- Email: [email protected]
The melting curve of helium from 4 to 25 K
New data for the melting curve o f He 4 for temperatures between 4 and 25 K agree with earlier results by Mills and Grilly and by Crawford and Daniels, but disagree with the relationship given by Glassford and Smith [CRYOGENICS 6 (1966) 193] who carried out equation of state studies on fluid and solid He4. The reasons for the disagreement are not clear, but can be interpreted in terms o f temperature scale inaccuracies which could have influenced all of their results.
The melting curve of helium from 4 to 25 K J.K. Krause and C.A. Swenson
Glassford and Smith 1 published in 1966 the results of thermodynamic measurements on fluid and solid helium for temperatures from 4.2 to 20 K. These experiments filled a gap which existed in the experimental equation of state data for helium. Recently, however, a question has been raised 2 as to the consistency of these data 1 compared with those of other workers. In particular, their melting pressure relation shows disagreement at higher pressures with earlier data of Mills and Grilly, 3 and with the more recent results of Crawford and Daniels. 4 The only other melting curve experiments in the temperature region between 4.2 and 14 K are those of Dugdale and Simon s for which the temperature scale was not well-defined. The melting pressure relation for a solid provides a continuous series of thermometric 'fixed points' to which the high pressure equation of state and heat capacity data for both the solid and the fluid can be referred. Constant volume specific heat measurements for solid helium, 6 for instance, do not usually involve direct molar volume determinations, but have used the sample melting temperature and the data of Grilly and Mills7 to obtain molar volumes along the melting line. The gap in the melting parameter data which Glassford and Smith 1 filled is related directly to thermometry difficulties in the 5 to 14 K temperature region. Mills and Grilly, 3 for instance, used He 4 vapour pressures from 2 to 5 K, then calibrated thermocouples up to roughly 14 K, hydrogen vapour pressures from 14 to over 20 K, and a calibrated platinum resistance thermometer at high temperatures. They estimate that their uncertainty could be as much as 0.1 K in the region where the thermocouples were used. Grilly and Mills7 limited their data-taking to temperatures where liquid baths were available; 2 to 5 K with He 4, 14 to 24.5 K with H2, 24.5 to 35 K with Ne. Crawford and Daniels 4 in turn restricted their experiments to temperatures above 13.8 K where they had a valid platinum resistance thermometer calibration. Glassford and Smith's ~ apparatus included a He4 gas thermometer which was calibrated near the He4 boiling point, 4.2 K. Unfortunately, their experiment did not incorporate any other form of thermometry, so their melting pressure versus temperature relation provides the primary basis for comparison with other results. We have used the high pressure heat capacity apparatus of Fugate and Swenson 8 in a manner similar to that used by Grilly and Mills7 and Glassford and Smith 1 to determine the melting pressure relation of He4 from 4 to 25 K. The most The authors are with the Physics Department, Ames Laboratory-ERDA, Iowa State University, Ames, Iowa 50010, USA. Received 23 January 1976.
