NBS MONOGRAPH 126

Platinum Resistance

Thermometry

Platinum Resistance Thermometry

55fc -

John L. Riddle, George T. Furukawa,

and Harmon H. Plumb

Institute for Basic Standards

National Bureau of Standards

»>. Washington, D.C. 20234

< i

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued April 1973
Library of Congress Catalog Card Number: 72-60003

National Bureau of Standards Monograph 126

Nat. Bur. Stand. (U.S.), Monogr. 126, 1 29 pages (Apr. 1972)

CODEN: NBSMA6

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price: S2.10, domestic postpaid: $1.75, GPO Bookstore. Stock No. 0303-01052

(Order by SD Catalog No. C13.44:126).
Contents

Page

1. Introduction 1

2. Background and basic concepts. 1

2.1. Thermodynamic temperature scale 2

2.2. Practical temperature scales 2

3. Platinum resistance thermometer construction 4

4. Using the thermometer 9

4.1. Mechanical treatment of SPRT 9

4.2. Thermal treatment of SPRT 10

4.3. Thermometer immersion 13

4.4. Heating effects in SPRT 13

5. Resistance measurements 17

5.1. Mueller bridge 18

5.1.1. Bridge ratio arms 19

5.1.2. Bridge current reversal 20

5.2. A-C bridge 20

5.3. Potentiometric methods 20

6. Calculation of temperatures from the calibration data and observed resistances 22

6.1. Temperatures from 0 to 630.74 °C 22

6.2. Temperatures belowf 0 °C 24

6.2.1. - 182.962 °C (90.188 K) to 0 °C (273.15 K) , 25

6.2.2. 13.81 to 90.188 K 26

6.2.2.1. 54.361 to 90.188 K 26

6.2.2.2. 20.28 to 54.361 K 26

6.2.2.3. 13.81 to 20.28 K 26

6.2.3. Calibration tables between 13.81 and 273.15 K (0 °C) 26

6.3. Errors of temperature determinations 26

7. Calibration 28

7.1. Triple point of water 29

7.2. Metal freezing points 31

7.3. Metal freezing point furnaces 35

7.4. Tin-point cell 37

7.5. Zinc-point cell 43

7.6. Oxygen normal boiling point 44

7.7. Comparison calibration between 13.81 and 90.188 K... 47

8. References 50

9. Appendixes:

A. The International Practical Temperature Scale of 1968 (adopted by the Comite

International des Poids et Measures) 52

B. Comparison of assigned fixed point values and interpolating formulae in the temperature ranges defined by the platinum resistance thermometer for ITS-27, ITS-48,

IPTS-48, and lPTS-68 62

III
Page C. Tables of difference in the values of temperature between the IPTS-48 and IPTS-68

and between the NBS-1955 and IPTS-68 in the temperature range 13 K to 630 °C... 64

D. Table of the function

and its derivative from t' = 0°C to i' = 650°C 74

E. Tables of values of the reference function and its derivative from 7"= 13 K to r= 273.15 K and over the equivalent range of temperature with the argument in

degrees Celsius 81

F. Analysis of the first derivatives at 0 °C of IPTS—68 platinum resistance thermometer

formulations above and below 0 °C 106

G. Derivation of differential coefficients for the analysis of errors in platinum resistance

thermometry 108

H. Calibration of Mueller bridges 113

I. Calibration of the linearity of a potentiometer 122

J. Guide for obtaining the calibration of standard platinum resistance thermometers... 124

iv
Platinum Resistance Thermometry

John L. Riddle, George T. Furukawa, and Harmon H. Plumb

The monograph describes the methods and equipment employed at the National Bureau of Standards for calibrating standard platinum resistance thermometers (SPRT) on the International Practical Temperature Scale of 1968 (IPTS-68). The text of the scale is clarified and characteristics of the scale are described. Several designs of SPRT's are shown and discussed in the light of the requirements and

recommendations of the text of the IPTS—68. Possible sources of error such as those due to the internal and external self-heating effects and the immersion characteristics of SPRT's are described in detail. Precautions and limitations for the mechanical and thermal treatment and for the shipment of SPRT's

are indicated, and a guide is given for those desiring the thermometer calibration services of NBS.

The description of equipment employed at the National Bureau of Standards for maintaining the IPTS-68 includes the triple point of water cell, tin point cell, zinc point cell, oxygen normal boiling

point comparison cryostat, the 13 to 90 K comparison cryostat, and the reference SPRT's upon which

the NBS-IPTS-68 in the region 13 to 90 K is based. Methods are given for calculating temperatures

from the calibration data and observed resistances; the propagation of calibration errors is discussed.

Supplemental information given in the Appendices includes the authorized English version of the text of the IPTS-68, tabular values of the reference function used below 0 °C, tabular values of the

differences between IPTS-68 and IPTS-48, analysis of the first derivatives at 0 °C of the IPTS-68

formulations, methods for calibrating potentiometers and Mueller bridges, and the derivation of the

coefficients used in the analysis of error propagation.

Key words; Calibration; calibration errors; cryostat; fixed points; freezing points; International Practical Temperature Scale; platinum resistance thermometer; thermodynamic temperature scale; thermometry; triple point.

1. Introduction

The National Bureau of Standards (NBS) has

the responsibility to establish, maintain, and

develop the standards for physical measurements

necessary for the nation's industrial and scientific

progress and, in cooperation with other national

laboratories, to establish international uniformity

of the basic physical quantitites. Dissemination of

these standards and the methodology of measure- ments is accomplished by appropriate publications, consultation, and calibration services. One

of the important activities is the realization, re- production, and transmission of the International

Practical Temperature Scale of 1968 (IPTS-68),

the present basis for assigning uniform values of

temperature. Standard platinum resistance thermometers are the prescribed interpolation instruments for realizing the IPTS-68 from 13.8 to 904 K;

they also serve as reference standards for calibrating many other types of thermometers, e.g., Uquidin-glass thermometers. Thus standard platinum

resistance thermometers are the base for an

extremely large fraction of the temperature meas- urements made in science and technology.

The information presented in this monograph is intended for those who use standard platinum

resistance thermometers, who wish to submit

such thermometers to the NBS for calibration, or who desire information that will aid in establishing

local calibration facilities. Good practice in the use

of these thermometers is emphasized to assist in

the realization of accuracy that is commensurate

with their capability, as well as to point out their in- herent limitations. This monograph describes the

calibration of standard platinum resistance thermometers at the NBS including the equipment,

techniques, and procedures. The discussion of the

IPTS-68, the defining fixed points, the interpolation

formulae, the design of standard platinum resistance

thermometers, and pertinent procedures which lead

to "state of the art" measurements are presented

in considerable detail.

2. Background and Basic Concepts

The development of the platinum resistance

thermometer resulted in an internationally acceptable practical temperature scale defined to give

1
values of temperatures close to those on the

thermodynamic scale. The birth of platinum

resistance thermometers as useful precision

instruments occurred in 1887 when H. L. Callendar

[16]' reported that platinum resistance thermometers exhibited the prerequisite stability and

reproducibility if they were properly constructed

and treated with sufficient care. In the next four

decades the platinum resistance thermometers

gained such wide acceptance that their use was

proposed and adopted by the Comite International

des Poids et Measures in 1927 in defining values on a practical scale of temperature, the first In- ternational Temperature Scale [15]. Since that

time improvements in the purity of platinum and

other materials of construction, in thermometer

design, and in calibration techniques have yielded

improvements in the precision, accuracy, and

range of temperatures that can be measured with

platinum resistance thermometers. The international scale has been redefined both to take advantage of these improvements and to bring the

scale more nearly into agreement with the thermodynamic scale.

2.1. Thermodynamic Temperature Scale

Temperature scales based on functions that can be derived from the first and second laws of

thermodynamics, and hence that give values of

temperature consistent with the entire system of

logic ("thermodynamics") derivable from these

laws, are said to be thermodynamic scales. The

present thermodynamic scale was established in

1954 by assigning the value 273.16 kelvins to the

triple point of water. [17] On the Kelvin thermodynamic scale, the values of temperature and the

designation of the appropriate units and symbols

have been adopted [51] as follows: The unit of

measurement on this thermodynamic scale is the

kelvin, indicated by the symbol K. The same name

and symbol are also used to denote an interval of

temperature. The designation of an interval of

temperature by "degrees Celsius" is also permitted.

The symbol of the value of temperature on the

Kelvin thermodynamic scale is T. The Celsius

thermodynamic scale is defined by the equation: ^=7"" 273. 15 K. The zero of the Celsius scale

is 0.01 K below the triple point of water. (Although

the value of the temperature of the ice point is close to and can be taken for all practical purposes

to be 0°C, the value of the ice point is no longer

defined as 0 °C.) The unit of a Celsius thermodynamic temperature is the degree Celsius, symbol

°C, which is equal to the unit of temperature on

the Kelvin scale. The symbols of the values of

temperature, t and T, contain the unit of temperature °C and K, respectively.

' Figures in brackets indicate the literature references on page 50.

2.2. Practical Temperature Scales

Accurate measurements of values of temperature on the thermodynamic scale are fraught with ex- perimental difficulties, yet aU values of temperature should be referable to the Kelvin thermodynamic scale. "Practical" temperature scales are intended to give values of temperature that are comparatively easily reproduced and, therefore,

utilitarian, but most are based on functions that are not related to the first and second laws of

thermodynamics. In contrast, vapor pressure scales,

e.g., that of ^He [13], are based on the laws of

thermodynamics as is also the radiation pyrometry

scale. Because many physical laws are based on thermodynamic temperatures, the values of temperature on any practical scale should be as close

to the thermodynamic scale as is possible or the

differences between a practical scale and the

thermodynamic scale should be appropriately

documented so that conversions from the practical

scale to the thermodynamic temperature scale are

possible.

Many practical scales have been used in the last

fifty years, but the only scales employing platinum

resistance thermometers that have had widespread use in the United States were those defined and sanctioned by either the International

Committee of Weights and Measures or the National Bureau of Standards. Successive scales that were defined by the International Committee of

Weights and Measures are the International

Temperature Scale of 1927 (ITS-27) [15], the

International Temperature Scale of 1948 (ITS^48)

[46], and the International Practical Temperature

Scale of 1948 with the text revision of 1960 (IPTS48) [48]. The differences in the definition of these

scales primarily arose from steps to improve the

reproducibility of the scale and the values of temperature between — 182.97 and 630.5 °C remained

essentially unchanged — the changes in the values of

temperature were less than the uncertainty of the

values on the scale at the time. These differences are summarized in Appendix B.

Because the ITS-27 [15] extended only 7 °C

below the normal boiling point of oxygen (— 182.962

°C), the National Bureau of Standards was motivated to develop a practical scale from the oxygen

point down to approximately 11 ^ [28]. This scale

has been variously referred to as the NBS provisional scale, the Hoge and Brickwedde scale,

and most recently as the NBS-39 scale; directly

related to this scale is the NBS-55 scale (T^nbs-ss = 7'nbs-39-0.01 K) [50]. These NBS scales have

had widespread use but, together with the previously

mentioned international scales, have now been

supplanted by the International Practical Temperature Scale of 1968 (IPTS-68) [51] (see Appendix A

for the complete text of IPTS-68 and Appendix C

for the difference in values of temperature).

The text of the IPTS-68 introduced the first major changes in the international practical scale

2
since 1927. Changes were made to extend the range

of the scale down to 13.81 K and to enhance its reproducibility as well as to improve its agreement

with the thermodynamic scale. The changes in

the values of temperature from the IPTS-48 to the

IPTS-68 are tabulated in Appendix C; included

in this appendix is a tabulation of the differences

between the IPTS-68 and the NBS-55 scale which

it replaced in the United States of America. Temperatures on the IPTS-68 may be expressed in

either kelvins or degrees Celsius. The symbols

and units of the international practical scale are

like those described earlier in this section for use with the thermodynamic scale. The symbol

for the value of temperature and for the unit of

temperature on the International Practical Kelvin

Scale are Tes and K (Int. 1968), respectively. For

the International Practical Celsius Scale they are correspondingly, tes and °C (Int. 1968). The relation

between Tes and ^68 is

t68=r68-273.15 K. (2.1)

The subscripts or parenthetical parts of the designation need not be used if it is certain that no confusion will result from their omission.

In the text of the IPTS-1968, platinum resistance

thermometers are defined as standard interpolation

instruments for realizing the scale from —259.34

to 630.74 °C (or from 13.81 to 903.89 K). The re- sistance values (or ratios of resistance values,

R{t)IR{0), where Rit) is the resistance at temperature t and R{0) the resistance at 0 °C) of any

particular standard platinum resistance thermometer are related to the values of temperature with

specified formulae. The constants of the formulae

are determined from the resistance values of the

thermometer at stated defining fixed points and,

usually, from the derivatives of the formula specified

for the temperature range immediately above. [Hereafter, platinum resistance thermometers that meet

the following specifications, based primarily on

the text of the IPTS-68, will be referred to as standard platinum resistance thermometers and abbreviated as SPRT: The platinum resistor shall be

very pure annealed platinum supported in such a manner that the resistor remains as strain-free as

practical; the value of /?(100)//?(0) shall not be less

than 1.3925; the stability of the thermometer shall

be such that, when the thermometer is subjected to thermal cycling similar to that encountered in the

normal process of calibrating it, the value of R{0)

does not change by more than 4xl0~^/?(0); the

platinum resistor shall be constructed as a four-lead

element and both the resistor and its leads shall be

insulated in such a manner that the measured re- sistance of the platinum resistor is not affected by

the insulator more than about 4X 10"^/? (0) at the

temperatures of calibration or use; and the four-lead

resistor shall be hermetically sealed in a protective sheath. (A SPRT that has a R{0) value of about

25.5 a will be referred to as a 25-0 SPRT.)] The

text of the scale assigns an exact numerical value

of temperature to each of the defining fixed points.

The defining points and their assigned values are

listed in table 1. The realization of the fixed points

Table 1. Defining fixed points of the IPTS-68

Assigned \alue of International Practical Equilibrium state Temperature W*

^68 (K) t68 rc)

Equilibrium between the soUd, liquid and vapor phases of equilibrium hydrogen (triple

point of equilibrium hydrogen) 13.81 -259.34 "0.0014 1207

Equilibrium between the liquid and vapor phases of equilibrium hydrogen at a pressure

of 33 330.6 N/m2 (25/76 standard atmosphere) 17.042 -256.108 "0.0025 3445

Equilibrium between the liquid and vapor phases of equilibrium hydrogen (boiling point

of equilibrium hydrogen) 20.28 -252.87 0.0044 8517

Equilibrium between the liquid and vapor phases of neon (boiling point of neon). 27.102 -246.048 0.0122 1272 Equilibrium between the solid, liquid and vapor phases of oxygen (triple point of oxygen)... 54.361 -218.789 "0.0919 7253

Equilibrium between the liquid and vapor phases of oxygen (boiling point of oxygen) 90.188 -182.%2 "0.2437 9911

Equilibrium between the solid, liquid and vapor phases of water (triple point of water)'' 273.16 0.01 Equilibrium between the liquid and vapor phases of water (boiling point of water)''''' 373.15 100 1.3925 %68

Equilibrium between the solid and liquid phases of zinc (freezing point of zinc) 692.73 419.58 2.5684 8557

^Except for the triple points and one equilibrium hydrogen point (17.042 K) the assigned values of temperature are for equilibrium

states at a pressure Po= 1 standard atmosphere (101 325 N/m^).

••The equilibrium state between the solid and liquid phases of tin (freezing point of tin) has the assigned value of r(i8 = 231. 9681 °C

and may be used as an alternative to the boiling point of water. At this value of temperature >F*= 1.8925 7109." The value of

t' = 231.929163 "C.

•^The water used should have the isotopic composition of ocean water (see sec. III. 4, Appendix A). "This value is slightly different from that given in the text of the IPTS-68 (see Appendix A).

3
and the use of interpolation formula are discussed

in detail in sections 7 and 6, respectively.

The IPTS-68 is defined by the text of the scale.

However, even if perfect experimental work were

possible, the values of temperature would not be

unique because of a lack of uniqueness in the definition of the scale itself. For example, the text of

the scale specifies the minimum quality of the

platinum to be used in the SPRT; yet different

samples of suitable platinum may not give exactly

the same value of a temperature between defining

fixed points. The different values of temperature

obtained would, however, be in accordance with

IPTS-68 because the samples of platinum are within the definition of the scale. Another source

of difference is the variation in the realization of

the fixed points where small differences in the purity

of the fixed-point samples could easily occur. This

spread of temperatures (the "legal" spread of the

scale) is the result of the looseness of the definition

of the scale; it is small compared to the usual errors

of experimental measurement, but does exist and is a practical necessity if the physical properties of

real materials are to be used in realizing the scale.

3. Platinum Resistance Thermometer

Construction

This discussion is primarily directed toward

thermometers that are suitable as defining standards on the IPTS-68; however, many of the techniques described are applicable to any resistance

thermometer. To be suitable as a SPRT as described in the text of the scale (Appendix A) the

resistor must be made of platinum of sufficient

purity that the finished thermometer will have a value of i?(100)/i?(0) not less than 1.3925 or a, defined as (/?(100)-/?(0))/100/?(0), not less than

0.003925. This requirement provides a scale that

is more closely bounded and is not unreasonable

since platinum wire of sufficient purity to yield the

above ratio is now produced in several countries.

The typical SPRT has an ice-point resistance of

about 25.5 Cl and its resistor is wound from about

61 cm of 0.075 mm wire. The wire is obtained "hard

drawn," as it is somewhat easier to handle in this

condition, but it is annealed after the resistor is formed. Although wires between 0.013 mm and 0.13

mm diameter are commonly used in industrial

platinum thermometers, experience shows that

finer wires tend to have lower values of i?(100)/i?(0)

(implying the presence of more impurities or strains).

The insulation material that supports the resistor

and leads must not contaminate the platinum during

the annealing of the assembled thermometer nor when subjected for extended periods of time to temperatures to which thermometer is normally

exposed. The insulation resistance between the

leads must be greater than 5 X lO^ O at 500 °C if the

error introduced by insulation leakage in the leads

of a 25-0 thermometer is to be less than the equivalent of 1 /Ltfi. For SPRT's the most commonly used

insulation is mica. The primary difficulties in the use

of mica are the evolution of water vapor at high

temperatures and the presence of the iron oxide

impurity which, if reduced, leaves free iron that will

contaminate the platinum. The most usual mica is muscovite (H2KAl3(Si04)3), sometimes caUed India

or ruby mica; it contains about 5 percent water of

crystallization. The dehydration starts at approximately 540 °C [41] and not only causes mechanical

deterioration of the mica but supplies free water

vapor that reduces the insulation resistance.

Phlogopite mica (H2KMg3Al(Si04)3) or "amber"

mica is used for thermometers that are expected

to function up to 630 °C. Although the room temperature resistivity of phlogopite mica is slightly

lower than that of muscovite, it does not begin to dehydrate until well above 700 °C [41].

Some designs employ very high-purity alumina or synthetic sapphire insulation (see fig. 1) [1, 18, 23,

52]. Fused silica has also been used [18]. Although

the properties of these materials are superior to mica at high temperatures, the materials are more

difficult to fabricate and, additionally, they must be

Figure 1. Synthetic sapphireformfor thermometer.

Synihetic sapphire disks are employed to keep the platinum wires separated in a "bird-cage" type SPRT. A centrally located, heavier gage platinum wire maintains the spacings between the disks [23].

4
(a) ^^mmt^ (b)

Figure 2. Construction of the coils ofsome thermometers.

(a) Coiled filament on mica cross [34).

(b) Sin^e layer filament on mica cross 147).

5
TWO TWIN-BORE

SILICA TUBES

PLATINUM OR GOLD

GLASS SHEATH

cm

Figure 2 — Continued. Construction ofthe coils ofsome thermometers.

{c) Coiled filament on twisted glass ribbon [49]. (d) Coiled filament in glass U-tube [2],

6
Figure 2 — Continued. Construction ofthe coils ofsome thermometers.

(e) Single layer filament around twin-bore silica tubing [3].

(f) Coiled filament on helically grooved ceramic. (Courtesy the Min.co Products, Inc.)

7
Figure 2 — Continued. Construction ofthe coils ofsome

thermometers.

(g) Filament threaded through four-hole ceramic tui)es. (Courtesy the Rosemount Engineering Company.)

of very high purity because migration of impurities

and the consequent contamination occurs much

more easily at high temperatures. In cutting either

mica or alumina great care must be taken to avoid

or eliminate traces of metal that originate from

either the cutting tool or from the metal clamps

that are used to support the insulation during

machining. A "carbide tipped" tool should be used

for the cutting process. Unfortunately, attempts to remove the metal chemically from the mica result

in contaminating the mica. Synthetic sapphire, however, can be chemically cleaned after machining.

The configuration of the resistor is inevitably the

result of compromise between conflicting requirements. The resistor must be free to expand and contract without constraint from its support. This

characteristic is the so called "strain-free" construction. If the platinum were not free to expand,

the resistance of the platinum would not only be a function of temperature but would also relate to the

strain that results from the differential expansion

of the platinum and its support. Seven methods of

approach toward achieving "strain-free" construction are illustrated in figure 2 [2, 3, 4, 5, 34, 47].

Because of the lack of adequate mechanical support,

the wire in each of these designs may be strained by

acceleration, e.g., shock or vibration. The thermal

contact of the resistor with the protecting envelope

or sheath is primarily through gas which, even if the

gas is mostly helium, is obviously poor compared to the thermal contact that is possible through many

solid materials. This poor thermal contact increases

the self-heating effect and the response time of the

thermometer. The designs shown in figures 2a, 2b,

and 2g suffer less in these respects than the others.

On the other hand, for calorimetric work the instrument of the lowest heat capacity is often

preferred.

The sensing elements of all SPRT's have four

leads (see fig. 2). The four leads define the resistor

precisely by permitting measurements that eliminate

the effect of the resistance of the leads. The resistor

winding is usually "noninductive," often bifilar, but

occasionally other configurations that tend to minimize inductance are used. This serves to reduce the

pick-up of stray fields and usually improves the performance of the thermometer in a-c circuits. (If the

resistor is to be measured using a-c, the electrical

time constant, i.e., reactive component should be

minimized.)

Because the junction of the leads is electrically a

part of the measured resistor, the leads extending

immediately from the resistor must also be of highpurity platinum; the lengths of these leads are often

as short as 8 mm. Either gold or platinum wire is employed in continuing these leads within the

thermometer. Gold does not seem to contaminate the

platinum and is easily worked. Measurement of the

resistor may be facihtated if the four leads are made

of the same material with the length and diameter

the same so that the leads have about equal resist- ances at any temperature within the temperature

range of the thermometer. This statement is also

applicable to the leads that are external to the protecting envelope. Figure 3 shows the arrangement of

thermometer leads near the head of three SPRT's.

The hermetic seal through the soft glass envelope

at the thermometer head is frequently made using

short lengths of tungsten wire, to the ends of which

platinum lead wires are welded. The external platinum leads are soft soldered to copper leads that are mechanically secured to the head. For d-c measurements, satisfactory external copper leads can be

made from commercially available cable. The cable

consists of twelve wires (No. 26 AWG solid bare

copper wire) each insulated with double silk windings; additionally, another double silk wrapping

encloses each group of three insulated wires. A

double silk braid and a tight, waterproof, polyvinyl

coating cover the entire four groups. After the

appropriate end insulation is removed, the ends of

the three wires in each group are twisted together to form a lead to which a lug is soldered. Stranded leads

which do not have individually insulated strartds

should not be used, as the breakage of a single strand

may cause "noise" in the resistance thermometer

circuits which is difficult to locate and eliminate. The

leads external to small capsule type thermometers

are usually solid copper wires, although wires of

other materials such as manganin are sometimes

used; the leads to the thermometer must be placed

in good thermal contact with the system to be measured by the capsule thermometer.

8
Figures. "Heads" of three SPRrs.

The upper head is that of a long stem thermometer. The internal leads are brought out through hermetic seals and are connected to external copper leads at the left. In the center is a capsule type SPRT with leads brought out through glass seals. Capsule body is platinum. The bottom capsule type SPRT shows the thermometer leads brought out through individual metal-ceramic-metal type seals. Capsule body is stainless steel.

4. Using the Thermometer

4.1. Mechanical Treatment of SPRT

A SPRT is a mechanically delicate instrument.

As discussed in the section on thermometer construction, the platinum wire cannot be rigidly

supported and at the same time be free to expand

and contract with temperature changes. Shock,

vibration, or any other form of acceleration may

cause the wire to bend between and around its supports, thus producing strains that change its temperature-resistance characteristics. Strains in

the platinum resistor normally will increase the

resistance and decrease the value of a. If a tap of

a thermometer with glass sheath is hard enough to be audible, but still not cause breakage, the action

wiU typically increase the triple-point resistance of a 25 n SPRT by an amount between 1 and 100 /xXl.

(A change of 100 (xQ. is equivalent to 0.001 °C in

temperature.) Thermometers that have received

repeated shocks of this kind through rough handling

over a one year period have increased in resistance

at the triple point of water by an amount equivalent

to 0.1 °C. Similar changes may be caused by using

the thermometer in a bath that transmits vibrations

to the thermometer or by shipping the thermometer

not suitably packed.

It is preferable to "hand carry" the thermometer

to maintain the integrity of its calibration. If the

thermometer must be shipped, it should first be

placed in a rigid and moderately massive container

that has been lined with material which softly con- forms to the thermometer and protects it from

mechanical shocks or vibrations. This container

should then be packed in an appreciably larger box

with generous room on all sides for soft packing

material that will absorb or attenuate the shocks

that might occur during shipment. Two reasonable

package configurations are shown in figures 4 and 5. In arranging storage for the thermometer in the

laboratory, a container should be used that minimizes or eliminates the possibility of "bumping"

the thermometer. This is a most reasonable precau9
tion when one considers the amount of handling that

many SPRT's receive during the course of many

measurements. This precaution may be even more

pertinent for SPRT's that are routinely used as a standard for calibrating other thermometers.

Figure 6 shows a storage arrangement employed

at NBS.

Care should edso be exercised in protecting the

thermometer from cumulative shocks that might be

received during insertion into apparatus. Figure 19

(sec. 7) shows the simple provisions made at NBS

for reducing mechanical shock when thermometers

are inserted into triple-point of water cells. The

polyethylene plastic tube at the top of the cell

guides the SPRT into the thermometer well without

"bumping", and the tapered entrance of the

aluminum sleeve near the bottom of the well then

guides the SPRT into the sleeve and onto the soft

polyurethane sponge at the bottom.

4.2. Thermal Treatment of SPRT

With exception of specific instances, SPRT's, as they are generally used, are not greatly susceptible

to damage from thermal shocks. In the case of

capsule type thermometers, the metal to glass seals

may be broken by rapid cooling, e.g., when a capsule

thermometer that has been at ambient temperature

is quickly immersed in liquid nitrogen. Sudden

exposure of capsule thermometers to temperatures

that are much higher than ambient seldom occurs

since they are rarely used above 100 °C. Another

specific instance of thermal shock is subjecting

long-stem SPRT's to temperatures that are well

above 600 °C and subsequently quenching them.

This treatment can mechanically damage the thermometer but, even if no visible damage has been

done, the calibration of the thermometer may have

been altered by the freezing in of defects that oc- curred during the quenching [24].

Figure 4. A methodfor packaging a SPRTfor shipment.

The metal case contains a SPRT snugly nested in polyurethane foam. The metal case in turn is protected during shipment by a tightly fitted polyurethane foam lined box.

10
Figure 5. Special containerfor SPRT shipment.

The wooden case contains a SPRT snugly nested in polyurethane foam. Each of the eight corners of this wooden case is attached elastically to the large metal container. Blocks

of polyurethane foam give additional protection in the event of "overtravel."

(courtesy the Space Division, North American Rockwell Corporation)

The upper temperature limit of a SPRT is re- stricted by the softening point of the material of the

protecting sheath, the temperature at which the

thermometer was annealed before calibration, the

evolution of water and other contaminants from the

sheath and insulators, and grain growth in the

platinum wire. The concomitant changes are a function of both time and temperature and most are predictable in only a quahtative way.

Thermometer sheaths of borosilicate glass soften

noticeably at temperatures above 500 °C, e.g., their use at 515 °C (1000 °F) is hmited to a few hours unless

they are specially supported to prevent deformation.

Fused silica sheaths should be used for measure- ments above 500 °C. Platinum grain growth has

been observed in SPRT's that have been used over a period of several hundred hours at 420 °C. Figure 7,

made with an electron microscope, shows in detail a short section of platinum wire from a thermometer

that was subjected to such a treatment. Grains

which are one to two wire diameters long and as large as the wire in diameter may be clearly seen. The growth of large grains results in a thermometer

that is more susceptible to calibration changes

caused by mechanical shock; consequently, the

thermometer may be considered unstable. The

evolution of water within the thermometer sheath at temperatures below 500 °C has been observed in a few thermometers but seems to be avoidable if sufficient baking and evacuation are furnished

during the fabrication of the thermometer. The

presence of water may become more conspicuous

when the thermometer is cooled, e.g., in a triple

point cell. The presence of water, either within the

thermometer or on the external insulated leads, will

be evidenced during electrical measurements by the

kick of the galvanometer when the current through

the thermometer is reversed. The kick stems from

the polarization of water on the insulation. The

decay time (the time for repolarization of water on

the insulation) of the galvanometer pulse is usually

greater than a minute when moisture exists in the

thermometer but is only of a few seconds' duration

with moisture on silk insulated leads, indicating

that the quantity and physical state of moisture and

the effect of moisture on the insulation inside the

11
Figure 6. A storage for SPRT's.

The thermometers are stored vertically by slipping into the wire braid sleeves. The

cover to the box has been removed only to show the arrangement of the wire braid

sleeves.

thermometer are different from those on the external

silk insulation.

Annealing may occur if the platinum wire was not

properly annealed during the manufacturing process

or (more Ukely) if it has been strained by mechanical

shock since it was last well annealed. If the strains are sufficiently severe, noticeable annealing will occur at temperatures as low as 100 °C. The annealing process does have somewhat beneficial aspects

because the thermometer tends to return toward the

metallurgical state that it presumably was in during

its previous caUbration. (At NBS all long-stem

SPRT's are annealed between 470 and 480 °C for 4

hours, removed from the furnace while stiU hot, and

allowed to cool in air at ambient temperature prior

to their cahbration). The comparison of the resistance

of the thermometer at a fixed point before and after

annealing will show a shift downward in resistance

and, usually, a shift upward in R{t)IR{0) (see sec. 6 on the computation of temperature).

In the measurements at temperatures above

500 °C the difficulties discussed above are more

likely to occur or be accelerated. More care must be

taken in baking and evacuating the thermometer

after assembly. For work near the upper temperature Hmit (630 °C), the SPRT's should have phlogopite mica or fused silica coil forms and a fused

sihca sheath. Also, care should be exercised to avoid the previously mentioned effect of freezing in

of high temperature defects when cooling the

thermometer.

Figure 7. Electron microphotograph ofplatinum thermometer wire subjected to a total ofseveral hundred hours at 420 °C.

Magnification: photograph (a) 172 X (b) 850 x. The photographs show; (1) grain growth; (2) migration. No attempt has been made to date to identify the white "flecks."

12
4.3. Thermometer Immersion

A thermometer is sufficiently immersed in the

bath when there is no heat flow between the sensor and its environment through the thermometer leads

or sheath that extend from the region of sensor. Regrettably, considering the ease with which the

adequacy of immersion can be checked, insufficient

immersion of a thermometer is still a very common

error. The test simply involves a equilibrium

temperature measurements at two depths of

immersion, while the bath is maintained at very

nearly constant temperature. If, after taking into

account any change in the temperature of the bath,

the temperatures at the different immersions do not

agree, an immersion problem exists. ExperimentaUsts are frequently led astray by the erroneous

idea that the required immersion is strongly temperature dependent. Figure 8 shows the difference

between the bath temperature and the temperature

indicated by the thermometer as a function of

immersion for two different thermometers. The

immersion characteristics are more clearly seen in

figure 9 where the same data are given in a semilog

plot. A linear relationship between the immersion

and the logarithm of the temperature difference is to be expected in simple cases and in practice this

is a very useful approximation in the usual region

of interest, namely, where the temperature difference

is small. Figure 9 shows that for thermometer G in an ice bath, the temperature difference is reduced

by a factor of 10 for each 3.3 cm of immersion.

For thermometer M, the temperature difference is reduced by a factor of 10 for each 1.4 cm of immersion. If the error due to immersion is to be less than

0.025 mK, the temperature difference must be

reduced by six orders of magnitude and, therefore,

both thermometers must be immersed in the ice

bath by six times the respective amounts cited

above, or a total of 19.8 cm and 8.4 cm, respectively.

Even if the temperature difference between the

ambient and the bath were only 2.5 degrees the

thermometer immersion could be decreased only

shghtly because, for the same accuracy, a reduction of the temperature difference by five factors of

ten is still needed.

The similarity of the radial conductance of heat

to or from the thermometer in the bath is important.

Figure 10 shows the immersion characteristics of

thermometers G and M in a triple-point of water

cell rather than an ice bath. The immersion characteristics have changed for both thermometers,

but particularly for thermometer M. This is caused

primarily by the increase in thermal resistance

radially from the thermometer to the ice-water

interface of the triple point cell: the sheath of

water that surrounds the thermometer within the

triple-point cell well and the glass wall of the

well contribute to the increased thermal resistance

(see also sec. 4.4). The immersion problem would

be even worse if the space between the thermometer well and the thermometer were filled with air

IMMERSION OF THERMOMETER TIP, cm

Figure 8. Immersion characteristics in an^ ice bath of two long stem SPRT's with different sheath materials and internal

construction.

The plot shows the relative temperature as a function of the depth of thermometer

immersion.

instead of water. Typical immersion data for

thermometers in tin and zinc freezing point cells are given in the discussion of that apparatus (see

sec. 7).

4.4. Heating Effects in SPRT

The measurement of resistance necessarily

involves passing a current through the resistor. The

resultant heating that occurs in the resistor and its leads raises their temperature above that of their

surroundings until the resistor element attains a temperature sufficiently higher than the surroundings to dissipate the power developed. A typical

steady-state profile of the radial temperature distribution caused by a current of 1 mA flowing in a 25 n SPRT is shown in figures 11 and 12. In figure

13
IMMERSION OF THERMOMETER TIP, cm

Figure 9. Immersion characteristics in an ice bath of two

long stem SPRT's with different sheath materials and internal

construction.

The data of figure 8 have been replotled to show the linear relationship between the logarithm of the relative temperature and the depth of thermometer immersion.

11, the internal heating effect of the thermometer,

i.e., between the platinum resistor and the outside

wall of the protecting sheath, at a given environmental temperature is a function only of the thermometer construction and the current and is, therefore, the same during both calibration and use. This

assumes that the thermometer resistor does not

move within its protecting sheath. At the ice point,

the internal heating effect may be measured by

direct immersion in the ice bath and is typically

between 0.3 mK/(mA)2 and 1.2 mK/(mA)2 for a 25 n SPRT. If the thermometer is used with the

same current that was used during its calibration,

the same internal heating effect occurs and no

error is introduced in the measurement.

As figure 12 indicates, there is also an external

heating effect, an extension of the heating effect

beyond the outside of the thermometer sheath

because the generated joule heat must flow to some

4 6 8 10 12 14 16 18 MMERSION OF THERMOMETER TIP, cm

Figure 10. Immersion characteristics in the triple point of water cell oftwo long stem SPRT's with different sheath materials

and internal construction.

The plot shows the relationship between the logarithm of relative temperature and

the depth of thermometer immersion. Comparison with figure 9 shows that the immersion characteristics of the thermometers tend to be poorer in the triple-point cell than

those in the ice bath. This change in the observed immersion characteristics is caused

primarily by the higher resistance to radial heat flow in the triple-point cell.

external heat sink. The total heating effect, i.e., the

combined internal and external heating effects, can easily be determined by calculating the resist- ance that would be measured at zero current; this can be performed either algebraically or graphically

as shown in figure 13. When it is desirable to determine the internal heating effect only, the experimental conditions must be such that the external

heating effect is negligible. This condition can be

closely approximated by directly immersing the

thermometer into an ice bath wherein the solid ice

particles are in contact with the thermometer sheath,

or into a metal (tin or zinc) freezing point apparatus

in which the metal freezes directly on the thermometer. (The metal must be remelted before complete

solidification or the thermometer will be crushed.)

14
Figure 11. Platinum resistance thermometers at 33 cm immersion in an ice bath.

Temperature profile from the middle of thermometer coil out to the ice bath with 1 mA

current.

A. Platinum coils of coiled filament thermometer; only coils of one side are indicated.

B. Platinum coils of single layer helix thermometer; only turns of one side are indicated. C. Borosilicate glass thermometer envelope.

D. Finely divided ice and water.

The recommended measurement and calculation

procedures are identical to those previously

described for determining the total heating effect

(see fig. 13). For the most precise work, aU resistance

measurements should be made at two currents and

use made of the value of resistance calculated for

zero current. Reiterating what was said earlier, an

error due to the heating effect is introduced if the

thermometer is not calibrated and used at the same

current or is not in good thermal contact with its surroundings.

The external heating effect may be reduced by

making the thermometer well a relatively close fit to the thermometer and placing a material of high

thermal conductance in the annular space between

the thermometer and the well. The material used

to fill the space must not undergo an exothermic

(or endothermic) reaction at the temperatures in- volved because such a reaction would additionally

change the temperature of the thermometer from

that of the surroundings. Examples of this difficulty

that have been experienced in the NBS calibration

laboratory are the slow decomposition of Hght

mineral oil (at 122 °C) and oxidation of a steel bushing (at 444 °C). Difficulties associated with this

type of reaction can be detected by comparing the

derived values of resistance at zero current that

have been obtained from measurements in which

(i) the questionable material was used and (ii) the

material was not present, or a better substitute was

employed.

Another source of heat flux to and away from the

platinum coil, and consequently a possible source

of error, is radiation. If the sensor can "see" a surface that is appreciably hotter or colder, the

power gained or lost by the resistor will result in

its temperature being changed. In the triple-point

of water measurements, radiation from lights in

the room which is incident upon the top of the ice

bath or triple point cell can easily produce an error

of 0.0001 K (see sec. 7 on triple-point cell). The

water-triple-point cell should, therefore, be immersed in an ice bath in which no extraneous

radiation from sources above room temperature can

reach the sensor of the SPRT. At a higher temperature (630 °C) an error as large as 0.01 °C can

occur if a clear fused silica thermometer sheath is employed because "light piping" takes place. In

this process radiation is conducted towards ambient

temperatures within the wall of the thermometer

sheath, being confined there by total internal

reflections. Any technique which would eliminate

these reflections would of course, eliminate this source of error. The error can be substantially

reduced by coating the exterior wall with graphite

paint as is depicted in figure 14 or by roughening

(by sand-blasting) the external wall.

15
Figure 12. Platinum resistance thermometers at 33 cm immersion in a water triple point cell. Temperature profile from the middle of thermometer coil out to the ice-water interface with 1 mA current. A. t'latinum coils of coiled hlament thermometer; only coils of one side are indicated.

B. Platinum coils of single layer helix thermometer; only turns of one side are indicated. C. Borosilicate glass thermometer envelope.

D. Water from ice bath.

E. Aluminum bushing (length not to scale). F. Borosilicate glass thermometer well. G. Ice mantle. H. Inner melt.

I. No inner melting; temperature profile relative to the temperature of outer ice-water interface.

J. Water in cell. K. Cell well (borosiUcale glass). L. Outside ice-water bath. M. Temperature profile of the ice mantle. N. Polyurethane sponge.

16
SQUARE OF THERMOMETER CURRENT,!^

Figure 13. Plot of SPRT resistance (temperature) versus the square ofthe thermometer current. The plot illustrates how the value of "zero current" thermometer resistance may be

obtained graphically or algebraically from measurements at two currents. Ro= resistance of SPRT at "zero current." R 1 and R2= resistances determined at currents ii and i-zy respectively.

TU U 1 Hi M U i i i i i 1 1 ! i I i i i U i I t i i i i i i i i I ! i I i U I ii 1

! 1 U U U

I

& . 1 2 3 8 ^ 4 ^ S

^^MiMiiLiihiiliiiliiiliiiliiiliiiliiiliiiliiiliiiliiilii^

Figure 14. Two methodsfor reducing radiation ^*piping" infused silica thermometer sheaths.

The sheath of the thermometer at the lop was given a matte finish by "sand blasting." "Aqua dag" is shown painted on the sheath of the SPRT at the bottom.

5. Resistance Measurements

In this section, the salient features of instrumentation used to measure the resistance of SPRT's

in the NBS calibration laboratory wiU be described

so that the laboratory's general electrical measure- ment procedures may be understood. The discussion will neither give details on other instruments

that cU-e available for measuring electrical resist- ance nor special design features of particular

instruments, nor a comparison of different measure- ment techniques. Some of the instruments will be

mentioned by name and literature references will

be cited to direct an interested reader to more

detailed information.

The most suitable methods for and the limitations on achieving both accurate and reproducible

resistance measurements depend on several

factors. Johnson noise [31], which is inherent in

any resistor (caused by the random movement of

electrons within a conductor), is given by:

e = 7.43 X 10-12 VfRiKfj, (5.1)

where e, is the thermal agitation voltage in volts,

A/ is the effective bandwidth in hertz, R is the

resistance of the conductor in ohms, and T is the

temperature of the resistor in kelvins. The constant

7.43 X 10-12 comes from 2VX, where k is the

Boltzmann constant. However, this frequently

mentioned limitation is seldom the predominant

source of uncertainty in resistance measurements

with SPRT's. For example, using a 25 -fl thermometer at room temperature with an observation

(averaging) time of 1 second, this noise is of the

order of 0.6 nV — slightly less than the signal

(1.0 nV) that results from a l-fj,Cl unbalance when

1 mA is flowing through the thermometer. Spurious

emfs, variations in contact resistance, mechanical

or electrical disturbance of the detector system.

17
and variations in thermometer lead unbalance,

are sources of "noise," one or more of which

contribute much more to the electrical measure- ment uncertainty than the Johnson noise in the

vast majority of measurement systems.

The signal level at the null detector caused by

the resistance unbalance can be raised by simply

increasing the current through the thermometer.

This may at times be a satisfactory solution; limitations are encountered, however, because increasing

the current rapidly increases the uncertainty due

to the self-heating of SPRT (see sec. 4) and may

even introduce significant error due to the self

heating of reference resistors in the bridge or potentiometer. The increased power dissipated in the

thermometer may produce additional difficulties

if the thermometer is used in a calorimeter system

with a small heat capacity.

In addition to the uncertainty of null detection in a reasonable period of time, there is also the problem

of referencing the unknown resistor to a single

resistance standard. A large number of arrangements have been proposed and used for this purpose.

They may be broadly classified as either bridge or potentiometer circuits that employ either alternating or direct currents with either resistors or inductive

dividers to establish the ratio of the unknown

resistor to a standard resistor. (A standard resistor

is defined, hereafter, as a stable resistor or a combination of stable resistors of known value.)

5.1. Mueller Bridge

At the National Bureau of Standards the traditional

instrument used with SPRT's is the Mueller bridge

[37, 38]. There have been several modifications

C c

1— —J

+ _+- COMMUTATING SWITCH CONTACTS-+ f

+ h NORMAL reverse''' *

Rt

Figure 15. Schematic of a Mueller bridge circuit in NORMAL

and REVERSE thermometer connections.

G is the null detector (galvanometer). Rt and R2 represent the ratio arm resistors. Kij represents the adjustable resistor. Rc andfir are the resistances of the "potential" leads of the SPRT. Rx is the resistance of the SPRT.

[21, 22, 26] of the bridge since its first appearance,

but this discussion will only cover its principal

features. A simple form of the bridge is shown

schematically in figure 15. It is basically the classical

equal-ratio-arm Wheatstone bridge with provisions

for interchanging (commutating) the leads of a fourlead resistor in such a way that the average of two

balances is independent of the resistances of the

leads. Referring to figure 15 (NORMAL), when no

current flows through the null detector, the voltages

ei and e-z are equal, and the bridge is said to be

balanced. The equation of balance is:

Rd, + Rc — Rx + Rt, (5.2)

where Rdi is the resistance of the variable decade

balancing resistor,

Rc is the resistance of a lead from the bridge to the thermometer sensing element,

Rx is the resistance of the thermometer sensing

element, and

/? r is a lead resistance similar to RcAfter commutating the leads (marked C, T, c, and t) to the positions shown in figure 15 (REVERSE) the

equation of balance is

Rd2 + Rt — Rx'^ Rc, (5.3)

where

Rd2 is the resistance of the adjustable-decade

balancing resistor that is required for the second

balance of the bridge.

The addition of eqs (5.2) and (5.3) yields the value of

/iA- in terms of/?Oj and/?D2:

or

Rdi + Rd2 — 2Rx

nx — —X—

(5.4)

(5.5)

An assumption made in the above equations is that

Rt and Rc are constant during the time required for

making the two balances.

Actually Rt and Rc need not be constant. If eqs

(5.2) and (5.3) are rewritten as:

RDi + Rcj — Rx + Rtj

and

(5.6)

Rd2^ Rt^ — Rx'^ Rc2 , (5.7)

and then added, there results

Rd,^Rd2-^{Rc,-Rt,)-{Rc2-Rt2)=2Rx (5.8)

or

18
Rx

Rd, + Rd, , iRc,-RT,)-iRc,-RT,) _ (5.9)

Thus, a sufficient condition for the measured resist- ance to be independent of the lead resistances is that the difference in the resistance of the two

potential leads is constant during the period of

observation. Equation (5.9) demonstrates that experimental emphasis could be placed on insuring that

the leads be of equal length and cross section and

that the temperature gradients between the leads be

constant or only changing slowly, rather than the

more difficult option of maintaining the temperature

of the leads constant.

Successful operation of the Mueller bridge is dependent upon the reproducibility and self consistency (linearity) of the adjustable resistor

indicated in figure 15 as Rd- The methods of ac- complishing this include thermostating the resistors

and employing special circuitry. (See fig. 1 of

Appendix H. A procedure for calibrating the

Mueller bridge is also described in Appendix H.)

The circuitry has been designed to reduce the

uncertainties that are associated with the variations

in the contact resistances of the "dry" switches.

For the 1-0 and O.l-O step decade resistors, the

switch contact resistances are placed in series

with the bridge ratio arms (Ri and R2), which are usually from 500 to 3000 O, so that the effect of the

possible variations in the contact resistances (about

0.0005 fl) can usually be neglected. For the measure- ment of the higher thermometer resistances this

arrangement introduces uncertainties which may

be significant. For a 25-fi SPRT the uncertainty of

0.0005 n in 3000 ft or 1 part in 6 X 10^ in the ratio

arm corresponds to only 1 (jlCI or 0.01 mK when

measuring 6 fl (near —183 °C) but increases to 10

fjSl or 0.1 mK when measuring 60 fi (near 350 °C).

The decades with steps of 0.01 fl or less are the

Waidner-Wolff shunted decades [38] which reduce

the effect of contact resistance in the switch by a

factor of 250 or more. The switches for the lO-fl

step decade and the commutator are directly in

series with the resistors in the adjustable arm of

the bridge and the thermometer resistor; they have,

therefore, mercury-wetted contacts. The mercurywetted switches in the Mueller bridges employed

at the National Bureau of Standards have an uncertainty of less than 2 juil when well maintained.

AH of the mercury contacts are normally cleaned

every day before use. The mercury is removed by

vacuum using a small polyethylene tip at the end

of a vacuum line with a mercury trap. Fresh mercury

is placed on each amalgamated contact. If the

entire surface of the contact is not wetted by the

new mercury, the surface is scrubbed (without

removing the mercury) with the flat end of a solid

copper rod about V4 inch in diameter until the

entire surface becomes wetted. The mercury is again removed and replaced with clean mercury.

The switch is then reassembled and operated

several times after which it is reopened to remove

any mercury which has splashed onto the surrounding surfaces. The switch is then finally assembled

for use. Switches with sliding contacts are exercised

every day before use by revolving them 10 or 20

times; this is particularly important for the 1, 0.1,

and 0.01 H decade switches. The sliding switches

are cleaned occasionally with a lint-free cloth,

either dry or moistened with benzene or "varsol".

(Carbon tetrachloride is not recommended as it frequently contains impurities which will result in

corrosion.) After cleaning, the contacts are lubricated with a light coating of pure petrolatum.

5.1.1. Bridge Ratio Arms

If the two ratio resistors, Ri and R2 in figure 15,

change so as to become unequal, this may be

compensated by adjusting the "tap" on the slidewire resistor joining Ri and R-z- Because this tap

is in the battery arm of the bridge, the variations of

its contact resistance are unimportant to the determination of thermometer resistances. The ratio arm

resistors are adjusted to be equal by varying this tap on the slide wire between i?i and R2 until the interchanging of the ratio arms does not change the

bridge balance. The accuracy of the ratio is limited

by the uniformity of the resistance of the copper

leads and switch contacts which are in series with

Ri and R2 (those connecting the 1-Cl and O.l-O

decades) and, of course, by the sensitivity of the

null detector.

Some versions of the bridge have incorporated

into the commutator additional switch contacts

which reverse the ratio arm resistors simultaneously

with each commutation of the thermometer leads;

the switch contacts of the 1-fi and O.l-O decades

and their leads to the commutator switch are, however, not reversed. The eff"ect of their variations

must stiU be considered in the determination of

the bridge resistance. Referring again to figure 15,

but now with the ratio arms Ri and R2 interchanged

in the reverse bridge connection, the equations of

bridge balance with the normal and reverse connections are, respectively:

and

Rd^+ Rc _Ri 1 + € Rt + Rx R2 1

Rd2 Rt _ R2 _ 1 Rc + Rx~Ri~ 1 +

(5.10)

(5.11)

where e— — 7?2)//?2- After combining eqs (5.10)

and (5.11) and eliminating Rt and Rc,

Rx

Rd, -\- Rd.,{1 + e)

2 + € (5.12)

By adding and subtracting {Rd,-\- Rd.^)!^ from the

right-hand side of eq (5.12) and combining, the

19
relation

(Rd, +Rd.,) (Rd, -RD^)e

2 + 2(2 + €) ^^-^^^

is obtained. Equation (5.13) shows that if the bridge

is operated to yield /?Dj =/?d2' then Rx—Rdi =Rd2^

without regard to the lack of equality of the ratioarm resistors. In practice, because the thermometer

leads {T and C) are usually made nearly equal,

and Rd^ will be only slightly different, typically

much less than 0.002 fl. Also, the equahty of the

ratio-arm resistors can be adjusted to better than

1 ppm with ease. Thus, the second term on the

right of eq (5.13) is completely negligible in normally

conducted Mueller bridge measurements which

utihze simultaneous commutation of the SPRT

leads and the ratio-arm resistors.

5.1.2. Bridge Current Reversal

The indicated balance of any d-c bridge is dependent upon the iR voltages across the elements of

the circuit and the spurious emfs. The equations

of bridge balance involve only resistance values;

therefore, the effects of spurious emfs must be

eliminated. This could be done by first observing the "indication" of the null detector with no bridge current, then balancing the bridge

with current to the same indication. The indication

of the null detector with no current includes the

effect of any spurious emfs; when the bridge is balanced to the same indication with current in

the bridge, the iR voltages are balanced if the effect

of the spurious emfs remains unchanged. However,

one should recognize that a change in the magnitude

of the thermometer current will change the thermometer temperature (due to self heating) and that

enough time must elapse before reading to allow the

thermometer to attain a steady thermal state. During this time the effect of spurious emfs may

change significantly. This problem can be simply

surmounted by reversing the battery current with

a snap action toggle switch, so that an essentially

continuous heating power is retained in the thermometer (hence, a uniform self heating effect). In

the process, the galvanometer or null detector

sensitivity is effectively doubled. The rate at which

the current reversals must be made is dependent

on the rate of change of the spurious emfs.

5.2. A-C Bridge

If the current is reversed sufficiently rapidly, it

is usually said that the bridge is an "a-c" bridge.

Bridges operating at 400 hertz have been built at the NBS based on a design by Cutkosky [19]. These

bridges were designed for use with SPRT's and

include special provisions for use with thermometers

having values of R{0) as low as 0.025 fi. The bridge

utihzes an inductive ratio divider that eliminates

the necessity of calibrating the bridge because the

initial uncertainty of the divider is about 2 parts in

10* and appears to be stable. Additionally, the

bridge requires only one manual resistance balance,

the phase angle balance being automatic, and

incorporates a built-in phase-sensitive null detector

with which 1 /xfi in 25 fl can be easily resolved.

Small deviations from balance can be recorded

continuously, the accuracy of recording these

deviations being limited primarily by the resolution

and linearity of the recorder. A small (usually less

than 10 fiil) error may be introduced in measuring

a 25-11 SPRT unless coaxial leads are used between

the bridge and the thermometer head. (The heads of

SPRT's have been modified to contain two BNC

coaxial connectors. The two leads from one end of

the SPRT coil were connected to the center "female"

contracts of the BNC receptacles and the two leads

from the other end of the SPRT were connected to the outer shells or the shield contacts.) For precision measurements the length of the pair of coaxial

leads should not be greater than 15 meters to Hmit

the dielectric losses of the shunt capacitance.

Preliminary measurements on 25-fi SPRT's indicate

that, if the leads do not affect the measurements,

the accuracy of the measured value in ohms of a thermometer element is limited by the accuracy to which the reference standard resistor is known.

However, in the accurate determination of the

resistance ratio, R{t)IR{0), the stabihty, rather

than the accuracy, of the reference standard is the

important requirement. Further work is in progress

at NBS to determine the dc-ac transfer characteristics of SPRT's.

5.3. Potentiometric Methods

The Mueller bridge and the Smith bridge methods

[6, 7, 27] are suitable for a four-terminal determination of thermometer resistance if the lead resistances

are relatively stable. When the lead resistances are

variable, e.g., in measurements at low temperatures

where the cryostat temperature is varying and the

thermometer resistance is small, potentiometric

methods become better suited to the measurements.

The potentiometric method of resistance measure- ment depends in principle upon the determination

of the ratio of iR voltages developed across the

SPRT and across a resistor of known resistance

that is connected in series. The ratio is determined

by comparisons with iR voltages that are developed

in a separate resistance network and usually a separate current supply.

One of the major drawbacks of the potentiometric

method has been the requirement of exceptionally

high stability of current in both the potentiometer

and in the SPRT circuits during the measurement

period. However, the recent availability of highly

stable current supplies and their continued improvement have made the potentiometric methods more

popular.

20
Three circuits for the more commonly used

potentiometric methods are illustrated in figure 16.

To eUminate the effect of spurious emfs in the

measurements employing circuits (a) and (b) the

currents ii and 12 are reversed. Four balances are necessary in each method, two with the current in one direction and two with the current reversed.

Four balances are also necessary employing

circuit (c) by reversal of the current I'l and at the connection between the voltages to be compared. The readings for the normal and reverse connections are averaged. By using high quahty

reversing switches along with current sources of

high resistance (to make the relative effect of small

changes in resistance associated with the switching

negligible compared to the total resistance of the

circuit), the current in the circuit can be made more

stable and the switch in series with the detector

need not be disturbed during the current reversal

process. Nevertheless, a shunt is needed across the

BA-<

^

BA+

vvw\A^A/wwww^\/w\^-

> —

Rp

^A<\AA-LvwWWWV\AA

I

Rs

BA-'

VWWAMAAAAA- >BA +

(a)

BA--

(b)

BA- vwvwwwwwwvww^-^ BA +

Ra

BA-

^AA(V^-pAAAA/WWWW

^ ^

Rp

BA—

BA +

(C)

Figure 16. Potentiometric circuits used to determine resistance values.

Rp, the resistance to be determined.

Rs, standard resistor of precisely known value. P, potentiometer or precisely known adjustable resistor, adjustable resistor. C, stable capacitor of low dielectric absorption, low absorption time constant, and high leakage resistance (polystyrene). G, null detector

11, independent stable current in the unknown resistance circuit.

12, independent stable current in the potentiometer or in the voltage transfer instrument circuit.

21
detector to protect it because of asynchronous

contact operation during the reversal. The discussion of the potentiometric methods to follow

assumes that the required balances are made with

current reversal.

Using circuit (a) of figure 16, the potentiometer P

is successively balanced against the voltages URp

and iiRs in terms of izPp and i2Ps-, respectively,

where Rp and Rg are the resistances of SPRT and

standard resistor, respectively, and Pp and Ps are

in the potentiometer resistance units. {Pp indicates

the potentiometer resistance when its iR voltage

is balanced against the SPRT; Ps indicates the

potentiometer resistance when balanced against a standard resistor.) Thus, with iiRp= i^Pp and

iiRs= i2Ps, the SPRT resistance is given by

Ps

Rp — TT ^s.

There is a wide variety of potentiometers that

employ essentially the circuit shown in 16(a). The

potentiometer iR voltages in some designs [43, 54,

55, 56] are developed with different currents in

certain decades; in others, the iR voltages are adjusted by varying, by means of the potentiometer

switches, the current through a fixed resistor.

These designs keep the resistance of the detector

circuit constant. In the "double potentiometer,"

which is designed to make two consecutive voltage

balances more conveniently, there are duplicate

sets of switches but only a single resistance network

to develop iR voltages. A requirement of the

potentiometer, whatever the design, is that the

"resistance units" or the "iR voltage units" that are developed in the instrument for voltage balance

be linear.

Two standard resistors are used in circuit (b). Current i2 is adjusted until izRs, — iiRsi'-, then P is adjusted until i-iPp = iiRp. The SPRT resistance is given by:

R P

Rs,'- (5.14)

In circuit (c), either current iz or resistance Ra is adjusted until izRA — iiRp; then P is adjusted until

12^.1 = iiPp- The thermometer resistance is given

directly hy Rp — Pp.

