has its source in the intrinsic properties of the standard (e.g. oscilla­ tions and fluctuations). Its standard deviation is a function of time because of the changes of properties of the standard. The distribution of this error is l

x

χ

(e).

(2.7)

Pe

The errors of a standard are conceptually difficult to describe. The opinion of the CIPM issued as Recommendation CI-1981 [178] is proof of this statement. The recommendation is not convincing, and polemical opinions have been publi­ shed [164, 177]. Therefore the reader of the present interpretation should also familiarize himself with the physical causes of errors (Subsection 3.3.3) and the general model of error (3.136). It should be stressed that the errors h (T) and g (T){e } are measurable using more accurate standards. Thus, the result presented in Fig. 2.6 may be interpreted as a systematic error distribution, namely the convolution of dis­ tributions (2.6b) and (2.7). In fact, it is the error distribution described by par­ ameters (2.19) and (2.22), where part of some of the components may be negli­ gibly small. x

i

1

2.2 PROPAGATION OF STANDARDS

45

ρ[Δ)\

-002 μ*-

-0.01

2

ο

0.01

Q02

Δ%

Δ

\ ^

Fig. 2.6. The error distribution in a manufactured series of 410 standard normal resistors (1000 Ω, class 0.02) [100].

The uncertainty of measure of the secondary standard is similarly expressed as A2 = w 2 - w 2 0 ,

(2.8a)

and {A2} = g2(T){e2}

+ h2(T).

(2.8b)

The location of the instant of comparison Τ in the scale of service time is different for each of the standards. For example, if the aim is to determine the measure of a secondary standard, then Tt > T09

T2 < T0,

(2.9a)

while if periodic checking is considered, then Γι > T0,

T2 > T0,

(2.9b)

where T0 is the instant when the nominal value of the standard is established, 7\ is the time of comparing against the standard Wl9 and T2 the time of comparing against the standard W2-

"Λ/ο

A

y

comparator Fig. 2.7. A scheme for comparing two standards in the process of determining the measure of a secondary standard.

STANDARDS A N D THE PROPAGATION OF STANDARDS

46

Ch.2

The standards are compared in the comparator, as shown in Fig. 2.7. The reading of the comparator is denoted as y> whereas the correct value y 0 of the reading is y0 = w2-wl

(2.10)

and is not known. 5 The comparator introduces an error of comparison Δ =y-y0,

(2.11)

which may be modelled by formula (2.2), and hence {A} = g0(e0}+h0.

(2.12)

When establishing a secondary standard is being considered, the comparison 6 result (measurement equation) is in the form w2 = Wi+y.

(2.13)

2.2.1 Single comparison Random errors have one random realization each. Substituting expressions (2.1), (2.2), (2.8) and (2.12) in the measurement equation (2.13), we obtain ^2o+^2«2+A

2

= WIO+GI^+Ai+^O+GOEO+Ao.

(2.14)

The measure of the secondary standard may be derived from the known compo­ nents of equation (2.14), including w10 and y = yo+goS0+h0. Hence W20 = Wio+y

= WIO+jo+goeo+Ao.

(2.15)

The correct result should be H>20corr

= "Ίο+^Ο,

(2.16)

which means that the derived measure of the secondary standard is burdened with the comparator error. The error equation is obtained by subtracting the correct value (i.e. equation (2.16)) from equation (2.14): g2s2+h2

= gi^+Ai+go^o+Ao.

5

(2.17)

Usually, when the uncertainty of the measure of each of the standards is <5, the difference between the measures of primary and secondary standards generally does not exceed a value of several δ. A comparator error of the order of 1% introduces a contribution of several hund­ redths of δ in the comparison error and is usually negligible. This feature results from the calculus of errors of indirect measurement, when the difference between the quantities being measured directly is very small [91]. 6 In the checking of two standards (2.9b), y =

H>2

Wi.

2.2 PROPAGATION OF STANDARDS

47

After the averaging operation is performed, the equation for systematic errors is obtained according to the definition of systematic error (4.28). Since all the com­ ponents in equation (2.17) are single realizations of random processes, their values are constant and E(x) — x. The value of the systematic error of the stan­ dard, calculated from equation (2.17), is h2 = h1-hgle1-g2e2+g0e0

+ h0.

