2.3.1.  Interpolating equations prescribed in the ITS-90 text

For describing the temperature dependence of the resistance of an SPRT, it is necessary to interpolate between the deviations $\Delta W=W_{T_{90}}-W_{\text{r}}\left(T_{90}\right)$ obtained at the temperature fixed points. ITS-90 specifies eight different interpolating functions ($\left(W-W_{\text{r}}\right)$ in dependence of $W$) to be used over the eleven SPRT sub-ranges. These are listed in Table 2, along with the fixed points used to calibrate the SPRT. Most are simple polynomials in $\left(W-1\right)$ and/or $\ln \left(W\right)$.

Table 2.  The sub-ranges, deviation functions and calibration points for SPRTs used to define ITS-90.

SPRT Sub-range	Deviation functiona	Fixed points
$\mathrm{13.803 3}\mathrm{K}$ to $273.16\mathrm{K}$	$a\left(W-1\right)+b{\left(W-1\right)}^2+\sum _{i=1}^5{c}_i{\left[\ln \left(W\right)\right]}^{2+i}$	e-H2, Ne, O2, Ar, Hgb
$\mathrm{24.556 1}\mathrm{K}$ to $273.16\mathrm{K}$	$a\left(W-1\right)+b{\left(W-1\right)}^2+\sum _{i=1}^3{c}_i{\left[\ln \left(W\right)\right]}^i$	e-H2, Ne, O2, Ar, Hg
$\mathrm{54.358 4}\mathrm{K}$ to $273.16\mathrm{K}$	$a\left(W-1\right)+b{\left(W-1\right)}^2+c{\left[\ln \left(W\right)\right]}^2$	O2, Ar, Hg
$\mathrm{83.805 8}\mathrm{K}$ to $273.16\mathrm{K}$	$a\left(W-1\right)+b\left(W-1\right)\ln \left(W\right)$	Ar, Hg
$-\mathrm{38.834 4}\mathrm{°C}$ to $\mathrm{29.764 6}\mathrm{°C}$	$a\left(W-1\right)+b{\left(W-1\right)}^2$	Hg, Ga
$0\mathrm{°C}$ to $\mathrm{29.764 6}\mathrm{°C}$	$a\left(W-1\right)$	Ga
$0\mathrm{°C}$ to $\mathrm{156.598 5}\mathrm{°C}$	$a\left(W-1\right)$	In
$0\mathrm{°C}$ to $231.928\mathrm{°C}$	$a\left(W-1\right)+b{\left(W-1\right)}^2$	In, Sn
$0\mathrm{°C}$ to $419.527\mathrm{°C}$	$a\left(W-1\right)+b{\left(W-1\right)}^2$	Sn, Zn
$0\mathrm{°C}$ to $660.323\mathrm{°C}$	$a\left(W-1\right)+b{\left(W-1\right)}^2+c{\left(W-1\right)}^3$	Sn, Zn, Al
$0\mathrm{°C}$ to $961.78\mathrm{°C}$	$a\left(W-1\right)+b{\left(W-1\right)}^2+c{\left(W-1\right)}^3$ $+d\left[W-W{\left(660.323\mathrm{°C}\right)}^2\right]$	Sn, Zn, Al, Ag
a  the constants $a$,$b$, etc., take different numerical values in each sub-range, except that for thesub-range $0\mathrm{°C}$ to $961.78\mathrm{°C}$ the coefficients $a$, $b$, $c$ are the same as used for the sub-range $0\mathrm{°C}$ to $660.323\mathrm{°C}$. The coefficient $d$ is determined from the measurement at the silver point. b  For the sub-range $\mathrm{13.803 3}\mathrm{K}$ to $273.16\mathrm{K}$, two of the calibration points are required to be inthe vicinity of $17\mathrm{K}$ and $20.3\mathrm{K}$ and are determined using a hydrogen vapour-pressure thermometer or an interpolating helium constant-volume gas thermometer (see Chapter 3 Vapour Pressure Scales and Pressure Measurements and Chapter 4 Gas Thermometry for details).

2.3.2.  Alternative interpolating equations for special applications

The ITS-90 interpolating equations (Table 2) can be written in many mathematically equivalent forms. Amongst the alternatives is a form, similar to Lagrange interpolation, that is useful for propagation-of-uncertainty calculations (Section 6.2), investigating the consistency of calibration data (Section 4.4.2), and understanding Type 1 non-uniqueness (Section 6.3.1) [White and Saunders 2007, White and Strouse 2009, White 2013]. It should be emphasised that the alternative equations are simply rearrangements of those prescribed in the text of the ITS-90; they give identical values in interpolations. Their solution is single-valued because the number of coefficients is equal to the number of equations. People being only interested in the calibration of SPRTs according to the ITS-90 can skip this alternative mathematics without any loss of information.

For the example treated in Section 2.1, the sub-range from the TPW to the freezing point of aluminium, the alternative interpolating equation is a sum of four functions each multiplied by one of the ITS-90 reference resistance ratios:

where the four interpolating functions containing ratio differences both in the numerator and denominator are:

Although these equations are not as compact as Equation (5), the interpolation is now expressed in terms of the interpolating functions Equation (12), which are actually the sensitivity coefficients for uncertainties in the fixed points (see Section 6.2). The functions have properties similar to a set of orthonormal basis functions, which simplify the manipulation of mathematical expressions for uncertainty. Note too that the interpolation Equation (11) now maps directly to $W_{\text{r}}$ rather than the deviations $W-W_{\text{r}}$, and Equation (12) are rational functions of the various $W$ values only (i.e., there are no $W_{\text{r}}$ values). All of the ITS-90 equations can be expressed in a similar form, see Appendix 1 and for the most commonly used sub-ranges of the ITS-90, Table 4 in Section 6.2.

The fact that the alternative interpolation equations are in reality equal to those prescribed in the text of the ITS-90, i.e. that they contain the same terms, which are only arranged differently, can be illustrated by the following equations for the sub-range from the TPW to the melting point of gallium:

An alternative interpolation equation is

with

Insertion of Equation (14) in Equation (13) yields with $W_{\text{H2O}}=1$:

By comparing Equation (15) with the rearranged interpolation equation of the ITS-90, $W_{\text{r}}=a+W\left(1-a\right)$, cf. Table 2, one obtains for the coefficient $a$ the known expression

The general form for all of the alternative interpolation equations is similar to Equation (11) [White and Saunders 2007, White and Strouse 2009, White 2013]:

where $N$ is the number of fixed points, $W_i$ are the resistance ratios determined at the fixed points, $W_{\text{r,i}}$ the corresponding reference resistance ratios, and $W_1=W_{\text{r,1}}=1$ are the resistance-ratio values at the TPW. Alternative interpolating functions for each of the ITS-90 sub-ranges are listed in Appendix 1, and for the most commonly used sub-ranges of the ITS-90 in Subsection 6.2, Table 4.

All of the interpolating functions satisfy the orthonormality property

That is, they take the value one for the fixed point after which they are named and the value zero at all of the other fixed points in the interpolation. (This property is easily verified for the functions in Equations Equation (12).)

If any function $g\left(W\right)$ can be interpolated exactly by the interpolating equations, then the function can be generated from samples $g_{W_i}$ at the fixed points using

and for all of the ITS-90 interpolations, this relation leads to the two identities

and

which are also useful for simplifying some expressions. The differences between the general ITS-90 interpolation Equation (17) and the two identities Equation (20) and Equation (21) generate two other useful forms for the ITS-90 interpolating equations:

which leads to simpler expressions for Type 1 non-uniqueness [White and Strouse 2009], and

which is an alternative way of writing the ITS-90 interpolations in terms of the deviations.