The "isolating potential comparator" method

described by Dauphinee [20] is an adaptation of

circuit (c) where the voltage iiRp is set up as i-zRA

and measured as iiPp. The method is shown in

circuit (d). The voltage iiRp appears across a high

quality capacitor C and is compared with the voltage

iiPp, P being adjusted until i\Rp=iiPp. The breakbefore-make, double-pole chopper switches the

capacitor alternately across Rp and then across P

between 20 to 80 times a second. Extraneous voltages are canceled by reversing the current and the

capacitor connections and averaging the second

reading with the first.

The linearity of the potentiometer can be calibrated by comparing successive steps of one decade

against the total {X) of the next lower decade.

See Appendix I for a method of calibrating

potentiometers.

6. Calculation of Temperatures from the

Calibration Data and Observed Resistances

The SPRT is the standard interpolation instrument

between the defining fixed points in the range

13.81 K to 630.74 °C. (See table 1 and Appendix A.)

The "constants" of the interpolation formulae that

relate the resistance of a particular SPRT to the

value of its temperature on the IPTS-68 are obtained

by resistance measurements at the defining fixed

points. At NBS the "long stem" SPRT's are usually

calibrated for application above 90 K; the capsuletype SPRT's are calibrated for use between 13 K

and 250 °C or occasionally up to 400 °C. The equipment and procedures employed at NBS to achieve

the fixed points are described in the next section.

The resistance measuring instruments employed at NBS and other instruments that can be used for

resistance measurements of SPRT have been

described earlier in section 5. This section deals

with the methods in use at NBS to obtain the

constants of the interpolation formulae from the

calibration measurements of an SPRT at the fixed

points. Methods of calculating temperatures from

the observed resistances, when the constants of the

interpolation formulae are known are also described.

At NBS aU evaluations of equations and calculations

pertaining to the SPRT are performed on a highspeed electronic digital computer (UNIVAC 1108).

6.1. Temperatures from 0 to 630.74 °C

From 0 to 630.74 °C the temperatures on the

IPTS-68 are defined by

hi = t' + M{t'), (6.1)

where

=0.045 (4) (4 -l)

{—\m.fs

(6.2)

and

a\R{Q)

R{t') -0 + <4)(4-l)- (6.3)

In the official English version of the text of the

scale (see Appendix A) the unit °C appears in

22
the defining equations, e.g.,

t68 = t' + 0.045

100 °c vioo °c

(419.58 °C 0 (630.74 °C 0

^'

(6.4)

In this discussion the equations will be simplified

by omitting the units. Similarly R{t°C) and

R{0°C) will be simplified to R(t) and R{0). In

addition, tes will, henceforth, be abbreviated to t. The constants R{0), a, and S are determined from

calibration resistance measurements of the SPRT

at the triple point of water (TP), the steam point or the tin point, and the zinc point. (The tin point is now employed at NBS.) The constants may be more

conveniently obtained from (6.1) and from the

relation, equivalent to equation (6.3), given by:

Wit')=Rit')IR{0) = 1 +At' +Bt" (6.5)

where R{t') is the observed resistance at the

temperature t' and R{0) is the resistance at 0 °C.

The constants A and B are related to a and S by

also,

^ = a(l + 8/100),

5 = - 10-4^8;

a^A + lOOfi,

8^-WBI{A + lOOfi).

(6.6)

(6.7)

(6.8)

(6.9)

Equations (6.3) and (6.5), appear to be of the same

form as the earlier formulations of the International

Temperature Scale [15, 46, 48]. But, the value of

temperature, t, on the IPTS-68 is not equal to t'

,

and the value of t' is not the value of temperature on the IPTS-48 because the definitions of the two

scales are different (see Appendix B).

The value t' , obtained from (6.3) or (6.5), may be

considered to be a first approximation to t, the value

of temperature on the IPTS-68. Equation (6.2)

gives the adjustment to be made to the value t' to yield t. This adjustment was included in the

definition of the IPTS-68 with the intention of

bringing the scale into closer agreement with the

thermodynamic scale. Therefore, at any given

hotness (except certain fixed points) the values of t' and t are different, but they both represent the same

hotness; accordingly, the resistances Rit') and

R{t) are equal. The observed SPRT resistance will,

hereafter, be indicated by R{t) . See Appendix D.

At the NBS, the SPRT that is received for calibration is first annealed between 470 and 480 °C for

about four hours and then allowed to cool in air at the ambient conditions. The caHbration measurements at the fixed points are made in the following

sequence: TP, zinc point, TP, tin point, and TP,

If any observed resistance is questioned, the meas- urement is repeated. Any additional tin or zincpoint measurements will usually be bracketed

before and after by TP measurements in order to check any change in the TP resistance that may

occur. Whenever the TP resistance of a SPRT

changes by more than 75 /xfi (depending upon the

design) in the course of caHbration the complete

sequence of calibration, including the annealing,

is repeated.

A measuring current of 1 mA through the resistor

of the thermometer is normally used. The sequence

of measurements with the Mueller bridge at each

of these fixed points is, unless additional measure- ments are found necessary, NRRN, where N =

normal and R = reverse connections of the bridge.

The readings are corrected for the "bridge calibration" and "bridge zero" and averaged (see sec. 5 on Mueller bridge resistance measurements).

For the tin and zinc-point calibration measurements, the "equilibrium temperatures" at the ex- ternal surface of the SPRT near the resistance

coil are computed and assigned to the observed

resistance values. The experimental equilibrium

temperature value is slightly different from the

IPTS-68 value of the fixed-point temperature; the

value of the equilibrium temperature at the outer

surface of the thermometer sheath is obtained by

correcting the IPTS-68 fixed-point temperature

value for the hydrostatic head of the liquid metal

column (also, for any significant deviation from 1 atm

gas pressure over the fixed-point cell) and for the

external heating effect of the SPRT in the particular

cell. In the case of the measurements in a triple

point of water cell, the assigned value of the temperature is adjusted for the temperature effect of

the hydrostatic head of the water column and ex- ternal heating effect of the SPRT.

The value of /?(TP) that is measured after the

/?(tin) measurement is employed to obtain W{tm);

in a similar manner, the value of /?(TP) measured

after the R(zmc) measurement is employed to obtain ^^^(zinc). The value of R(0) is calculated from

i?(TP) according to eq (6.5) by assigningfi = 0 for the

small temperature interval of the evaluation. The

value of ^ employed is 3.98485 X 10-^ "C"' , which is an average value for SPRT's calibrated at NBS in

the past few years. The uncertainty in the adjustment of the value fromi?(TP) to/?(0) in this manner

is about ± 1 X 10-8 /j(0) or ±0.25 fxCL for a 25 O

SPRT and is, therefore, negligible. The thermometer

constants A and B are obtained by simultaneous

solution of eq (6.5) from the values of W{tin),

f'(tin), and JF(zinc), ?'(zinc). The constants a

and 8 are obtained from A and B according to eqs

(6.8) and (6.9). The printed results of the computer

analysis of calibration data include, in addition to tables of W{t) or R(t) versus t, the observed resist- ances (adjusted for bridge calibration and bridge

23
zero) at the three fixed points, and the values of

R(0), a, 8, A, and B.

Several possibilities are available to the user for

computing temperatures and for checking the calibration of his SPRT. First, in eq (6.3) /?(0), a,

8 may be taken as invariant and t' calculated from

the measured R{t). Second, in eq (6.3) only a,

and S may be taken as invariant, requiring then the

determination of R(t)IR(0). The first method is equivalent to using a table of R versus t; the second

method is equivalent to using a table of /?(t)/R(0) versus t. Equation (6.3) may be rewritten as

' -^^«(100)-/?(0) + ^W^100~^^- ^^-^^^

In eq (6.10), as a third method, /?(100)-i?(0),

a and 8 may be taken as constant, requiring the

determination of R{t) —R{0) to ceilculate t' . The

fourth possibility seen with equation (6.10) is to take only 8 as constant which requires the determiR{t)-R(0)

nation oi „

.

_ . . . Ihe ultimate extension K(LW) — A (0)

of this procedure of reducing the SPRT constants

is not to assume that anything has remained constant and recalibrate the thermometer every time

it is used. In 80 years of platinum thermometry

each of the above assumptions or its equivalent has

been made by.investigators.

From the standpoint of accuracy the worst of

these assumptions is the first, which requires only

the measurement of R{t). This choice may be dictated, however, by convenience or economy. If

values of ^(0) can be obtained, the choice is not as

clear between using R{t)IR{0) or using Rit) — RiO)

to calculate the value of temperature. Experience

with a particular thermometer may help one make

the choice. Lacking this, one may be guided by the

experience of the NBS and the National Research

Council of Canada which indicates that below

500 °C the second assumption, namely, that

a and 8 are constant and using the value of /?(?)//? (0) to calculate a value of temperature, is the most

satisfactory. This assumption has the distinctive

practical advantage of being dimensionless (i.e.,

being independent of the size of the resistance

unit). If request is made at the time the SPRT is submitted to NBS for calibration, tables based on any one of the assumptions discussed in this paragraph, in a form suitable for linear interpolation, can be provided from the calibration data.

When many observations of R(t), such as in

specific-heat measurements, are to be converted

to t and a high speed electronic computer is available, eqs (6.1), (6.2), and (6.3) or (6.10) may be more

convenient than tables. Iteration methods are best

employed in the solution of either eq (6.3) or (6.10).

For example, an approximate t' can first be obtained

neglecting the term containing 8. Improved approximations are obtained by substituting successive

solutions into the complete equation, stopping when

t' does not change more than desired limits.

If it is desired to generate a table of values of

W(t) versus values off, entries for integral values of t at one-degree intervals seem quite useful. The error contributed by using simple linear interpolation

within the one-degree intervals is less than 10-'* °C

in the range from 0 to 630 °C. The cjdculation of the

table is most easily performed employing eqs (6.1)

and (6.5). The only difficulty is that (6.5) gives W

as a function of t' rather than t, hence if W is to be

tabulated as a function of t the value of t' corre- sponding to the value of t must be first obtained

and used in (6.5). By rearranging (6.1),

t' = t-M(t'). (6.11)

Below 472 °C the error introduced by replacing (6.11)

with t' = t-Mit) is less than 1.2 X lO"'* °C. The

error in the value of t' to be used may be reduced

below 10-5 °Q a single iteration of eq (6.11). A

very efficient equation for calculating the rath entry

for tables with intervals of less than 10 degrees is

f; = f„-M(f„-M(«„-M(C,)), (6.12)

Equation (6.12) does not require iteration and it may

be shown that it introduces an error of only

[dM{t')ldt']^ times the tabular interval.

6.2. Temperatures Below 0 °C

From 0 °C down to — 259.34 °C, the temperatures on the IPTS-68 are defined by

Wit)=W*(t) + MF(t), (6.13)

where W{t) = R(t)IR(0) is the ratio of the observed

resistance of the SPRT at temperature t to that at 0 °C and W*{t) is a reference value of the resistance

ratio given by the function

20

f = 5^i(ln r*(0)''. (6.14)

i=l

In this monograph the symbol W*{t) will replace the

symbol Wccj-esiTes) used in the official text of the

IPTS-68 to represent the reference (standard)

values of W. The values of W*{t) represent a

fictitious but not "ideal" platinum resistance

thermometer. The constants Ai of the reference

function are listed in Appendix A. The deviations,

AJF(f) of eq (6.13), are expressed by simple polynomial equations of specified form; the constants of

these equations are determined from measurements

of W{t) at the appropriate defining fixed points and

the corresponding values of W*{t). The intermediate

temperatures are determined from the corresponding measured values of W{t), the appropriate

polynomial equation for AW{t), and eqs (6.13) and

24
(6.14). (See Appendix A which lists the four

temperature ranges, the associated polynomial

equations, and the fixed points defined by the

IPTS-68 in the range -259.34 to 0 °C.) The text of the scale, given in Appendix A, states the deviation equations for use below the oxygen point as functions of T (in kelvins). This form of these equations will also be used in this monograph because

of their greater simpHcity. It follows that eq (6.13)

becomes

WiT)=W*{T)+ ^W{T), (6.13a)

and the reference function (6.14) becomes

20 r=273.15 + 2 r*(7'))'. (6.14a)

Procedures for obtaining the calibration constants

for the polynomial equations will be described later

in this section.

The calculation of W*{t) must be carried out

with a large number of digits. At NBS approximately 17 decimal digits are used. A table of

W*{t) at closely spaced values of t is given 'n Appendix E together with a short Fortran subroutine which will produce this table efficiently

(no more than two iterations per entry).

The following procedure may be employed to convert the measured values of W{t) to t. From

eq (6.13) is obtained

W*{t)=W(t)-^W{t). (6.13b)

When eqs (6.14) and (6.13b) are combined,

20 t='^Ai\Ii\_W{t)-^W{t)}^. (6.15)

i=l

To solve for t, an approximate value of t is first obtained by neglecting AJF(t) and evaluating eq

(6.15) v«th the measured W{t) only. (A closely

spaced table of W*{t) could be employed for this

evaluation.) The approximate value of t is used with

the appropriate polynomial equation to obtain an

approximate value of ^W{t) which is then employed

with W{t) in eq (6.15) (or table of W*{t)) to obtain t. The process is repeated until the value oit no longer

changes more than the desired limits.

Considering the complexity of the reference

function (eq (6.14)), a high speed electronic computer or a table compiled by such a computer at small enough temperature intervals (0.1 to 1 °C,

depending upon the value of t) to permit linear

interpolation is a necessity.

Although a table of W*{t) could be employed to obtain t from the measured value of W{t) in accord- ance with the procedure outlined above, a direct

interpolation in a table of t versus W{t) (measured)

is obviously more convenient. Such a table can be

generated from the relation given by eq (6.13).

Values of AJF(f) can be calculated at appropriate

values of t, using the polynomial equations obtained

from cahbration, and combined with the corresponding values of W*{t) to obtain W{t). (For tables furnished by NBS, see Appendix J.)

6.2.1. -182.962 °C (90.188 K) to 0 "C (273.15 K)

The polynomial deviation function specified by

the text of the IPTS-68 between -182.962 °C

(90.188 K) and 0 °C (273.15 K) is:

M^{t) = W{t)-W*{t)=A^t + C4Ht-m). (6.16)

The constants 4 and C4 in eq (6.16) are determined

by calibration measurements at the steam point,

the triple point of water, and the boiling point of

oxygen, or alternatively, at the zinc point, the tin

point, the triple point of water, and the boiling point

of oxygen. Calibration measurements at the oxygen

point and again at the TP are added to the sequence

of measurements described earher (sec. 6.1) for the

range 0 to 630.74 °C. The value of W{t) at the oxygen

point Js obtained from /{(oxygen) and /?(0). The

value of a (see eq 6.3)) is required from the meas- urements above 0 °C in order to evaluate the

constant of eq (6.16). From the definition of a,

a* = (r*(100)-l)/100

a = (r(100)-l)/100;

(6.17)

(6.18)

and

hence,

Ar(lOO) = r(lOO) - r*(100)= 100(a - a*). (6. 19)

(The a*(= 0.0039259668 °C-») is the value used in

forming the W\t) table [51].) Substituting eq (6.19)

in eq (6.16) with t=100 °C, there is obtained

Ai = a-a* = Ar(100)/100. (6.20)

The value of C4 is determined by measurement of

W{t) at the boiling point of oxygen. The deviation

^W{t)=W(t) — W*{t) at the oxygen point and the

value oi Ai from (6.19) gives the constant C4

AW{t)-A,t

f3(t-100)

(6.21)

where t is the oxygen normal boiling point temperature in °C.

The first and second derivative of the IPTS—68

reference function (eq(6.14) ) have the same value at 0°C as those of the function iV{t) obtained from eqs

(6.1), (6.2), and (6.3) with the thermometer constants a = 3.9259668 X lO-^ °C-i and 8 = 1.496334

°C. On the other hand, these derivatives of the temperature-resistance function for real SPRT's are not

in general continuous through 0 °C. The discontinuities are, however, very small and may be neglected. See Appendix F for detailed discussion of this

point.

25
6.2.2. 13.81 to 90.188 K

From 13.81 to 90.188 K, the SPRT's that are received at the NBS are not cahbrated at the defining fixed points established by phase equilibrium in accordance with the text of the IPTS-68,

but they are cahbrated by a procedure that is equivalent to the IPTS-68. The SPRT's are

CcJibrated at NBS by intercomparison with SPRT

standards that maintain the NBS-IPTS-68

(see sec. 7 for details). The temperatures of the

intercomparison cahbration measurements have

been selected so that the measurements, which are made at temperatures very close to those of the

defining fixed points, are supplemented by measure- ments suitably placed between these points. Usually sixteen measurements are made between

12 and 90 K. The data are analyzed by a method of

least squares and values of resistances corresponding to the defining fixed-point temperatures are evaluated. The values of W{T) are then computed

for the defining fixed-point temperatures: 13.81,

17.042, 20.28, 27.102, 54.361, and 90.188 K, and

apphed to eq (6.13) or (6.13a) to obtain the deviation

AJF(r) at each of these temperatures. The discussion to follow describes procedures for obtaining the

constants of the specified polynomial deviation

functions.

Below the oxygen point the deviation, LW{T),

of the value of W(T) from the- reference function,

W*(r), is defined with a succession of polynomial

equations, each covering a relatively short range.

Each of these deviation equations is described by

a three or four term power series in T (where T is the value of temperature on the International Practical Kelvin Scale: r=«-f 273.15 K) and is a smooth

downward extension of the deviation curve found

for the temperature range immediately above.

6.2.2.1. 54.361 to 90.188 K

In the range from the oxygen normal boihng

point (90.188 K) down to the oxygen triple point

(54.361 K) the deviation function is

AWiT)=A3 + B:iT+C3T^. (6.22)

The three constants A3, B3, and C3 are determined

from the values of AW{T) at the two oxygen

fixed points and the value of the first derivative of

the deviation equation (6.16) specified for above the

oxygen point.

(dAWit)

\ dt

--Ai-300C4fi + iCit\ (6.23)

where the t in eq (6.23) is the value of the oxygen

normal boihng point temperature (—182.962 °C).

At the oxygen point the first derivative of the deviation function (eq (6.22)) extending below the oxygen

boihng point is set equal to the first derivative of the

deviation function (see eqs (6.16) and (6.23)) above

the oxygen normal boihng point; i.e.,

dT dt 02? (6.24)

at f = -182.962 °C and 90.188 K. Thus, the

deviation functions and their first derivatives are forced to be equal at the point of joining.

6.2.2.2. 20.28 to 54.361 K

The deviation function defined by eq (6.22) is joined smoothly at the oxygen triple point

(r= 54.361 K) in a completely analogous manner

as described in section 6.2.2.1, with the deviation

function between 20.28 and 54.361 K given by

AW{T)=A2 + B2T+C2T^ + D2TK (6.25)

The constants are determined from the values of

AW{T) at the normal boiling points of equihbrium

hydrogen (20.28 K) and neon (27.102 K) and the

triple point of oxygen (54.361 K) and by equating

its derivative and the derivative of eq (6.22) at the

point of joining (54.361 K).

6.2.2.3. 13.81 to 20.28 K

A fourth deviation function

AW(T)^Ai+BiT+Cir- + DiT^ (6.26)

covers the range 13.81 to 20.28 K. The constants are determined from the values of AW{T) at the triple

point (13.81 K) and the normal boihng point (20.28 K)

of equihbrium hydrogen and at the temperature of

17.042 K (boiling point of equilibrium hydrogen at 25/76 atm) and by equating the derivative of eq (6.26)

and the derivative of eq (6.25) at the point of joining

(20.28 K).

6.2.3. Calibration Tables Between 13.81 and 273.15 K

(0 °C)

At NBS, the coefficients of the deviation functions obtained through the procedure outhned above

allow the values of AW{T) to be calculated over the

temperature range specified for each of the deviation functions. Values of W(T) for the thermometer

being cahbrated are then given by

W(r) = W*{T) + AW{T). (6.13a)

Values are calculated to provide tables of WiX) or R{T) versus T at equal temperature intervals

sufficiently small to permit hnear interpolation. In

the region 13 to 90 K the tabulations are usually

given at 0.1 K intervals and from 90 to 273 K at 1 K

intervals.

6.3. Errors of Temperature Determinations

The usefulness of a measured value of temperature depends strongly upon both the amount of its

26
uncertainty and the knowledge of the amount of its uncertainty. Reduction of the errors contributing to the uncertainty is primarily limited by the ingenuity

of the experimentalist; there is, however, an in- herent source of uncertainty or spread of temperature values on any practical scale which is predicated upon the use of real materials, even if "perfectly" calibrated thermometers are used and

no error is introduced in the measurements.

Unfortunately only scattered data are available from

which one could infer the degree of this ambiguity

or possible spread of values of the defined IPTS-68.

McLaren [33] did not find a distinctive difference

among the measured temperature values at 321

and 231 °C of eleven thermometers with a ranging

from 0.003921 °C-i to 0.003926 °C-K (Five of the

eleven thermometers met the requirements of

IPTS-68, i.e., a equal to or greater than 0.0039250

°C"i.) The sensitivity of McLaren's test was about

± 0.5 mK; his results indicate a spread of temperature values which are an order of magnitude smaller

than the spread reported earlier for seven thermometers of much less pure platinum (a from

0.003909 °C-' to 0.003925 "C"') by Hoge and

Brickwedde [29]. These latter thermometers,

while acceptable on the ITS-27 which was in use at the time of their work, were not made of sufficiently

pure platinum to meet the requirement of the

IPTS-68 that a equal or exceed 0.0039250 °C-K

The thermometers that were investigated by Hoge

and Brickwedde showed a spread of 7 mK between — 190 and 0 °C, and spread of 1.3 mK between 0 and

100 °C. The six thermometers measured between

100 and 444 °C had a maximum spread of 12 mK.

Data which would indicate the spread of interpolated

values of temperature on the IPTS-68 below 0 °C

due to the variations in presently acceptable

platinum wire is sparse and inadequate [11, 36, 44].

On the basis of existing data, Bedford and Ma [9]

estimated that the IPTS-68 is reproducible to ± 3 mK between 14 and 20 K, ± 1 mK between 20

and 54 K, ± 2 mK between 54 and 90 K, and ± 5 mK

between 90 and 273 K. It should be pointed out that,

in addition to unaccounted for variations in the

platinum wire, a spread in the temperature values

may also arise from incomplete definitions of the

materials that define fixed points (e.g., their isotopic

composition)

.

The remainder of this section wiU deal with errors

that arise both from errors of measurement by the user and from errors of calibration of the

thermometer.

Many of the same kinds of error sources plague

both the calibrator and the user. The errors may

be divided into two categories; first, a temperature

error, i.e., a difference between the actual thermometer temperature and the temperature of the point

of interest or reference temperature, and second, a resistance error, i.e., a difference between the

measured resistance and the resistance of the

sensor. For example, temperature error occurs

because of an unknown difference between the

temperature of the thermometer and the equilibrium

temperature reaUzed in a fixed-point apparatus used

for calibration. (The calibrator is also plagued, of

course, by any unknown difference between the

equihbrium temperature reaUzed in his fixed-point

apparatus and the equihbrium temperature defined

by IPTS-68.) Temperature error also occurs be- cause of unknown temperature differences between

the thermometer and the object of interest in the

user's apparatus. The experimenter must decide

where to place the thermometer to minimize the

magnitude of this error. Several possible sources

of temperature error are discussed in section 4. In considering the errors of thermometer resistance

measurements one must not only examine the re- producibihty and calibration of his measurement

equipment but also his measurement technique.

Sections 5, 7, and 4 indicate possible sources of

resistance or resistance ratio error. The following analysis describes the error in the

calculated value of temperature that results from

errors made in calibrating the SPRT at the fixedpoints. Any additional imprecision and inaccuracy

introduced by the user are assumed to be best

known by him.

At any value of temperature, t, the total differential oit for the SPRT is,

*=2(jt)*+S(^)'«^'- (6-27)

where ti is the temperature attributed to the calibration fixed point and Wi is the value of the resist- ance ratio, R(ti)IR{0), attributed to that temperature.

The analysis may be simphfied by converting any

error in the value of temperature attributed to the

calibrating fixed point to an equivalent error in W

at that fixed-point; then, dti of eq (6.27) vanishes

and eq (6.27) reduces to

where dWi now includes the contribution from

errors in the values of temperature as well as the

errors in the values of R{t)IR{0) at the calibration

points. By utilizing eq (6.28) the error in the value

of temperature t on the IPTS-68 caused by an error

in Wi corresponding to a positive unit error in f, was

evaluated as a function of temperature. Figure 17

shows the errors in the values of temperature that

would result from the calibration errors at each of

the fixed points. For each curve the calibration error

is taken to have occurred at only one fixed point with no cahbration error at the other fixed points.

The measurement at the triple-point of water

should be a part of platinum resistance thermometry

work and W{t), i.e., R{t)IRiO), should be used in

calculating temperatures (see sec. 6.1). The value

27
100 200 300 400

TEMPERATURE, DEGREES C (IPTS-68)

Figure 1 7. The error at various temperatures propagatedfrom errors made in the calibration ofa platinum resistance thermometer.

The curves show the error (i.e. , departure from the IPTS-68) in the values of temperatures caused by a unit positive (hotter) error in realizing the temperature (hotness) of one given

calibration point. The calibration at the triple point of water is assumed to have been made without error. The fixed point at which the error was made is indicated on the curve. The error curve depends not only upon the particular fixed point at which the error occurred, but also upon which other fixed points above 0 *C were employed in the calibration. The calibration at the other fixed point above 0 "C, indicated in parenthesis, is assumed to have been performed without error. A calibration error at the oxygen normal boiling point does not introduce an error in the measured values of temperature above 0 °C. The first derivatives of the ZINC(TIN) curve and the TIN(ZINC) curve are not continuous through 0°C.The curve marked Ro shows the error that would be introduced if the experimenter makes a unit positive error in realizing the temperature (hotness) of 0 "C and then calculates the value of a temperature from the value of R{t)lR{0 °C).

of W{t) is particularly sensitive to the errors of

i?(0) in applications at high temperatures where

any error of R{0) becomes amplified Figure 17

shows the error curve resulting from an error in

R{0) corresponding to a positive unit error in

temperature.

The estimated uncertainties of the calibration

measurements obtained at NBS are 0.002 or 0.003 K

at the oxygen normal boiling point, 0.0002 K at the

triple point of water, 0.001 K at the tin point, and

0.001 K at the zinc point. These uncertainties are estimates; hopefully, in the future, all of the desired documentation of these estimates will become

available. The experimenter could estimate his

overall error by statistically summing his measure- ment error with the calibration errors that can be

obtained from the curves on figure 17 and the above

estimate of calibration uncertainties. (For further

details see Appendix G.)

7. Calibration

The cahbration of a SPRT on the IPTS-68 in- volves the measurement of the resistance when the

thermometer is at the temperature of the prescribed

defining fixed points. This section deals with the

equipment, the preparation of fixed-point cells, and

the procedures employed at NBS to realize the

temperatures of the prescribed defining fixed points

for calibrating SPRT's.

In practice, thermometer calibrations are not

made at exactly the temperatures of prescribed

equilibrium conditions for the fixed points. For

small known departures from the prescribed conditions a temperature correction can satisfactorily be

made, e.g., such a departure may exist because of

the hydrostatic head at the level of the thermometer

resistor in calibrations that employ triple-point or freezing-point cells. (The effect of pressure devia28
8.3 cm

51 cm

N

Figure 18. Holder for capsule-type platinum resistance ther- mometers.

Calibration measurements are performed in the holder at the triple point of water,

tin point, and the normal boiling point of oxygen.

A. Elastomer tubing to helium gas source.

B. Thin (0.005") wall stainless steel tubing for purging the holder with helium gas before the vacuum tubing connector (/) is sealed.

C. Leads to measurement equipment (Mueller bridge). D. Sections of mechanical tie-down (brass), soldered to the stainless purge tube, for guiding and fastening the incoming thermometer leads. E. Hard wax for holding and sealing the 0.005" gold leads that extend down to the ther- mometer.

tions is given in the text of IPTS-68, Appendix A.)

Gross departures from the prescribed conditions

usually require extensive efforts to establish that

the adjusted temperature and resulting calibration are sufficiently close to the IPTS-68.

The calibration apparatus at the NBS has been

designed to be used with the great majority of

SPRT's. Capsule type SPRT's, however, are mounted in special stainless-steel holders (fig. 18)

before they are calibrated in fixed-point equipment

with deep thermometer wells.

7.1. Triple Point of Water

The triple point of water (0.01 °C) is the most

useful and important of the defining fixed points for

calibrating SPRT's. The virtues of regular thermometer measurements at the triple point of water are

so great that all but the most casual measurements

of temperature with a SPRT should include a refer- ence measurement at the triple point. The triple

point is realized in a sealed glass cell (fig. 19)

containing ice, water, and water vapor. When the

cell is in use it may be placed in an ordinary crushed

ice-water bath. Figure 19 shows the cell and icebath system used at NBS.

The cell is first immersed in the ice bath with the

mouth of the re-entrant thermometer-well above the

surrounding ice-water level. The well is thoroughly

dried, then fiUed with crushed Dry Ice^ and maintained full for about 20 min by replacing the sublimated Dry Ice. The initial freezing of water within

the triple-point cell wiU occur several degrees below

the triple-point temperature because water easily

supercools. When the water does start to freeze

fine needles of ice crystals (dendrites) are initially

formed and protrude from the wall of the well into

the liquid. The fine needles quickly cover the well

but soon disappear to form a clear coating of ice

(on the well) that will grow and become a 4 to 8 mm

thick mantle in about 20 min. It is important to keep the well completely fuU of Dry Ice during this

period. (If the Dry Ice level in the well is allowed to drop several inches and then the well is refilled, the

ice mantle is very apt to crack. The desired triplepoint temperature may not be achieved if a crack

2 Certain commercial materials are identified in this paper in order to adequately

specify the experimental procedure. Such identification does not imply recommendation or endorsement by the National Bureau of Standards.

F. "O" ring vacuum seal to "brass tube (W).

G. Thin wall stainless steel tube (7/16" o.d.) closed at the bottom.

H. Brass tube with lead seal at top and vacuum tubing connector (/) at the bottom.

I. Vacuum tubing connector for sealing the tube (G).

J. Polytetrafluoroethylene plastic lead spacers.

K. Insulated gold leads (4) passing through holes close to the outer diameter of the spacers (J) to attain good tempering. The lead insulation (not shown) is polytetra- fluoroethylene tubing cut into a helix and held in tension to eliminate buckling. The

four gold leads are welded at the bottom end to short sections of platinum leads.

L. Thin polytetraHuoroethylene sheet rolled into a cylinder to insulate the exposed

leads (near the connections to the capsule thermometer) against the stainless steel tubing. Connections between the short sections of platinum leads of the holder and the platinum leads of the capsule thermometer.

N. Capsule thermometer.

O. Aluminum sleeve to fit the thermometer and the stainless steel tube. The sleeve reduces the external heating elfert of the thermometer.

M

29
in the ice mantle extends from the well surface into

the surrounding liquid water.) In the process of

introducing Dry Ice into the well some of the Dry

Ice may be deposited around the top of the cell,

causing the water within the cell to freeze sohdly

across the top. The ice at the top of the cell should

be melted immediately to avoid the possible breaking of the cell glass. Whenever ice is frozen solidly across the top surface of the ceU water and a strong

bond is formed between the thermometer well and

the outer cell wall, any subsequent freezing of water

below the surface ice can result in sufficient pres- sure to rupture the cell. The surface ice can be

melted by raising the cell slightly and warming the

top of the cell with the hands briefly while gently

shaking the top of the cell sideways to "wash" the

region with the cell water which facilitates the

melting of the layer of ice. After 20 min no additional

Dry Ice should be added and the remaining Dry Ice

in the well should be allowed to subhme completely.

Finally, when the Dry Ice in the well is completely

gone, the cell is lowered deeper into the ice bath

and the well allowed to fill with water. If the cell is raised high enough to see the mantle during the

freezing process the magnification of the cyhnder of

water will give the impression that the cell is or is about to be frozen sohd with ice although the coating

of ice on the well may still be as little as 1 or 2 mm

thick. If the cell is inverted, the true thickness of

the ice may be seen. (The cell should not be inverted

after the "inner-melt," described in the next paragraph, has been made.) An immersion type cooler

may be used instead of Dry Ice for freezing the ice

mantle [25]. However, care must be taken to avoid

admitting the auxiliary heat-transfer Mquid, e.g., alcohol, into the surrounding ice bath.

A second ice-water interface is formed by melting the ice immediately adjacent to the well surface.

This is referred to as the "inner melt." The inner

melt is made by inserting a glass tube at ambient

temperature into the well for a few seconds. A test for the existence of the ice-water interface over the

entire interior surface of the mantle is to give the

cell a small rotational impulse and determine

whether or not the ice mantle rotates freely around

the axis of the thermometer well.

Because of some evidence [12] that the temperature of the triple-point cell is sometimes shghtly

low (the order of 2 X 10-^ °C) immediately after

freezing, the cell should be prepared at least one day prior to its use. The reason for this low initial

temperature and the subsequent gradual increase

during one or two days to a steady value is not

clearly established but is believed to be connected

with structural strains that are produced when the

ice is first frozen; presumably the strains are

relieved with time as the ice anneals. The magnitude of the lower initial temperature and the rate

of increase to a steady temperature value is dependent upon the specific technique that is employed in freezing the cell.

Figure 19 shows a triple-point cell immersed in an ice bath. In using the cell, a small soft plastic

sponge if) is first placed at the bottom of the well

to reduce the mechanical shock that the thermometer might otherwise experience when it is lowered.

Also, a closely fitting aluminum bushing (/) about

5 cm long is placed above the sponge (at the bottom

of the well) to reduce the external self heating of

Figure 19. Water triple point cell.

A. Heavy black felt shield against ambient radiation. B. Polyethylene tube for guiding the SPRT into the thermometer well. C. Water vapor. D. Borosilicate glass cell. E. Water from ice bath.

F. Thermometer well {precision bore). G. Ice mantle. H. Air-free water.

I. Aluminum bushing with internal taper at upper end to guide the SPRT into the

close-fitting inner bore.

J. Polyurethane sponge.

K. Finely divided ice and water.

30
the thermometer. In a half-inch well, filled only

with water for thermal contact between the SPRT

and the well, the external self heating of a typical

SPRT (7.5 mm O.D.) is about 0.2 mK/{mA)2.

The bushing reduces this heating by a factor of

five or more depending upon its fit with the well

and the SPRT. (See sec. 4 for the discussion on

the self heating of SPRT.) To eliminate the ambient

room radiation (from ceiling lights in particular) a heavy black felt cloth (A) covers the top of the cell

except for a hole through which the thermometer

may be inserted. The thermometer is precooled in

the ice bath that surrounds the cell before it is in- serted into the cell. A polyethylene plastic tube (B)

from the hole in the felt cloth to the re-entrant well

provides a guide for inserting the SPRT. Before

measurements are made a minimum of five minutes

is allowed to elapse (with the thermometer current on and the bridge nearly balanced) for the thermometer to attain thermal equilibrium. If an ice particle

is present in the weU near the SPRT resistor, an

error wiU occur in the calibration. The water in

the well must be free of ice before the SPRT is inserted. Routine measurements made in cahbrating thermometers in triple-point cells have an

estimated standard deviation of less than 0.14 mK.

Very careful work using two currents and extrapolating to the resistance value for zero current has

yielded an estimated standard deviation of less

than 0.04 mK.

7.2. Metal Freezing Points

Freezing points are advantageous because the

effect of a change in pressure on the temperature,

the value of dtjdp, is much smaller than that of

boiling points. In a metal freezing-point cell, the

temperature at the solid-liquid interface depends

upon the concentration and kind of impurities; also,

strains in the solid and grain size effect the temperature. The concentrations of impurities (solute)

existing at the interface depend upon the amount

and kinds of impurities in the sample, the amount

of sample frozen, and the rate of freezing. To

achieve a true temperature and phase equilibrium

the net rate of freezing (or melting) must approach

zero. This equihbrium condition may be illustrated

by the binary composition versus temperature

phase diagram shown in figure 20 in which the two

constituents are completely miscible in both the

liquid and solid phases, e.g., the system Ag-Au or Bi-Sb. The phase diagram of figure 20 is presented

for its simplicity and, since metal samples employed

in freezing-point cells are prepared by the zone

refining process, the remaining impurities (combined as a single component in the illustration) are expected to form solid solution with the major

component. (Most binary meted systems on which

data exist do not form a continuous series of solid

solutions; various degrees of immiscibility are usually found.) The figure shows the region of the

A 100 90 80 TO 60 50 40 30 20 10 0

0 10 20 30 40 50 60 70 80 90 100 B MOLE PERCENT

Figure 20 Binary phase diagram of a system that is completely miscible in both solid and liquid

phases.

The solidus and liquidus curves represent compositions, respectively, of the soUd and Uquid phases thai can coexist in equiUbrium. The

left side of the diagram shows that the solute concentration in the solid phase is less than that in the liquid phase, i.e., A < 1. The right side of the diagram, on the other hand, shows that the solute concentration in the solid phase is greater than that in the liquid phase, i.e., k > \.

31
composition and temperature where the solid and

liquid phases can coexist. The solidus curve indicates the temperature at which a solid solution of

any given composition would begin to melt; the

liquidus curve indicates the temperature at which

a liquid solution of any given composition would

begin to freeze (assuming no supercooling). Alternatively, when both solid and liquid phases are present

in equilibrium, the composition of each phase is given by the intersections of the temperature line

with the soUdus and liquidus curves, respectively.

An enlarged section of this phase diagram, showing

the effect of a small amount of B in nearly pure A,

is shown in figure 21a. If a completely melted sample

of composition Lq is allowed to cool under equilibrium conditions, no change in composition of

the liquid occurs until the temperature reaches the

Hquidus curve (Li at temperature Ti) at which

point solid (assuming no supercooUng) of the composition Si, given by the solidus curve at the same

temperature, is formed. In the case shown here, the

first solid contains a smaller proportion of B than

the liquid from which it was formed; as more solid

is formed the concentration of B in the liquid in- creases. When freezing under equilibrium conditions the composition of the soUd phase moves

from Si to Sn', the composition of the liquid phase

moves from Z-i to Ln. The last of the liquid solidifies

at Tn. Obviously the amounts of A and B in the

completely soHdified sample must be the same as

in the original liquid. Sn must, therefore, equal

Lo and Li and no segregation occurs. The temperature range of the "freeze" is from Ti to T„. Although in most systems the solutes lower the

freezing point, there are systems in which solutes

raise the freezing point; antimony in tin is an ex- ample of the latter [30]. Such a system is illustrated

by the extreme right portion of the equilibrium

phase diagram where B is the major constituent as

in figure 20. An enlarged section is shown in figure

21b. The analysis of the phase diagram is similar

to that given previously. The most notable differ- ence is that, as the diagram shows, the concentration of the solute is greater in the sohd than in

the liquid. The ratio (5/L) of solute concentration in the solid (S) to that in the liquid (L) is called the solute distribution coefficient (k) and,

as given in figures 20 and 21 is less than unity for

solutes that depress the freezing point and is greater than unity for solutes that elevate the

freezing point. Except for very dilute solutions,

depicted in figure 21, the solute distribution coefficient depends on concentration.

Phase diagrams are idealized conceptions of

systems at equihbrium (often based on relatively

httle data). For equilibrium freezing the crystallization process must proceed at such a negligible

rate that there is sufficient time for the diffusion of

impurities within the sohd matrix to achieve uniform distribution throughout the solid, i.e., there is

Lo

SOLUTE CONCENTRATION < SOLUTE CONCENTRATION

Figure 21 . Enlargement ofa binary phase diagram in the region ofhigh purity ofeach component.

The solute distribution coefficient k is taicen to be constant. Figure la) represents a phase diagram when k < I and figure (b) when k> 1. The heavy lines on the liquidus and solidus curves represent compositions of the liquid and solid phase during the equilibrium freezing process. The dashed line beginning at Si in (a) or Sz in (b) and extending downward represents the average composition of the solid phase during a semiequilibrium freezing process. The composition of the solid at the

freezing interlace is represented by the solidus curve and that ot the liquid phase m equilibnum with the Ireezmg interlace is represented by the hquidus curve. Cl he solute distri- bution coefficient may not be constant over the entire range of fraction frozen for semiequilibrium freeze; therefore, eq (7.1) may not be valid when^ approaches unity.)

32
no concentration gradient in the solid. Experimentally this rate of freezing is never reaUzed; however,

freezing rates can be achieved which, while large

compared to the diffusion rates in the solid matrix,

are small compared to the diffusion rate in the liquid.

This condition leads to a maximum segregation in

the solid but homogeneity in the liquid; this condition will be referred to in this monograph as semiequilibrium freezing. The results of semiequiUbrium

freezing are shown in the phase diagram of figure

21 and on the coohng curve of figure 22. The phase

diagram still represents the compositions at the

solid-liquid interface but the sohdus curve no longer

is the average composition of the solid phase; the

average composition of the solid is given by the

dotted hne beginning at Si in figure 21a. Compared

to an equiUbrium freeze, the freezing temperature

range for the entire sample is increased due to the

increased concentration of impurities at the interface near the end of the freeze. (This assumes no

eutectic is formed in the equiUbrium freeze.) For

the case of semiequilibrium freezing Pfann [40] ^ gives an expression equivalent to

SILo = k{l-g)'^-\ (7.1)

i_ or~—I

1 1 1 r

uj COMPUTED FREEZING CURVE FOR k =0.4 p -6 - EQUILIBRIUM FREEZE f SEMIEQUILIBRIUM FREEZE

0 .1 .2 .3 4 .5 .6 .7 .8 .9 1.0 FRACTION FROZEN

Figure 22. Comparison of computedfreezing curves for equilibrium and semiequilibrium freezing processes of a completely

miscible (in solid and liquid phases) binary system with

constant solute distribution coefficient {k=0.4).

The solid line represents the equilihrium freezing process and the dashed line the semi- equihbrium freezing process. In the equilibrium freezing process, the temperature, when the last trace of liquid freezes, is shown to be depressed one unit for the "sample";

the temperature, when the first sohd freezes, is shown to be depressed OA units. The

freezing curve for the semiequilibrium freezing process is shown relative to that of the equilibrium freezing process. (I'he solute distribution coefhcient may not be constant over the entire range of fraction frozen for semiequilibrium freeze; therefore, equation

(7.1) may not be valid wheng approaches unity.)

where Lo is the overall solute concentration and S

is the solute concentration of the freezing interface

after the fraction g of the original mass of liquid

has frozen. (The equation is not applicable for the

entire range of g. The derivation assumes the distribution coefficient k to be constant and the solute

diffusion rate in the solid to be zero.) Figure 23

gives curves of relative solute composition of the

freezing interface resulting from this expression

for various values of k. Figure 22 compares the curve for k=OA, replotted on a hnear scale, with

the curve that would be obtained for a corresponding equihbrium freeze. The semiequilibrium freeze

is shown to have a broader freezing range.

Sufficiently rapid freezing causes a departure

from the semiequilibrium freeze. As the velocity of

the advancing solid-Uquid interface increases, the

rate of solute rejection into the liquid at the interface increases. When the impurity can not be

uniformly distributed throughout the liquid by

either diffusion or convective mixing, the concentration of the solute builds up at the interface

as shown in figure 24b. When this occurs, the

effective segregation of the solute decreases (i.e.,

the effective value of k approaches 1). Because of

the increase in the solute concentrations at the

interface, the temperature of the interface becomes

lower. Figure 24c shows qualitatively the result of a freeze which is very rapid compared to the

impurity diffusion rate in the liquid (and for which

there is no other method of homogenization, e.g., stirring).

A notable example of nonequilibrium freezing in

metal freezing-point cells is the rapid freezing that

^ Pfunn [40] refers to semiequiUbrium freezing as normal freezing.

o

o

o

o

= 1 1 1 1 1 1

1 1 1 1 =

; -

; -

//I -

//I -

^ '/ -

k=2 ^

-^^i^n -

^ A\ ~

- I< = 0-'' M — ^ X ^ 'C^V// =

k = O.I

;><r // I I - v-x 'A -

I k = 0.04

k = O.OI

\ \ — \ -—— --"^

\ -

- \ 1 \

I k =0.004

\ _

\ =

\

\ SOLUTE CONCENTRATION IN SOLID

^

AT THE FREEZING INTFRFACE

\ \ \ \ DURING SEMIEQUILIBRIUM FREEZE \ V \ \ \

>

1 \ II 1

1

DURING EQUILIBRIUM FREEZE

1 1 1 1 1 1

.2 .3 4 .5 .6 .7 .8 .9 1.0 FRACTION FROZEN

Figure 23. Comparison of relative solute concentrations of the solid phase at the freezing interface during equilibrium and

semiequilibrium freezing processes of a completely miscible

(in solid and liquid) binary system with constant solute distri- bution coefficient k.

Curves are given for various values of solute distribution coefficient. The solid curve

represents an equiUbrium freezing process and the dashed curve represents a semi- equiUbrium freezing process. (The solute distribution coefficient may not be constant over the entire range of fraction frozen for semiequiUbrium freeze; therefore, eq (7.1) may not be vaUd when^: approaches unity.)

occurs after a supercool. When the completely

melted sample is cooled through the temperature

at which the phase diagram indicates that the

33
CO

o

(0)

1 1 1 1 1

EQUILIBRIUM FREEZE _

- ^ L 3

-

EFFECTIVE k

IS THE SAME

L

LIQUID

"

SOLID

^POSITION OF INTERFACE "

1 1 1 1 1 (b)

1

1 1 1 1 1

NONEQUILIBRIUM FREEZE

- l' \

' \ EFFECTIVE k

L

SOLID LIQUID

1

4—POSITION OF INTERFACE ~

1 1 1 1

0 12 3 4

DISTANCE FROM INTERFACE (ARBITRARY UNITS)

-2-10 1 2 3 4

DISTANCE FROM INTERFACE (ARBITRARY UNITS)

T

VERY RAPID FREEZE

IN THIS REGION THE EFFECTIVE k

APPROACHES UNITY

_L _L _L _L

FRACTION FROZEN

Figure 24. Dependence ofsolute concentration upon thefreezing rate.

(a) The concentration of solute in the region near the freezing interface during an equilibrium freeze with the solid-liquid interface advancing at a negligible rate. There are no concentration gradients in neither hquid or solid phases.

(b) The solute concentration near the interface during a nonequilibrium freeze with the interface advancing rapidly relative to the rate

of diffusion of the solute in the liquid. The solute concentration in the liquid at the interface increases until at steady state the flow of solute from the freezing hquid to the hquid at the interface equals the flow away from the liquid at the interface to the bulk of the liquid. The

solute concentration of the frozen solid is shown to be increasing with the advancing interiface.

(c) The solute distribution in the frozen solid approached by a very rapid freeze. The curve shows the initial transient rise of solute concentration of the solid to that corresponding to the solute distribution coefficient of unity.

solid should first appear (i.e., crosses the hquidus

curve) no solid appears. The first solid appears at a somewhat lower temperature (typically 0.02 or 0.06 °C lower in zinc, 1 to 25 °C lower in tin, water,

or antimony). The solid tends to grow most rapidly

into the cooler parts of the liquid until the released

heat of fusion raises the temperature of the liquid

to the equilibrium temperature of the composition

at the solid-liquid interface. The temperature to which a liquid supercools is not very reproducible even with the same sample. The amount of supercooling seems to depend on the purity of the sample,

the thermal history of the melt, the occurrence of

mechanical shock or vibration, and other effects

not known or understood. The system may be far

from equilibrium during the recovery from the

supercool; the solid-liquid interface advances very

rapidly and there is very little time for diffusion or

convection to homogenize the liquid. As a result,

the impurities rejected from the freezing liquid for

the case where A: < 1 become relatively concentrated

in the hquid in the region of the interface. When

the recovery from the supercool is nominally

complete, the high concentration of impurities in

the liquid at the interface will be reduced by diffusion and the temperature will rise until the flow of

impurities from the freezing liquid to the hquid at the interface is equal to the flow away from the

interface through the remaining "bulk" liquid.

Rapid freezing impHes a rapid transfer of heat.

Many authors [42, 45] have suggested that it may

be necessary for the temperature at the solidliquid interface to be significantly below the

equilibrium temperature if freezing is to take

place rapidly; i.e., the net rate of freezing is zero

at the equilibrium temperature and increases only

34
with decreasing solid-liquid interface temperatures.

The metal in the two phases at the interface thus

constitutes a "proportional" controller, supplying

heat on demand and having a temperature "offset"

from the equilibrium value if demand exists. This

picture is intuitively appealing and is in qualitative

agreement with the theory of reaction kinetics. But,

the crucial questions would seem to be what is the

rate of energy released per unit of temperature

departure from equilibrium and how is the measure- ment of this separated from other events briefly

outlined above that occur in nonequilibrium freezing? The difficulty of this measurement has prevented firm establishment of this model.

7.3. Metal Freezing Point Furnaces

The metal freezing-point cells are used in furnaces

of the design shown in figure 25. At NBS, two

furnaces are employed, one for the tin point and

the other for the zinc point. Except for the metal

sleeves employed at the top section of the furnace

core, the two furnaces are the same. The core of the

furnace is a stack of three cyhndrical coaxial blocks

of aluminum (top (G), center (L), and bottom (T))

that are thermally insulated from each other by

Fiberfrax paper and surrounded by a nichrome wire

heater (main heater, O) which extends the full length of the three blocks. The heater wire fits closely within each of the two holes of the 51 cm

lengths of two-hole oval alumina tubing. These

alumina tubings are very closely spaced around the

outer surface of the core (parallel to its axis) and

are held tightly against the outside of the blocks

with three Inconel "garters." (See the periphery of

the furnace core section drawings 1, 2, 3, 4, and 5.)

The garters were made of Inconel wire bent into a form resembling a continuing sine-wave about one

centimeter in amplitude and wavelength, then

rolled to slightly reduce the thickness, and welded

into a ring. The top and bottom blocks each have an

additional heater (F and U) consisting of nichrome

wire that passes through 98 mm lengths of two-hole,

round alumina tubings which were selected to fit closely in the 12 holes of each block. To minimize

the thermal time lags, the heater assemblies have

small clearances between the heaters and the

aluminum core blocks. (The leads extending through

the core to the heater of the top core block are gold.

The leads to the other heaters are heavy nichrome

wire.) Extending down through holes that run the

length of the three assembled blocks are six thinwall (0.13 mm) stainless steel tubes (B) nominally

3.2 mm in diameter. The tubes pass through 3.22

mm holes in the end blocks (see a and b of figure 25).

To permit the holes for these tubes to be accurately

positioned through the 30.5 cm long center block,

the center block was made in two cyhndrical pieces.

The end plate (Q) was attached later to the center

core block. Grooves that closely fit the stainless

steel tubes were milled on the outside of the inner

cyhnder which was then fitted tightly inside the

second outer cylinder. The two cylinders were

"shrunk together", a process that involved precooling the inner cylinder in liquid nitrogen and heating

the outer cylinder. The tubes are weUs into which

thermocouples are placed for controlling the

furnace temperature; they also serve as wells for

small exploratory resistance thermometers to determine the temperature distribution within the

furnace core. The entire core of the furnace,

including all three blocks and the heaters, slips into a stainless steel tube (A) 11.4 cm in diameter with

a 0.76 mm wall. Sheets of mica are wrapped around

the main heater near each end to center the core

within the tube, to enhance thermal contact between

the tube and the main heater near the ends, and to reduce undesirable heat convection currents. The

centering leaves a small annular air space between

the tube and the alumina insulators of the main

heater in the region of the center block; the thermal

contact of the end blocks with the surrounding

stainless tube serves as "thermal end guards" and

reduces the thermal gradients in the middle section

of the tube opposite the center block.

The weight of the core rests on a 0.51 mm wall

stainless steel tube (W) 11.4 cm long and 2.5 cm

in diameter. The space beneath the core is fiUed

with loose Fiberfrax insulation. The outside of the

furnace is a 35.6 cm diameter brass tube with end

plates and is filled as indicated in figure 25 with

J'iberfrax insulation. A rather widely spaced helical

coil of 9.5 mm copper tubing is soldered to the

outside brass tube (K) to permit water cooling; a comparable provision is made for the top end plate.

The water cooling is not used when the furnace is at the tin point but is helpful at the zinc point in

reducing convection currents around the furnace

that cause gradients in the head of the thermometer

which, in turn, may result in thermal emfs. The

thermal insulation is sufficient so that, even at the

zinc point with no water coohng, the outside of

the furnace is not "hot to the touch." To achieve

further thermal insulation, the furnace core is recessed from the top of the furnace as shown in

figure 25. A major path of heat loss from the top of

the core is along the 11.4 cm diameter stainless

steel tube that contains the core; to reduce the heat

loss the core was recessed and the tube was made

relatively longer.

The top core block (G) of the tin furnace was

designed to receive an aluminum sleeve (H)

10.2 cm long that is bored to fit as closely as possible

the glass cell that holds the tin sample. By using

removable metal sleeves a reasonable range of

glass cell diameters can be accommodated while

still achieving good thermal contact with the cell

above the crucible where the heat shunts are

located. The glass cell is of sufficiently uniform

diameter to achieve a suitably close fit. In the

design, allowance has been made for the differential

expansion of the aluminum and the glass cell. (The NBS glass cells are generally ground on the

35
a Q71 cm

Figure 25. Schematic

A. Stainless steel tube. 11.4cmo.d. X 0.76 mm wall.

B. Stainless steel tubes (6); 3.20 mm o.d. X 0.013 mm wall. C. Insulation, Fiberfrax mats.

D. Stainless steel tube.

E. Insulation, Fiberfrax mats and mica sheets.

F. Control heaters for top core block. G. Top core block. H. Sleeve (aluminum or Inconel) for metal freezing point cell.

I. Elevation of thermocouple junction (Chromel-P/Alumel), top core block. (See a.)

J. Insulation, Fiberfrax sheets. K. Brass shell, 4.8 mm thick.

L. Center core block. M. Insulation, bulk Fiberfrax. N. Elevation of thermocouple junction (Chromel-P/AIumel), center core block. (See a.) O. Main heaters, held on with Inconel "garters."

offurnace body and core. P. "Spider" for centering the freezing point cell. Q. Center core block end plate. R. Insulation, Fiberfrax sheets.

S. Elevation of thermocouple junction (Chromel-P/Alumel) bottom core block. (See a.) T. Bottom core block. U. Control heaters for bottom core block. V. Insulation, Fiberfrax mats and mica sheets. W, Stainless steel tube support, 2.5 cm o.d. X 0.51 mm wall x 11.4 cm long. X. Heater leads. Y. Posts for heater leads.

Z. To electric power.

a. Wells (B) for control thermocouples (see I, N, and S).

b. Wells (B) for testing the temperature profile of furnace core.

c. Leads from top core block heaters.

36
outside to a uniform diameter.) Because aluminum

sleeves bond to the aluminum top core block at the zinc point, spht inconel sleeves are employed

in the zinc point furnace. Also, because of the close

tolerances needed for good thermal contact, the

zinc-point cells are not permitted to cool to room

temperature in the furnace, otherwise the borosiUcate glass tube would be crushed owing to the

differential contraction between the aluminum top

end block and the glass.

The center core block (L) was bored to provide

approximately 1.0 mm clearance for the most

common (and largest) size of glass tubes, specifically

51 mm o.d.; this 1.0 mm annular space provides

the thermal insulation that reduces the heat transfer

to the crucible and metal sample during freezing

or melting experiments. The design also reduces

the need for close control or knowledge of the

furnace temperature in many operations.

Three Chromel-P/Alumel thermocouples (TC)

enclosed in alumina sheaths control the furnace

temperature. Two of the thermocouples (differential) are referenced to the temperature at the middle of

the center core block; the measuring junction of one is located approximately 1.2 cm up into the

top core block and that of the second is similarly

located down into the bottom core block. The third

TC is referenced outside the furnace and the measuring Junction is located in the middle of the center

block.

Each differential TC is connected in series with

a stable voltage source (powered by mercury cells)

which is adjustable between ±75 ptV (±1.5 °C).

The center block TC connects to a reference junction, which is self-compensating for changes in

room temperature, and to a voltage source that has

(i) a range of 15 mV, (ii) a stabihty of 0.01 percent,

and (iii) a reproducibility of setting of better than

2 juV. The combined output of each TC and voltage source is amphfied by a chopper type DC amplifier

whose output operates a "three-mode" controller.

The zero stability of the amphfier is better than

0.5 fxV. The output of each controller is an adjustable hnear (proportional) function of the input, its rate of change, and its time integral. The signal

from the controller operates a gate drive for a full- wave silicon-controlled rectifier which, in turn,

controls the power supplied to the heater. The

rectifiers do not cause detectable interference

with any other equipment (low level null detectors,

a-c and d-c bridges, etc.) in the room. Each aluminum block of the core is grounded by means of a heavy gold wire.

The control of the furnace was made very flexible

in that several combinations of manual and automatic controls may be selected for operating the

furnace, including the provision for independently

setting the temperature offset between the center

and top and between the center and the bottom

core blocks. A freeze is usually conducted with

each of the three heaters under the automatic

control of their corresponding TC's. A plot of

thermometer resistances (relative to that at the

center of the center block) as a function of depth

in the core of the furnace, while the furnace and a

tin-point cell was at 235 °C, is shown in figure 26.

These measurements were made with a small fourlead platinum resistance thermometer, 2.8 mm

diameter and 2 cm long, having a nominal ice-point

resistance of 50 O.

This furnace, which was designed and built at the NBS, is more sophisticated than necessary for

caUbrating thermometers at the tin and zinc fixed

points with reasonable (better than 0.(K)2 °C) accuracy. It was designed for special studies of the

freezing and melting phenomena of tin and zinc.

