2.1.  Helium vapour-pressure relations

The vapour-pressure relations for helium are of the form

Values of the constants $A_0$, $A_i$, $B$ and $C$ are given in Table 3 of the Text of the ITS-90 for the three ranges $0.65\mathrm{K}$ to $3.2\mathrm{K}$ for ³He; $1.25\mathrm{K}$ to $\mathrm{2.176 8}\mathrm{K}$ for ⁴He (⁴He II); and $\mathrm{2.176 8}\mathrm{K}$ to $5.0\mathrm{K}$ also for ⁴He (⁴He I)¹. The ⁴He equations coincide at the lambda point ($\mathrm{2.176 8}\mathrm{K}$, $\mathrm{5 041.8}\mathrm{Pa}$) with a first derivative $\text{d}\ln p/\text{d}{T}_{90}$ of $2.461{\mathrm{K}}^{-1}$. In deriving these equations, no constraints were placed at the second derivative, which is discontinuous at this point.

The upper temperature limits for the use of the two helium isotopes were chosen to be somewhat below the critical points, since the specification of the relationship and measurement of vapour pressures become matter of some difficulty as the critical point is approached. The lower temperature limit was caused by the fact that when the ITS-90 was established, it did not appear possible to measure widely pressures below $100\mathrm{Pa}$ with a relative uncertainty sufficient for reaching the desired temperature uncertainty of order $0.1\mathrm{mK}$, cf. Table 1 and Figure 1.

Figure 1 — Examples of uncertainties in pressure measurements using a capacitance diaphragmpressure gauge below $1.3\mathrm{kPa}$, and the equivalent uncertainties $\mathrm{\delta }{T}_3$ and $\mathrm{\delta }{T}_4$ in temperatures $T_3$ and $T_4$ for ³He and ⁴He vapour-pressure measurements, respectively [Rusby and Swenson 1980]. © Bureau International des Poids et Mesures. Reproduced by permission of IOP Publishing. All rights reserved.

The equations adopted in the ITS-90 are a restricted set of those derived by Rusby and Durieux (1984), which themselves were simplified forms of the equations of Durieux and Rusby (1983) that were approved by the CIPM (1982) following the introduction of the International Temperature Scale EPT-76 [Durieux et al. 1979], see ITS-90 Guide Chapter 1 Introduction. The unrestricted forms are useful for those choosing to use them at temperatures close to the critical points or below $0.65\mathrm{K}$.

2.2.  Hydrogen vapour-pressure relations

For the calibration of standard platinum resistance thermometers, see ITS-90 Guide Chapter 5 Platinum Resistance Thermometry, two calibration temperatures may be determined by boiling points ofequilibrium hydrogen, see Subsection 1. The two temperatures must lie within the ranges $16.9\mathrm{K}$ to $17.1\mathrm{K}$ and $20.26\mathrm{K}$ to $20.28\mathrm{K}$, respectively. The precise values have to be determined by the following vapour-pressure-temperature relations:

Equation (2) is valid for the range $16.9\mathrm{K}$ to $17.1\mathrm{K}$, and Equation (3) for $20.26\mathrm{K}$ to $20.28\mathrm{K}$.

3.1.  Systems immersed in liquid baths

In its simplest form, a ⁴He vapour-pressure thermometer consists of a vessel containing the liquid in equilibrium with its vapour, and the thermometer liquid is also the refrigerating liquid. This is possible because ⁴He can be transported in transport vessels and all impurities freeze out. (The only relevant impurity ³He has a negligible natural abundance.) A schematic illustration of a cryostat for calibrating capsule-type thermometers is given in Figure 2a. The vapour pressure is sensed via a tube inserted into the vapour space and terminating above the liquid surface. The pressure is usually regulated by means of a throttling valve in the pumping line (not shown). The pressure, and hence the temperature, is progressively reduced so as to prevent large temperature stratification. An electrical heater at the bottom of the liquid (not shown) can be used for reheating and, at low power levels, will promote mixing.