3.2.1.  Capsule-type standard platinum resistance thermometer

Capsule-type standard platinum resistance thermometers (CSPRTs) are typically used between $13.8\mathrm{K}$ and $30\mathrm{°C}$, but sometimes as high as $156\mathrm{°C}$, and very occasionally to $232\mathrm{°C}$. A schematic diagram of a typical design of a $25\mathrm{\Omega }$ CSPRT is shown in Figure 1. The platinum sensor is mounted on an insulating former and inserted into the sheath, which is a platinum or glass tube about $5\mathrm{mm}$ in diameter with a closed end. The four short ($30\mathrm{mm}$ to $50\mathrm{mm}$) platinum lead wires emerge through a glass seal at the open end of the sheath. Electrical connections can be made to the leads with ordinary soldering techniques; however, care must be taken to avoid straining the leads where they emerge from the glass seal as they are prone to breaking there. A common solution is to tie the (insulated) leads together above the top of the capsule and, if need be, ply them back from there. The capsule is filled with helium gas, usually at a pressure of about $30\mathrm{kPa}$ at room temperature, to ensure good thermal coupling between the wire and the sheath.

For calibration at the fixed points of mercury, water and gallium, the capsule may be housed in an adaptor made of a small copper sleeve to fill the gap between the capsule and the inner wall of the cell. The sleeve is attached to a stainless-steel capillary which leads up to room temperature. Two thermal shunts between the capillary and the cell, one close to the capsule and one near the top of the cell, reduce heat flow along the leads. Such an adaptor eliminates the problems associated with simply suspending the capsule from its wires.

The low temperature limit for the use of CSPRTs has been set at $13.8\mathrm{K}$ because at lower temperatures their resistance and sensitivity become inconveniently small. Also, the low-temperature characteristics become increasingly dependent on non-thermal resistance effects due to strain, lattice defects, and impurities. To compensate for the loss of sensitivity, measuring currents below $24.5\mathrm{K}$ are generally increased from the $1\mathrm{mA}$ usually used above that temperature, to about $5\mathrm{mA}$. At this level, because of the low values of resistance in this range (down to about $30\text{ m}\mathrm{\Omega }$), the self-heating should not exceed $0.2\mathrm{mK}$, and the minimum sensitivity is $\sim 30\mathrm{\mu V}/\mathrm{K}$.

The use of CSPRTs above $30\mathrm{°C}$ is limited by electrical leakage in the glass seal, particularly if the surface is contaminated, e.g. by solder flux, and the diffusion of helium through the glass. Calibrations up to $505\mathrm{K}$ (the freezing point of tin) are possible, though the results at this temperature are not likely to be of the best quality and the calibration at low temperatures may be adversely affected. Because of these limitations, CSPRTs cannot be annealed, so that any change in resistance due to mechanical shock or other influences effectively causes a permanent shift in the CSPRT resistance. Regular checks should be carried out to look for such shifts, either by measuring the ‘residual resistance’ in liquid helium, where the temperature coefficient of the CSPRT is very small, or the resistance at the triple point of water, which can be measured easily and accurately. The residual resistance measurement is often more effective for detecting small changes, however, as part of a cryogenic run, but care must be taken because the temperature dependence of the resistivity near $4\mathrm{K}$ is not zero [Tew et al. 2013]. If a significant change is found in the residual resistance it cannot be reversed. It may, as a first approximation, be satisfactory to correct for it by subtracting this resistance change from all measured values, but more usually a new calibration is indicated.

CSPRTs are usually used “totally immersed”, meaning that the entire capsule is immersed in the medium of interest; preferably inserted in a well in a copper block in the cryostat. Because thermal conductivity and electrical conductivity are closely related phenomena, the thermal conductivity of thermometer leads is relatively high at low temperatures. To avoid heat leaks through the leads, connections are usually made via long fine copper wires thermally anchored to the block, or another object at a similar temperature, to reduce or eliminate heat flow into or out of the thermometer. This is particularly important where the wires are in the cryostat vacuum space, with little heat exchange along their lengths. When the anchoring is done correctly, the effect of heat leaks can be reduced to well below $0.1\mathrm{mK}$ [Hust 1970, Kemp et al. 1976, Gaiser and Fellmuth 2013a]. Heat leaks also influence measurements in higher-temperature applications where CSPRTs are used at the fixed points of mercury, water and gallium. Immersion in air within fixed-point cells provides insufficient thermal coupling so close-fitting sleeves, a suitable oil, or an adaptor as discussed earlier, are needed.

3.2.2.  Long-stem standard platinum resistance thermometer

Long-stem standard platinum resistance thermometers (LSPRTs) are applied in the temperature range between the triple point of argon ($-\mathrm{189.344 2}\mathrm{°C}$) and the freezing point of aluminium ($660.323\mathrm{°C}$). A typical design is shown in Figure 2. The sensor element is platinum wire of $0.05\mathrm{mm}$ to $0.1\mathrm{mm}$ diameter, which may be wound onto the former in a variety of ways. The wound element is placed into a fused-silica tube with a length of $480\mathrm{mm}$ to $650\mathrm{mm}$ and outside diameter up to about $8\mathrm{mm}$. The tube is sandblasted or blackened along part of the lower length to reduce radiative heat transfer (light-piping) between the element and the head at room temperature. Some metal-sheathed SPRTs are also made (see comments in Sec 5.1.1). The former for the sensing element is usually made of fused silica, occasionally Vycor or ceramic, and in older thermometers, mica. The length of the element is typically $35\mathrm{mm}$ to $50\mathrm{mm}$, and it has a nominal resistance at the temperature of the TPW of $25\mathrm{\Omega }$, giving the SPRT a sensitivity of $0.1\mathrm{\Omega }/\mathrm{°C}$.

The upper temperature limit for LSPRTs having mica formers or mica insulation is $500\mathrm{°C}$ due to the release of water of crystallisation from the mica at higher temperatures. Once driven from the mica, the water condenses at temperatures near $0\mathrm{°C}$ causing electrical leakage and measurement errors. Above $660\mathrm{°C}$, LSPRTs with silica insulators exhibit an exponentially increasing electrical breakdown of the insulation that can lead to errors of a few tens of millikelvin at $960\mathrm{°C}$ (see Section 3.2.3). Mechanical problems can also arise due to the large difference between the thermal-expansion coefficients of platinum and fused silica, leading to mechanical deformation and short circuits either in the sensor element or between lead wires.

Figure 3 shows two of the most frequently used designs of the resistance element of LSPRTs. Because the plasticity of platinum increases sharply at temperatures above $200\mathrm{°C}$, the sensor element should have a design that prevents the coil from sliding along the former and causing short-circuiting of the wire turns. Four platinum leads with a diameter of $0.2\mathrm{mm}$ to $0.4\mathrm{mm}$ connect the resistance element to the external copper cable and connections. The lead wires are usually kept apart by silica (or mica) discs. In some designs, the leads may be placed into separate fused silica capillaries to prevent short-circuiting. The silica or mica discs also reduce convection within the thermometer.

To ensure the thermometer sheath is hermetically sealed at the head where the platinum wires exit the assembly, ‘graded’ seals are often used. The seals are designed to bond to the platinum wires to form an air tight seal, and are formed from layers of glass with graded linear coefficients of thermal expansion to match that of platinum where the seal joins the wire and match the fused silica where it joins the sheath. The platinum leads are connected in the head of the thermometer to 4-wire multi-strand copper cable to allow connection to the measuring instrument. The leads may terminate as plain wires, spades, banana plugs or BNC connectors. Sometimes a cable screen is used to connect the sensor or the thermometer head to a ground at the resistance bridge.