Tin and zinc freezes for calibrating SPRT's have

been performed very successfully in many laboratories with furnaces that contain a single copper

block and employed a manually controlled heater.

Such cahbrations have been performed in furnaces

that exhibit core temperature gradients of 1.5 °C

over the length of the crucible containing the sample

[53] and relatively long response times. Reafization

of the tin and zinc freezing points with reasonable

accuracy is principally dependent upon the high

purity of freezing-point samples and the use of

proper freezing and measurement techniques.

7.4. Tin-point Cell

The realization of either the tin-point or the

steam-point is necessary for the calibration of

SPRT's in accordance with the specifications set forth in the text of IPTS-68. The tin-point (231.9681

°C) has two distinctive advantages; first, the

temperature is much closer than the steam point

to the midpoint between the triple point of water

and the zinc point and, therefore, tends to produce

less average error in the calibration of the thermometer (see fig. 17), second, and of greater

practical importance, the solid-hquid equihbrium

temperature of tin is 8600 times less sensitive to pressure changes than the liquid-vapor equilibrium

temperature of water at 1 atm. For the freezing point of tin, rfi/rfp = + 4.3 X lO-^ °C/torr or + 2.2 X 10-5 °C/(cm column of liquid tin). Therefore,

knowledge of the pressure within ±1 torr is adequate

for determining the temperature of a tin-point cell. The tin sample for the freezing-point cell must be

of high purity (nominally > 99.999%) and the

freezing apparatus must be designed and operated

to interpose a solid-liquid interface of tin completely

around the resistance element and lower part of

the thermometer stem. The immersion of the

resistor must be sufficient to prevent sensible heat

flow from the resistor up the stem of the thermometer. In addition, the sample holder must be

chemically inert and not introduce any impurities

that would affect the freezing temperature of tin. At NBS the tin sample is contained in a closed

crucible of high-purity graphite. A re-entrant well.

37
0

4

8

g 12 3

CD

UJS

20

I-o 24

,xj 28

cm in. Oi32

S _ _

36

" 40

UJo

z

^ 44

(/I Q

48

52 20

12 -

UJO o to

O CD o

1

TIN- POINT FURNACE

TEST TEMPERATURE : 235°C

TEMPERATURE CONTROL SETTINGS OF TOP AND BOTTOM END BLOCKS RELATIVE TO THE CENTER OF THE CENTER BLOCK

AT, °C TOP BOTTOM SYMBOL

0 0 o + 0.2 -0.2 + 0.1 -0.3 X

O ox

OX D *

REFERENCE TEMPERATURE"

Ho

HOHO

0.1 °C

J I I L J L I I I J I I L

-50 0 50 MILLIOHMS150

Figure 26. Vertical temperature profile oftin-point furnace at 235 °C. The vertical distances are relative to the top of the top core block; the thermometer resistances are relative to that observed at 25.4 cm from the top of the top core block (close to the center of the center core block). The resist- ance of the thermometer at the reference temperature was about 95 VI. The resistance thermometer (2.8 mm

diameter x 2.0 cm long) sensed the average temperature of a section of well about 2 cm longalso of high-purity graphite, is screwed into the lid. The assembly is shown schematically in figure 27.

The cell assembly permits sealing the stem of the

thermometer, as well as the crucible with its charge

of tin, in helium gas which provides an inert atmosphere for both the crucible and the tin and additionally improves the thermal conductance to the

thermometer.

Because of the relatively high thermal impedance

of gas, even helium, considerable effort was made

to obtain small clearances at the heat shunts to keep the thermal impedance as low as possible.

This is practical with components of borosihcate

glass and graphite because their thermal coefficients

of expansion are similar. The outer borosilicate

glass cell was formed from precision bore tubing

and its external diameter was ground to fit the

furnace sleeve (see (H) fig. 25). The two graphite

heat shunts (G), shown in figure 27, were fitted

closely to both the thermometer guide tube (F)

and the glass cell (H). The shunts primarily serve

to conduct heat to the inner glass tube of the

tin-point cell, thereby improving the "immersion

characteristics" of the thermometer. (See the end

of this section for results of tests on the immersion

characteristics of a SPRT in the tin-point cell.)

The space between the heat shunts is loosely filled with high purity Fiberfrax (washed) insulation to eliminate convection currents and radiation losses

from the top of the crucible and the heat shunts;

it also vertically positions the heat shunts.

The equipment arrangement for filhng the cru- cibles is shown in figure 28. The bell jar (which

contains the metal sample and its funnel and

crucible) is evacuated to about 0.01 torr by a mechanical pump that is "trapped" with a molecular

sieve and the temperature of the enclosed assembly

is then raised to a few degrees below the tin point.

After 1 hour, the induction heater power is increased

so that the tin sample will melt and flow into the

crucible within about 15 min. Next, the sample is allowed to cool nearly to room temperature, the

crucible is removed from the bell jar, and the

crucible is slipped into the special borosilicate

glass cell, figure 29. The glass cell is then purged

with helium, lowered into the furnace that is used

for reahzing the freezing point, and the tin reheated

to about 250 °C. Subsequently, the glass cell is

38
raised out of the furnace and the graphite lid and

well assembly are smoothly and quickly pushed into

place. To perform the assembly easily the outside

diameters of the crucible and lid were made equal

and the inside dimensions of the glass tube only

slightly (perhaps 0.01 mm) larger. The glass tube

served to guide and correctly position the lid to cover the crucible. The lid is closely fitted to the

crucible to minimize the possibiUty of later con- tamination. Finally, the crucible is again allowed

to cool, removed from the special glass cell, and

assembled as a freezing-point cell such as is shown

in figure 27.

In the preparation for tin freezing point measurements, the tin-point cell is placed in the furnace

(held about 10 °C above the tin point) until complete

melting occurs. During the heating process the cell

temperature, which is monitored with a SPRT,

rises until melting begins, becomes nearly constant

until the melting is completed, and then rises

again. There is no reason to raise the temperature

of the tin more than a few tenths of a degree above

the melting point. In a furnace with large temperature gradients precautions should be taken to make

certain that the metal is completely melted. There

is no evidence to indicate that heating the tin to several degrees above its melting point in an

inert atmosphere is harmful to the sample.

After the melting is completed, the furnace is allowed to cool to the tin point. A thermometer

is inserted in the cell well and instrumented so

that a temperature range of 25 °C below the tin

point can be visually monitored within about 0.5 °C

by a person, standing beside the furnace, holding

a tin cell ready to be reinserted. A galvanometer

is used for this purpose. When the cell has cooled

to the tin point it is removed from the furnace. The

cell temperature, after a few seconds, will suddenly

decrease very rapidly, perhaps as much as 20 or 25 °C. As soon as its temperature ceases to drop,

the cell is immediately lowered into the furnace

again; subsequently, the cell temperature will

increase even more rapidly than it previously

decreased. If the cell temperature is examined in

more detail, the final approach to the tin point

(the last few ten-thousandths of a degree) wiU be

seen to take a few minutes (see fig. 30). When

the temperature plateau is reached, the SPRT

is measured. It may then be removed and a second

SPRT inserted. The second thermometer is preheated above the tin point so that its temperature, during insertion into the tin point cell,

slightly exceeds that of the cell. Figure 31 shows

the results of successive insertions of the same

SPRT in a single tin freeze.

Figure 27. Metalfreezing-point cell.

A. Platinum resistance thermometer.

B. To helium gas supply and pressure gauge.

C. Thermometer stem seal with silicone rubber.

D. Silicone rubber cap.

£. Insulation, washed Fiberfrax.

F. Thermometer guide tube, borosilicate glass. G. Heat shunt, graphite. H. Borosilicate glass cell.

I. Graphite cap (lid).

J. Graphite thermometer well. K. Metal sample.

L. Graphite crucible. M. Insulation, Fiberfrax paper.

39
^—7.6 cm —

I

Figure 28. Arrangement for filling the freezing-point cell with

metal sample by induction heating.

A. Borosilicate glass envelope.

B. Metal sample.

C. Graphite sample holder and funnel

D. Graphite crucible. E. Induction heater coils. F. 'O" ring groove.

G. Borosilicate glass stand. H. Slot for pumping out and gas purging.

I. Connection to vacuum and purified argon supply.

40
560 mm

153

mm

219

mm

560 mm

244

mm

H

K

Figure 29. Apparatus and method for

A. Stainless steel pusher rod. B. Cas seal with silicone rubber. Permits linear motion of the pusher rod {-€). C. Inlet for purified helium gas that is used in purging and maintaining positive pres- sure of the gas during assembly.

D. Silicone rubber cap. E. Stainless steel flange attached to the pusher rod for pressing against the graphite

lid during assembly.

F. Graphite lid for the metal sample celL G. Slit on the pusher rod. The two halves are sprung out to hold up the graphite thermometer well and Ud up while melting the metal sample.

H. Graphite thermometer well

I. Borosilicate glass tube.

J. Section of glass tube shrunk to fit the crucible and lid so that lid can be easily guided onto the crucible. K. Graphite crucible. L. Molten metal sample.

installing graphite thermometer well and lid in the graphite crucible containing

the molten, metalfreezing-point sample.

41
.4

.3 1.2

I

i-

.4

4T.3

-J L__i 1 I 1 1 1 - A

- -

-

Tilt DAiirT rnmiii/^ in\ir \

"A' TIN CELL REMOVED FHOH FURNJCE \ - B"CaL REINSERTED IN FURNACE FURNACE HELD l"C BELOW TIN POINT

\ -

u

1 1 1 1 1 1 1

24

22 J

20

16 S

14 s

12 1

10 g

08 S

06 g

04 1

02^

4 6 8 10 12 14 TIME AFTER START OF FREEZE, hrs

Figure 30. The freezing curve of tin obtained using an a-c bridge with the head of the SPRT adapted for coaxial

connectors.

The furnace was controlled to be 0.9 K below the tin point. The mass of tin sample was 1300 g.

"1 1

1 I

I" TIN POINT

1—I—I r "1

I I r

6 MINUTES

60MINUTES

J I L I I I I J L J \ L

TIME -

Figure 31. Consecutive measurements of the resistance of a SPRT in the same tin freeze.

(An a-c bridge was employed in the measurements.)

Following each equilibrium resistance observation, the SPRT was completely with- drawn from the tin-point cell, preheated close to the tin point in an auxiliary furnace, and reinserted into the ceU. The data shows that the preheated SPRT comes to equilib. rium in a very short time. In two of the cases shown, the temperature of the SPRT when

inserted into the cell was slightly above the tin point. (For comparison when the SPRT

was not preheated close to the cell temperature, see fig. 35.)

An experimental procedure similar to that

described above for observing the supercooling of

the tin may be used to check the preheating of the

SPRT before inserting it into the tin-point cell. The thermometer should be preheated and held

in the air over the tin cell while monitoring the

decreasing thermometer temperature. As a first approximation, the thermometer can be inserted

into the tin-point cell when its indicated temperature

is 30 °C above the tin point. If subsequent close

monitoring indicates that the thermometer temperature is rising, then the thermometer was too cold

when it was inserted. Several thermometers may be

consecutively calibrated during one freeze if each is properly preheated (see fig. 31). A simple check

E

uj" ^ 0.1

> 0.2 —

h-<

_J

LiJ O

UJ>

TIN-POINT CELL

0.1 mK

TOP OF CRUCIBLE

J I 1 \ I I I I I I I I I I I I I I I L

o o —6

o o

o- o

20 15 10 5 0 DISTANCE OF THERMOMETER TIP ABOVE BOTTOM OF THERMOMETER WELL, Cm

Figure 32. Immersion characteristic of a SPRT in a tin-point

cell. The data show that the indications of the SPRT are very nearly the same betv/een about 8 or 1 0 cm above the bottom and the bottom of the thermometer well.

on the temperature constancy during a freeze

is afforded by employing the same thermometer

in both the first and last measurement in a series

of SPRT calibrations. At NBS a single SPRT is set aside and only used for this purpose as an

experimental control.

The immersion characteristics of a SPRT were

tested in one of the tin-point cells that were

described earlier. The observed resistances are shown in figure 32 as a function of the distance

from the bottom of the crucible thermometer well.

The results show that the immersion of the SPRT

in the cell is more than adequate. This figure does

not, however, show the temperature gradient

expected from the change in pressure with depth.*

In the reduction of the calibration data the temperature at the point of immersion of the SPRT in

the tin-point cell is employed. The value of temperature assigned in the text of the IPTS-68 is adjusted

for the departure from the equilibrium conditions

specified for the fixed point; i.e., an adjustment of

temperature is made for the effect of the hydrostatic head of liquid tin and for any departure from

1 standard atm of the gas pressure in the cell. The high reproducibility of the tin point may be

seen in the results of tests (made at the NBS) that

compare the freezing points of a number of tin

samples. The average standard deviation of meas- urements on a given cell is ±0.05 mK from freeze to freeze (see fig. 33). All of the measurements are

plotted relative to the mean of samples 6C and 6E.

The individual measurements (one for each freeze) are shown as short horizontal bars; four freezes are represented for each cell with the exception of 6K

Recent work done at NBS indicates that the temperature gradient expected from

the change in pressure with depth is distorted or obscured when observations are made from within the graphite thermometer well if the solid liquid interface is not very close to and surrounding the well. Further work in this area is in progress.

42
on which only two freezes were performed. The

values are based on the determinations of R{t)IR{0)

that were obtained from extrapolations to zero thermometer current of observations of RUYs and /?(0)'s

at 1 and 2 mA. The /?(0's were measured 1.5 hours

after the initiation of a freeze. The /?(0)'s were measured immediately after every determination of /?(i)'s.

The samples identified as 5 were reported to be

99.999 percent pure [35]; the samples identified as 6 were reported to be 99.9999 percent pure [35]. The

sample identified 6M was accidentally overheated to a temperature above 500 °C.

7.5. Zinc-Point Cell

The equipment and procedures used at NBS for

filling jiraphite crucibles with zinc are identical to those employed for tin, except that after evacuation

at a temperature slif^htly below the tin point, an atmosphere of purified dry arjion is admitted into

the bell jar. Because of the relatively high vapor

pressure of zinc at the melting point (over 0.1 torr).

filling the graphite crucible by melting the sample in

vacuum is less practical. The power of the induction

heater is raised sufficiently to melt the zinc sample

(approximately 1280 g) in about 30 min. In spite of

the presence of argon gas, enough zinc vapor dif- fuses and is deposited on the surface of the bell jar

to cause difficulty in observing when the zinc is completely melted. The sample is allowed to cool

and the assembly of the zinc-point cell is carried out

in the same manner as the tin sample (see sec. 7.4).

To realize the freezing point of zinc with an accuracy of about ±0.002 °C the furnace need

not be very complex. However, a zinc sample

of high purity and the use of proper freezing-point

techniques are important. The experimental procedure for realizing the freezing-point of zinc

differs appreciably from that employed for tin. The

zinc is melted in the furnace while being monitored with a SPRT. The zinc freeze is initiated in the

furnace, unlike the tin freeze where the cell is removed from the furnace to initiate the freeze.

The furnace temperature is reduced so that the

0.2

0.1

0

-0.1

^-0.2

- 0.3

0.4

0.5

0.6

•0.7

T I I I r T I 1 I I 1 I r 1 I r "T 1 1 1 r

I LAB. ISTD.

T6G

6H

6C

>

6J d;6K

1

5D

61

6H

5F'

5B

J I I L

1970-1971 mmm of tin point cells - INDIVIDUAL OBSERVATION O AVERAGE

X OBSERVATION NOT IN AVERAGE

J L J I

I I I I L J I I I I L 20 25 30 35

RESISTANCE RATIO (Ro°c/f^4.2K^ ^^^"^

40 45

Figure 33. Comparison of tin-point freezing plateau temperatures with resistance ratios: RiO °C)/R(4.2 K).

The prefixes 5 and 6 indicate samples nominally 99.999 percent pure (Lot No. 6637) and samples nominally 99.999 percent pure (Lot No. 6779), respectively. The alphabetic character identifies ihe cell The prime ( ') identifies a second set of measurements on the same cell. LAB. STD. is a tin-point ceil that has been used for over four years at the NBS in the

calibration of SPRT's. Observations from four separate freezes are shown for each cell The plateau temperature selected was the reading 1.5 h()urs after the initiation of the freeze; the temperature corresponded to about 25 percent frozen. The temperatures of the SPRT during the previous hour differed typically 0.000()2 or 0.00003 K but never more than 0.0001 K from the selected temperatures of any measurements. The observed plateau temperatures are shown relative to the mean value obtained for the tin-point cells 6C and 6E. Cell 6M was accidentally overheated above 500 "C before the measurements. Only two measurements were obtained for cell 6K. (Resistance ratii) measurements were made by R. L. Powell at the Boulder Laboratories of the National Bureau of Standards.)

43
sample temperature, as measured within the well,

reaches the zinc-point temperature at a cooling

rate of 0.1 to 0.2 °C per minute. The zinc supercools

below the zinc point by 0.02 to 0.06 °C (as measured

by the SPRT) before the recalescence occurs. The subsequent rise toward the plateau temperature, while rapid at first, becomes very slow after

several minutes (see fig. 34). To hasten this process

the induced-freeze suggested by McLaren [32] is employed to form a second solid-liquid interface

immediately adjacent to the well. (The first solidliquid interface occurs at the crucible wall.) After

the zinc cell has completed the initial rapid freeze

following the supercool described above, two cold

thermometers are inserted into the well to freeze

the zinc around the well; a solid-liquid interface

that completely surrounds the well is necessary

and a single thermometer may not freeze a complete

coating of zinc on the well. The resistance of the

second thermometer will rise to a steady value

I (within 10-* °C) within a few minutes (typically

10 to 15 min) after insertion and wiU remain nearly

constant for a period of time that is determined by

the freezing rate of the outer solid zinc layer and

the purity of the zinc. (The rate of freezing, of course,

depends principally upon the heat loss to the furnace, i.e., the temperature setting of the furnace).

Freezes lasting as long as three days have been

observed at NBS, but 12 to 16 hour freezes are more typical. As with tin, SPRT's may be consecutively calibrated in a single zinc freeze if the

thermometers are preheated (see fig. 35). The preheating of the thermometer serves to minimize the

growth of the inner freeze on the well and thereby

extend the useful life of the freeze. Care must be

exercised to avoid melting a hole in the inner freeze

or loosening the mantle on the well so that it slides

to the bottom of the crucible. To ensure that no

melting and no excessive freezing occurs, the

thermometer is preheated to a temperature slightly

below the zinc point before inserting it into the cell. As with tin, a SPRT is set aside at NBS for checking

the constancy of the freeze, and it is used for both

the first and last measurements in each freeze.

The freezing-point measurements are normally not

conducted at exactly the specified pressure of 1 standard atmosphere and a correction is made for

this departure. The pressure at the interface

surrounding the thermometer resistor includes

both the gas pressure over the sample and the

pressure due to the head of liquid zinc. For zinc at the melting point the value of dtldp is +5.7 X 10"^

°C/torr or +2.74X10~5 °C/cm column of liquid

zinc [32].

The immersion characteristics of a SPRT were

tested in one of the zinc-point cells that were

described. The observed resistances are shown in

figure 36 as a function of the distance from the

bottom of the crucible thermometer well. The

results show that the immersion of the SPRT in

the cell is more than adequate.

7.6. Oxygen Normal Boiling Point

The oxygen normal boihng point cahbration is realized at the NBS by reference to the NBS-55

scale adjusted to correspond to the IPTS-68 (see

sec. 7.7). The temperature (hotness) assigned to the oxygen point, now maintained by SPRT standards, is 0.0019 K lower [8] than that maintained

previously on the NBS- 1955 scale [50]. This change

resulted from efforts made to achieve uniformity

4

3

2

I

65.560

9

8

7

6

54

3

2

I

65.5509

8

7

65.546

I I I I

|o.oi°c

I I I I I

JO-OOI"'

ZINC POINT FREEZING CURVE

°A° SUPERCOOL

"B" SUCCESSIVE INSERTION OF TWO THERMOMETERS (NO RECORDED VALUES) FURNACE HELD 0,9°C BELOW ZINC POINT

0 0.5 .1 .15 .2 2 4 6 8 10 12 14 TIME AFTER START OF FREEZE, hrs

18 uj

OL

UJ 16 H

UJ '5 i

14 1

UJ 13 I

I- 12

1

1

10 20

Figure 34. Thefreezing curve ofzinc obtained using an a-c bridge.

(The head of the SPRT had been adapted for coaxial connectors.) The furnace was controlled at 0.9 K below the zinc point. The mass of zinc sample was 1280 g. Two cold SPRT's

were inserted in the cell at "B" to freeze a coating of zinc around the graphite thermometer well.

44
n I \ I

I

\ I \ I

I

\

\ \\

^

\

\ r

ZINC POINT

12 MINUTES ImK r

60 MINUTES

I I I I J \ I L J \ L

TIMEFigure 35. Consecutive measurements of the resistance of a SPRT in the same zinc freeze.

(An a-c bridge was employed in the measurements.)

Following each equilibrium resistance observation, the SPRT was completely with- drawn from the zinc-point cell, exposed to the ambient temperature for one minute, and reinserted into the cell. In the actual procedure employed in calibration the SPRT is preheated close to the zinc point (see text). The data show that the SPRT

comes rapidly to equilibrium even when inserted into the cell relatively cold. (For comparison when the SPRT was preheated, see fig. 31.)

S

UJ

I-> 0.2

^03

~l I -4.4mK I I I

I s

"I rn I rn i I

I

ZINC -POINT CELL

O.lmK

-TOP OF CRUCIBLE o °

I I I I I I I I I I I I I I I 20 15 10 5 0

DISTANCE OF THERMOMETER TIP ABOVE BOTTOM OF THERMOMETER WELL . cm

Figure 36. Immersion characteristics of a SPRT in a zinc-point

cell. The data show that the indications of the SPRT are very nearly the same between

about 8 or 10 cm above the bottom and the bottom of the thermometer well.

in several national temperature scales; for details see reference [8]. Calibrations near the oxygen

point are performed by inserting the SPRT's to be

calibrated and an SPRT standard in the apparatus

shown in figure 37. In the preparation for calibration

the apparatus is evacuated and immersed in hquid

nitrogen to the level indicated. There are eight

thin-walled Monel wells (A) extending into the

copper block (P). Two of these wells are one-half

inch in diameter to accommodate capsule type

thermometers in holders. Thermometers are sealed

Figure 37. Cryostat for comparison calibration of thermometers

with standards at the oxygen normal boiling point.

A. Thermometer well; eight wells are located around the copper block. The SPRT's

are sealed at the top with molded silicone rubber.

B. Tube to vacuum pump to draw liquid nitrogen through cooling tubes. C. Tube to helium gas supply.

D. Manifold for distributing helium gas to the thermometer wells. E. Envelope (brass). F. Radiation shields (copper). G. Liquid nitrogen. H. Liquid oxygen. (Not employed in comparison calibrations.)

I. Vapor-pressure tube, to differential pressure diaphragm and manometer.

J. High vacuum line. K. Valve to control the liquid nitrogen input for cooling.

L. Top heat shields, to control thermometer well and thermometer stem temperatures. M. Heaters. N. Cooling tubes (thin-wall Monel). O. Oxygen bulb.

P. Copper block.

into the wells at the top by a molded band of very

soft silicone rubber. The thermometer wells are

filled with helium gas to a pressure that is shghtly

above atmospheric. The helium enhances the

thermal contact between the wall of the well and

45
Figure 38. Cryostat for the intercomparison of capsule type SPRT's.

46
the thermometer and prevents condensible gases

from entering the wells. The copper block (P) and

shields (F and L) are maintained under vacuum.

They are initially cooled by admitting nitrogen

gas and/or liquid through the valve (K) shown just

below the surface of the liquid nitrogen. The cool

vapor flows through coils (N) on the top shields and

coils on the massive center block, finally passing

through (B) to a large-capacity vacuum pump. After

cooling the block to a temperature near the oxygen

point, the valve is closed and the nitrogen gas in

the cooling coils is removed; the temperature of

the block is brought within 1 K of the oxygen point,

and the outer shields are controlled at the temperature of the block. Experience shows that the

temperature of the block can be maintained more

nearly constant by allowing the inner shields to "float" without heating at a temperature near that

of the block. [Heaters (M) on the copper block and

the inner shields are used, when desired, to attain

higher temperatures]. The outer shields are controlled to be at the temperature of the block by

means of one differential thermopile between the

top shield and the block, and a second thermopile

between the outer top shield and the outer side

shield. The signals from these thermopiles go to commercial low level chopper type d-c amplifiers

and thence to three-mode controllers; the output

signals from the controllers are then raised to power levels that are sufficient to supply the shield

heater (less than 1 W). The measurements on the

thermometer to be calibrated and the standard

thermometer are made simultaneously using two

Mueller bridges. The standard deviation of the

intercomparison is less than the equivalent of

0.1 mK.

7.7. Comparison Calibration Between

13.81 and 90.188 K

To determine temperatures below the oxygen

normal boiling point, the text of IPTS-68 (Appendix

A) assigns values to defining fixed points and prescribes the form of interpolation formulae. At the

time the IPTS-68 was adopted no national laboratory could satisfactorily realize all the fixed points.

The four temperature scales (e.g., NBS-55) upon

which the extension of the IPTS-68 below the oxygen point was principally based are each highly

reproducible. The text of the IPTS-68, therefore,

formally recognized that the use of these national

scales, adjusted by the published differences [8], will

give a close approximation to the IPTS-68 below

90.188 K. At NBS a reference group of capsule

type SPRT's maintains the "national scale" (NBS55). The NBS version of IPTS-68 (NBS-IPTS-68)

is achieved by making reference to the NBS-55

scale and utilizing the published differences between the NBS-55 scale and the IPTS-68 referred

to above. Thermometers are calibrated by intercomparison in a nearly isothermal copper block

with one of the thermometers of this reference

group. A second thermometer from the reference

group is also included in the comparison to serve

as a check on both the reference thermometer being

used and the comparison techniques. These two

thermometers will be referred to, henceforth, as the

first and second standards, respectively.

The comparison of capsule type SPRT's is performed in a cryostat that has been built and operated

ratlier like an adiabatic calorimeter. The apparatus

is shown in figure 38. The copper comparison block

(V), shown enlarged in figure 39, has wells for six

thermometers, two standards and four thermometers

under test. The schematic in figure 40 shows that the

first standard is wired independently of the other

five thermometers. The wiring arrangement employed for the five thermometers requires only

2n-\~2 leads for n thermometers, in this case, 12

leads, thus minimizing the number of leads and,

therefore, the heat transfer through the leads with

the surroundings. The first standard is electrically

isolated from the others. This permits the use of two

bridges to make simultaneous measurements on the

first standard and any one of the thermometers to be calibrated or the second standard. The resistance

measurements are made with two Mueller bridges.

A. Guide for directing transfer tube into liquid helium Dewar.

B. Metal bellows to permit differential expansion between the central tube and the supporting thermometer block well. C. Exit for the vacuum line and electrical leads to the thermometers, the thermocouples, and the heater on the copper thermometer block.

D. Supporting shelf. E. Demountable "O" ring seal to the well around the thermometer block.

F. Well around the thermometer block. G. Central tube for supporting the thermometer block. H. Seal for the electrical leads from the vacuum can that surrounds the shield and lower portion of the thermometer block well.

I. Line from vacuum can "O".

J. Demountable "O" ring seal to the liquid helium Dewar.

K. Glass stopcock to permit reevacuation of liquid helium Dewar.

L. Line for pumping the space within the liquid helium Dewar. M. Liquid helium Dewar.

N. Thermal tie-down for lead wires. O. Vacuum can that surrounds the lower portion of the thermometer block well and the thermal shield. P. Copper sleeve on vacuum can to maintain uniform temperature when liquid helium level is low. Q. Thermal tie-down for leads similar to N with a heater and five-junction Chromel-P/constantan thermocouple for temperature control. R. Copper ring on the thermometer block well with temperature control components similar to Q.

S. Heavy copper shield with temperature control components similar to Q.

T. Thermometer lead terminal block of anodized aluminum.

U. Capsule type platinum resistance thermometers thermally attached to copper block with vacuum grease. V. Copper thermometer block with holes for six thermometers, W. Location of thermocouple junctions placed on shield. X. Reentrant "thumb" in the bottom of well. "Thumb" contains reference junctions for thermocouples on R and S. Y. Heavy copper tail on vacuum can to reach liquid helium at a low level

47
Figure 39. Thermometer comparison block.

A. Stainless steel tube to support thermometer comparison block. B. Man ganin leads for thermometers and copper leads for block heater. C. Platinum leads of resistance thermometer.

D. Stainless steel well for thermometer block. E. Heater distributed along well "D".

F. Copper thermometer block with wells for six thermometers.

G. Heater for thermometer block. H. Platinum resistance thermometer.

I. Copper sleeve (attached to central supporting tube) with longitudinal grooves to serve as thermal tie-downs for manganin leads and thermocouples.

J. Brass plug to seal end of central support tube. K. Location of grooves in sleeve "I."

L. Reference junctions for thermocouples on shield (at "W" in fig. 38) and thermometer block well ring heater (at "R" in fig. 38). M. Closely fitted reentrant copper "thumb" (with a groove for venting gas).

ab cdef gh i) kl mn op

Figure 40. Schematic of wiring capsule-type platinum resist- ance thermometers in the copper comparison block of the

cryostat for calibration. To avoid excessive heating of the copper block, the bridge connections to the ther- mometers, wired in series, are made as follows:

SPRT Leads used

1 efgh

2 ghij 3 ijkl 4 klmn STD2 mnop

A. Cryostat leads. B. Thermometer connections made among them and with the cryostat leads. C. Schematic of the copper comparison block. D. Platinum resistance thermometers. STDi and STD2 are reference standards;

others are to be calibrated.

To minimize the possible change in the difference

of resistance in the potential leads, the leads from

the thermometers that pass up through the cryostat are of No. 26 AWG (0.404 mm diameter) manganin

wire insulated with a heavy coating of Formvar.

The choice represents a compromise among the

requirements of low heat conductivity, the resist- ance seen by the galvanometer, and lead-resistance

stabihty. The leads are "tempered" by being placed

in good thermal contact with the series of "thermal

tie downs" (N) shown in figure 41. The manganin

leads and copper leads for thermocouples and

heaters exit through a hard wax seal at (C) the

top of the cryostat and then join to heavier copper

leads. The external copper leads are thermally

insulated to minimize both the temperature change

of the leads and any possible gradient change

between leads. The resistances of the copper

sections of the leads and those of the manganin

sections of the leads were separately adjusted for

common equality to maintain the lead balance

with changing room temperature. The thermometer

leads are connected to the bridge through selector

switches that employ either mercury-wetted contacts or multiple all-silver contacts in parallel.

The cryostat is constructed so that the central

core, including the copper comparison block, is stationary while the remainder of the cryostat

assembly may be lowered to expose the block for

thermometer installation or removal. The only

seals required for sefding the cryostat are at room

temperature and consist of simple Viton "O" rings

(E).

N. Copper plug with longjludinal grooves to serve as thermal lie-downs for thermocouples. O. Vent.

P. Thermocouple leads to shield and thermometer block well ring heater. 48
Figure 41. Thermal tie-downs.

A. Central support tube (stainless steel) for thermometer block.

B. Rounded comer, tangent to the bottom of the grooves for wire. C. Round bottom grooves in cylinder "D" with wire cemented in place. D. Copper cylinder soldered (tin-lead eutectic) to the central support tube.

E. Copper cylinder fitted to "D" and held in place by pin "H."

F. Thermocouple lie-down.

G. Spring fingers of beryllium copper heavily silver plated and gold "flashed." H. 1.5 mm steel pin to hold cylinder "E" in position. L Pan head 2-56 brass screw to clamp thermocouple tie-down assembly to "E."

J. Insulating washer of 0.005 mm Mylar coated with vacuum grease.

K. Copper washer with pre-tinned groove for thermocouple wire.

L. Polyimide insulated thermocouple wire, placed at the bottom of the pretinned groove and "potted" in eutectic tin-lead solder. M. Epoxy insulation on screw.

The cryostat proper (fig. 38) is surrounded with

the usual arrangement—an inner Dewar flask (M),

which contains hquid helium, guarded by an outer

Dewar flask (not shown) that contains liquid

nitrogen. During the initial cool down, the inner

Dewar is filled with liquid nitrogen and the liquid

nitrogen vapor pressure is reduced to about

Vs atm. The vacuum can (O) of the cryostat is filled with ^He gas to a pressure of about 0.1 torr

to facilitate the cooling; under such conditions, the

thermometer block cools to about 65 K in approximately 12 hours (overnight). The liquid

nitrogen remaining in the inner Dewar is removed

by applying a small over-pressure of *He gas which

forces the nitrogen out through a 3.2 mm thin-wall

stainless tube (not shown in fig. 38) that extends

from the Dewar bottom to the room. (Precooling

the cryostat from about 80 K to about 65 K by

pumping on liquid nitrogen reduces the liquid

helium required for cooling by about 2 liters.) Six

to seven Uters of liquid helium are usually required

to cool the apparatus from 65 to 4.2 K and fill the

inner helium Dewar.

As a part of the calibration procedure the thermometer resistance readings are first made at about 4 K, without the use of temperature controls.

For the calibrations above 4 K, the ^He gas in the

vacuum can is removed to reduce the heat transfer

and the temperature of the comparison block is controlled by the use of three sets of five-junction

Chromel-P/Constantan diff'erential thermopiles

that sense temperature diff"erences between the

comparison block and the thermal shields. Appropriate automatic controls are employed to power

three separate heaters to reduce the indicated

temperature diff"erences to zero. When the space

in the vacuum can is evacuated to a pressure in the

10~^ torr range, most of the heat transfer to the

comparison block from the surroundings is (i) down through the wall of the well (F) and (ii) within

the well through the central thin-waU stainless

tube and electrical leads. Consequently, two sets

of thermopiles and heaters are affixed, one at (R) on the wall of the well and the other at (Q)

within the well. These two locations are about 15

cm above the comparison block. The third thermopile and heater are mounted on the heavy copper

shield which surrounds the lower end of the well

where the comparison block rests. The thermopiles

are aU referenced to the temperature of the compari- son block. If adequate temperature control of a system is to be achieved, the thermal lags of both

the heaters and the temperature sensors (in this case thermopiles) must be small. In addition, the

"immersion error" that is caused by heat conducted along the leads of the sensor should be small.

These characteristics were achieved in the cryostat

by employing the "thermal tie downs" shown in

figures 38 and 41. (The selection of Chromel-P/Constantan thermocouples was influenced by their low

thermal conductivity. There are other materials

that possess better thermoelectric characteristics

but their relatively high thermal conductivities were

considered to be disadvantageous.) The spurious

emf of each thermopile is balanced out with a simple

emf injector that is adjustable between ± 5 /x,V.

The adjustment for the spurious emf is made at 4.2 K after equilibrium has been attained with

^He gas still in the can. The calibration resistance measurements are made at 16 temperatures; approximately 12.20,

13.81, 15.426, 17.042, 18.661, 20.28, 23.691, 27.102,

33.917, 40.732, 47.547, 54.361, 66.303, 78.245,

84.216, and 90. 188 K. The values of W{T) at the

IPTS-68 defining fixed-point temperatures are obtained by least square analysis of the comparison

data (see sec. 6). To change from one calibration temperature to the next higher temperature, current is supplied to the heater that is wound directly on the

comparison block. Although the control of the

shields is adequate to track the temperature of the

block when it is rapidly heated at the lower temperatures (below about 40 K), the shields lag at higher temperatures when the block is being rapidly

heated because of the limitations of the power

amplifiers which drive their heaters. The time re- quired for the system to attain thermal equilibrium

after the heating of the block is stopped increases

with temperature. At 13.8 K the thermometer drift

wiU decrease to less than 10 //,ft/min in a minute.

49
while at 90 K more than 30 min is required to achieve

this drift rate. Measurements are considered unsatisfactory if the indicated drift rate is more than

5 [xCll min at temperatures below 20 K or more than

10 (xfllmin at temperatures above 20 K.

In order to achieve these low drift rates rather

quickly, particularly at temperatures above 30 K, it was found necessary to uniformly heat the lower

20 cm section of the well (F) when the comparison

block was being heated to the next higher calibration temperature. This was accompHshed by powering a heater wound around the well {E) as shown in

figure 39. An adjustable fraction (depending upon

the temperature) of the voltage applied to the block

heater is applied to the well heater to raise its temperature at the same time and rate as the block

temperature.

In the calibration of the capsule type SPRT's

the resistance measurements are carried out with

a continuous current of 1 mA through the thermometer and with a commutator switching order

of NRRNNRRN with the Mueller bridge. This

order of measurements is not the optimum one

(which is NRRNRNNR), but when a large number

of measurements must be made, such as in the

calibration of thermometers over this temperature

range, the pattern used is less subject to gross

errors. At approximately 90.2 K, measurements are made at two currents, 1 mA and 2 mA, to determine

the self heating effect of the SPRT. Although in- frequent, the glass to metal seal of a thermometer

may leak; this causes a loss of helium and conse- quent increase in the self-heating effect of the

thermometer. A thermometer having this defect

may be unreliable. The treatment of the data re- sulting from these measurements differs markedly

from those obtained above 90.188 K (see sec. 6).

The authors gratefully acknowledge the critical

review and many suggestions given by Martin L. ReiUy which have helped to improve this monograph. They also acknowledge the help of William

J. Hall who wrote the computer program for re- ducing the SPRT calibration data between 13.81

and 90.188 K and William R. Bigge who made the

measurements for some of the illustrations.

8. References

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Tech. (USSR), No. 6, 487-488 (June 1964), translation from Izmeritel, Tekhn. No. 6, 21-22 (June 1964).

[2] Barber, C. R., Platinum resistance thermometers of small dimensions, J. Sci. Instr. 27 , 47-49 (1950).

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[46] Stimson, H. F., The International Temperature Scale of 1948, J. Res. Nat. Bur. Stand. (U.S.) 42, 209-217 (1948)

RP1962.

[47] Stimson, H. F., Precision resistance thermometry and fixed points, in American Institute of Physics, Temperature,

Its Measurement and Control in Science and Industry,

Vol. 2, 141-168 (Reinhold Publishing Corp., New York,

N.Y., 1955).

[48] Stimson, H. F., International Practical Temperature Scale

of 1948. Text revision of 1960, J. Res. Nat. Bur. Stand.

(U.S.), 65A (Phys. and Chem.), No. 3, 139-145 (MayJune 1961).

[49] Strelkov, P. G., Borovik-Romanov, A. S., and Orlova, M. P., Thermodynamic studies at low temperatures. I. Measure- ment of temperatures between 12 and 300 °K, Zh. Fiz. Khim. 28, 345-352 (1954).

[50] Swindells, J. F., National Bureau of Standards Provisional

Scale of 1955, in Precision Measurement and Calibration. Temperature, Nat. Bur. Stand. (U.S.), Spec. Publ.

300, Vol. 2, 513 pages (Aug. 1968), p. 56.

[51] The International Practical Temperature Scale of 1968,

Adopted by the Comite International des Poids et Mesures,

Metrologia 5, No. 2, 35-44 (Apr. 1969).

[52] Toenshoff, D. A., An evaluation of some factors which may

affect the stability of birdcage thermometers, report of work conducted under Contract No. CST383 (National

Bureau of Standards) by Engelhard Industries (East

Newark, N.J. 07029), 44 pages, January 1, 1968.

[53] Unpublished test measurements at NBS with manually

controlled copper block furnaces suitable for conducting

metal freezing and melting measurements.

[54] White, W. P., Potentiometers for thermoelectric measure- ments especially in calorimetry, J. Am. Chem. Soc. 36,

1868-1885(1914).

[55] White, W. P., Einige neu Doppelkompensatoren. I. Thermokraftfreie Kontakte, mechanische Anordnungen und

Zubehorteile, Zeit. Instrumentenk. 34, 71-82 (1914).

[56] White, W. P., Einige neue Doppelkompensatoren. II. Die

elektrische Einrichtung. Zeit. Instrumentenk. 34, 107-

113, 142-151 (1914).

51
9. Appendixes

Appendix A.

The International Practical Temperature Scale of 1968*

Adopted by the Comite Intpriiational des Poids ot Mesiires

Received February 17, 1969

Foreword

The Comite International des Poids et Mesures (CIPM) at

its meeting in October 1968 agreed to adopt the International

Practical Temperature Scale of 1968 (lPT.S-68) in accordance

with the decision of the 13"' Conference (Jenerale des Poids

et Mesures, Resolution 8. of October 1967. This resolution gave

the CIPM permission to introduce a new Scale to replace the

existing IPTS-48 as soon as the Coniite Consultatif de Therniometrie (CCT) made its recommendations.

There were two important reasons for revising the rPTS-48. The first of these was the need to extend the Scale to lower temperatures to achieve a unification of the existing national

scales in the region 10 K—90 K. Secondly the Scale had not been revised significantly over the range from —183 ^C to 1063 ^C since its inception in 1927. and modern gas thermo- meter measurements showed that the Scale gave values con- siderably different from thermodynamic temperatures.

The CCT agreed on the basic data and methods of inter- polation for a new definition of the IPTS at its 8th meeting

held in Washington and Ottawa in September 1967. Provision was made to extend the Scale down to 13.81 K, the triple point

of equilibrium hydrogen, and for the use of the best known

values of thermodynamic temperatures. The value of c.^ u.sed

in the Planck equation was also revi-sed as a result of a better knowledge of the values of fundamental physical constants. The CCT ap))ointed a small sub-committee to prepare a text

of the definition of the IPTS-68 in acconlance with its recommendations, with instructions to resolve some points of detail. The final draft of the English text, which is reproduced below,

received the approval of all members of the CCT before it was

submitted to the CIPM. The official text of the IPTS-68 is the French text which was prepared by the Bureau Inter- national des Poids et Mesures from the English text, and is j)ul)lished as an annexe to Comptes rendus des .seances de la Trei/.ienie Conference Generale des Poids et Mesures.

C. R. B\RKEK

I. Introduction

The basic temperature is the thermodynamic

temperature, symbol T, the unit of which is the kelvin.

.symbol K. The kelvin is the fraction 1/273.16 of the

thermodynamic temperature of the triple point of

watei ' . The Celsius temperature, symbol t. is defined by

t=T~-To (1) where Tq = 273.15 K. The unit employed to express

a Celsius temperature is the degree Celsius, symbol "^C. which is equal to the kelvin. A difference of temperature is expressed in kelvins ; it may also be expressed

in degrees Celsius.

* The text in French of this Scale is published in Comptes

rendus de la Treizieme Conference Generale des Poids et Mesures, 1967—1968, Annexe 2. and Comite Consultatif de Thermometrie, 8'' session, 1967, Annexe 18.

1 13th General Conference of Weights and Measures ( 1967), Resolutions 3 and 4.

The International Practical Temperature Scale of

1 96S {IPTS-68) has been chosen in such a way that the

temperature measured on it closely approximates

the thermodynamic temperature : the difference is within the limits of the present accuracy of measurement.

The International Practical Temperature Scale of

I96S distinguishes between the International Practical

Kelvin Temperattire with the symbol 7^,8 ^'^'^ the

International Practical Celsius Temperature with the

synibol t^^\ the relation between Tgg and t^^^ is

«68=- ^68 -273.15 K. (2) The units of Tgg and t^^ are the kelvin. symbol K. and

degree Celsius, symbol °C. as in the case of the thermodynamic temperature T and the Celsius temperature t. The International Practical Temperature Scale of

I96S was adopted by the International Committee of

Weights and Measures at its meeting in 196S according

to the ])ower given to it by Resolution S of the 13th

General Conference of Weights and Measures. It replaces the International Practical Teniperatur(>

Scale of 194S (amended edition of 1960).

II. Uefinition ol the International

Practical Temperature Scale of IftGH^ (IPTS-6>i)

7. Principle of the IFTS-GS ami Defiiun'j Fixed Pointu

The IPTS-6S is based on the assigned values of the

temperatures of a number of reproducible e(|uilibriuni states (defining fixed ])oints) and on standard instruments calibrated at those temperatures. Interpolation

between the fixed point temperatures is provided by

formulae used to establish the relation between indi- cations of the standard instruments and vahu's of

International Practical Temperature.

The defining fixed points are established by realiz- ing specified e((uilibrium states between phases of

|)ure substances. These ('(luilibrium states and the

values of the International l'ra(^tical Tenifx iaf iirc assigned to them arc given in Table 1

.

The standard instrument used Irotn 13. SI K to 630.74 °(' is the platinum resistance tlicrmometcr. Tlic thermometer resistor must he strain-free, ainieah'd

pure platinum. The resistance ratio defitied by

W(T,,) 7^(7'e„)//;(273.l5 K) (3)

'- In this document Kelvin temperatures aw used, in general, below 0 C and Celsius temperatures arc u.sed abov(; 0 "(!. This avoids the use of negative values and conforms with

general usage.

52
36 The International Practical Temperature Scale of 1968 Metrologia

Table 1. Defining fixed points of the IPTS-€8'

Equilibrium state Assigned value of International

Practical Temperature

Equilibrium between the solid, liquid

and vapour phases of equilibrium

hydrogen (triple point of equilibrium

hydrogen)

Equilibrium between the liquid and

vapour phases of equilibrium

hydrogen at a pressure of 33 330.6 NVm''' {25/76 standard

atmosphere)

Equilibrium between the liquid and

vapour phases of equilibrium

hydrogen (boiling point of equilibrium hydrogen)

Equilibrium between the liquid and

vapour phases of neon (boiling point

of neon)

Equilibrium between the solid, liquid

and vapour phases of oxygen (triple

point of oxygen)

Equilibrium between the liquid and

vapour phases of oxygen (boiling

point of oxygen)

Equilibrium between the solid, liquid

and vapour phases of water (triple

point of water)"=

Equilibrium between the liquid and

vapour phases of water (boiling

point of water)"'"'

Equilibrium between the solid and

liquid phases of zinc (freezing point

of zinc)

Equilibrium between the solid and

liquid phases of silver (freezing point

of silver)

Equilibrium between the solid and

liquid phases of gold (freezing point

of gold)

13.81 • 259.34

Above 1337.58 K (1064.43 °C) the International

Practical Temperature of 1968 i.s defined hy the

Planck law of radiation vi^ith 1337.58 K a.s the refer- ence temperature and a value of 0.014 388 metre

kelvin for Cj.

2. Definition of the International Prtictical Temperature

of 1U6S in Difjerent Temperafiire Jiaage.t

a) The range from 13.81 K to 273.15 K

From 13.81 K to 273.15 K the temperature T^g

is defined by the relation

(5)

20.28 252.87

27.102 —246.048

54.361 — 21S.789

90.188 —182.962

0.01

» Except for the triple points and one equilibrium hydrogen

point (17.042 K) the assigned values of temperature are for equilibrium states at a pressure po = 1 standard atmosphere

(101 325 N/m^). In the realization of the fixed points small departures from the assigned temperatures will occur as a

result of the differing immersion depths of thermometers or

the failure to realize the required pressure exactly. If due

allowance is made for these small temperature differences, they

will not affect the accuracy of realization of the Scale. The

magnitudes of these differences are given in section III.

" The equilibrium state between the solid and liquid

phases of tin (freezing point of tin) has the assigned value of

<88 -= 231.9681 "C and may be used as an alternative to the

boiling point of water. The water used should have the isotopic composition of ocean water, see section III, 4.

where R is the resistance, must not be less than

1.392 50 at T^^ 373.15 K. Below 0 °C the resistancetemperature relation of the thermometer is found from

a reference function and specified deviation equations.

From 0 °C to 630.74 °C two polynomial equations

provide the resistance-temperature relation. The standard instrument u.sed from 630.74 to 1064.43 °C is the platinum- 10 "o rhodium/platinum

thermocouple, the electromotive force-temperature

relation of which is represented by a quadratic equation.

.17.042 —256.108 where W(T^^) is the resistance ratio of the platinum

resistance thermometer and H^ccT-esf^es) the resis- tance ratio as given by the reference function set out in Table 2^. The deviations MV(T^^) at the temperatures

of the defining fixed points are obtained from the

measured values of W{T^g) and the corresponding

values of TfccT-esl^V.s). see Table 4. To find AW(T^g,)

at intermediate temperatures interpolation formulae

are u.sed. The range between 13.81 K and 273.15 K

is divided into four parts in t;ach of which AW{Tf^^) is defined by a polynomial in T^g. The constants in the

polynomials are determined from the values of

AW{Tgg) at the fixed points and the condition that

there should be no discontinuity in dA W''(T'gg)/dTfi8 at the junctions of the temperature ranges. From 13.81 K to 20.28 K the deviation function is A W( T,,) ^^A, + B,T,, + C, + D, Tl, (5) where the constants are determined by the measured

deviations at the triple point of equilibrium hydrogen,

the temperature of 17.042 K and the boiling point of

equilibrium hydrogen and by the derivative of the

deviation function at the boiling point of e(iuilibrium

hydrogen as derived from Eq. (6). From 20.28 K to 54.361 K the deviation function is AW{T,,) == + B^T,^ + C\Tl^ + D^Tl, (6) where the constants are determined by the measuicd

deviations at the boiling point of equilibrium hydrogen, the boiling point of neon and the trijjle point of

oxygen and by the derivative of the deviation function at the triple point of oxygen as derived from

Eq. (7). From 54.361 K to 90.188 K the deviation function

is

273.16

373.15

692.73

1235.08

100

419.58

961.93

1337..58 1064.43

^3 + -^^3^68 "t" C'jTL (7) where the constants are determined by the measured

deviations at the triple point and the boiling point of

oxygen and by the derivative of the deviation function at the boiling point of oxygen as derivetl from

Eq. (8). From 90.188 K to 273.15 K the deviation function

^4^68 (^68 100 °V)

where t^^ -- 'jre„ ~ 273.15 K and the constants ar(> determined by the measured deviations at the boiling

point of oxygen and the boiling point of water'.

" For the relation between the [PT8-68 and tlio national

scales" from u liich it was in part derived see Appendix I.

^ If the freezing point of tin (.see Note ''. Table 1 ) is used as a fixed point insU^ad of the hoiling point of water, ir(100 ^(.')

for the platimim thermometer sliould be calculated from

Eqs. (9) and (10).

53
Vol. 5, No. 2, 1969 The International Practical Temperature Scale of 1968 37

Table 2. The reference function I^'^ccT-»8(?',g) for platinum resistance thermometers for

the range from 13.81 K to 273.15 K»

T'es = j^o + 2^ -^'P" '^'ccT-es [T,,)y]^ K (22)

Coefficients At:

i A, i Ai

0 0.27315 y 10' 11 0.767 976 358 170 845 8 X 10

1 0.250 846 209 678 803 3 X 10= 12 0.213 689 459 382 850 0 y 10

2 0.135 099 869 964 999 7 X 103 13 0.459 843 348 928 069 3

3 0.527 856 759 008 517 2 X 10= 14 0.763 614 629 231 648 0 X 10--1 4 0.276 768 548 854 105 2 X 102 15 0.969 328 620 373 121 3 X 10--2

5 0.391 053 205 376 683 7 X 10- 16 0.923 069154 007 007 5 X 10--3 6 0.655 613 230 578 069 3 X 102 17 0.638 116 590 952 653 8 X 10--4

7 0.808 035 868 559 866 7 X 102 18 0.302 293 237 874 619 2 X 10--5 8 0.705 242 118 234 052 0 X 102 19 0.877 551 391 303 760 2 X 10--7 9 0.447 847 589 638 965 7 X 102 20 0.117 702 613 125 477 4 X 10--S 10 0.212 525 653 556 057 8 X 102

The reference function Wccr-is(T is continuous at T^^ = 273.15 K in its first and second derivatives with the function W(t^^) given by Eqs. (9) and (10) for a =

3.925 966 8 x lO"' °C-i and S = 1.496 334 °C.

» A tabulation of this reference function, sufficiently detailed to allow interpolation

to an accuracy of 0.0001 K, is available from the Bureau International des Poids et Mesures, 92-S4vres, Prance. A skeleton tabulation appears in this text as Table 3.

b) The range from 0 °C (273.15 K) to 630.74 °C

From 0 "0 to 630.74 °C t^^ is defined by

^es = ^' + 0.045 (^) [^-i)

- 1 1 °C (9) 1,419.58 °C \^ 630.74 °C where t' is defined by the equation

t' = ^ mt') - 1] + 6 [^) [-^^ - l) (lOa)

where W{t') = ^^^4^ . The constants ^(0°C), a and (5 are determined by measurement of the resistance at the triple point of water, the boiling point of water

(or the freezing point of tin, see Note ^, Table 1) and

the freezing point of zinc. Eq. (10 a) is equivalent to the equation

W(t') ==1 + At' + Bf" , (10b)

where ^ = «(1 + djiOO °C) and £ = - 10-^ « <5 °Q,-^.

c) The range from 630.74 °C to 1064.43 °C

From 630.74 °C to 1064.43 °C iga is defined by the

equation = « -i- &<68 + c^is (11)

where E{t^^) is the electromotive force of a standard

thermocouple of rhodium-platinum alloy and platinum, when one jimction is at the t€mperature t^^ =

0 °C and the other is at temperature t^^. The constants

a, b and c are calculated from the values of E at 630.74 °C ± 0.2 °C, as determined by a platinum

resistance thermometer, and at the freezing points of

silver and gold.

The wires of the standard thermocouple shall be

annealed and the purity of the platinum wire shall be

such that the ratio W{iOO °C) is not less than 1.3920. The rhodium-platinum wire shall contain nominally

10% rhodium and 90% platinum by weight. The

thermocouple shall be such that the electromotive

forces i;(630.74 °C), E{Ag) and E{Au) shall satisfy

the following relations

7 Metrologia

^?(Au) = 10 300 (J.V ± 50 [xV (12)

^(Au) - ^?(Ag) = 1183 |xV

+ 0.158 [E{A\i) - 10 300 [xV] ± 4 (xV (13)

^;(Au) - £^(630.74 °C) = 4766 (JiV

-f 0.631 [E{Au) - 10 300 [xV] ± 8 [xV . (14)

d) The range above 1337.58 K (1064.43 °C)

Above 1337.58 K (1064.43 °C) the temperature T^^

is defined by the equation

L;i{'l\,{.\ii))

exp

C2 -

1

[A7'e,(Au)J

exp - 1

(15)

in which Lx{T^^) and Lx{Tgg{Au)) are the spectral

concentrations at temperature T^^ and at the freezing

point of gold, Tgg{Au), of the radiance of a blackbody

at the wavelength A = 0.014 388 metre kelvin.

III. Supplementary Information

The apparatus, methods and procedures described

in this section represent good practice at the present

time.

1. Standard Resistance Thermometer

A standard platinum resistance thermometer

should be so designed and constructed that the four- terminal resistance element is as free as possible from

strain and will remain so in use. Satisfactory resistors have been made with platinum wires of uniform dia- meter between 0.05 and 0.5 mm and with at least a

short portion of each lead adjacent to the resistor also made of platinum. A commonly used value of i?(0 °C)

is 1^ 25 ohms and the measuring current for such a thermometer is normally 1 or 2 mUliamperes. All thermometer components in close proximity to the

resistor must be clean and non-reactive with platinum.

' Since T^^ (Au) is close to the thermodynamic temperature

of the freezing point of gold and c, is close to the second

radiation constant of the Planck equation, it is not necessary

to specify the value of the wavelength to be employed in the measurements [see Metrologia 3, 28 (1967)].