CRYOGENICS. J U L Y 1976
important difference between the present experiments and previous experiments is that we have access to a well-defined temperature scale (see below) which is believed to be close to the thermodynamic scale. Thus, these measurements can be used through a comparison of melting pressures to investigate the adequacy of the thermometry in previous experiments. The Fugate and Swenson 8 calorimeter is a high pressure bomb which is connected to a high pressure system at room temperature by a vacuum jacketed stainless steel capillary. Various heaters and a heat switch make it possible to hold the bomb at any temperature above 1 K while maintaining all sections of the capillary at higher temperatures. Hence, it is possible to condense a small amount of solid helium in the bomb and then to warm or cool the bomb and sample while remaining on the melting curve. A 2.75 kbar (2.75 × 105kN m -2) Heise gauge in the high pressure line to the capillary and a calibrated germanium resistance thermometer on the bomb then allow melting temperatures and corresponding melting pressures to be determined. The present experiments were carried out with varying amounts of solid in the bomb and with very different heat fluxes (heat switch conductances) with no observable differences in the results. All data points were taken under static conditions, with the sample temperature held constant to 1 mK (10-3K) for at least five minutes before the pressure was read. The calibration of the 2.75 kbar Heise gauge was compared initially with that of a similar 6.6 kbar gauge. Both instruments were stated by the manufacturer to be accurate to 0.1% of full scale (3 bar and 7 bar, respectively), although they could be read with a precision three times greater than this. The calibrations of the two gauges differed in a roughly linear fashion, with the smaller pressure gauge reading too high by approximately 16 bar at 2.5 kbar. The present data can be made to agree with those of Mills and Grilly 3 and Crawford and Daniels 4 (see below) by arbitrarily decreasing the 2.75 kbar gauge pressures by 0.3 % (8 bar at 2.5 kbar). There is no indication from the fit of the empirical three parameter Simon equation to these data (see below) that systematic errors were introduced into the results by this arbitrary procedure which assumes a linear gauge calibration. The germanium resistance thermometer used by Fugate and Swenson 8 was calibrated in terms of a magnetic temperature scale. 9 This scale can be made consistent with the NBS Acoustic scale between 2.5 K and 20 K and with the currently adopted platinum resistance thermometer scale IPTS-68 for temperatures greater than 27 K by increasing all temperatures
413
obtained from it by 0.03 %)0 After this adjustment the differences with the other two scales below 27 K are less than 5 mK and are not significant for the present investigations. This adjusted scale (similar to TXAC' as given by Swenson 1°) has been used in the present work since it is believed to be thermodynamically consistent with T6s at all temperatures. Glassford and Smith's ~ gas thermometer scale which was based on the liquid helium vapour pressure relation Tsa gives temperatures which are lower than the adjusted magnetic temperatures by approximately 0.2 %, due to inconsistencies in the vapour pressure scale. We have.arbitrarily adjusted their temperatures upwards by this factor for comparison with the results. The calibration of the platinum thermometer used by Mills and Grilly 3 is not likely to differ from IPTS-68 by more than 10 mK, while Crawford and Daniels 4 used a thermometer which had been calibrated on this scale. Fig. 1 compares the results of Glassford and Smith, a Mills and Grilly, 3 and Crawford and Daniels 4 with the present data. The reference pressures for this plot were obtained by fitting by least squares the Simon melting relation to a combination of our results and those of Crawford and Daniels 4 to ensure consistency between 20 and 25 K. The resulting relation is P =
17.452 T l's4~sl -- 20.60 bar
l.O
?.0 ,0.0
12
. CD / x
8
I /MI,
/GS
6
a.8
x ~'~=
x D
v i
-4
0
-6
I
-10
l
GS smooth curve
-12
x
Alternote groups represent
-14
•
SuccessiveGS isochores
x ~
-16 -18
I
I
I
I
I
I
I
I
I
I
I
5
6
7
8
9
I0
II
12
1:5
14
15
x I
16
Temperoture, K Fig. 2 Differences between the temperatures given by Glassford and Smith 1 and the present scale f o r each o f their melting pressures. Alternate clusters of the symbols (e) and (+) represent data f o r successive isochores. The letters A through H correspond in order to decreasing specific volumes
(1)
The scatter of the present data is consistent with the precision to which the pressure gauge could be read, although the actual calibration of the gauge is a major limitation in the experiment. The agreement in shape with both the Crawford and Daniels 4 and Mills and Gritty a results is reasonable from 14 to 24 K and provides confidence in the normalization of our gauge readings. The temperature scale used by Mills and Grilly, a and also Grilly and Mills, 7 appears to be almost an order of magnitude better than they believed it to be. The smoothed Glassford and Smith ~ relation, plotted in Fig. 1, shows large deviations at both low temperature and at high temperature which are well outside their stated experimental uncertainties.
