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Study of the interpolation characteristics of a sealed low temperature gas thermometer
Study of the interpolation characteristics of a sealed low temperature gas thermometer K. Nara*, R.L. Rusby and D.I. Head Division of Quantum Metrology, National Physical Laboratory, Teddington, Middlesex, TW11 OLW, UK
Received 11 May 1990 A sealed gas thermometer with a pressure sensor attached has been developed and its interpolating characteristics have been studied. It was calibrated at three temperatures and a temperature scale was derived by quadratic interpolation, as is specified in ITS-90. This was compared with an established temperature scale, NPL75, between 5 and 24 K. The accuracy of the interpolation is found to be of the order of 1 mK between 5 and 14 K and 3 mK between 14 and 24 K. The systematic errors of the sealed gas thermometer are estimated and an assessment of the design is given.
Keywords: thermometers; measuring methods; instrumentation
Nomenclature
c, c, do(/~, do E
No, Pmax
r~ rd
so t
Reference capacitance Variable capacitance to be measured Gap between electrodes Gap at null pressure Young's modulus Total adsorbed helium (mol) Adsorbed helium (mol) Sealed pressure at 293 K Radius of moving capacitor plate Radius of diaphragm Effective area of electrode Thickness of diaphragm
max
w
x03 I"(/3 }'b(/3 }'~(73 2,(/3
Sealing temperature (293 K) Gap between capacitor plate and guard ring Bridge balance X(T)/[ 1 - X(T)] Y(T) of background Y(T) corrected by use of background Pressure parameter
Greek letters ~inax
Pg tIrnax
"
Maximum deflection of diaphragm Molar density of gas phase (mol m -3) Maximum stress
The International Temperature Scale of 1990, ITS-90, came into effect on 1 January 1990, replacing the International Practical Temperature Scale of 1968, IPTS-681,2. In this temperature scale the lower limit has been extended to 0.65 K using helium vapour pressure equations. Between 5 K and the triple point of neon (24.5561 K) ITS-90 specifies the use of a constant volume gas thermometer for interpolating values of temperature. The design and the associated error of the interpolating gas thermometer has been studied in detail by Steur et al. 3. The characteristics of the 3He constant volume gas thermometer were also studied by Pavese et al.4. In a conventional gas thermometer the gas bulb is held at the working temperature and the pressure measurement system is at room temperature. In this paper a sealed gas thermometer is described with a pressure sensor attached to the bulb. This gas thermometer, which is called the
'gas cell' hereafter, was filled at room temperature with 4He gas at about 8 atm* and sealed by an indium-tipped plug. The output of the gas cell was calibrated at three temperatures and a quadratic interpolation made. The interpolating error between 5 and 14 K was found to be of the order of 1 mK. The advantage of in situ pressure measurement in gas thermometry is obvious. When the pressure sensing system is placed at room temperature and the bulb at the working temperature, a pressure sensing capillary must be used to connect the two. As a result, two systematic errors are introduced by this capillary. The first is the dead space error: a finite amount of gas is present within the capillary. To make a correction, the temperature distribution along the capillary must be known. This applies to an interpolating gas thermometer as well as to an absolute one, where accuracies of 1 mK are sought. The other systematic
*Permanent address: National Research Laboratory of Metrology, 1 - 4 , Umezono 1-Chome, Tsukuba, Ibaraki 305, Japan
~I atm = 10 5 Pa
0011 - 2 2 7 5 / 9 0 / 1 1 0 9 5 2 - 07 © 1990 B u t t e r w o r t h - H e i n e m a n n Ltd
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Cryogenics 1990 Vol 30 November
Study of a sealed low temperature gas thermometer: K. Nara et al.
effect is the thermomolecular effect which, unlike the dead space effect, increases as the capillary diameter is reduced. Some compi'omise between these opposing requirements must be reached, unless the pressure is always sufficiently large that the thermomolecular effect can be neglected. Both of these errors will disappear if the bulb can be sealed at low temperature in such a way that the pressure of the gas in the bulb can still be measured. Astrov et al. 5 used a differential pressure transducer operating as a null detector between the bulb and pressure sensing capillary. In their design, the diameter of the capillary can be large enough to keep the thermomolecular effect negligible as this does not increase the dead space. In the present design the pressure sensing system was also placed at cryogenic temperature. Several designs of in situ pressure sensor have been reported. Most were developed to measure the pressure of liquid 3He along the coexistence curve with solid 3He6'7. In that situation an in situ pressure sensor must be used as the pressure of the coexistence curve has a minimum around 0.3 K and the pressure sensing capillary is blocked by solid 3He. To attain a high resolution in pressure, a capacitive transducer is used to detect the deflection of the diaphragm due to the pressure change. Van Degrift and co-workers describe a miniature sealed gas thermometer in which a tunnel diode oscillator was used to sense the displacement of the diaphragm 8. The performance was limited by the mechanical properties of the diaphragm. The present work was intended to demonstrate what could be achieved using a medium-sized gas cell with a conventional capacitive transducer incorporated in the base.
cell is empty) should be small and reproducible. This must be measured so that a correction can be made (if need be) to the gas cell output. The gas cell described here was designed to meet these requirements with modest complexity and give reproducible results with an accuracy of only a few tenths of a millikelvin. The design ~f the capacitance electrodes and the measuring circuit are also described. The gas cell was made by assembling several parts, as shown in Figure 1. The main cell is made of copper and has thick walls with holes for the thermometers. The base plate, which incorporates the diaphragm, is made of BeCu and is bolted to the main cell with an indium seal. The cell was filled through a small inlet at the top 9. The deflection of the diaphragm was detected by a capacitive technique, using the assembly of electrodes mounted on the base of the cell. The dimensions are given in Table 1. The deflection of the centre of the diaphragm e~ (m) in this case is given as 6
Experimental apparatus
By using 2.7 x 1012 for E, the following values are obtained: ~max= 97/xm and Om,x= 5 × 109 Pa, which is =40% of the yield stress of the BeCu.
3r 4 ec
o....
The main requirements for the gas cell are good stability and linearity of output. It is also important that the 'background' temperature dependence of the pressure transducer (i.e. the temperature dependence when the gas
.,
..,.
' T|ghtenlng
..... Figure
1
Copper Backing
Spring
(2)
In designing the gas cell, apart from achieving the required sensitivity, the main concerns were good mechanical
~--~
p , a t e
Bottom
4t 2 p
Design of gas cell
inlet
B--e
(l)
where ro (m), t (m), E and p (Pal are the radius and thickness of the diaphragm, Young's modulus and pressure, respectively. The maximum stress O'max ( P a l in the diaphragm occurs at the circumference and is given as
Structure of gas cell
Gas
16Et 3 p
- .... -
"
p,.,
?
plate
j ~ _ _ ~ ~ ] disc " ( C ~ , p l a t e - ~%..~ --",-~ .........