Below the lambda point, the phenomenal thermal diffusivity of ⁴He II ensures that under stable conditions no temperature gradients exist within the liquid. In this case the system of Figure 2a can be used for high-accuracy vapour-pressure measurements. The only limitations are the ability of the pump to reduce the temperature as far as required and, possibly, the appearance of thermomolecular effects at low pressures. The latter limitation is eliminated by use of a tube of inner diameter $10\mathrm{mm}$ or more equipped with appropriate radiation baffles. Above the lambda point, the convection mechanism in ⁴He I is a feeble one at these temperatures, and significant temperature gradients can be present in the liquid. Such gradients occur even when the temperature monotonically decreases with time. Their magnitude being likely to increase as the temperature is lowered. At $4.2\mathrm{K}$, the hydrostatic head (pressure increase due to the liquid column) causes a gradient of $12\mathrm{\mu K}/\mathrm{mm}$. Temperature differences perhaps become as high as $5\mathrm{mK}$ at the lambda point [Cataland et al. 1962]. On reheating, the temperature of the bulk liquid may respond only slowly if the pressure is allowed to rise, and gross gradients can then result.

Figure 2 — Schematic illustrations of systems for realizing⁴He vapour pressures: using a bath ofliquid (a), suitable for ⁴He II, and a bulb (b) for ⁴He I, unsuitable for ⁴He II. Thermal shields around the helium bath are not shown.

The difficulties with temperature gradients in the thermometer liquid ⁴He I can be avoided by mounting the thermometers in a copper block containing a vapour-pressure bulb (Figure 2b) that is independently supplied with helium, so that a liquid-vapour interface is contained within it. In a closed system, the liquid fraction will increase when the temperature is reduced, and the total amount of helium must be such that the bulb does not overfill. The supply tube (of stainless steel or another material of low thermal conductivity, and typically $2\mathrm{mm}$ inner diameter) is also the pressure-sensing tube. Where it passes through the surrounding liquid, it must be insulated sufficiently well to avoid condensation within it (cold spots). Light insulation is reported as being sufficient for this purpose, since cold spots tend to be self-stifling by virtue of the heat of condensation [Ambler and Hudson 1956]. However, a stainless-steel vacuum jacket is often used and may extend up to room temperature. In addition, copper cladding or electrical heaters on the sensing tube can be used. These have the advantage of keeping the sensing tube and vapour within it relatively warm, only reaching the liquid temperature just above the bulb, thereby reducing the aerostatic-head correction. For an exposed tube and a bulb at $4.2\mathrm{K}$, this correction may be about $0.5\mathrm{mK}$, compared with $0.1\mathrm{mK}$ to $0.2\mathrm{mK}$ that is typical for a vacuum-jacketed tube. For the latter one, however, the temperature distribution, needed for calculating the pressure head, can be difficult to ascertain. At the lower temperatures, aerostatic-head corrections are smaller in terms both of pressure and of the temperature equivalents, because of the rapid decrease of vapour density with decrease in temperature.

No radiation trap is shown inside the pressure sensing tube of Figure 2b. If this is $2\mathrm{mm}$ in diameter, radiative heat transfer down the tube (assuming that all the radiation emitted at room temperature is adsorbed in the bulb) would be about $1.4\mathrm{mW}$. The helium in the bulb is not likely to absorb much of this, while the cooper block could easily do so without setting up significant temperature gradients. A trap could be included near the bottom of the tube, but it would need careful design. A simple bend in the tube is unlikely to be effective, while any other system must be so constructed that no liquid can be held at that point. Even with a straight tube, it is possible for liquid to block the tube just above the bulb, leading to substantial measurement errors. A pressure pulse may dislodge such a block, but as a matter of design, the portion of the tube exposed to low temperatures should be short, or the vacuum jacket can be extended right down to the bulb. In the latter case, the heat conducted down the sensing tube, perhaps $0.1\mathrm{mW}$, can be readily absorbed in the block. But a baffle must be included in the vacuum jacket to intercept the radiation from room temperature components of the jacket. There will be some differential contraction between the sensing tube and the vacuum jacket. These must not touch, and an insulating spacer, which can also serve as the radiation trap, should be used to prevent this. Alternatively, bellows or a sliding seal at the upper end of the tube can prevent touching.