(2.18)

Allowing for the properties of the sources of those errors which are the compo­ nents in equation (2.18), the expected value and the variance of the systematic error of the secondary standard may be determined as E(h2) = E(hl)+glE(e1)-g2E(e2)+g0E{e0)

+ E(h0) = Αχ+ϊο,

(2.19)

because the mean values of the processes {ε} were assumed to be zero, and var(A2) = variAO-i-var(A 0)+gi+gi+^o(2.20) These results need some comment. The value of the systematic error Ax realized in the primary standard is an element of the set of systematic errors, characterizing the standards of the same type, under the same conditions and at the same time interval T— T0 from the beginning of their use. Such characteristics are rarely available, partly because the power of the sets of identical standards is not large enough for the statistical characteristics to be reliable. An example may be the error distribution shown in Fig. 2.6. The expected value of the set of systematic errors is usually near zero, though in the example from Fig. 2.6 it differs markedly from this value. Also the variance of Ax characterizes the set of errors of the stan­ dards of the same type, under the same conditions, at the same service time and for the same value of the reading y. The result of variance estimation includes the variance of the random error of the secondary standard as one of the components. This is natural, since in the process of comparison this error was recognized and accounted for when the value of the error was determined. Expression (2.18), which is a sum of random processes, is the basis for determination of the probability distribution of the error, systematic in origin but with an unknown value, as the distributions of these errors are known (according to assumptions (2.6)). Therefore, for a given confidence level, the interval of uncertainty of the measure of a standard may be calculated. Besides the uncertainty of measure mentioned, the secondary standard, when used, generates physically the random error W2r}

= g2{e2},

(2.21)

which causes the variance of the uncertainty of the measure of a standard to be var(w 2) = v a r { J 2 } = var(A 2 )+var[g 2 {e 2 }] = var(A 2 )+g| = var(A1) + var(A 0)+g? + 2gf+g§.

(2.22)

STANDARDS A N D THE PROPAGATION OF STANDARDS

48

Ch.2

Since the variance of the error of the primary standard is

varOvO = var {A,} =

(2.23)

var^)+gl,

a comparison of variances (2.22), (2.23) implies v a r ^ - v a r ^ ) = var(A 0 )+go+2gi > 0,

(2.24)

which means that the uncertainty of measure of the secondary standard is greater than the uncertainty of the primary standard. This fact is illustrated in Fig. 2.8. Moreover, the measure of the secondary standard, when related to the measure of the primary standard, differs by the value of the comparator error.

Si

comparator

W

the estimation |of the secondary^ standard characteristic

%{4

(b) primary standard

raw comparison result

comparison error

comparator error

secondary standard

"20 \

» 2{

\92{ε2}

Fig. 2.8. A diagram of propagation of a measure and uncertainty of a primary standard onto a secondary standard: (a) the estimation course, (b) measure and error components.

2.2 PROPAGATION OF STANDARDS 2.2.2 η-fold

comparison

49

with the same primary

standard

Let us consider a comparison process occurring over the time interval τ > r l i m, for which ρ χ (τ) = ρ 2 (τ) = ρ 0(0 = 0, but Α ^ Γ + τ ) = Α^Γ) and £ ι ( Γ + τ ) = gi(Γ) and similarly for A 2 , g2, h0,g0. If a mean value is accepted as a result of comparison, then instead of (2.14), η equations are obtained for the subsequent realizations of random processes fo}, {ε2}, {ε0}: w i 0+ G 2 « 2 i + * 2 =

Wlo+gle11+h1+y01+g0ε0ί+h0,

(2.25) Wio+gltln

+ lh

=

WlO+glZln

+ h+yon+goeon

+

ho.

By adding the equations and dividing by n, we obtain W20+G2

η

/ι

π

= wio+g1^^eu+h1+go^^je0i

+ h0 + ^ ^ J o i i=l

/=1

(2.26)

i=l

The nominal value of the secondary standard is not the same because η Wio

= w 1 0+

η

=

w

i o + g o j ^ ^ * o i

wio =

+ h0+

—

i=l

/=1

and for w

η

^\oi>

(2.27)

i=l

oo (2.28)

+ h0.

Wio+yo

After the correct value is subtracted from equation (2.26), the value of the sys­ tematic error is obtained η

h'2 =

η

η

ft-^^en+Ai-GA-^^eM+i-GO^eoi i=l

ί=1

+ Ao.

(2.29)

1=1

The expected value of the error is E(h'2) = h1+h0

(2.30)

and its variance is smaller than in the case of a single comparison, namely var(Ai) = var(A1)+var(A0) +

-(ii+d+iS).

(· ) 2 31

η

The error random in origin is g2 {ε2 }, so the variance of the uncertainty of measure of the secondary standard is var(zl 2) = var(A 1)+var(A 0) + -(g?+«2) + - ^ ^ g i η

η

2

( -32)

STANDARDS A N D THE PROPAGATION OF STANDARDS

50

Ch.2

Formula (2.8b), expressing the uncertainty of a secondary standard, is noteworthy after (2.29) is substituted in it. The error is the sum of 3n+1 components which are independent realizations of random processes. The number η is considerable, and depends on the number of comparisons. At least 2/z-f 1 of them belong to 7 similar populations. This fact forms the basis for inferring that the distribution of error A 2 is close to normal. The inference is based on the central limit theorem of probability. If the value of hi is significant, a normal distribution with a mean value different from zero is taken into consideration. 2.2.5 m-fold

comparison

with different

standards

M-fold comparison with different standards having the nominal values w W i o 2 > · · · > W i o m gives interesting results. The subsequent standards are characteri­ zed by the error components l 0 9 i

···>

ε

ιι>

£ιι>···>£ι»μ

A n , . . . , A l m.