~ ' ~
~
',o'
Cross-sectional and exploded view of gas cell. Scale applies to the cross-sectional view
Cryogenics 1990 Vol 30 November
953
Study of a sealed l o w temperature gas thermometer: K. Nara e t al. Table I
Physical dimensions of the gas cell
Main cell (internal dimensions) Diameter Length Capacity Surface area
30 mm 56 mm 39.6 cm 3 66.8 cm 2
Sealed pressure Pressure
8.5 x 10 ~ Pa at 293 K 350 tool m - 3
Density
BeCu diaphragm Thickness Diameter (diaphragm) Diameter (centre post)
0.5 mm 28 mm 8 mm
Quartz plate for the electrode Diameter
25 mm with rounded
Thickness Parallelism Roughness
1.5 mm 1 #m 1 /~m
edge
Moving electrode Diameter of centre electrode Inner diameter of guard electrode
20 mm 22 mm
Fixed electrode Diameter of electrode
25 mm
Quartz spacer Diameter Thickness
8 ± 0.1 mm 3.044 ± 0.001 mm
stability and temperature uniformity. The main cell, the middle plate and the bottom plate are made of oxygenfree high conductivity copper to minimize the thermal equilibrium time especially around 24 K. The inside of the main cell and the base plate are gold plated and polished to reduce the effect of the adsorption on the inner surface l°. After assembling four plates, a heat link is attached between the bottom plate and the base plate to reduce the thermal equilibrium time of the pressure sensing elements. The diaphragm and its support are machined from a single piece of BeCu, which is known to be well suited as a diaphragm at the low temperature 6. All of the spacers are made of fused quartz, the thermal expansion coefficient of which is known to be much smaller than that of most metals 1~-~3. The brass backing plate is used to bolt the electrode assembly together. It is tightened towards the base plate with three thin copper discs between base and backing plate. As these discs are placed opposite to the quartz spacers, there is no bending stress within the bottom plate. This design was found to improve the mechanical stability. Detailed assembly procedures are given later. Design o f electrode and m e a s u r i n g apparatus
In constructing a capacitive sensing system, we adopted a design in which both the capacitance transducer and the reference capacitance were mounted on the gas cell so that in measuring their ratio the effect of the thermal expansion would be approximately cancelled. Both pairs of elec-
954
Cryogenics
1990
Vol 30 November
trodes were made of the same materials and had a symmetric design. The circuit of Figure 2 was devised, after consultation with Dr B.P. Kibble of the Division of Electrical Science, National Physical Laboratory, UK. The electrodes were formed by depositing gold on the fused quartz plates with round edges. It was necessary to make a predeposition of AI to strengthen the 0.2 #m gold film. The moving electrode had a guard electrode around it to reduce the effect of the edge field and associated nonlinearity 14. To keep the guard electrode at the ground level, four coaxial cables were used instead of three. The four quartz plates with the electrodes were attached by silver paste to the base plate, both sides of the middle plate and the bottom plate. The operating frequency and voltage of the bridge were 3 kHz and 2 V, respectively, and its resolution was 0.1 ppm. This resolution corresponded to a temperature resolution of 0.16 rnK. E s t i m a t i o n o f s y s t e m a t i c errors
A temperature change in the gas cell gives rise to a change in the capacitance in the transducer and hence to a change in bridge balance X. Once the device has been calibrated, the cell temperature can be interpolated from a measurement of X. Various systematic effects in the cell and transducer give rise to corrections whose magnitudes must be estimated. These include the non-ideality and adsorption of the gas, the thermal and pressure expansion of the cell, non-linearities in the deflection of the diaphragm, edge effects and stray capacitance. These must be separately assessed. But since the device is to be used for interpolation between calibration points, corrections need not be made for each, as long as the effect is a weak function of temperature. Using the design parameters in Table 1, several systematic errors can be estimated and the results are given in Table 2. In this, linear interpolation between 5 and 24 K is assumed. For quadratic interpolation the errors would be smaller. In addition, some of the errors relating to the capacitors are expected to be cancelled in the selfcompensating design. An estimation of the errors and an assessment of the design are given in the Appendix. Experimental
procedure
C o n s t r u c t i o n of gas cell and b a c k g r o u n d measuremen t
The gas cell was constructed as follows. The main cell and the base plate with the diaphragm were assembled Table 2
Linear interpolation errors between 5 and 24 K Interpolation error (mK)
Adsorption of helium gas Non-ideality of helium gas Thermal expansion of gas bulb
0.05 1.0 0.1
Pressure expansion of gas bulb Edge effect Tilting effect of electrode Thermal expansion of
0.2 3.6 3.5
electrode
2.6
Study of a sealed low temperature gas thermometer: K. Nara et al.
with an indium O ring seal. Then the helium gas was pressurized to 8.5 atm through the inlet at the top. The pressure was chosen so that the effect of the non-ideality of the helium gas would not exceed 1.0 mK and that the maximum deflection of the diaphragm was =0.1 mm (References 15 - 17). After placing a mylar sheet temporarily on the electrode to protect its surface from damage, the middle plate was placed on the base plate with quartz spacers between them. The gap was adjusted by inserting 30/~m aluminium foils between the plates and the spacers. The mylar sheet was then removed. Apiezon N grease was used as an adhesive. The same procedures were followed for the bottom plate. After tightening the backing plate, the total system was left pressurized overnight to let the excess grease come out. The pressure was then released and the background effect was measured with the cell empty (this was also measured after the experiments with the pressurized cell). As the gap was adjusted with the helium gas at the same pressure as in the sealed run, the gaps did not need to be readjusted. The cell was suspended with cotton threads from the bottom of a temperature-controlled copper block and linked thermally to the block with copper braid. The whole assembly was placed in a thermal shield anchored to the copper block. The temperature of the gas cell was determined using three rhodium-iron resistance thermometers which had previously been used in the derivation of NPL 75 (Reference 15). The circuit shown in Figure 2 gives X, which is determined from the two capacitances Cx (variable capacitance) and Cr (reference capacitance). The balance condition is given as follows X-
l/Cx 1/Cr + 1/C~
(3)
As the inverse of the capacitance 1~CO has a linear dependence on the gap, it is more convenient to use a parameter Y defined as follows
Y = X/(1 - X)
(4)
= (l/Cx)/(l/Cr)
In the course of the vacant run, a spurious signal was found above 14 K when the temperature of the cell was changed. This was found to be caused by a temperature gradient between the pressure sensing plates and disappeared after thermal equilibrium was reached. This phenomenon strongly suggests that the present symmetric design reduces the background change by compensating for the thermal expansion of the construction materials. Pressurized m e a s u r e m e n t
After the background measurement had been made, the gas cell was warmed up, pressurized with helium to 8.5 atm and sealed using an indium-tipped plug. As the helium gas liquefies around the lambda point, the output of the gas cell Y(T) can be calibrated against the helium vapour pressure scale is. This calibration can be used to establish the relation between the bridge reading and the real pressure value. The pressurized run was then carried out between 4 and 24 K to establish the interpolation characteristics of the gas cell. The measurement procedure started at 4.2 K and finished at the same temperature to check the reproducibility of the system. The reproducibility between 14 and 4 K was of the order of 0.3 mK and a maximum change of 1 mK was found for thermal cycling between 4 and 24 K. As this change is of the same order as the change found in the vacant cell, the main cause of the reproducibility may be the mechanical instability between the constructing materials rather than the BeCu diaphragm.