The thermometer sheath in an LSPRT is usually filled with a mixture of inert gas plus oxygen at a partial pressure of about $2\mathrm{kPa}$. The total pressure is equivalent to one atmosphere at the maximum temperature of the thermometer’s application range. The low level of oxygen is sufficient to oxidise metallic contaminants and hold them at grain boundaries: they might otherwise migrate into the platinum lattice and affect the temperature dependence of the resistivity. At the same time, it limits the formation of an oxide film on the surface of the platinum wire (see Section 5.1.1).

3.2.3.  High-temperature standard platinum resistance thermometer

High-temperature standard platinum resistance thermometers (HTSPRTs) are designed for use in the temperature range up to the freezing point of silver ($961.78\mathrm{°C}$). Their design is very similar to that of LSPRTs, except that they use lower resistance sensors (with larger gauge platinum wire) in order to reduce the effects of electrical leakage at temperatures above the freezing point of aluminium. Figure 3 shows also one design of HTSPRT sensing elements.

At temperatures approaching the silver point, the electrical conductivity of the fused silica formers and discs, which make up the insulating components of thermometer, becomes appreciable. The effects shunt the sensing element leading to errors of several tens of millikelvin. By reducing the resistance $R\left(0.01\mathrm{°C}\right)$ from the $25\mathrm{\Omega }$ value used in LSPRTs to $0.25\mathrm{\Omega }$ the influence of the unwanted shuntingresistance is reduced by a factor of 100. In practice $R\left(0.01\mathrm{°C}\right)$ values ranging from $0.2\mathrm{\Omega }$ to $2.5\mathrm{\Omega }$ are used. A measuring current in the range $5\mathrm{mA}$ to $10\mathrm{mA}$ is typical for HTSPRTs, which partially compensates for the low resistance and loss of voltage sensitivity.

The shorter length of larger diameter wire ($0.3\mathrm{mm}$ to $0.5\mathrm{mm}$) used in the low-resistance sensors simplifies the construction and has other beneficial effects. Firstly, it improves structural stability of the element against the large difference between the thermal expansion of silica and platinum. The differential thermal expansion also means that about $1\mathrm{cm}$ of space must be left at the bottom of the sheath, when the thermometer is cold, to allow the platinum lead wires to expand. A second advantage of the thicker wire is that it reduces the surface/volume ratio of the wire, which slows the impact of contaminants, making the thermometer more stable, and reduces the effects of oxidation at the surface of the wire.

At the higher temperatures, metals and other potential contaminants of platinum become increasingly mobile and volatile, with some contaminants able to diffuse through the silica sheath. Cleanliness of the sheath and preventing metallic contamination of the sensor element is critical for long-term reliability of HTSPRTs.

4.4.1.  Calibration Procedures

For the lowest uncertainties, SPRTs should always be calibrated in their most reproducible state; that is, the resistance should correspond to the zero-current resistance when the SPRT is in an unoxidised state, and with an equilibrium concentration of vacancies. While this state is generally not entirely accessible to CSPRTs, it is the ideal operating state of LSPRTs and HTSPRTs.

Before the calibration begins, LSPRTs and HTSPRTs should be annealed until the $R\left(0.01\mathrm{°C}\right)$ value is stable; typically within $0.2\mathrm{mK}$ for a LSPRT, and $0.8\mathrm{mK}$ for a HTSPRT. Once the SPRT is stable, fixed-point measurements should be made progressing from the highest temperatures to the lowest temperatures. For LSPRTs and HTSPRTs and fixed points temperatures below and including the zinc point, the $W$ values for the fixed points should be calculated using TPW measurements made in the same oxidation state as during the fixed-point measurement. This means doing a TPW measurement within a few hours after the fixed-point measurement. If the slow cool-down from a high temperature is run overnight and continues much below $480\mathrm{°C}$, the SPRT should be re-heated to $480\mathrm{°C}$ for 1 hour, and then removed to room temperature before the TPW measurement is made.

Where corrections to fixed point resistance measurements are required, such as for self-heating, hydrostatic pressure, gas pressure, impurity, or isotope effects, they should be made to the measured resistance values, before the ITS-90 interpolating equations are applied.

4.4.2.  Consistency checks

SPRT calibrations involve a large number of measurements and corrections with considerable opportunity for operator mistakes, either in the signs of corrections, or operating conditions of the fixed points, or in the values of standard resistors. Therefore, it is helpful to include a number of checks in a procedure to build confidence and reduce the chances of making mistakes.

As much as is possible, validated software should be used for making the corrections and calculating the various $W$ values from the bridge ratio measurements and standard resistor values. The software should be validated using either synthetic data (including pressure corrections, etc.), or older data from a real SPRT that has been checked using independent software or a calculator.

Where possible, a sequence of fixed-point measurements should include repeat measurements or additional fixed points as a redundancy check. Some laboratories use the indium point as the redundant point for LSPRT and HTSPRTs, to check that the value of $W_{\text{r,In}}$ deduced from the measured $W_{\text{In}}$ is close to the ITS-90 reference value. Plots of the calculated deviation function $\Delta W$ versus $W$ should also be done. For temperatures above $50\mathrm{K}$, the curve should have weak quadratic shape and be smooth.

When calibrating a group of SPRTs, at least one SPRT should be included for which the calibration data are already known. Many laboratories employ check thermometers for specific fixed points. This helps to ensure that there are no major changes in the furnace operating conditions, and ensure the fixed-point cells have not developed leaks to atmosphere and have not become contaminated.

It is also helpful to calculate the parameter

for each fixed point. These $S_i$ values should be very similar for all fixed points above $50\mathrm{K}$, although there is a step down in crossing to temperatures above $273.16\mathrm{K}$ due to a small inconsistency in the ITS-90 definitions. (The use of the $S_i$ values for checking purposes has been proposed in [White and Strouse 2009] applying the alternative interpolation Equation (22). The near constancy of the $S_i$ values for each SPRT is a consequence of the resistance closely following Matthiessen’s rule. This rule states that the additional resistivity caused by lattice imperfections is nearly independent of temperature.)

The measurements of $W$ should be checked to ensure that the SPRTs meet the ITS-90 quality criteria Equation (2) and Equation (3); most SPRTs meet these criteria comfortably. Note that the two relations defined by ITS-90 are not quite consistent: an alternative but very similar requirement is that $S_i$ values from Equation (25), calculated for all fixed points, should be greater than 0.9994 [White and Strouse 2009]. This requirement has the advantage of being applicable to fixed points other than mercury and gallium. Additionally, a large drop in the $S_i$ value for the silver-point measurements is a more sensitive indicator of insulation breakdown than the ITS-90 quality criterion Equation (4).

4.4.3.  Reporting

The simplest and minimal option for the presentation of SPRT calibration data is to report the measured $W$ values for all of the fixed points, and their uncertainties. The uncertainties should be presented interms of resistance ratio, though it is helpful to give the equivalent temperature uncertainty. It is also essential to include the coefficients for the interpolating equations realisable using those fixed points as additional information.

It is also necessary to report the resistance value at the TPW for the SPRT. This value is required to track the stability of the SPRT with shipping to and from the calibration laboratory. This value of $R\left(0.01\mathrm{°C}\right)$ should not be used to calculate $W$ values: instead the user should measure the $R\left(0.01\mathrm{°C}\right)$ on their own equipment, as this minimises the propagation of uncertainties associated with the bridges and standard resistors, and real changes in the SPRT resistance.

Appendix 2 providesa table of typical resistance ratios and sensitivity coefficients at the fixedpoints, which are useful for calculating the temperature equivalents of measurement uncertainties.