E

im\

The components of the results of comparison Joi,

-..»^om;

£ o £ o i > ···>

go^oml

(2.33a)

y

l 9

are

...,ym

(2.33b)

Κ·

For subsequent comparisons, m equations are obtained =

^ 2 0 + ^ 2 ^ 2 1 + ^2

w 1 0i

+gneu

+h11+y01

+g0e0i

+h09

(2.34) W20+gl^m+h'l

=

;

WlOm+Sl

£

+ Alm+J 0m+g0

0m

By adding the equations and then dividing by m, the result obtained is m w

*°

+

g2

\ i

Σ

m

Έ- Σ

>

E2J+H

m

=

;=i

w

i

°

j

7=1

+

Σ

m (

g

—

So

j

E

i

j

)

-k

+

7=1

m +

i

Σ

£

Σ ^ Λ

+

7=1

m

(2.35)

ο ; + — Σ ^ + Λ ο -

7=1

7=1

We then have m m

m m

Ζ—1 7=1 m

:

7

In

Σ 7=1

Z—i 7=1

m W

l

0+ J

m

i

+

7=1

Σ7 = 1

Soj+Ho

>

These are the components eUf ε2ι and ε 2> generated by a secondary standard.

( 2

·

3 6 )

2.2 PROPAGATION OF STANDARDS

Ki=

51

^•Σ » ^Σ " " -* ΐΣ " * ^Σ Α

var(AiO =

~

+

y

III

g

(?

j

j

+

β

± . ( g l

*mmm*

)

2

g ^ + ± .

+

ΊΎΙ

ill

7=1

β

ο

βν+Αο

var(A u )+var(A 0 ).

y /

+

·

(l37)

(2.38)

•

7=1

A comparison with different standards causes the part ht of systematic error of the primary standard to be reduced ]/m times. The w-fold realization of random errors also reduces their value y'm times. 2.2.4 η-fold comparison with m different primary standards By repeating the calculations in Subsections 2.2.2 and 2.2.3, mn equations are obtained (n equations (2.34) for each j = 1, ..., m). By adding them and dividing by mn, the nominal measure of the secondary standard is derived: m

η

m

η

- ^Σ^ ^ΣΣ^ =

+

7= 1

7=1

m

ί=1

m

η

m

η

=i-Σ^ -άτΣΣ^ -hΣ*«Σ ^ °· < · +

+

7= 1

7=1

ι=1

m

η

ε

W

=

™*°"

Ίη

Σ

W

i

0 +1

Έη~

7=1

Σ Σ «· 7=1

2 39)

/=1

7=1

The correct nominal value of the secondary standard is m

+Α

·

Λ

(2 40)

ί=1

The value of the measure is burdened with the comparator error. The systematic error is calculated as before, giving the results m

η

/-=ι

/=ι

m

> --ssr Σ *« Σ

H

+

i Σ y=i

^ ΣΣ ^ °' ?ο

ε

7=1

+Α

m

η

y=i

/=ι

ΣΣ ^ ε2

+

·

(2 4ΐ)

1=1

with expected value m

wo=^-Σ ' ~ Αι +Α

ο

·

(2 42)

STANDARDS A N D THE PROPAGATION OF STANDARDS

52

Ch.2

and variance var(A 2") =

g

+

{g2+gl)+

2^ ^ l^

V

{

^^

2 )

7=1

Also, ,

,,

var(J 2 ") = var(A 2 )+g|.

(2.44)

The degree of reduction of the respective sources of errors may now be easily seen from the formulae. 2.2.5 m times larger measure of a secondary standard

8

Let the measure of the primary standards be wu =

(2.45)

and let the metrological properties be described by the error {4j}

=

(2.46)

*ij{*ij}+Aij

whose components are identical with those in (2.2). The metrological properties of the standard w2 are described by (2.8b), and the properties of the comparator by (2.12). The comparison m

H>2 =^*u+y

(2.47)

7=1

implies m

m

m

7=1

7=1

7=1

Aij+JO+^o^o+Ao.

(2.48)

8

Usually, the ratio of measures of two standards is approximately a natural number. There are two methods: (i) the comparison method is applied, with the value of k = m/n (coincidence method), i.e. nw2o — mw10, where m, η are natural numbers [91]; (ii) the primary standard is compared with η identical secondary standards for which wl0 = nw2o, or m identical primary standards are compared with one secondary standard, and then mw10

= H>20.

Physically realized operations of division or merging of the states of quantities are applied here. The operations are physically realized for a number of physical quantities, in accordance with the properties of metric scales. The properties of the first method are the consequence of the mode of measurement applied, as described in Chapter 5. For this reason, only the second method is discussed here.