Result Background m e a s u r e m e n t
The background measurement was made three times. The first measurement was made before the pressurized runs. The second and third measurements were made after the pressurized runs. As the gap was adjusted with a pressurized gas bulb, further adjustment was not necessary during these experiments. The background measurement gives the bridge balance Xb(T) for each run. From Xb(T), Yb(T) is calculated by using Equation (4). The normalized results are given in Figure 3. Measurements before and after the pressurized
1 o o o o
0
0; *t
-1
t
t
i =+ =+
C-.l
=+
~ -4
*+ t+
~- -5
~+
g÷ ,~ -7 Ratio
10
15
20
25
30
T(K)
transformer
Fused
Figure 2
o
5
/
quartz
Diagram of the capacitance m e a s u r e m e n t s y s t e m
Figure 3 Temperature dependence of the background: O, +, measurements before the sealed run; * , O, after pressurized run. Values at 4.230 K are used to normalize the results
Cryogenics 1990 Vol 30 November
955
Study of a sealed low temperature gas thermometer: K. Nara et al. runs are shown. Yb(T) is normalized with the value at 4.230 K. This background change amounts to a correction of 0.18 K at its maximum. For each curve, polynomial fits are made to correct the pressurized results. Yb(T) shows a linear dependence on T below 14 K but shows a clear deviation from linearity above 14 K, which limits the performance of the interpolation characteristics of the gas cell as is shown below. Vapour pressure cell Below 2.2 K, where the helium begins to condense, the cell behaves as a vapour pressure cell, the characteristics of which can be used to calibrate Y(T). This was normalized with its value at 4.230 K, and the background correction was made. The result, Yc(T), is plotted against vapour pressure in Figure 4, where a clear bend aroun'd 6000 Pa shows the change from vapour pressure cell to gas cell operation. The results in Figure 4 were fitted with a linear and a quadratic function of the pressure p (Pa) as follows
~(T)
(5)
= A + Bp + @2
The coefficients in the pressure equation were used to deduce a pressure parameter Z(T) from the data taken between 4 and 24 K. As Yc(4.230 K) = 1 for all runs, the coefficients A, B and C in Equation (5) are commonly used for all runs. Interpolation characteristics To estimate the interpolating characteristics of the gas cell, the bridge balance readings were measured at three calibration points and some points between them. Three calibration points were taken close to 5 K, the highest end of the helium vapour pressure scale, at 14 K, the triple point of equilibrium hydrogen, and at 24 K, the triple point of neon. Two runs were made to test the reproducibility of the interpolation characteristics. The bridge balance X(T) was converted to Y(T) by using Equation (4). Then the background correction was made by use of Yb(T). As there were three background runs and two sealed runs, six sets of results were obtained. Then the pressure parameter Z(T) was calculated by use of the parameters in Equation (5).
The temperature values were obtained by quadratic interpolation from the values of Z(T) (Pa) at 5, 14 and 24 K using the following equation
T = a + bZ(T) + cZ(T) 2
(6)
The temperature scale so derived was compared with the established temperature scale, NPL75, between 5 and 24 K. The results are shown in Figure 5, where two sets of sealed runs corrected by three sets of background (vacant) runs are shown. The accuracy of the interpolation is found to be of order 1 mK between 5 and 14 K and 3 mK between 14 and 24 K. Discussion
Errors in the results
Figure 5 shows the result of quadratic interpolation in terms of the pressure parameter Z(T), which was determined with a quadratic pressure function. It can be concluded that interpolation errors are of order 1 mK between 5 and 14 K and smaller than 3 mK between 14 and 24 K. Note also that the BeCu diaphragm itself shows excellent reproducibility after thermal cycling between 4 and 24 K. The difference in temperature errors, for example, between two sets of data for the same line type does not exceed 0.3 mK for all measured temperatures in Figure 5. From the above discussion it is concluded that good results can be obtained with an interpolating gas thermometer with a built-in capacitive diaphragm, but that the mechanical instability still introduced larger uncertainties than are desirable for high precision realization of ITS-90. Possible improvements of the gas cell are suggested below. Possible improvement of the gas cell The systematic errors shown in Figure 5 can be traced back to the complex temperature dependence of the background shown in Figure 3. Even a slight difference between runs 2 and 3 gives qualitatively different fits in Figure 5. Reducing the background change will contribute to reducing the fitting error. In Figure 3, a linear dependence of the background on temperature can be seen below 10 K and a clear devia-
1.013 1.012
='++ ..'.~ +
1.011
÷
~-~
÷
1.01 u 1.009 >-
2
£1
÷
of 0
÷
of of uJ-
1.008
0 ./
I
"q:%/.
1.007 ÷
1.006
÷
1.005
,
0
2000
-5
t
4000
6000
Figure 4
Pressure dependence of Yc(T), normalized with values at 4 . 2 3 0 K
Cryogenics 1990 Vol 30 November
4
8000
p(po)
956
.; f
-2
6
8
10
12
14
16
18
20
22
24
26
T(K) Figure 5
Errors of quadratic interpolation: +, x, the t w o sealed runs; , correspond to the background change +, * and o , respectively, shown in Figure 3
Study of a sealed low temperature gas thermometer: K. Nara et al. tion from linearity above 10 K. One of the possible causes of the deviation is the thermal expansion of the aluminium sheets inserted between the quartz spacers and the copper plates. In this case it would be desirable to adjust the gap differently. Another possible cause is the tilting of the electrode due to the thermal expansion of the adhesive between the quartz plate with the electrode and the copper plate, which might not be of sufficiently uniform thickness. As adhesive must be used between BeCu diaphragm and quartz plate with the moving electrode in the present diaphragm design, the layer of adhesive must be made thinner and more homogeneous. Although the cause of the linear dependence is not yet known, it does not introduce any interpolation error. Tilting of the electrode may also result from nonuniformity of the diaphragm. Therefore, the diaphragm should be designed to be as thick as possible to reduce the effect of machining inaccuracy.
of error, as listed in Table 2. The values of the fixed points are 5 and 24 K, at the upper end of the helium vapour pressure scale and the triple point of neon, respectively. The maximum interpolation error between the two fixed points is calculated. Effect of adsorption. Published data for the adsorption of helium on gold plate is used to calculate the effect of adsorption t°. The amount of helium adsorbed at 4.2 K is calculated to be 0.029 x l0 -7 mol. In this case, the ratio Nads/Ntot will be 2.13 X 10 -5. Assuming that Nads at 5 K is roughly half that at 4.2 K, the maximum linear interpolation error is 0.05 mK for the lower calibration temperature of 5 K. Non-ideality of helium gas. The non-ideality of the gas is usually expressed by the virial coefficients as follows p = pgRT[1 + B(T)pg + C(T)p 2 + • • • ]
Acknowledgement The authors would like to thank Dr K.H. Berry and Dr B.P. Kibble of NPL and Dr D.N. Astrov of VNIIFTRI for helpful discussions. K.N. was supported during his stay in the UK by the British Council.