I0
i ee -2
4
A major concern is with the reliability of the rather extensive equation of state data given by Glassford and Smith.1 Fig. 2 shows the difference between their stated temperature and the present temperature for the melting pressure data points on each of their isochores. The melting pressure relation which they give, the solid line, does not represent their results well at low temperatures. Indeed, the agreement between their actual results and the present data is reasonable through the set of points labelled D, or to 9 K. The higher temperature deviations, for which their temperatures are too low by up to 1%, cannot be understood from the description which they give of their experiment. The Dugdale and Simon s results show deviations of opposite sign but of twice the magnitude and with much greater scatter. In conclusion we believe that (1) represents the melting pressure relation for He 4 to -+ 0.01 K at temperatures from 4 to 25 K. The previous results of Mills and Grilly a are in agreement with this relation to roughly this accuracy in spite of the much larger uncertainty estimate which they give. The data of Glassford and Smith ~ must be used with caution. Presumably, Fig. 2 could be used to 'correct' their thermodynamic data if the discrepancies are associated solely with thermometry, but the basis for this assumption is not at all clear.
2 _
0 -2 -4
References I 4
I
I I I I I 6 8 I0
I 15
I 20
I
I I I I I1~(111 30 40 ,50 60
Temperature, K Fig. 1 A comparison between pressures calculated f r o m (1) and the present data (•), those o f Crawford and Daniels 4 (+), and the smoothed relations of Mills and Grilly 3 (MG) and Glassford and Smith L (GS). Actual melting pressures are indicated along the top of the figure, while the dashed line ( . . . . . . ) gives the pressure differences which at each temperature correspond to a temperature difference o f 0.01 K
414
1 2 3 4 5
Glassfotd, A.P.M., Smith, J.L. Jr. Cryogenics 6 (1966) 193 Angus,S. Imperial College, London, private communication Mills,R.L., Grilly, E.R. Phys Rev 99 (1955) 480 Crawford,R.K., Daniels, W.B.J. Chem Phys 55 (1971) 5651 Dugdale,J.S., Simon, F.E. Proc Roy Soc (Lend} A218 (1953) 291 6 Swenson,C.A., in Rare Gas Solids [J.A. Venables and M.L. Klein (eds)] (Academic Press, London, 1976)Ch 14 7 Grilly, E.R., Mills, R.L. Ann Phys 8 (1959) 1 8 Fugate, R.Q., Swenson, C.A. JLow Temp Phys 10 (1973) 317 9 Cetas,T.C., Swenson, C.A. Metrologia 8 (1972) 46 10 Swenson,C.A. Metrologia 9 (1973) 99
C R Y O G E N I C S . J U L Y 1976
The melting curve of helium from 4 to 25 K J.K. Krause and C.A. Swenson
Glassford and Smith 1 published in 1966 the results of thermodynamic measurements on fluid and solid helium for temperatures from 4.2 to 20 K. These experiments filled a gap which existed in the experimental equation of state data for helium. Recently, however, a question has been raised 2 as to the consistency of these data 1 compared with those of other workers. In particular, their melting pressure relation shows disagreement at higher pressures with earlier data of Mills and Grilly, 3 and with the more recent results of Crawford and Daniels. 4 The only other melting curve experiments in the temperature region between 4.2 and 14 K are those of Dugdale and Simon s for which the temperature scale was not well-defined. The melting pressure relation for a solid provides a continuous series of thermometric 'fixed points' to which the high pressure equation of state and heat capacity data for both the solid and the fluid can be referred. Constant volume specific heat measurements for solid helium, 6 for instance, do not usually involve direct molar volume determinations, but have used the sample melting temperature and the data of Grilly and Mills7 to obtain molar volumes along the melting line. The gap in the melting parameter data which Glassford and Smith 1 filled is related directly to thermometry difficulties in the 5 to 14 K temperature region. Mills and Grilly, 3 for instance, used He 4 vapour pressures from 2 to 5 K, then calibrated thermocouples up to roughly 14 K, hydrogen vapour pressures from 14 to over 20 K, and a calibrated platinum resistance thermometer at high temperatures. They estimate that their uncertainty could be as much as 0.1 K in the region where the thermocouples were used. Grilly and Mills7 limited their data-taking to temperatures where liquid baths were available; 2 to 5 K with He 4, 14 to 24.5 K with H2, 24.5 to 35 K with Ne. Crawford and Daniels 4 in turn restricted their experiments to temperatures above 13.8 K where they had a valid platinum resistance thermometer calibration. Glassford and Smith's ~ apparatus included a He4 gas thermometer which was calibrated near the He4 boiling point, 4.2 K. Unfortunately, their experiment did not incorporate any other form of thermometry, so their melting pressure versus temperature relation provides the primary basis for comparison with other results. We have used the high pressure heat capacity apparatus of Fugate and Swenson 8 in a manner similar to that used by Grilly and Mills7 and Glassford and Smith 1 to determine the melting pressure relation of He4 from 4 to 25 K. The most The authors are with the Physics Department, Ames Laboratory-ERDA, Iowa State University, Ames, Iowa 50010, USA. Received 23 January 1976.