A

3

2.2 PROPAGATION OF STANDARDS

53

The value of the nominal measure of the secondary standard is η H'20

m

=Σ

= ]Tj Wioj+yo+goZo+fio-

Wioj+y

7=1

(2.49)

7=1

The systematic error, with an unknown value, is then m *2

m

= X]

Σ

7=1

A

i i " " ^ ß 2 + ^ o ^ o + A

0

(2.50)

,

7=1

and the random error is g 2 {ε2 }. If the primary standards are of the same type and the conditions iu

= gi>

\s = \

for

y=l,...,m,

(2.51)

are met, i.e., the errors come from the same population and g2

h2

w2

Si

Ax

- ~ - Ä -=~ - Ä —

then

Ä m,

(2.52 )

m £ ( A 2) =

J ] * « 7=1

+

A

°

=

m

^ i

+

A

o

=

(2.53)

Αί+Αο,

and, writing h[ = mh1, g[ = / n ^ and varAi = m varAx, 2

var(A2) = mgf+mvarCA^+gi+^g+variAo) = — m

[^gf+w^iAiM+gi+eg+variAo) 2

2

= -i-[gi +var(A0]+g2 +^+var(Ao).

(2.54)

The results given above are interpreted as follows. If the relative error of the primary standards and the secondary standard are close, i.e., the absolute errors differ m times (2.52), the expected value of the systematic error of the secondary standard will be equal to the sum of the expected value of the errors of the primary standards (2.53). The case hx = 0 is very advantageous. In the case when a small value of the measure of primary standards is transferred onto a secondary stan­ dard, the variance of the error, systematic in nature, decreases m times (the first component in formula (2.54)). For example, if the limiting error of m = 10 primary standards is 0.01%, the part of this error in the error of the secondary standard is 0.01 /j/10 = 0.003%. The effect is better accuracy of the secondary standard as compared with the accuracy of the primary standards. This effect arises when hXJ = 0, but this condition is hardly verifiable. In other cases the effect is not so strong (see Fig. 2.9).

v

54

STANDARDS A N D THE PROPAGATION OF STANDARDS

Ch.2

Fig. 2.9. The influence of the systematic error of a primary standard on the error of the secondary standard. The mean value of the primary standard is (a) zero, (b), (c) different from zero.

Multiple comparisons with the same set of equipment leads to η equations (2.48) if the subsequent comparisons are performed at time intervals τ > r , assuring independent realizations of random errors. By making calculations as in Subsections 2.2.2 and 2.2.5, the nominal measure of the standard may be calcu­ lated: im

rn

η

(2.55) i=l

7=1

Also the systematic error in

m

η

η

π

y=i j=i ι=ι i=i /=i as well as its expected value and variance may be calculated. 2.2.6 m times smaller measure of a secondary standard The measure of a secondary standard is w m ^ w. (2.57) The comparison procedure is more complex here. Μ values of nominal measures of secondary standards should be determined, together with m esti­ mates of the uncertainty of the measures. A minimum of m independent compari­ sons must be made in order to obtain m independent equations for calculating the results. Because of condition (2.57), it is necessary to perform the compari­ son described by the measurement equation 2J

x

m

J]w - = w +y 2j

t

lt

(2.58)

2.2 PROPAGATION OF STANDARDS

55

Fig. 2.10. A scheme of comparisons made to obtain the measure of m secondary standards: (a) as related to the first standard, (b) comparisons of subsequent standards, (c) another combina­ tion of comparisons.

together with m — 1 other comparisons. Examples of comparison schemes are shown in Fig. 2.10. For the scheme shown in Fig. 2.10a, where each secondary standard is compared with the secondary standard denoted by j = 1, the system of equations is W l l

= w2J+yj9

j = 2, 3,

(2.59)

To avoid any improper mathematical transformations, the index / (/ = 1, ... ...,#*) is introduced, denoting the number of comparison. This allows distinct individual realizations of random errors of the standard W21. For / = 1, the comparison (according to equation 2.58) may be described as follows: m

]T

m W20./+

m

]T (g2je j)+ X] h 2

2j

= Wio+giei+Ai+^oi+goieoi+Aoi.

(2.60)

From (2.59) we derive ra — 1 equations describing the comparisons / = 2, 3, ... .. . m: W20j+g2je2Jl+f>2Ji

=

^ 2ο ι + ^ 2 ΐ β 2 1 Ι + Α 2 1 ~ ^ Ο ί - ^ Ο Ι « 0 | - Α θ 1 ·

(2.61)

56

STANDARDS AND THE PROPAGATION OF STANDARDS

Ch.2

After substituting m—1 equations (2.61) solved for w20J into (2.60), we obtain

(2.62) The assumed properties of the systematic error imply h2ij = h2J, h21i = h2i, and for / = 2, 3, ..., m we have h0i = h0 and g0i = g0. The last two components on the left-hand side are reduced. The nominal measure of the standard W21 is m