References
(AI)
where p° is molar density. If we use the expression of Berry 15 ~or B(T) and of Gugan and Michel ~ for C(T), they are given as follows 15-17 B(T) = (17.19 - 3 9 6 . 2 K / T - 48K2/T2) cm 3 mol -~ (A2) C(T) = (5420/T)cm 6 K mo1-2
(A3)
1 The International Practical Temperature Scale of 1968 Amended Edn 1975 Metrologia (1976) 12, 7 2 Preston-Thomas, H. Metrologia (1990) 27 3 3 Steur, P.P.M., Pavese, F. and Durieux, M. JLow Temp Phys (1987) 69 91 4 Pavese, F. and Steur, P.P.M. J Low Temp Phys (1987) 69 91 5 Astrov, D.N., Beliaasky, L.B., Dedikov, Y.A., Zacharov, A.A. and Polunin, S.P. Metrologia (1989) 26 151 6 Straty, G.C. and Adams, E.D. Rev Sci lnstrum (1969) 40 1393 7 Griffioen, W. and Frossati, G. Rev Sci lnstrum (1985) 56 1236 8 Van Degrift, C.T., Bowers, W.J. Jr, Wildes, D.G. and Pipes, P.B. lnstrum Soc Am Trans (1980) 19 15 9 Ancsin, J. Metrologia (1988) 25 221 10 Meyer, L. Phys Rev (1956) 103 1593 11 White, G.K. and Birch, J.A. Phys Chem Glasses (1965) 6 85 12 Touloukian, Y.S., Kirby, R.K., Taylor, R.E. and Desai, P.D. Thermophysical Properties of Matter Vol 12 Thermal Expansion, Metallic Elements and Alloys, IFI/Plenum, New York, USA (1976) 13 Touloukian, Y.S., Kirby, R.K., Taylor, R.E. and l)esai, P.D. Thermophysical Properties of Matter Vol 13 Thermal Expansion, Nonmetallic Solids, IFl/Plenum, New York, USA (1977) 14 White, G.K. C©'ogenics (1961) 1 151 15 Berry, K.H. Metrologia (1979) 15 89 16 Supplementary Information on IPTS68 and EPT76, BIPM Document, S~vres, France (1984) 17 Gugan, D. and Michel, G.W. Metrologia (1980) 16 149 18 Durleux, M. and Rusby, R.L. Metrologia (1983) 19 67
A sealed pressure of 8.5 × 105 atm at 293 K gives a molar density pg of 349 mol m -3. The linear interpolation error is calculated to be 1.0 mK.
Appendix
where rc and d(T) are the radius of the electrode and the distance between the electrodes, respectively. Substituting values from Table 1, the linear interpolation error is estimated to be 3.6 mK.
Estimation of systematic errors As the gas cell is normally calibrated at three points, corrections need not be made for each cause separately as long as the non-linearity is small and a smooth function of temperature. Here it is shown that the systematic errors are reasonably small and that three point interpolation is expected to be valid at cryogenic temperatures. In this Appendix, the output of the gas cell is assumed to be calibrated at two fixed points and the deviation of the output from the linear law is calculated for several sources
Thermal and pressure expansion of the gas volume. Thermal expansion changes the volume of the gas cell. By use of the reported thermal expansion coefficients, the effect of the thermal expansion of the gas bulb is estimated to be 0.1 mK (References 11-13). As the gas cell has a diaphragm at the bottom, the volume increase of the gas bulb due to deflection of the diaphragm is considerable, but the linear interpolation error is estimated to be <0.2 mK. Edge e f f e c t . As shown in Figure 2, the moving electrode is guarded with an electrode at ground level with a gap of w. The capacitance C(T) between the parallel plates with a guarded electrode is given as follows ~4 rcw ( ~1r +~ ) C(T) _ rc2 + re0 d(T) d(T) + 0.22w
(A4)
Tilting of the electrode. Since the quartz moving plate of the gas cell is attached to the diaphragm with adhesive, it is not easy to keep the pair of electrodes perfectly parallel. As the gap is adjusted by inserting aluminium sheets, the parallelism can be out by 30/~m on the edge. The tilting angle is then calculated to be about 0.003 rad in the worst case. Also, as the thickness of the diaphragm might not be uniform, the tilting angle might depend on
Cryogenics 1990 Vol 30 November
957
Study of a sealed low temperature gas thermometer: K. Nara et al. Table 3
Limiting condition for design parameters
Parameter
Limiting condition
Gas cell
Pmax ~max arnax t rd
Max. for moderate non-ideality of helium gas Min. for acceptable sensitivity with reasonable extra gap Max. (material limitation) Min. for homogeneity of the diaphragm Max. (space limitation)
8.5 x 105 Pa 100/~m 5 x 109 Pa 0.5 mm 10 mm
the deflection and also be as large as the 0.003 rad. The linear interpolation error is estimated to be as large as 3.5 mK.
Background temperature dependence. As the spacers and the plate with the electrodes are made of fused quartz, the temperature dependence of the gap is expected to be very small. Also, in the design of the gas cell, this effect is expected to cancel. However, there is still an uncancelled background as shown in Figure 4. This may partly be caused by the thermal expansion of the 30/zm aluminium sheets inserted between the copper plate and the quartz rods to adjust the gaps between the electrodes. The maximum systematic error is estimated below by assuming that the aluminium sheets are inserted between only one pair of electrodes. The capacitance is assumed to be proportional to SJd(T) where S~ and d(T) are the effective area of the moving electrode and the gap between the electrodes, respectively. S¢ depends on the temperature in the same manner as the quartz plate. On the other hand, the thermal expansion of the quartz spacers is overwhelmed by that of the aluminium sheet of 130/~m, as the thermal expansion of the fused quartz is negligible compared with that of aluminium in the design of the gas cell. The linear interpolation error is estimated to be as large as 3 mK. Assessment of gas cell design Temperature sensitivity of pressure transducer. To design the thickness and radius of the diaphragm, the maximum deflection e~,x at room temperature should be given. The gap between the electrodes do(T) is then set with an additional separation to avoid shorting between the electrodes. When the gas cell is pressurized at room temperature Tm~ with pressure Pr~x, the temperature sensitivity of the impedance ratio X is given as follows. dX
-- emax/d0
1
dT
(2 - Emaxp/doPmax)2
Zmax
958
Cryogenics 1990 Vol 30 November
(A5)
where the reference capacitance Cr is assumed to be same as the varying capacitance Cx at 0 K. If it is possible to adjust the gap do to be exactly the same as the maximum deflection em,x, the sensitivity above depends only on P/Pmaxand T~x, neither of which are controllable parameters. So, in an ideal case, the sensitivity of the gas cell does not depend on any adjustable parameters. In a real case, however, the sensitivity depends on the choice of the parameters due to the extra gap added to keep the electrodes from shorting. If the extra gap is chosen to be a technically controllable value of -~ 30 #m, the maximum deflection should be > 100 #m to avoid a loss of sensitivity of 30% or more. If the maximum deflection is too small, a non-negligible loss of sensitivity cannot be avoided. It is clear that there is an upper limit for maximum deflection for a given pressure, as the diaphragm must be thick enough to avoid a large tilt with the increase of deflection due to the lack of homogeneity of the diaphragm thickness. Therefore, the best value of the maximum deflection should be determined from the technical point of view.
Design conditions. Conditions which determine the design parameters are summarized below. The results for the gas cell show that the reproducibility of the BeCu diaphragm is good enough for it to be used as the pressure transducer. This is expected to be true as long as the maximum stress is well below the yield stress of BeCu. It has been shown that the maximum deflection of the diaphragm must be large enough to avoid a serious loss of temperature sensitivity. If insertion or removal of thin sheets to adjust the gap cannot be avoided, the maximum deflection might be > 100/~m. From the discussion of the effect of tilt, the diaphragm should be thick enough that it can be uniform. The requirements above cannot always be fulfilled for a given sealed pressure, as there is often a space limitation for rd, the diameter of the diaphragm. These contradictory conditions are summarized in Table 3, which also includes the design parameters of the gas cell.