CRYOGENICS. J U L Y 1976
important difference between the present experiments and previous experiments is that we have access to a well-defined temperature scale (see below) which is believed to be close to the thermodynamic scale. Thus, these measurements can be used through a comparison of melting pressures to investigate the adequacy of the thermometry in previous experiments. The Fugate and Swenson 8 calorimeter is a high pressure bomb which is connected to a high pressure system at room temperature by a vacuum jacketed stainless steel capillary. Various heaters and a heat switch make it possible to hold the bomb at any temperature above 1 K while maintaining all sections of the capillary at higher temperatures. Hence, it is possible to condense a small amount of solid helium in the bomb and then to warm or cool the bomb and sample while remaining on the melting curve. A 2.75 kbar (2.75 × 105kN m -2) Heise gauge in the high pressure line to the capillary and a calibrated germanium resistance thermometer on the bomb then allow melting temperatures and corresponding melting pressures to be determined. The present experiments were carried out with varying amounts of solid in the bomb and with very different heat fluxes (heat switch conductances) with no observable differences in the results. All data points were taken under static conditions, with the sample temperature held constant to 1 mK (10-3K) for at least five minutes before the pressure was read. The calibration of the 2.75 kbar Heise gauge was compared initially with that of a similar 6.6 kbar gauge. Both instruments were stated by the manufacturer to be accurate to 0.1% of full scale (3 bar and 7 bar, respectively), although they could be read with a precision three times greater than this. The calibrations of the two gauges differed in a roughly linear fashion, with the smaller pressure gauge reading too high by approximately 16 bar at 2.5 kbar. The present data can be made to agree with those of Mills and Grilly 3 and Crawford and Daniels 4 (see below) by arbitrarily decreasing the 2.75 kbar gauge pressures by 0.3 % (8 bar at 2.5 kbar). There is no indication from the fit of the empirical three parameter Simon equation to these data (see below) that systematic errors were introduced into the results by this arbitrary procedure which assumes a linear gauge calibration. The germanium resistance thermometer used by Fugate and Swenson 8 was calibrated in terms of a magnetic temperature scale. 9 This scale can be made consistent with the NBS Acoustic scale between 2.5 K and 20 K and with the currently adopted platinum resistance thermometer scale IPTS-68 for temperatures greater than 27 K by increasing all temperatures
413
obtained from it by 0.03 %)0 After this adjustment the differences with the other two scales below 27 K are less than 5 mK and are not significant for the present investigations. This adjusted scale (similar to TXAC' as given by Swenson 1°) has been used in the present work since it is believed to be thermodynamically consistent with T6s at all temperatures. Glassford and Smith's ~ gas thermometer scale which was based on the liquid helium vapour pressure relation Tsa gives temperatures which are lower than the adjusted magnetic temperatures by approximately 0.2 %, due to inconsistencies in the vapour pressure scale. We have.arbitrarily adjusted their temperatures upwards by this factor for comparison with the results. The calibration of the platinum thermometer used by Mills and Grilly 3 is not likely to differ from IPTS-68 by more than 10 mK, while Crawford and Daniels 4 used a thermometer which had been calibrated on this scale. Fig. 1 compares the results of Glassford and Smith, a Mills and Grilly, 3 and Crawford and Daniels 4 with the present data. The reference pressures for this plot were obtained by fitting by least squares the Simon melting relation to a combination of our results and those of Crawford and Daniels 4 to ensure consistency between 20 and 25 K. The resulting relation is P =