5.1.1.  Oxidation

SPRT sheaths are filled with a gas for improving the thermal coupling between the sensing element and the sheath. In the past, dry air was widely used, but nowadays the gas consists mainly of a mixture of an inert gas, often argon or helium, with a partial pressure of oxygen between about $2\mathrm{kPa}$ to $10\mathrm{kPa}$. An oxygen content of at least $1\mathrm{kPa}$ is necessary to prevent contamination of the sensing element by metallic impurities reduced from their oxides. On the other hand, effects associated with the oxidation of the platinum cause hysteretic changes in the SPRTs resistance. This effect is complicated further by its dependence on the partial pressure of oxygen, the operating temperature, the presence of impurities, and crystal size and orientation [Wang and Yeh 1998]. To reduce oxidation effects, the oxygen content should be as small as practical. A partial pressure of oxygen of about $2\mathrm{kPa}$ is a good compromise between the need for oxygen and its deleterious effects.

The chemical interactions between platinum and oxygen are complex, with as many as a dozen possible oxides and allotropes. The oxides most relevant to platinum thermometry are PtO, PtO₂ and Pt₃O₄ [Berry 1978, 1980, 1982a, 1982b, Seriani et al. 2006, Sakurai and Tamura 2011, Jursic and Rudtsch 2014]. Platinum oxidation has been investigated by variously applying calorimetry,

thermogravimetry, mass spectrometry, electron diffraction, acid solubility, and thermodynamic modelling [Seriani et al. 2006, Sakurai et al. 2008, Sakurai and Tamura 2011].

For partial pressures of oxygen of a few $\mathrm{kPa}$, the formation and dissociation of platinum oxides takes place with temperature-dependent rates as follows. Up to about $350\mathrm{°C}$, which is dependent on the partial pressure of oxygen, less than one monolayer of oxide is formed, so the oxide is described as two-dimensional. The rate of formation is slow, being just detectable after one hour at $200\mathrm{°C}$, and is still progressing after many tens of hours at $300\mathrm{°C}$. In saturation, its temperature equivalent of the effect amounts to about $1\mathrm{mK}$ to $2\mathrm{mK}$ in $25\mathrm{\Omega }$ thermometers.

At higher temperatures, the two-dimensional oxides dissociate. But at temperature up to about $550\mathrm{°C}$, a three-dimensional surface layer may slowly propagate deeper into the wire, apparently limited only by diffusion and without signs of saturation. The temperature equivalent of the resistance change may reach $10\mathrm{mK}$ or more in $25\mathrm{\Omega }$ thermometers, and it depends on the heat-treating period. This effect also depends on the partial pressure of oxygen in the exchange gas; both the rate of oxidation and the dissociation temperature decrease as the partial pressure is decreased [Ancsin 2003]. Studies performed by Berry (1980, 1982a, 1982b) and Sakurai and Tamura (2011) show that the formation of the three-dimensional oxide can usually be suppressed by applying partial oxygen pressures of only a few $\mathrm{kPa}$. Heat treating the SPRTs at $600\mathrm{°C}$ or higher will cause the three-dimensional oxide to dissociate and should largely restore the SPRT. However, extended heat treatment for very long periods (> > 10 hours) may be required, and in some SPRTs, the effect can be difficult to suppress. Sakurai and Tamura (2011) have also observed anomalous phenomena that indicate a kind of irreversibility associated with oxidation.

When oxidation effects occur in SPRTs, the relative change of the resistance ratio $W\left(T_{90}\right.$) is much smaller than that of $R\left(T_{90}\right)$. This is caused by the fact that oxidation and dissociation change the cross-sectional area of the platinum core, while the resistivity of the core remains unchanged. Berry (1980, 1982a) has developed a two-zone model with a relatively poorly conducting oxide film at the surface for describing the two-dimensional effect. The moderately good stability of $W$ with respect to oxidation reduces the problem to a manageable level in most cases. The main difficulty for high-accuracy measurements is that the measurements of $R\left(T_{90}\right)$ and $R\left(273.16\mathrm{K}\right)$ used for the $W\left(T_{90}\right.$) calculation must correspond to the same oxidation state, and hence $R\left(273.16\mathrm{K}\right)$ must be measured frequently if the oxidation is causing substantial drifts. Particular care may be necessary when the SPRT passes from an oxidising to a dissociating temperature range. If a frequent measurement of $R\left(273.16\mathrm{K}\right)$ is not feasible, dedicated experiments are required to investigate the magnitude of the effects and make a reliable estimate of the uncertainty.

5.1.2.  Impurities

In an ideal metal conductor, electrical resistance is caused mainly by the scattering of electrons due to the thermal motion of the metal atoms. Additional scattering of electrons caused by impurities gives rise to an additional contribution to the electrical resistance, which is approximately independent of temperature [Berry 1963]. Impurities cause irreversible changes in the resistance-temperature dependence of SPRTs, and are a main cause of long-term drift. The impurities may originate from the production of the wire, including insufficient purification of platinum, contamination during preparation of the sensing element or production of the thermometer. Impurities may also originate in the sheaths after the thermometer is assembled, by diffusion especially from metal sheaths, at temperatures above $450\mathrm{°C}$. Some impurities can diffuse through fused-silica sheaths at temperatures above about $900\mathrm{°C}$ [Marcarino et al. 1989]. For thermometers used at temperatures below $450\mathrm{°C}$, the long-term drift is usually very small, no more than $1\mathrm{mK}$ with many hundreds of hours of use [Berry 1962]. Above $450\mathrm{°C}$ the drift increases with increasing temperature and length of exposure and may reach $5\mathrm{mK}/100\mathrm{h}$ at the silver point [Berry 1966, Fellmuth et al. 2005]. The form and speed of drift depend on quantitative and qualitative composition of impurities in the platinum wire [Pokhodun et al. 2005].

5.1.3.  Strain and hysteresis

Elastic strain on the platinum wire, which causes a temporary distortion of the atomic lattice, modifies the resistance. From research on platinum strain gauges it is known that electrical resistance of platinum increases when the wire is under tension and decreases when under compression. Strain typically arises from differential thermal expansion between the platinum wires and its insulating supports, or from mechanical movement of the wire, or a mixture of both. Because it is a purely elastic effect, the deformation disappears on cessation of the mechanical forces. However, where mechanical and thermal causes combine, the effect gives rise to hysteresis, which becomes apparent with cyclic excursions in temperature of more than several tens of degrees.

The magnitude of the hysteresis is strongly influenced by the design of the SPRT, the materials used in its construction, and the degree to which the platinum is allowed to expand and contract relative to support structures. The effects are typically no more than $0.3\mathrm{mK}$ peak-to-peak, but can be as low as $0.1\mathrm{mK}$ peak-to-peak and range up to $1.8\mathrm{mK}$ peak-to-peak [Berry 1983].

5.1.4.  Vacancies and defects

In a manner, very much like the effects of impurities, defects in the crystal structure of the platinum lattice also increase the resistance of the wire. The defects may be induced thermally or mechanically.

5.1.5.  Moisture

Berry (1966) and Zhang and Berry (1985), established that the electrical insulation resistance of mica-insulated SPRTs deteriorates once the thermometer has been used much above $500\mathrm{°C}$ for an extended period. Marcarino et al. (1999) subsequently demonstrated an effect as large as $1\mathrm{mK}$ or more. The problem is caused by the release of water of crystallisation from the mica when the SPRT is exposed to high temperatures. Quartz insulated SPRTs may also exhibit a moisture effect due to the small quantities of water trapped within the sheath during manufacture, although the effect is usually much smaller than for mica. The effect of the water is greatest near $0\mathrm{°C}$, where it condenses or freezes on the internal surfaces of the SPRT. The effect decreases exponentially as the temperature moves away from $0\mathrm{°C}$ [Berry 1966]. As the temperature decreases below $0\mathrm{°C}$, the conductivity of the ice falls, while above $0\mathrm{°C}$ an increasing fraction of the water is vapour. At $-39\mathrm{°C}$, the effect is very much reduced and by $200\mathrm{°C}$ the effect is negligible.