54
38 The International Practical Temperature Scale of 1968 Metrologia

Table 3. Values of WccT-as{Tes}, accordiiu/ to the equation giv&n iTi Table fit intcji'Cil Vdliic-s of Tqs

1 ITccT-es C'es) 68 ' 68 IFcCT-68 (2^65) ' 68 K K K K K K

50 0.075 377 56 100 0.286 302 01 150 0.498 611 35 200 0.704 966 94 2.50 0.907 383 09

51 0.079 101 23 101 0.290 621 54 151 0..502 787 07 201 0.709 0.50 04 251 0.911 397 53 52 0.082 875 95 102 0.294 938 41 152 0.506 960 58 202 0.713 131 61 252 0.915 410 74

53 0.086 698 59 103 0.299 252 45 153 0.511 131 72 203 0.717 211 74 2,53 0.919 422 74 54 0.090 566 00 104 0.303 563 59 154 0.515 300 65 204 0.721 290 26 254 0.923 433 43

55 0.094 47515 105 0.307 871 83 1.55 0.519 467 .37 205 0.725 367 33 255 0.927 442 83 56 0.098 423 36 106 0..312 17710 156 0.523 6.31 80 206 0.729 442 88 256 0.931 451 01 57 0.102 407 74 107 0.316 479 39 1.57 0.527 794 09 207 0.733 516 90 257 0.935 4,58 05

58 0.106 425 83 108 0.320 778 ,56 158 0.531 9,')4 17 208 0.737 589 47 258 0.939 463 71 59 0.110 475 06 109 0.325 074 67 159 0.536 112 11 209 0.741 660 59 259 0.943 468 22 60 0.114 5,5312 110 0.329 367 65 160 0.540 267 92 210 0.745 7.30 26 260 0.947 471 52

61 0.118 657 89 111 0.333 657 51 161 0.,544 421 67 211 0.749 798 41 261 0.951 473 52 62 0.122 787 22 112 0.337 94416 162 0.548 573 36 212 0.753 865 18 262 0.955 474 30

13 0.001 230 61 63 0.126 93914 113 0.342 227 68 163 0..5,52 722 91 213 0.757 930 43 263 0.959 473 85 14 0.001 459 73 64 0.131 111 89 114 0.346 508 00 164 0.556 870 48 214 0.761 994 30 264 0.963 47219

15 0.001 745 41 65 0.1.35 303 63 115 0.350 785 19 165 0.561 (U6 06 215 0.766 056 72 265 0.967 469 31 16 0.002 094 74 66 0.139.512 84 116 0..3.55 0,59 10 166 0..565 1,59 58 216 0.770 117 70 266 0.971 46513

17 0.002,515 12 67 0.143 738 00 117 0.3.59 329 89 167 0.569 301 12 217 0.774177 30 267 0.975 459 80

18 0.003 014 28 68 0.147 977 73 118 0.303 597 54 168 0.573 440 76 218 0.778 235 45 268 0.979 453 25

19 0.003 599 62 69 0.1.52 230.58 119 0.367 861 99 169 0.577 .578 48 219 0.782 292 23 269 0.983 445 41 20 0.004 277 80 70 0.1,56 495 41 120 0.372 123 31 170 0..581 714 23 220 0.786 347 56 270 0.987 436 42

21 0.005 054 95 71 0.160 771 08 121 0.376 381 51 171 0.585 848 06 221 0.790 401 51 271 0.991 426 14 22 0.005 936 68 72 0.165 0,56 43 122 0.380 636 57 172 0.589 979 99 222 0.794 454 09 272 0.995 414 71 23 0.006 928 04 73 0.169 350 49 123 0.384 888 51 173 0.594 HO 08 223 0.798 505 23 273 0.999 401 99 24 0.008 03316 74 0.173 652 40 124 0..389 137 32 174 0.598 238 35 224 0.802 555 06 25 0.009 255 04 75 0.177 961 17 125 0.393 383 16 175 0.602 364 78 225 0.806 603 .52 26 0.010 595 85 76 0.182 276 05 126 0.397 625 94 176 0.606 489 31 226 0.810 6,50,54 27 0.012 056 90 77 0.186.596 28 127 0.401 865 67 177 0.610 612 08 227 0.814 696 25

28 0.013 639 01 78 0.190 921 07 128 0.406 102 42 178 0.614 733 10 228 0.818 740,59

29 0.015 342 61 79 0.195 249 92 129 0.410 336 28 179 0.618 ,852 29 229 0,822 783 64 30 0.017 167 68 80 0.199 582 12 130 0.414.567 09 180 0.622 969 72 230 0.826 825 31 31 0.019113 63 81 0.203 917 14 131 0.418 795 07 181 0.627 085 40 2.31 0.830 ,S65 61 32 0.021 179 44 82 0.208 2,54 45 1.32 0.423 020 15 182 0.u3l 199 39 232 0.834 904 61 33 0.023 363 43 83 0.212 593 44 1.33 0.427 242 33 183 0.635 3 li 64 233 0.838 942 24 34 0.025 663 35 84 0.216 933 88 134 0.431 461 69 184 0.639 422 13 234 0.842 978 57 35 0.028 076 45 85 0.221 275 23 135 0.435 678 31 185 0.643 530 94 2,35 0..847 013 ,53 36 0.030 .599 53 86 0.225 617 12 136 0.439 892 10 186 0.647 638 07 236 0.851 047 26

37 0.033 22916 87 0.229 9,59 10 1.37 0.444 103 22 187 0.651 743 52 237 0.8,55 079 63 38 0.035 961 55 88 0.234 .301 05 138 0.448 311 .59 188 0.655 847 30 238 0.859 110 69

39 0.038 793 05 89 0.238 642 48 139 0.452 517 30 189 0.659 949 47 239 0..S63 140 46

40 0.041 719 68 90 0.242 983 15 140 0.456 720 33 190 0.664 049 96 240 0.867 168 94

41 0.044 737 60 91 0.247 322 90 141 0.460 920 77 191 0.668 148 86 241 0.871 196 11 42 0.047 842 92 92 0.251 661 28 142 0.465 118 61 192 0.672 246 07 242 0,875 221 99

43 0.051 031 78 93 0.255 998 36 143 0.469 313 ,87 1 93 0.676 .341 76 243 0.879 246 57 44 0.0,54 300 36 94 0.260 333 69 144 0.473 ,506 60 194 0.680 435 77 244 0.88.3 269 94 45 0.057 644 86 95 0.264 667 18 145 . 0.477 096 82 195 0.684 528 25 245 0.8,^7 292 00 46 0.061 061 61 96 0.26S 998 70 146 0.481 884 ,59 196 0.688 619 13 24(> (1.S91 :U2 69 47 0.0G4 .540 79 97 0.273 328 07 147 0.486 069 85 197 0.692 70S 41 247 0.895 332 24 48 0.068 096 90 98 0.277 6,55 16 148 0.490 252 74 198 0.696 796 17 248 0.899 3.50 49 49 0.071 708 35 99 0.281 979 88 149 0.494 433 19 199 0.700 882 32 249 0.9<t:! ;!()7 44

50 0.075 377 56 100 0.286 302 01 1,50 0.498 611 .35 200 0.704 966 94 250 0.9(I7 383 09

During fabrication it is rccorn mended that the th(!rniomotor be evacuated wliilo at about 450 °( ' and then

filled with dry gaa and hcrtnctically sealed. It is desirable to have o.xygen present in the gas lilUng to ensure that trace impurities in the platinum will remain in an oxidised state. After completion, the

T.ablc 4. Values of It'cc'c-asC/'os)' (iccordimj to the data <]loeii in Table 2, at the fixed-point temperature's

Fixed point '^68 (K) '68 (^C) WCOT-OS

e-Hj triple 13.81 —2.59.34 0.001 412 06

e-H,^ 17.042 17.042 -256.108 0.002.5:54 44

(' 11., boiling 20.28 —252.87 0.004 4S5 17 No boiling 27.102 —246.048 0.012 212 72

trii)l(! .54.361 —218.789 0.091 972 52

0.^ boiling 90.1.88 -182.962 0.243 799 09

273.15 0 1 lljO boiling 373.15 100 l.,392.596 68

.Sn freezing .505.1181 231 .9681 1.892 570 86

rosistanco elemetit should he stabilized by heating at a leinp(\raturc higher tli.in its intended maximum

operating t(^mi)eratur(5 and in any ease not lower than

4r)U "C. Th(^ insulation resistance of the eompononis sup-

])orting the resistance clement and leads must be high

euougli to avoid signilicant shunting of the element.

For example, care must be taken to avoid condensation of water va,[)our between the leads at low

tomporatin-(;s, and intrinsic leakage in the insulators

tliemselves at high temperatures. The insulators are usually fabricated from mica, silica or alumina., and

these materi.ils normally give adequate intrinsic in- sulation up to 500 °C. ifowever, as the temperature

a])proaches f).30 the problem becomes more critical and errors of 1 mlv or greater may easily occur. In the

case of mica, there is the ad<.Utional difficulty that

significant amounts of water may be released dining

its exposure to tenii)eratutes above 450 °C, and unless

55
Vol. '), No. 2, 1969 The International Practical Temperature Scale of 1968 39

this moisture is removed by periodic pumping or by

a desiccant the insulation resistance will deteriorate

rapidly.

To ensure adequate stability in the resistance and

the temperature coefficients of resistivity, the resistor of a standard platinum resistance thermometer should

be maintained, as far as possible, in an annealed state. Added resistivity may arise both from the accidental

cold working that results from normal thermometer

handling and also as a result of rapid cooling when a thermometer is transferred rapidly from an environment above 500 °C to room temperature. This latter increase in resistance is due to quenched-in, nonequilibrium concentrations of vacancy defects and is retained as long as the thermometer remains below

200 °C. Much of the cold work and all of the quenched-in resistance may be removed by annealing at 500 °C for 30 min.

Significant errors can be caused by radiation loss from the thermometer by total reflection in the constructional components, particularly if these are of

sUica. Such loss in the sheath, but not in the internal

Table 5

Elject of -pressure on the jreezing-point temperatures of melals

Metal Freezing point

at 1 standard atmosphere

(°C)

Pressure coefficient kelvins per kelvins per atmosphere centimetre

of liquid

Mercury -38.862 +0.005 4 +0.000 071 Indium 156.634 +0.004 9 +0.000 033 Tin 231.9681 +0.003 3 +0.000 022 Bismuth 271.442 -0.003 5 —0.000 034

Cadmium 321.108 +0.006 2 +0.000 048 Lead 327.502 +0.008 0 +0.000 082

Zinc 419.58 +0.004 3 +0.000 027 TUitimony 630.74 +0.000 85 +0.000 005

components, can be suppressed by blackening the outer surface of the sheath (e.g., with a colloidal graphite suspension) or by sand-blasting the surface to produce a matt finish. The completed thermometer should be tested to

establish that the depth of immersion is sufficient to avoid heat conduction errors. An effective way of

doing tliis is to confirm that the apparent temperature

gradient in a metal freezing point is in agreement with

that to be expected from hydrostatic effects (see

Table 5). For temperatures below 90 K it is usual to use a

small platinum resistance thermometer, generally not

larger than 5 mm in diameter and 60 mm in length,

that can be totally immersed in a uniform temperature

zone with heat conduction down the leads being suppressed by attaching them to a suitable guard ring. In order to achieve good thermal contact between the

resistor and its surroundings the resistor is contained

in a thin sheath, commonly of platinum about 0.25 mm

thick, which is filled with hehum.

A useful criterion by which the efficiency of the

annealing and the reUability of the thermometer may

by judged is the constancy of its resistance at a

reference temperature. The temperatures of the triple

point of water (273.16 K) and the boiling point of

helium (4.215 K) are commonly used for this purpose.

The first of these is convenient for most high tem7»

perature thermometers, while the second is not only

often conveniently attained for thermometers built

into cryogenic apparatus but has the additional

advantage that the resistance is relatively insensitive to temperature variations. In practice it is found that

variations of resistance at the triple point of water for commercially produced high temperature thermometers should not exceed 4 x 10~* R (0 °C) (equivalent to !=« 1 mK above 40 K), and will not exceed

5 X 10~' R (0 °C) over a reasonable period of use for the very best thermometers when these are handled

with extreme care. For resistance thermometers used

only at temi^eratures of 100 °C or less, variations should

not exceed 5 x 10-' R{Q °C).

The small temperature rise of the thermometers

caused by the measuring current may be determined

by measurements at two currents.

2. Standard Thermocouple

Satisfactory standard thermocouples have been

made of wires of a uniform diameter between 0.35 and

0.65 mm. The thermocouple wires must be thoroughly

annealed in order to ensure constancy of e.ra.f. in use. For this purpose it is necessary to heat the platinum

wire to a temperature of at least 1100 °C and the

platinum-rhodium wire to 1450 °C. If the annealing is done before the wires have been mounted in their

insulators the completed thermocouple must bo

heated again to a temperature of at least 1100 °C until

its electromotive force has been stabilized and local inhomogcncities caused by strain have been removed.

When this has been satisfactorily accomplished the

thermocouple e.m.f. should not be changed by changes

in the temperature gradients along the ^vires ; it should

not change, for example, with increase of depth of

immersion in an enclosxire at a uniform temperature.

3. Pressure In practice pressures are usually determined by means of a mercury column. The mean density of pure

mercury at the temperature t^^ in a barometric

column supported by the pressure f being measured

is given, with sufficient accuracy over the temperature

range from 0°C to 40°C and for the pressures relevant

to these measurements, by the relation

G(py_-Ps,)_

^ ' [l-|-^(<e8-20°C) + 2J(<es-20°C)=]x \\-Jl^--p\

(16)

where

4-18 115 X 10-8°C-i, B = 0.8 X 10-8°C-^ = 4 X 10-" N-i m2,

Q (20°C,/3o) = 13 545.87 kg/m^ is the density of pure

mercury at t^^ =^ 20°C under a pressure =- 1 standard atmosphere (101325 N/m'^).

A sufficiently accurate value of the local gravity may

be obtained by using the Potsdam system and ap- plying a correction of — 14 x 10"^ m/s^ (—14 milligals)*.

Hydrostatic head pressures within the fixed points

* By Resolution 1 (1968). the International Coramittec of Weights and Measures decided that for mctrological purposes,

the value of the acceleration due to gravity at Potsdam,

the reference point of this System, shoulci be taken as 9.812 60 m/s^ instead of 9.812 74 ra/s-, the value adopted ini- tially.

56
40

cells cause small but significant temperature effects; these are summarized in Table 5.

4. Triple Point of Water

The temperature of the triple point of water can

be realized in sealed glass cells containing only water

of high purity and of substantially the isotopic composition of ocean water. The cells have an axial well

for the thermometers and the triple-point temperature

is obtained wherever the ice is in equilibrium with a

liquid-vapour surface. At a depth h below the liquidvapour surface, the equihbrium temperature igg between ice and liquid water is given by

<e8 = -4 + Bh (17)

where A = 0.01 °C and B = -1 x 10-* m-i °C. The

method recommended for preparing a triple-point cell consists of forming a thick layer of ice around the

thermometer well by cooUng from within, then melting

enough of this sheath, also from within, to produce a new water-ice interface adjacent to the well. During

the first hours following the preparation of the cell the

temperature measured in the thermometer well rises fairly rapidly by a few ten thousandths of a kelvin

becoming stable after from 1 to 3 days. This initial change of temperature is probably caused by the

growth of the ice crystals or by the slow disappearance

of strain in the crystals. A cell prepared in this way

and kept in an ice bath is capable of maintaining a temperature constant to about 0.0001 K for several

months. Even with cells from various sources, when

used in this way, the differences in the temperatures

obtained should in any case not exceed 0.0002 K. A

significant rise in temperature of the thermometer

above the triple-point temperature may be caused by

artificial light or sunhght falling on the ice-covered cell and it is therefore recommended that measurements

should be made with the cell suitably shielded from

radiation.

Variations in the isotopic content of naturally

occurring water are such that they will result in

detectable differences in the triple-point temperature.

Ocean water contains about 0.016 moles of deuterium,

^H, per 100 moles of hydrogen, ^H, and 0.04 moles of

I'^O and 0.2 moles of "0 per 100 moles of "O. This

proportion of heavy isotopes is substantially the

highest to be found in naturally occurring M'ater. Continental surface water normally contains about

0.015 moles of ^11 per 100 moles of ; water coming

from polar snow may occasionally contain as little as

0.01 moles of per 100 moles of m.

The operation of purifying the water may slightly

modify its isotopic composition and the isotopic

composition at an ice-water interface is slightly

dependent on the freezing technique.

An increase of 0.001 moles of per 100 moles of

corresponds to an increase of temperature of the

triple point of 0.000 04 K ; this is the difference between the triple points for ocean water and the nor- mally occurring continental surface water. The ex- treme difference in the triple-point temperatures of

naturally occurring water is 0.000 25 K.

5. Triple Point, 17.042 K Point and Boilin'j Point

of Equilibrium Hydrogen

Hydrogen has two molecular modifications, designated by the prefixes ortho and para, caused by

Metrologia

different relative orientations of the two nuclear spins

in the diatomic molecules. The equilibrium ortho para

composition is temperature dependent and at room

temperature is about 75% orthohydrogen and 25%

parahydrogen (so-called "normal hydrogen"). On

liquefaction the composition changes slowly with time

and there are corresponding changes in the physical

properties. At the boiUng point the equilibrium

composition is 0.21% ortho- and 99.79% parahydrogen and the temperature is lower than that of normal

hydrogen by about 0.12 K. The name equilibrium

hydrogen means in this document that the hydrogen

has its eqxiihbrium ortho-para composition at the

relevant temperature. In order to avoid errors in the

realization of these fixed points caused by indetermi- nate composition it is advisable to use equilibrium

hydrogen converted by the use of a catalyst such as

ferric hydroxide. Hydrogen of high chemical purity

should be used as may be obtained by diffusion through

palladium.

The temperature of equilibrium between solid,

liquid and vapour phases of hydrogen can be realized

by using a sufficient quantity of liquid hydrogen wth

some catalyst in a cavity in a copper block in which

platinum resistance thermometers are immersed and

which is surrounded by a vacuum space. The temperature of the block is reduced until the hj^drogen is solidified and then the temperature is allowed to rise slowly and the transition at the triple point is observed. The flat portion of the time -temperature curve can be constant to 0.0001 K for 30 min or more. The temperature of equilibrium between liquid

and gaseous hydrogen is normally realized by the

static method. In this method a cavity in a block of

metal of high thermal conductivity is maintained at a temperature close to the boiling point by immersing it in liquid hydrogen. In order to avoid temperature

gradients due to hydrostatic pressure the liquid hydrogen makes contact with the top of tlie block only, the

lower part being shielded by a vacuum jat kot. The

cavity contains a small quantity of very pure liquid

hydrogen together with some catalyst. The vapour

pressure of this hydrogen is transmitted by a thin tube

of low heat conduction connected to a manometer

outside the enclosure. Precautions must be taken to avoid direct radiation down this tube into the cavity

and to ensure that the tube at every point is at a higher

temperature than the temperature at the surface of

the liquid hydrogen in the cavity. Comparisons are made between the vapour-pressure thermometer thus

formed and platmura resistance thermometers immersed in closely fitting holes in the metal block adja- cent to the cavity.

The vaUdity of the measurements may be checked

by showing that the values obtained arc independent

of the ratio of the volume of the liquid hydrogen to the volume of the vapour in the cavity.

The temperature T^g as a function of the vapour

pressure of equilibrium hydrogen is given to an ac- curacy of a few millikelvins for the range from

13.81 K to 23 K by the equation

\gl- = A + ^l!- + CT,,-rDTl (18)

where

The International Practical Temperature Scale of 1968

57
Vol. 5, No. 2. WOO The International Practical Temperature Scale of 1968

A = 1.711 466, B^- -44.010 46 K,

C = 0.023 590 9 K-i, D = -0.000 048 017 K-^,

75„ 101 325 N/m2.

6. Boilin'j Point of Neon

The boiling point of neon can be realized in a manner similar to that described for hydrogen. The

normal isotopic composition of neon is 0.0026 moles of

"Ne and 0.088 moles of 22Ne per 0.909 moles of ^oNe. The temperature T^^ as a function of the vapour

pressure of neon is given to an accuracy.of +0.0002 K

for the range from 27 K to 27.2 K by the equation

27.102 + 3.3144 f-^- - 1

0.74 f^- iY]k

i.24(i^-ir

(19)

7. Triple Point atid Boiling Point of Oxygen

The triple point and the boiling point of oxygen

can be realized in a manner similar to that described

for hydrogen. Particular care must be given to the

purity of the oxygen in the vapour-pressure thermometer. The ygen is sufficiently pure when the

normal boiUng point remains constant with the

removal of successive fractions of vapour. The temperature as a function of the vapour pressure of

oxygen is given to an accuracy of ±0.0001 K for the

range from 90.1 K to 90.3 K by the equation

^68 — 90.188 + 9.5648

Po 3.69 1

K. (20)

<S'. Boiling Point of Water

The temperature of equilibrium between liquid

water and its vapour is usually realized by the dynamic

method with the thermometer in the saturated vapour.

For precise calibration it is preferable to use closed

systcTtis in which the boiler and manometer are con- nected to a manostat filled with air or preferably

heUum.

The boiler must be constructed in such a way that

contamination of the water is avoided. The thermometer must be protected from radiation emitted by

bodies which are at temperatures different from the

boiling-point temperature. If the equilibrium temperature has been obtained the observed temperature

(reduced to a constant pressure) will be independent

of the time elapsed, variations in the rate of heat

supplied to the liqmd, and the depth of immersion of

the thermometer.

A change in the proportion of deuterium in the

water produces a change in the boiling point of water

in the same direction as in the triple point but to about one third the extent. The temperature t^^ as a function of the vapour

pressure of wat^-r is given, to an accuracy of ±0.0001 K

for the range from 99.9 °C to 100.1 °C by the equation

100 + 28.0216 - 1 11.642

Po

1

7.1

Mutr.il.Miii

(21)

'.). Freezing Point.s of Tin and Zinc

Very reproducible temperatures can be realized by

observing the flat part of the temperature versus time

curve obtained during the slow freezing of very pure

metals.

The melting and freezing of tin and zinc may be

carried out in a crucible of very pure artificial graphite

(99.999°/, by weight) about 5 cm in diameter pro- vided with an axial thermometer well. The depth of

immersion of the thermometer in the metal must be

sufficient to eliminate the influence of thermal con- duction along the thermometer leads on the temperature of the sensitive element. It is convenient to contain the crucible and ingot of metal in a pyrex or

silica tube under an inert atmosphere and to heat it in a metal-block furnace.

The aim of the cooling technique in the determination of the freezing point is to ensure that the thermometer sensor is as nearly as possible enclosed by, and

is in thermal equilibrium with, a solid-Uquid interface

shortly after nucleation there should be present either a completed solid shell nucleated on and thickening

from the crucible wall or a completed solid mantle

induced on the thermometer well. The equilibrium temperature between soUd and

liquid metal varies slightly with pressure. The magnitudes of these variations are given in Table 5. High purity tin (99.9999% by weight) .supercools

by 20 K to 30 K before solidification when cooled from

the liquid state. The following technique has been

successful for nucleating an ingot of tin while avoiding

excessive undercooling of the freezing-point furnace.

Starting with the temperature a few kelvins above the

freezing point, the furnace is slowly cooled at about

0.1 kelvin per minute until the melt reaches its li<iuidus- point temperature ; the sample holder containing the

melt and a monitoring resistance thermometer is then

either withdrawn into the throat of the furnace or removed from the furnace entirely. Under either of

these conditions the sample cools rapidly; when the

rapid rise in temperature that indicates genera!

nucleation is detected the sample holder is (juickly

replaced in the furnace which is still cooling slowly.

Thereafter, as solidification proceeds slowly, a characteristic cooling curve for a high purity metal is realized

having a temperature plateau that is reproducible for a particular sample to ±0.0001 K for durations

dependent on the rate of furnace cooling.

High purity zinc (99.9999 by weight) is treated

in a somewhat different manner since it docs not

supercool excessiveh^ A thin layer of solid metal is produced on the central thermometer well by removing the thermometer when the melt reaches the liquidus temperature, and either cooling the thermometer

to room temperature and replacing it, or inserting a

silica rod in its place for about 30 s before rc})lacing

the thermometer.

A criterion for checking that the j)urity of a sample of zinc or of tin is satisfactory is that its melting

range is not more than 0.001 K.

JO. Freezing Points of Silver and (laid

The equilibrium temperatures between the solid and liquid phases of silver and of gold may be realized in covered crucibles of either very pure artificial

58
42 The International Practical Temperature Scale of 1968

Table 6. Secondary reference, poitds

Equilibrium state [iiternationnl I'nicl icil Tempera tun'

Equilibrium between the solid, liquid and vapour phases of normal hydrogen (triple point of normal

hydrogen)

Equilibruim between the liquid and vapour phases of normal hydrogen (boiling point of normal

hydrogen)

Va 68

,4 - 1.734 791, ^ = -44.623 68 K, C = 0.023 186 9 K-i, D -0.000 048 017 K-2

for the temperature range from 13.956 K to 30 K.

Equilibrium between the solid, liquid and vapour phases of neon (triple point of neon)

Equilibrium between the liquid and vapour phases of neon

^=4.61152, 5 = -106.3851 K, C = -0.036 833 1 K->, Z) = 4.248 92 < 10"* K-2

for the temperature range from 24.555 K to 40 K.

Equilibrium between the solid, liquid and vapour phases of nitrogen (triple point of nitrogen)

Equilibrium between the liquid and vapour phases of nitrogen (boiling point of nitrogen)

, P A B

'g^ = ^ + -jr- Pn •' 68 A = 5.893 139, B = -404.131 05 K. C

E = 72.5342 x 10-« K-^

+ Clg

T

-2.3749, D = -0.014 250 5 K-i,

for the temperature range from 63.148 K to 84 K.

Equilibrium between the liquid and vapour phases of oxygen

68 I C

'g; A = 5.961 546. E = 50.8041 X

A +

Po B " -467.455 76 K.

10-« K-2

+ DT^^ + ET'I,

-1.664 512, D -0.013 213 01 Kfor the temperature range from 54.361 K to 94 K.

Equilibrium between the solid and vapour phases of carbon dioxide (sublimation point of carbon dioxide)

^68 = 194.674 H- 12.264 - ij—9.15^^ - ij^ K

for the temperature range from 194 K to 195 K.

Equilibrium between the solid and liquid phases of mercury (freezing point of mercury)-'

Equilibrium between ice and air-saturated water (ice point)

Equilibrium between the solid, liquid and vapour phases of phenoxybenzene (diphenyl ether)

(triple point of phenoxybenzene)

Equilibrium between the solid, liquid and vapour phases of benzoic acid (triple point of benzoic acid)

Equilibrium between the solid and liquid phases of indium (freezing point of indium)*

Equilibrium between the solid and liquid phases of bismuth (freezing point of bismuth)"

Equilibrium between the solid and liquid phases of cadmium (freezing point of cadmium)"

Equilibrium between the solid and liquid phases of lead (freezing point of lead)" Equilibrium between the liquid and vapour phases of mercury (boiling point of mercury)

^RR — 356.66 -I- 55.552 fP--l)- 23.03 (-1-1)% 14.0 (^--iV

\Po ' \Po I \Po I °c

for p ^ 90 X 10'N/m2 to 104 x 10' N/m^.

Equilibrium between the liquid and vapour phases of sulphur (boiling point of sulphur)

^BR — 444.674 + 69.010 Ji 1 - 27.48 1 -19.14 1 \Po

for - 90 X 10^ N/m2 to 104 x 10* N/m^.

Equilibrium between the solid and liquid phases of the copper-aluminium eutectic Equilibrium between the solid and liquid phases of antimony (freezing point of antimony)'

Equilibrium between the solid and liquid phases of aluminium (freezing point of aluminium)

Equilibrium between the solid and liquid phases of copper (freezing point of copper)

Equilibrium between the solid and liquid phases of nickel (freezing point of nickel)

Equilibrium between the solid and liquid phases of cobalt (freezing point of cobalt) Equilibrium between the solid and liquid phases of palladium (freezing point of palladium)

Equilibrium between the solid and liquid phases of platinum (freezing point of platinum)

Equilibrium between the solid and liquid phases of rhodium (freezing point of rhodium)

Equilibrium between the solid and liquid phases of iridium (freezing point of iridium)

Equilibrium between the solid and liquid phases of tungsten (temperature of melting tungsten)

(27)

13.9.56

20.397

(23)

24.555

(24)

63.148 77.348

(25)

(26)

(28)

717.824

(29)

821.38

903.89 933.52 1357.6 1728

1767

1827 2045

2236

2720

3660

259.194

252.753

-248.595

-210.002

-195.802

194.674 - 78.476

234.288 - 3H.862

273.15 0

300.02 26.87

395.52 122.37 429.784 156.634

544..592 271.442

594.258 321.108

600.652 327.502

629.81 356.66

444.674

548.23

630.74

660.37

1084..-I 1455

1494 1554

1772 1963 2447

3387

» See Table 5 for the effect of pressure variations on these freezing points.

59
Vol. .1, No. •>. IWi'J The International Practical Temperature Scale of 1968 43

graphite, or ceramic material or vitreous silica. If

grapliit^* i.s used it i.s advisable to prevent the access of

air to the cniciljle so that tlu? tjrapliite is not oxidised.

IVl()lt(^n silver should \w. f)rotcctcd so as to prevent the

soluliun of o.vy>^(^n into it and the consequent de- pression of the freezing point.

The ingot of metal should he heated to a uniform

temperature a few icelvins above the melting point

of the metal and then cooled slowly. The thermocouple to be calibrated, mounted in a protecting tube

of suitable refractory material, ^vith refractory insulators separating the two wires, is immersed in the molten

metal which is then allowed to freeze. The depth of

immersion of the thermocouple in the metal must be

sufficient to eliminate the influence of thermal conduction along the wires of the thermocouple.

That the equilibrium temperature is obtained can

be checked by the following criteria : the electromotive

force of the thermocouple .should be independent of

small variations in the depth of immersion in the ingot

of metal during successive freezes, and should remain

constant for at least 5 min during one freeze. A blackbody at the freezing point of gold is re- quired for the establishment of the reference temperature for radiation temperature measurements. For the

realization of ^uch a blackbody the crucible containing

the gold mitCt be modified to provide a uniform

temperature enclosure immersed in the gold. A blackbody enclosure is more readily achieved if the material

employed for its construction has already a liigh

emissivity and therefore graphite is very suitable for

this purpose.

11. Secondary Reference Points

As well as the defining fixed points of the IPTS-68

given in Table 1 other reference points are available.

Some of these points and their temperatures on the

IPTS-68 are given in Table 6. Except for the triple

points and the vapour pressure-temperature equations

each temperature is that for a system in equilibrium

under the pressure of one standard atmosphere.

Appendix I.

History of the Development

of the International Temperature Scales

;

Differences between the IPTS-68 and the IPTS-48

The International Temperature Scale was adopted

in 1927 to overcome the practical difficulties of the

direct realization of thermodynamic temperatures by

gas thermometry and to unify the existing national

temperature scales. It was introduced by the 7th

General Conference of Weights and Measures with the

intention of providing a practical scale of temperature

which was easily and accurately reproducible and

which gave as nearly as possible thermodynamic

temperatures.

The International Temperature Scale was revised

in 1948. The experimental procedures by which the

Scale was realized remained substantially unchanged,

but two amendments were made to the definition of

th(; Scale resulting in appreciable changes in the

numerical values assigned to measured temperatures.

Th(! change in the value of the temperature of the

freezing point of silver from 960.5 °C to 960.8 °C

changed temperatures measured with the standard

thermocouple (range 630 °C to 1063 °C) ; the maxiniuni

difference was about 0.4 K near 800 °C. The atloption of

the value of 0.014 38 metre kelvin instead of 0.014 32

metre kelvin for the radiation constant C2 changed all temperatures above the freezing pointof gold, while the use of the Planck radiation formula instead of the Wien

formula affected the very high temperatures. Temperatures above the freezing pointof gold were decreased, for example, by 2.2 K at 1500 °C and by 6 K at 2000 "C. Also at this revision, in order to secure international

uniformity of nomenclature, the 9th General Conference of Weights and Measures decided to abandon the

word "Centigrade" and its French equivalent "Centesimal" in favour of the name "Celsius". That is "^C!" was now regarded as the abbreviation of "degree

Celsius".

Table 7. Estimated uncertainties of the assigned values oj the defininy fixed points in terms of thermodynamic temperatures

Defining fixed point A.ssigned Estimated

value uncertainty

(K)

Triple point of equilibrium

hydrogen

17.042 K point

13.81 K 0.01 17.042 K 0.01 Boiling point of equilibrium

hydrogen 20.28 K 0.01 Boiling point of neon 27.102 K 0.01 Triple point of oxygen .)4.361 K 0.01 Boiling point of oxygen 90.188 K 0.01 Triple point of water 273.16 K Exact by

definition Boiling point of water 100 "C 0.005 Freezing point of tin Freezing point of zinc

231.9681 °C 0.015 419.58 0.03 Freezing point of silver 961.93 °C 0.2 Freezing point of gold 1064.43 °C 0.2

An amended edition of the 1948 Scale was adopted

by the 11th General Conference of Weights and

Measures under the new title "International Practical

Temperature Scale of 1948 (amended edition of I960)",

the numerical values of temperature remaining the same as in 1948. The new edition incorporated the new definition of the kelvin given by defining the

thermodynamic temperature of the triple point of

water as exactly 273.16 K (10th General Conference

of Weights and Measures 1954, Resolution 3). It was

also recognized at this time that the IPTS no longer

represented the thermodynamic tcmperatmo as

closely as possible and the text included a section on

the differences between them. The TPTS-()8 has bo(Mi

devised to bring these differences within the limits of

accuracy to which the thermodynamic tein|)eratur('s are known (see Table 7) and to extend the Scale to lower temperatures.

The IPTS-68 differs from the IPTS-48 in tho

following ways. The lower limit of the Scale is now

13.81 K instead of 90.18 K. The values assigned to the

defining fixed points are modified where necessary to conform as nearly as possible to the thcrmotlynamic

temperatures, the only points remaining unchanged

being the triple point of water, which is permanently

fi.xed by definition, and the boihng point of water. The

interpolation instruments remain the same as before

but the standard platinum resistance thermometer is

60
44 The International Practical Temperature Scale of 1968 Metrologia

Table 8 Approximate diljerencts (t68 'is)', in kelvina, between the values of temperat are given by thelPTS oj 19fi8 and the IPTS of 1948

t op

68 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100

-100 0.022 0.013 0.003 -0.006 -0.013 -0.013 -0.005 0.007 0.012 - 0 0.000 0.006 0.012 0.018 0.024 0.029 0.032 0.034 0.033 0.029 0.022

t °c 68 0 lU 20 30 40 50 du TA 80 90 100

0 0.000 -U.UU'* -0.007 -0.009 -0.010 -0.010 f\ Ai i\ A AAQ -0.006 -0.003 0.000 100 0.000 0.007 0.012 0.016 0.020 A AOK A A-)(l 0.034 0.038 0.043 200 0.043 \).\r± 4 0.051 0.054 0.058 0.061 u.uo / 0.069 0.071 0.073 300 0.073 (\ f\'7A 0.075 0.076 0.077 0.077 A f\nn A i\'in u.u/ / 0.077 0.076 0.076 400 0.076 v.viO 0.075 0.075 0.074 0.074 A Ci'lA U.U/4 A ATfC 0.076 0.077 0.079 500 0.079 0.085 0.089 0.094 0.100 A 1 1 0.126 0.137 0.150 600 0.150 U.lDO 0.182 0.200 0.23 0.25 A -yo A •? 1 U..>1 0.34 0.36 0.39 700 0.39 0.42 0.45 0.47 0.50 0.53 0.56 O.o8 0.61 0.64 0.67 800 0.67 0.70 0.72 0.75 0.78 0.81 0.84 0.87 0.89 0.92 0.95 900 0.95 0.98 1,01 1.04 1.07 1.10 1.12 1.15 1.18 1.21 1.24 1000 1.24 1.27 1.30 1.33 1.36 1.39 1.42 1.44

/ °c, '68 ^ 0 lUU 200 300 400 500 DUU 800 900 1 000

1000 1.5 1.7 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 2000 3.2 3.5 3.7 4.0 4.2 4.5 4.8 5.0 5.3 5.6 .5.9 3000 5.9 6.2 6.5 6.9 7.2 7.5 7.9 8.2 8.6 9.0 9.3

now required to have PF(100 °C) at least equal to 1.3925 instead of 1.3920. Over the temperature range

from 90.188 K to 273.15 K the Callendar-Van Dusen

equation is no longer used for interpolation, instead

the reference function W^ccT-es (Tes) ^ employed.

Above 0 °C, the Callendar equation is modified so that interpolated values of temperature conform more

nearly with thermodynamic-temperature values. Finally the latest value of Cg, namely 0.014 388 metre kelvin

is introduced in the Planck equation for determining

temperatures above the freezing point of gold. The

effect of all these changes is summarised in Table 8

which gives the differences between the values of

temperature derived from the IPTS-68 and the IPTS48. In the range from 13.81 K to 90.188 K the IPTS-68

is based on the average of four "national scales" and

on chosen "best" temperatures for the defining fixed

points. These national scales are each defined in terms of platinum resistance thermometers calibrated against a gas thermometer and are highly reproducible.

The differences between the IPTS-68 and the national .scales are published in Metrologia, Vol. 5, p. 47,

(1969). This allows the use of the national scales and

those (lifforonoes to j{ivc a close approximation to the

1I'TS-«S. Tho toxt of this dofinition of the IPTS-6K confoi'iUH with tho (looision of tho KUli Conoral ConfcroncH^

of Woights and Measures to denote the unit of therinodynaniio temperature by tho name "kolvin", symbol

K, and to denote a tc^injunaturo interval by tho same

unit and symbol or b^' "degree Celsius" or ' "C".

Appendix II. Practical Scales of Temperature

for Use over the Range from 0.2 K to 5.2 K

Temperatures can be derived from measured vapour

pressures of *He and ^He. The upper limits for use are

set by the critical points of the gases (5.2 K for ''He and 3.3 K for ^He) and the lower limit by the vapour

pressures becoming too low for practical measurement.

The "1958 ^He scale" and the "1962 ^He scale" are the recommended scales' in whicli temperatures are denoted by T^^ and T^gg respectively.

The "1958 *He scale", recommended in 1958 by

the International Committee of Weights and Measures,

is defined by a table of ^He vapour {>ressures versus temperature (Comite Consultatif de Thermometrie, 5e

session 1958, p. T 192 and Proces-Verbaux CIPM.

26-A, 1958, p. T192)^

The "1962 ^Ile scale", recommended by the Inter- national Committee of Weights and Measures in 1962,

is defined by an equation giving the vapour pressure

of ^He as a function of temperature (Comite Consultatif de Thermometrie, 6^ session 1962, p. 184)^ In the temperature range between 0.9 K and the

critical temperature of ^He the temperatures T^g and

Tfl2 *ire believed to be in agreement to within 0.3 mK.

' Recent meaaurements by the acoustic thermometer give

toinporatures higher than those of the He vapour pressure scales; t ho difference at tlie boiling ])oint of ^He is about 0.008 K.

" An e.\})iuulc(l form of this table together with auxiliary

information i.s given in the .T. Res. Nat. Bur. Standards t)4.4.1 (1060).

" A table of values and information on measuring vapour

presaiinis are given ii\ the J. Res. Nat. Bur. Standards (>8 .-\,

.547, .5,59, .567. 579 (1964).

Offprint from ^Metrologia"

Journal International de Metrologie scientifique — International Journal of scientific Metrology

Internationale Zeitschrift fiir wissenschaftliche Metrologie. Vol. 5, No. 2, April 1969, Pages 35—44

Springer- Verlag, Berlin Heidelberg New York

61
Appendix B. Comparison of Assigned Fixed Point Values and Interpolating

Formulae in the Temperature Ranges Defined by the Platinum Resistance

Thermometer for ITS-27, ITS-48, IPTS-48, and IPTS-68

(a) Comparison of fixed point values

Defining fixed points ITS-27 °C ITS-4S °C IPTS-4« °C

IPTS-68

K

TP e-H. BP e-H,, 25/76 atm. NBP e-H2 NBP Ne

TPO2

NBP O2

Ice point

TP H2O

NBP H2O

FP Sn

FP Zn

NBP S

Some secondary fixed points

SP CO2-

FP Hg...

Ice point.

TP H2OFP Sn...

FP Pb...

FP Zn...

NBP S__

FP Sb-__

— 182.97 — 182. 0.000 0

100.000 100

444.60 444.

-78.5 -78.

-38.87 -38.

0. 231.85 231. 327.3 327. 419.45 419.

630.5 630.

182.97

-259.34

-256.108

-252.87 -246.048

-218.789

- 182.962

0.01 0

100 100

231 419.505 419

444.6

-78.5 -78

-38.87 -38

0.000 0

231.91 327.3 327

444.674

630.74

13.81 17.042 20.28 27.102

54.361 90.188

273.16

373.15 505.1181

692.73

194.674

234.288

273.15

600.652

717.824

903.89

TP= triple point, BP= boiling point, NBP= normal boiling point, FP= freezing point, SP= sublimation point.

62
Appendix B. Comparison of Assigned Fixed Point Values and Interpolating

Formulae in the Temperature Ranges Defined by the Platinum Resistance

Thermometer for ITS-27, ITS-48, IPTS-48, and IPTS- 68—Continued

(b) Comparison of temperature ranges and interpolating instrument and formulae

ITS-27 ITS-4e IPTS-48 IPTS-68

R{\00)IR(0) of Pt. g 1.390 > 1.3910 g 1.3920 g 1.3925

Temperature Range -190 toO°C - 182.970 to 0 °C - 182.97 to 0 °C 13.81 to 273.15 K

Intprnolatinff Pormiila

Calibration Temperatures-

^ quartic

NBP02,ice point,

NBP H2O, & NBP S

^ quartic

NBP O2, ice point,

NBPHiO.&NBPS

^ nnartir

NBP02,TPH20,

NBPHaC&NBPS

or FP Zn

rpiprpnf^p fiinptinn

plus deviation

^ polynonlials

TPe-H2,BPe-H2

(25/76 atm), NBP

e-H2,NBPNe,

TP02,NBP O2,

TPH20,NBPH20

or FPSn,&FPZn.

Temperature Range 0 to 660 °C 0 to 630.5 °C 0 to 630.5 °C 0 to 630.74 °C

Interpolating Formula

Calibration Temperatures

quadratic

ice point, NBP H.O,

& NBP S

^ quadratic

ice point, NBP H2O,

& NBP S

^ quadratic

TP H2O, NBP H2O,

& NBP S

Quadratic plus * cor- rection function

TP H2O, NBP H2O

or FP Sn, & FP Zn.

^ Quartic interpolation formula: R(t) = R{0)[l+At + Bf^ + C {1-100)1^].

" Quadratic interpolation formula: R(t)=R(0) {l+At + Bt^).

*^ Reference function

20 7'68= 21 ^'[In ^* (^)V (Values of are given in Appendix A.)

i = 0

Deviation polynomials

13.81 to 20.28 K:

20.28 to 54.361 K:

54.361 to 90.188 K: -182.%2 to0°C:

where 7'=7'68, t = tes, W*=W

AW(T)=W{T)-fP'*(T)

AW{D = A,+B,T+C,r + D,T^

AW{D = A2 + B^T+CzT^ + DiT^

AW{T^ = A3 + B3T + C3T^

\W(t) = A4t + C4tHt-m °C)

- oo» - -00, CCT-68' ^Quadratic interpolation formula plus correction function: R{t')IR (0) = I -\- At' ^ Bt"^ plus

0.045 (y^) (y^-l) (419.58 °C~0 (630.74 °C"0"

63
Appendix C. Tables of Difference in the Values of Temperature Between

the IPTS-48 and IPTS-68 and Between the NBS-1955 and IPTS-68 in the

Temperature Range 13 K to 630 °C

The values given are the calculated diflferences

between values of temperature on the IPTS-48

and on the IPTS—68, and, below the oxygen point,

between values on the NBS-55 scale and on the

NBS-IPTS-68 (see sec. 7.7). The calculations

above the oxygen point are based upon the differ- ences of the scales as defined in their respective

texts (see Appendix A). The assumption is made

that the value of a temperature on each scale is uniquely defined by its text. (In contrast, refer to the last paragraph of sec. 2.2.) The further as- sumption is made that, on the IPTS-48, the value

419.505 °C recommended by the text of the IPTS-48

for the freezing point of zinc is exactly correct. The coefficients employed were a = 0.(X)39259665,

8=1.491356, and /3 = 0.1109843 on the IPTS-48.

These values were derived from the reference function (W*) and the corresponding values of a and

8 given in the text of the IPTS-68 (see Appendix A).

The calculation of the differences between Vcdues on the NBS-55 scale and the IPTS-68 are based

upon the estimated differences published by

Bedford, Durieux, Muijwijk, and Barber (Metrologia 5, 47 (1969)). The calibration of a thermometer

on the NBS-55 scale was first adjusted by these

pubhshed diflferences. The difference (AJF) between each of these adjusted values and the cor- responding value of the reference function (W*)

was then smoothed by a least squares method using

the nearest five points. These smoothed values were added to the reference function values and the

resulting table subtabulated at 0.1 K intervals by

cubic interpolation. The differences between these

results and the calibration on the NBS-55 scale are tabulated here as the estimated differences

between the NBS-55 scale and the IPTS-68.

The two scales formerly used, i.e., the IPTS-48

and the NBS-55 scale, as reahzed and maintained

at the National Bureau of Standards, were con- tinuous through their point of joining, the normal

boiling point of oxygen. Bedford, et al. (loc. cit.),

however, estimated that the "oxygen point" upon

which these scales were based was too hot by

0.0019 K. The correction to be appUed to values on

the NBS-55 scale to get values on the NBS-IPTS68 scale is, as previously stated, based on the estimates of Bedford, et al. and, therefore, includes a

correction for their estimated error in the hotness

of the NBS "oxygen point." In contrast, the cor- rections to be used to adjust values on the IPTS-48

to values on the IPTS-68 are based only upon the

diflferences in the definitions of the two scales.

This tacitly assumes that all fixed points are perfectly realized. The result of using the different

bases for the temperature diflferences in the two

ranges (above and below the oxygen point) is a

discontinuity in the estimated corrections at the

oxygen point.

64
The Corrections to Values of Temperature on the NBS-55 Scale. (The correction (Teg — Tnbs-ss)

is given with the argument in Tnbs-55»)

T Corr. T Corr. T Corr. T Corr. K mK K mK iK mK K mK

15 .

0

2. 04 20. 0 8. 94 25.0 7 . 76

15.1 2. 36 20. 1 8. 96 25.

1

7 . 71

15.2 2. 61 20. 2 8. 95 25.2 7 . 65

15 .

3

2. 90 20. 3 8. 92 25.3 7 . 61

15.4 3. 19 20.,4 8. 90 25.4 7 . 55

15.5 3. 47 20..5 8. 88 25.5 7 . 50

15.6 3. 72 20..6 8. 87 25.6 7 . 47

15.7 4. 02 20.,7 8. 84 25.7 7 . 42

15.8 4. 35 20..8 8. 85 25.8 7 . 39

15.9 4. 69 20..9 8. 89 25.9 7 . 36

16. 0 5. 02 21

.

. 0 8. 90 26.0 ry '7 o 7 . 32

16.

1

5. 31 21

.

. 1 8. 92 26.

1

ry o o 7 . 28

16. 2 5. 55 21

.

.2 8. 90 26. 2

ry rjc 7 . 25

16.3 5. 73 21

,

.3 8. 88 26. 3

ry r\T 7 . 23

16.4 5. 92 21

,

.4 8. 86 26.4 7 . 21

16.5 6. 10 21

,

.5 8. 87 26.5 ry o r\ 7 . 20

16.6 6.,27 21

,

.6 8. 86 26.6 ry 1 ry 7 . 17

16.7 6. 48 21 .7 8.,87 26.7 7 . 14

16.8 6.,72 21 .8 8.,90 26.8 7 . 12

16.9 6.,97 21 .9 8.,89 26.9 7 . 10

17. 0 7,,13 22 .0 8.,89 27. 0

ry no 7 . 08

17.

1

7. 27 22 . 1 8.,89 27.

1

ry r\ 7 . 06

17.2 7. 45 22 .2 8..88 27. 2

ry r\ A 7 . 04

17.3 7.,67 22 .3 8,.87 27.3 ry r\ cr 7 . 05

17.4 7. 89 22 .4 8..84 27.4 7 . 06

17.5 8.,04 22 .5 8,.83 27.5 ry r\r7 7 . 07

17.6 8.,17 22 .6 8 .84 27.6 ry r\ O 7 . 08

17.7 8.,27 22 .7 8 .82 27.7 rr r\r\ 17.8 8. 33 22 .8 8 .82 27.8 1 . iU

17. 9 8,,41 22 .9 8 .81 27.9 ry T T

( . 11

13.0 -12. 01 18. 0 8. 50 23 . 0 8 .81 28.0 ry n r>

1 . 12

13.1 -10. 63 18.

1

8.,59 23 . 1 8 .80 28.

1

ry -I '7

1 . io

13.2 -9. 27 18. 2 8. 65 23 .2 8 .75 28. 2 1 . 14

13.3 -8. 13 18. 3 8.,69 23 .3 8 .69 28.3 1 . 14

13.4 -7.,07 18.4 8.,76 23 .4 8 .62 28.4 ry 1 ry (.11

13.5 -6.,11 18.5 8..84 23 .5 8 .57 28 .

5

T on

13.6 -5. 29 18.6 8..86 23 .6 8 .55 28. 6 1 . do

13.7 -4.,45 18.7 8,.87 23 .7 8 .50 28.7 ry QC 1 . 20

13.8 -3.,64 18.8 8,.87 23 .8 8 .46 28.8 1 . 2o

13.9 -2,.88 18.9 8 .90 23 .9 8 .41 28.9 1 . oU

14.0 -2..20 19. 0 8,,90 24 . 0 8 .36 29 .

0

( . o2

14.

1

-1

,

.61 19.

1

8 .92 24 . 1 8 .31 29 .

1

( . o4

14.2 -1,.14 19.2 8 .94 24 .2 8 .24 29 .

2

•7 "ZQ

/ . OO

14.3 .68 19.3 8 .96 24 .3 8 .17 29.3 ry jtr\

1 . 42

14.4 .23 19.4 8 .87 24 .4 8 .11 29.4 ry A

I . 4b

14.5 .17 19.5 8 .95 24 .5 8 .04 29.5 ry r- (~\

14.6 .51 19.6 8 .96 24 .6 7 .99 29.6 '7 C y1

I . 04

14.7 .88 19.7 8 .99 24 .7 7 .95 29 .

7

(.Of

14.8 1 .28 19.8 8 .98 24 .8 7 .89 29.8 7.61

14.9 1 .68 19.9 8 .96 24 .9 7 .83 29.9 7.65

15.0 2 .04 20.0 8 .94 25 .0 7 .76 30.0 7.69

65
The Corrections to Values of Temperature on the I\BS-55 Scale. (The correction (Tes — Tnbs-ss)

is given with the argument in Tnbs- ss*) — Continued

T

K

Corr. mK

T

K

Corr. mK

T

K

Corr. mKT

K

Corr. mK

30 .

0

rr

I .. oy OO A. u ± ± . 0*7 UO 4UAn A. U lO . AO4/C 40A R . A, U 15 . 12

30 .

1

rr

1 ., 1 o OO . J. J. X . 141 A 4UA A . 1 lO ACi . 4b 40A R . 1 15 . 09

30 .

2

rr

1 . rrc OO O. <C 11 . <cb /I A 4U o. d 10 . ouRn A R 40 . o 15 . 06

30 .

3

rr

( . rrry

. 1 I oO "Z

. o 11 . "70 oy >1 A 4(J

'7

. O 10 RT . Oo A C 40 .

'7o 15 . 02

30 .

4

rr

I . oO . 4 1 1 . 01 y1 A 4U . 4 10 . RT0 / 40 . yl

, 4 14 . 99

30 .

5

rr

1 ., 82 OO . o 1 1 . czbo yl A 40 . 5 liD . oy yl C 45

.

5 14 . 96

30 .

6

rr

1 .. 85 OO . b 1 1 . f o

A A 40 . 6 io CO . bd A CZ 45 ., 6 14. 92

30.7 rr7

,

. 89 OO rr

. I 11 . PO" 8 / 40 . 7 lo C A. b4 A CZ 45

.

rr

. 7 14. 89

30 .

8

rr

1 ., 92 OO p. 8 11 . QOy8 40 . 8 10 cc . bb A C 45 . o8 14 . 85

30 .

9

rr

I .. y 1 OO Q. y

1 oId . 1 AlU 4U . y 10 . CQby yl K 45 ., y 14. 82

31 .

0

oo .. uu ob A. (J1 oId . dd A 1 41 A ID . /

1

yl C 4b

.

A, U

"x A 14. 80

31 .

1

oo .. Do ob . 1 Id LAo4 A T 41 . 1 10 r7A

. 1 4

A C 4b

.

, 1

1 A rrrr 14 . 77

31 .

2

Qo . AC ob o. <c 1<C . 4bA C A T 41 o. d 10 . (

b

A C4b

.

o. d

T A rr A 14.74

31 .

3

Qo

.

AQ . ijy Ob"ZC •z

. o 1<C . OORP A 1 41 . O 10 ,. TP( o

A C 4b

.

o

^ A rr 1 14.71

31 .4 £5 .. io OO . 4 Lc. 1 O ., DC? 41 . 4 lO 7Q 4b

.

A

. 4 14 . 69

31 .

5

Q . X£3 OO . o 1 <c .. PIOl 41A 1

. 0 lO . pnou 40Ad .. O

1 A ^ C 14 . 65

31 .

6

p OO OO7fi c. D 1<C .. C31 41A ^

. cb lO . pnou 40/IC . c. b 14 . 62

31 .

7

Qo

.

0'7

. C. 1 OO rr

. 1 lO .. U/C 41 rr lO . pnou ylC 40 . rr

. 1 14 . 5y

31 .

8

p . Ol OO Q. O lO .. 1<0 41A 1

. op lO . PIOl 4b/IC ., pO

1 yl cr c 14 . 56

31 .

9

p . O ( OO Q 1

"7 lO . 9*7

. cO 41 Q. y lO . olPI 4b/I C

.

. Qy

1 A CZ A 14 . 54

32 . 0 po , A O . 4/; O 1 A. V 1

"7 lO . "79 A O 4(d lO . olPI 4/IT1 .. u

1 A C A 14 . 50

32 .

1

p , 4t5 O"ZTf . 11

"7 lO . 4/i AO A o 4<c . 1 10 . oupn 4/IT( .. 1 14 . 48

•7 0 O 32 .

2

p . Do "ZT O ( O. <C1

"7 lO . 0<c ^ o 4d o. d 10 . oupn 4/IT( . o. dt A A A 14 . 44

•7 0 '7 32 .

3

p . oy "ZT O / •z

. O

1

"7 lO . C9b<c A O 4/;•7

. o 10 . ry yIT 4 / ,. o

T A A 14. 42

•7 *^ >l 32 .4 pO . o4 "ZT O ( . 4 lo . r 1

yl O 42 . 4 lo TQ . ry yl rr 4 f ,. 4

T A ^7 r\ 14 . 39

32 .

5

po TA "ZT O f . 0 lo PA . oU A O 42 . 5 10 TQ . (y yIT 4 / .. 5 14. 35

32.6 oo

rrc "ZT O f . b lo PQ , 8y A O 42 . 6 lo rrrr

. 1 1

yl T4 / .. b 14 . 32

32.7 oo

OO O 1 rr

. ( lo QP . y8 42 . 7 lo rrc

. IO yl "7 4 / . rr

. I 14 . 28

32 .

8

po OP . o8 O / o. o 14 -. ACUb A O 42 . 8 10 rr A

. 1 4

yl T4 f . o. 8 14. 25

32 .

9

po . yo ZTO f . y 14 . 10 42 a. y lo TT . / O

yIT 4 1 .. y 14 . 21

•7 '7 f\ 33 .

0

y AO OOZP . u 14 . O A

. <d4 43 . 0 lo rr 1

. 1 1

A O 48

.

A. U 14. 17

'7 "7 T 33 .

1

1 A Oo"ZQ . 1 141 /I

, "79

. od A T 43 . 1 10 CP . b8 yl O 48

.

. 1 14 . 13

•7 '7 O 33 . 2 y . ±y "ZQ Oc3 Q. <c 141 A . "70

. oy 43 o. d 10 . CTb 1 48A P

.

o. d 14 . 08

•7 '7 '7 33 .

3

y . d. ( OC3 ZP o 141 A . /IT . 4 / yl 43 . o

T R 10 ,. CRbO 48A P

.

. o 14 . 03

33 .

4

y . OO "ZP Oi3 . 4 14 , f\A 04 yl T 43 . 4

1 R 10 .. bo 48A P

.

A

. 4 13 . 98

"7 C 33 . 5 Q . 40 "ZP 0£3 .. O 14 ,. cnbu yl 'Z 43 . O

1 R 10 . RQ . oy A O 48 . 5 13 . 93

33 .

6

Q . 04 OOZP ,. b 14 ,. ccbb 43 . bT R 10 .. ObRC 48A P

.

c, b 13 . 88

33 .

7

Qy . DO Oo"ZP ,, ( 141 A , '7'7

. f o 43 rr

. 1 1 R 10 . RT , Oo 48A P

.

rr

. 1 T "7 O "7 13 . 83

<53 . o y . 1 O OO"ZP ,. po 141 A .. PAou A T 43 o. 8

1 R 10 . R 1

, 01 48yl P

.

p8 13 . 78

•7 "7 r\ 33 .

9

y PT . OO Oo"ZP .. y 141 /I.. obPC yl T 43 . y lo

.

A O 48 A O 48 . y 13 . 74

33 . 0 y . QOy<c oy"ZQ

.

A 141 A .. QOyic /I yl 44 A lo

.

A C 40 yl Q4y

.

AU 13 . 68

34 . 1

1 niU AO "ZQ oy

.

, 1

1 A 14 ., QTy / A A 44 . 1 lo

.

A O 4d yl C\ 4y

.

1 13 . 62

34 .

2

1 nlU . io "ZQ oy

.

o lt3 . AO , Ud 44 . 2 lo

.

3y yl ri 49 . 2 13 . 56

•7 >1 '7 34 .

3

ID O T. 23 •ZQ oy

.

•73 lo

.

AOU8 44 . 3 15 . 36 49

.

3 13 . 50

34.4 10 . 34 3y

.

4 15

.

14 44.. 4 15

.

33 49

.

4 13 . 45

34. 5 10 .45 39. 5 15. 19 44..5 15. 30 49. 5 13.39

34. 6 10 .57 39. 6 15. 24 44,.6 15. 27 49. 6 13.33

34.7 10 .68 39. 7 15. 29 44.,7 15. 23 49. 7 13.27

34.8 10 .81 39. 8 15. 34 44.,8 15. 19 49. 8 13.21

34.9 10 .92 39. 9 15. 38 44.,9 15. 16 49. 9 13.16

35.0 11 .03 40. 0 15. 42 45.,0 15. 12 50. 0 13.09

66
The Corrections to Values of Temperature on the ]\BS-55 Scale. (The correction (Teg— Tnbs-ss)

is given with the argument in TnBS-55') — Continued

T Corr. T Corr. T Corr. T Corr. V mK K mK K mK K mrv.