While a vapour-pressure bulb contained in a copper block is preferable for normal helium, ⁴He I, this is not the case for superfluid ⁴He II. With ⁴He II in the bulb, a superfluid film would creep up the walls of the sensing tube, would evaporate at some higher temperature, and would then reflux back to the bulb. This action can result in a measurement error of several millikelvins [Sydoriak et al. 1964] due in part to pressure gradients in the tube and in part to temperature gradients in the bulb. In vacuum-isolated systems, an orifice in the lid of the helium bulb can reduce the film flow, see Section 3.2. Continuity of measurements on passing through the lambda point is a good test of any design. By contrast, the system of Figure 2a works well below the lambda point. In this system, the superfluid film evaporates as part of the cooling process and never affects the pressure sensing. Clearly a dual system, in which both a vapour-pressure bulb and a bath pressure-sensing tube are provided, would enable the complete ⁴He range to be covered, and would also allow investigations of the differences between the two realizations to be made.

3.2.  Vacuum-isolated systems

Most of the realizations of helium vapour-pressure scales were performed applying systems, in which copper thermometer blocks containing vapour-pressure bulbs were suspended inside a vacuum jacket [Sydoriak et al. 1964, El Samahy 1979, Rusby and Swenson 1980, Meyer and Reilly 1996, de Groot et al. 1993 and 1997, Hill 2002, Engert and Fellmuth 2003, Engert et al. 2007, Shimazaki et al. 2011,<Sparasci2011/>,Sparasci et al. 2011b]. Sydoriak et al. (1964) have performed extensive comparisons of the vapour pressures of ³He and ⁴He prior to the derivation of the 1962 ³He vapour-pressure scale and equation, cf. [Quinn 1990]. These comparisons were performed in an apparatus designed to reduce the number and magnitude of corrections associated with the refluxing film in the ⁴He pressure-sensing tube and the attached bulb. Problems with ³He are its high cost and the need to take into account of contamination with ⁴He. Because of its cost, ³He is usually kept in a closed system and repeatedly used. As typical examples, the vapour-pressure systems of Rusby and Swenson (1980) and Engert et al. (2003) and (2007), respectively, are described in detail in the following. In both cases, the vacuum jacket is immersed in a liquid ⁴He bath. Nowadays preferably cryostats designed around closed-cycle cryocoolers are applied for high-accuracy low-temperature thermometry. Examples of modern systems are given in [Steele 1997, Hill and Steele 2003, Sakurai 2003, Nakano et al. 2007, Pavese et al. 2011, Sparasci et al. 2011a, Yang et al. 2011, Shimazaki et al. 2011, Pavese and Molinar Min Beciet 2013]. These cryostats allow measurements to be performed for extremely long periods (months), uninterrupted by disturbances usually caused by refilling liquid refrigerant. Up to now, vapour-pressure systems without ⁴He baths are not realised. In realising such system, the main design principles for the vapour-pressure part discussed below have to be considered.

A combined ³He and ⁴He vapour-pressure cryostat, somewhat simpler than that of Sydoriak et al. (1964), has been described by Rusby and Swenson (1980) and was used by them for the re-determination of the vapour-pressure relations, see Figure 3. The copper thermometer block containing the helium bulbs was suspended inside a vacuum jacket, which was surrounded by liquid ⁴He at $4.2\mathrm{K}$. A single $50{\mathrm{cm}}^3$ bulb of ⁴He was used for cooling and sensing the vapour pressure. The problem of film flow was avoided by including an orifice of $0.6\mathrm{mm}$ diameter in the lid of the bulb. Such film as flowed through the orifice was soon evaporated and thereby contributed to cooling. The pressure-sensing tube joined the $6\mathrm{mm}$ diameter pumping tube some $40\mathrm{mm}$ higher up and so was not affected by film refluxing. The pressure drop across the orifice and along this section of the tube was negligible at temperatures above $1.4\mathrm{K}$. A larger orifice could have permitted accurate measurements to still lower temperatures. The ⁴He bulb contained a spiral of copper foil to promote temperature uniformity (see the authors’ discussion of measurements above and below the lambda point).

Figure 3 — A schematic diagram of the cryostat used by Rusby and Swenson [1980] for CMNmagnetic thermometry and for ³He, ⁴He I and ⁴He II vapour-pressure measurements. © Bureau International des Poids et Mesures. Reproduced by permission of IOP Publishing. All rights reserved.