SPRTs affected by moisture typically exhibit long (> 10 minutes) settling times, and sometimes instability at the TPW. They will also exhibit hysteresis due to the migration of moisture within the sheath, which may be confused with oxidation effects. The effect may be sensitive to the operating frequency of the resistance bridge, so it is sometimes detectable by changing the frequency (current-reversal times for dc bridges) [Marcarino et al. 1999].

The presence of moisture can be assessed by using dry ice to cool the upper end of the SPRT sheath with the SPRT in a water triple point cell. The dry ice condenses the moisture away from the electrically conducting elements of the SPRT. The observed change in the triple-point resistance typically ranges from undetectable for good quartz SPRTs to $100\mathrm{\mu K}$ or more for older mica-insulated SPRTs.

5.1.6.  High-temperature insulation breakdown and contamination

At sufficiently high temperatures, all of the insulating materials used in SPRTs show the thermistor-like decrease in electrical resistivity characteristic of large-band-gap semiconductors. The resistance decreases exponentially with temperature, causing significant effects in $25\mathrm{\Omega }$ SPRTs at temperatures above $700\mathrm{°C}$ and inducing errors as large as $400\mathrm{mK}$ at $960\mathrm{°C}$. Berry (1995) and earlier Zhang and Berry (1985) investigated the effect and demonstrated a complex dependence on a variety of influence variables, including:

• The electrical operating conditions, including any ground and screen configuration of the measurement circuit, grounding or screening of components in the fixed-point furnace, any dc polarising voltage, and the time that the SPRT is subjected to the polarising voltage.

• The structure of the thermometer, including the insulator material, the geometry of the insulator material and the platinum winding, and contact between the platinum winding and the insulator.

• Thermal operating conditions including the temperature distribution along the thermometer sheath and the thermal history.

There are also peculiar effects, similar to the electrical charge and discharge of batteries, associated with the insulation [Berry 1995, Moiseeva 2005]. White et al. (2007) and Yamazawa et al. (2007) showed that most of the observed complexity is explained by the influence of metal-semiconductor diodes (also known as Schottky-barrier or point-contact diodes) formed at the points of contact between the platinum and the silica insulators in the SPRT. It also seems likely that the conduction in fused silica is ionic rather than electronic, being due to impurities, which explains the battery-like effects. In practice, the leakage effect is far more complicated than a single conductance in parallel with the SPRT resistance. There are conductances distributed between each of the leads along the full length of the thermometer, each subject to different voltages and temperature profiles. Additionally, there are shunt resistances and voltage differences between the SPRT and the furnace.

Evans (1984) estimated the magnitude of both internal and external (through the sheath) leakage, and showed that guarding helps suppress some leakage resistance effects for several major models of thermometer. Berry (1995) showed that a dc offset of the correct polarity (approximately +6.4 V between thermometer wires and ground), applied to the measurement circuit or to conducting screens, allows to reduce significantly the conductance of the shunt (the resistance is increased). Some resistance bridges have the facility for introducing the offset. Recent experiments [Widiatmo et al. 2013] suggest this may be a way of switching the platinum-silica diodes off so that the leakage resistance effect can be measured.

Measurements of the shunt resistance by many workers, e.g. [Pokhodun et al. 1990, Berry 1995, Yamazawa and Arai 2003, 2005, Moiseeva 2005], suggest that typical values for a lumped shunt resistance, in the absence of a dc bias, for thermometers with quartz insulation are in the range $0.5\mathrm{M\Omega }$ to $20\mathrm{M\Omega }$ at $960\mathrm{°C}$, which produce leakage effects of $3\mathrm{mK}$ to $0.08\mathrm{mK}$ for a sensor with $R\left(273.16\mathrm{K}\right)=0.25\mathrm{\Omega }$. Note that this effect may increase or decrease following calibration, when thethermometer is in use, depending on the temperature profiles along the thermometer sheath. These observations suggest a standard uncertainty of the order of $1\mathrm{mK}$ at $960\mathrm{°C}$ for $0.25\mathrm{\Omega }$ thermometers. For thermometers employing glass and/or mica insulation, the leakage conductance is much greater and becomes apparent at lower temperatures with errors of about $1\mathrm{mK}$ at $600\mathrm{°C}$.

During the calibration of HTSPRTs at the silver freezing point, the diffusion of silver through quartz may lead to contamination of the platinum sensing element. Different attempts to utilize passive barriers (glassy carbon crucible, sapphire thermowell, platinum sheath) and dc electric fields to protect the HTSPRTs from silver have been described [Hill 2015].

5.2.1.  Conduction and immersion effects

Static temperature-measurement errors are caused by the sensing element of the SPRT not being in direct thermal contact with the object of interest. The temperature indication of the thermometer is in practice affected by the many thermal conductances between the sensor and neighbouring objects. These include the axial thermal conductances of the measuring leads and the sheath, which extend to room temperature for LSPRTs and HTSPRTs, and the radial thermal conductance, which is influenced by the thermal conductivity of the exchange gas and the sheath and the heat-transfer coefficient between the sheath and its environment. A low radial thermal conductance also means a high self-heating effect, see Section 5.3.3.

The thermometer immersion problem has been investigated theoretically in [White and Jongenelen 2010, Gaiser and Fellmuth 2013a and 2013b] and experimentally, for instance, in [White and Jongenelen 2010, Riddle et al. 1973]. It has been shown that after the whole sensing element is immersed in the temperature environment to be measured, the static temperature-measurement error decreases exponentially with further immersion. The immersion characteristic can be used experimentally to estimate the residual errors. A thermometer is sufficiently immersed when there is no detectable change in the indicated temperature with additional immersion in a constant-temperature environment.

For fixed-point measurements, the hydrostatic pressure effect must be considered (see Chapter 2 Fixed Points), the magnitude of which is given for the different fixed-point substances in Table 2 ofthe text of the ITS-90. Usually the results of measurements of a fixed-point temperature are considered as acceptable, if the SPRT immersion curve corresponds to the linear relation, describing the effect of the hydrostatic pressure on the temperature of the solid-liquid interface, over a distance of at least $\left(3-5\right)\mathrm{cm}$ from the cell bottom. The uncertainty due to the immersion effect can be evaluated with the following equation [Strouse 2005]:

where $N$ is the number of measurements of the immersion characteristic, $h$ is the distance along the axis of the cell, and $\text{d}{T}_{90}/\text{d}h$ is the hydrostatic-pressure coefficient given in the text of the ITS-90. The determination of an immersion curve requires high-accuracy measurements of very small temperature differences. It is preferable to make measurements at two measuring currents with subsequent extrapolation of the results to the zero power in order to eliminate the influence of changes in the heat-exchange conditions when moving the SPRT in the re-entrant well of the cell. It may be preferable to make measurements on insertion of the SPRT, rather than on its withdrawal, because on withdrawing the SPRT the heat transfer fluid (air, water, oil) flows down into the measurement zone, with potentially harmful thermal effect, whereas expelling the fluid on insertion does less harm (though any vertical movement will create some disturbance to the temperature profile along the stem, which must be allowed to subside).