Received 11 May 1990 A sealed gas thermometer with a pressure sensor attached has been developed and its interpolating characteristics have been studied. It was calibrated at three temperatures and a temperature scale was derived by quadratic interpolation, as is specified in ITS-90. This was compared with an established temperature scale, NPL75, between 5 and 24 K. The accuracy of the interpolation is found to be of the order of 1 mK between 5 and 14 K and 3 mK between 14 and 24 K. The systematic errors of the sealed gas thermometer are estimated and an assessment of the design is given.
Keywords: thermometers; measuring methods; instrumentation
Nomenclature
c, c, do(/~, do E
No, Pmax
r~ rd
so t
Reference capacitance Variable capacitance to be measured Gap between electrodes Gap at null pressure Young's modulus Total adsorbed helium (mol) Adsorbed helium (mol) Sealed pressure at 293 K Radius of moving capacitor plate Radius of diaphragm Effective area of electrode Thickness of diaphragm
max
w
x03 I"(/3 }'b(/3 }'~(73 2,(/3
Sealing temperature (293 K) Gap between capacitor plate and guard ring Bridge balance X(T)/[ 1 - X(T)] Y(T) of background Y(T) corrected by use of background Pressure parameter
Greek letters ~inax
Pg tIrnax
"
Maximum deflection of diaphragm Molar density of gas phase (mol m -3) Maximum stress
The International Temperature Scale of 1990, ITS-90, came into effect on 1 January 1990, replacing the International Practical Temperature Scale of 1968, IPTS-681,2. In this temperature scale the lower limit has been extended to 0.65 K using helium vapour pressure equations. Between 5 K and the triple point of neon (24.5561 K) ITS-90 specifies the use of a constant volume gas thermometer for interpolating values of temperature. The design and the associated error of the interpolating gas thermometer has been studied in detail by Steur et al. 3. The characteristics of the 3He constant volume gas thermometer were also studied by Pavese et al.4. In a conventional gas thermometer the gas bulb is held at the working temperature and the pressure measurement system is at room temperature. In this paper a sealed gas thermometer is described with a pressure sensor attached to the bulb. This gas thermometer, which is called the
'gas cell' hereafter, was filled at room temperature with 4He gas at about 8 atm* and sealed by an indium-tipped plug. The output of the gas cell was calibrated at three temperatures and a quadratic interpolation made. The interpolating error between 5 and 14 K was found to be of the order of 1 mK. The advantage of in situ pressure measurement in gas thermometry is obvious. When the pressure sensing system is placed at room temperature and the bulb at the working temperature, a pressure sensing capillary must be used to connect the two. As a result, two systematic errors are introduced by this capillary. The first is the dead space error: a finite amount of gas is present within the capillary. To make a correction, the temperature distribution along the capillary must be known. This applies to an interpolating gas thermometer as well as to an absolute one, where accuracies of 1 mK are sought. The other systematic
*Permanent address: National Research Laboratory of Metrology, 1 - 4 , Umezono 1-Chome, Tsukuba, Ibaraki 305, Japan
~I atm = 10 5 Pa
0011 - 2 2 7 5 / 9 0 / 1 1 0 9 5 2 - 07 © 1990 B u t t e r w o r t h - H e i n e m a n n Ltd
952
Cryogenics 1990 Vol 30 November
Study of a sealed low temperature gas thermometer: K. Nara et al.
effect is the thermomolecular effect which, unlike the dead space effect, increases as the capillary diameter is reduced. Some compi'omise between these opposing requirements must be reached, unless the pressure is always sufficiently large that the thermomolecular effect can be neglected. Both of these errors will disappear if the bulb can be sealed at low temperature in such a way that the pressure of the gas in the bulb can still be measured. Astrov et al. 5 used a differential pressure transducer operating as a null detector between the bulb and pressure sensing capillary. In their design, the diameter of the capillary can be large enough to keep the thermomolecular effect negligible as this does not increase the dead space. In the present design the pressure sensing system was also placed at cryogenic temperature. Several designs of in situ pressure sensor have been reported. Most were developed to measure the pressure of liquid 3He along the coexistence curve with solid 3He6'7. In that situation an in situ pressure sensor must be used as the pressure of the coexistence curve has a minimum around 0.3 K and the pressure sensing capillary is blocked by solid 3He. To attain a high resolution in pressure, a capacitive transducer is used to detect the deflection of the diaphragm due to the pressure change. Van Degrift and co-workers describe a miniature sealed gas thermometer in which a tunnel diode oscillator was used to sense the displacement of the diaphragm 8. The performance was limited by the mechanical properties of the diaphragm. The present work was intended to demonstrate what could be achieved using a medium-sized gas cell with a conventional capacitive transducer incorporated in the base.
cell is empty) should be small and reproducible. This must be measured so that a correction can be made (if need be) to the gas cell output. The gas cell described here was designed to meet these requirements with modest complexity and give reproducible results with an accuracy of only a few tenths of a millikelvin. The design ~f the capacitance electrodes and the measuring circuit are also described. The gas cell was made by assembling several parts, as shown in Figure 1. The main cell is made of copper and has thick walls with holes for the thermometers. The base plate, which incorporates the diaphragm, is made of BeCu and is bolted to the main cell with an indium seal. The cell was filled through a small inlet at the top 9. The deflection of the diaphragm was detected by a capacitive technique, using the assembly of electrodes mounted on the base of the cell. The dimensions are given in Table 1. The deflection of the centre of the diaphragm e~ (m) in this case is given as 6
Experimental apparatus
By using 2.7 x 1012 for E, the following values are obtained: ~max= 97/xm and Om,x= 5 × 109 Pa, which is =40% of the yield stress of the BeCu.
3r 4 ec
o....
The main requirements for the gas cell are good stability and linearity of output. It is also important that the 'background' temperature dependence of the pressure transducer (i.e. the temperature dependence when the gas
.,
..,.
' T|ghtenlng
..... Figure
1
Copper Backing
Spring
(2)
In designing the gas cell, apart from achieving the required sensitivity, the main concerns were good mechanical
~--~
p , a t e
Bottom
4t 2 p
Design of gas cell
inlet
B--e
(l)
where ro (m), t (m), E and p (Pal are the radius and thickness of the diaphragm, Young's modulus and pressure, respectively. The maximum stress O'max ( P a l in the diaphragm occurs at the circumference and is given as
Structure of gas cell
Gas
16Et 3 p
- .... -
"
p,.,
?
plate
j ~ _ _ ~ ~ ] disc " ( C ~ , p l a t e - ~%..~ --",-~ .........