A ³He cooling chamber was provided, and measurements of ³He vapour pressures were made with a small $1{\mathrm{cm}}^3$ bulb ($5\mathrm{mm}$ diameter by $60\mathrm{mm}$ long) and a sample of purified gas. The vapour-pressure tubes were $2\mathrm{mm}$ in diameter at temperatures up to $4.2\mathrm{K}$, and $6\mathrm{mm}$ (for ⁴He) and $5\mathrm{mm}$ (for ³He) above this. These dimensions give thermomolecular effects, calculated from the Weber-Schmidt equation (see Section 4.2), equivalent to $1\mathrm{mK}$ at $1.25\mathrm{K}$ and $0.65\mathrm{K}$ for ⁴He and ³He, respectively, with the effects increasing rapidly at lower temperatures. If tubes with larger diameters are used at higher temperatures, the temperature of the junctions must, of course, be known, but this is desirable in any case for the calculation of the aerostatic head effect. The vapour-pressure sensing tubes used by Rusby and Swenson passed through the ⁴He bath and the aerostatic head was consequently quite large (equivalent to values as large as $0.6\mathrm{mK}$). Small lengths of yarn were inserted into the tubing at room temperature to reduce thermal oscillation effects. An alternative could be to enlarge the tubes. Cold spots were not evident in measurements at $4.2\mathrm{K}$, but measurements of pressures above atmospheric were made only when the liquid level in the main helium bath had fallen below the top of the vacuum jacket, and with the bath pressurised to a maximum of $0.2\mathrm{MPa}$ (absolute). In this or any similar system, the tubing that is at room temperature but is open to the cryogenic area must be clean, as any desorbed gases will diffuse into the cryostat and be re-adsorbed there. The pressure gradient due to this diffusion can be significant at low pressures.

A special design of a ³He vapour-pressure thermometer was used by Engert et al. (2003) and (2007) to reduce drastically the uncertainty of the corrections associated with the aerostatic head and the thermomolecular pressure difference. The principal design of the different pressure-sensing tubes is shown in Figure 4. The upper parts, extending from room temperature down to the inner vacuum can (IVC) flange, are made of stainless steel. Below the $80\mathrm{K}$ level, they are vacuum isolated. The temperatures of the tubes at the IVC flange level can be varied from $4.2\mathrm{K}$ to $30\mathrm{K}$. Inside the IVC, i.e. from the IVC flange level to the vapour-pressure bulb located in the experimental platform, two vertical copper tubes are connected with three horizontal stainless steel tubes. The temperatures of the copper tubes can be stabilised in the range from $0.6\mathrm{K}$ to $20\mathrm{K}$ using heaters and thermal links to the $1\mathrm{K}$ pot and the mixing chamber (MC) of the dilution refrigerator, respectively. The temperature differences between the main stainless-steel tube above the IVC flange level, the copper tubes, and the bulb are concentrated on the interconnecting horizontal tubes. These tubes have a small angle of inclination to avoid condensation of liquid in parts other than the bulb. The inner diameters of the largest pressure-sensing tube are as follows, with the typical temperature distribution in parentheses: $35\mathrm{mm}$ ($300\mathrm{K}$ to $80\mathrm{K}$), $17\mathrm{mm}$ ($80\mathrm{K}$ to $8\mathrm{K}$), $5.4\mathrm{mm}$ ($8\mathrm{K}$ to $2.2\mathrm{K}$), $4\mathrm{mm}$ ($2.2\mathrm{K}$), $2\mathrm{mm}$ ($2.2\mathrm{K}$ to $1.4\mathrm{K}$), $3\mathrm{mm}$ ($1.4\mathrm{K}$), $2\mathrm{mm}$ ($1.4\mathrm{K}$ to the temperature of the bulb). For this tube, the temperature equivalent of the thermo-molecular pressure difference has been estimated to be smaller than $0.03\mathrm{mK}$ at $0.65\mathrm{K}$ applying the approximated Weber-Schmidt equation (see Section 4.2). The temperature equivalent of the overall aerostatic-head correction at $10\mathrm{kPa}$, i.e. at $1.7\mathrm{K}$, is of order of $0.5\mathrm{mK}$ with a standard uncertainty of $0.015\mathrm{mK}$. The temperature stabilisation of the tubes at specified levels prevented thermal oscillations caused by condensation and evaporation of liquid as well as the formation of cold spots.