The influence of poor thermometer immersion on fixed point measurements can also be assessed by applying to the furnace a small, $\sim 1\mathrm{K}$, temperature oscillation with a period of at least 30 min while observing the plateau [Fahr and Rudtsch 2008]. The amplitude of the observed fluctuations in the plateau temperature provides a direct measure of the coupling between the SPRT and the furnace. Note that the value of the thermal coupling constant varies throughout the melt or freeze of the cell [Fahr and Rudtsch 2008]. The difference between the furnace temperature and the fixed-point temperature can be determined from the monitoring SPRT once the cell has stabilised after the freeze.

The difference between the mean temperature of the sensing element and that of the solid-liquid interface depends on several factors: the depth of the thermometer immersion into the fixed-point cell, the presence of a continuous interface along the height of the fixed-point substance in the cell, the SPRT design, as well as heat exchange in the thermometer well and outside the cell. The influence of each of these factors can be evaluated by analysing the change of the thermometer immersion curve after a change in particular conditions during the fixed-point realisation. The depth of thermometer immersion into the cell is generally considered to be the main factor influencing the static temperature-measurement error. Above room temperature, the necessary depth increases with increasing fixed-point temperature up to about $450\mathrm{°C}$. Then it decreases slightly as heat exchange by radiation improves. At higher temperature especially, longitudinal radiation baffles and measures to inhibit radiation piping through the sheath are essential, see Section 5.2.2. For different SPRT designs, the necessary immersion depth may differ by as much as $10\mathrm{cm}$. Typically, about $20\mathrm{cm}$ to $25\mathrm{cm}$ of immersion (in addition to the length of the sensing element) in an isothermal zone is required to reduce the uncertainty component to $0.1\mathrm{mK}$ at $400\mathrm{°C}$.

Collections of experimentally determined immersion curves are available within reports on Key Comparisons performed with LSPRTs at the TPW (CCT-K7 [Stock and Solve 2005, Stock et al. 2006]) and at the other fixed points in the range $84\mathrm{K}$ to $933\mathrm{K}$ (CCT-K3 [Mangum et al. 2002]). In several cases, the curves do not follow the linear behaviour corresponding to the hydrostatic-pressure coefficient, even near to the maximum possible immersion. These departures increase the uncertainty estimate from Equation (28). The analysis of possible causes for such deviations is somewhat easier for the TPW because cells can be checked visually to confirm the thermometer well is completely surrounded by a solid-liquid interface and the temperature field around the cell is homogeneous along its height. The immersion curves of an SPRT were obtained for 23 different TPW cells with the immersion depth varying from $205\mathrm{mm}$ to $289\mathrm{mm}$ [Stock and Solve 2005], where the completeness of the interface was visually noticed. The analysis of immersion curves is more complicated for the other fixed points because the position of the interface is invisible and it is difficult to control the homogeneity of the temperature field along the cell and around the thermometer sheath. The following tests could be helpful:

• Improvement of the thermal contact between the thermometer sheath and the re-entrant well by introducing sleeves or contact liquids in the gap.

• Check of the influence of the temperature around the cell, e.g. by adjusting the temperature periodically, which may allow corrections for disturbing influences [Fahr and Rudtsch 2008]. Large influences may be caused by thermal links between the thermometer well and the outer wall of the cell. In the beginning of a freeze such links may consist of connecting spaces filled with liquid fixed-point substance and at the end of the freeze through spaces filled with solid substance [White and Mason 2011].

• Application of different methods for initiating an inner solid-liquid interface around the thermometer well, on which the completeness of the interface depends [Ivanova et al. 2013, White and Mason 2011]. In particular, it depends on the starting point of the initiation after observation of supercooling, on the duration of the cooling with a rod inserted in the thermometer well, and on the thermal conductivity and heat capacity of the rod that is used. A complete outer interface at the wall of the cell forms gradually. The time necessary for this process is dependent on the temperature around the cell.

• Check if thermometers that have transparent (glass, quartz or sapphire) sheaths are protected from intrusive, or escaping, radiation. (Metal sheathed thermometers need no further radiation shielding, but the heat conduction down the sheath is usually such as to require deeper immersion.)

The experimental immersion curve is in general not a reliable estimate of the temperature profile along the axis of the thermometer well because a displacement of the thermometer leads to a change of the profile due to its influence [Ilin 2003]. However, the slope of the immersion curve is an indicator of the closeness of the measurement conditions to the thermal equilibrium between the solid-liquid interface and the thermometer. During the measurements, not only the thermometer immersion depth must be taken into account, but also the design of the furnace or bath and the chosen modes of its operation.

A computer simulation of the heat exchange in a zinc cell during the phase transition has shown that lifting the LSPRT by $10\mathrm{cm}$ may lead to a change in the temperature in the lower end of the thermometer well by $0.7\mathrm{mK}$ [Batagelj 2004, Batagelj et al. 2005]. Another simulation yielded a relationship between the change in the temperature of the upper heater of the furnace and the temperature profile along the axis of the thermometer well at the freezing points of tin and zinc [Ivanova and Ilin 2005]. Furthermore, the influence of the position and the dimensions of the sheath area, over which the surface is roughened to decrease radiation effects, see Section 5.2.2, on the heat removal along the thermometer was estimated. This allowed assessment of the difference between the interface temperature and the temperature detected by the sensing element of an LSPRT [Batagelj 2004, Batagelj et al. 2005].

In capsule-type SPRTs, there is often an additional effect due to background heat fluxes, and static temperature-measurement errors are caused by the heat flow $\mathrm{\varphi }$ via the sensing electrical leads and the thermometer [Gaiser and Fellmuth 2013a]. To evaluate the errors, two thermometer parameters are needed: the thermal resistance $R_{\text{T}}$ of the CSPRT between the glass seal at the thermometer head and the sheath, and the so-called reduction factor $V=\left(T_{\text{T}}-T_{\text{S}}\right)/\left(T_{\text{H}}-T_{\text{S}}\right)$ ($T_{\text{T}}$ mean temperature of the sensor element, $T_{\text{S}}$ sheath and $T_{\text{H}}$ head temperature). These parameters have to be determined in dedicated experiments. The error caused by the heat flow is given by $T_{\text{T}}-T_{\text{M}}\approx V\mathrm{\varphi }{R}_{\text{T}}$ since the difference between the temperature of the measuring object $T_{\text{M}}$ and $T_{\text{S}}$ can usually be made vanishingly small.

5.2.2.  Radiation effects

A significant source of error, especially at high temperatures, is heat flux to and from the platinum coil, via thermal radiation. If the sensor is in line of sight of a surface that is appreciably hotter or colder than that being measured, the power gained or lost by thermal radiation will result in a change of its temperature. In TPW measurements, thermal radiation from incandescent lights in the room, which may be incident upon the emerging thermometer sheath or the inner wall of the triple point cell, can produce an error of up to $0.2\mathrm{mK}$ [McLaren and Murdock 1966, Riddle et al. 1973]. The TPW cell should, therefore, be immersed in a bath in which no extraneous radiation from sources above room temperature can reach the sensor of the thermometer. If this precaution is omitted, even with radiation entry restricted to the top of the cell, a thermometer may require an immersion below the vapour-liquid level of $35\mathrm{cm}$ (which is seldom available), much more than the immersion of $15\mathrm{cm}$ for a thermometer with excellent immersion characteristics, in order for it to achieve an accuracy approaching $0.05\mathrm{mK}$. For high-resolution measurements, such as TPW cell comparisons, opaque radiation shields can be used to cover the SPRTs and the top of the cells to confirm the absence of any effect.