OU . U 13 09 55 0 7.07 60 .0 - .80 65 .0 — . uy

. ± 13 02 55. 1 6.88 60 . 1 - .88 65 . 1 — . U4

o DU . /c 12 96 55. 2 6.70 60 .2 - .95 65 .2 . Uo

OU . o 12 90 55. 3 6.51 60 .3 -1 .03 65 .3 . Ui3

OU . 4 12 83 55. 4 6.33 60 .4 -1 .10 65 .4 . 14

OU . o 12 77 55. 5 6. 13 60 .5 -1 .16 65 .5 . on<cu OU . D 12 70 55. 6 5.95 60 .6 -1 .22 65 .6 . tCO OU . ( 12 63 55. 7 5.76 60 .7 -1 .26 65 .7 . o<c OU . o 12 56 55. 8 5.57 60 .8 -1 .31 65 .8 . oo

OU . y 12 48 55. 9 5.38 60 .9 -1 .36 65 .9 . 44AA

Ol . u 12 41 56. 0 5.20 61 .0 -1 .39 66 .0 . 4i3

Ol . 1 12 33 56. 1 5.02 61 .1 -1 .42 66 . 1 Rzl . 04

Ol . <c 12 25 56. 2 4.83 61 .2 -1 .45 66 .2 . OoRR

Ol . o 12 18 56. 3 4.65 61 .3 -1 .47 66 .3 . DO

01.4 12 09 56. 4 4.46 61 .4 -1 .49 66 .4 . D / Ol . o 12 01 56. 5 4.28 61 .5 -1 .50 66 .5 . 1 <c 01 . D 11 93 56. 6 4.10 61 .6 -1 .51 66 .6 I'd

01 . / 11 84 56. 7 3.92 61 .7 -1 .51 66 .7 . ou

01 . c5 11 74 56. 8 3.74 61 .8 -1 .51 66 .8 Q.A . o4

01 . y 11 65 56. 9 3.56 61 .9 -1 .51 66 .9 . O (

0<c . U 11 55 57. 0 3.38 62 .0 -1 .50 67 .0 . yi

0<C . 1 11 45 57. 1 3.21 62 .1 -1 .48 67 . 1 . QAy4

0<d . d 11 35 57. 2 3.03 62 .2 -1 .47 67 .2 QP

0/c . o 11 24 57. 3 2.86 62 .3 -1 .45 67 .3 1 . Ul

0<c . 4 11 13 57. 4 2.70 62 .4 -1 .43 67 .4 1 . UO

0<c . 0 11 02 57. 5 2.53 62 .2 -1 .40 67 .5 1

1

. ORuo

0(C . D 10 90 57. 6 2.36 62 .6 -1 .37 67 .6 1 . U ( RO '7 0<; . ( 10 77 57. 7 2.19 62 .7 -1 .34 67 .7 1

1

. OPUO

O/C . £5 10 65 57. 8 2.02 62 .8 -1 .30 67 .8 11

. npuo

o<c RO . Qy 10 52 57 9 1.86 62 .9 -1 .26 67 .9 1 . yjzj OoR'7 n. u 10 38 58 0 1.70 63 .0 -1 .22 68 .0 11

. npuo

ooR"^ 1. 1 10 24 58 1 1.55 63 .1 -1 .17 68 . 1 11

. npuo

OoR"? O. <c 10 11 58 2 1.39 63 .2 -1 .13 68 .2 1

1

. nTU ( ooR"^ . O 9 97 58 3 1.24 63 3 -1 .08 68 .3 11 nfi . UD

OOR*^ A. 4 9 82 58 4 1.10 63 4 -1 .03 68 .4 1

1

. nRuo

OoR'Z R. 0 9 66 58 5 .95 63 5 - .98 68 5 1 n'7 1 . uo

OOR'7 . CD 9 51 58 6 .81 63 6 - .93 68 6 11

. mUl

R T T Oo . ( 9 35 58 7 .68 63 7 - .88 68 7 QQ

R "7 O Oo . o 9 19 58 8 .54 63 8 - .82 68 8 . yo

R T Q Oo . y 9 02 58 9 .41 63 9 - .76 68 9 . yo

04R /I r\. U 8 85 59 0 .28 64 0 - .70 69 0 PQ

04R /I 1

. 1 8 69 59 1 .15 64 1 - 64 69 1 . PRoo

04Ryl O. d 8 52 59 2 .02 64 2 - 58 69 2 . pnOU

R/1 '7 04 . o 8 34 59 3 -.10 64 3 - 52 69 3 . 7R1 O

04R/1 /I. 4 8 16 59 4 -.21 64 4 - 45 69 4 . ( 1

04R/1 R. 0 7 98 59 5 -.31 64 5 - 40 69 5 . KRDO

04R/1 C. D 7 .80 59 6 -.42 64 6 33 69 6 . OORP

04R/1. T1 7 .62 59 7 -.52 64 7 28 69 7 . 0<C RO

R >1 o 04 . o 7 .43 59 8 -.62 64 8 22 69 8 A R . 40

54.9 7 .25 59 9 -.71 64 9 15 69 9 .38

55.0 7 .07 60 0 -.80 65 0 09 70 0 .31

67
The Corrections to Values of Temperature on the NBS-55 Scale. (The correction (Teg — T^bs-ss)

is given with the argument in T^bs -55.) — Continued

T Corr. T Corr. T Corr. T Corr. K mK K mK K mK K mK

70 .0 .31 75 .0 -5 .59 80,.0 -9. 03 85 ,0 -2.91

70 . 1 .23 75 . 1 -5,.71 80,.1 -9. 01 85 .1 -2.69

70 .2 .15 75 .2 -5,,84 80,.2 -8. 98 85,.2 -2.47

70 .3 .06 75 .3 -5,.96 80,.3 -8. 95 85,,3 -2.24

70 .4 — .02 75 .4 -6,.08 80 .4 -8. 91 85,.4 -2.00

70 .5 — .10 75 .5 -6,,20 80 .5 -8. 88 85,.5 -1.76

70 .6 — .20 75 .6 -6,.31 80 .6 -8. 82 85 .6 -1.52

70 .7 — .29 75 .7 -6 ,43 80 .7 -8. 77 85 .7 -1.27

70 .8 — .39 75 .8 -6,.55 80 .8 -8. 71 85 .8 -1.02

70 .9 — .49 75 .9 -6,,67 80 .9 -8, 65 85 .9 -.76

71 .0 — .59 76 .0 -6 .78 81 .0 -8,,59 86 .0 -.50

71 .1 - .70 76 . 1 -6 .89 81 .1 -8,,52 86 .1 -.24

71 .2 — .81 76 .2 -7 .00 81 .2 -8, 44 86 .2 .02

71 .3 — .92 76 .3 -7 .11 81 .3 -8,,36 86 .3 .29

71 .4 -1 .02 76 .4 -7 .21 81 .4 -8,.28 86 .4 .56

71 .5 -1 . 13 76 .5 -7 .31 81 .5 -8..19 86 .5 .83

71 .6 -1 .25 76 .6 -7 .42 81 .6 -8,,10 86 .6 1.11

71 .7 -1 .36 76 .7 -7 .51 81 .7 -8.,00 86 .7 1.38

71 .8 -1 .48 76 .8 -7 .61 81 .8 -7.,90 86 .8 1.65

71 .9 -1 .60 76 .9 -7 .71 81 .9 -7..80 86 .9 1.92

72,.0 -1 .72 77 .0 -7,.80 82 .0 -7..70 87 .0 2.20

72., 1 -1 .85 77 . 1 -7,.89 82 .1 -7..59 87 . 1 2.47

72..2 -1 .96 77 .2 -7 .98 82 .2 -7,.48 87 .2 2.75

72,.3 -2 .08 77 .3 -8 .06 82 .3 -7,,36 87 .3 3.02

72,.4 -2 .20 77 .4 -8 . 14 82 .4 -7 .24 87 .4 3.30

72,.5 -2 .32 77 .5 -8 .22 82 .5 -7 .12 87 .5 3.57

72,,6 -2 .44 77 .6 -8,.30 82 .6 -6 .99 87 .6 3.84

72,,7 -2 .58 77 .7 -8 .37 82 .7 -6 .86 87 .7 4. 12

72,.8 -2 .71 77 .8 -8,.44 82 .8 -6 .73 87 .8 4.39

72,.9 -2 .84 77 .9 -8,.51 82 .9 -6 .59 87,.9 4.65

73,.0 -2 .97 78 .0 -8,.57 83 .0 -6 .46 88 .0 4.92

73,.1 -3.,10 78 . 1 -8,,63 83 .1 -6 .32 88 . 1 5.18

73,,2 -3.,23 78 .2 -8,,69 83 ,2 -6 .17 88 .2 5.45

73,.3 -3.,36 78,.3 -8,,74 83,,3 -6 .02 88 .3 5.70

73,,4 -3.,50 78,.4 -8,,78 83,.4 -5 .88 88,.4 5.96

73,,5 -3.,64 78 .5 -8,,83 83,,5 -5 .72 88,,5 6.22

73,.6 -3..78 78,.6 -8,,87 83,,6 -5 .56 88,.6 6.47

73,.7 -3,.91 78,,7 -8,,91 83.,7 -5 .40 88,.7 6.72

73. 8 -4. 04 78,.8 -8.,94 83 ,8 -5 .24 88. 8 6.96

73..9 -4.,17 78.,9 -8. 97 83.,9 -5 .07 88. 9 7.21

74. 0 -4. 30 • 79. 0 -9. 00 84,,0 -4 .90 89. 0 7.45

74. 1 -4. 43 79. 1 -9. 03 84,,1 -4,.72 89. 1 7.68

74. 2 -4. 56 79. 2 -9. 05 84,,2 -4 .54 89. 2 7.92

74. 3 -4. 69 79. 3 -9. 07 84, 3 -4,.34 89. 3 8.15

74. 4 -4. 82 79. 4 -9. 08 84.,4 -4,.16 89. 4 8.37

'7A 0 —4

.

yo 0 Do QAC54 , 0

•z0 , cjy

.

D 0 . «JO 74. 6 -5. 08 79. 6 -9. 09 84. 6 -3 .76 89. 6 8.79

74. 7 -5. 20 79. 7 -9. 08 84. 7 -3 .56 89. 7 9.00

74. 4 -5. 34 79. 8 -9. 07 84. 8 -3,,35 89. 8 9.20

74. 9 -5. 46 79. 9 -9. 06 84. 9 -3,,13 89. 9 9.40

75. 0 -5. 59 80. 0 -9. 03 85. 0 -2,.91 90. 0 9.59

68
Estimated Corrections to Values of Temperature on the IPTS of 1948. (The correction (tes" ^48)

is given with the argument in t4g.)

top

Corr. t°c

Corr. op

top

Corr. optopCorr. op

-150 ,012630 -100 022386 -50 .028618

-149 .012963 -99 023166 -49 .028183

-148 .013216 -98 023921 -48 . 027734

-147 .013392 -97 024650 -47 . 027272

-146 .013491 -96 025352 -46 . 026798

-145 .013514 -95 026027 -45 . 026311

-144

_'

.013464 -94 026675 -44 .025814

-143 .013343 -93 027295 -43 . 025305

-142 . 013152 -92 027886 -42 . 024785

-141 . 012893 -91 028449 -41 . 024256

-140 .012571 -90 028983 -40 . 023717

-139 .012186 -89 029488 -39 . 023170

-138 . 011741 -88 029963 -38 . 022613

- -137 . 011240 -87 030410 -37 . 022049

-136 . 010685 -86 030827 -36 . 021478

-135 .010079 -85 . 031215 -35 . 020899

-134 . 009425 -84 . 031574 -34 . 020314

-183 . 007936 -133 . 008725 -83 031904 -33 .019722

-182 .009822 -132 . 007984 -82 032204 -32 .019125

-181 .011224 -131 _ .007203 -81 . 032476 -31 .018523

-180 .012195 -130 .006386 -80 032720 -30 .017917

-179 .012785 -129 .005536 -79 032934 -29 .017306

-178 .013040 -128 _ . 004656 -78 . 033121 -28 .016691

-177 .013003 -127 . 003748 -77 033280 -27 .016073

-176 .012712 -126 .002815 -76 . 033411 -26 .015453

-175 .012202 -125 .001860 -75 - 033515 \J \J \J \J JL \,J -25 .014829

-174 .011507 -124 . 000887 -74 033593 -24 .014205

-173 .010656 -123 . 000104 -73 03"^ 644 -23 . 013578

-172 . 009676 -122 . 001108 -72 033669 —22 . 012951

-171 . 008592 -121 -L J. 002123 -71 033668 —21 . 012323

-170 . 007427 -120 003148 -70 -20 . 011696

-169 .006200 -119 . 004179 -69 033591 -19 .011069

-168 .004931 -118 . 005214 -68 033517 -18 .010443

-167 .003637 -117 . 006250 -67 033418 -17 .009819

-166 .002332 -116 . 007287 -66 033296 -16 .009196

-165 .001029 -115 . 008321 -65 033152 -15 . 008577

-164 -.000258 -114 . 009351 -64 032985 -14 . 007960

-163 -.001520 -113 . 010375 -63 032797 -13 . 007347

-162 -.002747 -112 .011390 -62 032587 -12 .006739

-161 -.003930 -111 . 012396 -61 032357 -11 .006136

-160 -.005063 -110 . 013390 -60 032106 -10 . 005537

-159 -.006138 -109 . 014370 -59 031837 -9 . 004945

-158 -.007152 -108 .015337 -58 031548 -8 . 004360

-157 -.008099 -107 . 016287 -57 . 031240 -7 . 003782

-156 - . 008975 -106 . 017220 -56 030915 -6 . 003213

-155 -.009779 -105 . 018134 -55 030573 -5 . 002652

-154 -.010506 —104 019028 -54 030?]

3

-4 . 002100

-153 -.011156 -103 .019901 -53 . 029838 -3 .001559

-152 -.011727 -102 .020753 -52 . 029446 -2 .001029

-151 -.012218 -101 .021581 -51 .029040 -1 .000511

-150 -.012630 -100 .022386 -50 .028618 0 .000005

69
Estimated Corrections to Values of Temperature on the IPTS of 1948. (The correction (t68~t48)

is given with the argument in t4g.) — Continued

t°c

Corr. ctLCorr. ctcCorr. LtcCorr. C

0 + . 000005 50 010366 100 . 000000 150 n o n /I

1 000487 51 010321 101 . 000341 151 n or\oo T

2 000966 52 010269 102 . 000686 152 . Udld (o

3 001432 53 . 010209 103 . 001034 153 . U^l r do

4 001884 54 010141 104 . 001386 154 . 022174

5 . 002324 55 • 010065 105 .001742 155 . 022625

6 002750 56 009981 106 . 002101 156 . 023077

7 003164 57 009890 107 . 002464 157 . 023528

8 003565 58 , 009791 108 . 002830 158 O '7 Pi O 1

. 023981

9 003953 59 , 009685 109 . 003200 159 no y1 /ITT

10 004329 60 , 009571 110 . 003572 160 . U,c;4ooo

11 004692 61 , 009450 111 . 003948 161 AO K T T O

12 , 005043 62 , 009322 112 . 004327 162

13 , 005382 63 , 009187 113 . 004710 > 163 AOCO y1 /I 14 r\ r\r~ T7 r\r\

. 005709 64 . 009045 t ^ A 114 . 005095 164

15 , 006024 65 . 008896 115 . 005483 165 . U/c f i4y

16 , 006328 66 . 008740 116 . 005873 166 AO'7CAO

17 . 006619 67 /-\ /-\ o errrr

. 008577 117 . 006267 167 AOOAC A 18 r\ ^ tn r\ rs

. 006899 68 . 008408 118 . 006663 168 AOOC A^

19 . 007167 69 . 008232 119 y^ y^ y—

»

. 007062 169

20 . 007424 70 . 008050 120 . 007463 170 AOQ A r\Ci

. u<;y4uy

21 , 007670 71 . 007861 121 . 007867 171

22 . 007905 72 . 007666 122 . 008274 172 . 030311

23 . 008128 73 . 007465 123 . 008682 173 . 030761

24 . 008341 74 . 007258 124 . 009093 174 . 031211

25 . 008543 75 . 007044 125 . 009507 175 . 031660

26 . 008734 76 . 006825 126 .009922 176 . 032109

27 . 008914 77 . 006600 127 . 010339 177 . 032557

28 ~-. 009084 78 . 006369 128 . 010759 178 r\ '7 ^7 r\ r\ a

. 033004

29 . 009243 79 . 006133 129 . 011180 179 . Ooo4ol

30 . 009392 80 . 005891 130 . 011604 180 . Ooooyb

31 . 009531 81 . 005643 131 .012029 181 . 0o4o41

32 . 009660 82 . 005390 132 . 012456 182 . 034 f OO

33 . 009779 83 . 005132 133 . 012884 183

34 . 009888 84 . 004868 134 .013315 184 . 03ob /

0

35 . uuyyoY 85 . 004599 135 . 013747 185 . Uoolld

36 . 010076 86 . 004325 136 . 014180 186 . 036552

37 . 010156 87 . 004046 137 . 014615 187 . 03Dyyi

38 . 010226 88 . 003763 138 . 015051 TOO 188 . 03 r4<cy

39 . 010287 89 . 003474 139 . 015489 189 . 03 (abb

40 . 010339 90 . 003180 140 . 015928 190 A TOT A T

. 03o30i

41 . 010381 91 . 002882 141 .016368 191 A '7 00"7

. 038736

42 . 010414 92 . 002579 142 .016809 192 . 039169

43 . 010439 93 — .002272 143 .017251 193 o '7 o o n

. 039601

44 . 010454 94 . 001960 144 .017695 194 O j4 O O '7 T

. 040031

45 . 010461 95 . 001644 145 .018139 195 . 040460

46 . uiu4oy 96 . 001323 146 . 018584 196 A /f AOOO . U4Uooo

47 .010448 97 .000999 147 .019031 197 .041314

48 .010429 98 .000670 148 . 019478 198 . 041739

49 .010401 99 .000337 149 .019925 199 .042162

50 .010366 100 .000000 150 . 020374 200 . 042584

70
Estimated Corrections to Values of Temperature on the IPTS of 1948. (The correction (fes" *48)

is given with the argument in t4g.) — Continued

t Corr. t Corr. t Corr. t Corr. °c °C °C °C °c °C "C °C

200 . 042584 250 .061015 300 .072611 350 . 076876

201 . 043004 251 .061320 301 . 072765 351 . 076897

202 . 043422 252 .061621 302 .072916 352 .076915

203 . 043839 253 .061921 303 . 073065 353 .076931

204 . 044254 254 .062217 304 .073210 354 . 076945

205 . 044668 255 .062510 305 .073353 355 . 076957

206 . 045079 256 .062801 306 . 073492 356 . 076967

207 . 045489 257 . 063088 307 . 073629 357 . 076975

208 . 045897 258 . 063373 308 . 073763 358 . 076981

209 . 046303 259 .063655 309 . 073893 359 . 076985

210 . 046707 260 .063933 310 .074021 360 . 076987

211 .047109 261 .064209 311 . 074146 361 . 076987

212 .047510 262 . 064482 312 . 074267 362 . 076985

213 . 047908 263 . 064752 313 . 074386 363 . 076981

214 . 048304 264 .065019 314 .074502 364 . 076976

215 . 048698 265 . 065283 315 .074615 365 . 076969

216 .049091 266 . 065544 316 . 074726 366 . 076960

217 . 049481 267 .065802 317 . 074833 367 . 076949

218 . 049869 268 . 066057 318 . 074937 368 . 076937

219 .050255 269 .066309 319 .075039 369 . 076923

220 .050638 270 .066558 320 .075138 370 . 076907

221 .051020 271 . 066804 321 . 075234 371 . 076890

222 .051399 272 . 067047 322 . 075327 372 . 076871

223 .051776 273 . 067287 323 .075418 373 .076851

224 .052150 274 . 067524 324 .075505 374 . 076830

225 .052523 275 . 067758 325 . 075590 375 . 076806

226 . 052893 276 . 067989 326 . 075673 376 . 076782

227 .053260 277 .068217 327 . 075752 377 . 076756

228 .053626 278 . 068441 328 . 075829 378 . 076729

229 . 053988 279 . 068663 329 .075903 379 .076701

230 . 054349 280 . 068882 330 . 075975 380 . 076671

231 . 054707 281 . 069097 331 . 076043 381 . 076640

232 .055062 282 . 069309 332 .076110 382 . 076608

233 .055415 283 .069519 333 .076173 383 . 076575

234 .055766 284 .069725 334 . 076234 384 . 076541

235 .056114 285 . 069928 335 . 076293 385 . 076505

236 . 056459 286 .070129 336 . 076349 386 . 076469

237 .056802 287 .070326 337 . 076402 387 . 076432

238 .057142 288 .070520 338 . 076453 388 . 076394

239 . 057480 289 .070711 339 .076501 389 . 076355

240 .057814 290 . 070899 340 . 076547 390 .076315

241 .058147 291 .071084 341 . 076591 391 . 076274

242 . 058476 292 .071266 342 . 076632 392 .076233

243 .058803 293 . 071444 343 . 076670 393 .076191

244 .059127 294 .071620 344 . 076707 394 .076148

245 . 059449 295 .071793 345 . 076741 395 .076105

246 . 059768 296 .071962 346 . 076772 396 . 076061

247 . 060084 297 .072129 347 . 076802 397 .076016

248 .060397 298 .072292 348 .076829 398 . 075971

249 . 060707 299 . 072453 349 . 076854 "ZQQ oyy . U f oy/cb

250 .061015 300 .072611

71

350 . 076876 400 .075880
Estimated Corrections to Values of Temperature on the IPTS of 1948. (The correction (f68"~*48)

is given with the argument in t48.) — Continued

t"CCorr. °C

t "C

Corr. °C

t-c

Corr. °C

t"C

Corr. "C

400 . 075880 450 . 074257 500 .079211 550 .100515

401 . 075834 451 . 074264 501 . 079440 551 .101189

402 . 075788 452 . 074273 502 . 079675 552 .101874

403 . 075741 453 . 074284 503 .079917 553 . 102570

404 . 075694 454 . 074298 504 .080166 554 . 103278

405 . 075648 455 . 074316 505 . 080422 555 . 103998

406 . 075600 456 . 074336 506 . 080684 556 . 104730

407 . 075553 457 . 074359 507 . 080954 557 . 105474

408 . 075506 458 . 074385 508 .081231 558 .106231

409 . 075459 459 . 074414 509 .081514 559 . 106999

410 .075412 460 . 074446 510 .081805 560 . 107780

411 . 075366 461 . 074482 511 .082104 561 . 108573

412 .075319 462 . 074521 512 .082410 562 . 109379

413 . 075273 463 . 074563 513 . 082723 563 .110198

414 . 075227 464 . 074609 514 . 083044 564 .111030

415 .075181 465 . 074658 515 . 083373 565 .111874

416 . 075136 466 .074711 516 .083710 566 .112732

417 . 075091 467 . 074767 517 . 084054 567 .113603

418 . 075047 468 . 074828 518 . 084407 568 .114487

419 . 075004 469 . 074892 519 . 084768 569 .115385

420 . 074961 470 . 074960 520 .085137 570 .116296

421 . 074918 471 .075032 521 .085514 571 .117220

422 . 074877 472 .075108 522 . 085899 572 .118159

423 . 074836 473 .075188 523 . 086293 573 .119111

424 . 074796 474 . 075273 524 . 086696 574 .120078

425 . 074757 475 .075361 525 .087108 575 .121058

426 . 074719 476 . 075455 526 . 087528 576 .122053

427 . 074682 477 . 075552 527 . 087957 577 .123062

428 . 074647 478 . 075654 528 . 088395 578 . 124086

429 .074612 479 . 075761 529 . 088842 579 .125125

430 . 074578 480 . 075872 530 . 089299 580 . 126178

431 . 074546 481 . 075988 531 . 089764 581 . 127245

432 . 074515 482 .076109 532 .090239 582 . 128328

433 . 074486 483 . 076235 533 . 090724 583 . 129426

434 . 074458 484 . 076366 534 .091218 584 .130539

435 .074431 485 . 076502 535 .091722 585 .131668

436 . 074406 486 . 076644 536 .092236 586 .132812

437 . 074383 487 . 076790 537 . 092759 587 . 133972

438 . 074361 488 . 076942 538 .093293 588 .135147

439 . 074341 489 . 077099 539 . 093837 589 . 136338

440 . 074323 490 . 077262 540 . 094391 590 . 137545

441 . 074307 491 . 077430 541 . 094955 591 . 138768

442 . 074293 492 . 077604 542 .095530 592 . 140008

443 . 074281 493 . 077784 543 .096115 593 . 141264

444 . 074271 494 . 077969 544 .096711 594 . 142536

445 . 074263 495 .078161 545 .097317 595 . 143825

446 . 074257 496 . 078359 546 . 097935 596 .145130

447 . 074254 497 . 078562 547 . 098563 597 . 146453

448 . 074252 498 . 078772 548 .099203 598 . 147792

449 . 074254 499 . 078988 549 . 099853 599 . 149148

450 . 074257 500 .079211 550 .100515 600 .150522

72
Estimated Corrections to Values of Temperature on the IPTS of 1948. (The correction (tea" ^48)

is given with the argument in t4g.) — Continued

t CoiT. t Corr. t Corr. t Corr. L L. Of or* op or* op

r\ r\ 600 . 150522 bUo . lb<;l4o bib . 174953 CO/1 1 flQnnc

601 . 151913 coo buy 1 CZCOT . Iboboo b± f

. 176640 COK . 13UOOO

602 . 153322 biU . ibb/cob bio . 178346 coc b/cb

603 . 154748 bll T CCQf\Q biy . 180073 COT b<i / . 1 j*iDl 1

604 .156192 612 . 168399 620 .181819 628 . 196522

605 . 157654 613 . 170009 621 . 183585 629 . 198400

606 .159134 614 . 171637 622 . 185372 630 .200406

607 .160632 615 . 173285 623 . 187178 631 . 202381

73
Appendix D. Table of the Function

M(t') 0.045

(^^QQ (

^

l) (419^58 1) ( 630.74 °C ^) °^

and its derivative* from t' = 0 °C to f' = 630 °C

at za

(s = 100 "C, z = 419.58 °C,a = 630.74 °C)

t'L

M{t')

L

dM(t')ldt' t'c

M(t')

L

dM(t')ldt'

0 .000000 -.000450 50 . 009124 .000040

1 — . 000444 -.000437 51 . 009080 .000048

2 — .000875 -.000425 52 . 009029 .000055

3 . 001294 -.000413 53 . 008971 . 000062

4 . 001701 -.000401 54 . 008906 . 000068

5 . 002095 -.000389 55 . 008834 . 000075

6 . 002478 -.000377 56 . 008755 . 000082

7 . 002849 -.000365 57 . 008670 . 000089

8 . 003208 -.000353 58 . 008578 . 000095

9 . 003555 -.000342 59 . 008480 . 000101

10 . 003891 -.000330 60 . 008375 . 000108

11 . 004215 -.000319 61 . 008264 . 000114

12 . 004528 -.000307 62 . 008147 .000120

13 — . 004830 -.000296 63 . 008024 .000126

14 — .005121 -.000285 64 . 007895 .000132

15 — . 005401 -.000274 65 . 007760 . 000138

16 — . 005670 -.000264 66 . 007619 . 000144

17 — .005928 -.000253 67 — . 007473 .000149

18 — .006176 -.000242 68 — . 007320 .000155

19 . 006413 -.000232 69 — . 007163 .000160

20 — .006639 -.000221 70 . 007000 .000166

21 — .006856 -.000211 71 . 006831 .000171

22 — .007062 -.000201 72 — . 006657 .000176

23 — .007258 -.000191 73 — . 006478 .000182

24 — . 007444 -.000181 74 — . 006294 .000187

25 . 007620 -.000171 75 — . 006105 .000192

26 — . 007787 -.000162 76 . 005911 .000196

27 — .. 007944 -.000152 77 . 005713 . 000201

28 . 008091 -.000142 78 . 005509 .000206

29 — .008229 -.000133 79 . 005301 .000210

30 . 008357 -.000124 80 . 005088 .000215

31 . 008476 -.000115 81 . 004871 .000219

32 , 008586 -.000106 82 . 004649 .000224

33 — ., 008687 -.000097 83 . 004423 .000228

34 , 008780 -.000088 84 .004193 .000232

35 ,008863 -.000079 85 .003959 .000236

36 008937 -.000070 86 , 003720 .000240

37 009003 -.000062 87 , 003478 .000244

38 009061 -.000053 88 ,003231 . 000248

39 009110 -.000045 89 002981 .000252

40 009151 -.000037 90 002727 .000256

41 009183 -.000029 91 002470 .000259

42 009208 -.000020 92 002209 .000263

43 009224 -.000012 93 001944 .000266

44 009233 -.000005 94 001676 .000270

45 009234 .000003 95 001404 . 000273

46 009227 .000011 96 001130 .000276

47 009212 .000018 97 000852 .000279

48 009190 .000026 98 000571 .000283

49 009161 .000033 99 000287 .000285

50 009124 .000040 100 000000 . 000288

74
Appendix D. Table of the Function — Continued

M(t') 0.045

(^^QQ oq)

(

Ij

( 419.58

and its derivative* from t' = 0 X: to t' = 630 °C

t' 630.74 °C

* dMjp^Om^^^,, _ ^ ^ ^ ^ 2(ffi + sa + za)t' - sza] at s' za

(s = 100 'C, z = 419.58 "C, a = 630.74 °C)

t' °c

M{t')

°C

dM(t')ldt' t' "C °CdM{t')ldt'

100 -.000000 .000288 150 .016527 . 000345

101 .000290 .000291 151 . 016872 .000345

102 .000582 . 000294 152 .017217 .000344

103 . 000878 . 000297 153 .017560 . 000343

104 .001176 .000299 154 .017904 . 000343

105 . 001476 .000302 155 .018246 .000342

106 . 001779 .000304 156 .018588 .000341

107 .002085 . 000307 157 .018929 .000341

108 . 002393 .000309 158 .019269 . 000340

109 . 002703 .000311 159 .019608 .000339

110 .003015 .000313 160 . 019947 .000338

111 .003330 .000316 161 .020284 .000337

112 . 003646 .000318 162 .020620 .000336

113 . 003965 .000319 163 .020956 .000335

114 .004285 .000321 164 .021290 .000334

115 . 004608 .000323 165 .021623 .000332

116 . 004932 . 000325 166 .021955 .000331

117 .005257 . 000327 167 . 022285 .000330

118 .005585 .000328 168 .022614 .000328

119 .005914 .000330 169 . 022942 .000327

120 . 006244 .000331 170 .023268 .000326

121 .006576 .000333 171 .023593 .000324

122 .006909 . 000334 172 .023916 .000322

123 . 007244 . 000335 173 . 024238 .000321

124 . 007579 .000336 174 . 024558 .000319

125 .007916 . 000337 175 . 024876 .000318

126 .008254 .000339 176 .025193 .000316

127 . 008593 .000340 177 .025508 .000314

128 . 008933 . 000340 178 .025821 .000312

129 . 009274 . 000341 179 .026132 .000310

130 . 009616 . 000342 180 . 026442 .000308

131 . 009958 . 000343 181 . 026749 .000306

132 .010302 .000343 182 . 027054 .000304

133 .010645 . 000344 183 . 027358 .000302

134 .010990 . 000345 184 . 027659 .000300

135 .011335 .000345 185 . 027958 .000298

136 .011680 . 000346 186 .028255 .000296

137 .012026 .000346 187 . 028550 . 000294

138 .012372 . 000346 188 . 028843 .000291

139 .012718 . 000346 189 .029133 .000289

140 .013065 . 000347 190 .029421 . 000287

141 .013411 . 000347 191 . 029707 . 000284

142 .013758 . 000347 192 . 029990 .000282

143 .014105 . 000347 193 .030271 .000280

144 .014451 . 000347 194 . 030549 .000277

145 . 014798 . 000347 195 . 030825 .000275

146 .015144 . 000346 196 .031098 .000272

147 .015491 . 000346 197 .031369 .000269

148 .015837 . 000346 198 .031637 .000267

149 .016182 . 000345 199 .031902 . 000264

150 .016527 .000345 200 .032165 .000261

75
Appendix D. Table of the Function — Continued

M(t')- 0.045 (iQQo(.) (iQQ (419.58 °C ^) (630.74 °C ^) °^

and its derivative* from t' = 0 °C to = 630 °C

* dl\^^0^^^^,^ _ + ^ _,. + 2(sz + sa + za)l' - sza] at za

(s = 100 t, 2 = 419.58 -C, a = 630.74 "C)

M(t') dM(t')ldt' t' M(t')

°C

dM(t')jdt

°c °C °c

200 .032165 .000261 250 .041170 . 000088

201 . 032425 .000259 251 .041256 .000084

202 . 032682 .000256 252 .041339 .000080

203 . 032937 .000253 253 .041417 .000076

204 . 033188 .000250 254 .041491 .000072

205 . 033437 .000247 255 .041561 .000068

206 . 033682 .000244 256 .041627 .000064

207 . 033925 .000241 257 . 041689 .000060

208 .034165 .000238 258 . 041746 .000056

209 . 034402 .000235 259 .041800 .000052

210 . 034636 .000232 260 .041849 . 000047

211 . 034866 .000229 261 .041895 . 000043

212 .035094 .000226 262 .041936 .000039

213 .035318 .000223 263 . 041973 .000035

214 .035540 .000220 264 .042006 .000031

215 . 035758 .000216 265 . 042034 .000027

216 . 035972 .000213 266 .042059 .000022

217 .036184 .000210 267 . 042079 .000018

218 . 036392 .000207 268 . 042095 .000014

219 . 036597 .000203 269 .042107 .000010

220 . 036799 .000200 270 .042114 .000005

221 . 036997 .000197 271 .042117 .000001

222 .037192 .000193 272 .042116 -.000003

223 . 037383 .000190 273 .042111 -.000007

224 . 037571 .000186 274 .042102 -.000012

225 . 037756 .000183 275 . 042088 -.000016

226 . 037937 .000179 276 . 042070 -.000020

227 .038114 .000176 277 . 042048 -.000024

228 . 038288 .000172 278 .042021 -.000029

229 . 038459 .000169 279 .041990 -.000033

230 . 038625 .000165 280 .041955 -.000037

231 . 038789 .000161 281 .041916 -.000042

232 . 038948 .000158 282 . 041872 -.000046

233 .039104 .000154 283 . 041824 -.000050

234 .039256 .000150 284 . 041772 -.000054

235 . 039404 .000147 285 .041715 -.000059

236 . 039549 .000143 286 .041654 -.000063

237 . 039690 .000139 287 .041589 -.000067

238 .039827 .000135 288 .041520 -.000072

239 . 039960 .000131 289 . 041446 -.000076

240 . 040090 .000128 290 .041368 -.000080

241 .040215 .000124 291 .041285 -.000085

242 . 040337 .000120 292 .041198 -.000089

243 . 040455 .000116 293 .041107 -.000093

244 . 040569 .000112 294 .041012 -.000097

245 . 040679 .000108 295 .040912 -.000102

246 . 040785 .000104 296 . 040809 -.000106

247 . 040887 .000100 297 . 040700 -.000110

248 . 040986 .000096 298 . 040588 -.000115

249 .041080 .000092 299 . 040471 -.000119

250 .041170 .000088 300 .040350 -.000123

76
Appendix D. Table of the Function — Continued

M{t') 0.045 oj

(

l) (419.58 °C ^)( 630.74 °C

1 °C

and its derivative* from t' = 0 °C to t' = 630 °C

' dMW) 0.015

[it'' - 3(s + z + a)t'' + 2(.sz + sa+ za)t' - sza]

dt' za

'

(s = mX.,z = 419.58 °C,a= 630.74 X.)

t' M{t')

°C

dM(t')ldt' t' M(t') dM{t')ldt'

°c "C °C

300 . 040350 -.000123 350 .029063 -.000322

301 .040225 -.000127 351 . 028740 -.000325

302 .040095 -.000132 352 .028412 -.000329

303 .03996 -.000136 353 .028082 -.000332

304 .039823 -.000140 354 . 027748 -.000336

305 .039681 -.000144 355 .027410 -.000339

306 .039534 -.000149 356 . 027069 -.000343

307 . 039383 -.000153 357 .026725 -.000346

308 .039228 -.000157 358 . 026378 -.000349

309 . 039069 -.000161 359 .026027 -.000352

310 . 038906 -.000166 360 . 025673 -.000356

311 . 038738 -.000170 361 .025316 -.000359

312 .038566 -.000174 362 .024955 -.000362

313 .038390 -.000178 363 .024592 -.000365

314 .038210 -.000182 364 .024225 -.000368

315 .038025 -.000186 365 .023855 -.000371

316 . 037837 -.000191 366 . 023482 -.000375

317 . 037644 -.000195 367 .023106 -.000378

318 . 037447 -.000199 368 . 022727 -.000381

319 . 037247 -.000203 369 .022345 -.000384

320 . 037042 -.000207 370 .021960 -.000386

321 .036832 -.000211 371 .021572 -.000389

322 .036619 -.000215 372 .021181 -.000392

323 .036402 -.000219 373 . 020787 -.000395

324 .036181 -.000223 374 .020391 -.000398

325 .035955 -.000227 375 .019992 -.000401

326 .035726 -.000231 376 .019590 -.000403

327 . 035493 -.000235 377 .019185 -.000406

328 .035255 -.000239 378 . 018778 -.000409

329 .035014 -.000243 379 .018368 -.000411

330 . 034769 -.000247 380 .017955 -.000414

331 .034520 -.000251 381 .017540 -.000416

332 . 034267 -.000255 382 .017122 -.000419

333 .034010 -.000259 383 .016702 -.000421

334 . 033749 -.000263 384 .016280 -.000424

335 . 033484 -.000267 385 .015855 -.000426

336 .033215 -.000271 386 .015427 -.000429

337 . 032943 -.000274 387 .014997 -.000431

338 . 032667 -.000278 388 .014565 -.000433

339 .032387 -.000282 389 .014131 -.000435

340 .032103 -.000286 390 .013695 -.000438

341 .031816 -.000289 391 .013256 - . 000440

342 .031524 -.000293 392 .012815 -.000442

343 .031229 -.000297 393 .012373 -.000444

344 .030931 -.000300 394 .011928 - . 000446

345 .030629 -.000304 395 .011481 -.000448

346 .030323 -.000308 396 .011032 -.000450

347 .030013 -.000311 397 .010582 -.000452

348 . 029700 -.000315 398 .010129 -.000453

349 . 029383 -.000318 399 . 009675 -.000455

350 .029063 -.000322 400 .009219 -.000457

77
Appendix D. Table of the Function — Continued

M(.')= 0.045 (3^- 1) (41^- 1) (^3oW l) -C

and its derivative* fix>m t' = 0 °C to t' = 630 ^

= —j— [it''-3{s + z + a)t'' + 2{sz + sa+za)t' -sza]

za

(s = IOO° z = 419.58 t, a = 630.74 "C)

°c

M(t'

)

dM(t')ldt'

°c

M(t') dM(t')ldt'

AC\C\ 009219 -.000457 450 . U X 1 1 «C<J —

.

000477

008761 - . 000459 4R1 n 1 R9m — 00047fi

AC\0 OOS'^Ol - . 000460 4S9 — 00047R

AC\X 007840 - 000462 *too . niUXOXQX fil m — 000474

yi n >1 4U4 — 000464 4R4 ftOft — 000477

/I - 000465 . UX / — 000471

4Ub >i nc 006447 — 000466 ftOO . UX / OD

/

— 000470

— 000468 4R7 ftO / . UXOUOD — 0004fiP

4Uo nn^si

1

— 000469 400 . UXoOUo — 0004fi7

A rvQ — 000471 40i? . uxoyoy — 0004fiR

A'X Ci 4iU — 000472 40U . uxy4oo — 0004fi'7

411 - 000473 4fil ftOX . nU X"0"0 1 QPQR — 0004fil

41<c —

.

000474 4fi9 — 0004fiO

410 > OO'^l W X 49^^ - 000475 090^1

4

— 0004RR

ft J. ft 002673 - . 000476 4fi4 091 971 — 0004Sfi

ft J.O 002196 -. 000477 091 79R — 0004'i'^

ft J.D , 001718 -.000478 0991 7R — 0004'i1

4.1 7 001239 -.000479 4fi7 099fi9fl - 000449

ftXo 000760 -.000480 09*^07^ — 000447

41 Q . 000279 - . 000481 469 09*^^99 - 000444

490 fr<£U . 000202 V w \J \y #w - . 000482 470 - 000442

ft<iX 000684 -.000482 471 *t 1 X 09440^ — 0004'^9

ft<c<c . 001167 - . 000483 479 094R44 — 0004*^7

49T 4^0 w W J. X - 000484 47*^ ft 1 0 n9'=;97Q — 0004*^4

494 4<c4 - 000484 474 09R71

9

. UcO / X/C — 0004*^1

49C^ 00?fi1

9

- 000485 47ft f 0R 09f;i 49 — 00049Q

49C . 00"^! \J \J \J A. 04\J ''t — 00048'^ 47f^ . 09^^RfiQ U/cDOOy — 00049fi

4iC 1 00'=;i=iR9 - 00048^ 477 4 / ( — 00049*=? 00407^ — 0004ftfi 447Pf C . UiC (4X0 — 000490

4^y nn4Rfii 4 f y . \J<. fOOO 00041

7

4oU . 00^047 UUtJiJft / 4oU n 0 0 0 /1

0

OOO/l 1 X . UUU4XO

40l nnsR'^7 4cSl . U/£oboy — 00041

0

— . UUUftX

U

>1 TO 4»3/c >4Q9 4C5iC . Ucyubo — 000407

4oo — On04Rfi 400 . Uicy4 f 0 — 00040'^

4o4 A "ZA — 0004Rfi 4o4 . Urfiyo (

4

— 000400

435 . 007479 -.000486 485 030272 -.000396

436 . 007965 -.000486 486 030666 -.000392

437 .008451 -.000486 487 031057 -.000389

438 .008937 -.000486 488 031443 -.000385

439 .009423 -.000485 489 031826 -.000381

440 .009908 -.000485 490 032205 -.000377

441 .010392 -.000484 491 032580 -.000373

442 .010877 -.000484 492 032950 -.000369

443 .011360 -.000483 493 033317 -.000364

444 .011843 -.000483 494 033679 -.000360

445 .012326 -.000482 495 034036 -.000355

446 .012807 -.000481 496 034390 -.000351

447 .013288 -.000480 497 034738 -.000346

448 .013768 -.000479 498 035082 -.000342

449 .014247 -.000478 499 035422 -.000337

450 . 014725 -.000477 500 035756 -.000332

78
Appendix D. Table of the Function — Continued

M(O=0.045 (j^(3^-l 419.58 °C

and its derivative* from f' = 0 to t' = 630 °C

630.74 °C

1 °C

^^'= -^[4('-' - 3(s + z + a)t-' + 2(sz + sa + za)t' - s at za

(s = 100 'C, z = 419.58 "C, a = 630.74 X.)

t' °cM(t')

°C

dM{t')ldt' t' °c

M(t')

°C

dM{t')ldt'

500 035756 -.000332 550 — .044316 .000030

501 036086 -.000327 551 — . 044281 .000040

502 036410 -.000322 552 - . 044236 .000050

503

_' 036730 -.000317 553 - .044181 .000060

504 037044 -.000312 554 -

,

.044116 .000070

505 037353 -.000306 555 - . 044041 .000080

506 037657 -.000301 556 - . 043956 .000090

507 037955 -.000296 557 - .043860 .000101

508 _ 038248 -.000290 558 - . 043754 .000111

509 038535 -.000284 559 - . 043637 .000122

510 038817 -.000279 560 - .043510 .000133

511 _ 039092 -.000273 561 - . 043371 .000144

512

_'

039362 -.000267 562 .043222 .000155

513 039626 -.000261 563 - .043062 .000166

514 039884 -.000255 564 — . 042890 .000177

515 040135 -.000248 565 — . 042707 .000189

516 040381 -.000242 566 — .042513 .000200

517 040620 -.000236 567 _ .042307 .000212

518 040852 -.000229 568 _ .042090 .000223

519 041078 - . 000223 569 _ .041861 .000235

520 , 041298 - . 000216 570 _ .041620 .000247

521 041510 - . 000209 571 _ .041367 .000259

522 , 041716 - . 000202 572 _ .041102 .000271

523 . 041915 -.000195 573 _ .040825 .000283

524 . 042107 - . 000188 574 _ .040535 .000296

525 . 042292 -.000181 575 _ .040233 .000308

526 . 042470 -.000174 576 — .039918 .000321

527 . 042640 -.000167 577 — .039591 .000334

528 . 042803 -.000159 578 — .039251 . 000346

529 . 042959 -.000152 579 -

,

. 038898 .000359

530 . 043106 -.000144 580 -

,

,038532 . 000373

531 . 043247 -.000136 581 ,038153 .000386

532 . 043379 -.000128 582 -

,

. 037761 .000399

533 . 043504 -.000121 583 -

,

, 037355 .000412

534 _ . 043620 -.000113 584 .036936 .000426

535 . 043729 -.000104 585 ,036503 . 000440

536 . 043829 -.000096 586 ,036056 .000454

537 .043921 -.000088 587 035596 . 000467

538 . 044004 -.000079 588 ,035121 .000482

539 . 044080 -.000071 589 ,034633 . 000496

540 ,044146 -.000062 590 034130 .000510

541 . 044204 -.000053 591 033613 .000524

542 . 044253 -.000045 592 033081 .000539

543 . 044293 -.000036 593 032535 .000554

544 . 044324 -.000027 594 031974 .000568

545 . 044346 -.000017 595 031398 .000583

546 . 044359 -.000008 596 030807 .000598

547 . 044362 .000001 597 030201 .000614

548 . 044356 .000011 598 029580 .000629

549 . 044341 .000020 599 028944 .000644

550 . 044316 .000030 600 028292 .000660

79
Appendix D. Table of the Function — Continued

M(t') 0.045 (

t' 100 'c) (lOO 419.58 X. ^) V 630.74 °C -Ij °c

and its derivative* from (' = 0 to t' == 630 °C

• am (I } V.WO

dt' za

[4t'^ -3(s + z + a)t'^ + 2(sz + sa + za)t' —sza\

(s == 100 2 = 419.58 a =630.74 10

°C

M(t')

°C

dM(t')ldt' t' °cM(t')

"C

dM{t')ldt'

600 028292 000660 616 ^.015648 .000926

601 —

.

027624 . 000675 617 -.014713 . 000944

602 —

.

026941 .000691 618 -.013760 . 000962

603 —

.

026241 . 000707 619 -.012789 . 000980

604 -. 025526 . 000723 620 -.011800 .000998

605 024795 . 000739 621 -.010793 .001016

606 024047 . 000756 622 - . 009767 .001035

607 023283 . 000772 623 -.008723 .001053

608 022503 . 000789 624 -.007660 .001072

609 ,021705 .000806 625 -.006579 .001091

610 .020891 .000822 626 - . 005478 .001110

611 .020061 .000839 627 -.004359 .001129

612 .019213 .000856 628 -.003220 .001148

613 .018348 .000874 629 -.002062 .001168

614 . 017465 .000891 630 -.000884 .001187

615 .016566 .000909 631 .000313 .001207

80
APPENDIX E. Tables of Values of the Reference Function and Its Derivative

from T=13 KtoT==273.15K and over the Equivalent Range of Temperature

With the Argument in Degrees Celsius

W*{T) and its derivative are defined by the

expressions

20 T= 2 Ai[\n JF*{T)y

i=0

and

dW*{T) 1 dT 1 20

where T=Tes, W*{T)=WCCT-edT^»), and the

values of Ai are:

i i Ai

0 0 273 15 X 103 11 0.767 976 358 170 845 8 x 10

1 0 250 846 209 678 803 3X103 12 0.213 689 459 382 850 0x 10

2 0 135 099 869 964 999 7x 103 13 0.459 843 348 928 069 3

3 0 527 856 759 008 517 2X102 14 0.763 614 629 231 648 0 x lO-i 4 0 276 768 548 854 105 2X102 15 0.969 328 620 373 121 3 x IO-2 5 0 391 053 205 376 683 7x 102 16 0.923 069 154 007 007 5 x 10-3 6 0 655 613 230 578 069 3X102 17 0.638 116 590 952 653 8 x lO-"

7 0 808 035 868 559 866 7X102 18 0.302 293 237 874 619 2 x lO-'* 8 0 705 242 118 234 052 OX 102 19 0.877 551 391 303 760 2 x 10-' 9 0 447 847 589 638 965 7X102 20 0.117 702 613 125 477 4 x lO-s 10 0 212 525 653 556 057 8x 102

81
The function W*{T) and its derivative were evaluated and printed employing the following FORTRAN

routine:

I FOR WSTAR, WSTAR

DIMENSION TX(1200)

INTEGER SCALE, SCALED

DOUBLE PRECISION W , WW ( 1200 ), DWDTT (1200 ) , WLOG , A ( 20 )

,

IDTDW , DWDTB4 , DWDTNX , CHORD , T , TC , STEP , SIZE , TO , TSTART

READ (5, 98) ( I , A ( I ) , N=l , 20

)

98 FORMAT (I2,D24. 16)

1 READ ( 5 , 99 , END=70 ) (TEND , STEP , SCALE

)

99 FORMAT ( F8 . 0 ,D4 , 0 , A6

)

STEP=(-ABS(STEP)

)

IF ( SCALED. EQ. SCALED) GO TO 10

SCALE=SCALE

SIZE=1

.

IF ( SCALE. EQ. ' DEC C) TO = 90

IF ( SCALE . EQ . ' KELVIN ' ) T0=273 . 15D00

IF ( SCALE. NE. ' DEG F' ) GO TO 5

SIZE=. 5555555555555555D00

T0=32

.

5 TSTART=STEP*IDINT( TO/STEP)

DWDT=1/A(1)

W = 1 + ( TSTART-TO ) *SIZE

10 IEND= ( TEND-TSTART ) /STEP+1 .

5

IF (TEND.GT.TO) GO TO 69

IF (lEND.LT.l) GO TO 5 CHORD=DWDT

DWDTNX=DWDT

W=W-CHORD*STEP*SIZE

DO 35 1 = 1, lEND

T=TSTART+ ( 1-1 ) *STEP

TX(I)=T

TC=(T-TO)*SIZE

W=W+CHORD»STEP»SIZE

DWDTB4=DWDT

15 WLOG=DLOG(W)

T=A(20)*WL0G

DO 20 J=l,19

20*T = WL0G* (A(20*-J)+T)

DT=T-TC

W=W-DT*DWDTNX

IF ( ABS ( DT ) - . 000001 ) 25 , 15 , 15

25 WW(I)=W

WLOG=DLOG(W)

DTDW=20»A(20)

DO 30 J=l,19

30 DTDW=(20-J)»A(20-J)+WL0G*DTDW

DTDW=DTDW/W

DWDTT(I) =DWDT

DWDTT ( I ) = DWDT

CH0RD= ( 3

. *DWDT-DWDTB4 ) /2

DWDTNX=2 . DWDT-DWDTB4

35 CONTINUE

JEND=IEND+50

IPAGES= ( IEND-2 ) /lOO+l

IF (IPAGES.LT.3) GO TO 55

IT100=M0D ( IDINT ( TEND/STEP ) , 100

)

IF(ITIOO.EQ.O) GO TO 55

ISPACE=MOD (

( 100-ITlOO ) , 50

)

82
LINES=MOD{ (100+ITlOO) ,100)

IPAGES=IEND-LINES-2 ) /lOO+l

IF (LINES-50) 40,40,52

40 PRINT 101, SCALE

IF(ABS(IT100)-50.EQ.O) GO TO 50

DO 45 J=1,ISPACE

45 PRINT 199,

50 JEND=IEND+ISPACE-50

PRINT 201, (TX(J) , WW(J) , DWDTT(J) , J=IEND, JEND, -1)

GO TO 55

52 PRINT 102, SCALE, SCALE

IEND=IEND+ISPACE

JEND=IEND-ISPACE+1

PRINT 201, (TX(J-50) ,WW(J-50) ,DWDTT(J-50) , J=IEND , JEND , -1

)

IEND=IEND-ISPACE

JEND=IEND+ISPACE-50

PRINT202, (TX(J) ,WW(J) ,DWDTT(J) , TX{ J-50 )

( WW ( J-50 ) ,DWDTT(J-50) , J=

IIEND, JEND,-1)

55 IEND=JEND-50

JEND=IEND-50

DO 60 K=1,IPAGES

PRINT102 , SCALE , SCALE

PRINT202, (TX(J) .WW( J) ,DWDTT( J) ,TX(J-50) , WW(J-50) , DWDTT( J-50) , 1IEND.JEND,-1)

IEND=JEND-50

JEND=IEND-50

60 CONTINUE

TSTART=TEND

65 GO TO 1 69 PRINT 999,

70 STOP

199 FORMAT (IX)

100 FORMAT ( IHl/ 2X ' TEMP .

' 5X ' WSTAR ' 5X ' DERIVATIVE ' 5X / IX , A6 , 4X

1 'FUNCTION '5X'X 1000 '7X ) 102 FORMAT (1H1/2(2X' TEMP. ' 5X 'WSTAR ' 5X ' DERIVATIVE ' 5X) /2 ( IX . A6 ,4X

1 'FUNCTION '5X'X 1000 '7X))

101 FORMAT (IHl/ 38X'TEMP. ' 5X ' WSTAR ' 5X ' DERIVATIVE ' 5X / 38X,A6,4X

1 'FUNCTION '5X'X 1000 '7X ) 200 FORMAT! IX, 0PF6 . 1 , 2X , FIO . 8 , 3X , 3PF9 . 7 , 6X ) 201 FORMAT( 38X, 0PF6 . 1 , 2X , FIO . 8 , 3PF9 . 7 , 6X ) 202 F0RMAT(2(1X,0PF6.1,2X,F10.8,3X,3PF9.7,6X)

)

999 FORMAT (IH, •*****CAN NOT COMPUTE TABLE AS REQUESTED. RUN STOPPED'

1/lX, 'HERE. CHECK TABLE DISCRIPTION CARD *»«' ) END

XQT WSTAR

1 0 . 2508462096788033D003

2 0 . 1350998699649997D003

3 0.5278567590085172D002

4 0 . 2767685488541052D002

5 0.3910532053766837D002

5 0.6556132305780693D002

7 0 . 8080358685598667D002

8 0. 7052421 182340520D002

9 0.4478475896389657D002

10 0.2125256535560578D002

11 0.7679763581708458D001

12 0.2136894593828500D001

13 0.4598433489280693D000

14 0 . 7636146292316480D-01

15 0.9693286203731213D-02

16 0.9230691540070075D-03

17 0.6381165909526538D-04

18 0.3022932378746192D-05

19 0.8775513913037602D-07

20 0.1 17702613 1254774D-08

83
IPTS-68 Reference Function and Its First Derivative With the Argument in Kelvin

T dW*{T)ldT T dW*(T)jdT

K W*{T) X 1000

K-' K X 1000 K-'

n .00174542 .3163198

1

. X .00177737 .3227158

1 ^ O .00180996 .3291831

1 .00184321 .3357224

J. \j > A .00187711 . 3423344

\j .00191168 .3490197

PD .00194692 .3557792

X o . 7 .00198283 .3626135

X o .. Qo .00201944 .3695234

±o q . 00205674 .3765094

XD1 P. ., nu .00209474 .3835722

XD1 P... X .00213346 .3907123

XD .. o<c .00217289 . 3979300

XD .. o .00221305 .4052258

XD c /i .00225394 .4125998

XD ,, o .00229557 .4200522

XD1 P. , p.

. D . 00233795 .4275832

X D .. 1 .00238109 .4351925

X1 PD . a, o . 00242499 .4428801

X D . q . 00246967 .4506458

1X 71 . n .00251512 .4584892

X / .. X .00256137 .4664099

X / . o . 00260841 .4744074

X I . o .00265625 .4824812

X ( .. 4 . 00270491 .4906307

X I ,. O . 00275438 . 4988551

X ( , c. D .00280468 .5071538

X ( , rr

. 1 .00285581 .5155260

1 T1 ( ,. oQ .00290779 .5239710

X ( , q. y .00296061 . 5324879

13.0 .00123062 .2033212 X O ,. u .00301429 .5410760

13.1 .00125120 .2082522 Xo ,. X .00306883 . 5497344

13.2 .00127228 .2132663 X o , o. c .00312424 . 5584624

13.3 .00129386 .2183622 X o

,

z. o .00318052 . 5672591

13.4 .00131595 .2235382 X o

.

. ^ .00323769 .5761238

13.5 .00133857 .2287929 X o

.

. o .00329575 .5850558

13.6 .00136171 .2341247 xo

.

c. D .00335470 . 5940542

13.7 .00138539 .2395323 XO .. 1 .00341456 .6031185

13.8 .00140962 .2450143 XO .. O . 00347533 .6122479

13.9 .00143440 .2505696 XO . q .00353701 .6214418

14.0 .00145974 .2561970 xy

.

. U .00359962 .6306996

14.

1

.00148564 .2618957 . 1 .00366316 .6400207

14.2 .00151212 . 2676649 o. <c . 00372763 . 6494046

14.3 .00153918 .2735041 "7

. O . 00379304 . 6588507

14.4 .00156682 .2794127 ly

.