(2.63) and the correct value is (2.64) In formula (2.63) it was accepted that the values w1 compared in the first measurement are m times greater than the values compared in the rest of the measurements, for which the properties of the comparator are different. The values of h01 and h0 are different as well as the values of g0i and g0. The value of the systematic error h2i is obtained by subtracting correct value (2.64) from equation (2.62):

(2.65) The expected value of the systematic error is

(2.66) and the variance is

(2.67)

2.2 PROPAGATIO N O F STANDARD S

57

The parameter s o f th e standard s W 22 > 3 > . ·. , W 2m ar e calculate d fro m equatio n (2.61), a s i n Subsectio n 2.2.1 , becaus e o f th e propertie s alread y determine d fo r the standar d W 21. For j = 2 , 3 , m, wzoj

=

(2.68 )

Wioi-yj,

A2j- = h 2l-h0+g2ls2li-g2jS2ji-g0e0i.

(2.69 )

Thence £(*2j)

= £

2 1

-Ä0

(2.69a )

and var(A2j.) = var(A 2 1)+ var(A 0)+ 2 g | + s g , where A Also

21

(2.69b )

an d var(A 2 1) ar e expresse d b y formula e (2.66 ) an d (2.67) , respectively .

var(zl 2 j) = var(A 2 1)+var(A 0 ) + 3gl +gl -

(2.70 )

Theschem eo fcomparison s applie d prefer s t o som e exten t th e secondar y standar d W2l9 sinc e th e uncertaint y o f thi s standar d i s smalle r b y th e rando m erro r o f th e compared secondar y standard s an d th e rando m erro r o f th e comparato r i n a single comparison . On interpretatio n o f (2.67) ,i t i s see n tha t th e rando m error s occurrin g a t ever y comparison ar e a significan t par t o f th e uncertaint y o f th e standard . Therefore , the secondar y standar d No . 1 shoul d b e th e standar d wit h th e smalles t rando m error, i.e. , wit h th e smalles t valu e o f g 2. Multipl e comparison s b y th e sam e schem e will caus e par t o f th e rando m error s t o diminish , b y a rati o whic h ca n b e determi ned fro m th e derive d formulae , knowin g th e value s o f th e component s o f error . To determin e th e measur e o f th e secondar y standard , th e othe r schem e o f comparisons ma y b e applied , a s i n Figs . 2.10b , c . Th e schem e ma y includ e η > m comparisons by any method. In this way, more results than unknowns are obtain­ ed, and the accuracy of the measures of secondary standards determined and of their errors is improved. In the case discussed, groups of standards containing 2, 3 or more elements may be compared. Thus, one comparison causes several sources of random errors to be realized at the same time. The effect of diminishing the variance of random error will depend on the properties of the error of the sum, as shown in the example (see Tables 4.6 and 4.7). The problem discussed is connected with the mutual comparison of the ele­ ments of a group standard, e.g. the comparison of the elements within one stan­ dard. It is known that such intercomparison does not allow estimation of changes of measure of a group standard. However, it allows estimation of the change of measure of elements, referred to the group measure [190]. Such information

58

STANDARDS A N D THE PROPAGATION OF STANDARDS

Ch.2

is sufficient for checking the elements of the group standard and for elimination of the elements with measures departing from the mean value. The practical procedure [168, 177] prefers comparison schemes of the type 'each element with each other'. Such a scheme produces symmetrical results, easy to process. As an effect of applying the above ways of transferring the measure of a stan­ dard, a system of standards with different values of measure may be created based on one standard only. The uncertainty of measure in this case is determined with respect to the way in which the measure is established. Figure 2.11 shows an standard

working standard

2

m wQ

Χ

mwQ, Δ2

m •4

I class standards

mw0 ,Δ3

m

II class standards

Τ •A

Λ"

J

i

m

4

Τ "

III class standards

checking instruments

measunng instruments Fig. 2.11. An example of a hierarchy of standards.

example of the structure of a system of standards of one quantity, for which the manner of division and merging described in Subsections 2.2.5 and 2.2.6 was applied. The diagram is simplified. One standard is shown at each level, while the national standard (etalon) consists of several standards: — the fundamental standard, — the comparison standard, — the witnessing standard, — the reference (working) standard.