Figure 4 — ³He vapour-pressure thermometer built at the Physikalisch-Technische Bundesanstalt(PTB) [Engert et al. 2003 and 2007]: On the left: Schematic sketch showing the principal design of the pressure-sensing tubes (IVC inner vacuum can, MC mixing chamber of the dilution refrigerator). The thermal links are indicated by the zigzag lines. On the right: Photograph of a vapour-pressure bulb with a horizontal stainless-steel part of the pressure-sensing tube and different capsule-type rhodium-iron resistance thermometers installed. Figure reproduced with the permission of AIP Publishing.

Three quite different vapour-pressure bulbs were used, the volume-to-surface ratios of which vary by an order of magnitude. A significant influence of this ratio could be ruled out. At the bottom of each of the vapour-pressure bulbs, a layer of pressed copper ($<60\mathrm{\mu m}$) or silver ($70\mathrm{nm}$) powder ensures good thermal contact between the liquid helium and the body of the bulb. For all bulbs, the effect of the filling level with liquid ³He as well as the effect of different high-purity ³He samples ($\mathrm{99.999 9}$ atom % nominal purity) on the determination of the vapour pressures was estimated to be of the order of about $0.01\mathrm{mK}$ temperature equivalent. Because the cryogenic set-up allowed the variation of the temperature distribution along the pressure-sensing tubes, it was possible to check the influence of heat input from the tubes into the bulbs. The temperature of the individual low-temperature parts of the tubes was changed stepwise for each vapour pressure measured. The extrapolation to zero of the dependence of the bulb temperature on the temperature difference between the bulb and the lowest vertical part of the tube enabled the determination of vapour-pressure values for zero heat input. Furthermore, this made the data highly repeatable. Applying the optimised vapour-pressure bulb shown in Figure 4, a repeatability of a few $0.01\mathrm{mK}$ temperature equivalent was obtained for the results of an individual run. The repeatability of the ³He vapour-pressure realizations for all nine runs performed was about $0.12\mathrm{mK}$ standard deviation. This value includes the influence of all different designs of the bulbs and tubes. In the last four runs using the same optimised set-up, the repeatability was of order of $0.03\mathrm{mK}$. For the temperature range from $0.65\mathrm{K}$ to $1\mathrm{K}$, the complete uncertainty budget for the calibration of rhodium-iron resistance thermometers against the ³He vapour-pressure scale is given in Table 2. The pressure measurement was performed applying capacitive diaphragm gauges (CDGs), which were calibrated from $50\mathrm{Pa}$ to $1.3\mathrm{kPa}$ against the vacuum primary static-expansion standard of the PTB.

Table 2.  Uncertainty budget for the calibration of rhodium-iron resistance thermometers against the ³He vapour-pressure scale at PTB [Engert et al. 2007] (CDGs capacitive diaphragm gauges, EP experimental platform). $\Delta T$ stands for temperature difference. Uncertainty values are in $\mathrm{mK}$.

4.1.  Primary standards and transducers for pressure measurements

Pressure measurements are required for the realization of the helium vapour-pressure scales, the boiling points of hydrogen and the interpolating gas thermometer, see ITS-90 Guide Chapter 4 Gas Thermometry. Table 1 summarises helium and hydrogen vapour-pressure data and allows themeasurement requirements to be calculated. It shows that in order to cover the complete range for ⁴He, it is necessary to measure absolute pressures from $100\mathrm{Pa}$ to $200\mathrm{kPa}$, with standard uncertainties of $0.1\mathrm{Pa}$ to $15\mathrm{Pa}$ (relative $0.1\%$ to $75\mathrm{ppm}$, $\mathrm{ppm}$ means parts per million), respectively, to achieve $0.1\mathrm{mK}$ uncertainty in $T_{90}$. The needed relative uncertainties are less wide ranging, varying from $30\mathrm{ppm}$ for hydrogen at $20.3\mathrm{K}$ to $0.1\%$ for ³He at $0.65\mathrm{K}$. For gas thermometry, the range is $30\mathrm{ppm}$ at $3\mathrm{K}$ to $4\mathrm{ppm}$ at $25\mathrm{K}$. A measuring instrument with a constant relative uncertainty is thus more suitable than one with a constant absolute uncertainty.