, 4 .00385940 . 6683587

14.5 .00159506 .2853906 19..5 . 00392671 . 6779280

14.6 .00162390 .2914375 19. 6 . 00399499 . 6875581

14.7 .00165335 .2975537 19.,7 . 00406423 . 6972487

14.8 .00168342 .3037392 19.,8 .00413444 . 7069993

14.9 .00171410 .3099944 19.,9 .00420563 .7168096

15.0 .00174542 .3163198 20.,0 . 00427780 . 7266791

84
IPTS-68 Reference Function and Its Derivative With the Argument in Kelvin — Continued

T

K

dW*(T)ldT

X 1000

K-'

T

K W*{T) X 1000 K-'

on nu 00427780 7266791 25. 0 0099R'=i0fi 1 2810697

on J. 00435097 7366074 25. 1 00938376 1 2929735

9n/£U . 0<c 00442513 7465942 25. 2 00951365 1 3048932

C\) o 00450029 7566391 25. 3 00964474 1 3168281

90<cu . /I 00457646 7667415 25. 4 00977702 1 3287776

00465364 7769012 25. 5 00991049 1 • 3407410 KJ X \J 1 X ^—

9n a.o 00473184 7871176 25. 6 01004517 1 • 3527177 <_/ fw 1^1 1 9n I 00481106 7973902 25. 7 01018104 1 • 3647070 \^\j^^ 1 \j 1 ^y on QO 00489132 ,8077186 25. 8 01031811 1 • 3767085 KJ 1 Xl/ 1 \J \mf\J on Q 00497261 ,8181023 25. 9 01045638 1 3887215

91 nU 00505494 ,8285406 26. 0 01059585 1 4007456

91 1

J. 00513832 ,8390330 26. 1 01073653 1 4127801

0

1

o/C 00522275 ,8495789 26. 2 01087841 1 • 4248246 A A^ XV— X W

0

1

c,± . O 00530824 ,8601776 26. 3 01102149 1 . 4368786

on /I 00539479 , 8708284 26. 4 01116578 1 4489416

0

1

cO 00548240 .8815306 26. 5 01131128 1 4610132

0

1

D 00557109 .8922834 26. 6 01145799 1 4730930 X 1 KmJ yJ ^J \^ 9

1

( 00566086 .9030861 26. 7 01160590 1 4851804

91 oo 00575171 .9139377 26. 8 01175502 1 * 4972751 X ^*J 1 tfO 1 «^ 91 Q 00584365 . 9248375 26. 9 01190536 1 . 5093766

99 n 00593668 . 9357846 27. 0 01205690 1 . 5214845

OO 00603081 . 9467780 27. 1 01220965 1 . 5335984

99 o 00612604 .9578167 27. 2 01236362 1 . 5457180

99 z

. O 00622237 . 9688998 27. 3 01251880 1 . 5578427

99/C/C .. 4A 00631 98? .9800263 27. 4 01267519 w X *v vy I \J X ^ 1 . 5699723

99 . O 00641 838 .9911951 27.,5 01283279 1 . 5821063

99 c. D 00651 806 1 .0024053 27.,6 01299161 1 . 5942443

99 ry

. / 00661 886 1 .0136557 27.,7 • 01315164 vy X »—/ X O X \J^X 1 . 6063859

99 . po 00672079 1 . 0249453 27,,8 01331289 1 . 6185307

OOC.C, Q 00682385 1 .0362729 27, 9 01347535 1 . 6306783

9"^ (CO n 00692805 1 . 0476376 28.,0 01363902 1 . 6428283

9"^

. X 00703338 1 .0590382 28., 1 01380391 1 . 6549803

01. o. <c 0071 3986 1 . 0704735 28.,2 01397002 1 . 6671338

9"^ z

. o 00724748 1 .0819427 28,.3 01413734 1 . 6792885

9"? A

. 4 0073'=i6?5 1 . 0934444 28.,4 01430588 1 . 6914438

9*^ c

. O 0074661

7

1 . 1049777 28..5 01447563 \-/ ^ X X 1 KJ \JKJ 1 . 7035995

9T . o 00757724 1 .1165414 28 ,6 01464660 1 .7157549

9"^

. 1 1 .1281345 28 ,7 01481878 1 . 7279097

9*7 (CO Q. o 1 . 1397560 28 .8 01499218 1 . 7400633

9*7 (CO Q. y 1 . 1514048 28 .9 01516679 1 . 7522154

9/1 (C4 n. u 1 . 1630800 29 .0 01534262 1 . 7643654

9/1 <c4 . X . 0081 \J \J O d. %J'^005 \J \J \J 1 . 1747804 29 . 1 01551966 1 .7765129

9/1 <c4 o. (C . \J\JO J. J. 1 .1865051 29 .2 01569792 1 . 7886573

9/1 . o 1 .1982532 29 .3 . 01587740 1 . 8007982

0/1 <c4 . 4 nnftf=in77fi 1 .2100238 29 .4 01605808 1 .8129350

9/1 /c4 . o 1 .2218159 29 .5 01623998 1 . 8250672

9/1 <c4 . D 1 .2336287 29 .6 01642310 1 . 8371944

94 00887608 1 .2454613 29 .7 . 01660742 1 .8493158

24 .8 .00900122 1 .2573128 29 .8 .01679296 1 .8614311

24 .9 .00912754 1 .2691826 29 .9 .01697971 1 . 8735397

25 .0 .00925506 1 .2810697 30 .0 .01716767 1 . 8856409

85
IPTS-68 Reference Function and Its Derivative With the Argument in Kelvin — Continued

T

K W (1 } dW*{T)jdT

A iUUU

K-'

T

J\. W (I)

dW*(T)ldT

X iUUU

K->

30 . 0 .01716767 1 . 8856409 35.,0 . 02807645 2 .4686391

30 . 1 .01735684 1 . 8977343 35., 1 .02832386 2 . 4796248

30 . 2 .01754721 1 .9098192 35.,2 . 02857237 2 .4905766

30 . 3 .01773880 1 .9218951 35,,3 .02882197 2 .5014945

30 .4 .01793159 1 .9339615 35.,4 . 02907267 2.,5123779

30 . 5 .01812559 1 .9460176 35.,5 . 02932445 2 .5232267

30 . 6 .01832080 1 .9580630 35.,6 . 02957731 2 . 5340406

30 .7 .01851720 1 .9700970 35.,7 .02983125 2 .5448194

30 .8 .01871481 1 .9821191 35. 8 .03008627 2 .555562^

30 .9 .01891363 1 .9941286 35.,9 . 03034237 2 . 5662706

31 . 0 .01911364 2 .0061250 36.,0 . 03059953 2 . 5769426

31 . 1 .01931485 2 .0181076 36. 1 . 03085775 2 . 5875786

31 . 2 .01951726 2 .0300758 36. 2 .03111704 2 .5981783

31 . 3 .01972087 2 .0420291 36. 3 .03137739 2 . 6087417

31 .4 .01992567 2 .0539668 36. 4 .03163879 2 .6192684

31 .5 .02013166 2 . 0658884 36. 5 .03190124 2 . 6297583

31 . 6 .02033884 2 . 0777932 36. 6 . 03216474 2 .6402113

31 .7 .02054722 2 .0896806 36. 7 . 03242928 2 . 6506273

31 .8 .02075678 2 .1015501 36. 8 . 03269486 2 .6610059

31 . 9 .02096753 2 .1134010 36. 9 . 03296148 2 . 6713472

32 . 0 .02117946 2 .1252328 37. 0 . 03322913 2 .6816510

32 . 1 .02139257 2 . 1370449 37. 1 . 03349781 2 .6919171

32 . 2 .02160687 2 . 1488366 37. 2 . 03376751 2 .7021455

32 . 3 .02182234 2 . 1606075 37. 3 . 03403824 2 .7123360

32 .4 .02203899 2 .1723570 37. 4 . 03430998 2 . 7224886

32 . 5 .02225681 2 . 1840844 37.,5 . 03458273 2 .7326031

32 .6 .02247580 2 . 1957893 37. 6 . 03485650 2 . 7426794

32 .7 .02269597 2 .2074710 37. 7 .03513127 2 . 7527175

32 .8 .02291730 2 .2191291 37. 8 . 03540704 2 . 7627173

32 .9 .02313979 2 .2307631 37. 9 . 03568381 2 . 7726787

33 .0 .02336345 2 . 2423723 38.,0 . 03596158 2 .7826016

33 . 1 .02358827 2 .2539562 38.,1 . 03624033 2 . 7924860

33

,

. 2 .02381424 2 .2655145 38. 2 . 03652007 2 .8023319

33

,

. 3 .02404137 2 . 2770464 38. 3 . 03680080 2 .8121391

33.,4 .02426965 2 .2885517 38. 4 .03708250 2 .8219077

33

.

. 5 .02449908 2,.3000297 38. 5 .03736518 2 .8316376

33

.

.6 .02472965 2 .3114801 38. 6 . 03764882 2 .8413287

33

.

.7 .02496137 2 .3229023 38.,7 . 03793344 2 .8509810

33

.

,8 .02519423 2 .3342959 38.,8 .03821902 2 . 8605946

33

.

,9 .02542823 2..3456605 38.,9 . 03850556 2 ,8701693

34

.

0 .02566336 2..3569956 39.,0 . 03879305 2 . 8797052

34

.

1 .02589963 2,.3683009 39., 1 .03908150 2 .8892022

34

.

2 .02613702 2 . 3795758 39,.2 . 03937089 2 . 8986603

34

.

3 .02637554 2 .3908200 39,.3 . 03966123 2,. 9080795

34

.

4 .02661518 2..4020332 39,.4 .03995251 2,.9174598

34. 5 .02685595 2 .4132149 39,.5 . 04024472 2., 9268012

34. 6 .02709783 2,. 4243648 39,.6 . 04053786 2. 9361038

34. 7 .02734082 2,.4354825 39,.7 . 04083194 2. 9453674

34. 8 .02758492 2..4465677 39,.8 .04112694 2. 9545921

34. 9 .02783013 2..4576200 39,,9 .04142286 2. 9637779

35. 0 .02807645 2.,4686391 40,.0 .04171969 2. 9729248

86
IPTS-68 Reference Function and Its Derivative With the Argument in Kelvin — Continued

T

K W*(T)

dW*{T)ldT

X 1000

K-'

T

K W*(T)

dW*(T)ldT

X 1000

4UACi , n .04171969 2 .9729248 45 . U . 05764487 •7O

•ZPl O'ZCV

4UAC\ ,. 1 .04201744 2 .9820329 45 . 1 . 05798335 'ZO TOO

A

TTQ . ooo4o /

y

/in o. /C .04231610 2 .9911020 45 . 2 .05832255 •zo . oyobuiy

/in 7

. o .04261566 3 .0001324 A CZ. 45 . 3 . 05866247 'Zo

/I noOT) PQ . 4U2 ( <ioy

A n A

. 4 .04291612 3 .0091238 45 . 4 . 05900310 •7o

A noPT PP . 4Uyolc50

Ar\ , O .04321748 3 .0180765 A C 45 rr

. 0 . 05934443 '7o . 41bC3 ( lC5

., D .04351974 3 .0269903 A C

. b . 05968647 zO /lOTPPTQ

AH .04382288 3 .0358654 A C 40 rr

. ( .06002921 zO /iTnPCTo

^n p .04412691 3 .0447017 A C 40 , o. O . 06037264 •7O . 4oA TTPnQC ( ttuyb

An Q .04443182 3 .0534993 A C 40 , . 06071677 •7O/I /I /I "71 I^T

. 444 f lOO

n . 04473761 3 .0622582 A C 4b

.

. U . 06106158 TO . 4010040

/1

1

. 1 . 04504427 3 . 0709783 4b

.

. i . 06140708 •zO . 40c341b A c;P/l 1 C71 A 1 o. /C .04535180 3 . 0796598 A C 4b

.

, <c .06175327 o . 4b0<cl/c0

ftl Al . 7. O .04566020 3 .0883026 A d 4b

.

•2O . 06210013 zo . 4/IT!( 1

Q'Tl y ( 1 Po

41 . A. 4 . 04596946 3 .0969069 4b

.

, 4 . 06244766 zO . 4 ( 00340

41 ., o .04627958 3 .1054726 4b

.

. o . 06279586 •z0 . 4000011

41 ,. D .04659055 3 .1139997 A C: 4b

.

, b . 06314473 zO 4QOn'71 1 A^41 .. 1 .04690238 3 .1224883 A C 4b

.

rr

, ( . 06349427 •zO . 4i/004417 4QPfi44Q

A^41 . p .04721505 3 . 1309384 A C 4b

.

p, O . 06384446 "ZO . OUO/C/C/CO

4141 . . 04752856 3 . 1393500 A Cl 4b . Q, y .06419531 •zO . Ol 1 1 ooy

A O 4<c . n . 04784292 3 . 1477233 A "7 4 ( ., U . 06454681 •zO . oio<cby<c

A O . 1 .04815811 3 . 1560581 A *y 4 [ .. 1 . 06489896 •zO . 0<c4 / OOO

/I o o. /C .04847413 3 . 1643546 A4 ( .. 2 .06525176 •zo

c-z 1 1 "71 P . OOll ( lO

/I o T

. O .04879098 3 .1726127 47

.

. 3 . 06560520 •zo . Oo ( Obso

/I o 4<c .. 4 .04910865 3 .1808326 A T4 { .. 4 . 06595927 •7o . 040i/OUO

A O 4/; . O .04942714 3 . 1890142 A4 ( ., 5 . 06631398 zO . OOUcOb

(

A O 4/C . b . 04974645 3 .1971576 A T4 ( .. b . 06666932 "ZO . 00b04b0

/I o 4/C . rr

, / .05006657 3 .2052628 47

.

rr

. 1 .06702529 •zO ccopni T . ObcOulo

/lO 4/C . p .05038750 3 .2133299 4 1 o. o .06738188 zO RCQno no

4<c AO . Q .05070924 3 .2213589 4 ( .. y . 06773909 "ZO . O 1 O/C UO I 40/IT . n .05103177 3 .2293498 A O 4o . u . 06809692 •zO . OOlOOl RQl TRl 7

(

40 .. 1 .05135511 3 .2373027 A P 4o . 1 . 06845536 zO . OO 1 4040

4"^ 40 . o. c .05167923 3 .2452176 A O 4o O

. <C . 06881441 zO . Oi'0041 z) 40/I "T

.

•2

, o .05200415 3 .2530945 A O 4o '7

. o .06917407 •zO . 0-3i/004<&

40/I. /I, 4 .05232985 3 .2609335 A O 4o A

. 4 . 06953433 zO . DU00310

404T .. O .05265633 3 . 2687347 A O 4o . O . 06989519 •zO . Ol lOOOO

40/I. c, D .05298359 3 . 2764980 A O 4o . b . 07025664 "ZO

CI '7Rnn'7 . Ol 1 DUU

1

4o/IT , rr

. 1 .05331163 3 .2842236 A O 4o rr

. 1 .07061869 zO . 0/i04U0U

A T 4o . p .05364044 3 .2919114 A O 4o o. o .07098132 "Z

. D<; cj/c ( uo

A "7 4o . Q .05397001 3 .2995615 A O 4o . y . 07134454 •zO . RTRi DOOl nTT UOO

44/I A , n. u .05430035 3 .3071739 A C\ 4y . u . 07170834 zO R4nQni R

/I /I 44 .. 1 .05463145 3 .3147488 49 . 1 . 07207272 "ZO . 04D000(C

A A 44 o. /i .05496330 3 .3222860 49 . 2 . 07243767 'ZO . bo<coy4o

44A A .. 6 .05529590 3 .3297857 49 . 3 .07280320 "ZO . OOOUOy/£

/I A 44 . 4 .05562925 3 . 3372479 49 . 4 . 07316929 •zO . bbo 1 4y

1

44/I /I. 0 .05596335 3 . 3446727 49 . 5 . 07353595 •zO . OOi/O I Ol

44 .6 .05629819 3 .3520602 49 .6 .07390316 3 . 6749683

44 .7 .05663376 3 .3594102 49 .7 . 07427094 3 .6805266

44 .8 .05697007 3 .3667230 49 .8 . 07463927 3 .6860509

44 .9 .05730710 3 .3739985 49 .9 . 07500815 3 .6915414

45 .0 . 05764487 3 .3812367 50 .0 . 07537757 3 . 6969982

87
IPTS-68 Reference Function and Its Derivative With the Argument in Kelvin — Continued

T1K W*{T) X 1000

K->

i

T

K. dW*{T)ldT

XIOOO

K'

ou . n 075*^7757 zO •JO 55 Oq44751

q

. Us7*±*4 1 oXTy 3 9291588

ou . 1X 07574755 •zO 709491 "Z 55 . X 0q4RRR'70 3 9330396

ouRfl . . u f ox xouo "Zo . 707R1 1 \J 1 ox 09U3 55 p 0q59R1 7q 3 9368927

o n7fi4ftqi

1

zo 71 -zi C70 55 z. o . 0q5R55RR VJ^ODOODO 3

.

9407182

*± •z 71 R4R9R 55 4 09R04994 3

.

9445161

zo

7P'Z779'z OO55 R. O 09R4445R 3

.

9482867

p. n77fin544 zo 79QO'ZR7 OO55 . RO 09RR'^959 3

.

9520301

71 n77q7ftfii •zo 7'Z49R91 OO55 7 0979 '^49R 3

.

9557464

o •zo 7*^94494 OO55 R 097R7074 3. 9594356

so 9 o 744R070 OO55 q 09809687 3 9630981

0 0791 0199 •zo 749771

R

. 1 f oxo 0\J 5R 0 09849"^"^fi 3

.

9667339

O J. . X 07947fi44 •zo 754R9'^q 0\J 5R 1

. X 09RR9099 3 9703430

•SI oc 07qS5P1 ft o"Z 7RQRR'ZR OO5R p 09991 747 3 9739257

uj-

.

zo 0ft099R49 o"Z

. 7R4QT 1 D41 JX 07U 1 OO5R z. o n99R1 ROO 3 9774821

*± OROfiORI

R

o•z OO5R 4 1 0001 999 3 9810123

OftOqft940 "Z 77/1RRR9 OO5R R. O . XVJVJ*±XX<CU 1 0041 1 90 3 9845165

c . OR! uoxoouxo ^fiOl •z 77Q7QR7 OD5R c. D . 1XUUOW^yOO 00R09R7 3 9879947

«J X . 7 . OR! UOX7'^R7R 1 oooo z 7R/1RQ7'Z OO5R n. f 101 90RR0 3 9914472

«J X . oQ 0R91 1 707 z OD5R . RO 1 01 R0R1

9

. XVJXDVJOX/C 3

.

9948740

ox . q 0R94qfi97 o•z 7Q4'ZQRR OD5R q . X1 0900777 \J/Zf\J\J III 3

.

9982753

s? 0R9R75q5 o"Z 7QQpn91 O57I 0 . X1 0940777 \Ji^*±\J III 4. 0016511

s? 0R'^95fil 1 "Z on'zq7'Z7 O571 1

i X 1 09R0R1

0

4, 0050017

p 0R'^fi'^fi74 o•z R0R71 40 O571 p . X\JOtZi\JO 1 0*^9 0R771 1 4. 0083273

S? "7

. 0R401 wO'xWX 7R5 f oo "Z R1 7499q O57f . •z. o . X\JO\J\J^ 1 0'^R09771 1 4. 0116278

SP 4 0R4'^qq4'^ o"Z R1 R1 OOR O571 . 4 1 0401 1 09 4. 0149033

s? . 0R47R1 \JO*± r OXI47f o"Z R997479 O571 R. O . X\J*±^X<t 1 0441 97R1 o 4, 0181542

C 0R51 R'^qR o•z R97'ZR9R O571 . c. o 1 04R1 479 4. 0213805

s? 7 0R554Rq4 o"Z R'ZI QA7R O571 7 1 0591 709 4 0245823

s? » ORSq^O'^R o•z R'ZRROl R O57I . o. o . X\JO\jX^\j*± 1 05R1 9fi4 4 0277598

s? q ORR"^! 494 zo ^iA^ OPRO OS71 . q 1 0R09957 4 0309130

0 nRRfiqRR7 "Z R4RR1 7Q OORR . . XUO^^OOc 1 0R49RR9 4 0340422

1

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RylQQROi^ OORR .. J. . XUOOjC^OO 1 ORR9Q'ZR 4 . 0371474

9 0R74RR5R "Zo . O044x<c ( OORR . p 1 079'^79R 4 0402288

. o 0R7R5499 "Z

. 00001410 OORR . •z

. o . X1 07R774'^ U I OO 1 ^O 4 0432865

< 4 0RR940'^9 o . ODOxOOl? RR'ZI RRQ OORR . 4 . XUOv*i±^7X 1 0R041 91 4 0463206

. \J 0RRR9RRR "Z

. OO / OfCi? J. OORR .. oR . XU0*4**00^ 1 OR44RR9 4 0493314

\J<J .. ao ORqOl "^R"^ "Zo . O ( 10410 OOc^R .. D . XVJOOOX 1 ORRRl 771 1 4 0523188

• 7 0Rq401 9"^ "Z R7R1 PA'Z OORR . rr

. 1 1 099R71

R

. X \J^c,o t Xo 4 0552830

» 0R97R905 o•z

. RRO'Z77R OOUO I/O OORR . o, o . XU^DO^OO 1 09RR9R'^ 4 0582243

9 0901 77*^0 o"Z RRylROI A OORR . q 1 1 OORRRO 4 0611426

54 0 0905fi5Q7 . o"Z ooQ7qcq . ooo / ^?o^ 03Rq . . X1 1X 047R0R f OvO 4 0640381

54 1 0909550fi o"Z RQPQCl "Z 03Rq .. X . XXVJOOXOO 1 1 0RR1 RO 4 0669110

54 p 091 '^445R •z

. RQ70Q'77 Oi7 ( Ul7 I ( 03Rq . p . xxx^coo**** 111 9RR44 4 0697614

54 091 7'^44ft z QOl POt^P 03Rq . "Z

. o . X111X X0.7000 R9RRR 4 0725893

54 A. *± 0991 94R0 "Zo 03 . A. ft . XXc,X\j<C^o 1191 09QR A*±

.

»j*± . 09951 5R4 zo

QHQ'Z'Z'ZQ oyRQ

.

R. o1 1 PR1 OR'Z . XXc.oXo'Oo 4 . 0781786

09990RR7 zo . yxoooo4 oyRQ

.

. D . ±

1 j.<cyxooy 1 PQ1 RRQ 4 . 0809402

. 7f •zo . y

1

1 o4oo RQoy

.

rr

. 1 1 1 'Z'ZOCRO . XiOO<CDO(C 4 . 0836798

54..8 .09369014 3 .9213134 59..8 . 11373532 4 . 0863978

54..9 . 09408247 3 .9252501 59..9 .11414410 4 . 0890941

55.,0 . 09447519 3 .9291588 60..0 .11455314 4 . 0917688

88
IPTS-68 Reference Function and Its Derivative With the Argument in Kelvin — Continued

T

K

dW*(T)ldT

X 1000

K-'

T

K

dW*(T)ldT

X 1000

60 . 0 . 11455314 4. 0917688 65

.

0 . 13530364 4 . 2007677

60 . 1 .11496245 4. 0944222 65. 1 . 13572380 4 . 2025038

60 .2 .11537203 4. 0970544 65. 2 . 13614414 4 . 2042243

60 .3 .11578186 4

.

0996654 65. 3 . 13656464 4 . 2059294

60 .4 .11619196 4. 1022555 65. 4 . 13698532 4 . 2076192

60 .5 .11660231 4

.

1048246 65. 5 . 13740617 4 . 2092937

60 .6 . 11701292 4

.

1073730 65. 6 . 13782718 4 . 2109531

60 .7 . 11742379 4

.

1099008 65. 7 . 13824836 4 . 2125975

60 .8 .11783490 4

.

1124081 65. 8 . 13866970 4 . 2142270

60 .9 .11824627 4. 1148950 65. 9 . 13909120 4 . 2158416

61 . 0 . 11865788 4

.

1173616 66. 0 . 13951287 4 . 2174415

61 . 1 .11906974 4

.

1198081 66. 1 . 13993469 4

y^ -I y^ y^ y^

. 2190268

61 . 2 . 11948184 4

.

1222345 66. 2 . 14035667 4

y^ y^ y^ /^r^

. 2205976

61

,

.3 . 11989419 4

.

1246411 66. 3 . 14077881 4 . 2221539

61 .4 . 12030677 4. 1270279 66. 4 . 14120110 4

y^ y^ p^ y^ »—

. 2236958

61 .5 .12071959 4. 1293951 66. 5 . 14162355 4

y^ y^ y^

. 2252236

61

.

.6 .12113265 4. 1317428 66. 6 . 14204615 4

y-v y^ ^^f^fVI^ ^

. 2267371

61

,

.7 . 12154594 4. 1340709 66. 7 . 14246889 4

y^ ^9 y^

. 2282367

61

.

.8 .12195946 4. 1363799 66. 8 . 14289179 4

y^ y^

. 2297223

61

,

.9 .12237321 4. 1386696 66. 9 . 14331484 4 . 2311939

62 . 0 . 12278719 4. 1409403 67. 0 . 14373803 4 , 2326518

62

,

, 1 .12320140 4., 1431921 67. 1 . 14416137 4,. 2340961

62 . 2 .12361583 4

.

. 1454250 67. 2 . 14458485 4,. 2355267

62 . 3 . 12403049 4

.

, 1476393 67. 3 . 14500847 4

,

. 2369439

62 .4 . 12444536 4.. 1498349 67. 4 . 14543224 4

,

. 2383476

62 . 5 . 12486045 4.. 1520121 67.,5 . 14585614 4

,

. 2397380

62..6 . 12527576 4.. 1541709 67. 6 . 14628019 4., 2411152

62..7 .12569129 4.. 1563115 67. 7 . 14670436 4., 2424793

62 .8 .12610702 4.. 1584339 67., 8 . 14712868 4

.

2438303

62..9 . 12652297 4.. 1605384 67., 9 . 14755313 4. 2451683

63 , 0 .12693913 4.. 1626249 68..0 . 14797771 4

.

2464935

63,. 1 . 12735550 4 . 1646937 68., 1 . 14840243 4. 2478058

63 .2 .12777207 4 . 1667448 68.,2 . 14882727 4

.

2491055

63 .3 .12818884 4 , 1687783 68..3 . 14925225 4

.

2503925

63 .4 .12860582 4 . 1707944 68,.4 . 14967735 4.

y-\ 1— 1 y^ y^ f^r -l 2516671

63 .5 .12902300 4 . 1727931 68.,5 . 15010258 4. 2529291

63 .6 . 12944038 4 1747746 68,.6 . 15052794 4

.

2541788

63 .7 . 12985796 4 . 1767390 68,.7 . 15095342 4. 2554163

63 .8 . 13027573 4 . 1786864 68,.8 . 15137902 4. 2566415

63 .9 . 13069369 4 . 1806169 68 . 9 . 15180475 4

.

2578547

64 .0 .13111185 4 . 1825306 69 . 0 . 15223059 A4

.

2590558

64 . 1 .13153020 4 . 1844276 69 . 1 . 15265656 4

.

2602449

64 .2 . 13194874 4 . 1863081 69 .2 . 15308264 4

.

2614222

64 .3 . 13236746 4 . 1881721 69 .3 . 15350884 A4

.

2625877

64 .4 . 13278637 4 . 1900197 69 .4 . 15393516 4

.

2637415

64 .5 . 13320546 4 . 1918511 69 .5 . 15436159 4

.

2648838

64 .6 . 13362474 4 . 1936663 69 .6 . 15478813 4. 2660144

64 .7 . 13404420 4 . 1954655 69 .7 . 15521479 4. 2671336

64 .8 . 13446383 4 . 1972487 69 .8 . 15564156 4. 2682414

64 .9 . 13488365 4 .1990161 69 .9 . 15606844 4. 2693379

65 .0 .13530364 4 .2007677 70 .0 . 15649543 4. 2704231

89
IPTS-68 Reference Function and Its Derivative With the Argument in Kelvin — Continued

T

K lv^(l ) dW*(T)ldT

X 1000

K->

]TK W*(T)

dW*(T)ldT

X 1000

K-'

70,.0 . 15649543 4.2704231 75 .0 .17796116 4 .3119570

70,.1 .15692252 4 . 2714973 75 .1 .17839239 4 .3125623

70,.2 .15734973 4.2725603 75 .2 . 17882367 4 .3131598

70,.3 .15777703 4.2736124 75 .3 .17925502 4 .3137496

70,,4 . 15820445 4.2746535 75 .4 . 17968642 4 .3143316

70,.5 .15863197 4 . 2756838 75 .5 .18011789 4 .3149061

70,.6 .15905958 4.2767033 75 .6 . 18054940 4 .3154730

70,.7 . 15948731 4.2777121 75 .7 . 18098098 4 .3160323

70,.8 .15991513 4.2787103 75 .8 . 18141261 4 .3165842

70..9 .16034305 4.2796980 75 .9 .18184430 4 .3171287

71

.

.0 .16077107 4.2806751 76 .0 . 18227604 4 .3176658

71..1 .16119918 4.2816419 76 . 1 . 18270783 4 .3181957

71

.

.2 .16162739 4.2825983 76 .2 . 18313968 4 .3187183

71

.

.3 .16205570 4 . 2835445 76 .3 .18357157 4 .3192336

71..4 .16248410 4 . 2844805 76 .4 . 18400352 4 .3197419

71..5 .16291260 4.2854064 76 .5 . 18443552 4 .3202431

71..6 .16334118 4.2863222 76 .6 . 18486757 4 .3207372

71

.

.7 . 16376986 4.2872280 76 .7 . 18529967 4 .3212243

71..8 .16419863 4.2881240 76 .8 .18573181 4 .3217045

71..9 . 16462749 4.2890101 76 .9 .18616401 4 .3221778

72..0 . 16505643 4 . 2898864 77 .0 . 18659625 4 . 3226442

72..1 . 16548546 4.2907530 77,. 1 . 18702854 4 .3231039

72..2 .16591458 4.2916101 77,.2 . 18746087 4 .3235568

72..3 . 16634378 4 . 2924575 77,.3 . 18789325 4 .3240030

72.,4 .16677307 4.2932954 77,.4 . 18832567 4 . 3244426

72..5 . 16720244 4.2941240 77,.5 . 18875814 4 . 3248755

72..6 .16763190 4 . 2949431 77,.6 . 18919065 4 .3253020

72,.7 .16806143 4.2957529 77,.7 . 18962320 4,.3257219

72,.8 .16849105 4.2965535 77..8 .19005579 4,.3261354

72,.9 . 16892074 4 . 2973449 77..9 . 19048842 4.. 3265424

73,.0 .16935052 4.2981273 78.. 0 .19092110 4..3269431

73,,1 . 16978037 4.2989005 78.. 1 .19135381 4.. 3273375

73,,2 .17021030 4 . 2996649 78,,2 . 19178657 4.,3277257

73 .3 . 17064030 4.3004202 78.,3 .19221936 4.,3281076

73 .4 .17107038 4.3011667 78, 4 . 19265219 4., 3284832

73 .5 .17150053 4.3019045 78.,5 .19308505 4.,3288529

73 .6 .17193076 4.3026335 78.,6 . 19351796 4. 3292164

73 .7 .17236106 4.3033539 78.,7 . 19395090 4. 3295738

73 .8 . 17279143 4.3040656 78. 8 . 19438387 4. 3299253

73 .9 .17322187 4 . 3047688 78. 9 . 19481688 4. 3302707

74 .0 .17365238 4.3054635 79. 0 . 19524993 4. 3306104

74 . 1 . 17408296 4.3061498 79. 1 .19568300 4. 3309441

74 .2 .17451361 4.3068277 79. 2 .19611611 4. 3312720

74 .3 . 17494433 4 . 3074974 79. 3 . 19654926 4. 3315941

74 .4 .17537511 4.3081587 79. 4 . 19698243 4. 3319105

74 .5 .17580596 4.3088119 79. 5 . 19741564 4. 3322212

74 .6 . 17623687 4.3094570 79. 6 . 19784888 4. 3325262

74 .7 . 17666785 4.3100939 79. 7 . 19828214 4. 3328256

74 .8 . 17709889 4.3107229 79. 8 . 19871544 4. 3331194

74 .9 .17753000 4.3113439 79. 9 . 19914877 4. 3334077

75 .0 .17796116 4.3119570 80. 0 .19958212 4. 3336906

90
IPTS-68 Reference Function and Its Derivative With the Argument in Kelvin — Continued

T

" \i ) dW*(T)ldT

X 1000 K->

T

K " \' )

dW*{T)ldT

X 1000

K'

80. 0 . 19958212 4. 3336906 85. 0 .22127523 4. 3416637

80. 1 .20001551 4. 3339679 85. 1 .22170939 4. 3417150

80. 2 . 20044892 4

.

3342399 85. 2 . 22214357 4. 3417627

80. 3 .20088235 4. 3345065 85. 3 . 22257775 4. 3418067

80. 4 .20131582 4

.

3347678 85. 4 .22301193 4. 3418471

80. 5 .20174931 4. 3350238 85. 5 .22344612 4. 3418839

80. 6 .20218282 4. 3352745 85. 6 . 22388031 4. 3419171

80. 7 .20261636 4. 3355200 85. 7 .22431450 4. 3419467

80. 8 . 20304993 4. 3357603 85. 8 . 22474870 4. 3419728

80. 9 .20348351 4. 3359955 85. 9 .22518289 4. 3419955

81. 0 .20391713 4. 3362257 86. 0 .22561709 4. 3420146

81. 1 . 20435076 4. 3364507 86. 1 .22605130 4. 3420303

81. 2 . 20478442 4. 3366708 86. 2 .22648550 4. 3420427

81. 3 .20521809 4. 3368858 86. 3 .22691971 4. 3420516

81. 4 .20565179 4. 3370959 86. 4 .22735391 4 3420572

81. 5 .20608551 4. 3373012 86. 5 .22778812 4 3420594

81. 6 .20651925 4 3375015 86 6 .22822232 4 3420584

81. 7 .20695301 4 3376970 86 7 . 22865653 4 3420541

81. 8 . 20738679 4 3378877 86 8 .22909073 4 3420465

81. 9 . 20782059 4 3380737 86 9 . 22952494 4 3420357

82. 0 . 20825441 4 3382549 87 0 . 22995914 4 3420217

82. 1 . 20868824 4 3384315 87 1 .23039334 4 3420045

82. 2 .20912209 4 3386034 87 2 . 23082754 4 3419842

82 3 .20955596 4 3387706 87 3 .23126174 4 3419608

82 4 . 20998985 4 3389333 87 4 . 23169593 4 3419342

82 5 .21042375 4 3390914 87 5 .23213013 4 3419047

82 6 .21085766 4 3392450 87 6 .23256431 4 3418720

82 7 .21129160 4 3393942 87 7 .23299850 4 3418363

82 8 .21172554 4 3395389 87 8 .23343268 4 3417977

82 9 .21215950 4 3396791 87 9 . 23386686 4 3417560

83 0 .21259348 4 3398151 88 0 . 23430103 4 3417114

83 .1 . 21302747 4 3399466 88 1 .23473520 4 3416639

83 .2 . 21346147 4 3400739 88 2 . 23516937 4 3416135

83 .3 .21389548 4 3401968 88 3 . 23560352 4 3415602

83 .4 .21432951 4 3403155 88 4 .23603768 4 3415041

83 .5 . 21476354 4 .3404301 88 5 .23647182 4 3414451

83 .6 .21519759 4 . 3405404 88 6 . 23690597 4 3413833

83 .7 .21563165 4 . 3406466 88 7 . 23734010 4 3413188

83 .8 .21606572 4 . 3407487 88 8 . 23777423 4 3412514

83 .9 . 21649980 4 . 3408466 88 9 . 23820835 4 3411814

84 .0 .21693389 4 . 3409406 89 0 . 23864247 4 3411086

84 .1 .21736799 4 .3410304 89 1 . 23907657 4 3410332

84 .2 .21780210 4 .3411163 89 2 . 23951067 4 3409551

84 .3 .21823621 4 .3411983 89 3 . 23994476 4 3408743

84 .4 .21867034 4 .3412764 89 4 . 24037885 4 3407909

84 .5 .21910447 4 .3413505 89 5 . 24081292 4 3407048

84 .6 .21953861 4 .3414207 89 6 . 24124699 4 .3406162

84 .7 . 21997275 4 .3414871 89 7 . 24168105 4 .3405251

84 .8 .22040690 4 .3415498 89 8 . 24211509 4 .3404314

84 .9 .22084106 4 .3416086 89 .9 .24254913 4 .3403352

85 .0 .22127523 4 .3416637 90 .0 .24298316 4 .3402365

91
IPTS-68 Reference Function and Its Derivative With the Argument in Kelvin — Continued

T

K W*{T)

dW*(T)ldT

X 1000

K->

T

K

dW*(T)ldT

X 1000

90 . 0 . 24298316 4 .3402365 95 .0 . 26466721 4 .3325507

90 . 1 . 24341718 4 .3401353 95 . 1 .26510045 4 .3323498

90 . 2 . 24385119 4 .3400317 95,.2 .26553368 4 . 3321474

90 . 3 . 24428519 4 .3399257 95,.3 .26596688 4 .3319435

90 . 4 . 24471917 4 .3398172 95..4 .26640007 4 .3317379

90 . 5 . 24515315 4 .3397063 95,.5 . 26683323 4 .3315309

yu . 6 . 24558711 4 .3395931 95.,6 . 26726637 4 .3313224

y(j . 7 . 24602107 4 . 3394775 95.,7 . 26769950 4 .3311124

90 . 8 . 24645501 4 .3393596 95. 8 .26813260 4 .3309009

yu . 9 . 24688894 4 .3392394 95. 9 .26856568 4 .3306879

yi . u . 24732286 4 .3391169 96. 0 . 26899873 4 .3304735

yi . i . 24775676 4 .3389921 96. 1 .26943177 4 .3302577

yi . 2 . 24819065 4 .3388651 96. 2 . 26986479 4 .3300404

yi . 3 . 24862454 4 .3387358 96. 3 . 27029778 4 .3298217

91 . 4 . 24905840 4 .3386043 96. 4 .27073075 4 .3296017

91 . 5 . 24949226 4 .3384706 96. 5 .27116370 4 .3293803

91 . 6 . 24992610 4 .3383348 96. 6 .27159663 4 .3291575

91 . 7 .25035992 4 .3381968 96. 7 .27202953 4 .3289333

yi . 8 . 25079374 4 .3380566 96. 8 . 27246241 4 .3287078

yi . y . 25122753 4 .3379143 96. 9 .27289527 4 .3284810

y2 . u . 25166132 4 .3377700 97. 0 .27332811 4 .3282529

y^ . i . 25209509 4 .3376236 97. 1 . 27376092 4 .3280234

y/d . 2 . 25252884 4 .3374751 97. 2 .27419371 4 . 3277927

y2 . 3 . 25296258 4 .3373245 97. 3 . 27462648 4 .3275607

y^

,

. 4 . 25339631 4 .3371720 97. 4 .27505922 4 . 3273274

92 ,. 5 . 25383002 4 .3370174 97. 5 .27549195 4 .3270929

92 .. 6 . 25426371 4 .3368608 97. 6 . 27592464 4 .3268572

92 . rr, 1 . 25469739 4 .3367023 97. 7 . 27635732 4 .3266203

92 . o, O . 25513105 4 .3365418 97. 8 . 27678997 4,.3263821

92 . 9 . 25556470 4 .3363794 97. 9 .27722259 4..3261427

93 . U . 25599833 4 .3362150 98. 0 .27765520 4,.3259021

93

.

1 . 25643194 4 .3360487 98. 1 . 27808777 4..3256604

93

.

2 . 25686554 4 .3358806 98. 2 . 27852033 4,.3254176

93

.

3 . 25729912 4 .3357106 98. 3 . 27895286 4..3251735

93

.

4 . 25773268 4 .3355388 98. 4 . 27938536 4., 3249283

93

.

5 .25816622 4 .3353651 98. 5 .27981784 4.,3246821

93

.

6 .25859975 4,.3351896 98. 6 .28025030 4., 3244347

93 . 7 .25903326 4,.3350122 98. 7 . 28068273 4.,3241852

93 . 8 . 25946675 4,.3348332 98. 8 .28111514 4.,3239366

y3

.

y . 25990023 4 .3346523 98. 9 .28154752 4. 3236859

y4

.

(J . 26033368 4,. 3344697 99. 0 .28197987 4. 3234341

y4

.

1 . 26076712 4,.3342853 99. 1 .28241220 4. 3231813

y4

.

2 . 26120054 4,.3340992 99. 2 . 28284451 4. 3229276

94

.

3 . 26163394 4,.3339115 99. 3 . 28327679 4. 3226727

94

.

4 . 26206732 4,,3337220 99. 4 .28370904 4. 3224168

94

.

5 . 26250069 4,.3335309 99. 5 .28414127 4. 3221599

94. 6 .26293403 4.,3333381 99. 6 . 28457348 4. 3219020

94. 7 .26336735 4.,3331436 99. 7 .28500565 4. 3216432

94. 8 .26380066 4.,3329476 99. 8 . 28543780 4. 3213833

94. 9 .26423394 4.,3327499 99. 9 .28586993 4. 3211225

95. 0 . 26466721 4.,3325507 100. 0 .28630203 4. 3208607

92
IPTS-68 Reference i uii^iion dnn lis Derivative With the Argument in Kelvin — Continued

T dW*{T)ldT T dW*(T)ldT

X 1000

K W*(T) X 1000

K • K

100 0 . /COOOU<tuO *± . 0<CUODU / 150 . 0 .49861134 4. 1769430

101 0 . O

^11 PIOl QPP SCO 151 . 0 .50278711 4.1746153

102 . 0

•7 1 ^AACi'J 152 . 0 .50696058 4.1723135

103 . 0

"7 1 pci ACi 153 . 0 .51113175 4. 1700371

104 0 . OUOODOOl/ A . OUc/ "^nQTP( <c 1

1 R

D

154 0 .51530066 4.1677859

105 0 . OU 1 O ( XO^l /I *± . OUD I I lO 155 0 .51946733 4. 1655591

106 0 156 0 .52363178 4.1633567

107 0 . 0±D^ I c50 1 . OUU ( OUl 157 0 . 52779405 4.1611778

108 0 . 0;CU ( ( OO f . I DO/Cl 158 0 .53195415 4. 1590222

109 0 . 0<cOU '^PRD'y/lfiT' ( 'ID / A OOiA^AAO 159 0 .53611210 4.1568895

110 n . Oc^OD ( DO fl . <cyi41<;<c PQ1 .41 PO 160 0 . 54026793 4.1547790

111 0 . OOODO 1 PPPPR1

P

. /COO(CDl <C 161 0 .54442167 4.1526903

112 0 . PPROQCP <coouyD<c 162 0 . 54857332 4.1506230

113 n '^^ppp'ycp PP1 QQ 1

T

. /coi y/ci ( 163 0 .55272292 4.1485767

114 n . O'^lDOUoU 1 A PTPT^I "7

. <C ( O ( 41 ( 164 0 . 55687048 4.1465508

115 '7Rn'7$5R1 C ^ / OODUU 165 0 .56101603 4 . 1445449

116 . OOOUOi7iO *± P'70'7Pm 166 0 .56515958 4.1425586

117 '7RQ'7PQQO ft pcQoriKo /c Dy<cuo<c 167 0 .56930115 4.1405914

118 . oDoOy / 04 pcKn'7Pi /cDDUool 168 0 . 57344077 4.1386427

119 . OD ( oDxiUU A PCPPPI <Cb<c001DPi 169 0 . 57757844 4.1367123

120 n . O / <cl<cool <coy ( oou 170 0 .58171420 4. 1347998

121 0\J . O ( DOC3±4cS /I /cODDUyO 171 0 . 58584805 4.1329045

122 n . OOUDODOo <coo4iyou 172 0 .58998001 4.1310262

123 n fi <cOU4UOO 173 0 .59411011 4.1291644

124 0 700 T 'ZTZ c 4A O /I '7'7 "7 '7R <c4 1 oooo 174 0 . 59823835 4.1273188

125 TQTIXQ '7 T C . oc/oo£3olb 4 O AAOQ.'^A <c44<coo4 175 0 . 60236475 4.1254889

126 '7Q'7C;ORQ'7 >1 fl Q/11 <c41<cObo ORC7 176 0 . 60648933 4.1236743

127 0 . ^UlODOuo 4 0'7POR'7'7 177 0 .61061210 4.1218746

128 0\J . flUDlU<i44 4 0'7RO'7C7 178 0 .61473308 4.1200896

129 0 T n'7'7CO>d . 4HUooD<c4 4A O "70 70 <cO(CO<;0U 179 0 .61885229 4.1183188

130. . 4140D / lU 4 /c/cy4UUb 180 0 . 62296973 4. 1165619

131

.

n . flics ( youo 4 <c<cbOUo f 181 0 . 62708542 4.1148185

132. 0 /ip^fiom o ft O0'7C7/1 <c<cobo4oP 182 0 .63119937 4.1130884

133. n .dOTO/l 0'7'7 4 . /d<iU ( y40 183 0 .63531160 4. 1113711

134. . 4oi4Di A'Z.'\ Ad^ TO

(

d 4A . O<ci1 TQOOQ fyo^iy 184. 0 .63942211 4. 1096663

135. nu . 4oOd f ool 4 . o 1 Konn7 <ilO<i(JUo 185. 0 . 64353093 4.1079738

136. 0u 4 . o 1 o >i /I Tn 186. 0 . 64763807 4.1062933

137. n 4A . o noTo '7

1

<cUy f ^ol 187,. 0 . 65174352 4.1046244

138. n . 44ooiiOo 4 . O HTno 0£^

{ U<cob 188. 0 . 65584732 4. 1029669

139. nu . /I K O K T TOO 40/d0i / (CO 4A . <cU4oboO 189. 0 . 65994946 4.1013205

140

.

0 . 40b A RC70n'70 / (CUo/d 4 . Om<cUl TOTP i (L to 190. 0 . 66404996 4. 0996849

141

.

n . 4DUy£:U (

4

4 . 1 QQ 1 O 1

'7 lyyi^iio 191. 0 . 66814884 4. 0980600

142. 0u . 4bt3l /! 1 1 PKT loo ( 4A . 1 Q/^P^ A "70 iybo4oy 192. 0 .67224609 4 . 0964453

143. 0 A ^0*7 1 TO A . 4DyolOo4 4 . 1 Q7QQRR lyoyyoo 193. 0 . 67634173 4. 0948408

144. 0u . 4 / OOUbO f A4 . 1 Q 1 yl TKT ly 14 ( O

/

194. 0 . 68043577 4. 0932461

145. 0u .^11 byboU 4 . 1 OOQO A A looyo44 195. 0 . 68452823 4. 0916611

146. nu . 4oloo400 4 . 1 OO C O 1 o lobO<il/; 196. 0 .68861910 4. 0900855

147. . 4obUbyoo A4 . 1 o yi noco lo4Uooo 197. 0 . 69270840 4. 0885191

148. 0 . 4QnPRO'7'7 4iyU(CO<c 1 o

/I 41 . loiD 1 PI RTTQ ( (

y

198. 0 . 69679614 4. 0869618

149

.

0 .49443322 4. 1792971 199. 0 .70088233 4. 0854132

150

.

0 .49861134 4. 1769430 200. 0 . 70496697 4 . 0838734

93
IPTS-68 Reference Function and Its Derivative With the Argument in Kelvin—Continued

T

K

dW*(T)ldT

X 1000 K-'

T

K W*{T)

dW*(T)ldT

X 1000

K'

200 0 7049S697 4. 0838734 250

.

0 90738312 4

.

0150756 WX*^v 1 201 0 70905008 4. 0823419 251

.

0 91139757 4 0138199

202 0 71313166 4. 0808187 252

.

0 91541076 4

.

0125672

203

.

0 71721172 4. 0793037 253

.

0 91942270 4 0113173

204

.

0 72129027 4. 0777966 254. 0 92343340 4 0100702

205

.

0 72536731 4. 0762973 255

.

0 92744285 4 0088257

206

.

0 72944286 4. 0748056 256. 0 93145105 4 B 0075837 1 ^^^^ 1 207

.

0 73351693 4. 0733215 257. 0 93545801 4 0063439

208

.

0 73758951 4. 0718447 258. 0 93946374 4 0051063

209

.

0 74166062 4. 0703751 259

.

0 94346823 4 0038708

210

.

0 74573026 4. 0689126 260

.

0 94747148 4 0026371

211

.

0 74979844 4. 0674572 261

.

0 95147350 4 0014051

212

.

0 75386518 4. 0660085 262

.

0 95547429 4 0001747

213

.

0 75793046 4., 0645666 263

.

0 95947385 3 9989456

214

.

0 76199431 4,,0631313 264., 0 963472 1

8

3 9977178

215 0 7660567"^ 4.,0617025 265

.

0 96746929 3 9964910

216 0 7701177? 4.,0602801 266 0 971 4651

7

3 9952650

217 0 77417729 4., 0588640 267

.

, 0 97545982 3 9940397

218

.

0 77823545 4,. 0574541 268.. 0 97945325 3 9928147

219

.

0 78229220 4,.0560503 269 . 0 98344545 3 9915902

220

.

0 78634755 4 .0546525 270

,

. 0 98743643 3 . 9903656

221

.

0 79040151 4 .0532606 271 , 0 99142618 3 . 9891409

222

.

0 79445408 4 .0518746 272 . 0 99541471 <—' —/ v./ X ^X 1 X 3 . 9879159

223

.

0 79850526 4 .0504942 273 . 0 99940201 3 . 9866902

224

.

0 80255507 4 .0491195 273 . 15 1 00000000 3 . 9865063

225

.

, 0 • 80660350 \J \J \a/ \J \J 4 . 0477504

226

.

, 0 81065057 4 . 0463867

227

.

, 0 81469628 4 .0450285

228

.

, 0 81 874063 4 . 0436755

229

.

, 0 82278363 4 . 0423277

230

.

, 0 82682528 4 .0409851

231

.

, 0 83086560 4 . 0396475

232

.

, 0 83490458 4 .0383149

233 , 0 83894223 4 .0369871

234 . 0 84297856 4 .0356641

235 . 0 84701356 4 .0343459

236 . 0 85104725 4 .0330323

237 . 0 85507963 4 .0317232

238 . 0 85911070 4 . 0304186

239 . 0 86314047 4 .0291183

240 . 0 86716894 4 .0278223

241 . 0 . 87119611 \-f 1 X X «^ W X X 4 .0265305

242 . 0 87522200 4 . 0252428

243 . 0 87924660 4 .0239590

244 . 0 88326992 4 .0226792

245 . 0 88729196 4 .0214031

246 . 0 891 31 27"^ 4 .0201307

247 .0 .89533222 4 .0188618

248 .0 .89935045 4 .0175964

249 .0 .90336742 4 .0163344

250 .0 .90738312 4 .0150756

94
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius

t°c W*(t)

dW*(t)ldT

X 1000

°C '

f °C W*(t)

dW*(t)ldT

X 1000

-260 .0 .00126168 . 2107490 -255 .0 .00309642 . 5540898

-259 .9 .00128300 .2158042 -254 .9 .00315227 .5628522

-259 .8 .00130484 . 2209403 -254 .8 .00320900 .5716830

-259 .7 .00132719 .2261558 -254 .7 .00326661 .5805814

-259 .6 . 00135007 . 2314492 -254 .6 .00332511 . 5895467

-259 .5 .00137349 .2368191 -254 .5 . 00338452 . 5985782

-259 .4 .00139744 . 2422640 -254 .4 . 00344483 .6076751

-259 .3 .00142194 . 2477828 -254 .3 .00350606 .6168368

-259,.2 .00144700 . 2533743 -254 .2 .00356820 .6260627

-259 .1 .00147262 .2590375 -254 . 1 .00363127 .6353522

-259 .0 .00149881 . 2647716 -254 .0 .00369527 . 6447048

-258 .9 .00152557 . 2705758 -253 .9 .00376022 .6541199

-258,.8 .00155293 . 2764497 -253 .8 .00382610 .6635970

-258..7 .00158087 .2823930 -253 .7 .00389294 .6731357

-258,.6 .00160941 . 2884054 -253 .6 .00396073 .6827354

-258,.5 .00163855 . 2944869 -253 .5 .00402949 .6923958

-258,.4 .00166831 . 3006377 -253 .4 .00409921 .7021165

-258,,3 .00169868 .3068581 -253 .3 .00416991 .7118970

-258, 2 .00172968 .3131483 -253 .2 .00424159 .7217370

-258,, 1 .00176131 .3195089 -253 .1 .00431426 .7316359

-258,, 0 .00179358 .3259405 -253.,0 . 00438792 .7415936

-257.,9 .00182650 . 3324437 -252,.9 . 00446258 .7516094

-257,,8 .00186007 .3390193 -252,,8 .00453825 .7616831

-257,,7 .00189431 . 3456678 -252.,7 .00461492 .7718143

-257.,6 .00192921 .3523901 -252. 6 .00469261 .7820023

-257. 5 .00196479 .3591870 -252. 5 .00477132 .7922469

-257. 4 .00200105 .3660590 -252. 4 .00485106 .8025475

-257. 3 .00203800 .3730068 -252. 3 .00493183 .8129036

-257. 2 .00207565 .3800312 -252. 2 .00501364 .8233146

-257. 1 .00211401 .3871326 -252. 1 .00509650 .8337801

-257. 0 .00215308 .3943114 -252. 0 .00518040 . 8442993

-256. 9 .00219288 .4015681 -251. 9 .00526536 .8548717

-256. 8 .00223340 .4089030 -251. 8 .00535138 .8654965

-256. 7 . 00227466 .4163162 -251. 7 . 00543846 .8761731

-256. 6 .00231667 .4238079 -251

.

6 .00552661 . 8869007

-256. 5 .00235942 .4313780 -251. 5 .00561584 . 8976786

-256. 4 .00240294 .4390265 -251. 4 .00570615 .9085058

-256. 3 .00244723 . 4467532 -251. 3 .00579755 .9193817

-256. 2 .00249230 .4545578 -251. 2 .00589003 .9303052

-256. 1 .00253815 .4624399 -251. 1 . 00598361 .9412756

-256. 0 . 00258479 .4703991 -251. 0 .00607829 .9522917

-255. 9 .00263223 . 4784348 -250. 9 .00617407 .9633528

-255. 8 . 00268048 . 4865465 -250. 8 .00627096 . 9744577

-255. 7 . 00272954 .4947336 -250. 7 .00636896 .9856055

-255. 6 . 00277943 .5029952 -250. 6 .00646808 .9967951

-255. 5 .00283014 .5113308 -250. 5 .00656832 1 .0080255

-255. 4 .00288170 .5197395 -250. 4 .00666969 1 .0192956

-255. 3 .00293409 .5282205 -250. 3 .00677218 1 . 0306044

-255. 2 .00298734 . 5367731 -250. 2 .00687581 1.0419507

-255. 1 .00304145 .5453965 -250. 1 .00698057 1 .0533335

-255. 0 .00309642 .5540898 -250. 0 . 00708648 1 .0647516

95
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius — Continued

t °c W*(t)

dW*(t)ldT

X 1000

<°C

dW*{t)ldT

X 1000

°c->

-250. 0 00708648 1 0647516 -245 , 0 01388681 1 .6610569

-249. 9 00719352 1 0762039 -244 . 9 . 01405353 1 .6732110

-249. 8 00730172 1 . 0876895 -244 .8 01422146 1 . 6853661

-249. 7 00741106 1 . 0992071 -244 .7 . 01439060 1 .6975216

-249. 6 00752156 1 . 1107558 -244 . 6 . 01456096 1 . 7096772

-249. 5 00763322 1 1223343 -244 . 5 . 01473254 1 .7218324

-249. 4 . 00774603 1 . 1339418 -244 , 4 . 01490533 1 . 7339866

-249. 3 . 00786000 1 . 1455771 -244,. 3 . 01507933 1 .7461396

-249. 2 . 00797515 1 . 1572392 -244 . 2 . 01525455 1 .7582907

-249. 1 . 00809145 1 . 1689271 -244 . 1 . 01543099 1 . 7704395

-249. 0 . 00820893 1 . 1806398 -244 . 0 . 01560864 1 .7825855

-248. 9 . 00832758 1 . 1923763 -243 .9 . 01578751 1 . 7947282

-248. 8 . 00844741 1 . 2041358 -243 .8 .01596759 1 . 8068671

-248. 7 . 00856841 1 . 2159172 -243,.7 .01614888 1 .8190017

-248. 6 . 00869059 1 . 2277198 -243 .6 . 01633139 1 .8311315

-248. 5 . 00881395 1 2395426 -243 . 5 01651511 1 .8432558

-248. 4 . 00893850 1 2513847 -243 ,4 . 01670004 1 . 8553743

-248. 3 00906423 1 2632455 -243 . 3 01688618 1 . 8674863

-248. 2 00919115 1 2751240 -243 . 2 01707354 1 .8795912

-248. 1 00931926 1 2870195 -243 . 1 01726210 1 .8916886

-248. 0 00944856 1 2989314 -243 , 0 01745187 1 . 9037778

-247. 9 00957904 1 3108588 -242 . 9 01764286 1 .9158583

-247. 8 00971073 1 3228011 -242 , 8 01783504 1 . 9279295

-247. 7 00984361 1 3347576 -242 .7 01802844 1 . 9399908

-247. 6 . 00997768 1 . 3467277 -242 . 6 01822304 1 .9520417

-247. 5 01011295 1 3587108 -242 . 5 01841885 1 .9640815

-247. 4 . 01024942 1 3707063 -242 . 4 01861586 1 .9761096

-247. 3 . 01038709 1 3827136 -242 . 3 01881407 1 .9881255

-247. 2 . 01052597 1 3947322 -242 . 2 01901348 2 .0001285

-247., 1 .01066604 1 .4067616 -242 , 1 01921410 2 .0121180

-247.,0 .01080732 1 . 4188011 -242 . 0 01941591 2 .0240935

-246.,9 . 01094980 1 . 4308504 -241

,

. 9 01961891 2 . 0360544

-246,.8 . 01109349 1 . 4429090 -241 , 8 01982312 2 .0480000

-246,.7 . 01123838 1 . 4549764 -241 . 7 02002851 2 . 0599297

-246 .6 . 01138448 1 4670521 -241

,

. 6 02023510 2 .0718429

-246 .5 . 01153179 1 4791358 -241

.

, 5 02044288 2 . 0837391

-246 .4 . 01168031 1 4912269 -241

,

,4 02065185 2 ,0956176

-246 .3 . 01183004 1 5033250 -241

,

3 02086200 2,, 1074779

-246 .2 01198098 1 5154297 -241 2 02107334 2,.1193193

-246 . 1 01213312 1 5275407 -241 02128587 2,.1311413

-246 .0 01228648 1 5396575 -241 0 02149957 2.. 1429433

-245 .9 . 01244106 1 5517797 -240

.