59

2.2 PROPAGATION OF STANDARDS

No physical limitations, other comparison methods, etc. have been taken into account. Such a structure of the system of standards is characteristic of additive quantities. Another group consists of quantities with a complex structure of the system of standards. For example, temperature reproduced in the International Practical Temperature Scale (IPTS) has several reference states distinguished as the defi­ ning fixed points. The continuous standards used for reproduction are the plati­ num resistance thermometer, PtRh-Pt thermocouple and the radiation effect. Other examples are the system of time standards and the system of inductivity standards, with different frequencies and comparison methods. The third group is made up of derived quantities. The system of standards is based here on a standard of a given quantity and on the additive structure of lower-level standards, as well as on standards whose measure is determined by an indirect method, as in a definition of the quantity. There are at least two different ways for determining the measure of a standard at a given level in the hierarchy. An example may be the volume standard: the measure may be deter­ mined by comparison with a higher-level standard or by an indirect method, using length standards. A similar situation occurs for the pressure standard, for the calculable capacitance standard, etc. A number of physical and technical limitations to the realization of selected reference states exist. The problem of working out the real structure of the hierarchy of standards is complex, because of the difficulty of formulation of the optimality criterion and because of changes of the data on the properties of standards, the maintenance costs, verification costs, etc. The problem of the multiplication factor of the uncertainty of measure of the standards at neighbouring levels in the hierarchy and the problem of the respective number of levels in the hierarchy are the most frequently discussed questions [13, 112]. The first question may be answered on the basis of formulae derived above if the ratio of the systematic and random errors is known. For the ratio 1:1 and for negligible comparator errors, it may be seen (for example, from formula (2.32) that the ratio of variances var(Zl 2)

var(J0

v a r ^ ) + v a r ( / * 0 ) + (1 /n) (g\ +gg) + [(n + l)/n]g 2 2

var (/*i)+g

2

(2.71)

varies from 2 (for η = 1) to 1 (for η -• oo) as g x = g2. This problem has been discussed in detail by Crown [14], but for another aspect of errors. For a known quotient of errors and for a required accuracy of instruments, a number of levels in the hierarchy may be derived [107]. Other authors have analysed the problem of propagation of errors in the aspect of erroneous estimation of an allowable error of the secondary standard [53, 134]. The number of levels in the hierarchy of standards is calculated from the assumed

60

STANDARDS A N D THE PROPAGATION OF STANDARDS

Ch.2

level of hazard. The problem is more definite when the costs of calibration are taken into account [107]. Examples of the hierarchies established for specific applications of instruments may be found in the bibliography [34, 65]. Keeping the measures unified and conforming to international agreements is a task for the state, which is performed by the state administration and author­ ized institutions. The state administrations of measures maintain the national standards, perform comparisons and attestations of standards, establish systems of standards, etc. In the regulations published by the state administration of measures, diagrams showing the hierarchy of standards are periodically included. According to needs, the measurement staff also perform the tests and check that measurement apparatus is consistent with the national system of standards. In this way, a reference to international standards is established. 2.3 The Certified Reference Materials The formulation of images of reality must include qualitative and quantitative phenomena. The qualitative image of matter is expressed in the following categones: — the elements forming matter on a macroscopic scale; — chemical compounds; — mixtures of different elements and compounds; — organic and biological bodies; — particles and quanta building atoms and atomic nuclei; — the phenomena of interaction of matter at rest or in motion; — forms of energy and the transformations of energy; — the phenomena connected with the influence of energy on matter; — different qualities of matter and energy described by the definitions of physical quantities. Some of the qualitative categories are easy to identify, while others are not. The measure of distinction of the qualitative categories is the number of physical qualities and the states which these qualities may assume. The qualities must be observable and measurable. The existing qualitative categories have their images in the domain of abstraction. The images are expressed by names and different symbols, and they are one-to-one mappings of reality. A qualitative category is described by a denumerable number Ν of the qualities Ql9 . . . , Q N > and by the sets of states {q^}9 {qN} assumed by these qualities. The number yV increases as the reality is recognized. Two categories A and B: A (2.72a) = A{Ql9 ...,Qi9 ...,QN,ql9 · Β = B(Ql9 ...9QJ9

...9QM9ql9

...9qM)P

(2.72b)

2.3 CERTIFIED REFERENCE MATERIALS

61

are distinct when for at least one and the same quality QIA = QJB, i = 1, ..., yV, j = 1, ..., M9 the sets of the states are disjoint, which may be written as { i u } n {qJB} = 0 ; G m = QJB(2.73) Condition (2.73) may be fulfilled by more than one quality. The categories are then easier to distinguish. For readily understandable practical purposes, the qualities which are more easily observed and measured and which satisfy condition (2.73) are used to distinguish the qualitative categories. In intentional human activity, it is a necessary to apply standards for a consi­ derable number of qualitative categories in the places where different materials are manufactured and used, for quality control, routine checks and scientific research. The standards are samples of materials properly prepared for use, called certified reference materials (CRM). Because of the activity of the physical envi­ ronment and because of the ageing process, the life of these standards is limited. Moreover, in most cases the measurement of qualitative features causes some consumption of the standard material. Thus, it is necessary that the CRM can be restored to their former state and consumption can be compensated. According to the classification by Plebanski [100], there is a demand for the following kinds of standards: — analytical standards, — standards of ultra-pure substances, — standards of physical properties, — standards of technical properties, — standards for health protection and natural environment protection, as well as for agriculture (e.g. standards for clinical laboratories, standards of pesticide mixtures, biological standards, etc.). The certified reference materials are the physical bodies which reproduce, with a determined accuracy, those of their properties for which the materials were certified by a suitable institution [100]. Although the definition is similar to the one used for the standards of measurement units, the rank of the CRM is not as high. The reasons for this fact are as follows. 1. The qualitative category reproduced by a certified reference material is defined through measurable qualities and through the states of those of the qualities for which the standards of units of measure exist. 2. No primary certified reference material is necessary to determine a qualita­ tive category or a measure of a certified reference material which expresses this category, because the measure may be determined by measurement of the physical properties of the material. 3. A certified reference material reproduces the measure of the standard over a limited range of physical and chemical properties. These properties are certified.