At higher temperatures, radiation escaping via the SPRT sheath will cool the sensor element and can lead to errors as large as a few tenths of a millikelvin at $420\mathrm{°C}$ (zinc point) rising to several tens of millikelvin at $962\mathrm{°C}$ (silver point). This heat loss is due to “light piping” through the sheath due to total internal reflections within the wall of the thermometer sheath. This error can be substantially reduced by roughening the lower part of thermometer sheath from just above the sensor for some $15\mathrm{cm}$ to $20\mathrm{cm}$ by sand blasting, or by coating this part of the sheath with graphite paint [Widiatmo et al. 2006, Žužek et al. 2014]. The same heat-loss mechanism can affect the apparent temperatures of fixed point cells that have fused silica thermometer wells. These are less easily dealt with by sand blasting or painting but it is a fairly simple matter to arrange that they ‘see’ only opaque surfaces substantially at their own temperature, and not suffering net loss or gain of heat.

5.3.1.  Resistance bridges

There are two basic types of thermometry resistance bridges in common use: ac bridges using a sinusoidal sensing current in the frequency range $10\mathrm{Hz}$ to $90\mathrm{Hz}$, and so-called dc bridges, which use a low-frequency square-wave (ac) sensing current, in the range $0.01\mathrm{Hz}$ to $10\mathrm{Hz}$. The bridges typically have a resolution of six to nine digits and a specified uncertainty in resistance ratio between $2\times {10}^{-8}$ and $4\times {10}^{-6}$, corresponding to equivalent uncertainties in temperature measurements near the TPW ranging from about $5\mathrm{\mu K}$ to $1\mathrm{mK}$. Some of the sources of error within thermometry bridges and bridge calibration methods are summarized in [White 2003]. Rudtsch et al. (2005), Strouse and Hill (2003), and White (1997) give additional information on typical performance of some resistance bridges. The errors in bridges can be classified as noise, large scale or ‘integral’ non-linearities (INL), and short-range or ‘differential’ non-linearities (DNL).

The resolution of a bridge is determined by a combination of the thermal noise generated in all resistances and amplifiers, and quantisation error due to the conversion of the signal to a digital reading [White and Edgar 2013]. The resolution is readily measured as the standard deviation when measuring a single stable resistance. Uncertainty due to resolution can be reduced by decreasing the operating bandwidth of the bridge or, equivalently, increasing the measurement time or averaging many results.

Noise in the resistance measurement may also arise from electromagnetic interference (EMI), particularly electrical noise coupled from ac furnace controllers, electric motors, and other ac resistance bridges operating nearby on the same carrier frequency. In a bridge with resolution limited by thermal noise, the standard deviation of measurements will change as the square root of the bandwidth or measurement time. If EMI due to ac machinery is present then often the standard deviation may change more rapidly or not at all. Digital interface connections between the bridge and a computer for data acquisition can also cause interference which arises from current flow caused by differing ground potentials between the two devices; this can be overcome by using optical interface connections.

INL are those non-linearities that change slowly and smoothly over a wide range of bridge readings. These effects are caused by finite input and output impedances, power coefficients of resistors, finite gains of amplifiers, and stray capacitance and resistance of sub-circuits of the bridge. They tend to be well modelled by quadratic or cubic functions of bridge reading. With many bridges, the INL effects are below the manufacturers’ accuracy specification and, therefore, less significant than many other sources of uncertainty in the temperature measurement. INL errors larger than the bridge manufacturer’s specification are usually indicative of a faulty sub-circuit within the bridge or poor adjustment. Additionally, because the ITS-90 is defined in terms of resistance ratio, only the non-linear bridge errors (offsets, quadratic, and cubic terms) propagate and affect the temperature measurement.

Further, if the same bridge is used for the calibration and use of an SPRT, then the SPRT interpolating equations largely correct the INL errors of the bridge. In such cases the INL correction and uncertainty can be omitted from the uncertainty analysis.

DNL are commonly caused by errors in the transformer windings or analogue-to-digital converters. They may occur at every change in reading and are generally too complex to measure, model and correct. The effects tend to be below the manufacturers’ specifications, but can increase with time and mechanical vibration as transformer cores lose magnetic permeability, and ADC circuits drift out of adjustment. DNL are the dominant source of uncertainty in resistance measurements for high resolution or differential measurements (e.g., fixed-point comparisons and self-heating measurements). The magnitude of the resulting uncertainty can be inferred from the standard deviation of residual error in the calibration curves for bridges.

5.3.2.  Reference resistor

The two main contributions to the uncertainty in the value of the standard resistance, $R_{\text{S}}$, are caused by the calibration uncertainty (the value of the resistance) and the temperature dependence of $R_{\text{S}}$. The effects can be evaluated using a model of the temperature dependence of the resistance:

where $T_{\text{bath}}$ is the temperature of the oil or air bath used to maintain the resistor during use, $T_{\text{cal}}$ is the temperature of the bath used to maintain the resistor when it was calibrated, and $\mathrm{\beta }$ is the temperature coefficient of the resistor at $T_{\text{cal}}$. $\mathrm{\beta }$ is normally expressed as the fractional change per degree; i.e.,

with typical values for good-quality resistors within the range $\pm 5\times {10}^{-6}/\mathrm{°C}$. If the resistor is not maintained at its calibration temperature, then a correction can be applied using Equation (29). The cumulative uncertainty in the resistance value due to these terms is

where $u\left(R_{\text{S},\text{cal}}\right)$ is the calibration uncertainty as supplied by the electrical calibration laboratory, $u\left(T_{\text{bath}}\right)$ is the uncertainty arising from fluctuations in the resistor-bath temperature, and $u\left(\mathrm{\beta }\right)$ is the uncertainty in the temperature coefficient of the resistor. Where $T_{\text{bath}}$ and $T_{\text{cal}}$ are nominally equal, the uncertainty in the temperature coefficient can usually be ignored, as can any quadratic temperature dependence of $R_{\text{S}}$ in Equation (29). Also, where the same resistance bridge is used for the two resistance measurementsused to calculate the $W\left(T_{90}\right.$) value (Equation (1)), uncertainty in the value of $R_{\text{S}}$ can be ignored and only the stability of $R_{\text{S}}$ needs be considered. The uncertainty in the $R_{\text{S}}$ does need to be considered where the $R\left(0.01\mathrm{°C}\right)$ value of a thermometer is reported.

The calibration value for the standard resistor, $R_{\text{S},\text{cal}}$, is nearly always the dc value. If the resistor is to be used with an ac bridge, standard resistors with a low ac-dc difference should be used [Wilkins and Swan 1970]. For frequencies below $100\mathrm{Hz}$, ac-dc differences arise from stray inductance and capacitance, dielectric absorption, and eddy current effects. Errors in dc and low frequency measurements can arise from thermoelectric EMFs produced as a result of Peltier heating at the connections [Kirby and Laubitz 1973], but these are usually small. Some standard resistors exhibit relative ac-dc differences as large as ${10}^{-5}$. With resistors of the Wilkins’ design, the relative uncertainty due to ac-dc difference, $u\left(\mathrm{\delta }{R}_{\text{S},\text{ac-dc}}/R_{\text{S}}\right)$, see Table 3 in Section 6.1.1, is below ${10}^{-7}$.