9 02171446 2., 1547247

-245 .8 . 01259684 1 5639069 -240

.

8 02193052 2. 1664849

-245 .7 . 01275384 1 5760388 -240

.

7 02214775 2., 1782235

-245 .6 .01291205 1 5881748 -240

.

6 02236616 2., 1899397

-245 .5 . 01307147 1 6003146 -240

.

5 02258574 2. 2016331

-245 .4 . 01323211 1 6124579 -240

.

4 02280649 2. 2133031

-245 .3 . 01339396 1 6246042 -240

.

02302840 2. 2249492

-245 .2 .01355703 1 . 6367530 -240. 2 .02325147 2. 2365708

-245 . 1 .01372132 1 . 6489041 -240. 1 .02347571 2. 2481674

-245 .0 . 01388681 1 .6610569 -240. 0 .02370111 2. 2597386

96
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius — Continued

t °c W*(t)

dW*{t)ldT

XIOOO

°c->

t °C W*(t)

dW*{t)ldT

X 1000

°c-'

-240..0 .02370111 2 . 2597386 -235 ,0 . 03638008 2.,7974138

-239..9 . 02392766 2 .2712837 -234 ,9 .03666031 2..8072404

-239.,8 .02415536 2 . 2828024 -234 ,8 .03694152 2..8170283

-239.,7 . 02438422 2 . 2942942 -234 ,7 .03722371 2., 8267775

-239.,6 . 02461422 2 . 3057584 -234 .6 . 03750688 2.. 8364880

-239,,5 . 02484537 2 .3171947 -234 .5 .03779101 2.,8461597

-239, 4 . 02507766 2 . 3286027 -234 ,4 .03807611 2.. 8557927

-239. 3 .02531109 2 .3399819 -234 ,3 .03836217 2.. 8653868

-239, 2 . 02554565 2 .3513318 -234 .2 . 03864919 2.. 8749421

-239, 1 .02578135 2 .3626520 -234 ,1 . 03893716 2,. 8844585

-239, 0 .02601818 2 .3739421 -234 .0 .03922608 2 .8939361

-238, 9 .02625614 2 .3852018 -233 ,9 .03951594 2,. 9033747

-238. 8 .02649522 2 . 3964305 -233 ,8 .03980675 2,.9127745

-238, 7 . 02673543 2 . 4076280 -233 ,7 . 04009850 2,.9221354

-238, 6 . 02697675 2 .4187939 -233 .6 .04039118 2,, 9314574

-238, 5 .02721918 2 .4299277 -233 .5 . 04068479 2 , 9407404

-238. 4 . 02746273 2 .4410292 -233 .4 . 04097932 2 . 9499846

-238. 3 . 02770739 2 .4520979 -233 .3 .04127478 2 .9591898

-238. 2 .02795315 2 .4631337 -233 .2 .04157116 2 . 9683562

-238. 1 .02820001 2 .4741362 -233 .1 . 04186845 2,. 9774837

-238. 0 . 02844798 2 .4851050 -233 .0 .04216665 2 , 9865723

-237. 9 . 02869703 2 .4960398 -232 ,9 . 04246576 2 .9956221

-237. 8 . 02894718 2 .5069405 -232 ,8 . 04276578 3 . 0046329

-237. 7 . 02919842 2 .5178066 -232 ,7 . 04306669 3,.0136050

-237. 6 . 02945074 2 . 5286380 -232 .6 . 04336850 3 .0225383

-237. 5 . 02970415 2 . 5394344 -232 .5 . 04367120 3 .0314327

-237. 4 . 02995863 2 .5501956 -232 ,4 . 04397478 3,. 0402884

-237. 3 .03021419 2 .5609212 -232 .3 . 04427925 3 ,0491053

-237. 2 . 03047081 2 .5716111 -232 .2 . 04458460 3 . 0578836

-237, 1 .03072851 2 . 5822652 -232 .1 . 04489083 3 ,0666231

-237, 0 . 03098726 2 . 5928830 -232 ,0 .04519793 3 , 0753239

-236,,9 .03124708 2 . 6034646 -231 .9 .04550589 3 , 0839861

-236. 8 .03150796 2 .6140096 -231 .8 . 04581472 3,,0926096

-236.,7 .03176988 2 . 6245180 -231,.7 . 04612441 3 ,1011946

-236.,6 .03203286 2 . 6349895 -231..6 . 04643496 3 , 1097409

-236..5 . 03229688 2 . 6454240 -231..5 . 04674636 3 .1182488

-236..4 .03256194 2 .6558212 -231..4 . 04705861 3 .1267181

-236,.3 . 03282804 2 .6661812 -231..3 . 04737170 3,.1351490

-236,.2 .03309518 2 . 6765038 -231..2 . 04768564 3 ,1435415

-236 , 1 . 03336334 2 . 6867888 -231..1 .04800041 3 . 1518955

-236 ,0 . 03363253 2 . 6970360 -231.,0 .04831601 3.. 1602111

-235 ,9 . 03390275 2 .7072455 -230, 9 . 04863245 3.. 1684884

-235 .8 . 03417398 2 .7174171 -230, 8 . 04894971 3., 1767274

-235 .7 . 03444623 2 .7275506 -230. 7 . 04926779 3., 1849282

-235 .6 . 03471949 2 . 7376460 -230. 6 . 04958669 3.,1930907

-235 .5 . 03499376 2 . 7477032 -230. 5 . 04990641 3., 2012150

-235 .4 .03526903 2 .7577222 -230. 4 .05022694 3.,2093011

-235 .3 .03554530 2 .7677028 -230. 3 . 05054827 3.,2173491

-235 .2 .03582257 2 .7776450 -230. 2 .05087040 3.,2253591

-235 .1 .03610083 2 . 7875486 -230. 1 .05119334 3. 2333310

-235 .0 . 03638008 2 .7974138 -230. 0 .05151707 3. 2412648

97
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius — Continued

t r*(()

dW*(t)ldT

X 1000

t °C W*(t)

dW*{t)ldT

X 1000

-230 . 0 .05151707 3 . 2412648 -225 .0 . 06863481 3 . 5905076

-229 . 9 .05184159 3 .2491608 -224 .9 .06899417 3 . 5965674

-229 . 8 .05216690 3 .2570187 -224 .8 .06935412 3 .6025922

-229 . 7 . 05249299 3 . 2648389 -224 .7 . 06971468 3 .6085818

-229 . 6 .05281987 3 .2726211 -224 .6 . 07007584 3 .6145364

-229 . 5 .05314752 3 .2803655 -224 .5 .07043759 3 . 6204562

-229 . 4 . 05347594 3 . 2880722 -224 .4 . 07079993 3 .6263411

-229 . 3 .05380513 3 .2957411 -224 .3 .07116286 3 .6321912

-229 . 2 .05413509 3 .3033724 -224 .2 .07152637 3 . 6380067

-229 . 1 . 05446580 3 .3109660 -224 . 1 .07189046 3 . 6437876

-229 . 0 . 05479728 3 .3185221 -224 .0 .07225512 3 . 6495340

-228 . 9 .05512951 3 . 3260405 -223 .9 . 07262036 3 . 6552461

-228 . 8 . 05546248 3 .3335215 -223 .8 .07298617 3 .6609237

-228 . 7 .05579621 3 .3409650 -223 .7 .07335255 3 . 6665672

-228 . 6 .05613068 3 .3483711 -223 .6 . 07371948 3 . 6721765

-228 . 5 . 05646588 3 . 3557398 -223 .5 . 07408698 3 .6777517

-228 . 4 .05680182 3 .3630712 -223 .4 . 07445503 3 .6832930

-228 . 3 .05713850 3 .3703654 -223 .3 . 07482364 3 . 6888004

-228 . 2 . 05747589 3 . 3776222 -223 .2 .07519279 3 . 6942740

-228 . 1 . 05781402 3 . 3848420 -223 . 1 . 07556249 3 . 6997139

-228 . 0 .05815286 3 .3920245 -223 .0 . 07593273 3 .7051203

-227 . 9 . 05849242 3 .3991700 -222 .9 .07630351 3 .7104931

-227,. 8 . 05883269 3 .4062785 -222 .8 . 07667483 3 .7158326

-227

,

. 7 .05917368 3 .4133499 -222 .7 . 07704668 3 .7211387

-227,. 6 .05951536 3 . 4203845 -222 .6 .07741906 3 .7264117

-227

,

, 5 . 05985775 3 .4273822 -222 .5 .07779196 3..7316515

-227., 4 . 06020084 3 . 4343430 -222 .4 .07816539 3,, 7368584

-227. 3 . 06054462 3 .4412671 -222 .3 .07853933 3, 7420323

-227

.

2 . 06088909 3,.4481544 -222,,2 .07891379 3., 7471734

-227

.

1 .06123425 3,.4550051 -222,. 1 . 07928877 3. 7522819

-227. 0 .06158009 3,.4618192 -222..0 . 07966425 3. 7573578

-226

.

9 .06192661 3,.4685967 -221,,9 .08004024 3. 7624011

-226

.

8 . 06227381 3., 4753378 -221, 8 .08041673 3. 7674122

-226

.

7 .06262168 3., 4820424

.

-221. 7 . 08079372 3. 7723909

-226

.

6 .06297022 3. 4887106 -221

.

6 .08117120 3. 7773375

—226

.

5 .06331942 3. 4953426 -221. 5 .08154918 3. 7822520

-226

.

4 .06366928 3. 5019382 -221. 4 .08192765 3. 7871346

-226

.

3 .06401981 3. 5084977 -221. 3 .08230661 3. 7919854

-226

.

2 . 06437098 3. 5150211 -221. 2 .08268605 3. 7968044

-226

.

1 . 06472281 3. 5215084 -221. 1 .08306597 3. 8015918

-226

.

0 . 06507528 3. 5279597 -221. 0 . 08344637 3. 8063478

-225

.

9 . 06542840 3. 5343750 -220. 9 . 08382724 3. 8110723

-225

.

8 .06578216 3. 5407545 -220. 8 . 08420858 3. 8157656

-225

.

7 .06613655 3. 5470982 -220. 7 . 08459039 3. 8204278

-225

.

6 .06649157 3. 5534062 -220. 6 . 08497266 3. 8250588

-225. 5 . 06684723 3. 5596785 -220. 5 .08535540 3. 8296591

-225

.

4 .06720351 3. 5659152 -220. 4 . 08573860 3. 8342284

-225

.

3 . 06756041 3. 5721164 -220. 3 .08612225 3. 8387671

-225

.

2 . 06791793 3. 5782821 -220. 2 .08650635 3. 8432753

-225. 1 . 06827607 3. 5844125 -220. 1 . 08689090 3. 8477530

-225. 0 . 06863481 3. 5905076 -220. 0 . 08727590 3. 8522003

98
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius — Continued

t °c W*(t)

dW*(t)jdT

X 1000

°C-'

t°C W*(t)dW*{t)ldT

xlOOO

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-220,,0 . 08727590 3 .8522003 -215 .0 .10703128 4 .0386911

-219.,9 . 08766134 3 .8566175 -214 .9 . 10743530 4 .0417606

-219.,8 . 08804722 3 .8610046 -214 .8 . 10783963 4 . 0448065

-219..7 . 08843354 3 .8653617 -214 .7 . 10824426 4 . 0478289

-219,.6 . 08882029 3 . 8696890 -214 .6 .10864919 4 .0508280

-219.,5 . 08920748 3 . 8739866 -214 .5 . 10905443 4 .0538038

-219..4 .08959509 3 . 8782546 -214 .4 . 10945995 4 . 0567565

-219..3 .08998313 3 .8824931 -214 .3 .10986578 4 .0596863

-219. 2 .09037159 3 . 8867023 -214 .2 .11027189 4 .0625932

-219., 1 . 09076046 3 . 8908823 -214 .1 .11067829 4 . 0654773

-219. 0 .09114976 3 .8950332 -214 .0 .11108498 4 . 0683390

-218. 9 . 09153947 3 .8991550 -213 .9 .11149196 4 .0711781

-218. 8 .09192959 3 .9032481 -213 .8 .11189922 4 . 0739949

-218.,7 .09232012 3 .9073125 -213 .7 .11230676 4 . 0767895

-218.,6 .09271105 3 .9113482 -213 .6 .11271458 4 .0795621

-218.,5 .09310239 3 .9153555 -213 ,5 .11312267 4 .0823127

-218..4 . 09349412 3 .9193345 -213 .4 .11353104 4 .0850415

-218..3 . 09388625 3 .9232852 -213 .3 .11393968 4 . 0877486

-218.,2 . 09427878 3 . 9272080 -213,.2 .11434859 4 . 0904341

-218., 1 . 09467169 3 .9311027 -213,. 1 . 11475776 4 . 0930982

-218.,0 .09506500 3 . 9349697 -213,.0 .11516721 4 .0957410

-217.,9 . 09545869 3 9388089 -212..9 .11557691 4 .0983625

-217.,8 . 09585276 3 .9426206 -212. 8 .11598688 4 . 1009631

-217.,7 . 09624721 3 . 9464048 -212.,7 .11639710 4 . 1035427

-217..6 . 09664204 3 .9501618 -212. 6 .11680759 4 .1061014

-217.,5 . 09703724 3 .9538916 -212. 5 .11721832 4 .1086395

-217.,4 . 09743282 3 . 9575943 -212. 4 .11762931 4,.1111570

-217..3 . 09782876 3 .9612702 -212. 3 . 11804055 4,.1136540

-217..2 .09822507 3,.9649193 -212. 2 .11845204 4,.1161308

-217..1 . 09862174 3,. 9685417 -212. 1 .11886378 4,.1185874

-217..0 .09901878 3,. 9721376 -212. 0 . 11927576 4..1210238

-216..9 .09941617 3., 9757072 -211. 9 .11968798 4. 1234404

-216..8 .09981392 3..9792505 -211. 8 . 12010045 4. 1258370

-216 .7 .10021202 3., 9827677 -211. 7 .12051315 4. 1282140

-216,.6 . 10061047 3.,9862589 -211. 6 .12092609 4. 1305714

-216,.5 . 10100927 3., 9897242 -211. 5 .12133926 4. 1329093

-216,.4 . 10140841 3. 9931638 -211. 4 .12175267 4. 1352278

-216,.3 . 10180790 3., 9965778 -211. 3 .12216631 4. 1375271

-216,.2 . 10220773 3., 9999664 -211. 2 .12258018 4. 1398074

-216,. 1 . 10260789 4. 0033296 -211. 1 . 12299427 4. 1420686

-216,.0 . 10300839 4., 0066676 -211. 0 .12340859 4. 1443109

-215,.9 . 10340923 4. 0099806 -210. 9 . 12382313 4. 1465345

-215,.8 .10381039 4. 0132686 -210. 8 . 12423790 4. 1487394

-215.,7 .10421188 4. 0165319 -210. 7 . 12465288 4. 1509258

-215,.6 . 10461370 4. 0197704 -210. 6 . 12506808 4. 1530938

-215,.5 .10501583 4. 0229845 -210. 5 .12548350 4. 1552435

-215 .4 . 10541829 4. 0261741 -210. 4 .12589913 4. 1573750

-215 .3 . 10582107 4. 0293394 -210. 3 .12631497 4. 1594884

-215 .2 . 10622416 4. 0324806 -210. 2 . 12673103 4. 1615839

-215 . 1 . 10662756 4. 0355978 -210. 1 . 12714729 4. 1636615

-215 .0 .10703128 4. 0386911 -210. 0 . 12756376 4. 1657214

99
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius — Continued

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-205. 2 . 14776541 4. 2458324 -200. 2 . 16913562 4 2977373

-205. 1 .14819005 4. 2471512 -200. 1 .16956543 4 2985150

-205. 0 . 14861484 4. 2484572 -200. 0 .16999532 4 2992838

100
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius — Continued

t °c W*(t)

dW*(t)ldT

X 1000

°c->

t °C r*(t)dW*(t)ldT

X 1000

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200. 0 .16999532 4. 2992838 -195. 0 .19157018 4. 3275324

-199. 9 . 17042529 4. 3000436 -194. 9 .19200296 4. 3279174

-199. 8 . 17085533 4. 3007946 -194. 8 . 19243577 4. 3282962

-199. 7 . 17128545 4. 3015367 -194. 7 .19286862 4. 3286688

-199. 6 . 17171564 4. 3022701 -194. 6 .19330150 4. 3290353

-199. 5 .17214590 4. 3029948 -194. 5 . 19373442 4. 3293958

-199. 4 . 17257624 4. 3037108 -194. 4 .19416738 4. 3297503

-199. 3 . 17300664 4. 3044183 -194. 3 . 19460037 4. 3300987

-199. 2 . 17343712 4. 3051172 -194. 2 . 19503340 4. 3304413

-199. 1 . 17386767 4. 3058077 -194., 1 . 19546646 4. 3307780

-199. 0 . 17429828 4. 3064898 -194. 0 .19589955 4. 3311088

-198. 9 . 17472896 4. 3071636 -193.,9 .19633268 4. 3314337

-198. 8 .17515971 4. 3078291 -193.,8 . 19676584 4.,3317530

-198. 7 . 17559053 4. 3084864 -193,.7 . 19719903 4.,3320665

-198. 6 .17602141 4. 3091354 -193,.6 .19763225 4., 3323744

-198. 5 . 17645236 4. 3097765 -193,.5 .19806551 4,,3326766

-198. 4 . 17688336 4. 3104094 -193,.4 . 19849879 4..3329732

-198. 3 . 17731444 4. 3110344 -193,.3 .19893210 4,.3332643

-198. 2 . 17774557 4. 3116515 -193 .2 . 19936544 4 .3335498

-198. 1 . 17817677 4. 3122607 -193 . 1 . 19979881 4 ,3338299

-198. 0 . 17860802 4. 3128620 -193,.0 .20023221 4,.3341046

-197. 9 . 17903934 4. 3134557 -192,,9 .20066563 4,. 3343739

-197. 8 . 17947071 4. 3140415 -192,.8 .20109908 4,. 3346378

-197. 7 .17990215 4. 3146198 -192 .7 .20153256 4,. 3348964

-197. 6 . 18033364 4. 3151904 -192 .6 .20196606 4,,3351498

-197.,5 . 18076519 4. 3157535 -192,.5 .20239959 4..3353979

-197.,4 .18119679 4. 3163092 -192,.4 .20283314 4..3356408

-197..3 .18162845 4.,3168574 -192 .3 .20326672 4 . 3358786

-197..2 .18206016 4. 3173982 -192,.2 .20370032 4,. 3361112

-197,. 1 . 18249193 4. 3179317 -192 . 1 .20413394 4,.3363388

-197,.0 . 18292375 4., 3184579 -192 .0 . 20456758 4,.3365614

-196,.9 . 18335562 4.,3189768 -191 .9 .20500125 4 . 3367789

-196 .8 . 18378754 4.,3194887 -191 .8 . 20543494 4 . 3369915

-196 .7 .18421952 4.,3199934 -191,.7 .20586865 4 .3371991

-196 .6 .18465154 4.,3204910 -191,.6 .20630238 4 .3374019

-196 .5 . 18508361 4,.3209816 -191,.5 .20673613 4 .3375999

-196 .4 .18551574 4,.3214653 -191,.4 .20716990 4 . 3377930

-196 .3 . 18594791 4.,3219420 -191,.3 .20760369 4 .3379813

-196 .2 . 18638012 4,,3224118 -191,,2 .20803750 4 .3381649

-196 . 1 . 18681239 4 . 3228749 -191,,1 .20847132 4 . 3383437

-196 .0 . 18724470 4 .3233312 -191.,0 .20890516 4 .3385180

-195 .9 . 18767705 4,. 3237807 -190.,9 .20933902 4 . 3386875

-195 .8 . 18810945 4 .3242236 -190.,8 .20977290 4 .3388525

-195 .7 . 18854190 4 . 3246599 -190.,7 .21020680 4 .3390129

-195 .6 . 18897439 4 .3250896 -190. 6 .21064070 4 . 3391688

-195 .5 . 18940692 4 .3255128 -190. 5 .21107463 4 . 3393202

-195 .4 . 18983949 4 .3259294 -190. 4 .21150857 4 .3394671

-195 .3 .19027210 4 .3263397 -190. 3 .21194252 4 .3396095

-195 .2 . 19070476 4 . 3267436 -190. 2 .21237649 4 . 3397477

-195 . 1 .19113745 4 .3271412 -190. 1 .21281047 4 .3398814

-195 .0 .19157018 4 .3275324 -190. 0 .21324447 4 .3400108

101
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius — Continued

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, 4 . <c<c;40oibU 4 ,. o4i3;DU<c —182 ,. 4

O/lCO'ZOA/l . /c4b(COoU4 4 . ooy4ioo XXQA 1 QQ

iO1 QT( .. O OO/iQCRTQ . <c<c4ybo (

y

A4 ,. 04i304D —182 , 3

O A CCTT QO . <c4bb ( lyo 4/I 7^7QoQQQ

iO1 QT1 . Q. c . OOR'^QQQQ <c<cooyyyy 4 , '74onnRR —182

.

, 2

O ^TT ARQA . <c4 ( iuoyu 4A TTQl TQ/l . Ooy i ( 04

—iO^ QT1 .> i . <c<cOoo4<cU A "^AOC^OOOi 04<CU<C<Ci3 —182

,

, 1 O .^TR'ZQQ T

. /ci4 f ooyoi 4 . 0013U040 ^"^QORylQ —iO1 QT1 .. u . (C/cb/cbo4U /I 4 ,. 04<C,U0D3 '^40n'^CQ TOO —lo2 ,. u

O/ITQT^ZTT . <;4 / y f O ( 1 4 . OOOc7<cOO ^^QQOQQ

—XOD . Q . oocTnocn <C(Cb ( U/cbU /I

. 04/CU4 74C>n4TC1 D

TOT —lol

.

o

. y

O,^ Q/1 ATCA . /;4o4U ( bU 4 . OOOOUU ^xQQnnT1 ——XOD 1 QR ., QO . (Cc. 1 iobOi A4 ,. 04/CU040 TOT —lol . o. o

O^QQ/IT /IT

. (C4oo4i4 f 4/I 7'7QCTn'^ . OOOD / UO ——XOD 1 QR . T OOTRTl

m

. (C/C I O 1 iUi 4 ,. 04<cUOO "^.^ODRQT( TOT —lol . rr

, 7

O/iaoTR^Z"? . dnrdd 1 ooo 4A '^'^QR'^TQ . OOOOO (

o

——XOD 1 QR . D . QOQnnRoo <c«cOUUO<c/C 4A , 7/1 onRQ'^ . O4<cU0yu —lol

.

6 O^QTAQT Q . <c4y f uyio 4 "ZQ/inOQ . ooo4u<iy —XOD 1 QC . OR . <i/C04oy4o 4 ,. o4(;U0bb 7/1 OARCC -181

.

5 ORAT A'Zril

. icDUi4oUi 4/I '7'ZQOCCA . OOo/cbbU

XOD 1 QC . A

. OOQQTTC? <C(c;oo / obo 4 , 7/1 OARAT . o4<cUoU I TOT —181

.

4

0RARTC0^7 . dOUD 1 OOO 4

7*70 T OTA . 008i<i lU

XOD 1 QC . zo OOQ'znTO A 4

,

7 vi O A /I T R

. o4<;U4io TOT -181

.

3 OR T A T AC yl

. <COlUiUb4 4

Z ZTQO RO . OO f yoo8

— XOD 1 QC . o/C 00QTy10n>! . <C(Cy 1 4/c:U4 4

,

7 y1 o AOQ T TOT -181

.

2 ORT A A A A'7

. 25144443 4

Z ZTO yl O R

. 33 r 8425

— XOD 1 QC . i . <;oUi / b/d4 4 . 3420135 T O T -181

.

1

OCT OfTOOO

. 25187820 4 . 3376971

T QC — iob

.

U . (£oUbiU44 4

,

7 /I T QQ >1 O . o4iyy4o -181

.

0 . 25231197 4

Z 'ZTR y1 QC . 30 f 54yb

— T QK ioo . Qy . (ioiU44b4 A4

,

7 /I T QTOQ . o4iy I d^ -180

.

9 . 25274571 yl 4

Z ZT y1 AA T

. OO MUUi

1 Qc: —ioo

.

QO . -coi4 /oo4 4

•Z A "y d A TQ , o4iy4 ?

y

-180

.

8

O C '7 T fTO A C

. 25317945 /I 4

7 ZTO y1 OR . OO ( <i485 —ioo 1 QR . ly

1 OTl QT "ZAT 4

•Z A 1 Q T QO . o4iyiyo T O -180

.

7 . 25361316 y1 4

7 "ZT AQ A Q . OO ( oy4y

— ioo 1 QR . D . /CO<co4 i dd 4 ,. o4iooo ZA T QQQT

/

T O -180

.

6 . £o4U4oo (

yl 4Z^ZCQ'ZQyl

. Oobyoy4

iOO 1 QR . O

O '7 r>TO T /I T

. (do/c f oi4i 4 .3418545 -180

.

5 oc A A a r\c c

. 25448055 4 . 3367818

—ioo 1 QR . A4

0*7 '70T RRQ . <;oo<diooy 4 .3418173 -180

.

4 . 25491422 4 .3366223

——XOO 1 QR . rOO'Z'ZC^QTT . <coob4y ( ( 4 . 3417772 T O —180

.

3

ORR^Z yITOO . ^OOo4 loo 4 . 3364608

-185. 2 . 23408395 4 . 3417341 -180. 2 .25578151 4 . 3362975

-185. 1 .23451812 4 .3416881 -180. 1 .25621514 4 .3361321

-185. 0 .23495228 4 .3416391 -180. 0 . 25664874 4 . 3359649

102
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius — Continued

t °c W*(t)

dW*(t)ldT

X 1000

°C"'

t °C W*(t)dW*{t)ldT

X 1000

°c-'

-180. 0 . 25664874 4,. 3359649 -175 .0 .27830405 4 .3255391

-179. 9 .25708233 4,.3357959 -174 .9 .27873660 4 .3252957

-179. 8 .25751590 4 .3356249 -174 .8 .27916911 4 .3250511

-179. 7 . 25794945 4,.3354522 -174 .7 .27960161 4 .3248053

-179. 6 .25838299 4 .3352775 -174 .6 . 28003407 4 .3245585

-179. 5 .25881651 4 .3351011 -174 .5 .28046652 4 .3243105

-179. 4 .25925001 4 .3349229 -174 .4 . 28089894 4 .3240615

-179. 3 . 25968349 4 . 3347430 -174 .3 .28133133 4 .3238114

-179. 2 .26011696 4 .3345612 -174 .2 .28176370 4 .3235602

-179. 1 .26055041 4 . 3343777 -174,. 1 .28219604 4 .3233079

-179. 0 . 26098383 4 .3341925 -174,.0 .28262836 4 .3230546

-178. 9 .26141724 4 .3340056 -173,.9 .28306065 4 .3228003

-178. 8 .26185064 4 .3338169 -173,.8 . 28349292 4 .3225448

-178. 7 .26228401 4 .3336267 -173,.7 .28392516 4 .3222885

-178. 6 . 26271736 4 . 3334347 -173,,6 . 28435738 4 .3220311

-178. 5 .26315069 4 .3332411 -173,,5 . 28478957 4 .3217727

-178. 4 .26358401 4 .3330459 -173,,4 .28522173 4 .3215133

-178. 3 . 26401730 4 . 3328489 -173. 3 . 28565387 4 .3212530

-178. 2 . 26445058 4 .3326505 -173,,2 . 28608598 4 .3209917

-178. 1 . 26488383 4 .3324505 -173, 1 .28651807 4 .3207295

-178. 0 .26531707 4 .3322488 -173. 0 .28695013 4 . 3204663

-177. 9 .26575028 4 .3320456 -172. 9 .28738216 4 .3202022

-177.,8 .26618348 4 .3318409 -172. 8 . 28781417 4,.3199371

-177.,7 .26661665 4 .3316346 -172. 7 . 28824615 4,.3196712

-177.,6 . 26704980 4 .3314269 -172. 6 .28867810 4,,3194044

-177,,5 . 26748294 4 .3312176 -172. 5 .28911003 4 .3191366

-177,,4 .26791605 4 .3310068 -172. 4 .28954193 4,.3188680

-177,.3 .26834914 4 . 3307946 -172. 3 . 28997380 4,.3185986

-177,.2 .26878221 4 .3305809 -172. 2 . 29040565 4,.3183283

-177,. 1 .26921525 4 .3303658 -172. 1 . 29083747 4,.3180571

-177,.0 . 26964828 4 .3301492 -172. 0 .29126926 4,.3177851

-176 .9 .27008128 4 .3299313 -171

.

9 .29170103 4,.3175123

-176 .8 .27051427 4 .3297119 -171. 8 .29213276 4,.3172386

-176 .7 . 27094723 4 .3294912 -171. 7 . 29256447 4,,3169642

-176 .6 .27138016 4 .3292691 -171. 6 .29299616 4, 3166889

-176 .5 .27181308 4 .3290455 -171. 5 .29342781 4,,3164129

-176 .4 . 27224597 4 .3288207 -171. 4 . 29385944 4,,3161361

-176 .3 . 27267884 4 . 3285945 -171. 3 .29429104 4,,3158585

-176 .2 .27311169 4 . 3283671 -171. 2 .29472261 4.,3155801

-176 . 1 . 27354452 4 .3281383 -171. 1 .29515416 4. 3153010

-176 .0 . 27397732 4 .3279082 -171. 0 .29558567 4. 3150212

-175 .9 .27441010 4 . 3276768 -170. 9 .29601716 4. 3147407

-175 .8 . 27484286 4 . 3274442 -170. 8 . 29644862 4. 3144593

-175 .7 .27527559 4 .3272103 -170. 7 .29688005 4. 3141774

-175 .6 .27570830 4 .3269752 -170. 6 .29731146 4. 3138947

-175 .5 . 27614098 4 . 3267389 -170. 5 . 29774283 4. 3136113

-175 .4 . 27657365 4 .3265013 -170. 4 .29817418 4. 3133271

-175 .3 . 27700628 4 .3262625 -170. 3 . 29860550 4. 3130424

-175 .2 . 27743890 4 .3260226 -170. 2 .29903679 4. 3127569

-175 . 1 . 27787149 4 .3257814 -170. 1 .29946805 4. 3124708

-175 .0 . 27830405 4 .3255391 -170. 0 .29989928 4. 3121841

103
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius — Continued

°c

dW*(t)ldT

X 1000

°c-> "C W*(t)

dW*(t)jdT

X 1000

°c->

-170. 0 . 29989928 4. 3121841 -120 .0 .51175723 4,. 1696979

-169. 0 .30421002 4. 3092826 -119 .0 .51592580 4,. 1674503

-168. 0 .30851783 4. 3063249 -118 .0 .52009214 4 . 1652272

-167. 0 .31282265 4. 3033185 -117 .0 .52425626 4,.1630283

-166. 0 .31712445 4. 3002706 -116 .0 .52841820 4 . 1608530

-165. 0 .32142318 4. 2971877 -115 .0 .'53257798 4 . 1587009

-164. 0 .32571882 4. 2940758 -114,.0 .53673561 4 .1565715

-163. 0 .33001133 4., 2909406 -113 .0 .54089113 4 . 1544643

-162. 0 . 33430069 4., 2877872 -112,.0 . 54504455 4 . 1523789

-161. 0 . 33858690 4. 2846206 -111

,

.0 .54919589 4 .1503148

-160. 0 . 34286993 4. 2814449 -110,.0 .55334518 4 . 1482715

-159. 0 . 34714978 4. 2782645 -109,.0 . 55749244 4 . 1462487

-158. 0 .35142646 4. 2750828 -108,,0 .56163769 4 . 1442458

-157. 0 .35569995 4.,2719035 -107,.0 . 56578094 4 . 1422623

-156. 0 . 35997027 4., 2687295 -106,,0 .56992222 4 . 1402979

-155. 0 . 36423741 4.,2655639 -105,.0 .57406154 4 .1383521

-154. 0 . 36850140 4., 2624092 -104..0 . 57819893 4 . 1364243

-153. 0 . 37276224 4., 2592676 -103,,0 . 58233440 4 . 1345144

-152. 0 . 37701994 4.,2561416 -102..0 . 58646796 4 .1326217

-151. 0 . 38127452 4.,2530330 -101..0 .59059965 4 . 1307459

-150. 0 .38552601 4., 2499434 -100,.0 . 59472946 4 .1288866

-149. 0 . 38977442 4., 2468745 -99.,0 . 59885742 4 . 1270433

-148. 0 .39401977 4., 2438278 -98.,0 .60298355 4 .1252157

-147. 0 . 39826208 4. 2408044 -97,,0 .60710786 4 . 1234034

-146. 0 .40250138 4., 2378054 -96.,0 .61123036 4 .1216060

-145. 0 . 40673770 4.,2348319 -95.,0 .61535108 4 .1198231

-144.,0 .41097106 4. 2318846 -94.,0 .61947002 4 .1180544

-143.,0 .41520148 4. 2289643 -93.,0 .62358719 4 .1162996

-142.,0 .41942899 4.,2260716 -92.,0 .62770262 4 . 1145582

-141.,0 .42365363 4.,2232070 -91. 0 .63181631 4 .1128299

-140..0 . 42787542 4.,2203709 -90. 0 .63592828 4 .1111145

-139.,0 .43209438 4. 2175636 -89. 0 .64003855 4 .1094117

-138.,0 .43631056 4., 2147855 -88. 0 .64414711 4 .1077210

-137..0 .44052396 4.,2120366 -87. 0 . 64825399 4 .1060422

-136,.0 . 44473464 4. 2093170 -86. 0 .65235920 4 . 1043750

-135,.0 . 44894261 4. 2066270 -85. 0 . 65646275 4 .1027192

-134,.0 .45314790 4. 2039663 -84. 0 . 66056464 4 .1010745

-133 .0 .45735055 4. 2013350 -83. 0 . 66466490 4 . 0994405

-132,.0 .46155058 4. 1987329 -82. 0 . 66876353 4 .0978171

-131 .0 . 46574803 4. 1961599 -81. 0 . 67286054 4 .0962040

-130 .0 .46994291 4. 1936158 -80. 0 . 67695594 4 .0946010

-129 .0 .47413527 4. 1911002 -79. 0 .68104974 4 . 0930078

-128 .0 .47832512 4. 1886132 -78. 0 .68514196 4 . 0914242

-127 .0 .48251250 4. 1861542 -77. 0 .68923259 4 .0898500

-126 .0 . 48669744 4. 1837229 -76. 0 .69332166 4 . 0882850

-125 .0 . 49087996 4. 1813191 -75. 0 .69740917 4 . 0867290

-124 .0 .49506009 4. 1789423 -74. 0 .70149512 4 .0851817

-123 .0 . 49923785 4. 1765922 -73. 0 .70557953 4 . 0836431

-122 .0 .50341328 4. 1742684 -72. 0 .70966241 4 .0821129

-121 . 0 . 50758640 4. 1719704 -71. 0 .71374376 4 .0805910

-120 .0 .51175723 4. 1696 79 -70. 0 .71782359 4,. 0790772

104
IPTS-68 Reference Function and Its First Derivative With the Argument in Celsius —Continued

i' °cdW*(t)ldT

X 1000

t °C W*{t)dW*(t)jdT

X 1000

°c-'

-70. 0 .71782359 4 0790772 -20 0 .92002439 4 . 0111301

-69. 0 .72190192 4 0775712 -19 0 .92403489 4 . 0098834

-68. 0 . 72597874 4 0760731 -18 0 .92804416 4 . 0086393

-67. 0 .73005407 4 0745825 -17. 0 .93205217 4 . 0073976

-66. 0 .73412791 4 0730995 -16. 0 .93605895 4 . 0061582

-65. 0 .73820027 4 0716238 -15. 0 . 94006449 4 . 0049209

-64. 0 .74227116 4 0701553 -14. 0 . 94406879 4 . 0036856

-63. 0 .74634058 4 0686939 -13. 0 .94807186 4 . 0024522

-62. 0 .75040855 4 0672394 -12. 0 .95207370 4 012205

-61. 0 .75447506 4 0657918 -11. 0 .95607430 3 . 9999902

-60. 0 .75854013 4 0643509 -10. 0 .96007368 3 9987614

-59. 0 .76260377 4 0629166 -9. 0 .96407183 3 9975337

-58. 0 . 76666597 4 0614887 -8. 0 . 96806875 3 9963070

-57 0 . 77072675 4 0600673 -7. 0 . 97206444 3 .9950811

-56 0 .77478611 4 0586521 -6. 0 .97605891 3 9938559

-55 0 . 77884405 4 0572432 -5. 0 .98005215 3 9926310

-54 0 .78290059 4 0558403 -4. 0 . 98404417 3 y»14065

-53 0 . 78695573 4 0544434 -3. 0 . 98803497 6 yyuioly

-52 0 .79100948 4 0530523 -2. 0 .99202454 3 yooyo fi

-51 0 .79506184 4 0516672 -1

.

0 .99601288 6 yo ( / 321

-50 0 .79911282 4 0502877 0 .1 . 00000000

-73 yo65063

-49 0 .80316242 4 0489138

-48 0 .80721065 4 0475455

-47 0 .81125751 4 0461827

-46 0 .81530302 4 0448252

-45 0 .81934716 4 0434730

-44 0 .82338996 4 0421259

-43 0 .82743142 4 0407841

-42 0 .83147153 4 0394472

-41 0 .83551031 4 0381154

-40 0 . 83954777 4 0367884

-39 0 . 84358389 4 0354661

-38 0 . 84761870 4 0341486

-37 0 .85165219 4 0328357

-36 0 . 85568437 4 0315273

-35 0 .85971525 4 0302233

-34 .0 . 86374482 4 0289237

-33 .0 .86777310 4 0276283

-32 .0 .87180008 4 0263371

-31 .0 . 87582577 4 0250500

-30 .0 .87985018 4 0237668

-29 .0 . 88387331 4 0224875

-28 .0 .88789516 4 0212120

-27 .0 .89191573 4 0199401

-26 .0 . 89593504 4 0186718

-25 .0 .89995308 4 0174069

-24 .0 .90396985 4 0161454

-23 .0 .90798537 4 0148870

-22 .0 .91199963 4 0136318

-21 .0 .91601263 4 0123795

-20 .0 .92002439 4 0111301

105
Appendix F. Analysis of the First Derivatives at 0 °C of IPTS-68 Platinum

Resistance Thermometer Formulations Above and Below 0 °C

The four specified deviation functions of the

IPTS-68 in the interval 13.81 K to 273.15 K are formulated to join smoothly by constraining the

values of the first derivatives of the formulations

above and below each point of joining to be identical

at this point (see the IPTS—68 document). However, the values of the first derivatives of the formulations above and below 0 °C of real SPRT's are

not constrained to be equal at 0 °C. This discussion

will show that the magnitude of this difference is negligible for SPRT's that have suitable thermometer constants. All SPRT's received at the NBS for

calibration in the past few years had suitable

constants (see the last page of this Appendix).

The IPTS—68 reference function in Celsius

20 t=^Ai[\n W*{t)V °C (1)

i=i

was formulated so that its first and second derivatives would have the same value at 0 °C as obtained

from

—

«(7)(7-')(7-)e-)-

and

,=i[r(,')-i]+6(f)(^-i)x (3)

with thermometer constants

U) =A4t + CitHt- 100). (5)

The constants of eq (5) are determined from the deviations at the oxygen point and the steam point.

When differentiated with respect to t, there re- sults from eqs (4) and (5), respectively.

dW(t) dw*{t) , d^W{t)

dt dt dt (6)

and

dAWjt)

dt = ^4 + 4C4«M«-75). (7)

At 0 °C, the reference function (eq (1)) on differentiation yields

dW*{t)\ 1 dt /o Ai (8)

and eq (7) yields

(9)

When eqs (6), (8), and (9) are combined, there

results

dW(t)\ 1 ^ .

dt /OB Ax

(10)

a = 3.9259668x10-3/ °C

and

where the subscript OB refers to the formulation

below 0 °C. The constant A4 is obtained from the

value of r(100 °C), i.e., from

8 = 1.496334 °C.

(The above selected values of a and 8 given in the

text of the IPTS—68 will, henceforth, be referred to as a* and 8*, respectively. In eqs (1), (2), and (3),

W*{t) = WccT:-Gs{t%») and t = t68 (for convenience),

5= 100 °C, z = 419.58 °C, a = 630. 74 °C, and W{t') =

W{t)=R{t)IR{0).) The second term of eq (2)

represents a "correction" to be made to the temperature value t' to obtain a temperature value t that is a closer approximation to the thermodynamic

scale. The values t' and t represent the same temperature (hotness). For real SPRT's the derivative

of the formulation between - 182.962 °C and 0 °C

is obtained from the definition of the IPTS—68 in

this temperature range

W{t) = w*{t)^^w{t)

and the deviation function

(4)

and

r(100 °C) = 100 a+1

r*(100 °C) = 100 a*+l.

(11)

(12)

The value of a in eq (11) can be obtained from meas- urements at the triple point of water and the steam

point or at the triple point of water, tin point, and

zinc point (a=[R(lOO)-R 0)/lOOi?(0)). Combining equations (11) and (12) with eq (5) {t —

100 °C) there results

a — a*.

When eq (10) and (13) are combined,

dW{t)\ 1 ^

dt / OB -^1 a

(13)

(14)

106
The derivative of the formulation between 0 and

630.74 °C is obtained by differentiating eq (2) and

the equation

there results, respectively,

^^^^0_045 ,^ at za

(15)

-\-2{sz + sa-{- za)t' — sza], (16)

and

dW{t)

dt' = A + 2Bt'. (17)

Equation (15) is equivalent to eq (3) with the thermometer constants A = a{l + 8/100 °C) and B =

-10-4 oQ-2_ At t = 0°C, t'^0 °C, and eqs (16)

and (17) become respectively.

^) =1-^= 0.99955

dt'Jo 100

(18)

and

dW{t)

dt' Jo

Combining eqs (18) and (19),

= A. (19)

(dW{t) \ ^(dWit) \ /dt\ ^ A

\ dt JoA \ dt' Jo \dt)o 0.99955' ^

'

where the subscript OA refers to the formulation

above 0 °C.

By subtracting eq (20) from eq (14), the difference

in the derivatives at 0 °C of the two formulations

becomes

dWjt)

dt

\ /dW{t) \ ___^ /dW(t) \

Job \ dt JoA \ dt Jo Ai

+ a — a*-

From the relations

^ = a(l + 8/100 °C)

0.99955

(21)

(22)

and

fl = -10-4a8°C-2 (23)

A* and B* are obtained by substituting a* and 8*

for a and 8, respectively. By substituting A* and

B* in eq (15), differentiating, and combining with

eq (16), there results

^dWHty^ A*

OA 0.99955' (24)

an expression that is very similar to eq (20). When

eq (24) is compared with eq (8) and because the two

derivatives are equal.

A I 0.99955* (25)

By substituting eq (25) for l/Ai in eq (21), there

results

Although A

dWjt)

dt

dW{t)

= a. — a*-\-

A*-A

0.99955' (26)

, 1 can be evaluated from eq (26),

^ at Jo

the conversion of ^* and A in terms of B* and B

results in an equation that is simpler to evaluate.

The substitution of the expression (22) for A* and

A and then of the expression (23) for the a8 product

results in

dW{t)\ 0.00045(a*-a)-100(fi*-iB)

0.99955

(27)

The a values of SPRT's calibrated at NBS in the

past few years range between 3.925 and about

3.927X 10-3 °C-i and the B between -5.872 and

-5.877 X 10-7 oC-2; both a* and B* are about the

average of the above limits. Therefore, a* — a is about ±lXlO-6°C-i and B-B* is about ±2.5 X

10-10 °C-2. (Note that "100" in eq (27) has the unit

°C associated with it.) When these values are substituted in eq (27)

dWjt)

dt

3x 10-8°C-i. (28)

This may be taken as the maximum expected discontinuity. Any variation of this discontinuity is mostly dependent on the constant B. Inasmuch as

{dW{t)ldt)o is about 4X 10-^ °C-\ the relative discontinuity in the first derivatives of the two formulations at 0°C is expected to be less than 8X 10"^.

The discontinuity is neghgible for most purposes.

107
Appendix G. Derivation of Differential Coefficients for the Analysis of Errors

in Platinum Resistance Thermometry

This appendix deals with the propagation of

errors in the temperature determination that results

from the errors of calibration of a SPRT. The

differential coefficients (the rate of change of the

values of temperature with respect to the change

of the resistance ratios, /?(f)/i?(0), measured for

calibration) are derived for the calibration measure- ments at the steam point, tin point, zinc point, and

oxygen normal boiling point (NBP).

The total error, in the value of temperature that

is obtained from measurements employing a SPRT,

is the sum of the error introduced by the calibration measurements and the error from the experimenter's own measurements. The experimenter

must determine his own observational error from a

careful evaluation of his measurement techniques.

He must also know the errors that may be introduced in the values of temperature caused by

possible errors in the thermometer calibration

measurements at the NBS. He must also be aware

of possible errors from an unknown change in the

cahbration.

The variation among the SPRT's is yet another

source of uncertainty; i.e., even though the measure- ments with the SPRT's are made at the same

temperature (hotness), they do not yield exactly the

same values of temperature. A number of scattered

data do exist [1,2,3,4], but there are no systematic

high-precision measurements on the intercomparison of SPRT's that employ modern SPRT's and

measuring equipment.

The errors of calibration at NBS may be separated

into two types: (1) errors of deviation from realizing

the fixed points and (2) errors of resistance measurements. The total differential of t for the SPRT is given by

where = temperature of the defining fixed points,

i.e., for this discussion the oxygen NBP, the triple

point of water, the steam point or the tin point,

and the zinc point. The Wi refers to the measured

resistance ratios R{ti)IR(0) corresponding to the

fixed points. The dti or 8ti refers to a variation in

the fixed-point temperature (hotness) experimentally

realized, e.g., the deviation of the temperature of a

tin-point cell. The differential coefficient | —

1

\dtiJ

represents the rate of change in the value of temperature with respect to a change in the ith fixed-point

temperature (hotness); the coefficients may be

derived from the prescribed interpolation formulae

using the values of defining fixed temperatures.

The differential coefficients are temperature dependent, i.e. , =f{t).

The dWi or 8Wi refers to a variation in the

resistance ratio measurement, e.g., the error of

the resistance ratio measurement process with the

SPRT at NBS. The differential coefficient {—^

represents the rate of change in the value of temperature with respect to a change in the measured

resistance ratio at the ith fixed point; the coefficients are derived from the prescribed formulae that relate

the measured resistance ratio to the constants of

the SPRT. Similar to \ ^r\, [t^^t] is also tempera-

\dti) \dWil

ture dependent.

Some preliminary and limited investigations at NBS on the realization of the tin point and the

triple point of water indicate that these fixed

points are reproducible within ±0.1 mK. This

value, although obtained employing the same

thermometer over a short interval of a few days

to determine the error attributable to the variations

in the temperatures of the fixed-point cells, includes

both the variations in the temperature {bti) of

many freezes in different cells and the variations

in the measurement process {&Wi). Over a period

of several months the reproducibility of measure- ments on different SPRT's em.ploying the same cell

is estimated to be about ±0.2 mK. This value

likewise includes any variations in the temperature

of the cell as well as the variations in the measure- ment process. Thus, the errors of the realization

of the fixed-point temperatures and the errors of

calibration resistance measurements are difficult to separate. Therefore, the analysis of error propagation has been simplified by combining the two

sources of errors in calibration measurements.

Equation (1) then simplifies to

In the expression the change or error bti in the value

of temperature has been replaced by a corresponding change or error bWi. At present, the total

uncertainty (which includes St; and bWi of eq (1)) of

the calibration measurements is estimated to be

2 or 3 mK at the oxygen NBP, and 1 mK at the tin

point, and 1 mK at the zinc point.

The differential coefficients ^^^^^

have been

derived separately for two ranges of temperature

108
defined by the SPRT, the range 0 to 630.74 °C and

the range -182.962 (90. 188 K) to 0°C (273.15 K). In

the first range (0 to 630.74 °C), the coefficients have

been derived for two types of calibration measurements: (1) at the triple point of water, the steam

point, and the zinc point and (2) at the triple point

of water, the tin point, and the zinc point. In the

second range, the differential coefficients have also

been derived for two types of calibration measurements: (1) at the oxygen NBP, the triple point of

water, and the steam point and (2) at the oxygen

NBP, the triple point of water, the tin point, and the

zinc point.

The error dt in the value of temperature, that

would arise from an error corresponding to a

positive unit temperature (hotness), was calculated

as a function of the temperature employing the

derived differential coefficients; i.e., the function

was evaluated as a function of the temperature.

When the calibration error at a particular fixed

point was considered, the calibration measurements

at the other fixed points were taken to be correct.

The following cases have been calculated and

plotted in figure 17.

(1) The range 0 °C to 630.74 °C

(a) calibration measurements made at the triple

point of water, the steam point, and the zinc point;

calibration error occurring only at the steam point

or only at the zinc point,

(b) calibration measurements made at the triple

point of water, the tin point, and the zinc point;

calibration error occurring only at the tin point or only at the zinc point.

(2) The range -182.962 °C (90.188K) toO°C {273.15 K)

(a) calibration measurements made at the oxygen

NBP, the triple point of water, and steam point;

calibration error occurring only at the oxygen NBP

or only at the steam point,

(b) calibration measurements made at the oxygen

NBP, the triple point of water, the tin point, and the

zinc point; calibration error occurring only at the

oxygen NBP, or only at the tin point, or only at the

zinc point.

In the range —182.962 to 0°C, the cahbration errors

at the oxygen NBP for the cases 2a and 2b yield the

same temperature error 8t; therefore, the analysis

of only one of these cases is given.

As a separate analysis, the case of an error in the

realization of the triple point of water by the experimenter has been calculated; the calibration of the

SPRT at NBS was considered perfect and an error

of resistance measurement 8R corresponding to a

positive unit temperature at the triple point of

water was considered to have been made by the

experimenter. The error 8R in the triple point of

water measurements was taken to be propagated in

the determination of the resistance ratio R{t)IR{0).

The results of the analysis are also plotted in

figure 17. The following sections deal with the details of the

derivation of the differential coefficients that were employed to avaluate the temperature error function

given by eq (3) for the cases cited in the previous

paragraphs. In the discussion, the variations in the

measurements at the fixed points will be described

in terms of the resistance ratio, Witi) = R{ti)IRiO)

,

the ratio of observed resistance at the fixed-point

temperature t, to that at 0°C. The variations in

W(ti) will be taken to be centered in R{ti). Obviously, any error in R{0) will be reflected in W{ti);

thus, the present discussion is also applicable to possible errors in the determination of ^(0).

l.O°C to 630.74 °C

From 0 to 630.74 °C, the temperature t on the

IPTS-68 is defined by

+ 0.045(9

and

<' = i(r(0-l)+6(f)(^-l)-C, (5)

where

W{t'i)^Witi)^Riti)IRiO) (6)

s=100T, z = 419.58°C, a = 630.74°C, and t = tGs (for convenience). The second term on the right side

of eq (4) will, henceforth, be defined as

The function Af(f') represents a correction to t' to obtain the temperature t that is a closer approximation to the thermodynamic scale. The values t'i and ti, therefore, represent the same temperature

(hotness) and R{t'i) = R{ti). Also, eq (5) is equivalent

to

Wit')^l+At' + Bt'\ (8)

where

^ = a(l + 8/100°C) (9)

and

B = -10-%S°C-2 (10)

The constants of the IPTS-68 eqs (5) and (8) are determined by measurement of the SPRT resistance

at the triple point of water, the steam point, and the

zinc point or the triple point of water, the tin point,

109
and the zinc point. To simplify the symbols and

terminology, the following definitions are made:

(1) it')^W{t')-l=At' + Bt'\

I.e.

(Os=1Fs-l = [R{100)IR{0)]-l

(0T= Wt-1 = [R (231.9292)/i? (0)] - 1„

(11)

(12)

(13)

and

coz=rz-l = [i? (419.58)//? (0)]-l. (14)

The differential coefficients will be derived first for the SPRT formulation with cahbration measure- ments at the triple point of water, the tin point,

and the zinc point. The differential coefficients

sought are

.aojT/ \ao)z

Because w is a function off', the conversion

d(D

(15)

must be made. From eq (4),

+ 2(5Z + 5a + za)t' — sza]

at s^za

= 1 +

dMjt')

dt' (16)

Equation (16) will be employed to convert dt'ldcj to

dtldb).

From eq (11),

and

ojT = At'+ Bt'J

coz = A4 + Bt'z\

(17)

(18)

where tj is the defined tin point temperatur^ and

is the defined zinc point temperature. (According

to the definition of the IPTS-68, i;=f6s^- 0.038937

and t^ = t6s^; henceforth, the "prime" symbol will

not be employed in the designation of a fixed-point

temperature.) Solving for A and B, there results

and

A = {(OTtl - 0Jz4)ltTtz{tz - It), (19)

B =^ (ojztr - (DTtz)ltTtzitz - tj ) . (20)

Substituting in eq (11) for A and B, there is obtained

CO {t') =

{oirtl — 0)zt^)

trtzitz ~ It)

t' +

{(Oztr — 0)Ttz)

trtzitz ~ tr)

(21)

The differential coefficients

dt'

and

dt' are

\da>T/ \do)zJ

obtained by impUcit differentiation of eq (21); for

(t)T, eq (21) yields

ty tzt

f2

dt' = 0

trtzitz ~ tr)

^[ioirtl — (Dzt\) -\-2i(x)ztT — oirtz)t']

trtz itz ~ tr)

After transferring, eq (22) becomes

dt' ^

tzt'jtz-t')

d(t}T (Drtzi^t' — tz) — coztri^t' — tr)

dt'

Similarly to obtain -—, the differentiation of eq (21)

(22)

(23)

yields

trt"-

d(t)z

It'

trtzitz ~ tr)

d(i)2

^ [icjrtl — coztj) +2ia)ztr— 0)Ttz)t']

trtzitz ~ tr)

After transferring, eq (24) becomes

dt' trt'itr — t')

dcoz (OztTi2t' — tr) — 0)Ttzi2t' — tz)

dt'^0. (24)

The desired differential coefficients

dt

and

(25)

dt

dcor dojz are obtained by combining eqs (23) and (25), respectively, according to eq (15), with eq (16); there

results, respectively.

dt

d(t)q

and

dt

d(x)z

dMit')

dt'

tzt' itz- 1')

[(ji)rtzi2t' — tz) —0)ztTi2t' — tr)]'

(26)

14

dMit') trt'itr-t')

dt' J [(DztTi2t' - tr) —0)rtzi2t'—tz)]'

(27)

Equations (26) and (27) represent the rate of change

of temperature t with respect to a change in oir and

(t)z, respectively, at the temperature t' . Although

110
both equations (26) and (27) can be converted to terms of t by employing the relation given by eq (4), the formulations given are simpler and practical to use. After calculating dtjdoiT or dtldojz at t' , then t' can be converted to t according to eq (4). The error in the temperature t that results from

the error in the tin point calibration is

8t =

dt_

8(1)1 (28)

the error from the zinc point is

dt

8t = 8(DzFigure 17 shows the error that the experimenter

would encounter for errors of 8o)t or 8(i)z corresponding to a unit positive error of temperature (hotness).

The temperature in t' has been converted to t in

the figure.

The derivation of the differential coefficients for

the SPRT formulation with calibration measure- ments at the triple point of water, steam point,

and zinc point follow the same procedures outlined

above. The final equations may be obtained directly

by substituting ojs for cjt and ts for It in eqs (26) and

(27); the results are

dt

d(t)s

and

aojz L

dMjt')

dt'

tzt'{tz-t')

[cjstzi2t' -tz) b)ztsi2t' -ts)]

(30)

dMjt')

dt'

tst'{ts-t')

[(ozts{2t' - ts) - (Ostz{2t' - tz)]

(31)

The errors in the temperature t that result from the

errors in the calibration at the steam point and the

zinc point are given by

8f =

dt

8(i)s (32)

and

8t

\da)z/i'

8(az. (33)

Figure 17 shows the error that the experimenter

would encounter for errors of 8(i)s and 8a»z corre- sponding to a unit positive error of temperature

(hotness).

2. 90.188 K {-182.962 °C) to 273.15 K (0 °C)

From - 182.962 to 0 °C the temperature t on the

IPTS-68 [5] is defined by the relation

Wit)^JF*it) + AW{t),

where

r(o=/?(0/«(o),

(34)

(35)

and W*it) is the resistance ratio given by the refer- ence function in Celsius

20 t^-^Ailln W*it)]

(29) From eq (36)

8t= {dtldW*{t))8W*{t),

and from eq (34)

8W*(t)=-8MV{t).

(36)

(36a)

(36b)

For the derivative dtldW*{t), see Appendix E. The experimenter must determine his observational

error 8fF t). The deviation ^W{t) is represented

(in this temperature range and at 100 °C) by the

function

MF{t)=A4t + C4tHt- 100 X), (37)

and the constants of the deviation equation are determined from the calibration measurements at the oxygen NBP, triple point of water, and the

steam point. The deviation at 100 °C, Ar(lOO),

may also be obtained from the calculated JF(IOO)

based on the calibration of the SPRT at the triple

point of water, the tin point, and the zinc point.

The differential coefficients will be derived for both

cases. For convenience, the symbols and terminologies

used in the text of the IPTS-68 have been abbreviated as follows:

and

B^*(0 = B^CCT-68 (tes),

W{t) = WM,

AWit) = AW(tes).

(38)

(39)

(40)

(41)

In addition, the temperature of the fixed points will

be indicated as:

to = oxygen NBP,

ts = steam point,

tr^tin point (tr= fesj.- 0.038937 °C).

Ill
and

tz = zinc point

and the deviations \W{t) at the fixed points will

be indicated as ^Wo= Wo- (deviation at the oxygen NBP)

(42)

AWs= Ws — W* (deviation at the steam point).