62

STANDARDS A N D THE PROPAGATION OF STANDARDS

Ch.2

This fact limits the applications of the CRM to an instrument in which the certified property is the basis for identification. The CRM and the standards of units of measure are alike in: (1) the way in which the measure of standard and the accuracy of measure are expressed; (2) propagation of some of the CRM standards by comparison of primary and secondary standards; (3) the purpose and the mode of applying the standards and the procedure of attesting and certification. The qualitative category of a certified reference material is expressed verbally by the name of the material, or by generally accepted symbols denoting the chemical constitution of the material, and by the state of a non-dimensional quan­ titative feature (one, or a few) from the set <0,1>. The state of a quantitative feature may be expressed as a percentage or a quo­ tient of quantitative states, for example, as a quotient of mass and volume. The quantitative feature is defined and indirectly measurable by an absolute method. The measure of a CRM standard is a quantitative feature called the concentration or mole (or molar) fraction (of weight or of volume). The measure includes every component of the certified reference material existing within the uncertainty of measurement. The properties of this measure are identical with the properties of the measure of physical quantities, and may be treated as a vector measure. The restrictions imposed on a measure of CRM standard have the form of component balance: η

(2.74) where qi is the fraction of the ith component of the material. The inaccuracy of measure of a CRM standard concerns the components ien and the compo­ 9 nents ι > n. The inaccuracy of measure Aqi may be modelled by a multidimensional nonstationary random process (3.136) {A} = g{s}+h, (2.75) in which g, h are the functions of such quantities as spatial dimensions of the sample, time, temperature, etc., i.e, describe the changes of the properties of the sample material as an effect of diffusion, dissolving of the materials, oxidation, etc., and {ε} is a stationary random process describing the non-homogeneity of the sample material along the chosen coordinates of the sample, e.g. the surface 9

This way of presenting the measure of a CRM standard is valid for some standards only. In other ways, for example when the components are added to stabilize the properties, the full constitution is not given.

63

2.3 CERTIFIED REFERENCE MATERIALS

or volume coordinates. Thus the uncertainty of measure of a CRM standard changes after a time interval T. The inaccuracy of the measure of a CRM standard is expressed by metrolo­ gical characteristics, of the classes 4 or 5 (see Section 4.4). A small (usually negli­ gible) fraction of the random error expressed by g{e} is characteristic for the CRM. The value of this error may not always be determined unambiguously. The reason for this is that in the process of reproduction of the state of the CRM standard, only a part of the surface or volume of the standard is active and the measure is the mean value for the active area. The reproduction of a measure in an instrument with a different active area of the standard makes the result of averaging different. The systematic error h has two components: — the error of nominal measure of the CRM standard (the error is systematic in origin, but its value remains unknown); — the instability error caused by ageing of the standard. The ways of determining the measure of the CRM standard and its inaccuracy may differ, depending on the kind of CRM standard considered and its manu­ facturing technology. The most typical cases will be examined here. Standard gases. The standard gases, which are the standards of physical properties with various applications, are prepared by measuring [110]: (a) volume at constant pressure; (b) pressure at constant volume; (c) flow rate of components A, B, C of a mixture, and by mixing of the components. The components of the mixture should have a known chemical constitution. The measure of this standard is the volumetric fraction of the components of the mixture, expressed as percentages. The measure is derived from the quantities measured directly XA=?±xl0O,%A, XA = ¥±-x

or

100,

or

XB =

XA = ^

χ 100,

χ 100, %B,

or

(2.76a)

etc.,

(2.76b)

where VA + VB + VC + ·.. = V9 PA + PB + PC+

·.· = P.

(2.77a) 2 77B

(- >

The uncertainty of measure of a CRM standard is determined on the basis of errors of measurement of the quantities defining the measure of the standard.