~ ' ~
~
',o'
Cross-sectional and exploded view of gas cell. Scale applies to the cross-sectional view
Cryogenics 1990 Vol 30 November
953
Study of a sealed l o w temperature gas thermometer: K. Nara e t al. Table I
Physical dimensions of the gas cell
Main cell (internal dimensions) Diameter Length Capacity Surface area
30 mm 56 mm 39.6 cm 3 66.8 cm 2
Sealed pressure Pressure
8.5 x 10 ~ Pa at 293 K 350 tool m - 3
Density
BeCu diaphragm Thickness Diameter (diaphragm) Diameter (centre post)
0.5 mm 28 mm 8 mm
Quartz plate for the electrode Diameter
25 mm with rounded
Thickness Parallelism Roughness
1.5 mm 1 #m 1 /~m
edge
Moving electrode Diameter of centre electrode Inner diameter of guard electrode
20 mm 22 mm
Fixed electrode Diameter of electrode
25 mm
Quartz spacer Diameter Thickness
8 ± 0.1 mm 3.044 ± 0.001 mm
stability and temperature uniformity. The main cell, the middle plate and the bottom plate are made of oxygenfree high conductivity copper to minimize the thermal equilibrium time especially around 24 K. The inside of the main cell and the base plate are gold plated and polished to reduce the effect of the adsorption on the inner surface l°. After assembling four plates, a heat link is attached between the bottom plate and the base plate to reduce the thermal equilibrium time of the pressure sensing elements. The diaphragm and its support are machined from a single piece of BeCu, which is known to be well suited as a diaphragm at the low temperature 6. All of the spacers are made of fused quartz, the thermal expansion coefficient of which is known to be much smaller than that of most metals 1~-~3. The brass backing plate is used to bolt the electrode assembly together. It is tightened towards the base plate with three thin copper discs between base and backing plate. As these discs are placed opposite to the quartz spacers, there is no bending stress within the bottom plate. This design was found to improve the mechanical stability. Detailed assembly procedures are given later. Design o f electrode and m e a s u r i n g apparatus
In constructing a capacitive sensing system, we adopted a design in which both the capacitance transducer and the reference capacitance were mounted on the gas cell so that in measuring their ratio the effect of the thermal expansion would be approximately cancelled. Both pairs of elec-
954
Cryogenics
1990
Vol 30 November
trodes were made of the same materials and had a symmetric design. The circuit of Figure 2 was devised, after consultation with Dr B.P. Kibble of the Division of Electrical Science, National Physical Laboratory, UK. The electrodes were formed by depositing gold on the fused quartz plates with round edges. It was necessary to make a predeposition of AI to strengthen the 0.2 #m gold film. The moving electrode had a guard electrode around it to reduce the effect of the edge field and associated nonlinearity 14. To keep the guard electrode at the ground level, four coaxial cables were used instead of three. The four quartz plates with the electrodes were attached by silver paste to the base plate, both sides of the middle plate and the bottom plate. The operating frequency and voltage of the bridge were 3 kHz and 2 V, respectively, and its resolution was 0.1 ppm. This resolution corresponded to a temperature resolution of 0.16 rnK. E s t i m a t i o n o f s y s t e m a t i c errors
A temperature change in the gas cell gives rise to a change in the capacitance in the transducer and hence to a change in bridge balance X. Once the device has been calibrated, the cell temperature can be interpolated from a measurement of X. Various systematic effects in the cell and transducer give rise to corrections whose magnitudes must be estimated. These include the non-ideality and adsorption of the gas, the thermal and pressure expansion of the cell, non-linearities in the deflection of the diaphragm, edge effects and stray capacitance. These must be separately assessed. But since the device is to be used for interpolation between calibration points, corrections need not be made for each, as long as the effect is a weak function of temperature. Using the design parameters in Table 1, several systematic errors can be estimated and the results are given in Table 2. In this, linear interpolation between 5 and 24 K is assumed. For quadratic interpolation the errors would be smaller. In addition, some of the errors relating to the capacitors are expected to be cancelled in the selfcompensating design. An estimation of the errors and an assessment of the design are given in the Appendix. Experimental
procedure
C o n s t r u c t i o n of gas cell and b a c k g r o u n d measuremen t
The gas cell was constructed as follows. The main cell and the base plate with the diaphragm were assembled Table 2
Linear interpolation errors between 5 and 24 K Interpolation error (mK)
Adsorption of helium gas Non-ideality of helium gas Thermal expansion of gas bulb
0.05 1.0 0.1
Pressure expansion of gas bulb Edge effect Tilting effect of electrode Thermal expansion of
0.2 3.6 3.5
electrode
2.6
Study of a sealed low temperature gas thermometer: K. Nara et al.
with an indium O ring seal. Then the helium gas was pressurized to 8.5 atm through the inlet at the top. The pressure was chosen so that the effect of the non-ideality of the helium gas would not exceed 1.0 mK and that the maximum deflection of the diaphragm was =0.1 mm (References 15 - 17). After placing a mylar sheet temporarily on the electrode to protect its surface from damage, the middle plate was placed on the base plate with quartz spacers between them. The gap was adjusted by inserting 30/~m aluminium foils between the plates and the spacers. The mylar sheet was then removed. Apiezon N grease was used as an adhesive. The same procedures were followed for the bottom plate. After tightening the backing plate, the total system was left pressurized overnight to let the excess grease come out. The pressure was then released and the background effect was measured with the cell empty (this was also measured after the experiments with the pressurized cell). As the gap was adjusted with the helium gas at the same pressure as in the sealed run, the gaps did not need to be readjusted. The cell was suspended with cotton threads from the bottom of a temperature-controlled copper block and linked thermally to the block with copper braid. The whole assembly was placed in a thermal shield anchored to the copper block. The temperature of the gas cell was determined using three rhodium-iron resistance thermometers which had previously been used in the derivation of NPL 75 (Reference 15). The circuit shown in Figure 2 gives X, which is determined from the two capacitances Cx (variable capacitance) and Cr (reference capacitance). The balance condition is given as follows X-
l/Cx 1/Cr + 1/C~
(3)
As the inverse of the capacitance 1~CO has a linear dependence on the gap, it is more convenient to use a parameter Y defined as follows
Y = X/(1 - X)
(4)
= (l/Cx)/(l/Cr)
In the course of the vacant run, a spurious signal was found above 14 K when the temperature of the cell was changed. This was found to be caused by a temperature gradient between the pressure sensing plates and disappeared after thermal equilibrium was reached. This phenomenon strongly suggests that the present symmetric design reduces the background change by compensating for the thermal expansion of the construction materials. Pressurized m e a s u r e m e n t
After the background measurement had been made, the gas cell was warmed up, pressurized with helium to 8.5 atm and sealed using an indium-tipped plug. As the helium gas liquefies around the lambda point, the output of the gas cell Y(T) can be calibrated against the helium vapour pressure scale is. This calibration can be used to establish the relation between the bridge reading and the real pressure value. The pressurized run was then carried out between 4 and 24 K to establish the interpolation characteristics of the gas cell. The measurement procedure started at 4.2 K and finished at the same temperature to check the reproducibility of the system. The reproducibility between 14 and 4 K was of the order of 0.3 mK and a maximum change of 1 mK was found for thermal cycling between 4 and 24 K. As this change is of the same order as the change found in the vacant cell, the main cause of the reproducibility may be the mechanical instability between the constructing materials rather than the BeCu diaphragm.