Compared with the high-level realizations of the pressure scale, the requirements (in terms of room temperature capability for pressure measurement) of vapour-pressure thermometry are not overly stringent. In the pressure range of interest here, the primary pressure standards of the national metrological institutes are based on liquid-column manometers and pressure balances. Their typical relative uncertainties range from $100\mathrm{ppm}$ at $100\mathrm{Pa}$ to a few $\mathrm{ppm}$ at $200\mathrm{kPa}$ [Miiller et al. 2002, Pavese and Molinar Min Beciet 2013]. The smallest relative uncertainty achieved with pressure balances above $70\mathrm{kPa}$ amounts to $0.7\mathrm{ppm}$ [Zandt et al. 2015].

An overview of transfer standards for absolute pressure measurements is given in Chapter 8 of the book Modern Gas-Based Temperature and Pressure Measurements [Pavese and Molinar Min Beciet 2013]. From the standpoint of the various operating principles, they are grouped as follows: Piezoresistive transducers, optical transducers, force-balance transducers, capacitance transducers, and vibrating-structure transducers. Criteria for the selection of an appropriate pressure measuring device are the pressure range, the uncertainty, the resolution, the dependence of the signal on temperature, linearity and hysteresis. Considering these criteria, the application of the two primary pressure standards, namely liquid-column manometers and pressure balances, see below, can be recommended for vapour-pressure thermometry. As transfer standards, non-rotating (force-balanced) pressure balances and capacitive diaphragm gauges (CDGs) are suitable. On the contrary, quartz-Bourdon tube transducers should not be used because the diffusion of helium into the quartz tube causes large drifts.

A well-defined calibration method should be applied to make sure that a transducer is good enough to be used as a transfer standard at the desired uncertainty level. For checking the metrological characteristics of the transducer, the stability of the transducer output signal at zero pressure should be carefully determined for a long time. Calibration shifts are frequently dominated by the zero signal shift. The calibration should be carried out with repetitive tests for both increasing and decreasing pressures, made at different times. Full-scale pressure drift should also be determined in order to understand whether some fatigue or creep effects may influence the readings.

4.2.  Thermomolecular pressure difference

A thermomolecular pressure difference (TMPD), also called thermal transpiration, will result from a temperature change along the pressure sensing tube if the diameter of the tube is not very large compared with the mean free path of the gas particles. The pressure at the higher-temperature end (frequently at room temperature) will be greater than the cryogenic bulb pressure due to this effect. The magnitude of this pressure difference depends on (i) the temperatures at both ends of the tube, (ii) the gas properties, (iii) the absolute pressure value, and thus (iv) the existing flow regime (viscous, intermediate, Knudson), (v) the tube diameter, (vi) the material of the tube, and (vii) the state of its internal surface, where (vi) and (vii) determine the accommodation coefficient.

A detailed overview of the existing empirical and theoretical models for describing the TMPD is given in [Pavese and Molinar Min Beciet 2013]. Unfortunately, it is concluded that a straightforward and elementary discussion of the effects does not exist. A widely used model equation has been developed starting with Weber and Schmidt (1936), generalised by McConville (1972), and approximated by Swenson (1989):

where $p_{\text{h}}$, $p_1$, $T_{\text{h}}$ and $T_1$ refer to the pressures and temperatures at the high and low temperature extremities, respectively, of a tube of diameter $r$.

Though it is recommended to apply Equation (4) for correcting for the TMPD in vapour-pressure thermometry, it should be considered that relative differences between calculations and experimental results are typically of the order of $20\%$. Lower uncertainties can be achieved only by performing in-situ investigations.

For the case of ³He and ⁴He vapour-pressure measurements, for a tube of constant diameter, more than $90\%$ of the TMPD occurs between liquid nitrogen and room temperature. The magnitude of the effect can, therefore, be considerably reduced by employing a tube with two or more sections increasing in diameter from cold to hot [Sydoriak et al. 1964]. However, in the extreme case of ³He vapour-pressure measurements at $0.65\mathrm{K}$, this requires extreme diameters as used in [Engert et al. 2007]. At this temperature, a uniform diameter of $5\mathrm{mm}$ would still necessitate a correction of about $1\mathrm{mK}$ temperature equivalent, with an uncertainty of at least $0.2\mathrm{mK}$ without in-situ investigations.