17.452 T l's4~sl -- 20.60 bar
l.O
?.0 ,0.0
12
. CD / x
8
I /MI,
/GS
6
a.8
x ~'~=
x D
v i
-4
0
-6
I
-10
l
GS smooth curve
-12
x
Alternote groups represent
-14
•
SuccessiveGS isochores
x ~
-16 -18
I
I
I
I
I
I
I
I
I
I
I
5
6
7
8
9
I0
II
12
1:5
14
15
x I
16
Temperoture, K Fig. 2 Differences between the temperatures given by Glassford and Smith 1 and the present scale f o r each o f their melting pressures. Alternate clusters of the symbols (e) and (+) represent data f o r successive isochores. The letters A through H correspond in order to decreasing specific volumes
(1)
The scatter of the present data is consistent with the precision to which the pressure gauge could be read, although the actual calibration of the gauge is a major limitation in the experiment. The agreement in shape with both the Crawford and Daniels 4 and Mills and Gritty a results is reasonable from 14 to 24 K and provides confidence in the normalization of our gauge readings. The temperature scale used by Mills and Grilly, a and also Grilly and Mills, 7 appears to be almost an order of magnitude better than they believed it to be. The smoothed Glassford and Smith ~ relation, plotted in Fig. 1, shows large deviations at both low temperature and at high temperature which are well outside their stated experimental uncertainties.
I0
i ee -2
4
A major concern is with the reliability of the rather extensive equation of state data given by Glassford and Smith.1 Fig. 2 shows the difference between their stated temperature and the present temperature for the melting pressure data points on each of their isochores. The melting pressure relation which they give, the solid line, does not represent their results well at low temperatures. Indeed, the agreement between their actual results and the present data is reasonable through the set of points labelled D, or to 9 K. The higher temperature deviations, for which their temperatures are too low by up to 1%, cannot be understood from the description which they give of their experiment. The Dugdale and Simon s results show deviations of opposite sign but of twice the magnitude and with much greater scatter. In conclusion we believe that (1) represents the melting pressure relation for He 4 to -+ 0.01 K at temperatures from 4 to 25 K. The previous results of Mills and Grilly a are in agreement with this relation to roughly this accuracy in spite of the much larger uncertainty estimate which they give. The data of Glassford and Smith ~ must be used with caution. Presumably, Fig. 2 could be used to 'correct' their thermodynamic data if the discrepancies are associated solely with thermometry, but the basis for this assumption is not at all clear.
2 _
0 -2 -4
References I 4
I
I I I I I 6 8 I0
I 15
I 20
I
I I I I I1~(111 30 40 ,50 60
Temperature, K Fig. 1 A comparison between pressures calculated f r o m (1) and the present data (•), those o f Crawford and Daniels 4 (+), and the smoothed relations of Mills and Grilly 3 (MG) and Glassford and Smith L (GS). Actual melting pressures are indicated along the top of the figure, while the dashed line ( . . . . . . ) gives the pressure differences which at each temperature correspond to a temperature difference o f 0.01 K
414
1 2 3 4 5
Glassfotd, A.P.M., Smith, J.L. Jr. Cryogenics 6 (1966) 193 Angus,S. Imperial College, London, private communication Mills,R.L., Grilly, E.R. Phys Rev 99 (1955) 480 Crawford,R.K., Daniels, W.B.J. Chem Phys 55 (1971) 5651 Dugdale,J.S., Simon, F.E. Proc Roy Soc (Lend} A218 (1953) 291 6 Swenson,C.A., in Rare Gas Solids [J.A. Venables and M.L. Klein (eds)] (Academic Press, London, 1976)Ch 14 7 Grilly, E.R., Mills, R.L. Ann Phys 8 (1959) 1 8 Fugate, R.Q., Swenson, C.A. JLow Temp Phys 10 (1973) 317 9 Cetas,T.C., Swenson, C.A. Metrologia 8 (1972) 46 10 Swenson,C.A. Metrologia 9 (1973) 99
C R Y O G E N I C S . J U L Y 1976