Result Background m e a s u r e m e n t
The background measurement was made three times. The first measurement was made before the pressurized runs. The second and third measurements were made after the pressurized runs. As the gap was adjusted with a pressurized gas bulb, further adjustment was not necessary during these experiments. The background measurement gives the bridge balance Xb(T) for each run. From Xb(T), Yb(T) is calculated by using Equation (4). The normalized results are given in Figure 3. Measurements before and after the pressurized
1 o o o o
0
0; *t
-1
t
t
i =+ =+
C-.l
=+
~ -4
*+ t+
~- -5
~+
g÷ ,~ -7 Ratio
10
15
20
25
30
T(K)
transformer
Fused
Figure 2
o
5
/
quartz
Diagram of the capacitance m e a s u r e m e n t s y s t e m
Figure 3 Temperature dependence of the background: O, +, measurements before the sealed run; * , O, after pressurized run. Values at 4.230 K are used to normalize the results
Cryogenics 1990 Vol 30 November
955
Study of a sealed low temperature gas thermometer: K. Nara et al. runs are shown. Yb(T) is normalized with the value at 4.230 K. This background change amounts to a correction of 0.18 K at its maximum. For each curve, polynomial fits are made to correct the pressurized results. Yb(T) shows a linear dependence on T below 14 K but shows a clear deviation from linearity above 14 K, which limits the performance of the interpolation characteristics of the gas cell as is shown below. Vapour pressure cell Below 2.2 K, where the helium begins to condense, the cell behaves as a vapour pressure cell, the characteristics of which can be used to calibrate Y(T). This was normalized with its value at 4.230 K, and the background correction was made. The result, Yc(T), is plotted against vapour pressure in Figure 4, where a clear bend aroun'd 6000 Pa shows the change from vapour pressure cell to gas cell operation. The results in Figure 4 were fitted with a linear and a quadratic function of the pressure p (Pa) as follows
~(T)
(5)
= A + Bp + @2
The coefficients in the pressure equation were used to deduce a pressure parameter Z(T) from the data taken between 4 and 24 K. As Yc(4.230 K) = 1 for all runs, the coefficients A, B and C in Equation (5) are commonly used for all runs. Interpolation characteristics To estimate the interpolating characteristics of the gas cell, the bridge balance readings were measured at three calibration points and some points between them. Three calibration points were taken close to 5 K, the highest end of the helium vapour pressure scale, at 14 K, the triple point of equilibrium hydrogen, and at 24 K, the triple point of neon. Two runs were made to test the reproducibility of the interpolation characteristics. The bridge balance X(T) was converted to Y(T) by using Equation (4). Then the background correction was made by use of Yb(T). As there were three background runs and two sealed runs, six sets of results were obtained. Then the pressure parameter Z(T) was calculated by use of the parameters in Equation (5).
The temperature values were obtained by quadratic interpolation from the values of Z(T) (Pa) at 5, 14 and 24 K using the following equation
T = a + bZ(T) + cZ(T) 2
(6)
The temperature scale so derived was compared with the established temperature scale, NPL75, between 5 and 24 K. The results are shown in Figure 5, where two sets of sealed runs corrected by three sets of background (vacant) runs are shown. The accuracy of the interpolation is found to be of order 1 mK between 5 and 14 K and 3 mK between 14 and 24 K. Discussion
Errors in the results
Figure 5 shows the result of quadratic interpolation in terms of the pressure parameter Z(T), which was determined with a quadratic pressure function. It can be concluded that interpolation errors are of order 1 mK between 5 and 14 K and smaller than 3 mK between 14 and 24 K. Note also that the BeCu diaphragm itself shows excellent reproducibility after thermal cycling between 4 and 24 K. The difference in temperature errors, for example, between two sets of data for the same line type does not exceed 0.3 mK for all measured temperatures in Figure 5. From the above discussion it is concluded that good results can be obtained with an interpolating gas thermometer with a built-in capacitive diaphragm, but that the mechanical instability still introduced larger uncertainties than are desirable for high precision realization of ITS-90. Possible improvements of the gas cell are suggested below. Possible improvement of the gas cell The systematic errors shown in Figure 5 can be traced back to the complex temperature dependence of the background shown in Figure 3. Even a slight difference between runs 2 and 3 gives qualitatively different fits in Figure 5. Reducing the background change will contribute to reducing the fitting error. In Figure 3, a linear dependence of the background on temperature can be seen below 10 K and a clear devia-
1.013 1.012
='++ ..'.~ +
1.011
÷
~-~
÷
1.01 u 1.009 >-
2
£1
÷
of 0
÷
of of uJ-
1.008
0 ./
I
"q:%/.
1.007 ÷
1.006
÷
1.005
,
0
2000
-5
t
4000
6000
Figure 4
Pressure dependence of Yc(T), normalized with values at 4 . 2 3 0 K
Cryogenics 1990 Vol 30 November
4
8000
p(po)
956
.; f
-2
6
8
10
12
14
16
18
20
22
24
26
T(K) Figure 5
Errors of quadratic interpolation: +, x, the t w o sealed runs; , correspond to the background change +, * and o , respectively, shown in Figure 3
Study of a sealed low temperature gas thermometer: K. Nara et al. tion from linearity above 10 K. One of the possible causes of the deviation is the thermal expansion of the aluminium sheets inserted between the quartz spacers and the copper plates. In this case it would be desirable to adjust the gap differently. Another possible cause is the tilting of the electrode due to the thermal expansion of the adhesive between the quartz plate with the electrode and the copper plate, which might not be of sufficiently uniform thickness. As adhesive must be used between BeCu diaphragm and quartz plate with the moving electrode in the present diaphragm design, the layer of adhesive must be made thinner and more homogeneous. Although the cause of the linear dependence is not yet known, it does not introduce any interpolation error. Tilting of the electrode may also result from nonuniformity of the diaphragm. Therefore, the diaphragm should be designed to be as thick as possible to reduce the effect of machining inaccuracy.
of error, as listed in Table 2. The values of the fixed points are 5 and 24 K, at the upper end of the helium vapour pressure scale and the triple point of neon, respectively. The maximum interpolation error between the two fixed points is calculated. Effect of adsorption. Published data for the adsorption of helium on gold plate is used to calculate the effect of adsorption t°. The amount of helium adsorbed at 4.2 K is calculated to be 0.029 x l0 -7 mol. In this case, the ratio Nads/Ntot will be 2.13 X 10 -5. Assuming that Nads at 5 K is roughly half that at 4.2 K, the maximum linear interpolation error is 0.05 mK for the lower calibration temperature of 5 K. Non-ideality of helium gas. The non-ideality of the gas is usually expressed by the virial coefficients as follows p = pgRT[1 + B(T)pg + C(T)p 2 + • • • ]
Acknowledgement The authors would like to thank Dr K.H. Berry and Dr B.P. Kibble of NPL and Dr D.N. Astrov of VNIIFTRI for helpful discussions. K.N. was supported during his stay in the UK by the British Council.