64

STANDARDS A N D THE PROPAGATION OF STANDARDS

Ch.2

This is done as in the method for indirect measurement, because the uncertainties of the quantities measured directly are known. The sources of uncertainty of measure are: (1) the uncertainties of the quantities measured directly; (2) contamination of the component gases; (3) changes of temperature during the process of measurement; (4) departures from Dalton's law at higher pressures [122]; (5) badly mixed components, lamination of the mixture (when the densities of the components differ considerably), condensation of the components, etc. In most of the cases, the direct finding of the measure of the standard gas ensures better accuracy than in case of direct measurement of its constitution by analytical methods. The electrolyte standard. The most common low-concentration electrolyte standard is an aqueous solution of potassium chloride [100], A weighed amount of KCl is added to a measured volume of distilled water. The measure of the standard is calculated from the formula m_ mg KCl (2.78) χ = c = Ύ' IH O " 2

The uncertainty of the measure is the effect of: (1) uncertainty of the direct measurements of mass and volume. As the defining equation (2.78) is in the form of a quotient, i c = » m + *v, (2-79) where δ is the relative error of the quantities denoted by the indices; (2) uncertainty of the purity of the components; (3) instability of the chemical constitution, caused by dissolving the atmos­ pheric components (mainly C 0 2 ) and the vessel material.

Standards obtained by preparative chromatography. The effect of the eluate con­ centration as a function of time is applied here, as shown in Fig. 2.12. Over the

Fig. 2.12. The principle of extracting pure components from a chromatographic eluate. A, B, C are the components of the output sample.

2.3 CERTIFIED REFERENCE MATERIALS

65

time interval t1 — t29 the eluate is the component A of the mixture which is being separated in the chromatographic column. The purity of the component depends on the time interval (tl9t2}9 since for t < t1 and t > t2 the eluate contains the carrier gas or another component, due to the dissipation of the interface in the equipment beyond the column and because the interface in the column is not sufficiently sharp. A pure component of the mixture being separated is obtained as a CRM standard. The kind of component obtained is determined by the value of its retention time. The purity of the standard may be verified by chromato­ graphic methods. Determination of the measure of a CRM standard based on a sample. Some CRM standards are obtained by metal or metal alloy melts. The number of samples taken from one melt is N. The measure of a standard is determined from investigation of η <^ Ν samples [119]. The output material is characterized by the spatial hetero­ geneity of its constitution and properties (the coordinates of this space are usually geometric ones). The heterogenetity may be described by a model (2.75), i.e., a multidimensional, non-stationary random process (more precisely, by as great a number of processes as the number of qualities characterizing the standard). For example, in the case of a metal alloy, the measure of a standard is determined by m components and by their concentrations. The sample is described by m 3-dimensional processes {qA = gj(x9y9z){Sj}

+ hj{x9y9z)9

7 = 1 , ...,m.

(2.80)

The processes are usually correlated. The stationary process {ej} describes the fluctuations of concentration due, for example, to wrong mixing of the components. The function h describes various stratifications due to sedimentation, degasing, etc., occurring before freezing of the alloy. The realization of the process qj is one sample, and the geometric dimensions of the sample are the limitation for the horizon of observation of the processes {ej}. The number of realizations depends on the expected character of the function hj. To identify the heterogeneity, the methodology described in Subsection 4.5.1 may be applied. The measure of a standard is the mean value of measures of the samples, corrected by the interpolated value of the systematic heterogeneity h. The measure of uncertainty is the sum of the component gj {ej }, of the uncertainty of determination of the function hj and of the measurement error of the states of quality qj9 i.e., (2.81) After adding these processes, the uncertainty limit is calculated as a confidence interval at a given confidence level. This uncertainty is characteristic of a certified reference material, which is a single realization of the process {Aj}. The error of

66

STANDARDS A N D THE PROPAGATION OF STANDARDS

Ch.2

the CRM standard is systematic in origin, but has an unknown value. The way of calculating the expected value of uncertainty and its variance is similar to that described by formulae (2.19), (2.20). Application of the CRM standards created with this method allows reduction of the uncertainty of measurement. This may be done by: 1. Application of η CRM standards from the same population, and taking the mean value as a result. The uncertainty of the measure introduced by components 1, 2 and the part gq{sq} of component 3 in formula (2.81) decreases η times. The component hq remains unchanged. The mean values of elements 1 and 2 are zero and E{Aqj} = hq. 2. The component hq may be reduced by applying m standards whose measures are determined with m sets of instruments for which hq = 0. There is an analogy here to the examples described in Subsections 2.2.1-2.2.6. The international I S O requirements [73] were issued for certified reference 11 materials, like the IUPAC requirements [11] which apply to pure substances. Institutions certifying the CRM (in order to eliminate the systematic errors of instruments) conduct comparisons of CRM standards, like those for the standards of measures. Certified reference materials play an important role in metrology. Not only do they make the measurements simpler, but also some of them are the sources of constant and strictly determined physical quantities. Quite frequently these states are defined on the basis of a type of pure material (e.g. water, pure metals for reproduction of temperature at fixed points, etc.). The certified reference materials make studies of the instability of some measurement systems possible [96], even when a nominal measure is not known precisely. It is sufficient to ensure that the given quantity is constant. Therefore, the certified reference materials may be regarded as pseudo-standards of some of the physical quantities, i.e., as pseudostandards of measures. 10

10 11

The International Organization for Standardization. The International Union of Pure and Applied Chemistry.