In bridges based on dc current comparators (see for example [McMartin and Kusters 1966]), the current $i_0$ through the standard resistor changes with bridge reading. This causes variable power dissipation in the standard resistor and an error with a cubic dependence on the measured resistance ratio, $X=R\left(T_{90}\right)/R_{\text{S}}$:

where $P$ is the power dissipated in the standard resistor, and the term involving the derivative is the power coefficient. Electrical calibrations of standard resistors are normally carried out at a specified power. The power coefficient is the product of the temperature coefficient and the thermal resistance between the resistor windings and the environment, so good quality standard resistors are generally quite bulky devices. The Wilkins’ type resistors [Wilkins and Swan 1970] have a power coefficient of less than $4\times {10}^{-6}/\mathrm{W}$. The error is then much less than $1\mathrm{\mu K}$. But this error can be considerably larger if a purpose-built standard resistor is not used. If sufficient time is allowed for the resistor to stabilise after current changes (tens of minutes), the effect is compensated by the self-heating correction for the SPRT.

Considering also the uncertainty components due to ac-dc differences and the power dissipation, from Equation (31) it follows:

5.3.3.  Self heating

Measurements with resistance thermometers necessarily involve passing a current through the thermometer sensing element, which gives rise to Joule heating. As a result, the sensing element warms to a slightly higher temperature than that of the object or environment whose temperature is to be measured. The warming continues until the temperature difference is sufficient to dissipate the heat and the steady state is reached. Considering SPRTs, most of the heat flows by conduction through the filling gas to the sheath, though there is some convection and also conduction through solid components. This internal so-called “self-heating” is fixed by the design of the thermometer, but there is also an external effect between the sheath and the heat sink (for example, the liquid-solid interface in a fixed-point cell) [Riddle et al. 1973], which in a well-designed experiment can be made relatively small.

The self-heating effect depends on the thermal conductance between the sensor and the measurement environment, and is therefore a useful indicator of whether good thermal contact has been achieved, and subsequently for routine checks. Influencing factors are the design of the sensor element, the thermal conductivity of the exchange gas, and the heat transfer to the heat sink, i.e. the solid-liquid interface of the fixed-point material. Thus, the magnitude of the effect has to be checked individually for each experimental setup. It is not alone determined by the parameters of the SPRT. The measurement current must be chosen so as to achieve good resolution without excessive self heating. For $25\mathrm{\Omega }$ SPRTs the current is often $1\mathrm{mA}$ (dissipation of $25\mathrm{\mu W}$ at $0\mathrm{°C}$) and the self heating effect in a TPW cell is typically between $0.2\mathrm{mK}$ and $4\mathrm{mK}$. With modern resistance bridges, a temperature resolution of $<0.1\mathrm{mK}$ is easily achieved. Self heating has usually a typical variation with temperature, showing a maximum slightly above $200\mathrm{°C}$.

The self-heating effect is determined by making measurements at two currents and calculating the temperature rise from the change in measured resistance. The user then has a choice of whether or not to apply a correction for the effect. If a thermometer is measured using the same current as was used in its calibration, and if the external conditions for the SPRT are also similar, then the effect will be similar and the thermometer can be used without making corrections. This is convenient and is common practice in secondary-level calibrations and measurements.

For precise work, self-heating corrections must be applied [Batagelj et al. 2003a, 2003b]. Since the power dissipated increases as the square of the current, the resistance at zero current, $R_0$, can be calculated from resistance measurements $R_1$, and $R_2$ made using measuring currents $i_1$ and $i_2$ using the following relation:

Commonly, the current ratio $i_2/i_1$ is chosen to be $\sqrt{2}$. In that case a simple measurement sequence $i_1$: $i_2$: $i_1$ allows the zero-current resistance to be determined from the corresponding measurements $R_1$: $R_2$: $R_3$ as $R_0=R_1+R_3-R_2$. Pearce et al. (2012, 2013) have shown that the uncertainty in theextrapolation to zero current is minimised if the current ratio is 2 and the measurement time is apportioned in the ratio 4:1:4.

The zero-current bridge ratio $X_{\text{meas}}^0$ is calculated corresponding to Equation (34). Its uncertainty can be estimated using the equation:

In Equation (35), $u\left(\mathrm{\delta }{X}_{\text{DNL}}\right)$ is the uncertainty component due to the DNL error between the measurements of $X_{\text{meas}}^1$ and $X_{\text{meas}}^2$ with the two currents $i_1$ and $i_2$, $u\left(\mathrm{\delta }{X}_{\text{noise}}\right)$ is the component due to electrical noise, which is assumed to have the same order of magnitude for both currents, and $u\left(i_1,i_2\right)$ is the estimated covariance associated with $i_1$ and $i_2$. If the relative errors of the two currents are equal, i.e. if they are fully correlated, the second term vanishes.

5.3.4.  Connecting cables and lead resistances

Figure 4 shows a simplified equivalent circuit for the measurement of a 4-terminal resistance, which includes the finite shunt resistance of the connecting cables. The model can also be used to characterise the input and output impedances of the bridge, hence the following nomenclature: $R_{\text{out}}$ is the resistance shunting the current source $i_0$, and $R_{\text{in}}$ is the resistance shunting the voltage measurement $V_m$. The equivalent circuit also includes the lead resistances $R_{\text{L}}$ in each of the four leads to the measured resistance. Analysis of Figure 4 shows that the error in the measured resistance ratio (the bridge reading), $X$, is well approximated by

where

Thus, the finite resistances, $R_{\text{out}}$ and $R_{\text{in}}$, give rise to an error proportional to the square of the resistance ratio. With care, cable shunt resistances can normally be maintained at greater than ${10}^{10}\mathrm{\Omega }$, ensuring the effects are negligible. Significant effects can be observed in very long cables, or unscreened cables immersed in oil (the standard resistor bath), and with some coloured PVC cables. With ac bridges, dielectric loss in the cables gives rise to a similar error that increases as the square of the resistance and increases linearly with frequency. Dielectric loss can be a major problem in low-temperature measurements where many meters of lead wire is used for thermal anchoring. Operating an ac bridge at lower frequencies or using a dc bridge can significantly reduce the effects. The effects of lead resistances are usually negligible in comparison to the effects of the finite cable shunt resistance since $R_{\text{L}}\le R_{\text{S}}$ in most resistance measurements.

In ac bridges, the measurements can also be affected by the stray capacitance of the connecting cables. Any capacitance in parallel with $R\left(T_{90}\right)$ (Figure 4) causes the impedance ‘seen’ by the resistance bridge to be

where $\mathrm{\omega }=2\mathrm{\pi }f$ is the angular frequency, $C$ is the cable capacitance (about $100\mathrm{pF}/\mathrm{m}$ for coaxial and twisted pair cable), and $j=\sqrt{-1}$ represents a $90\mathrm{°}$ phase shift. The real part of Equation (37) is the resistance seen by an ideal ac thermometry bridge; the imaginary part contributes to the quadrature component of the bridge signal, which must be balanced or rejected by the bridge. The real part of Equation (37) contributes a resistance ratio error with a cubic dependence on the measured resistance ratio:

Normally, the effect of stray capacitance is negligible. However, because of the square-law dependence on capacitance, in measurement systems with multiplexers and long leads, the capacitance may overload the quadrature-balance capability of the bridge or lead to relative errors in resistance ratio as large as ${10}^{-4}$ (many millikelvin). AC bridges used at low temperatures may also be affected by the self-inductance of any long high-resistance lead wires used for thermal anchoring. In principle, the measured resistance is independent of the inductance (unlike the case for capacitance). However, the large quadrature signal can overload the bridge leading to excessive noise.

For dc resistance bridges the effect of stray capacitance is complicated by the many harmonics in the ‘square-wave’ sensing current, but normally below the limit of detection. Where detectable, it is apparent with changes in the current-reversal times and settling times [Zhang and Berry 1985].

In systems with short leads, most cable effects can be reduced to negligible levels with care. If the effects are significant they are often indicative of fault conditions.