References
(AI)
where p° is molar density. If we use the expression of Berry 15 ~or B(T) and of Gugan and Michel ~ for C(T), they are given as follows 15-17 B(T) = (17.19 - 3 9 6 . 2 K / T - 48K2/T2) cm 3 mol -~ (A2) C(T) = (5420/T)cm 6 K mo1-2
(A3)
1 The International Practical Temperature Scale of 1968 Amended Edn 1975 Metrologia (1976) 12, 7 2 Preston-Thomas, H. Metrologia (1990) 27 3 3 Steur, P.P.M., Pavese, F. and Durieux, M. JLow Temp Phys (1987) 69 91 4 Pavese, F. and Steur, P.P.M. J Low Temp Phys (1987) 69 91 5 Astrov, D.N., Beliaasky, L.B., Dedikov, Y.A., Zacharov, A.A. and Polunin, S.P. Metrologia (1989) 26 151 6 Straty, G.C. and Adams, E.D. Rev Sci lnstrum (1969) 40 1393 7 Griffioen, W. and Frossati, G. Rev Sci lnstrum (1985) 56 1236 8 Van Degrift, C.T., Bowers, W.J. Jr, Wildes, D.G. and Pipes, P.B. lnstrum Soc Am Trans (1980) 19 15 9 Ancsin, J. Metrologia (1988) 25 221 10 Meyer, L. Phys Rev (1956) 103 1593 11 White, G.K. and Birch, J.A. Phys Chem Glasses (1965) 6 85 12 Touloukian, Y.S., Kirby, R.K., Taylor, R.E. and Desai, P.D. Thermophysical Properties of Matter Vol 12 Thermal Expansion, Metallic Elements and Alloys, IFI/Plenum, New York, USA (1976) 13 Touloukian, Y.S., Kirby, R.K., Taylor, R.E. and l)esai, P.D. Thermophysical Properties of Matter Vol 13 Thermal Expansion, Nonmetallic Solids, IFl/Plenum, New York, USA (1977) 14 White, G.K. C©'ogenics (1961) 1 151 15 Berry, K.H. Metrologia (1979) 15 89 16 Supplementary Information on IPTS68 and EPT76, BIPM Document, S~vres, France (1984) 17 Gugan, D. and Michel, G.W. Metrologia (1980) 16 149 18 Durleux, M. and Rusby, R.L. Metrologia (1983) 19 67
A sealed pressure of 8.5 × 105 atm at 293 K gives a molar density pg of 349 mol m -3. The linear interpolation error is calculated to be 1.0 mK.
Appendix
where rc and d(T) are the radius of the electrode and the distance between the electrodes, respectively. Substituting values from Table 1, the linear interpolation error is estimated to be 3.6 mK.
Estimation of systematic errors As the gas cell is normally calibrated at three points, corrections need not be made for each cause separately as long as the non-linearity is small and a smooth function of temperature. Here it is shown that the systematic errors are reasonably small and that three point interpolation is expected to be valid at cryogenic temperatures. In this Appendix, the output of the gas cell is assumed to be calibrated at two fixed points and the deviation of the output from the linear law is calculated for several sources
Thermal and pressure expansion of the gas volume. Thermal expansion changes the volume of the gas cell. By use of the reported thermal expansion coefficients, the effect of the thermal expansion of the gas bulb is estimated to be 0.1 mK (References 11-13). As the gas cell has a diaphragm at the bottom, the volume increase of the gas bulb due to deflection of the diaphragm is considerable, but the linear interpolation error is estimated to be <0.2 mK. Edge e f f e c t . As shown in Figure 2, the moving electrode is guarded with an electrode at ground level with a gap of w. The capacitance C(T) between the parallel plates with a guarded electrode is given as follows ~4 rcw ( ~1r +~ ) C(T) _ rc2 + re0 d(T) d(T) + 0.22w
(A4)
Tilting of the electrode. Since the quartz moving plate of the gas cell is attached to the diaphragm with adhesive, it is not easy to keep the pair of electrodes perfectly parallel. As the gap is adjusted by inserting aluminium sheets, the parallelism can be out by 30/~m on the edge. The tilting angle is then calculated to be about 0.003 rad in the worst case. Also, as the thickness of the diaphragm might not be uniform, the tilting angle might depend on
Cryogenics 1990 Vol 30 November
957
Study of a sealed low temperature gas thermometer: K. Nara et al. Table 3
Limiting condition for design parameters
Parameter
Limiting condition
Gas cell
Pmax ~max arnax t rd
Max. for moderate non-ideality of helium gas Min. for acceptable sensitivity with reasonable extra gap Max. (material limitation) Min. for homogeneity of the diaphragm Max. (space limitation)
8.5 x 105 Pa 100/~m 5 x 109 Pa 0.5 mm 10 mm
the deflection and also be as large as the 0.003 rad. The linear interpolation error is estimated to be as large as 3.5 mK.
Background temperature dependence. As the spacers and the plate with the electrodes are made of fused quartz, the temperature dependence of the gap is expected to be very small. Also, in the design of the gas cell, this effect is expected to cancel. However, there is still an uncancelled background as shown in Figure 4. This may partly be caused by the thermal expansion of the 30/zm aluminium sheets inserted between the copper plate and the quartz rods to adjust the gaps between the electrodes. The maximum systematic error is estimated below by assuming that the aluminium sheets are inserted between only one pair of electrodes. The capacitance is assumed to be proportional to SJd(T) where S~ and d(T) are the effective area of the moving electrode and the gap between the electrodes, respectively. S¢ depends on the temperature in the same manner as the quartz plate. On the other hand, the thermal expansion of the quartz spacers is overwhelmed by that of the aluminium sheet of 130/~m, as the thermal expansion of the fused quartz is negligible compared with that of aluminium in the design of the gas cell. The linear interpolation error is estimated to be as large as 3 mK. Assessment of gas cell design Temperature sensitivity of pressure transducer. To design the thickness and radius of the diaphragm, the maximum deflection e~,x at room temperature should be given. The gap between the electrodes do(T) is then set with an additional separation to avoid shorting between the electrodes. When the gas cell is pressurized at room temperature Tm~ with pressure Pr~x, the temperature sensitivity of the impedance ratio X is given as follows. dX
-- emax/d0
1
dT
(2 - Emaxp/doPmax)2
Zmax
958
Cryogenics 1990 Vol 30 November
(A5)
where the reference capacitance Cr is assumed to be same as the varying capacitance Cx at 0 K. If it is possible to adjust the gap do to be exactly the same as the maximum deflection em,x, the sensitivity above depends only on P/Pmaxand T~x, neither of which are controllable parameters. So, in an ideal case, the sensitivity of the gas cell does not depend on any adjustable parameters. In a real case, however, the sensitivity depends on the choice of the parameters due to the extra gap added to keep the electrodes from shorting. If the extra gap is chosen to be a technically controllable value of -~ 30 #m, the maximum deflection should be > 100 #m to avoid a loss of sensitivity of 30% or more. If the maximum deflection is too small, a non-negligible loss of sensitivity cannot be avoided. It is clear that there is an upper limit for maximum deflection for a given pressure, as the diaphragm must be thick enough to avoid a large tilt with the increase of deflection due to the lack of homogeneity of the diaphragm thickness. Therefore, the best value of the maximum deflection should be determined from the technical point of view.
Design conditions. Conditions which determine the design parameters are summarized below. The results for the gas cell show that the reproducibility of the BeCu diaphragm is good enough for it to be used as the pressure transducer. This is expected to be true as long as the maximum stress is well below the yield stress of BeCu. It has been shown that the maximum deflection of the diaphragm must be large enough to avoid a serious loss of temperature sensitivity. If insertion or removal of thin sheets to adjust the gap cannot be avoided, the maximum deflection might be > 100/~m. From the discussion of the effect of tilt, the diaphragm should be thick enough that it can be uniform. The requirements above cannot always be fulfilled for a given sealed pressure, as there is often a space limitation for rd, the diameter of the diaphragm. These contradictory conditions are summarized in Table 3, which also includes the design parameters of the gas cell.