(43)

When the deviations at the oxygen NBP and the

steam point are applied to eq (37), there results

Aro =^4fo + C4ig(io-100) (44)

^Ws = A^ts + C^tl{ts-\QQ). (45)

(For simplification the °C unit after 100 and 75 will

be deleted in some of the equations from 44 through

51.)

The expressions for ^4 and C4 become

^Wstl{to-lm - Mf'otUts - 100)

A4=—T7-. (46)

and

C4

tstlito- 100) -totlits- 100)

tstlito- 100) -totlits -100)' (47)

The differential coefficients ( ^

.

and /^^^(^)\

\ dAJVo / V d^Ws I are obtained by differentiation after combining

eqs (46) and (47) with eq (37); there results

dAWjt) _ -ttl{ts-\00) ^ tdHt-lOO)

dAWo tstlito-lOO)-totlits-100)

(48)

When ts= 100 °C is substituted in expression (48),

there is obtained

dAWit) t^it-lOO)

dAWo ^3(^^,-100)

For dAWit)/dAWs,

dAWit) _ ttlito - 100) -totHt- 100)

dAWs tstlito - 100) -tot^^ its -100)

(49)

(50)

and, when is =100 °C is substituted wherever

simplification can be made,

dAWit) ^tUlito- 100) -tHt- 100)]

dAWs tstlito- 100)

^^^^

Figure 17 shows the error the experimenter would

encounter for calibration errors of 8AW0 or 8AWs

corresponding to a unit positive error of temperature.

If the tin and zinc point calibrations are used, eqs

(46) and (47) must be modified. Expressing Ws and

W* of eq (43) in the form of eq (8), there is obtained

where

and

AWs=iA-A*)ts+iB-B*)tl (52)

^* = a*(l + 8*/100°C) (53)

5*=10-4a*8*°C-2, (54)

where a* (3.9259668 X lO-^C) and 8* (1.496334

°C) are the constants of the lPTS-68 reference

function above 0 °C. The constants A and B are determined from calibration measurements at the

tin and zinc points. By introducing the expressions

for A and B from eqs (19) and (20) in eq (52), there

results

AWs^

0)ztj

.trtzitz — tr)

-A* ts

+

Oiztj 0)Ttz

trtzitz — tr)

B* (55)

Equation (55) is then substituted for AWs in eqs

(46) and (47). The eq (49) for the differential coefficient of dAWit)/dAWo remains unchanged. Because

the true value of AWs is independent of whether the

calibration is at the steam point or at the tin and

zinc points, the error function (dAWit)ldAWo)8AWo

for the case 2b will be the same as that for case 2a

with the steam-point calibration.

Instead of substituting eq (55) in eqs (46) and (47)

and performing a long algebraic process, the total

derivative of AWs can be obtained in terms of cot and (jjz, i.e..

rfAr.= (^)d„,+ (^)rf<«. ,56)

\ dcjT I \ 00)Z I

The expression (56) is employed to obtain dAWs,

i.e.,

6Ar.= (^)6..+ (^)a.., ,57)

and the error function f^^ . ^[

^ \8AWs is then evalubAWs

ated as

dAWit)-= dAWjt)

dAWs

+

8(1)7dAWs

ddiz

8(xiz (58)

112
The coefficient , . nf. is given by eq (51) with eq (55)

substituted for AJFs. The differential coefficients

and are obtained irom eq (55). Ihus,

d(OT d(i)z

c)AWs tits tztl

dcjT tftzitz — ^7") trtzitz — tr)

which simplifies to

^^Ws ^ tsjtz - ts)

_

d(i)T tritz — tr)

I< or^ , eq (55) becomes

(59)

(60)

d(t}z

dAWs -tbs

+

trtl

d(t)z tftzUz ~ tr) trtzitz ~ tr)

which simplifies to

dAWs ^ tsjts- tr)

_ do)z tzitz — tr)

(61)

(62)

Combining eqs (60) and (62) with eq (58), there is obtained

dMV{t): tsitz - ts)

tritz

+

tr)

ts jts- tr)

tzitz- tr)

8(oz (63)

Figure 17 shows the error the experimenter would

encounter for errors of 8(or or Swz corresponding

to a positive unit error of temperature.

References to Appendix G

[1] Belyansky, L. B., Orlova, M. P., Sharevskaya, D. I., and

Astrov, D. N., Investigation of the Resistance-Temperature Properties of Platinum for Resistance Thermometry

over the Range from 90 K to 273 K, Metrologia 5, No. 4, 107-111 (1969).

[2] Hoge, H. J. and Brickwedde, F. G., Intercomparison of Plati- num Resistance Thermometers Between— 190° and

445° C. J. Res. Nat. Bur. Stand. 28, 217-240 (February

1942) RP1454.

[3] Moussa, M. R. M., Van Dijk, H., Durieux, M., Comparison

of Platinum Resistance Thermometers Between 63 K and

373.15 K, Physica 40, 33-48 (1968).

[4] Sharevskaya, D. I., Orlova, M. P., Belyansky, L. B., and

Galoushkina, G. A., Investigation of the ResistanceTemperature Properties of Platinum of Resistance Thermometry over the Range From 14 K to 90 K, Metrologia 5, No. 4, 103-107 (1969).

[5] The International Practical Temperature Scale of 1%8.

Adopted by the Comite International des Poids et Mesures,

Metrologia 5, No. 2, 35-44 (April 1%9).

Appendix H. Calibration of Mueller Bridges

1 . The Mueller Bridge

The Mueller bridge is designed for measurement

of temperatures in conjunction with four-terminal

standard platinum resistance thermometers (SPRT).

This bridge is basically the equal arm Wheatstone

bridge but with important modifications. One of

these modifications permits the SPRT connections

to be commutated so that the lead resistances of

the SPRT can be canceled by averaging the two

bridge readings (see sec. 3.3). Other modifications

include a provision for adjusting the ratio arms to equality and for reducing the uncertainties due to variations in contact resistance. In. the process of

determining the thermometer resistance the

bridge current is reversed so that the effect of any

steady spurious emfs are eliminated. The current

reversal method also doubles the apparent null

detector sensitivity.

The Mueller bridges employed in the Platinum

Resistance Thermometer Laboratory of the NBS

have additional features including the simulta- neous interchange of the ratio arms when the SPRT

connections are commutated. Any errors arising

from a small inequality of the ratio arms are thus

made neghgible when the two bridge readings are averaged. The schematic circuit diagram of this

form of Mueller bridge is shown in figure H-1.

For this discussion the upper two bridge arms

will be referred to as the ratio arms, the lower left arm of the bridge plus the 0.1 and 1-ft decades as the rheostat arm, and the lower right arm as the

thermometer or the SPRT arm of the bridge.

This bridge has seven twelve-position and one

four-position measuring dial switches; the 0.00001,

0.0001, 0.001, 0.01, and O.l-O dials have positions - 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and X (10) and the 1 and

lO-H dials have positions 0, 1,2, 3, 4, 5, 6, 7, 8, 9,

X (10), and Y (11). The IGO-Q dial has the four

positions 0, 1, 2, and 3. Thus, resistances between 0

and 422.1111 Q can be measured with the bridge.

The 10 and lOO-O dial switches have mercury-wetted

contacts; the 0.00001, 0.0001, 0.001, and 0.01-dial

switches, which have "dry" wiping contacts, have

Waidner-Wolff resistance elements to reduce the

effect of variations in the switch contact resistance.

The 0.1 and 1-Q dial switches have dry wiping contacts which are in series with the relatively high

resistance of the ratio-arm resistors to minimize the

effect of variations in the contact resistance. The

113
RATIO ARM

ADJUSTMENT

BATTERY

4n worn jm J BRIDGE CURRENT

b'^^'^'V^'^^f'*'^^^ ADJUSTMENT

BRIDGE RATIO ARM

RESISTORS

TO CURRENT

REVERSING

SWITCH

ZERO

RATIO MEASURE

SWITCH

(IN MEASURE

POSITION)

MAIN DOUBLE

COMMUTATOR

(IN NORMAL

CONNECTION)

TO PLATINUM

RESISTANCE ^^'^ THERMOMETER

Figure H-1. Schematic circuitry of Mueller bridge employed in the Platinum Resistance Thermometry Laborcaory at the National

Bureau of Standards.

The resistors within the dashed lines are thermostated.

114
0.1 -H dial switch operates "subtractively". A

mercury-wetted, three-position switch permits the

selection of 500 or SOOO-H ratio arms or the "off'

position. In the "off" position both ratio arm resistors are disconnected from the lower sections of the

bridge. The operation of the bridge for adjustment of

the ratio arms, for determination of the bridge zero, or for measurement of a SPRT resistance is selected

by means of a three-position, ratio-zero-measure

switch, which is a dry wiping contact switch for the

older models of this Mueller bridge but a mercurywetted switch for the newer models. A mercurywetted switch (commutator switch) commutates the

SPRT connection and simultaneously interchanges

the ratio arms. The principal resistors of the bridge,

shown enclosed in the shaded portion of figure H-1

are thermostated at about 35 °C.

Terminals are provided for batteries and for a current reversing switch. A miUiammeter located in

the battery circuit indicates the current through the

SPRT (one-half of the bridge current). Variable

resistors are incorporated in the bridge for adjusting

the current. Terminals and variable resistors for

damping are provided for the null detector. Special

terminals and an adjustable resistor permit the

equalization of the potential lead resistances of the

SPRT.

2. Auxiliary Equipment for Calibrating Mueller

Bridges

To calibrate the Mueller bridge, the following

special equipment (other than null detector, current

reversing switch, and batteries that are necessary

in regular resistance measurements) is required:

(1) Adjustable decade resistance box that is suffi- ciently stable during the time required to intercompare two combinations of resistances (dial

settings) of the Mueller bridge. At NBS a second

Mueller bridge serves as this auxiliary adjustable

resistor. (To avoid confusion the auxiliary Mueller

bridge will, henceforth, be referred to as the

auxiliary adjustable resistor.)

(2) Standard resistor of 10 fl (preferably) or 100 Cl with a recent calibration to convert the

bridge resistance into terms of the national resist- ance unit. Duplicate resistors will help reduce the

calibration uncertainties.

(3) A thermometer and an insulated enclosure or thermostatted oil bath for the standard resistors.

3. Preliminary Preparation

3.1. Bridge Thermostat

The bridge resistors should have been thermostatted at a constant temperature at least one week prior to calibration. Safety provisions should

be made to avoid any large deviation from the

thermostatted temperature; e.g., auxiliary control

and battery power should be provided to maintain

the bridge temperature whenever the main power

line is disrupted. The bridge should be recalibrated periodically and whenever the temperature control fails.

3.2. Switches

The old mercury should be removed from the

mercury-wetted switches and the studs and links

amalgamated with clean mercury. The studs of

the dry wiping contact switches should be cleaned

and lubricated with pure petroleum. The spaces

between the wiping switch blades should be freed

of accumulated dirt.

3.3. Bridge Ratio Arms

Either the ratio arm resistors of 500 or 3000 il

may be selected; the equality of the ratio arms is adjusted separately by means of the slide-wire

resistors which have the resistance range of

about 0.02 percent of that of the ratio arm re- sistors. The "tap" (contact) on the slide wire

resistor is in the battery circuit so that any variations in the contact resistance has no effect on the

bridge balance. The choice of either 500- or 3000-11

ratio arms is based upon the resistance of the SPRT

to be determined, the voltage supply available, and

the insulation resistance between the voltage

supply and the bridge components. (One is particularly concerned with leakage between the

voltage supply and the null detector or thermometer.)

The contact and lead resistances associated with

the 0.1- and 1-fl dials is in series with the ratio arm

resistors. For a given variation in the contact

resistance of these switches, the uncertainty of a bridge reading due to these switches and their

leads is greater for the 500-0 ratio arm resistors

than for the 3000 Cl resistors. Problems with insulation leakage, on the other hand, increase with

both the value of the voltage supply and the resist- ance of the ratio arms. To adjust the ratio arms to equality, the threeposition, ratio-zero-measure dial switch is placed

in the "ratio" position. In this position, two auxihary resistors (each of about 98 fi) are each con- nected to the rheostat and SPRT arms of the bridge;

also, the 10- and 100-fi resistors are "by-passed."

The SPRT should be disconnected. The "tap" of

the slide-wire resistor is initially placed near the

midpoint. The bridge is balanced with the ratio

arms in one position and balanced again with the

ratio arms interchanged. The bridge dials are then

set on the average of the two balance readings and

the slide wire adjusted to balance the bridge. The

setting on the slide-wire resistor should be checked

by balancing the bridge again with the two positions of the ratio arms. The ratio arms have been

adjusted to equality when the two bridge readings

are the same. (All bridge balance readings, unless

stated otherwise, are performed with current

reversal.)

115
3.4. Bridge Zero

A residual resistance exists in the rheostat arm

even with all measuring switches set on zero be- cause of the resistances of the leads that connect

the resistor net work and the resistances of the

switch contacts. The Waidner-Wolff elements contribute also to the residual resistance. Therefore,

in the bridge design a resistor (about 0.98 fi) is placed in series with the 0.1 -O decade resistors to balance the rheostat and SPRT arms of the bridge

when all dials are set to zero. The remaining deviation from a perfect balance is the "bridge zero."

The bridge zero must be subtracted algebraically

from the dial readings of all measurements.

To determine the bridge zero, the bridge ratio arms are first equahzed (see sec. 3.3) and the threeposition, ratio-zero-measure switch is then placed on the "zero" position. The bridge zero is the

average of the bridge balance readings obtained

with the commutator switch in normal (N) and

reverse (R) positions. If the contact resistance of

the three-position selector switch varies between the

N and R readings, the bridge zero will be in error.

If the selector switch is considered unreliable,

which it may be if a dry wiping contact switch is used for a selector switch, the bridge zero is determined by connecting a special shorting link made

by soldering three copper wires (about AWG 18)

into the form of an "H." The "H" should be about

5 cm high and 10 cm wide. One short vertical wire

of the "H" is connected between the terminals

C and T; the other short vertical wire of the "H"

is connected between the terminals c and t. When

using this special shorting hnk to determine the

bridge zero the three-position selector switch should

be in the "measure" position. The bridge zero is the average of the bridge balance readings obtained

with the commutator in normal and reverse

positions.

3.5. Resistance of Leads Between the 1- and 0.1 -fl Decade Resistors and the Ratio Arm Switch

The resistors of the 0.1- and 1-fl step decades are enclosed, along with other principal resistors, in a thennostatted aluminum block. The resistances of

the leads from these resistors to the dial switches

must be sufficiently close so that their differences

have neghgible effect on the ratio of the ratio arms. The double commutator that interchanges the ratio

arm resistors does not interchange the resistances

of the leads from the resistors of the 0.1- and 1-fl step decades, the switch contact resistances of these

decades, nor the resistances of the leads from the

dial switches of these decades to the ratio arm

reversing switch (see bridge circuit diagram,

fig. 4-1).

The design of the bridge permits the separate

determination of the degree of equahty of the leads

from the 0.1- and l-fi decade resistors to their dial

switches and of the leads from these switches to the

ratio arm reversing switch. The "check" terminals

are employed to test the degree of equahty of the

resistor lead plus the switch contact resistances;

the resistances of the leads between the dial

switches and the ratio arm reversing switch is determined by measurements using another bridge.

By introducing branch points at locations a and c

of the ratio-arm reversing switch, the • degree of

equality of the combined leads may be determined

(see fig. H-2). The discussion to follow in this

section (3.5) wiU assume that a pair of wires from

points a and c are available at special terminals on

the bridge. If the special wiring is not available, an equivalent circuit may be achieved by removing

the double commutator and placing amalgamated

heavy copper hnks from a to b and c to d. Wires

are attached to the hnks close to the a and c studs

of the switch and are then connected to a doublepole, double-throw (DPDT) all copper knife switch

as shown schematically in figure H-2. Amalgamated

hnks for the SPRT and battery are also placed on

the double commutator studs corresponding to resistance measurement in the N connection (see

fig. H-1). If the special wiring and terminals are

available on the bridge, the double commutator is placed in the N position. The null detector is con- nected to the Mueller bridge through the DPDT

switch so that the switch can be connected to the

RATIO ARM

ADJUSTMENT

REVERSING

SWITCH

BATTERY 7^

Figure H-2. Schematic circuitry for testing the equality of the Mueller bridge leads between the ratio arm reversing switch

{part of the main commutator switch) and 0.1-Cl and

decades.

The contact resistances of the decade switches are included as part of the bridge leads.

116
leads from positions a and c of the ratio-arm re- versing switch or to the bridge galvanometer posts

(see figs. H-1 and H-2). The ratio-zero-measure

dial switch is placed on the "measure" position and

the auxiliary adjustable resistor is connected as a four-terminal resistor to the SPRT posts. (At NBS

the auxiliary Mueller bridge is prepared for this and

subsequent bridge calibration applications by shorting the "check" terminals and by placing its commutator in the normal position, the galvanometer damping resistors in the open positions, the

ratio-zero-measure switch in the "measure"

position, and the ratio arm selector switch in the

"off' position. Three heavy gauge copper wires are connected from the C and T posts and the shorted

"check" terminals of the auxiliary bridge to the

T, C, and c posts, respectively, of the bridge to be

calibrated. In the above circuit, when the dials of

the auxiliary Mueller bridge are set to zero, approximately equal resistances are introduced into the

lower two arms of the Mueller bridge to be calibrated; the O.l-fi decade operates subtractively

(see fig. H-1)).

Instead of testing all possible combinations of

the switch positions of the 0.1- and 1-fI decades,

the maximum difference between the lead re- sistances is determined by comparing the 0 step

of the 0.1-fl decade with cdl steps of the l-O decade

and 0 step of the 1-fl decade with all steps of the

O.l-H decade, totaling 22 comparisons. With 0.1-

and decade dials on 0 and the DPDT switch

thrown to the a-c position, the bridge is balanced

(reading 1) employing the lower dials and the

auxiliary adjustable resistor. The DPDT switch

is then thrown to the galvanometer post position

and the bridge is balanced (reading 2) using the

lower dials only. The difference in the balance

readings is, for all practical purposes, the difference

in the lead resistance. The process is repeated with

the l-fl decade on the 1 position. A sample of the

complete test data is summarized in table 1. (The

tabular values given are only for descriptive

purposes and do not represent any actual data.)

The maximum error in the bridge reading is, in

74 74

proportional parts, 590'^^^"^ 3000 ^ 10~^,

depending upon the choice of ratio arm resistors.

For this particular (fictitious) bridge the results

demonstrate that in the determination of high

resistances the 3000-0 ratio arms should be employed, also that significantly large error may occur

in the determination of small resistance differences

in which the 0.1- and 1-fl decade switches cU-e moved.

After determining the differences in the lead

resistances the auxiliary connections are removed

and the Mueller bridge is placed in the normal

operating condition.

Table 1. Comparison and determination of the maximum difference in the lead resistance between the 0.1- and 1-ohm decades and

the ratio arm reversing switch

1-fl decade with the 0.1-fl decade on 0 0.1-fl decade with the 1-fl decade on 0

Dial Reading 1 Reading 2 Difference Dial Reading 1 Reading 2 Difference

Position /Ltfl /ifl IjXI Position IjSI /xfl

0 142 136 + 6 0 237 247 -10

1 521 508 + 13 1 217 206 + 11 2 506 500 + 6 2 225 239 -14

3 517 506 + 11 3 273 259 + 24

4 511 501 + 10 4 280 260 + 20

5 496 502 -6 5 256 277 -21

6 499 507 -8 6 247 263 -16

7 501 498 -3 7 250 244 + 15 8 500 489 -11 8 312 326 -16

9 497 511 -14 9 327 320 + 7 X 504 488 -16 X 317 311 + 6

Y 502 517 -15 -1 222 234 -12

1 ohm decade 0.1 ohm decade

Maximum + 13 Minimum — 16

Difference 29

Maximum

Minimum

Difference

+ 24

-21

45

Sum of difference=74

74 74 Error= Bridge Reading X —X lO-^n or Bridge Reading X —— X 10-6 n

500 oUUU

117
3.6. Calibration of the Mueller Bridge in Terms of

the National Unit of Resistance

To calibrate the Mueller bridge in terms of the

national unit of resistance the X positior of the

10-fi decade and/or the X position of the 1-fl decade are compared with recently calibrated

lOO-fl and/or lO-fl standard resistors, respectively.

The choice of the comparison depends upon the

stability of the standard resistors and the bridge

decades that are most used in the measurements.

On the basis of absolute resistance, resistors of

higher resistance are usually less stable. In the

NBS Platinum Resistance Thermometry Laboratory the Mueller bridge is usually compared with

10-fl standard resistors. (The bridge resistors may

all be expressed in terms of, for example, the re- sistance of O.lx being exactly 1 bridge unit and, if the SPRT resistance ratio R{i)IR{0) is employed

in all temperature evaluations, the bridge unit is satisfactory. The resistance in terms of the national

unit is, however, necessary for some comparisons,

e.g., comparison of the measured R{0 °C) of the

SPRT with that obtained at the National Bureau of

Standards.)

TheX position of the 1-fi decade is compared with

a lO-Cl standard resistor by measuring the standard,

as a four-terminal resistor, directly with the Mueller

bridge. The following table 2 gives an example of

the results obtained. (One or two figures more than

are experimentally realized are carried in the

calculations for internal consistency.)

Table 2. Comparison of X position of dial with a 10-Q. standard resistor

(Standard resistor at 26.4 °C)

Ohms

Standard at 25 °C 10.00026,4

Temperature correction to 26.4 °C -0.0000047

Standard at 26.4 °C 10.0002567

Bridge reading, average of N and R

X.0010491

Bridge zero (subtract) 0.0001832

Bridge reading corrected for bridge

X. 0008659

Standard resistor minus bridge reading

corrected for bridge zero -0.0006092

Resistance of 1a- 10 - 0.0006092 0

4. Calibration of the Mueller Bridge Decade

Steps

The ratio arms of the Mueller bridge must be

adjusted to equality before calibrating the 0.1 or

1-fl decade steps. The double commutator is then

placed in the "normal" position and the auxiliary

adjustable resistor is connected as a four terminal

resistor to the SPRT posts. As previously stated,

all bridge balance readings are obtained with current

reversal.

In principle the resistance steps of the Mueller

bridge are calibrated internally by comparing each

step of one decade with the X position of the next

lower decade. The differences between the resistors can be obtained conveniently in terms of the lower

decades. If upon analyzing the calibration data

from the higher resistance decade down to the lower

resistance decades the latter decades should

show significant corrections, the differences ex- pressed in terms of them could then be corrected

and the calibration data reanalyzed (see step (18)).

The procedure used at the NBS Thermometry

Laboratory for calibrating the Mueller bridge is as

follows:

(1) Set to 1 on the 1-fl decade of the bridge and to 0 on the 0.1-fi decade and use the lower decades

to balance an appropriate resistance in the auxiliary

adjustable resistor. Galvanometer (detector) deflections are interpolated to obtain fractions of the

0.00001 -a step.

(2) Without disturbing the external circuit, change

the bridge setting to 0 on the l-fi decade and to X on the 0.1-fi decade and again use the lower

decades to balance the bridge. The dials of the

auxiliary adjustable resistor should be adjusted so

that the lower decades are near zero for the lesser

of (1) or (2) bridge balances.

(3) Repeat (2) and then (1). (4) Set to 2 on the 1-Cl decade of the bridge and to 0 on the 0.1-fi decade and balance with a corre- sponding resistance in the auxiliary adjustable

resistor, as in (1).

(5) Change the bridge setting to 1 on the 1-fl decade and to X on the 0.1-fl decade and proceed

as in (2) and (3). (6) Continue this process until each of the l-fl steps has been compared with the X position of the

0.1-fl decade.

(7) The observations under (1), (2), and (3) yield

the relation:

l, + ai = 0.1x + 6i, (1)

where the Arabic numeral index indicates reference

to the resistance change due to that individual

step in the decade, the Roman numeral index X

indicates reference to the total resistance change

due to changing the decade switch from position 0 to position X (or 10), and where ai and bi are the

corrected averages of the respective lower decade

readings, including galvanometer interpolations,

which were obtained in the balancings of the bridge.

The relation (1), when transferred to

li-0.1x=6i-a, = ci, (2)

118
indicates the difference in resistance between

step 1 of l-H decade and X of the 0.1-11 decade.

Equations similar to (2) are obtained from (4),

(5), and (6) leading to the remainder of the set:

1 — U.lx — C2

1 — U.lx — C3

1

,

U. 1 X — C4

15 -0.1x = C5

le -0.1x = C6

I7 -0.1x = C7

Is -0.1x = C8

I9 -0.1x = C9

lio — 0.1x = Cio

Table 3 summarizes the resuhs of such a comparison

between the l-fi decade and 0.1 x- In the twodigit notation of the dial pairs that are given in

column 1, the first digit indicates the dial position

of the decade being compared and the second digit

the dial position of the next lower decade. The

values listed in the second column correspond to the resistance differences, the c's, of eqs (2) through

(3).(8) Summing eqs (2) and (3), there is obtained

10 10 2li-10X0.1x = 2^" (4)

i=l i=l

but

10 2:i*=ix. (5)

i=l

The comparison of 1 x with a lO-fl standard resistor

showed that

lx= 10 - 0.0006092 n. (6)

(See table 2, sec. 3.6). Combining eqs (4), (5), and

(6),

Table 3. Calibration of /-fl decade by comparison with O.lx

Dial IJl d.1 Uiner- Corrected INearest

p3irs cncc CIyj. 11 fjt^L uO Z, U. 1 fill U.l fill A I ..0

10-OX -359 -593.6 -593.6 -594

20- IX -298 -532.6 -1126.2 -1126

30 -2X -408 -642.6 -1768.8 -1769

40 -3X -406 -640.6 - 2409.4 -2409

50 -4X -385 -619.6 -3029.0 -3029

60 -5X -368 -602.6 -3631.6 -3632

70-6X -376 -610.6 -4242.2 -4242

fin 7Y OZV.O — wOo

90 -8X -414 -648.6 -5511.4 -5511

X0-9X -346 -580.6 -6092.0 -6092

2 differences = — 3746

0.1*= (10 - 0.0006092- (-0.0003746))/10 = 1 -0.00002346 fi.

(11) The 0.1-fl decade steps are compared with

O.OIa" of the next lower dial following the procedures

given in (1), (2), (3), (4), (5), and (6). A set of relations given by (2) and (3) is then obtained. Table 4

shows the results of a comparison.

(12) A relation similar to (4) is given by

10 10 2 0.1 — 10X0.01;,= 2 Ci, (8)

i=l 1=1

where

10 £ c,= -0.0000318 n and

i=l

10 £ o.ii=o.iA'= 1 -0.00002346 n

i=i (see step (8) and table 3).

There results similar to relation (7)

0.1x=(l0 - 0.0006092-^ Ci)/10. (7)

^ 1=1 '

10 Fromtable3, ^ c, = - 0.0003746 and, therefore,

i=l 0.1;f= 1-0.00002346 0. The relation (7) is shown

evaluated at the bottom of table 3.

(9) Each step of column 2 (table 3) is corrected

for the real value of O.lx according to eqs (2) and

(3) in terms of the national resistance unit. Column

3 lists the corrected values.

(10) The stepwise cumulative corrections of the

dial positions are listed in column 4 and rounded

to the nearest 0.1 fiSl in column 5. The last entry

of columns 4 and 5 must agree with the value of

Ix obtained by comparison with the standard resistor. This serves as a check on arithmetic operations.

0.01a'= (1-0.00002346- (-0.0000318) )/10 = 0.1 + 0.00000083 0. (9)

(13) Each dial step of column 2 (table 4) is cor- rected for the real value of 0.01^- in terms of the

national resistance unit. Column 3 lists the cor- rected dial steps. Column 4 lists the stepwise

cumulative corrections and column 5 lists values of

column 4 rounded to the nearest 0.1 /ifi.

(14) The O.Ol-H decade steps are compared with

O.OOIa: of the next lower decade following the same

procedures given in (1), (2), (3), (4), (5) and (6). The

measurements are analyzed according to (7), (8),

(9), (10), (13) and (14). The results of such a comparison are summarized in table 5. The O.OOl-fl and

0.001 -fi decade steps are cahbrated in the same

manner; the results are given in tables 6 and 7,

respectively.

119
Table 4. Ca'ibration of 0.1 -D, decade by comparison with O.Olx

Dial Differ- Corrected Nearest

pairs ence 0.1 yll 2, 0.1 tiQ, 0.1 /Afi compared 0.1 IX.Q. 10-OX -51 -42.7 -42.7 -43

20-lX -07 + 1.3 -41.4 -41

30 -2X -17 -8.7 -50.1 -50

40 -3X -42 -33.7 -83.8 -84

50 -4X -59 -50.7 -134.5 -134

60 -5X -42 -33.7 -168.2 -168

70 -6X -28 - 19.7 — 187.9 — 188 80 -7X -34 -25.7 -213.6 -214

90 -8X -28 -19.7 -233.3 -233

X0-9X -10 -1.7 -235.0 -235

2 differences = — 318

0.01a-= (1-0.00002346- (-0.0000318) )/10 = 0.1 + 0.00000083 ft.

Table 5. Calibration of O.Ol-O, decade by comparison with

O.OOlx

Dial Differ- Corrected Nearest

pairs ence 0.1 fin X, 0.1 ^ft 0.1 jLift compared 0.1 fj.n 10-OX + 7 + 16.8 + 16.8 + 17 20-lX -19 -9.2 + 7.6 + 8

30 -2X + 13 + 22.8 + 30.4 + 30

40 -3X -19 -9.2 + 21.2 + 21 50 -4X -30 -20.2 + 1.0 + 1 60 -5X -4 + 5.8 + 6.8 + 7 70 -6X -5 + 4.8 + 11.6 + 12 80 -7X + 6 + 15.8 + 27.4 + 27 90 -8X -27 -17.2 + 10.2 + 10 X0-9X -12 -2.2 + 8.0 + 8

2 differences = -90

0.001a- = (0.1 + 0.00000083 -(- 0.0000090) )/10 = 0.01 + 0.00000098 ft.

Table 6. Calibration of 0.001-VL decade by comparison with

0.0001 X

Dial Differ- Corrected Nearest

pairs ence 0.1 /ift 2, 0.1 /Lift 0.1 /Lift compared 0.1 /Aft 10-OX + 2 -0.4 -0.4 0

20-lX + 4 + 1.6 + 1.2 + 1 30 -2X + 2 -0.4 + 0.8 + 1 40 -3X + 4 + 1.6 + 2.4 + 2 50- 4X + 1 -1.4 + 1.0 + 1 60 -5X + 4 + 1.6 + 2.6 + 3 70 -6X + 3 + 0.6 + 3.2 + 3 80 -7X + 4 + 1.6 + 4.8 + 5 90 -8X + 4 + 1.6 + 6.4 + 6

X0-9X + 6 + 3.6 + 10.0 + 10

2 differences = + 34

O.OOOlx = (0.01 + 0.00000098 - 0.0000034) /lO = 0.001-0.00000024 ft.

Table 7. Calibration of 0.0001 -CI decade by comparison

with O.OOOOlx

Dial Differ- Corrected Nearest

pairs ence 0.1 fiil 2, 0.1 iiift 0.1 /Lift compared 0.1 fiQ.

10— ox + 1 + 0.5 + 0.5 + 0

2U— IX +

1

— 1.5 — 1.0 — 1 30 — 2X — 1 — 1.5 — 2.5 — 2 40 —3X 0 — 0.5 — 3.0 — 3 50 — 4a + 2 + 1.5 — 1.5 — 2 60 -5X -1 -1.5 -3.0 -3

70- 6X + 1 + 0.5 -2.5 -2

80 -7X + 1 + 0.5 -2.0 -2

90 -8X 0 -0.5 -2.5 -2

YAAU — OVVA + 1 + 0.5 -2.0 — oz

2 differences = + 3

0.00001jf = (0.001 - 0.00000024 - 0.0000003)/10

0.0001-0.00000005 ft.

Table 8. Calibration of 0.00001 ft decade

Dial Differ- Corrected Nearest

pairs ence 0.1 /Lift 2, 0.1 /Lift 0.1 fdl

compared 0.1 fiil 10-OX + 2 + 1.6 + 1.6 + 2 20-lX + 1 + 0.6 + 2.2 + 2 30-2X 0 -0.4 + 1.8 + 2 40 -3X 0 -0.4 + 1.4 + 1 50 -4X 0 -0.4 + 1.0 + 1 60 -5X -2 -2.4 -1.4 -1

70 -6X 0 -0.4 -1.8 -2

80 -7X + 2 + 1.6 -0.2 0

90 -8X + 1 + 0.6 + 0.4 0

X0-9X 0 -0.4 0 0

2 differences == + 4

0.000001;f = (0.0001 - 0.00000005- 0.0000004)/10

0.00001 - 0.00000004 ft.

(15) The 0.00001-ft decade is calibrated by the

method of galvanometer deflection. Set the 0.00001-

n dial on zero with the bridge nearly balanced by

appropriately setting the auxiliary adjustable re- sistor. Step to position X and record the galvanometer deflection. Assume 1/10 of the deflection to correspond to O.OOOOOIa- w^hich will be adjusted to the true value in the manner as the previous X's were adjusted. Set the O.OOOOl-fi dial back to zero

position and step to the 1 position. Record the deflection difference with respect to the imaginary

O.OOOOOIa:. Balance and repeat with other steps of

the dial. The results of a sample data are summarized in table 8. The sum of the galvanometer

deflections corresponds to 0.0001-0.00000005 H

(see the bottom of table 7).

120
(16) The lO-fi decade is calibrated by comparison

with the X position of the l-Cl decade following the

procedures (1), (2), (3), (4), (5), (6), and (7). The results

of the comparison are summarized in column 2 of

table 9. The resistance of Ia- is 10-0.0006092 ohms

(see table 2). Column 3 (table 9) lists the dial steps

corrected to the national resistance units. Column 4

hsts the stepwise cumulative corrections.

(17) The 100-fl dial steps are cahbrated by comparison with lOx- Column 2 to table 10 summarizes

the resuhs. The resistance of 10^= 100-0.003717 fl (see table 9). The column 3 (table 10) lists the cor- rected dial steps and column 4 lists the stepwise

cumulative corrections.

(18) AU differences, i.e., c's of eq (2), in the dial

readings should be examined to determine whether

they are changed significantly when the dial readings

are corrected according to the cahbration. If the

correction is significant, then a second analysis

should be made employing the corrected differences.

(19) The decade steps of the sample data have

been compared with the X of the next lower decade

in terms of the lower decade steps (see column 2 of

tables 3 to 10). The decades from O.OOOOl-ft to l-O

are compared in terms of the O.OOCXIl-fi decade dial

positions; the lO-fl decade and the three 100-fl

steps are compared in terms of the 0.00001- and

0.0001 -n steps. The 0.001-il decade dial was used at one occasion with the lOO-fl dials (see table 10). In

as much as the lower dials have very small corrections, the bridge resistors including the auxiliary

adjustable resistor should be adjusted so that the

comparison measurements would involve changes in

dial positions of the less significant dials wherever

possible (see step (2)). By following this procedure

the calibration correction to the differences (c's of

eq (2)) can be made small or negligible and a second

analysis (step (18)) of the calibration data based on

the corrected values of c's may be avoided. In the

sample cahbration data the corrections to the lower

dials are in the range of interpolated values of the

least significant dial (0.00001 -O). The corrections to the differences (c's) are negligible.

(20) The results of calibration given in tables 3 to 10 are summarized in table 11 for rapid reference in

making bridge dial corrections. Corrections of less

than one ^t^l may be taken as zero without significantly increasing the uncertainty of the resistance

measurements.

(21) For further information reference may be

made to the following:

(a) Brooks, P. P. B., Cahbration procedures for

direct-current resistance apparatus, Nat. Bur.

Stand. (U.S.), Monogr. 39, 53 pages (March 1, 1962).

(b) Brooks, P. P. B., Calibration of Mueller Thermometer bridges, 39 pages (October 1965), from

the National Bureau of Standards List of Publications LP38 (August 1966), Calibration Procedures

of the NBS, Electrical Reference Standards

Section.

Table 9. Calibration of 10-il decade by comparison with Ix

Dial peiirs Difference Corrected

compared p.n,

10 -OX + 289 -320 -320

20 -IX + 227 -382 -702

30 -2X + 287 -322 -1024

40-3X + 015 -594 -1618

50 -4X + 167 -442 -2060

60 -5X -009 -618 -2678

70 -6X + 272 -337 -3015

80 -7X + 576 -033 -3048

90 -8X + 267 -342 -3390

X0-9X + 282 -327 -3717

1a-= 10 - 0.0006092 n.

Table 10. Calibration of 100-fl dial steps by comparison

with I Ox

Dial pairs Difference Corrected

compared 1x0.

10 -OX -945 -4662 -4662

20 -IX -1710 -5427 - 10089

30 -2X + 973 -2744 - 12833

10a = 100 - 0.003717 ft.

Table 11. Summary of Mueller bridge calibration

Bridge corrections in micro-ohms

Dial Bridge decades

posi- tion100 10 1 0.1 0.01

0.001 0.0001 0.00001

1 - 4662 - 320 - 59.4 - 4.3 + 1.7 0 0 + 0.2 2 - 10089 - 702 -112.6 - 4.1 + 0.8 + 0.1 -0.1 + 0.2 3 - 12833 -1024 -176.9 - 5.0 + 3.0 + 0.1 -0.2 + 0.2 4 -1618 -240.9 - 8.4 + 2.1 + 0.2 -0.3 + 0.1 5 -2060 -302.9 -13.4 + 0.1 + 0.1 -0.2 + 0.1 6 -2678 -363.2 -16.8 + 0.7 + 0.3 -0.3 -0.1

7 -3015 -424.2 -18.8 + 1.2 + 0.3 -0.2 -0.2

8 -3048 -486.3 -21.4 + 2.7 + 0.5 -0.2 0

9 -3390 -551.1 -23.3 + 1.0 + 0.6 -0.2 0

X -3717 -609.2 -23.5 + 0.8 + 1.0 -0.2 0

121
Appendix I. Calibration of the Linearity of a Potentiometer

The equipment required for calibrating the

linearity of a potentiometer is:

(a) A source of adjustable stable voltage, e.g., a second potentiometer or an adjustable well regulated electronic voltage supply.

(b) A detector of sufficient sensitivity. (For the

resolution of 1 fiCl at 1 ma in the SPRT a sensitivity

of one nanovolt is needed.)

The calibration procedure, analogous to that of

the Mueller bridge (see Appendix H), is the comparison of successive voltage steps of one decade

against the X position of the next lower decade.

The sensitivity of the detector should be sufficient

to permit interpolation of the last dial of the potentiometer. The current reversal procedure will

double the detector sensitivity and cancel any

steady extraneous emfs.

1. Calibration of the Decade Switches

Let j A, be the difference in nanovolts between the

ith step of decade j and the X position of j+l

(next lower) decade, i.e.

,

(1)

where V is the voltage associated with the dial position. (The decade j=l is the most significant decade.) The average difference between each of the

steps of decade j and X of decade 7 + 1 is:

1 10 ^^= E A= (jF^-10j+.Fv)/10. (2)

The difference between each observed jA, and jA is

corrections on "decade 1," the most significant dial

with steps nominally 0.01 V, of a six-dial potentiometer. The record of the differences between dial positions and X of the next lower decade is given in nanovolts, but the actual laboratory record is recorded as galvanometer deflections and the sensitivity.

The procedure for determining the linearity of

the lower decades is the same as that for decade

1, i.e., the dial positions of decade 2 are compared

with the X position of decade 3 and so forth. Tables

of differences jA;, differences jS, from the mean

jA, and the deviations from linearity jDn are constructed as shown in table 1 for decade 1. The linearity of the least significant decade is determined from galvanometer deflections. The

galvanometer deflection corresponding to each

step is compared with the average deflection for

the decade (see eq (3)). Since the least significant

decade can tolerate relatively large deviations in

the resistance elements, usually its linearity is found to be adequate for most measurements.

The above calibration data yield the deviation

from linearity of a dial position in terms of the aver- age step of the decade (see eq (5)). The overaU lin- earity of the measuring decades is determined in

terms of the linearity of the dial positions of a decade

and the relation of its X position with the average

step of the next higher decade (eq (2)), i.e., lA or

6.7 nV (see table 1 for the deviation of the average

step of decade 1 from X position of decade 2). The

X position of decade 2 is 6.7 nV smaller than the

average step of decade 1; each step or the average

step of decade 2 is, therefore, 6.7/10 nV smaller.

The total deviation from linearity of the nth step of

decade 2 is the sum of the deviations from linearity

within the decade (as determined from comparison

1

10

1=1

Table 1. Example of calibration of potentiometer dial I steps = nominally 0.01 V

= {iVi-jVi_,)-jVxllQ. (3)

Equation (3) shows that the calibration process re- duces to the comparison of each step of the decade

with the average step of the decade. The deviations

from linearity jD,, at the nth step of decade j is the

algebraic sum of the deviations from the average

difference of the 1st to nth steps inclusive, i.e..

(4)

^ i [j^i-^f.A^-jVn-niV,l\Q. (5)

i = l i=l

In eqs (4) and (5), jDio = 0. Table 1 shows an ex- ample of calibration data and calculations of linearity

Position i Si ,z>„ dial 1 nanovolts nanovolts nanovolts

1 21 14.3 14 2 33 26.3 41 3 -47 -53.7 -13

4 -52 -58.7 -72

5 34 27.3 -44

6 43 36.3 -8

7 62 55.3 47 8 -11 -17.7 29 9 27 20.3 50 X -43 -49.7 0

2 iA, = 67 nV.

(=1

,A = 6.7 nV.

122
of the 2d decade steps with the X position of the

3d decade) and the deviation between decades 1 and 2, i.e.,

2e« = 2/>« — niA/10 (6)

2en is the total deviation from linearity of the rath step of decade 2. The total deviation from linearity of the mth

step of decade 3 is:

3€m-3Z)»>-m[2A/10+,A/100]. (7)

The total deviation from linearity of the kth step

of decade 4 is:

4e/t = 4D, - ^[3 A/10 + 2 A/ 100 ^- 1 A/ 1000] . (8)

For the pth step of decade 5 the total deviation

from linearity is:

5gp =^Dp-p [4A/IO + 3A/ 100 + 2A/ 1000 + , A/ 10000]

.

(9)

The term containing 1 A/ 10000 or even 2A/IOOO may

be insignificant. The deviations from linearity of even lower decades are calculated as an extension

of the above.

2. Calibration of the Range Switches

In contrast to the Mueller bridge, potentiometers

have range switches with xl, xO.l, or even xO.Ol

factors. Any deviations from the nominal factors are determined by comparing the X position of decade

1 on the xO.l range and 1 position of decade 1 on the

xl range corrected for linearity. The xO.Ol range is compared in the same manner with xO.l range.

2.1. To Check the xO.l Range Factor

With the range switch set on x\ position, adjust

the external voltage source until it is balanced

against the step 1 of the first 0=1) decade. When

a balance is achieved, change the range switch to xO.\ position and the first decade to step X Since

the step 1 of the first decade is 1 ppm too high

(from the calibration given in table 1), the dial set- tings should be XOOOOl on the six-dial potentiometer

being employed in the example given here. This

gives an accurate 10 to 1 ratio for the dial settings.

If the potentiometer current with the range switch

set on JcO.l position is too high, then (using current

reversals) the galvanometer deflections will have

the same sense with the increase in the dial setting.

If the current is too low, then the galvanometer deflection will have the opposite sense with the in- crease^ in the dial setting. If, e.g., galvanometer

deflection is equivalent to + 10 nanovolts, the range

factor correction becomes + 1 X 10~^.

2.2. To check the xO.Ol Range Factor

The xO.Ol range is compared with the xO.l range

in the same manner as above between xO.l and xl

ranges. The range factor correction for the :c0.01

range relative to the xl range is then the combined

correction factors (very nearly the sum of the cor- rection factors) for the xO.l range and the xO.Ol

range relative to the xO. 1 range.

3. Combined Decade Switch and Range Switch

Corrections

The potentiometer voltage equation with calibration correction is:

V=F{1 + K) {Er + E2 + E3 + E, + E, + Ee

where

+ €1 + 63 + 63 + €4 + €5 + ee)

F = range factor xl, xO.l, or xO.Ol

K = range factor correction for jcO.l orxO.Ol ranges,

zero for xl range

Ei through nominal dial readings for the decades

ei through ee = total deviations from linearity of the

dial settings.

The above calibration procedure is suitable for a "simple" potentiometer with one working battery.

When the battery current is branched (spht-circuit),

for example in the Wenner voltage-difference potentiometer [1], the ratio of the currents in the two

branches must be precisely adjusted before comparing a decade in one branch with a decade in the

other and checked periodically during this intercomparison.

Instruments with more than one potentiometer in

tandem, for example the White potentiometer [4,

5, 6] and other more recent designs [3], require precise standardization of the currents when the decades between two potentiometers are compared.

For application of universal ratio sets to potentiometer cahbration see NBS Monograph 39 by Brooks [2].

References to Appendix I

[1] Behr, L. , The Wenner potentiometer, Rev. Sci. Instr. 3 , No. 3, 109-120(1932).

[2] Brooks, P. P. B., Calibration procedures for direct-current resistance apparatus, Nat. Bur. Stand. (U.S.), Monogr. 39,

53 pages (March 1, 1%2).

[3] Schaeffer, W. H., The six-dial thermofree potentiometer. Instruments and Control Systems 34, No. 2, 283-291

(February 1%1).

[4] White, W. P., Potentiometers for thermoelectric measure- ments especially in calorimetry, J. Am. Chem. Soc. 36,

1868-1885 (1914).

[5] White, W. P., Einige neue Doppelkompensatoren. I. Thermokraftfreie Kontakte, mechanische Anordnungen und Zubehorteile, Zeit. Instrumentenk. 34, 71-82 (1914).

[6] White, W. P., Einige neue Doppelkompensatoren. II. Die

elektrische Einrichtung, Zeit. Instrumentenk. 34, 107-

113, 142-151 (1914).

123
Appendix J. Guide for Obtaining the Calibration of Standard Platinum Resistance Thermometers at the National Bureau of Standards

The following guides are intended for users and

those immediately connected with the application

of SPRT's who are interested in obtaining calibration services at the NBS or who are planning to purchase SPRT's.

The SPRT calibration equipment and procedures

employed by NBS are for four-terminal thermometers that meet the IPTS-68 specification of quality

(see sec. 2) and that are constructed within certain

external physical dimensions and immersion requirements. To obtain caUbration services within the regular time schedule, the SPRT's must conform to these specifications. Any deviations from these specifications will result in additional cost, time, or re- jection from calibration, e.g., the calibration of an

over-sized SPRT is prohibitively expensive.

1. Inquiries and Purchase Orders for the SPRT

Calibration Service

1.1. Inquiries and purchase orders shall be di- rected to the attention of:

Temperature Section, Heat Division

B04 Physics Building

National Bureau of Standards

Washington, D.C. 20234

1.2. Attention is called to paragraphs (d), (e), and

(f) of Section 200.100 in NBS Special Publication

250, Calibration and Test Services of the National

Bureau of Standards (1970 edition) which deal with

shipping, insurance, and risk of loss or damage of

instruments sent to NBS for calibration and test. These three paragraphs are quoted below for con- venience of reference.

(d) Shipping and insurance coverage instructions should be

clearly and legibly shown on the purchase order for the calibration or test. The customer must pay shipping charges to and from NBS;

shipments from NBS wiD be made collect. The method of return

transportation should be stated and it is recommended that return shipments be insured, since NBS will not assume liability for their loss or dameige. For long-distance shipping it is found

that air express and air freight provide an advantage in reduction

of time in transit. If return shipment by parcel post is requested

or is a suitable mode of transportation, shipments will be prepaid

by NBS but without covering insurance. When no shipping or insurance instructions are furnished, return shipment will be

made by common carrier collect and uninsured.

(e) NBS will not be responsible for the risk of loss or damage

to any item during shipment to or from the Bureau. Any arrange- ments for insurance covering this risk must be made by the customer. Return shipment will be made by the Bureau as indi- cated in paragraph (d) of this Section. The purchase order should

always show the value of the equipment and if transit insurance is carried by the customer this fact should be stated.

(f) The risk of loss or damage in handling or testing of any

item by NBS must be assumed by the customer, except when it is determined by the Bureau that such loss or damage was

occasioned solely by the negligence of bureau personnel.

1.3. The user of the SPRT shall assume the re- sponsibility to provide the following information

either directly or through his procurement group.

If any one item of the requested information is lacking, the caUbration will be delayed. To minimize the "paper work," all necessary information

should be forwarded in a single communication.

1.3.1. Manufacturer, model number, and serial

number of the SPRT.

1.3.2. The temperature range in. which the

SPRT will be employed.

1.3.3. Specific calibration desired. (See under

section 4 the list of test services that are available.)

1.3.4. Specify the type and quantity of thermometer calibration tables desired. (See under

section 4.3 the list of tables that are available.)

1.3.5. If more than one type of calibration service, e.g., the calibration of standard resistors is also

being purchased from NBS on the same purchase

order, enough copies of purchase orders should

be prepared so that one copy can accompany

each type of instrument to be tested.

1.3.6. If insurance of the return shipment is desired it must be requested. Such insurance is recommended.

2. Shipment ofSPR T to NBSfor the Calibration

Service

2.1. Direct shipments of SPRT by common

carrier to the attention of:

Temperature Section, Heat Division

B04 Physics Building

National Bureau of Standards

Route 70S and Quince Orchard Road

Gaithersburg, Maryland 20760

2.2. Precautions should be exercised to protect

the SPRT from mechanical shock that would alter

its calibration. In shipment, the SPRT must be

softly supported within a rigid case but must not be

free to "rattle" within the case. The packing material

must, therefore, be resilient and not become compacted. An outer case must be sufficiently rigid and

strong to withstand the treatment usually given by

shippers. Styrofoam is not sufficiently rigid to be used

as an outer case. The SPRT's will not be returned in

containers that are obviously unsuitable, such as those closed by nailing; a wooden box should be

closed with screws. Suitable containers will be

provided, for a fee, whenever the SPRT shipping

container is not satisfactory for re-use. The user shall assume the responsibility of instructing his shipping

department concerning packaging and the desired

carrierfor shipping his SPRT.

Two methods suitable for packaging the SPRT

for shipment are illustrated in figures 4 and 5 of

section 4.7.

124
3. Requirements to Qualifyfor Regular Calibration

Schedule

3.1. Quality:

The SPRT must be a four-terminal resistor that

conforms with the IPTS-68 specifications for construction practice and quality (see Appendix A).

3.2. Dimensions:

3.2.1. Long stem type SPRT

3.2.1.1. The maximum length of the resistance element shall be 5 cm. The entire element

must be located near one end of the SPRT.

3.2.1.2. The maximum outside dimension of

the SPRT sheath shaU be 7.52 mm (0.296") for a minimum length of 46 cm (18").

3.2.1.3. The maximum overall length of

glass-sheathed SPRT's shall be 1 meter. The maximum overall length of SPRT's that have sheaths of

thin metal is limited by the height to the ceihng of

the NBS calibration laboratory; SPRT's up to 4

meters can be accommodated.

3.2.1.4. The suitabiHty of the SPRT for

calibration at NBS also depends upon the immersion

characteristics of the thermometer. Although the

thermometer may meet the preceding requirements

of dimension, the immersion characteristics of the

thermometer may be too poor for calibrations that

employ the existing equipment at NBS.

3.2.2. Capsule type SPRT

3.2.2.1. .The maximum length of the resist- ance element shall be 5 cm. The element must be

located at one end of the SPRT.

3.2.2.2. The maximum over-all length of the

SPRT shall be 8 cm.

3.2.2.3. The sheath, at the end that contains

the resistant element, shall be reasonably uniform

and shall not exceed 5.72 mm (0.225") outside

diameter for a minimum length of 5 cm and maximum

length of 6 cm.

3.2.2.4. The outside diameter of any part of

the SPRT, including the head v^rhere the four leads

emerge through a hermetic seal, shall not exceed

9.6 mm (0.38"); the length of the enlarged section

shall not exceed 2 cm.

4. Calibration Services

4.1. Long stem SPRT

4.1.1. Calibration at the triple point of water, tin

point, and zinc point. A table will be furnished with

entries at 1 -degree intervals between —50 and

H 500 °C.

4.1.2. Calibration at the triple point of water,

tin point, zinc point, and oxygen normal boiling

point. A table will be furnished with entries at 1-degree intervals between —183 and +500 °C.

4.1.3. Thermometers which will withstand

temperatures above 500 °C, such as quartz sheathed

thermometers, calibrated under 4.1.1 or 4.1.2 will

be furnished tables to 631 °C.

4.2. Capsule type SPRT

The capsule type SPRT's are not generally

calibrated above the tin-point temperature at the

NBS. Because of the uncertain electrical insulation

characteristics between the leads at the zinc point,

any calibration performed with the capsule type

SPRT's at the zinc point wiU not be of as high a

quality as is obtained with the long stem SPRT's.

Instead, the thermometer constant B is assumed

on the basis of the values of B for other SPRT's

calibrated at NBS during the past few years.

4.2.1. Calibration at the triple point of water

and tin point. A table will be furnished with entries

at 1 -degree intervals between — 50 and + 250 °C

.

4.2.2. Calibration at the triple point of water,

tin point, and oxygen normal boihng point. A table

will be furnished with entries at 1-degree intervals

between -183 and +250 °C.

4.2.3. Calibration at the triple point of water,

the tin point and a comparison calibration between

13 and 90 K. A table will be furnished with entries

at 0.1 -degree intervals between 13 and 90 K and at 1-degree intervals between 90 and 600 K.

4.3. Tables

The user has the choice, as a part of the calibration service, as to the form of table for his thermometer.

4.3.1. The argument (the temperature) can be

°C,°F,orK.

4.3.2. The function can be the resistance ratio

R{t)IR{0), the resistance Rit), or the resistance

difference R{t) — R{0) . For most applications the

resistance ration R{t)IR{0) versus temperature is recommended.

4.3.3. If no specific request is made, the table

furnished for the long stem SPRT will be the resist- ance ratio Rit) IR{0) versus temperature in °C. The

table furnished for the capsule type SPRT will be

the resistance versus temperature in K whenever

the comparison calibration between 13 and 90 K is requested; otherwise the table furnished will be

the resistance ratio R{t) IR{0) versus temperature

in °C.

5. Non-Standard Platinum Resistance

Thermometers

A very special configuration of platinum resistance

thermometer may be desired for certain applications. For the cahbration of such thermometers

the user may contact the NBS (see 1.1.) concerning

the possibility of a calibration at NBS or advice on

the calibration of the thermometer in the user's

laboratory employing a SPRT that has been calibrated at the NBS. The information on dimensions

and wiring configuration should be furnished with

the inquiry. If the calibration can be performed at NBS, the fee for the special cahbration will be

furnished.

Calorimetric type platinum resistance thermometers, which have the resistance element wound on

a flat mica sheet and sandwiched between two

I 125
mica sheets and then inserted into a flattened tube,

do not meet the specifications of the IPTS-68. A

caHbration of these thermometers may be furnished

by NBS through special arrangement.

Inquiries concerning the calibration of other resist- ance thermometers, such as germanium resistance

thermometers, are welcomed. The NBS will at- tempt, whenever possible, to meet reasonable

calibration requests of services; the cost of such

services will be negotiated. Appropriate discussion

should take place directly between the user and the

NBS technical staff.

126
FORM NBS-114A (1-71)

U.S. DEPT. OF COMM. 1 . PUBLICATION OR REPORT NO. 2. Gov't Accession

BIBLIOGRAPHIC DATA NBS-MN-126

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Platinum Resistance Thermometry

5. Publication Date April 1973

6. Performing Organization Code

7. AUTHOR(S)

John T,. Riddle, freorge T. Furukawa and Harmon H. Pliimh

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as 9.

13. Type of Report & Period

Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The monograph describes the methods and eq_uipment employed at the National Bureau of

Standards for calibrating standard platinum resistance thermometers (SPRT) on the

International Practical Temperatiire Scale (lPTS-68) . The official text of the scale

is clarified and characteristics of the scale are described. Several designs of

SPRT's are shown and discussed in the light of the requirements and recommendations

on the text of the IPTS-68. Possible sources of error such as those due to the internal and external self-heating effects and the immersion characteristics of SPRT's

are described in detail. Precautions and limitations for the mechanical and thermal

treatment and for the shipment of SPRT's are indicated, and a guide is given for those

desiring the thermometer calibration services of NBS . The description of equipment

employed at the National Bureau of Standards for maintaining the IPTS-68 includes the

triple point of water cell, tin point cell, zinc point cell, oxygen normal boiling

point comparison cryostat, the 13 to 90 K comparison cryostat , and the reference

SPRT's upon which the NBS-IPTS-68 in the region 13 to 90 K is based. Methods are given for calculating temperatures from the calibration data and observed resistances;

the propagation of calibration errors is discussed. Supplemental information given in

the Appendices includes the authorized English version of the text of the IPTS-68,

tabular values of the "reference function" used below 0 °C, tublar values of the

differences between IPTS-68 and IPTS-U8, analysis of the first derivatives at 0 °C of

the IPTS-68 formulations, methods for calibrating potentiometers and Mueller bridges,

and the derivation of the coefficients used in the analysis of error propagation.

17. KEY WORDS (Alphabetical order, separated by semicolons) Calibration; calibration errors ; cryostat;

fixed points; freezing point; International Practical Temperature Scale; platinum

resistance thermometer; Thermodynamic Temperature Scale; thermometry; triple point.

18. AVAILABILITY STATEMENT

PTI UNLIMITED.

1 1 FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF PAGES

129

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

22. Price

$2.10 domestic

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